 Okay, so we're nearing the point where we will be able to get the friction factor for turbulent flow and using the Darcy Weisbach equation we'll be able to figure out head loss for turbulent pipe flow. So we're looking at turbulent flow and what we're going to do we're going to use the velocity profile that Milliken came up with in 1937 and that was for the velocity profile in the overlap region. So this was the region where both laminar shear stress and turbulent shear stress were important and we need to do a transformation here of the coordinate system. Milliken was doing this for flat plate boundary layer channel flow but what we want to do is we want to transform the y-coordinate into a radial coordinate and if you think about at r equals capital R that would be so if this is our pipe and we have the pipe wall there that's the thickness so we have a little r here and out here this is a little r equals capital R and that would be the radius so looking at it in this manner if we have r minus r so on the outer wall so at r equals capital R we get y equals 0 so basically what we're doing is we're re-transforming that so that this is kind of like y coming in that way away from the pipe wall which is what it should be for pipe flow and the relationship that we will use is the one that Milliken proposed now one thing that I should say is even still to this day people are debating these profiles and the constants and the universality of the relationships that result people are arguing whether or not this relationship would apply to flat plate boundary layers or open channel flows or what the coefficients are that's all taking place because we're getting better and better data enabling us to be able to study this profile in more detail for the purposes of what we're doing here however we're not really going to worry about that because the main thing Matt Prandtl and Von Karman and everybody else who studied this were trying to do where they were trying to come up with a functional relationship with which to collapse experimental data and that's the approach we're going to take so we're going to use Milliken's profile this one here and with that we're able to calculate an average velocity I'm not going to go through the math I'll just show the final result of that the u star remember that's the friction velocity u star is tau w over rho square root okay so we have that and the constants that we're going to use here although they can be tweaked as I mentioned before we'll use these ones in which we once in Milliken originally had now with that what we can write is the average velocity divided by the friction velocity is approximately okay so that's where we get to what can we do with that well it turns out that we can actually do quite a bit and we're going to focus in on this term here and we're going to take the friction velocity from up here and we're going to rewrite that in terms of our friction factor which is what we're after the friction factor relationship came out of our early dimensional analysis of this problem a few lectures ago so rewriting the left hand side and here we have our wall shear stress this came out of the dimensional analysis and so with that we're able to continue on and rewrite this in terms of our friction factor F which is a good thing because that's what we're after so that means we have friction factor on the left hand side of the equation let's get it on the right and we'll be in business so taking a look at the right hand side we're going to work on the term that was in the natural logarithm now I'm going to play a little bit of trickery here I'm basically introducing a V and a V I should be a little careful this here is kinematic viscosity so don't get that confused with my V's hopefully or not that's a V that's a V I've introduced it into the numerator and denominator so that we can simplify this a little and what we're going to do we notice that the first term is one half of the Reynolds number based on diameter and then we have the friction factor which is what we're after and with that we can then rewrite Milliken's velocity relationship and we get something that looks like this and I've now transformed from a natural logarithm to a base 10 log so this profile is something that Ludwig Prentel came up with in 1935 and so he adjusted the constants to provide a better fit and that will be the next equation that I will write so Prentel further reduced that or recast it in the following form okay so we have this relationship here and this is good because we're getting something that has the friction factor in it has the Reynolds number it is a transcendental equation meaning that in order to solve it you have to iterate because the friction factor appears on both sides there was one other contribution that needed to be added however and that had to deal with the fact that not all pipes were smooth some pipes are rough and consequently we needed to take into account the pipe roughness effects that's what we're now going to do it's a bit of an adjustment to this equation so it turns out for turbulent pipe flow the pipe wall roughness has an impact on the friction factor and we can classify the surface roughness based on a ratio which has our friction velocity in it so in a way this is kind of like a Reynolds number when you think about it because we have a length scale and we have a velocity and we have the kinematic viscosity in the denominator but if that number is less than 5 that would be a hydraulically smooth pipe if the number ranges between 5 and 70 the number being our length scale times the friction velocity divided by the kinematic viscosity this is transitional and it results in moderate Reynolds number effects so what's happening here is as you increase Reynolds number it will have a moderate impact on our friction factor F and then finally if this number which is basically a Reynolds number for our roughness if it is greater than 70 actually let me write that over here if it is greater than 70 then we have what we refer to as being fully rough flow and here we can see the friction factor is independent of the Reynolds number so with that you can integrate that in and that basically modifies Prandtl's equation to include surface roughness and what we end up with is the famous cold brook white equation which is used for estimating the friction factor in turbulent pipe flow again it's a transcendental equation meaning that you need to iterate if you want to solve it and sometimes referred to as being the cold brook white equation very famous one in fluid mechanics now this equation you can either solve it directly by going through the trial and error iteration or you can go and look at plots of it and this is plotted in the moody diagram which we looked at earlier for laminar flow so what we're going to do we're going to go back and take a look at the moody diagram and we're going to look at what happens when you have the moody diagram and you add pipe data for increasing roughness of pipe so let's take a look at that now so what we have here is moody diagram with a laminar flow curve on it and then we have smooth pipe curve you can see it there and then we start putting curves with variable epsilon over d so surface roughness and we have the smallest surface roughnesses and then we have it increasing and as the surface roughness gets bigger and bigger or greater and greater the friction factor goes up which we would expect and then finally with very very rough pipes and and so that is the moody diagram that's a diagram that we use quite extensively within mechanical engineering in order to give us the friction factor and estimating the pressure drop within pipe flow or you could estimate other things how you could be doing sizing and determining the diameter or you could be determining the volumetric flow rate going through a given pipe flow configuration so with that we're almost at the end we come back to the Darcy Weisbach equation enabling us to come up with an expression for the turbulent head loss which is what we would use in the energy equation so if you remember back we looked at the Darcy Weisbach equation and all we do here is we put in the value of the friction factor for turbulent flow that we either get from the moody diagram or from the Colbrook white equation and we get that now there is one final relationship that I want to leave you with and this is an approximation for the friction factor that does not require the iteration due to the transcendental nature of the equation and so this is an approximation and so with this equation and you'll notice there is no friction factor on the right-hand side of the equation this would give you a starting point perhaps by which you could estimate the friction factor and then after you get it from here you could go to the Colbrook white and then you'll be closer to being converged but anyways that's another relationship that you can get the friction factor for turbulent pipe flow from so that concludes friction factor we've looked at friction factor for laminar pipe flow and we looked at it for turbulent pipe flow we came up with different relationships Colbrook white the analytic expression for laminar flow and finally the moody diagram those are all things that you can use when you're solving pipe flow problems in engineering