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1950
The application of the relaxation method to the
solution of steady-state heat transfer through a
concentric square insulator
William Edward Simpkin
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Simpkin, William Edward, "The application of the relaxation method to the solution of steady-state heat transfer through a concentric
square insulator" (1950). Masters Theses. 4896.
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THE
LIC TION OF THE
TION
THOD TO
THE SOLUTION OF STEADY-ST TE HEAT TRANSFER
TH 0 TGH
C
TRIC SQ,
I SULATOR.
BY
vITLLIAM EUNARD SDlPKDJ
THESIS
submitted to th
SCHOOL OF MINE
•
faculty of the
D vIETALLURGY OF 'lm UNIVERSITY OF
SSOURI
in partial fulfillment of the work required for the
Degree of
MASTER OF SCIENCE IN MEGHANI CAL ENGINEERING
Rolla, Missouri
1950
ii
ACKNO
OOEMENTS
The author is indebted. to Dr. A. J. Miles for the suggestion of and the assistance in the solution of this thesis
problem.
The author also wishes to take this opportunity to
express his appreciation to various members of the faculty
for their marginal assistance and to relate his gratitude to
his friends and loved ones for their encouragement and sponsorship.
111
TABLE OF COnTENTS
Acknowledgements ...............•..
List of Illustrations
Page
ii
iv
List of Tables....................
v
Introduction.......
1
Statement of Problem......
7
Review of 1Ji terature. .. . . . .... .. ..
9
Origin of Relaxation Method
10
Di s c u s s ion. . . . . . . . . . . . . . . . . . . . . . . .
12
Initial RssQTIptions
13
Heat-Flow Evaluation
26
Results of Problem Solution......•
30
Conclusions •......................
31
Discussion of Solution Accuracy .••
32
Summ.ary. . . . . . . . . . . . . . . . . . . . . . . . . . .
34
Applications for the Results •..•..
35
Bibliography.................•••.•
36
Vi ta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
-0-
iv
LIST OF ILLU TRATIONS
Fig.
1.
Page
Sketch of Infinitesimal body in
Rectangular Co-ordinates ......•...........
2.
2
A Plot showing the eight symmetrical
portions of concentric squares ...•.•..••.. 14
3.
One-eighth Section for Case I' ............. 15
4.
One-eighth Section for Case II ............ 18
5.
One-eighth Section for Case III ........... 20
6.
One-eighth Seotion for Case IV ............ 22
7.
One-eighth Secti on for Case
Plate
1.
V ••••••••••••• 24
Page
Cs in terms of the ratio of the width
of the outside to the inside square .....•. 30
v
LI 3 T OF TJJ3L:::S
Table
I. Tabulation of the Relaxation
f
0 l~
I.................... 17
Ca s e
Tc:bulation of -etc; ::'(-Jlaxation
T""
_ . ...1..- •
fOl'"
Case II
Tabulati on of
III.
19
tJH:~
Rel::.x'3tion
for Case III
IV.
Tabulation of
for
V.
Page
'J3se
t~e ~elaxat~on
TV...................
Tabulation of the
fo:(' 'Jas""
VI.
21
~elaxation
V....................
Determin:J.tiol] of
23
TJeb
25
t :1'low from
Temperature Distribution
27
VII. Determination of Seat Flow through
Concentric Conductors ........•
VIII.
Tabulati on of
OS"
• • • • • • • • • • • • • • •
28
29
INTRODUCTION
1
The basic equation for the flow of heat by conduction
is given by Fourier's law which states that the instantaneous
rate of heat flow, ~, is equal to the product of three factors: the area normal to the path of heat flow, A; the temperature gradient, dt; which is the rate of chan e of tempercirx
ature, t, with respect to the length of path, x, in which the
heat is flowing; and the dimensional constant, k, the conductivity of the conducting medium.
Fourier's law expressed
mathematically is as follows:
~.
-kill
dQ
(1)
dx
where dQ is the amount of heat flowing in a differential time
dQ, adjusted to indicate positive heat flow in the direction
of decreasing temperature by the minus sign.
Z
dz
x
dx
Fig. 1.
Sketch of infinitesimal body in
rectangular co-ordinates.
The rate of heat flow, ~ hereafter referred to as dq,
dQ
into the element in the x direction according to Fourier's
law is (see
d.
ig. 1.)
i" =-
K"
ely J. 'L
(~)
2
The heat flowing out of the element in the x direction is
r~t
~ (;;>t)J.: t1
cl tx"=-1<..c:Lyclz Lix
+~
~ ~J .
