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>> OK, a quick review. We derived the Bernoulli equation in Chapter 3. It's one of the most basic equations in fluid mechanics. It has a number of assumptions about that must be satisfied. They're listed here. Number one is a steady state equation. Number two, the fluid is incompressible and inviscid. Inviscid means that the effects of viscosity of the fluid can be neglected. And finally, it applies along a straight line. And we worked an example
0:51
problem last time from using Bernoulli equation. And we're going to work a few more today. OK. I've drawn one of the schematics for problem 3.24 on the board, 3.24, yeah. You've got problems in that area too, 3.36, 3.19, so it's somewhat similar. Let me read the problem statement from the textbook. A pitot-static tube, now we're going to talk about what that is in a few minutes, but for right now, this thing is called-- a pitot-static tube is used
1:40
to measure the velocity of helium, well it's not here, in a pipe, that's easy. The temperature and pressure are 40 Fahrenheit and 25 PSIA. OK. A water manometer, that means there's water here shown by the diagonal line filled with water. Connected to the pitot-static tube indicates a reading of 2.3 inches. OK. So we've got 2.3 inches. Determine the helium velocity, which is V in the pipe. OK, so. All right, so as-- We're in Chapter 3. We
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know we're probably supposed to use Bernoulli equation, so it's not a big guessing game about the equation. The equation is going to be Bernoulli equation. Why? Because there's a velocity in the Bernoulli Equation. There's the Bernoulli Equation. But we said last time there are three different variations of that Bernoulli equation. They're all the same Bernoulli Equation but they're just three different forms. This is called the head form. Everything
2:59
in here is in terms liters per feet. This one is pounds per square foot, our kPa possibility. This one, P over rho version. OK. So we can use any one of the three. I think I'm going to use this guy right here, the third one here. Yeah. OK. So the first thing is pick one of the equations you want to use. Number two is you got to show a point 1 and a point 2. OK. I said last time you typically select points 1 and 2 as-- first of all, if you want
3:42
to find something somewhere, that should be a point. The other thing is if you know a lot of stuff somewhere, that should be a point. So I want to find something somewhere. I want to find velocity. So I'm going to say-- I'm going to call this point 1. That's the velocity at that location. Now comes what-- another point where I know something. Well, I'll tell you what I'm going to do. I'm going to take this as point 2. Just in front-- Now, let
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me just cover this is. This is an open-ended tube, open-ended tube bent like this, with a U, this is a U-tube manometer because it has a shape of a U, a U-tube manometer. But they're different fluid in it, this is helium, a gas, this is water, a liquid. OK? And then, it's attached here. You drill a hole in the pipe and attach it right here. This one sticks down the pipe and it points right directly towards a fluid coming toward it. OK. Now,
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I'm going to write the Bernoulli Equation from point 1 to point 2, P1 over gamma. Plus V1 over 2g, plus Z1, now I do point 2. OK. Why can I say that? Because these three terms are a constant. No matter where along the streamline you are, these three terms are the same constant. So whether I'm at point 1 or point 2 or in the middle here somewhere, those three values are the same constants. OK. Now, I start to make things simpler. Z
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is the elevation of ups and down. I don't know, take it by the Blackboard. Z1 from here up to here, Z2 from here up to here, notice they're the same, it's horizontal. So Z1 equals Z2. OK. Here, here, gone. Now comes the little bit tougher part. At point 2, the velocity is zero. Point 2 is called the stagnation point. Point 2 is a stagnation point. V2 is zero. V2 gone. Why? Well, you know, here it is, when I get to this point here, can the
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fluid flow up this way? No, no, look there's water in there. If the fluid flowed in there, it would blow the water up. No, it doesn't. The fluid is static in that tube. The fluid is static in the tube, which means if it's not moving here, it's not moving here, it's not moving there at the entrance of the tube, that's my point 2. So the fluid at point 2 has zero velocity. Here's my streamline, horizontal, one to two. Here's my streamline. OK. The
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equation now has reduced to something much, much simpler. P1 over, let's do the V1 squared. V1 squared over 2g is equal to V2 minus P1 over gamma. You got to do your bookkeeping correctly. There are two gammas, so you better differentiate those gammas. One is water in a manometer and one is helium in the pipe. Gamma W is water. Gamma with no subscript is what's in the pipe, helium. OK, helium. OK. So let's-- first of all, this is helium
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and we need gamma, let's go ahead and solve this guy out here then. Our density, P over RT. OK. And let's see what our-- P. Well, we have to go through the pressure, 25 PSIA, isn't it? Yeah, 25. Did I write that down? Let's see. Oh, let's put it over here. T in the pipe and P in the pipe. OK. He says, I'll read it again, the temperature and pressure of the helium is 40 degrees at 25 PSIA. OK. So, 25 PSIA, 144, conversion factor. It's
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again, pounds per square foot, pounds per square foot, divided by R for helium. We go to the inside front cover under liquids-- Gas department, oh not inside front cover, that's air. So let's go to the back of the book if you can't find-- If it's not water or air, go to the back of the book. Let's see. We're here [inaudible]. It is in the front. Let's double check that. I think it's in the front. Oh, it was in the front, yeah.
