In an extrusion process, copper wire emerges from the

Chapter , Problem 7.72

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In an extrusion process, copper wire emerges from the extruder at a velocity Ve and is cooled by convection heat transfer to air in cross flow over the wire, as well as by radiation to the surroundings. (a) By applying conservation of energy to a differential control surface of length dx, which either moves with the wire or is stationary and through which the wire passes, derive a differential equation that governs the temperature distribution, T(x), along the wire. In your derivation, the effect of axial conduction along the wire may be neglected. Express your result in terms of the velocity, diameter, and properties of the wire (Ve, D, , cp, ), the convection coefficient associated with the cross flow , and the environmental temperatures (T, Tsur). (b) Neglecting radiation, obtain a closed form solution to the foregoing equation. For Ve 0.2 m/s, D 5 mm, V 5 m/s, T 25 C, and an initial wire temperature of Ti 600 C, compute the temperature To of the wire at x L 5 m. The density and specific heat of the copper are 8900 kg/m3 and cp 400 J/kg K, while properties of the air may be taken to be k 0.037 W/m K, 3 105 m2 /s, and Pr 0.69. (c) Accounting for the effects of radiation, with 0.55 and Tsur 25 C, numerically integrate the differential equation derived in part (a) to determine the temperature of the wire at L 5 m. Explore the effects of Ve and on the temperature distribution along the wire

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