 In this lecture what we're going to do is we're going to take a look at coming up with a relationship for the friction factor and see if you recall at the end of the last lecture we came up with the Darcy Weisbach equation which was an equation that enables us to quantify the pressure drop within pipe flow and within the Darcy Weisbach equation was this friction factor term here and what we're going to be doing in the next few segments we're going to look at it for both laminar as well as turbulent flow and we want to be able to find a way to be able to quantify what this friction factor is once we are equipped with it we can calculate pressure drop in pipes we can do sizing problems flow rate diameters a lot of different engineering applications provided that we can come up with a relationship for the friction factor F so what we're going to do we're going to start with the governing equations the equation that we'll begin with is continuity and we'll express continuity and polar coordinates or cylindrical coordinates which would be adapted to a pipe system now in fully developed pipe flow we know that there will be no swirl the swirl would be velocity in the azimuthal or circumferential direction nor is there variation in this direction consequently we can write v theta equals partial theta equals zero so any term involving partial partial theta or v theta would be zero and with that looking at continuity this term disappears and we are dealing with fully developed flow where we're saying that the velocity is not changing as a function of axial position and consequently that term drops as well and we have fully developed flow and for fully developed flow the characteristic we said was that the velocity was a function of radial location alone it does not change with axial position and so with that continuity reduces to something quite simple so if we integrate that equation we get our VR is equal to the constant of integration in order to determine what that constant of integration is let's apply the boundary conditions so one boundary condition would be at our equals capital R which is the outer wall remember what we're dealing with here is a boundary condition if this is our pipe and this is radial location is from the center line so our equals capital R would be the outer wall and at that point we know that the velocity in the radial direction that would be the no flow boundary condition is equal to zero and with that a VR at the wall is equal to zero and we have this result here that tells us that that constant is zero and therefore VR equals zero everywhere in the flow so that reinforces the notion that you is equal to you of our only so that's what continuity tells us now let's move on to the momentum equation and we'll be looking at the x component of momentum so this is the momentum equation then and what we're going to do we're going to take a look at the different terms here to begin with we're dealing with steady flow so that term disappears we said there is no v theta or derivatives with respect to theta and consequently those disappear we said the radial velocity was zero everywhere so this term has radial in it that disappears and we said that the velocity does not change in the x direction that we have fully developed flow so that term disappears and that term disappears so all of a sudden continuity reduces to a fairly simple equation and what I'm going to do let me just do a little bit of an expansion with this term here before we rewrite it I'm going to pull the viscosity inside of the derivative so what I'm going to do is rewrite this as 1 over r partial partial r and I'm going to pull the viscosity because viscosity is a constant we can do this and we'll have r mu du by dr and the reason why we're doing this is because mu du by dr that is tau and so what we will do is rewrite this equation as 1 over r partial by partial r and then we will have our tau and the reason why we're doing that is because tau will be something that we're after in order to get the friction factor tau being the shear stress and and so that's the logic behind making that substitution and that rearrangement so let's carry on and rewrite what we have for the momentum equations we have zero so we get that another substitution we can make gx is equal to g sin phi now what was phi remember long last lecture we had our pipe and we had the centerline orientation here phi was the angle that the pipe made with respect to the horizontal and we also had a coordinate system drawn we had z1 and we had z2 and then x was in that direction and we're going to make a little bit of a flip here with sine convention I'll get to it in a moment but it relates to how x and z relates and and so what we can do I will rewrite this now with the substitution and we have an equation 1 over r and pressure is only changing as a function of x it doesn't change as a function of radial location at a given x location pressure will be constant across that cross-section area so I change that to a total derivative and then and what I've done I made another little bit of a trick here I put this x in because we are putting the rho gx sine phi inside of the derivative operator that's why I've added an x there so if that's a little confusing that's the mathematical trick behind that and the next thing we're going to do we're going to make a sine change here and the way that we're going to be able to justify that is as z decreases x increases or counter as x increases z decreases and consequently I'm going to be able to change the sine here and I can write this or rewrite it as d dx p plus rho gz so I've now used the x sine phi in order to get and replace that with a z and I flipped the sine because of this relationship here so what we've now done we've rewritten our equation so we have that let's go to the next slide and we'll put that up at the top because we're going to work with that now this here and varies with respect to r only and this here varies with respect to x or z only and we can use a bit of a mathematical formulation and we did this we we're working with streamlines stream functions and potential flow but let me write that out again okay so the mathematical aside here if we have two functions we have f of x and g of x and if we're taking the derivative of those sorry that should be g of y so we have f of x and g of y and we're taking the derivative that gives us some function f of x and g of y these functions are equal if and only if they are the same constant and and so you can look back here as basically that is what we have we have something that is a derivative with respect to r and something derivative with respect to x and consequently as we integrate either the left or the right hand side it's going to be a constant and that constant has to be equal so let's take a look at applying this mathematical aside to what we have for within the pipe so if we integrate the left hand side of the equation and then we know with the boundary condition tau is equal to zero at the center line of the pipe and that tells us that C1 is equal to zero what we're left with is tau is equal to r times some constant and that would be equal to rc so we're going to take this now back up into this equation here and I'm going to evaluate the left hand side so we get that for the left hand side to tau over r substituting that back in and we can solve for tau then okay so that's a lot of work to come up with a relationship for tau and pipe flow that's a sheer stress but what we're going to do in the next segment we're going to take that and apply it to laminar flow and that will be the method by which we come up with an expression for the friction factor and laminar flow once we get to turbulent we won't be able to use this relationship and we have to use other relationships for turbulent but we're on our way to figuring out what the friction factor is for laminar flow