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There it is, yeah. OK. Helium, we have in English, gas constant, helium there it is, 1.242, 1.242 10 to the 4th. And T, don't forget T is absolute, absolute, 460 plus 40 degrees R. The units for helium in the back of the book, what pounds per slugs degree R. And let's see, when I do that, I add up density, 5.80 10 to the minus 4. Slugs predicts what? OK. There you have it. But always check your units, always check your units. Because that
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sometimes is what makes the big difference is you don't check your units. Always check your units. Make sure it's going to be right. OK. We have rho, we need gamma. So gamma equal rho g. OK, that's it, I can get that. I'll replace that as rho G up there. What's left? Well, there's another g right there. They're going to cancel out anyway, OK, fine. I need P2 minus P1. OK. Now comes-- this is Chapter 3. Back to Chapter 2, P2 minus P1, U-shaped
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manometer. OK. Start at point 2, P2. What do we do? We go up. How far up? To that black dot. Minus gamma helium times that distance, don't forget what we said in Chapter 2. If you've got a gas and a liquid in this manometer, pretty much you can neglect the density of the gas compared to the liquid. You can forget about it. What we're going to say then is we're going to say the pressure at 2 in the helium right here, here, is the same as it
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is there. OK. Now I know the pressure here. Now what do I do? I go up. How far up? H minus gamma of water times H. That gives me the pressure right here at the helium of water interface. Now Chapter 2, that's a static fluid, static fluid in there. Everything is static. Nothing moves in that tube. That's the same pressure there in helium as it is there in helium. OK, I'm there now. Now I go down in the helium plus. Gamma helium times
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this distance. I don't care, I'm going to neglect it because I know the density of helium is much, much less than the density of water. And what's it equal to then? That's equal to P1. So what is P2 minus P1? OK, P2 minus P1 is then equal to gamma water times H. So put all that stuff back up here. V1 squared over 2g is equal to P2 minus P1 gamma water times H divided by gamma. OK. And my gamma here is rho times g. G's cancel out. Gamma
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water is-- We said what was 2.3 inches divided by 12-- I'm sorry H of water, 2.3 divided by 12. All the numbers are there. Rho is up there, 2 is down here, square root. When you're done, you get V1, the velocity of helium is like 203 V per second, 203 V per second. OK, that's called-- Let's go back in here again. We're going to do this. OK. When we got this equation, this is in pressure terms, pressure terms, just a quick review. This is called
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a static pressure, P. This is called the dynamic pressure. This is called the hydrostatic pressure. And everything together is called the total pressure. OK? Here we go again, static pressure, dynamic pressure, hydrostatic pressure, total pressure. So, at point 2-- Stagnation by the way, these two guys here are called stagnation pressures. OK, stagnation pressures. So how many pressures are there? One, two, three, OK. The quote for-- These two make five, five
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different pressures. Know what the words mean. So if you get a problem that says the stagnation pressure is this or the static pressure is this, you know what I'm giving you what problem. OK. If-- OK, we'll go up here. This point out here is a stagnation pressure. This point right here is a static pressure. OK. So this pressure tap here measures static pressure, this pressure here, stagnation pressure. Bring the fluid to rest, stagnation, bring it to
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rest. OK. Most people don't-- Why do they do this? Why do they do this? To find the velocity at a point in a pipe or a duct, an HVAC duct, like the HVAC ducts in the room are a big water pipe. If you want to find the pressure or the velocity somewhere, you could do this. OK? Static pressure tap, stagnation pressure tap. But people-- Oh, by the way, did somebody lose a ruler here? Like my [inaudible]. >> That was actually mine. >> OK. >> Thank you. >> He found it for you. Thank [inaudible].
