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A boundary residual method with heat polynomials for solving unsteady heat conduction problems
A Boundary Residual Method with Heat Polynomials for Solving Unsteady Heat Conduction Problems ~~H.YAN~,~.F~KuTANI
and A.KIEDA
Department of Mechanical Engineering, Doshisha University, Kyoto 602, Japan
In this paper, we present a new method for solving unsteady heat conduction problems, which is based on a time-space boundary residual method with heat polynomials. More specifically, it employs an integral least squares criterionfor the initial and boundary residuals so as to determine the unknown coefficients in a trial expansion of heat polynomials. Though it treats only one-dimensional cases, the present approach shows a good applicabilityfor such heat conduction problems.
ABSTRACT:
I. Introduction In general, unsteady or transient heat conduction problems can be solved typically by purely analytical methods such as the separation of variables technique and the Green’s function method which are described in detail in (1) and, alternatively, by purely numerical methods such as the finite difference technique developed early by Emmons (2), Dusinberre (3) and others. As is well known, these solution techniques have their own features, advantages and disadvantages. For example, purely analytical methods are considered to be rather poor in the practical applicability for engineering problems. On the other hand, the finite difference technique which is widely applicable for engineering purposes, is not always economical. In the present paper, we shall propose a new semi-analytical technique based on a boundary residual method, which employs heat polynomials originally defined by Rosenbloom and Widder (4). In order to clarify its fundamental performance, the formulation deals with only one-dimensional transient problems. However, it can be easily extended to two- or three-dimensional cases.
ZZ.Problem Definition In the present paper, we restrict ourselves to one-dimensional transient heat conduction problems illustrated in Fig. 1, where temperature changes occur in a solid plate of width I, isolated at one end and heated or cooled by a fluid at the other end. One-dimensionality is insured by the fact that the cross-sectional area perpendicular to the direction of the heat flow is sufficiently large or substantially infinite.
QThe
FranklinInstitute 001W032/83$3.00+0.00
291
H. Yano, S. Fukutani and A. Kieda
Fluid
FIG. 1. One-dimensional
The governing
equation
heat conduction.
for this case is de’ a% at,="=
(1)
where t’ designates the time, x’ the positional coordinate along the heat flow, 6’ the temperature in the solid, and a the thermal diffusivity. Equation (1) can be rewritten in the dimensionless form
ae a28 y$=ax’
(2)
with
and
where Q; and 0; represent respectively the fluid temperature and a representative value in the initial solid temperature distribution. In the present problem, the following initial and boundary conditions are Journalof 292
the Franklin Instate Pergamon Press Ltd.
A Boundary Residual Method with Heat Polynomials presupposed
: O=f(x)
ae
-BO
ax’
at t=O,
at x = 1,
with
where B is referred to as the Biot number, and c( and ,I denote respectively transfer coefficient at x = 1 and the thermal conductivity of the solid.
the heat
III. Heat Polynomials The heat polynomials were defined by Rosenbloom and Widder (4), and their mathematical properties investigated in detail by them. According to Rosenbloom and Widder, the heat polynomial of degree n can be defined as
= (-~J”‘~H, where [ *] means the greatest integer of “*“, and H,( .) represents Hermite polynomial of degree n. The heat polynomials v,(t,x) can also be defined by the generating function exp (xz + tz’) = fj 5 un(t,x), n=o .
(7)
which clarifies that on(t, x) are solutions of (2). Additionally, it is obvious from (7) that
~4G,x) =no,-,(t,x), & u,(t, x) = 2
ZV. Approximate
un(t, x) = n(n - l)u, _ 2(t, x).
(8) (9)
Method for Solution
With the heat polynomials uj(z,x) as fundamental functions, we assume an approximation e^, to the solution 8 in the interval of T < t < T + 1 (where T is any Vol. 316, No. 4, pp. 291-298, Printed in Great Brain
October
1983
293
H. Yano, S. Fukutani and A. Kieda integer) in the following
form : 8,(Z, X) =
~
C~‘Uj(Z,X),
0 ~ Z ~ 1,
(10)
j=O
with z = t - T, where Cy’ are unknown present time-space boundary residual
parameters. The least square criterion method is to minimize the functional
in the
(11) where so, s1 and s2 represent the residuals on the respective three boundaries and TZ of the hatched time-space region illustrated in Fig. 2. With the aid of Eq. (8) these residuals can be expressed as &()= f
To, rI
cjuj(o,x)-8,-,(1,x),
(12)
j=O
with
e^-,UA-f(x), (13)
.zl = f j Cjuj_ l(z,O), j=O
and ~2
=
f
Cj{B~j(r, 1) +jrj-
(14)
r(r, l)}.
j=O
The minimization is to be made i = 1,2,..., m. Therefore
with respect
ar -=O, Xi
i=1,2
T
to the unknown
,...,
parameters
m,
Ci,
(15)
T+l
t
t---+7 FIG. 2.