The net gain or the amount of heat stored in the element is
the difference between the two.
J.'t,,:: .,li: - J.f: = - \<." ~ "'J[ f)( ~).,l~
In the same manner it may be shown that the heat stored due
to heat flow in the y direction is
tr =- k..'( .,luI. z [~ (~ ) tA:t]
and in the z direction
;)t')tA. ,1
All. = -""..l)( J.'( [-h, (n ZJ.
.A.
The rate at which heat is being stored in the element may
then be expressed as
I
:.
e>t...:; .l"", + J..1'(+ ~"7.
~ k,.J.'( J.1 ~(I<.~)+~(Ky~)+?i(~
~
~)~ .
T is is equal to the rate of heat accummulation within the
element or
•
erel' is the density, and c is the specific heat of the
material.
Therefore we may equate the two expressions with
1
cl"c:lycl.zpc Z9 =,l)(J.ytA.7.[~'c''SXtL+ K~~" + \<.f. 'Sl"J
the following e;rttion resul tinge ~~t
;)-t
~'t
If the conductivity is the same in all directions, then the
above expression can be reduced to
~t
I<. (~t
~'a.t
J'lt")
~:',oc.
where
-!Pc
':)(.. +-
W
+~
(2)
is a dimensional constant for each material and is
designated by the symbol oC, and is called the thermal dif-
3
fusi vi ty.
In the steady state condition of heat transfer, temperature does not vary with time, consequently ~ is zero.
There-
fore we obtain
~~t
~ ){~
+
~"t: +~. . t"
~ "'(~
~ 7..
:.
0
(3 )
•
which is the general equation for
he~t
conduction in the steady
state, for a homogeneous medium.
In the problem at hand, the medium is homogeneous and the
flow is two dimensional and steady, hence equation (3) reduces
to
,
Equation (2) is a general
quation for all heat-conduc-
tion problems in homogeneous media, and is the result of applying Fourier's
law·(equ~tion
1).
Therefore it may be seen
that the solution of all conduction problems
re~uires
a solu-
tion of this differential equation which will meet the boundary conditions.
This solution may be simple as in the cas
of one dL,ensional hest flow in the steady state, and in two
and three dinlensional steady-state heat flow involving circles or spheres.
However for most irregular shapes the so-
lution of this differential equation has not been made because of the difficulty involved.
s a result of this oiffi-
culty, recourse has been made where possible to graphical,
statistical, experimental, or analogous methods.
Graphical analysis was one of the first attempts for
the solution of the difficult LaPlacian equation.
Early work
was done by Binder(l) in the field of electromagnetism.
This
4
(1) Binder, L. Munich, 1911
was the forerunner of the Scillrridt(2) graphical analysis meth(2) Schmidt, E., Berlin, 1924
ode
The methods as proposed by Moore(3) in the field of elec-
(3) Moore, A. D.; Fundamentals of electrical design, McGrawHill Book Co., Inc. 1927
tromagnetism give workable La lacian solutions, and valuable
data on thermal properties of heterogeneous systems.
The
text of Me dams(4) is an indispensable reference in the heat
(4)
!eAdams, V. H., Heat Transmission,
cGraw-Hill Book Co.,
Inc. 1942
transfer field and presents some results obtained by graphical analysis.
Closely allied with the graphical methods are the analogous methods.
There are several under the classification of
electrical analogies.
A
network type of analogue was devel(5 )
oped by Paschkis and
aker
(5) Paschkis, V. and
aker, H. D., Transactions
primarily for transient heat
.S.N..E.
Vol. 64, No.2, pp 105-112 (Feb. 1942)
flow, but could undoubtably be used for determinations of
steady-state conditions.
In this analogue the heat-flow path,
whether one-, two-, or three-dimensional, is represented by
a number of electrical resistors arranged in the form of a
5
built-up network to simulate thermal resistance of the original form.
The flow of direct current through the network is
equivalent to the heat flow conditions.
Langmuir developed a geometrical analogue in 1913 which
utilizes an electrical conducting bath in a shallow tank for
model study of various heat-flow shapes.
electrolyte represents the 8i gl
ial.
conduct'ng liquid
isotropic homogeneous mater-
With an alternating-current electrical potential est b-
lished between the parallel wall electrodes, the equivalent
of steady-state heat-flow conditions between constant temperature walls was set up.
By means of an electrical probe,
isopotential l±nes representing isothermal lines of the 801id body could be established.
