>> So here are pumps in a series. So we have a flow rate coming in. And we'll call this "pump a." And then series means there's a pump b. And, of course, because they're in series, in steady state, the flow through pump a must equal the flow through pump b. Conservation of mass. Now, we have the pump head. So pump head is, there's two in series, the pump head 1 plus the pump head, that's called "a," pardon me. Pump head for pump a plus the pump head for pump b. You know, it looks pretty similar to pipes in series. Pipes in series, qa equal qb. Pipes in a series, the total head loss due to friction is equal to head loss due to friction in pipe a plus the head loss in friction due to pipe b. So, yeah, they're similar. That's good news. Now, pumps in parallel. Okay. Okay. Flow rate, we'll call this "qa," "qb." Now we have a flow rate coming in, q. And now we have two pumps, pump a and pump b. qa, qb. Just like pipes in a series. Conservation of mass. The flow rate that comes in, q equal qa plus qb. The pump head in pump a, pump head a, equal pump head b. They must be equal, because they're in parallel. We're going to look only at identical pumps. They won't be two different kinds of pumps. That's another level of complexity. So we'll just focus on these problems for homework or in-class, whatever. They'll always be the same pump, the same size, the same physical pump. Okay. Two equations. Conservation of mass, energy. Conservation of mass, energy. Okay? So we start off, and let's do series first. All right. So here's the curve. This is a flow rate, q, again. This is the pump head, hp. And let's say that this is pump a. These are in series. So pump a. So if they're in series, here it is, at any given flow rate q, at any given flow rate q, you add the pump heads together. You add them. I'm going to take a flow rate of 0. Here's the pump head for one pump. Add it to the pump b flow rate of 0. There. Take this flow rate. There's the head of pump a. Add it. There's the head of two of them together. Take these two, right here, take that distance. Add it. There. Take this one, here to here. Add it. There. Take this one, 0. Add it to 0. 0 plus 0, 0. Now connect the dots. There's two pumps in series, identical pumps. How do we do it again? Okay. You choose arbitrarily a flow rate q. You take the head of one pump at that flow rate q. Let's say the head is ten meters, ten meters. Add it. 10 plus 10 is 20. That's 20 up there. So you take the head for one pump, and you add it for two pumps, identical. That's how you construct the curve for our two pumps in series. Okay? Now we take two pumps in parallel. Same thing. We take q over here. We take hp up here. Okay. Here's pump a. Now the rules change. It says now what you do, the pump heads are the same, but you add the q's. Okay. So I'll start here. I'll make something up. I don't know. Let's just say 50, just for fun. Okay. The pump heads are 50. Add the flow rates. What's the flow rate through pump a? 0. The flow rate through pump b? 0. 0 plus 0 is 0. Okay? So we're still here. Okay? But now that's our starting point. Now, we take another, choose a pump head now. Here we chose a q. Now you choose a pump head, horizontally. What does it say? Add the flow rate through pump a to the flow rate through pump b. They're identical pumps. Here's the flow rate through pump a. Add it. Here's the flow rate through pump b. There they are. That's the first dot, second dot, pardon me. Guess another hp from the graph. Here's the flow rate through pump a. Add it to flow rate through pump b. There. Take the one down here, at this pump head. This is a flow rate through pump a. Add it to the flow rate through pump b. There. Now connect the dots. Two pumps in parallel. Okay. That's not there yet. There's ten pumps in a series? That's okay. You can do that. Just multiply it by a factor of ten. Okay. So there's how you add pumps in series and parallel. Now, we talked about the operating point before. What's the operating point? It's the intersection point between the pump head curve, and the system head curve, graphically, where the two lines intersect. We said previously that if you had q and here's hp. Here's the pump curve, hp. Here's the system curve, h system. The intersection point is called the "operating point." It gives you a flow rate, and it gives you a head. That's the flow rate that would occur if you put that pump in that particular system. Where does the system curve come from? The geometry of the problem. Are there two reservoirs separated by 50 meters, 1,000 feet of pipe, one-meter diameter, the pipe is made of commercial steel? All of those things go into the system curve. Are there minor losses? If there are, it goes in the system curve. So that's the system. This is a pump you put in the system. Where do they intersect? That's called the "operating point." You should be close to the maximum efficiency to be efficient in your pump operation. So you want to have