- Email: [email protected]
General solutions of the heat equation in finite regions
Int. 1. Engng
Sci., 1972, Vol. 10, pp. 577-591.
GENERAL
Pergamon
Press.
Printed
inGreat Britain
SOLUTIONS OF THE HEAT IN FINITE REGIONS
EQUATION
M. D. MIKHAILOV Institute for Mechanical and Electrotechnical Engineering, Sofia, Bulgaria Abstract- Using a finite integral transform an analytical solution is found for a large class of heat transfer problems. This solu$on is obtained in form of infinite series and contains quasi-steady and transient terms. It is shown that the Olcer’s conductive heat transfer solution[2,3] is a special case of the general solution obtained. The latter can also be applied when studying heat transfer in laminar and turbulent flows of Newtonian and non-Newtonian fluids in pipes and ducts, temperature development in the entrance region of MHD channels and elsewhere. 1. INTRODUCTION
the famous fundamental Fourier’s work [ 11, an immense number of papers and books appeared on the analytical heat transfer theory. Digital computers made these complicated solutions accessible for technological applications. The information flow in this field is growing lately more and more. It is sometimes easier to solve and tabulate a problem as a new one rather than to find out whether its solution is published. Therefore it seems quite useful to find a general solution, which could easily permit to obtain numerous special cases with practical application. In [2,3] Olcer presented such general solutions of the unsteady heat conduction in a finite homogeneous region of arbitrary geometry and initial conditions with general boundary conditions, including the boundary conditions of the first, second and third kind (prescribed surface temperature, prescribed heat flux and Newtonian convections) or any combination of these three. This solution was reported again in[4] and used by Olcer to solve a number of particular problems [5,6,7]. Recently the analytical heat transfer theory is concentrating on more complicated, compared with conductive heat transfer cases, problems, namely: heat transfer in laminar and turbulent flows of Newtonian or non-Newtonian fluids in pipes, ducts; heat transfer by Hartmenn’s flow in thermal entrance region etc. In [8] and [9] the author presented a one-dimensional and three-dimensional solution of a broad class of heat transfer problems, including the above-mentioned as special cases. The present paper is a further development of these results in a form, which contains in itself as a special case the results obtained by &er. FOLLOWING
2. STATEMENT
AND
SOLUTION
OF THE PROBLEM
Subject of the present study will be the partial differential equation cp(T)W(M)
aT(g,7,=
div [(k(M) grad T(A4, T)] +
rP(~)ww) -PW)lTef,
7) +pwf,7)
(1)
with the initial condition
TW, 0) =f,(W 577
1J.E.S.
Vol.
IO No. 7-A
(2)
578
M. D. MIKHAILOV
and the general boundary condition (3) The solution of the Sturm-Liouville
problem
div [k(M) grad JI(ZW1-t [P’w(M) A(N)
-_p(M)]JI(M)
a+(N) ,,+Bw)w)
= 0
(4)
=0
is granted for known. Let $$(M) and t&(M) be two eigenfunctions, values pi and pj
corresponding
to two different eigen-
I-&(W+CI,(W
= P(W&(W
-div
[k(M) grad$i(Wl
/-+(WJI,(W
=p(M)ljrj(W
-div
[k(M)
(61
gradWW1
(7:
where Am
+B(N)$i(N)
=O
and
A(fv)T
+B(N)JII(N)
= 0.
(8:
Multiplying (6) by I,!+(M), subtracting from the result the multiplied with e*(M) equation (7) and integrating the result obtained, after taking into account the well. known Gauss Theorem, which transforms the volume integral into a surface one [lo] one gets
From the boundary conditions (8) follows VJfW
GiCN>
&I,(N)
T
aJij(N)
=
O
(10
an
i.e. the eigenfunctions, corresponding to different eigenvalues (pi # pj) are orthogona with a weight w(M) . To solve equation (1) at the conditions (2) and (3) the finite integral transform Fi(T) = J w(M)&(M)T(M,r)
dV
(11
V
is to be used. It follows, from the orthogonal&y of the eigenfunctions expanded into a series as T(M, 7) = I:1 W&W%(W
I,IJ~ (M) , that T (M, 7) can bc (12
General solutions of the heat equation in finite regions
579
where Gil=
j- w(M)g(M)
dV
(13)
V
and (12) is sometimes called the inversion formula for (11). Let us carry out the transition to boundary pI + pj in equations (9). For (13) the following expression is obtained:
G;’ =
& * 1k(N)
(14) JIiow
8
-
an
Multiplying (1) by I&(M), subtracting from the result the multiplied with T(M, T) equation (4) and integrating the result obtained, one gets, as in (9),
(P(T)v+[pf-P(T)]Fi(T) =I k(N) s
+
I $iOW’W,
7) a’.
