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1993-11-10
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H E A T D O C U M E N T A T I O N
Description:
X30/AURORA SPACECRAFT HEAT SHIELD DESIGN
This program designs the heat shield for the X30 Aurora Spacecraft.
The shield is assumed to be of rectangular shape with square core.
Simple graphical representation of the steady-state heat distribution
is provided plus a table of numerical values. The program is adaptable
to other types of heat shield design problems unrelated to the X30
problem. Source for unix and DOS targets, GNU license.
Copyright: October, 1993 by Bill J. Wigginton, All Rights Reserved.
GNU Licensing agreement: see files 'license.txt' and 'copying.txt'
This software with sources may be freely
distributed and recompiled subject to the
license agreement.
mailing address:
William R. and Bill J. Wigginton
185 N. West Temple #311
Salt Lake City, Utah 84103-1562
email address : c-wwj@math.utah.edu (1993) or gustafson@math.utah.edu
GNU PUBLIC LICENSE.
This program "HEAT" is released as copyrighted material under the GNU
PUBLIC LICENSE:
NO WARRANTY
Because HEAT is licensed free of charge, absolutely no warranty is
provided, to the extent permitted by applicable state law. Except
when otherwise stated in writing, Bill Wigginton provides HEAT "as
is" without warranty of any kind, either expressed or implied,
including, but not limited to, the implied warranties of
merchantability and fitness for a particular purpose. The entire
risk as to the quality and performance of the program is with you.
Should the HEAT program prove defective, you assume the cost of all
necessary servicing, repair or correction.
In no event unless required by applicable law will Bill Wigginton
and/or any other party who may modify and redistribute HEAT be liable
to you for damages, including any lost profits, lost monies, or other
special, incidental or consequential damages arising out of the use
or inability to use (including but not limited to loss of data or
data being rendered inaccurate or losses sustained by third parties
or a failure of the program to operate with programs not distributed
by Bill Wigginton ) the program, even if you have been advised of the
possibility of such damages, or for any claim by any other party.
NO COST?
This program is provided free of charge to individuals and
educational institutions. Money is not requested.
X30/AURORA SPACECRAFT HEAT SHIELD DESIGN
PROBLEM:
Design a heat shield with a liquid core to achieve a given
heat distribution at equilibrium.
Design parameters are identified and a graphical representation
of the equilibrium heat distribution is given.
Approach:
Take a cross section of the shield at some point of interest. Define
the shape of the shield in terms of a grid. (See Figures 1 and 3.)
Determine boundary point locations and values. Using an iterative
technique compute the temperature values at the grid points within
the shield wall until the results are sufficiently accurate.
F I G U R E 1.
Heat Shield Cross Section
150 Degrees
UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU
s/ / / / / / / / / / / / / / / / / / / / / / / / / / / / s
s / / / / / / / / / / / / / / / / / / / / / / / / / / / /s
s/ / / / / / / / / / / / / / / / / / / / / / / / / / / / s
s / / / / / / / / /CCCCCCCCCCCCCCCCCCCC / / / / / / / / /s
s/ / / / / / / / / C C/ / / / / / / / / s s|
s / / / / / / / / /C -350 Degrees C / / / / / / / / /s s|
s/ / / / / / / / / C C/ / / / / / / / / s s|
s / / / / / / / / /CCCCCCCCCCCCCCCCCCCC / / / / / / / / /s s|
s/ / / / / / / / / / / / / / / / / / / / / / / / / / / / s
s / / / / / / / / / / / / / / / / / / / / / / / / / / / /s s|
s/ / / / / / / / / / / / / / / / / / / / / / / / / / / / s s|
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
3600 Degrees
U = Upper or top edge temperature
B = Bottom temperature
C = Core temperature
s = Side temperature
/ = Area in which temperatures are computed
Example:
In 1993, NASA is building and designing a "spaceplane". This is a
vehicle that can fly in space and also have powered flight including
take-offs in the atmosphere. It is called the X30. An atmoshperic
only prototype has been named the Aurora.
One problem that is currently being worked on is withstanding the
heat upon reentry. Because of the size and weight of the plane its
under side will remain at extremely high temperatures (3600 degrees
F.) for an excessive length of time. The tiles currently used on the
space shuttle are inade- quate. A new material must be developed.
To help cool the outer "skin", "cores" containing liquid hydrogen or
liquid helium will be imbedded in this new material. The internal
temperature of these cores is expected to be about -350 degrees F.
