HEAT AND MASS TRANSFER

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3) Significance of temperature-dependent thermal conductivity Learning Objective: Solving problems with variable physical properties using both exact solutions and average properties Problem Background For In class problem #H3, you found the average thermal conductivity assuming that it varied according to the function k=k,(1+B7). (b) To determine the temperature distribution in the wall, we begin with Fourier's law of heat conduction, expressed as 0--K(T) dT where the rate of conduction heat transfer and the area A are constant. Separating variables and integrating from x = 0 where TO)=T, to any x where 7(1) = T. we get ["Qar=-4[*kat Substituting k(T) = k/(1+B7) and performing the integrations, we obtain Ox=-Ak[(T - T) + P{T2 - T})/2] Problem 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) = k (1 + BT) where ko = 38 W/m-K and B=9.21 x 10- K-!. 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: 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. O = 4. 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