that point close to the peak efficiency if you can. Over here. Okay. Here's a system curve, h system. If you only have one single pump, here's the operating point. Typically, why do you put pumps in series? Well, you can see what happens here. You develop a much bigger head, a much bigger head. So, typically, there's always exceptions, but, typically, you put pumps in series that generate a high head. When they want to pump oil out of an oil well, and the oils way down deep in the earth, and it's not coming out very fast, and you want to pump it out of there, you're going to be pumping from a great distance down there, a great distance down. Your head to pump is going to be big. You've got to pump oil from way down there up to the surface. If you want to pump out of a high head situation, you put pumps in a series. Typical, in maybe an oil well. Typical, maybe 20 to 25 pumps in series. Not two. I said 20 to 25 pumps in series. Why don't they just use one big pump? Well, if you want to use one big pump, you've got to drill a hole, I'm going to make something up, you've got to drill a hole five feet in diameter. Oh, my gosh. How do you drill a hole that deep into the ground to get the oil out? No, no, no. You drill a small hole, and you put 25 pumps on one shaft. And those 25 pumps are in series. And they pump oil out of great depths of oil wells to the surface. They use pumps in series. So now you take the case of pumps in parallel. I'll make this up. Let's say that this is the system curve for that. Here's a single pump. Okay. Here's two pumps in parallel. What did you just do? Wow, did I increase the flow rate dramatically? The flow rate was here. Now the flow rate is here. Did the head change by much? No. Did the flow rate change by much? No. Did the head change by a lot? Yes. There's always exceptions. But, typically, this is kind of the rule. If you want to develop more flow, put the pumps in parallel. If you want to develop a higher head, put the pumps in series. Okay. So that's kind of the rules of how you operate with these guys. And now we'll take an example on that, and see how that works out. Let's see here. Okay. I'm going to take these two first. Okay. So the example I'm going to take, and I'm going to put that I think in the middle panel for right now. Okay. Okay. Between two reservoirs. All right. So let's go ahead and make that between two reservoirs. Let's see what we're supposed to find here. Okay. We know that. All right. So we'll take him. Let's make the lower reservoir on the left. Upper one on the right makes it more realistic. So we'll take this reservoir here,-- z1. Okay. And let's make sure that I call that by z1. Yeah. Here it is. Okay. Okay. Let's see if I've got that here. Yeah, we'll kind of, I've got it right here. Yeah. This is good. Okay Let's see what we've got then. This is what's given in the problem statement. We're given a pump pumping water up to an upper reservoir. Okay. z2, z2. We're given that delta z is 15. Okay. We're given that d is 300 meters, pipe diameter, 300 millimeters. And the length of the pipe is 70 meters. And the friction factor is 0.025. And for minor losses a summation of the case is 2.5. There's elbows in there. There's entrances in there. There's exits in there. There's all kinds of things in there. So we don't know. He wasn't specific on that. But we know that that's what it was. Okay. So 15. All right. So, first thing, find, I guess the [inaudible], yeah, find the flow rate and the pump head. So find the flow rate q, and the pump head. And I'll draw the picture for the pump over here. Yeah. So he gives us the picture for the pump head. So let me draw that pump head picture. Starts out at 21, 2, 3, yeah, 23. Let me verify that. One, two, three. Okay. q versus pump head starts out at 22.3. Okay. And then the flow rate 0, 0.1, 0.2, 0.3 cubic meters per second. Okay. And when it gets down to .3, it's about 16.3. It's down here roughly to 16. So here's 10, here's 20, here's 30, 40, and so on. So this is given to you. That's given to you from the manufacturer of the pump. That's a pump head curve as a function of a q. Okay. So, now, let's do part a, for a single pump. Okay. Single pump. Okay. Here's a system curve, delta z plus f l over d, v squared over 2g plus summation of k's times v squared over 2g. All that system curve depends on is how far apart the reservoirs' free surfaces are, how big is the pipe, how long is the pipe, what's the pipe made out of, and what are the minor losses in the line. Elbows, valves, entrances, exits. Things like that. Okay. So this guy is equal to our difference in elevation of them, which is 15 I guess, right, 15 plus, I'm not going through all of that, it's 85 q squared. Replace the v with the q. q, q equal va. So replace the v with the q divided by the area. We know the area. It's pi d squared divided by 4. There it is right there. So get rid of that v. Why? Because I've got a q in that graph. I don't want