>> Okay. Just an update so you know in our course syllabus surprisingly we're a little ahead of the game. So that's good news because we can spend a little more time on our last topics, compressible flow, which is a fairly difficult topic, and then boundary layers and drag in external flows. So I think that's a good thing, but just so you know, we're not corresponding to the course syllabus day by day. We're going to start compressible flow here today. And then we'll spend about 45 to 50 minutes on this, and about 20 to 25 minutes on review for Wednesday's midterm. Okay. Chapter nine. Compressible flow in the White textbook. We start out and compressible flow depends quite heavily on some equations from thermal. So I'm going to put down what equations are important from thermal, and then we'll use those equations in chapter nine. What I'm saying also is that for our second midterm, which covers this topic, these equations probably should be on your equation sheet for that second midterm. Okay. So pretty much we're going to look at perfect gasses in compressible flow. Compressible flow, of course, is where compressibility effects are important in fluids. In ME 311, the first fluids one class, we pretty much treated the liquids as incompressible. Water. Oil. Whatever. Incompressible. We pretty much treated gasses, especially air, as incompressible. And that's -- That can be done. Air can be considered incompressible if the Mach number is less than about 0.3. Less than about 0.3. The Mach number. But when you get to fluids two and you get to compressible flow, now it's not an incompressible fluid anymore. It's a compressible fluid. So we have to kind of shift gears in a way. We're going to pretty much be looking mainly at perfect gasses. Okay. We know from thermal the definition of a perfect gas is one that will base a perfect gas equation state P over rho equal RT. So I'll put down a number of equations which come from thermal that are going to be important in the compressible flow chapter. Okay. Let's see the second one here. Yep. There it is. Change in internal energy is equal to C sub V DT. Specific heat at constant volume. And then we have HU plus PV from thermal. And we have -- I'll put these just right here. C sub P and C sub V. Only functions of temperature. And we also have C sup P minus C sub V equal R. The gas constant R. These things are in the tables in the back of the book too. C sub P for the gas, C sub V for the gas, R for the gas, whatever the gas might be. Carbon dioxide. Air. Nitrogen. And then we also have the ratio of the specific heat given by K. C sub P over C sub V. And C sub V. R over K minus 1. C sub P. KR. K minus 1. Okay. The change in U. U2 minus U1 is C sub V. T2 minus T1. We're going to assume that the gas we're looking at has constant specific heats. So when we integrate C sub V DT, the C sub V's constant, pulled outside the [inaudible] sign, we get that. And then we also have our equation for H that we're going to need. H2 minus H1. C sub P. T2 minus T1. And then from thermal we have the change in entropy. TDS. And another form of it. And then we have for a perfect gas my DS solve for DS. Or the other form in terms of DH. Okay. And then we integrate that DS. Temperature absolute. Or in this form. Let's just say and. We know in the perfect gas equation of state that temperature's absolute, that pressure's absolute. Got to be. So everything you see here, that pressure's absolute, that temperature's absolute. Not gauged. Not relative. Okay. If process is isentropic, S2 equal S1. So if that's the case we get these equations. Okay. I think that's all we need there. If you put these two guys together, you end up with this. That's a three minute quick review of what we need in thermal for compressible flow. You probably -- These equations we'll use for derivations and for homework solutions and things like that. Okay? Okay. Now that gets us to the material we want to get to. And the first thing we're going to do is look at the speed of sound in a compressible fluid. Perfect gas. Okay. So here's the ground. This is a wave. Okay? Sound wave. See, it's moving in a region where the air is still, the velocity 0. The density is rho. The pressure is P. So this sound wave is moving in to a region of still air. On the -- And of course the observer is staying on the ground. Somebody does this. You hear the noise. How did you hear the noise? I don't see anything. How'd it get to my ears? Well, there's a sound wave. And that sound wave passed through the air until it got to your ear where