00:01
And this problem, we have a on then a system where we have a compressor here.
00:06
We're dealing with air here.
00:07
E guess i didn't write that down anywhere, but this is all air, all right, it in blue, just for front.
00:16
So we have a compressor here, and the idea is that we have this tank here.
00:21
It's like a reservoir.
00:22
Um, and we restore compressed air in here.
00:25
So we poland air here, compress it and store it into this tank, which is then, you know, under high pressure.
00:33
And obviously that would mean kind of this valve here is closed.
00:37
After we get it up to enough pressure, we close this valve and we open this valve, and then we run it through a carbon eso.
00:45
It's basically it's a way of storing energy, right? basically storing compressed air.
00:50
Um, that then can then, you know, do some work.
00:53
And again, i should say, there's some, you know, work coming out of this turban.
00:59
Of course.
00:59
So what we can do then, is, um well, let's analyze this thing.
01:07
So what they tell us is we have a volume in this in this tank of 500,000 cubic meters.
01:20
Right.
01:20
So that is a that is a big tank.
01:25
So, um yeah, that's ah, cube root of 500.
01:34
I see.
01:34
See if this guy can do cube roots.
01:40
Oh, i am so eight by eight by eight by 88 m by 8 m by 8 m tank would be the the size of that size of that.
01:54
Is that right? why should i let me see here? just put the three zeros in here.
01:58
No, no, another zero.
02:01
Get under there.
02:02
Another zero.
02:05
No.
02:06
Yeah, that's right.
02:07
It should be.
02:07
It's an 80 by 80 by 80 m tank.
02:11
Yeah, that's like that's let's take a water tower.
02:15
Anyway, this is a this is one huge tank.
02:20
Um, yeah, because i was thinking that i was how i was doing it in my head was that take the cuba to this and take the cuba to this, but the cuba to this is 10.
02:28
So we need to multiply the cube root of this by 10 on.
02:31
So that's why we get 70 70.
02:34
About 80.
02:36
80 cubic.
02:36
80 meters on the to aside, that is a huge tank.
02:44
Um, i wonder if they really thought about the numbers they were putting in my wonder thing, we meant that make that should have been leaders, because that is a huge tank 80 m on the side, you know? anyway, um hadn't really thought about how big this actually waas anyway, so initially, this tank is that 100 killer paschal's and has is a 20 degrees c.
03:10
So basically surrounding temperature.
03:13
Um, e guess you could you could use, um there's some cases where you could basically have caves or natural reservoirs where you can store compressed air.
03:25
Eso we might have something like that where this is like a man made thing.
03:30
It's something like a cave or a kravis are, you know, some kind of void in the earth where you can store stuff or abandoned mineshaft? maybe.
03:41
Um anyway, so let's see, here we are told, what else? it's a nice entropic compressor.
03:50
So the entropy at two here is the same as the entropy.
03:54
It one, um, coming in at one is basically at atmospheric conditions.
04:00
And then we pressurize this thing up to 600 killer pascal's, and it remains at 20 degrees c.
04:09
All right, so then we can figure out.
04:13
Okay, how much mass was in here to begin with? so this is a rare so we have our air properties here, and we wind up having, um, about 595 times 10 to the third kilograms, 595,000 kg again.
04:32
A lot of mass because, well, we have a lot of volume.
04:36
Um, now, let's see here.
04:41
Um, okay, then we have the mass afterwards, and obviously we have a much higher mass because we have a much higher pressure.
04:55
And that winds up being, you know, about what is that? three point? about 3.6 million kg.
05:04
Um, so you know ah, lot.
05:07
A lot of air in here, but again, it's a big volume now, since it's ice and tropic.
05:15
Um, and an ideal gas.
05:17
We can use this formula here for the temperature at the outlet.
05:23
And now what we need to dio is because, you know, things are changing kind of continuously over time, as we as we, um you know, compress this so way.
05:37
Don't know what the really the intellect pressure is here, right? because it is going up.
05:41
We know it started at 100.
05:43
You know what's ending and not 600 but we don't really know.
05:47
It's a zoo we saw from a previous problem.
05:50
Taking an average here would give us a rough estimate, and it would be a little simple analysis, but it probably you know, it might not be all that accurate, but it would give us a rough estimate.
06:04
Just taking an average aziz, the intellect, conditions.
06:08
We know the temperature stays the same, but we're gonna do a little more detailed analysis on this.
