22 - Convection - Correlations for Heat Transfer Coefficient

Since the existence of convective heat transfer between the surface and the fluid dictates that the fluid temperature must continue to change with \(x\), one might question whether fully developed conditions ever can be reached. The situation is certainly different from the hydrodynamic case, for which \((\partial v/\partial x)=0\) in the fully developed region. In contrast, if there is heat transfer, \((\partial T/\partial x)\) at any radius is not zero. Introducing a dimensionless temperature \((T_w-T)/(T_w-T_m)\), condition for which this ratio becomes independent of \(x\) are known to exist. Although the temperature profile \(T(r)\) continues to change with \(x\), the relative shape of the profile does not change and the flow is said to be fully developed. Instead of \((\partial T/\partial x)=0\), the condition is \[\boxed{\frac{d}{dx}\left[\frac{T_w(x)-T(r,x)}{T_w(x)-T_m(x)} \right] = 0}\] i.e., \((T_w-T)\) changes in the same way as \((T_w-T_m)\).