10. Consider the equilibrium heat conduction problem with conductivity, k(u), depending on the temperature u, de...

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10. Consider the equilibrium heat conduction problem with conductivity, k(u), depending on the temperature u, de (k(u) du). dx u(0) = = 0, u(1) = 10. = The next simplest choice for k(u) after the constant case is to assume that the conductivity depends linearly on temperature, i.e. k(u) ko(1 bu) where b and ko are positive constants. This corresponds to a substance where the conductivity decreases as the temperature increases. (a) Show, by solving the equilibrium 1-D heat equation, that u(x): = 1 = 0, 1 1 - 26(10 - 50b)x if b< 1 10 (Make certain discarding the positive sign solution of the quadratic equation is justified.) (b) Sketch the graph of the equilibrium temperature. Also sketch the con- stant conductivity case on the same diagram. (c) Discuss the physical significance of the condition b≤. 10. Consider the equilibrium heat conduction problem with conductivity, k(u), depending on the temperature u, de (k(u) du). dx u(0) = = 0, u(1) = 10. = The next simplest choice for k(u) after the constant case is to assume that the conductivity depends linearly on temperature, i.e. k(u) ko(1 bu) where b and ko are positive constants. This corresponds to a substance where the conductivity decreases as the temperature increases. (a) Show, by solving the equilibrium 1-D heat equation, that u(x): = 1 = 0, 1 1 - 26(10 - 50b)x if b< 1 10 (Make certain discarding the positive sign solution of the quadratic equation is justified.) (b) Sketch the graph of the equilibrium temperature. Also sketch the con- stant conductivity case on the same diagram. (c) Discuss the physical significance of the condition b≤.

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To solve the equilibrium heat conduction problem with linearly temperaturedependent conductivity we need to solve the differential equation ddx ku ddx ... View the full answer