Fluidization Engineering

Dr. Subrata K. Majumder

Department of Chemical Engineering
Indian Institute of Technology, Guwahati

Lecture – 01
Introduction

Welcome to the massive online course on Fluidization Engineering sponsored by the

Ministry of Human Resource Development, Government of India. Today this course will be
discussed on fluidization engineering.

(Refer Slide Time: 00:51)

Now, what is fluidization? The fluidization is a process by which a bed of solid particle is
converted from a static solid-like state to a dynamic fluid-like state, by means of a flow of gas
or liquid.

Here see one or two video regarding this fluidization then, you will understand what is this
fluidization? Here see initially, the bed is of full of solid particles which is rest without
moving any bar any gas or liquid is supplied from the bottom of this column of the solid bed
you will see the solid particles will try to lose from its rest position due to this flow of gas or
liquid. And it will get fluidized state; that means, called dynamic fluid like state.
Now, in this case another example if we increase the gas or liquid velocity from the bed we
will see another different type of fluidization phenomena occurs. In this case gas is coming
from the bottom as a dispersed phase of bubbles; that means, here this white space here we
are seeing in the video that this space of white space these are nothing but the empty space
without solid particle. And here the solid particles are entering by this gas bubbles from the
bottom to the top and this gas is dispersed at a dispersed phase of bubbles from the bottom of
the bed.

Whereas in this case the left video you will see there is no formation of bubble. Why?
Because the gas velocity or liquid velocity it is supplied from the bottom of this column in
such way that and here a minimum gas velocity or liquid velocity is maintained. So, that the
solid particle just to become fluidized. So, this minimum fluidization state is called minimum
fluidization and the velocity at which this minimum fluidization condition is maintained is
called incipient velocity or incipient fluidization velocity or minimum fluidization velocity.
Beyond this minimum fluidization velocity you will see there are several other different types
of pattern of this fluidization will occur.

The question is that then you are getting this type of fluidization we will discuss later and
also what is the different application of the fluidization system based on this dynamic fluid
like state. Now, before coming to that point we will discuss that this fluidization is nothing,
but multiphase system phenomena multiphase phenomena. What is that multiphase system?
Multiphase system means more than one phases will be coming in contact to each other and
governs to some processes in chemical and biochemical industries and what is that then
multiphase system. You know that phases different phases like gas liquid and solids, now
combination of these phases forms multiphase system.

Now, this suppose gas and liquid the two phases, these two phases this two phases. Again this
gas and solid, again this liquid and solid liquid liquid and gas liquid solid there are different
combination of these systems and this different combinations are called the multiphase
combination system.
(Refer Slide Time: 04:58)

Now, this fluidization is one of the multiphase systems. Now fluidization basically occurs in
two phase system and three phase systems in two phase fluidization you will see here the two
phase fluidization means here see two phase fluidization is gas solid combination and liquid
solid content combination. Here gas is coming contact with solids and there is an application
of gas solid reactions or any other application like adsorption, absorption etcetera. Similarly
liquid and solid liquid is coming with solid in contact and in three phase fluidization gas
liquid and solids three phases are coming in contact.

Now, in case of gas and solids you will see or gas liquid solid, the gas or liquid acts as a
continuous and solid act as a discrete medium or you can say the dispersed phase medium.
Now in two phases this gas is a continuous phase and solid has a discrete phase whereas, in
the three phase systems the gas and solid both maybe as a dispersed phase and liquid as a
continuous phase. Even gas will be continuous phase, but liquid and solid will be discrete
phases. Now, depending on the application of the process it will be applied in a three phase
fluidization system.

Now, if suppose gas is supplied in the liquid medium and now this gas will be dispersed as a
dispersed phase of bubbles now bubbles will be forming through a distributor, now the
distributor will hold some holes with different sizes. Based on that size the different sizes of
the bubbles will form. We will discuss later on the distribution process of the gas and the
fluidized bed.
Now, see this gas will be dispersed this dispersed phase of bubbles in that case liquid phase
would continuous where in the liquid solid also would be discretized and this solid and liquid
will be forming as a slurry. Now, this solid and liquid medium will be acting as a slurry
medium and gas will be disperse medium.

Now, there are different aspects of this application of this gas liquid solid injection
sometimes gas shaded a slurry reactor, maybe you will see that Fischer Tropsch synthesis
they are in presence of catalyst particles the formation of the carbon monoxide and hydrogen
mixture this is gas production that is applied in slurry bubble column reactor. So, that is one
application of three phase fluidization system.

Whereas two phase fluidization like gas and solid they are suppose adsorption of some gas in
a solid medium. Even adsorption of organic gas into the solid even adsorption of liquid into
the solid also there, even gas is adsorbed on the liquid on liquid and solid operation there
liquid is absorbed in the solid. And also there are some other applications like granulation
coating in that case gas solid operations with important liquid solid fluidization is very
important. In the gas solid operations we will see there is a solid will be dispersing in the gas
medium in such way that there will be some uniform mixing of the bed so that the fluid like
behavior of this solid bed will be applied for the different processes.

(Refer Slide Time: 09:06)

Now, this why does this fluidized solid bed behave like a fluid. Now, you will see any fluid
whenever it will behave it will have some properties like it will seek some level. Of course,
whenever this fluidized bed will behave like a fluid it will seek its own level like bed height.

Even when in this case you will see in the fluidized bed compared to this fluid like behavior
if you insert some object with a lower density then a bed density you will see there will be
floating of that object or bobbing up and down if pushed downwards in the bed. So, this is
one property by which you can say that this fluidization bed solid or solid bed behave like a
fluid.

Another example of course, this fluidized beds have a static pressure head due to the gravity
and levels between two similar fluidized bed equalized their static pressure heads. The
hydrostatic pressure which rises with the depth in the bed also, as like that suppose in a liquid
of course, the fluidized hydrostatic pressure will be depending on the depth of the bed.

(Refer Slide Time: 10:36)

Now, contacting method, you will see there are several ways or modes of contacting of the
solid liquid or solid liquid gas in the fluidized fluidization bed or fluidization operation. Here
batch processes there in that case will you see in this gas, in this case the solid is moving due
to the supply of gas the solid will be supplied as a batch wise whereas, in the cocurrent
process or concurrent modes in that case gas and solid input will be supplied in a continuous
mode, continuous mode and it will be moving in the same direction of the bed. And then
cocurrent means here both gas and solids will be moving in the same direction.
And countercurrent in this case if gas and solids are there the solids will be flowing opposite
to that gas flow and if it is liquid solid then solid will be flowing opposite to the liquid and
also in the case of counter current operation will see for gas liquid solid operation the gas and
liquid will be flowing opposite to each other in a base of solid. Where the solid particles if
any catalyst particle or solid particles is lighter than the liquid then we will see due to their
buoyancy of the solid particles will try to move up whereas, liquid will be moving
downwards. So, in that case the due to the downward movement of the liquid solid particles
will try to move downward against its buoyancy. So, this type of fluidization process is called
inverse fluidization.

Cross current another one type of operations, like here the in this case the gas or liquid will
be flowing in the direction perpendicular to the movement of the solids or solid is moving
cross currently to the direction of the fluid or gases.

(Refer Slide Time: 13:07)

Now, you see there are different patterns based on the gas velocity or liquid velocity that is
applied to the fluidized bed. You will see some are batch operated some are transport
operated. Now, batch operated systems are like fixed bed bubbling bed churn turbulent bed
fixed bed means here it is not as that that fluidizing condition, but you will see if just simply
this fluidization operation is maintained at a minimum gas velocity the solid will be fluidizing
from its rest just from its rest at a minimum fluidization condition. So, that minimum
fluidization condition whenever fluidized bed is operated it is called particulate fluidized bed
or it is called homogenous fluidization condition.

And you will see if you gradually increase the gas velocity or liquid velocity. Here see if you
increase this gas or liquid velocity the fluidized bed is operated under bubbling condition.
From at that condition the gas will be flowing from the bottom of the column to the
distributed as the dispersed phase of bubbles. So, this type of phenomena is called bubbling
fluidization.

And another is called if you increase again the gas velocity and if it is the narrow fluidized
bed then what will happen the gas will be dispositive dispersed bubbles, but size of the gas
will be something different than previous one. If the size of the bubbles is almost near about
the diameter of the column of this fluidized bed then you will see that type of condition is
called slugging fluidization. This slugging may be axial and flat you will see sometimes the
bubble which is bigger in size and almost equal to size of its diameter then it will go
vertically in a chain wise. So, this type of it is called axial fluidization. And if the size of
these bubbles, the size of this bubble is flatter, like if the gas bubble size occupy or gas
bubble is occupied whole cross sectional area of the bed then it will move up just occupying
the whole cross sectional area and this type of slugging is called flat slugging.

Even if you increase more gas velocity there in the bed then you will see there will be a
intermixing of the gas and solid in such way that there will be formation of churn inside the
bed. So, this type of bed operation or pattern of this fluidization is called churn turbulent
pattern of the fluidization. And here this is the churn turbulent fluidization. Even if you
increase the size of the increase the velocity you will see there will be fast fluidization in this
case the settling velocity of the solids, if they are suppose any settling velocities there vt then
20 times higher than if it is there this gas velocity then we will see fast fluidization occurs,
even if you increase more than 20 times of the settling velocity of the solid particles you will
get pneumatic transport type of fluidization operation.

This fast fluidization and pneumatic transport fluidization you will see there will be no
formation of individual shape or bubbles, but there will be a discontinuous shape of bubbles
and there will be sometimes will see the continuous gas phase will be occurring inside the
bed. And also the porosity of the solid particles inside the bed will be higher in case of
pneumatic transport, but here this dilute medium of the bed of pneumatic transport will be
having in the process.

(Refer Slide Time: 17:43)

Now, see what are the basic elements of fluidization bed? Of course, you should know that if
suppose any fluidization operation occurs in the bed this is called fluidized bed; that means,
the bed where the fluidization process takes place. Now in that fluidization fluidized bed you
will see some components of this fluidized bed will be having basic components you can say
there will be having someone distributor through which gas will be distributed and what is
that internal heating or cooling system will be there because for some operations you will see
the medium to be of course, controlled at a certain temperature and pressure. So, that a
certain temperature to be maintained and that temperature to control of course, it will be
heating at a certain temperature. And also this heating of this medium to be controlled by
externally heating so there are two provisions that heating conditions that internal and
external provision of the heating.

And also the solids whenever it will be loaded in into the solid bed you will see there will be
a provision to supply this solids into the bed. So, this is your solid feeds provision and there
will be a feeder of the solids. And another if you want to supply the feed at liquid feed inside
the bed there will be some distributor of liquid inside the bed by which you can supply the
liquid into the bed. And there are some other this is called this part is called shell this shell
part and inside this shell this of fluidization occurs this called totally fluidized bed.
And another one important component is called cyclones these cyclones are being used
because if solid particles is coming up through the gas there is generally fine particles whose
size is very fine then small, then it will be coming out by this gas and of course, these fine
particles to be separated now. This cyclone separator is being used to separate the solid
particles. And these cyclone separators are used internally sometimes or maybe externally,
but some sometimes this both externally and internally cyclone separators are being installed
because of more separation of the finer particles they are. And then after separation these
solid particles again will be reused or in the bed.

And then what is that another component is blower of course, by which gas will be supplied
and before going to that fluidized bed this gas will be maintained certain temperature and that
maintaining temperature of course, on heat exchanger to be used for heating this gas medium.
And extra that is solid optics of course, there will be some prohibitions by who is the solid
particles to be given up taken off from this bed for reuse or maybe refreshment or maybe
which are not being used for that after use, maybe if it is combustion then solid particles
which are not being again used it would be taken off from the bed. So, that is why solids
optic option of course, will be there.

Now, gas is distributed from the bottom through distribute, there are several different kinds of
distributor of course, you will have we will be discussing later on the distributor part. The gas
or liquid with a minimum velocity which passed through the bed via the empty space
between the particles will cause the minimum fluidization. So, these are the basic elements of
fluidization bed. So, there are operation, they of course, there are some other accessories of
course, that depending on the operation and that is installation and also design of the fluidized
bed that depends on.
(Refer Slide Time: 22:20)

Now, what are the different layouts of the fluidized bed? Basically outs you will see
sometimes some fluidized bed will be circulating fluidized bed, the circulating fluidization
bed sometimes it is being you after fluidization if the solid particles will be reused or
recycled. So, that is what this called circulating fluidization bed. Some fluidized bed will be
internally recycling after separation of the solid particles by cyclone separators and it will be
used as a turbulent fluidization bed where high rigorous fluidization due to the high velocity
of the solid or liquid inside the bed to be maintained. So, that turbulence will occur there. In
intense mixing of the solid and gas will be there inside the bed that of course, will be used for
specific application.

Now, laterally staged bed also it is a one that is pattern or layouts you can say in this case
here in this direction solids are being supplied and loaded and after drying it would be
coming out. So, laterally it is actually supplied the solid particles are literally moving
compared to that what is that gas and liquid. And also particle staged bed also it is one
important layout in this case solid particles would be loading from the top of the bed and gas
will be supplied from the bottom of the bed, and it will be stage wise it will be supplied. Also
there other provisions like twin type bed and also protein fluidized bed even in the fluidized
bed you will see whenever solid particles will be recycling that will down comer will be used
and also when it will be fluidized to be called a riser, in the riser of course, the solid particles
will be fluidizing.
(Refer Slide Time: 24:10)

Now, question is that what the advantage of this fluidization there are so many things
regarding that there are some applications of course, will be there, but what are the
advantages. You will see this if you apply the solid liquid operation or solid gas operations or
gas liquid solid operations for specific chemical processes and like drying process drying
processes, now solid will be drying in the fluidized bed. So, in that case ease handling of the
solid particles and also there will be good mixing of the solid particles during this operation,
and also continuous operation you can do in a fluidized bed otherwise in the conventional
way like in the atmosphere open sources open just field. If suppose PDs kept for drying then
it will be best wise that you cannot continuously doing this operation, but in a fluidized bed
they due to the mixing of the solid particles you will see for a certain temperature if you dry it
immediately you can get the dried solid particles there.

So, there it is possible for continuous operation. Even for some other chemical operations it is
seen that there will be higher mass transfer coefficient and also there will be a distribution of
heat for that because of inter mixing of the solid particles with the fluid then it will give the
higher heat transfer coefficient. It is also good for exothermic reactions and also we will see
there are more contact efficiency more contact residence time also inside the bed. So, these
are the advantages even for large scale operation you can do of any chemical or biochemical
industry or any physical operations, in large scale operation pilot plant scale operations you
can do by this fluidized bed.
But of course, every system though it has some advantages there will be some disadvantage
also.

(Refer Slide Time: 26:22)

What are those disadvantages? Disadvantages like sometimes it is difficult to get the plug
flow operations this plug flow, plug flow operations is actually suitable for higher yield for
chemical processes. So, in that case the fluidized bed sometimes there will be back mixing, so
non ideal flow even plug flow phenomenal not be actually feasible in this fluid bed
operations. Of course, there are several provisions are made to get it plug flow because some
baffle or some other provisions are made so that the plug flow operations can be possible in
the fluidized bed.

Some other disadvantage like of course, initial capital cost is so high to design this fluidized
bed, and sometimes back mixing also is a very crucial factor whenever it is fluidized some
solid particles whenever it will be going up and it will of course go down due to its weight.
And some of course, segregation of this higher solid particles coarser solid particles it will be
moving down even not only that any size of these particles it will be moving inside the bed in
a circular motion. So, there will be some back mixing when you will not get the plug mixing
there, so back mixing is sometimes is not actually suggested for getting the intensified yield
of the processes. So, back mixing sometimes hinder the yield of the process.

Entering problem also entrainment sometimes some spine particles will clog this some, what
is that devices like cyclone separators, even the distributor of the gas in the fluidized bed. So,
this type of problem may be problematic in the fluidized bed operation. Of course, the fine
particles is coming out because of its low buoyancy and low size and this entrainment of the
solid particles of course, a hinder the process efficiency.

Size problem of course, there will be you have to make the you have to design this fluidized
bed in a bigger size sometimes for of course, the it is not suitable for some specific
applications if you are making it large way, there will be some you know static zone, there
will be some dynamic zone, there will be mixing not mixing properly there are huge cost. So,
there will be some problem in that case.

Complex hydrodynamics of course, you will see you cannot predict that hydrodynamics
precisely they are inside the bed what is happening there. You will see there will be
sometimes solid liquid interactions, liquid liquid interactions, if there is any bubble is
forming inside the bed bubbling fluidized bed then bubble bubble interactions some you
cannot predict that bubble size accurately what is going on. Even in dynamic way it is very
difficult to predict, but still we are scientist or economic persons they are developing different
models for that.

Even some other what is the intermixing criteria or attrition how it will be happening, how it
will be minimized and what are the other heat transfer mass transfer coefficient that will be
related to this hydrodynamic aspect, what is the different flow regimes, how we can regulate
this flow regime like particular to bubble, bubbling fluidized bed it is very difficult to get that
uniform flow pattern inside the bed. And also errosion of course, the important the high
velocity or fast fluidization of the coarser particles is trying to break into smaller particles and
again it will be entrained and it will be coming out from the top of the bed. So, there will be a
loss of fine particles. Of course, there is a high costly catalyst particle it is broken down then
it will be very problematic in place.

So, these are the disadvantage we can say we can summarize here some difficulty in plug
flow capital cost back mixing entrainment problem, size problem, complex of hydrodynamics
inside the bed errosion problem is there, sudden pressure loss of course, they are during the
operation because of sudden; that means, interaction of the bubble or any other fluid bed
particles inside the bed there will sudden fluctuation sudden pressure loss inside the bed
which will may the creative problem inside the bed for its operation.
(Refer Slide Time: 31:21)

Now, of course, it is advisable to tell of historical perspectives. Now, all the operations in
fluidized bed has categorized into three cycles like first cycle, second cycle and this is third
cycle. In first cycle actually generally around 1970 to 75 what happens regarding this
fluidization phenomena. So, there are different investigators they have reported, different
investigations based on their experimental work and different of course, information that you
can get here.

Now, first cycle like 1970 to 75 informs the phenomenology of the fluidization, even for
1975 to 1995, 2000 you can say some other phenomenology of the fluidization even after that
2000s the phenomenology in different way in the fluidization operation are reported by
various investigators.

Now, in this case 1970 around 1970 to 75 will see first development is the structural two
phase models of this fluidization operation. And then going on and then there are other
several important investigations, like what would be the hydrodynamics of the bubbling
fluidized bed, what are the models, how it can be modeled and what are the size of the
bubbles and based on the size distribution, how these bubble bubble interactions inside the
bed, how these bubble bubble interactions actually will affect the process of the process in
fluidization.

And also in 1975 onward to 1995 and 2000, you will see there are several things actually
which are very important like what are solids actually that are been used whether the size will
be what will be the size of that particles, so what should be the classification of the size
whether it is coarser whether should be very fine particles to be used. So, Gelder actually he
actually classified the solid particles different way like abcd different types of classifications
of the solid particles. So, classification even then what would be the diameter effect of the
solid particles, even fast fluidization whether this coarser particles to be used or not even if
we use the finer particles whether this agglomeration will happen or not that also has been in
a reported in different investigator as in different way.

Now, it will be discussed above one by one later on also. This classification of the powder
how this powder classification will effect this different flow pattern of the fluidization to be
discussed later on. And then structural of course, there are different type of you know that
fluidized bed and also the different shape of the particles how to be effect on that process in
of course, it is important even if there any hydrodynamics like is there any fluctuation of the
pressure heat transfer operation mass transfer operations what will be the effect of fluctuation
of the pressure and how this fluctuation of the pressure will enhance this or if it is decreased
or in the fast fluidization, how it will affect the process yield that is our efficiency of the
fluidized bed will be discussed later on and also it is reported in the literature.

Now, essential of full phase diagram of the, of course, this simulation of the fluidized bed.
So, full phase diagram even what are the full pattern that has been discussed in 1975 2000s.
And 2000 onward there are several other different phenomena other fluidization has been
reported now this meso scale suspension structure issue also it has been reported in the
fluidization operation. Multi solids and particle size distribution what will be the effect on the
particle size distribution for the modeling of the fluidized bed. Even charging and discharging
issue is there any effect of charging and discharging issue whenever applied in for a
particular process whether it will effect on the performance of the reactor or not.

Even particle particle interaction issues very important if one particle is interacting to another
particle whether this adsorption or any other mass transfer operation will effect or not that
also been reported. Even gas phase mixing and reduction retention time is very important of
course, any process yield depends on the mixing of the phases more mixing or back mixing
sometimes retard this process and also back mixing also not suitable some operations. But
sometimes mixing is very important back mixing also very important suppose for uniform
distribution of the heat inside the bed in a back mixing is very important. And also retention
time of course, this adsorption suppose this adsorption retention time is very important how
long these solid particles will be inside the bed that will effect the performance of the reactor.

(Refer Slide Time: 37:01)

Let us come to that what are the different application of the fluidized bed. Different
applications of the fluidized bed if you see that we are getting the different like solid liquid,
liquid solid, gas liquid solid there are several applications. If we categorize to these things as
solid catalyzed gas phase reactions you can see is done in fluidized bed like fluid catalytic
cracking, reforming operation Fischer-Tropsch synthesis like this, and what is the phthalic
and maleic anhydride formation or production in the fluidized bed, acrylonitrile and aniline
production in the fluidized bed, chlorination and bromination of the hydrocarbons in the
fluidized bed, polyethylene and polypropylene production in the fluidized bed, even
oxidation of sulfur dioxide and sulfur trioxide inside the bed it is being done in the fluidized
bed.

Some other operations some other applications you can say gas solid reactions, like roasting
of ores like zinc sulphide, copper sulphide, nickel sulphides all are roasting from the ores and
different products of that ores are coming after fluidization. Combustion and incineration coal
combustion it will give the different gaseous products from the coal incineration of the solid
waste it will give you the different of products by fluidization gasification, coking, pyrolysis,
carbonization, calcinations, limestone phosphates, aluminum hydroxide all are being calcined
in a fluidized bed.
Fluorination of uranium oxide is very important in atomic energy section that is done in
fluidized bed. Fluid coking is very important which is being done in fluidized bed reduction
of iron oxide is being done in fluidization bed, even catalyze catalyst regeneration is very
important which are also being by which is being done by fluidization operation.

(Refer Slide Time: 39:14)

Commercial applications you will see other commercial applications like gas phase non
catalytic reactions like natural gas combustion is very important which is being done in
fluidization phenomena. Gas liquid solid now fluid bed catalytic cracking, hydro treating,
hydro processing, biochemical processes, cultivation of microorganism, all are these
biochemical applications are being done in fluidization fluidized bed. Even hydrotreating,
hydro processing are being feasible to do in fluidization, in fluidized bed.
(Refer Slide Time: 39:53)

Other operation like physical processes drying of particles, coating of surfaces, granulation,
heat treatment, medical beds, filtration, blending, classification, particle classification that is
being done in a fluidized bed. These are all physical operations like dying of particles in a
fluidized bed coating of the surface like coating of the tablet us in pharmaceutical industries
even you know that heat treatment by fluidization operation blending these are the some
applications physical applications which have been done in fluidized bed.

(Refer Slide Time: 40:31)

Now, some key terminology of course, you have to know some key terminology like attrition.
What is that attrition? Attrition is nothing but the breakdown of particles. Actually whenever
at high velocity solids being fluidized in the bed you will see there will be a breaking of solid
particles this breakdown of the solid particles are called attritions. Then choking, what is the
choking? Choking means now collapse of dilute phase suspension in fluidized bed into dense
space flow as the gas velocity is reduced at constant solids flow. So, if the gas velocity is
reduced at constant solid flow you will see sometimes this dilute phase will be collapsing in
dense space. So, this type of phenomena is called choking.

Circulating fluidized bed what is that circulating fluidized bed? The circulating fluidized bed
is nothing, but is the when the solid particles have been send around in loop continuously
with no offer interface within the bed; that means, a solid particles are recycled to the bed.
So, the configuration will be done in such a way that configuration intended to same particles
around in loop continuously with no upper interface within the bed. So, this type of is called
circulating fluidized bed.

And then what is downer? Down is one important terms that this is one type of column where
the particles are made to fall to under gravity usually with cocurrent gas flow. Distributor or
grid this is very important crucial one design aspect that the gas or liquid which are being
supplied from the bottom of the bed of course, it will be a distributor. Now this distributor of
course, it will be a some support plate at bottom which introduces the gas to the bottom of the
bed and supports the weight of the bed when it is shut down. Now this distributor of course,
consist consist of holes now this holes will decide what will be the size of the bubbles will be
formed inside the fluidizedbed. So, it depending on the size of the bubbles and it will be
designed in such way different type of distributor are being used in the bed.

What is that elutriation? This elutriation is nothing, but a tendency for the fine particles to be
preferentially entrained from the reactor entering from the reactors.
(Refer Slide Time: 43:24)

(Refer Slide Time: 43:31)

What is fast fluidization? This is one terms fast fluidization, this is one type of flow regime of
the fluidization bed; however, there is a relatively dense suspension, but no distinct upper
surface and a superficial velocity generally at least 3 meter per second are being actually
maintained so that you can get the fast fluidization. Of course, we will learn that different
onset of the pattern of the fluidization later on.

What is fines? Fines on sub fines a generally particles smaller than 37 micrometer in diameter
it is smallest regulars ship size. What is free mode? You will see in the fluidized bed one
region to the extending from top of beds are placed at the top of reactor vessel. So, this is the
region extended bed that is from some location of the bed to the top, this is a free board
where the very dilution of the solid particles or bed will be there.

(Refer Slide Time: 44:42)

Interstitial gas, what is that interstitial gas? Gas between the particles gas between the
particles in the dense suspension it is called interstitial gas that is how much volume of gas is
occupying between those particles in whenever bed is in operation, so interstitial gas.

And what is porosity? Porosity is nothing but whatever the volume fraction of the gas in the
bed in a given region as a whole or only inside the particles sometimes used interchangeably
with the voidages. So, this is porosity what are the fraction of gas in the bed or given region
as a whole or only inside the particles.

What is riser? Riser that is also one type column this is a fluidized column where the particles
are carried upwards by the gas with no making any distinction bed surface.
(Refer Slide Time: 45:45)

Segregation, what is that segregation? Segregation is nothing but the separation of the solid
particles from its size like tendency for particles to gather in different zones according to their
size or you can say density. Solids what is they used for actually synonymously you can say
with particles, this is this terms is being used in fluidized bed. And superficial velocity of
course, this is very important whenever you are going to correlate with various variables of
course, you have to consider the superficial velocity. What is that superficial velocity? This is
the gas flow rate divided by the column cross sectional area of course, this is empty cross
sectional area there is nothing inside the column that you have to consider. So, this is the
superficial velocity.
(Refer Slide Time: 46:41)

And then what is this transport disengaging zone? Transport disengagement zone is the
region in the freeboard beginning at bed surface in which particle flux decreases with height
and above which the entrainment is independent of height. So, this is one important that is
depending or that is being used this term is being used for entrainment purpose. And voidage
or void fraction what is that fraction by volume of suspension or made which is occupied by
the fluid.

(Refer Slide Time: 47:19)

Now, we have learnt lot of things about that what is this fluidization, what is the fluidized
bed, what are the different pattern of the fluidization, and what are the advantages, what are
the disadvantages of the fluidized bed and what are the different terminologies, what are the
different layouts, all these things we have studied. Now what is actually to learn in this
fluidization engineering course? Of course, you have to know something more about this
fluidization what is that you have to know the design proper design of efficient fluidization
system.

So, of course, for designing of efficient fluidization bed you have to know some
hydrodynamics. What is that hydrodynamics? Hydrodynamics means fundamentally you can
say that what is that particle classification which is being actually taking part in a role for
enhancing or just changing the design parameters. Minimum fluidization condition what is
that and how does it depends on different operating variables now what would be the flow
regime. So, that this flow regimes depending on that your other different factors or not, what
is the distribution mechanism of phases, what is the entrainment characteristics, what is the
phase interactions, what is the size distribution of the particles, what is the mixing of the
phases inside the bed, what is the attrition that you have to know.

What is magnetic and what is these there any acoustic field effect on the particle size in the
bed of course, you have to know and also what is the scale of issues all these things you have
to know.

(Refer Slide Time: 49:08)

And other thing is that except this hydrodynamic some other transport processes of course,
you have to know like what is the heat transfer characteristics inside the bed, what is the mass
transfer characteristics inside the bed, and what is that how this fluidized bed can be modeled
or simulated based on this hydrodynamic and transport phenomena of this fluidized bed.

(Refer Slide Time: 49:33)

Now, see this is one performance sheet you can say the performance of the reactor what are
the different factors that effect the performance of the fluidized bed. Now, we will see like
some variables, like design variables, like operating variables, like physico chemical and
thermodynamic properties that will effect on the performance of the fluidized bed.

Now, of course, along with these variables of course, this fluidized bed the dynamics of the
fluidized bed and the transport phenomena of course it is very important to study and know
and then there is design variables like a spudger design, reactor geometry, what should be the
reactor geometry reactor internals catalyst, activity size and concentration heat transfer duty
and others. What are the other operating variables like gas flow rate, liquid flow rate, gas and
liquid recycle rate, what are the free temperature and composition, what is the catalyst
renewal rate, what is the pressure and other several variables that will be included. And
different variables of course, effect on different fluid dynamics and transport characteristics
inside the bed.

Like see here bubble formation growth, coalescence, removal of bubble size destruction, even
gas hold up distribution liquid recirculation phenomena even liquid turbulence and back
mixing is there or not, even you will see catalyst recirculation agglomeration all these
phenomena, even liquid solid mass transfer flow regime. All this phenomena depends on the
factors like design variables and operating variables. Of course, detail depends on the spudger
design, reactant geometry, even gas flow rate, liquid flow rate, size of the particles type of the
particles, even pressure frictional pressure what is the frictional, frictional pressure all these
important variables that effect on these characteristics.

Now, if you consider the reactor performance you have to of course, consider these fluid
dynamics and transport phenomena and also kinetics of this fluidized bed. So, all the kinetics
transport phenomena, fluid dynamics, depending on basically variables like geometric
variables what is the size of the bed, what the size of the particles, what is the size of the
column, and what is the length, what is the breadth, if it is two dimensional, if it is
cylindrical, what is the diameter and also you can say the is there any internals are being used
like any bubbles for internals to reduce the mixing, is there any provisions are being used or
not and what is the size of that provisions that size also will effect on the hydrodynamics and
transport characteristics of the fluidized bed.

Even what is that operating variables like or dynamic variables you can say gas flow rate,
liquid flow rates, even gas and liquid recirculation rate, this will effect on the hydrodynamics
and transport characteristics. Even other thermodynamic variables like what is that pressure
temperature that will also change the hydrodynamic characteristics of the fluidized bed.

So, thank you for this class. Next class we will be discus with some other topics.
 Fluidization Engineering
Dr. Subrata K. Majumder
Department of Chemical Engineering
Indian Institute of Technology, Guwahati

Lecture – 02
Particle Properties and Classifications

So welcome to this massive open online course on fluidization engineering. Today, lecture 2
on particle properties and classifications. So, in fluidization, particle properties and it is
classifications play an important role for the behavior and also the performance of the
fluidization in a fluidized bed.

(Refer Slide Time: 01:04)

So what are these actual particle properties? Thus, particle properties are important because
you know that the different sizes shape and strength and true density particle outline also zeta
potential acting in such a way that the mobility of the particles inside the bed that will govern
the efficiency of the fluidized bed also different size and the shape of the particles also
important for that.

So, you see that these properties like; size, shape, strength, porosity, true density and acts the
roll on flow ability, reactivity, caking even segregation behavior in the fluidized bed what
should the absorption and adsorption characteristics that depends on the particle properties,
what is the caking properties inside the bed? And what should be the attrition and friability,
of course, it is very important for the fluidization. So, we have to know these particle
properties before starting the hydrodynamics of the fluidization engineering or in a fluidized
bed, of course, we have to know.

(Refer Slide Time: 02:24)

And what is the test for a particle? So, that you can know what should be the size and shape
of the particles? And what should be the distribution of the particle? And also, what should
be the particle density? How will this particle density behave? All these things to be known,
so particle size distribution being done by sieve analysis or laser light scattering technique.
Whereas particle density, of course, you cannot use a bulk density for the purpose-specific
purposes mostly, it is being used as a true solid density of the particle for analysis.

And also, what should be? What should be the shape of the particles that can be obtained by
microscopic just images? You have to take some images from the by microscope and then
analyze what should be the shape, size. You can analyze by taking the image by microscopy
by enlarging or magnifying the images and analysis by some software. And also, another
important property of the particle, which is called dustiness, will give you the whether these
particles will make any dust; that means, here, if any fine particles are forming, then a dust
reformation, of course, will be there.

In that case, because of the ejection velocity that dustiness will create a problem, and also
sometimes clogging and coagulation also possible for a particular fluid velocity by it is
ejection through the distributor and all different types of distributor that can be used in the
bed. Now it can be measured in such a way that that what should be the gas ejection velocity
depending on the gas velocity it will be measured or estimated the dustiness.

(Refer Slide Time: 04:27)

And also, other properties like; what should be the particle and moisture interactions, of
course, any particle whenever it is being used in fluidized bed will contain some moisture.
What will be the equilibrium moisture that contains the particles? And also, what should be
the other properties of the interaction of the moisture with the particles? You will see some
other properties like saturation moisture content what should be the saturated moisture
contained in the particles and also it should be needed to know because the maximum
moisture material can retain without becoming a slurry, so it is required to know for a
particular analysis, and also moisture absorption or desorption is a very important factor here
you will see powders tendency to gain or lose moisture at a constant temperature and is it is
important to understand caking behavior.

And another important factor is dust, extension, moisture, so it is nothing but the material the
property material property by which you can say whether this material will emit dust or not.
So, moisture at which this moisture does not emit any dust this is called dust extension
moisture, another important factor is called transportation moisture limit, of course, this
transportation moisture limit will evaluate the characteristics of the liquefaction behavior of
the material. So, these are the important properties for particle moisture interactions or to
know the characteristics of the particle, and these are like suspicion moisture content
moisture sorption or desorption you have to know, dust extinction moisture you have to
know, and transportation moisture limit also you have to know.

(Refer Slide Time: 06:37)

You will see some other properties like; particle strength and durability it depends on particle
material, of course, the durability of the particle whenever in fluidization operation particles
will be used you see sometimes the particles will be broken down at a high velocity, so in that
case, the strength of the particle is important for this purpose. Now a particle breakage force
as a function of the time required to understand breakage strength of the pellet us whatever it
is being used whether this is the this will form any agglomerates or any finished products will
be good in nature or not that also will be depending on the strength of the particles.

So, if it is more strength, then, of course, it will be the particle breakage will be less. So, here
sometimes, you will see the finer particles you will see to break it very problem, whereas the
hard particle it is some particles are easily broken down some particles are you are not that
much extinction of durability what it is to it is original condition.

And drop shatter, your measurement of freeze conditioning agent’s effectiveness on

preventing coal caking is very important here. Critical for coal bunker designed for freezing
conditions with draft shutter is a very important factor for the fluidization, now unconfined
compressive strength. This is another important property of the particle strength, so you see
the rock breakage strength required for crusher or other communition equipment selection.
So, unconfined compressive strength you have to measure before starting the fluidization
operation.

(Refer Slide Time: 08:41)

Another important this is a very crucial factor that particle size and concentration because
you will see the size of the particles will make what should be the flowability of the
fluidization. Now, different patterns of the fluidization depend on this size; if there is a
smaller or finer or it is a coarser particle you will see a different pattern of the fluidization
behavior characteristics that changes based on the particle size, sometimes very fine particles
are not suitable for fluidization sometimes some coarser particles are being actually suitable
for a specific application in the fluidization operation, you cannot use where very coarser
particles that will be very tough to fluidize also very fine particles also it is not suitable
because it makes some cohesiveness nature or making a clock inside the bed or you can say
agglomeration formation also this sometimes possibility there too in the fluidization.

So, this particle size and it is concentration has a direct influence on reactivity or dissolution
rate like; catalysts and tablet’s in that case these are used for reactivity or dissolution rate in
the fluidization operation.
(Refer Slide Time: 10:15)

Another ok, important thing is that; how will you represent the particle size? There are
different way to represent to the particle size like here sphere of same maximum length
sometime it is being used to represent the size of the particles, and you will sometimes see the
minimum particle diameter as a sphere of the same minimum length it is being represented to
show that flow behavior needs in the fluidized bed. And also the sphere of the same weight;
of course, you will see if you are using the particle of the same weight and what will be the
size of the different particles? And if you are using the same weight, then the sphere of the
same equivalent weight it will represent what will be the size of the diameter also diameter or
size of the particle.

And also the sphere of the same volume, it is represented by the same volume of the sphere if
you are represented in what will be the equivalent diameter of that particle. Also sphere of
same surface area it is very important you are just comparing that it the different shape of the
particles are there we are comparing that the surface what will be the surface area and if you
make it the equivalent spherical surface area then what should be the size of the bubble that
you can represent it? That is represented by ds.

Another important sieve sphere passing the same sieve aperture, then what should be the
aperture of the sieve? That represented by the particle size also or this type sieve aperture will
represent the particle size. Generally, we measure in the operation mechanical operation lab
how to measure the particle size by the sieve or screen. And however, for irregular shape
particles, it is not always appropriate where the size in the so as to where the size in at least
one dimension can differ significantly from that of the other dimensions, here see one figure
here this is the one shape of the particle see here this is rectangular, this is cylindrical, and
this is spherical.

So, all these different shapes of the particles you will see you have to make it one equivalent
size of the sphere, so that what should be the diameter of the sphere and if you make the same
volume or by same surface or by maximum length or minimum length or the sphere having
the same sedimentation rate, if you are using this type of concept to make the equivalent
particle diameter you can have. So, we will show later on that what should be the different
definition of the particle how to measure? And how to estimate the different mean of the
particle we will show later on.

(Refer Slide Time: 13:13)

And then particle size distribution, of course, for a mixture of the feed, you will see if your
solid particle feed is there, you will not get the same all the same size in the mixture. So,
there are different sizes of the particles that will be in the mixture, maybe within a certain
range.

Now, what should be that? What should be the number of particles within a certain range of
the size, if you divide the different classes then you will see there will be a different number
of particles will be under in the certain classes. So, if you make it n number of classes, what
will in each class within a maximum and minimum actually aperture, or you can say a
diameter range you will get certain y number or z number of particles will be there. So, for
perfectly monodisperse in that case, every single particle has exactly the same dimensions if
you are using, it will consist a statistical distribution of particles of different sizes, it is normal
to represent this distribution in the form of sometimes in frequency terms and some
sometimes in cumulative terms.

This is here frequency terms the x-axis it will be in your diameter of the particle in y-axis it
will be the RDF, RDF means a relative density of the frequency here what will be the
suppose particle up in a particular bean at in a particular class, how many numbers of
particles out of a total number of particles then it is called a relative frequency function or the
cumulative density frequency function that is called. So, here, in this case, you will see the
distribution will be like this, and from this distribution, you can say in which particle
diameter will have the maximum number of particles inside in the sample.

And also a cumulative representation of the distribution is important in this case; in this class
what will be the number it will be just summing of successively and then it will be
represented as cumulative distribution function and the function will be profile the functional
profile will be the lie with these, and here that is. So, is our particular particle diameter what
should be the, a number of particles also you can obtain from the distribution. So, the particle
size distributions are classified into; weighted distributions, number weighted distributions,
volume-weighted distributions and intensity weighted distributions, this intensity weighted
distributions mean suppose if you are by analyzing the particle size which it is certain color
for a certain color any image or dynamic way, then how this intensity of the color will change
according to it is size of this particle it will give you the; some distribution.

Volume weighted distribution is also important what will be the volume of that particular size
of the particles will give in a percentage that is number weighted distributions like what will
be the number in a particular class of the particles that will be given by a distribution. And
weighted distributions, of course, it will be what will be the weight? What is the mass percent
of the solid particles in a sample that will be represented by a distribution? So, this is very
important if you are having all the sizes are the same in the sample then you will get the only
one-line distribution of the particles; whereas, if there is a mixture of different sizes particles,
you will get this type of distribution.
So, if suppose this distribution is very narrow. What does it mean? This; all the particles will
within the very short range of the particle sizes if it is wider in range; that means, here, the
sample consists the very minimum to the very maximum sizes of the particles in the sample.
So, from the distribution, you can obtain what will be the particle, this particle size
distribution is very important for analysis the hydrodynamics or flow behavior of the particles
or any process yield that depends on this particle size distribution.

If you are using finer particles, of course, the surface area of the particles will be more, in that
case, the catalyst activity of the fine particles should be more there may be mass transfer will
be more; whereas, the but larger particles or medium size particles there if you use there will
see some heat transfer operation mass transfer operations even some other drying operations
also it will be more favorable. So, you have to know what should be the particle size
distribution ok.

(Refer Slide Time: 18:26)

So, this is important for the fluidization operation. If we see this diagram size distribution
variance here increases and here it is size decreasing. What does it mean? They are the two x
scales; if I represent the fluidization characteristics, you will see if we just consider this one.
This is a low expansion sluggish. What does it mean here? The size of the particles is very
large whereas, the size distribution of variance will result in a low. So, in this case, the low
expansion sluggish inside the bed will be there; that means, sluggish will be more whereas, so
there will be no expansion of the solid particles is possible inside the fluidized bed.
Another important aspect is the particle and powder fluidization where you will see the size
distribution if it is there is a mixture of different sized particles you will see, the size
distribution will vary, in that case, the wider size distribution may be possible, but all the
particles are very smaller in range, so in that case, it will exist here, but it may change if you
decrease the size, so in this case whatever size if you decrease the, even more, finer particles
as a finer you will see smoother fluidization you can obtain relative to this fluidization in this
case.

And also here, if you consider these very bigger particles mixed with a finer particle, in this
case, the segregation in the fluidized bed will be feasible, and that means, easily you can
segregate these particles so that this distribution will be in this case you will see some bigger
particles should be and smaller particles the distribution will be in such way that you can
easily identify these particles, what would be this particle? What would be this particle? What
is the fraction of this particle?

