Flat plate heat transfer constant flux model

[kramersteve99]

Hello,

I need help understanding the constant heat flux flat plate parallel flow transfer problem. What I'd like to know is what is the temperature distribution of the plate for designing a heatsink.

According to the model in most textbooks: Qdot"(x) = h(x)*(Ts(x)-Tf), where Qdot" is the local heat transfer rate per unit length and width and Tf is tbe incoming free stream temperature. Local convection coefficient decreases as the thermal boundary layer thickness increases and reduces the thermal gradient at the wall. Nu(x) is the temperature gradient at the wall at x and is a function of Re and Pr and is related to h(x) by Nu(x) = h(x)*x/k. In the isothermal plate derivation there is the result h(x) ~ x^(-1/2). Qdot" = h(x)*(Ts - Tf) is the local heat flux. This means infinite heat transfer per unit area at the leading edge of the plate for an isothermal plate. Of course the integrated area approaches 0 so tbe average integrated heat transfer allows you to define an average h for a given x.

For uniform heat flux Nu(x) is given as Nu(x) = .453*Re(x)^(1/2)*Pr^(1/3). It is stated this is greater than the isothermal case I would assume due to the higher plate temperature needed to maintain constant Qdot"(x) as the thermal boundary layer thickness increases. h(x) is solved from Nu(x) and then applied to Newtons law of cooling Qdot"(x) = h(x)*(Ts(x) - Tf). So Ts(x) can be solved for as Ts(x) = Tf + Qdot"/h(x).

My question is, what is the temperature at the leading edge of the plate? According to this model T(0) = Tf, but I think this is not a reasonable result. Doesn't it seem that a plate heated at a high rate will have a high leading edge temperature if convection rate is low?

Ultimately I'd like to calculate the plate temperature distribution to be able to couple the surface with radiation heat transfer and calculate the system results.

Thanks,
Stephen

 

[tkordyban]

This constant heat flux does not seem to give a realistic result at the leading edge because it is based on unrealistic assumptions:
1. the heat flux is absolutely uniform (how do you get that in real life, especially at the edges?"
2. the flat plate has no heat conduction within itself.

Because the heat transfer coefficient varies with distance from the leading edge, but the heat flux is uniform, the temperature of the plate has to vary with distance from the leading edge. In any real plate, if there is a temperature difference, heat will conduct from the high temperature location to the lower temperature location. Conduction would tend to make the leading edge hotter than T(f). But if we allowed conduction to happen, then the assumption of uniform heat flux would be broken, and we couldn't solve the math (or at least it wouldn't come out so neatly.)

Use the constant flux case as an approximation of a real flat plate, which will behave something like that if it has very low conductivity and the heat source is relatively uniform.