 Okay, we're going to solve another example problem involving the calculation of pressure drop in pipe flow. So I'll begin by writing out the problem statement. Okay, so we have one mile of pipe that is three inch internal diameter and we're told that it is wrought iron with a epsilon or surface roughness of 0.046 millimeters and within that pipe is flowing water at 20 degrees C and seven meters per second is the average velocity and we're told to compute the head loss in feet and pressure drop in psi. So we have two ways of going about this. We can use the British units or use SI. I will convert to the SI units because I am more comfortable with those units and then looking at values for water. Okay, so we have the dynamic viscosity, we have the density and we are told the velocity is seven meters per second and so the first thing we want to do, we want to calculate the Reynolds number for the pipe flow and so substituting in the values, we have all the values for the Reynolds number and we find that we're at 5 times 10 to the 5, 5.5. Wow, let me write it out. So 532,000 is the Reynolds number and therefore that's greater than 2,300 and we have turbulent flow. So what does that mean? That means that when we do our pressure drop calculations, we have to use the friction factor for turbulent flow and there are a number of different ways that we can go about doing that and what we'll do before we get into that, let's look at the roughness. So for the wrought iron, we have epsilon 0.046 millimeters and that gives us an epsilon over diameter of 6.04 times 10 to the minus 4 and we'll estimate that as 0,0,0,6,0. So these two values we will be using, we'll be using epsilon over D and we'll be using the Reynolds number to compute the pressure drop or the friction factor for this particular case. So what we're going to do to begin with, we're going to go to the Moody diagram and we will use an epsilon over D of 0.0006 and an RE based on diameter of 530,000 and we'll read the Moody diagram and estimate the friction factor from the Moody diagram. So let's take a look at the Moody diagram. So here we have the Moody diagram and the Reynolds number we said was 530,000 so we come down, this is 5, that would be 12345, 5.3 times 10 to the 5. So we're someplace in here is where we will go into the Moody diagram and then our epsilon over D was 0.0006. So looking here, that is going to be this curve here. Consequently, we're going to want to follow this curve and I'll dash it out so that we don't get lost. So we want to follow that curve and then we want to move up. So moving up directly. Okay, so that means that the value that we want to read is someplace in about there. Now going over, I'll change colors of pen and I won't use blue because the lines are blue. Let's try green. So we have this one here and that goes there, there. Carefully coming over and then we come out about there. So that's 15678. We estimated the friction factor to be about 0.018. So F is equal to 0.018. So by reading the Moody diagram, we can write out F equals 0.018 and that would be our first estimate. Now that is using Moody. Now there are other ways that we can go about doing this calculation and some of them do not involve reading the Moody diagram. Others would involve using equations. So let's take a look at those different techniques next. One of them is using the Holland equation and this is the equation that did not require iteration. And so writing out the Holland equation for the friction factor and plugging in values so we can solve for the right hand side and that's going to be equal to 1 over the square root of the friction factor and with that we can solve for F to be 0182. So that's pretty close to what we got with the Moody diagram. So Holland gives us that. Moody gave us 0.018. Now this is an approximate at it's not a perfect estimate and none of them really are perfect. They're probably all about plus or minus 10% because we're fitting empirical data from experiments. So that is the technique by Holland and then another technique is the Colbrook-White equation. And this equation is transcendental meaning that we have to iterate. So computationally this is the one that takes the most time. However on a computer this one would be fairly easy to do. In an exam this one would be one that would make students sweat because they have to do iteration after iteration until they converge. Okay so how do we solve this? We see that we have friction factor there and we have friction factor there. The way that you do it is you just do trial and error. And so what I'm going to do is I'm going to develop a table here, f value left hand side, right hand side, and then the difference between the two. And I'm going to put in numbers and we'll take a look at where it takes us. Okay so we have the numbers here. We have our friction factor, the left hand side of the equation, the right hand side, and then the difference between the two. So that would be left hand side, minus right hand side. And so when we get near the root of this equation we'll have a zero crossing and consequently the sign of the difference should change. We see that takes place down here. So we could go through and do more trial and error with the values of the friction factor in order to estimate. Or you can go ahead and just do interpolation. And if you do interpolation what you find is the friction factor comes out to be 0, 1, 8, 3. So really not too far off what we got with the Haaland equation or is it far off from what we got from the Moody diagram. So that's another technique. This is one that would work well with the computer code. But to do it by hand I would either use the Moody diagram or the Haaland equation. That seemed to be very efficient and compact and gives you the answer directly. So what we're going to do, we have these different estimates. We have from the Moody diagram we got 0.018 from the Haaland equation we got 0182. And then just now using the cold brook white we got 0.0183. So I'm just going to say f equals 0, 018. And we're going to use that to continue on with the calculation for this problem of estimating the pressure drop in this pipe that we're dealing with. So now that we've estimated the friction factor what we can do is we can go ahead and using the Darcy Weisbach equation we can estimate the head loss for turbulent flow. And when we plug in the numbers here and then converting it to feet because that's what we were asked to do for the question. So there is one part of the answer we were told to compute the head loss in feet and the pressure drop in PSI. So this would give us the first half of the answer. Now for the pressure drop what we're going to do is we're going to go to the energy equation. So let's write out the energy equation. And remember we have the kinetic energy coefficient alpha that we have in the energy equation when we're dealing with pipe flow. And given that we're dealing with turbulent flow the value of alpha is going to be 1. Okay so alpha 1 equals alpha 2 equals 1.0 because we're dealing with a turbulent pipe flow. V1, we have fully developed flow so V1 is equal to V2 equals 7 meters per second that's not changing. We know that, well we don't know they didn't tell us anything about elevation change so let's assume Z1 is equal to Z2. And with that what we get is this equation and this term goes with that term, that term goes with that term. So we're just left with the pressure differential and the head loss due to friction in the pipe. So what we end up with is P1 minus P2 divided by rho of the water is equal to GHF. So we can then take that and solve plugging in the value of HF we can solve for the pressure drop or the delta P in this pipe. So the pressure drop is going to be 9.295 MPa. Now they told us to compute this in PSI. And so this is a conversion that you should have memorized and that is you multiply this by 14.7 PSI divided by 101 325 and this is going to be Pascal so I'm going to have to put this as times 10 to the 6. But when we do that we get 1349 PSI and this is the conversion basically what we have for atmospheric pressure in either Pascal's or PSI. So with that we get delta P is equal to 1349 PSI and we get the head loss due to friction in the pipe is 3114.8 feet. So that gives you an example of calculating pressure drop in turbulent pipe flow. We saw that there are three techniques. There is the Moody diagram. There is the Holland equation and there is the Coldbrook white equation and the Coldbrook white was the one that required iteration. So that was not the most efficient. The Holland equation on the other hand was quite compact and actually kind of a nice way of doing it. The Moody diagram is good as well but it is prone to errors by having to read the table and you need to have the table there. So if you're doing things quick and dirty I would say Holland is probably the way to go. Coldbrook white is good if you're doing with the computer and Moody is also good if you are doing it quick and dirty. I should get rid of that error thing but it is prone to error because you're reading a table or the figure. So that is pressure drop with turbulent pipe flow.