- Email: [email protected]
A note on a classical problem in the mathematical theory of heat conduction
Int. J. Engng Sci. Vol. 5, pp. 3948.
Pergamon Press 1967. Printed in Great Britain
A NOTE ON A CLASSICAL PROBLEM IN THE MATHEMATICAL THEORY OF HEAT CONDUCTION H. NUTTALL Department
of Theoretical Mechanics, University of Bristoi, England Communicated
by R. BERKER
A~ct-Two-dime~ional steady heat flow in a rectangular corners of the region implies an infinite rate of heat transfer. give rise to this condition are never realized in a mat physical estimate of the rate of heat flow may be calculated for more 1. STATEMENT
region with a temperature discontinuity at the The boundary conditions of the problem which situation. A method is given whereby a simple realistic boundary conditions.
OF THE
PROBLEM
WE define the steady state temperature at a point (x, y) of the region as fI(x, 7). The problem basic to the discussion in the sense that more general boundary conditions may be included by superposition, is to determine the temperature distribution and heat ffow in a rectangular region where the boundary conditions are as shown in Fig. 1. y ,0=$
b _-,
0
--
- x b R=O
8=0’
-II
The differential equation to be satisfied is
V20_a20 a20
-=0, for -dX2Cay2
06 Ixj
(1.1)
The boundary conditions are,
(1.2) where 0 is a constant.
H. NUTTALL
40
A formal solution of the problem using Fourier series is ( - 1)”cosh(2s + l)ny/2n. cos(2s + 1)7rx/2a (2s + l)cosh(2s + l)nb/2a For a rigorous demonstration to Carslaw and Jaeger [l].
(1.3)
that this is the solution of the problem the reader is referred
The rate of heat flow into the region across the faces I,v]=b is given by (1.4) where K is the thermal conductivity. When applied to 6’as given by (1.3) we have I6KO m tanh(2s + l)rcb/2a Y=-c (2s+l) 0
(1.5)
The series in (1.5) is clearly divergent, the physical explanation lying in the discontinuity of 0 at the corners of the region. This temperature discontinuity implies infinite temperature gradients at the corners of the region which makes the integral in (1.4) infinite. 2. DISCUSSION
OF SOLUTION
FOR
A SEMI-INFINITE
STRIP
A closer insight into the behaviour of 0 in the corner regions of the finite rectangle is obtained by considering a semi-infinite strip having the same boundary conditions as the finite rectangle except that 0 is zero at infinity. This leads to a closed solution and it is reasonable to suppose that the general form of the temperature distribution at the corners will give a qualitative picture of the behaviour for the case of the finite rectangle.
Y'
-e=o
”
FIG. 2
A note on a classical problem in the mathematical
.4 conformal
41
transformation z’=sin
transforms
theory of heat conduction
2,
region I, Fig. 2, into region II.
A second transformation z’-1 w=log-=u++u z’+ 1 transforms
the region 1I of Fig. 2 to the region 111. zL.-
1
I I
w=u+iv=log-
z’f I
+iarg-,
z’-
1
z’+ 1
v=tan-’
(2.1)
= tan- ‘20 where
tan 0 = cos x/sinh y.
Hence
If the temperature
at the base of the strip is 0
(2.2)
=-
4KO
‘I2 dx -+cc lt s 0 cosx
(2.3)
Equation (2.2) exhibits the behaviour of 8 and shows clearly the way in which the temperature gradient tends to infinity at the corners of the strip. 3. MODIFIED
It should
be remarked
BOUNDARY
CONDITIONS
in passing that the problem
with boundary
e=e,,
06 Ix\
IYI =h,
e=e,,
o< Jyj
1x1 =a,
c=-e,, is equivalent
to the problem
treated
in section
1 with 0, - e2 =O.
conditions
42
H.NUTTALL
It has been shown that these boundary conditions imply an infinite rate of heat flow. While boundary conditions may exist in a real physical situation which resemble those specified above, the model breaks down at the corners where temperature discontinuities will not exist in a real situation. The mechanical or chemical engineer is more interested in rates of heat transfer than in temperature distribution. Our concern in the remainder of this paper is to establish as simply as possible a means of estimating the rate of heat flow through arectangular section where the boundary conditions resemble the idealized ones stated above but where the discontinuity at the corners is removed.
x
FIG. 3
We consider a rectangular such as
section as in Fig. 3. It is realistic to suppose that the faces
can be maintained at a constant temperature
and that the faces
1x1 =a can be maintained at a constant temperature B=O for OG]yl
in which 0 falls from 0 to 0 as y varies from b to b--E.
