Q. 3.16

Reynolds–Colburn Analogy Applied to Laminar Boundary-Layer Flow

Consider a heated plate of length L and constant wall temperature $\rm T_w$, subject to a cooling air-stream $\rm(u_∞, T_∞)$. Find a functional dependence for $\rm q_w(x)$.

 Approach Assumptions Sketch • Reynolds– Colburn analogy • Thermal Blasius flow (see Sect. 5.2) • Constant properties

Verified Solution

Rewriting Eq. (3.73) we have:

$\rm\frac{1}{2} C_f(x)=St_x\,Pr^{2/3}\quad\text{for}\quad 0.6\lt Pr\lt 60$                                 (3.73)

$\rm \frac{{q}_{\mathrm{w}}({x})}{\mathrm{\rho \;c_{p}u_{\infty}(T_{w}-T_{\infty})}}=C_{\mathrm{f}}\;{\mathrm{Pr}}^{-\;2/3}$                                       (E.3.l6.1)

From Example 5.1 we can deduce that

$\rm C_f\sim Re_x^{-1/2}$                         (E.3.16.2)

where actually $\rm C_f=0.664/\sqrt{Re_x}$ as shown in Sect. 5.2. Now, with everything else being constant

$\rm \mathrm{~{q}_{w}}(x)\sim\frac{{ K}}{\sqrt{{ x}}}\,,\qquad K=⊄$                                (E.3.16.3a,b)

Graph:

Comment: The wall heat flux from the plate surface decreases nonlinearly with plate distance because of the increasing Re(x), or better, the larger thermal boundary-layer thickness, $δ_{th}$ (x) , and hence milder wall temperature gradients (see Eq. 3.63).

$\rm\vec{\bf q}=-k\,\nabla\,T$                                 (3.63)