ME422
FEM Homework #7
Distributed: January 24, 2011
Due: January 31, 2011
Problem 1: Consider quenching of a steel slab. Initially, the slab is at a
temperature T0 . It is suddenly plunged into oil at Toil .
20W
20W
2W
k
ρ
cp
ho
Toil
T0
W
=
43 w/mK
= 7801 kg/m3
=
473 J/kgK
= 2000 W/m2 K
=
20 C
=
850 C
=
0.01 m
These are realistic properties for steel, and fairly reasonable values for the
convection coefficient for quenching. Based on your engineering intuition, is
the centerline temperature more likely to drop to about 450 C in (a) 1 day
(b) 1 minute or (c) 1 microsecond? Describe any experiences you have had
that back up your answer. (Just give me a few lines here– you don’t have to
write a long description. Pretend you are trying to explain your reasoning
to your boss.)
1
Problem 2: Now do this problem in ANSYS Workbench. Because the slab
has three planes of symmetry, we can model just one-eighth of the entire
slab. (This saves a whole lot of elements!)
(a) For your one-eighth model, you should have three faces with convection
and three faces that are insulated. Sketch the one-eighth model and show
the boundary conditions on your sketch.
(b) Run the one-eighth model in Workbench.
• You will need to move the “Transient Thermal” application to your
toolbox. I believe that you will find that most of the changes for transient analysis are fairly intuitive.
• Be sure that you refine the mesh enough to get at least three or four
elements through the narrow dimension of the slab.
• Rather than using the default values for the time stepping, use a total
time of 10 seconds and a time step size of 0.1 seconds. Do this in
“Analysis Settings”:
– Set “Auto Time Stepping” to Off.
– Set “Number of Steps” to 1.
– Set “Current Step Number” to 1.
– Set “Step End Time” to 10s.
– Choose “Define by”, “Substeps”
– Set “Number of Substeps” to 100
– Leave “Time Integration” “On”
Now solve it.
• Workbench can then be used to make some really nice animations of
the temperature profiles. Print the plot of the temperature contours
at time=1 second so I’ll know that you were able to get this to run.
Make sure that the results you get are reasonable, given your answer
to Problem 1.
(c) Now click on “Solution Information”, which gives you a really old-fashioned
looking printout. Click in the printout window and type control-F, use the
search function to find the word “THETA” in the document. What value of
2
θ was used with the θ method for the time integration of this problem inside
ANSYS?
Problem 3: If we want to make a simplified model of the slab, we can just
model what happens along the midline (as shown in the sketch below.) Use
2 linear finite elements along with Euler backward time integration to solve
for the temperatures in the slab as a function of time.
W
slab
convection (oil)
ho
T2
T3
T1
T oil
(a) Find the 3×3 stiffness matrix for the problem. Don’t forget the convection
term.
(b) Find the 3×3 mass matrix.
(c) Find the forcing vector (3 terms).
(d) What is θ for Euler Backward integration?
(e) For Euler Backward integration, we can write
K̂T|n = F̂
Find K̂ and F̂ for this problem.
(f) Solve for T as a function of time and plot T1 (t), T2 (t), and T3 (t). Run
oil
your calculation until T1 ≤ T0 +T
.
2
(g) Does your answer seem reasonable compared to your estimate from Problem 1, and with your Workbench solution from Problem 2? Fix something if
it doesn’t seem correct.
3