In this video we are going to talk about what an entrance length for pipe flow means.
Here I have let's say some reservoir.
It is connected to a pipe and water is going to flow from that reservoir through this pipe.
I have some imaginary wall preventing this wall from doing so at this point.
This region that the water flows into this pipe is going to be termed the entrance region.
This entrance region covers the point from which the water enters the pipe to the point
which it reaches what is called fully developed flow.
We will talk about that in a second.
Let's focus on what is occurring within this entrance region.
As water starts to flow into the pipe we get a velocity profile that is pretty much uniform.
We would expect that through out the diameter of this pipe the velocity would be pretty
consistent.
We do know that as the fluid moves through this pipe the viscous effects of the pipe
acting on the fluid therefore then the no slip clause will start creating what is known
as a boundary layer.
This boundary layer will start forming and get larger as we move down the pipe.
This is going to happen every where throughout the pipe.
Eventually these boundary layers will come together and where they come together we get
what is known as fully developed flow.
For pipe flow we know that this has a parabolic velocity profile.
What fully developed flow means is at this point the velocity in the pipe direction which
we can call the x direction no longer changes with x.
Where before we had pretty uniform flow we see that as we move down this pipe from the
entrance region we get a smaller velocity towards the wall since at the wall it is 0.
The within this viscid core where we are not seeing the effects of the wall yet we still
have pretty uniform flow and we start forming our parabolic shape.
Because of mass continuity since the velocity of the fluid near the wall is 0 and small
the velocity in the middle of this pipe has to increase to make up for the decrease close
to the wall.
Again we can look at a further point down the pipe and again we see that the velocity
in the middle will start getting faster to make up for the velocity decreasing near the
walls.
Eventually these viscous effects of the boundary layer affect all of the fluid and when we
get to that point we have reached our fully developed flow.
As we mentioned in terms of terminology this inviscid core is where we have a uniform velocity
distribution and viscous effects are negligible.
Within the boundary layer the viscous effects are not negligible.
They are very significant hence why we have a change in velocity with radial distance.
Once the boundary layers come together we get to fully developed flow.
Remember that is where the velocity in the x direction is only a function of radius.
As we know the shape of these velocity profiles is going to be dependent on whether the flow
is laminar or turbulent.
Therefore we would expect the length of this entrance region if you call this L_e to be
a function of the flow type.
We know that flow can usually be characterized by the Reynolds number.
As with many other properties of pipe flow we can't always characterize something by
just an equation and so using something like similitude gives us an idea of how dimensionless
parameters might be related to each other.
This dimensionless parameter of the entrance length can be divided by the pipe diameter
and this is going to be a function of the Reynolds number.
For laminar flow with e Reynolds number of less than 2100 for pipe flow this typically
correlates very well with the following relationship.
We take 0.06 times the Reynolds number.
However for turbulent flow it is not quite as straight forward.
Turbulent flow this dimensionless entrance length is equal to 4.4 times the Reynolds
number to the (1/6) power.
What is interesting is if you look at the relationship for laminar flow we will have
some kind of line for the entrance length that looks like the following.
For turbulent flow we get a different line since it has a different relationship but
you see that for low turbulent flow regimes we can get a shorter entrance length than
for high laminar regimes.
The reason we bother to calculate the entrance length is because we want to know at what
point in our pipe do we reach fully developed flow since most of the relationships we look
at are for that kind of characteristic flow.
In another screencast we will look at calculating the entrance length for different types of
fluid flow.