 In the previous segment what we did is we started with the governing equations X momentum continuity and energy and we went through a non-dimensionalizing process. Well you can use those same equations and use them to come up with the solution for the temperature and velocity profile in the case of a vertically heated flat plate and what results you can come up with a solution that then enables you to get the new salt number and so for laminar flow so this would be for a vertically heated flat plate constant temperature but if you go through the analysis what happens is you would use a similarity solution and it results in a nonlinear ordinary differential equations a series of them two of them and those equations would be solved using numerical methods and with that within the equations what we have they're related to the Grashof number or the solutions would be and note here the Grashof number is expressed as a function of X so remember our coordinate system that we have imposed for this vertically heated flat plate X going in this direction so we're looking at the Grashof number at some X location and what we can do with these solutions is that the solutions will be derived or obtained through the numerical methods and numerical integration and then in order to relate that to the new salt number you have to go through essentially a curve fitting procedure and and fitting functional forms to the data and so one expression for the new salt number that results through that process is the following expression and again there will be different relationships that exist depending upon what type of curve fitting is being used so that would be an expression that would be obtained for the new salt number now that gives us the local value of the new salt number very much like what we saw for the flat plate when we had force convection and so what that is doing is that would be quantifying the local heat flux at some location X and so that would be location X there now quite often we're interested in obtaining the average convective heat transfer coefficient across a plate H bar and so in order to do that what we would have to do is integrate the local heat transfer coefficient and so looking functionally I won't go through it but if you recall the new salt number is H X over K and with the relationship that we just have put here we see that we have Grashof number to the one fourth while Grashof number is X cubed so that is X to the three fourths is the new salt number relationship and so with that we can say that the new salt number relationship is X to the three quarters and then notice that we have the X here so if I want to isolate for H that means that H is going to then be a function of that will be one actually it'll be K but K is a constant so we don't have to worry too much about it but it would be over X to the one quarter or X to the minus one quarter and so with that if we want to obtain and so in here we're looking at H of X and if we want to obtain the average convective heat transfer coefficient across the plate we would integrate it over whatever the length of the plate would be and we would do H of X DX just like we did earlier for the flat plate and with this minus one quarter when we integrate that what we end up with is a relationship that looks in the following manner so it would be four thirds of the value of H at the end of the plate so if this is Y and this is X so let's say this is our plate here X equals L we would evaluate H at X equals L and then H bar or the average value for the entire plate is then just four thirds H at X equals L which would be the convective heat transfer coefficient at the end of the plate so that is what we get and this is in the case of an isothermal plate that we were looking at where we said that this was T wall and T infinity was out here and we said T wall was greater than T infinity so that is the solution that we can obtain and we had the functional form for the new salt number there are different ones that exist in different books depending upon how the curve that has been performed but the main thing to remember four thirds H X equals L and this is for a laminar it's not for the case of a turbulent flow so we have not gone through the transition process yet