What is the friction fraction of this particle from the distribution itself you can easily identify,
at a different location if you measure the size distribution of course if you take the samples
and also analyze then you will get the size distribution and from this size distribution you will
be able to say this sample will have more finer particles or these samples will have more
coarser particles. So, in this way, you can segregate the solid particles by fluidization by
taking it is or analyzing it is size distribution. Also, you will see if you decrease the size even
more; that means, here if the less than suppose 30 microns what will happen, the
cohesiveness of the particles will be there because intermolecular attraction will be more, in
that case, the fluidization is very difficult.

So, for finer particles, very finer particles, if you have all the finer particles, are the same in
size, the particles the size description will be narrower. So, in that case for finer particles with
narrower distribution, it will be it will not be suitable for fluidization whereas for whereas, it
can be suitable if you just mix with some other coarser particles with that there will be
intermolecular attraction will be changed and also if you use different size different nestled
particle also likes of positive or cationic particles, of course, you will use, you will sometimes
see the positive ion particles or ion particles will bind with the negative ion particles and
there it will be fluidized.
So, in that case, some specific example of the fluidization it may be a suitable for having this
type of particle size distribution with this cationic and anionic particles also, sometimes
elutriation is very important here important in that case also you have to know how much
elutriation is possible from the bigger particles or if you initially use all the bigger particles
and after elutriation, after break down of the particles and if you analyze that how many solid
particles are being broken down and what will be the size distribution of that? Then from
these, you can at least interpret what should be the elutriation rate? What should be the
elutriation efficiency? You can analyze it by the particle size distribution.

(Refer Slide Time: 22:59)

And for the distribution, of course, you have to analyze something not only the simple
distribution, no it will not be simple distribution. Here you have to know what should be the
mean of that particle size or over a size of the population that you have to move what should
be the median size where 50 percent of the population is below or above that you have to
know and also another important statistical parameter is called mode the size with the highest
frequency, of course, you have to know. If you have the highest frequency; that means, that
particles will be more in number in the sample and also means sometimes these lower range
to higher range if you are making mean; that means if you there are different type of means
you will get; sometimes arithmetic, sometimes surface mean sometimes volume mean
volume surface means there are different types of means definition will be shown later on,
but you have to know what should be the mean size of the particles by which you can analyze
the further efficiency or simulation purpose or modeling purpose you have to know the
particular size of the particles ok.

So, you have to use the mean size of the particles. So, this statistical parameter for the size
description you have to analyze by mean made in and more, these are the most important
statistical term there are some other statistical parameters are there, but so these are these
threes. They are mostly used for fluidization purposes.

(Refer Slide Time: 24:31)

Now, what is the mean of a different particle? That is being used for fluidization that you
have to know, what is the definition of the means here? You will see a mean diameter you
can define in a different way, general equation of a defining these it is like;

What is p? P for particle; d for diameter and m for these coefficients m for some other
coefficients, so I want to infinity means here individual particles; number 1, number 2,
number 3, number 5, n number of particles will be there. So, if there is n number of particles
and if each particle will have it is own diameter like; D pi is the i-th number of particles will
have the diameter of dp.
So, in that case, if you are defining this d p mean like here if you are considering that mean
diameter. What will be the mean arithmetic linear mean? Arithmetic linear mean is defined
by m is equal to 1 and N is equal to 0, so; that means, here a d p10, dp10 means here it will be as
per this general definition here 1 by N is equal to sum into a summation of what is that? Are
summation of I is equal to 1 to N, N i to dpi, here this is called the arithmetic mean similarly,
d20, d30, d21, d31, d32, d43 these are the different notation for the define definition of the mean
of the particles in a sample.

So, some are d10 it is called arithmetic linear mean, d20 it is called surface mean, even d 21 it is
called surface diameter mean, even d32 it is called Sauter mean, even d43 it is called d broke
mean and also d30 is called volume means. And this arithmetic means here it is a respective
definition. It is given in the slides. So, how will it be defined? And also all this mean
diameter actually for analysis, for different processes are being used specifically like;
arithmetic linear mean generally is used for evaporation purpose and d 20; that means, the
surface mean it is being used for absorption, d30 that is volume min it is being used for any
hydrology of a processes and also surface diameter mean like adsorption, even volume
diameter mean it is the d31 it is I generally being used for the analysis of evaporation or
molecular diffusion with the samples.

And Sauter mean it is very widely used mean for the samples of different sized particles, this
is for efficiency studies of like mass transfer gas-liquid reaction when gas-liquid-solid
reactions in the fluidized bed, this particle mean are being represented by this Sauter mean
this is d32; that means, here volume to surface mean diameter, d43 de broke this is also one
important for fluidization this is for combustion if you are using combustion in a fluidized
bed what should be the equivalent particle diameter is being they are produced then you can
analyze by this a d43, d43here; that means, here m is 4 and n for 3 here 34 volume even surface
square; that means here ah, so this is defined in this way.

So combustion, equilibrium and spray system these means are very widely used, so these are
the different definitions of the means you can use for different purposes, but whenever you
are using a fluidized bed that is the; d32, d43 and these 3’s are very widely driven it is
suggested to use for different operations.
(Refer Slide Time: 29:22)

Now, another important to analyze that particle size distribution by percentiles, for volume-
weighted particle size distribution you will see such as those measured by lesser depiction it
is often actually convenient to report parameters based upon the maximum particle size for a
given percentage a volume of the sample here. Suppose in c particle size and this is volume
person this particle size d50 or d10 or d90 this nothing, but 50 percent d50 here means fifty
percent of the particles are volume percent is like this and also here d90 percent of the
particles are size in d90 like this micrometer. So, percentiles are defined as X, where X is the
parameter usually this is being used for diameter d for diameter and alpha is the distribution
weighting example; suppose n for a number here and v for volume I for intensity and beta is
the percentage of sample.

Below this particle size example like 50 percent, sometimes written as a decimal fraction that
is 0.5. So, here very important the sample below this particle size example: 50 percents of
this; that means, here within this diameter of the particles like X alpha B here in this case
these particles represent the 50 percent of the volume of the particles will be a diameter less
than this particle size. And here 90 percent of the, in this case, this 90 percent of the particles
in the samples will have the diameter of the particles less than this d90 ok.

Similarly, the most common percentiles reported are Dv 10, Dv 50 and Dv 90, as illustrated
in the frequency and cumulative plots in the figure like this, here the this is the 80 percent of
the samples will have the particle size and frequency over overall of the or the total volume is
80 percent of the volume will contain the number concentration of the particles the 80 percent
or number concentration will be within this region of 10 percent to here 10 percent. So, this is
the representation of the percentiles in the first particle size distribution. This is also one
important sometimes if you are going to analyze the particle size by what is that the laser
depletion method to like a million particle size analyzer they are also it is being analyzed by
what is the volume percent? What is the number percentage? You can convert it from volume
percent is to number percent to volume percent if you know the particle size also and within a
certain range of the, what will be the percentage of the solid particles will be in the
distribution that you can read from the distribution of the parameter.

(Refer Slide Time: 32:39)

And then, this particle shape this is also important all the particles you do not get the same in
size, of course, some particles will be more in diameter even some particles will be
rectangular, some particles will be cylindrical, some particles will be only just rod-like some
particles will be even irregular in shape, so these shape of the particles also very important to
analyze the fluidization behavior in the fluidized bed.

Now, in that case, how? This particle shape will also effect on the flow behavior in the
fluidized bed or performance of the fluidized bed like reactivity and solidity like
pharmaceutical actives in that case also it may vary the reactivity because of the size of the
particles, sometimes uniform particle size will be mostly actually favorable for uniform
mixing and also yield of the processes. And also, ceramics inter property is important. In that
case, if you use a different shape of particles, this center properties, of course, will change,
and also powder flow and handling you cannot have the same flowability with a different
shape of the particles.

Even you will see the texture and feel if you use the different shapes of the particles, the
texture of the particles in the fluid as a fluid ingredient will vary. So, particle shape
parameters sometimes in the 2 d projections you can measure by the aspect ratio this aspect
ratio is nothing, but what will be the width of the particles whatever be the length of the
particle by which you can represent the particle shape, or you can say the particle shape, you
can see the particle shape parameter by which you can analyze the flow behavior inside the
fluidized bed.

(Refer Slide Time: 34:45)

And then particle outline, the outline of the particle it will provide the information about
properties such as surface roughness, for conducting particle outline parameters a concept
known as convex hull perimeter is used convex hull perimeter you have if you know then by
this convex hull perimeter will give you the outline of the outline parameter of the particles.
Here see in the remaining the convex hull perimeter some parameters based on it can be
defined such as; convexity or solidity, where this convexity is defined as what will be the
ratio of convex hull perimeter and actual perimeter.

Suppose this is the irregular shape particles here, what should be the perimeter? Perimeter
just considering what that is, the picks of this hull of this objects, so just joining this point of
this peak of this different point of this particle and what should be the perimeter this is called
convex hull perimeter whereas, the actual perimeter will be if you just go through this like it
is exactly, wherever is their curvature if you consider the length of that due to that curvature
this will be the actual perimeter.

So, if you divide this convex hull perimeter by actual perimeter of the object, then you will
get the convexity, so this convexity is the one perimeter one sorry parameter or particle
outline to represent. Another important is solidity that is defined by the area bounded by
actual perimeter by the area bounded by a convex hull perimeter, so in this case what will be
the actual perimeter, of course, you can obtain the and from which you can calculate what
should be the area covered by this perimeter.

And also, what should be the area bounded by convex hull perimeter. So, those things are not
the same, actually, so there will be area differences. So, what should be the area of this
convex hull perimeter and area of the X 1 perimeter? If you just make it ratio, you will get the
solidity. So, this solidity, of course, will be is less than of course 1, because this area bounded
by the convex hull maybe is greater than the area bounded by the actual perimeter.

Circularity is also one important parameter used for particle characteristics. This circularity
depends on the, of course, that what will be the actual perimeter and what will be the
perimeter of an equivalent area circle? So, if you are having, this is suppose this is one object,
and this the what will be the actual perimeter, and if you consider that this perimeter has a
parameter of a circle then what should be the area of that circle, just making it as what is that?
2 pi r is equal to this perimeter and what is that area is pi r square. So, what would be the r of
that? So, you can calculate the circularity from this portion.
(Refer Slide Time: 38:38)

Now, another important zeta potential, this generally be used your; a liquid, solid operations
in a fluidized bed or in gas-liquid-solid operation in a fluidized bed, zeta potentialities are
actually it will measure the magnitude of the electrostatic or charge whether it is repulsive or
attractive, how much charge is there for repulsion and attraction between particles in a liquid
suspension.

So, zeta potential will give you it is measurements now; that means, here, this zeta potential
will give you the what should be the magnitude or degree of the electrostatic charge, which is
being actually for the repulsion or attraction between the particles. And this is of course,
important because sometimes the stability of the solid particles inside the bed for liquid-solid
or a solid, liquid gas operation it is very important because you have to anyway stabilize the
system in a fluidized bed for stability for better dispersion or you can say uniform dispersion
with the stability you have to know this zeta potential.

And whether these particles will attract some other particles or not, it will make coagulation,
or it will make any what are those bigger particles by conjugating with a negatively charged
particles or other charge particles that depends on the stability the potential, can be applied to
improve the formulation of a dispersion, emulsion and suspension of course if you know the
zeta potential you will be able to know how much dispersion will be there, how much
emulsion what is the characteristics of the emulsions? What is emulsified? What extent of
this emulsification is possible, and also is it making any the stable suspension or not you can,
and you can interpret by the zeta potential.

Now, what is that zeta potential on how to estimate that zeta potential? This zeta potential we
are Smoluchowski equation is important here. In that case, the zeta potential is defined as

What is e? e is nothing, but mobility this mobility means here this whenever solid particles
are kept in a solution whatever be these solid particles will try to move aside by attraction and
repulsion of the relatively with other particles also. So, in that case, if you are making it free
there will be no mobility if only due to the settling velocity will move down, but if there are
different sizes particles, differently charged particles if you put into the sample what will
happen one particle will attract another, and another particle will repulse another particle

So in that way, the mobility will change because of this nature of the ion of the particles. So,
this mobility you have to measure, and then, what will be the extent of that mobility? That
can be measured by the instruments specific instruments are being used. It is also which are
size are also several equipments is a standard method to measure this mobility and also what
will be the viscosity? That is, the eta is called dynamic viscosity of the dispersion medium in
which the solid particles are dispersed.

And  is zeta potential and epsilon r is the dielectric constant of the dispersion medium, and
epsilon 0 is the permeability of the free space. Ah. So, these parameters will give you the
particularly the zeta potential; you will see there are some ranges of the zeta potential and
what will be the effect of this range of zeta potential on the stability behavior of the colloid
medium inside the bed. So, from 0 to plus-minus 5, it may be plus it may be minus, of course,
based on the ion of the particles and also mobility. So, in this case 0 to a  5, it will give you
the repeat coagulation or flocculation,  10 to  30 you will see incipient instability and from
 30 to  40 it will give you moderate stability, from  40 to  60 it will give you good
stability, for more than  60 it will give you the excellent stability.

So, if you are getting more excellent stability, you have to use the particles in such a way;
that zeta potential will be more than 60. So, these are there in the picture the different
particles, of course, having different surges some positive some negative and whenever you
are using a surface charge and in surrounding this, there will be a positive ion and even
another layer if you are putting that some other negative ion will come to this positive ion
particles. There will be attraction and so their different layer by making a population of the
different ions and from which you can obtain the mobility or potential of the particles, just
from move one point to another point and from which you can be able to measure or estimate
the zeta potential.

(Refer Slide Time: 44:45)

Now, important classification of the particles; now whatever particles that you are you are
using that is very important, how can it be classified? Of course, you have to know the in
fluidization purpose, whether this classification of the particles, whether it is the smaller
particles or finer particles, or you can support the particles that. So, 1973 geld art; he has
given the standard segregation or standard way of representation of the particle by classifying
in a different group, and he categorized the particles based on the diameter and the relative
density of the particles, the particles to represent that different groups of particles and
segregate that in different groups based on their size.

(Refer Slide Time: 46:04)

 .

And he actually identified four regions in which the fluidization character can be distinctly
defined. What are those four regions of the particle class? He has defined the four regions
like C, A, B, D. What is this C means? Cohesive groove; that means, here the solid particles a
size will be ranged between 0 to 30; that means, up to 30 micrometers you can define this
particle as cohesiveness of particles, this is very fine particles and because of these fine
particles this you know that van der Waals force or this attractive force will be more higher
and because of which you will see that the cohesiveness nature will exist in this group.

So, that is why this is categorized as C group; C means, cohesive. An example like: flow
particles whatever in market commercially available you see the particles are very fine it will
be less than 30 microns in size because this and also whenever you will add some water in the
particles that the particles will you will see you very difficult to segregate from each other.
So, that is why it is called more cohesiveness in nature.

And here, in this case, this attraction between the particles will be very high. And another
type it is called Aeratable, Aeratable means here in this case you will see if you allow some
gas through this space of these particles you will see the gas will be escaping from it is space
between the particles. So, this aeration possible by these particles, in this case, the size of the
particles will be within the; a range of 30 to 100 micrometers, so like milk powder, you can
easily flow, you can easily fluidize this milk powder by just by variation.

So this is called, why is it called Aeratable? In this case, a size will be within the range of 30
to 100 micrometer, and generally, low particle density are being categorized in this case to
1.5 gram per cc, another one is bubbling I think I have shown that that bubbling fluidization,
so bubbling fluidization occurs on the B type particle; this B type particle means the size
range will be 100 to 1000 micrometer; like sand if we are using sand particles sand particle
within the range of 100 to 1000 micrometer you will see the formation of bubble inside the
bed.

So, in this case, the attraction force will be not as much higher than the C type particles or B
type particles, another type it is called D type in this case the particle size will be very a
coastal; that means, the size range will be is greater than 1000 micrometer like coffee beans
the size is greater than 1000 micrometer. So, in this case, the spout able nature of the
fluidization occurs, you will see the gas will be making a channel through the space of the
particles and also there will be the certain thrust of the cash particles and also the movement
of the solid particles downward and upward, and in a certain fashion, the gas should be
moving up. So, in that case, the; this is represented by this spout able type of diameter or
spout able type of type of particles like coffee beans, the drying grains or peace roasting of
coffee beans you will see the gasifying coals also this type of particles are being used for that
operation and it is feasible.

(Refer Slide Time: 50:03)

Effect of particles and the fluidization quality, how this a particle size based on these forces
will affect the fluidization quality. And if you see this if you decrease the particle size and if
you increase the particle size, in this case, the agglomeration behavior will increase because
of this change of the particles if you decrease the particle size agglomeration will decrease
whereas, if you increase the particle size this individual particle behavior will increase.

So, points so this is actually basically that how this force will be applied in the fluidization
quality and also the size, it is matters and what should be the difference as well, so these are
the things basic things that you have to know what will be the particle size, how it will
behave on the fluidization. So, next class, we will discuss the geld art different classification
of the particles of powders and others this size of these particles or classify the size of these
particles how it will change the flow pattern of the fluidization.

Thank you.
 Fluidization Engineering
Dr. Subrata K. Majumder
Department of Chemical Engineering
Indian Institute of Technology, Guwahati

Lecture – 03
Particle/Powder Classifications

Welcome to massive open online course on Fluidization Engineering. Today the lecture is on
particular powder classification.

(Refer Slide Time: 00:40)

What is powder? The powders are subdivided solids which are classified according to the size
of their constituent particles of you ranges from 1.25 micrometer to 1.7 millimeter in
diameter. A powder is a dry and these are special subclass of granular materials, although the
terms powders are sometimes used to distinguish separate classes of material and it is
produced by grinding, crushing or disintegration of a solid substances.
(Refer Slide Time: 01:29)

Of course, the powder formulation is the main important task before going to the fluidization
operation. Now, a good powder formulation of course, depends on the how the particle size
are distributed whether it is uniformly or not that is very important to know. And if the
particle size distribution is not uniform the powder can be segregated according to the
different particle sizes which may result in inaccurate dosing or you can say inconsistent
performance. If the particle size distribution is uniform that is all the particle size in the
powder sample are same then you can say it will be a good powder.

Now, uniform particle size distribution ensures an uniform dissolution rate if the powder is to
dissolve and of course, it ensures an uniform sedimentation rate if the powder is used in a
suspension and it minimizes the stratification when powders are stored or transported or
processed for fluidization operation. Now, the reducing the size of the particles in the powder
is important job because the reduction of this particle size will give you more uniform size
distribution in the sample of powder.
(Refer Slide Time: 03:18)

The process of producing of the particle size is called comminution. These comminution is of
3 types generally trituration, pulverization by intervention and the levigation.

Now, these 3 processes are being used to reduce the particle size for processing of this
fluidization operation for different chemical and biochemical industries because the chemical
process even in biochemical processes generally in pharmaceutical products of course, the
making of powders in fluidization operation the depends on the size of the particles. And how
uniformly the size would be made by reducing this particle size depends on its performance
or to make the powder formation even the tablet formation in the fluidization operation. Even
sometimes the tablet coating that has been done in the fluidization on fluidized bed depends
on the size of the particles. More finer particles if you are making in a powder sample of
course, there will be the more surface area and more uniform size will be formed even after
coating.

Now, what is then trituration method by which you can see the bubbles sorry, this powders
particle will be reduced? Now, this is the continuous rubbing or grinding of the powder in a
mortar with a pestle. So, this is basically a grinding process and pulverization by intervention
is used with hard crystal and powders that do not crash or triturate easily. So, those who are
not crush or triturate easily; that means, the crystal and powders that you have to make the
size by polymerization or intervention a suitable.
The first step is to use an intervening solvent for this pulverization. The solvent which are
being used generally are alcohol or acetone and these step of course, this intervening step will
dissolve the compound of this solvent and the dissolve powder is then mixed in a mortar or
spread on an ointment slab to enhance the evaporation of the solvent.

As the solvent evaporates the powder will recrystallize out of solution as fine particles. Now,
levigation is another one important methods by which you can reduce the size of the particles
in the powder. Now, it is the process by which it reduce the particle size by triturating it in a
mortar or spatulating it on an ointment slab or pad with a small amount of a liquid in which
the solid is not soluble; that means, that means, those who are non soluble solid those soluble,
non-soluble solid can be actually done by levigation process to make it finer.

The solvent should be of course, somewhat dispersed such as mineral oil or glycerin to make
these particle size lower. Now, the sizing of the particles can be done also by screening
initially. So, after screening you will see different classified particles will be forming. That
classification can be defined in different way sometimes the particles would be called as very
coarse sometimes it will be called as coarse and some are moderately coarse some are fine
and some are very fine.

(Refer Slide Time: 07:06)

Now, very coarse particles are being generally defined when the particle size is greater than
1000s micrometer and this is being obtained by screening when the mesh size number of the
screen is controlled 2 to 10. And coarse sized particles are obtained or is defined when its size
is within the ranges of 355 to 1000 micrometer and this can be obtained the mesh size
number of 20 to 40. And moderately coarser particles are being defined when the particle size
is in 180 to 355 micrometers and this is obtained by 40 to 80 mesh. And fine particles which
are being defined as 125 to 180 are obtained by screening by 80 to 120 mesh size number.
And of course, very fine numbers particles which are defined if the particle size is less than
125 micrometer, generally 90 to 125 micrometers particles are being defined as very fine
particles and this is obtained by the mesh size number of 120 to 200.

So, if you increase the number of mesh size you will see the number of; that means, size of
the particles will reduce. So, here in this case you will see in this case here one see table is
given that mesh size number in the slides mesh size number corresponding to this what is that
millimeter in size of the mesh opening size. So, mesh opening size depends on these mesh
size number. And there is a correlations is being obtained here by MOS will be equals to
21584 into msn to the power minus 1.08. That means, here MOS; that means, mesh opening a
size is inversely related to the mesh size number and mesh size number is here that like 2 to
200 generally the standard size whereas, mesh opening size within the range of this 9.52 to
0.074 millimeter in a range whereas, in 9520 microns to 74 microns. So, this is the correlation
say MOS can be obtained directly if you know the mesh size number as by this correlation.

And this screening methods there are 3 types of screening generally that is air swept
screening, when pneumatic screening and vibrating screenings are there.

(Refer Slide Time: 10:50)

Now, of course, the size of particles in the powder whatever you are getting that powder may
not be the same, all the particles may not be same in size, sometimes some particles would be
within a certain range some other particles will be within other ranges. So, there are different
sizes of particles would be there, but is particles may not be the same in shape also. Same in
shape sometimes some particles will be in circular sometimes particles will be in cubical,
some particles may be in the shape of needle like sometimes some particles would be within
the shape of a like sphericity you can some crystalline.

There are various shapes of particles will be there and some spherical also of course, will be
there. But you have to know that what should be the actually equivalent size of that particles
if you are having different shape of particle in the sample. Now, you have to consider that
equivalent size. So, to get the equivalent size some parameter of course, you have to know by
which you can say that how much non spherical it will be and it will be converted to the
spherical one.

Now, this sphericity is one measure by which you can say how these particles would be very
closely related to the shape of a particle of a perfect to spherical one. Now, spherical is
defined as sphericity is defined as that the ratio of surface area of a sphere that having the
same volume as the particle by the surface area of the particle. And here this the equation is
given that

1/3 2/3
Surface area (A s ) of a sphere having the same volume as the particle  (6V p )
s  
Surface are of the particle (A p ) Ap

where, this Vp is called the volume of the particle Ap is called the area of the particle, surface
area of the particle.

Now, for a spherical particle of diameter suppose d then V p will be is equal to that is volume
of the sphere will be is equal to Vp = d3/6 whereas, Ap that is surface area will be equals to
Ap =  d2. So, if you divide these volume by surface area as per these equation given here in
the equation pi is equals to these then you will get that pi s will be equals to 1. So, for
spherical particle the sphericity will be equals to 1. Whereas, for non spherical particle
suppose the cube of width is a, now what should be the volume of that q it will be equals to a
cube whereas, surface area of the cube will be equals to 6 a square.
So, if you substitute this Vp and Ap in this equation then you will get the sphericity of that
cube will be equals to 0.806, even some other non-spherical particles also you will get in this
way the different is sphericity. So, before going to the fluidization operation you have to
consider the particle size as an equivalent particle size by multiplying the sphericity with the
original or actual particle size.

Now, how to analyze this particle size in the powder? There are different methods or by
different equipments are commercially available by which you can also you can analyze with
the particle size.

(Refer Slide Time: 14:29)

Now, there are several methods like sent tributes electrical sensing even lesser a diffraction
method, even light scattering method, even scanning electron microscopic method there are
various other methods also available by which you can analyze the size of the particles
directly in the by the equipment also and the equipment is being actually designed based on
this concept.

Now, sedimentation is one principle by which that centrifuge action, centrifuge by centrifugal
action that you can analyze how much actually size will be there. Even electrical sensing
method of course, it will sense that what will be the characteristics length of the particles and
by which you will be able to get this size. Even this depends on the, that is electrolytic
behavior in the suspension. And also laser diffraction that depends on the how much light is
scattering that is angular or some other way that by which you can obtain the particle size and
optical microscope of course, this is magnifying the server side and then analyzed by the
software and scanning electron microscope also is there directly image analysis software will
be there by use the taking the image and magnifying and then analysis by which you can get
the particle size.

Now, different methods of course, they have some ranges of that is particle size that can
obtain. Now, centrifuge that is by sedimentation method you can obtain the particle size
within the range of 0.01 to 40 micrometer whereas, electrical sensing that you are getting that
will be that you are getting the particle size within the range of 1 to 240 micrometer. Laser
diffraction will be within the range of 0.04 to 2000 micrometer. For a light scattering that is
generally it is being mono dispersed the spheres in that case almost the size will be very low
that is 0.003 to 3 micrometer.

So, whereas, optical microscope it will be magnifying 2x that is 0.52 micrometer 2x

micrometer that is based on the design of the microscopic operation. And then scanning
electron microscope will give the here the size based on the resolution of the picture. Now, it
will be depending on the resolution and then 10 micro meter nanometer resolution that
whatever the principle you can get it. And more details of this method you can get it from the
book like particle size measurements and fundamentals practice quality by Henk Merkus that
is Springer 2009, they published very nice books for that you can go further for more
information.

Now, powder, you have to classify this powder in different way based on the size of the
particles. 1970 according to Richards and Brown, five general categories of the powders are
being obtained or they have actually a reported the five categories of the powder based on the
size of the particles.
(Refer Slide Time: 17:56)

Now, they have classified this powder like ultrafine powder, a superfine powder, granular
powder, even a granular solid and then broken solid, based on the size range of the particles.
And they have defined this ultrafine powder as when the particle size will be within the range
of 0.1 to 10 micrometer, sorry where 0.1 to 1.0 micrometer. Whereas, superfine powder is
being defined as 1 to 10 micrometer and granular powder is defined as 10 to 100 micrometer,
whereas this granular solid will be within the range of 0.1 to 30 millimeter and broken solids
which are being very high in size in range that is 3 to 10 millimeter range.

So, these are the powders that you can classify and based on these classification of the
powders of course, the performance of the fluidization depends. Of course, these way this is
one of the way to classify the powder there are other way that is very common that is called
Geldart classification of powder.
(Refer Slide Time: 19:24)

Geldart (1973), he classified the powders based on the particle diameter and the relative
density difference between the fluid phase and the solid particle. And he categorized these
powders into 4 regions in which the fluidization character can be distinctly defined.

(Refer Slide Time: 19:49)

And he actually reported this 4 types of classification and denoted by C A B and D.

Now, C means what? This is very cohesive in nature; that means, fine particles are there very
fine particles are considered in this case the particle size will be 0 to 30 micrometer like
example flow. Now, this 4 particle; that means, since the particle size is very low; that means,
they are inter molecular force between the solids particles that is fine particles are very high
and because of which the cohesiveness nature will be there. And this vender walls force are
more dominantly acting on this fine particles.

Another type is called A type particles. This A type means aeratable, aeratable here A is for
aeratable and C for cohesive. This aeratable particles actually will be within the range of size
range of 30 to 100 micrometer here of course, the intermolecular force between the particles
will be relatively lower than the cohesive type of particles.

In this case like example milk powder and the density of the particles will be almost equal to
1.4 gram per cc. So, in this case since the particle attraction that is intermolecular attraction,
or inters particle attraction is relatively lower. In this case the particles which are moving
upward in the fluidized bed there will be a particulate bed formation not exactly that some
bubbles will be forming some other very space that is porosity will be more higher, and of
course, there will be no clogging formation said the bed that is not obtained by this aeratable
type powder.

Whereas beat a powder; that means, here the bubbling formation inside the bed if you are
working with this bit a particle, now bubbles will form and to be going upward and during
this upward movement of the bubble at the top of this power, at the top of this bed these fine
particles would be swipe away from the surface and. So, bubbling phenomena will be
occurred by this B type powders, B here bubbling for B. So, the size range of the particles of
this powder will be 100 to 1000 micrometer like sand. Here also the particle density will be
within the range of 1.4 to 4 gram per cc.

Now, another term that is D type particle, D type powder. D type sphere means spoutable in
this case the particle size will be is greater than 1000 micrometer like coffee beans. So, with
this D type of particles it is generally being suitable for fluidization operation for drying or
gains, even a drying, a peas roasting of coffee beans, even gasification of coals, even roasting
of metal ores. So, these operations in the fluidize bed of course, you have to choose the
particle size within the ranges of these D type powders so that you can get easily the
spoutable pattern of the fluidized bed and it will be suitable for drying purposes. That is
physical operation are very spoutable for D type a powders, with D type powders.

So, these are basically 4 types of powders are classified by Geldart (1973). So, based on these
different types of powders the performance of the fluidize bed will be actually estimated and
also the efficiency of the fluidize bed depending on this particle size and also other factors.
Now, what we told that there will be some intermolecular force. So, intermolecular force
between the particles of course, there will be high for queasy when lower than A lower for B
type and then even workforce beta even work for D type particles.

(Refer Slide Time: 24:26)

Now, how these forces acting depending on the particle size for different fluidization
operation even the ranges of the fluidization operation based on this attraction of this
intermolecular particle attraction and. Now, how it will be related? Generally 3 types of
courses are acting (Refer Time: 24:48) bar the fluidization operation is being done with these
particles of 4 types. Now, those are Fd, Fd means here drag force and Fg ,

Fg here it is called gravitational force and also c is called cohesive or Van der Waals force.

Now, this Fd is of course, is related to the a particle diameter this particle diameter of course,
to the power 1 to 2 it varies and g, gravitational force of course, depends on the cube of this
particle diameter and Fc, Fc means cohesive force it depends on the particle diameter. Now,
how it will be? Depends that is inversely proportional to the particle diameter if you increase
the particle diameter you will see the cohesiveness will be reduced. But if it is size is reduced
then more cohesive force will be there intermolecular force Van der Waal force will be more
higher compared to the other forces.

Now, these 3 forces if you add then this total force that will be represented by F. So, here this
Fd is drag force it is nothing, but

that means, here this is Fd drag force is directly related to the fluid density and the velocity of
the fluid and of course, the projected surface area of the particle. Now, C D is called drag force
it will be discussed later on how it is related to the velocity of the fluidization operation and
of course, it will be related with the Reynolds number of the operation it will be shown later
on.

Now, this gravitational force of course, is the relative density of the fluid and the particle and
Fc is called quasi force that is inversely proportional to the particle diameter this related by
this

What is A is called Hamaker? Hamaker constant and Z is called distance between the surface
of the two contacting particles.

Now, here see this F by this relative force that is total force relative to the drag force in the y
axis and in the x axis is called particle size. Now, how it will be related this graph how see if
saw total force is decreasing. Total force is decreasing initially for a particle diameter up to
certain value and then it will increase and it depends on the particle that is powder type for
Geldart C type particles you will see this total force will certainly decreases very steeply
when particle size increases. But for the other types of like delta b and delta d type particles
they are you will see different other forces the domain of the other forces will be in such a
way that it will be higher like Fg minus Fd in the B type particle Fg minus Fd. Here in this case
you will see within this region there will be certain variation of this with respect to particle
size. Whereas, this Fg by Fd it will be increasing with respect to particle size here and it will
be lower for Geldart A part, A type particle where it will be higher in Geldart B type particle
higher to be Geldart B type particular is be more higher.

Now, in this case another type that is called if suppose the particle size is very low; that
means, here dp in this case the tendency to make the agglomeration in the fluidization
operation that is with c type particles. So, is F c by Fd in that case it will be higher relative to
the other particles. In this case dp star is at a particular certain diameter by means you can get
that there are some forces cohesive forces or Fd force or Fd force are similar.

(Refer Slide Time: 29:16)

Now, particle diameter at which the cohesive force becomes nearly the same as the drug force
it will be represented by the d star. Now, this effect of particle size and fluidization quality
how it will be depending. Now, you will see if you reduce the, if you reduce the particle size
here if you reduce the a particle size; that means, dp if for less than that dp star then you will
see there will be effect of structure of the particle, even how where the how what is the shape
of the particle that is very important because there is a cohesiveness nature some surface will
be made in such a way that the two surfaces are joined together in such a way that the
cohesiveness force is acting, acting and it will very difficult to separate out. So,
agglomeration phenomena will be in hence there and dominate in this case.

Whereas, if dp just slightly less than dp star in that case like C type particles here the cohesive
force will be more or greater than F d that is drag force here. So, still here cohesiveness
nature will be pertaining. So, agglomeration behavior of course, pertains in this region, this
region. Whereas, at the d p star; that means, here particle diameter at which the cohesive a
force cohesive force are equal to the drag force in this case unique particle size of course, will
be represented by this phenomena when these two forces are equal.

So, in this case you will see either you are getting the cohesiveness force behavior or some
other behavior like in the individual particle behavior, in this all the particles will be moving
separately from each other. So, this is the transition point or transition force by which you can
get the fluidization operation that is called particulate bit. It is just sliding the particles from
each other without making any agglomeration whereas, if d p is greater than dp star in that case
you will see that cohesiveness force will be less than the drag force. There drag force will be
dominating in that case fluid particle interaction will part in and if you are using the mixture
of A and B particles they are you will see more suitable fluidization occurs. But they are of
course, the liquid slurry system, liquid and solid slurry system there if you use high viscous
fluid then again there be a drag force will be more higher more higher and because of
cohesive you will see some phenomena of the fluidization will hindered.

And then if you increase particle size; that means, here if more greater than d p star then that
case the drag force should be far far higher than that cohesiveness cohesive force or bender
wall force. In that case intermolecular attraction forces will be very low and this happened
only for D type particles like for drying operation that is why the drying operation will be
feasible this type of particles because there will be a separate movement of the particles
possible in the fluidization operation without making any agglomeration.
(Refer Slide Time: 32:41)

And then of course, we can then based on the performance of the fluidization which region or
which particles will give you the better fluidization operation that also can be obtained. Now,
based on the Geldart powder classification they have actually make they made a one, they
made one map by the by considering the relative density of the particle to the fluid and the
particle diameter.

The y axis see here the relative density is particle diameter to the gas diameter sorry gas
density to particle density here rho p minus rho g, here rho p means particle diameter and rho
g means gas density rho p means density of the particle. So, in the x axis is d p the particle
diameter and here some regions of C regions A regions B regions and D regions these are the
regions based on this relative density.

Now, this C regions at any point if you just consider here you will get the you will get the C
type particles; that means, if particle diameter is 20 micron and if you are considering here
these 20 microns the dense relative density will be somewhere here above 500 kg per meter
cube. Now, these regions will refer the C type powder and it is hardly used for your
fluidization operation. Whereas, catalytic reactions will fever in case of a type powder in this
case the since the cohesiveness nature of the powder is less than this C type, so it will be
more useful for catalytic reactions and event for combustion and gasification reactions will be
more favorable for B type and for drying and other poly ethylene production D type products
are.
So, this is one map from which you can calculate which at which region you are going to if
any you are having any particle with a size then you and also what will be the relative density
in which medium you are going to operate the fluidization, then you can obtain of course,
which fluidization operational to which particles will be favorable. Like if you are having the
100 size particle micrometer and you are using the some gaseous medium, in gaseous
medium of density of different and particle relative density here then you can say it is the A
type powder and it will be most favorable for catalytic reactions. Even also if you want to do
the combustion gasification the favorable reason B then you can get of course, if you are
having the relative density of the powder and what would be the particle size you can make it
200 micrometer then it will be favorable for you to do fluidization.

So, A means here aeratable whereas, minimum fluidization for bubbling will be greater than
minimum fluidization that is U mf and material has a significant deaeration time for the like
FCC catalyst and for B type like bubbles above Umf. That means, minimum velocity for
fluidization for bubbling will be equals to minimum fluidization velocity like based on 500
micrometer sand it is suitable for bubbling and C means cohesive flour fly ash, D means
spoutable wheat, 2000-micron polyethylene pellets these are the examples.

So, Geldart they made a map of particular powders based on the relative density and the
particle diameter in this way. So, it will be helpful to identify which particles to be suitable
for getting what type of fluidization phenomena. Now, let us discuss the characteristics of the
different group of particle for its operation for fluidization.
(Refer Slide Time: 37:07)

You see characteristics of group C power particle that is powder you can say. So, this is
cohesive in nature that is very difficult to fluidize and also the channeling occurs channeling
occurs you see here in this video, there you see how channeling occurs here in this case just
see the video one video here see fluidization occurs and then this chunk of solid particles is
moving up and then it will be collapsed and after that it makes the channeling through which
the gas is flowing upward.

So, this happens because of the very cohesiveness of the particles. If you increase the gas
velocity you will sees the channeling will be more channeling will be forming and also the
depth of the channeling will be of course, change. And here you will see another operations
you have seen this fluidization. In this case the inter-particle forces how it will actually affect
the fluidization. And also other characteristics like this mechanical powder compaction even
prior to prior to fluidization, greatly affected the fluidization behavior of the powder, even
after the powder had been fully fluidized for a while.

So, and also you will see another important aspect of this characteristic of group C is that
saturating the fluidization air with humidity because this type of particles will carry more
moisture and it reduced to the formation of agglomeration and greatly improved fluidization
quality. If you make the saturation level in such that is minimum moisture contained in this
particle then you will you can improve the fluidization by just lowering the agglomeration
phenomena inside the bed. And of course, this C type powder in that powder particle size will
be in the range of 0 to 30 micrometer, like example flour, cement etcetera the best example
for this type of particle.

(Refer Slide Time: 39:28)

Now, characteristics of group A, here see what video here see how this fluidization occurs
this group A type particle the size range is 30 to 100 micrometer, here this particulate bed
fluidization occurs. Here there is no you will see bubble formation during this operation and
the surface of the bed is tilting and also there will be a and this is aeratable because you will
see gas will be or air may be will be passing through the solid space in the bed easily, the
smoothly the gas is passing through the space of the solid. So, that is why it is called
aeratable.

And here of course, characterized it characterized by the small particle and low density if you
have the more density then you will not get this particular type of bit. Large bed expansion
before bubbling starts of course, you will see there will be a bed expansion will see some bed
expansion video. And gross circulation of the powder even if only a few bubbles are present
here you will see these particles gross circulation of the powder you can get here.

Large gas mixing of course, you will get for better fluidization operation the mixing of
course, is one important phenomena and gas exchange state between bubbles and emulsion or
of course, will be very high in this case. And bubble size is reduced by reducing average
particle diameter and then the particle size of course, will be within the range of 30 to 100
micrometer. So, this is the characteristics of the group A particles for this of course, that you
can say that this type of group A is very important for catalytic reactions.

Now, bubbling that means, characteristics of group B what is that it will be bubbling; that
means, whenever you are working with this type of particles for fluidization you will get the
you will see the formation of bubbles will be there. That will bubble forms with a certain
diameter, after the size of the bubbles will change and at the bottom of this fluidized bed the
size of the bubbles will be lower.

(Refer Slide Time: 42:10)

Whereas, whenever it will be coming up as per this video here bubbles will be a moving up
and two bubbles whenever it will be coming to each other there will be a coalitions of these
two bubbles and make it a bigger bubbles. And at the top of this surface of this fluidized bed
you will see the bubbles are just breaking into our collapse in such a way that solid particles
will be swept away from the label.

Now, this is basically this bubbling fluidized bed yeah, solid particles of course, you have to
use within the range of 100 to 1000 micrometer. If you use a more than 1000 micrometer you
will not get this type of bubbling phenomena and of course, cohesiveness cohesive type or
particles also you cannot get the bubbling phenomena only this type of particles within a
certain operating range you can get it.
So, solid recirculation rates of course, will be a smaller in this bubbling phenomena and gas
back mixing of course, will be lower in this case because bubbles will be forming and it will
be a carrying the particle or, every the particle in the one direction and gas exchange rate
between the bubbles and emulsion will be smaller. If you increase the diameter of the column
of course, the back mixing may enhance of course, exchange of these particles and the
bubbles of course, will be there. A gas exchange means suppose some particles some gases
will be inside the bubble and also inside the particle, then there will be a chance of
exchanging the gas between the particles and emulsion of course. And of course, the inside
the bubble sometimes some gas of course will be there, there also there will be exchange of
the gas phase from the bubbles to the emulsion.

Now, bubbles size almost independent of the particle diameter. If you are increasing the
particle there may not be changed the bubble size there. Particle diameter will be within this
range like sand particle if you are making the same particulate within the range of 100 to
1000 micrometer and may make the fluidization with a certain velocity above this minimum
fluidization velocity then you will get this type of bubbling phenomena fluidized bed.

(Refer Slide Time: 44:43)

And the characteristics of group D that is called spoutable type of fluidized bed you have see
this video how spoutable phenomena is coming. Here see, important through the center of the
bed the gaseous is just coming out just making a channel with a certain diameter and see how,
but in this case you will see the solid particles also moving downward at the exhausting to the
wall of the column.

Now, in this case the spouting you have see how this gas or you can say that solid particles
will be moving upward in such there will be like you will that is a fall of like solid particles
and making the spout like this. And in this case of course, the character is either very large or
dense particles you can use or bubble of course, coalitions rapidly and flow to large size of
course, will be there. Bubbles rise more slowly than the gas percolating through the emulsion,
if your den space has a low voidage and particle diameter of course, will be greater than 1000
micrometer like coffee beans, wheat and lead etcetera.

