flat plate isoflux convection - how to calculate temperature

Question

I am an engineer trying to design a simple 1D program for evaluating the temperatures of a multi-layer heatsink that includes convection and radiation heat transfer from the external surface.

For a known heat input to the bottom layer, I want to calculate the bottom surface temperature of the first layer given the material properties, and the radiation and convection properties of the external surface.

In heat transfer textbooks I've found the flat plate model with constant heat flux model to be applicable. In this model, the temperature of the external surface is allowed to vary but heat rejection from it is a constant rate in the flow direction. In an example it is shown that the maximum plate temperature occurs at the trailing edge, and can be found from Newton's law of cooling by Ts(L) = Tf + Qdot"/h(L), where Ts(L) is the trailing edge surface temperature, Tf is the incoming free stream temperature [K], Qdot" is heat input rate per unit area [W/m^2], and h(L) is the convection coefficient at the trailing edge [W/(m^2*K)], calculated from the Nusselt number fluid conductivity and plate thickness.

I would like to couple this model with one that includes radiation from the external surface. For this I will solve for the temperature of the external surface at each point and then iteratively calculate the combined heat transfer until it matches the heat input.

My question is how accurate is the surface temperature calculation using the iso-flux condition with thermal boundary layer theory? According the model, the leading edge of the plate should be at the incoming free stream temperature, and have infinite heat transfer rate. This isn't reasonable because it seems a plate strongly heated in a low convection stream would have a leading edge temp greater than the free stream. What then is the accuracy of the model vs the distance from the edge?

Does anyone have any insight into the accuracy of the model or what I can use, or what references I can review to improve accuracy for this calculation?

Thanks, Stephen

Answer 1

The equation is saying that the temperature at the trailing edge will be whatever it has to be in order to keep the heat flux equal to the prescribed value under an assumption of 1D heat transfer. This is as good as your assumption that the heat flux is 1D, which is probably a good assumption at the trailing edge of the flat plate because:

• by this point the boundary layer is growing more slowly, so
• $h$ isn't varying much with $x$, so
• $T$ isn't varying much with $x$, so
• there isn't much of a temperature difference to drive heat flow in the stream-wise ($x$) direction within the heat sink, so
• the heat transfer is approximately 1D

The other source of inaccuracy will be the correlation that you use to calculate the Nusselt number. You'll probably have to trace the references back to the original authors of the empirical correlations to quantify the error, but, since many of these correlations are pretty old, I would guess that they have lasted because they are accurate enough for engineering work.

As for the idea that $h=\infty$ at the leading edge... there is no boundary layer there, so heat transfer will be enormous. It's not a problem because the leading edge also has infinitesimal area; the infinite heat transfer per area and the infinitesimal area will cancel out in such a way that the heat transfer at the leading edge is equal to the prescribed value. As for the case you describe, if we assume that the heat flux remains approximately 1D, then - because the leading edge has infinitesimal length - there is no opportunity to heat the fluid, so the $T$ at the leading edge does equal the free stream temperature. It's one of those infinity times zero paradoxes.