In this screencast we are going to work through an example problem where we calculate the
entrance length to reach fully developed flow.
Here we have water that enters a 4 inch diameter piping system from a reservoir with a volumetric
flow rate of 50 gallons per minute.
We want to determine whether the flow is laminar or turbulent.
If during the winter the water temperature is 45 degrees Fahrenheit and if it reaches
65 degrees Fahrenheit during the summer what is the length of the pipe that is necessary
to reach fully developed flow?
The question first is asking us to determine whether the flow through the pipe is laminar
or turbulent.
We have the water entering the pipe from this reservoir and it will reach a fully developed
state after some entrance length.
This entrance length will be dependent on whether or not we have laminar or turbulent
flow.
Therefore it is important to determine that first.
Recall for pipe flow that we calculate the Reynolds number as the density of the fluid
times the average velocity through the pipe times the characteristic length, in this case
it is diameter, over the viscosity of the fluid.
We can also write this as the velocity times the diameter over the kinematic viscosity
of the fluid depending on which viscosity we use.
From our problem statement we are given a diameter of 4 inches and a flowrate Q of 50
gallons per minute (gpm).
We know based on the volumetric flow and the cross sectional area of the pipe that we can
calculate the average velocity.
We are given the diameter so the only thing we need to look up for water is the density
and viscosity or the kinematic viscosity.
I am going to write down first the kinematic viscosity at both 45 degrees Fahrenheit and
65 degrees Fahrenheit.
Looking these up I see 1.536*10^-5 ft^2/s at 45 degrees and 1.131*10^-5 ft^2/s for 65
degrees.
Now that we have those values and we can take our diameter and convert from inches to feet.
We get 0.33 feet.
Lastly to calculate the Reynolds number we are going to need the average velocity.
That is going to be the average flow rate over our cross sectional area.
We are going to take our 50 gpm and divide by the cross sectional area in this case of
pi times our radius squared.
Our radius is going to be 0.167 ft squared.
However our units do not match up.
We need to convert our gpm into feet cubed per second This is going to give us 0.111
ft^3/s.
We can plug that into our velocity calculation and we should get a velocity of 1.28 ft/s.
Now we have our velocity, our diameter, and our kinematic viscosities to calculate our
Reynolds numbers.
Our Reynolds numbers at 45 degrees is 2.77*10^4.
If we repeat this calculation we get 3.76*10^4 for 65 degrees.
Indeed for pipe flow a Reynolds number greater than 4000 indicates turbulent flow.
In both cases we have turbulent flow.
To calculate the entrance length of these two profiles we would use the following relationship
where the entrance length over the diameter of our pipe is going to equal 4.4 times the
Reynolds number to the power of 1/6.
If we set this up for our first flow of 45 degrees we would have an entrance length of
8.06 ft.
It would take 8.06 ft of pipe to reach fully developed flow.
If we repeat this for flow at 655 degrees we get 8.49 ft.
As we increase the temperature we would expect a higher entrance length into the pipe because
our Reynolds number increased due to the lower viscosity of the fluids.
Hopefully this gives you an idea on how to calculate the entrance length to reach fully
developed flow for pipe flow.