In this case this figure see how this a group D particles will be fluidizing here this video also
will show you or this nice fluidization operation you will see. This is of course, with a
particle send particles with what that these detail particles, you can do this operation with
coffee beans wheat also if its size is greater than 1000 micrometer; that means, by 1
millimeter in size. You will see it is there one, see for low velocity of the gas will see there
will be no a fluidization operation this is almost fixed bit condition after the certain velocity
that means, if you increase the velocity beyond its minimum velocity you will see the
fluidization operation will occur.

Now, this starts to the fluidization operation here and the gas is very hard to flow this gas
through this particle, but of course, it will be there of course, you have to maintain the gas
velocity in such a way that the gas will be flowing through the space of the solid particles and
making a spout like this. This is of course, see this in the video how this fluidization occurs.

Like above picture also the same way this with a peddy that had been, it has been done and
then this spoutable nature is formed in this picture. Other important thing at that before going
to that fluidization operation different forces that acting now how these forces will be
balancing and from who is how to calculate the drag force and drag coefficient is very
important to know.
(Refer Slide Time: 48:38)

Now, forces on particles from gas or liquid that is particle fluid interaction is very important
aspect of fluidization you have to know the different forces in different regions of flow rate,
different ranges of flow rate if you are operating the fluidized bed very low flow velocity then
it is called of course, Reynolds number if it is less than 1, it is called stokes range of
operation; that means here the drag force would be higher that case it depends on the
viscosity of the fluid and the size of the particles of course, other important factor is the
velocity.

Now, since in stokes range the velocity will be the terminal velocity or settling velocity and
in that case what will be Stokes drag force that will be calculated by

where is it  is called viscosity, v is called velocity and x is the characteristics length of the
solid particle. If it is spherical of course, this characteristics length will be diameter of the
particle. And here in this case for turbulent drag if suppose the Reynolds number is greater
than 1, even more than Reynolds in this case particle Reynolds number of course, we are
talking about if it is not the stokes condition then you will get this drag force as that depends
on the surface area of the particles and the kinetic energy acting on the particles. This kinetic
energy is
and CD is called here the proportionality constant of this kinetic energy by which you can get
the drag force it is called drag coefficient. So, this turbulent drag you can calculate by this
equation as

This is f is the fluid density.

Here see here this balancing of this force F and v here, this F is that is drag force is acting
upward, whereas this velocity if it is that will be in the downward and also the settling
velocity will be acting downward here, that is in favor of gravity it will be acting.

(Refer Slide Time: 51:19)

Now, particle settling of course, at settling condition you will see the balance of these two
forces one is called drag force or you can say and the stokes drag force and another is called
gravitational force that is m into g. This m, this mass of the particle and this g is the
gravitational acceleration and then F is equal to m dash to g and where the settling velocity is
v which is represented by vs, s for settling. So, here stokes drag force is
and that would be equals to your apparent weight of the solid particles. This apparent it will

be calculating as , this is the volume of the particle and then what will be the relative

density of the particle that is . So, this is called mass of the solid and this g where
it will be weight of the solids.

So, this is stokes drag force will be balancing by this weight of the solid. So, from who is you
can calculate what will be the vs, vs means settling velocity. So, at the equilibrium condition;
that means, that when the solid particles will stay by balancing these two phases or two forces
then you can of course, easily calculate the settling velocity from this equation.

(Refer Slide Time: 53:01)

Now, here Reynolds number of course, will be less than 1, but if Reynolds number is greater
than 1 or if suppose Reynolds number will be more higher or at any Reynolds number you
can say beyond this stokes condition you can calculate this settling velocity like this here
balancing these two forces. What are this balancing two forces here? F is the force acting on
this particle whenever it will be a moving inside the bed and then it will be calculating as m
dot m dash to g here m dash is nothing, but here relative mass of the a solid compared to that
fluid here ms minus mf, ms means mass of the solid, f means mass of the fluid. So, here this is
the relative mass and our effective mass of the solid into g that is gravitational acceleration.
So, this will be equals to total effective weight of the solid when it will be moving with any
Reynolds number.

Now, from this equation you can calculate just substitute the value of this C D; that means, this
force will be calculating this is not as the Stokes drag force this is that other than a drag force,
other than Stokes drags force this is at any Reynolds number it will be depending on the
kinetic energy of the fluid by which these solid particles is moving. So, in this case it will be

So, that will be equals to this effective weight of the solids.

Now, from this two forces you can calculate this v s squared equals to this. So, v s will be is
equal to root over of this. So, you can directly calculate what will be the terminal velocity or
you can say settling velocity of the solids at any Reynolds number from this equation.

(Refer Slide Time: 55:11)

Now, see particle settling if you are representing this particle settling velocity normalized
particle settling velocity suppose at any velocity, if you are operating the fluidization
operation with this solid particles what will be the velocity of the fluidized fluidization at
which this fluidization occurs that is represented v and what do the settling velocity then v by
vs it is called normalized settling velocity.
Now, at low density region you will see this little v t; that means, normalized settling velocity
does not change; that means, here the volume concentration if you are maintaining very low
then there will be no change of this normal settling velocity here and normal setting. But
increasing the solid concentration or volume concentration in the fluidized bed you will see
there will be a peak of this normalized at a certain volume concentration whereas, this peak
will decrease, this peak will decrease gradually if you increase the volume concentration. So,
this normalized settling velocity will be changing with respect to the volume concentration.

(Refer Slide Time: 56:23)

Now, drag coefficient how to calculate just balancing the drag force and the effective weight
of the solid in the fluidized bed from which you can get the drag coefficient. So, drag
coefficients it will be equal to drag force by area. What will be the fluid stress is there. So,

drag force is F and this area this is projected area that is , this x is the characteristics length
or diameter of the particle and this is the fluid stress is nothing, but the kinetic energy here

Now, this CD for turbulent flow of course, this will be F will be is equal to turbulent
condition, at turbulent condition what will the force drag force is applied and this is generally
remains almost constant at 0.44 a turbulent conditions whereas, and the Stokes range of the
operation that is Reynolds number if it is less than 1 you will see this drag coefficient will be,
will be is equal to 24 by Re. How it will be here.

that will be equal to just after rearranging you can get it this is 24 by this Reynolds
number.

What will be the F turbulent? This is nothing, but

This Reynolds number is nothing but, this Reynolds number is nothing but here rho f density
of the fluid into velocity of the particle at which it moves not settling velocity here and into
characteristics length divided by eta; that means, viscosity of the medium. If it is gas then gas
density, gas viscosity if it is liquid of course, it will be the liquid viscosity.
(Refer Slide Time: 58:29)

So, this normalized settling velocity will be changing with respect to the volume
concentration some important dimensionless number of course, population operations is
important here like Reynolds number, drag coefficient, Weber number, even Eotvos number.
These are actually very important dimensionless improved by use you can express the
fluidization phenomena physically.

Now, Reynolds number if a response is of course, this is a ratio of inertia force to the viscous
force, but they are fluidized bed you will see whether inertia force will be dominant to the
viscous force or not. Of course, if you are operating the fluidized bed with the gas flow then
of course, inertia force will be higher than the sorry inertia force will be higher than the
viscous force. And here CD this drag coefficient actually signifies the ratio of drag gravity to
inertia of force whereas, Weber number it will give you is there any surface tension effect is
there or not. So, Weber number is nothing but the ratio of inertia force to the surface force or
surface tension force or capillary force you can say.

Inverse number will give you the ratio of gravitational to the capillary force of course, by
which you can say whether the quadratic part of that shape of the bubble will be there, what
will be the whether it will be circular or whether it will be the elongated bubble will be
forming that depends on the surface tension and also other size of the particle even other
operating conditions.
(Refer Slide Time: 60:07)

Now, important commercial catalyst particles that zeolite, cracking, for cracking operation,
silicon dioxide and aluminum oxide mixture that is for cracking operations cobalt
molybdenum and aluminum oxide this is suitable for hydro treating, even nickel aluminum
oxide hydrogenation operations important. Iron aluminum oxide when potassium oxide these
are very suitable for ammonia synthesis fluidization operation. Vanadium pentoxide is most
important for partial oxidation and even platinum gauze is used for ammonia oxidation in
fluidized bed.

(Refer Slide Time: 60:41)

Some other commercial are developed catalyst particles are cobalt vanadium, even cobalt
thiocyanates, even other different derivatives of this cobalt formation, cobalt and vanadium
chelates like this cobalt phthalocyanines, thiocyanates, this type of commercial develop
catalyst particles are being used in industry.

(Refer Slide Time: 61:11)

Hydrocracking developed catalyst like HC-140LT, HC-205LT, HC-120LT even HC-185LT,

generally it these are being used for in Honeywell Company for their hydro cracking in
fluidized bed.

Thank you.
 Fluidization Engineering
Dr. Subrata K. Majumder
Department of Chemical Engineering
Indian Institute of Technology, Guwahati

Lecture – 04
Minimum Fluidization Velocity: fluid-solid System

Welcome to massive open online course on fluidization engineering. Today’s lecture will be
on minimum fluidization velocity of a fluid-solid system. Before going to that minimum
fluidization velocity, we have known something about terminal velocity of the particle
because this terminal velocity of the particle has an immense rule.

(Refer Slide Time: 00:57)

Whether the fluidization is occurring or not because beyond this terminal velocity of course,
fluidization will start so, what is that terminal velocity of the particle in the fluidized bed or
fluidized manner that you have to know. This is basically the velocity at which to the particle
falls freely through a fluid, and at the terminal velocity, the weight of the object, or particle
you can say here is exactly balanced by the upward buoyancy force, and of course, it will be
balanced by the drag force.

And here see if you consider based the weight of the particle as W and a buoyancy force is
denoted by Fb here and drag force is represented by F D then this weight of the particle will be
balanced by the summation of these two forces like Fb and FD Fb means buoyancy force and
FD means drag force what is drag here drag is nothing, but a force acting opposite to the
relative motion of any particle moving with respect to a surrounding fluid.

Now, what is that how to calculate this weight of the particle W this weight of the particle is
nothing, but mass into gravitational acceleration how to calculate mass if you know the
volume of particle-like what will be that volume of the particle if you know the effective
diameter of the particle d p then pi by 6 into d p cube it will be the volume of the particle and
density of the particle is rho p then volume into density then it will give you the mass of the
particle then the mass of the particle into gravitational acceleration g it will give you the
weight; that means, that force that will be acting downward that is the under gravity.

Fb is the buoyancy force. what is buoyancy force? buoyancy force is the force that is applied
on a particle by the fluid what is that? this buoyancy force is nothing but how much volume
of fluid is displaced by this particle of course, the volume of the fluid will be displaced by the
volume of the particle. What will be the fluid that is being displaced by the particle is

 3
Fb  dp f g
6

and then this mass volume that will be considered for fluid volume by which the buoyancy is
acting on the particle.

Now, this volume into density of the fluid, then you will get the mass of the fluid then into g
this gravitational acceleration. This mass into gravitational acceleration it will give you the
buoyancy force here this mass will be the how much mass of the fluid that is displaced by the
solid particle and then FD, FD there is drag force two types of drag force you will get
sometimes you will see for very laminar region.

That means flow is under terminal condition means the fluid is acting on the particle when
the particle is going downward under it is that is normal gravitation normal gravity that is
called a terminal condition; that means, they are the velocity whatever it is it will less than it
will be less than 0.2 for Reynolds numbers. Reynolds number if Reynolds number of the
particle if it is less than 0.2 then the drag force will be calculated as this by stokes flow that
will be equals to 3 into pi into mu into d p into terminal velocity.

Under this terminal velocity what will be the drag force, this will be
FD  3 d p ut for Re p  0.1

and this is applied only for Reynolds number if it is less than 0.1 what is that Reynolds
number is nothing, but this what will be the in inertia force under this the terminal velocity. It
will be inertia force, and it will be the ratio to the viscous force. Reynolds number is nothing
but the inertia force to the viscous force.

Now, this Reynolds number of the particle will be calculated based on this particle diameter
here. So, Rep will be is nothing, but

 f ud p
Re p 


now if suppose Reynolds number is greater than 0.2 or it is generally for Kunii and
Levenspiel told that that it will be if it is less greater than 500 then this drag force will be
calculated by this here this drag force will be directly related to the kinetic energy of the fluid
at which this moving under this terminal velocity. From

W = Fb + FD

 3  1 
d p  p g  d 3p  f g  CD  f ut2 d p2
6 6 2 4

Now, what will be the a is the projectional area of the fluid particle and U t is the terminal
velocity of the solid particle that will be is equal to

 gd p2 
ut   ( p   f )
18 

1
 f ut2
2 this is called kinetic energy by which this solid particle seal will be moving
downward or upward or any way that is under this terminal velocity.

Now this drag force the drag force is directly related to this kinetic energy projectional area
of the particle, then if what will be the then const proportionality constant that proportionality
constant will be is equal to CD that is called drag coefficient. So, FD will be is equal to
 1 
CD  f ut2 d p2
2 4

This will be equal to the drag force. Drag force this is under the condition of Reynolds
number is greater than 500.

Now, how to calculate this projectional area if it is a fluidical particle, of course, this
projectional area of the particle will be is equal to that is cross-sectional area of the particle;

 2
dp
that means, 4

 2
dp
This is your cross-sectional area of the particle if it is fluidical of course, this will be 4

Now, if we if we equalize this force is as per this equation

W = F b + FD

(Refer Slide Time: 08:19)

  3
dppg
if we substitute here for W that will be is equal to 6 and Fb. Fb will be equal to this

 3
dp f g
here 6 and then FD this FD is the drag force if we consider it as a under that is Stokes

3 d p ut
flow that means, here if Rep is less than 0.2 then it will be .

Now, if we simplify it or rearrange it, then you will get the terminal velocity under this
Stokes flow that will be

 gd 2 
ut   p (  p   f ) 
18 

So, here of course, this by this equation you can calculate what should be the terminal
velocity under this stokes flow this stokes flow of will considered if Re p less than 0.2 here see
one particle this on this particle this is one W force is acting and then what is thus drag force
is acting and buoyancy force acting.

This buoyancy force and W; that means, the weight of the particle if it is going downward, of
course, it will be relative velocity will be W minus F b this is called effective weight of the
particle or you can say the apparent weight of the particles this then this apparent weight of
the particle will be balanced by the drag force this drag force the direction of the drag force
will be opposite to the particle motion. If you balance say a similar way, it is nothing, but w
minus Fb will be is equal to the apparent weight.

Sometimes in different references, because they are directly related as the apparent force will
be equals to drag force. The apparent force will be calculated as W minus F b what will be the
relative force of weight and buoyancy force, this by this way we can calculate then terminal
velocity under this stokes flow that if Reynolds number is less than 0.2
(Refer Slide Time: 10:39)

Now, if Reynolds number is greater than 500 that is it is not stokes flow; that means, under
turbulent flow, then, of course, you have to substitute the drag force as

1 
CD  f ut2 d p2
2 4

This is your drag force at turbulent condition; that means, Reynolds number of the particle if
it is greater than 500

Now, this sees Reynolds number how it is defined here. This is

 f ud p
Re p 


Here, mu is the viscosity rho F is the density of the fluid u is the velocity d p is the particle
diameter.

 3
dppg
Now, again if you substitute this W as this a weight of the particle as 6 then you will
see this will be your volume this will be your density and volume into density will give you
the mass into g that is the weight of the particle and this is buoyancy force again the same
way, this buoyancy force how much volume will be displaced by the particle into it is density
it is into gravitational acceleration this will be buoyancy force and drag force under this
condition of Rep is greater than 500.

After rearrangement and simplification, you will see this terminal velocity at this terminal
turbulent flow will be calculated as here just

1/ 2
 4 gd p (  p   f ) 
ut   
 3CD f 

In this way a turbulent condition you can calculate what should be the terminal velocity of the
particle.

(Refer Slide Time: 12:40)

And then, we will represent this terminal velocity and also particle diameter as a
dimensionless form. What will be the dimensionless particle diameter and terminal velocity?
Why we should do this sometimes some references, you will see that it is easier to actually
represent the flow resign and also the flow behavior by this dimensionless number.

We can define this dimensionless number of this d p that was particle diameter as dp star this
dp star will be

1
  g  s   g  g 3
d P*  d P  
 2 
here rho s is the solid density and rho g is the gas density if you are considering the fluid as a
gas then you have to say we are you have to consider this gas density as rho g whereas, mu is
square mu square is also the gas viscosity if it is air then of course, it will be air viscosity. So,
dp will be defined in this way dp star. This is a dimensionless diameter.

After rearrangement you can get it here this totally here

1
  g  s   g  g 3 1
d P*  d P    Ar 3
 2 

Iit will be Archimedes number to the power 1 by 3 here this portion dp cube into rho g nto rho
s minus rho g into g by mu square it is called as Archimedes number already we have defined
earlier this Archimedes number. This Archimedes number here it will be

1
1
3 3
Ar   CD Re2P 
3

4 

this is one form.

Now, u star u star; that means, here you can define it as u into some dimensionless number
here is rho g square by mu into rho s minus rho g to the power 1 by 3 then it will be Re p by

1
1
3 3
Ar   CD Re2P 
3

Archimedes number to the power 1 by 3 and that will be is equal to 4 

In this way this u star will be defined what is that U t star Ut star means what will be the
dimensionless number of the terminal velocity this terminal velocity of this dimensionless
form will be equals to like here this

1
 
18 2.335  1.744 s 
ut*    , 0.5   S  1
 d*  2 * 0.5
 P
d 
 P 

This terminal velocity you can represent in terms of this dimensionless number of particle
diameter as this of course, it will be valid or can be calculated within the range of 0.5 to 1 for
sphericity and then here Ut star will be is equal to then
 1
 
*  18 0.591 
ut   , S  1
  d *  2  d *  0.5 
 P P 

This will be power if sphericity will be equals to 1 like for a spherical particle.

Here it is important to know that what will be the terminal velocity at it is dimensionless form
what should be the particle diameter at it is dimensionless form. This will be actually used
later on for a different aspect.

(Refer Slide Time: 16:16)

And then, terminal velocity this terminal velocity, you can represent this terminal velocity
graphically also.

Now, if we represent this terminal velocity by this graph shown here the slide see here in the
x-axis in the x-axis the dp star is denoted as

1
  g  s   g  g 3

d P*  d P  
 2 

and y axis this is Ut star that is for terminal velocity what should be that Ut into this, by this
how this Ut that is terminal velocity will be related to the particle diameter you will see here
Ut star if you consider the if you consider here.
See the Ut star will be changing with dp for different sphericity you will see for sphericity is
equal to 1, this line will give you the different data of terminal velocity for different d p that is
particle diameter and also for different sphericity will get different terminal velocity, and
from the graph, it is observed that if you decrease the sphericity; that means, if it has deviated
from the spherical particle then you will see the terminal velocity how it will be changing
with respect to the particle diameter, of course, the terminal velocity will increase if d p is
increased.

That means particle size if it is increased then terminal velocity will increase of course, if the
size is increased that very; that means, how fast this particle will fall down freely and for
different other shapes like 0.23 you will see this type of terminal velocity, from this graph
you can easily calculate you can identify the number of terminal velocity at a different
particle of size the different particle diameter.

(Refer Slide Time: 18:31)

Now, let us discuss about the particle fluid interaction now you will see whenever a particle
is flowing in a fluidized bed, or it is falling in a fluid medium freely then you will see the
liquid and particle there will be interaction now there will be some force acting on the particle
whenever it will be moving or falling freely.

We have already seen that there are two types of forces are acting the particle on the particle,
like buoyancy force wind drag force we will see buoyancy force already we have discussed
and stokes there will be if it is there if the particle is falling freely with the laminar flow; that
means, if Reynolds number is less than 0.2 then you will see there will be a force acting on
the particle it is called drag force that drag force is a calculated as

FDStokes  3 d p u

And whereas, this drag force will be acting whenever it will be moving with Reynolds
number is greater than 500 also, in that case, this drag force is called turbulent drag force this
turbulent drag force the will be calculated as

1
FDTurbulent  CD  f u2 A
2

of course, you will see there will be a force acting whenever it will be moving down or
upward based on the particle size also and also what will be the velocity on that the depends
on.

Now what should be the drag coefficient then from the drag force of course, you will be able
to calculate the drag coefficient in the previous slides we have seen that here and the previous
slide here you will see this is the drag force and this drag force is related to the kinetic energy
acting on the particle by which the particle is moving and also the projectional area of the
particle a and this proportionality constant is called drag coefficient.

(Refer Slide Time: 20:59)

Now, more precisely you can say this drag force will be defined as this drag force divided by
area; that means, here projectional area by fluid stress by fluid stress; that means, here this

 d p2 / 4
drag force is represented by FD and here projectional area is the this is dp is the

 f u2 / 2
particle diameter, and fluid stress will be represented as by . This will be is equal to
fluid stress or you can say kinetic energy acting on the particle here.

Now, by this, you can calculate from this equation, you can calculate what should be the drag
coefficient now this drag coefficient for the turbulent flow you will see it will be calculated as
what will be the FD turbulent and then

FD ,turbulent
1
CTurbulent
D 
 d / 4  f u2 / 2
2
p

here you will see u square will be high. So, that the Reynolds number of the particle will be is
greater than 500.

In that condition, it is seen that this drag coefficient will remain almost constant, and this
constant value will be equal to 0.4. At the turbulent condition, the drag coefficients will
remains constant this is will be equals to 0.4 whereas, in the stokes condition; that means, if
Reynolds number is less than 0.2 then you will see the drag coefficient as per this equation it
will be represented by this quantity like this C D stokes will be equals to 24 by Re p here it will
be depending on the Reynolds number of the particle of course, here it will not be remained
constant of course, within this range of this Reynolds number that is less than 0.2 if it is
suppose 0.1 this will be 0.01.

In that region that is if Reynolds number will decrease then you will see C D will increase; that
means, drag coefficients inversely proportional to the Reynolds number. By this equation,
you can calculate what should be the drag coefficient in the stokes condition and also
turbulent condition.

Drag coefficient, you will be able to calculate now different authors. They have actually
obtained different drag coefficients from their experimental data, and they made the
correlations in a different way to represent the drag coefficient.
(Refer Slide Time: 23:44)

And here see Haider and Levenspiel 99 they have proposed a correlation for drag coefficient
and they have represented the correlation as the CD will be equals to

24
CD  1  (8.1716e 4.0655S ) Re0.0964
P
 0.5565 S

Re P


 
73.69 e 5.0748S Re P
, for s  1
6.2122 S
Re P  5.378e

From this observe from this correlation what to see in that if Re p is greater than 0.2, we can
easily calculate what should be the drag coefficient from this correlation and this correlation
from this correlation it is saying that this CD not only depends on the Reynolds number it
depends only on also depends on the sphericity of the particle see how this sphericity of the
particle actually effect on a CD you can calculate it from this correlation.

If you have different sphericity and a different velocity of course, diameter then you can use
this equation to calculate the drag coefficient even for spherical particles if suppose phi s is
equal to one then very simple here from this you can directly obtain that correlation obtained
the value of CD from this correlations are putting the phi s is equal to 1.

Finally, you can get this equation to this equation if we put the phi s is equal to 1 then C D will
be is equal to 24 by Rep plus this. From this equation you will be able to calculate what
should be the drag coefficient of course, this drag coefficient correlation is limited to the flow
if Rep is greater than 0.2 you cannot apply this correlation for stokes condition; that means,
Rep and; that means, Reynolds number if it is less than 0.2.

Now, let us see 1 example here is that they calculate the terminal velocity for the sharp
irregular sand particles if there is a regular sand particle then what should be the terminal
velocity and this, of course, this irregular sand particle is allowed to fall in a air medium; that
means, to air then what should be the terminal velocity.

(Refer Slide Time: 26:30)

The density of the air will be taken as 1.2 into 10 to the power minus 3 gram per cc and
viscosity is 1.8 into 10 to the power minus 4 this is at standard condition at 20 degree
centigrade whereas, this d p star what is the dimensionless particle diameter is 160
micrometer and density of the solid is given here 2.60 gram per cc a sphericity is taken as
0.67.

Now, if you calculate the dp star as this here this is not dp star it will be dp it will 160 then you
will see dp star will be is equal to 7.28 and U t star will be is equal to 1.294 and then finally,
terminal velocity will be equals to 88 centimeter per second.

You can use the equation that has already been shown earlier that how to calculate the ut.
(Refer Slide Time: 27:42)

And then Ut star from Ut star you can calculate Ut as here 88.88 centimeter per second and
then we know now what will be the terminal velocity of the particle. Before going to that
minimum fluidization condition that is why this terminal velocity is required because
minimum fluidization will occur beyond this terminal velocity of the particle.

Now, of course, balancing the terminal velocity; that means, here the force that is net force
what is the acting downward you have to balance that net force to get the minimum
fluidization now this minimum fluidization what is that minimum fluidization; that means,
the velocity at which all the particles are just suspended by upward flowing gas or liquid by
balancing the terminal velocity of the particle.

Now, at this velocity, you will see the frictional force between particle and fluid just
counterbalances the weight of the particles, then only you will get the minimum fluidization
the pressure drop through any section of the bed about equals the weight of the fluid and
particles in that section. So, in any section, you will see a pressure drop to any section of the
bed. It will be about equal the weight of the fluid and particles in that section.

It is happening only for minimum fluidization conditions. The bed is considered to be just
fluidized and is referred to as an incipiently fluidized bed or bed at minimum fluidization.
This minimum fluidization is just balancing or counterbalancing the weight of the particles.
(Refer Slide Time: 29:39)

And then what should be the net force acting on the solid particle at this minimum
fluidization condition you will see here on particle this will be to be fluidized now what are
the force now 1 is buoyancy forces acting on this particle and that will be calculated that is
the volume of fluid displaced by the particle volume into fluid density and then into
gravitational isolation, you can calculate the buoyancy force

And what are the other forces that will be the apparent weight of the particle now this
apparent weight is nothing, but what will be the actual weight of the particle is this is

W – Fb = LA(1-)(p- f )g

what is this? this is not a single particle here. The total amount of particle here what will be
the weight of the particle in the fluidized bed.

Now, what will be the l is the length of the fluidized bed. A is the cross-sectional area of the
bed. epsilon is the void fraction means other than particle what will be the space there, and
then rho p is the density of the particle rho F is the density of the fluid and W A is the apparent
weight.

Now, if this is weight, this is weight means solid weight in the bed; this is the length of the
bed into a cross-sectional area that will give you the volume of the bed. Then 1 minus epsilon
means what will be the volume of the solid particles and into the density of the particle this
will give you the mass of the particle, and into g this will give you the weight of the particle
and buoyancy force, of course, it will be what will be the ala; that means, the volume of the
bed into 1 minus epsilon means this is the solid volume now this solid volume will displace
the water volume or fluid volume or gas volume you can say this gas volume if it is in the
gaseous medium then into g this will give you the buoyancy force.

And what should be the apparent force is nothing, but W minus F b, W means weight and this
buoyancy force this is the apparent weight is nothing, but here is just you will just subtract
this buoyancy force from it is weight then you will get the apparent weight. Apparent weight
is nothing, but W – Fb = LA(1-)(p- f )g

(Refer Slide Time: 32:26)

And from this apparent weight of the particles, of course, you will get the minimum
fluidization.

Now, how to get you to have to balance this apparent weight of the particles with a drag force
that is acting upward acting by upward moving gas. This will be the drag force will be
balanced by this apparent weight of the particle, that you can get the minimum fluidization.

Now, how to calculate the drag force by upward moving gas this will be if you know the
frictional pressure drop across the bed and if you know the cross-sectional area of the bed,
then drag force will be easily calculated here as drag force as fictional pressure drop across
the bed into it is the cross-sectional area. This will be your drag force here. So, this is the total
drag force.
What will be the frictional pressure drop that will give you the drag force, this frictional
pressure drop into the cross-sectional area this will be your drag force and what will be the
apparent weight that is W that will be is equal to this

Wa  ALmf (1   mf )(  s   g ) g

How to actually this will give you the volume of the bed at it is minimum height into what

1   mf
will be the fraction of solid here at this minimum fluidization condition then it will
give you the total volume of the particle inside the bed volume into then what will be the rho
s minus rho g this would be the apparent density or apparent weight from which you can
calculate rho s minus rho g into g.

This already calculated in the previous slides how to obtain this apparent weight. If you
balance this; that means, if you equalize these two forces of this, what that drag force and the
apparent weight you will get the condition of you will get the condition of minimum
fluidization is.

(Refer Slide Time: 34:51)

Pb
Lmf
Now, how to calculate the frictional pressure drop, of course, this is called frictional
pressure drop per unit length of the minimum fluidized height that will be is equal to here this
will be
Pb
 (1   mf )(  s   g ) g
Lmf

this can be obtained just by rearranging this previous equation now here delta p b is pressure
drop, Lmf is equal to minimum fluidizing height epsilon mf is the minimum void fraction and
rho s is the density of the solid and rho g is the density of the gas.

Now, at this onset of fluidization; that means, minimum fluidization condition the void age is
a little larger than in a packed bed condition of course, from this packed bed condition you
are just going to suspend the particles to get to the minimum fluidization. What should be the
minimum void age to get this minimum fluidization? Here actually, this minimum void age
will correspond to what should be the void age in the packed bed that the minimum
condition.

At the onset of the fluidization the void age is the little larger than in a packed bed almost
equals to the little bit higher, but you can omit you can neglect also you can directly consider
the what should be the porosity or what should be the void age in packed bed condition
actually corresponding to the loser state of the packed bed of hardly any weight here that is
why this a little larger than in packed bed condition this void age.

Now, thus umf you can estimate from the random packing data or better still 1 should measure
it experimentally since this is the relatively simple matter, of course, you can calculate to the
void age experimentally otherwise you can directly add this minimum condition you just
consider what should be the void age at packed bed condition there will be hardly an error for
this if you are taking into consideration of packed bed void age here, but it still for the
accurate result you can, of course, measure the void age at this minimum fluidization
condition.

Now, measurement of the pressure drop before going to have this minimum fluidization
velocity of course, you have to know frictional pressure drop then drag force you have to
calculate already you know that what should be the apparent weight of that then you have to
balance these 2 forces then you will get the minimum fluidization velocity.

Now, what should be the frictional pressure drop how to calculate or how to measure also
you have to know.
(Refer Slide Time: 37:50)

Now, the frictional pressure drop can be measured experimentally which can be expressed as
here. This measured pressure drop will be is equal to there are 2 parts of the measured
pressure drop you can measure it by a manometer or by any electrical electronics device like
you know that pressure transducer you can use that to calculate the accurate pressure drop
across the column.

This measured pressure drop will have 2 parts one is frictional pressure drop another is the
hydrostatic pressure drop, now if you consider the gas-solid fluidization you will see the
hydrostatic pressure, of course, this density of the fluid as a gas here it will be very small and
also this hydrostatic part is very negligible compared to the frictional parts.

Only here you can say that measured pressure drop you can calculate directly from the
frictional pressure drop; that means, the frictional pressure drop is equal to the measured
pressure drop here you will see an equation 2 it is seen that measure pressure drop is delta p

  f Lm g
for bed for the frictional part and also this is nothing, but hydrostatic pressure drop
this plus sign stands for up-flow fluid the last term may be appreciable for flowing liquids,
but it can safely be ignored for flowing gases unless one is dealing with the deep bits at high
pressure of course, that high pressure you have to consider this hydrostatic pressure also.
Now, in most cases with gases you can directly obtain the bed pressure drop by considering
the frictional pressure drop. This frictional pressure drop of the bed will be equals to the
measured pressure drop; this is shown in equation number 3.

(Refer Slide Time: 40:07)

Now, at the minimum fluidization of course, the onset pressures drop a fluidization starts and
the packed bed condition you will see this diminishes. So, at the minimum fluidization just
diminishing the packed bed condition and then this minimum fluidization can be then
estimated from the frictional pressure drop now this frictional pressure drop of course, you
can measure it experimentally or without measurement also you can consider this frictional
pressure drop you can estimate the frictional pressure drop from the Ergun equation.

Ergun equation will give you directly the frictional pressure drop in a fluidization bed here.
This is the Ergun equation. By the Ergun equation you can calculate the frictional pressure
drop per unit length as this here the two parts of this Ergun equation 1 is called this is inertia
this is inertia, and this is called a viscous this is viscous layer and this is inertia layer inertia,
this is inertial and this is viscous and inertia.

Now, you will see this Ergun equation is giving you the frictional pressure drop per unit
length. This is a function of viscosity and velocity and particle diameter, even a sphericity.
This Ergun equation you can directly use to calculate the frictional pressure drop.
(Refer Slide Time: 41:45)

Now, equating this equations 1 and 4, 1 can write here this part this

(1   mf )(  p   f ) g

that will be equals to this pressure drop by the Ergun equation; now if you rearrange this
equation, then you will get this final form of this equation now you can denote this big
equation in different ways also.

(Refer Slide Time: 42:20)

And like this you can simplify this as this equation by

 1.75 (1   mf )
 mf3  s
 Re 2
p , mf   150
 mf3  2s
 Re p,mf   Ar

Now, see this equation becomes the quadratic equation of the Reynolds number at the
minimum fluidization condition. This is here see, if we equalize the drag force that is drag
force that is calculated from the Ergun equation with the apparent weight of the bed then you
are getting the quadratic equation of the Reynolds number at it is minimum fluidization
condition.

Where this Ergun number sorry Ergun or this is the not Ergun which is called Archimedes
number Archimedes number or sometimes some references they are represented it as Galileo
number. This Archimedes number is defined as

 f (  p   f )d p3 g
Ar 
2

and minimum Reynolds number is the

 f umf d p
Re p ,mf 


If you solve this quadratic equation you will get the equation of equation for reynolds number
for minimum fluidization condition. Solve this quadratic equation to get the Re mf from this
minimum Remf you will get the minimum velocity. This if you know these Re mf from this
equation of 7 and then just after that you can calculate the u mf from this here what should the
umf then umf will be is equal to umf from this Reynolds number it will be equals to

umf  (Re p ,mf  ) / (  f d p )

From this, you will be able to calculate what will be the minimum fluidization velocity.
(Refer Slide Time: 45:03)

And then what should be the void age that minimum fluidization of course, to get to get the to
get the minimum fluidization velocity you have to know the minimum porosity. minimum
porosity is represented by epsilon mf this epsilon mf how to calculate this epsilon mf this
epsilon mf you can calculate it from or you can consider from the packed bed condition
otherwise you can directly consider this equation which is given by Wen and Yu (1966) that
is

3 1
 s mf 
14

from which is you can calculate for from the minimum porosity here or minimum void age
from this equation.

 mf  mf
depends on the shape of the particle for spherical particles is usually 0.4 to 0.4 5 and
you will see the if the sharp particle is there with the sphericity of 0.67 for 60 micron particle
the sphere void fraction will be is equal to 0.60 whereas, for anthracite coal of sphericity 0.63
you will get the void fraction of about 0.5 to at 400 micrometer in size. This void fraction
depends on the particle size and also it is shape.
(Refer Slide Time: 46:43)

Now, once you know the minimum voidage of the minimum fluidized bed then you will be
able to calculate what will the length of the minimum fluidized bed this length of the
minimum fluidized bed can be calculated from the mass of the bed. what will be the mass of
the bed this will be the what will be the solid particle and also what will be the volume of the
fluid what will be the volume of the solid then you have to add these now how to calculate
here see mass of the bed will be calculated as density of the bed into the volume of the bed.

Now, you can calculate it from this now if you consider only the mass of the fluid then it will
be the density of the fluid into a volume of the fluid and what should be the mass of the solid.
Mass of the solid will be equals to the density of the solid into the volume of the solids. From
this, you will be able to calculate what should be the mass of the bed now once you know the
mass of the bed if you divide this mass of the bed by the what is that by this area into the
density of the particle then you will get this here this minimum fluidized height.
(Refer Slide Time: 48:08)

Now, special cases in the special cases of means whether it is very small particles or not, of
course, you will see in the special cases of very small particle the first term of the right-hand
side of equation 7 equation 7; that means, see our equation 7 uses see here this equation 7
what is the quadratic equation of the Reynolds number to get the minimum fluidization you
will see that the first term of this right-hand side of this equation 7 can be neglected for the
very small particles.

In this case, the inertial effect is negligible compared to the other effect. If Reynolds number
is less than 20, then you can directly calculate the minimum fluidization from these by
neglecting the right-hand side part of the equation 7 whereas, for very large particle whereas,
for very large particle this Reynolds number if it is greater than 1000 first one will be
dominating and hence you can calculate the minimum fluidization velocity from this
equation.

 mf
So, in this case, you need, of course, the value of and the a sphericity.
(Refer Slide Time: 49:33)

And then if you do not know the value of epsilon mf or sphericity you can still estimate the
minimum velocity for a bed of irregular particles as follows in this case you have to first
write the rewrite the equation 7 as this here this equation 7 and in this case as I considering or

1.75
K1  3
 mf s
denoting by here k1 and k2 . this k1 is defined as here and k2 will be

(1   mf )
K 2  150
 mf3  2s

Here this equation 7 is represented in terms of k1 and k2 these two parameters. From these,
you can calculate what will the minimum fluidization condition for Reynolds number of these
here it will be as a function of k1 and k2 and also it is Archimedes number. From this, you can
calculate the minimum fluidization velocity.

Now, k 1 and k 2 stayed nearly constant for different kinds of particles over a wide range of
condition; that means, if r e is within the range of 0.01 to 4000 then you will get the k 1 and k2
within this range now different investigators they got the k 1 and k2 value here see Wen and
Yu they have represented this k1 and k2 has 24.51.

These are actually experimental data they have obtained these from the experiment and
experiment of course, this whatever this k1 and k2 is coming from this equation it depends on
  mf
this and  s , with different minimum This void fraction depends on the particle size and
of course, with different sphericity they have done different experiments and from their
experiment they got this k1 and k2.

Now, using these values for k1 and k2 then the minimum fluidization velocity can be obtained
as follows here see for force particle; that means, see particle size if it is greater than 1000
micrometer.

(Refer Slide Time: 52:10)

Then you can calculate the minimum fluidization velocity from this equation here in this case
k1 and k2 are taken from this Chitester et al. 1984. They got this, and it will be k1 will be equal
to 20.24 and k2 will be equals to 1161.94.

Whereas for fine particles see particle size is less than 100 micrometer then you have to
calculate the minimum fluidization from this equation, in this case, k 1 will be is equal to
24.51 little bit higher than the earlier one whereas, this k 2 will be is equal to again this k 2
value is higher than this earlier one it will be 651.96 this has been given by Wen and Yu.

Different investigators they are getting different results for different sizes particles here if
particle size increase or decrease then of course, accordingly this k 1 and k 2 will be
increasing or decreasing, in this case, is a minimum fluidization velocity is the most
important measurement you need for design the and you need to focus of a tremendous
amount of experimental condition and data for wide variety of condition for calculating the
minimum fluidization velocity.

(Refer Slide Time: 53:32)

Now, for gas fluidization, the Wen and Yu correlation is ofter taken as being most suitable
for particle larger than 100 micrometer. Wen and Yu correlation is very suitable if it is if
particle size is greater than 100 micrometer whereas, the correlation of Baeyens (1974) he has
proposed in this case will be the best for particle less than a hundred micrometers. This will
be your correlation of Baeyens that you can directly use for calculating the minimum
fluidization velocity if your particle is being used or you are using particle, which is size less
than 100 micrometer 100 micrometers.

This will be a suitable correlation to calculate the minimum pressure; there are other different
investigators. They have calculated, or they have developed with the different correlations for
minimum fluidization velocities.
(Refer Slide Time: 54:29)

These are some correlations that are given here in the slides that you can follow for
calculating the minimum fluidization velocity.

(Refer Slide Time: 54:39)

Now, what are the factors that affect the minimum fluidization velocity you can see some
geometric variables some variables as the physical properties some are thermodynamic
conditions now geometric variables like bed diameter bed height particle size distributor hole
diameter through which the gas is distributed, or liquid is distributed now fluid density fluid
viscosity fluid surface tension slurry concentration all are the factors that affect on the
minimum fluidization velocity even some thermodynamic condition at a higher pressure, of
course, you will see the minimum fluidization velocity will be higher whereas, the
temperature at a lower temperature it will be the higher fluidization minimum fluidization
velocity. These are the factors that affect the minimum fluidization velocity. The next class
will be the one example that will be given to you for minimum fluidization.

Thanks for all today.

 Fluidization Engineering
Dr. Subrata K. Majumder
Department of Chemical Engineering
Indian Institute of Technology, Guwahati

Lecture – 05
Minimum Fluidization Velocity: Liquid-solid & gas-liquid-solid System

So, welcome to the massive open online course on fluidization engineering. Today’s lecture
will be on minimum fluidization velocity in the liquid-solid system and gas-liquid-solid
system.

(Refer Slide Time: 00:40)

So, you know that as a recap, we can say that minimum fluidization velocity for the gas-solid
system can be calculated from this equation that this is the quadratic equation of Reynolds
number for minimum fluidization condition.

This is a function, of course, Archimedes number and or sometimes it is referred to as Galileo

number and this minimum fluidization velocity depends on different factors like geometric
variables, physical properties and also the thermodynamic conditions.
(Refer Slide Time: 01:20)

Now, you know that for the gas-solid system, this minimum fluidization system also is
represented in a different way in terms of some parameters K 1 and K2. So, this K1 and K2 are
actually defined as the term that is obtained from the Ergun equation some parameters and
this form of this equation in terms of K 1 and K2 that are valid within the range of Reynolds
number 0.001 to 4000 s.

(Refer Slide Time: 02:00)

And using this value of K 1 and K2, the minimum fluidization velocity can be obtained for
different particle numbers, and if particle number is greater than one particle sorry particle
size is greater than 100 micrometers, then you can use this equation to calculate the minimum
fluidization velocity. Whereas, for if the particle size is less than 100 micrometers, then you
have to calculate the minimum fluidization velocity based on this equation. So, in this case,
this K 1 and K 2 are, of course, different for different ranges of particle size.

