In the production of sheet metals or plastics, it is
Chapter 7, Problem 7.28(choose chapter or problem)
In the production of sheet metals or plastics, it is cus-tomary to cool the material before it leaves the produc-tion process for storage or shipment to the customer.Typically, the process is continuous, with a sheet ofthickness ? and width Wcooled as it transits the dis-tance Lbetween two rollers at a velocity V. In thisproblem, we consider cooling of an aluminum alloy(2024-T6) by an airstream moving at a velocity u?incounter flow over the top surface of the sheet. A turbu-lence promoter is used to provide turbulent boundarylayer development over the entire surface.(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, derive a differential equation thatgoverns the temperature distribution along thesheet. Because of the low emissivity of the alu-minum, radiation effects may be neglected. Express your result in terms of the velocity, thickness, andproperties of the sheet (V,?, ?, cp), the local con-vection coefficient hxassociated with the counterflow, and the air temperature. For a known temper-ature of the sheet (Ti) at the onset of cooling and anegligible effect of the sheet velocity on boundarylayer development, solve the equation to obtain anexpression for the outlet temperature To.(b) For ??2 mm, V?0.10 m/s, L?5 m, W?1m,u??20 m/s, T??20?C, and Ti?300?C, what isthe outlet temperature To?
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