An electrical geometrical analogue was presented by
Kayan(6) in 1945.
This method is of the geometric type pri-
Kayan, C. F., Transactions
.S.M.E. Vol. 67, No.8,
pp713-718 Nov. 1945
marily designed for steady-state one- and two-dimensional
studies.
Conductive sheets of various electrical resistanc-
es were used simulating different thermal conductivities of
diff rent materials in a memb r to be studied.
The basic
electrical analogy of this method is the same as the Langmuir method; equivalent temperature conditions f r a heat-flow
path can be determined through analysis of an electrical-flow
path io which the component resistances
shi _ bet'.'Jeen theoselves as the
t~ler!c3.1
h~v~
the swne relation-
resist,nces.
An elec-
a~d
tricel probe is also used
a "contour map" of isopotential
lines (vlhich re:,r83cnt isotherr,lal lines) is drawn.
It "Jas
tor e.
:G)Cint8(~
out ::,y :Prantl, in 1903, th9.t the equations
l}'J::lo~e11eou~:lelilbrQne
are of trlc LaFlacian form.
This
6i ~;covor:)T was I113.de as a result of hi s attempts to find a ,solution for the
equat~0ns
of torsional stress analysis.
1937, Christopherson and Southwell
(7) Christopherson, l.
~.
( '1 )
In
indicated that every
and 30uthwell, .• V., Proceedings
os the Royal 30ciety of London, Series A, Vol. 168,
pp 317-350 (1938)
plane-potential problem has a membrane analogue and that PrantIts analogue is merely a particular example.
Southwell(S)
(8) Ibid
applied this analogy to two-dimensional heat-flow problems.
The deflection of the membrane represents, to some scale,
the temperature distribution throughout the member under consideration; that is, the hole in the plate over which the
membrane is stretched must be geometrically similar to the
shape of the member under investigation, and the differences
in elevation of the various parts of the model must represent,
to some scale, the temperature change between those parts.
The boundary conditions must be known and the model constructed accordingly.
STATEMENT OF PROBLEM
8
It is the object of this thesis to express the relationship between the heat transfer through a square insulator of
variable inside-outside dimension ratios, in terms of a concentric round insulator of the same inside-outside diameter
ratio, by means of a continuous curve.
athematically speak-
ing, this thesis will have a plot of a variable coefficient,
cs' to be used
in the following relationship, in terms of
the ratio of the width of the outside square to the width
of the inside square.
REVI:>
OF LITERATURE
10
The relaxation method, basically a trial and error procedure, is a means of satisfying a set of given boundary conditions, and internal characteristics for plane- or spacepotential problems.
The earliest foundations for the relax-
ation method were made in 1932 by cross(9) in the field of
(9) Cross, Hardy, "Analysis of continuous frames by distributing fixed end moments."
(paper No. 1793) Trans. Am.
Soc. Civ. Eng. 96 (1932)
stress analysis.
Shortly thereafter in England, Southwell(lO)
(10) Southwell, R. V., stress-calculation in frameworks by
the method of systematic relaxation of constraints.
Part III.
Froc. Roy. Soc. of Lon. A, Vol. 153 pp 41-
76, (1935)
devised a system of 'relaxing' the load from one support on
a continuous bema for the solution of the stress in the beam
--from which the relaxation method derived its name.
Inde-
pendently, Cross(ll) used a similar nQ~erical method of
80-
(11) Cross, Hardy, Analysis of flow in networks of conduits
or conductors.
Univ. Ill. Bull. No. 286, (1936)
lution for a pipe-line network.
Throughout the late thirties
application was made in this country and in ·ngland of this
method for 'indeterminate' mechanics problems.
The first application of the relaxation method to heat
transfer was made by Emmons(12) in 1943.
In his words, "Ex-
11
(12) Emmons, H.
~.,
The numerical solution of heat-conduction
problems, Trans. A.8.M.E., Vol. 65, No.6, pp 607-615
(1943)
cept for the simplest geometric shapes and boundary conditions,
the relaxation method is far superior to analytical methods
of solution in point of tiffie required to reach a given desired accuracy.
The relaxation method has the further adv nta es
of permitting the calculator to put into the calculation all
the physical intuition he
~ay
have about the problem and, at
the same time, to know at each step just how seriously his
solution still differs
fro~
the correct answer.
By an exten-
sion of the nethoc, one-, two-, and three-dimensional transient-heat-flow problems are easily solved.
