>> Dr. John Biddle: 1 minus little R over big R quantity squared. That's [Inaudible]. So now we know if it's [Inaudible] into the [Inaudible] will be parabolic. The next step we did last time was to get the flow rate Q. We integrated over [Inaudible] times the differential area. We talked about that last time what that was. It was [Inaudible]. We have different area. Like this. There's our [Inaudible]. There's another area. We've got a circular area, [Inaudible] squared. We've got the surface area and we got the differential area where we have the pipeline [Inaudible]. This is the radius R. Circumference times DR and DR is [Inaudible]. Circumference 2 Pi R. So the differential area of 2 Pi R, DR. Okay so anyway we integrated that from the centerline, R equals zero out to the outside radius. Here's a picture of the outside radius goes to here, capital R. When you do that, you end up with this equation. This is this equation. Okay, I think that's about where we left off last time. Now the next step. Yeah I think I did this too last time. [Inaudible] equals 2B. Yeah okay. So what we said last time was [Inaudible] of B equal Q over A. We put that Q up there, divided by the area, Pi big R squared and we get the average velocity, LPD squared over 32 UL. Note when we did this last time we compared that to BC. That the average velocity just BC over 2. Okay, that's I think where we stopped off last time. Now we're going to solve this guy here for Delta B. We're going to divide this by one-half row B squared. That back in chapter three, yeah, chapter three. That was called the dynamic pressure. One row B squared. So divide that by one-half row B squared. And when you do that, this is what you end up with. [Inaudible] is row BD over U. Now here's Delta B multiplied to by one-half row B squared over two. That's that one. There's Delta B. This thing, this was the definition. This is the definition. We define F by this equation. This F is called the friction factor. So if we want to find the pressure drop in a pipe, we can take the length, divide by the pipe diameter, multiply it by the density of the fluid flowing, multiply it by the average velocity squared, divide it by 2 and times a factor, which we call a friction factor. Now if it's [Inaudible] you can see in [Inaudible] what F is. There's F. So far [Inaudible] F equals 64 over [Inaudible]. Now friction factor [Inaudible]. So it's important to know. And again, friction factor is something we've invented, we define it. Why did we invent it? Because it makes it easy to calculate the pressure drop in the pipe with this concept of a friction factor. And [Inaudible] is very simple. It's 64 divided by the [Inaudible]. Now we can also divide this by row G. So divide this guy by row G. So our row G is Gamma so Gamma, Delta B, over Gamma is equal to FL over D. The rows cancel out. The G downstairs. B squared over 2 G. Units are feet, are meters. And we know call that the head loss due to friction. The head loss due to friction. And where do we use that? That's an equal sign there. We use that of course in the energy equation from chapter five. Okay? The energy equation, chapter five. There it is. Now let's just review that equation again. It's sometimes called the modified Beroulli, except now we allow, compared for Bernoulli's equation, we allow many other things to happen. First that's the head by the pump. This is the head developed by the turbine. This L stands for the length of a piece of pipe. This is the head loss due to pipe friction. This is called the minor losses due to elbows and fittings and valves. They're all this is called the head form of the energy. Because every terms has units of feet or meters. This HL now, we used to say this was called the losses. Just called losses, L-O-S-S-E-S. We call it that. Now we've got a way to find the losses. There it is. I got to find though the friction factor. That's the key. I know how long the pipe is. I know it's diameter. It still gives me the flow rate. I can find the velocity. But it comes down to finding F. So that's what it amounts to. We have to find F and if we have [Inaudible] it's very easy. There's the equation for F. Okay, let's work an example problem using F. Otherwise the reason why we write the energy equation like that, on the right hand side of the board is because everything on the left hand side is energy that is coming into the fluid. The subscript one means entering. A pump does what? Adds energy. So the fluid brings in energy. The pump adds more energy. Now where does the energy go? Some of it goes up with the fluid leading. There it is. A turbine can take energy out. That's what a turbine does. Losses of course reduce the energy available. [Inaudible] of course reduce the energy available. So that's why it's nice to write that equation in that form. Because you could then talk about it in words and not just in symbols like that. Okay, so our example then horizontal pipe. What if we had a horizontal pipe, a horizontal pipe of constant diameter? A horizontal pipe of constant diameter in compressible flow? Continuity, think compressible. Z1, Z2, the same. Horizontal pipe, Z1, Z2. Okay, we're left with what? P1 is clearly P2. Yeah, pressure is high coming in. Delta P divided M equal if there's no pump. If there's no turbine to bind. If we neglect minor losses, gone. Then and only then do we get that guy. Okay. That's where it comes from down. Just so you know where it came from. That equation there is valid for what again? Incompressible horizontal pipe. No pumps, no turbines, neglect minor losses. If that's not true, go back up there. That H of L. This same thing. But you don't have this equation. See this equation right here? H of L is equal Delta Gamma, that's where it comes from. Just be careful when you use that guy. That thing is always true. That thing is always true. This thing is sometimes true. When it's sometimes? Well here we go again. No pumps, no turbines, neglect minor losses. An incompressible horizontal pipe. So this example is incompressible horizontal pipe or specific 8.5. Okay velocity, [Inaudible] velocity. 