Show all the steps leading to the integral balance equation (18.6) in the text.

Use the following boundary conditions and verify Eq. (18.7). (i) At \(y=0, T=T_{\mathrm{S}}\). (ii) At \(y=\delta_{\mathrm{t}}, T=T_{0}\). (iii) At \(y=\delta_{\mathrm{t}}, \partial T / \partial y=0\). (iv) At \(y=0, \partial^{2} T / \partial y^{2}=0\).

What is the rationale for the fourth condition above? Using the cubic profile, derive Eq. (18.9).

1/4 [ (T - Too) dy] vx (T-T)dy | dx qw (18.6) PCp