UNIT II

The law of conservation of mass. According to the law of

conservation of mass for a steady flow system, the rate of mass
entering the flow system is equal to that leaving as accumulation
is either constant or nil in the flow system under steady
conditions.

[In a steady state system the values of the quantity and variables
of the system do not change with time].

Consider a flow system (a stream tube of varying cross-section)

as shown
As the flow cannot take place across the walls of the stream tube,
the rate of mass entering the stream tube must be equal to that
leaving. Let u1 , ρ1 and A1 be the average velocity of the fluid, the
density of the fluid and cross-section area of the tube at the
entrance, and let u2 , ρ2 and A2 be the corresponding quantities at
the exit of the tube. Assume that the flow to be potential flow and
the density to be constant in a single cross-section.
Rate of mass entering the flow system = ρ1 u1 A1

Rate of mass leaving the flow system = ρ2 u2 A2

Let ·m be the rate of flow in mass per unit time (mass flow rate
of the flowing fluid).
Under steady flow conditions,

m = ρ1 u1 A1 = ρ2 u2 A2
it follows for a stream tube,
m = ρ u A = constant

is known as the equation of continuity. It is applicable to

compressible as well as to incompressible fluids. In the case of
incompressible fluids, ρ1 = ρ2 = ρ.
 Average Velocity
When the flow is not potential flow, the local velocity of fluid will
vary from point to point within a given single cross-section of the
conduit (tube/pipe). The average velocity (u) of the entire fluid
stream flowing through the cross-sectional area A is defined by

u = ·m /(ρ.A)

If ·m is expressed in kg/s, ρ in kg/m3 and A in m2 , then u will be

having the units of m/s. The average velocity is also equal to the
ratio of the volumetric flow rate to the crosssectional area of the
conduit.
where Q is the volumetric flow rate. If Q is expressed in m 3 /s
and A in m2 , then u will have the units of m/s.
 The mass flow rate through the conduit is related to the
volumetric flow rate by the relation

·m =Q.ρ
where ρ is the density of the flowing fluid.

Mass Velocity
In the case of flow of compressible fluids, it is a usual practice to
use the mass velocity. The mass velocity does not depend upon
temperature and pressure when the flow is steady. The mass
velocity of a fluid is the ratio of the mass flow rate of the fluid to
the crosssectional area of the conduit.

Mass velocity = G = . m/ A
If ·m is expressed in kg/s and A in m2 , then G will have the units
of kg/(m2.s)

Stream Line and Stream Tube

A stream tube is a tube of small or large cross-section which is
entirely bounded by stream lines. It may be of any convenient
cross-sectional shape and no net flow occurs through the walls of
the stream tube.
 BERNOULLI EQUATION
the Bernoulli equation without friction can be derived on the
basis of Newton's second law of motion (force is equal to the
rate of change of momentum) for potential flow.

Let us consider an element of length ∆L of a stream tube of

constant cross-sectional area as shown

Let us assume that the cross-sectional area of element be A and

the density of the fluid be ρ. Let u and P be the velocity and
pressure at the entrance (upstream), and u + ∆u, P + ∆P are the
corresponding quantities at the exit (downstream).
The forces acting on the element (treating the element as a free
body) are

The force from the upstream pressure = PA

The force from the downstream pressure normal to the cross-
section of the tube = (P + ∆P) A

The force from the weight of fluid [i.e., the force of gravity
acting downward (taken as negative)] = ρA ∆L.g

The component of this force acting opposite to the direction of

flow is ρ A∆L g cos θ.

Rate of change of momentum of the fluid along the fluid

element
 = ρ u A ∆u
According to the Newton's second law of motion,

Sum of all forces

acting in the direction of flow = Rate of change of momentum
of a fluid

PA – (P + ∆P) A – ρ A ∆L g cos θ = ρ . u.A ∆u

– ∆P.A – ρ A ∆L g cos θ = ρ . u.A ∆u

∆P.A + ρA ∆L g cos θ + ρ.u.A ∆u = 0

∆P/ ρ∆L + g cos θ + u.∆u/ ∆L = 0

cos θ = ∆Z/∆L

1 /ρ ∆P/ ∆L + g ∆Z /∆L + u ∆u/ ∆L = 0

dP /ρ + g dZ + d (u2 /2) = 0

P /ρ + g Z + u2/ 2 = constant
 Kinetic Energy Correction
In the previous discussion, it is assumed that the velocity u to be
constant over the area A. But in actual practice, the velocity varies
over a single cross section. The velocity of the fluid is zero at the
wall surface and maximum at the centre of the pipe. Hence,
allowance must be made for the velocity profile in the kinetic
energy term. This can be done by introducing a correction factor α
into the kinetic energy term.

