Heat conduction in a long square duct

Problem 3 Heat conduction in a long square duct (50 points) A long square duct (outside: 0.2m0.2m; inside: 0.1m0.1m ) is filled with a heat-generating material that provides for a uniform volumetric generation rate of q=3105W/m3. The square duct is exposed to air at Tair=300K with the convection coefficient h=100W/m2K. Both the duct wall and heat-generating material have the same thermophysical properties: =1000kg/m3;Cp=2500J/kgK;k=25W/mK. Initially the entire square duct including its content is at a uniform temperature Ti=300K. We are interested to find the temperature distribution in the square duct (both within the duct wall and inside the duct) at time t and at steady state. To solve this problem, you will have to use the finite difference method. Let us make a grid size x=y =0.05m and label the temperatures T1,T2,T3,T4,T5,T6 at 6 different locations as shown in the diagram. a) Use energy balance to derive the finite difference equations for T1,T2,T3,T4,T5,T6. (18 points) [Note: you must use energy balance to derive the finite different equations for all six temperature nodes. You will not get the full credit if you simply used the finite difference equations given in the lecture note or textbook.] b) Specify an appropriate t for use to determine these temperatures. Briefly discuss your reasons for the specified t. (4 points) c) Find the temperatures T1,T2,T3,T4,T5,T6 at time t=1h and at steady state. (12 points) d) Find the maximum temperature in the square duct and the time to reach the maximum temperature. (3 points) e) Find the heat transfer rates per unit length q ' from the square duct to air at time t=1h and steady state (8 points) f) How do you know if your estimated temperatures and heat transfer rates are accurate or not? Briefly discuss how you may improve the accuracy or reduce the errors in using the finite difference method. (5 points) Note: clearly show the methods and steps you used in finding the temperatures and q '. If you use Excel for the calculation, please also submit your Excel file