This new "skin" or material then must be able to withstand
temperatures ranging from -350 to +3600 degrees F. This suggests
some type of composite or sandwich material. But before a composite
can be designed, it is necessary to determine the heat distribution
in the material. Thus enters the "Heat Equilibrium" program.
Assuming the cabin temperature must be maintained at 150 degrees F.
to advoid damaging instruments and people we can design a model as
follows.
Take a rectangular cross section that contains at its horizontal
center a cooling core. Using a grid with horizontal and vertical size
of add integers < 80, scale the section to the grid. Input the
initial values:
Top temperature = 150 == U
Core temperature = -350 == C
Bottom temperature = 3600 == B
Percent at bottom temperature = 1
Find the heat distribution at equilibrium in the cross hatched area.
By changing the shape, size and verical location of the core, heat
stress requirements of the material can be determined.
This plot is a rectangle with a square core. It doesn't "look" like it
because the letters are taller than they are wide. To get a proper
scale set a printer so that the number of lines per inch are the same
as the number of characters per inch.
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
CCCCCCCCCCCCCCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBCCCCCCCCCCCCCCCCCCCCCCCCCCCC
DDCCCCCCCCCCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBCCCCCCCCCCCCCCCCCCCCCCCCDD
DDDDDDDCCCCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBBBCCCCCCCCCCCCCCCCCCDDDDDDD
DDDDDDDDDDDCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBCCCCCCCCCCCCCDDDDDDDDDDD
DDDDDDDDDDDDDCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBCCCCCCCCCCCDDDDDDDDDDDDD
EEEDDDDDDDDDDDCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBCCCCCCCCCDDDDDDDDDDDEEE
EEEEEEDDDDDDDDDCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBCCCCCCCCDDDDDDDDDEEEEEE
EEEEEEEEDDDDDDDDCCCCCCCBBBBBBBBBAAAAAAAAAAAAAAABBBBBBBBBCCCCCCCDDDDDDDDEEEEEEEE
FFEEEEEEEEDDDDDDDCCCCCCBBBBBBAAAAAAAAAAAAAAAAAAAAABBBBBBCCCCCCDDDDDDDEEEEEEEEFF
FFFFEEEEEEEDDDDDDCCCCCCBBBBBAAAAAAAAAAAAAAAAAAAAAAABBBBBCCCCCCDDDDDDEEEEEEEFFFF
FFFFFFEEEEEEDDDDDDCCCCCBBBBAAAAAAAAAAAAAAAAAAAAAAAAABBBBCCCCCDDDDDDEEEEEEFFFFFF
GFFFFFFEEEEEEDDDDDCCCCCBBBBAAAAAAAAAAAAAAAAAAAAAAAAABBBBCCCCCDDDDDEEEEEEFFFFFFG
GGFFFFFFEEEEEEDDDDDCCCCBBBAAAAAAAAAAAAAAAAAAAAAAAAAAABBBCCCCDDDDDEEEEEEFFFFFFGG
GGGGFFFFFFEEEEEDDDDCCCCBBBAAA AAABBBCCCCDDDDEEEEEFFFFFFGGGG
