A common annoyance in cars in winter months is the formation of fog on the glass surfaces which blocks the view. A practical way of solving this problem is to blow hot air or attach electrical resistance heaters to the inner surfaces. Consider the rear window of a car that consists of 0.4 m thick glass (k = 0.84 W/mK and a = 0.39 ´ 10^-6 m²/s). Strip heater wires of negligible thickness are attached to the inner surface of the glass, 4 cm apart. Each wire generates heat at a rate of 10 W/m length. Initially the entire car, including the windows, is at the outdoor temperature of T₀ = -3° C. The heat transfer coefficients at the inner and outer surfaces of the glass can be taken to be h_i = 6 W/m²K and h_o = 20 W/m²K.
 
 

Using the explicit finite difference method with a mesh size of Dx = 0.2 cm along the thickness and Dy = 1 cm in the direction normal to the heater wires, determine the temperature distribution throughout the glass 15 minutes after the strip heaters are turned on. Also, determine the temperature distribution when steady-state conditions are reached.
 
 

Notes:

1. For a description of the explicit and implicit methods refer to pages 248-263 of the textbook. Table 5.2 on p. 252 lists the stability criteria for various node configurations. You need to take these criteria into account when choosing a time step Dt. Your time step needs to be chosen so that the Fourier number Fo (as defined by eq. 5.72 on p.249) satisifies all of the stability criteria, i.e. at the interior, side and corner nodes. If even one of these stability criteria is not satisfied, the solution will diverge.
2. When you submit your project, please include:

• all equations used along with any heat balance diagrams (10% of your grade)
• clearly labeled plots with at least 10 contour levels (20%)
• a hardcopy of your code; if you use the built-in formulas in Excel then a copy of the representative equations as you typed them in (10%)
• brief comments about the temperature distribution and its main features (10%)

• the numbers on your Excel spreadsheet
• your files on disk

1. Remember one thing: this project is different from the first project because it is a time dependent problem instead of a steady state one. In the first project, the iterations represented better and better approximations to the steady state solution; in this project, the iterations represent actual temperature distributions at different times. You will need to use substantially different code for this project; instead of a loop which terminates when convergence is reached you need to march forwards in time. You may work with the same team member or switch or go solo. Please list the contribution of each team member