A note on a classical problem in the mathematical
theory of heat conduction
43
If this dist~bution of temperature is known, the problem is a Dirichlet problem and 0 and the rate of heat Row may be determined. In general however, 0 will be unknown along the element of surface as specified by (3.1) and the problem would be made unique by specifying some boundary condition relating the surface temperature and its normal derivative to the manner in which heat is being transmitted across the element to the external environment [2]. It is not the purpose of the discussion to seek a rigorous solution for a prescribed set of boundary conditions along the surface (3.1). In practice the true nature of this boundary condition may not be well defined. We shall show that if 8 is small and we assume the temperature 0 to vary linearly from value 0 to Q along the surface element (3.1), it is possible to calculate the rate of heat flow fairly easily and that the value obtained is not unduly sensitive to the size of a within what might be regarded as a practical range of size. 4. SOLUTION
WITH
MODIFIED
BOUNDARY
CONDITIONS
We therefore seek a solution of the differential equations
with boundary conditions,
e=o,
It is clearly seen by the use of Fourier series that such a solution is
+ $ a, cos(2s + l)ny/2b mcosh(2s f l)nx/2b
=9-b++,
From (4.2) and (4.1) 40 * ( - lY cosh(2s + I f(nyf2a) scos(2s + I)(ltx/2a) 0=-T (2s + l)cosh(2s -I-l)(rrb/2a)
b-s
(4.1)
(4.2)
44
H. NUTTALL
The rate of heat flow into the block per unit length is dx which when applied to (4.3) gives nb
tanh(2s + 1): - tanh(2s + 1)~;; 1 --
2b m tanh(2s + 1~(~#~2b)sin(2~+ t)(as/2b) r: (2s + 1)” x.5 0
(4.4)
The numerical valuation of this expression for given ratio b/a requires the summation of the series
Let us call n~/2h=s
and denote the sum of the series by S(X). Since 1 - nnx =flog 2(1 -cosx) cDol cos
= ‘ilog
1 1 -&log 2( 1 - cos x) 2( 1 - cos 2x)
=gog
( > cotZ
2 ’
whence
but s(n/2)=+$-
. . . =0*9159656,
and S(x) = 0.9159656 + +j:,z log(cot.$d~
(4.5)
A note
on a classical
problem
in the mathematical
theory
45
of heat conduction
where (4.6) where B, denotes Bernoulli’s (4.5) and (4.6) give
number.
% siN~;s~~~~Pb =y
(
If we denote
nqi)+ 0.78539821
log cot-
the ratio c/b by A, x =ne/2b =d/2,
+O.l 076607A3
+ 0.0185945A5 + 0~00345~’ + 0.00067925A9 + . . . Since tanh(2s + l)[na/26], tanh(2s + l)[rcb/2a] converge rapidly to unity values of s, the rate of heat flow q as given by (4.4) is easily obtained. The following table shows the values of q obtained From (4.4) the value of q may be expressed as
with
(4.7) increasing
for varying values of ,I and ratio u/b.
q =k(a/b, A)K@ where k(a/b, A) is tabulated
in Table
1.