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>> Thanks man. >> Here's a-- This is a pitot-static tube. This is a pitot-static tube. You put that thing in a pipe like that. OK. OK, like that. In the front, there is a hole in the nose of the tube. That's a stagnation point. Like in front of your car is a stagnation point where the velocity comes to zero near the grill. On a missile or aircraft, there's a point in the front where the flow comes to zero velocity, that's a stagnation point.
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I don't care if it's a-- it was a bus, there's a point in the front where the velocity is zero, that's a stagnation point. OK, that's a stagnation point. There is [inaudible] in there. This pressure tap up here measures the stagnation pressure. This side, apart this side, that tap measures that. Now there's also static pressure taps. They're hard to see but they're around the circumference, right here at my fingertip. There's little
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tiny holes drilled in there about eight of them. Around the tube, little tiny holes drilled, eight of them. OK? The streamlines of the-- This is an air [inaudible]. Streamlines go like for this, OK? So the holes are placed normal to the streamlines, that's how you measure static pressure. The holes you drill are drilled normal to the streamlines. That measures static pressure. Now this is a tube within a tube. There's a big outside tube,
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inside here, there's a smaller tube. The inside tube carries the pressure from the nose. The outside tube carries the pressure from the static taps. This one, put it in a duct. This one, the high pressure, goes over here. The low pressure goes over here. Put it on a U-tube manometer. And then measure that delta p, measure delta p, put it up here and find the velocity anywhere you want in the tube or the duct, like this. Put it anywhere you want
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it. So these guys are used frequently. We have one set in the fluids like large lamp, they come in different sizes. There's a little one. In different sizes. There's big ones and there's smaller ones. That's not the only way to measure velocity, but it's one that's used many times. Other things are called hot wire anemometers. There's one little turbine [inaudible] that spins and you hold it outside in the wind. It will tell you the wind speed,
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the little propeller blade spins with the wind and you can get the velocity that way. So, yeah, that's just one way to get velocity. What is a good example of a Bernoulli equation? OK. So know how to do this calculation right here. That is important, so perhaps just doing that. Why didn't I add 14.7? Because he told me it's absolute. That means I don't need to add 14.7, just 25. That is absolute already. OK. Now let's take a look at 361. There are--
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In this chapter, there's a total of what we're looking at of 14 examples. That's one reason we chose this textbook because there are so many work examples in this textbook. So between the problems I've worked in class, the problems worked out in that chapter and the problems you do for homework, you'll have a library of probably 50 work problems, 50 work problems in the second exam. That more problems you see work, the less you're likely to be surprised
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on exam. The more problems you see worked and especially worked it yourself, you can [inaudible] the problems worked out all day long and not to come in exam. But if you go through in great detail, then you'll be ready for an exam that way. OK. So let's take the other one I want to work, 3.61. OK. We'll put that, I didn't put the schedule right here, 3.61, OK. Water is siphoned from a tank. This is called a siphon. That's a way to measure
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velocity over there. Now we're going to do a siphon. OK. So here's the water tank. OK. The siphon is at the bottom. Comes up, goes over, down, comes up this way. OK. That's open there. OK. Let's put some dimensions on this. I think that's eight feet, eight and three. So this from here to here is eight feet. And this distance from the free surface down here is three feet. OK. Purpose of a siphon is to drain a tank without drilling
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a hole in it. OK. So you put a tube in there and you take it up the top, put it down here. And do you need a pump? You're not supposed to. It does it all by itself. Wow, isn't that neat? Yeah, does it all by itself. OK. So let's keep reading. Determine the flow rate from the tank, so find. Now-- There are different things you can find. Number one, somebody can ask you to find V. V stands for velocity. We just did it over here. And you might know
