To control the drag force or the rate of heat transfer to a surface, a technique called

Question:

To control the drag force or the rate of heat transfer to a surface, a technique called transpirational cooling is employed. The situation is shown in Figure P12.13. The surface of the substrate to be cooled is perforated and positive or negative pressure is applied to induce flow through the surface.

a. Using our integral method of analysis, show that the hydrodynamic boundary layer thickness is given by:
\[\begin{aligned}& v_{o}>0 \quad x=\frac{7 v_{\infty}}{15 v_{0}} \delta_{h}-\frac{7 v}{10 v_{\infty} v_{o}^{2}} \ln \left[1+\frac{2 v_{o} \delta_{h}}{3 v}\right] \\& v_{o}]\end{aligned}\]

b. Plot the boundary layer thickness as a function of position along the plate for:
\[v_{\infty}=1 \mathrm{~m} / \mathrm{s} \quad v_{o}= \pm 0.001 \mathrm{~m} / \mathrm{s} \quad T_{s}=T_{\infty}=300 \mathrm{~K} \quad L=20 \mathrm{~cm}\]

c. What are the implications of your result with regards to:
(1) Controlling the heat transfer coefficient?
(2) Controlling the friction factor? image text in transcribed