Chapter 5
Mechanical Micro Processes In Fluid
Objectives
❖Develop an understanding of the forces resisting the motion of particle and
❖Provide methods for the estimation of the steady velocity of the particle relative to the fluid.
• Mechanics of particle motion
• The drag forces
• Flow of fluid through a granular beds, Packed beds and fluidized beds
MOTION OF PARTICLES IN A FLUID
▪ Many processing steps, especially mechanical separations, involve the
movement of solid particles or liquid drops through a fluid.
▪ Examples:
• the elimination of dust and fumes from air flue gas
• The removal of solids from liquid wastes
▪ The fluid may be flowing or at rest.
▪ The dynamics of a solid particle in a fluid depends on many parameters and the
inter-relation among these parameters is pretty complex
Let us therefore start with a simplified analysis of the phenomenon
making the following assumption:
• The shape and size of the particle are defined (if not mentioned take as
Spherical particle of diameter dp).
• The particle is Non pores and incompressible, insoluble in the fluid and does
not react chemically with it.
• Fluid has constant density and viscosity.
• Effect of interfacial condition between solid and fluid on the particle dynamics
is neglected.
• Free settling particle.
• Boundary effect ( wall effect ) neglected.
Mechanics of particles motion
▪ The movement of a particle through a fluid requires an external force acting on the
particle.
▪ This force may come from a density difference between the particle and the fluid or it
may be the result of electric or magnetic fields.
▪ Gravitational or centrifugal forces arise from density differences
Three forces act on a particle moving through a fluid
I. The external force (gravitational or centrifugal)
II. The buoyant force(which act parallel with the external force but in opposite direction)
III. The drag force (appears whenever there is relative motion between the particle and
the fluid)
Equations for one-dimensional motion of particle through fluid
▪ The drag force acts to oppose the motion and acts parallel with the direction of movement
but in the opposite direction.
• Equation for one dimensional motion of particle through fluid
𝑚𝑑𝑢
ΣF = 𝑑𝑡 = 𝐹𝑒 − 𝐹𝑏 − 𝐹𝐷 ………………… 5.1
Where
m is mass of particle moving through a fluid
Fe is external force
u is velocity of the particle relative to the fluid
Fb buoyant force
FD drag force
➢ The external force can be expressed as
𝐹𝑒 = 𝑚𝑎𝑒 ………………… 5.2
➢ The buoyant force is from Archimedes' principle (the product of mass of fluid displaced by the
particle and acceleration from the external force)
𝑚ρ𝑎𝑒
𝐹𝑏 = ……………………. 5.3
ρ𝑝
➢ The drag force is
𝐶𝐷 𝑢02 ρ𝐴𝑃
𝐹𝐷 = ……………………… 5.4
2
Where CD is dimension less drag coefficient
Ap is projected area of particle
uo =u
➢ Substituting the forces from the above equations gives
……….5.5
Motion from gravitational force
➢ If the externa force is gravity, ae is g
……………….. 5.6
Motion in centrifugal field
➢ The acceleration from a centrifugal force from circular motion is
……………… 5.7
Terminal velocity
➢ In gravitational settling g is constant and the drag always increases with velocity
➢ The acceleration decreases with time and approaches zero
➢ The particle reaches a constant velocity, which is the maximum attainable under the
circumstances, and which is called terminal velocity.
……………. 5.8
➢ In motion from centrifugal force, the velocity depends on the radius
➢ The acceleration is not constant if the particle is in motion with respect to the fluid.
➢ In many cases du/dt is small in comparison with the other two terms of eq. 5.7
➢ So du/dt neglected and the terminal velocity is given by
..………… 5.9
Drag Coefficient
▪ To use the above equations the numerical values of the drag coefficient CD must be
available
▪ Drag coefficient is a function of Reynolds number
▪ A different drag coefficient Vs Reynold number relation exists for different shape and
orientation.
Restrictions
• The particle must be a solid sphere,
• it must be far from other particles and from the vessel walls and
• it must moving at its terminal velocity with respect to the fluid.
Free settling
• The process when the particle is at sufficient distance from the boundary and from
other particles so that its fall is not affected.
• The particle concentration is less than 0.2 vol % in the solution.
Hindered settling
• If the motion the particle is impeded by other particles (which will happen when the
particles are near each other even though they may not be colliding) the process is
called hindered settling.
