This screencast is about the difference between
the entrance region and the fully developed
region in a pipe.
So if we look at this picture right here of
a pipe.
If you notice we have a constant velocity
going in, and then as soon as it hits the
pipe wall, because we have the now slip condition
at the wall.
Another words the fluid takes the velocity
of the wall, which is 0.
What it created are things called boundary
layers, and so these boundary layers, which
go along the pipe create a viscous region.
However we have here in the center, an invisid
core.
So this continuous until the boundary layer
meet right here, and as soon as they meet
and become one viscous area we now have what
is known as fully developed flow.
So fully developed flow means that du/dx equals
0 or another words u does not change with
respect to x.
It may change, it does change with respect
to r, but not with respect to x.
Another interesting thing about this entrance
region is in the invisid core, because there
are no viscous affects we can apply Bernoulli's
to any 2 points i that core.
So now lets look at an example.
So lets say we have water it is at 20 degree
C, and it enters a lead pipe with a diameter
of 10 cm, and a length of 8 m.
The volumetric flow rate is 15
85 gal/min, and what we want to find is that
Entrance length, as well as figure out does
the the flow become fully developed.
So what does that mean?
What it means is if the entry length is greater
then 8 m the entire pipe is still the entrance
region, and we don't have any fully developed
flow.
So the 2 governing equations for this entry
length depend on the flow regime, So this
one right here is for laminar flow, and this
one is for turbulent flow.
So how do we determine whether we have laminar
or turbulent flow/ Lets take a look.
If you notice both of these depend on the
Re number.
So lets calculate the Re number.
So the Re number is rho(density)*velocity*
diameter divided by the viscosity.
We don't have velocity.
However what we do have is a volumetric flow
rate.
So we can write the Re number in terms of
Q, and look up our properties of water at
20 degree C, and so our viscosity is 1.002x10^-3
N*s/m^2.
Our density equals 998 kg/m^3/ We rewrite
our Q, which is 15.85 gal/min, and we want
to put that in the same units.
So 1 min is 60 seconds, and there are 0.0037
m^3 per 1 gallon, and we therefore end up
with 0.001 m^3/second.
Our diameter is 10 cm, which is 0.1 m and
so we can calculate our Re numbers using these
properties, which is 12860.
So what tells us whether it is laminar or
turbulent?
Well anything over a Re number of 4000 in
a pipe is considered turbulent.
Therefore we use that the entry length is
4.4 times Re^1/6 times the diameter, which
is 2.125 m.
So that means that in our pipe, which is 8
m long.
As soon as we get to 2.125 m the flow here
is now is fully developed.