TABLE 1. VALUES OF k(u/b, A)
a/b
a/b
l/l&
114 14.8887
14.8699
314 14.7365
l/200
16.6538
16.6348
l/500
18.9871
l/l,ooo l/10,000
alb
4
0
14.4666
413 14.0042
13.1046
4/l 11.3572
16.5016
16.2316
15.7694
14.8702
13.1226
18.9682
18.8348
18.5648
18.1024
17.2030
15.4556
20.7523
20.7334
20.6000
20.3300
19.8676
18.9682
17.2208
26.6158
26.5969
26.4635
26.1936
25.7312
24.8318
23.0844
l/2
alb
l/l
211
a/b
It is useful to note that for the range of values of i listed in the above table, the values If in equation (4.7) for small values of of k(a/b, A) may be obtained by a simple expression. .I we neglect terms in A3 and higher powers, replace cot(rry4) by 4/7r3, and rearrange the terms in (4.4); then for small 1
H. NUTTALL
46
If we write I = lo-“, then equation (4.8) becomes k(a/& R)=k(a/b)+5.8635~,
(4.9)
where the value of k(a/b) for the several values of the ratio (a/b) are given in Table 2. TABLE2
l/4
l/2
314
l/l
413
2/l
4/l
3.1615
3.1425
3.0093
2.7393
2.2770
1.3777
-0.3699
a/b k(0)
5. DISCUSSION
OF THE
CASE @
WHERE
THE
TEMPERATURE
FALLS
FROM
TO 0 NONLINEARLY
It may be that the temperature drop will not be linear in the region 1x1=a, b--E< IyI< 6, but may vary as shown in Figs. 4(a) and (b). A
FIG. 4(b)
FIG. 4(a)
In case 4(a),
[email protected](Y>,wherefI (Y)k 0,
b-&y<
b, .fi(b)=fl(b-&)=O.
In case 4(b), 8=o[l-(E+j
+
@f,(y),
where _fdy> G 0,
b-s
_f2(b)=f2(b--e)=O.
It will be clear that for these cases the solution to our problem is determined by superposing upon the solution (4.3) harmonic functions which satisfy the boundary conditions, o< IXlGz,
e=o,
on IxI=a,
I4
o=o,
on IxJ=a,
b--E< lyl
6 =Of,(y)
case 4(a)
0 = Ofi
case 4(b).
on
jyl=b,
A note on a classical problem in the mathematical
theory of heat conduction
47
It follows from the Mean Value and Maximum Modulus theorems of harmonic functions that the superimposed solution arising fromf;(y) is at all points greater than or equal to zero. This implies a nett inflow of heat into the region along the sides 1x1=a arising from the positive temperature distribution fi(y). Similarly by the same reasoning there is a nett outflow of heat along the sides 1x1=a, arising from the negative distributionf,(y). Thus for the case 4(b) the rate of flow through the region exceeds that calculated upon the assumption of a linear drop over a length E, while in case 4(a) the rate of flow is less than that of the linear assumption. If the departure from linearity is not too great and if tangents to the temperature curves are drawn at the point A and produced to meet the line 1x1=a, at B as in Figs. 4(a) and (b), it will be evident that calculations of heat transfer based upon the linear temperature drops AB Fig. 4(b) and AB Fig. 4(a) will be upper and lower bounds respectively for any curve of temperature intermediate to Ii(y) and f2(y). As an illustration we consider a temperature drop of the form (j=@
[
1_(b-y) -+a(b-y)(b-y-E)
.
E
1
In this expression a@--W-y-s)
is fi(r)
or A(Y)
according as to whether the sign of a is negative or positive. The slope (&/dy) at y= b is given by de
0 dy
0
b=;(l
+a.?).
For a = &(1/2.s’) this gives values 30/2s and O/2&. Thus the rate of heat flow for this case will be bounded between the linear calculations corresponding to 1=2&/3b and 1=2&/b. An indication of the magnitudes of the difference between the upper and lower bounds is given in Table 3 below where the bounds are tabulated for two cases e/b = l/200, e/b = l/1000. The lower bounds for these two cases are given by rows 1 and 3 of Table 1. The upper bound values are calculated from the equation (4.9) and Table 2 for values x=log,,3oo=2~47712,
x=log,,
1500=3~17609. A
b
FIG. 5
48
H.
NUTTALL
TABLE 3
a/b
k(al&) upper E/b
314
l/l
413
2/l
411
17.6671
17.5339
17.2639
16.8016
15.9023
14‘1.547
16.6538
16.6348
16,5016
16.2316
15.7694
14.8702
13.1226
14.8887
14.8699
14.7365 _.~
14.4666
14.0042 _--
13.1046
11.3572
2 I -7845
21-7655
2 I .6323
21.3623
20.9~
20.~7
18.2531
20.7523
20.7334
20.6000
20.3300
19.8676
18.9682
17.2208
18.9871
IX.9682
18.8348
18.5648
18.1024
17.2030
15.4556
l/4
l/Q
17.6861
~-
--
4alb)
.= l/200
linear k(alb) lower ___-
-____-
k(albl
upper Ma/b) linear Ma/b) lower
E/h = l/l000
if the exact nature of the departure of the temperature drop from linearity were known the rate of heat flow could, in principle, be calculated by the method used for the iinear case. However, this is unlikely to be known and the above calculation demonstrates that the rate of heat transfer as calculated on the assumption of linear temperature drop is not unduly sensitive to a moderate departure from linearity. It will be apparent that the method which has been discussed within the context of heat conduction theory is applicable to a class of plane potential problems for rectangular regions with similar boundary conditions and finds analogous application in the fields of electrostatics and hydrodynamics. REFERENCES [I] CARSLAWand JAEGER,Cu~~~r~~~~of Heat in Solids, 121 Z&id.p. 18.
p.
