 Oil is flowing through 100m of pipe with a roughness of 0.06 mm and a rate of 0.342 cube metres per second. The density of the oil is 950 kg per metre, oil is flowing through 100m of pipe with a roughness of 0.06mm at a rate of 0.342 cubic metres per second, the density of the oil is 950 kg per cubic metre, and the kinematic viscosity of the oil is 2 x 10 to the negative 5th m2 per second. If the allowed friction head is 8 meters across this length of pipe, determine the required pipe diameter. So since I know the friction head, I can actually jump to our definition of friction head, which we are using only major losses. Therefore it's going to be F times L over D times V squared over 2 times gravity. I know the length is 100 meters. I know gravity. I can relate the velocity to the volumetric flow rate and plug that into my relationship. At which point I have F times L over D times 4 squared, volumetric flow rate squared, divided by 2 times pi squared times diameter to the fourth power times gravity. I know L. I'm solving for D. I know gravity. I know volumetric flow rate and I know HF. So D is going to be the fifth root of F times L times 4 squared times volumetric flow rate squared divided by 2 times pi squared times gravity times the friction head. The only unknown at this point is F, which if we have laminar flow, F is just going to be 64 divided by the Reynolds number, and if we have turbulent flow, we can either use the Kohlbrock equation or the chart. Since I don't know the friction factor and I don't have enough information to calculate the friction factor, I'm going to have to guess and check, meaning I'm going to guess an F value, I'm going to use that F value to calculate a diameter, I'm going to use that diameter to calculate a velocity, I'm going to use that velocity to calculate a Reynolds number, I'm going to use that Reynolds number to calculate a new F value. So my guess and check process is going to start with an F value, then I'm going to calculate a diameter, and then I'm going to calculate a velocity, and then I'm going to calculate a Reynolds number, which I can use to determine F. And in the last couple of examples, I have used the fact that my TI-89 can handle the Kohlbrock equation, but just to try to make this as hands-on as possible, let's limit ourselves to just using the chart. So I'm going to do everything with simple calculations as much as possible, the sort of thing that you might use if you were limited in your calculator usage, like on the FE, and then I'm going to use the chart for the lookup from Reynolds number to F. And just like in the previous example, I'm going to split my iterations into columns. So my calculation for F is going to be this equation, and in the interest to make this as simple as possible, I'm going to plug in everything except for F. So I'm going to leave F out of it and plug in everything else so that I'm left with a constant and then after the one-fifth power. And that constant is going to be the value of everything inside those brackets, except for F, and then take that quantity to the one-fifth power. So that quantity is going to be L times 4 squared times volumetric flow rate squared divided by 2 times pi squared times gravity times the allowed friction head, then everything is going to be taken to the one-fifth power. And what I should get out is a quantity in meters. So I want everything inside of this to end up being meters to the fifth power, so I have 100 meters times 4 squared times the volumetric flow rate in the problem which was 0.342 squared meters cubed per second squared, which is going to be meters to the sixth, second squared divided by 2 divided by pi squared, and then 9.81 meters per second squared if we assume standard gravity, and eight meters was the allowed friction head. So I have meters times meters to the sixth, which gives me meters to the seventh, then I subtract two. So I'm left with meters to the fifth, second squared cancels second squared, so the only unit left will be meters to the fifth. Then when I take that to the one-fifth power, I'm going to be left with a quantity in meters. And that quantity is going to be 100 times 4 squared times 0.342 squared divided by 2 times pi squared times 9.81 times 8. And I'm going to wrap this entire thing in parentheses and add to the one-fifth power to the end. So I get out a constant that is 0.655265 meters. So equation number one is going to be, I guess I should say equation number two maybe? The equation for diameter is 0.655265 meters times f to the one-fifth power. Then I'm going to use that to calculate a velocity, which is going to involve the volumetric flow rate. So I'm going to take four times the volumetric flow rate, which was given. 0.342 cubic meters per second divided by pi and then divided by our shiny new diameter. So if I leave that out, what I'm left with is going to be a quantity in cubic meters per second, which means when I divide by diameter squared, I'm going to get out meters per second. So 4 times 0.342 divided by pi is 0.43545. So velocity is 0.43, 0.435448 cubic meters per second times d to the negative 2 power. Then once I have velocity and diameter, I can calculate a Reynolds number. That Reynolds number is going to be density times velocity times diameter divided by dynamic viscosity, which can also be written as velocity times diameter divided by kinematic viscosity. Let's see what we know about the fluid. We know it's density and we know the kinematic viscosity, which means that I'm going to want to use this form of the equation. And at that point, I will know v and I will know d. So I can just write Reynolds number is equal to the kinematic viscosity that we were given raised to the negative first power. So that would be 2 times 10 to the negative fifth raised to the negative first power. And we get 50,000. So 50,000, which I'm actually going to write as 5 before, times velocity times diameter. And the units on that 50,000 are going to be seconds per square meter. Then once we have the Reynolds number, we can apply our Reynolds number with respect to diameter and a relative roughness to the Moody chart. We have a roughness of 0.06 millimeters. So our relative roughness is going to be something that we calculate. Perhaps I'll add that as a step five just to really demonstrate how we are breaking this up. So since my diameter will be in meters, I want my relative roughness to be expressed in meters. So that when I divide them, I end up with a unitless proportion that was 0.06 divided by 1,000. That's going to be a quantity in meters times diameter to the negative first power. And then my chart lookup will be step six now. I'll use my Reynolds number and my new relative roughness