We are going to assume this is a closed system.
It's not a very bold assumption here because we have essentially no other
information. So we're going to assume that this is a
closed system. Why is that important?
It's important because it tells us that the mass in this system is a constant.
Which tells me that this expression, which is in terms of the absolute volume,
is equally valid for the specific volume. So take, for example, each side of this
expression and divide it by the mass raised to the n power.
[SOUND] [INAUDIBLE]. [SOUND] Okay.
Well this term is simply the specific volume.
[SOUND] Right. So, that's important because our problems
statement gives us information in specific volumes not in absolute volumes.
So we look at this expression. We say, hey, look.
I know state one's pressure and specific volume.
I know the specific volume at state 2. Plug and chug through this problem and
you can determine what this, what the pressure is at the end state for
[INAUDIBLE], the end state of step 1 or the process, the process from 1 to 2.
So, that pressure, if you go ahead and plug in the numbers here, will be 246
kilopascal. And make sure, you know, you all remember
how to do this type of solution but it should be pretty straightforward.
So what we know now is that the volume at state two is 0.02, the specific volume is
0.02 and the pressure is 246 kPa and so I can now label state 2.
A polytropic process looks sort of like it shouldn't be double-valued down here.
So, just expand my doc here. looks like a curve.
and probably can't see the level of curvature unless you have a large space
that you're covering. So this is a Polytropic process.
[SOUND]. Okay.
And I can label it equally either absolute volume or specific volume.
Cool, we've got from step one to step two.
Now we need to get from step two to step three.
What do we know? Well, we know that the air is subject to
a constant pressure process, so we know it's going to be somewhere along a
constant pressure line. And what we need to decide is will my
line move to the right or will my line move to the left to get to state three?
Well we're told the final specific volume is equal to the initial state, and that's
state 1. [SOUND] So the volume at state 3 is equal
to the specific volume at state 1. Okay.
There we go, right? We move across.
And that's the process diagram. Two, two processes.
But this is the complete process diagram from one to two and two to three.
So, hopefully that was pretty straight forward.
And you can see how you're going to have to be very careful about interpreting the
information that, that's given to you. Now, in thermodynamics classes, we, we do
a lot of pre-chewing your food for you. So we give you the information that you
need. In the real world, often you'll have
either an underdefined or an overdefined system.
So that's the hardest part of these problems is understanding what
information to use, and what information can you neglect or ignore in your
analysis. So again in this [INAUDIBLE] in this
class, we'll really set things up, so that you can really focus on the
analysis. But understand, just writing the problem
statement is quite a challenge in the real world, often.
Okay. Having said that, now we have our process
diagram. Let's go ahead and determine the specific
work for each step in the overall process.
So I'm going to move on to the next screen here, so we have enough space to
work with. And all we need to do is remember that,
okay, this is expansion and compression process.
We already saw that there's a compression step, that's one to two, and there's an
expansion step, that's two to three. We need to treat each of these steps
individually, and then we can add them together.
To determine the work transfer for this process what we need to remember is that
there are two steps. And what we can't do is bundle the steps
together. Now we know that we have expansion and
compression work. So that's what we're trying to find.
And remember, we're asked to find the specific work.
So the work normalized by the massive air in the systems.
So we'll denote that as a lower case w. And that's equal to the expansion and
compression work. And again, if we were to just calculate
the work from one to three, that would be incorrect.
This is a two-step process. Work transfer is path dependent.
So in reality, the total work is given by the sum of the contribution from the two
steps. So from one to two, and from two to
three. We sum the expansion work transfer.
Expansion and compression work transfer contributions.
Okay. So let's look at each of these steps one
at a time. So we'll start with the work transfer
from one to two. And remember, we need to understand how
the pressure varies as a function of specific volume.
And in that first part of that process, we're told that we have a polytropic
compression process. And we know that that means that the
pressure and the volume vary using this proportionality.
Pressure times volume to the n equals a constant.
So if we solve for pressure we know that, we have some constant divided by volume
to the n. We can take this expression and plug it
into the first, the work transferred term for the first part of this two step
process. So we do that here, so we have a constant
volume to the n and that n is known differential of a specific volume.
Okay. So let's go ahead and plug in the value
for n, which is 1.3. Go ahead and do that integration and what
we end up with is an expression that looks like the, let's see, constant.
We have a negative sign upfront, because remember, our work our sign convention
for work transfer. And we know that the compression process,
compression is work in. Okay.
1 divided by volume to the 0.3 power and we're going to evaluate that from, again,
state 1 to state 2. And if we go ahead and say hey this
constant can be determined by using either the information at state 1 or
state 2, whichever one is fully defined. And in this problem of course P1 and the
pressure in the specific volume at state 1 are fully defined.
So we can go ahead and substitute that in here to the 1.3 power, divided by 0.3.
And we have 1 divided by [SOUND] the specific volume raised to the 0.3 power
at state 2 minus [SOUND] this parallel expression for the state 1 condition.
And then all of these values are known. We plug in our numbers and we get.
an answer for the work from one to two, the specific work transfer of minus 3.08
kilojoules per kilogram. And again this is less than zero which
confirms our, confirms our understanding that work in is negative.
That's our sign, our sign convention. [SOUND].
Super. So now we go onto step two to three.
And that's, of course, much easier because, again, we need to understand how
the pressure varies as a function of specific volume.
But that was a constant pressure process, so we can just pull the pressure outside
the integral and we get [SOUND] nice simple expression for those specific work
transfer for that second part of that process.
And again we know all of these volumes, we know the pressure.
Again this is the pressure we can either use the pressure at state 2 or at state
3. We'll go ahead and call that P2.
plug in those numbers and we get 4.92 kilojoules per kilogram for the specific
work transfer from two to three. This number is positive, it's an
expansion process. So, again, this confirms our sign
convention, which says, expansion is work out.
Expansion, work out. And so that is a positive value.
Now we're asked to find the total work for both steps, so all we do is add these
two together. And that 4.92 plus the minus point, 3.08.
And what we find is that the net work for these two step process, two, two
processes and sequence is positive. So we have a net work out of the system
and kilojoules per kilogram units. So net work out.
So we've learned a lot in this process and some of it implicit and some of it
explicit. We've used our sign convention, that's
great. We see, we've used our definition of
expansion and compression work. Okay.
So we've seen some actual numbers here. We saw that the system has some intrinsic
signs associated with it, whether or not it's work in for a compression step or if
it's work out for an expansion step. And we've also seen that you cannot
evaluate this system, you can't answer the question which is the overall work
for the process, by just going from state 1 to 3.
So you've done this example for a polytropic compression process, now what
I want you to do. s draw on a PV diagram, what would a
polytropic expansion process look like?