A 1-inch diameter wire of 10-ft length (which has an internal resistance of 3 kΩ, and thermal conductivity of 300 W/m-°C) is subjected to a current of 10 amperes. This heat generation caused the internal and external temperatures of the wire to be Ti and To (in K), respectively. A. Calculate the net volumetric heat generation (W/m3) in terms of current, wire resistance, and total volume of the electric wire. B. Illustrate the physical model describing the heat transfer of the abovementioned system. Indicate the control volume (differential volume) in terms of the transfer area and distance. Specify the elements/components needed in establishing the shell balance. Indicate the boundary conditions and illustrate the direction of transfer. C. What is the unsteady state heat shell balance for the system? D. Derive the steady state differential heat transfer relation describing the system. E. Using general terms (variables), derive the steady state temperature profile as a function of the radial transfer of heat in the electrical wire. F. What are the appropriate boundary conditions to solve for the constants of integration? What are the units of the constants of integration in the temperature profile? G. Derive an expression for the average temperature of the wire in terms of the temperature profile of the electric wire.
A 1-inch diameter wire of 10-ft length (which has an internal resistance of 3 kΩ, and thermal conductivity of 300 W/m-°C) is subjected to a current of 10 amperes. This heat generation caused the internal and external temperatures of the wire to be Ti and To (in K), respectively.
A. Calculate the net volumetric heat generation (W/m3) in terms of current, wire resistance, and total volume of the electric wire.
B. Illustrate the physical model describing the heat transfer of the abovementioned system. Indicate the control volume (differential volume) in terms of the transfer area and distance. Specify the elements/components needed in establishing the shell balance. Indicate the boundary conditions and illustrate the direction of transfer.
C. What is the unsteady state heat shell balance for the system?
D. Derive the steady state differential heat transfer relation describing the system.
E. Using general terms (variables), derive the steady state temperature profile as a function of the radial transfer of heat in the electrical wire.
F. What are the appropriate boundary conditions to solve for the constants of integration? What are the units of the constants of integration in the temperature profile?
G. Derive an expression for the average temperature of the wire in terms of the temperature profile of the electric wire.
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