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this stuff from physics, fine. Maybe you had thermal, maybe, maybe not. If you have, this is review but that's OK. V is velocity. OK. Now comes a flow rate. OK. This HVAC duct in here, you think those guys when they analyzed those HVAC duct say the flow rate in there is 50 gallons of air per minute. Are you kidding me? Fifty gallons of air per minute? No, you know the language they speak? CFM or maybe liters per second. No, that's called volumetric
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flow rate. So number two, Q, volumetric flow rate OK. To calculate it, you take the velocity times an area. Velocity, feet per second. Area, square feet. Cube, cubic feet per second or cubic feet per minute, CFM. That's called volumetric flow rate. Now if we multiply that by the density, we get mass flow rate. That's rho AV or rho times Q. Pounds mass per second, kilograms per hour, whatever. Oil pipeline carries so many pounds per hour. A steam power
28:20
plant, the line going to the turf and carry so many pounds per hour of steam. They don't say they carry so many CFM. They don't give you the velocity. They give you the mass flow rate in these power plants or chemical plants. If it's HVAC work, they give you the volumetric flow rate, Q. And once in a while, if they're analyzing flow in a river or something, you might want to know the velocity. You want to find V, because various times you want
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to find each of these three things. It depends on who you work for and what your job is. OK. This particular job says, he wasn't very specific, he said find the flow rate. He means to find Q. OK. I would be more specific on the exam. I would say find the volumetric flow rate for the mass flow rate. There they are, all three of them. OK. And then he says, and also find the pressure at points 1, 2 and 3, 1, 2 and 3. OK Now I'll give you point
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1, 2, 3 in there. OK, so. Have any of you taken anything out of a container? What do you do first? Like how do you do it? >> Putting a hose in like a water troughs. >> OK. >> Fill the hose up with the water completely. >> The hole completely, right that's the key, right? >> Yeah. >> Get everything under water. Get water in there first, right. >> Yeah. >> Then what do you do? >> And then, you cap off the end so you doesn't lose or-- >> Put your finger on one end then raise it out.
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>> And raise it out and then put it underneath the-- >> The level-- level of reservoir, perfect. OK, you're expert at [inaudible]. Yeah, see I hope not. [ Laughter ] OK, right. [Inaudible] steps, one, two, three, exactly right. So you know, you just don't put the tube and then say water is going to come out of here. You know it's not going to come out. No. Put it all underneath that, get them full of water, put your finger over,
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then raise it high here and put it down below this and that makes finger up, all up, so it would come out. OK. Good. Good. OK. We're in Chapter 3, we must have to use Bernoulli's, obviously, OK, is it steady state. Not exactly, which is probably a good point. We just check this, 361-- 361, OK. Yeah, OK. Let me check something else here. It doesn't always say this and I think the next, the next problem, I'm going to work this [inaudible]. I'm going
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to still accept this and see problem-- OK, yeah, problem 380. Let me check from 3-- I'm going to read problem 380 to you. This is a problem with a tank [inaudible] water coming out. He says, the tank is large. So what? The tank is large. They don't always say that. If they don't say it, assume it in Chapter 3. He won't always say it. Sometimes he will. Problem 380 says the tank is large. This problem just says the tank. OK, why is it important
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to say the tank is large? Because if I were Bernoulli's, it got to be steady state. And if I-- if I put this in the [inaudible] that they'll drain in a few seconds, well, there level is going to go down fast. Illegal, illegal. That water should stay the same. If this is big enough and that's small enough, it will be the same, relatively speaking. OK, so that's-- That's how we satisfy that guy. If you don't see the word large tank in a problem in Chapter
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3, you assume there's a large tank, otherwise it's not steady state, you can't be [inaudible]. Number two, incompressible. Yeah, it's water. Inviscid. OK, we have to neglect friction. Is that easy to do? Yes and no. It's not bad here but when we get in those two, there's going to be some viscous effects. There's going to be may be the pressure about due to viscosity in that two. But in this chapter we're going to neglect it, come back in three
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weeks and we'll talk about [inaudible] for viscosity. OK, so yeah, we know there is some viscous effects in here that we're going to ignore for right now, OK. Along the streamline. Well, not yet but it's going to be. OK, here we go again. I'm going to have to start somewhere, OK. So I'm going to call that point 1. Now I tell you where he put points to, 1, 2 and 3. He put [inaudible] change mine here, he call that bottom right there. One of the zero,
