PHGN 450 / EGGN 502
Fall 2005
Project 8: Sprayforming
Due November 11
Version: 11/07/05
INTRODUCTION
The goal of this project is to determine simulate the temperature of a sprayformed part
as a function of position and time.
Schematic of the sprayform tool process
4 electric arc
spray guns
6-axis robot
pyrometer
freeze cast
master
The following idealizations are used:
A single spray gun is used. Its height is xed, and it delivers heat to the surface in a
Gaussian distribution.
The build up metal on the ceramic is ignored in the thermal problem.
The raster pattern of the spray gun is a xed, rectangular sweep. The rst pass is
centered on one edge, the second path reverses the direction and is 1/3 of the way
up the plate, the third pass reverses the direction again and is 2/3 of the way up the
plate, and the fourth pass reverses the direction once more and is centered at the top
edge of the plate. The pattern is then rotated 90 degrees and repeated.
Within the nondimensionalized setting derived in class, the spray gun moves across
a unit square. Within this setting, the velocity of the gun is v = 0:05, the initial
temperature of the plate is zero, the Gaussian distribution of heat can be described
by a function of the form:
r = 100 exp[ 200(x
x0 )2 ] exp[ 200(y
y0 )2 ]:
Within the nondimensionalized setting derived in class, the ambient temperature is
zero, and the convective heat transfer coe cient equals 2:0.
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The edges of the plate are treated as being perfectly insulated, so heat cannot leave
the plate laterally.
APPLICATION
Step 1 (5 points)
State the relevant PDE for modeling the temperature of this system. Nondimensionalize
these equations and provide an explanation of how you accomplished this.
Step 2 (10 points)
Derive a set of nondimensional, algebraic equations that approximate the solution to this
PDE. You may certainly draw on the notes provided in class.
Step 3 (30 points)
Construct an algorithm to solve your algebraic equations using a Alternating Direction
Implicit Crank-Nicolson method. Print out a copy of your code. Make sure that it is
documented well enough for a reader to follow your approach. The points associated with
this section are awarded for the clarity, commenting and logical development of your code.
Step 4 (30 points)
Implement ADI C-N algorithm. Make sure that all elds and parameters are clearly
identi ed. You are to provide an summary that includes:
(1) ve snapshots of the temperature distribution ;
(2) the maximum and minimum temperatures at the end of one full raster cycle;
(3) an analysis that supports your claim that you have used time and spatial steps
su ciently small so that your maximum and minimum temperatures are within 10% of the
exact values.
Step 5 (15 points)
Now try to make your algorithm as fast as possible. Use timing commands to determine
how fast your code runs the problem above. You must meet the accuracy goal of Step 4.
State the clock speed of your computer. Now take the product of calculation time and
clockspeed. State this nal gure of merit for the e ciency of your code. Also state, for the
record, whether you used Fortran, Mathematica etc. to get this number. The people with
the fastest codes, for each programming platform, will be announced in class and given a
superb award (that may look similar to a physics trading card).
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STEP 6 (10 POINTS)
Explain how you can derive a method for checking your algorithm that is along the lines
of what we did using body forces for the elasticity problem.
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