13 centimeter diameter type. At a flow rate, Q, 0.2 to 0 cubic meters per second. I want to find head loss in 100 meter length of pipe. Okay. There it is, head loss. Head loss, H of L. It's horizontal, incompressible, constant diameter. There it is. No pumps, no turbines, no minor losses, H of L. Delta P over again, which is F over D, B squared over G. First of all, pretty much always in a fluid problem, flow in a pipe or a tube or a duct, always calculate the [Inaudible] number. You have to know that to start with. VD over new. Okay, let's get the velocity up here. The velocity is Q over A. So we have 0.020 divided by the area, Pi over 4B squared. Diameter 15 centimeters. Velocity 1.13 meters per second. Okay, 1.13. Now if you want to, you could use--, we have an equation which was in terms of Q for the [Inaudible] number. You could do that too. That's best a shortcut way to do it but I found the velocity first [Inaudible] later on. We'll see. Times the diameter, diameter is 15 centimeters. Divided by [Inaudible] given, 6 times 10 over. So the [Inaudible] 283. Less than 2100, so it's [Inaudible]. Okay, so now we know, yeah okay [Inaudible] now I can get the friction factor. Friction factor, 64 over [Inaudible] number. Friction factor, .226. Now I plug it into the Delta P equation. F226, the length of the pipe, 100 meters. The diameter of the pipe, .15 meters. B1.13 squared divided by 2G, 9.83 meters. What I did to [Inaudible]. Okay, so forward flow. That's how we get that. Now we take a little more difficult one. Because most of the flows in the real world are not laminar, most of them are turbine. So now you got to figure out the way to get the turbine pressure down the pipe. Okay. We'll let's, by the way this equation is still going to hold whether the flow is turbine. The only thing we can do is this guy right here. Because he came from here and he came from here and he came from here. And it came from Newton's Law this constant. So this is the one you cannot use for turbine flow. But this equation is still true. This equation is still true. Except now we need F for turbine flow. Let's just review real quick laminar flow. Laminar flow pretty much if you follow a fluid particle as it moves down in a laminar flow field, it pretty much is straight. Now if you go to a turbine flow field, if you greatly saturate it, just to show you on the board, that's what it does. See the word laminar comes from lamina. What does lamina mean? Lamina means less surfaces. A laminate flooring for your kitchen or whatever, flat surface. A textbook, these pages are like lamina, they can slide over each other very smoothly. They slide, that's laminar. So the fluid motion, you pretty much go in a U direction. X direction. The U velocity. And they don't [Inaudible] they go straight like this. They don't hit each other too much. Now when you get the turbine flow, you get a little B component of velocity. U goes this way. V goes normal to it. You get a little B component. So this molecule starts to bounce around like this as they go down. They bouncing like this and of course two cars on a freeway, one going like this, one going like this, guess what happens? Metal on metal. Friction slows the car down. Guess what happens to molecules in the water? They bump into each other. They exchange momentum and friction develops and they start to slow down. This is rougher. It's got more reaction so that's what's happening here. There is a B component of velocity in turbine flow as opposed to a nice, smooth flow in laminar flow. Okay let's get on turbine flow. Turbine flow. To divvy up the values they didn't go to any theoretical variation like we did for laminar flow. No it's very, very difficult, nearly impossible. So what they did is they took pipes and it's described in here. They took pipes and they--, different diameters and they glued sand on the inside surface of the pipe. Now they glued very precise diameter sand particles not random sand particles. Very precise diameter sand particles onto the inside of a pipe to go to a rougher and rougher pipe. And then they made the flow turbulent and then they looked at the results of this pipe roughness. So they artificially roughened the inside surface of a pipe with sand particles and glued [Inaudible]. But anyway, that's what they did. Now they didn't get an equation. What they did was is they took all the data and different people massaged it and worked with it and they ended up with something called a Moody Diagram. Now here's what a Moody Diagram