HEAD LOSS DUE TO FRICTION IN CIRCULAR PIPE
The Darcy equation gives the head loss in turbulent flow for a circular pipe.
hf = 4 f l x c2
d 2g
Where:
f is the friction factor
l is the length of pipe being consider
d is the pipe diameter
c is the mean velocity of the fluid
The friction factor f depends on the velocity of flow, the pipe diameter, the fluid density and viscosity and the roughness of the pipe.
One mean of calculating the friction factor (for turbulent flow is smooth pipes) is to use the Blasius equation:
f = 0.079/Re 1/4
This is accurate within + 5% for smooth pipes at Reynolds numbers less than 100 000
In order to calculate losses for a larger range of Reynolds numbers and pipe roughness we need to use the Moody Chart. This is a graph of the friction factor f against Reynolds numbers for various values of k/d, where k is a measure of the wall roughness and d is the pipe diameter.
From the Moody chart we can see that there are 3 regions which can be considered when attempting to gain a value for f.
- Laminar flow – f = 16/Re (Up to Re = 2000)
- For pipe roughness k value less than 0.001, the Blasius equation can
d
f = 0.0079
Re1/4
3. For high k values or high Reynolds numbers we need to use the
d
appropriate k value and Reynolds number on the chart.
d
Example
Water with a coefficient of dynamic viscosity of 1.4 x 10-3 NS/m2 flows along a pipe 50mm diameter. If the pipe has an absolute roughness of 0.0000 7m, calculate:
1. The head lost due to friction per km of pipe
2. The power required to overcome friction per km of pipe
(a) For a flow rate of 3 litres/min
(b) For a flow rate of 40 litres/min
1(a) For a flow rate of 3 litres/min
Calculate the Reynolds number to determine whether the flow is laminar or turbulent
Q = _____3___ = 5 x 10 -5 m3/s
1000 x 60
C = Q
A
= 5 x 10 -5 x 4
π x 0.052
= 0.025 m/s
Re = c d
μ
= 1000 x 0.025 x 0.05
1.14 x 10-3
= 1096
This is less than 2000, therefore the flow is laminar, and we can use the equation
f = 16
Re
= 16 = 0.0146
1096
hf = 4 f l c2
2 g d
= 4 x 0.0146 x 1000 x 0.0252
2 x 9.81 x 0.05
= 0.037 m of water
2(a) Power to maintain flow = Q γ h
= 5 x 10-5 x (1000 x 9.81) X 0.037
= 0.018 W
1(b) For a flow rate of 40 litres/min
(i) Q = 40 = 6.67 x 10-4 m3/s
1000 x 60
C = Q = 6.67 x 10-4 x 4 = 0.34 m/s
A π x .052
Re = c d
μ
= 1000 x 0.34 x 0.05
1.14 x 10-3
= 14912
This is significantly greater than 2000, therefore the flow can be assumed turbulent. Therefore we can use a combination of the Moody Chart and Darcy equation.
Using the Moody Chart, for Re = 14912 and relative roughness =
0.00007 = 0.0014
0.05
f = 0.0078
hf = 4 f l c2
2 g d
= 4 x 0.0078 x 1000 x 0.342
2 x 9.81 x 0.05
= 3.68m
2(b) = Q γ h
= 6.67 x 10-4 x (1000 x 9.81) x 3.68
= 24.1 W
headloss/docs/sci/DC/MGS