your eardrum responded and you heard the sound. Okay. So you're this stationary observer, and the sound wave went past you. On the other side of the sound wave things change. The velocity now is a very small differential velocity, DV. The densities change, rho plus D rho. The pressures change, P plus DP. On the other side of the sound wave. For the observer this is an unsteady problem. The sound wave passes. Wait a few seconds. It's quiet again. To make it a steady process, we attach the observer to the wave. Okay? This is unsteady. So now we attach the observer to the sound wave, and the observer is standing with this. Okay. I'm the observer. I'm sitting on the sound wave. I see air approaching me at that speed, C, where C is the speed of sound. So I see air approaching me at a speed C. The density is rho. The pressure is P. What's leaving the backside of my control volume, the dashed line, okay, is a relative velocity. C minus DV. The pressure there is the same or the density is just like the above picture. P -- Rho plus D rho. P plus DP. Now this is steady. Observer fixed to sound wave. Our textbook calls the speed of sound A. Other books call it C. I like C because A can mean -- When you write a sentence, A can mean the speed of sound or it can mean A in the sentence. Okay? Apples and oranges. So -- So when you read your textbook you're going to see the symbol A, but I like to use the symbol C. So just so you know, the board work versus the textbook work, okay. Okay. So there we are fixed to the sound wave. Sound wave is a very weak wave. It's a weak wave. What does weak wave mean? Well, you can look at two kinds of waves. Number one's a sound wave. When I talk to you, I don't blow your eardrum out. You're on the deck of an aircraft carrier. FAT flies over 100 feet over the deck. It will blow your eardrums out. That's not a weak wave. That's a very strong compression wave. So right now we're talking about sound waves. Later on we'll talk about shockwaves. Okay? Because they're strong. Big change in pressure. In this one there's a small differential change in pressure, a differential change in velocity, a differential change in density. Small. Okay? There -- Yes. The pressure difference is really, really small in there. The -- They pass through our ears pretty quickly. Bam, and it's done. But we're riding with it now. Okay. So there's that picture. What are we going to do? Okay. Same old, same old. Fluid mechanics. We're going to apply continuity, and momentum. All right. First equation. Apply continuity for that control volume. Conservation of mass. Steady state. What comes in the left goes out the right. Okay. I'm going to call this area on both sides A. This is my control surface. This is my control volume. There is the area A normal to the velocity C. What comes in the left? We know M dot N equal M dot out. What's M dot? Rho AV. Rho A. Velocity C. What goes out over here? Density here. Velocity here. Area here. Let's -- I think I put a -- Yeah. That's okay. Rho plus D rho. C minus DV. DV. Area is A. Of course the area on both sides cancels, cancels, gone. Rho C equal rho C plus CD rho minus rho DV minus D rho DV. Rho times C here. Rho times DV here. Minus sign. C times D rho here. Plus sign. DV, D rho, minus sign. Okay. Both sides of the equation cancel, cancel. Zero. Equal. This guy has a differential D rho. It's got a differential DV. He's got two differentials. D rho, DV. If this number's small, and that's small, this guy is really small. Two small numbers multiplied together. I'm going to ignore them, right? That's typical of what we do. We say, "Let's ignore higher [inaudible] differentials." Okay. Gone. We're left with C, D, rho, minus rho, DV. Okay. Now we see -- Okay. That's conservation of mass. I'm going to write X momentum. Okay. First thing. Left-hand side momentum. Fluids one. Summation of the force is acting on the control surface. Okay. Pressure force times area. Pressure times area gives me the pressure force. Okay. Which way is it pointing? To the right. I define my coordinate system positive to the right, positive up. P times A. On the left pressure force is always a compressive stress on the control surface. It always points inward to the control surface. The pressure force on the right points this way. Negative. Negative. Pressure. P plus DP. Area. A. Pressure force. Equals. The right-hand side. Momentum flux. M dot times velocity. Coming in. Okay. Coming in. M dot times velocity. Okay. Rho AC. That's M dot. M dot is rho times A times velocity. Rho times A times velocity. Going