 In this lecture we're going to begin by taking a look at the energy balance for fluid flow in a pipe. And so this is a pipe with either a constant surface temperature that is different from the fluid or with a constant heat flux, but in any case there's heat transfer on the surface. So I'll begin by drawing out a schematic of a pipe and then we'll apply an energy balance to that pipe. Okay, so here is a schematic of our pipe and what we have is we have a pipe with fluid coming from the left to the right. The fluid begins at the inlet I and then eventually at X equals L it goes to the outlet O, but what we're going to do, we're going to consider a section of the pipe in the middle where we have heat transfer taking place. And so we're going to say that we have convective heat transfer coming in and we don't know the exact nature of that but we do know that it is through convection so it could either be constant temperature, constant heat flux or a combination for right now. But the way that we will quantify that is we are going to have some surface heat flux in watts per meters squared and then we're going to multiply it by the area or the wetted area on the inside of the pipe and that is going to be the perimeter of the pipe multiplied by dx and dx is the length of our little differential element that we are applying the control surface to. So we have fluid coming in at a bulk temperature Tm and leaving at a different temperature so it's either becoming hotter or cooler but there is a dtm and then we also have the pressure multiplied by nu that is the specific volume and then on exit we have the pressure times specific volume plus the differential change of the PV term. So to express that what we have is Tb that's the bulk temperature mixing cup temperature it is also equal to Tm and so that is the bulk or mean temperature and the other thing that we have is nu and if you remember from thermodynamics that's where we use specific volume quite often it is one over the density and so that is specific volume. Alright so that is the system that we are going to perform an energy balance on. We're going to perform an energy balance on the control surface so let's go ahead and start that process. So to begin with remember we said that we have some form of convective heat transfer around the perimeter so energy is coming in that way and then we have energy flowing into our system on this side because we have flow coming in here so energy is flowing in and at the same time we have energy flowing out through the fluid and all of this needs to equal zero because we're operating a steady state and if you're wondering where I'm getting these terms from essentially this is just the first law of thermodynamics and the CVT that is our internal energy PV is the flow work because we have an open system where we have mass crossing our control boundary and so that is where that formulation is coming from but now with this expression what we note is first of all some of the terms are going to cancel out quite quickly most importantly this and this cancel out and then what we're left with for the differential amount of heat being transferred through convection is equal to the mass flow rate times the following and we have that expression there now what we're going to do we're going to work with expressing this and this in other forms and we're going to take advantage we could have a liquid or a gas but let's consider if we have a gas for right now so we know through the ideal gas equation we have that expression and here that is the relationship between c sub p and c sub v and so what we can do we can make a substitution here first of all for the PV term and then noting this the CV plus R that is just CP T M and consequently we can rewrite Q convection as being the following so that becomes one expression that we can work with and we will be working with that as we go through now if we're dealing with a liquid and so what this is saying is that if we've been dealing with the liquid the specific volume of the liquid is very small and for a liquid CP is equal to CV and consequently we would result in the same equation for a liquid so although we made the assumption of the ideal gas it equally we could have gone through for a liquid and have come up with the same so that means that this expression here applies for both a liquid and a gas which is good because that's what we'll be using it for now what we can do we can take that differential equation and we can integrate from the inlet to the exit and we obtain this expression and I'm going to box it because we will be using this expression later on in this lecture and the last thing we can say is we can look at Fourier's law and if all of the heat is being transferred along the inside wall of the pipe we can write out Fourier's law in this manner now this here that T wall that could also be expressed as T surface and then this is the fluid temperature TM or the bulk fluid temperature so this is what we have been looking at we've been finding ways of obtaining this and quite often we have relationships for a new salt number for either a laminar or a turbulent flow and we can determine the value of H but the thing that we now need to ask is how can we handle this temperature difference for different types of problems and what I'm referring to is that if we have a constant wall temperature for example how is the bulk temperature going to vary as a function of position along the pipe and if we have constant heat flux how will this vary as a function of position along the pipe and so consequently we we have a bit of a challenge when we're dealing with pipe flow in terms of being able to resolve this term here and that is what we are going to be looking at in this lecture finding a way or ways to be able to handle the temperature differential term for convective heat transfer in pipe flow so that's what we'll be going in the next couple of segments