Entrance Region & Fully Developed Flow Video Lecture | Fluid Mechanics for Mechanical Engineering

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.