Air enters a square duct through a 1 - \(\mathrm{ft}\) opening as is shown in Fig. P9.19. Because the boundary layer displacement thickness increases in the direction of flow, it is necessary to increase the cross-sectional size of the duct if a constant \(U=2 \mathrm{ft} / \mathrm{s}\) velocity is to be maintained outside the boundary layer. Plot a graph of the duct size, \(d\), as a function of \(x\) for \(0 \leq x \leq 10 \mathrm{ft}\) if \(U\) is to remain constant. Assume laminar flow.

Figure P9.19

U = 2 ft/s 1 ft d(x) 2 ft/s