3.
For similarity between geometrically similar flows (assuming large Re), we have the following pump
scaling laws:
1
Q
Q
3
D
2
3
D
1
gH
(11.23)
2
gH
(11.24)
1
2
2
D
2
2
D
2
1
2
W
(11.25)
W
1
3 D 5
3 D 5
2
1
1
2
(since 1 = 2, 1 = 2

Now lets examine the turbulent core region. One simple model, known as Prandtls Mixing Length
Hypothesis, assumes that the fluctuating velocities, e.g. u and v, are approximately equal to some typical
eddy length, l, multiplied by the velocity gradient, d

6.
The turbulent kinetic energy, K, is defined as the kinetic energy of the normal turbulent fluctuations:
1
K 2 uiui
(9.24)
The turbulent kinetic energy is used in models that examine the balance of energy associated with
turbulent motion (beyond the sco

1.
Entrance Region
inviscid core
pipe diameter, D
entrance
fully developed flow
length, L
The shaded regions are where viscous stresses are
important (the boundary layer).
The flow in the entrance region is complex and will not be investigated here. Exper

Now lets examine the Navier-Stokes equation in the z-direction:
u
u
u
u
z
z
u
z
1
u
1
p
u
u
z
u
r
z z
r
r r
z
r
z
r
2
z
(10.7)
f
t
z
r
u
2
2
r
z 2
z
2
We can simplify this equation using our assumptions:
u
u
u
z
u
1
u
1
p
u
z
u
u
z
z

3.
We can determine stresses using the constitutive relations for a Newtonian fluid. The shear stress that the pipe
walls apply to the fluid, w, is:
(10.13)
u
R dp 4
w
2 dz
R
where u is the average velocity in the pipe. Note that an alternate method for

3.
Fully Developed Turbulent Circular Pipe Flow
Turbulent Flow in a Smooth (but Frictional) Pipe
The volumetric flow rate in a smooth pipe for turbulent flow may be estimated by integrating the time
averaged velocity profile, modeled using the Law of the

Turbulent Flow in a Very Rough Pipe
The roughness of the pipe walls can significantly affect the friction factor for turbulent flows (roughness
has a negligible effect on the friction factor for laminar flows). Recall from the Law of the Wall that the
tim

Notes:
1.
For Reynolds numbers less than 2300, one may use either the analytical expression for the friction
factor:
f
D
64
Re
D
or the Moody chart.
2.
Reynolds numbers between approximately 2300 and 4000 correspond to the transitional regime
between lam

Notes:
1.
Experimental curve fits indicate that: K 0.41 and c = 5.0.
2. Comparing laminar and turbulent pipe flows we observe that the turbulent velocity profile (a
logarithmic curve) is blunter than the laminar profile (a parabolic curve).
laminar
turbul

The vorticity transport equation is an alternate expression of the Navier-Stokes equations. Consider the
Navier-Stokes equations for an incompressible fluid with constant dynamic viscosity:
Du
= p + 2 u + f
Dt
Divide through by the density, , (note that i

Example:
1.
5
Using the Moody chart, determine the friction factor for a Reynolds number of 10 and a relative
roughness of 0.001.
2.
What is the friction factor for a Reynolds number of 1000?
3.
What is the friction factor for a Reynolds number of 10 in a

u
=0
t
Taking the dot product of Eqn. (4.131) with a little length of line, dx, that is along either a streamline or
a vortex line gives:
dp
+ 1 2 u u + G dx = ( u ) dx
(4.132)
Since the vector (u) is perpendicular to both the streamline and the vortex l

1.
Introduction to Turbulence
Lets consider the following simple experiment (this thought experiment is similar to the famous dye
injection experiment performed by Osbourne Reynolds). At a particular point in a pipe flow, lets measure
a velocity component

2.
Time-Averaged Continuity and Navier-Stokes Equations
Since the time-varying velocity data shown in Section 1s example appears to consist of a fluctuating part
superimposed on a mean value, lets make the following definitions. First, express the instant

