Chapter 7, Problem 7.28P

In the production of sheet metals or plastics, it is customary to cool the material before it leaves the production process for storage or shipment to the customer. Typically, the process is continuous, with a sheet of thickness $\delta$ and width W cooled as it transits the distance L between two rollers at a velocity V. In this problem, we consider cooling of an aluminum alloy (2024-T6) by an airstream moving at a velocity $u_{\infty }$ in counter flow over the top surface of the sheet. A turbulence promoter is used to provide turbulent boundary layer development over the entire surface.

1. By applying conservation of energy to a differential control surface of length dr. which either moves with the sheet or is stationary and through which the sheet passes, derive a differential equation that governs the temperature distribution along the sheet. Because of the low emissivity of the aluminum, radiation effects may be neglected. Express

your result in terms of the velocity, thickness, and properties of the sheet $\left(V,\delta ,\rho ,c_p\right),$ the local convection coefficient $h_x,$ associated with the counter flow, and the air temperature. For a known temperature of the sheet $\left(T_i\right)$ at the onset of cooling and a negligible effect of the sheet velocity on boundary layer development, solve the equation to obtain an expression for the outlet temperature $T_o.$

2. For $\delta =2\text{mm,}V=0.10\text{m/s,}L=5\text{m,}W=1\text{m,}{u}_{\infty }=20\text{m/s,}{T}_{\infty }=20°\text{C,}$ and $T_i=300°\text{C,}$ what is the outlet temperature $T_o?$