Building on the previous problem allows us to look at predicting the temperature distribution for an arbitrary heat-flux specification at the surface. Here we can solve for the temperature distribution providing the heat transfer coefficient can be written in the following form:

\[h\left(x_{o}, x\right)=f(x)\left(x^{a}-x_{o}^{a}\right)^{-g}\]

Fortunately, our solution for the unheated starting length fits into this form and leads to the following expression for the temperature profile:

\[T_{s}(x)-T_{\infty}=\frac{0.623}{k_{f}} \operatorname{Pr}^{-1 / 3} \operatorname{Re}_{x}^{-1 / 2} \int_{0}^{x}\left[1-\left(\frac{x_{o}}{x}\right)^{3 / 4}\right]^{-2 / 3} q_{s}^{\prime \prime}\left(x_{o}\right) d x_{o}\]

Assuming a constant surface heat flux, determine an expression for the Nusselt number. How much of an increase do we obtain in the heat transfer coefficient over the constant surface temperature case?

Previous problem

Consider flow over a flat plate where the surface of the plate has a temperature distribution. Here we are solving the situation for the unheated starting length but once the plate is heated, the steady-state temperature distribution obeys:

\[T_{s}=T_{\infty}+b x\]

The heat flux for an arbitrary wall temperature distribution is given by:

\[q_{s}^{\prime \prime}=\int_{0}^{x} h\left(x_{o}, x\right)\left(\frac{d T_{s}}{d x_{o}}\right) d x_{o}+\sum_{i=1}^{n} h\left(x_{o i}, x\right) \Delta T_{s i}\]

where \(\Delta T_{s i}\) represents any abrupt jumps in surface temperature. The idea here is that we break the calculation up into a series of unheated starting length problems and the sum them up to get the final heat flux.