(Refer Slide Time: 02:38)

And, also there are various correlations proposed by various investigators like here one
important correlation that is given by Baeyens, 1974, for gas-solid operation. Then the
minimum fluidization velocity can be obtained from this correlation. This correlation has
been developed from the experimental data within the range of size less than 100
micrometers.

Other different correlations that are given by Baeyens and Geldart even Ergun in terms of this
what is that Archimedes number along with the equation of this Reynolds number and Leva
et al. he has given these correlations and even Goroshko et al., they have proposed to this
equation.
(Refer Slide Time: 03:32)

They are several other different correlations that are given by different investigators for
calculating the minimum fluidization velocity, and those correlations have been developed
based on the experimental data.

Now, what should be the minimum fluidization velocity for a liquid-solid system? As per
Thonglimp, 1984, the correlation for the liquid-solid system they have developed based on
the concept of whatever it is developed for the gas-solid system. It is also represented by this
Archimedes number, and this is your minimum fluidization condition what should be the
Reynolds number, and this Reynolds number at the minimum fluidization condition will be
obtained from this equation. But, as this Archimedes number will be depending or defined as
in this equation.

So, Thonglimp, 1984, he has proposed this concept to calculate the liquid-solid minimum
fluidization velocity.
(Refer Slide Time: 04:33)

And, in this case, another important recently developed correlations which are given by Li et
al., in 2016, how to calculate the minimum fluidization velocity for the liquid-solid system.

Now, for this, of course, this concept also has been taken based on the concept of whatever it
is for the gas-solid system. This is the Ergun equation part by which you can calculate the
frictional pressure drop, and that will be balanced by the apparent weight of the bed. So, from
this equivalence, you can calculate the minimum fluidization velocity for the liquid-solid
system. And, here, of course, this now this minimum porosity for this minimum fluidization
condition, or you can say minimum voidage that depends on the shape of the particle. For

 mf
spherical particles, this that is minimum voidage minimum voidage will be within the
range of 0.4 to 0.45.

Otherwise, you can calculate the minimum voidage from this equation; this epsilon mf will
almost equal to

1/3
 1 
 mf  
 14 s 

This
 s is nothing, but the sphericity. So, other than spherical particle, you can calculate the
minimum voidage from this equation. After getting this minimum voidage you have to
substitute that minimum voidage here in this equation and this equation and calculating the
minimum velocity just by solving this equation.

(1   mf ) 2 l umf 2
(1   mf ) l umf
150  1.75
 mf3  2s d p2  mf3  s dp
   p (1   mf )  l  mf  g

(Refer Slide Time: 06:25)

Now, what should be the minimum fluidization velocity in a conical fluidized bed with the
solid-fluid system? Now, see there are different types of fluidized bed system; some conical
fluidized bed is also one important, they are some advantages of this conical fluidized bed
system. Here, you will see that of course, from this bottom how conical part will be there at
this location and from this conical system you will see there are two diameters that are D b that
is upper bed surface diameter and this is the lower part this conical portion that is D i at the
entrance what should be the diameter here.

So, this is the particle and particle here, and this is the fluidization, and this is the bed, fixed
bed condition, and after fluidization, how this fluidization occurs and what should be the
minimum fluidization for this solid-fluid system in this term conical fluidized bed. So, here
you will see for this conical fluidized bed, of course, giving advantage because this
circulation pattern will be changing here inside the bed. So, that the mixing may be more
intense mixing can be obtained by this conical fluidized bed and also the solid circulation this
conical fluidized bed are more important due to the solid circulation inside the bed, and it will
be very much beneficial for the physical operations like coating like drying purpose all those
things.

Now, what should be the minimum fluidization velocity in this conical fluidized bed system,
and here also this is a quadratic equation for the umf here. So, that is also obtained by, or this
is proposed by Zhou et al., 2009. So, he or they developed this equation by balancing this
momentum and then obtain this equation for minimum fluidization velocity and in this case
these one important aspect that you have to know the minimum voidage for this solid-fluid
system here; this minimum voidage should be calculated based on this here

0.16u g
 mf 
 mf (ug  ulmf )

Here, in this case, solid-fluid means here, you can say that in this case, liquid also will be a
very important gas-liquid-solid system. So, in this conical fluidized bed gas-liquid-solid

 mf
system is more important here. So, what this gas-liquid-solid system is will be equals to
these.

This is based on the equation that is proposed what should be the mixture velocity here for
gas and liquid here and this equation will be a valid only for if the ratio of gas velocity to the
mixture velocity of gas and liquid is less than equals to 0.93 and in this case conical fluidized
bed system, of course, you have to calculate the diameter of the bed as an equivalent diameter
at this minimum fluidization condition and then minimum equivalent diameter can be
calculated from this what should be the volume fraction of the solids inside the bed.
(Refer Slide Time: 09:50)

Now, let us see an example of the gas-solid and gas-liquid system. So, let us see if a fluidized
bed is operating with air at a minimum fluidized condition with a particle of cubes 0.02 meter
on a side. The bulk density of the fluidized bed here is 980 kg per meter cube. The density of
the solid cube is given here, and the density of the air is given as 1.18 kg per meter cube, then
the viscosity of air is given like this. So, what should be the void fraction of the bed? How to
calculate the effective diameter here and also how to determine the sphericity of the cubes
and what should be the minimum fluidization velocity, if you are using water at 20-degree
centigrade and a bed diameter of 0.15 meter.
(Refer Slide Time: 10:46)

So, under this condition, let us calculate this. Now, how to calculate this void fraction? This
void fraction can be calculated; first of all, you have to calculate the total volume of the bed.
Total volume means here total volume of the bed means the volume of the fluid and the
volume of the solid, and then the weight of the bed will be equal to the weight of fluid and
then the weight of solids.

Now, what should be the weight of the bed will be equals to that density of the bed into the
volume of the bed. Now, if we can also what should be the volume of the, what should be the
mass of the fluid inside the bed that will be volume and that is a product of volume and
density of the fluid. Similarly, for a mass of the solid should be equal to the density of the
solid and into the volume of the solids, but in this case, you will see there will be the solid
mass will be much greater than the fluid mass. So, you can neglect this fluid mass related to
the solid mass. So, it should be then what should be the mass of the bed. You can directly
calculate what should be the mass of the solid.

So, if you know these mass of the bed by moving this mass of the solid, then you can
calculate what should be the void fraction here. So, the void fraction will be equals to the
volume of bed minus volume of solid divided by volume of bed; that means, here what
should be the amount of volume for space here except solids in the bed out of total bed
volume that will be your void fraction. So, this void fraction then
 Vbed  Vsolid
ε
Vbed

So, here this for this V solid, you can substitute here this portion, and then finally, you can
get

bed
  1
 solids .

1 minus this rho bed; rho bed means here what be the bed density effective density it will be
980 kg per meter cube as per your problem and then what the rho solid it is coming here 0.35.

(Refer Slide Time: 13:26)

Now, void fraction at minimum fluidization condition you can calculate from this equation
given by when and you that this from this equation. So, from this equation,

3 1
 s mf 
14

you can get this minimum void fraction will be equals to 0.445. Whereas, in this without, that
means, except minimum void fraction it is coming to 0.35, for minimum void fraction it will
be higher than this actual fluidizing a condition void fraction is a 0.35 and effective diameter
of the particle you can calculate from this here, first you have to calculate if it is cubical then
you have to calculate the volume of the cube, then it will be equivalent to the spherical
particle then it will be d p is equal to the 0.025.

Whereas, the sphericity of the particle then phi s is equal to the surface area of the sphere,
having the same volume and divided by this surface area of the particle A p. So, finally, you
can get what should be the void, what should be the sphericity of the particle it will be equal
to 0.81.

(Refer Slide Time: 14:44)

Now, for a coarse particle, of course, if dp is greater than 100 micrometer, then the minimum
fluidization velocity can be calculated from this equation like here

Re p ,mf  [(28.7) 2  0.0494 Ar]1/2  28.7

So, this 28.7 this portion that is obtained from the parameters of this Ergun equation.

Now, first, you have to calculate what should be the Archimedes number here. Archimedes
number, if you substitute the different variables in this equation, then you will get the
Archimedes number is this one now 342247.7 then after that you substitute this Archimedes
number here in this equation you will get the minimum Reynolds number for this minimum
fluidization high condition that will be equals to 104.45. Now, once you know this minimum
Reynolds number, which is defined as rho f into u mf; u mf is the minimum fluidization
velocity than from this equation, you can calculate what should be the minimum fluidization
velocity it will be exactly 3.15.

(Refer Slide Time: 16:12)

Now, if we use this minimum porosity as 0.445 and the sphericity of 0.81, then you can get a
different way different you can get different fluidization velocity at this minimum condition
based on this equation where you have to calculate first K 1 and K2 here you have to use the
equation which is the function of these parameters K1 and K2. Now, this Rep,mf will be is equal
to this, is a function of K 1 and K2. Now, if you substitute this K 1 and K2 value, here then you
can get the Reynolds number at this minimum fluidization condition as 92.38. So, we are on
this. You can calculate a umf, that means, minimum fluidization velocity from this equation
that will be equal to 2.78 meters per second.

Whereas, without using this minimum voidage of 0.445 and the sphericity of 0.81 if you use
this previous equation, then we are getting 3.15. So, what will be the deviation from these if
you use this K1 and K2 parameter? So, we have seen that it will be an almost 11.7 percent
error. So, almost it will be 12 or near about 12. So, we can accept this one also.
(Refer Slide Time: 17:47)

Now, if the fluidization did with fluid as water instead of air, that means, a liquid-solid and
liquid system then we can get this here according to that equation here instead of here rho g it
will be rho f; that means, rho l density of the liquid must have a density of the gas. So, if we
substitute the density of the liquid instead of density of air or gas here, then you can get the
Archimedes number here this whereas, this if you substitute this Archimedes number in this
equation,

Re p ,mf  [(28.7) 2  0.0494 Ar]1/2  28.7

then you can get the Reynolds number at this minimum fluidization condition at this, and
from this equation, you can calculate what should be the then minimum fluidization velocity,
and it is coming almost 0.076.

What we observed here from these if we use water instead of air, you will see that minimum
fluidization velocity is drastically reduced here, of course, because of this viscosity. So, only
water has a higher viscosity than the gas that is why this minimum fluidization velocity will
be reduced.
(Refer Slide Time: 18:59)

And another example; calculate the minimum fluidizing velocity for the fluidized bed
operating with his sharp sand particles and with air under the following properties. Here
fluidizing gas is the ambient air, and density is 1.2 kg per meter squared per meter cube and
viscosity 0.00018 kg per meter second, whereas solid is sharp sand particles of diameter 75
micrometers and phi s that is sphericity it is given as 0.767, and the density is 2600.

(Refer Slide Time: 19:40)

Now, compare these values of minimum fluidization velocity calculated by different

 mf
correlations. So, if we use here, is equal to 0.45 as per the equation given by when an u
and the sphericity as 0.67, and then you can calculate what should be the parameter K1 and
what should be the parameter K2. So, from these parameters K1 and K2, we can calculate what
be the Reynolds number at this minimum fluidization condition. So, if we once know this
Reynolds number at this minimum fluidization condition, we can obtain the minimum
fluidization velocity from this Reynolds number as this one.

So, you are very interesting that we can calculate the minimum fluidization velocity, which is
depending on different variables like porosity, like the minimum void fraction, like the
density of the fluid, like the viscosity of the fluid and particle size. Here, in this case, the
particle size is very small, that is why your minimum fluidization velocity is coming almost
near about 0 not exactly 0, but will be very low.

(Refer Slide Time: 20:53)

Now, as per correlation of Baeyens, if we substitute all variables here in this correlation and
then we can get the minimum velocity as 0.000385, whereas, in the previous case we have
calculated the minimum fluidization velocity at 0.000395, here it is coming for 0.0019 sorry
0.000395, and in this case, it is coming 0.000385. So, 385, 395, it is nothing that is one hardly
2.6 percent error.
(Refer Slide Time: 21:29)

Whereas, if we calculate the minimum fluidization velocity by different correlations like

correlations of Leva, you can get the minimum fluidization velocity 0.699 whereas, Goroshko
et al., based on their correlation we get the minimum fluidization velocity and we so in this
table it is given the minimum fluidization velocity obtained by different correlations here, and
from this correlations, we see that we can get the mean value of 0.650 and what should be the
deviation from the different correlation we can calculate and then it is seen that every
deviation from the mean is not that much sub significant. So, any correlation you can use for
the design aspect of this minimum fluidization calculation.
(Refer Slide Time: 22:27)

Now, minimum fluidization in the gas-liquid-solid system. So, up to this, you have calculated
the gas-solid and liquid-solid. So, you can easily calculate what will the minimum
fluidization velocity, but, in case of the gas-liquid-solid system, they are the interaction of the
solid particles with not only single gas or single liquid, here, both the gas and liquid
simultaneously will be there. So, in this case, the frictional pressure drop in case of gas
whatever it is, they are we have not actually considered the hydrostatic pressure there, but in
this case, of course, one should consider the hydrostatic pressure of course.

So, in this case, the gas-liquid-solid system, the major pressure drop here it will be is equal to
frictional pressure drop along with the hydrostatic pressure drop. So, this hydrostatic pressure

drop is this can be calculated of

Ph  m gH . Here, m will be the mixture density. How to
calculate this mixture density? Mixture density will be equals to rho l epsilon l plus rho g
epsilon g plus rho s epsilon s.

Now, what is the epsilon l; epsilon l is nothing, but the volume fraction of the liquid in the
gas-liquid-solid three-phase systems and epsilon g is the gas volume fraction in this total

phase of gas-liquid-solid and what should be the solid fraction that will be denoted by
s ,
and this is the volume fraction for solid. So, this mixer volume depends on this volume

g g
fraction or gas-liquid-solid. So, this will be your volume of liquid is , it will be volume
of gas density of the capability density of the gas and
 s s it will be the effective density of
the solid. So, the total density of this liquid gas and solid will be equals to the mixer density
of the three-phase system here.

 m   l  l   g  g   s s

So, if you know that mixer density at this fluidization system, then, of course, you can
calculate what should be the hydrostatic pressure there. So, once you know this hydrostatic
pressure and what should be the frictional pressure drop. So, the frictional pressure drop will
be equals to measured pressure drop minus delta P h; that means, here hydrostatic pressure
drop.

(Refer Slide Time: 25:09)

Now, if voidage of the bed, that means, only fluid part if you know then what should be the
density of the fluid; that means, here fluid means solid and gas mixture, not be considered as
solid here, so, only just considering the gas and liquid here. So, only these two parts, suppose
liquidized bed, this is the part is here; this is the gas and liquid, and this is solid. So, this is
solid, and this is the gas flask liquid. So, if we consider only these gas and liquid what should
be the voidage of this void fraction of this gas and liquid; that means, what should be the
volume that is occupied on this gas and liquid?
So, if we represent that mass volume fraction of this gas and liquid, epsilon, then we can say
here rho f into epsilon that this rho f is nothing, but, mixture density of fluid only; that means
the density composite density of the gas and liquid here. So, this density into this void
fraction of gas-liquid this will be equals to rho g epsilon g plus rho l epsilon l, g for gas l for
liquid here.

So, what should be the density of the liquid here? It will be

 g  g  l  l
f 


and this epsilon is that total volume fraction of the gas and liquid portion only and this f
stands for here fluid mixture and the fractional holdups of gas and liquid in the gas-liquid
mixture that can be represented respectively by then we have if some if we take this part; that
means, here

g
g 


and then it will be equals to what will be the fraction of this part, and then what should be this
part epsilon g this volume fraction of the gas to the total volume of gas and liquid it will be
your fractional holdup of this gas here, and for liquid, of course, it will be is equal to

l
l  1   g 
 .

If you add this alpha g and alpha l; alpha g plus alpha l that will be equals to 1.0.

l   g  1

So,

l  1   g
. So, therefore, we can say that

 f   g g  l (1   g )
(Refer Slide Time: 28:06)

Now, for a homogeneous mixture of gas and liquid, if we consider there will be a
homogeneous mixture and gas in liquid in gas-liquid-solid system then the superficial fluid
velocity of the homogeneous gas-liquid mixture can be calculated as here u sh, that will be
the mixture velocity or liquid and gas.

Here s stands for superficial, h stands for homogeneous and usl, that means, superficial liquid
velocity, in this case, it will be calculated as what should be the volumetric flow rate of liquid
per by cross-sectional area of the bed. This will be your superficially liquid velocity, whereas,
the superficial gas velocity is defined as what should be the volumetric flow rate of gas if you
divided by its cross sectional area of the bed then you can get the superficial gas velocity by
this equation.
(Refer Slide Time: 29:05)

Now, the homogeneous fluid density of the gas-liquid mixture, then what should be the
homogeneous density? Here, of course, you can say as per the rho f this would be nothing,
but rho f then it will be rho g epsilon g plus rho l epsilon l divided by total volume fraction of
gas and liquid mixture in the bed. So, if we simplify this equation by neglecting these gas
mass related to this liquid mass, then you can get this homogeneous density of the fluid inside
the bed as

 g  g  l  l l  l
h    (1   g ) l
 

Now, the frictional pressure drop for the homogeneous mixture as per Wallis, 1969, that it
can be expressed as here. In this case, of course, this frictional pressure drop will be a
function of homogeneous mixture velocity and the density of the mixture, as well as what
should be the volume fraction of the liquid or and gas, that means, the fluid total fluid inside
the bed and also what should be the projectional area of the solid particles inside the bed.

So, if we know the frictional pressure drop from this Wallis equation here of course, we can
equalize this homogeneous frictional pressure drop of this equation with the total mass of the
bed then from which you can calculate the minimum fluidization velocity for the gas-liquid-
solid system. Now, see this is the Wallis equation there is one factor that is called friction
factor. This friction factor, of course, will be in between the particle and the, of course,
particle surface or solid surface to the gas and liquid.

So, it will be represented by this

Pf 1 
 (1   ) f f  s Ap   huh2 
L 2 

f f s
means fluid and solid surface. Now, this solid surface, including the wall, also, here the
friction is with the solid and the homogeneous mixture of the gas and liquid. The buoyant
weight of the solid particle is supported by this upward homogeneous fluid drag on the
particle. This is your fluid drags by what is calculated with a frictional pressure drop, and
then we can say this buoyant weight will be balanced by this frictional pressure drop as this

Pf
 g (1   )(  s   h )
L

at this gas-liquid-solid fluidization condition.

(Refer Slide Time: 31:50)

Now, if we equate these two equations, then, of course, we will get the minimum fluidization
velocity. Before going to that, you have to, of course, calculate the frictional projected area of
the particles because this is required for calculating the frictional pressure drop, or you can
say that a fluid drag by this Wallis equation. So, for this what should be the A p; that means,
your projectional area of the particle, surface area per unit volume, you can say also this will
be calculated by

2
( / 4)d eff 3 3
Ap  3
 
( / 6)d eff 2deff 2 s d p

 s is the sphericity.

Now, when the liquid velocity increases and gas velocity remains constant, the ratio of
epsilon l by epsilon increases, and so, density will increase. Thus in contrast with the 2 phase
fluidized bed, this frictional pressure drop decreases and also what should be the drag
coefficient or you can say friction factor f fs that will, of course, will be depending on this
porosity or a void fraction of the gas and liquid mixture here

f f s
  n
fs

and this is of course, is a function of void fraction and the friction factor, if we consider only
solid particles there not gas and liquid.

So, in that case for single fluid-solid system this n fact, n is the coefficient, n is equal to 4.7
whereas, for gas-liquid-solid system n will be is a function of superficial gas velocity and that
can be calculated from this equation

n  5.7  8usg

So, once we know these equations and you have to substitute in Wallis equation, then you can
get the fluid drag by this Wallis equation.

Pf 1 
 (1   ) f f  s Ap   huh2 
L 2 
(Refer Slide Time: 33:40).

Now, what should be the individual solid phase friction factor, which is required here in this
equation of equation 16. So, you can calculate this individual solid phase friction factor a
phase which is developed by Rowe (1961) for a three-phase flow system as this

24
fs 
Re

1  0.15 Re0.687  for Re  103

 0.44 for 103  Re  105

1/ 2
4 g  s 

ut   s d p   1
 3 fs 
 (1   g ) l 


Here, this equation valid for if

Re , that means, here Reynolds number based on terminal
velocity if it is less than 1000 whereas, it will be only 0.44 if it is more than 1000 and less
than 10 to the power 5.

Then, fs you can calculate for the single solid phase friction and
Re , that means, the
Reynolds number based on the terminal velocity will be equals to
 ut  h d p ut (1   g ) l d p
Re  
h l

here
h is the viscosity of homogeneous mixture of gas and liquid and u is the terminal
t

velocity of the solid and rho is the mixture density of gas and liquid only and d p with the

particle diameter and from which you can calculate the our
Re and if you substitute this l

(1   g ) l g
as , then you can get here this Reynolds number this is a function of again this ;

g
, is the gas of void fraction that is fractional holdup of gas inside the bed.

So, this Reynolds number, of course, will be changing with respect to how much gas is
obtained how much gas is actually occupying in the bed. So, fractional gas holdup of the
system that will enhance, that will affect the Reynolds number and which directly will be
related to this what is the friction factor here.

Now, what should be the terminal velocity? Of course, this terminal velocity will be that
already we have shown earlier also that

1/ 2
4 g  s 

ut   s d p   1
 3 fs 
 (1   g ) l 


This fs, of course, this is a friction factor which is again is a function of Reynolds number is a
function of again terminal velocity. So, this is the non-linear equation for terminal velocity
here. By trial and error method, you have to find out the terminal velocity from this equation.
(Refer Slide Time: 36:15)

Now, for any gas, the liquid velocity at which the bed is operated, the gas holdup in the three-
phase bed can be predicted as per Marquardt, 1963. He has developed this correlation for
calculating the gas holdup in the gas-liquid-solid three-phase system. So, this is a function of
superficial liquid velocity particle diameter, even solid density, even superficial gas velocity,
the density of the gas and liquid, viscosity of the liquid, and also the bed diameter.

So, all those parameters, here, if you substitute these parameters or variables here in this
equation, then you will get the fractional gas holdup in this three-phase system. Otherwise,
you can calculate this gas holdup by experimentally. So, here the gas holdup alpha g can be
experimental obtained by the phase isolation techniques.

What is that phase isolation techniques? At the running condition you have to observe how
much gas-liquid-solid label inside the bed is up to that, and suddenly if you stop this
operation you will see the gas inside the bed will disengaging, and it will be dissolved at the
top, and after a certain time after degassing all the gas here there will be a clear liquid-solid
mixer height inside the bed, then if you know that clear liquid height here as L c and the mixer
height as Lm then you can obtain the gas holdup experimentally by this equation L m minus
Lc by Lm.

What is that? See, alpha g is nothing, but the gas volume upon total mixer volume. So, this
you can represent it as the gas volume as mixer volume minus clear liquid volume here. Now,
this mixer volume will be calculated as A into L m. Lm is the minimum fluidizing height there,
and A is the cross-sectional area.

So, this is the volume of bed at the minimum fluidization condition and this AL c this is the
volume after degassing all the gases in the gas-liquid-solid fluidized bed, and if you subtract
this then you will get the total gas volume here whatever occupied in the gas-liquid-solid
fluidized bed, and this AL m is the, what will be the minimum fluidization condition, what
will be the total volume of gas-liquid-solid mixer. So, from which you can calculate what
should be the gas holdup by this isolation technique. Otherwise, you can directly without
doing experiments you can calculate this alpha g from this Marquardt equation.

g
Now, see what is that if you use this Marquardt equation here, for the minimum

 g ,m
fluidization condition, then it will be , the minimum fluidization condition. Then, here it

will be uslm at the minimum fluidization condition, here again, m f with a minimum
fluidization condition, here again, it will be considered as minimum fluidization condition.
So, to calculate the minimum fluidization condition again with a function of again minimum
fluidization liquid and solid-gas velocity, then it will be a non-linear equation.

(Refer Slide Time: 39:27)

So, now, from equation 13 and 14, if you consider or if you take into account the expressions
for homogeneous mixer fluid velocity and homogeneous fluid density and the Ap from the
equation 15 and the friction factor from equation 16 for a certain gas velocity then the
following equation can be obtained for the frictional pressure drop and the porosity as like
this.

Pf
So, here delta H is will be equals to this

Pf 3(1   ) f s (1   g ) l (usl  usg ) 2

  (1   ){ s  (1   g ) l }g
H 4 s d p n

we are, and this is nothing, but what with the apparent weight of the bed herein under this
fluidization condition. So, after rearranging this equation 22 you can get epsilon; that means,
what will be the void fraction that is occupied by the mixture of gas and liquid only not solid
that will be depending on the particle diameter and also the volume fraction of the gas and
also density of the liquid gas and liquid velocity inside the bed. So, this will be your void
fraction that is gas and liquid here or for the gas-liquid-solid system.

(Refer Slide Time: 40:56)

Now, equation 22, together with equation 15 to 20, one can obtain the minimum superficial
fluidization velocity of the liquid here.
 1/2
n
 4 s d p mf g{ s  (1   g ,mf ) l } 
usmf ,l    usg
 3 f s (1   g , mf ) l 

So, this will be as for liquid here. So, this will be obtained after rearrangement, and again it is
a function of usg here. So, see you usmf here, that means, minimum fluidization condition what
should be the minimum liquid velocity by which you can get the minimum fluidization
condition for the gas-liquid-solid system.

Now, this is again is a function of see here mf and this mf is a function of this is again the
liquid velocity and gas velocity. So, that liquid velocity will be considered here to see liquid
velocity will be considered as a minimum fluidized condition and then here alpha gmf at
minimum fluidization condition what should be the fractional gas holdup that can be obtained
from this equation that is given by Zhang et al.,1998.

0.16usg
 g ,mf 
 mf (usg  usmf ,l )

This is again a function of gas velocity and the liquid velocity. Now, this liquid velocity you
have to consider at minimum fluidization condition. Now, if you substitute here in this
equation for alpha g in equation 24, then you will see the minimum fluidization velocity for
liquid is again a function of minimum fluidization velocity for liquid.

So, this is a non-linear equation you have to solve this non-linear equation or optimize this
non-linear equation for liquid velocity by different optimization techniques or by trial and
error method you can calculate what should be the minimum velocity for liquid in the gas-
liquid-solid system.
(Refer Slide Time: 42:44)

Then from for this of a minimum voidage, you have to obtain here, again by this Wen and Yu
model, what should be the minimum voidage at this minimum fluidization condition that is a
function of sphericity of the solid particle.

1   mf
3
 11
 mf  s2

Or

1
3
 14
 s
mf
(Refer Slide Time: 43:06)

Now, different investigators, they have developed different correlation from their different
experimental data and obtain the minimum fluidization condition for a liquid-solid system,
gas-liquid-solid system, even the gas-solid system also. We have already shown different
correlations for minimum fluidization conditions for gas and solid system, even liquid and
solid systems. In this case, the liquid gas and solid system, the minimum condition can be
obtained from that gas and solid and liquid and solid system also. If you know the two-phase
a minimum fluidization condition, you can obtain the three-phase fluidization condition by
this equation of Begovich by this equation here.

So, this is here the minimum fluidization velocity for the gas-liquid-solid system, and this is
your minimum fluidization condition for liquid and solid systems. So, this is a function of
this gas velocity, the viscosity of the liquid, particle diameter, density of the solid and density
of the liquid and this minimum fluidization velocity for liquid and the solid system it is a
function of again Archimedes number which is a function of a few densities, particle density,
again viscosity of the liquid and the diameter of the particle.

So, once you know this Archimedes number you can obtain this minimum velocity in case of
liquid and solid system and if you know the minimum velocity in case of liquid and solid
system you can substitute here and also with the other parameters, other variables you can get
the minimum velocity of gas-liquid-solid fluidization from this equation 27.
(Refer Slide Time: 45:06)

Now, Zhang et al. (1988), they have developed the minimum fluidization velocity for a three-
phase fluidization system based on the concept of the gas-solid system. Here, it is called the
gas-perturbed liquid model. So, again they have obtained the minimum Reynolds number
based on the equation here

This Archimedes number is defined in a different way here. So, instead of gas density, they
have considered liquid density here. So, from this equation, you can calculate that what
should be the Reynolds number at this minimum condition and this is a function of again the
minimum voidage of gas, which is a function of gas velocity and the minimum liquid velocity
in this three-phase fluidization system.

So, this model actually applied to this three-phase fluidization involving Newtonian liquids,
which equates the liquid-buoyed weight of the solids per unit bed volume to the frictional
pressure gradient, which is given by the Ergun packed equation and also it is applied to the
liquid-solid part of the incipiently fluidized bed. So, this Zhang et al. model can also be used
to calculate the minimum fluidization velocity in the gas-liquid-solid system.
(Refer Slide Time: 46:39)

Li et al., very recently they have developed another one correlations to calculate the
minimum fluidization velocity in the gas-liquid-solid system which is given by this

Regmf  (33.7 2  0.0408 Arlg ) 0.5  33.7

this Archimedes number based on the density of the liquid and gas in a composite way. So,
here this the definition of this Archimedes number here as

 g (  s  l ) gd p3
Arlg 
 g2

we will see that only if we consider the gas and solid system the density of the gas here, it
will be considered, whereas, in liquid and solid system if we use the same concept the as per

Li et al., it is defined as
l instead of  g .

So, here this Archimedes number, that means, for liquid and gas system here as

 g (  s  l ) gd p3
Arlg 
 g2
but this viscosity of gas will be considered here instead of liquid viscosity. For gas-solid
system, it is seen that this Remf is a function of again Archimedes number, but only it is based
on gas density, but other terms you have to see 33.7, here also 33.7. All these are same, all

g l . So, this should be

the coefficient also are same. So, instead of here it will be coming
noted.

(Refer Slide Time: 48:28)

Now, for low liquid velocity, Zhang et al., also given another equation to calculate the
minimum fluidization velocity; now turning now to that closely related situation where there
is a small liquid flow, far less than that required to fluidize the solids, revert to previous
equation inserting just mf here then you can get this equation for Reynolds number.

Now, this applicable for superficial velocity only if it is less than equals to 0.98 and it is valid
only 0.98 to 1, not it is then earlier for gas-solid system earlier model they have shown that it
will be applicable only up to 0.93, but here it is applicable up to 1. So, here this mf, that
means, here minimum voidage for gas it is again a function of this angle, at different angle
orientation of the fluidized bed, how this gas holdup will be changing that has been
incorporated by this equation here.
So, this equation will give you at any angle of the inclination of the bed you can calculate to
directly the minimum gas volume fraction here. So, if you know the minimum gas volume
fraction, then you can get the minimum fluidization velocity from this equation.

Note that, here if x is equal to 1 that means, ratio of gas velocity to the mixture of gas and
liquid equals to 1, that means, if u sl equals to 0, that means, there is no flow of liquid. This
above equation yields that mf into mf is equal to 0.406, which implies that since mf equals to
1 for this condition, this mf  mf is equal to 1 a reasonable value for the minimum
fluidization voidage of impermeable spheres. So, this is very important to be noted down.

(Refer Slide Time: 50:46)

Li et al. (2016), they have developed another type of correlation like here; this is actually
developed based on the dimensional analysis incorporating all the variables in their
experiments. Now, they have taken it, and they have dimensional they have done the
dimensional analysis and obtained these groups as Archimedes number, Froude number, even
ratio of hydraulic diameter to the particle diameter, even what should be the height of the
fluidized bed and also what should be the diameter of the fluidized bed and also what should
be the ratio of surface tension of the liquid to the surface tension of the water.
So, from those, they got this dimensionless group and done the regression analysis with this
dimensionless groups with the experimental data, and they got the equation for its coefficient
0.103 and 0.7933 and other coefficients like this here in this equation it is given, and from
this equation, you can calculate what will be the minimum fluidization velocity, but this is
applicable only if Re Lmf usually less than 1.

Whereas, if ReLmf is greater than 1, but less than 120, they have developed another correlation
based on the same dimensionless groups, and they got different coefficients like this by
multiple regression analysis. So, this equation also can be used to calculate the minimum
fluidization velocity.

In this case, you will see this H s is the static bed height part of the static bed height, D h is the
inner diameter of the fluidized bed or hydraulic diameter you can consider here if it is 2-
dimensional bed and sigma is the surface tension and Froude number is denoted by Fr which
is defined as Fr = ug2/gdp.
(Refer Slide Time: 52:52)

Now, minimum fluidization velocity in a conical fluidized bed in a three-phase system, they
have done some experiments with the conical flask also this three-phase system, and they got
this minimum fluidization velocity which is expressed by the minimum Reynolds number and
again they have considered some parameters here as K1 and K2, but this K1 and K2is a
function of the diameter in the inlet and the surface of the bed.

So, K1 is defined as

and K2 is the nothing but

and also mf this minimum gas void fraction is a function of again this gas velocity and liquid
velocity, and this is applicable only for if the ratio of gas velocity to the mixture of gas and
liquid velocity is less than 0.93.

Now, then this equation can be applied if you know this K1 and K2 for this conical type of
fluidized bed and if you know the ratio of this gas and liquid velocity inside the bed you can
easily calculate the minimum fluidization velocity for the conical fluidized bed in a three-
phase system. Thank you for today’s lecture. The next lecture will be given on that other part
of this fluidization engineering.

Thank you.
 Fluidization Engineering
Dr. Subrata K. Majumder
Department of Chemical Engineering
Indian Institute of Technology, Guwahati

Lecture – 06
Flow regime and its map: Gas-solid Fluidization

Welcome to the massive open online course on fluidization engineering, today the
lecture will be on flow regimes and its map, especially on gas-solid fluidization.

(Refer Slide Time: 00:39)

So, what is flow regime or pattern, basically the multiphase flow behaviors that are
affected by interfacial tension, wetting characteristics of the fluid on the channel wall, or
you can see bed wall, and also contact angle and the exchange of mass momentum and
energy between the phases? And this flow regime dictates the flow behavior and shape
of the interface between phases in a multiphase mixture, which is commonly referred to
as the flow regime, or sometimes this is called flow pattern.
(Refer Slide Time: 01:29)

What are the factors that affect this flow regime? The flow regime of multiphase flow in
a fluidized bed, that depends on several factors like dynamic variables what are the
dynamic variables like what are the flow rates in the fluidized bed flow rates of gas and
liquid, even if any other suppose phase like solid if it is continually flowing in into the
bed then the solid velocity also will be a one dynamic variables here. And geometric
variables of course, what should be the size of the bed, that is what is the diameter of the
bed what is the cross-sectional area of the fluidized bed, and what is the particle
diameter, what is the pore diameters of the distributor to which the fluid is distributed
into the fluidized bed.

And thermodynamic variables like what is the pressure inside the bed, how it can be
changed and or if I change the pressure inside the bed then what should be characteristics
of the fluidized bed, and what should be the different hydrodynamic and transport
processes inside the bed that depend on the pressure and another thermodynamic variable
it is called temperature, of course, the temperature is the main effect here some time
since the drying operation in the fluidized bed you have to maintain certain temperature,
and also this increases the temperature or decrease the temperature, how this fluidization
characteristics will change that depends on the temperature variation.

And the other different transport processes, like mass transfer heat transfer, depends on
the variation of the temperature that is controlled in the fluidized bed. So,
thermodynamic variables like pressure in temperature are the main important factors
here. Physical property is, of course, any fluidized bed, whether it is gas solid or gas
liquids and solid multiphase fluidized system, of course, the physical properties like
density surface tension and viscosity of the fluid medium, it will affect the fluidization
characteristics like flow regimes. Suppose you, if you change the properties of the fluid
and also other dynamic geometric and thermodynamic variables, then you will, of
course, have the different types of flow regimes inside the bed.

The boundary of the two regimes is sometimes referred as the flow regime transition of
course, one regime to another regime you will get to make the to make change of one
flow regime to another flow regime, the what should be the barrier, what should be the
point from which you can get another flow regime that point is called transition point or
the certain reason certain that is profile from that profile onward are beyond that point
you will get the different type of flow characteristics.

So, the transition point that is called flow regime transition, that also very important in
this case, you have to know in which flow regimes actually this fluidization operation is
being carried out. And different flow regimes, you will get the different behavior
applications different, hydrodynamic characteristics different, mass transfer
characteristics inside the bed.

(Refer Slide Time: 05:26)

Now, see the diagram here in this bed. There are different types of flow regimes or flow
patterns in the gas-solid fluidization system. Here initially, you will see this is called a
fixed bed; that means, here, the solid particles are not being suspended at this gas
velocity.

So, here it is a fixed bed operation. So, we cannot say this is a fluidized bed condition.
So, this is the before that fluidization operation; after a certain change of gas velocity
beyond this fixed-rate operation you will see, there will be a change of characteristics of
the fluid bed inside the fluidized bed, like if you increase certain minimum gas velocity
then you will see this type of particulate or you can say homogeny homogenous
fluidization operation. Here you will see there will be a significant change of the level of
the bed where there will be some tilting or you will see that where we shape of the bed;
that means, here beyond a certain minimum velocity of this fixed bed operation, you will
see certain this fluid is going to the suspended here inside the bed. So, this is here one
operation; this is called particulate regimes.

So, whole this fluidization gas-liquid gas-solid fluidization system, you can be divided
into two parts here batch operated and transport operated flow regimes. Now batch
operated flow regimes here you will see some flow regimes like homogenous or
particulate you can say, sometimes bubbling bed and also churn turbulent fluidized bed.
And under this gas-operated system, you will see gas-operated fluidized bed this
bubbling fluidized bed, slugging fluidized bed, and also the churn turbulent fluidized bed
we will have different characteristics and different hydrodynamic characteristics. And
also, the transport operator here see first fluidization and pneumatic transport. There are
two types of fluidization flow regimes, and all this flow regimes will be changing if you
just changed the gas velocity inside the bed, but you cannot say that all this flow regimes
will be changing only based on the gas velocity no.

If you fixed some gas velocity and if you change the other variables like a geometric
variable if you like our that particle diameter if you change the what is that physical
properties of the system, if you see different gas with higher density gas has been you
will get a different type of flow regimes also. So, let us see at a certain gas velocity here,
particulate bed, the characteristics of the particulate bed here the just particles are tries to
you are trying to suspend inside the bed, and there will you know bubble formation here.
Whereas, in this case, if you increase a certain amount of gas velocity, you will see there
will be a formation of a certain gas, and there will be a certain shape of the bubble here.

So, here gas will be distributed through a distributor, and from the distributor, there will
be a formation of bubble inside the bed. So, this is this type of phenomena is called
bubbling fluidized bed; and when if you increase a little bit the gas velocity this bubbling
fluidized bed will behave and will form a bubble in such way that the bigger bubbles will
form and those bubbles are going up through the center of the bubbles, as a large the
bubbles. And this large bubble, sometimes it will be almost equal to the cross-sectional
area of the bed that is the diameter of this bed will be equals to the diameter of the bed.
So, if it is diameter, then, of course, you will see slug you will form even sometimes we
will see if it is not the same the diameter with the fluidized bed, then you can get the plug
flow type fluidized bed here.

So, there will be a plug and a slug. So, this is under the bubbling fluidized bed fluidizing
condition. Even if you increase the fluid velocity or you can say the gas velocity, there
will be a see the churn turbulent condition will from here, the bubbles will be forming or
you can say voidage will be there, but that voidage will not be exactly the bubble shape,
and it will sometimes be a long longitudinal bubble here long bubble and remain in
between this two bubbles there will be a channel of the solid particles or channel of the
gas along with the solid particle, in such a way that there will be a churning phenomenon
inside the bed. So, this is called churn turbulent fluidized bed and slugging fluidized-bed
you will see you will obtain very narrow tubes narrow beds, in that case, slugging
condition you can get.

Whereas the churn turbulent flow, you cannot get in the narrow tubes. In that case, you
have to have the diameter of the column at least five centimeters. Otherwise, it will be
very difficult to get this churn turbulent. Because this churn turbulent will actually
follow, there will be an internal circulation of the internal cell circulation of the fluid
particles and solid inside the bed, and there will be churning condition formation. So, this
is this gas-operated whereas, in transport operated cases if you increase further this gas
velocity you will see, there will be very dilute solute that is the concentration of the solid
that distribute inside the bed in such way that maximum portion will be the gap or you
can say the void inside the bed. So, here, but smaller particles also will be moving up a
small amount of particles will be moving up.
But this will be first fluidization, and particles will try to move up as early as possible
along with the gas. And in this case, of course, the fluidization condition will be far more
than this minimum that is the settling velocity of these particles or you can say the
terminal velocity of the particles and pneumatic transport here very dilute phase of solid
particles all the solid particles will try to go up. It will be coming out from the top, and it
will be again reused even for first fluidization. Also, you can use or reuse these solid
particles here.

So, in pneumatic transport, there be no bubble formation specifically the shape of bubble
formation will not be there, and there will be very dilute the concentration of solid
particles, and it will be flowing up many a case. So, in this case, at least 20 times larger
than the terminal velocity of the fluidization bed inside the bed should be there. So, here
you can get that from the fixed bed, we can get different fluidization or different
fluidized regimes like homogenous bubbling, churn turbulent first fluidization pneumatic
transport.

Now, to get the minimum fluidization, of course, you have to maintain; you have to
maintain certain a gas velocity that is called an incipient condition, and that incipient it is
called incipiently fluidized bed or minimum fluidization condition. So, of course, to get
the fluidization that minimum fluidization velocity should be required for, you will be
discussing how to calculate the minimum fluidization condition later and also. So, in this
case, we are then we are getting the different fluidization, different fluidized fluidization
regimes, based on this solid and gaseous system. And you can get other different types of
flow regimes, if you the fluidized bed is operated with the liquid and solid and also gas
and liquid and solid.

So, we will be discussing subsequent in a lecture that what are the different types of flow
regimes will be there in the fluidized bed.
(Refer Slide Time: 13:58)

Now, let us consider let us discuss what should be the wherein characteristics of the
different flow regimes in the fluidized bed. Now homogenous or particulate fluidization,
this is as the gas velocity increases, the bed expands smoothly in a homogenous manner.
So, in this case, you will see there will be a gap between two particles; there is small gap
between two particles through which the gas will be flowing upward without making any
bubble inside the bed or either making churning condition without making any dilute
phase, without making any other slug condition you will get the simply homogenous
flow inside the bed.