out. Okay. Rho, of course, changes. Rho AC. Times the velocity. Velocity DV. Okay. Okay. C minus DV, pardon me. This is M dot velocity in. M dot velocity out. Okay. That's how you write momentum. Why isn't this rho plus D rho? And why isn't that the velocity C minus DV? Because it's right up here. The mass that comes in the left goes out the right. M dot on the left equal M dot on the right. Take the easy way out. What's M dot in the left? Rho AC. What's M dot out the right? Rho AC. Okay. Here's where people always mess up. What's the sign? Okay. Which way is the velocity vector? To the right. Positive. Which way is the dot product of the velocity vector and the area vector? The velocity vector points in. The area vector points out. Minus sign. Dot product. Minus sign. Okay. Velocity vector to the right. Plus sign. Velocity vector. Minus sign. Dot product velocity dotted with area. Minus sign. Okay. Right-hand side. Don't forget the area vector always points outward normal to the control surface. Area vector points that way. Area vector points that way. Area vector points that way. Area vector points that way. Which way is the velocity vector? Plus sign. Which way is the area vector? That way. Take the dot product. What's the angle between them? 0. Sign? Cosine 0. Plus 1. Plus. Plus sign. Okay. There it is. Obviously some things are going to cancel out. This guy, if I do it right, I'll get it right. M dot's rho AC times the velocity is C so it's squared. Okay. So we reduce this down. Here's a PA. There's a PA. They go out there. Okay. So we get -- Oh. There's an area vector there. I mean area, area, area. Cancels out all three terms. This simplifies, okay, about two lines. Simplified to get DP equal rho C DV. I'll do it right here. Okay. Combine these to get -- This is what I get then when I combine these guys. DP equals C squared D rho. Or C is equal to the square root of D DP D rho. Okay. You put on here subscript S [inaudible] entropy's constant. Okay. Isentropic. Isentropic. What's isentropic mean again, from thermal? Adiabatic reversible. Is it Adiabatic? Is there a lot of heat generated? I don't think so. I don't think my voice is generating a lot of heat in here. No. Is it reversible? Yeah. Kind of. It's reversible. It's such a gentle wave. Differential changes in properties that we treated as if it were reversible. Is the shockwave reversible? Oh, my gosh, no. It generates tons of heat. Tons of heat. Lots of friction. So it's not reversible. No. No. Shockwave is -- It's a shocking thing, you know. It's not isentropic. My voice in your eardrum is pretty much isentropic. Okay? I could scream and it still wouldn't matter. I can't reach mach [inaudible] my voice. So okay. So okay. We put the S on there. Why? Because we're going to use this guy down here. If the [inaudible] is isentropic, that's true. If it's true, we can do this guy right here. Okay. So we can take this guy right here. And we can say DP D rho. Because that's what I need. DP D rho. This is P rho to the minus K. Okay? So okay. You differentiate that guy. You can do that. You get K. P over rho. Perfect gas. KRT. There's two lines missing there. You can differentiate that guy, though. Minus KP to the rho to the minus K minus 1. That's it. Put that result in to here. Okay. That's the speed of sound we're going to use. The speed of sound can be calculated as the square root of K times R. And [inaudible] the gas. Is it air? Is it nitrogen, carbon dioxide? Times the absolute temperature of the gas. It's a perfect gas. Okay. The process assumed to be isentropic because it's a very weak pressure wave. Okay. So now we have an equation for the speed of sound. And what is it a function of? Well, it's a function of absolute temperature. What does that mean? The hotter the temperature, the faster the speed of sound. The colder the temperature, the slower the speed of sound. Okay. So let's then define. Go back and we talked about the mach number in the dimensional analysis chapter. So recall the mach number. And that was defined as the local speed divided by the speed of sound. So let's do a simple little problem. So here's the ground. And let's assume that there's some aircraft that's flying. Let's see. Where the -- Okay. Here it is. Sea level. So let me take sea level here and show this picture here. The velocity of that aircraft at sea level. I'm going to make something [inaudible]. 580 miles per hour. F-18 flying over the ocean. Dot sea level. I want to find its mach number. Okay. Mach number. V over C. C is the local speed of sound. V is the velocity of the aircraft. I can't use miles per hour. I'm going to convert to feet per second. 