In the last segment what we did is we came up with a differential equation that enables us to try to determine what happens with the bulk temperature as we go along the length of a pipe and so what we're going to do we're going to begin by looking at the constant heat flux boundary condition so if you recall constant heat flux boundary condition that means that the heat transfer per unit area is going to be a constant along the wall of the pipe and so with that let's take a look at our differential equation for the bulk temperature in the pipe so this is the expression that we had derived it was one part of that equation and looking at this equation well we just said that the heat flux is a constant if we're dealing with a pipe of uniform cross-sectional area the perimeter of the pipe is not going to change with length or position so position X is in that direction and mass flux is always conserved within any kind of closed duct flow and consequently m dot is conserved and if we assume that the temperatures are not varying that significantly we can then make an approximation that Cp is a constant so with that what that tells us is that that is equal to a constant and it is not a function of X so that's an interesting result it's telling us that the temperature gradient is going to be a constant so what does that mean that means that temperature is going to change at the same rate with position so let's integrate this equation and we'll plot up the temperature profile in a moment which will help us see what is going on but to integrate this expression the first term is a constant and then we will have a constant of integration C1 and so what we need to do we need to apply a boundary condition so what we're going to do we're going to say at X equals zero the temperature is the bulk temperature on inlet TMI and that will be our boundary condition and so with that boundary condition what we can do is we can rewrite this expression in the following manner so that becomes the temperature distribution for constant heat flux and with that what we're going to do let's take that and plot it so we have TMI this is X equals zero and what we can see from the equation let me write it out again well it's just a constant slope and this is the slope and so consequently what we can do is we can say that the temperature the bulk temperature is going to look something like that it's a constant slope curve a linear curve now what else can we extract out of this well the other piece of information looking at our pipe so here's our pipe we have we've just determined a way to express the bulk temperature in the pipe as a function of position we have the case of constant heat flux so we have qs double prime equals constant another piece of information that we're interested in is what about the wall temperature what is happening with the wall temperature so Tw as a function of X what is going on there and we can shed some light on that by looking at Fourier's law so looking at Fourier's law in watts per meter squared we have this expression and we know that this here is a constant because we have a constant heat flux boundary condition if we're in the fully developed flow region outside of entrance region effects then H as well is going to be a constant for pipe flow and so what that tells us is that this difference also needs to be a constant so T wall minus TM is equal to a constant and so if we look back at our plot so if we take this and bring it up here what that tells us is that the wall temperature has to be a constant amount larger than the bulk temperature and it as well will be a linear variation where the difference here is related to the heat flux divided by the convective heat transfer coefficient and so that would be an expression for T wall of X so that gives us some insight into what is happening with both the bulk temperature as well as the wall temperature under the case where we have a constant heat flux boundary condition and what we'll be doing in the next segment we'll take a look at what happens when we have a constant wall temperature versus a constant heat flux boundary condition


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