ME 422
FEM Homework #3
Distributed: December 13, 2012
Due: December 20, 2012
Consider the sphere shown below, which experiences natural convection on
its outside. It also has internal heat generation, (we could imagine this occurs
due to electrical heating).
.
q’’=
n h(Ts − Ta )
D = 1m
h = 4 W/m 2C
Ta = 20 C
k = 4 W/mC
g = 4x103 W/m3
Problem 1
Let us try to get some idea of what the temperature in this sphere could
be. If the sphere has a diameter D=1m and an internal heat generation
g=4×103 W/m3 , how many Watts does this sphere generate? What is the
surface area of the sphere? If we have an ambient temperature of 20C, and
a convection coefficient of 4 W/m2 C, how hot does the surface of the sphere
have to be to dissipate the generated Watts of power?
Problem 2
Now that we have some idea of what the answer could be, let us try this
problem on Workbench. Go ahead and make a 3-D model using a solid
sphere, even though that is like smashing a gnat with a wrecking ball. Since
this is a thermal problem, you will need to use “Steady State Thermal” from
1
the toolbox, rather than Static Structural. Once the analysis runs, check
your answers for the surface temperature against the results from Problem
1. The two answers should be quite close. If they don’t agree, fix the one
that’s wrong until they do agree. Use a plane to slice the sphere to show
the temperatures through the cross-section. Print and turn in a plot of your
temperature results.
Problem 3
a. The problem can be reduced to a one-dimensional conduction analysis
with a varying area (in this case spherical). State the governing differential
equations and boundary conditions for the problem. (You might consider
using a zero heat flux boundary condition at the center of the sphere.)
b. Derive the analytical solution for the temperature, as we did in class for
a rod and a cylinder. Plot (on a computer) the temperature as a function of
radius for the numerical values given in the figure. I intend that this answer
be close to the answers for Problems 1 and 2. If they don’t agree, fix the
one(s) that are wrong until they do agree. Print and turn in the hardcopy
of the plot, as well as the code that was used to create the plot.
Problem 4
a. Derive the weak form for this particular problem. (i.e. Put the particular
area and particular boundary conditions into the weak form.)
b. Show that the stiffness matrix for a typical 2-node finite element which
has node one at x = x1 and node two at x = x2 is:
4πk 2
(x + x1 x2 + x21 )
K=
3Le 2
1 −1
−1
1
(Note: this is a perfectly lovely time to use Maple, if you wish.)
c. Show that the forcing vector due to internal heat generation for that same
typical 2-node finite element is:
⎡
x2
⎤
πL3e g ⎣ 6 L12e + 4 Lx1e + 1 ⎦
F=
x2
3
6 L12 + 8 Lx1e + 3
e
(Note: another appropriate time to use Maple.)
2
d. Use five 2-node elements to discretize the sphere. Assemble the global
matrix equations (6 equations in 6 unknowns) from the elemental stiffnesses
and forcing vectors. Apply the boundary condition terms to the matrix.
e. Solve the finite element matrix equations. You may use MATLAB but do
not use Maple. Plot your answers. Once again, I would expect your answers
to be quite similar to those from Problems 1-3. If your answers don’t agree,
fix the one(s) that are wrong until they do agree. Print and turn in the
hardcopy of the plot, as well as the code that was used to create the plot.
3