The kinetic energy term would be written as αu 2 /2 . For the

flow of a fluid through a circular cross-section, α = 2 for laminar
flow and α = 1 for turbulent flow.
 Correction for Fluid Friction
The Bernoulli equation is derived for the frictionless fluid.
Therefore, it must be corrected for the existence of the fluid
friction whenever boundary layer forms. Fluid friction is an
irreversible conversion of mechanical energy into heat. Thus, the
quantity P/ρ + u2 /2 + gZ is not constant but always decreases in
the direction of flow.

The Bernoulli equation for incompressible fluids is corrected for

friction by adding a friction term on the R.H.S.

P1/ ρ + g Z1 + α1 u12 / 2 = P2 /ρ + g Z2 + α2 u22 / 2 + hf

Each term involved in the Bernoulli equation has the units of

J/kg.
 PUMP WORK IN BERNOULLI EQUATION
Assume that a pump is installed in the flow system between the
stations 1 and 2 as shown

Let Wp be the work done by the pump per unit mass of fluid.

Let hfp be the total friction in the pump per unit mass of fluid

The net mechanical energy delivered to the flowing fluid is the

difference between the mechanical energy supplied to the pump
and frictional losses within the pump. i.e., Wp – hfp . But to
obtain the net mechanical energy (net work) delivered to the
fluid, instead of using hfp, a pump efficiency designated by the
symbol η is used. It is defined as
 Wp – hfp = η Wp

η = (Wp – hfp)/ Wp

Since η is always less than one, the mechanical energy delivered to

the fluid (ηWp ) is less than the work done by the pump. The
Bernoulli equation corrected for the pump work between stations 1
and 2 is thus given by

P1/ ρ + g Z1 + α1 u12 / 2 + η Wp = P2 /ρ + g Z2 + α2 u22 / 2 + hf

 FLOW OF INCOMPRESSIBLE FLUIDS IN PIPES
Shear-stress distribution in a cylindrical tube
Consider the steady flow of a fluid through a horizontal tube.
Imagine a disk-shaped element of fluid, of radius r and length dL,
concentric with the axis of the tube as shown. Let P and P + dP be
the fluid pressures on the upstream and downstream faces of the
disk respectively. Assume fully developed flow and the density of
fluid to be constant.
The sum of all the forces acting on this disk shaped element of
fluid is equal to zero as the flow is fully developed.

Pressure force on the

= Pπr2
Upstream face of the disk

Pressure force on the

= (P + dP) π r2
Downstream face of the disk

Shear force opposing flow

acting at the outer surface of the = shear stress × cylindrical area
disk due to the viscosity of fluid

= τ . (2πr dL)
 ∑F=0
+ π r2P – π r2 (P + dP) – 2π r dL τ = 0
– π r2 dP – 2π r dL τ = 0
πr2 dP + 2πr dL τ = 0

Dividing by πr2 dL, we get

dP/dL +2τ/r= 0

In steady-state laminar or turbulent flow, the pressure at any given

cross-section of a stream tube is constant, therefore dP/dL is
independent of r. For the entire cross-section of the tube, Equation
can be written by taking τ = τw at r = rw, where τw is the shear stress
at the wall of the tube and rw is the radius of the tube.

dP/dL +2τw/r w= 0
 τ w/rw = τ/r

Relation between Skin friction and Wall shear

The Bernoullis equation can be written over a definite length, ∆L
for the complete stream. Let P1 be the pressure at the upstream of
length ∆L and P2 be the pressure at the downstream of length ∆L
such that P1 > P2 , so that ∆P is the pressure drop over the length
∆L.
The term ∆P is commonly used for pressure drop, i.e., P1 – P2
(inlet/upstream pressure –outlet/downstream pressure) .
If P1 = P, then P2 = P – ∆P
In this case, Z1 = Z2 = 0 and u1 = u2. The friction that exists in
this case is the skin friction between the wall and the fluid
stream (hfs). Skin friction is the tangential friction associated
with a fluid flowing over a smooth surface. it is only the skin
friction that exists.

Then, the Bernoulli equation over the length ∆L becomes

P/ρ =(P – ∆P)/ρ + hfs

hfs =∆P/ρ

each term has the units of J/kg.

The Bernoulli equation, over the length ∆L, when each term in it is
expressed as the head (in m of flowing fluid) is

P/ρg =(P – ∆P)/ρ g +hfs/g

hfs/g =∆P/ρg

hfs/g represents the head loss due to friction.

