>> Okay, so last time we met we were talking about converging nozzles and we were talking about how a converging nozzle is always subsonic. You might reach Mach one at the throat under certain conditions, but you'll never reach supersonic flow in a converging nozzle. Now we also talked about a converging diverging nozzle in which it goes from subsonic to supersonic and it leaves the exit at supersonic conditions. That's called a supersonic design converging diverging nozzle. So we look at the roadmap and I'm going to put the roadmap here. This is p over p knot and this is the throat. This is the exit, this is zero and this is p over p dot of one. This is .528 if k is 1.4. Okay, so.This is called the back pressure region. This is the exit pressure here. This is a shockwave here. It's probably best to call these guys one and two. One is the upstream values in front of the shockwave. Two the downstream values in back of the shockwave. T stands for throat, p knot t knot reservoir pressures and temperature. Okay so let's go through the specific conditions here for this example problem. P knot is 300kpa, t knot is 500k. The area of the throat is one square meter, the area where the shock is is two square meters and the area of the exit is three square meters. We're going to call the exit area three. So the shock is assumed to be very very thin, so A1 and A2 are the same thing. They're both two square meters and the exit area is A3, that's three square meters. And the fluid is air and we're asked to find oh various things, I'm not going to write them all down there, just too many. But we're asked to find many many things. Alright, so the first thing is to decide where we are on the roadmap. Okay here's the roadmap, no flow. If the back pressure is equaled to the reservoir pressure, nothing causes flow. So if the back pressure equal the reservoir pressure, no flow. As we reduce the back pressure, we start to get flow and it looks like this. I'll draw a couple curves here. If we keep reducing the back pressure, we eventually are going to reach a Mach number at the throat equal one at the magic number .528 for k of 1.4. Since the back pressure is still up here somewhere, it kicks back and it goes back to subsonic. Or if the back pressure's way down here, it can go on down here where it's supersonic. Subsonic here, supersonic here. Now if there's a shockwave somewhere, then it looks like this. So there's our picture. We know what line we're on on the roadmap. I'm going to erase these to show you where we're at. We're not up here. There's a shockwave, we're not here, there's a shockwave, we're not here, there's a shockwave, we're not here, there's a shockwave, we're not here, there's a shockwave. There we are, that's where we are. A shockwave is at the diverging portion of a nozzle, okay. Now we do our roadmap here. From here to here, the Mach number's one at the throat. Up here the Mach number's less than one subsonic. From here to just in front of a shockwave, just in front of a shockwave, supersonic. Downstream side of the shockwave, we know across the shockwave the flow goes from supersonic to subsonic. And outside here of course it's still subsonic. So there's our flow regimes. Now comes where do you go to find numbers. Subsonic isentropic, table B1. Supersonic isentropic, table B1. Across the shock, table B2. Subsonic isentropic, table B1. So there's where you go to find the answers you need. What table do you go to? B1 subsonic, B1 subsonic, supersonic, across the shock, B2, back to subsonic isentropic, table B1. Alright, so let's then find out we know that the Mach number, the throat's one. Okay, why do I know that? There's a shockwave. Anytime there's a shockwave in the diverging portion of the nozzle, there's going to be Mach one at the throat, okay. At location one, alright so one at location one there. Upstream is shock. A1 over A1 star. A1 is two square meters, A1 star is one square meter.What does this star condition mean? It means where the Mach number reaches one.So at the throat is where the Mach number has reached one. So a throat is the same as a star one. Why is it a star one? Don't forget in any subsonic region, a star does not change. In any region isentropic a star does not change. A star is the same from my left-hand to my right-hand, a star is the same. Okay, so a star one is two. I go to table B1. And I have a Mach number in there. Oh here we are. On not one, just a Mach number. P over p knot. Row over row knot.T over t knot. A over a star.Okay, go down here. Mach number there. A over a star is one. And let's see okay. I'm trying to find a over a star of two. So here it is, two. Whoops there's one down here too. In that table B1, the isentropic tables, there are two places where you're going to find a over a star equal two, but here's the key. This one up here is subsonic flow. This one down here, supersonic flow. What do I have at point one? Here's point one. What do I have? Supersonic flow at point one.So which one do I want? I want the supersonic Mach number. That guy is 2.20.Okay, now I've got Mach one. So table B1, Mach number at one equal 2. 