Examples – Internal Flows
These worked examples have been fielded as homework problems or exam questions.
Worked Example #1
(a) The average flow velocity is calculated from the volume flow rate, i.e.,
(b) The Reynolds number of the flow in the pipe is
so in this case the pipe flow will be turbulent because the Reynolds number is greater than 2,000 and we will need to use the Moody chart to find the friction factor .
(c) We now need the pressure drop and head loss along the pipe. To use the Moody chart we also need the relative surface roughness, which is
From the Moody chart for a Reynolds number of 84,900 and a relative roughness of 0.00087 (using interpolation) we have . Therefore the pressure loss over the length of pipe is
The corresponding head loss over this pipe is
(d) The pumping power required will be
Worked Example #2
(a) The crosssectional area of the pipe is
The mass flow rate is given as 125~kg hr = 0.0347~kg s, i.e.,
so the average flow velocity in the pipe is
The Reynolds number based on pipe diameter is
Notice that this Reynolds number is the laminar regime (socalled Couette flow) so the friction factor is given by
The pressure drop is given by
(b) The equivalent head loss will be
(c) The pumping power can be determined from
Worked Example #3
This pressure drop is a direct consequence of the action of viscous effects. The fluid is oil (dense and thick) so we expect the flow to be at very low Reynolds number (i.e., laminar), so from laminar pipe flow theory we have that
Notice that for a given length of pipe the pressure drop is proportional to the viscosity of the fluid and to the flow speed (so the faster we try to move a given fluid the larger the pressure drop will be) but inversely proportional to the square of the pipe diameter. The average flow velocity is
We can now check to see that the flow is laminar by calculating the pipe Reynolds number, i.e.,
so it is very low and the expectation of laminar flow is verified confirming the use of the correct equation for the pressure drop. The volume flow rate through the pipe will be
Substituting in the known values gives
The corresponding mass flow rate of oil will be
Worked Example #4
The pressure losses associated with the pipe flow are determined using the equation
where is the DarcyWelsbach friction factor. The friction factor depends on the flow Reynolds number and the roughness of the pipe material. Hence, the pressure loss in a pipe is also a function of the Reynolds number and the roughness of the pipe. Remember that this pressure loss is caused by the frictional effects of viscosity, and loss is irrecoverable and appears as heat.
(a) The Reynolds number of the flow can be determined using the equation
(b) The pressure loss can be determined using the equation
where in this case the friction factor for the laminar flow in a smooth pipe is
Therefore
and so
(c) The pumping power required to overcome this pressure drop is
where the volume flow rate is
Therefore
(d) The head loss can be determined using
So, we would likely need to specify a pump that has a head capability of at least 185~.
Worked Example #5
To calculate the power we need to find the pressure loss . To use the appropriate equation for the pressure loss, we need to know whether the flow is turbulent or laminar. For which, we need to know the Reynolds number
For the above equation, the average flow velocity can be calculated based on the flow rate of the oil. Flow rate of the oil is
Now the Reynolds number can be calculated, i.e.,
This value of Reynolds number is larger than 4,000. Hence, the flow is turbulent. For a turbulent flow, the pressure loss can be determined using the equation
Here, the friction factor is a function of relative roughness and Reynolds number. For the wrought iron pipe, the relative roughness
Consulting the Moody chart for and , we can find the friction factor, which is about . The pressure drop in the pipe can now be determined calculated, i.e.,
Required power to overcome this pressure loss is
Worked Example #6
(a) In this case we are dealing with a duct that has a rectangular crosssection, so we first need to find the hydraulic diameter , which is
where is the crosssectional area and is the perimeter of the section. We have a square crosssection of side so the equivalent hydraulic diameter is
We are given a hint to consult the Moody chart so the flow through the pipe is likely going to be turbulent. The equation for the pressure loss through a pipe is
where is the friction factor, still to be determined, which depends on the Reynolds number and pipe roughness and its value will come from the Moody chart.
The average flow velocity can be found from the flow rate and crosssectional area. The pipe area is
so the average flow velocity is
To use the Moody chart we need the corresponding Reynolds number, which in this case is
where we find the density and viscosity of air are at ISA conditions.
Because in this case then the flow through the duct is going to be in the fully turbulent regime. We also need the relative pipe roughness, which is
Therefore, from the Moody chart the friction factor is
Returning to the calculation of the pressure drop then
(b) The power required to move the air through this part of the duct and overcome this pressure drop is
so
Worked Example #7
(a) The volume flow rate, is given by
The average flow velocity can be calculated using
Calculating the Reynolds number gives
For a smooth pipe the roughness factor, mm and . Obtaining the friction factor from the Moody chart gives . So, pressure drop along the pipe can be calculated using
and substituting values gives
(b) The pressure at the pump located at point 1 is then the required pressure at the last sprinkler plus the pressure drop
Substituting values in the above equation gives
(c) The corresponding head loss can be determined using
So, we would likely need to specify a pump that has a head capability of at least 121.3~, which would be a fairly large pump.
Worked Example #8
(a) The general equation for the pressure drop is
where is the friction factor, which depends on the Reynolds number and pipe roughness. The average flow velocity can be found from the flow rate and pipe area. The pipe area is
so the average flow velocity is
To use the Moody chart we need the Reynolds number, which in this case is
We also need the relative roughness, which in this case is
Therefore, from the Moody chart it can be found that
Returning to the calculation of the pressure drop then
The corresponding head loss can be determined using
(b) The pumping power required to overcome this pressure drop is
where the volume flow rate is
Therefore
Worked Example #9

 Determine the hydraulic diameter of the pipe.
 Determine the average flow velocity though the pipe.
 Determine the Reynolds number of the flow in the pipe.
 Is the flow in the pipe laminar or turbulent? Explain.
 Determine the friction factor of the pipe.
 Determine the pressure drop and head loss through the pipe.
1. The crosssectional area of the rectangular pipe is
so the hydraulic diameter is
2. The average flow velocity in the pipe is determined from the volumetric flow rate and the crosssectional area, i.e.,
3. The Reynolds number based on the hydraulic diameter is
4. Because the Reynolds number is greater than 4,000, the flow in the pipe will be turbulent.
5. The relative surface roughness is
Using the Moody chart, the friction factor for this Reynolds number and relative roughness is about 0.023. Some small error in reading the chart is acceptable.
6. The pressure drop over a pipe length of 200 ft is
and inserting the known values gives
The corresponding head loss is