 In the last segment we came up with an equation that characterized the shear stress and pipe flow as a function of the pressure as well as the body force and we were doing that coming up with that relationship because we want to try to find the friction factor and so what we're now going to do we're going to take the equation that we derived in the last segment and we're going to look at it specifically in terms of laminar flow so writing out the equation that we had we had tau and I'm going to make a substitution here of du by dr the definition of the shear stress okay so we have this equation now what we're going to do we're going to integrate that equation and we have a boundary condition that the velocity is going to be zero on the outer wall and from that we come up with an expression for the velocity profile so we get that and now applying our boundary condition we can rewrite that so we get this expression here for the velocity now it is still written in terms of the change in pressure as a function of axial position as well as the body force term but we can play with this and extract a few things that enable us to determine the friction factor to begin with what we can do is we can write out an expression for what the maximum of this is going to be so this would be maximum velocity in the pipe flow another thing that we can do if we integrate it with respect to area we can come up with an expression for the volumetric flow rate in the pipe and another thing we can say is that the average velocity is the symbol v and that is q the volumetric flow rate divided by the cross-sectional area by comparing q and u max we find that the average velocity for this flow is just one half u max and this is for laminar flow which enabled us to make that original substitution for tau is mu du dr other things we can write out wall shear and the reason why we're interested in wall shear is because tau was the thing that enables us to determine the friction factor and so let's take a look at that and see where we can go with it now what I'm going to do the term in here I'm going to go back to the u max and we're going to make a substitution from the u max expression so that gives us an expression for tau on the wall and now what we're going to do we're going to take that and we're going to look at the expression that we had from when we did dimensional analysis and came up with the expression for the friction factor so what I am now going to do we have this tau wall here but I'm going to use the expression for tau wall that we just had I will also use the expression for you max I will use the fact that V the average velocity is one half of you max and finally I'll use the fact that the diameter little d is two times the radius and with that we can rewrite this expression and we can rearrange and we get this expression here and looking at these terms this is just one over the Reynolds number so therefore the friction factor for laminar flow is equal to 64 divided by the Reynolds number based on diameter and so that's a very nice clean result that we get and that is the friction factor for laminar flow now what we do in fluid mechanics is we have these functional relationships but we also have a diagram that we use for getting friction factor you can use equations or the diagram and this is called the moody chart or the moody diagram and so what we're going to do we're going to take a look at what this looks like within the moody diagram and then as we go along and we get the relationships for the turbulent flow condition we will add them in but let's begin looking at the moody diagram for laminar flow and the moody diagram begins with Reynolds number plotted in a logarithmic scale on the horizontal and then to the vertical axis we add the friction factor and so there you can see the friction factor and then when we put in the relationship that we just solved for you can see the green line and then you extend that directly and that goes into the transitional flow regime so that is the original or early construction I should say of the moody diagram and we will revisit that once we have an expression for the friction factor for turbulent flow but that is a diagram that engineers use quite often they either use the functional relationship or the moody diagram so what we now have is we have an expression for the friction factor for laminar flow and we can write this in terms of the head loss for laminar flow so we come back and this was using the Darcy Weisbach equation which we derived at the end of the last lecture and so we can say h due to friction and laminar writing out the Darcy Weisbach equation and what I'm going to do now is I'm just going to make substitutions and a little bit of rearrangement we're going to be doing some manipulation here we will also use the definition of the average velocity which is volumetric flow rate divided by area so in the area of our pipe is pi d squared over 4 making that substitution we end up with an expression for the head loss in laminar flow expressing it now in terms of flow rate so there we go and that gives us head loss laminar flow you can then use that in the energy equations to solve for pressure drop and we also have an expression for the friction factor in laminar flow so what we're going to do next is we're going to dive into turbulent flow and I can promise you that things will not be as neat and clean it will be a little bit more complex but we'll look at turbulent flow and then we'll come up with the expression that enables us to estimate the friction factor in turbulent pipe flow