Question

A long wire with circular cross section and diameter D = 1 mm is submerged in an oil bath with temperature To 20。C. The wire is initially at temperature T = 20。C, the same temperature as the oil. A current is passed through the wire, resulting in a volumetric heat generation of q - 1.5 x 108 W/m3. The convection coefficient is h - 500 W/m2 - K, and the properties of the wire are ρ-8000 kg/m3, Cp-500 J/kg-K, and k-20 w/m . K. We are interested in the wire temperature as a function of time. a) Justify use of the lumped capacitance approximation for this problem b) Neglecting thermal radiation, calculate the steady state temperature of the wire. c) Still neglecting radiation, write an energy balance for the wire, and derive an analytical expression for the temperature as a function of time in terms of the given problem parameters. (Note that this requires solution of an ordinary differential equation. It will be convenient to use the change of variables 0 (t)- T(t) - Tss, where Tss is the steady state temperature found in (b). You may leave your solution in terms of 0 and 6i, where e is the initial value.) d) How long does it take for the wire to reach a temperature that is within 1 °C of the steady state temperature? e) Now drop the assumption that radiation can be neglected, and assume that the wire (with emissivity E 0.9) exchanges radiative energy with its surroundings at To. Write dT an energy balance for the wire that includes radiation, and give an expression for -- in dt terms of the problem parameters. f) Solve for the steady state temperature of the wire when radiation is accounted for. g) Solve numerically (e.g., using Matlab) for the temperature as a function of time for the case with radiation, and again calculate how long it takes for the wire to reach a temperature that is within 1 °C of the new steady state value. Please include a printout of any code you write as an appendix with your submission.

Answer 1

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