So, this will be happening in a condition which is called homogenous fluidized bed. The
top surface of the fluid bed is well defined with small scale motion of the solid particles
here, of course, this minimum fluidization condition you will see the top surface of the
fluid bed will be well defined, and it will behave like a liquid here if in this case the
surface of the bed, of course, will be tilting or it can it may change according to the
operation. And then also it will be well defined, and there will be a little bit motion of the
solid particles at this homogenous fluidization regime the solid particles have little
tendency to aggregate also, in this case, there will be no actual formation of agglomerate
inside the bed, and in throughout the cross-section, we will see there will be a uniform
distribution of the gas through space between. The solid particle that is called voids in
the bed.
And a further increase of gas velocity bubbles is formed; in this case, in this particulate
fluidized bed with the same particle, you will see if you just changed the gas velocity a
little bit higher, and you will get this. So, there will be a change of phenomena from this
particulate to the bubbling phenomena. Inside the bed, there will be a formation of a
small amount of small bubbles, and it will be going upward at a certain velocity. And the
particulate fluidization occurs in the range of gas velocity that will be, of course, that will
be higher than the minimum fluidization velocity, but it should be less than the
fluidization condition of the bubbling fluidized bed.

So, here UMA, we are denoted this minimum fluidization velocity, and UMB is the
minimum fluidization velocity for bubbling condition. So, in this case, you will see this
particulate bed fluidized regimes can be defined only when the fluidized bed is operated
within this minimum fluidization form to this bubbling fluidization condition, and this
pattern is sometimes called as a homogenous fluidized pattern. So, this is the
characteristics of the homogenous or particulate fluidization.

(Refer Slide Time: 17:27)

Whereas bubbling fluidization, of course, you have to increase the gas flow rate beyond
this particulate fluidized bed so that you can have large instabilities or some instabilities
in such a way that there will be a formation of bubbles and channeling of the gas that
will follows inside the bed. And higher flow rates, of course, the movement of the solids
becomes more vigorous in that bubbling condition, and in this bubbling fluidization
condition whenever bubbles will moving bubbles will move up, they are, of course, the
bubbles will carry some solid particles from its behind by making way. So, and also from
the top there will be just; there, the bubbles will move a side that the solid particles from
its top portion and whenever bubbles will move up at the surface it will be collapse and
solid particles will be throwing out at the surface.

So, this type of phenomenon will happen inside the bubbling fluidized bed. So, there will
be some bubbles will form and those bubbles will be forming from the distributor region
the bubbles will be very smaller, and whenever it will come to the top we will see the
bubbles, will coalescence to each other, and it will make big bubbles and after
coalescence and then it will go up very fastening. So, initially, bubbles will move slowly
because its size is very small near about the distributor, in it depends on the whole
diameter of the distributor, and the size of the bubble depends on the whole diameter of
the distributor and because of which you will get the bubble size very small.

But whenever it moves up due to the coalescence of two or three bubbles at a time, there
will be a formation of big bubbles, and it will go up very fastly because of their
buoyancy effect. So, bubble size increases as gas velocity increases; of course, you will
get the different sizes of bubbles as a function of gas velocity. If the bubbles become
larger, the phenomena of a slug are also developed. So, of course, if you are making high
gas velocity, then you will see the two bubbles will come to each other and making big
bubbles, and if these bubbles are almost equaled to the diameter of the bed, then it will
make one slug. Even flag also it will be there because the sometimes two bubbles
whenever it is coalescence making a big bubble, it may be approximately 85 percent of
the cross diameter of the bubble, then it will go up.

But it will be very large and as a slug; that means, as a plug it will be moving up. So, it
will be plug flow; and slugging the gas bubbles or void coalescence and grow as they
rise.
(Refer Slide Time: 20:46)

So, in that case, it will be forming slugging fluidization. In a small diameter did bed, you
will see the bubbles may become large then up to spread across the bed. So, in that case,
we will see the slugging tendency will be there in the bed. And if the bubbles are actually
vertically elongated and move axially and move axially the phenomena some time it is
called axial slagging. So, there are two types of slugging will be there flat and axial
slugging.

So, if most of the bed cross-section is a captured by the bubble or void the phenomena is
called plug slugging and so, that is why there if axial slugging means there, of course, the
bubbles are vertically elongated whereas, the plug slugging the bubbles will be
horizontally elongated, and it will occupy most of the cross-section of the bed.
(Refer Slide Time: 21:55)

Whereas turbulent fluidization, in this case, small bubbles or void or form. So, the solid
particles, of course, clusters, and it will move rapidly by forming elongated, or you can
say that form an elongated bundle of slugs.

So, the, in this case, you will see some clusters of particles will be forming, and that
clusters will move upward and sometimes this clusters whenever it will be collapsing or
it will be just reaching to or to the bottom of another slug, it maybe elongated and bundle
of slugs will form. And the top surface of the bed in this flow pattern, of course, it is very
difficult to distinguish, because you will see there will be the unstable condition of this
turbulent fluidization, and at the same time solid and the gas both will be flowing in such
a way that the particle and gas that is particle fluid interactional will be higher, and
because of fluids that you will not get the particular shape of the surface inside the bed
and it will be very unstable in this case. The turbulent fluidization pattern will starts
when bubble coalescence and the break up reach dynamic balance.

Of course, there will be a simultaneous happening of coalescence and breakup of the

bubble inside the bed, but you thing this turbulent fluidization. So, for smooth bubbling
condition smooth coalescence, but breakup will be less whereas, in the turbulent
condition because there are what is that intermixing and also the internal circulation of
the fluid that will sometimes this stuff the uniform movement of the bubbles, and it may
break the bubbles and also bubble-bubble interaction will grow up, and also coalescence
also because of that it will be there.

So, simultaneous operation simultaneous actually process of coalescence and break up

will be there inside this turbulent fluidization. And in a fluidized bed with an increase in
the gas flow, in this case, the most unstable condition will occur, and of course, this
correspondence both the beginning of the transition of bubbling and slugging to turbulent
fluidization and to bubbles of maximum size here.

(Refer Slide Time: 24:51)

And first fluidization in this case particles are transported out the top, of course, the gas
velocity will be so high, at least 20 times of the terminal velocity of the particles there.
So, if 20 times of the terminal velocity of the gas, you will see particles will not be
coming back to its original position, and it will move up, and it will be transported out
the top, and that transported out solid of course, it will be separated from the gas mixture
by somewhat is that separator, and after that, it will be again used for this fluidization.

So, in this case, particles are transported by the gas here, back mixing of the solid
particles will be less because of high kinetic energy here. And in this case, you will see
the must be replaced by this by adding a solid set or near the bottom of the bed. Of
course, you have to add the solids here near the bottom of the bed otherwise if you
cannot actually add this solid at the certain height because at this operating condition, if
you supply from a certain height of the bed, it will go immediately to the top and it will
not having that much of retention time inside the bed.

And the clusters of the particles move, of course, downward near the wall, while gas
containing dispersed particles moves upward in the interior. Of course, it will be they are
if you are using very fined particles there will be sounds to form of clusters, or you can
say agglomeration and different sizes of clusters will be there inside that bed, that
depends on the gas velocity and also the weight ness of the solid particles they are.

And if there is the big size of the clusters or form inside the bed, of course, it will not be
reached immediately to the stop, sometimes because of this weight. It may go back to its
previous position; that means there will be a tendency of formation of circulation to its
downward, and that is why the downward movement of the solid particles near the wall.
Because to get back to it is it will have enough space if, for the very narrow tube, you
may not get this type of backup movement or downward movement of the solid particles
inside the bed. So, to get this backward, that means, downward movement of the solids,
there will be a plus diameter of the fluidized bed.

And the solids in the first fluidized bed may typically occupy up to 25 percent of the bed
volume and is in a state of extreme turbulence marked by extensive refluxing of dense
strands and packets of particles there. So, we are the. It is a very important point to note
that in the first fluidized bed occupies at up to 25 percent of the bed. So, otherwise, it is
very difficult to get this type of fast fluidization condition.
(Refer Slide Time: 28:56)

Now, pneumatic conveying fluidization in this case homogenous dilute phase of the
solid-fluid is observed in this case, and there will be no bubble formation. There will be
known what that chunk churning condition is. There will be you know what is that first
fluidization condition here.

Only the solid particles will straight forward going up, and it will be coming out from the
top of the bed. The particles are fully suspended in the gas, and particles travel like a
piston with relatively high-pressure drop and high feed rate of particles here. So, there
will be you will see there will be a velocity of the solid, or you can say the same flow
rate of the solids along moreover the gas velocity harm. Of course, in this case also more
than 20 times of terminal velocity of the solids, the fluidized bed will be operated here.
Now all particle spread in this transported all particle that is spread in the spread that will
be transported out the top of the bed as a lean phase here.

So, in this case, this pneumatic conveying fluidization, of course, it will depend on the
velocity and the solid content of the airflow, and different transport states may occur in
this case. And there will see there will be different transport states like suspension flow
or dilute phase transport intermediate flow or you can say strand transport, and dense
phase dilutes or you can say dune transport phase here, and also dense phase plus
transport, of course, will be there as per figure it is there.
So, here this will be your suspension flow or dilute phase transport, and this is b here
intermediate flow or strand transport here; you will see this how the solid particles are
moving, here flow pattern in is section will be almost the same here. Whereas in the see
that is called dune transport here in this case the flow pattern will not be same in the
middle section maybe or the depending on the gas velocity and solid particles type, it
depends on the also cross-sectional area of the bed that how the solid particle should be
transporting here. And also g is called plug flow here some plug or bubble formation will
be there in the bed region of dense portion will be formed here instead this pneumatic
conveying fluidized bed.

So, this happens because of this cross-sectional area of the fluidized bed.

(Refer Slide Time: 31:52)

And suspension flow dilute phase transport at high velocity is the solid particles move
through the line distributed uniformly across the cross-section of course; the particles in
fact, against each other or against the bed wall, will be there. And intermediate flow or
strand transport the velocity is reduced, while the solid content remains constant, you
will see the energy of the flow, that will not be sufficient to hold the entire solid mass
that is suspended in the bed.

And some of the solid particles you will see slide along the bottom of the pipe in the
form of strands, and the raised are transport trade in suspension above the strands. Or as
dense phase dune transport if the velocity is reduced further you will see the solid
particles move like a dune and particles are moved over the summit of the dune and are
deposited on its sheltered side if the velocity is reduced further in insipient plugs may be
formed from the dunes which occupies a major part of the cross-section of the bed.

(Refer Slide Time: 33:22)

And dense phase plug transport, in this case, a very low velocity, the material occupies
the entire cross-section of the pipe and plugs are formed, and plugs advanced slowly, if
the air compressor does not have sufficient fresher, of course, plug transport may quickly
lead to blockage of the pipeline. So, this will, of course, this regime, of course, will be
having the beyond this 20 times of the terminal velocity of the solid particles.
(Refer Slide Time: 33:58)

And then spouted fluidization, this is another important fluidization regime here, this
pattern represents a mod wherein comparatively coarse, uniformly sized solids are
contacted by gas.

In this case, you will see that solid size or particle size will be a little bit higher than the
other particle. So, in this flow pattern, gas forms a single opening through the distributor
and which in which some particles flow and fall to the outside. At a higher gas flow rate,
you will see hesitation becomes more violent, and the movement of the sides becomes
more vigorous, and the bed does not expand must beyond its volume at minimum
fluidization. Now, in this case, you will see the principal features of flow regimes here.
So, we are getting different flow regimes like fixed bed here that is before particulate
fluidization regime.
(Refer Slide Time: 35:08)

There will be some region that is u range the velocity that is U mf is minimum fluidization
velocity, Umb the bubbling fluidization this for minimum bubbling fluidization condition,
in this bed expands smoothly in a homogenous manner top surface well defined the small
scale particle motion will be there whereas, bubbling regimes in this case beyond this
minimum bubbling fluidization velocity whereas, the less than the minimum velocity for
slugging condition. In this case, gas voids form near distributors the bubbles will
coalesce and grow rise to surface and brake through.

Whereas the slug flow bubble size approaches column cross-section, the top surface rises
and collapses with regular frequency here. And the turbulent fluidization flow regime
these cases because you will see pressure fluctuation gradually decreases until the
turbulent fluidization flow regime is reached. And turbulent regime small gas voids and
particle clusters dirt to and flow top surface is very difficult to identify. And first
fluidization no upper surface to bed and particles transported out the top in clusters and
must be replaced there, and pneumatic conveying in this case now a bed of course, you
will see that particular bed that will be some level, it will not be having in your the
pneumatic conveying.

All particles spread in a transported out the top as a lean phase there.
(Refer Slide Time: 36:49)

Now let us classify this different fluidization condition as for the fluidization regime, and
also based on the classification of the particle. Now I think we have discussed in the
previous or earlier lecture there four different types of gilded particles like C A B and D.
Now you will see if you consider the c particles and if you increase the gas velocity then
how the, what type of fluidization regime you can expect in your fluidized-bed. Suppose
here this particle type is C type; that means, very cohesive in nature very fine particles
here interparticle attraction is very high, and there will be a cohesiveness nature because
of which you may expect in the fluidization bed as channeling and also you can have the
flow regime of turbulent churning fluidization with this fine particles, and also you can
get the first fluidization of course, if the gas velocity if it is greater than 20 times of
terminal velocity.

And you can get the pneumatic transport which will be of course, greater than 20 times
of terminal velocity of the particles here. So, this C type with this C type particle, you
may expect this channeling turbulent channeling first fluidization pneumatic transport.
Whereas, either particle here, the particle size is a little bit higher than this cohesiveness
that we have earlier discuss that greater than 100 micrometers. In this case, you can get
the smooth fluidization bubbling fluidization, whereas, in C type particles, you may not
get the bubbling fluidization in this type of fluidization. You can get the bubbling
fluidization here.
You can get the turbulent churn turbulent fluidization and also the first fluidization, but it
is very difficult to get pneumatic transport with these particles. So, this. So, all this flow
regimes you get only by increasing the gas velocity with fixing this solid particles time
whereas, with b type particles in this case you get bubbling fluidization, and turbulent
fluidization, but you cannot get this what the smooth channeling fast fluidization on
pneumatic transport type fluidization with this B type particles because here the size
higher the size matters is. So, but D type particles here only two types of fluidization you
may except here bubbling fluidization and turbulent churn turbulent fluidization you
cannot get any smooth channeling fast or pneumatic transport fluidization.

So, only the d type particles applicable to get this fast oh sorry turbulent and the bubbling
fluidization. So, it is very difficult to get these other types of fluidization by switching
one particle type, but you can infer what type of fluidization regime can get from this can
be obtained from this particle from this knowledge.

(Refer Slide Time: 40:42)

Now, flow regime map and its transition very important here, grace 1986 based on a lot
of experiment with a different type of particles and with varying gas velocity in the void
range of course, with here they got different results of different fluidization phenomena
and they have actually suggested one map by representing the transition of different flow
regime, based on their experimental data. And they have represented the transition of
different flow regime or different patterns flow patterns inside the fluidization bed, by
just choosing to the choosing the axis in a different way, to represent it graphically in this
case you have to see this here flow regimes in this case very important in this case you
will see the x-axis is dp star, dp star is what is that non-dimensional particles diameter
which has made which has been suggested in this way this will be d p into; that means,
here

rho g into rho s minus rho g by mu square into z to the power 1 by 3 or you can say d p
star is equal Archimedes number to the power 1 by 3.

Whereas this y-axis that is u star, u star is the non-dimensional velocity of the gas which
is defined by that is

Now if you see this figure here we are getting a different type of phenomena here you
first we will see this C A B and D, these are the different type of particles here and this
lines this is the lines these lines form line will be representing the transition line or the
data on this line will give you the transition data, from beyond which you can get the
what that different flow pattern is. Here see here very interesting here is that here if you
consider the c here c in this case, you will see some flow pattern map, here this is called
this region is called this is spouted bed this is bubbling a fluidized bed this region is
bubbling fluidized bed, and here pneumatic transport fluidized bed and this region is
called fast fluidization bed.

So, here see here if you consider c particles this is the c particles flow regime here, what
happened if you consider the c particle here or the point, any point at this point if you go
you will see at this location what is this flow regime here, this is the point where you can
get at this point like suppose dp star is equal to 1, dp star is equal to 1 if you go up here
you can get this one, this here this is your minimum fluidization condition, and this is
you terminal velocity profile and at this point. So, this d p star you can get one terminal
velocity and here if you go this and if suppose u star is somewhere here one what will
happen this is the beginning of the first fluidization condition and then beyond which
after they suppose up to10, here you will see at this location, and you can get the first
that is pneumatic transport condition.

Now, if you considered the B particle here and if you considered the d p star is equal to
this like what is the value here like suppose 10 then it is 11, then dp star is equal to 11
what will happen you can get up to this here any point here whether it will be spouted or
some other that you can; obviously, you can take from this regime that it will be the
spouted region here. Because here in this case at this any location you just pinpoint here
g by this combination of this point you can get here spouted. Even if you considered
here, suppose this is what that? This d p star is equal to suppose eleven, and you can get
this bubbling fluidization only when u star is equal to 1. So, this is the phenomenon. So,
from this graph you can read actually or before going to the experiment, you can have
some rough idea from this flow pattern map whether you are getting the bubbling
fluidized bed or not or you are getting a sliding condition, or you are getting pneumatic
condition or you are getting spouted beds, that you can, of course, I have some idea from
this flow regime.

Any before going to start any experiment, you can, of course, suppose you want to have
the bubbling fluidization condition, then what you have to do, you have to select b type
particle first select B type particles. So, if I select the b type particle, and if you want to
offer it at one point, what is this? This is the d p start equal to 10. Then you have to have
the velocity at least minimum velocity should be here at this point. So, this is your point
from which you can get if this b particle what should be the velocity of the gas by which
you can get the bubbling fluidization condition; and other also if you want to get the
spouted beds like you have the pd.

So, the type is D type where dp star is equal to this, and then you want to get the spouted
bed, suppose you want to get this sprouted bed at this bubble and for similar gas velocity
what should be the velocity from which you can calculate this should be your velocity by
which you can get the spouted bed. So, in this way, before going to start your
experiment, you know and if you want to apply this fluidization operation for a specific
operation, and if you know that the operation you will be favorable in bubbling
condition, then the operation condition you can select from this fluidization map also.
(Refer Slide Time: 48:26)

Other types of fluidization regime here you can say here some gas-operated here Bi and
Grace, 1995. So, they have given this fluidization map.

Here gas-operated and transport operated two types of fluidization map they have
presented here, and these are the lines you can get the transition points and also here this
is the transition point these are the transition points or transition line here beyond which
you can get the fluidization operation whereas, below this line or less than this point you
can get the packed bed at a different what is that boundary of the particles here. Now this
in x-axis it is represented by Archimedes number to the power 1 by 3, and in y axis this
is U star, U star here this is defined as

Here again, this at this suppose here type A and C this is a mixture of a particle A and C,
and if you go this here in this region you can get this some minimum fluidization or
particulate fluidization and beyond which at this location, these are the locations in you
can get the bubbling fluidization this location means at any point you select, and you
read the data of this Ar and U star and from which you can have the idea what should be
the value of gas velocity and what should be the value of Archimedes number. From
which you can get what should be the diameter of the size if you know the all other
parameters constant here and also the turbulent fluidization this is here also this is the
transition line beyond which you can get the tabular transportation and also below this
point you can get the turbulent fluidization. So, these are the transition points. These are
the transition points. These are the transition points.

So, these are the transition points and the data with this transition points, they have
developed they have suggested some correlations from which you can get the transition
data; that means, that that particular value, you can get either this packed bed or suppose
particulate bed, that depends on the operating condition and other different variables of
course. Suppose to get the pneumatic condition. So, in that case of what should be the
minimum velocity you can get from this correlation, what should be the minimum
velocity to be maintained so, that you can get the pneumatic transport condition here. So,
in this case, you can, from these correlations, these are the correlation that has been
developed from this experimental data, from which you can get the minimum
fluidization condition.

So, we will be discussing what should be the minimum fluidization condition, and the
different flow regime transition will be discussed later. And also and here, of course, you
will see one term is here Uc, Uc see is a very interesting point, this Uc is the corresponds
to the point where the standard deviation of differential pressure fluctuation reaches a
maximum. In the fluidized bed, you will see whenever it is operated under certain
operating conditions the fluctuation of the pressure inside the bed, of course, will be
there. Now, this pressure from this pressure fluctuation you can also obtain what type of
flow regimes there in the bed.

Generally, the standard deviation of differential pressure fluctuation when it will be

reaching a maximum, you may expect there will be a choking that is called pneumatic
transport there. And the critical velocity that is se, which is defined as the point where the
solids begging to be entrained significantly inside the bed at this particular fluidization
operation, then it will be called as A-C; that means, here the significant entrainment will
be occurring. Now this very important, but whenever you are considering you are
working with the transport operated fluidization bed you have to consider the fluidization
velocity as A B which will be of course, the beyond the minimum fluidization and you
have to add this minimum fluidization along with the operating fluidization condition
there.
And another important point the bubbling fluidization of how you can get when exactly
at which velocity you can get, or you can expect to the bubbling fluidization there. You
can get the minimum fluidization velocity for bubbling condition by this correlation here.
See here see 33 into dp into rho G by mu c to the power 0.1

for does it signify it signifies that if particular dp, if you are just keeping it constant and if
you increase the density of the gas you will see the bubble velocity; that means, here the
velocity of the gas inside the bed will be so, higher that you can get the operation of the
bubbling fluidization, but what should be the minimum. You just if you considered that a
r and particle size is 100 micrometer and here if you substitute here the d p is equal to 100
microns, that is you have to convert to the meter and then if you substitute a g that is 1.2
and by g means here what should be the viscosity of the water here.

So, in this case, viscosity 0.00018. So, if you substitute, you will get the minimum
bubbling condition. So, you can directly apply to get the minimum bubbling condition
this range here. So, you can easily calculate before starting the bubbling fluidization
condition operation.

(Refer Slide Time: 55:00)

Another important fluidization map for flow regime map here given in different way;
that means, here then x axis you will see this is

the Cd into Reynolds number to square whole to the power 1 by 3

this is one parameter and also here another parameter here Re into C d to the power 1 by 3
what is Cd? Cd is nothing, but the drag coefficient will be discussed later and also even
previous earlier class also what is drag coefficient has been discussed. So, this is the drag
coefficient. So, here in the x-axis, you will see for different particles here group C, even
mixture of groups A and C, group A and group B, and group D Geldart group D
particles.

For this type of particle, you will see different flow regimes, different flow patterns you
can get by this axis. Here these are the dotted lines are represented by the low cast of u
and f; that means, at this point, you can get the minimum fluidization to come, what is
that packed bed to the particulate fluidized bed. Whereas, this line will give you the
terminal velocity of the particles here. Now if you consider any group c particles here c
at this what is that if the value of this x-axis, this x is equal to this then and here y, y is
equal to this then you will get this value for a particular x is equal to 1 and then here
what should be the value of what will be the flow regime you may expect here.

So, this regime is nothing, but see the locus of u tr from transition; that means, the
transport flow regime. So, here at this. So, even you can get transport regime whereas, in
this position, you can get the turbulent fluidization condition, and with this y value of
this, you can get the, what is that bubbling and bubbling fluidization condition and slug
in fluidization condition also. So, this is one way to represent the different flow regime
maps here.
(Refer Slide Time: 57:23)

Now, what should be the transition from particulate to the bubbling fluidized bed? The
equation here is given that is developed by Geldart and Abraham.

0.1
 
umb  33d p  g 
 g
 

So, to get the minimum bubbling fluidization velocity, you can use this correlation to
calculate for minimum bubbling fluidization condition.

(Refer Slide Time: 57:45)

 Minimum Bubbling Velocity
For minimum bubbling velocity also there are other correlations.

u mb 2300 g0.13  0.52 exp(0.72P45 m ) where P is the frac

 less than 45 micro
u mf d p0.8 (  s   g ) 0.93
For FCC catalyst of s
It is important from which you can calculate the minimum bubbling fromgas velocity ug = 3umf
the fluidization
form
condition. Here, in this case, this is the function of the density of the gas viscosity
In Group
particle diameter, A powders
solid when
density and also Umb
there will > Umf
be a what is that abubbles are constant
fraction of powder,

coalescing,
which has the particle and a ismaximum
diameter stableforbubble
45 micrometer generally size
FCC catalyst of sizeis5 to
achieved.
for good quality, smooth fluidization.
100 micro at gas velocity, Ug is equal to 3Um f bubbles begins to form here. In group a
powder when U m b; that means, minimum obligation condition is greater than minimum
fluidization is constantly splitting and coalescing and maximum stable bubble size is

Through
achieved. splitting and coalescence,
Thisbubbles
makes good achieve a maximum
qualities smooth fluidization here,stable
and also size,
through splitting and
effectively
coalescence, independent
of course, the bubbles achieveofa gas velocity
maximum orand it will be
stable size
vessel
effectively size.gas velocity or vessel size non bubbling expansion of a fluidized
independent
bed of group a powder as the gas velocity is gradually increased to there.
non-bubbling expansion of a fluidized bed of Group A powder
(Refer Slide Time: 58:59)
as the gas velocity is gradually increased
Or the transition from bubbly to slugging fluidization of course, in this case. this
minimum fluidization condition for slugging is um a is equal to 0.07 into gD to the power
0.5,

from this correlation, you can get what should be the minimum velocity for slugging
condition and then minimum velocity you can calculate from this equation.

(Refer Slide Time: 59:18)

And other criteria for slugging also here, it is seen that there are several other criteria to
get the slugging fluidization here. If velocity terminal velocity is by gdp; that means it is
called cloud number if it is greater than 140.

Then you see there will be a stability of upward flow of a uniform unbounded suspension
you can expect here no allowance for wall effects will be there. If this number is greater
than 0.35, of course, you will see there will be slug stability based on the equation of
here Harrison et al. they have produced here. And this criterion will give you slug
stability based on the empirical evidence and this correlations here this correlations will
give you the slug postulated not to be able to rise faster than porosity waves into of
course, these correlation has been developed by smith 1978 whereas, Guedes and
Carvalho (1981), they have observed or some mechanism of slug stability based on the
modified Harrison equation and they have developed one criterion for which you can get
the slugging condition based on bubble splitting from the real wall.
 Slugging Transition
So, these are the criteria other criteria from which you can except the different slugging
conditions.

(Refer Slide Time: 60:42)

When the size of the bubbles is greater than about one-
third of the diameter of the equipment their rise velocity is
controlled by the equipment and they become slugs of gas

Slugging is attended by large pressure fluctuations. and so it is

generally avoided in large units since it can cause vibration to
the plant. If the bed is sufficiently shallow slugging is unlikely to
occur at any velocity.
If the bed is deeper than this critical
slugging will not occur provided the
height then slugging will occur when
criterion in Equation (A) is satisfied.
the gas velocity exceeds Ums given by
This criterion works well for most
Now, this slugging transition you come this isEquation
the condition(B)from(Baeyens
which you can and Geldart,
get the
powders.
slugging transition. And if the bed is different 1974).
then this critical height, then the slugging
will occur when the gas velocity exceeds u m s given by the equation B here this. This is
your minimum height of the fluidized bed, now if the bed is different than this critical
(A)by this equation.
height, then slugging will occur then the gas velocity exceeds Umf given

(B)

And

And slugging will not occur provided the criteria in equation A is satisfied these criteria
work well for most powders.
(Refer Slide Time: 61:20)

And this is the condition from conditioner or transition of slugging to churn turbulent
condition, and from this transition, you can expect for the type of either the slugging or
turbulent of from slugging to turbulent how can I get to operate.

(Refer Slide Time: 61:38)

And this is your transition from churn turbulent to first fluidization, of course, and by
this equation, you can calculate to get the minimum churn turbulent flow or just to start
the first fluidization.
Re se  1.53 Ar 0.50 (2  Ar  4 106 )

(Refer Slide Time: 61:52)

And transition for pneumatic transport fluidization of course, at a fixed solid flux, the
transport operator fluidized bed experiences bubbly slug flow turbulent flow or first flow
fluidization before assuming pneumatic transport. Now according to Bi and Grace 1995,
you will see a bubble-free dense phase flow regime may also exit for group A particle.
So, in such cases, the flow patterns of the transport planner completely determined by the
relative velocity between the gas and the particles phase rather than the superficial gas
velocity.

The minimum fluidization velocity under this condition how to denoted by u m v m f all
other velocities under continuous condition are represented by capital here latter B
instead of small letter u use for based operated fluidized bed can be estimated.
(Refer Slide Time: 62:50)

And the minimum fluidization for pneumatic transport condition you can get from this
equation here.

Gs mf
Vmf  umf 
 p (1   mf )

Where

umf  g d p 1
 0.5
 27.2
g  27.2  0.0408 Ar 
(Refer Slide Time: 62:59)

Pneumatic transport condition here minimum pneumatic bubbling condition here, this is
your minimum pneumatic transport bubbling fluidization and minimum pneumatic
slugging condition also here.

(Refer Slide Time: 63:11)

So, this is the case transition for pneumatic transport fluidization from which you can get
this type of equation from which you can calculate what should be the choking velocity
in the pneumatic transport condition.
 Gs ch
Vch  uch 
 p (1   ch )

 ch   bch  (1   bch ) mf

 Bc  0.30 Ar 0.04 (2  Ar  4  106 )

(Refer Slide Time: 63:22)

What is choking? Choking actually as a generally used to describe phenomena that occur
when there is an abrupt change in the behavior of a gas-solid conveying system. The
choking point, therefore, has been characterized by the formation of slugs of plugs or
there will be is there any severe instability or not. Three types of choking generally are
observed in fluidization operation accumulative choking, classical chocking, blower or
stand-pipe induced choking.
(Refer Slide Time: 63:55)

So, different choking. So, we will give you different phenomena.

(Refer Slide Time: 63:59)

And there are various definitions of the choking that observed in the fluidized bed that
observe.
(Refer Slide Time: 64:10)

And here classical choking, of course, you can get the apparent relative velocity at
choking that can be calculated by this equation here.

Gs ,cc cc
us ,cc  ucc 
 p (1   cc )

(Refer Slide Time: 64:17)

The transition from humongous to dilute to core annular, this is very important
correlations you can directly get from homogenous to the annular dilute flow by this
equation.

( gd p )0.347 (Gs /  g ) 0.310

ump  10.1
(d p / D)0.139 Ar 0.021

The only thing is that you have to change the solid flux here, and also you can change the
diameter of the pipe if you change then you can get the annular condition of the
fluidization.

(Refer Slide Time: 64:46)

So, next class from what do will be discussing some other part of the fluidized bed, like
the implication of what would be the ratio of this minimum fluidization to minimum
bubbling condition how it will be affecting in that ok.

Thank you.
 Fluidization Engineering
Dr. Subrata K. Majumder
Department of Chemical Engineering
Indian Institute of Technology, Guwahati

Lecture – 07
Flow regime and its map:
Liquid-solid & Gas-liquid-solid Fluidization

Welcome to a massive open online course from fluidization engineering, today’s lecture
part will be on flow regime, and it is a map basically on a liquid-solid and gas liquid
solid fluidization system.

(Refer Slide Time: 00:45)

Now already, we have discussed what the definition of flow regime or pattern is. So, this
flow regime or pattern means this is the flow behavior and shape of the interfaces
between phases in a multiphase mixture, which is commonly referred to as the flow
regime or flow pattern.

The flow regime of multiphase flow in a fluidization bed that we have discussed, that
depends on several factors like dynamic variables, geometric variables, thermodynamic
variables and physical properties of the features dynamic variables like flow rate of the
gas-liquid or even if solid is in continuous mode, then what will be the solid velocity and
then geometric variables means column diameter or bed diameter bed cross sectional
area volume of the bed, even what would be the particle diameter and the geometry of
the particle, like, what should be the shape of the particles?

What is the sphere city? All those variables are coming under geometry variables and
thermodynamic variables are like pressure, temperature; how this pressure and
temperature will affect the fluidization behavior in a fluidized bed and mostly the
physical properties of the phases that is density of the fluid, viscosity of the fluid and if
the gas-liquid-solid system, what should be the surface tension of the liquid that we will
affect the fluidization behavior inside the bed.

If you change the density, of course, the behavioral change viscosity highly viscous
fluid, it is very difficult to flow are getting different flow regime inside the bed; even if
you are changing the particle diameter that is coarser, we will not get the specific flow
regimes for a particular operation, then you have to change is that accordingly the flow
regime we will change, and application will be different, and also pressure affect there
will be at high pressure if it is operated at high pressure is the minimum fluidization
velocity will be more in that case.

Now, if we have different flow regimes, of course, we have discussed in earlier classes
or earlier lectures that what are the different flow regimes in the gas-solid system and
what should be the flow regime transition it is actually nothing but the boundary of the
two regimes, but then what should be the than flow regime made for the map for liquid-
solid or gas-liquid-solid fluidization system.
(Refer Slide Time: 04:07)

Now, liquid-solid flow in that case flow regime for this operation you will generally get
2 types of flow regimes, one is colloidal dispersive flow regime, and another is
homogenous flow regime; this colloidal dispersive flow regimes, in this case, conveying
of find spherical particles as a suspension without any turbulence, by Brownian
molecular movement with Reynolds number of the solid particle should be less than 10
to the power minus 6.

So, in that case, the colloidal dispersive flow regimes will have occurred, and this type of
flow regimes you will get only for the flow velocity, that you can say the velocity of the
fluid inside the bed it is very low that is even over than the strokes flow regimes;
homogenous flow regime you will get that flowing of find nonspherical particles of low
density, that hold suspension by hydraulic posters and in this homogenous suspended
flow regime, actually is resulted at a little turbulence in a range of Reynolds number 10
to the power minus 620.1. So, in this case, everywhere inside the bed, the concentration
of the solids will be almost similar, or you can say the same throughout the column, and
in this case, the liquid velocity, of course, will be a little bit higher than the setting
velocity of the solid particle.

We see in this diagram are given or this picture or given of the first one is for a colloidal
dispersive flow regime, in this case, the particles are moving with the Brownian motion,
whereas there is no turbulence in this case; but sometimes we will see some particles are
making a colloid with like unstable colloid there making and coagulation to each other
and then it will be moving as Brownian motion. Whereas, a stable colloidal dispersion, in
that case, there will be no oh actually coagulation where no actual agglomeration will be
forming between the particles and homogenous flow regimes they are we will see there
may be a some to cog may coagulations; since the homogenous flow, but depends on
what should be the velocity inside the bed.

For higher velocity, the breaking of this coagulate action or in consider agglomeration
will be a little bit lower tendency related to the other velocity. So, these are the things
that homogenous flow regimes, how it will be there inside in the picture it is shown here.

(Refer Slide Time: 07:34)

Now, of course, it has been discussed, what should be the minimum velocity for gas-
solid fluidization system in particulate bed that we have already obtained from the
balance of frictional pressure drop to the bed. So, in the same way here also we can get
the minimum velocity for liquid-solid or homogenous flow regimes for this liquid-solid
system.

In this case, also you have to find out what should be the frictional pressure drop from
the Ergun equation;
Ergun equation here is this is the part for Ergun equation by which you can calculate to
the frictional pressure drop,

(1   mf ) 2 l umf 2
(1   mf ) l umf
150 3
 1.75
 mf s2 d p2 3
s  mf dp

whereas this

[  p (1   mf )  l  mf ]g

this part is called the weight of the bed or apparent weight of the bed, so this to be
balanced so that at the minimum velocity for liquid-solid homogenous flow, you can get
the minimum velocity as a umf here. Whereas this epsilon mf is nothing but the minimum
fluidization velocity, this minimum fluidization velocity again you can get from this

3 1
 s mf 
14

So, from the equation you can get what should be the minimum porosity or minimum
void age inside the bed for this liquid and solid system and this epsilon m f minimum
fluidization void age depends on the shape of the particles, for spherical particles it will
be the range in 0.4 to 0.45, so those things we have already we have discussed earlier.
So, this is the way to find out or estimate the minimum fluidization velocity for a liquid-
solid homogenous flow regime. Now flow regimes of solid-liquid flow, you can say
another flow regime is very important. It is called Pseudo homogenous flow regime.
(Refer Slide Time: 10:07)

In this case flow of particles need some more turbulence, so that the particles will be
suspending with very low velocity, but with the little turbulence and the Reynolds
number range for this will be 0.1 to 2 and of course forgetting this turbulence you have
to apply more velocity compared to the minimum and also what is the homogenous flow
regime system.

And of course, for this more velocity are the particles in this Pseudo homogenous flow
regimes are getting suspended, and to keep suspended you have to maintain that velocity,
of course greater than the homogenous flow regime velocity; a certain degree if
segregation in this case also can be permitted to get this you know homogenous flow
regime and to maintain equal conditions for solids of any density at the his at this
Reynolds number range of 0.1 to 2, the ratio of the settling velocity of the particle and
the fluid velocity must remain constant in this case. So, if you are maintaining the ratio
of settling velocity and this liquid velocity in this way, you can expect this there will be a
uniform suspension of the particle inside the bed, with this range of Reynolds number.

So, here is the Pseudo solid-liquid flow scene. In this picture that here there will be no
actual uniform concentration inside the bed because there will be some void age. So, that
will be distributed inside the column that is not uniform throughout the column also, and
also particles may be agglomerated inside the bed, but this agglomeration also may be in
such way that, so that little bit nearer to the value of this uniformity of the solid particles
is compared to the homogenous flow regimes and other them that is called heterogeneous
flow regime.

(Refer Slide Time: 12:44)

In this case, of course, you will see the disorder mixer flow of the particles inside the
bed, in this case the velocity should be little bit higher than this Pseudo homogenous
flow; in this case the Reynolds number should be greater than 2 and segregation of the
particle, of course, will be seen by this flow regime and in this heterogeneous flow
regimes.

Lower velocities can, low can sometimes result to convince by siltation and finally to the
so-called critical deposition velocity inside the bed; in this case, the solid particles begin
to settle out at this point, the pressure drop of the mixture will be minimum. So,
heterogeneous flow regimes you can get it from Reynolds number is greater than 2, with
little bit higher turbulence here.

So, see this picture here so this is turbulent, you cannot observe here what should be the
exactly uniformity or uniformity pattern inside the bed, there will be a internal fluid
circulation inside the bed the solid particles and since the Reynolds number is greater
than 2 there will be a more turbulence inside the bed and the ship of the particles may not
be the same inside the bed or you can operate this heterogeneous flow regimes, we poster
particles with high more than higher liquid velocity also.
(Refer Slide Time: 14:35)

And now see rheological behavior and the solid-liquid flow regimes inside the bed, here
see in the table into seen that homogenous flow regime you can get for the Newtonian
fluid if it is concentration; that means, the concentration of the solid particle inside the
bed is less than 25 percent by volume and you can get the transition from homogenous to
Pseudo homogenous flow if the Reynolds number of the particle is greater than 0.1.
Whereas Pseudo homogenous also for this Newtonian flow you have, that means the
viscosity will be linear with respect to liquid and gas. In this case, there is no gas, so only
liquid. So, Newtonian fluid if is less than 25 percent by volume, you can get the Pseudo
homogenous flow, but here the Reynolds number will be is greater than 0.1 and less than
2.

Whereas heterogeneous flow behavior, in this case, Newtonian flow behavior also you
can get if the concentration is less than 25 percent volume, but Reynolds number should
be greater than 2; this Reynolds number is defined based on the terminal velocity of the
solid-like here rho p into up this is here Res you can say r e s will be is equal to rho p or
rho s into u settling velocity into d p particle diameter divided by viscosity of the liquid,
so this is Res. So, based on these Res you can just divide what should be the flow
behavior with a concentration of less than 25 percent. Now at a certain concentration
which is greater than 30 percent, the slurry starts begin as a non-Newtonian fluid, what is
that Newtonian and non-Newtonian fluid Newtonian and non-Newtonian fluid actually
you know that whenever stress liquid, stress is directly proportional to the shear rate then
you can get it is a Newtonian fluid.

Whereas this shear stress if it is not directly related to the shear rate, then it may be non-
Newtonian fluid; we have also learned in different fluid mechanics books also you can
also get you just go through more fluid mechanics books they are the how Newtonian
and non-Newtonian fluids are classified and differentiated. So, based on this shear stress
and shear rate relationship you can classify, you can say if suppose shear stress is
denoted by tau, then tau should be directly related to the shear rate like du by dy; here du
is nothing but the change of velocity with respect to the length.

Suppose if there is a flow is flowing like this in a pipe, then what should be the y that is
here in r you can say here. In this y-direction or you can say in the r direction, here then
the velocity what should be the change of velocity in this direction, then the gradient of
this velocity this radial or you can say the axial direction this will be represented by du
by dy and this is called shear rate and the shear stress is directly related to this share rate
and this proportionality constant is called mu this mu is called the viscosity of the fluid.

Now, if it is directly related that means, it is Newtonian. This type of relationship is

actually obtained for the like water, so this called Newtonian fluid. If it is not directly
related to this shear rate then, that means here tau if it is related like this k in to du by dy
to the power n; that is a non-linear relationship, then this k and n we will give you the
what is the tendency of what should be the behavior of the fluid.

Now, if k and n are changing, k is not mu; that means, here then n is not 1 then you can
see it will be non-Newtonian; that means, n maybe is less than 1 or greater than 1. So, it
depends on the type of fluid, so those fluids will be called non-Newtonian fluid. So,
whenever liquid and solids will be flowing in a fly for in a column, of course, it will
behave like a non-Newtonian fluid also if the concentration of the solid inside the bed is
greater than 25 percent. So, in this case, the viscosity, of course, will change viscosity. It
will be highly viscous, and that viscosity will be related to the viscosity of that is a
Newtonian fluid.