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OK so I'm going to call that point 1, so it's going to be point 2, 3 and 4. I've changed the problem statement because I-- He puts point zero here. I don't like doing that. So that's point 1, here's the point 2, here's point 3. Oh here's point 3, pardon me. There's point 3, there is point 4, there is point 5, OK. Now I know with the cycling, water is going to come out of here. That's the sight of this purpose in life, water comes out in
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here. I know, I want to find that-- that velocity V5. Because I know Q is V times A and what A, the area of pipe right there where my fingertip is. OK, so I've got to find V5. OK, fine. Over there ahead of time, V1, what I do is [inaudible] over here, what I do [inaudible] OK from what 1, 2 or 1 and 2. What's my equation? There it is. OK, with my streamline. Right there. OK. Let's write the equation down. There are these-- I'm sorry, there are these
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right there. Let's put a checkmark by what we know. P1 up here, again, this typically means atmosphere, so P zero big, I-- But I know it. V, he said a large tank, therefore the velocity from the free surface is assumed to be zero. Got it? Z1, got it. Z2, got it. V2, I don't know what the velocity is in there. P2, I don't know what the pressure is just at the pipe in the-- just at the pipe in the-- This is not fluid statistics. It's not gamma
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times this because the fluid is moving at point 2. So it's not Chapter 2, fluid statistics, it's Chapter 3. I got a problem. I got one equation, two unknowns. No, that didn't work, try and try again. One to 3. Do I know the velocity of 3? No. The pressure of 3? No. Four, do I know the velocity of 4? No. Do I know the pressure at 4? No. Try and try again. Do I know the velocity of 5? No. Do I know-- Here it comes, the pressure of 5?
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What it [inaudible] to atmosphere. Oh yeah, one equation, one unknown. From where to where? One to 5. OK, so P1 over gamma, V1 squared over 2g, Z1 equal P2-- P5, pardon me, P5 over gamma V5 squared over 2g plus Z5. OK, P1 zero, V1 zero, x of 2, atmosphere at 5, P5 zero. OK there it is. V5 squared equal 2g, Z1 minus Z5. OK, let's see if I found that one. Yeah, there is 13.9 feet per second. Take the square root out. Let's just make that right. OK,
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now let's try the next step. Oh, I didn't get the Q so I'll get Q. OK, two-inch diameter hose, two-inch diameter hose. Given [inaudible]. OK now. Yeah. >> When we're doing Z1 minus Z5, that is going to be positive 3 because if you just do zero. >> Yeah I'm calling, I'm calling-- >> The data at the bottom [inaudible]. >> OK, could we use [inaudible]. Yeah. OK, there's answer part A. Now I'm going to do part B. OK, P2, there is 2. Now I'm-- go back to 2. So I'll start again. There we go. Here
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is zero, zero, Z1. Yeah, 8. P2, unknown, find it. OK, steady state conservation of mass. Same diameter and height. What comes in here goes up here at the same velocity, steady state. You know, I think you-- It's pretty straightforward but, OK. Mass that comes in at 2 in the pipe equal mass that goes out at 5. Rho AB. Is the density the same at 2 and 5? Yes, it's water. Is A the same at 2 and 5? Yes, it's a constant area pipe. Therefore,
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the rho and the A cancel out, which includes then that V2 equal V5. OK. Z2 we call zero. Put the numbers in, no gamma. So the pressure at 2, 312 pounds per square foot. [Inaudible] absolute. To answer that, you say, well what's this pressure 1, 0 gauge and absolute? And the answer is gauge. So P1 is gauge, P5 is gauge. Well, OK, 13. What is this interesting? Looks like Z1 equals Z3. That correct. Z1, Z3. P1 zero. V1 zero. Well, OK. So, P3-- and
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P3 [inaudible] minus V3 squared over 2g. OK, here it comes again. Steady state, constant diameter incompressible V2, equal V3, equal V4, equal V5. Let's say [inaudible] V5. And let's see what pressure that was, P3. Yeah. OK, minus 187, OK. Let's take the last one right here [inaudible]. Obviously 1, 4. P1 over gamma plus V1 squared over 2g. If you want, you could write it from 2 to 3 Bernoulli. Write 2 to 4 the Bernoulli. Write from 2 to
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5 the Bernoulli. So, you know, there's lots of choices. Some are easier than others. In this case it doesn't matter too much because these guys are all straightforward, the 4. P1 zero, V1 zero. Why didn't-- I didn't point out there. I'm going to-- I'm going to call this h just for fun. This is not part of the problem in the textbook but I'm adding something to it. OK. So we're going from 1 to 4, so I'm going to call-- I want to call Z1 zero.
