Chapter 6, Problem 6.13P

The heat transfer rate per unit width (normal to the page) from a longitudinal section, $x_2-x_1,$ can be expressed $q_{12}^{\text{'}}={\bar{h}}_{12}\left(x_2-x_1\right)\left(T_s-T_{\infty }\right),$ where ${\bar{h}}_{12}$ is the average coefficient for the section of length $\left(x_2-x_1\right).$ Consider laminar flow over a flat plate with a uniform temperature $T_s.$ The spatial variation of the local convection coefficient is of the form $h_x=C{x}^{-1/2},$ where C is a constant.

1. Beginning with the convection rate equation in the form $dq\text{'}=h_{x}dx\left(T_s-T_{\infty }\right),$ derive an expression for ${\bar{h}}_{12}$ in terms of $C,x_1,$ and $x_2.$
2. Derive an expression for ${\bar{h}}_{12}$ in terms of $x_1,x_2,$ and the average coefficients ${\bar{h}}_1$ and ${\bar{h}}_2,$ corresponding to lengths $x_1$ and $x_2,$ respectively.