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Electrical Analogy of Fluid Flow
We have realized that for incompressible, laminar pipe flows through pipes, the
volumetric flow rate is a function given by:
R 2  dp 
Q
 
8  dz 
dp
 0 along the z-direction, pressure must drop along the length of the pipe.
dz
Over a length, L, of the pipe
Since
dp p 2  p1

dz
L
(2)
(1)
L
R 2 p1  p 2
 Q

8
L
Now, imagine that this above pressure drop (p1 - p2) is the driving force for the flow
through the pipe of length L. This is similar to the potential difference or voltage in
an electrical circuit. We know that applying a voltage, V, across two points results in
a current flow, i, given by the Ohm’s law:
i
V  i R el
or
R el 
V
i
(2)
(1)
V
where Rel = Electrical resistance of the
circuit
For the corresponding fluid flow analysis, replace V by (p1 - p2), i by the volumetric
flow rate Q, and Rel by Rfl:
 R fl 
p1  p 2 8L 128L


Q
R 2
D 2
where D = 2R = Pipe Diameter
This Rfl is called the resistance in fluid flows. This is frictional in nature and it
depends on the fluid viscosity and geometrical parameters like length and diameter
of the pipe. The frictional resistance results in frictional losses of pressure head.
This analogy is an important one to understand the concept of “Head Loss” in fluid
flows, which is defined more formally below. For now, we notice that:
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p1  p 2  Q 
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128L
D 2
 D 2 
Since Q  VA  V 
 4 


(A = Pipe cross sectional area)
 p1  p 2  V32L
or,
p1  p 2
64 V 2 L
64 L V 2


 
 
VD 2 D Re D D 2


[ Re D 
 VD
]

The above expression shows that the loss of flow energy per unit mass between
points (1) and (2) in a pipe flow may be expressed as:
p1  p 2
L V2
f  

D 2
where f = friction factor =
64
for laminar flows.
Re D
The above expression is called the Darcy-Weisbach formula. It relates the loss of
L
pressure head to the friction factor,
, and the average velocity of the flow. A more
D
general form of this formula is derived from the Engineering Bernoulli equation
below. The plot of f vs. ReD shown below is called the Moody Diagram and is plotted
for a large range of Reynolds numbers.
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Moody Diagram
As we can see from the plot the far left region corresponds to laminar flow, where
the friction factor is given by f = 64/ReD derived before. It turns out that for
turbulent flows f is not only dependent on ReD, but a roughness factor  also. The
roughness is measured as a protruding length from the smooth wall of the pipe.
Thus “relative roughness,”  , which is a parameter in the Moody Diagram for
D
different pipe materials in turbulent flows, is a non-dimensional quantity. For
“smooth pipes” the relative roughness is considered to be zero.
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