Head loss due to friction = h’fs =hfs/g =∆P/ρ g

(frictional head loss)
eliminating ∆P
hfs =((2 τw)/(ρrw ))∆L =((4 τw)/(ρ D) )∆L

where D is the diameter of the pipe

 THE FANNING FRICTION FACTOR (f)
It is especially useful in the study of turbulent flow. It is defined as
the ratio of the wall shear stress to the product of the kinetic
energy of fluid and the density.
f =(τw)/(ρ.u2/2) =(2 τw)/ρu2
τw = f ρ u2/2
Substituting the value of τw from Equation
hfs =(4 f ρ u2 ∆L)/2 ρ D

h’fs =(4 f ρ u2 ∆L)/2 D

hfs =∆P/ρ
 hfs =∆P/ρ =(4 f L u2)/2 D

Whenever we have to calculate the head loss due to friction, then

Equation modifies to :

h'fs =hfs/g =∆P/ρ g =(4 f L u2)/2 g D

∆P =(4 f ρ L u2)/2 D

The pressure drop due to friction in a pipe for turbulent flow

can be calculated from Equation and is known as the Fanning
equation.
 LAMINAR FLOW IN CIRCULAR PIPE
The velocity distribution for Newtonian fluids can be obtained
through the definition of viscosity. Equation can be rewritten as

 
du
dr
The negative sign in the above equation is incorporated to take into
account the fact that in a pipe, u (velocity) decreases as r (radius)
increases.
Rearranging Equation
du 

dr 
Substituting the value of τ from
 du w
 .r
dr rw 

w
du  .rdr
rw 
Integrating Equation with the boundary condition : At r = rw : u =
0, we get
u r
w

0
du  .rdr
rw  r w
 w
u .rw  r 
2 2

2rw 
The maximum value of the local velocity (umax) is located at the
centre of the pipe. At the centre of the pipe, r = 0 and u = umax .
Thus, from Equation, we get

 w .rw
umax 
2
Substituting for τw as
 p.rw
w 
2L
 w .rw
umax 
2

p.rw2
umax 
4 L

p.D 2
umax 
16 L
Dividing Equations

w
u .rw2  r 2 
2rw 

 w .rw
umax 
2

u   r 
2

 1    
umax   rw  
 
The average velocity u of the entire stream flowing through any
given cross-sectional area (A) is defined by

1
ua  u.dA
A
w
u .rw2  r 2 
2rw 

As A = π r2w and dA = 2 π r dr = area of elementary ring of radius

r and width dr
 rw
w
ua  3 .rw2  r 2 .r.dr
rw  0

 w .rw
ua 
4
ua
0.5
umax
Eliminating τw by replacing it by ∆P
 p.rw
w 
2L
p.D 2
ua 
32L

32Lua .
p 
D2

32 Lua .
p  2
D
Equation is the Hagen-Poiseuille equation. The Hagen-Poiseuille
equation is useful to determine experimentally the viscosity of a
fluid by measuring the pressure drop and the volumetric flow rate
through a tube of a given length and diameter. This equation
also useful for the calculation of the pressure drop due to friction in
laminar/viscous flow if the viscosity is known.
The head loss due to friction in laminar flow is given by

' h fs
p 32 Lua .
h 
fs  
g  .g  .g .D 2

We know that
  w .rw
ua 
4
 w .D
ua 
8
Rearranging Equation

8ua
w 
D
Substituting the value of τw from Equation fanning friction
 2  8u 
f  2  
 .u  D 

  
f 16 
 D. .u 

16
f 
 Du 
 
  
16
f 
N Re
 TURBULENT FLOW IN PIPES
The turbulent flow is characterised by the presence of eddies (eddy
- a circular movement of, say, water causing a small whirlpool).
The eddies are of various sizes and they coexist. The
fluctuations/oscillations are present in all the three directions. The
fluctuations cause the mixing of different fluid portions by lateral
movement. The turbulent flow is of a great importance because it
brings about mixing of the fluid elements which results in higher
rates of heat and mass transfer.

Fanning friction factor as a function of the Reynolds number for

turbulent flow through a smooth pipe is given by

1
f

4 log N Re . f  0.4
Equation (7.117) is known as the Nikuradse equation. This
equation requires a trial and error procedure for estimating the
friction factor.
The other empirical relation that can be used for estimating the
friction factor from the Reynolds number for turbulent flow is

0.078
f 
N Re 0.25
 EFFECT OF ROUGHNESS
Our foregoing discussion was restricted to smooth conduits. In
turbulent flow, a rough pipe results in a larger friction factor than
that with a smooth pipe, for a given Reynolds number. If a rough
pipe is made smooth, the friction factor is reduced and ultimately a
stage will come when further smoothening of the pipe does not
reduce the friction factor for a given Reynolds number. The pipe is
then said to be hydraulically smooth.