Time-space coordinate system for the present formulation. Journal
294
of the Franklin Institute Pergamon Press Ltd
A Boundary
Residual Method with Heat Polynomials
which leads to :
AijCj = di,
i = 0,1,2,.
(16)
. .,m,
j=O
with
Aij = (Vi(O,X)3Vj(O, X)), + ij(‘J- I(z, O),Vj_ I(z, 0)),
+B2(vd~, 11,VAT,1))r+jB(vi(T, + iW+
l), vj- I(T, l)),
I(73 I), Vj(r, I)), + ij(vi- I(z, l),
vj_
I(z,
l)),,
(17)
and (18) where
s s 1
,= and
uv dx,
(19)
uv dz.
(20)
0
1
(u, v>, = From the definition
0
(6) for heat polynomials,
x =
1 i+j+
1’
(21)
10, where2i!i j!or j is odd,
Oh vj(z,
O))x
=
I
(i/2)!(j/2)!(i+j+2)’
(22)
where i and j are even. WI
WI
(23) with 1 aik = (i_2k)!k!’ and
(24) With known CcT ~ “, the unknown parameters CT’, j = 0, 1, . . , m can be determined by (16). TherefAre, the parameter sets Cr’, T = 0, 1,2,. . . , can all be computed from Vol. 316, No. 4, pp. 291 298, October Printed in Great Britain
1983
295
H. Yano, S. Fukutani
and A. Kieda
T = 0 on in a successive manner, time intervals.
and thus 8,(r, x) can be obtained
in all the desired
V. Numerical Discussions In the case of a uniform distribution of initial solid temperature f(x) = 1 with Biot number B = 1, we obtained or(r, x) with m = 2 and 4. Figure 3 shows the results of temperature distributions at t = 0.5, 1 and 2 in comparison with the exact solutions by the separation of variables technique described in the literature (5). Figure 4 indicates the results of temperature variation with time at x = 0. It can be considered that these results generally agree well with the exact solutions except near t = 0 in Fig. 4. As is clear from Fig. 5, which presents the results of solution errors A0 for various values of m at x = 0 and t = 0, we can obtain a good approximation with m = 8 even near t = 0. Consequently, considering that the entire temperature distribution 8r(r, x) in any desired time interval can be obtained with good accuracy using very small numbers of expansion terms, the present method is a very efficient approach for solving unsteady heat conduction problems.
VI. Conclusions A new approach for solving one-dimensional unsteady heat conduction problems is presented. It is based on a so-called time-space boundary residual method with
I .”
0 l
-
m=2 m=4 Exact solution
0.6 -
x
FIG. 3. Temperature
distributions
with B = 1 at various times. Journalof
296
the Frankhn Institute Pergamon Press Ltd.
A Boundary Residual Method with Heat Polynomials
FIG. 4. Temperature
variation
0.15
with time t at x = 0 (B = 1).
0
0.10 /
1
0 0
0
0
0.05
A0
0
C)
7
8
0,
:,“rP,” t,, 0
1
2
3
4
5
6
m
FIG. 5. Solution
errors A0 for various values of m at x = 0 and t = 0.
Accurate numerical results are then obtained with small numbers of expansion terms. Therefore, the present technique is much more efficient than the ordinary finite difference methods. Furthermore, there is a possibility of generalizing this method to two- and threedimensional problems. heat polynomials.
References (1) H. S. Carslaw
Oxford,
and J. C. Jaeger, “Conduction 1947.
Vol. 3L6, No 4, pp. 291-298, Printed in Great Britam
October
of Heat in Solids”, Oxford University
Press,
1983
297
H. Yano, S. Fukutani
and A. Kieda
(2) H. W. Emmons, “The numerical solution of heat-conduction problems”, Trans. ASME, Vol. 65, pp. 607-615, 1943. (3) G. M. Dusinberre, “Numerical methods for transient heat flow”, Trans. ASME, Vol. 67, pp. 703-712, 1945. (4) P. C. Rosenbloom and D. V. Widder, “Expansions in terms of heat polynomials and associated functions”, Trans. Am. Math. Sot., Vol. 92, pp. 22&266, 1959. (5) W. H. Giedt, “Principles of Engineering Heat Transfer”, Van Nostrand, New York, 1957.