166 (2nd edition). Oxford University Press (1900).
(Received 27 October 1965)
R&m&---Un Ccoulement de chaleur & deux dimensions, stabilisk, dans une region rectangulaire, avec des discontinuitCs de temperature dans les coins, implique une vitesse de transmission de chaleur infinie. Les conditions aux limites, necessaires pour faire apparaitre cet &at, ne sont jamais physiquement r&a&sables. L’auteur donne une mCthode suivant laquelle il est possible, par ie calcul, de faire une estimation de la vitesse de la transmission de chaleur pour des conditions aux fimites plus realistes. Zusammenfassung-Ein zweidimensionaler stationlrer Wirmefluss in einer rechteckigen Zone, in deren Ecken WLrmestosstellen auftreten, setzt voraus, dass die WLrmeleitungen mit unendlich grosser Geschwindigkeit vor sich geht. In der Natur sind die Randbedingungen fiir einen derartigen Zustand niemals gegeben. Hier wird unter der Annahme mehr reatistischer Randbedingungen eine Methode angegeben, mit der sich die Geschwindigkeit der W~rme~~~ragung ann~herungsweise und einfach berechnen &sst. Russo termico uniforme bidimensionaie in una regione rettangolare con una discontinuita di temperatura negli angoli della regione comporta un ritmo infinito di dispersione termica. Le condizioni limite de1 problema the portano a tale condizione non si verificano mai in una situazione fisica reale. In quest0 articolo si propone un metodo con il quale b possible calcolare una semplice stima de1 flusso termico per condizioni limite piti realistiche. Sommario-Un
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Pergamon Press 1967. Printed in Great Britain
A NOTE ON A CLASSICAL PROBLEM IN THE MATHEMATICAL THEORY OF HEAT CONDUCTION H. NUTTALL Department
of Theoretical Mechanics, University of Bristoi, England Communicated
by R. BERKER
A~ct-Two-dime~ional steady heat flow in a rectangular corners of the region implies an infinite rate of heat transfer. give rise to this condition are never realized in a mat physical estimate of the rate of heat flow may be calculated for more 1. STATEMENT
region with a temperature discontinuity at the The boundary conditions of the problem which situation. A method is given whereby a simple realistic boundary conditions.
OF THE
PROBLEM
WE define the steady state temperature at a point (x, y) of the region as fI(x, 7). The problem basic to the discussion in the sense that more general boundary conditions may be included by superposition, is to determine the temperature distribution and heat ffow in a rectangular region where the boundary conditions are as shown in Fig. 1. y ,0=$
b _-,
0
--
- x b R=O
8=0’
-II
The differential equation to be satisfied is
V20_a20 a20
-=0, for -dX2Cay2
06 Ixj
(1.1)
The boundary conditions are,
(1.2) where 0 is a constant.
H. NUTTALL
40
A formal solution of the problem using Fourier series is ( - 1)”cosh(2s + l)ny/2n. cos(2s + 1)7rx/2a (2s + l)cosh(2s + l)nb/2a For a rigorous demonstration to Carslaw and Jaeger [l].
(1.3)
that this is the solution of the problem the reader is referred
The rate of heat flow into the region across the faces I,v]=b is given by (1.4) where K is the thermal conductivity. When applied to 6’as given by (1.3) we have I6KO m tanh(2s + l)rcb/2a Y=-c (2s+l) 0
(1.5)
The series in (1.5) is clearly divergent, the physical explanation lying in the discontinuity of 0 at the corners of the region. This temperature discontinuity implies infinite temperature gradients at the corners of the region which makes the integral in (1.4) infinite. 2. DISCUSSION
OF SOLUTION
FOR
A SEMI-INFINITE
STRIP
A closer insight into the behaviour of 0 in the corner regions of the finite rectangle is obtained by considering a semi-infinite strip having the same boundary conditions as the finite rectangle except that 0 is zero at infinity. This leads to a closed solution and it is reasonable to suppose that the general form of the temperature distribution at the corners will give a qualitative picture of the behaviour for the case of the finite rectangle.