plotted. On the Y-axis they are plotting the friction factor, F. Now let me give you one and then I'll just go put it on the board. So if you don't want to copy it, don't copy it. Anyway, that's the friction factor. And this is the Reynolds number. So what I will do is give it to you now so you can look at it. Okay, [Inaudible]. So this Moody, I'm just going to sketch on the board again, mention some things about it. We're plotting F on the X-axis. It starts down there as 0.008 and it goes up to 0.10. This is 0.01. Yeah, okay. 2, 3, 4, 5, 6, 7, 8, 9. That's 0.01. Rows number 10 squared. I'll just start at 10 squared. [Inaudible], 4th, 5th, 6th. Laminar flow. There are probably about 2100. Here's 100. Now this is a long, long axis first of all. Long, long axis. Both axes are long. So here's 100, 1,000, 10,000, 100,000, million. So laminar flow of 2100. 2,000, 10,000, 2,000, 3,000, 4, 5, 6, 7, 8, 9, 10 up there. 2100. In that range in laminar flow the friction factor is 64 or the Reynolds number. When you plot that on long, long axes, that's about one. Plot it on long, long axes, guess what you get? A straight line. Okay, there it is. Here down to about .03. So there's a line at FD equals 64 over the Reynolds number. Laminar flow. But now you see another family of lines and they start out up here at about .003 and this is 4, roughly 4,000. So this 4,000 and these lines go something like this. I'm just going to sketch them right here. And these lines are labeled on the right hand axes. You can either say little E or epsilon. Take your choice. I'll use little E. E over D. The relative roughness. E over D, these lines are E over D. This guy is called the pipe roughness. This one of course is the pipe diameter. Where do you get the pipe roughness? In the little legends. In the little legend it says the pipe is cast iron or galvanized iron or drawn tubing or concrete. Here's the value of little E. So they give you the value of little E for different pipe materials. Notice concrete. Really, really rough compared to a wrought iron type. Of course concrete is rough. [Inaudible]. It feels rough. Now taking on a PVC pipe, [Inaudible] PVC pipe. Wow, it's [Inaudible]. Copper, wow is it [Inaudible]. This line here is labeled a smooth pipe line. PVC can be approximated most of the time as smooth. Drawn copper tubing could be approximated most of the times as smooth. This is a little bit beyond that they hydraulically smooth. But normally in this class, if you hear something like PVC or copper, look at drawn tubing. Oh my gosh. What are there? Six zeroes in five feet? Yeah, it's really pretty smooth. So yeah we're going to assume a pipe like that is smooth. Yeah. So how did they get those lines like that? Don't forget they got this one over here. They got this one from theory. These guys they didn't get from theory. They got them during what? Gluing sand particles very carefully on the inside of a tube. And the bigger the sand particles were, the rougher the tube was. This is high roughness. This is no roughness. So we want to get F in turbine flow to put this equation right here. Or up here. Then we have to get the relative roughness divided by the diameter. Find what line we're on. Get the Reynolds number, go up here with the Reynolds number until you find out what line you're on. Then go across horizontal until you get to the F axis and that's the value of the friction factor. Okay. Let's see just for your own information, I gave you three examples on the bottom of that page with the Moody Chart on it. So make sure you're reading the Moody Chart regularly. One is for galvanized iron pipe, one is for a riveted seal pipe, the other one is for a drawn tubing pipe. And I got the points labeled there to show you on the Moody Chart where those points are, to give you the friction factor in the left hand side of the page there. So that's examples of how you get the friction factor. Okay, this is called [Inaudible]. You can see where we start to be flat around here. You can draw a dash line here. This is where it starts to be flat. Everything to the right of that dash line, these lines are horizontal. Horizontal. This region is labeled in our textbook completely turbulent. The region to that side of the line, the region up here to here is labeled transition. Transition region between these two dash lines. Transition region. The region from here to here of course, laminar. The region between these two dots from here to here, critical. So there are four regions of the Moody Diagram. Four regions of the Moody Diagram. You wouldn't need the Moody Diagram for laminar flow because you don't really use the Moody Diagram for laminar. You do it mathematically 64 divided by Reynolds number. As long as you read in the log chart. But where you need it is over here of course or you could find some perfect equation that might work but you get that as a function for different things. Okay, so let's say you're in the laminar region. It depends on well, on let's say you're in the complete [Inaudible] turbulent region. That depends on E over D only. Finding one pipe right here. That diamond cap won't change along here where it's horizontal. And what's the region called where it's horizontal? The completely turbulent region. Now take the transition. That depends on, okay, now it depends