So, that effective viscosity maybe as per Einstein’s theory that you can say that viscosity
of the non-Newtonian fluid for slurry system
that will be is equal to l; that means, not without slurry you can viscosity of pure liquid
then into 1 plus here 2.5 into 5 means volume percentage of the solid inside the bed and
2.5

here this will be your effective slurry viscosity inside the bed. So, in this way you can
calculate what should be the flow behavior inside the bed, now if you use run the
fluidization with the higher concentrated solid particles then have to calculate the slurry
viscosity not yet there Newtonian and not as a Newtonian it will be non-Newtonian to it
flow, and the viscosity will be something different.

Now, solid particles can be conveyed upward if the transport condition is well satisfied,
that means when fluid velocity extends the terminal or settling velocity of the solids here.

(Refer Slide Time: 21:54)

See if settling velocity as hindered settling, this course hindered settling velocity, in this
case, this hindered settling will be is equal to

n
us  uset  1   s 

here this epsilon s is nothing, but the solid concentration by volume. So, this hindered
settling you can calculate that means, here if a particle concentration; that means, is
slurry concentration you can say that the bed is the higher than this individual settling
velocity of the particles, may be reduced by the interaction of the other particles, in that
case, effective hindered settling velocity, will be is equal to this settling you set a
particular; that means, single particle settling velocity into 1 minus epsilon to the power
n.

Where this epsilon s you can obtain what should be the ratio of solid volume to the total
volume inside the bed.

Vs
s 
Vs  Vl

So, vs is nothing, but the solid volume inside the bed and this v s plus vl are nothing but
the total volume inside the bed of solid and liquid, respectively. So, from this ratio that is
called this is nothing, but a fraction this is by volume you can calculate what should be
the solid concentration by volume inside the bed and here settling velocity for individual
particle, or you can say that single-particle settling velocity that can be of course
obtained from the that is the balancing by the buoyant force and weight of the solid and it
is drag whenever it is flowing under this settling velocity. So, from this condition, you
can calculate this settling velocity for the single-particle by this here.

So, uset will be is equal to

1/2
 4 d p ,50 (  p  l ) 
uset  g
 3 CD l 

Now in this case, you see Cd is called drag coefficient. Already we have discussed in
earlier lectures that how to calculate the drag coefficient, and here this d p,0.50 is nothing,
but the particle diameter at 50 percent. What does it mean that is grain size or particle
size at 50 percent passing sieve, what are the size of the particles are those particles
which are coming down to the sieve will be considered as d 50 here, and that d 50 will be
considered for this settling velocity of this particle and here this n is coefficient, this
coefficient depends on the Reynolds number of the particle that is the given by Weber,
1974, and this will be is equal to

n  4.15 Re p0.063

So, if you know the Reynolds number by this equation here, Reynolds number is
 l uset d p ,50
Re p 
l

Once you know this Reynolds number, what should be the coefficient for n that can be
calculated from this equation, and once you know this n and u set and this epsilon easily
can be calculated what should be the hindered settling inside the bed. Now to get these
solid particles conveyed in upward, you have to maintain this liquid velocity inside the
bed, would be must greater than this hindered settling velocity. So, that is why ul should
be is greater than us there, and to calculate this u settling velocity, of course, another
important equation for drag coefficient will be calculated based on this Reynolds number
here this relationship is given.

So, from this equation, what should be the criteria for getting the solid particles in the
liquid medium to be conveyed upward? So, in this way, you can calculate this hindered
settling and then criteria for liquid velocity.

(Refer Slide Time: 26:22)

Now, Liang 1997 they have observed different types of flow regimes of solid-liquid
system, say according to Liang et al. they have actually observed that there should be
some conventional fluidization regime in the liquid-solid system; whereas, other than
conventional fluidization regime like circulating fluidization regime even transport or
hydraulic conveying regimes also are observed, in this case as they told that if liquid
velocity beyond the minimum fluidization velocity; that means, Umn that you can
calculate that has been shown earlier the regime becomes conventional fluidization
regimes.

It is also for the solid-liquid system, not commonly gas-solid system. So, the gas-solid
system also Umn, but liquid-solid system Umn if you are considering, but it will be
calculated in different just considering the liquid properties instead of gas properties.
Now circulating fluidization regime in this case, particles will recirculate to maintain
certain particular hold up at a certain liquid velocity, beyond the conventional
fluidization regime. So, in this case, circulating fluidization in the main important point
is that you have to that this is calculation will be they are at a certain particle hold up
inside the bed, that particle holdup means nothing but this volumetric concentration of
the particle inside the bed.

That can be calculated again by this, what should be the volume of solid particles out of
the total volume of solid and liquid inside the bed. So, this will be your particle hold up
here inside the bed. Now, this from this of course, you have to maintain constant this
particle concentration or particle hold up at a certain liquid velocity inside the bed; then
only other circulating fluidization regime can be obtained and transport or hydraulic
conveying regime, in this case, this flow regime we will start if the liquid velocity is
further increased to transition velocity to get this transporting or conveying mechanism
in the fluidized bed.

The radial flow, in this case, will be nonuniformity and which will become insignificant
in the transport or hydraulic conveying regime. So, this is very important that you have
to increase the velocity in such a way in a certain manner. So, that it will be beyond the
transition velocity from the circulating fluidization regime, so, for that may be
nonuniform, that is the instability manner instability behavior of the fluidization bed will
be observed inside the bed.

Now, Liang et al. also they have given one flow regime map for this liquid-solid
fluidization system, where they have represented this flow regime map as for that Kunni
and Levenspeil model based on this u star and dp star definition.
(Refer Slide Time: 30:14)

So, here dp star they have mentioned in an x axis again, it is defined as

1/3
*   g (  s  l ) 
d  dp  l
p 
 l2 

and in y axis they have defined this u star as

1/3
*  l2 
u  ul  
 l g (  s   l ) 

It is very interesting that here there are liquid velocities instead of cash velocity, and also
this density of the liquid is considered instead of gas velocity for gas-solid fluidization
velocity. So, in this case, since it is liquid solids, the density and viscosity of the liquid
will be considered. So, if you see this graph here for flow regime map for liquid-solid
fluidization, you are just observing here that this profile this locus; this dotted line this
locus is represented for minimum fluidization velocity in this liquid solid system, that
minimum fluidization velocity is being calculated based on that the balance of the
frictional pressure drop, which is being calculated by organ equation or by experimental
data with the apparent weight of the liquid-solid bed here.

So, this equation you can get, so this here may is a function of this dp star and this line
this dotted line is represented for the terminal velocity or settling velocity of the particle
inside the bed in a liquid-solid system. Now below this terminal velocity, you will get
this conventional fluidization resigned and with this dp star. So, this is your conventional
fluidization system for this liquid-solid fluidization; whereas, beyond this terminal
velocity there will be a certain velocity and beyond that velocity also you will get a
different type of flow regimes, sometimes up to this region will give you the circulating
fluidization regime; you can beyond this critical velocity that is represented by ua.
Beyond this critical velocity, you can get the transport regime by which the solid particle
should be transported upward by this high liquid velocity beyond this ua.

Now, this important point is that here you can get different flow regimes just simple
from this map, by calculating what should be that? Now, if you want to have a certain
flow regime condition now, what should be the used are based on the fixed dp value.
Suppose here 1 point is there, now the question is that you have this dp star is fixed like
what is that dp particle diameter and what is that liquid density and gravitational
acceleration of particle, particle concentration is also fixed and the density of the particle
is there certain, and density of the liquid is their viscosity of the liquid, you know then
you can calculate what should be the dp star.

Now, once you know this dp star as this like here, suppose this point what should be the
d p star here some value and corresponding to this value u star it is almost 1000 you
know sorry it is 10; then this ten from this ten that means, this u star will be is equal to
10 it u square is equal to 10, then from this what should be the u l just divide if this
portion of this then u star divided by this; that means, 10 by this value then you will get a
what should be the value of ul, then how to operate the fluidized bed with this liquid
velocity; so that you can get this transport regime. Now suppose if you want to get the
conventional fluidization regime here and with a certain liquid velocity, then what should
be the particle diameter to be taken here, so that you can get the conventional fluidization
regime.

Now, if you are liquid velocity is only 1; that means, here sorry u star is 1 then
corresponding what should be u star here and then u l if it is known to you then, what
should be the particle diameter you can calculate from this value. So, from this flow
regime, you can get the idea what should be the particle size to be taken to get this
particular fluidization regime or if you are particle size is fixed, then what should be the
fluid velocity inside the bed to be maintained to get this particular fluidized regime. So,
you can calculate from this fluidization map. Now, of course, you have to know what
should be the transition of this different fluidization regimes at the bed, now minimum to
conventional fluidization regime of course, you have to you have to maintain the liquid
velocity in such a way that you are just going to change the minimum fluidization to it is
convention of fluidization with this liquid solid system.

(Refer Slide Time: 36:37)

Now you have to calculate the minimum velocity; that means, if your dp star within the
range of 3.50 to 100, then what should be the minimum fluidization velocity.

1/3 1/3
 d 3 g 2 (  s  l ) 2    g (  s  l ) 
ul ,mf  0.0223  p   0.123  l  ; 3.50  d *p  100
 l  l   l2 

Now, to get the conventional fluidization velocity, you have to take them or you have to
consider the velocity of the liquid just greater than this minimum fluidization velocity.
So, from this correlation, you can calculate or you can get the idea of what should be the
velocity for liquid for this convention of fluidization, another transition for conventional
fluidization to circulating fluidization regime. Suppose if you want to work with a
circulating fluidization regime with this liquid, solid system, then we have what should
be the or you have an idea what should be the velocity of the liquid to be maintained
inside the bed so that I can get the circulating fluidization regime.
So, for that you have to calculate critical velocity or transition velocity by this correlation
and this correlation will give you what should be the critical velocity for liquid; now you
just to increase the velocity or you have to take the velocity of the liquid just beyond this
critical velocity so that you can get circulating fluidization regime and then how to get
the transport fluidization regime or transport or hydraulic conveying regimes what this
liquid solids slurry system in the fluidized bed. Here also, one correlation is developed,
and it is given by Liang et al., 1997 so that now you can get the hydraulic conveying
regime just by these criteria. In this case, also you have to calculate the velocity from this
correlation like by knowing the viscosity of the liquid particle diameter even solid
density and calculating by this equation, what should be the value if this value is if you
are liquid velocity is greater than this value, then you can say your fluidization will be
operating with the hydraulic conveying regime.

Now, coming to the point here, the flow regime in the 3 phase fluidized base system, you
will see the flow regime whatever regimes we got from liquid-solid or gas-solid system;
the flow regime for 3 phase system, of course, will be something different then this
earlier 1. Now, in this case, 3 phase system you are gas liquid solid 3 phase will be
interacting to each other and it will change the flow regime map and also flow regime
transition because of that simultaneous interaction of the gas liquid solids inside the bed.

(Refer Slide Time: 39:59)

Now you see diagram here shown this regime map, in this case, there are several flow
regimes are shown here dispersed bubble flow, discrete bubble flow, coalesced bubble
flow, slug plug flow regime, churn flow, bridging flow and annular flow. So, all these
flow regimes are generally observed in gas-liquid-solid fluidized bed just by changing
the liquid velocity and gas velocity inside the bed.

Now if you keep the gas velocity constant and if you increase the or decrease the liquid
velocity, you will get this dispersed bubbly flow regimes here; that means, for low gas
velocity and low liquid velocity, there is a possibility to get this dispersed bubble flow.
In this case, bubbles are dispersing in a uniform fashion, and all the bubbles are almost
uniform in size throughout the bed and gas should be dispersed, as a dispersed phase of
these uniform bubbles with this solid particles also and then if you increase the liquid
velocity for this low gas velocity you will get another dispersed bubble flow it is called
discrete bubble flow; in this case, the concentration of the bubbles will be lower than the
earlier 1, so dispersed bubble flow.

In this case, the coalescence of the bubbles may not be possible, but they are separately
moving up to the top of the column. So, all the bubbles indivisible bubbles will be
moved up there will be no interaction, but they were very that is less interaction will be
there, but there may be a chance of interaction of solid particles and cash particles and
liquid particles, but bubble interaction will be a little bit lower; whereas, in coalescence
bubbles in this case interaction between bubbles and the solid particles will be higher.
So, in this case, whenever the interaction of the gas bubbles will be there may be a
chance of coalescing the bubble.

Coalescence means what two bubbles or three bubbles will come together and make it is
bigger bubbles. So, they will come to each other, and they will conjugate, and to will
become bigger 1. So, in this case, the formation of the bubbles will be totally
nonuniform; that means, here some bubbles will be bigger some bubbles will be finer,
and there will be a continuous actually coalescence of the bubbles will be there, those
bubbles will be coalescence they will go very fast due to their buoyancy effect and those
are not coalescence they are moving slowly to the up and whereas, the solid particle
sometimes will take part to the interaction that maybe hinder the coalescence part.
Sometimes the in-between two bubbles, if there is a solid particle, then coalescence may
not be possible.
So, they may clock to the solid particles and go off, but whereas if there are no particles
in between 2 bubbles there immediately coalescing and go up. So, this type of
phenomenon is called coalescence bubble flow. Whereas, if you increase a little bit this
gas velocity by keeping this liquid velocity at fixed, then you can get this another type of
flow regime it is called slug or flag flow; in this case, you will see big bubbles will move
just was sent to the top along with some smaller bubbles and with the solid particles. So,
in this case, the bubbles will be nonuniform in shape not will be spherical; it may
sometimes form this bullet shape bubbles. So, this type of is called slug flow.

But this let us say if sometimes if this bullet is just the bullet cross-section, the lower part
of the bullet cross-section will be occupying total cross-section of the or almost equal the
cross-section of the bed, when you will see it will the flat slug and also flat slag with the
bullet nose, so this type of bubbles will be moving up as per this so here this is called this
slug flow. Whereas, if bubbles are bigger but it will be not exactly the cross-sectional
area or the diameter it is diameter is not the same as the diameter of the bed and but it is
a ledger, and it is the shape is like that one slug then it maybe it is called as the plug
flow. Whereas, if you are increased more velocity then what will happen you will get the
churning phenomena inside the bed that is called churn flow.

In this case, the bubbles should not give you the special form of round shape or what it
is, but maybe it will be elongated, and this gas will be spreading inside the bed like this
shown in a figure like this here. So, this is called churn flow. In this case, high turbulence
will be there more interactional be there, here the continuous coalescence and break up
will be there inside the bed. So, this is called churn turbulent flow and bridging flow in
this case, you will see bubbles are moving upward, but they are some form of the breeze
type breeze like bubbles and that is why it is called this breezing flow here; in between
two bubbles there will be a u density of the solid particles will be more higher inside the
bed.

Whereas this annular flow if you increase the gas velocity again, at a certain liquid
velocity you will get the annular flow in this case to the center of the bed, you will see
gas will be flowing continuously without breaking of that surface; that means, here
continuous gas flow will be there, but surface between solid and liquid or surface
between gas and liquid will be in such a way that there will be no uniform surface
formation, there will be some zig-zag surface formation inside the bed, so it is called
annular. That means a central core region of the bed. The gas should be moving up,
whereas, adjacent to the wall, the liquid-solid slurry will be flowing. So, this type of
phenomenon is called annular flow.

So, in a three-phase fluidized system, we can give this7 type off low regime inside the
bed, based on the gas velocity and liquid velocity changing. Now let us discussed the
behavior of this different flow regime; now, what is that now discrete bubble flows?
What is that this discrete bubble flow pattern predominates at low gas and liquid
velocities.

(Refer Slide Time: 47:33)

The flow pattern is characterized by the small bubbles with relatively uniform size
distributions, the bubble size, and it is distributed in the discrete bubble flow pattern that
is influenced by the gas distributor.

At low flow velocity when a small amount of gas is introduced into the column or bed
will see small bubbles appear and the number of bubbles and the gas hold up, of course,
are also a very small and bubbles do not have sufficient time to coalescence; since they
are a separation distance is large in this case, and of course, this distance will be larger
compared to the size of the bubble. The bubble frequency increased if your gas velocity
increased. Of course, this will be linearly proportional and the bubble of course; that is
why it is called bubble frequency increased linearly with the gas velocity here. Now in
the case of dispersed bubble flow, the dispersed bubble flow pattern is encountered at
higher liquid velocities; the bubble flow pattern is characterized by the small bubbles
with more uniform in size distribution then discrete bubble flow pattern.

(Refer Slide Time: 49:16)

As a result, you will see the liquid turbulence will be a little bit higher than the earlier
one, at high liquid velocity, dispersed bubble flow exists because the solid sold up of this
3 phase fluidized bed decreases towards 0 as liquid velocity is increased. As reported by
Zhang et al. 1997, they observed that in the case of air water and the particles with 1.5-
millimeter glass beads, the discrete bubble flow pattern might not exist there.

(Refer Slide Time: 50:03)

Coalescences bubble flow in this case, and in this case, this flow pattern is restricted to a
very narrow gas velocity range, so depending on the bed diameter.

Now, larger bubbles of wider size distribution are encountered with increasing gas
velocity, as there population increase the distance between individual bubble decreases
and small bubbles sometimes travel with lower buoyancy, and in this case, the overall
behavior of the multiphase mixer is represented as a coalescence bubble flow, just by
coalescing of the smaller bubbles to make it larger bubbles and bubble coalescence is
more intensive in an air-water system containing glass beads less than 2.5 millimeters in
diameter.

Then the corresponding gas-liquid system since the particles increase both the emulsion
phase viscosity and the density. Here important that see why coalesced bubble flow
observes that here viscosity physical properties of the fluid is very important. In this
case, if you increase both the viscosity and density of the slurry inside the bed, you can
have the different phenomena of the bubble coalescence, the tendency, or you can say
that intensity of the coalescence of the bubble inside the bed will be higher.

The pattern, of course, the gas passes through the liquid-solid mixture has bubbles of
irregular shape and larger in size, and this differs from the 1.5-millimeter glass bit
system where bubbles are spherical or spherical caps in the coalesced bubbles flow
pattern. So, these flow patterns depend on the shape of the bubbles; also, this coalescence
behavior depends on the shape of the bubbles. In this case, you will see the coalescence
phenomena will be easier for spherical compared to the nonspherical bubbles;
sometimes, caps type bubbles will give you the which is your coalescence compare to the
other type of other shapes of bubbles of course. So, this coalescence behavior depends on
particle diameter; also, for finer particles, the coalescence behavior will be higher than
the course are particles.

Slag flow as the gas velocity is increased further when bubbles become larger and more
Elongated, and some bubbles cross-sectional dimensions approach the diameter of the
column, so this phenomenon.
(Refer Slide Time: 53:05)

So, that is why slug will form the appearance of Taylor bubbles; that means, bullet shape
bubbles indicate that the flow pattern has changed to or slug flow; the slug flow pattern
spans a wide range of gas velocity here, at low liquid velocity the onset of slug flow is
almost independent of liquid velocity, while at the high liquid velocity the transition
from dispersed bubble flow to the slug flow is a function of the liquid velocity. When if
you increase the superficial liquid velocity inside the bed, the transition gas velocity
from this dispersed phase to the slug flow that will increase and churn flow.

(Refer Slide Time: 53:59)

In churn flow, the tailor bubbles are distorted and as the gas velocity again is increased
the distorted Taylor bubbles become elongated and if liquids flux between successive
Taylor bubbles become thinner than you will get the successive Taylor bubbles
coalescence and forms larger bubbles and you will get the churning condition inside the
bed.

(Refer Slide Time: 54:21)

Whereas bridging flow, in this case, you will see a liquid extends over the co regions of
the column at high gas velocity, and this flow pattern, of course, liniment the liquid film
occupying the annular region of the column in any cross-section.
(Refer Slide Time: 54:49)

In case of annular flow we will see the continuous gas region at the course, surrounded
by the continuous liquid region at the wall will happen and in some cases at high gas
velocity small bubbles may in train in the annular region of solid liquid emulsion; there
will be no bubble or solids or liquid bridge at the centre of the vertical pipe or column or
bed in 3 phase system.

(Refer Slide Time: 55:23)

Now, here another one example of flow regime map of fluidization for this 3 phase
instance are given, here see this flow regime is represented by the superficial gas velocity
and the liquid velocity; the superficial velocity gas velocity in the x-axis and superficial
liquid velocity in the y-axis. Now, if you change these gas and liquid velocity, you will
observe different flow regimes in this 3 phase fluidized bed, and this fluidization map
flow regime map is actually made by Zhang et al., 1997 with air, water and steel shot of
diameter 1.2 millimeters.

Now, this flow regime map it will represent of these it is actually telling you, what it will
tell you that what should be the dispersed bubble flow for a certain gas velocity and
when you can expect or when you cannot expect and what should be the transition from
1 flow regime to another flow regime.

Now, suppose if you want to run any fluidized bed, which is an air-water or air water and
this steel shot of diameter 1.2 millimeters in your lavatory; now you want to run this
fluidized bed with as slug flow then, what should be the gas velocity what should be the
liquid velocity that you can obtain from this flow regime map. How to? Let us consider
this gas velocity you have 1 column fluidized bed column in this case, one distributor
will be there through which gas will be distributed, and this is gas and again liquid also
to be supplied from the bottom here; now you want to actually get the phenomenon of
slug flow inside the bed.

Now, coming to this flow regime map here past, consider any location here for this slug
flow let it be here it gas velocity is equal to 1; if gas velocity is equal to 1 if your
maintaining the gas velocity is 1.0 meter per second. Now if you are fixing the point here
yet this gas velocity, then what should be the liquid velocity to be maintained or to be
supplied in the bed at which liquid velocity you can get this slugging condition let it be
here; if you are maintaining the liquid velocity inside the bed as 10 to the power minus 1;
that means, 0.1 meters per second then you can have this slugging behavior inside the
slugging bed.

Even fixing this gas velocity, if you decrease the gas velocity may be here; in this case,
then again, you can get the slugging behavior. So, by fixing the gas velocity if you
change the gas velocity below these point or above these point of 0.01, you can get a
different combination of the gas-liquid velocity to get this slugging condition because
you are here in this location of slug flow; you can get at this point also the slug flow.
Here what should be the gas velocity and what should be the liquid velocity here. So,
from this condition, you can get what should be the liquid and gas velocity so that you
can get the slugging condition.

Now you want to get the breezing flow. What to do now you have to maintain the gas
velocity here at this point here, suppose here this is 1 2 3 4 this is your suppose 4. So, 4
meters per second if you are just maintaining the gas velocity as 4 meters per second,
then what should be the liquid velocity to get this breezing flow here; suppose this is the
point. So, if you maintain this 0.1 liquid velocity and 4 meters per second gas velocity,
then you can get the breezing flow. So, this is a nice way to identify which flow regime
you can expect for your operation.

Now you can control you can change the different flow regimes or suppose if you want
to apply the fluidization operation for a particular application, where this flow regime is
more suitable.

Suppose, in this case, the churn turbulent flow for highly mixing conditions to get this
supposes reaction to any reaction. So, you have to maintain this gas velocity as 2 meters
per second by changing the liquid velocity, like this here from 0 to up to this here also.
So, different ways different combinations of the liquid and gas velocity; if you are
changing, you can get the different flow regime map.

So, this is the map by which you can just obtain what flow regime you can expect from
your bed. Now, what are the different transitions, of course, this transition to get the
dispersed bubble to calls this bubble? So, this correlation will be helpful to calculate the
transition, or you can say the boundary at which gas or liquid velocity to be maintained
and coalescence to slug flow, of course, the liquid velocity should be less than this
correlation whatever it is.
(Refer Slide Time: 61:40)

So, for low gas velocity for high gas velocity, this correlation will be helpful to get this
minimum velocity to have this slug flow come coalescence flow.

(Refer Slide Time: 61:52)

Similarly, for other transition of flow regimes have slug to churn turbulent flow and
churn to bridging flow, you can calculate the minimum gas velocity for which you can
get the transition of the different flow.
(Refer Slide Time: 62:07)

Regimes now transition of flow regime bridging to annular, also this is one diagram from
which you can calculate to get this bridging to annular flow what should be the liquid
velocity or gas velocity to be maintained. So, next class on what will be discussing other
topics like pressure drop and other things. So, this is the flow regime map and flow
regime different flow regimes you can expect for a liquid-solid and gas-liquid-solid
system. So, that is for all today. Thank you, the next lecture will be continuing with other
topics.

Thank you.
 Fluidization Engineering
Dr. Subrata K. Majumder
Department of Chemical Engineering
Indian Institute of Technology, Guwahati

Lecture – 08
Frictional pressure drop in fluidized bed; fluid-solid system

Welcome to the massive open online course on fluidization engineering, today’s lecture
will be on frictional pressure drop in fluidization bed for the fluid-solid system. So, what
is the role of pressure drop in fluidization?

(Refer Slide Time: 00:42)

Of course, this pressure drop is an important design parameter, whenever you are going
to design any fluidized bed you have to consider this factor of frictional pressure drop
inside the bed and of course what should be the energy dissipation inside the fluidized
bed that depends on the frictional pressure drop inside the bed. And also, it helps in
modeling the system and forms the basis of an assessment of the performance of the
fluidized bed. And also, this frictional pressure drop will identify the flow regimes, for
basically is that different pressure drops will give you different types of flow regime
maps. So, changing the frictional pressure drop, you will get different flow regimes, and
also, based on that, you will see that there will be a change of frictional hold up in the
fluidized bed; it is not frictionally fractional hold up in the phase.
Change the bubble size in a bubbling fluidized bed. Of course, the pressure drop has an
important role in the size of the bubbles that are formed in the bubbling fluidized bed,
and I, in this case, if the bed pressure drop is increased, the bubble size reduces. So, it
has an important role in the size of the bubble and because this bubble size, of course,
will affect the interfacial surface area from which or through which the mass transfer
occurs from this bubble phase to the fluidized phase. A decrease of the overall
volumetric mass transfer coefficient with pressure at a constant mass flow rate, of course,
we will change.

The decrease of the liquid axial dispersion coefficient happens with an increase of
pressure in a void range of spherical gas and liquid velocity. So, we have seen that there
is a role of pressure drop inside the fluidized bed, by which we can get different
phenomena of fluorescents by which we can get the energy dispersion inside the
fluidized bed. We can say what should be the performance of the fluidized bed based on
a mass transfer and heat transfer also and then what should be the bubble size
distribution inside the bed? Of course, it depends on the pressure drop.

(Refer Slide Time: 03:31)

And also, we will see that when a fluid flows through a bed of particles in a column of or
in a fluidized bed, it will exert drag force upon the particles resulting in a pressure drop
across the bed.
Pf
 g (1   )(  s   f )
L

At the fluid approach, velocity is increased, the pressure drop is magnified. The pressure
drop across the bed; it is seen that it will remain constant even with further increase in
the fluid velocity at a particular flow regime. Generally, for particulate flow regimes
where bubbling is not possible only just through the space of the particle bed, just
gaseous flowing upward, in that case, it is called the particulate flow regimes. So, in the
particular flow regimes, the pressure drop remains constant even if you increase the gas
velocity or fluid velocity inside the bed. Also, this will be equivalent to the effective bed
of the bed per unit area in the bed.

So, this is the friction or pressure drop per unit length that is nothing, but the efferent
weight or effective weight of the bed for the unit area here and you will see here.

(Refer Slide Time: 05:01)

In this figure, the effect of pressure is shown; how the minimum fluidization velocity
will change with pressure drop, it is seen that minimum fluidization velocity will
decrease with an increase in pressure drop. Here these are the trends shown, but also this
trend will change if the particle size is changed; if suppose particle size is increased you
will see minimum velocity also will increase here at any point uses observed, that here
this is the point that at the particle diameter of 10 mm here what should be the minimum
fluidization velocity for the particular pressure drop of 4000 here kilo Pascal. Even if you
change this particle size to this 5 millimeter, you will see for the same pressure drop you
will get a lower value of minimum fluidization.

Whereas, in this case, if you see that if you keep on decreasing the particle size for the
same pressure drop, and if you do the fluidization operation and you will see the
minimum fluidization will decrease. Whereas at the higher pressure, it is seen that at any
particular diameter, there will be no significant change of minimum fluidization velocity
and all, but for lower pressure, it is seen that there will be a significant change of
minimum fluidization velocity as per particle size. In this case, why for lower a pressure
you will see there will be that other factor, of course, the velocity of the fluid if you
change the velocity accordingly, you will see there will be a change of pressure drop.

So, if you change the pressure drop to this lower value accordingly, the minimum
fluidization also will affect. Now here, in this case, is void is at minimum bubbling.
Now, if you are considering the bubbling flow fluidization regime in this case, the
fractional void age or you can say fractional gas hold up inside the bed will decrease if
you increase the particle size in micrometer.

So, in this case, also the voidage will increase with respect to pressure. So, if you
increase the pressure, the void age will increase, whereas if you increase the particle
diameter, it will decrease. The pressure effects them on gas-solid fluidized bed behavior;
it is seen that we can observe from this fluidization behavior, what should be the
minimum fluidization velocity at different pressure, and also what should be void age.

So, all this pressure drop at minimum fluidized or other minimum fluidization condition
or other conditions it is required to know because for the design of a fluidized bed in
which way in which flow regime it will be acting and based on that you can say what
should be the pressure drop from this operation.
(Refer Slide Time: 08:47)

Another important point is that we should know then what should be the basic principle
for pressure drop analysis; there are several models to analyze the pressure drop inside
the bed.

Now, the basic principle is that model for the pressure drop in the riser of fluidized bed
may be developed on the basis of Bernoulli force balanced equation. So, as per
Bernoulli’s equation, you will see the total force will be the total kinetic total energy will
be constant. So, based on that you can obtain what should be the pressure drop by
changing the velocity and also if you change the elevation of this fluidized-bed or level
of the mixture of the fluid and solid particles inside the bed, what should be the pressure
drop also that depends on the level of the bed.

So, what will be the total energy that should remain constant as per Bernoulli’s theory?
So, based on these fluids balance equation from Bernoulli’s equation, the total pressure
drop babe obtained or friction or pressure drop may be obtained based on that question.
Now the total pressure drop may be defined as the changes in the kinetic energy of the
solids and gas, the potential energy and the solid-gas interphase and intra-phase friction.
So, different types of pressure contribution to the total pressure drop.

Now, one is that kinetic energy of the solids and gas, one contribution and the potential
energy another contribution and the friction between solid and gas or you can say that
solid and fluid interface and also the interaction between the particles, because of that
there will be a change of energy and for which the frictional pressure drop or total
pressure drop will change. We will discuss as later on; how different contribution will
give the total pressure the drop and what should be the percentage of that contribution
will be calculating.

So, in other words, the pressure drop inside the bed may be due to the friction head and
acceleration of the solids and gas. So, pressure drop inside the bed, of course, there will
be a tree component maybe it is those are due to the friction; that means, frictional
pressure drop, due to the head that is called static or hydrostatic pressure drop and due to
the acceleration of the solids and gas inside the bed that will accelerative pressure drop.

(Refer Slide Time: 11:44)

What should be the mathematical then expression to analyze this total pressure drop? So,
on the basis of momentum balance on gas-solid suspension for steady-state, the pressure
drop in the riser can be mathematically expressed as this is dp by dz, that will be equals
to that is total pressure drop, which is measured in the fluidized bed operation.

 dp   dp   dp   dp 
        
 dz total  dz head  dz  acceleration  dz  friction

So, it should be measured pressure drop this is per unit length that is dp by dz and what
is this is the head pressure drop, what should be the hydrostatic pressure drop inside the
bed that depends on the level of the fluid mixture inside the bed, that is called hydrostatic
pressure drop on that is it is called head pressure drop.

 dp 
 dz   [  p (1   )   f  ] g
  head

And another component is accelerative pressure drop; what does it mean? If suppose
there is a change of velocity inside the bed of the solid particles and gas, there will be a
change location wise or you can say region wise or you can say pointwise there will be
change or velocity.

There will be no uniform velocity inside the bed because of which there will be some
energy formation or energy that is distributed in such a way that is all some acceleration
of the solid particles, so due to which there may be an accelerative pressure drop. Now
another is very important that frictional pressure drop; this frictional pressure drop means
here this friction between the fluid and solid friction between fluid and particle, now
friction between fluid and wall, friction between solid and solid particles, frictional
between a solid and the column wall or bed wall. So, there may be different frictions, of
course, it will be there. So, this frictional pressure drop will be due to the friction
between phase and wall and between phases.

So, these three components are very important to know that how this pressure, how this
total pressure is contributed by these different components of this pressure drop; now let
us see what is that static pressure drop in the bed, this is static pressure drop is the
pressure drop due to the change in potential energy of the gas and solids inside the bed.
Now here see this is a pressure drop equation here, this is the pressure drop due to the
change in potential energy of this gas and solids. Now here this rho p into 1 minus
epsilon g into g this is due to the pressure drop due to head of this solid bed and then rho
f into epsilon into z this is the pressure drop due to the head of this fluid inside the bed.
So, this is the total head pressure. This is nothing, but that is that you can say this is the
rho m into gz.

Here rho m is nothing, but the density, effective density inside the bed, this effective
density depends on the void age inside the bed. Now this is the particle density this is rho
p into 1 minus epsilon; 1 minus epsilon is nothing but what is the solid volume inside the
bed, and this fraction then total this will be the fraction of the solid volume inside, so the
solid volume inside the bed and then rho ip into epsilon is nothing, but the actual volume
of the fluid inside the bed. So, this will be your mixture volume into a mixture volume,
and then to density, it is called that what is that mass effective mass or you can say
effective mass per unit volume or you can say the mixture density inside the bed.

So, mixture density into z this will be your head pressure. Now, this is p is nothing
pressure, z is length here, and rho p is the solid density, rho f is the fluid density, epsilon
is nothing, but operating average void inside the bed.

(Refer Slide Time: 16:24)

Now, what should be the acceleratory pressure drop; how to calculate this accelerative
pressure drop inside the bed? The accelerative pressure drop is the pressure drop due to
the acceleration of the particles from the inlet of the riser or you can say in the fluidized
bed to the point, where the particles attain the maximum velocity in the riser or in the
bed; this riser means nothing but the fluidized bed.

 dp  du p usf d (usf /  )
    p (1   )u p   f 
 dz acc. dz  dz

So, here accelerative pressure drop can be calculated as here this will be how this particle
velocity will change with respect to z, and what should be the here the particle velocity.
So, what would be the momentum change here due to the particle velocity change inside
the bed, and what should be the momentum change due to the velocity change inside, the
velocity of the fluid change inside the bed.

So, this part is nothing, but the momentum change due to the velocity change of the
fluid, and this is the moment change particle due to the particle velocity inside the bed,
from the inlet of the fluidized bed to the point where the particle certain it is maximum
velocity. Now it is important to note down here that when the solid-gas two-phase flow
in the fluidized bed reaches a fully developed state, that means, everywhere the velocity
change will be constant.

So, in that case, you can say the accelerative pressure drop should be zero; that means dp
by dz of acceleration; it will be 0. So, for the fully developed state of the fluidized bed,
the accelerative pressure drop will not be considered, it will be neglected and what
should be the frictional pressure drop.

(Refer Slide Time: 18:30)

Now, the frictional pressure drop is due to the friction between phases and friction
between wall of the fluidized bed; now there are three types of frictional force developed
generally the frictional force between gas and solid, the frictional force between the solid
and the walls of the bed, the frictional force between gas and the wall of the bed.

 dp   dp   dp 
     
 dz  friction  dz  fluid  wall  dz  particle  wall
Now, this tend this frictional pressure drop will be equal to the frictional pressure drop
between fluid and wall, and the frictional pressure drop between particle and wall, the
friction between gas and solid may be negligible due to the lesser concentration of the
solids in the inside the bed. So, here you can neglect the frictional pressure drop between
solid and gas if it is an only gas-solid operation.

But in case of liquid-solid operation, you cannot neglect this 1, in that case, the
concentration of the solid or concentration with the there very important factor. So, for
gas for gaseous medium gas solids fluidization, you can neglect whereas liquid-solid
here you can, of course, consider that a frictional pressure drop between the friction
between a particle and fluid. So, let us see that frictional pressure drop between particles
and wall how it can be calculated.

(Refer Slide Time: 20:15)

Now, this frictional force per unit volume between the solid phase and wall can be
calculated by fanning equation, which can be written for gas solid fluidized bed as like
this here, this is the frictional pressure drop between particle and wall that will be is
nothing but here 2 into fp, is nothing but friction factor fanning friction factor per
particle, this is colliding with the wall and then 1 minus epsilon into rho p into u p square
by dbed.

 dp  (1   )  p u 2p
   2 fp
 dz  particle  wall dbed
So, this frictional pressure drop between particle and wall as per Fanning equation
depends on the particle velocity inside the bed, what should be the bed diameter and
what should be the density of the particle and of course there will be the void age inside
the bed; now we can say from this equation what should be the frictional pressure drop
because of the collide collision or because, of the friction between particle and wall. So,
there is no universal formula for the calculation of actually friction factor here f p;
however, several correlations that you can use or estimate or to calculate this friction
factor that has been proposed by different investigators. So, we will see how to calculate
this friction factor for particle here.

(Refer Slide Time: 22:02)

Let us see how to calculate the friction factor for particle flow. So, this friction you can
calculate from this correlation that is developed by Young 1998, based on their
experimental work

1.021
(1   )  (1   ) Ret
2
 usg / 
f p  1.025 10   for  1.5
 3  Re p  ut

0.979
(1   )  (1   ) Ret
3
 usg / 
f p  3.15 10   for  1.5
 3  Re p  ut
  f ur d p
Re p 
f

usf
ur   up


 f ut d p
Ret 
f

 gd p2 
ut   (  p   f )
 18 

24
CD  1  (8.1716e 4.0655S ) Re0.0964  0.5565S

Re P 
P


 
73.69 e 5.0748S Re P
, for s  1
6.2122 S
Re P  5.378e

here it is seen that this friction factor depends on the Reynolds number of particle and the
Reynolds of the terminal velocity of the particle, of course this factor is you know where
that is the epsilon that is what is the void age that is an important factor that governs this
friction factor inside the bed for the particle; now it is seen that this correlation will be
valid only for if the ratio of actual gas velocity to the terminal velocity of the particle if it
is less than 1.5.

Whereas for this ratio, if it is greater than 1.5, you have to calculate this friction factor
from this correlation here, again this correlation is a function of this Reynolds number of
particle and Reynolds number of terminal Reynolds number of the particle based on
terminal velocity. Now how this Reynolds number of the particle is defined is nothing
this rho f u r into dp by mu f, what is this rho f? rho f is nothing, but the fluid density, u r
is nothing but the relative velocity of the solid particles relative to the fluid velocity;
now, here u r is nothing but here what is this u sf by epsilon what is s for superficial
velocity f for fluid either may be liquid or gas, now for gas-solid fluidized bed you have
to consider here u s g instead of f it will be g.

So, ufg is nothing but the superficial gas velocity. So, if you divide the superficial
velocity by the void age you will get the average gas velocity inside the bed, or you can
say actual gas velocity inside the bed; now what should be the actual gas velocity and
what is the particle velocity if you just subtract it you will get the relative velocity of the
particles, through which at which this particle should be moving upward inside the
fluidized bed or particle fluidized with this relative velocity.

Now, this is the what should be the terminal Reynolds number, if this is Ret is defined as
rhof ut dp by muf, in this case this, of course, the velocity will be considered as terminal
velocity, others are same here rho f is the fluid velocity muf is the fluid viscosity here.
So, f for fluid may be liquid or gas, if it is gas-solid operation, then f will be represented
by g.

If it is liquid and solid fluidized bed then f will be represented by l and what should be
the terminal velocity then to calculate this Reynolds number at terminal velocity; this
terminal velocity again you can calculate based on this Stokes equation this

 gd p2 
ut   ( p   f )
18 

and also you can calculate this terminal velocity whenever fluidizing condition if it is if
Reynolds number is greater than 0.1, then you have to calculate this terminal velocity
from this equation. But they are, of course, it will be related to this drag coefficient. Now
this drag coefficient is related to this the Reynolds number that we have already
discussed earlier when a lecture that how to calculate the drag coefficient and how what
is that drag coefficient.

So, you can calculate friction factor from this correlation by using this terminal velocity
and the relative velocity and what should be the Reynolds number of particles, what
should be the Reynolds number at the terminal velocity of the fluidized bed.
(Refer Slide Time: 26:33)

Now, what should be the frictional pressure drop between fluid and wall, friction
between fluid and wall; now suppose gas-solid operation what should be the frictional
pressure drop between gas and fluidized bed wall.

 dp   f (usf /  ) 2
   2 f f
 dz  fluid  wall dbed

24
ff  ; Re f  2300
Re f

0.079
ff  ; Re f  2300
Re0.25
f

 f usf dbed
Re f 
f

Now, frictional force per unit volume between the fluid phase and the bed wall is given
by fanning’s formula, as the following equation for fluidized bed as here this fluid wall
that is the again as per fanning equation you can calculate what should be the fluid wall
frictional pressure drop. So, in this case, this friction factor will be represented by ff, here
t is only the between fluid and wall gas and wall or liquid and wall if you are considering
gas and gas solid-gas and wall.
So, here ff into epsilon rho f into us f by epsilon this is your actual a gas velocity or liquid
velocity inside the bed. So, here this is the fanning equation from which you can
calculate; now for this again, you have to calculate the friction factor of fluid flow. So,
this friction factor of fluid flow you can calculate from the equation if it is the laminar
flow you can directly calculate a ff will be equals to 24 by Ref, this Ref if it is less than
2300 then you can directly use this equation, this will be equal rho f usf dv d bed by mu f
, mu is the viscosity.

So, ff at Ref is greater than 2300 you can calculate from this Blasius equation, here this
0.079 why ref to the power 0.25. So, from this equation from this two correlations
Blasius you can calculate what should be the frictional friction factor, fanning friction
factor and then after substituting this friction factor here with other parameters like rho f
and velocity of the fluid and the diameter of the bed along with the void age, if you know
then you can calculate what should be the fluid wall frictional pressure drop.