Y'
-e=o
”
FIG. 2
A note on a classical problem in the mathematical
.4 conformal
41
transformation z’=sin
transforms
theory of heat conduction
2,
region I, Fig. 2, into region II.
A second transformation z’-1 w=log-=u++u z’+ 1 transforms
the region 1I of Fig. 2 to the region 111. zL.-
1
I I
w=u+iv=log-
z’f I
+iarg-,
z’-
1
z’+ 1
v=tan-’
(2.1)
= tan- ‘20 where
tan 0 = cos x/sinh y.
Hence
If the temperature
at the base of the strip is 0
(2.2)
=-
4KO
‘I2 dx -+cc lt s 0 cosx
(2.3)
Equation (2.2) exhibits the behaviour of 8 and shows clearly the way in which the temperature gradient tends to infinity at the corners of the strip. 3. MODIFIED
It should
be remarked
BOUNDARY
CONDITIONS
in passing that the problem
with boundary
e=e,,
06 Ix\
IYI =h,
e=e,,
o< Jyj
1x1 =a,
c=-e,, is equivalent
to the problem
treated
in section
1 with 0, - e2 =O.
conditions
42
H.NUTTALL
It has been shown that these boundary conditions imply an infinite rate of heat flow. While boundary conditions may exist in a real physical situation which resemble those specified above, the model breaks down at the corners where temperature discontinuities will not exist in a real situation. The mechanical or chemical engineer is more interested in rates of heat transfer than in temperature distribution. Our concern in the remainder of this paper is to establish as simply as possible a means of estimating the rate of heat flow through arectangular section where the boundary conditions resemble the idealized ones stated above but where the discontinuity at the corners is removed.
x
FIG. 3
We consider a rectangular such as
section as in Fig. 3. It is realistic to suppose that the faces
can be maintained at a constant temperature
and that the faces
1x1 =a can be maintained at a constant temperature B=O for OG]yl
in which 0 falls from 0 to 0 as y varies from b to b--E.
A note on a classical problem in the mathematical
theory of heat conduction
43
If this dist~bution of temperature is known, the problem is a Dirichlet problem and 0 and the rate of heat Row may be determined. In general however, 0 will be unknown along the element of surface as specified by (3.1) and the problem would be made unique by specifying some boundary condition relating the surface temperature and its normal derivative to the manner in which heat is being transmitted across the element to the external environment [2]. It is not the purpose of the discussion to seek a rigorous solution for a prescribed set of boundary conditions along the surface (3.1). In practice the true nature of this boundary condition may not be well defined. We shall show that if 8 is small and we assume the temperature 0 to vary linearly from value 0 to Q along the surface element (3.1), it is possible to calculate the rate of heat flow fairly easily and that the value obtained is not unduly sensitive to the size of a within what might be regarded as a practical range of size. 4. SOLUTION
WITH
MODIFIED
BOUNDARY
CONDITIONS
We therefore seek a solution of the differential equations
with boundary conditions,
e=o,
It is clearly seen by the use of Fourier series that such a solution is
+ $ a, cos(2s + l)ny/2b mcosh(2s f l)nx/2b
=9-b++,
From (4.2) and (4.1) 40 * ( - lY cosh(2s + I f(nyf2a) scos(2s + I)(ltx/2a) 0=-T (2s + l)cosh(2s -I-l)(rrb/2a)
b-s
(4.1)
(4.2)
44
H. NUTTALL
The rate of heat flow into the block per unit length is dx which when applied to (4.3) gives nb
tanh(2s + 1): - tanh(2s + 1)~;; 1 --
2b m tanh(2s + 1~(~#~2b)sin(2~+ t)(as/2b) r: (2s + 1)” x.5 0
(4.4)
The numerical valuation of this expression for given ratio b/a requires the summation of the series
Let us call n~/2h=s
and denote the sum of the series by S(X). Since 1 - nnx =flog 2(1 -cosx) cDol cos
= ‘ilog
1 1 -&log 2( 1 - cos x) 2( 1 - cos 2x)
=gog
( > cotZ
2 ’
whence
but s(n/2)=+$-
. . . =0*9159656,
and S(x) = 0.9159656 + +j:,z log(cot.$d~
(4.5)
A note
on a classical
problem
in the mathematical
theory
45
of heat conduction
where (4.6) where B, denotes Bernoulli’s (4.5) and (4.6) give
number.