850 feet per second. Okay. First thing I'm going to find. C. Two ways to find it. Number is you can go ahead and take the square root of KRT. K is 1.4. For air, R is 1,716. Okay. They came from properties of air in the back of the book. Let's go to properties of air. Oh. Okay. Yeah. That's right. That's good. Okay. I don't know if they gave us that. Oh yeah. Here it is, I think. Maybe. Maybe they did. Nope. They didn't give us the value in English units. SI. So that's what they are in English units. We have to find a temperature. Okay. So we go to the back of the book and we say, "Table A-6, Properties of the Standard Atmosphere, Table A-6." Properties of standard atmosphere. Okay. T. Z is the elevation, and then temperature in Kelvin. Z in meters, and temperature in Kelvin. So that's the only table you've got, and you've got to convert to feet, to meters, and Fahrenheit to [inaudible] and to Kelvin. Whatever. No big deal. Sea level. Okay. He's got it. He's got it. He's got a zero there. Okay. I have to shift these things over. Okay. There's zero. 288.16. 288.16. Okay. So convert that guy. I did. The temperature, let's see, there it is. 59 degrees Fahrenheit. 519 degrees R. 460 plus 59. Convert to absolute. Put those guys in there, K, R, absolute temperature. And the speed of sound at sea level for the standard atmosphere is 1,117 feet per second. Got it. The mach number is equal to the speed of the aircraft, 850, divided by the local speed of sound. So our mach number at sea level is 0.519. Is that a compressible flow problem? Oh, yeah. If the mach number is greater than 0.3, we've got to include compressibility effects. Yep. It is. What does the word local speed mean? It means the speed at the location where the aircraft is. Okay. 580 miles per hour. What does local speed of sound mean? That's the speed of sound at the location of the aircraft. Right there. What's the elevation above sea level? 0. Where do I go over here? 0. Now if you'd rather, there's another column in there called -- He calls it A. I call it C. Don't forget. Oh yeah. He gives it right in there. You don't need to go through here because he tells you the speed of sound at sea level is this number. Convert that number, meters per second, to feet per second, and you would end up with 1,117. Okay. So two choices. Put it in the equation yourself and find it or go to that table A-6 and it's in the last column on the right-hand side. Speed of sound depending on the altitude you're at above sea level. So his mach is about 0.52. Now you take that same jet fighter and you go up to 40,000 feet. Okay. I fly at the same speed. It's a calm day. This is a ground -- It's called the ground speed. 580 miles per hour. Okay. Mach number is equal to V over C. Mach number is equal to the velocity. It's the same. 850 divided by -- Okay. Now it's got to be the speed of sound at 40,000 feet. Just so you know, at 40,000 feet, with the standard atmosphere, the temperature up there is a mighty cold 67.6 degrees Fahrenheit. Absolute 392 degrees Rankine. Of course it is, you know, at the outside of an aircraft flying at 30,000 feet. You know it's cold out there. Yeah. You go up to 40,000 feet, it's really cold out there. The air. What happens? Oh, here you go. You plug it in there. Now you've got 392 compared to 519 sea level. So now your denominator here, what does that mean if it's colder? Lower temperature? Speed of sound goes down. Speed of sound went down. Aircraft velocity, still the same. 580 miles per hour. The mach number at 40,000 feet? 0.60. This guy says, "Hey, Joe down there at sea level, guess what my mach number is? 0.6." Joe says, "Oh, no. Mine's only 0.52. You're going faster than I am." No. You're not. They're both going the same speed. The mach meter isn't giving you a correct indication. The mach meter is giving you the mach number, but your actual ground speed is the same for both aircraft. So, you know, it changes. The velocity is the same, but the mach number's different because the temperature of the air is different at those locations. Okay. Now let's go ahead and take another -- That is what we need now to get the speed of sound and the mach number. Okay. So let's take a look at one more thing before we go in to our review. We're in good shape. Consider a flow. From point one to point two in a flow field. Okay. Point one. Is point two. All right. We're going to consider point two to be a stagnation point. This was covered in fluids