 Okay, so this is the last segment of this lecture and what we're now going to do, we're going to take a look at trying to determine the convective heat transfer coefficient for turbulent pipe flow. And just like what we saw for the boundary layer, the laminar boundary layer, we were able to come up with analytic expressions or it is possible, well not analytic, but it would be through numerical integration and you get coefficients. For turbulent either flat plate or in this case pipe flow, it's very very difficult and so what we need to do is rely on empirical data and what we're going to do, we're going to use the Colburn analogy which is something that we saw for the boundary layer flow. Now the Colburn analogy expressed in the following manner and if you recall it relates convective heat transfer coefficient to skin friction measurements and that was for the case of a flat plate. So what we're going to do, we're going to come up with the skin friction equivalent for pipe flow. And so what we'll be doing is we'll be using the Moody friction factor and this friction factor can either be obtained from the Moody diagram or it could be obtained from something like a Colbrook white equation where you have to iterate and you would then determine the friction factor. But if you recall from the friction factor, we can relate it to pressure drop and in this expression L is the length of the pipe or the section that this pressure drop pertains to. So what we're going to do, we're going to look essentially at a force balance for a pipe flow and so what we have, we have shear stress on the wall multiplied by the outer area and the length of the pipe and that is going to be balanced by the pressure, the pressure forces within the pipe and the pressure forces are going to be acting over the area, the cross-sectional area of the pipe and if we have a pressure gradient then I'll multiply it by the same length L. And so there we have essentially what amounts to being a force balance of shear stress with pressure gradient or pressure forces. This should be d0 squared, sorry about that. And this equation then translates into the following and so what I'm going to do, I'm going to approximate this knowing our delta p so the pressure drop over a given length and we have this expression. What I'm going to do is I'm going to combine this with the expression that we had using the friction factor from back here. So combining those two equations, before I do that however I'm going to look at the definition of the skin friction coefficient and the reason why I'm doing that is because with our colburn analogy we have the skin friction coefficient. So let's take a look at the definition of the skin friction coefficient CF. That is the wall shear stress divided by the dynamic pressure evaluated using the mean velocity in the pipe and now what I can do, I can take the expression here for the wall shear stress and substitute it in for the skin friction expression. So expressing the skin friction coefficient using our value for tau w and now what I'm going to do, I'm going to take our expression for delta p that we had from the moody friction factor. So what I've done there is I'm just pulling in this expression here and now I can do a lot of cancellations so we get that. Skin friction coefficient is the friction factor divided by 4. I can then take that and put it into the colburn analogy and we obtain the following. It's just the friction factor divided by 8. So with that what we can do is we can find empirical relationships for the friction factor within pipe flow realizing that it is a function of the roughness of the pipe wall but I'll just take one of the expressions. So this is an empirical expression and with that we can then come up with an expression for the new salt number based on diameter. So using nothing but the friction factor which is measured from the pressure drop in pipe flow, we can come up with an expression for the new salt number. So this is entirely based on fluid mechanic measurements and the colburn analogy. Now what people have done is they've taken this expression and they've used it or tested it out to see how well it matches heat transfer data and it turns out that it's good but it's not perfect and so people have used that functional form and have fine tuned it to come up with a better expression for the new salt number. So taking that basic functional form for the new salt number and using direct heat transfer measurements results in the following expression for the new salt number based on diameter. This expression is referred to as being the Dittus-Bolter equation and it is a very useful equation for computing the convective heat transfer coefficient in turbulent pipe flow. Now there are other expressions that are a little bit better that cover wider ranges of temperatures and parental numbers and things like that but this one actually turns out to be quite the useful one because it's quite compact, not all that complicated. I notice we have an exponent n and n will depend on whether or not we have heating or cooling. If we have a case of heating where the walls or wall temperature is higher than that of the fluid then in that case n is 0.4 and if we have cooling where the wall temperature would be cooler then n is 0.3 and the last thing is that the properties in this equation are evaluated at tb or tm so the bulk or mean temperature. Okay so that is an expression for computing the convective heat transfer coefficient in turbulent pipe flow and what we did is we took advantage of the coal burn analogy and the wealth of information that exists in computing pressure drop in pipe flows and enabled us to come up with an expression that was close to this but then this one was fine tuned using experimental measurements and that is the Dittus-Bolter equation.