Now lets take the same approach with the Navier-Stokes equations:
2
i
(9.11)
ui p ui f
u
u
k
t
x
k
x
x 2
i
i
k
To help with the upcoming analysis, lets re-write the left-hand-side using the continuity equation:
2
u
u
i
u
u
u
i
k
t
p
k
u
i
i
x
i
x
k
(

If we compare Eqn. (9.18) to the instantaneous Navier-Stokes equation, we see that an extra term appears
on the right hand side with the same dimensions as the laminar shear stress term:
(9.19)
uu
i k
xk
These terms are referred to as Reynolds stresses (i

3.
Near a wall, the Reynolds shear stresses are small due to the wall restricting the random motion of the fluid.
This region is termed the viscous sub-layer and
(9.21)
viscous sub-layer
laminar
Far from the wall, turbulent motion dominates. This region i

5.
Other Losses
The loss due to the viscous resistance caused by the pipe walls is referred to as a major loss. Pressure
losses may occur due to viscous dissipation resulting from fluid interactions with other parts of a pipe
system such as valves, bends,

Elementary Pump Theory
To determine the work that the pump does on the fluid passing through it, well use the Moment of
Momentum Equation which relates torque to momentum fluxes.
Consider flow through the following rotating pump:
Vrb2
front view of pump i

Substituting and noting that U2 = r2gives:
2
r Q cot
r
H
2
added to
2
2
CV
g 2 r b
g
2
2
2
r
H
2
added
to CV
(11.9)
cot 2 Q
g 2 b
g
2
theoretical head rise across an idealized centrifugal pump
Notes:
a.
Equation (11.9) is an equation of a line.
H
2 > 90

Example:
An idealized centrifugal water pump is shown below. The volumetric flow rate through the pump is 0.25
3
ft /s and the angular speed of the impeller is 960 rpm. Calculate the power required to drive the pump.
55
960 rpm
11 in.
3 in.
0.75 in.
3
0.2

Example:
Determine the NPSHA for the following system
P
H
SOLUTION:
Choose point 1 to be on the surface of the tank and point 2 to be just upstream of the pump.
Apply the EBE from 1 to 2:
p
V
p
V
2
2
g
2g
z
z H L12 H S 12
2 g
2g
1
p1 patm
where
p2 ps

2.
Net Positive Suction Head (NPSH)
Along the suction side of the impeller blade near the pump inlet are regions of low pressure.
suction side, region
rotation
blade
of low pressure
direction
fluid velocity at inlet
relative to blade
If the local pressure

where
r u dV
dt
0 (steady flow)
CV
r u u
rel
dA r2 e r V2 sin 2 e r cos 2 e r2 e V2 sin 2 2 r2 b2
CS
velocity of fluid
exit area
exiting fluid velocity w/r/t ground
exiting the CV
r 2 r V
2
2 2
cos
2
2
V
sin 2 r b e
2
2 2 z
(Note that there is no

The problem could have also been worked out using velocity polygons. Since the absolute inlet velocity
has no tangential component:
H
U2Vt
where
2
r2
U
2
g
and
W
m gH
m g r2Vt 2
W
g
Use a velocity polygon at the exit to determine Vt2.
Vrb2
V2
Vrb2

3.
Pump Similarity
Most pump performance data (H-Q curves) are given only for one value of the pump rotational speed and
one pump impeller diameter. Is there some way to determine the pump performance data for other speeds
and diameters without requiring

Notes:
1.
The power required to drive the impeller is:
Won CV Ton CV
Won CV m 2 t2 r V rV
1
t1
but U1 = r1 and U2 = r2 so that:
Won CV
t1
t2
1
m2U V U V
In terms of the head added to the fluid:
H
Won CV
U2Vt 2
(shaft head)
(11.4)
added
to CV
U1Vt1
m g