Figure drag coefficient for a spheres
Figure drag coefficient for different shape of particle
Motion of spherical particles
• If the particles are spheres of diameter Dp
Substitution of m and Ap gives the equation for gravity settling of spheres
..………… 5.10
• At low Reynold numbers, the drag coefficient varies inversely with NRe,p
𝐹𝐷 = 3𝜋𝜇𝑢𝑡 𝐷𝑃
24
𝐶𝐷 =
𝑁𝑅𝑒,𝑝
𝑔𝐷𝑃2 (𝜌𝑝 − 𝜌)
𝑢𝑡 =
18𝜇
• The last equation is known as Stokes’ law and it applies for particles Reynold number less than 1.0
• At NRe,p =1.0, CD =26.5 instead of 24 and since the terminal velocity depends on the square root of drag
coefficient
• For 1000< NRe,p <200,000 the drag coefficient is approximately constant and the equations are
𝐹𝐷 = 0.055𝜋𝐷𝑝2 𝑢𝑡2 𝜌
𝐶𝐷 = 0.44
𝑔𝐷𝑝 (𝜌𝑝 − 𝜌)
𝑢𝑡 = 1.75
𝜌
• The last equation is Newton’s law and applies only for fairly large particles falling in gases or
low viscosity fluids.
• The terminal velocity ut varies with Dp2 in stokes’ law range whereas the Newton’s law range
varies with Dp0.5
Criterion For Settling Regime
• To identify the range in which the motion of the particle lies, the velocity term is
eliminated from the Reynold number by substituting ut
• For stokes’ law range,
……..……..5.11
• If stokes’ law is to apply NRe,p must be less than 1.0
• To provide a convenient criterion K, let
……..……..5.12
• Then, from eq. 5.11 NRe,p =(1/18)K3
• Setting NRe,p =1.0 and solving gives K=181/3 =2.6
• If K is less than 2.6 stokes’ law applies
• For Newton’s law range NRe,p =1.75K1.5
• Setting this equal to 1000 and solving gives K=68.9
• Thus if 68.9<K<2360, Newton’s law applies
• For ranges out of stokes’ law and Newton’s law the terminal velocity is calculated by eq. 5.10
Hindered settling
• In hindered settling, the velocity gradients around each particle are affected by the presence
of nearby particles, so the normal drag correlations do not apply.
• The particle in settling displace liquid, which flows upward and makes the particle velocity
relative to the fluid greater than the absolute settling velocity
• For a uniform suspension, the settling velocity us can be estimated from the terminal
velocity for an isolated particle using the empirical equation of Maude and Whitmore
• Exponent n changes from about 4.6 in stokes’ law range to about 2.5 in the Newton’s law
region.
• For very small particles, the calculated ratio us/ut is 0.62 for ɛ=0.9 and 0.0095 forɛ=0.6
• For large particles the corresponding ratios are us/ut =0.77 and 0.28
• The viscosity of a suspension is also affected by the presence of the dispersed phase.
• For suspensions of free flowing solid particles, the effective viscosity µs may be estimated
from the relation
• …………5.13
• Equation 5.13 applies only when ɛ>0.6 and is most accurate when ɛ>0.9
Flow through fixed and fluidized bed
• The use of packed and porous media is frequently encountered in process industries.
• Packed beds are used for
❖ Waste water treatment
❖ gas-liquid absorption,
❖ distillation/ fractionation of liquid mixtures,
❖ liquid-liquid extraction and
❖ catalytic/noncatalytic reactions
Fluidization
• Fluidized beds are used widely in chemical processing industries for separations, rapid
mass and heat transfer operations, and catalytic reactions.
• when a liquid or gas is passed at very low velocity up through a bed of solid particles, the
particles do not move and pressure drop is given by Ergun Equation
• If the fluid velocity is steadily increased, the pressure drop and the drag on individual
particles increase, and eventually the particles start to move and becomes suspended in the
fluid.
Conditions for fluidization
• Consider a vertical tube partly filled with a fine granular material
• The tube is open at the top and has a porous plate at the bottom to support the bed and
distribute the flow uniformly over the entire cross section
• If the particles are quite small, flow in the channels between the particles will be laminar
and the pressure drop across the bed will be proportional to the superficial velocity 𝑉ത𝑜
Figure pressure drop and bed height vs. superficial velocity for a bed of solids
• As the velocity is gradually increased, the pressure drop increases but the particles do not
move and the bed height remains the same
• At certain velocity, the pressure drop across counterbalance the force of gravity on the
particles or the weight of the bed
• Any further increase in velocity causes the particles to move (point A )
• Some times the bed expands slightly with the grains in contact, since just a slight increase
in ɛ can offset an increase of several percent in 𝑉ത𝑜 and Δp constant.
• With further increase in velocity, the particle become separated enough to move about in
the bed, and true fluidization begins (point B)
• Once the bed is fluidized, the pressure drop across the bed stays constant, but the bed
height continues to increasing flow.
• If the flow rate to the fluidized bed is gradually reduced, the pressure drop remains
constant, and the bed height decreases, following the line BC.
• However, the final bed height may be greater than the initial value for the fixed bed (solid
dumped in a tube tend to pack more tightly than solids slowly settling from a fluidized
state)
• On starting up again, the pressure drop offsets the weight of the bed at point B, and this
point rather than point A, should be considered to give the minimum fluidization velocity,
𝑉ത𝑜 M.
Minimum fluidization velocity
• The minimum velocity at which a bed of particles fluidizes, is a crucial parameter needed
for the design of any fluidization operation.
