Problem 2) Significance of temperature-dependent thermal conductivity Learning Objective: Solving problems with variable physical properties using both exact solutions and average properties Problem Background (IC #H3) For In class problem #H3,

you found the average thermal conductivity assuming that it varied according to the function k=ko(1+BT). (b) To determine the temperature distribution in the wall, we begin with Fourier's law of heat conduction, expressed as Q=-k(T) AT dx where the rate of conduction heat transfer Q and the area A are constant. Separating variables and integrating from x=0 where 7(0) = T₁ to any x where 7(x) = T, we get || ₁ dx = -Afr Substituting k(T) = ko(1+BT) and performing the integrations, we obtain Qx=-Ak[(T-T₁) +B(T²-T)/2] k(T)dT/nHW Problem 2 statement: Consider a 2-m-high and 0.7-m-wide bronze plate whose thickness is 0.1 m. One side of the plate is maintained at a constant temperature of 600 K while the other side is maintained at 400 K, as shown in Fig. 2-66. The thermal conductivity of the bronze plate can be assumed to vary linearly in that temperature range as k(T) = ko(1+BT) where k = 38 W/m-K and B=9.21 x 10-4 K-¹. Disregarding the edge effects and assuming steady one-dimensional heat transfer, determine the rate of heat conduction through the plate. There are two ways to calculate the heat flux across the bronze plate; try them both as described below: a) Use the integrated version of Fourier's Law assuming a constant k, but use the average k. b) Use the exact solution derived using the function above when integrating the differential energy balance with the function above as described below (in this reference, Q q, in the notation we've been using). c) Do you get the same answer or not? What seems to be the best way to calculate the flux?