% siN~;s~~~~Pb =y
(
If we denote
nqi)+ 0.78539821
log cot-
the ratio c/b by A, x =ne/2b =d/2,
+O.l 076607A3
+ 0.0185945A5 + 0~00345~’ + 0.00067925A9 + . . . Since tanh(2s + l)[na/26], tanh(2s + l)[rcb/2a] converge rapidly to unity values of s, the rate of heat flow q as given by (4.4) is easily obtained. The following table shows the values of q obtained From (4.4) the value of q may be expressed as
with
(4.7) increasing
for varying values of ,I and ratio u/b.
q =k(a/b, A)K@ where k(a/b, A) is tabulated
in Table
1.
TABLE 1. VALUES OF k(u/b, A)
a/b
a/b
l/l&
114 14.8887
14.8699
314 14.7365
l/200
16.6538
16.6348
l/500
18.9871
l/l,ooo l/10,000
alb
4
0
14.4666
413 14.0042
13.1046
4/l 11.3572
16.5016
16.2316
15.7694
14.8702
13.1226
18.9682
18.8348
18.5648
18.1024
17.2030
15.4556
20.7523
20.7334
20.6000
20.3300
19.8676
18.9682
17.2208
26.6158
26.5969
26.4635
26.1936
25.7312
24.8318
23.0844
l/2
alb
l/l
211
a/b
It is useful to note that for the range of values of i listed in the above table, the values If in equation (4.7) for small values of of k(a/b, A) may be obtained by a simple expression. .I we neglect terms in A3 and higher powers, replace cot(rry4) by 4/7r3, and rearrange the terms in (4.4); then for small 1
H. NUTTALL
46
If we write I = lo-“, then equation (4.8) becomes k(a/& R)=k(a/b)+5.8635~,
(4.9)
where the value of k(a/b) for the several values of the ratio (a/b) are given in Table 2. TABLE2
l/4
l/2
314
l/l
413
2/l
4/l
3.1615
3.1425
3.0093
2.7393
2.2770
1.3777
-0.3699
a/b k(0)
5. DISCUSSION
OF THE
CASE @
WHERE
THE
TEMPERATURE
FALLS
FROM
TO 0 NONLINEARLY
It may be that the temperature drop will not be linear in the region 1x1=a, b--E< IyI< 6, but may vary as shown in Figs. 4(a) and (b). A
FIG. 4(b)
FIG. 4(a)
In case 4(a),
[email protected](Y>,wherefI (Y)k 0,
b-&y<
b, .fi(b)=fl(b-&)=O.
In case 4(b), 8=o[l-(E+j
+
@f,(y),
where _fdy> G 0,
b-s
_f2(b)=f2(b--e)=O.
It will be clear that for these cases the solution to our problem is determined by superposing upon the solution (4.3) harmonic functions which satisfy the boundary conditions, o< IXlGz,
e=o,
on IxI=a,
I4
o=o,
on IxJ=a,
b--E< lyl
6 =Of,(y)
case 4(a)
0 = Ofi
case 4(b).
on
jyl=b,
A note on a classical problem in the mathematical
theory of heat conduction
47
It follows from the Mean Value and Maximum Modulus theorems of harmonic functions that the superimposed solution arising fromf;(y) is at all points greater than or equal to zero. This implies a nett inflow of heat into the region along the sides 1x1=a arising from the positive temperature distribution fi(y). Similarly by the same reasoning there is a nett outflow of heat along the sides 1x1=a, arising from the negative distributionf,(y). Thus for the case 4(b) the rate of flow through the region exceeds that calculated upon the assumption of a linear drop over a length E, while in case 4(a) the rate of flow is less than that of the linear assumption. If the departure from linearity is not too great and if tangents to the temperature curves are drawn at the point A and produced to meet the line 1x1=a, at B as in Figs. 4(a) and (b), it will be evident that calculations of heat transfer based upon the linear temperature drops AB Fig. 4(b) and AB Fig. 4(a) will be upper and lower bounds respectively for any curve of temperature intermediate to Ii(y) and f2(y). As an illustration we consider a temperature drop of the form (j=@
[
1_(b-y) -+a(b-y)(b-y-E)
.
