Chapter 3
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 |
Step-by-Step
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)
