23 - Convection - Analogies between Heat and Momentum Transfer

Prandtl assumed that the flow field consisted of two layers, a viscous sublayer where the molecular diffusivities are dominant, that is, \[\varepsilon_m \ll \nu \qquad \text{ and } \qquad \varepsilon_h \ll \alpha\] and a turbulent zone where the turbulent diffusivities are dominant, that is \[\nu \ll \varepsilon_m \qquad \text{ and } \qquad \alpha \ll \varepsilon_h \qquad \text{and} \qquad \varepsilon_m = \varepsilon_h = \varepsilon\] Using the above in the equations for momentum and heat transfer in each layer, and using the definitions of \(h\) and \(f\), we get \[\boxed{\text{St} = \frac{h}{\rho C_P v_m} = \frac{f}{2}\left(\frac{1}{1+5\sqrt{f/2}(\text{Pr}-1)}\right)}\] This reduces to Reynolds analogy for \(\text{Pr}=1\).