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See I can change my data anytime I want as long as I use the same data in that-- datum in that equation. So I'm going to call my datum here now. So Z1 is zero. On 1, 2, up my datum up here. OK, what is Z for? Zero. What is Z3 minus h? What is Z2 minus h plus 8? You can put your datum any way you want. OK. Z4 is h [inaudible]. P4 don't know it, V4 I know it. So P4, P4 gamma times minus V5 squared over 2g minus h. OK. Now notice
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that the pressure is getting lower and lower, matter of fact it's negative here, OK? If we raise this point 4 up high enough, the pressure is going to get low enough so that the water will start to vaporize and we get vapor bubbles in there, and then that it work anymore. The cycle won't work anymore. And the pressure at which it starts to vaporize, you go to the inside front cover, underwater. Vapor pressure is listed right there, vapor
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pressure inside front cover. So if I want to know the height at which this won't work anymore, the siphon won't work anymore, I go at the inside front cover. I get the vapor pressure [inaudible] in solution 2. I put that pressure in here. I know V5, I know gamma, I know g. Find h. [Inaudible] high enough up here, the siphon won't work anymore, it will stop working. Yeah. >> What does it mean when you have the negative pressure inside the tube?
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>> OK, if I've got water-- Let's say your household pipe, I hope not. But if the pipes are in your ceiling and they get a pinhole leak in it, will air-- Will water come out of there or will air go in that pipe, you think? >> Air will go in or-- >> No. Many people have flooded their house because their pipes in their roof are old in their attic are old with corrosion and they get a pinhole leak in it, call the insurance
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company. My rug is full of water, my drywall is wet. No water will come [inaudible]. That pressure is about 60 or 50 PSI. It will come out of there. Now, sometimes in a pump that draws water in, if there's a leak on the inside, then air might be sucked in because the pressure is below atmospheric, the pressure is below atmospheric. So they're sometimes, this load comes up, other times the gas comes in. So that's negative pressure is-- what's the absolute
49:26
pressure? I don't know, this guy here plus 14.7. OK, the pressures in here, I'll read it to you. The pressures in here, vapor pressures, it says right there in the column, absolute, absolute, those pressures are absolute in there. So, you know, watch out. That's gauge, OK. So, you have to watch out your units, absolute [inaudible]. Yeah. So, yeah, ahuh, that's what happens. The pressure gets so low that the-- the liquid water, it gets vapor
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bubbles in it and then it won't work anymore. It's somewhat similar to your, your gasoline line, or say about vapor lock. I thought gasoline is a liquid. Where did vapor come from in my gas line? Where did it come from then? What turned the vapor-- >> Low pressure. >> Or? Either low pressure or a high temperature. Yeah. High temperature? Oh yeah, oh yeah. It could turn gasoline into vapor. Vapor lock, no more gasoline is pumped to the engine.