20, got it.I also have p over p knot 0.0935, p over p knot. So P1 over p knot one 0.0935.Solve for P1. So P1 comes out to be 28.06kpa. So now I've got the pressure in front of the shockwave. Now I go to the shock tables from table B2. Across the shock, okay table B2. Okay table B2, we've got Mach number one. We've got Mach number two, we've got P2 over P1. That's static pressure. We've got row two over row one, which happens to be, it can be show it's the same thing as V1 over V2. You can prove that.And then we have T2 over T1 and then we have p knot two over p knot one. We're not done yet and then we have a star two over a star one. Okay I know Mach number one, 2.2. I go the table Mach number one, 2.2. Downstream Mach numbers subsonic five, four, seven, one. So it went from supersonic to subsonic. Five, four, seven, one. Okay, let's see what's next. That's the Mach number there, okay. We've got, we can get P2, P2 over P1 Five point four eight. The pressure really goes up across the shock, 5.48. So P2, I know P1, I just found P1. So P2 comes out to be 154kpa. Okay I'm going to get p knot two over p knot one, .628. Okay, this is all from table B2, pardon me, table B2. Okay I know p knot one, p knot one equal p knot 300. So p knot two.Is equal to 188. Kpa.Okay, so we've got that. Okay now, next P2 over p knot two. I know P2, its right over here, 154 over 188. >> Professor.>> Ah huh. >> What does knot two represent? >> When you go across the shock, it takes energy. The stagnation pressure is a measure of how much energy the fluid has. To fight the shock, to push the molecules through the shockwave takes energy. So this energy goes down. What was p knot one? Three hundred, it went down to 188 because it took a lot of energy to push that molecule through the shockwave. That's so the energy the fluid has is a p knot is a measure of the energy the fluid has. This guy's ratio is .819. I go to table B1. P2 is going to be what now? Be careful see, P2 is subsonic, it says go to table B1. Okay, go to table B1, look for a pressure ratio of what? Eight, one nine. So eight, one, nine's going to be subsonic. There's a pressure ratio. A over a star, 1.255. Table B1.At P2 over p knot two equal .819. I get A2 over A2 star equal 1.255. So I get A2 star equal 1.592 square meters. A star changes over a shockwave. In isentropic region a star remains constant. A star changes over a shockwave. What does that mean? That means once you cross the shockwave, that is now subsonic, now it's subsonic. If you want to reach Mach one, what do you have to do? You have to go down to an area here, which is 1.592 square meters to reach Mach number equal one from where you were here. What's the Mach number there? Mach number two. Did I get it? I guess I got it here somewhere, let's see. No I got Mach one, Mach two, yeah, yeah. Mach no I didn't get Mach one, but that's Mach oh there's Mach, here's .547. Mach number .547, to get to Mach number one make the area go down to 1.592. That's what it means as far as seeing the change in the a star two, three, exit pressure, back pressure. Okay back to here. Okay, where were we? Down there, got the a star two. Alright, now notice it again, does a star change in isentropic region? No, that's a star three. Does p knot change in isentropic region? No, p knot two, 188. P knot three, 188. Stay the same. Okay, go over to here. At location three, A3 over A3 star equal. One point eight, eight, four. Table B1, why? It's isentropic, read the roadmap. Table B1, is location three subsonic or supersonic? Look at the roadmap, subsonic. Go here to table B1. Find the A3 over A3 star over here. Okay, that would be over here. From table B1, Mach number at three then. Zero point three, three o subsonic. P3, 188 times .928. That .928 is P3 over p not three. Table B1. P3, the pressure at three comes out to be 174kpa. But that's also equal the pressure, back pressure because P3 and p back are the same thing. There's P3, there's p back, they're the same thing. One more, find T3.T knot equal t knot one equal t knot two equal t knot three. Remember the stagnation temperature does not change in an isentropic region or across a shock. T knot does not change in an isentropic region or across the shock. Okay it's the same. So if I want to find T3, I go back here where the Mach number three's .330, table B1. It's subsonic, it's up here somewhere. I know the Mach number, it's .33. I go over here, T3 over T3 knot. And I read nine, seven, eight, two. So I solve for T3. The exit temperature, 489 kelvin. If I want to find the mass flow rate I can do this. Or I can do this or I can do this or I can do this. That was our equation in there and I think that was for the maximum mass flow rate. And I think did I have that and I think I do. Yes, here. There it is, 0.6847. P knot a star divided by the square root RT knot. Do I have p, wait no that