 Okay, what we're going to do in this section is we're going to take a look at flow in a circular pipe and we're going to look at a little differential element of a circular pipe and what we'll do is we'll apply the energy equation to that section and we will also apply the control volume analysis specifically the momentum equation along the axis of the pipe and with that we're going to come up with a relationship that is going to move us one step closer to being able to resolve pressure drop within pipe flow. So let me begin by drawing the schematic and then we'll work through the energy equation and the momentum and then we're going to do some dimensional analysis. Okay, so here is a section of pipe and we have fluid flowing within the pipe from position 1 down to position 2 and we do have a gravity vector and so the weight of the fluid it could be driving the flow as well in addition to pressure and we can see that there's a linear shear the pipe itself makes an angle feed with respect to the horizontal so I've drawn out on the right hand side here some angles and these are important because they lead to certain trig relations such as the gravity vector and delta z is equal to delta l sine phi which we'll be using in our analysis so with that that is the schematic that we'll be referring to so we have fluid flow coming in this direction and moving out of the pipe down here and what we're now going to do we're going to apply the energy equation and then we will apply the momentum equation in the direction of x so let's begin with the energy equation. So first of all it's steady flow so q1 is equal to q2 and that is equal to a constant and the mean velocity we will denote that by capital V in the analysis that we're doing here and v1 is equal to v2 given that the cross-sectional area of our pipe is not changing so looking at the steady flow energy equation so that is the steady flow energy equation we're dealing with fully developed flow so we're away from any entrance regions so what that means is that the velocity profile is not changing as a function of x and with that alpha 1 is going to equal alpha 2 be it laminar or turbulent we haven't specified that yet and v1 will also equal v2 given that the flow rate is constant we can make that assertion as well so with that what we can do is we can rewrite our steady flow energy equation I'm going to rewrite it and I am going to isolate the head loss term onto the left-hand side of the equation so from our energy equation a number of terms cancel out given that v1 and v2 are the same we can take that out and that out and what we're then left with is just the pressure terms and the elevation terms and we can then solve for hf is equal to that so what we're going to do we're going to park that for a couple of moments and after we look at the momentum equation we'll come back to this equation and we will couple it in so let's move on to the momentum equation using control volume analysis now we're dealing with steady flow so the first term will come out of the equation and looking at the x direction actually I should get rid of the x there because we haven't specified it yet we will now so we're now going to look at the x direction of the momentum equation and what we have for the forces on the right hand side it's going to be a balance of pressure we will have our body force and then we have the shear stress around the perimeter and then on the right hand side we have the momentum flux in and out and that term isn't really going to be all that interesting because the we're dealing with a very very small differential element and the momentum flux in is basically identical to the momentum flux out but let's work on the force terms on the right hand side okay so what we can see the first term here this is our pressure balance and multiplied by Pyra squared the area cross-sectional area then we have the body force term and I've corrected to give us Gx so the gravitational vector oriented into the direction of the pipe flow X and then that is multiplied by the volume of the cylinder to give us the mass and then finally we have shear stress and that is multiplied by the circumference of the pipe cross-section multiplied by the length delta L and the shear force is going in the opposite direction from either the body force or the pressure and then on the right hand side of the equation given that we're dealing with a small differential element what we're seeing is that the mass flux is not really changing at all nor is the velocity and consequently the momentum flux in and out are identical the velocity profiles not changing there so that term disappears and essentially what we have is a balance between pressure and including the body force and viscous shear and what we're now going to do we're going to make another substitution and then try to simplify this a little so pulling from those little diagrams that we drew earlier and rearranging when we do that we get that term there and what I'm going to do I'm going to divide through by rho g and pi r squared and when you do that you end up with delta z so we're dividing through by this rho g pi r squared so delta z by itself and I'm going to take the tau w over to the right hand side so we get that there but if we look at this this term here going back to our energy equation that is the exact same term which we said was equal to hf so delta z plus pi over rho g we can say that that is also equal to hf and that's from comparing energy with momentum okay so what does that do for us well we have this relationship now between the head loss and our shear stress we're now going to use dimensional analysis and so we saw the Buckingham Pi technique earlier on in this course the variables that we're dealing with here the important variables we can say shear stress is going to be a function of rho v mu d and epsilon epsilon is a quantification of the roughness of the pipe internal roughness because you can have smooth pipes and you can have rough pipes and they will definitely have different pressure drops so if we take that and we go through and do dimensional analysis much like we saw earlier with Buckingham Pi we can come up with the following we can come up with the following relationship and here what we're doing we're introducing a new constant and this is going to be something that varies depending upon Reynolds number and the roughness of the pipe but this is called the Darcy friction factor and when you're estimating pressure drop in pipes it's all about determining the Darcy friction factor that's really the main obstacle or the main goal but right now it's just some non-dimensional constant that we don't know much about so what I'm going to do I'm going to make some substitutions here I'm going to use the thing that came about when we balanced the energy equation with the momentum equation and then I am going to use the result that we obtained for tau w I will rewrite that and and the Darcy friction factors so that gave us the following so that's what came out of dimensional analysis and I'm now going to basically couple these two together the densities cancel so we get this equation let's clean it up a little bit more so we get this equation here and this is an equation that relates the head loss that we have in our energy equation to this yet to be determined friction factor and this is known as the Darcy wise back equation and a few comments about this equation so we see it's valid for either laminar or turbulent there is nothing in the derivation that we did that would assume one or the other our next step is to determine the form of the function f for flow and when I say for flow that could either be laminar or it could be turbulent and we will find that f does vary depending if it's laminar or turbulent and what this term does is it represents head loss in a pipe due to friction okay so that's the Darcy wise back equation a very important equation in pipe flow and what we will find is we will come up with a way to be able to determine what the friction factor is for laminar flow that one is kind of clean but when it gets to turbulent flow it's going to get a little more messy and that's because we have to rely very heavily on experimental data but that's what we'll be doing in the next lecture and we'll be working our way through and determining what the friction factor is and we'll come up with the equations for estimating it for either laminar or turbulent flow