~or
one-dimension-
al problems, the relaxation method is identical to the graphical :nethod of Schmidt. ,,(13)
tl5) Schmidt, E. op.cit. P 4
In the words of the writer, the relaxation method is a
trial and error guessing for which the right guess can be identified, and then a long arm process of manual summation as analo ous to mathematical integration.
However, regardless of
the labor involved, it is believed that the relaxation method
is an invaluable tool and will solve considerably more practical problems than the claculus except to the esoterica of
higher mathematics.
DISCUSSION
13
In order to develo
a curve for the value of
s as d scribed in the statement of the problem, it was decided that
C
five yoints would be a minimum to determine the curve.
To
deternline the value of these five points, separate solutio s
for each case was necessary.
It was arbitrarily decided to
give the inside width of each square a value of twelve units,
and in the five cases, have a conductor thickness of one, two,
three, four, an
six units respectively.
This gave an out-
side-inside ratio of widths of 1.167, 1.333, 1.500, 1.667,
and 2.000 respectively.
As may be seen from
symmetrical portions.
mine th
ig. 2, a concentric square has eight
Therefore it was necessary to deter-
temperature distribution in only one of these sections
to evaluate the heat flow by the relaxation method.
one thousand degree temperature difference between
the inside isothermal surface and the outside isothermal surface was arbitrarily chosen for its convenience in calculation.
Initial .ssumptions:
1.
teady-flow condition.
2. Constant conductivity, at k
3. On
1 Btu/hr-ft-oF.
foot axial len tho
Case I, a conductor
ig. 3.
*
0
unit thickness is illustrated in
The accompanying relaxation is given in Table I.
In the other four cases, the descriptive sketch ane its relaxation follow.
Since the value of ~ is equal to one, the heat flow
through any portion of the network to a given point is equal
14
'.
-
f----------- -
------4------
T
I
I
Figure 2. A Plot showing the eight symcetr10al
portions
or
oonoentrio squares.
15
to the summation of the temperature differences.
Under stead-
y-flow conditions, no heat maybe stored or taken from any internal point.
Therefore the temperature distribution may be
obtained by giving the point
temperature~
such values that
the net heat flow, Q, at 3ny point is equal to zero.
For the
purposes of this problem, considering the one thousand degree
overall temperature difference, the relaxation was continued
until the Q at any point was within tlO or less, thus giving
an accuracy of
~O.025%
on the points considered.
This how-
ever can not be considered to be the accuracy of the determination, since an infinite number of points would have to
be relaxed to obtain this accuracy.
Isothermal
1000·
a
-. b
1
2
3
4
5
6
Isothermal 0°
789
o
10
11
12
13
14
15
Figure 3. One-eighth Seotion for Case I.
.....
oa
TABLE I
Tabulation ot the Relaxation tor Square with X2 - 1.167, as per Fig.
Xl
Points
b2
Q.
t
0
0
0
0
220
215
212
211
t
Q.
440
430
424
422
-80
-20
-4
1
b5
b4
b3
t
Q.
85
-20
-6
0
460
485
480
479
t
Q.
12
16
1
485
490
494
t
Q.
b8
b7
b6
11 492
3 496
1-498
Q.
t
11 494
6 497
1 499
Q.
bID
b9
t
11
495
3.5 498.
2
499
~
t
Q.
8 496
6 497
1.5 498.5 1.5 499
2 499
2 499
Q.
t
Points
bll
Q.
b12
t
2.2 499
b13
b14
Q.
t
Q.
t
1.7 499.2 1.2 499.5 -.5 500
Q.
t
b15
Q.
t
o
500
....
.....:)
Isothermal
1000
o
a.
_.-
-
£
1
1
2
-
3
4
Figure 4.
_----- ._- -- -. ---- -- --------
0
b
-------------
_
Isothermal
56
One-ei~hth
..
o
7
8
c
9
Section for Case II.
~
CD
TAm.E II
Tabulation of the Relaxation for Square with !2 • 1.333, as per Fig.
Xl
Points
b2
t
Q
0 190
0 205
0 207
b3
Q
95
10
1
t
380
410
414
b4
t
Q
65 425
15 445
1 450
b5
Q
-22
1
-2
t
485
487
b7
b6
Q
9
-1
1
t
493
495
Q
7
3
t
496
498
b8
t
Q
4 498
2 499
+
-b9
Q
-4
t
500
...,
1lO
Isothermal
/
1000
o
I
---; a
~--------------_.