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Engine stops. Up here, you get high enough. Guess what happens. No more pumping action. Water stop flowing out. Anyway, that's sidelight just so you know. We've taken all these [inaudible] at once, free surface, end of the pipe, exit the pipe, same elevation. Why is the pressure different here? And it is of course, because this is on the free surface, this is inside the pipe. That's what that point 3 is meant to show you. They have the same elevation,
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but the pressures in those two are different. They're P1 zero to P3, if we got P3 there, right there. My [inaudible], they are different. OK, what if I-- what if I put my point 1 here or what if I put my point 1 there. Does it matter where on the free surface I can put my point? Now because they are, we're going to go down there like that. So it doesn't matter where you put your points. You say, I don't know if the water molecule goes down
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like that. You know, maybe the water molecule-- I don't know, maybe it does this to get down there. I say I don't care, just draw the line, OK? The main thing is, we're on a streamline. Where is the streamline? Oh, I think it looks like that but maybe like that. Does it matter? No, no. OK. So don't worry about where you put point 1 on a free surface. OK. Let's see, anything else here we discussed? I don't think so right now. Anything else? OK, siphon. Let's
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take a look at another. And if you get confused in a problem or an exam, what do you do? You look at that picture and you try and figure out where you know the most stuff and what you want to find and that I try 1 to 2, that didn't give me the velocity. One to three? No. One to four? No. One to five? Yes. You keep trying. Sometimes you'll automatically see it that way. OK, 381. Water flows steadily downward in the pipe shown. OK, and this here.
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Static pressure tap on the left [inaudible]. Static pressure tap on the left, from up here. OK, let's see what we [inaudible]. OK. This is oil. This is the gravity, 0.7. OK. This is water, so gamma water. OK, let's see, 1.5 meters up here. And here to here, tubes. I think that's all we need here. Determine the flow rate. OK, that's-- You know the flow rate, V times A. Number 1, got to find V. How [inaudible]. OK, I think it's pretty apparent
55:58
where you're going to put points 1 and 2. I mean, it can't be two random. I said, I don't know, I like-- I like that point. That's neat. And you know what, I think I'll put point 1 here at the point 2, the exit. No, first of all-- first of all, that's not an exit. That's not an exit to air. That doesn't come from here. When I draw a pipe like that, I mean the pipe goes on up and down. How far from down? I don't know, but a long way. So
56:38
that's not an exit. If it's an exit, I would put down here P atmosphere and put up here P atmosphere. But that's not an exit. That's a continuous pipe. OK, so you don't want to randomly put point 1 and point 2, some like that. You say where is this manometer attached? Oh, this point right here. And, oh, this point right here. So I'm going to call this 1 and I'm going to call that 2. It's very similar to statics. If you don't choose a good free
57:09
body diagram, you're going to stumble around for a while until you find it. Likewise here, if you don't choose points 1 and points 2, good. You're going to stumble around until you finally get the answer. That's going to take a while. So, points-- choosing points 1 and 2 is very important in this. OK, here we go again. Over there. OK, got to find the velocity. There's two unknown velocities, there's V1 and V2. Matter of fact, if we look
58:05
at that, we might say, you know what, I'm going to find-- I'm going to find P1. This goes back back to our Chapter 2 where we have a pipe like this with a velocity here. And we attach this glass tube up to here, OK, this is water for instance. And that pressure at the pipe centerline was gamma times h, gamma times h. By the way, is the water moving in the glass tube? Of course not. Is the water moving in the pipe? Of course it is. That's
59:02
fluid stacks. This is where to it moves. So, what's P1? Drill a hole in the side of the pipe, put a glass tube on it. Water goes up one and a half meters. P1 is gamma one and a half m. And we're going to call that gamma water, so we're going to-- demonstrating this water environments problem. OK, now we'll get P2. OK, P2. OK, it's a manometer, water here, OK, go up here two times gamma water plus. Now this is gamma of oil. Again, one
59:59
and a half gamma oil, gamma oil. So, P2 minus P1 is equal to 2 gamma water minus 1.5 gamma water plus 1.5 gamma oil. P2, obviously yeah, [inaudible] V1. Zero point five gamma water plus 1.5 gamma water divided by gamma water P2 minus P1 divided by gamma water, [inaudible] 0.5 gamma water divided by gamma water plus 1.5 gamma oil over gamma water equals 0.5 specific gravity of water plus 1.5 specific gravity of oil. Specific gravity of water. >> One.