298
Journal
of the Franklin Institute Pergamon Press Ltd.
and A.KIEDA
Department of Mechanical Engineering, Doshisha University, Kyoto 602, Japan
In this paper, we present a new method for solving unsteady heat conduction problems, which is based on a time-space boundary residual method with heat polynomials. More specifically, it employs an integral least squares criterionfor the initial and boundary residuals so as to determine the unknown coefficients in a trial expansion of heat polynomials. Though it treats only one-dimensional cases, the present approach shows a good applicabilityfor such heat conduction problems.
ABSTRACT:
I. Introduction In general, unsteady or transient heat conduction problems can be solved typically by purely analytical methods such as the separation of variables technique and the Green’s function method which are described in detail in (1) and, alternatively, by purely numerical methods such as the finite difference technique developed early by Emmons (2), Dusinberre (3) and others. As is well known, these solution techniques have their own features, advantages and disadvantages. For example, purely analytical methods are considered to be rather poor in the practical applicability for engineering problems. On the other hand, the finite difference technique which is widely applicable for engineering purposes, is not always economical. In the present paper, we shall propose a new semi-analytical technique based on a boundary residual method, which employs heat polynomials originally defined by Rosenbloom and Widder (4). In order to clarify its fundamental performance, the formulation deals with only one-dimensional transient problems. However, it can be easily extended to two- or three-dimensional cases.
ZZ.Problem Definition In the present paper, we restrict ourselves to one-dimensional transient heat conduction problems illustrated in Fig. 1, where temperature changes occur in a solid plate of width I, isolated at one end and heated or cooled by a fluid at the other end. One-dimensionality is insured by the fact that the cross-sectional area perpendicular to the direction of the heat flow is sufficiently large or substantially infinite.
QThe
FranklinInstitute 001W032/83$3.00+0.00
291
H. Yano, S. Fukutani and A. Kieda
Fluid
FIG. 1. One-dimensional
The governing
equation
heat conduction.
for this case is de’ a% at,="=
(1)
where t’ designates the time, x’ the positional coordinate along the heat flow, 6’ the temperature in the solid, and a the thermal diffusivity. Equation (1) can be rewritten in the dimensionless form
ae a28 y$=ax’
(2)
with
and
where Q; and 0; represent respectively the fluid temperature and a representative value in the initial solid temperature distribution. In the present problem, the following initial and boundary conditions are Journalof 292
the Franklin Instate Pergamon Press Ltd.
A Boundary Residual Method with Heat Polynomials presupposed
: O=f(x)
ae
-BO
ax’
at t=O,
at x = 1,
with
where B is referred to as the Biot number, and c( and ,I denote respectively transfer coefficient at x = 1 and the thermal conductivity of the solid.
the heat
III. Heat Polynomials The heat polynomials were defined by Rosenbloom and Widder (4), and their mathematical properties investigated in detail by them. According to Rosenbloom and Widder, the heat polynomial of degree n can be defined as
= (-~J”‘~H, where [ *] means the greatest integer of “*“, and H,( .) represents Hermite polynomial of degree n. The heat polynomials v,(t,x) can also be defined by the generating function exp (xz + tz’) = fj 5 un(t,x), n=o .
(7)
which clarifies that on(t, x) are solutions of (2). Additionally, it is obvious from (7) that
~4G,x) =no,-,(t,x), & u,(t, x) = 2
ZV. Approximate
un(t, x) = n(n - l)u, _ 2(t, x).
(8) (9)
Method for Solution
With the heat polynomials uj(z,x) as fundamental functions, we assume an approximation e^, to the solution 8 in the interval of T < t < T + 1 (where T is any Vol. 316, No. 4, pp. 291-298, Printed in Great Brain
October
1983
293
H. Yano, S. Fukutani and A. Kieda integer) in the following
form : 8,(Z, X) =
~
C~‘Uj(Z,X),
0 ~ Z ~ 1,
(10)
j=O
with z = t - T, where Cy’ are unknown present time-space boundary residual
parameters. The least square criterion method is to minimize the functional
in the
(11) where so, s1 and s2 represent the residuals on the respective three boundaries and TZ of the hatched time-space region illustrated in Fig. 2. With the aid of Eq. (8) these residuals can be expressed as &()= f
To, rI
cjuj(o,x)-8,-,(1,x),
(12)
j=O
with
e^-,UA-f(x), (13)
.zl = f j Cjuj_ l(z,O), j=O
and ~2
=
f
Cj{B~j(r, 1) +jrj-
(14)
r(r, l)}.
j=O
The minimization is to be made i = 1,2,..., m. Therefore
with respect
ar -=O, Xi
i=1,2
T
to the unknown
,...,
parameters
m,
Ci,
(15)
T+l
t
t---+7 FIG. 2.