,/
.. ~ -.... -- - . b
/
/
o
~
1
//
//
I.
2
:3
4
5
Isothermsl
6
O·
7
•
8
"d
9
10
Figure 5. One-eighth Section tor Case III
N
o
I\l
Points
02
Q
0
0
-2
-2
0
2
t
80
85
88
90
03
t
Q
-5 160
30
0 175
5
11
-5 179
-3
0
1
3
4
0
3
-1
0
05
04
Q,
t
67
25
-4
3
225
240
255
-4
-1
10
2
-2
-1
258
e
180
0
181
1
2
260
261
3
4
5
1
2
262
Q
20
35
-3
4
7
-5
0
2
4
5
1
3
4
0
1
05
t
272
280
297
Q,
301
302
30
8
3
8
12
15
-5
-2
0
1
2
3
303
0
300
07
t
293
305
315
320
4
321
Q,
20
12
28
-3
1
6
-6
-5
-3
0
1
09
08
t
305
310
313
325
328
Q.
-5
3
-4
-2
0
3
-1
0
t
325
Q.
t
3
330
332
0
330
331
010
1
2
Q,
-5
t
333
Isothermal
~,'
/iL__ - _.'
/
1000
-
._ _
~
I
I
,
/ :
I
I
i
-~----:
I
I
I
-
I
-_. .
~.---,--_.-
'--'r---
~-T----
I
//'
--lJ
"
i
c
I
,
I
I
I
I
•
J
I
i l l
2
j,
---_., --i
/------
1
b
t
I
I
I
_
"I
I
~
I
Ii
._-
a
Ii
I
:
L.t
I
1--.
0
Z
4
I
5
Figure
6
~
I
Isothermal
7
'
j
0
8
o
J
I
I
J
i
I
:
I
,.
i
9
10
11
d
e
12
One-eighth Section tor Case IV
l\)
{\)
TABLE IV
i 12 •
Tabulation of the Relaxation tor Square with
Points
b4
Q,
-162
-40
0
70
40
60
0
5
10
6
-?
-3
1
-2
t
503
480
480
440
455
475
480
482
485
483
481
480
Points
04
Q.
-23
-47
33
43
-1
14
1
6
2
-12
-8
-2
-3
t
320
300
310
315
317
318
315
312
311
60
71
100
46
23
0
1
0
-2
-3
-1
t
80
70
85
95
100
104
103
t
655
620
Q,
-14
-44
53
83
?8
-42
8
5
-I?
-2
-3
--1
610
630
660
650
652
656
650
649
648
Q
140
90
_ 100
50
40
22
10
4
-1
-73
-13
-6
-3
-2
112
72
52
30
17
5
1
-1
-2
-3
0
t
95
235
250
130
115
40
10
0
-5
-5
-2
-3
680
650630
660
680
700
705
708
?10
712
710
709
t
380
400
405
412
418
421
423
Q
-2
33
31
2
-3
5
-1
-2
?10
730
700
710
715
?30
732
53
47
47
35
35
0
-5
-2
9
7
5
3
1
731
t
450
455
453
454
453
Q,
140
100
40
30
20
10
3
-2
-16
-9
-1
d5
195
198
200
198
196
194
Q.
22
0
3
-2
9
1
-1
735
740
741
b10
t
742
745
Q
0
1
-1
4
1
2
t
Q,
749
-6
-3
-2
t
430
440
460
465
470
4?5
478
480
t
488
Q,
2
10
6
4
6
-3
-2
490
t
10
20
40
25
18
7
2
0
-2
0
480
t
498
Q.
-1
0
t
Q
-8 500
330
180
190
195
200
207
210
211
210
209
20?
d2
Q
-40
-20
30
10
6
0
-2
485
488
492
494
495
t
Q.
ell
010
Q
03
?50 -560
40
20
30
24
8
2
+2
-2
0
-2
-3
0
-4
746
09
~
t
Q.
?42
08
-
b11
'7.32
t
50
40
35
45
47
52
478
t
215
220
222
220
222
221
Q.
-23
2
0
2
-3
-2
t
240
-235
236
d9
d8
d7
d6
t
192
b9
t
733
737
Q
8
15
-15
3
8
3
0
2
0
07
410
405
403
402
Q
t
120 -10
0
130
3
140 10
147
3
152 -1
154
8
153 -3
1
0
152 -1
b8
t
Q,
06
d4
Q,
b?
Q,
05
Points
d3
Q,
b6
b5
1.66?, as per Fig.