1:01:30
>> Specific gravity of oil. He told me 0.7. So that's the nice way to lay it out step by step, don't take shortcuts. OK. So, let's go ahead then and find out-- OK, so now I've got those guys taken care of down there. OK, so Z1. I'll call to my [inaudible] Z2 zero. My choice. Z1, 2 meters. OK, everything is known except two things. I don't know V1, I don't know V2. One equation, two unknowns, I'm stuck, can't solve it. Not that, OK, can't
1:02:33
solve it. So, I need another equation. That's one equation two unknowns. I cant write anymore Bernoulli's. There's no more points on there to write other Bernoulli Equations. It won't help me to find another Bernoulli Equation. There's no more logical points to choose. OK. So, Bernoulli is out. OK, here we go. When you get stuck, think about this guy. Conservation of mass. Do you think the mass flow rate of 1 is the same as 2? Of course
1:03:08
it is. It's incompressible, steady state. Is mass being stored in there? No. Isn't it incompressible? Sure, there's water. Question. Yeah. >> Why is it not useful to add another Bernoulli? >> Like from where? Give me an idea of where. >> Like above the oil, at the top of surface-- at the top of surface [inaudible]. >> OK, there. Is the fluid oil moving there? >> No. >> And don't like Bernoulli's. >> Bernoulli is going to be written where? Along-- along the what? Along what? >> A streamline.
1:03:36
>> A streamline. And along the streamline, what happens? There's a velocity. So if we don't want to write it from 2 up to the oil, that's fluid stacks, that's Chapter 2 or in Chapter 3. OK, so let's rewrite this. Go ahead, now, missing link, conservation of mass. We're going to do official conservation of mass in about two weeks but for right now this is just kind of a low level conservation of mass, the simplistic one, the one we finish
1:04:08
all the lead. We got a garden hose, five gallons per minute come out the other-- the end, the nozzle. Do you think five gallons per minute comes in from your house water supply? Of course. You're not going to-- You know, water is not being, you know, created out of nothing. So yeah, you believe that. OK, m dot 1 then must be equal m dot 2. What's m dot? I wrote it down. Rho1 A1V1 equal rho2 A2V2. What's the fluid? Water. Is it incompressible? Yeah,
1:04:50
we assume it is. What does that mean? The densities are the same. Rho 1 equal rho 2. Here comes the [inaudible] now. Are the areas the same? No, no. Pi V1 squared divided by 4. Pi V2 squared over 4. Cancel out the pis, cancel out the 4's. Which is the-- Let's write in terms of a 1, yeah. So, V2 then is to D1 over D2 quantity squared times V1. So, wherever you see in that equation a V2, replace it with that. So, replace that V2 with D1 over
1:05:53
D2 times V1. Now, what are you left with? One equation, one unknown. What's the unknown? V1. So, from here, solve for V1. [ Inaudible Remark ] >> That D1 over D2 is squared, right? >> Oh yeah. Yeah, thanks [inaudible]. But, no I didn't get an answer. That-- I want to set the equations up. Try Bernoulli's, fine it's true. You're stuck. You can't get to V1 or V2. If you're stuck, look at conservation of mass. Good. Now, there are two equations
1:06:38
with two unknowns, solve them for V1. Once you get V1, then Q equal V1A1. OK. Now, we're going to come back with one more problem a week from today but I just want to mention for Monday now. For Monday, were going to review the midterm. I'm going to go over some problems that I saw people have difficulty with on homework or office hour questions. I'll try and address those on Monday. If you have any specific problems you want me to
1:07:17
look at Monday, homework or whatever, examples on the textbook, email me and say, "Professor Biddle, will you please look at problem 285. I have questions about it." And if I can, if there's enough time, we'll do. I'll do my stuff first and then time [inaudible] I'll look at the stuff you want me to look at and go over in class especially. OK, good stopping point, so we'll see you next Monday.
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