GGGGGFFFFFFEEEEDDDDCCCCBBBAAA AAABBBCCCCDDDDEEEEFFFFFFGGGGG
HHGGGGGFFFFFEEEEDDDDCCCBBBAAA AAABBBCCCDDDDEEEEFFFFFGGGGGHH
HHHGGGGGFFFFEEEEDDDDCCCBBBAAA AAABBBCCCDDDDEEEEFFFFGGGGGHHH
HHHHGGGGGFFFFEEEEDDDCCCBBBAAA AAABBBCCCDDDEEEEFFFFGGGGGHHHH
IHHHHHGGGGFFFFEEEDDDDCCBBBAAA AAABBBCCDDDDEEEFFFFGGGGHHHHHI
IIHHHHHGGGGFFFFEEEDDDCCCBBAAA AAABBCCCDDDEEEFFFFGGGGHHHHHII
IIIIHHHHGGGGFFFEEEDDDCCCBBAAA AAABBCCCDDDEEEFFFGGGGHHHHIIII
IIIIIHHHHGGGGFFFEEEDDDCCBBAAA AAABBCCDDDEEEFFFGGGGHHHHIIIII
JJIIIIHHHHGGGGFFFEEDDDCCBBAAA AAABBCCDDDEEFFFGGGGHHHHIIIIJJ
JJJIIIIHHHHGGGFFFEEEDDCCBBAAA AAABBCCDDEEEFFFGGGHHHHIIIIJJJ
JJJJIIIIHHHHGGGFFFEEDDDCCBBAA AABBCCDDDEEFFFGGGHHHHIIIIJJJJ
KJJJJIIIIHHHHGGGFFEEEDDCCBBAA AABBCCDDEEEFFGGGHHHHIIIIJJJJK
KKJJJJJIIIHHHHGGGFFEEDDCCBBAA AABBCCDDEEFFGGGHHHHIIIJJJJJKK
KKKKJJJJIIIHHHHGGFFFEEDDCBBAA AABBCDDEEFFFGGHHHHIIIJJJJKKKK
LKKKKJJJJIIIHHHGGGFFEEDDCCBAA AABCCDDEEFFGGGHHHIIIJJJJKKKKL
LLKKKKJJJJIIIIHHGGGFFEEDCCBAA AABCCDEEFFGGGHHIIIIJJJJKKKKLL
LLLKKKKKJJJIIIIHHGGGFFEDDCBAA AABCDDEFFGGGHHIIIIJJJKKKKKLLL
LLLLLKKKKJJJJIIIHHGGFFEEDCBAA AABCDEEFFGGHHIIIJJJJKKKKLLLLL
MMLLLLKKKKJJJJIIIHHGGFFEDDCBA ABCDDEFFGGHHIIIJJJJKKKKLLLLMM
MMMLLLLLKKKKJJJIIIHHGGFFEDCBA ABCDEFFGGHHIIIJJJKKKKLLLLLMMM
MMMMMLLLLKKKKJJJIIIHHGGFEEDBAAAAAAAAAAAAAAAAAAAAAAABDEEFGGHHIIIJJJKKKKLLLLMMMMM
NMMMMMLLLLLKKKKJJJIIHHGGFEEDCBBBBBBBBBBBBBBBBBBBBBCDEEFGGHHIIJJJKKKKLLLLLMMMMMN
NNNMMMMMLLLLLKKKJJJIIIHHGFFEDDDCCCCCCCCCCCCCCCCCDDDEFFGHHIIIJJJKKKLLLLLMMMMMNNN
NNNNNMMMMMLLLLLKKKJJJIIHHGGFFEEEEEDDDDDDDDDDDEEEEEFFGGHHIIJJJKKKLLLLLMMMMMNNNNN
ONNNNNNMMMMMLLLLKKKKJJIIIHHGGGFFFFFFFFFFFFFFFFFFFGGGHHIIIJJKKKKLLLLMMMMMNNNNNNO
OOONNNNNNMMMMMMLLLLKKKJJJIIHHHHGGGGGGGGGGGGGGGGGHHHHIIJJJKKKLLLLMMMMMMNNNNNNOOO
OOOOONNNNNNNMMMMMLLLLKKKJJJIIIIIHHHHHHHHHHHHHHHIIIIIJJJKKKLLLLMMMMMNNNNNNNOOOOO
OOOOOOOONNNNNNNMMMMMLLLLKKKJJJJJJIIIIIIIIIIIIIJJJJJJKKKLLLLMMMMMNNNNNNNOOOOOOOO
PPPOOOOOOOOONNNNNNMMMMMLLLLLKKKKKKKKJJJJJJJKKKKKKKKLLLLLMMMMMNNNNNNOOOOOOOOOPPP
PPPPPPPOOOOOOOOONNNNNNMMMMMMLLLLLLLLLLLLLLLLLLLLLLLMMMMMMNNNNNNOOOOOOOOOPPPPPPP
PPPPPPPPPPPPOOOOOOOOONNNNNNNMMMMMMMMMMMMMMMMMMMMMMMNNNNNNNOOOOOOOOOPPPPPPPPPPPP
QQQPPPPPPPPPPPPPPPOOOOOOOOOONNNNNNNNNNNNNNNNNNNNNNNOOOOOOOOOOPPPPPPPPPPPPPPPQQQ
QQQQQQQQQQQQPPPPPPPPPPPPPPPPOOOOOOOOOOOOOOOOOOOOOOOPPPPPPPPPPPPPPPPQQQQQQQQQQQQ
QQQQQQQQQQQQQQQQQQQQQQQQQQQQPPPPPPPPPPPPPPPPPPPPPPPQQQQQQQQQQQQQQQQQQQQQQQQQQQQ
QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ
With convergence value = 1.00000E-01 convergence
was achieved in 86 iterations.
