Mech 302 Heat Transfer HW4

Note: This homework requires numerical solution of the unsteady heat equation in 2D rectangular domains. A sample MATLAB script is posted on the web page. The script solves an unsteady version the problem in Example 4.3 on pg. 251 in your book. You may want to modify this MATLAB script for this HW. You have two weeks for this homework.

1. (Problem 4.181 in the Book (slightly modified for unsteady heat transfer)) A heat sink for cooling computer chips is fabricated from copper (ks = 400 W/mK), with machined microchannels passing a cooling fluid for which T = 25oC and h = 30,000 W/m2K. The lower side of the sink experiences no heat removal, and a preliminary heat sink design calls for dimensions a = b = ws = wf = 200 microns. A symmetrical element of heat path from the chip to the fluid is shown in the inset.

a. Using a uniform grid, determine the corresponding heat rate q’ to the coolant per unit channel length in W/m for a maximum allowable chip temperature Tc,max = 75oC. Assume that the initial temperature is T(x,t=0) = Tc,max in the sink.

b. Perform a grid convergence study for this case and show grid convergence for your results. To do this, start with a uniform grid and refine it successively until the results become independent of the grid refinement.

2. (Problem 4.184 in the Book (slightly modified for unsteady

heat transfer) A long bar of rectangular cross section is 60 mm-by-90 mm on a side and has a thermal conductivity of k = 1 W/mK. One surface is exposed to a convection process with air at 100oC and a convection coefficient of 100 W/m2K, while the remaining surfaces are maintained at 50oC. Assume that the temperature is initially uniform at T(x,y,t=0) = 50oC in the bar.

a. Discuss about the time evolution of temperature distribution in the bar.

b. Determine the steady temperature distribution in the bar.

c. Determine time evolution of the total amount of heat flux per unit width from the air to the bar.

d. Show grid convergence.