on Reynolds and E over D. It depends on both things, the Reynolds number and the relative roughness. E over D is called the relative roughness. It's dimensions. F is dimensionless. The Reynolds number is dimensionless. This is a dimensionless chart. We engineers love dimensionless charts and fluid mechanics and heat transfer, wow, they're must be 25 or 30 dimensionless parameters. The no salt number, the Stanton number, the [Inaudible] number, the Reynolds number. Blah, blah, blah. The friction factor, the relative roughness, those are all dimensionless parameters. So this is a dimensionless chart plotted on long log paper. Sometimes people have a lot of difficult reading a long log axis. If this is 10 squared and this is 10 and this is 10 cubed, okay, find 105. Reynolds number 105. Because if you can't read this axis, you're not going to get the F value right. 105, what this one? 100. What's that number? 10. What's that number? 1,000. Now don't say this. I think that's 500. No, no. Here's what it is. If you take the distance from here to here. Take a ruler and measure this distance, and take 70 percent of it. You got that? 70 percent of it. Don't think linearly, think logarithmically. That's how the skills make it. There's 500. It's not in the middle. No, it's 70 percent of the distance between my two fingers. That's where you put the 5 times 10 squared. That by the way, is 5 times 10 squared. What is 9 times 10 squared? There. Where is 2 times 10 squared? There. What's that? 200. What is 105? Right there and boy you better read those guys right. It's not rocket science. You know, it's not calculus. No. Not [Inaudible]. No. It's how you read a log shield reasonably accurate. So be careful. Some people when they look like this they really have a problem reading those scales. Okay, let's go back here again. In the series, oh this big time series on smooth pipe, very steep curve. Very steep curve. If you [Inaudible], some people think okay, 110,000, 1.1 times 10 to the fifth, there's two-tenths to the fifth. Okay, right there. It's right there. So be careful. It can really get you if you're not careful. Alright. I didn't do the critical region yet. I didn't do critical region. No, I didn't do it and I'm not going to do it. Look at the Moody Chart. What do you see there? I see a blank region. There's nothing there. The lines end. That's right. The lines end. What does that mean? It means don't even attempt to get a friction factor in the critical region. Well what if I design something and it's in the critical region? I say well you should have been an accountant then. [Inaudible]. But you're not an engineer. No engineer designs something to operate in the critical region and guess why? Because to the left is laminar flow and to the right is turbulent flow and since we don't know where we are, your neck is on the chopping block. Don't say well I think it might be laminar or you know, I'm pretty sure it's turbulent. I was pretty sure I think. It means you are in trouble. No it just means engineers don't design anything to operate in that region. Now we know we've got to go past it to get from no flow when you turn the pump on, no flow. You're going to be turbulent. We go past it but we don't sit there. We don't design something to sit there. Because we don't know what's happening. For instance, the flow can be laminar pipe in a factory, in a plant in a pipe and then the [Inaudible] and over there 20 feet away, is a big compressor. A big compressor. The compressor doesn't operate continuously. It goes on with the pressure. The air goes down in the line. The compressor kicks on. The flow is forced to vibrate. Guess what happens? You're running the risk of tripping back laminar flow into turbulent flow just because the compressor went on. Which means you the engineer don't know what is going to happen. So the lesson is, don't design anything near that region. Stay away. Say I'm going to make it exactly 2,100. Oh man. You're [Inaudible]. Don't do that. That's not a magic number. Four textbooks, our [Inaudible] textbook says 2,300. Some textbooks say 2,000. Some textbooks say 2,100. There's no magic number. It's around 2,100 plus or minus a couple hundred. So you know, don't think it as magically as 2,100. It's going to kick up to critical. No, no, no. Is 4,000 magic? No it's not magic. Now here is the rule in our class, yes magic number. If you get it on an exam, 2,050, okay it's laminar flow. You get 4,001, it's turbulent flow. So in our class, yeah, there are magic numbers. In the real world, no, they're not magic, they're negotiable. Where's your pipe? Inside the plant, outside the plant. How close is it to a diesel engine? Blah, blah, blah. Things like that. Okay. So that's our turbulent. Here's what it depends on. Now there are various fit equations to fit that data. Some are more accurate than others. We're going to use one equation that fits that data. And it's called modified Colburn equation. So this is for our entire non-linear and non-laminar flow. You could use the Colburn formula. It's called a modified Colburn. Sometimes called a [Inaudible] equation but we'll call it a modified Colburn. 