A ranges from -350.00000 to -117.64706
B ranges from -117.64706 to 114.70587
C ranges from 114.70587 to 347.05881
D ranges from 347.05881 to 579.41174
E ranges from 579.41174 to 811.76465
F ranges from 811.76465 to 1044.11755
G ranges from 1044.11755 to 1276.47046
H ranges from 1276.47046 to 1508.82336
I ranges from 1508.82336 to 1741.17627
J ranges from 1741.17627 to 1973.52917
K ranges from 1973.52917 to 2205.88208
L ranges from 2205.88208 to 2438.23511
M ranges from 2438.23511 to 2670.58813
N ranges from 2670.58813 to 2902.94116
O ranges from 2902.94116 to 3135.29419
P ranges from 3135.29419 to 3367.64722
Q ranges from 3367.64722 to 3600.00000
The following is a rectangular core with approximately the same volume
of coolant as the previous model.
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
CCCCCCCCCCCCCCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBCCCCCCCCCCCCCCCCCCCCCCCCCCCC
DCCCCCCCCCCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBBBCCCCCCCCCCCCCCCCCCCCCCCCD
DDDDDDCCCCCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBCCCCCCCCCCCCCCCCCCDDDDDD
DDDDDDDDDCCCCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBCCCCCCCCCCCCCCDDDDDDDDD
DDDDDDDDDDDCCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBCCCCCCCCCCCDDDDDDDDDDD
EEEDDDDDDDDDCCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBCCCCCCCCCCDDDDDDDDDEEE
EEEEEDDDDDDDDCCCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBCCCCCCCCCDDDDDDDDEEEEE
EEEEEEEDDDDDDDCCCCCCCBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBCCCCCCCDDDDDDDEEEEEEE
FEEEEEEEDDDDDDDCCCCCCBBBBBBBBBBBBBBAAAAAAAAABBBBBBBBBBBBBBCCCCCCDDDDDDDEEEEEEEF
FFFEEEEEEDDDDDDCCCCCCBBBBBBBBBAAAAAAAAAAAAAAAAAAABBBBBBBBBCCCCCCDDDDDDEEEEEEFFF
FFFFFEEEEEDDDDDDCCCCCBBBBBBBAAAAAAAAAAAAAAAAAAAAAAABBBBBBBCCCCCDDDDDDEEEEEFFFFF
GFFFFFEEEEEDDDDDCCCCBBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAAABBBBBBCCCCDDDDDEEEEEFFFFFG
GGFFFFFEEEEEDDDDCCCCBBBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAAABBBBBCCCCDDDDEEEEEFFFFFGG
GGGFFFFFEEEEEDDDDCCCBBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABBBBCCCDDDDEEEEEFFFFFGGG
GGGGGFFFFEEEEDDDDCCCBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABBBCCCDDDDEEEEFFFFGGGGG
HGGGGGFFFFEEEEDDDCCCBBBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABBBCCCDDDEEEEFFFFGGGGGH
HHHGGGGFFFFEEEDDDCCCBBAAA AAABBCCCDDDEEEFFFFGGGGHHH
HHHHGGGGFFFEEEEDDCCCBBAAA AAABBCCCDDEEEEFFFGGGGHHHH
IHHHHGGGFFFFEEEDDDCCBBAAA AAABBCCDDDEEEFFFFGGGHHHHI
IIHHHHGGGFFFEEEDDDCCBBAAA AAABBCCDDDEEEFFFGGGHHHHII
IIIHHHHGGGFFFEEEDDCCBBBAA AABBBCCDDEEEFFFGGGHHHHIII
IIIIHHHHGGGFFFEEDDCCCBBAA AABBCCCDDEEFFFGGGHHHHIIII
JJIIIHHHGGGFFFEEEDDCCBBAA AABBCCDDEEEFFFGGGHHHIIIJJ
JJJIIIHHHGGGFFFEEDDCCBBAA AABBCCDDEEFFFGGGHHHIIIJJJ
JJJJIIIHHHGGGFFEEDDCCBBAA AABBCCDDEEFFGGGHHHIIIJJJJ
KJJJJIIIHHHGGGFFEEDDCBBAA AABBCDDEEFFGGGHHHIIIJJJJK
KKJJJJIIIHHHGGFFEEDDCCBAA AABCCDDEEFFGGHHHIIIJJJJKK
KKKJJJJIIIHHGGGFFEEDCCBAA AABCCDEEFFGGGHHIIIJJJJKKK
LKKKJJJJIIIHHGGGFFEDDCBAA AABCDDEFFGGGHHIIIJJJJKKKL