one. A stagnation point is a point where the velocity comes to rest at 0. So fluid is brought to rest. V2 equals 0. It's brought to rest adiabatically. No heat transfer. The equation we get. It's equation 923 in the textbook. C sub PT plus V squared over 2. And that's the temperature at 1, but just call it general. Is equal to C sub P times T naught. Plus -- I guess we should -- The stagnation point. The velocity. In my picture here T and V are at any point in the flow field. T naught and V naught are at the stagnation point, at point two. So point two corresponds to the subscript 0 stagnation point. This guy's general which corresponds to point one. Where'd this come from? This is energy. I'll just give you a real quick -- C sub PT. I probably erased it. Maybe I didn't. C sub PT is a change in enthalpy. This is the enthalpy at that point. This is the enthalpy at that point. Okay? This is the kinetic energy. We ignore the change in Z. Okay? So there's a number of assumptions in here. It comes from the energy equation of a fluid. But we're going to solve that for stagnation point. So okay. Let's go ahead and do that. Okay. Rewrite. And we get -- If we rewrite this, T naught over T. V squared over 2. C sub PT. Plus 1. But for a perfect gas. C sub PT. Okay. So T naught over T. I'm going to put this down here then. So I took the [inaudible] equation. I worked on it, and I assumed a perfect gas. And I end up with that guy. Okay. This is for an adiabatic process. This guy here. Okay. Let me show you where that guy there came from. There it is. Okay. I'll put it up here. I'm going to get that C sub P. Okay. R equals C sub P minus C sub V. Got that? That's earlier on. Probably over there somewhere. Yeah. Right there. Okay. A C sub P over C sub V. Okay. So take that R divided by C sub P. 1 minus C sub V over C sub P. This is equal to -- And there's my K. Okay. C sub P and C sub V. So when you bring in the value of K, you solve this guy for C sub P. KR over K minus 1. This is 1 minus 1 over K. This is K minus 1 over K. So what is C sub P? Put C sub P over here. C sub P is K times R. Got it. Divided by what? K minus 1. Got it. That guy goes in there, and you get this. Okay. What is K times R times T? It's over there probably. Speed us up right there. There it is. Square root of KRT. This is the square root. Squared. So it's C squared. Okay? K minus 1. K minus 1. Put it in there. You have to go through all that thermal stuff to do it. I'm just touching the surface. I'm just telling you that's what you've got to do. Compressible flow is very mathematically oriented. Okay. But that's the proof of that guy, in case you want to know where that -- how you went from there to there. There it is. Okay. Now I'm going to erase this because I need this space right here. Okay. Okay. And I probably erased what I needed. Yeah. I think I did. Yes, I did. But over here I had those equations. I'll write them again. Okay. Equations 926. What are they for? Isentropic. Got it. I'm going to get rid of that T naught over T, and I'm going to replace that with a P naught over P. There's a T. What's point two? T naught. What's T1? T. T naught, T. T naught, T. There it is. Take that guy right there, and change that guy so it becomes P naught over P. And there's -- There's a rho naught over rho. This is a T naught over T. This is a rho naught over rho. Convert that guy to rho naught over rho. Those are the three equations that we will be using later on too. And what are they supposed to give you? Okay. Here's what they give you. In a flow field from any point one to a point two where point two is the stagnation point, if you want to find the stagnation temperature, use this equation. If you want to find the stagnation pressure, use this equation. If you want to find the stagnation density, use this equation. K depends on the gas. Air. 1.4. M is a mach number. T is a temperature. P is a pressure. Rho is a density. At some other point in the flow field. Point one in my picture. Okay. >> Professor. For the rho equations, the last one, why is that it's to the power of 1 [inaudible]? >> Yeah. These two guys here. What is I divide one by the other? Do the Ks cancel out? That's the game. The little math game in there, you know. You're going to find out when you do that that you're left with K minus 1 only. 1 divided by K minus 1. Yeah. Like I say, there's a lot of math between the lines to go through every gory detail. Which we don't have time to go through every gory detail. But that's a good point. Thank you. Okay. Good. So this is going to