1 over square root F. Looks like you can put them in the chart on some kind of computer code obviously. You got to [Inaudible] equation. Minus 1.8, log base 10. V over D, 3.7. Let's just make it simpler. So you can use this to calculate F. It still takes calculation obviously. And that works anywhere to the right of my piece of paper. And to the right of my piece of paper. Here from V of D, it's a smooth pipe then of course the roughness is zero. Okay. And our textbook tells you how to read it here. Okay. A word of caution is in order concerning the use of the Moody Chart or the equivalent formulas. Because of various inherent inaccuracies involved on certainty and relative roughness, on certainty experimental data to produce a Moody Chart, the use of several place accuracy in pipe flow problems is usually not justified. As a rule of thumb, a ten percent accuracy is the best expected. That's the best. It goes downhill from there. So the best you can get by reading a Moody Chart or using the equation is within ten percent of the actual one. And of course that's only for brand new pipe. The pipe has been in the line for five years, all bets are off. The roughness has increased dramatically because of hard water deposits on the inside of the pipe for instance. Your pipes in your home, hard water in the western region of the U.S., oh yeah, what does that do? The deposits calcium makes the surface really rough and it makes the opening smaller and smaller with time as you get more and more calcium. So yeah that's a big problem. This is only for designing new pipes. Okay. Now look at the friction factor for laminar flow. Is that equation valid for oil? Of course it is. Water? Yeah. Air? Uh huh. Carbon dioxide? Yeah. A winch pipe? Sure. Six-inch pipe? Yeah. Rectangular duct, HPC duct? Uh huh. 30 foot water pipe? Yeah. Okay? Is it valid for PVC? Yeah. Is it valid for cast iron? Uh huh. Concrete? Yeah. You get the point. There's not a special equation for different diameters. For different flow rates? Flow rate two gallon per minute? Ten gallon per minute? 30 gallons per minute? Does it work for all of that? Yeah it works for all of that. No matter what the pipe is made out of. Well here's a Reynolds number. Here's what the Reynolds number is. First of all it's a function of the pipe diameter. Second of all, it's a function of a flow rate. Third of all, it's a function of what the fluid is. Those are the three things that the friction factor depends on. So there's only one equation for everything. Now you go to turbulent flow. Now I add one more parameter. This dimension I said, E over D to that one. So now you tell me what kind of pipe it is. Stainless steel. Okay. I get E over D. Tell me it's flow rate. 100 gallons per minute. Okay, I got a V now. Two-inch pipe. Got it. What it's carrying? Water. Got it. So I know everything I need. What kind of pipe it is, what's it's diameter, what's the flow rate, what's the fluid? Based on that, I can go over here and I can find the friction factor. Is there 10,000 pieces of paper? Let's see. I'm trying to find stainless steel. Oh here it is. Now I'm trying to find stainless steel, too much diameter. Oh there's a Moody Chart for that right there and there. No, there's not 10,000 sheets of paper. One for each fluid, one for each diameter. One for each flow rate. No, no, no, no. Guess how many sheets of paper I need to give you the friction factor for any pipe material, any pipe flow rate, any fluid in the pipe? I'll tell you. One. I wish I invented the diagram. I love the sound of that. No I'm sorry. I couldn't do it. Preston Moody did that back then. So yeah, now people don't realize that. Are you kidding me? One sheet of paper for friction factor, for everything us engineers can imagine? That's why everybody graduates from fluid class better know the Moody Diagram. Everybody who graduating from a ME 218. [Inaudible] materials what should they know about a certain diagram? >> [Inaudible] >> Dr. John Biddle: The what? >> [Inaudible] >> What's the one with the circle? >> Dr. John Biddle: You got it. You got it. Are there equations for doing that? Yes there are. Why do we still teach the more [Inaudible]? Because we engineers love the visual impact of a diagram like that. And why would you use the equations? Because the equation came from that. So number one, you should learn how to read this. And this is just then what you put onto the Moody Diagram to solve for that. But that diagram. It would be interesting. I just love that diagram. You know, that's the way we are. We're that kind of people. Now the manuals say oh that I love that equation. I know, I love them too. They're good people. But they're not engineers okay? But we love other things. That's why I love that Moody Diagram. Everything is on that thing. [Inaudible]. That's called a dimensionless diagram. We engineers love dimensionless numbers. You don't plot the flow right here, the velocity, you don't plot the diameter, it's all divided by different things. Here and here. And F, dimensionless. That equation right there. It's a dimensionless equation. Nothing has dimensions. Okay. So that's the diagram that you want to use to get the friction factors for homework and other problems. You'll have one on the final exam. So you'll have that in your answer. Okay, I think a good stopping point. Next time I'm going to bring in. Here it is. We're going to bring the [Inaudible]. Next time we're going to discuss the minor losses.