06:15
So we know that the mass flow right into b is just, um, changing mass pretty, you know, time the mbt it's almost the definition of, you know, the change in the rate of mass change in here is three you made of mass flowing into here.
06:33
Now, um, conservation of energy.
06:38
The first law, we can write it in again differential form again because this is continually changing.
06:48
We can then use idea of gas, um, equations of state and assuming constant capacities, we can again put manipulate some things with, you know, using either gus.
07:03
And in the end, we could get everything these differentials in terms of rate of change of pressure.
07:10
Hey, so that we can we could figure out what the you know, knowing q dot and knowing these, these air all you know, we know all of these, and we know, um uh, yeah, we know everything there.
07:25
So do we know everything there? yeah.
07:30
We do know everything there.
07:31
This is this should this is also the b.
07:34
Um, well, maybe not in blue.
07:39
Okay, then we go.
07:40
We be.
07:41
So we actually know all of these coefficients.
07:44
So and if we knew this, we could figure out what the rate of change of pressure is, but we don't know this, um and and it isn't zero.
07:59
Um, if it were zero, um, then we have no rate of change of pressure, and things would be basically wouldn't work.
08:08
So we need some heat transfer in here.
08:11
Well, basically, we couldn't keep the temperature down.
08:13
Basically, what still work if we insulated this, but we have a higher temperature, obviously.
08:19
So, um, again constant heat capacities.
08:25
Then what we can do is not now, because we have everything as ah, as a time derivative.
08:33
We can integrate.
08:34
Um, i multiplied through by d t and then integrated.
08:38
So here we have an integral over from the initial pressure to the final.
08:41
Likewise here.
08:42
E could have combined these two integral.
08:44
They didn't.
08:45
Um and then we have an integral of the rate of heat transfer times dt, which just gives us the total heat transfer.
08:54
Assuming that, you know that initially it was zero.
08:57
Um, and so we just accumulate the heat going in, um, these things conveniently well, i know why i didn't didn't combine these.
09:09
These are conveniently not a function of pressure so that integral is easy.
09:15
This thing here, um, and again, i did a little bit of well, i used we had to use this.
09:22
Right? so this, actually so we see here that we have t here.
09:31
The temperature in here is a function of pressure, and we know, um, so we have we have this relationship of the temperature, so it's the temperature of the stuff coming in.
09:48
We know the temperature in the end, but we don't know kind of this temperature is changing.
09:55
Um, and the temperature of the stuff coming in is changing, but we're getting heat transfer going out, keeping it in the end, so we don't know exactly the timing of all this.
10:05
Right? so maybe the temperature got high in here.
10:09
This was insulated, and then we let it cool off.
10:12
Um, but in the end, this temperature he transferred gets his temperature down.
10:17
But we don't know what the temperature going in is so that that this tea here is the temperature of the stuff flowing in.
10:25
And but we know that because this is ice entropic.
10:29
No, that that is a function of the pressure that is, you know, at this inlet port.
10:36
So we can write this t here using this equation.
10:39
Basically, i got rid of the two because we don't really i shouldn't really have even written that too, because we know what we know.
10:47
It's true.
10:49
Um, um, yeah, that would have to to make sense there, because it is through the inlet state.
10:59
So, um, this we could have written as let's see here.
11:08
Oh, i know why this is started out of that.
11:13
This shouldn't be.
11:13
I was thinking something here was weird because this should be a one t one.
11:20
Here we come.
11:23
So, uh, plugging and this this we can actually write this t two.
11:30
But t two is a function of pressure, okay? and we don't know what this pressure at two is.
11:40
So we put that in and we, um now we have an integral and we see that basically it z not an article, because have p to the, you know, some power here so we can do the integral you know, easily enough.
11:55
Um, you know, pull this down and you know all that stuff.
12:00
So what we wind up getting is, um, this, uh, this term, this factor here where again i've used some ideal gas formulas.
12:11
Thio, thio change, you know, change some things, get rid of this t one.
12:17
And now we have v and r and again messed around with this a little bit.
12:24
Um, and in the end, we we can simplify things, and we get this expression.
12:30
So p two, you know, and we know p one so basically integrated from state one to state to this is a little bit weird because i'm not talking about state one to stay, to hear.
12:44
I'm talking about the turtle internally in there.
12:47
Thank your state one to state tube from b one to b two...