to determine an F value, which takes me back to step two. So again, to recap, my guessing and checking process here is to make up for the fact that I can't solve two equations and two unknowns when one of the unknowns comes from the chart. Because I don't know enough information to calculate a Reynolds number, I can't look up an F value, which means that I can't calculate a diameter directly. So our procedure is going to start with a guess value. We're guessing an F. We're using that to calculate a diameter, which I have simplified as much as possible to try to represent what you would do if you had a simple calculator, or honestly, if you just wanted to save time in the typing process if you had to type them every time. Then you're calculating a velocity from that diameter, a Reynolds number from the velocity and the diameter, a relative roughness from the diameter you're using, Reynolds number and relative roughness to determine an F value from the chart. So like last time, I'm going to be assuming fully rough with my first guess. But unlike last time, I don't know which relative roughness to use yet. So I'm going to have to pick an F value semi-arbitrarily. And just for fun, let's start with an F value of about 0.03. That's about in the middle. So my first guess is just in the middle of the range, just so I have somewhere to start. And then step two is to take that new F value, take it to the 1 fifth power, and multiply by 0.655265 meters. So that's 0.0655265 times F raised to the 1 fifth power. Come on calculator, you can cooperate. 1 fifth power. And I'm evaluating that at an F of 0.03. And I get a diameter of 0.032497 meters. And then I'm taking that quantity, raised to the negative 2 power and multiplying it by 0.435448. And I get a velocity of 412.34 meters per second. Then for step four, I'm taking D and I'm taking F and I'm multiplying them by 5 times 10 to the fourth 5E4 multiplied by V multiplied by D. Evaluated where V is equal to 0.03. Excuse me. Evaluated where V is equal to 412.338. D is evaluated with a value of 0.0325. So we get a Reynolds number of 669,985. Then step five, we take 0.06 divided by 1,000 multiplied by that number raised to the negative 1st power. 0.06 divided by 1,000 times the diameter from earlier to the negative 1st power. Evaluating that at a D value of 0.0325. And we get a relative roughness of 0.00185. Armed with a relative roughness and a Reynolds number, I can go look up stuff on the chart. First of all, I recognize that this is going to be more easily written in exponential notation. So 6.7 times 10 to the fifth. The first thing I'm going to do is draw a big vertical line at 6.7 times 10 to the fifth. So 10 to the fifth is this range here. I recognize that one is here. This is 1 and 1 half, 2, 2 and 1 half, 3, 3 and 1 half, 4, 4 and 1 half, 5. 5 and 1 half, 6, 6 and 1 half, 7, 7 and 1 half, 8, and so on. Since my Reynolds number is 6.7 times 10 to the fifth, I want to use right about here. So 6.7 times 10 to the fifth. And then my relative roughness is 0.00185. 0.00185 is going to be about 85% of the way between 0.001 and 0.002. So I'm going to eyeball 0.00185 as being there. On my vertical section, I'm going to 85% of the way. So if this is half, this is 3 quarters, 85 is right about there. So I'm going to draw a horizontal line and try to place that about the best I can right about there. That seems good. And that's going to correspond to an F value of 0.0212345. It's about 0.0228. 0228. Let's call it that 0.0228. So F is equal to 0.00228. And with that, I can go into iteration number 2, which I'm going to call red 0.0228. And then I can use that to calculate a diameter and then a velocity and then a Reynolds number and then a relative roughness and a chart lookup. So calculator, I am in need of your services once again. Diameter is going to be this equation using an F value of 0.0228. And we get a diameter of 0.030761 and then a velocity using that new number as our diameter. So 0.001951 meters per second. Oh, excuse me, that's a relative roughness. That makes more sense. 0.001951 and then a velocity number going to be this equation using our new diameter, which was this number here. So we get a velocity of 460.181 meters per second. And then our Reynolds number going to be 50,000 times those two numbers. So probably faster to just type 50,000 times 460.181 times 0.037061. Our Reynolds number is now 707,787, which is equal to 7.1E5. So I'm going to locate 7.1 times 10 to the fifth on the chart using a vertical line again. So 7.1 is going to be right about here-ish. And my new relative roughness was 0.0019. So 0.0019 is going to be very close to 0.002. So I'm going to choose right about there. And I'll draw another horizontal line. We're getting relatively close to convergence. So my ability to move the red line is going to be all the difference in the world here. So that's about 0.023 now. Let's call it that. So f is now 0.023. And that's probably good enough for convergence. But I do a third column. So let's do it a third time just to double check. We should get approximately 0.023 back out again. I'm going to call this gold. Actually, you know what? Let's go with blue. 0.023. And again, I want to calculate a diameter and then a velocity and a Reynolds number and a relative roughness. And then a chart will look up to see if my f value matches. So diameter calculation was this one now using 0.023. Diameters now 0.030815 meters. And then my velocity is going to be 458.58. And our Reynolds number is going to be 50,000 times those two numbers, 50,000 times. And it gives me 706,551, which is still about 7.1E5. I don't really have the ability to go any more granular on my chart look up than that. So essentially the same Reynolds number and then our relative roughness is going to be 0.0019. So we have the same input conditions, essentially the same relative roughness and the same Reynolds number. I don't have the ability to get any more precise on the chart than I already was in iteration number two. I can't tell the difference between 0.00195 and 0.001951 on the chart. In fact, I probably can't tell the difference between 0.0019 and 0.0018. So this is going to give me the same output since it has the same input, therefore, convergence. So convergence was really achieved after iteration number two. And then we did it a third time even though it wasn't explicitly necessary just to confirm that it wasn't changing. So I can say my diameter is either 0.03076 or 0.03081. I don't have any more accuracy than that. So I'm going to call the answer to the question a diameter of, let's go with 30.8 millimeters. So if the maximum allowable friction head is 8 meters, the diameter has to be 30.8 millimeters.