22 - Convection - Correlations for Heat Transfer Coefficient

By assumption, \(q_w=h(T_w-T_m)=\text{ constant}\).

For thermally developed flow, \(h = \text{ constant}\). Then, \[\frac{d}{dx}(q) = \frac{d}{dx}\bigl[h(T_w-T_m) \bigr] = 0\] i.e., \[0 = h\left(\frac{dT_w}{dx} - \frac{dT_m}{dx} \right) \qquad \Longrightarrow \quad \boxed{\frac{dT_w}{dx} = \frac{dT_m}{dx}}\]