Homework 10: ChE 2216
Due Thursday, April 22nd 2010, 6:00 p.m.
1. (10 points) Determine the steady state temperature distribution in a thin metal slab 10
cm on a side, insulated in the z direction both above and below, so that heat transport
(to a good approximation) is zero in the z direction.
T(x,0) = 25 + x, while T(x,10) =
35 + x.
The slab is also insulated at the T(0,y) line, and at T(10,y) the temperature
equals 35(sin(
π
y/10)+1)+y.
Hint: To make it easier, you should determine a
transformation U(x,y) = T(x,y) - f(x,y) that still obeys the heat equation, but has mostly
homogeneous boundary conditions, and should also think about what the form of the
equations are in each direction.
Plot the Fnal solution in Matlab, and call this graph
HW10Q1.pdf
2. (10 points) Consider a heated metal plate 12 cm by 12 cm.
In the z direction, the
plate is thin and insulated. The temperature at the top, right, and left edges are
maintained at 22 °C.
The bottom edge is insulated, except for a very small area
(less than 0.05 cm) a distance 3 cm from the left side which is suffering from a point
±aw, where the temperature is 700 °C.
Solve the steady state temperature
distribution in this plate. To do this:
a) Write code which calculates the matrix arising from the Fnite difference
representation of this system.
You should make it general for number of steps.
b) Solve the system of equations to Fnd the temperature distribution for spacings of 1
cm, and 0.5 cm; plot these distribution in Matlab in 3D plots named HW10Q2a.pdf
and HW10Q2b.pdf.
Be sure to add labels on the axes of the Fle.
Comment on any
differences between the two in comments in your code.
Put your code in an .m Fle called HW10Q2.m.
You can make other function m-Fles if
you wish, but make sure you check them in and that the Fle HW10Q2.m calls them,
generates, and plots the data.