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 δ and width W cooled as it transits the distance L between two rollers at a velocity V. In this problem, we consider cooling of plain carbon steel by an airstream moving at a velocity u[infinity] in cross flow over the top and bottom surfaces of the sheet. A turbulence promoter is used to provide turbulent boundary layer development over the entire surface. By applying conservation of energy to a differential control surface of length dx, which either moves with the sheet or is stationary and through which the sheet passes, and assuming a uniform sheet temperature in the direction of airflow. Required:
Derive a differential equation that governs the temperature distribution, T(x), along the sheet.