give us a good starting point, and we're ahead of the game on compressible by almost a week which is great. We need that time at the end because we've lost 2 holidays in a 10 week quarter. So we're going to be moving fast in the last five weeks. We moved pretty fast in the first five weeks too. Okay. So let's talk about -- Let's talk about the midterm coming up on Wednesday. Okay. It will be the full 75 minutes. Okay? There will either be three or four problems, depending on how long they are. Three or four problems. You'll be given a data package when you come in for the midterm. You'll put your name on the data package. You'll turn it in at the end of the test with your exam. I keep the data packages. I turn it back to you for the second midterm with your name on it. You turn it back in at the end of the exam. And then I give it to you before the final exam in class time. And you use it, and you turn it back in when you're done. So it will be your data package. You can mark it up if you want. You get it back for the first midterm, the second midterm, and the final. The one on Blackboard website is almost correct, but I had to add a Moody chart. So the one you'll get for the exam has a Moody chart attached to it because it's not what's on Blackboard. So only difference is there will be a Moody chart attached which is not on the Blackboard example of the data package. You're allowed to bring in with you one equation sheet, both sides, 8 and a half by 11. Anything on that is legal. Anything on it is legal. Okay. Homework problems. Example problems from the textbook. Stuff I had in class. You should of course try and put all the equations I boxed in class on there plus whatever else you might have used for homework. Any equation used for homework or I used up here to solve a problem on the board as an example should be on that equation sheet. Let's see. What did I miss? Anything? I don't think so. Okay. So here again this stuff should be pretty fresh in your mind. You went -- Oh. You went through the example that the first midterm from the Fall quarter -- So you know what I asked there. But here's kind of, you know, what we're covering. So first midterm review. And if you have any questions, I've got a two hour office hour tomorrow. I've got a one hour office hour on Wednesday. Okay? So three hours you can come and see me with questions. Obviously series pipes, parallel pipes, a combo problem, series, parallel. Branching pipes. So we worked one of each of those in class, and you had some for homework. We looked at two configurations about the series and parallel. Here's one and the water flows down in pipe one, and the water flows down in pipe two. This is pipe one. This is pipe two. Then we have one here, and here, and here. Water flows down there. Water flows down there. Here's pipe one. Here's pipe two. You can carry it on and on. I'll get tired, but I'll do one more for you. Here's one here. Here's one here. Comes down here, branching. You get the point. Series pipe. Parallel pipes. Series, parallel, pipes, one of each. It could be -- I guess I'll go through them. Why not? One. Two. Put a pressure gauge here. P1. Put a pressure gauge here. P2. Okay. Given P1 and P2, find the flow rate Q. Given Q, find the pressure drop across pipe one. Find the pressure drop across pipe two, given Q. Here's a pressure gauge here. Here's a pressure gauge here. Pipe one. Pipe two. Pressure gauge here. P2. Flow rate comes in here. Q1. Q. Flow rate goes out here. Q. This flow rate, Q1, this flow rate, Q2. This flow rate Q1. This flow rate Q2. Flow rate there's Q. Flow rate there's Q. Those -- This kind of problem. That kind of problem. For what? Series pipes, parallel pipes, series/parallel pipes. Now the branching pipes, I'm not going to go through that. We did a problem in class. You had a problem in the homework like that. But there won't be more than three reservoirs. Okay? You could have more than three reservoirs, but for homework you only had three. For the exam, the max you can have is three. I won't do any more than three. So three pipes. I'll just tell you one more time. If you don't know Q when you start, if somebody gives you the pressure drop here and says, "What is Q?" Somebody gives you the difference in elevation and says, "What is Q1 and Q2?" You guess the friction factors. You go through it and you get the velocities and the pipes, and then you get the Reynolds number in those pipes. Go to the Moody chart. See if the F value from Moody chart is what you guessed. If not, you say in words, "I would repeat