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Prove that the one-dimensional steady-state temperature distribution for a fin with a uniform cross-sectional area of diameter d and length L is given by:

d²T/dx² = ∂²T/∂x²

User Fmaccaroni
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Final answer:

The one-dimensional steady-state temperature distribution for a fin is described by a second-order differential equation representing the rate of heat conduction along the fin's length, assuming a constant thermal gradient and thermal conductivity.

Step-by-step explanation:

The steady-state temperature distribution in a one-dimensional fin is predicted by the heat diffusion equation which assumes a constant thermal conductivity and negligible radiative and convective heat transfers. The equation is given by the second-order differential equation d²T/dx², where T is the temperature, and x is the position along the fin's length. The expression d²T/dx² = ∂²T/∂x² indicates that the change in temperature with respect to time is governed by the spatial derivative of the temperature, assuming a steady state where the temperature change over time is zero.

For a fin with uniform cross-sectional area, the rate of heat transfer by conduction is directly proportional to the thermal gradient along the fin's length, and, in a steady state, this gradient is constant. Given that the cross-sectional area A of the fin is related to its diameter d by the equation A = (πd²)/4, the fin's thermal conductivity k, and the heat transfer by conduction is described by the equation dT/dx = -(P/KA), where P denotes the perimeter of the fin through which heat is being transferred.

User SAMIR RATHOD
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