E
1
In this expression a@--W-y-s)
is fi(r)
or A(Y)
according as to whether the sign of a is negative or positive. The slope (&/dy) at y= b is given by de
0 dy
0
b=;(l
+a.?).
For a = &(1/2.s’) this gives values 30/2s and O/2&. Thus the rate of heat flow for this case will be bounded between the linear calculations corresponding to 1=2&/3b and 1=2&/b. An indication of the magnitudes of the difference between the upper and lower bounds is given in Table 3 below where the bounds are tabulated for two cases e/b = l/200, e/b = l/1000. The lower bounds for these two cases are given by rows 1 and 3 of Table 1. The upper bound values are calculated from the equation (4.9) and Table 2 for values x=log,,3oo=2~47712,
x=log,,
1500=3~17609. A
b
FIG. 5
48
H.
NUTTALL
TABLE 3
a/b
k(al&) upper E/b
314
l/l
413
2/l
411
17.6671
17.5339
17.2639
16.8016
15.9023
14‘1.547
16.6538
16.6348
16,5016
16.2316
15.7694
14.8702
13.1226
14.8887
14.8699
14.7365 _.~
14.4666
14.0042 _--
13.1046
11.3572
2 I -7845
21-7655
2 I .6323
21.3623
20.9~
20.~7
18.2531
20.7523
20.7334
20.6000
20.3300
19.8676
18.9682
17.2208
18.9871
IX.9682
18.8348
18.5648
18.1024
17.2030
15.4556
l/4
l/Q
17.6861
~-
--
4alb)
.= l/200
linear k(alb) lower ___-
-____-
k(albl
upper Ma/b) linear Ma/b) lower
E/h = l/l000
if the exact nature of the departure of the temperature drop from linearity were known the rate of heat flow could, in principle, be calculated by the method used for the iinear case. However, this is unlikely to be known and the above calculation demonstrates that the rate of heat transfer as calculated on the assumption of linear temperature drop is not unduly sensitive to a moderate departure from linearity. It will be apparent that the method which has been discussed within the context of heat conduction theory is applicable to a class of plane potential problems for rectangular regions with similar boundary conditions and finds analogous application in the fields of electrostatics and hydrodynamics. REFERENCES [I] CARSLAWand JAEGER,Cu~~~r~~~~of Heat in Solids, 121 Z&id.p. 18.
p.
166 (2nd edition). Oxford University Press (1900).
(Received 27 October 1965)
R&m&---Un Ccoulement de chaleur & deux dimensions, stabilisk, dans une region rectangulaire, avec des discontinuitCs de temperature dans les coins, implique une vitesse de transmission de chaleur infinie. Les conditions aux limites, necessaires pour faire apparaitre cet &at, ne sont jamais physiquement r&a&sables. L’auteur donne une mCthode suivant laquelle il est possible, par ie calcul, de faire une estimation de la vitesse de la transmission de chaleur pour des conditions aux fimites plus realistes. Zusammenfassung-Ein zweidimensionaler stationlrer Wirmefluss in einer rechteckigen Zone, in deren Ecken WLrmestosstellen auftreten, setzt voraus, dass die WLrmeleitungen mit unendlich grosser Geschwindigkeit vor sich geht. In der Natur sind die Randbedingungen fiir einen derartigen Zustand niemals gegeben. Hier wird unter der Annahme mehr reatistischer Randbedingungen eine Methode angegeben, mit der sich die Geschwindigkeit der W~rme~~~ragung ann~herungsweise und einfach berechnen &sst. Russo termico uniforme bidimensionaie in una regione rettangolare con una discontinuita di temperatura negli angoli della regione comporta un ritmo infinito di dispersione termica. Le condizioni limite de1 problema the portano a tale condizione non si verificano mai in una situazione fisica reale. In quest0 articolo si propone un metodo con il quale b possible calcolare una semplice stima de1 flusso termico per condizioni limite piti realistiche. Sommario-Un
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