• The details of the minimum velocity depend upon a number of factors, including the shape,
size, density, and polydispersity of the particles.
• The density, for example, directly alters the net gravitational force acting on the particle,
and hence the minimum drag force, or velocity, needed to lift a particle.
• The shape alters not only the relationship between the drag force and velocity, but also the
packing properties of the fixed bed and the associated void spaces and velocity of fluid
through them.
• An equation for the minimum fluidization velocity can be obtained by settling the pressure
drop across the bed equal to weight of the bed per unit area of cross section, allowing the
buoyant force of the displaced fluid:
∆𝑃 = 𝑔 1 − 𝜀 𝜌𝑃 − 𝜌 𝐿 …………. 5.14
• At incipient fluidization, ɛ is the minimum porosity ɛM thus
∆𝑃
= 𝑔 1 − 𝜀𝑀 𝜌𝑃 − 𝜌 …………. 5.15
𝐿
• The Ergun equation for pressure drop in packed beds can be rearranged to
…………. 5.16
• Appling eq. 5.16 to the point of incipient fluidization gives a quadratic equation for the
minimum fluidization velocity 𝑉ത𝑜 M
…………. 5.17
• For very small particles, only the laminar flow term of the Ergun equation is significant
• With NRe,p <1, the equation for minimum fluidization velocity becomes
…………. 5.18
• In many applications of fluidization, the particles are in the range 30 to 300µm.
• However, fluidization is also used for particles larger than 1mm
• In the limit of the laminar flow terms becomes negligible, and 𝑉ത𝑜 M varies with the square
root of the particle size
• For NRe,p >103
…………. 5.18
• For low Reynold number, ut and 𝑉ത𝑜 M both varies with Dp2, (ρp - ρ) and 1/µ
so the ratio ut/ 𝑉ത𝑜 M depends mainly on the void fraction at minimum fluidization
…………..5.19
• For large particles (NRe,p >103)
…………..5.20
Types of fluidization
• Beyond 𝑉ത𝑜 M the appearance of beds fluidized with liquids or gases is often quite different
Fixed Bed
• For sufficiently low rates of flow, fluid passes through the void space between particles
without disturbing them. This case where the bed of particles remains in place is referred to
as a “fixed bed”.
Particulate fluidization
• When fluidizing sand with water, the particles move farther apart and their motion becomes
more vigorous as the velocity is increased, but the average bed density at a given velocity is
the same in all sections of the bed.
• Characterized by a large but uniform expansion of the bed at high velocities.
Figure : The regimes of fluidization as a function of the fluid velocity.
• Fluidization phenomenon may also be illustrate by plotting pressure drop vs. the superficial
velocity
✓ The zone OA represents the static bed
porosity or void fraction ɛ of the bed
remain constant even if fluid velocity
increases.
✓ At point A, the bed becomes unstable and
a minor movement and readjustment of particles begin
✓ Until point B instability of the bed continues as the fluid velocity increases point B is the
point of minimum or incipient fluidization
✓ From point B, the bed begins to expand with increasing fluid velocities (the porosity of the
bed increase)
✓ At point C fluidization is complete and all the particles are in motion. From point C to D
pressure drop remains constant
✓ At point D, the fluid velocity is large that it becomes equal to the terminal free settling
velocity of the particle. As a result the particle get carried by the fluid to outside the
column.
✓ This entrainment of solid particles in the outgoing fluid starts at D and get complete at E,
when all the particles are being carried off by the fluid and the porosity of the bed becomes
unity.
✓ The column now accts as conveyor
Particulate fluidization
• For particulate fluidization the expansion is uniform, and Ergun equation, which applies to
fixed bed
• Assume the flow between the particles is laminar
…………….5.21
• The expanded bed height obtained from ɛ and the values of L and ɛ at incipient
fluidization using
…………….5.22
Figure bed expansion for particulate fluidization
• For particulate fluidization of large particles, in water, the expansion of the bed is expected
to be greater than that corresponding to eq. 5.21 since the pressure drop depends partly on
kinetic energy of the fluid .
• The expansion data can be correlated by the empirical equation
…………….5.23
• To predict the bed expansion, m is estimated using the Reynold number at the minimum
fluidization velocity
Figure variation of porosity with fluid velocity in a fluidized bed
Bubbling fluidization
• For bubbling fluidization, the expansion of the bed comes mainly from the space occupied
by gas bubbles
• In the following derivation, the gas flow through the dense phase is assumed to be 𝑉ത𝑜 M
times the fraction of the bed occupied by the dense phase, and the rest of the gas flow is to
be carried by the bubbles
…………….5.24
• Where fb = fraction of bed occupied by bubbles
ub = average bubble velocity
• Since all of the solid is in the dense phase, the height of the expanded bed times the
fraction dense phase must equal to the bed height at incipient fluidization.
LM = L(1- fb) …………. 5.25
• Using eq. 5.24 and 5.25
…………. 5.26
• An empirical equation for bubble velocity in fluidized bed is
…………….2.27