(Refer Slide Time: 29:03)

Then total frictional pressure drop will be a summation of these two types of frictional
pressure drop.

 dp   dp   dp 
 dz     
  friction , tot  dz  particle  wall  dz  fluid  wall
(1   )  p u 2p  f (usf /  ) 2
 2 fp 2ff
dbed dbed
One is particle wall frictional pressure drop another is fluid wall frictional pressure drop;
now this particle wall frictional if you substitute the value of particle wall frictional
pressure drop and the fluid wall frictional pressure drop then you will get these equations
here. So, total frictional pressure drop will be is equal to

 dp   dp   dp   dp 
       
 dz total  dz  head  dz acceleration  dz  friction
du p usf d (usf /  )
 [  p (1   )   f  ] g   p (1   )u p  f
dz  dz
2
(1   )  p u p  f (usf /  ) 2
 2 fp 2ff
dbed dbed

So, the total pressure drop will be is equal to head pressure drop, acceleration pressure
drop, and friction of pressure drop. So, if you substitute the value or equation for those
three parts of this head acceleration and friction, then finally, you can obtain these
equations.

So, from the situation, you can say what should be the total pressure drop, but if you
know the total pressure drop from the experimental value and this equation is nothing but
the theoretical value, if you mess it of course, it will be some near about or with the least
error what should be the value that you can obtain. So, without doing experimental also
you can say what should be the total pressure drop inside the bed; once you know the
particle volume once you know the particle density, once you know the fluid density and
other parameters, of course, you can calculate from this equation.
(Refer Slide Time: 30:50)

Now, then total pressure drops what should be the total pressure drop across the full
length of the riser,

 du p usf d (usf /  ) 
L  [  p (1   )   f  ] g   p (1   )u p  f 
 dz  dz dz
Ptotal  2 2
0
 (1   )  u
p p  f (u sf /  ) 
 2 fp 2ff 
 dbed dbed 
2
 u 
 [  p (1   )   f  ] gL    p (1   )u 2p   f sf 
  

 (1   )  p u 2p  f (usf /  ) 2 
 2 fp 2ff  L
 dbed d bed
 

It will be having, what you have calculated that will be based on the pressure drop per
unit length, that the for the whole length of the fluidized bed then you have to integrate
that pressure drop from 0 to length of the bed.

This length of the bed actually not the actual length of the bed, it will be the actual length
up to which level the gas-liquid solid or gas-solid or liquid-solid level will be there; that
means, level of the fluid mixture inside the bed that will be your length here. So, if you
integrate this, then you will get this equation, so finally, you will get this equation for
calculating the total pressure drop for the whole length of the fluidized bed. So, it
depends on the particle density, it depends on a fluid density, it depends on void days, it
depends on particle velocity, it depends on fluid velocity, it depends on diameter of the
bed, it depends on density of the fluid; so there are several variables that will affect the
pressure drop inside the bed.

(Refer Slide Time: 32:17)

Now, see in this diagram or figure the different pressure drop contribution to the total
pressure drop, how it is actually changed this total pressure drop, how it will be changing
along with this other contribution; now see if you represent this total pressure drop by
this yellow line and then static pressure of by this brown line and then accelerative
pressure drop by this green line and then blue line and then frictional pressure drop by
the white line, then you can see here the percentage contribution of these static
accelerative and frictional pressure drop.

How it is coming now see the frictional pressure drop will increase with respect to the
superficial gas velocity and accelerative pressure is also increase with the superficial gas
velocity; whereas, the static pressure drop we will decrease with respect to the superficial
gas velocity, why if superficial gas velocity increase then, of course, void age, of course,
will increase and that void gas void age will increase; whereas, as the solid load solid
head will be increase.

So, overall the effect of the head will decrease with respect to gas velocity, whereas the
frictional and accelerative pressure drop will increase with respect to superficial gas
velocity, because of the particle interaction and fluid-particle interaction, n if you dual pa
wall particle interaction for which the frictional pressure drop will increase in that case
and total pressure drop will be immense. Now if you consider any just line here at a
certain superficial gas velocity, what should be the percentage here of this and what will
be the percentage of that is an accelerative pressure drop and what will be the percentage
of this total pressure drop you will get here, this the total pressure is 100x here. So, that it
is supposed this 1 is like here this case and what should be the value here it is maybe 18,
and this is maybe 10 12 13, and remaining 1, of course, will be is equal to head pressure
drop.

So, in this way, you can calculate what will be the percentage contribution. So, here if it
is supposed 20 percent and then here it will be what will be the percentage here at this
pressure drop, then here it will be of course less than 20 percent; whereas the head
pressure would be higher but relative to the previous 1. So, in this way, you can calculate
what will be the pressure for different parts, then you can calculate from this diagram.

(Refer Slide Time: 35:25)

Now, frictional pressure drop based on mixture model another model you can calculate
the frictional pressure drop, in this case, particles have significant impacts on the fluid
flow field of fluid-solid two-phase flow in fluidized bed; now it is difficult really to
measure the fluid wall friction and the particle wall friction separately. So, in this case,
most investigators measured the combined friction between fluid-solid suspension and
column wall, so we can calculate this fluid solids suspension and the column wall.

But it is very difficult to calculate the fluid wall friction and the particle wall friction
separately; as a result, a different approach can be made from the common approach to
separately calculate this fluid wall and the particle wall friction pressure loss, the fanning
friction equation. Fanning friction equation for single fluid flow, in this case, can be
used, to define this mixture friction factor calculation between the fluid-solid suspension
and the bed wall, let us see how to calculate this one.

(Refer Slide Time: 36:42)

Again the friction factor based on the fanning equation, here in this case mixture modern
means you have to consider this gas liquid-solid as a homogeneous mixture or if it is on
the gas-solid particles and gas-solid homogeneous mixture will be there, and if it is
liquid-solid then you can say the liquid-solid homogenous mixture; now that will be
represented by m.

So, as per Fanning equation, this dp by dz friction will be equal to

 dp  2 f m  mum2
  
 dz  friction dbed
So, in this case, fm is nothing but friction factor by considering the mixture of gas and
solid or liquid and solid, in this case, rho m is the density of the mixture of gas and solid
and here um is nothing but the mixture velocity of gas and fluid gas and solid or fluid and
solid. So, rho m is defined as

 m   f    p (1   )

what is this rho f into epsilon rho f is nothing but the density of the fluid either gas or
liquid into epsilon means what is the volume fraction of the fluid inside the bed and 1
minus epsilon is nothing but volume fraction of solid inside the bed into particle density.
So, here this is per unit volume what should be the total mass this it is here, then what
will be the mixture density you can calculate from this equation.

How to calculate these mixture velocities? Mixture velocity is nothing, but what should
be the mass flux of the solid mixture flux, mixture flux of the gas-solid inside the bed
divided by the density of the mixture;

Gm   f usf  Gs
um  
 m  f    p (1   )

what will be the Gm? Gm is nothing, but here what should be the mass flux of the fluid
and what should be the mass flux of the solid. So, solid mass flux is nothing, but what
should be the mass rate of the solid per unit cross-sectional area, so that will be your
mass flux here. So, mass flux is nothing but mass divided by time divided by cross-
sectional area.

So, this is the mass flux, so here the fluid mass flux is nothing but epsilon into rho f into
usf whereas, solid mass flux is denoted by Gs, Gs is nothing but a mass of solid into us
mass of solid into the velocity of the solid. So, in this and divided by rho m already
defined here, this rho m will be equal to rho f epsilon plus rho p into 1 minus epsilon.

So, this way you can calculate this mixture a velocity inside the bed, once you know the
mixture velocity inside the bed, you just substitute here in this friction fanning equation
for friction a null pressure drop, then you can get the final value of this frictional
pressure drop by this equation. Now for this f m is required f m is nothing but the friction
factor for mixture flow can be calculated this here
 4
fm  ; Re m  2300
Re m
0.079
 ; Re m  2300
Re0.25
m

, whereas Rem will be is equal to less than 2300 this will be, and Re m will be is equal to
greater than 2300 that will be equals to fm will be equals to 0.079 by Rem to the power of
0.25.

Whereas. Rem will be defined as the mixture density rho m mixture velocity and mixture
viscosity.

 mum dbed
Re m 
f
(  f usf  Gs )dbed

f

So, mixture viscosity is nothing but the fluid viscosity here because gas and gas viscosity
is very negligible compared to the fluid here you can say for liquid-solid operation only,
this fluid viscosity or liquid viscosity will be considered. So, in this case, f m to be
calculated here, and then from the Fanning equation, you can obtain what should be the
frictional pressure drop.

(Refer Slide Time: 41:28)

Now, for first fluidization or that means, the fast it will be is corrected in first
fluidization here Rem is greater than 2300; that means, under turbulent condition, then fm
can be calculated from this equation from this correlation.

0.079  0.25
f
fm  0.25
; Re m  2300
0.25
dbed    f usf  Gs 
So, once you know this fm from this correlation by substituting very various variables,
you can directly calculate what should be the frictional pressure drop from this equation.

1.75
 dp 

0.158 0.25
f    f usf  Gs 
  1.25
 dz  friction dbed [  f    p (1   )]

(Refer Slide Time: 42:01)

Therefore based on the mixture model, the final form of total pressure drop can be
calculated as like here, by integrating this equation final form of abbreviation you can get
from this equation here.
  du p usf d (usf /  ) 
L  [  p (1   )   f  ] g   p (1   )u p  f 
dz  dz
Ptotal   2
dz
0
 2 f 
m m mu 
 
 dbed 
 u2 
 [  p (1   )   f  ] gL    p (1   )u 2p   f sf 
  

 0.158 0.25    u  G  1.75 
f f sf s
 1.25
L
 d bed [  f    p (1   )] 
 

So, this equation will give you what should be the total pressure drop based on the
mixture model.

(Refer Slide Time: 42:24)

Now, actual solid concentration how to calculate because, it depends this is the very
important point because, you have to calculate anyway what should be the change of
solid velocity actual change of solid velocity inside the bed and what should be the void
age inside the bed, once you know solid volume fraction inside the bed. So, this can be
calculated from the total pressure drop also. So, epsilon s to be calculated here

1  dp   dp  
s      f g    
( p   f ) g  dz total  dz  friction 
and if you know the total pressure drop and if you know the frictional pressure drop and
head, of course, you can calculate the what should be the solid fraction inside the bed at
dynamic condition.

(Refer Slide Time: 43:15)

Now, see in this figure it is seen that how the frictional pressure drop will change based
on the flow regime of the fluidized bed, now in the fixed bed, of course, this frictional
pressure drop will be as per organ equation it will be directly related to the fluid velocity
up to up to a certain point; that means, here where the just fluidization starts up to that
this frictional pressure drop is just related here, this is where fixed bed condition.
Whereas, beyond this fixed bed condition.

If you increase the gas velocity or fluid velocity how this frictional pressure drop if
changing for particular fluidized bed, it is almost unchanged or very that is a small
change of this pressure drop here almost remain constant; whereas, if you increase little
bit that gas velocity of fluid velocity that means, there bubbling flow rate fluidization
regime these frictional pressure drop will be reduced.

Again from this bubbling fluidized bed the fluid frictional pressure drop if will gradually
decrease with respect to the fluid velocity; now it will be up to joint turbulent condition,
but beyond this joint turbulent condition first fluidization it is seen that frictional
pressure drop will decrease and again it will remain constant and after a certain height,
you will see that for a pneumatic condition it will be higher pressure drop, in that case,
the dilute solid particle there a very small amount of particle interaction of and particle
wall interaction will be there, and maximum part will come now the due to the high
velocity of the fluid and the friction between the fluid-wall.

So, this from this figure how actually frictional pressure drop will change with respect to
fluidized map it can be easily understood, and it is seen that frictional pressure drop, of
course, will be going to reduce or going to decrease with the gas velocity or liquid
velocity whereas, for a pneumatic condition it will be reverse order it will increase with
respect to gas velocity.

(Refer Slide Time: 45:35)

Now, effects of operating conditions the frictional pressure drop, we will see if you
increase the solid velocity it is seen that frictional pressure loss increases and that the
increase in rates rises with solid circulation rates; there will be a solid circulation and
because of which if you increase the solid circulation rate inside the bed, you will see the
frictional pressure drop will be there because solid interaction and solid wall interaction
will be higher.

And if you increase the gas velocity, of course, the frictional pressure losses or pressure
drop increases with superficial gas velocity or solid as circulation rate, solids
concentrations, of course, the frictional pressure drop increases with solid concentration.
If you increase the solid concentration for final particles, then you will see there will be
viscosity increases, and in that case, it is in that frictional pressure drops increases
because of an increase of viscosity inside the bed.

Now, particle diameters also have an effect in the frictional pressure drop, and it has the
different impact on the frictional pressure losses under different also superficial gas
velocity, the reason that it is seen that for lower superficial gas velocities if it is less than
7 meter per second, the particle diameter almost have no influence on the frictional
pressure losses.

Whereas under higher superficial gas velocities, if it supposes greater than 10 meters per
second, the frictional pressure loss with Geldart’s A particles, it is seen that it will be
greater than those with Geldart’s B particles and the difference increases with superficial
gas velocity.

(Refer Slide Time: 47:31)

Now, let us seen an example how to calculate this different part of frictional pressure
drop, how to calculate the terminal velocity, how to calculate the drag coefficient of the
fluidized bed, how to calculate the friction factor of gas-solid and mixture and how to
calculate he frictional pressure drop between particle and wall how to calculate the
frictional pressure drop between gas and wall, how to calculate the what should be the
percentage concentration of the different parts of the total pressure drop inside the bed.
Let us see, let us consider bubbling fluidized bed whose height is 1.6-meter height and
diameter is 0.08 meter and it is operated by air; that means, air solid at and air superficial
velocity is 0.11 meter per second and the particle is sand particle whose effective
diameter is 110 micro meter, given that the air density is 1.2 kg per meter cube and the
density of the sand particle is 2600 kg per meter cube, void age of the bed is given 0.55;
now consider that the particle is circulating with it is terminal velocity in the bed, the
sphericity of the particle is 0.85.

(Refer Slide Time: 49:05)

Now here what should be the different part of this, first of all, calculate the terminal
velocity based on the stokes law you know dp particle diameter, you know the viscosity
of the fluid that is the air here you know the density of the particle you know the density
of the gas.

So, how to calculate this terminal is let us substitute this value you will get this terminal
velocity is 0.0952 meter per second, whereas what should be the relative velocity of the
particle inside the bed; now this is your superficial gas velocity it is given here 0.11
meter per second the porosity or void age is given 0.55, this is your actual gas velocity
inside the bed and this is the particle velocity; here this particle velocity will be
considered as the terminal velocity inside the bed, that means beyond this terminal
velocity, it will be fluidized.
So, what would be the minimum this is the particle velocity is the terminal velocity
beyond which it will be moving up. So, this will be u rel, and because of that, it will be
circulating, and so this will be your relative velocity of the gas inside the bed. Now what
should be the Reynolds number based on terminal velocity, you just substitute the rho g
that is the density of the gassed in the terminal velocity of the particle diameter and
viscosity of the gas then you will get this is your Reynolds number this is less than 0.1.

So, it is the stokes region, of course, then terminal velocity should be this 1, and if Re p is
equal to 0.0768, again this is based on the gas velocity relative velocity of the gas here
and then what should be the Reynolds number of the particle.

(Refer Slide Time: 51:06)

Once you know this terminal velocity relative velocity terminal Reynolds number and
the particle Reynolds number, then you substitute here what should be the C D value; CD
is nothing but 24 by Rep because the Stoke’s region, of course, this will be equal to 24 by
a Rep this is nothing but 312.313. So, this is your direct coefficient and what would be
the particle friction factor, this particle friction factor ap will be is equal to what this is
3.15 into 10 to the power, there are two equations for calculating this particle. The
friction factor, of course, this is the first one. If it is supposed this is the ratio of this gas
velocity and terminal velocity is greater than 1.5 then only you have to calculate this one.
We have calculated that this Usg by epsilon by ut it is coming is greater than 1.5, so you
have to use this correlation. So, once you put the value of these Re t Rep and epsilon here,
then, of course, you will get this value of particle friction inside the bed as 0.0205.

(Refer Slide Time: 52:13)

Then what should be the Reynolds number for gas shear it will be rho g, here it will be
based on the gas velocity superficial as gas velocity inside the bed and the bed diameter
instead of particle diameter and the viscosity of the gas it is coming 58.67, of course, it is
laminar flow then you have to calculate fg as this 4 by Reg is equal to 0.0682, again here
this Re m as per mixture model; that Rem to be calculated as rho m into u m into d of bed
that is the diameter of the bed divided by the viscosity of gas.

So, once you substitute this value, you will get this Reynolds number will be equals to
1.10 into 10 to the power 5 and based on these at this Reynolds number what should be
the friction factor of the mixture, this will come here as per this since it is turbulent
region then we will get this value of 0.0043.
(Refer Slide Time: 53:18)

Now, once you know this value of this different part of this frictional pressure drop, like
dp by dz particle wall will be is equal to 2 f p is equal to. So, you know this f p you have
calculated you know rho p you know up you know bed diameter. So, finally, it will come
like this 5.422 and then here it will be of course Newton per meter cube and then gas
wall frictional pressure drop will becoming like this, after substitution of this value and
friction total, the frictional pressure drop will be is equal to than frictional pressure drop
of particle wall and frictional pressure drop of the gas wall, then you will get this total
value of this here as 5.4674.

And then the frictional pressure drop based on mixture model, it will come as 2 f m into
rho m into um square by the diameter of the bed, here rho m you know that rho m you
know um you know and fm you know after substitution you can calculate it will come as
5.794. Whereas, based on individual interaction to the wall of the bed, then you will see
the total frictional pressure drop is coming 5.4674, whereas it is for mixture model it is
coming 5.794 almost same here there will be little error for this.
(Refer Slide Time: 55:01)

So, we can consider the mixture model also. And here head; the head will be is nothing
but here mixture density into gravitational osculation it is coming like this, it will be
Pascal per meter and here dp by dz head it will be is equal to this 10.634 Pascal per
meter, and then total frictional pressure drop here it will be is equal to this 11496.2873
just summation of this 3 component of this total pressure drop.

And as per the mixture model, this total frictional pressure drop is coming here
11496.4992; there is an error is only 0.0018 percent. Now what should be the percentage
contribution this is the total pressure drop, and these are the contribution of different
parts of this frictional pressure drop, it is seen that hydrostatic part is the maximum one
that is 99.89 percent. So, whereas the accelerative portion is very low 0.058 and
frictional pressure drop is also very small. So, compared to the hydrostatic pressure, it is
coming very low; these are this accelerative frictional pressure drop. Now, this frictional
pressure drop once you know, put the total measured pressure drop, also you can
calculate the frictional pressure drop by subtracting this accelerative and head pressure
drop.
(Refer Slide Time: 56:38)

Now, the same example with fluid as water, let us consider this water instead of gas, now
the same here the height of the bed is this and what is that diameter of the bed is 0.08 and
water density only will change, and others remain same. So, what should be the different
components of this calculation?

(Refer Slide Time: 56:59)

Terminal velocity will come here, and relative velocity Reynolds number based on
terminal velocity, Reynolds number based on particle diameter it is coming like this.
(Refer Slide Time: 57:10)

And then what should be the drag coefficient is coming 13.97 within this range of
Reynolds number of particle here and fp again here this ratio is coming greater than 1.5.
So, it will be 2.80 as per that variable change of density, and the viscosity and other
things as per liquid just water.

(Refer Slide Time: 57:30)

So, again Reynolds number for liquid Reynolds number for this friction factor for liquid
as per fanning equation and then what is the Reynolds number for mixture, it will be
coming as 571.52 again the friction factor based on mixture model it is coming this one.
(Refer Slide Time: 57:53)

And then individual component this frictional pressure drop then it will come 0.1152 and
also for the liquid wallet it is coming 2.225 friction, then total frictional pressure drop it
will coming 2.340, just summation of these two; and then as per mixture model it is
coming 2.468 it is also almost same.

(Refer Slide Time: 58:20)

So, after substituting this value, you will see that the head component is coming like this,
and the accelerative component is coming, and this total, then pressure drop will be
coming like this.
Whereas this here frictional pressure drop is coming to this total pressure drop were
based on the mixture model, it is coming this one. So, the error is again very negligible,
but here also, you will see that hydrostatic pressure drop will be high, irrelative to this
accelerative and frictional pressure drop.

Now you will see if you increase the density of the fluid, of course, this frictional part
will be higher. The frictional part will be higher and accelerative part also, in this case,
the relative to the gas here. But a frictional pressure drop, let us see what should be the
frictional pressure drop as earlier a portion contribution here.

So, frictional this contribution is 0.048, whereas in this paper, liquid 0.01. So, it is less in
case of liquid, whereas in the case of gas, there will be higher. So, by this example, you
can say that the how the frictional pressure drop will change, what will be the terminal
velocity of the drag coefficient, how the total pressure drop is contributed to the different
components then a portion can calculate, what will be the total pressure drop inside the
bed by this concept.

So, next class or next lecture we will discuss again with that gas-liquid-solid system,
how to calculate the frictional pressure drop, total pressure drop all those things and you
can go through this portion through your textbook, but some portions you will not get
different, but you can get it from this only concept it will be useful for you so.

Thank you for today’s part.

 Fluidization Engineering
Dr. Subrata K. Majumder
Department of Chemical Engineering
Indian Institute of Technology, Guwahati

Lecture – 09
Frictional pressure drop in fluidized Bed: Gas-liquid-solid system

Welcome to this massive open online course on fluoridation engineering. Today’s lecture
will be on frictional pressure drop in a fluidized bed in a gas-liquid-solid system. So, in
earlier lectures, we have discussed the frictional pressure drop in the fluidized bed
system with a gas-solid and liquid-solid system. So, in this lecture three-phase system
will be discussed whatever the frictional pressure drop, based on the gas-liquid-solid
movement; that will be discussed in this lecture.

(Refer Slide Time: 01:05)

So, we know that that frictional pressure drop in any system has three parts. One is that it
is frictional due to the liquid-solid interaction. Frictional due to the solid wall interaction,
and frictional due to the liquid wall interaction. And with this 3-fictional pressure drop of
course, that the main contribution of this frictional pressure drop due to this interaction.
Other than this interaction three-phase fluidized system also have the other two parts that
are head pressure drop and acceleration pressure drop.

So, the total pressure drop in this gas-liquid solid system can be estimated by this total
pressure drop that will be equal to pressure drop due to the fluid head, and pressure drop
due to the frictional pressure drop that is friction between gas-solid friction between
liquid-solid friction between wall a solid and friction between gas-solid and liquid wall.
So, there are other equation like acceleration pressure drop whenever the phase is
moving inside the bed with a certain velocity if there is an acceleration of this movement
of the liquid; that means, they are velocity gradient will be there or radial direction there
will be a change of velocity or fluctuation of this velocity because of which some
development of the pressure drop inside the bed. And so, the total pressure drop will be
equal to head pressure drop frictional pressure drop and acceleration pressure drop.

 dp   dp   dp   dp 
        
 dz total  dz head  dz  friction  dz  acceleration

Now, what is that? Head pressure drop, or you can say this can be written as a
hydrostatic pressure drop also. Now this hydrostatic pressure drop is nothing but what
should be the pressure drop due to the gravity of the mixture of the fluid. Now, this
gravitational force of the mixture of the fluid this mixture of the fluid will be denoted by

 dp 
   m g
 dz  head

here rho m is the mixture of mixture density of the fluid and g is the gravitational
acceleration. So, where this rho m is the composite fluid density of three phases, which is
equal to

 m  l  l   g  g   s  s
  g l  g l   s  s

This is nothing but the portion of liquid. This is nothing but a portion of gas. This is
nothing but the portion of solid here. So, this will be the total what will be that mass of
the gas-liquid-solid out of the total volume, though this can be represented by this. So,
this epsilon l epsilon g and epsilon s are the volume fraction of the liquid gas and solid,
respectively. Now, if we consider that only liquid and solid or liquid and gas or solid and
gas, if we divide it into two parts by combining either of the ways; so here if we consider
that this liquid and gas as one phase. So, in that case, what should be the density of that
phase or what should be the portion of the density for those that would be rho s rho g l;
that means, the density of the gas-liquid and what will be the volume fraction of that gas-
liquid mixture there that will be represented by epsilon g l. And here this one this called
rho s epsilon s is nothing but the what is that what will be the density of the solid inside
the bed.

And here epsilon s you can calculate this epsilon s; that means, a fraction of the solid
inside the bed will be is equal to just you have to weight the solid of the made before
entering before inserting into the bed the water to the solid of the part of the what will be
the weight of the solid or mass of the solid if you divide it by density of the solid, then
you will get what should be the volume of the solid inside the bed. And what will be the
total volume? That will be equal to the cross-sectional area into l, l is the characteristics
length this characteristics length will be the height of the gas-liquid mixture. So, here this
will be your total volume of the gas-liquid mixture, and this is the volume of solid here.
So, this volume of solid divided by volume of total gas-liquid mixture that will give you
the volume fraction of the solids inside the bed. So, from this portion, we can calculate
from this head pressure by this mixture density of the phases inside the bed.

(Refer Slide Time: 06:24)

Now considering gas and liquid as one homogeneous space and solid is another phase.

So, if suppose considering that epsilon f epsilon f is the bed volume fraction of the gas
and liquid except solid, then we can write here

 g l  g l   g  g   l  l
  g  g  l  l
 g l 
 g l

this will be your density of the gas and liquid mixture. Here in this figure, if we consider
this total bed and if we consider this total bed here of gas-liquid-solid, and if we divide it
into two parts of this portion this is, this portion will be as a gas-liquid mixture and this
in this portion only solid. So, this this this portion or this fraction volume fraction will be
the present or a gas-liquid mixture volume fraction, and this will be the solid volume
fraction.

Ws /  s
s 
AL

Now so, what should be the effective density in this gas-liquid mixture? That will be
represented by rho g l into epsilon g l. Rho g l in the density of the gas-liquid mixture
and epsilon g l is the volume fraction of gas-liquid mixture. So, this will ultimately be
coming as what is that as per that mixture density is here rho z epsilon g; that means,
what is the gas portion that is gas density effective gas density and what will be the
effective liquid density. Summation of this density will be represented over that what
will be the effective gas-liquid mixture density. So, from this equation, we can rearrange
it as

 g l   g g  l (1   g )

So, here the g l will be represented by the only the gas-liquid mixture. And this f f is
nothing but f is here representing the fluid, fluid means here gas-liquid mixture here the
frictional, then hold saw holdup of gas and liquid in gas-liquid mixture can be
represented respectively by here. So, in this case, alpha z will be equal to

g
g 
 g l

What does it mean? This alpha g is the gas volume fraction inside the bed when we are
considering the gas-liquid portion.

So, in the gas-liquid portion, what should be the volume fraction of gas in gas-liquid
mixture. So, that will be defined as alpha g. And this is actually mathematically can be
expressed as epsilon g divided by epsilon g l. What is this? Epsilon g is the volume
fraction of gas out of the total volume fraction of the gas-liquid mixture inside the bed.
And then what should be the then remaining portion in the gas-liquid mixture that is
liquid? And what should be the liquid volume fraction inside the bed? That will be just

l
l  1   g 
 g l

because here alpha l plus alpha g that will be equal to 1. This is fraction nothing but
fractions. So, alpha l is nothing but liquid volume fraction in the gas-liquid mixture or
the portion of the gas-liquid-solid fluidized bed. That will be is equal to then epsilon l
divided by epsilon g l.

So, therefore, we can write this epsilon g l from this equation 5, that rho g l will be equal
to then what will be that rho g, what is that? Rho g alpha g rhog alpha g plus rho l into 1
minus alpha g. What is this rho g l? Rho g l is nothing but the effective density of the
gas-liquid mixture inside the total 3 phase fluidized bed.

(Refer Slide Time: 10:12)

Now, if we consider that homogeneous mixture of gas and liquid, the actual fluid
velocity of this homogeneous gas-liquid mixture inside the bed will be represented by
this ug-l. ug-l is nothing but the actual fluid velocity of the homogeneous gas-liquid
mixture. That will be represented by what will be the actual gas velocity plus what
should be the actual liquid velocity.

usg usl
u g l  
g (1   g )

Now, what would be the actual gas velocity? How to calculate if you know that epsilon
is that is if you know that what should be the velocity of the gas inside the bed, that is
represented by that is superficial gas velocity. How to calculate the superficial gas
velocity that? What will be the volumetric flow rate of the gas you divided by cross-
sectional area of the bed, then you will get the superficial gas velocity? And if you divide
the superficial gas velocity in this portion of gas-liquid mixture by this volume fraction
of the gas, then you will get the actual gas velocity.

Now, what should we then? Actually, liquid velocity actual liquid velocity in that portion
of the gas-liquid mixture of these 3-phase fluidized bed will be is equal to what will be
the superficial liquid velocity, and what should be the volume fraction of the liquid
inside the bed. If you divide it this superficial liquid velocity by this volume fraction of
liquid that will be 1 minus epsilon g, then you will get this actual liquid velocity inside
the bed. Now, what should be the then total or actual fluid velocity of the homogeneous
mixture of this gas? And liquid just you have to sum up these two actual gas velocity and
liquid velocity.

Now, what is the homogeneous fluid density of this gas-liquid mixture is you know that
you rho g l that will be is nothing but the what will be the portion of the gas and liquid
divided by out of total volume of this volume fraction of this gas-liquid mixture, then it
is simply

 g  g  l  l l  l
 g l    (1   g ) l
 g l  g l

So, this is the homogeneous fluid density that you have already calculated based on the
in the previous slides that how to calculate this a density of the gas-liquid mixture.
(Refer Slide Time: 12:41)

And once we know these mixture density of the gas-liquid portion of the total 3 phase
fluidized bed, and if you know the density of the solid inside the bed if you know the
volume fraction of gas and liquid inside the bed, and also what should be the fractional
volume of gas or gas or liquid or solid inside the bed of course, you would be able to
calculate what should be the total frictional pressure drop inside the bed.

Now, this frictional pressure drop portion, because this total pressure drop is that the
contribution of frictional pressure drop head pressure drop and accelerative pressure
drop. Now let us consider this frictional pressure drop first. So, gas-liquid-solid flow, in
that case, frictional pressure drop will have these three parts. One is the frictional
pressure drop due to particle wall friction. And one frictional pressure drop would be
fluid wall friction, and one pressure drop would be fluid particle friction. There are 3
components of this frictional pressure drop.

 dp 
 dz  (i.e. total frictional pressure drop)
  fr ,t
 dp 
  (i.e. particle  wall frictional pressure drop)
 dz  p  w
 dp 
   (i.e. fluid  wall frictional pressure drop)
 dz  f  w
 dp 
   (i.e. fluid  particle frictional pressure drop)
 dz  f  p
So, this particle wall frictional pressure drop would be represented by this dP by dz at
pw, p for particle w for wall. So, it will be a particle wall frictional pressure drop and f
for fluid and w for wall. F means fluid here fluid means here, gas and liquid mixture,
whereas fluid and particle there will be gas-liquid mixture interacting with the particle.
In that case, what should be the frictional pressure drop whenever there will be
interaction between fluid and particle. So, these three components of this frictional
pressure drop will give you the total frictional pressure drop in the gas-liquid-solid
system in the fluidized bed.

(Refer Slide Time: 14:45)

Now, this frictional pressure drop between particle and wall.

Let us see how to calculate this frictional pressure drop between particle and wall. So,
frictional force per unit volume between the solid phase and wall; can be calculated by
fanning equation which can be written as

 dp   s  p u 2p
   2 f pw
 dz  particle  wall dbed

So, by this equation, you can calculate what with the frictional pressure drop whenever
particle wall interaction will be there. So, in this case here c f p w. F pw is representing
the friction factor whenever there will be a particle wall interaction. And epsilon s is the
volume fraction of solid inside the bed rho p is the density of the particle inside the bed.
And up, is nothing but the particle velocity inside the bed. And d bed is nothing but the
diameter of the bed. So, as per this Fanning equation, you can calculate what will be the
particle wall a frictional pressure drop once you know this particle velocity once you
know the friction factor between particle and what wall.

There is no universal formula actually for the calculation of friction factor. Like this f p
w; however, several correlations have been proposed by different investigators. So, from
those investigators, we can, we can calculate what it should be the friction factor. And
then, finally, if you substitute this friction factor in this equation, then you will be able to
calculate what should be the particle wall frictional pressure drop.

(Refer Slide Time: 17:01)

Now this calculation of the friction factor for particle flow f p. Now, different
investigators, they got they if they proposed two different correlations to calculate or to
predict this friction factor of this particle wall interaction.

1.021
(1   g l )  (1   g l ) Ret
2
 u g l
f pw  1.025 10   for  1.5
 g l 3  Re p  ut

0.979
(1   g l )  (1   g l ) Ret
3
 u g l
f pw  3.15  10   for  1.5
 g l 3  Re p  ut
  g l u p d p
Re p 
 g l

u p  u g l  ut

 g l ut d p
Ret 
 g l

 gd p2 
ut   (  p   g l ) 
18 g l 

So, in this case, of course, this Young 1978 is very important and acceptable
correlations, or it is widely accepted because these correlations include all the operating
conditions as well as the particle size, and that is why these correlations can be used to
calculate this particle wall friction factor. Now there are two conditions of these
correlations they have proposed one if this ratio of gas-liquid velocity to the terminal
velocity of the particle if it is 1.5, then you have to use these correlations. Whereas, if
this ratio of gas-liquid velocity to this terminal velocity is greater than 1.5, you have to
use this second equation of this fanning friction factor.

In this case, this friction factor depends on the volume fraction of gas-liquid mixture
inside the bed, and the terminal velocity of the particle and this what is that the Reynolds
number of the particle that is actually because of the what will be the this is the based on
the particle velocity and the density of the gas-liquid mixture, whereas the terminal
velocity is the density of the gas-liquid mixture, but it will be the based on the terminal
velocity of the particle. So, R e p is nothing but the Reynolds number of the particle
which is defined as

 g l u p d p
Re p 
 g l

rhog is the density of the gas, if you are using gas, if you are using liquid then you have
to use of course, this or since it is a gas-liquid mixture then you have to use gas-liquid
mixture density here u p is the particle velocity and d p is the particle diameter, and
viscosity of the gas-liquid mixture is mu g l. Not here the mixture density of the gas-
liquid-solid inside the bed. Here you have to consider the viscosity of gas-liquid mixture.
Whereas this u p u p is what whenever particle is fluidized with the certain gas velocity,
you will see the particles will move with a certain relative velocity, relative to this
terminal velocity of the particle. Now, what should be this gas-liquid mixture velocity
inside the bed? That will be u g l whereas, the terminal velocity of this particle is u t. So,
terminal velocity generally, this it will we move downward whereas, in the fluidized bed
with a certain gas-liquid mixture velocity, it will move upward. So, what should be the
effective velocities they are for the particle? It will be the effective upward velocity will
be is equal to

u p  u g l  ut

this u t is the terminal velocity.

So, u p should be calculated part of the relative velocity of the gas relative to the terminal
velocity of the particle. Now what should be the terminal velocity of the particle that
already we have calculated if any single particle is suspended in the bed then as per
stokes law that terminal velocity can be calculated here by this here g d p square divided
by 18 mu gas-liquid mixture; that means, the viscosity of the gas-liquid mixture into
what should be the particle density you have to subtract it subtract the gas-liquid mixture
density to this particle density then you will get this terminal velocity.

So, once you know this terminal velocity and once you know this relative velocity of the
particles, you will if you know this Reynolds number of this particle and this turners
number based on the terminal velocity, and if you substitute here, you will get this
friction factor. Of course, before substitution, you have to calculate this what will be the
ratio of this gas-liquid mixture velocity and what should be the terminal velocity. If you
are having this ratio less than 1.5 then you have to use this first equation. If you are
having this ratio is greater than 1.5 then of course, you have to use this equation.

So, be careful which equation to be suitable for your calculation based on your problem.
(Refer Slide Time: 21:56)

Now, the frictional pressure drop between fluid and wall. Now this fluid and one. Now
previous 1 is what is that fluid and particle now this is fluid and wall, now frictional
force per unit volume between fluid phase and a bed wall is given by again Fannings
formula if we use this fanning equation again the following equation can be written.
Now, this dp by dz is equal to the fluid wall of friction or pressure drop. This is due to
that on the interaction of this fluid and wall.

Now, here it will be again

 dp   g l  g l u g l 2
   2 f f w
 dz  fluid  wall dbed

So, from this equation, you can calculate what should be the fluid wall frictional. In this
scale case, f f w is nothing but the friction factor between fluid and wall. And here it
depends on this gas-liquid mixture density a mixture velocity inside the bed. And of
course, by this equation, you have to calculate this fluid wall friction factor. Now for this
again, you have to know what should be the friction factor, whenever there is an
interaction between fluid and wall.

So, in this case, also you can use whether this flow is a laminar flow or turbulent flow
based on this; also, you can calculate directly here.
If f f w is 24 by Reynolds number in actually inversely proportional to the Reynolds
number. If Reynolds number of the gas-liquid mixture is less than 2300 you can directly
calculate what should be the f f w here by this

24
f f w  ; Re g l  2300
Re g l

Where is this Re gl is defined as the mixture density of this gas-liquid mixture, and
mixture velocity of the gas-liquid mixture what will be the bed diameter and apart from
the viscosity of the gas-liquid mixture.

So, from this equation here of Reynolds number, you can calculate what with the friction
factor between fluid and wall. Similarly, if Reynolds number of this gas-liquid mixture is
greater than 2300, of course, you can calculate this friction factor from the Blasius
equation here

0.079
f f w  ; Re g l  2300
Re0.25
g l

In this case again, here, you will get this friction factor between fluid and wall at this
turbulent condition. But what is that? Gas-liquid mixture velocity that is nothing but the
summation of the actual gas and liquid velocity inside the bed. That will be defined as
that already, we have defined earlier slides that this gas-liquid mixture velocity will be is
equal to actual gas velocity plus actual liquid velocity here.
(Refer Slide Time: 25:05)

Now, the frictional pressure drop between fluid and now particle, fluid, and particle here.

Now, in this case, the friction pressure again you can calculate from the homogeneous
mixture as per Wallis you can calculate these fluid particle mixture pressure a frictional
pressure drop because of this interaction of fluid and particle here like this will be is
equal to

 dp  1 
   (1   g l )CD , f  p Ap   g l u g2 l 
 dz  fluid  particle 2 

This is as far wall is a Wallis,1969 model that you can also calculate fluid particle
friction of pressure drop because of their interaction. Here cd is nothing but drag
coefficient between fluid and particle, and Ap is the projection of cross-sectional
projection and area of the particle, and what is that u gl? u gl is nothing but gas-liquid
mixture velocity

usg usl
u g l  
g (1   g )

and rho g l is nothing but the gas-liquid mixture density.

So, from this equation, you will be able to calculate what should be the fluid particle
frictional pressure drop.
(Refer Slide Time: 26:16)

Now, when the liquid velocity increases and the gas velocity remains constant, you will
see the ratio of epsilon g and epsilon g l; that means, volume fraction of liquid to the
volume fraction of gas-liquid mixture will increase. And so, in this case, gas-liquid
mixture density will increase. Thus, in contrast to the 2-phase fluidized bed you can say
that frictional pressure drop per unit length will decreases. Now the surface area per unit
volume here, in this case, A p can be expressed as here.

( / 4)d eff2 d eff2 (s d p ) 2

Ap  2
 2
 2
( / 4)dbed L dbed L dbed L

This, what will be the surface area per unit volume inside the bed? For this A p that will
be again required for this Wallis equation. Here to calculate this frictional pressure drop
between fluid and particle.

So, Ap should be calculated in this way that would be is equal to phi sdp whole square
divided by dbed square into L. So, this will be is the surface area per unit volume here.
Now the frictional fact the friction factor this f p for fluid-solid and the friction factor for
single-particle Ap are interrelated. So, in this case, it is very important to be noted that
the friction factor when there will be a fluid and particle interaction, and what should be
the friction factor only one single particle is moving in a fluid medium.
So, in that case, you can get this friction factor by this equation of these. Rho C D f p by
CD dP that will be is a function of epsilon g.

CD , f  p (8 usg 5.7 )
  g l
CD , p

So, this is directly related to this friction factor this is this drag coefficient. So, you it is a
function of the gas-liquid volume fraction inside the bed. And also what should be the
gas velocity inside that depends on. So, once you know this C D f P and CD p then you
will be able to calculate what should be the C Dp inside the bed. And then, once you
know this CDp, then what should be the frictional pressure drop you can calculate by this
equation of equation 13.

 dp  1 
   (1   g l )CD , f  p Ap   g l u g2 l 
 dz  fluid  particle 2 

Now, to calculate these CD f P and also C D P, you have to use some correlations from
which you can calculate directly what will be the drag coefficient between fluid and
particle.

(Refer Slide Time: 29:05)

Now, individual solid phase friction factor fp or you can say drag coefficient fp can be
calculated from the correlation developed by this row 1961 for 3 phase flow.
 24
CD , p 
Ret

1  0.15 Ret0.687  for Ret  103

 0.44 for 103  Ret  105

that will be that can be used this way. And of course, this also will be in a certain range if
terminal velocity is less than 10 to the power three or you can say one thousand, then you
have to use this cd as this equation. Whereas this if from term Reynolds number based on
terminal velocity if it is one thousand to 10 to the power of 5, then you have to use this
that is turbulent condition this drag coefficient would almost remain constant that already
been discussed in earlier lecture notes. That will be constant at 0.44. A Reynolds number
for the homogenous flow based on terminal velocity will be defined as what will be the
terminal velocity, what will be the density of the gas, liquid mixture, what will the
density diameter of the particle, and this.

ut  g l d p ut (1   g ) l d p
Ret  
 g l l (1   g )   g g

So, from this definition of this Reynolds number at it is the terminal velocity of the
particle you can get this equation directly if you substitute this ut or u t here and then

 gd p2 
ut   (  p   g l ) 
18[ l (1   g )   g g ] 

And here, the viscosity of the gas-liquid mixture, you can use this mixture viscosity as

 g l  l (1   g )   g g

So, this one is the effective viscosity of the gas-liquid mixture inside the bed. And this
terminal velocity will be defined as this here, for this 3-phase system here, it will be
equal to this. In this case density on viscosity will be based on this gas-liquid mixture
and the particle, what is this? So, based this is as per this equation 18, you can calculate
what should be the terminal velocity inside the bed.