Q
4
-1
2
iTl4
0
t
242
243
Q.
8
1
2
dl0
t
244
246
Q
4
2
t
247
248
dl1
Q.
-8
t
250
1'>
u.
/:I
, . :
1000 0
Isothermal
I
I I'
,
.
I
, I
r-j
;
-.-r----.------
-r --.
a
--. - ~
b
;
I
,
;
i
.
I
I
I
I
i
- -
i
i
---" -
~.
I
I
I
-
-
I
- - - - ---
---'-1- -
I
c
______: . ----r----"----II
d
-
,
. I
"
/11 --- -r---------- - ----.
// 1 I I t : ..
/r--~II -ji--T-t--r--r--~
I
I
I
2
3
4
5
6
7
I
I
l
I
.
!
I
I
I
I
,
I
;
I
I
1 e
- - - . -..--.-
I
_.
.
I ;t
8
'
,
I
Isothermal
1
--- - "-------- .-----J-_ _.
i
I
.J -- --- -- -- -
I
,
-
I
I
I,
_ _..i.-,_ _ "
----._-
I
I
.r
_
--- +_. -- - -
I
...._~-~
'I
i
I
1
__
t
g
00
9
10
11
12
13
Figure 7. One-eithth Section for Case V.
l\)
~
TABLE V
Tabulation ot the Relaxation tor Square with !2 • 2.000, as per Fig.?
X1
Points
B6
t
590
610
615
Q.
-80
20
10
5
-4
-4
613
611
b7
t
480
490
Q.
45
15
-35
-5
-10
-9
-4
-5
-7
485
483
480
477
t
740
Q.
-5
20
0
5
0
-7
-1
-3
Points
06
745
t
790
795
Q.
17
-3
-7
3
0
-4
07
793
bl0
t
Q.
1
2
0
-2
-4
815
b12
bll
t
Q.
1
-2
-1
-3
t
Q,
575
-13
-23
-28
-7
-12
-17
-2
-7
-8
-6
-8
09
08
570
565
563
561
t
622
Q.
-5
-11
-16
0
-5
-10
-15
-10
-2
-2
-6
825
t
Q.
7 828
3 .829
2
618
615
612
611
410
a
349
640
638
-9
-3
-5
406
t
662
Q,
-2
-5
-4
-3
80
81
85
10
0
6
Points
e12
t
330
Q,
3
t
120
Q.
140
60
5
7
132
80
14
9
4
6
3
1
2
Q
-5
t
455
450
446
t
Q
478
6
-7
-12
2
-3
-8
-4
-4
-5
475
472
471
e7
t
Q
240
5
-45
-25
-5
-5
235
~30
227
t3
t
20
333 -10
0
2
Q
-15
-10
-2
-5
-4
-5
-6
t
10 265
10
3 263
2
1
Q,
t
492
490
488
486
Q,
Q
20
0
5
36
19
4
3
140
145
150
152
Q
-8
10
-2
-1
t
156
159
Q
2
-1
t
300
240
220
210
208
206
204
495
0
-2
-4
-3
-4
t
35
40
41
t
60
Q
-5
6
5
64
t13
t
Q
t
162
163
:3
164
165
0
Q
-5
t
167
Q,
20
6
1
:3
d13
t
498
Q
-2
-3
t
500
Q
-4
494
-----
Q
8
13
-7
-2
---
ell
el0
t
308
310
309
t
Q.
-2
-6
1
322
t
328
327
Q.
-3
-1
320
292
291
t6
t5
t4
t12
tIl
tl0
t
t
Q
e9
t
300
295
Q.
-25
-12
-7
-12
-6
-2
-3
d12
d11
e8
Points
t9
-5
661
dl0
181
Q.
d4
Q
t
667 -260
-120
-20
0
-10
-2
-4
6
Q
653
d9
t
472
465
460
Q.
-43
-25
-15
-20
-25
-29
-12
-17
-2
-1
-4
t2
e13
388
385
381
380
U'
e6
t
180
Q,
380
390
IV
a5
e4
t
Q.
160
20
-10
-6
-10
-4
-2
-8
013
t
665
Q.
-1
-2
-
e3
t
832
05
t
833
Q.
-1
012
Points
Q,
b13
t
Q.
-2
-1
011
t
658
655
Q.
-7
-1
-3
-5
-1
-2
d8
t
445
425
420
115
Q,
t
420 -48
360 -35
-20
-10
-15
355
5
010
t
646
642
Q.
