Consider a solid bar of length L with variable cross-sectional area given by A(x), where x...

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Consider a solid bar of length L with variable cross-sectional area given by A(x), where x = [0, L] is distance along the bar. If the sides of the bar are insulated and if A(x) varies slowly, it is reasonable to make an approximation in which the heat flows only in the direction. Consider the rate of change of the total thermal energy between arbitrary positions x = a and x = b to derive a heat equation for the bar in terms of the temperature u(x, t). You may assume the density, heat capacity and thermal conductivity are all constants (i.e. the bar has constant thermal properties) and that there is no heat source. Consider a solid bar of length L with variable cross-sectional area given by A(x), where x = [0, L] is distance along the bar. If the sides of the bar are insulated and if A(x) varies slowly, it is reasonable to make an approximation in which the heat flows only in the direction. Consider the rate of change of the total thermal energy between arbitrary positions x = a and x = b to derive a heat equation for the bar in terms of the temperature u(x, t). You may assume the density, heat capacity and thermal conductivity are all constants (i.e. the bar has constant thermal properties) and that there is no heat source.

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The question asks us to derive the heat equation for a onedimensional bar with a variable crosssectional area Ax assuming that heat flows only in the ... View the full answer