this video addresses a thermodynamic concept
of flow work.
To imagine it consider this tube there is
a pipe with a large diameter up front and
it constricts to a smaller diameter, and fluid
flows into this pipe entering from the left.
There is a high pressure that forces it through,
and then it exits through the narrow tube.
So There is some high pressure to the left,
and atmospheric pressure to the right.
So it is accelerated as it flows into this
narrow constriction.
You need to find a control volume, I am just
going to draw in red here.
It is in the inside of the walls of this apparatus,
and believe it or not there is work actually
entering and leaving this control volume,
and we call it flow work.
The gist of flow work is that there is some
energy required to push the fluid into this
apparatus from the left, and there is some
work required to push the fluid agents atmospheric
pressure or effectively pushing atmospheric
pressure out of the way.
To conceptualize the flow work being done
on the system at this up stream point.
Imagine a syringe there is what is called
a plunger, and if you push on the plunger
if there is fluid in here it will drive the
fluid from left to right, and out the nozzle.
So if I draw the flow streams we wee flow
flowing from left to right.
For that to be happening there must be some
force acting on the plunger, and that force
if you were displacing or ejecting water from
this syringe you have to apply some force
using your thumb to push in the plunger.
Lets say that we squeeze the plunger distance
L from left to right, and I just applied a
force over a distance, and I did some work
on that system.
Lets call that work, flow work.
So lets say w flow, and that is equal to the
force applied, multiplied by the displacement
and distance L. On the left side of the plunger
there is some low pressure.
On the right side there is some high pressure,
that the plunger is pushing agents to drive
the fluid out, So w can say that the net pressure
if I am atmospheric on the left, and atmospheric
in the exit.
The net pressure is just this high pressure
in here.
I can rewrite the force, as simply the pressure
times the cross-sectional area of this plunger
and multiply that by L, and I still got the
flow work.
that I did.
The volume that this cylinder is simply the
area times the length.
So I can say it is the pressure times the
displacement of the volume us the flow work
that was done.
Lets look at the rate at which flow work is
done.
We will call it flow dot, and that will equal
the time derivative of the forces times the
length.
If we say that the pressure is constant or
the force is constant I can pull that out
of the derivative, and I have got the force
times the derivative of the length with respect
to time.
Again writing the force is simply the force
times the area, PAdl/dt, were dL/dt is now
the rate at which the plunger is being depressed
from left to right.
The area is a constant.
So I can put that into the partial, and that
is equal to P times dv/dt.
The rate at which flow work is being done
is the pressure on the up side of the plunger
multiplied by the volumetric flow rate of
fluid flowing through this devise.
What is interesting is the fluid does not
know if it is being pushed by a plunger or
if it is being pushed by its up stream neighboring
molecules.
An open flow processes we have molecules pushing
the fluid left to right, and to characterize
that lets say instead of the plunger for example.
Say that we have P times dm/dtV, The specific
volume is simply the volume, and I will move
this up here to emphasize that we are dealing
with the open system now.
So the pressure times the volumetric flow
rate.
The specific volume here is just a constant.
So I can pull that out of the differential.
I have the pressure times the speific volume
times dm/dt, or I can rewrite this dm/dt as
m dot time PV (Specific volume), and in an
open flow system you see this an awful lot
this mass flow rate times PV, and some times
it is called PV work.
So in the plunger example we had to do work
to drive the fluid through by pushing the
plunger.
Lets not forget that the fluid also had to
do work by pushing the atmosphere out of the
way.
So to conceptualize this/ Lets look again,
there is an up stream plunger looking from
left to right.
So forces on the up stream plunger, and I
will call this f1.
There is also a down stream plunger it is
representing atmospheric pressure in this
case.
If we assume there is a vacuum or no pressure
out here.
Then in this case there is no force acting
from left to right is atmospheric pressure.
I will call that F2.
Now I will draw a control volume around this
system, and the control volume contains the
shaft of both plungers.
What will happen is that the upstream of the
fluid does work on the fluid, and it depresses
that plunger from left to right, and the down
stream plunger is also moving from left to
right, but it moves a greater distance, and
it moves a greater distance because any displacement
up here results in a larger displacement down
here, because of velocity of the fluid is
greater as it flows through this constriction.
So just like the open case.
In this case there is flow work being done
on the system on the left and the system is
doing flow work on the right, and this case
it is agents a plunger but in often it would
be agents the atmosphere if it was a free
jet, or its down stream neighbors if there
was fluid flowing down stream.
I can conceptualize these with the two pistons,
because their cross-sectional areas are the
same the pressures are the sames.
The pressure pushing from left to right on
this piston would be the same as the pressure
pushing from left to right on this piston.
So positive work being done on the left.
The system doing work on the right, and work
being done on the system on the left.
As it is demonstrated here flow is being done
on the system on the left, and the system
doing flow work on the right.
For this example the amount of flow work done
on the system on the left just balances the
amount of flow work the systems did on the
right.
As a final example flow work occurs even very
simple geometries.
Here I got a pipe that the diameter is constant
through out.
The fluid flows from left to right, and if
I draw a control volume again I am going to
enclose the walls of the pipe and all the
fluid inside of it.
Believe it or not flow work is being done
on this control volume and the fluid within
the control volume is doing flow work on its
down stream neighbors, and it maybe no surprise
but the amount of flow work done on the system
is equal to the amount of flow work at the
system does.
Again I can conceptualize this with the two
plungers.
You will see the force of the plunger entering
the control volume on the left.
It will equal the force of the plunger leaving
the control volume on the right, because the
cross-sectional areas and the pressures are
the same.
Flow work being done on the system on the
left.
The system doing flow work on the right.