So, once you know this terminal velocity, of course, for this terminal velocity whenever
a gas-liquid-solid system, you need to calculate this alpha g, Alpha g is what? This alpha
g is nothing but the volume fraction of the gas inside the fluidized bed for the portion of
the gas-liquid mixture. Now how to obtain this volume fraction inside the bed? So, you
can you can obtain you can estimate this volume fraction experimentally just by phase
isolation technique inside the bed, suppose at any operating condition what should be the
gas-liquid-solid mixture hide inside the bed?

(Refer Slide Time: 32:05)

If we are we are able to know what should be the mixture hide inside the bed by this if
you consider this; we bed and this is the gas-liquid mixture gas-liquid-solid mixture
inside the bed, and if you suddenly stop this operation and you will see the gas inside the
bed it will be dissolved at the top of the gas and there will be after a certain time there
will be a clear liquid height inside the bed.

So, from this clear liquid height and the gas-liquid-solid mixture height inside the bed,
you can calculate what should be the volume fraction of the gas inside the bed. Now how
to do it? What you have to do that you have to first calculate what will be the gas volume
inside the bed? And what with the mixture volume inside the bed? Now from this ratio
you can calculate what will be alpha epsilon g here now this epsilon g. So, from this gas
volume, how to calculate this gas volume is nothing but mixture volume minus clear
liquid volume inside this.

This is the total mixture, gas-liquid mixture, volume, and this is the only liquid-solid
mixture volume. So, if you substitute this liquid-solid mixture volume from this total
gas-liquid mixture volume, you will be able to calculate what will be the gas inside the
bed gas volume inside the bed. Now from this mixture volume, you can calculate A into
Lm. What is the A? A is the cross-sectional area Lm is the gas-liquid mixing height
inside the bed gas-liquid-solid mixing height inside the bed. Then this will be your
mixture volume. And this is what A Lc? A is the cross-section area and the Lc is nothing
but the clear liquid-solid height inside that bed. And this is your total volume.

So, from this, if you cancel this a from this numerator and numerator, then you will get
this Lm minus Lc divided by Lm. From this portion, you will be able to calculate what
should be the volume fraction of gas inside the bed once you know this volume fraction
inside the bed. So, out of this total gas-liquid mixture volume, what will be that portion
that will be calculated as alpha g. Now as far as per the suggestion of Marquardt, 1963
one correlation as per the experimental work they have developed this volume fraction of
the gas inside the bed, and they have developed one correlation based on the different
operating parameters like gas was sliding velocity or can the superficial liquid velocity
and gas velocity, and also what will be the diameter of the bed and under the density of
the solid particle diameter.

3.464 109 usl0.66 (s d p )0.50  s2.30

g  3.74
 usl  0.06
1  1.74   L0.43  l   g  l0.08 dbed 0.23
 usl  usg
 

So, with those variables so, they have developed these big correlations, and from these
correlations, you can calculate directly what should be the volume fraction of gas inside
the bed.
(Refer Slide Time: 35:40)

And now so, how to calculate once you know this volume fraction of gas-liquid mixture?
And what should be the friction factor of the particle? And if you know the alpha g; that
means, the volume fraction of gas inside the bed, and what should be the volume fraction
of the gas in the gas-liquid mixture portion? And if you know the gas-liquid mixture
velocity inside the bed, from this, you just calculate you can calculate what should be the
frictional pressure drop whenever there are an interaction between fluid and particles.

So, from this equation 21, you will be able to calculate a fluid particle frictional pressure
drop.

 dp  3(1   g l ) f p (1   g ) l u g2 l
   (8 u  5.7)
 dz  fluid  particle 4 s d p g lsg
(Refer Slide Time: 36:33)

Now once you know this 3 part of this frictional pressure drop, total frictional pressure
drop will be equal to frictional pressure drop because of this particle wall interaction plus
fictional pressure drop because of this fluid and wall interaction and the frictional
pressure drop because of fluid and particle interaction. So, then if you just substitute to
the respective value of this frictional pressure drop, and then you will be able to calculate
will be calculated the total frictional pressure drop.

 dp   dp   dp   dp 
       
 dz  friction , tot  dz  particle  wall  dz  fluid  wall  dz  fluid  particle

(1   g l )  p u 2p
 2 f pw for particle  wall
dbed
 g l  g  l u g l 2
 2 f f w for fluid  wall
dbed
3(1   g l )CD , p (1   g ) l u g2 l
 (8u 5.7)
for particle  fluid
4 s d p g lsg
(Refer Slide Time: 37:09)

And our frictional pressure drop based on considering solid plus liquid as a slurry as one
phase and gas as another place. So, you can calculate the frictional pressure drop on
either way. Either you are considering gas and liquid mixture as one phase, and solid is
another piece. Or you can consider this solid and liquid mixture as a slurry as one phase,
and gas is another phase. So, you can calculate the frictional pressure drop based on
considering solid-liquid mixture as a slurry as one phase and gas as another phase. Let us
consider here this dp by dz as a slurry gas and wall. So, in this case,

 dp  2 f m  mum2
  
 dz  slr  g , w dbed

Now, here this portion will be considered as this slurry effective slurry density, and this
would be the effective gas density. And viscosity will be defined as effective viscosity
here as a slurry that is a liquid and solid mixture, and here it will be the effective gas
viscosity. And mixture velocity, which will be here this one is the actual gas velocity of
the gas and actual velocity of the slurry here. And the Reynolds number will be defined
as the mixture velocity and mixture density based on this and this mixture viscosity. And
then here again this what should be the f m; fm means the friction factor because of this
slurry gas mixture. So, in this case, slurry gas mixture. So, fm should be calculated as 24
by Rem where Rem should be 2300. Whereas, if the Reynolds number is greater than
2300, then you have to calculate this friction factor of this mixture of gas and slurry. This
will be 0.07 mined by Rem to the power 0.5.

Remember this here Rem is the main factor 2 just to get these to get this mixture of
friction factor. Now, this Rem will be changed because of this viscosity density and the
and the velocity of the mixture here. So, the viscosity of the slurry you can calculate
from this direct relationship either, or you can use other viscosity formula for this slurry.
Now this slurry viscosity here it is only liquid and solid. You see, sometimes this the
slurry viscosity will increase or will change because of the concentration of the solid
inside a solid of the slurry. Now, if suppose you are the volume fraction is suppose 2
percent, you will see the viscosity of the slurry will be something else.

So, what should be the effective viscosity compared to the pure liquid viscosity? So, if
you add some solid particles there and make the slurry, then you will see the viscosity of
the liquid will change accordingly. Now that depends on the concentration of the slurry,
the concentration of the slurry concentration of the particle inside the particle of the
slurry system. So, this slurry viscosity will be equal to

 3 slr 
 slr  l 1  
 1  1.923 slr 

What is this epsilon slr? Epsilon s l r is nothing but the volume fraction of the solid
particles of this slurry.
(Refer Slide Time: 41:20)

Now, what should be accelerative pressure drops if you are considering that slurry and
gas mixture? So, an accelerative pressure drop is the accelerative pressure drop is due to
the acceleration of the particles from the inlet to the point where the particles attain the
maximum velocity in the bed. So, accelerative pressure drop would be equal to

 dp  du p d (u g l )
    p (1   g l )u p  (  g  l  g l ) u g l
 dz  acc. dz dz
2
  usl u  
   p (1   g l )u 2p   g l  g l   sg  
  (1   g )  g 
   

up is the particle velocity into dup by dz. dup by dz is nothing, but for the change of
velocity across the that is along the length of the fluidized bed, and then plus what should
be the momentum change of this gas-liquid a mixture inside the bed.

d (u g l )
So, that will be equal to (  g l  g l ) u g l And u p is nothing but the relative
dz
velocity of the particle; that will be calculated just subtracting the terminal velocity from
the gas-liquid mixture velocity inside the bed. So, finally, you can get after
rearrangement and substitution of this value here you can get
 2
 u 
  p (1   g l )u 2p   g l  g l  usl  sg
 dp   
    
 dz acc.   (1   g )  g 
   

So, this is the actually mixture velocity of gas-liquid in inside the bed. Now, in this case
you will see when gas-liquid 2 phase flow in the fluidized bed in this riser section, where
that gas-liquid-solid mixture is rising up the reaches fully developed state, then
accelerating pressure drop will be neglected. Or it should be become 0, because they are
there will be no change of velocity inside the bed at this fully developed flow.

So, because of ways you have to neglect this accelerative pressure drop. Now let us do
an example for this gas-liquid mixture pressure drop. So, once you know these 3
components of this pressure drop and summation and substitute the value, you will get
this frictional pressure drop in 3-phase fluidized bed also.

(Refer Slide Time: 43:53)

Now, if we consider the gas-liquid-solid fluidized bed of a cross-sectional area of 50.26-

centimeter square, and it is operating with an airflow rate of 5 l pm that is liter per
minute and water flow rate of 500 liters per hour with a catalyst particle of size 100
micrometer. And a sphericity of 0.96, then the mass of the solid intake in the bed is
considered as 2.855 kg, and the density of the particle is given as 3950 kg per meter
cube.
Under this condition, you have to calculate what should be the volume fraction of the
gas, calculate the terminal velocity of the particle, calculate the relative velocity of the
gas-liquid mixture. How to calculate the particle wall, part fluid wall, and a fluid particle
frictional pressure drop? And also calculate what with a pressure drop due to the
acceleration, what will be the percentage contribution of the frictional pressure drop to
the total pressure drop or to be the percentage contribution of the head pressure drop to
this total pressure drop what will be the percentage contribution of the accelerative
pressure drop to the total pressure drop?

Now, you have to calculate based on the adhere equation. Now the volume fraction of
the gas you can directly calculate from these correlations. Now, this epsilon g is equal to
you can do you have to substitute what will be the different variables here in this case.
So, if you substitute the different variables like usl; that means, a superficial liquid
velocity you can calculate the superficial liquid velocity from this the flow rate of the
liquid here if it is 500 lph this is liter per hour you have to convert this 500 lp h to the
meter cube per second.

Then if you know this volumetric flow rate of this meter q per second, you have to divide
these by these cross-sectional areas of the bed. How to calculate the cross-sectional area?
That is given already otherwise if the diameter is given, you have to calculate what
should be the cross-sectional area. Now, if you know this cross-sectional area and a
volumetric flow rate of the liquid simply, you can calculate what will a superficial liquid
velocity inside the bed.
(Refer Slide Time: 46:17)

And phi is sphericity it is given to as per this particle shape it is given 0.97, and then DP
the particle size also. You can calculate the particle size, or you have to know by particle
size analyzer working with the particle size if it is given then you have to use this particle
size directly here. And rho s the density of the particle is given as per your problem is
3950. And what is the usl, you know that usg usg also, you can calculate this usg from
the volumetric flow rate of the gas. And if you divide this volumetric flow rate of the gas
by the cross-sectional area of the bed, then you can calculate what will be the superficial
gas velocity inside the bed. Then this is usg and then l the length of the bed. This is the
effective length of the bed height; that means, gas-liquid mixture height. If it is they are a
certain height, this is maybe they are a certain height; then, you have to calculate what
will be the epsilon g. And rho l minus rho g rho l is the density of the liquid rho g the
density of the gas to substitute they are gas here you can use that air as a gas-liquid
mixture that is, or it is given that some other gases then you have to use this gas density
of that other gases.

And viscosity of the liquid and diameter of the bed. The diameter of the bed, if it is given
the cross-sectional, you can calculate the diameter of the bed by just what will the cross-
sectional area by pi by 4 d bd square from which you can calculate the bed. Now once
you know this data, then you can directly substitute here and indirectly gate what should
be the epsilon g; that means, epsilon g is nothing but the volume fraction of the gas
inside the bed. That is here 5 percent of volume fraction. So, the column fraction of the
bed will be a volume fraction of 0.05. For this terminal velocity of the particle, how to
calculate this terminal velocity of the particle, then as per strokes equation you can
calculate this terminal velocity gdp square by 18 mu g l into rho p minus rho gl.

So, it will be is equal to 0.001973 meters per second. And then what should be the
mixture of gas-liquid velocity? Then if you know these, u s g, and if you know this usl,
then you can directly calculate what will be the mixture velocity of gas-liquid. We said
here if alpha g you can you have to convert this alpha g, from this epsilon g this epsilon g
is that gas-liquid in the total friction of this is the volume fraction of gas in the total gas-
liquid mixture.

Now, what should be the alpha g? Alpha g is nothing but epsilon g divided by epsilon
that will be your alpha g. So, alpha g, you have to calculate, and if you substitute this
alpha g here, you will be able to you will able to calculate.

(Refer Slide Time: 49:24)

What will be the mixture gas-liquid mixture velocity? And then substitute this and also
you have to calculate this f f p w again by this equation by fanning equations, and then
what should be this? This if you want to calculate this by substituting there are different
data, then you will be able to calculate this fluid wall frictional pressure drop, fluid-
particle frictional pressure drop, accelerative pressure drop, and this head pressure drop.
Now, see from these this particle wall pressure drop is 21.814; however, this fluid wall is
17.721; this is lesser than this particle wall. Whereas fluid particle is very negligible
0.0127, it is very small compared to the other pressure drop. Whereas, the head pressure
drop is the maximum amount here, and accelerative pressure drop is also here an
important factor. Because here in this case and the fluid may not be reaching to the fully
developed flow. So, in that case, you will get some accelerative pressure drop.

Now, in this case, you will see that different portions of this 21.814 is for particle wall
17.72 is for fluid wall and 0.013 for fluid particle, and 13419 is for head pressure drop
and the accelerative pressure. Total is this total pressure drop is the summation of these
three components of this frictional pressure drop. And then contribution you will see if
you get this contribution here as the percentage here particle wall contribution. And this
here fluid wall and this fluid particle contribution is very less this negligible. Whereas,
the head pressure is drop is the maximum amount of contribution by this hydrostatic
pressure drop.

So, in this way you can calculate what will be the total pressure drop if you know these
different variables only this gas velocity liquid velocity frictional pressure drop and also
what is the friction factor, and what should be the velocity of the particle what will be the
terminal velocity of what will be the other parameters if you know then you can directly
calculate this.

(Refer Slide Time: 52:00)

Now see here this different contribution in different pressure drop in different flow
regimes you see here, different flow regimes we have calculated here, different flow
regimes how it will be they are.

So, dispersed block flow here this is gas superficial velocity here 0.013 and liquid
superficial velocity this and in this dispersed bubble flow, and this discrete bubble flow.
And if you are considering coalescence here this slug churn turbulent bridging annular
and transport, then you will see a different contribution of this frictional pressure drop at
different operating conditions here.

So, in this case, you will see that here very interesting that here in transport cases here,
this hydrostatic pressure drop is negligible compared to this what is that fluid and wall
pressure drop. So, in the transport regime, the case hydrostatic pressure drop will be less.
In that case, will see the dilute phase the part fluid and particle will be more interaction
there. And whereas, in a bubble in dispersed or discrete bubble flow regimes, you will
see that there will be the main contribution of this head pressure.

Now, this head pressure will be gradually decreasing with the different flow regimes,
from dispersed this bubble flow to the transport, whereas this head pressure accelerative
pressure drop is also increasing, increasing, increasing from this dispersed bubble
through the transport flow. And other contribution of this frictional pressure drop it is as
per this here, you will see this particle wall contribution again will be increasing with the
transport from the dispersed bubble flow. And here a fluid particle, in this case, disposed
to this bridging and annular flow this bridging and annular flow there will be a certain
contribution whereas, this transport and other there will be a very negligible contribution
in this case.

But churn turbulent the highest contribution for this fluid particle frictional pressure
drop.
(Refer Slide Time: 54:43)

Now let us see this frictional pressure drop analysis by different models. Frictional
pressure drop analysis with the different models can be done. In this case, liquid and
solid mixture as a slurry, you have to consider one phase, and that will be heavier pace,
and gas-phase took the lighter phase. Let us consider in the next class what should be
these analyses of frictional pressure drop by different models. So, in the next class, we
will be discussing that. So, that is for all today this lecture.

Thank you.
 Fluidization Engineering
Dr. Subrata K. Majumder
Department of Chemical Engineering
Indian Institute of Technology, Guwahati

Lecture – 10
Analysis of Frictional Pressure Drop in Fluidized Bed By Different Models

Welcome to the massive open online course on Fluidization Engineering. Today’s lectures on
analysis of friction of pressure drop in a fluidized bed by different models. So, in the previous
lecture, we have discussed how to estimate the frictional pressure drop in a three-phase
fluidized bed. And also, we have given example that what will be the different contributions
of different portions of frictional pressure drop and also total pressure drop that has already
been discussed and also given example.

Now, in this lecture, how to analyze this frictional pressure drop by the different models.
These models may be correlation models and from the basic or there may be some
mechanistic models. So, let us see what will be the different models by which we can analyze
the frictional pressure drop in the fluidized bed.

(Refer Slide Time: 01:28)

So, for this analysis, different models, of course, the models will be based on the concept of
two-phase flow. Considering two-phase flow of the gas-liquid-solid system as; in this case
the liquid plus solid mixture will be considered as a slurry phase and it will be considered as a
phase 1 and which is heavier phase and to gas that will be considered as a phase 2 that is
lighter phase.

So, three-phases will be converted to two-phase just by considering liquid and solid mixture
as one phase and gas is another phase. And there are several models to analyze the frictional
pressure drop in two-phase flow Lockhart- Martinelli model is the basic model that is
developed in 1949.

(Refer Slide Time: 02:27)

And based on this concept of Lockhart-Martinelli model now what should be the frictional
pressure drop or what should be the multi; what should be the frictional pressure drop if we
know the single-phase pressure drop inside the bed?

So, if we do this frictional pressure drop of this two-phase pressure; in this case two-phase
means here what? Slurry is the one phase, gas is the another phase. So, frictional pressure
drop of these two-phase; that means, in the three-phase of this fluidized bed, then you can
write this frictional pressure drop of these two-phase will be is equal to

Pftp   sl2 Pfosl

what does it mean? This delta P fosl is nothing, but the frictional pressure drop of the single-
phase slurry. And phi sl is nothing, but the multiplication factor or Lockhart-Martinelli
multiplier for this slurry phase..
So, if we multiply this multiplier of this phi s l square to the single-phase pressure drop then
you can have this three-phase pressure drop. Even if you consider the single-phase pressure
drop of gas, then what should be the multiplication factor by which you can get the total
frictional pressure drop in the bed?

Pftp   g2 Pfog

Now, this you have to remember this Lockhart-Martinelli model is basically for the frictional
pressure drop analysis. So, this delta P ftp based on the single-phase of slurry what if you
multiply it by this phi sl square, you will get this three-phase pressure drop. Even if you
multiply this phi g square, this is the multiplication factor of the gas phase to this single-
phase pressure drop of gas; then, you can get what should be the total pressure drop of total
frictional pressure drop inside the bed.

Another parameter of this Lockhart-Martinelli model is X; what is this X? X is nothing, but

this multiplier ratio of this gas to slurry. So, in this case, it will be equal to

g  P 
X   fosl 
 sl  Pfog
 

So, in this case the frictional pressure drop due to the slurry and due to the gas, so the X is
nothing, but the root over of ratio of this frictional pressure drop of single-phase slurry and a
single-phase gas.

So, from this, you can get the total pressure; now, this phi s l is the parameter and X is also
one parameter. These are Lockhart-Martinelli parameters. Now from this Lockhart-Martinelli
parameter, you will be able to calculate probably the frictional pressure drop. Now how to
obtain this phi s l; this multiplication factor or this parameter X? Now different investigators
they have developed the different correlation based on the different operating variables from
the two-phase operation in the bed ah; maybe in the bubble column reactor, in the slurry
bubble column reactor or any other two-phase system; they have actually calculated this
frictional pressure drop and they have developed the multiplication factor and the Lockhart-
Martinelli model based on their experimental data.
Now, to calculate these multiplication factor of course, you have to move some parameter
here; what should be this X parameter that is Lockhart-Martinelli parameter? And for this you
have to know the fictional pressure drop of single slurry phase..

(Refer Slide Time: 06:59)

And for these, you have to calculate this frictional pressure drop of the single-phase by the
Fanning equation like this here. So, it will be

2 o f ouo2 z
Pfo 
dc

what is this rho 0? rho 0 is the single-phase density f 0 is the single-phase frictional friction
Fanning friction factor, u 0 is the single-phase velocity and delta z is the effective length of
the gas-liquid mixture inside the bed, and this is the column diameter or bed diameter.

And this f 0 is

f o  16 / Reo

if it is laminar flow. If

f o  0.079 / Reo0.25

this Reynolds number will be based on the single-phase and it is for turbulent condition.
So, this is for laminar-laminar condition and this is turbulent-turbulent condition and this is
for transition transition; that means, your laminar if Reynolds number of single-phase is less
than 2300 and if turbulent flow; that means, if Reynolds number is greater than 4000 or the
transition flow means here the Reynolds number in between these 2300 to 4000. So, you can
calculate this friction factor this if your operating condition is in the laminar flow or in the
turbulent flow or in the transition flow.

Once you know this Reynolds number, you be able to calculate this friction factor at different
operating conditions. Now this phi s l is shown is actually developed this phi s l square; that
means, here multiplier Lockhart-Martinelli multiplier here or two-phase multiplier you can
say that this will be is a function of X; X is the Lockhart-Martinelli parameters, this is
nothing, but the root over of ratio of frictional pressure drop of the single-phase of liquid to
gas or here in this case slurry to gas. And here another one parameter is C.

So, this is actually again you have to calculate or you have to obtain from the experimental
data; just by fitting these equations by the experimental data. Now how to actually obtain this
C here? See if you know this X by this frictional pressure drop of single-phase and if you
know this phi s l square; how to know this phi s l square? Because you know this total
pressure drop delta P of ftp three-phase flow and if you know this delta P of s l single-phase
flow, then this is the ratio now this ratio is defined as phi s l square.

So, this is your experimental data, and this is your, of course, experimental data. So, from this
experimental data or you can get the single-phase from the Fanning equation. If you know
this and if you divide these two pressure drop or if you are getting this ratio, this will be
represented as phi s l square. So, this phi s l square is known by the experimental data and
then X is also known from the Fanning equation and from the pressure drop. And then if you
feed this phi sl versus X here, maybe this from this graph here if you are drawing or if you
are plotting this data as the X; then you will get this type of relationship. If you are fitting this
you will get this what is that C value from this equation.

 C 1 
 sl2  f ( X )  1   2 
 X X 

So, just by least square method you can fit this equation then you can get what should be the
value of constant this C; so you can use either this single-phase slurry or you can use gas
phase single-phase gas phase; from who is also this phi g square you can get or you can relate
the phi g square with the X as

 g2  f ( X )   1  CX  X 2 

So, here again, the same C X will be obtained; now, this C depends on what are these
laminar-laminar condition, turbulent conditions, or transition that depends on how you are
calculated this f also and also your operating condition.

Another also see shown they had given or that X square this Lockhart-Martinelli parameter is
a function of the ratio of liquid to gas mass flow rate. And then, the density ratio of gas to
liquid and the viscosity ratio of liquid to gas here and this lambda

a b c
2
 m   g   l 
X  l     
 m g
   l   g 

and a, b, c those are constants that can be obtained from the experimental data.

Now, X you know that X you can calculate from the experiment also; how to get this? If you
know, if you do the experiment on the single-phase, then what should be the pressure
frictional pressure drop? And if you do the experiment to the single-phase of gas then also
you have to have the measurement data other. So, from this measurement, you can calculate
this; otherwise, you can calculate what will be the Fanning equation.

So, you can get it from Fanning equation also this relationship; once you know this X, once
you know this C then you will be able to calculate what should be the phi s l square. Once
you know this phi s l square then you can easily calculate what should be the total frictional
pressure drop just by multiplying this phi s l square to this single-phase pressure drop of the
slurry.

So, from this equation, you will be able to calculate how to or you can analyze what should
be the frictional pressure drop of this three-phase flow. So, this is one way; this is a Lockhart-
Martinelli model based on that you can analyze this here. Now if we know the experimental
data; what actually to be known? What actually to be find out?.

You have to find out this C value, if you are just if you are fitting these experimental data
with this type of equation, then you have to obtain these parameter C that is one coefficient
this is unknown parameter this will be obtained by the experimental data and just by fitting
with this experimental.

Otherwise, you can directly correlate with the different operating variables by dimensional
analysis and you will get different groups, dimensionless groups and then you just correlate
with this phi s l square and then you can get the phi s l square; once you know this phi s l you
can calculate the total frictional pressure drop here.

Pftp   sl2 Pfosl

(Refer Slide Time: 14:18)

Now, Bankoff 1960; he has developed another correlation based on this Lockhart-Martinelli
concept. Here they told that this two-phase frictional pressure drop or in this case here three-
phase frictional pressure drop by just considering the liquid-solid has one slurry and gases has
another phase.

 dp  7/4  dp 
    Bf  
 dz  f ,tp  dz  sl

So, in that case, he has proposed one correlation that this single-phase pressure drop; if you
multiply it by this a factor of phi B f to the power 7 by 4; then you can get this total frictional
pressure drop inside the bed. So, this phi B f is nothing, but the multiplier a Bankoff; so,
Bankoff multiplier. So, it is called if you are calculating these Bankoff multiplier from the
experimental data and if you are correlating this experimental data with this model of this phi
B f; here this phi is the function of X; X is nothing, but what is that? Mass quality of the
phases; this mass quality of the phases here X will be is equal to here mass of mass of gas by,
mass of slurry; slurry means here liquid and solid mixture.

3/7
1   g    
 Bf  1    1   1  x  sl  1 
1  x    sl     g  

So, X and here gamma is one parameter this gamma again is a function of this density ratio of
gas to slurry and also this mass quality for the gas of gas slurry mixture.

1
     1  x    g 
   c1  c2  g   1    
   sl     x    sl  

So, here you have to calculate now you have to find out now what should be the gamma
value or what should be the c 1 and c 2 coefficients for this gamma. This c 1 and c 2 are the
coefficients which will be found out from the experimental data. So, this coefficients c 1 and
c 2 are unknown parameter here; if you fit these parameters and by optimization method or
by least square method, if you feed this data with the experimental data then you will be
obtain you will you will get this c 1 and c 2 of parameters.

Now how to do this? Only thing this one is known to you, this is experimental and this is also
experimental, then this divided by this; you will get this phi B f. So, phi B f you can get you
can correlate this phi B f to this model here and X is known to you; only gamma, gamma is
nothing, but c 1, c 2 and this c 1 and c 2 you have to obtain from the just by fitting with
experimental data.
(Refer Slide Time: 17:20)

Another model is Baroczy Baroczy 1966 model; here in this case, they have introduced
property index like I p in terms of viscosity and density of the phases. So, they have defined
this property index as the ratio of viscosity of slurry to the gas here and also the ratio of
density of slurry to gas. So, here they have defined this as

0.2
   
I p   sl  /  sl 
 g  g
   

Now, based on their concept, they proposed that these three-phase pressure drop or two-phase
pressure drop will be is a function of this property index. and the function of single-phase a
pressure drop; that means if you multiply the single-phase pressure drop with this function of
property index, then their then two-phase or three-phase frictional pressure drop you can get.

Now, then 1 by I p will be then from this you can get it just this divided by this; this is single-
phase pressure drop to the slurry. So, 1 by I p is related to this here single-phase pressure
drop of gas to the single-phase pressure drop of slurry.

 p f   p f  1
  /  
 z  g  z sl I p

 p f   p f 
   f (I p )  
 z tp  z  sl
So, again you have to find out what should be the function of this property index from the
experimental data.

So, this one is your experiment; this one is your experiment, and then what should be the f I
p? And then again this f I p that you have to model that you have to make one correlation for
this I p in terms of different operating variables; now this operating variable will be maybe
what is that geometric variable, may be physical properties, maybe are some other what is
that thermodynamic conditions there are different operating conditions that will give you the
change of this I p.

(Refer Slide Time: 19:41)

Another model is Wallis 19696 model what happened here;

 p f  2 
p f 
   W  
 z tp  z  sl

They have also developed one correlation based on that Lockhart-Martinelli concept again
this multiplier here phi w is one multiplier concept; this phi w is the multiplier of the single-
phase pressure drop.

Then you will get these two three-phase or two-phase pressure drop here. Then you have to
correlate this phi;
 0.25
   g   sl   g 
  1  x sl
2
W  1  x 
 g g
  

w square that is here of multiplier as a function of this mass quality of the gas and also
density of the slurry, density of the gas viscosity of the gas; so in this way this correlation.
So, if you know this slurry density; if you know the slurry viscosity, if you know the mass
quality of the gas and other physical properties, then you can directly obtain this equation.

And then if you substitute these, you will see if you multiply the single-phase slurry frictional
pressure drop and multiplied this multiplier then you will get the three-phase pressure drop.
Now, this this this correlations actually predicts the frictional pressure gradient well for the
annular flow pattern. So, if is there any annual fluidized bed, you can apply this Wallis model
to predict, to analyze this frictional pressure drop of this three-phase flow inside the bed.

(Refer Slide Time: 21:18)

Chawla 1973 she has developed the same concept of this frictional pressure drop by
considering the multiplier has phi Chawla.

 dp   dp 
   Chawla  
 dz  f ,tp  dz  g

And in this case, they have correlated this phi Chawla as this mass quality and one parameter
is called here this S that is the sleep parameter this sleep parameter is defined as 1 by lambda
into this is a function of Reynolds number of gas, Froude number of mixture and the density
ratio of slurry to gas and the viscosity ratio of slurry to gas.

n

m  1 x  g 
Chawla  x 1  S   
  x   slr 

1
S
1  x n2 n3 
 (Re g Frm ) n1   sl /  g    sl /  g  
 x 

Frm  m t2 / ( gd c  m2 )

Re g  m g d c /  g

So, in this case here, this parameter is very important. This parameter depends on the
different operating conditions. And here, this Reynolds number of gas is defined as m dot gdc
by mu g whereas Froude number is this total mixture flow rate square divided by gdc into rho
m square. So, why this Froude number is defined; if you know this Froude number, you can
calculate and if you substitute here and if you are making correlations to experimental data
with this S; then you will be able to calculate what should be the total of frictional pressure
drop.

Now, what is that here? In this case, this phi Chawla to be obtained from the experimental
data. And then just fitting with this model with this experimental value of phi Chawla and
you can get this value of S and which is correlated directly with the physical properties and
geometrical properties and other operating conditions.

And this, of course, this del p; del p by del z or dp by d z of gas single-phase it is very
important in this case Chawla has taken a single-phase pressure drop of gas and that can be
obtained by experiment either or you can calculate it from this Fanning equation of this.

 dp  2 f g m t 2
  
 dz  g dc  g

For that f g is required that f g is a function again Reynolds number, it is greater than 2300; of
course, it will be transition if it is greater than 4000, it will be turbulent and if it is less than
2300; of course, you have to consider that laminar flow design and based on which you have
to calculate this gas-phase pressure drop.

(Refer Slide Time: 23:48)

Now, Friedel 1980, he has also developed the based on the same Lockhart-Martinelli
concept; here again he has taken this frictional pressure drop of single-phase pressure drop.

 dp   dp 
    Frd  
 dz  f  dz  sl

And he has multiplied this multiplier of phi Friedel and to get this total frictional pressure
drop. Now, this the multiplier; he has proposed this correlation of this multiplier into two
components like is one E another is the you like this lambda into FG by Froude number and
Weber number to the power this.

2  FG
 Frd E
Fr Wesl0.035
0.045
m

So, according to this what should be this E and lambda; E and lambda are both are two
parameters here in this case E is directly related to this ratio of the slurry to the gas density
and the friction factor of gas to slurry and also it is the mass quality of the gas.

 sl f g
E  (1  x) 2  x 2
 g f sl
It is seen that this E value is increased if increases E value is increased; if mass quality is
decreased; mass quality is decreased.

Whereas if it increases, so, for the same mass quality if increase where the density of the
slurry, then, of course, this value will increase. So, this E is the one component which will be
contributed by this a property of a fluid and what is that friction factor in the Fanning.
Another important is that here lambda into FG; F is defined as

F  (1  x) 0.224 x 0.78

This F is directly related to this mass quality of the gas and G; G is the function of ratio of
slurry to gas density, ratio of gas to slurry density and the viscosity of the viscosity ratio of
gas to slurry here.

0.91 0.19 0.70

   g   g 
G   sl    1  
 g
    sl    sl 

So, G can be calculated from this correlation, F will be calculated from this correlation and F
r m that is Froude number of the mixture will be defined as

m t2
Frm 
gdc  m2

Weber number is defined as

m t2 dt
Wesl 
 m

And rho m will be defined as

1
 x 1 x 
m    
 g 
 sl 

so this will be the mixture density of gas and slurry. So, once you know this E, F, G and
Froude number and Weber number of the slurry you; will be able to calculate what should be
the multiplier factor of this Friedel, then multiply this multiplier with this single-phase slurry
pressure drop you will be able to calculate what should be the frictional pressure drop.
Then you have to compare with experimental data of own experiment with this friction
factor. Then how much deviation from your experiment to this model you will be able to
calculate. If it is any modification of this model is required based on your experimental data,
you can also do just by fitting these unknown parameters with your experimental data or you
can directly correlate this F r d that is phi F r d to your in terms of your different operating
variables just by dimensional analysis.

And then, fitting with experimental data with these different dimensions groups just by
having the dimensional analysis and fitting with experimental data with these dimensional
groups by regression multiple regression analysis, you can easily obtain the or you can easily
predict the multiplier of this Friedel.

(Refer Slide Time: 27:49)

Another module is Gharat and Joshi model actually they have analyzed the frictional pressure
drop and they have suggested that this total frictional pressure drop in this three-phase system
or two-phase system based on the two contributions one is single-phase frictional pressure
drop. Another is the pressure drop due to the additional turbulence that made by these other
phases.

Pf ,tp  Pf , sp  Pf , AT

From one single-phase and then pressure drop will be developed by turbulence like this; if is
there any bubble system. Then, of course, this bubbly system bubble will just enhance the
turbulence even some other provisions; if it is made in the fluidized bed some other like
intensification of the bed by providing other devices like baffale or other internal devices, we
are adapt in the bed. Then additional turbulence may produce, and because of which this
frictional pressure drop will change.

So, they have considered those additional turbulences by this delta P f AT; AT means
additional turbulence here. So, this frictional pressure drop will be and the single-phase
frictional pressure drop without turbulence and then if is there any turbulence made by the
different provisions; what will be the frictional pressure drop. Now, this single-phase
frictional pressure drop; you can obtain it from the Fanning equation that is simple;

2 f f ,sp usp2  sl z
Pf , sp 
dc

this Fanning equation you have this is Fanning friction factor again to be calculated from the
what is the Blasius equation. And then additional frictional pressure drop that is that by
additional turbulence that will be calculated in that.

2 f f , AT us2  sl z
Pf , AT 
dc

In this case f AT; that means, friction actor due to this additional turbulence will be function
of that is the concentration of the solid, concentration of the slurry inside the bed. And this
will be is equal to

f sp
f tp   f AT
 sl2

and this f AT; f AT this is the friction factor due to the additional turbulence.

Now, you have to calculate this overall friction factor from the mixture velocity of the fluid
inside the bed. Then what will be the overall friction factor from which if you know this
single-phase friction factor and the slurry concentration, what should be the friction factor by
this additional turbulence just by fitting this situation here?

Now, you see that overall friction factor will change here like this; if you are changing this
epsilon s l and if you get this overall friction factor as this, then you will see there will be data
like this here. So, this type of equation straight line you will get; now from this straight like
you will see here 1 by epsilon s l square; if you are considering then here f s p will be your
slope; f s p will be your slope and then f AT will be the intersection that will give you your f
AT..

So, from this ah analysis of this overall friction factor from the experimental data, you will be
able to calculate what should be the friction factor due to these additional turbulences.

(Refer Slide Time: 31:39)

Now, this additional turbulence actually depends on the velocity fluctuation inside the bed.
Now that velocity fluctuation maybe in the y direction, z direction or x direction; if your
fluidized bed in such a way that your y directional velocity fluctuations is the dominant one,
then this f AT due to this is dominant velocity fluctuation of this y direction, it will be

2
 uy 
f AT  2 
 us 

What is this u s? u s is nothing, but u s is nothing, but the solid velocity and here this is the
fluctuating velocity at the y direction.Now, for heterogeneous flow you see this u y dash that
is turbulent velocity or you can say velocity fluctuation due to this turbulence and it will be is
equal to
 1/3
1    g usl 
uy   gl  usg    g us  
3    sl  

This is actually in this heterogeneous flow. This fluctuation velocity will you obtain from the
energy balance.

So, this energy balance will give you this direct this fluctuating velocity now this see this
fluctuation velocity again is a function of a u s g; that means, superficial gas velocity and
superficial liquid velocity. Of course, there will be if it is slurry system; you have to have the
concentration of the liquid inside the bed. And also what would be the solid velocity inside
the bed; and then this l is called this mixing length inside the bed the within which range of
this fluctuation will be there and then it will be defined as this

l   dc

depends on this column diameter or there is bed diameter.

This lambda is a parameter which is the function of again this slurry velocity and then the
volume fraction of the gas and the gas velocity. So, from this correlation,

  0.17  0.19usl  0.24 g  0.015usg

you will be able to calculate this psi value and from which you will calculate this mixing
length inside the bed and from this mixing length you will be able to calculate what will be
the fluctuating velocity or friction velocity you can say sometimes it is called friction
velocity; sometimes it is called fluctuating velocity, sometimes it is a turbulent velocity.

And then, from this turbulent velocity and the solid velocity inside that bed, you will be able
to calculate what will be the friction factor due to this additional turbulence. Now, additional
turbulence if you calculate from this here; the overall friction factor and if you are messing
with these if it is unknown parameter here lambda, lambda you can directly obtain from this
overall friction factor.

And if you can correlate again with this liquid velocity or slurry velocity or gas velocity here,
you will be the deviation from this. If there is a significant deviation, then you have to modify
this model by just fitting the experimental data and having this coefficient different from
modification. So, in this way, you can analyze what should be the friction factor inside the
bed due to this fluctuating velocity and also additional turbulence.

(Refer Slide Time: 35:10)

Now, for homogenous flow this u dash y;

u y  1.5 g us

that means, here fluctuating velocity is directly related to this effective velocity of the solid.
So, it is seen that it is 1.5 times of this effective velocity of the solid. And this u s; this us
solid velocity will be the this is called sometimes slip velocity, and it will be represented as
this

usg u
us    sl
 g 1  g

This is the actual velocity and us l by 1 minus epsilon g is called actual liquid velocity..

So, from which you will be able to calculate this u s value or relative velocity or slip velocity;
this sleeve velocity will give you the fluctuating velocity in the homogenous mixture. And
then phi l square to be calculated here this two-phase friction factor to the single-phase
friction factor and this two-phase friction factor will be is equal to single-phase friction factor
and this friction factor due to the additional turbulence..
And then this is what this friction factor due to this single-phase or you can calculate this ah
multiplier of the single-phase slurry phase that is in terms of the concentration of the slurry
and the friction factor of this additional turbulence to this friction factor of a single-phase
flow inside the bed. And also, what should be the ratio of slip to the single-phase slurry
phase.

So, you can directly obtain this multiplier of this slurry by this correlation.

Pf ,tp Pf , sp  Pf , AT

l2  
Pf ,sp Pf ,sp

Another way to calculate this in terms of these fluctuating velocity like this here;

2
2 1 2  uy 
  2
sl  
 sl f sp  usl 

here one term is the u s by u s l instead of u s l by u s l here it will come only the fluctuating
velocity to the slurry velocity this is fluctuating velocity instead of u s here; so, you can
calculate this phi s l.

(Refer Slide Time: 37:21)

So, for more details, you can get this different frictional pressure drop to analyze and to
predict from your experimental data from these two books from these books here this Kunii
and O. Levenspiel books, you can follow. And also this Fluidization and Fluid Particle
Systems that is by Yang 2003, you can get more information about the different correlations
models given by me that is your Hydrodynamics and Transport Processes of Inverse Bubbly
Flow; here there it is the extensive actually of analysis extensive way that is represented of
different fictional pressure drop correlations that you can get more information from this
relationship.

So, we can obtain from this lecture that how to analyze the frictional pressure drop by
Lockhart-Martinelli model, how to analyze this frictional pressure drop in the fluidized bed
by Friedel model by different other models also. In this case, you have to take care of taking
the concept of this Lockhart-Martinelli, which is actually basically for two-phase flow. Here
we can apply these two-phase flow for three-phase operations only just by taking this liquid
and solid as one phase and gases is another phase because this Lockhart-Martinelli concept
basically was from the; flow of horizontal flow of gas and liquid.

So, here instead of liquid, you are considering this slurry of liquid and solid, whereas, in the
horizontal flow, we are considering the vertical flow. Of course, all the parameters, whatever
this Lockhart-Martinelli parameters, multiplication or multiplier Lockhart-Martinelli, you can
say or two-phase multiplier or three-phase multiplier and different investigators, they are
naming in different way. So, these parameters will be obtained from the experimental data of
three-phase bed.

Then once you know this three-phase flow and three-phase frictional pressure drop just by
analyzing with this concept of Lockhart-Martinelli model; you will be able to predict the
three-phase frictional pressure drop at different operating condition also. By this frictional
pressure drop correlations, you can scale the system of this fluidized bed ah with different
operating variables.

Of course, this variables or you can say parameters Lockhart-Martinelli parameters that is fix
that is the ratio of frictional pressure drop of that is slurry to the gas and ah, but the multiplier
that is obtained that you have to make a correlation in terms of different operating
parameters; it will be better just by regression analysis with the different of my experimental
data and making general correlation for this. And then you can apply it with different
operating conditions, even after without doing experiment you can calculate what should be
the frictional pressure drop by that model.
So, I think it will be helpful to calculate this two-phase and three-phase frictional pressure
drop by different models and ah. So, up to this we have a discussion this liquid-solid
frictional pressure drop, gas-solid frictional pressure drop, even gas-liquid-solid frictional
pressure drop in fluidized bed. So, next lecture onward, we will be discussing what should be
the actually mechanism of distribution of gas inside the bed that will be discussed, and how
this distributor actually play important role for different hydrodynamic parameters.

Thank you for this lecture.