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_∞ 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.

 

(a) 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, derive a differential equation that governs the temperature distribution, T(x), along the sheet. Consider the effects of radiation, as well as convection, and express your result in terms of the velocity, thickness, and properties of the sheet (V, δ, ρ, c_p, ε), the average convection coefficient associated with the cross flow, and the environmental temperatures (T_∞, T_sur)

(b) Neglecting radiation, obtain a closed form solution to the foregoing equation. For δ = 3 mm, V = 0.10 m/s, L = 10 m, W = 1 m, u_∞ = 20 m/s, T_∞ = 20°C, and a sheet temperature of T_i = 500°C at the onset of cooling, what is the outlet temperature T_o? Assume a negligible effect of the sheet velocity on boundary layer development in the direction of airflow. The density and specific heat of the steel are ρ = 7850 kg/m³ and c_p = 620 J/kg ∙ K, while properties of the air may be taken to be k = 0.044 W/m∙K, v = 4.5
× 10^-5 m² /s, Pr = 0.68.

(c) Accounting for the effects of radiation, with ε = 0.70 and T_sur = 20°C, numerically integrate the differential equation derived in part (a) to determine the temperature of the sheet at L = 10 m. Explore the effect of V on the temperature distribution along the sheet.