equation only for choke flow. This equation choked or not choked, doesn't matter. This guy, only choke flow. Is he easy? Sure, I was given p knot, I was given t knot. I know a star. What a star is that? That's the a star at the throat, that's a star at the throat. One square meter, yeah one square meter. If you do that you end up with a I think I've got an answer on that, let's see. Five, thirty-seven kilograms per second. So this is really a complete problem. These guys with shockwaves in the diverging portion of a nozzle are really really really very complete that way. You go everywhere. You go from table B1 subsonic to table B1 supersonic back to table B1 subsonic. Across the shock table B2, everything. A star changes across the shock, p knot changes across the shock, t knot doesn't change across a shock. All kinds of rules like that you have to go through to determine where you're going and sometimes it gets hard to see where you're going. You can do all this by equations, but you can kind of get the idea. It's pretty tough stuff. It really helps to have a roadmap there and here especially. And those tables, that really helps to see where you're going and how you're going to get there. Okay let's do another one. This is in chapter nine, 957. So let me pull out that problem from our textbook and see what that one looks like, 957. Okay, air flows from a tank through a nozzle converging diverging into a standard atmosphere. The normal shock stands at the exit of the nozzle. Okay, so now a different picture. The problem 957. Said the shock is now at the exit plane. So now the shock is right here. Okay, let's see what we're given. We're given the fluid is air, okay. I am not given p knot, unknown. I'm given t knot 270, 373. It's hard to greasy [assumed spelling], so 373k. Air, area of the throat 10 square centimeters. I'm not given that, yeah I'm given that. A1 is 12.5 square centimeters. Exit area is the same as out there. Right, got them all. I'm going to call this one, I'm going to call this two across the shock. Okay, let's see what else. Oh p back is atmospheric pressure 101kpa. Okay, find p knot and m knot. We weren't given the reservoir stagnation pressure p knot. Okay, back to here again. Our roadmap now goes down here, shockwave at the exit and goes out here. Here's p back, here's P1, here's P2. Okay, is it choke flow? Yeah we want anytime there's a shockwave, we know we have choke flow. Okay so we know we have choke flow. Is that the maximum mass flow rate? Yes that's the maximum mass flow rate and I'm just going to put this here so we know only use that when you have choke flow. Otherwise you can find the mass flow rate anywhere you want. At the throat, at one, at two by row av. Okay back to here. There's our roadmap, so here we go again now. Let's see from here out to here. Let's just draw that and we'll draw that from there to there, supersonic table B1. Across the shockwave table B2, out here subsonic. Okay, there's our roadmap. Alright, we start out and we say okay now I know this is a picture what's going on. So I'm going to take A1 since I know it and a throat and take the ratio. So first A1 over a throat is equal to 12.5 divided by 10. One point two five, got it. Table B2, shock table, Mach number one is equal to 1.60, 1.25. So my area ratio is 1.25. Is that supersonic? Yeah, it's a shockwave supersonic, my roadmap shows me. So I go in there with this area ratio 12 point, 1.25 pardon me and I end up with a Mach number of 1.60. So I know the Mach number now, it's 1.60. I also know that P2 over P1 is equal to 2.820, 2.820. Okay, so and I know P2 equal p back. And the back pressure's 101kpa given to me. P2 is a back pressure. Okay, so now I can solve for P1. So P1 is equal to 31.82kpa.Okay, at the [inaudible] number equal 1.60, that's a Mach number one. P1 over p knot two, three, five, three, table B1. Why is that? Because when I'm in .1, I'm in table B1 region. Okay, go to table B1. P1 over p knot, that. Did I find P1?Yeah I found P1's right there. So solve for p knot. So p knot is equal to P1 divided by .2353 and that gives me 135.2. So that's how I find the pressure in the reservoir in order to cause those things to happen, the shock at the exit plane. Part two, find the mass flow rate. Well we know since its choke flow. You might as well use the easy equation. M dot max is equal to 6847 times a star p knot divided by the square root rt knot and a star is the throat area, 10 square centimeters. I didn't [inaudible], but that's its pretty straight forward obviously. So again you have to use table B1 and table B2 because there's a shock somewhere in the diverging portion of the nozzle. In this case the shock is at the exit plane and the pressure across the shock P1 on the upstream side, P2 on the downstream side. So two examples with a converging diverging nozzle. One, a shockwave is somewhere in the middle. Next one the shockwave is at the exit. So we're going to stop here on that and we'll pick it up then next time.
