 In this segment what we'll be doing is taking a look at the natural or free convection on a vertical flat plate under the condition of constant heat flux. So in the previous segment we had correlations for either laminar turbulent vertical flat plate or very large diameters cylinders and those were all for isothermal flows and in this case with constant heat flux sorry isothermal plate temperature I should say so the plate was at a fixed temperature T wall or T surface but here with constant heat flux what we find is that the temperature of the plate will vary with position and so we need a way to be able to deal with that so that's what we're gonna be looking at in this segment so what we're going to be looking at it turns out that you can use the vertical isothermal plate equation the correlation that we looked at in the last segment for laminar flow so there were two relations that we have for laminar flow there's also a third one that extended between laminar and turbulent but we'll look at the laminar one here assuming an isothermal wall temperatures that's what we did in the previous segment and a vertical flat plate now it turns out that you can approximate constant heat flux quite accurately or close enough if you replace the temperature difference because if you're calling any of the correlations that we were looking at we have the Grashof number times the Prandtl number which is the Rayleigh number and embedded within there is this temperature differential now when your wall temperature is constant this is easy to evaluate but if the wall temperature is changing as it would be in constant heat flux as you go up the vertical surface the temperature will change we need a way to approximate that and it turns out that you get good approximation if you take the temperature difference at the middle of the wall so what are we looking at here if this is our wall I remember we had the x-coordinate going in this direction y going in this direction we have natural convection developing so let's say that this is the length here so that is the vertical extent of our wall what we're looking at doing here is taking delta t at this point and that this point would be at L over 2 so x equals L over 2 so what we're looking for we're trying to find this delta t at L over 2 and it turns out that if we use this temperature and we bring that into our that would be the Grashof Prandtl number relationship which is also Rayleigh which we have embedded within that we had the temperature wall minus the t infinity if we replace this for that t wall minus t infinity in that expression using the laminar expression isothermal wall vertical flat plate we'll get a pretty good approximation so what I'm going to do is we're going to spend this segment trying to figure out how to determine this delta t L over 2 so let's go through that now and so essentially what we're looking at is how do we approximate that so the place that we're going to start we're going to begin with Newton's law of cooling and we've seen that before many many times in the course and what I'm going to do I'm going to express the heat transfer in watts per square meter some dividing both sides by the area and then we have h and I'm going to call the temperature difference delta t and for an isothermal plate that this is pretty easy because t wall doesn't change t infinity doesn't change but for constant heat flux it does and so we now have to resolve that because it will be a function of x but from this what I can do is I can write out that the convective heat transfer coefficient can be expressed as watts per square meter divided by our temperature differential delta t so let's park that for a moment and we'll come back and use this equation in a moment when we look at the next part and the next part what we're going to do remember I said that if you use the laminar relationship so this is assuming that we would have laminar flow on our constant heat flux surface and remember the requirements for that was Grashof-Prandtl which is equal to the Rayleigh number would be 10 to the 9 or less and so that was our approximation that we used for laminar flow and with this we saw relationships for the new salt number we saw a number of different correlations but this was the one that had these coefficients in it and I'm just going to use this one because what we are going to do in our analysis is just look at orders of magnitude and relationships between different variables and I could have taken a more complex one but the bottom line here what we're looking for is the fact that x will be raised to the power one quarter as well delta t due to that term there which if you recall that was the power that we said for laminar flow so it was turbulent that would be one third and so that would be a little bit different so with this what we're going to do we're going to sub in for the convective heat transfer coefficient here we're going to use Newton's law of cooling from just what we looked at so let's go ahead and do that now okay so we get this equation here what I've done I've subbed in for the Rayleigh number the Grashof-Prandtl this here is our convective heat transfer coefficient H and what I'm going to do I want to come out with a relationship from this equation between x and delta t and so what I'm going to do is I'm going to do a little bit of rearranging and we'll work towards this relationship between delta t and x so what I've done here is I'm looking at the proportionality between the left-hand side and the right-hand side where we have x and delta t and I've gotten rid of all the other variables that are here because we're we're looking at the trends in the relationships between x and delta t so from this I can reduce this a little bit further so we come up with this relationship here we find that delta t is proportional to x to the power one-fifth and what we're going to do we're going to look and remember this is our vertical plate x is going in this direction this up here would be x equals l down here this would be x equals 0 and we're interested in what is going on at x equals l over 2 and of course we have our thermal boundary layer and velocity boundary layer developing over this plate and we're considering a case of constant heat flux so what we have is we have a constant q coming in and that would be our q double prime that we have coming into the wall and so what I want to do I want to evaluate delta t at some arbitrary x and I will also want to evaluate delta t at l over 2 because that's where we said if we know that temperature difference we can put it into the laminar correlation and they get the value for the convective heat transfer coefficient so let's go ahead and do that so now that you notice I put an equal sign in here and that's because I've taken the relationship at two different places and you would assume that the relationship the constants would be pretty much the same and I should probably put this is a little bit of an approximation but it's pretty close what I want to do with this equation I want to isolate for delta t l over 2 because that's what we're looking for here so let's go through the process of doing that so with this we come up with an expression for delta t l over 2 as a function of delta t at some x location and the x location would depend upon where you have information so what we can write is that if you do know delta t at a certain x so if you do know delta t at a certain x for example at x equals l perhaps you know the temperature difference at that point then you can determine delta t at l over 2 which you then put into your correlations and you can evaluate the convective heat transfer coefficient and then the heat loss from this vertical surface now with that if you have a scenario where you do not know delta t at x that would probably be one where you would need to guess that and then go through trial and error and iterations as we do in many other heat transfer type problems but if you do know the delta t then you can go forward and it's kind of a straightforward calculation once you get this value of delta t at l over 2 you then plug that into your new salt number relationship for the case of laminar isothermal vertical flat plate and you can obtain the convective heat transfer coefficient so that's how you handle the case of a vertical flat plate under constant heat flux conditions