v. I want q in my equation. So get rid of the v there and there. And put in terms of q. I got it. Done it. Right there. Now, I plot the system curve on the same graph with the pump head curve. I plot this guy. When q equals 0, where does he start, 15. Okay. Got it. q equals 0. We start right here. Plot the point. This is what I did, actually. If I can find my notes here. Let's see if I can find what I did here. You just go ahead and guess some values of, yeah, this one right here. Okay. There's 100, nope, I don't have 100. Yeah. I guess I didn't put those values down. That's okay. We can go ahead, and just show you how it looks. I'm going to guess q equal .1, .1. Okay. Put that guy here. q squared, .1 squared point, .01. Okay. Boom, boom, all right, 15.85. So here it goes up to here. Let's see at 3 where is it. It's about, oh, okay, it's about 22 and a half. At .3, it's up at 22 and a half. Right about here. Okay. You guess some q's. You put the q's in here. And you plot the system curve. So what I did here, h system. Okay. First of all, you write the system curve like this. You know where it came from, the energy equation. Okay. You put in everything. You guess some q's. You plot the curves. Where they intersect, that's where the pump is going to operate. There it is. That's the operating point. If you draw it like I did, in engineering green, engineering green, I plotted it, and you go down here,-- You're going to get a value of q, 0.23. All right. So now I know. Single pump, q operating, 0.23 cubic meters per second. The pump head. There's a pump head. The pump head is 19.5 meters. Okay. One pump. That's enough. That's review. We did that before, one pump. We're not interested in one pump. We're interested in multiple pumps. Okay. So now we're going to say, second part of the problem, delta z is still 15. But now I put two pumps in parallel, two pumps in parallel. Here it is right here. Pumps in parallel. Here it is right here. Single pump here. Two pumps in parallel. Okay. What do I do? Okay. I choose some values. I'm going to change this now. I choose some values on the pump head curve. Choose q equals 0. Okay. So I chose q equal to 0. Okay. What's the flow rate 0? What's the flow rate through pump a? 0. What's the flow rate through pump b? 0. And together, 0 plus 0 is 0. Okay. Start out there, 0. Choose randomly. Pick a number on the y axis. There. What's the flow rate of .17? .17 plus .17, .34. Find .34. It's out here, .34. Got it. How about this point right here at that value of pump head? I don't know what that is. Maybe this was 20. That might be 13, 14. .3, .3. .3 plus .3, .6 out here. A big, fat black dot, .6. Connect the points. Here goes through here goes through there. Good. Two pumps in parallel. Where's the operating point? Where the two curves intersect. Right there, operating point, two pumps in parallel. Okay. Now if you go down directly down,-- The flow rate, I'm pretty good on this board, I can't believe it, there's q, operating point. It's .29. Part b, two pumps in parallel. q operating point is equal to 0.29. The pump head at the operating point, the pump head, 22.2. It's really close. There's the pump head, 22.2. But you have to be a little bit cautious here, because that's the flow rate when the two pumps are in parallel. Okay? There they are, two pumps in parallel. If the total flow rate is .29, guess how much goes through a? Half. How much goes through b? Half. That's the key. Okay? There it is right there. What's that q coming in? .29. What's that guy? .29 divided by 2. What's that guy? .29 divided by 2. Okay. Flow rate's half. How about the pump head? 22.2. Check it out. 22.2, 22.2. Conclusion. At the operating point, the pump head for both pumps is the same, but the q's add to get the total flow rate q. Okay.? Now, we do one more. Two pumps in parallel. Okay. I'm going to redraw this now. Yeah, I will. Okay? Let's see. Leave him on there. Single pumps okay. Okay. Get rid of him. Get rid of him. We'll start fresh. Okay. Okay. Get rid of him. Get rid of him. Get rid of him, and him, and him. We're almost there now. We don't want this guy. We don't want this guy. Okay. Okay. So now let me read the third part. This is a three-part problem. Part one said put a single pump in there. Got it. Part b said put two pumps in parallel. Got it. Part three says now we have, for part c, we have the pump layout, discharge and head for, delta z now is 25. So now delta z equal 25. Okay. Delta z is 25. So we go back to our system equation. What's delta z? 25, 25. Okay. So now we draw the curve. Start when q equals 0, when q equals 0. h is 25. And now you guess q. Yeah, go ahead. Which one now? This one? Yeah. >> Do they add any length to the height or the friction pad [inaudible]-- >> They don't. That pipe is so long you might add two meters or something like that. But, no. I mean, that length of pipe, 70, is this piece plus this piece plus this piece plus this piece, this piece, this