A roughness parameter (k) is a length representing the magnitude

of surface roughness. Relative roughness is the ratio of the
roughness parameter to the diameter of pipe, i.e., k/D. The
friction factor dependency on the surface roughness is given
through the relative roughness. For low values of Reynolds
number (NRe < 2100), the friction factor is independent of the
surface roughness. For large values of Reynolds
number (NRe > 2500), the friction factor is also a function of the
surface roughness. At very large values of Reynolds number, the
friction factor is independent of NRe but is a function of the surface
roughness.
 FRICTION LOSSES
When a fluid is flowing through a straight pipe, only skin friction
exists and so far we have considered only this kind of friction.
Whenever there are disturbances in a fluid flow path due to a
change in the direction of flow or a change in the size of the pipe or
due to the presence of fittings and valves, friction is generated in
addition to the skin friction, i.e., additional friction losses occur.
When the cross-section of the pipe changes gradually to a new
cross-section or when there is a gradual change in the direction of
flow, disturbance to the normal flow pattern can be small and the
amount of mechanical energy loss as friction is negligible.
If the change is sudden, an appreciable amount of mechanical
energy is lost as friction (heat). Similarly, the presence of fittings
and valves disturbs the normal flow pattern and this can cause
friction losses. All these losses should be included in the friction
term, hf , of the Bernoulli equation (Equation 7.66).
Now, we will deal with the friction losses occurring as a result of
a sudden enlargement or contraction of the cross-section of the
pipe and the losses due to fittings.
 Friction Loss from Sudden Enlargement
If the cross-section of the pipe is suddenly enlarged (increased),
as shown in Fig. 7.18, there is a friction loss (mechanical energy
loss) due to eddies which are greater at the point of sudden
enlargement than in the straight pipe. Here the fluid stream
separates from the wall of the pipe and issues in the enlarged
section as a jet, which then expands and consequently fills the
entire cross-section of the larger pipe. The space between the
expanding jet and the pipe wall is filled with eddies (fluid
vortices) and a large amount of friction is generated there.

The friction head loss from a sudden enlargement (expansion)

of the cross-section of the pipe for turbulent flow is
proportional to the velocity head of the fluid in the smaller pipe
and is given by
 u12
h fe K e
2. g

where Ke is a proportionality constant and known as the

expansion-loss coefficient and u1 is the average velocity of fluid in
the smaller pipe/upstream pipe.
The expansion-loss coefficient is calculated by the following
relation :
2
 A1 
K e  1  
 A2 
where A1 is the cross-sectional area of the smaller pipe and A2
is the cross-sectional area of the larger pipe.
 Friction Loss from Sudden Contraction
When the cross-section of the pipe is suddenly reduced, as shown
in Fig. 7.19, the fluid stream gets separated from the wall of the
pipe and a fluid jet is formed. This jet first contracts, at a short
distance from the sudden contraction, in the smaller pipe and then
expands to fill the entire cross-section of the smaller pipe.

The cross-section of minimum effective flow area at which the

fluid jet changes from a contraction to an expansion is known as
the vena contracta. Upto the vena contracta, the fluid is
accelerated and losses are very small. But beyond it, the velocity
decreases as the flow area increases and conditions are
equivalent to those for a sudden expansion. The space between
the wall and the jet is filled with eddies (as shown in Fig. 7.19).
 The friction loss or frictional head loss from a sudden contraction
of the cross-section of the pipe is proportional to the velocity head
of the fluid in the smaller diameter pipe. It is given by the relation
u22
h fc  K c
2.g
where Kc is a proportionality constant and is known as the
contraction-loss coefficient, u2 is the average velocity in the
smaller or downstream pipe.
For turbulent flow, Kc is given by the following relation :
 A2 
K c 0.4 1  
 A1 
where A1 is the cross-sectional area of the larger pipe/upstream
pipe and A2 is the crosssectional area of the smaller
pipe/downstream pipe.
 Friction Losses in Fittings and Valves
Various types of fittings and valves are used in industrial piping
systems to change the direction of flow, for connecting pipes of
different diameters, etc. and valves are used to control the flow or to
stop the flow of a fluid. Fittings and valves disturb the normal flow-
line and cause friction and may lead to greater frictional loss than
that caused by the straight pipe. The frictional loss due to fittings
and valves may be given either in terms of the velocity head or in
terms of the equivalent length.
The frictional head loss in terms of the velocity head is given by
the following equation :

2
u
h ff  K f 1
2. g
 The frictional loss in J/kg is given by
2
u
h ff K 'f 1
2

where Kf is the loss factor for the fitting and u1 is the average
velocity of the fluid in the piping leading to the fitting.

hf = total friction loss in J/kg

2
 4 fL ' ' '  1u
 D  K c  K e  K f  2