The heat transfer rate per unit width from a longitudinal section, x_{2}-x_{1}x 2 −x 1 , can be expressed as q_{12}^{\prime}=\bar{h}_{12}\left(x _{2}-x_{1}\right)\left(T_{s}T_{\inf ty}\right)q 12 = hˉ12 (x 2−x 1)(T s−T [infinity]), where \bar{h}_{12} h-12 is the average coefficient for the section of length \left(x_{2}-x_{1}\right)(x 2 −x 1 ). Consider laminar flow over a flat plate with a uniform temperature T_{s}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. (a) Beginning with the convection rate equation in the form d q^{\prime}=h_{x} d x\left(T_{s}-T_{\infty}\right)dq =h x dx(T s −T [infinity] ), derive an expression for \bar{h}_{12} hˉ12 in terms of C, x_{1}x 1 and x_{2}x 2 (b) Derive an expression for \bar{h}_{12} h-12 in terms of x_{1}, x_{2}x 1 ,x 2 , and the average coefficients \bar{h}_{1} hˉ1 and \bar{h}_{2} hˉ2 , corresponding to lengths x_{1}x 1 and x_{2}x 2, respectively.