piece. Okay. You don't really include these losses in here. Really, what it is, from here to here, and from here to here. That's the 70. This plus that. These guys, it's assumed the pump just pretty much, the pipe was in the pump real quick. So you don't throw that. That's minor, typically, minor. Guess q equal .1. Okay. .1 squared, .01. Dah, dah. 25 and 8, 33.5, 33.5. Let me get it drawn correctly. Okay. So if I go out to .3, I'm roughly at 32 and a half, .3 I'm roughly at 32 and a half. About there. Okay. At .2, I'm about at 26, .2 I'm roughly at 26. Okay. And at 0, it's 25. Oh, no, it's, pardon me, 27, 28. At there is 28 out there. So it's up here. All right. So here's a system curve. Here's the pump curve. I say, okay, now. That single pump, when I turn it on, what's the flow rate going to be? And I say, gee, I don't think I can tell you, because supposedly it's at the operating point. And the operating point is where the system curve intersects the pump curve. Conclusion. That pumps going to sit there, and it's going to rotate, because you've got it plugged into the power supply. But it's not going to move any water. It's not going to work. It's not going to work. Okay. So my system curve is too high. I've got to get more system head. Let's see now. What was a conclusion? If I wanted to get a bigger system head, what do I do? I add the pumps in series. Okay. Conclusion is I have to add another pump in series. Okay. There's one pump. How do we do that? Here's the game again. Guess a q, get an h pump, double it. Stop. Guess a q, double it. Stop. Guess a q, double it. Stop. Guess a q, 0, double it, 0. Got it. Go over here. h pump, 22.3 and the flow is 0. 22.3, double it, 44.6, 44.6. My first point at 1/10. Just choose them arbitrarily, double it. There. At 2/10 double it. There, with the ruler, if you want, or read the numbers off the axis. 3/10 here. Stop. There. Connect the dots. There. Now this is two pumps in series. Now you say to yourself, "Do they intersect?" And the answer is yes, they did now. Two pumps are going to work. Here's the operating point. Two pumps in series. And if we do that, and we go down on the curve, we get q. It should be, if I did it right, it would be-- When you plot it on engineering green, you'll find out, it's .3. So here. From graph operating point, q operating point is equal to 0.30 cubic meters per second. The pump head operating point is equal to, let's see if I can do my graph relatively correctly, 32.7. Yeah, not bad. h, two pumps operating point, two pumps in series, 32.7 meters. Here's a picture for part c. Okay. Now comes a question. For a single pump, one pump, for pump a, what's the flow rate? Ah, it's got to be him. Why? Right there, pumps in series. There it is. Okay. Here comes the other question now. For a single pump, what's the pump head for a single pump? Oh, okay. They add together. They're equal. Okay? They add together, not equal. They add together. Okay. So 32.7 equal h pump head for a pump a plus pump head for pump b. For a single pump, divide it by 2. Okay. Divide it by 2. Okay. Now, just so we know, you could also be given an efficiency curve, the efficiency curve, for pump a or pump b. They're identical. Efficiency curve. And, so, let me redraw, because I want to be closer to where I'm operating this than I am right here. That's fine. Let's do it this way. Okay. So for a single pump, what's the efficiency for the single pump, or what's the flow rate for the single pump there in series? .3. Flow rate for a single pump, .3. What's the efficiency? Right here. There's the efficiency right there. What if I want to get the power into the pump from a motor? So find input power. I don't have an input power on the graph. I don't have that. But I do know this. The efficiency of the pump equal w .pump out divided by w .pump in. I want that guy. The power into the pump from the motor. Okay. I can do it. w power into the pump equal w power out. Okay. What's that called again? The water horsepower. It's gamma q hp divided by the pump efficiency. So I go to the graph with the flow rate .3. What's the flow rate? .3. I get the pump head. I know the efficiency. I know gamma. I solve for the power n. So that's how I get the power n. If I'm given an efficiency curve on that graph, I can do that. Yeah. So there's different approaches. This is a good example, because we start off with a single pump, we then put two pumps in parallel. We solve that. And then we say, well, what if we try and pump water to reservoirs 25 meters separation? A single pump's not going to do it. You need two pumps. Are you going to put them in series or parallel? The rule of thumb is if you want a higher head, put them in series. We did that, and we found what the flow rate was, we found the pump head, and we found the power into a single pump, w .pump in. Okay? Any questions on that? Okay. Good stopping point. That finishes our pump story. So we'll see you next Monday then.