LLKKKJJJJIIIHHGGFFEEDCCBA ABCCDEEFFGGHHIIIJJJJKKKLL
LLLKKKKJJJIIIHHGGFFEEDCBA ABCDEEFFGGHHIIIJJJKKKKLLL
LLLLKKKKJJJIIIHHGGFFEDCBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABCDEFFGGHHIIIJJJKKKKLLLL
MLLLLKKKKJJJIIIHHGGFFEDCBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBCDEFFGGHHIIIJJJKKKKLLLLM
MMMLLLLKKKJJJIIIHHGGFFEDDCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDDEFFGGHHIIIJJJKKKLLLLMMM
MMMMLLLLKKKKJJJIIHHHGGFEEEDDDDDDDDDDDDDDDDDDDDDDDDDDDEEEFGGHHHIIJJJKKKKLLLLMMMM
NMMMMMLLLLKKKJJJIIIHHGGFFFEEEEEEEEEEEEEEEEEEEEEEEEEEEFFFGGHHIIIJJJKKKLLLLMMMMMN
NNNMMMMLLLLKKKKJJJIIIHHGGGGFFFFFFFFFFFFFFFFFFFFFFFFFGGGGHHIIIJJJKKKKLLLLMMMMNNN
NNNNMMMMMLLLLKKKKJJJIIIHHHHGGGGGGGGGGGGGGGGGGGGGGGGGHHHHIIIJJJKKKKLLLLMMMMMNNNN
ONNNNNMMMMMLLLLKKKKJJJIIIIIHHHHHHHHHHHHHHHHHHHHHHHHHIIIIIJJJKKKKLLLLMMMMMNNNNNO
OOONNNNNMMMMMLLLLLKKKKJJJJIIIIIIIIIIIIIIIIIIIIIIIIIIIJJJJKKKKLLLLLMMMMMNNNNNOOO
OOOOONNNNNNMMMMMLLLLLKKKKKJJJJJJJJJJJJJJJJJJJJJJJJJJJKKKKKLLLLLMMMMMNNNNNNOOOOO
OOOOOOONNNNNNNMMMMMLLLLLLKKKKKKKKKKKKKKKKKKKKKKKKKKKKKLLLLLLMMMMMNNNNNNNOOOOOOO
PPOOOOOOOOONNNNNNNMMMMMMLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLMMMMMMNNNNNNNOOOOOOOOOPP
PPPPPPOOOOOOOOONNNNNNNNMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMNNNNNNNNOOOOOOOOOPPPPPP
PPPPPPPPPPPOOOOOOOOOONNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNOOOOOOOOOOPPPPPPPPPPP
QQPPPPPPPPPPPPPPPPOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOPPPPPPPPPPPPPPPPQQ
QQQQQQQQQQQPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPQQQQQQQQQQQ
QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ
QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ
With convergence value = 1.00000E-01 convergence
was achieved in 82 iterations.
A ranges from -350.00000 to -117.64706
B ranges from -117.64706 to 114.70587
C ranges from 114.70587 to 347.05881
D ranges from 347.05881 to 579.41174
E ranges from 579.41174 to 811.76465
F ranges from 811.76465 to 1044.11755
G ranges from 1044.11755 to 1276.47046
H ranges from 1276.47046 to 1508.82336
I ranges from 1508.82336 to 1741.17627
J ranges from 1741.17627 to 1973.52917
K ranges from 1973.52917 to 2205.88208
L ranges from 2205.88208 to 2438.23511
M ranges from 2438.23511 to 2670.58813
N ranges from 2670.58813 to 2902.94116
O ranges from 2902.94116 to 3135.29419
P ranges from 3135.29419 to 3367.64722
Q ranges from 3367.64722 to 3600.00000
NOTE:
Lest one think that the problem is solved, there are other
considerations. The core temperature will rise over distance. The
speed of the coolant may create enough friction to raise the
temperature. The internal pressure must be considered. The length of
the cooling cores must be optimized. The number and size of the
compressors. The size, shape and number of cores must be optimized.
The expansion and contraction of the "skin". Heat distribution is
simply one piece of a very large problem. The greatest problem is
developing the material itself.