the above calculations with these new F values." F1 new equals fill in the blank. F2 new equals fill in the blank. But don't do any more work. Don't iterate. Go through it one time and tell me what your next F guess would be. Okay. Now let's go to pump performance curves. Okay. We talked about typically centrifugal pumps. Okay. It would be centrifugal pumps. This could be, for instance -- Let me do it this way. Okay. Let me just show you. I'll do one over here. You can be given the pump head curve. Q versus the pump head. And you could be given the pump efficiency curve. Okay. That's a pump performance type graph. Pump head versus flow rate. Efficiency versus flow rate. Or you can be given one like I had before where this is the pump head, and these are the efficiency lines here. Okay. And then down here is the power in to the pump from what? A motor. Yeah. Power in to the pump. There were different pump [inaudible] diameters, but I'm not going to give that. Looks like that. And then down here is the NPSH. Okay. So again one curve there. NPSH. Those are all pump performance curves. So just be aware of the different ways that pump performance can be given to you. A graph like that with efficiency or maybe a graph like that with efficiency. Okay. Obviously find -- Let's just stick with the pumps. All right. With the pumps, know how to operate with NPSH. So NPSH. Data. And we have that given. It's on here. It's on most of these guys on this graph here. It's stuck up here at the top. So it separates it out of the way, but the graph's have NPSH on them. What is that supposed to give you? Well, if you're worried about cavitation, then you better use NPSH. And you saw the [inaudible] midterm I did ask that. You might be asked to find the pump's specific speed which tells us -- which tells us what kind of pump should I select for a given job. Should I select a radial pump, a centrifugal pump, an axial pump, or maybe a max flow pump? Okay. It depends on the value of that N sub S which pump you're going to maybe select. And then you want to know the system curve. How do you find it? Well, you write the energy equation from point one to point two. Here's point one. Here's point two. Velocity at point one? 0. Velocity at point two is 0. Pressure at point one? 0. Pressure at point two? 0. What's the difference? Elevation. I call it A and B. that's one and two. That's A. That's B. What drives the flow? The difference in elevation of the free surface, ZA minus ZB. Okay. That's the system curve. Is there some friction in that pipe? Oh, yeah. That's HF1. Is there friction here? Yeah. That's HF2. Do to the HFs add? Yes. They do. Do the HFs add? No. They don't. HF1 equal HF2. What's the friction loss here from point A to point B? HF1 plus HF2. Don't use HF3. Or you could. The friction loss from A to B. HF1 plus HF3. What isn't it? What it isn't is the friction loss from A to B is HF1 plus HF2 plus HF3. No. You're wrong. Watch my pen. Some water goes that way. What's that going to be? HF1 plus HF2. Or some water goes this way. But this is what water doesn't do. Comes down here, goes through here, goes back through here, and back out. No. Doesn't do that. Doesn't do that, believe me. Believe me. Doesn't do that. Okay. Okay. So know how to draw the system curve. Are there minor losses? Is there a valve, an elbow, in there? Okay. Include them. K times -- [Inaudible] Ks times V squared with 2G. Plot the system curve on the pump head curve. Where the two curves cross is called the operating point. OP. That's where the pump is going to operate. From that, you can get the pump efficiency. You can get the pump flow rate. And you can get the pump head. Okay. Operating point. Now pumps in series. Why? Because I want a high head. Or you can do pumps in parallel. Why? Because I want a bigger flow rate. Typically. Typically. It doesn't always work that way, but you can see a general rule of thumb is that. Pumps in series to get a high head. Pumps in parallel to get a high flow rate Q. And that's it. That's enough, believe me. That's enough. Okay? Okay. Now we are -- I have 10 minutes more before the end of class. Some people say, "I can't make your office hour, Professor [inaudible]." And so on and so forth. Well, okay. Here's your time. I can stand around for a while now and talk to you. If you have any questions on that practice exam or anything about the first midterm, hang around. I'll be here to talk to you. So if not, we'll see you on Wednesday or office hours Tuesday or Wednesday from 10:30 to 11:30. Okay. See you later.
