In the production of sheet metals or plastics, it

Chapter 7, Problem 7.31

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In the production of sheet metals or plastics, it iscustomary to cool the material before it leaves the pro-duction process for storage or shipment to the customer.Typically, the process is continuous, with a sheet ofthickness ? and width Wcooled as it transits the distanceLbetween two rollers at a velocity V. In this problem, weconsider cooling of plain carbon steel by an airstreammoving at a velocity u?in cross flow over the top andbottom surfaces of the sheet. A turbulence promoter isused to provide turbulent boundary layer developmentover the entire surface.TurbulencepromoterWTiToPlain carbon steelSurroundings, TsurxLVuTAir(a) By applying conservation of energy to a differentialcontrol surface of length dx, which either moveswith the sheet oris stationary and through whichthe sheet passes, and assuming a uniform sheettemperature in the direction of airflow, derive a dif-ferential equation that governs the temperature dis-tribution, T(x), along the sheet. Consider the effectsof radiation, as well as convection, and express yourresult in terms of the velocity, thickness, and prop- erties of the sheet (V,?, ?, cp, ?), the average con-vection coefficient associated with the crossflow, and the environmental temperatures (T?, Tsur).(b) Neglecting radiation, obtain a closed form solutionto the foregoing equation. For ??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 Ti?500?C at theonset of cooling, what is the outlet temperature To?Assume a negligible effect of the sheet velocityon boundary layer development in the directionof airflow. The density and specific heat of the steelare ??7850 kg/m3and cp?620 J/kg?K, whileproperties of the air may be taken to be k?0.044W/m?K, ??4.510?5m2/s, Pr?0.68.(c) Accounting for the effects of radiation, with ??0.70 and Tsur?20?C, numerically integrate the dif-ferential equation derived in part (a) to determinethe temperature of the sheet at L?10 m. Explore theeffect of Von the temperature distribution alongthe sheet