Time-space coordinate system for the present formulation. Journal
294
of the Franklin Institute Pergamon Press Ltd
A Boundary
Residual Method with Heat Polynomials
which leads to :
AijCj = di,
i = 0,1,2,.
(16)
. .,m,
j=O
with
Aij = (Vi(O,X)3Vj(O, X)), + ij(‘J- I(z, O),Vj_ I(z, 0)),
+B2(vd~, 11,VAT,1))r+jB(vi(T, + iW+
l), vj- I(T, l)),
I(73 I), Vj(r, I)), + ij(vi- I(z, l),
vj_
I(z,
l)),,
(17)
and (18) where
s s 1
uv dx,
(19)
uv dz.
(20)
0
1
(u, v>, = From the definition
0
(6) for heat polynomials,
1 i+j+
1’
(21)
10, where2i!i j!or j is odd,
Oh vj(z,
O))x
=
I
(i/2)!(j/2)!(i+j+2)’
(22)
where i and j are even. WI
WI
(23) with 1 aik = (i_2k)!k!’ and
(24) With known CcT ~ “, the unknown parameters CT’, j = 0, 1, . . , m can be determined by (16). TherefAre, the parameter sets Cr’, T = 0, 1,2,. . . , can all be computed from Vol. 316, No. 4, pp. 291 298, October Printed in Great Britain
1983
295
H. Yano, S. Fukutani
and A. Kieda
T = 0 on in a successive manner, time intervals.
and thus 8,(r, x) can be obtained
in all the desired
V. Numerical Discussions In the case of a uniform distribution of initial solid temperature f(x) = 1 with Biot number B = 1, we obtained or(r, x) with m = 2 and 4. Figure 3 shows the results of temperature distributions at t = 0.5, 1 and 2 in comparison with the exact solutions by the separation of variables technique described in the literature (5). Figure 4 indicates the results of temperature variation with time at x = 0. It can be considered that these results generally agree well with the exact solutions except near t = 0 in Fig. 4. As is clear from Fig. 5, which presents the results of solution errors A0 for various values of m at x = 0 and t = 0, we can obtain a good approximation with m = 8 even near t = 0. Consequently, considering that the entire temperature distribution 8r(r, x) in any desired time interval can be obtained with good accuracy using very small numbers of expansion terms, the present method is a very efficient approach for solving unsteady heat conduction problems.
VI. Conclusions A new approach for solving one-dimensional unsteady heat conduction problems is presented. It is based on a so-called time-space boundary residual method with
I .”
0 l
-
m=2 m=4 Exact solution
0.6 -
x
FIG. 3. Temperature
distributions
with B = 1 at various times. Journalof
296
the Frankhn Institute Pergamon Press Ltd.
A Boundary Residual Method with Heat Polynomials
FIG. 4. Temperature
variation
0.15
with time t at x = 0 (B = 1).
0
0.10 /
1
0 0
0
0
0.05
A0
0
C)
7
8
0,
:,“rP,” t,, 0
1
2
3
4
5
6
m
FIG. 5. Solution
errors A0 for various values of m at x = 0 and t = 0.
Accurate numerical results are then obtained with small numbers of expansion terms. Therefore, the present technique is much more efficient than the ordinary finite difference methods. Furthermore, there is a possibility of generalizing this method to two- and threedimensional problems. heat polynomials.
References (1) H. S. Carslaw
Oxford,
and J. C. Jaeger, “Conduction 1947.
Vol. 3L6, No 4, pp. 291-298, Printed in Great Britam
October
of Heat in Solids”, Oxford University
Press,
1983
297
H. Yano, S. Fukutani
and A. Kieda
(2) H. W. Emmons, “The numerical solution of heat-conduction problems”, Trans. ASME, Vol. 65, pp. 607-615, 1943. (3) G. M. Dusinberre, “Numerical methods for transient heat flow”, Trans. ASME, Vol. 67, pp. 703-712, 1945. (4) P. C. Rosenbloom and D. V. Widder, “Expansions in terms of heat polynomials and associated functions”, Trans. Am. Math. Sot., Vol. 92, pp. 22&266, 1959. (5) W. H. Giedt, “Principles of Engineering Heat Transfer”, Van Nostrand, New York, 1957.
298
Journal
of the Franklin Institute Pergamon Press Ltd.