-11
-2
-5
-10
-5
0
-1
-2
d7
d6
Q,
t
360 -155
300
5
290
0
-5
-10
0
285
279
-5
-16
-2
-5
-7
-160
-30
-10
-15
-5
-13
-2
b9
742
Points
d5
Q,
b8
t
80
86
88
Q
35
1
-1
0
f8
f7
t
100
110
Q.
35
15
-4
o.
t
115
125
130
129
t
Q
35 130
10 140
0 143
25
The heat flow may be determined from the temperature
distribution by multiplying the difference in temperature
between the series of points next to the outside surface,
and the points along the outside surface by the kA (=1) facx
tor.
One-half of the temperature difference is used for the
center line calculation, an axis of
s~nmetry,
since half of
the heat flow along that line must be contributed to each of
the sides.
The summation and tabulation of the data for each
of the five cases is given in Table VI.
In order to express the heat-flow through the hollow
square conductor in terms of the heat-flow through a conoentrio circular conductor, of the same outside-inside dimensional ratio, it was necessar] to evaluate the radial heat flow
through the circular conductors, using the
s~~e
diameter ra-
tios, and considering the conductivity to be one, an one thousillld degree temperature difference, and a one foot in length
section, as before.
This data is calculated and tabulated
in Table VII.
The value of c s ' the ratio of the heat-flow through the
square to the heat-flow through the ciroular conductors, is
calculated and tabulated in Table VIII for each of ttle five
cases.
A oontinuous curve showin.g the value of
Os
in
ter~IlS
of
the ratlo of the v'Jidths of the outside to the inside square
(~) is shown on Platt I.
27
TABLE VI
Determination ot Heat Flow trom Temperature D1stribution
Caee I, from Table I
q • 8(211+422+&?9+494+498+499+499+499+499+499+499.2+499.5+
500+ix500) • 50,?74 Btu/hr-tt
Caee II, trom Table II
q • 8(20?+414+450+48?+495+498+499+!x500) • 25,408 Btu/hr-tt
Caee III. trom Table III
q • 8(90+181+262+303+321+328+331+332+ix333) •
• 18,520 Btu/hr-tt
Case IV, trom Table IV
q • 8(52+103+152+194+221+236+243+246+248+1x250) •
• 14,560 Btu/hr-tt
Case V. trom Table V
q • 8(20+41+64+88+110+129+143+152+159+163+165+
ix167) • 10,544 Btu/hr-tt
28
TABLE VII
Determination of Heat Flow through Concentrio Conduotors
Case I, with r2/rl m 1.167
q. 2nk(tZ -t,} • 6.28(1000} _ 40,700 Btu/hr-rt
In 1.167
In (r2!rl)
Case II, with r2/rl - 1,333
q -
6,28(1000} • 21,840 Btu/hr-rt
In 1,333
Case III, with r2/rl - 1.500
q -
6i~8i~gggl
- 15,490 Btu/hr-ft
Case IV, with r2/r, • 1.667
q -
6.28 1000) • 12,300 Btu/hr-rt
In 1.667
Case V. with r2/rl • 2.000
q • 6.28(10001 a 9070 Btu/hr-rt
In 2.000
29
TABLE VIII
Tabulation of
Case I
C
s·
q
a
50,774
1 246
q 0
• ~O,'oO·
~a
• 26,408 • 1.210
21,840
·
Case II
Os •
q •
Case III
° • 18,520
8
1~,490
• 1.196
Case IV
0
14,560
8 • 12,300
a
1.18'
Oa8. V
Os •
10,544
9,070
• 1.162
as
30
•: I
t
~
d;
II
l~~
11
'1
I
.......
"
t
It,.
Ii
J
.
CO CLUSIONS
3
From the plot on Plate I, it may be seen that the heatflow through the square conductor approaches that through the
circular conductor as the outside-inside
es.
~idth
ratio increas-
This is what one would expect, since the geometry of the
thick square more closely approximates the concentric thick
walled circles.
The only available indication as to the accuracy of the
heat-flow throueh the square conductor that could be found
was in ar- empirical formula as given by Jakob. (14)
(19) Jakob,
M~x,
The agree-
and Hawkins, George A., Elements of heat
transfer and insulation, 2nd edition 1950, John VTiley
P 39
ment with this form was within the order of +3%.
The relaxation of the various internal points was carried
on until the Q for that point was within ~lO, which would
mean the temperature was within three degrees, or
0.3%.
