Please turn in this assignment (and all future assignments for this course) by emailing them to me . Please use the subject of "Physics 222 HW7" for these assignments (if you use the link above, the subject will be added automatically). Please attach the source code to any programs that you write for this assignment. If you write more than one program, attach each separately. When answering the questions themselves feel free to answer directly in the body of your message (or attach your answers). I may take off points for assignments that are not sent to me in the correct form.
In this problem you will write a program to find the steady-state temperature profile of a window (or wall). The physical concept being explored here is heat conduction. The relatively simple equation that we will be dealing with for heat flow is:
P/A = -k dT/dxwhere P/A is the power per area (or energy per time per area) flowing through a surface, k is the conductivity and dT/dx is the temperature gradient in the direction the heat is flowing.
To solve this equation numerical we will convert the differential equation to a difference equation. The idea behind a difference equation is that derivatives are approximated as small (but non-infinitesimal) Δ's. So here we have
ΔT = - ((P/A)/k) Δx.In order to apply this difference equation in a program we have to convert the Δ's into differences between variables. In this case I am going to give you the temperature on the outside of the window. To find the temperature a small distance from the outside of the window we would say:
T_{1} = T_{0} + (((P/A)/k)) Δx.(Note that by starting at the outside and working our way into the house, we have flipped the sign on the heat flux term - if you prefer you can think of P/A as negative).
We can generalize this scheme to find the temperature at any place along the window based on the temperature at the previous location:
T_{i+1} = T_{i} + ((P/A)/k_{i+1}) Δx.The last equation is the one that you will want to translate into code for this program. Your program should contain a loop which iterates through to find the temperature based on the temperature at the previous location. Your program should save the temperature at all locations using an array. Note that you will also need an array which keeps track of the location (x) for each temperature. You will to use an equation similar to:
x_{1+1} = x_{i} + Δx .Your program should output the temperature at the end of each layer of material and create a plot of your results with PLPlot. Note that you should use lines [ pls->line(npts, x, y); ] in the plots instead of points. Also, note that in this case there is no need to send all of your data points to the plotting program, since with so many data points you would create very large plot files. Instead just plots the values at the edge of each material - the first and last point in each layer. Listed below are the conditions that you should test your program with. You should save these conditions directly in your code. Use a Δx of 1.0x10^{-5} m and an outside temperature of -20.0 degrees C for all runs.
P/A | k_{1} | width_{1} | k_{2} | width_{2} | k_{3} | width_{3} |
W/m^{2} | W/m/deg. C | m | W/m/deg. C | m | W/m/deg. C | m |
0.0 | 0.84 | 0.0055 | 0.018 | 0.003 | 0.84 | 0.0055 |
400.0 | 0.84 | 0.0055 | 0.018 | 0.003 | 0.84 | 0.0055 |
400.0 | 0.84 | 0.0055 | 0.84 | 0.003 | 0.84 | 0.0055 |
400.0 | 0.84 | 0.0055 | 0.024 | 0.003 | 0.84 | 0.0055 |
400.0 | 0.84 | 0.0055 | 0.018 | 0.0055 | 0.84 | 0.0055 |
The cases above correspond to several physical situations. The material with the highest conductivity is glass, the material with the middle conductivity is air, and the material with the lowest conductivity is argon gas. The first situation is a window with no heat source. In the second case there is a heat source and a window that has two panes of glass with a smaller argon gap between the glass layers. In the third case the argon is replaced by glass. In the fourth case the argon gas is replaced with air. In the last case a larger argon gap is used. Note that the assumptions for these cases are simplistic. We have assumed infinitely long walls made of only the listed materials. Also, we have assumed no convection.
For this program, turn in the output of your program running for each of the cases. Comment on the results of each run of the program. Do the results seem reasonable? Why or why not?