Mathematical Overview:
F I G U R E 2.
Heat Shield Profile with Liquid Core
Top Edge Temperature
____________________
| |Side
| ______ |Temperature
| | Core | |
~~~|~~~~~~|~Temp~|~~~~~~|~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
| |______| |Bottom portion of surface kept
| |Temperature at Bottom Temperature
|____________________| _____________________
F I G U R E 3.
Heat Shield Profile Defined on a Grid
C O L U M N S
1 2 3 4 5 6 7 8 9 10 11
1 U U U U U U U U U U U Top
2 s i i i i i i i i i s
3 s i i i i i i i i i s Side
R 4 s i i C C C C C i i s
O 5 s i i C C i i s
W 6 B i i C C i i B ~~~~~~~~~~~~~~~~~~
S 7 B i i C C i i B Percent
8 B i i C C C C C i i B kept at
9 B i i i i i i i i i B Bottom bottom
10 B i i i i i i i i i B temperature
11 B B B B B B B B B B B __________________
U = Top boundary points (user supplied)
B = Bottom boundary points (user supplied)
C = Core boundary points (user supplied)
s = Side boundary points (computed)
i = Points within the wall (or the domain) that are computed
Basic Equation:
Read Uxx as the second partial derivative of T with respect to x.
Read T[new](i,j) as the new value of T at point (i,j).
[...] = superscript, (...) = subscript in this usage.
The heat gradient at equilibrium is described by the Laplace Equation
Uxx + Uyy = 0
where T is the temperature at any point (x,y) in the domain. The
Laplace Equation (a special case of the Poisson Equation) can be
solve iteratively by an accelerated Gauss-Seidel method
T[new](i,j) = aT[avg](i,j) + (1-a)T[old](i,j) where
T[avg](i,j) = .25 * ( T[new](i-1,j) + T[new](i,j-1) +
T[old](i+1,j) + T[old](i,j+1) )
T[...](i,j) is the temperature at point (i,j)
i,j are row and column values within the defining grid (See Figure 3.)
a is an acceleration factor between 1 and 2
Domain:
The domain of the solution method is bounded externally by the shield
shape and is bounded internally by the shape and location of the
core. That is the i's in Figure 3 are the domain. Points outside the
shield boundary are not defined and hence not computed. The same is
true of the points within the core.
Shape:
The shape is defined on a 79 x 79 grid. The external shape of the
shield and the shape and location of the core is specified by the
user. The shapes may be a square, circle or rectangle. The shield
shape begins at (1,1). The internal shape location is normally
centered within the shield but may be moved by the user. The size of
the shape, in our example (Figure 3) an 11 x 11 square shield with a
5 x 5 core is controlled by the user and cannot be larger than the
basic grid. The shape of a circle is approximated by asking for is
diameter. The boundary points are then calculated by:
1. Determining the center of the circle.
2. Calculating a boundary row and column for each row in the
top left quadrant using r*r = x*x + y*y.
3. Since circles are symmetric the boundary points of the other
three quadrants are determined from the first quadrant values.
Define:
row,col = a boundary point within the grid
colm = mirror image of col about the center column
rowm = mirror image of row about the center row
d = diameter of the circle
rc = center row for the circle
cc = center column for the circle
r = radius
x = 1/2 the chord length for the row in question
y = integer 1 to rc
1. rc = (d-1)/2; cc = (d-1)/2
2. for y = 1 to rc
x = sqrt(r*r-y*y) rounded to the nearest integer
col = xc-x
row = rc - y
3. for y = 1 to rc
colm = d-col+1
rowm = d-row+1
boundary points are
for top left quadrant = row, col
the top right quadrant = row, colm
the bottom left quadrant = rowm,col
the bottom right quadrant = rowm,colm
Boundary Values:
All external surface and internal core values (U,B and C) are
specified by the user. (See Figure 3.) The points s are a linear
extrapolation between the upper and lower boundary points.
prcnt = percent of the shield surface held at the bottom temperature;
incr = temperature increment;
row = row number;
N = total number of rows
n = number of rows not immersed
r = row being processed
s(r) = value of s on row r;
n = (1 - prcnt) * N rounded to the nearest integer
incr = (U - B) / n
s(1) = U
for r = 2 to n
s(r) = s(r-1) + incr
The points i within the domain are given an initial value equal to
the value of the boundary points for that row.
i(r) = s(r) or
i(r) = B for all valid columns on row r.