In most cases the Q was in the order of from 0 to ~4, which
would. give an error of one degree, or 0.1%.
This gave an ac-
curacy in the heat-flow calculation of from 0.2 to 0.8%, since
the temperature of the first inside row of points varied from
80 to 500°F.
However this accuracy could not be attained un-
less a great nwnber of points had been used.
With the number
of points used, it is thought that the accuracy was within
one percent.
The relaxation method is quite applicable to the solution -of difficult heat transfer problems.
This numerical
33
analysis has the added advantage of the incorporation of variable conductivity, and of directional conductivity.
The
solution by this method requires only a nominal amount of
time.
The solution of Case V, having fifty points to relax,
took approximately seven hours.
This length of time was con-
siderably shorter than what would have been normally necessary, because of previous practice.
In conclusion, it may be stated that it is believed that
the results presented in this thesis are accurate to well
within the normal accuracy of heat-flow calculations.
35
A brief review is made of the solution of heat conduction problems by means of electrical an
and by graphical methods.
membrane analogies,
The origin of and various applica-
tions of the relaxation method of solution is discussed.
The solution of a continuous concentric square heat conduction problem is made by numerical analysis.
The relation-
ship between the heat conducted through the concentric square
conductor and that cnducted through a concentric circular
\
conductor is determined and indicated in one continuous curve.
The accuracy of this determination is surveyed and coneluded to be within plus or minus one percent.
In summary, the author would like to bring to your attention the applications for the above work in the fields of
heat-flow determinations in furnace, kiln, tunnel, and duct
36
BIBLIOGRAPHY
1. Books:
Dusinberre, G. M., The Numerical Analysis of Heat
~low.
McGraw-Hill Book Co. pp 53-59 (1949)
Jakob,
~ax,
Heat Transfer, Vol. 1, John Wiley pp 365-
373 (1949)
Jakob, Max and Hawkins, G. A., Elements of Heat Transfer and Insulation, 2nd Edition, John Wiley, P 39
(1950)
l:oore, A. D., Fundamentals of Electrical Design, McGrawHill, p 73-77 (1927)
McAdams, W. H., Heat Transmission, 2nd Edition, Mc rawHill, P 1-25 (1942)
Southwell, R. V., Relaxation Uethods in "l"":ngineering Science, Oxford University Press pp 1-4 (1949)
2. Periodicals:
Christopherson, B. A., and uouthwell, R• • ,
Methods Applied to
ngineering Problems.
of the Royal Society of London,
elaxation
Proceedings
eries A, Vol. 168,
pp 317-350 (1938)
Emmons, H.
"
robl .ms
0_
The Numerical Solution of Heat-Conduction
Trans. of the A. S. M.E. Vol. 63, No.6,
pp 607-615 (1943)
Kayan,
0
F., An
lectrical Geometrical Analogue for
Complex Heat Flow.
Trans. of the
A.S.~oE.
Vol. 67,
37
No.8, pp 713-718 (1945)
Paschkis, IT. and Baker, H. D., A Method for Determining
Unsteady-state Heat Transfer by means of
Analogy.
n Electrical
Trans. of the A.S.N;.E. Vol. 64, No.2,
pp 105-112 (1942)
Unpublished Material:
Mann, H. T.,
Dete~mination
of the Economic Feasibi1-
i ty of Producing Marginal Gas '.'lJells.
Thesis, 1issouri
chool of Mines and Metallurgy, Rolla, Missouri (1950)
Tilson, L. H., The
pplication of the
embrane
to the Solution of Heat Oonduction Problems.
issouri School of
alogy
Thesis,
ines and .etallurgy, Rolla,
s-
souri. (1948)
zwierzchowski,
ethod to the
., The Application of the Relaxation
olution of Problems Involving t e
Fluids through Porous Media.
of lines and
low
Thesis, Missouri School
etallurgy, Rolla,
issouri. (1949)
38
VITA
The author was born July 26, 1926, at Hannibal, Missouri.
His high-school education was obtained at the Hannibal High
chool, and was completed in June 1944.
Pre-engineering work
was taken at the Hannibal-LaGrange College in Hannibal, Nissouri.
First entrance to the
~issouri
allurgy was made in September 1946.
~chool
of Mines and Met-
He was graduated in June
1948 with the degree of Bachelor of Science in Mechanical Engineering from this institution.
concurrent wit
f
e- ntrance for graduate work,
an instructorship in the Mechanical Engineering
Department, was made in September, 1948.