(1%
V
From (3) and (5) A (N) and B(N) obtained, one gets
are determined.
h,(N) -
After summing up the results
CWiW)/anl
A(N)+B(N)
*
(16)
Substituting the result in (15), for Ti (7) is obtained in ordinary, first order, linear differential equation, which is easily solved, using the transformed according to (11) initial condition (2). Using this solution in the inversion formula (12), the desired solution of the problem is obtained as follows:
+
J
&(M)P(M,T)
b
If p(M) = 0 and B(N)
dl’ d7 . I
I
(17)
= 0 (second order boundary conditions), then p = 0 and and eigenfunction of the Sturm-Liuville problem (4)-
I,IJ~ = const, are also eigenvalue
M. D. MIKHAILOV
580
(5). Therefore, in this case an ~ditional term, co~esponding to the zero-eigenvalue, appears in the solution and (17) takes the form
~li(M)f’(M,
+
7) dV
(18)
s V
Consider further the special cases ~(7) = 1 and p(r) = 0, where equations (17) and ( 18) take the form:
(19)
These expressions (19) and (20), although representing exact solutions, are not in a convenient form for practical purposes. In the case where the time dependent source functions f(N, 7) and P(M, 7) are given by their Taylor series expansion in time or by exponential lotions, the solutions should be booed into sums of quasi steady and transient terms readily applicable for the purpose of nail evaluations.
581
General solutions of the heat equation in finite regions 3. ZERO-ORDER
SOLUTIONS
For faster convergence of the series (19) and (20) the time integral contained integrated by parts with respect to time. Then the solution ( 19) becomes
WWW,
0) dv
is
1 (21)
where
(22) is called according to 6lcer [2] pseudo-steady zero-order solution. It is easy to show, that the expression (22) is a solution to the differential equation div [k(M)gradT,(M,
0
T)]--p(M)To(M,7)+P(M,7)=
(23)
at the boundary condition
(24) And indeed, after multiplying (23) by $i (M) , and subtracting from it equation (4) multiplied by To (M, T) , the result obtained is integrated using the finite integral transform (1 l), and, as in (15), one gets:
Toi = f
s
k(Nlf(N,
TpW) - hWWan1 A WI +Bw)
ds+
I
V
$i(M)P(M,
1
7) dV . (25)
Substituting (25) in the inversion formula (12) we come to the expression (22). Therefore, the pseudo-steady zero-order solution can be obtained by solving directly equation (23) at the boundary condition (24). The solution (20), valid at second-order boundary condition, after integrating by
582
M. D. MIKHAILOV
parts the time integral contained, can be written, as in (2 l), in the form:
T* CM,7) = T,*CM,T) +
2 G&(M)
exp (-p?r){
1 w(M)rj~~(M)fo(M) dV
i=l
V
1
+
s w(WdI’
{I v
w(Wf,(W
k(N)f(N, r) d&Y A(N)
dV+ 0
s
V
+
I
dV
f’(M,T)
V
I
dr
(26
1
where I,*
= 5 GRi(M)$[ i=l
s
(27
V
is a pseudo-steady zero-order solution. For the case discussed p(M) = 0, B(N)
I
dV]
f k(Nlf(N,r)~dS+IP(M,~)~i(M)
w(M)&(M)
dI’=-5
V
= 0 i.e. [a$i (N)/an]
.
1 k(N)y ”
= 0, therefore we ge
dS.
(28
s
Taking into account (28), after multiplying both parts of the expression (27) b w(M) and integrating with the volume, we get the following condition, which is to b met by the pseudo-steady zero-order solution at second-order boundary conditions IV w(M)T$(M,7)
dV=
It can be shown, as in (22), that the expression equation k(N)
f(N, mdS+ 7)
s V
P(M,7)dV
0.
(29
(27) is a solution to the differentia
1
=div[k(M)gradT,*(M,~)l
+f’(M, 7)
(30
at the conditions (29) and (31
Therefore the pseudo-steady zero-order solution TZ (M, T) can be obtained solving directly equation (30) at the conditions (29) and (3 1).
b
583
General solutions of the heat equation in finite regions
The zero-order solutions obtained (21) and (26) are particularly convenient the source functionsf(iV, 7) and P(M, 7) are not functions of the time. 4. HIGHER-ORDER
when
SOLUTIONS
An even faster convergence of the series (19) and (20), the time integral contained is q-times integrated by parts with respect to time. Then (19) can be written in the form
+
I
&CM)
V
dV ““p:T~~ O) 1
(-l)q+l Texp I
+ /$c9+1,
k(N) aq+YV, 7) $f W) -
(&)
[I
0
+
$*(M)
V
a++1
s
A (W
[W*,(Whl a +Bw)
1I
aq+lp(~, 7) dV dr aTq+l
(32)
where
(33) is called according to &er [ 111 pseudo-steady solution of order m. It can easily be shown, that the expression (33) is a solution (when m # 0) of the differential equation w(M)
aTm--1 WV7) = div [k(M) grad T,(M, T)] -p(M)T,(M, a7
T)
(34)
at the boundary condition A
(N)
dTmW97) +B(N)T,(N, an
7) = 0.
(35)
And indeed, after multiplying (34) by I&((M) and subtracting from it equation (4) multiplied by T, (M, T) the result obtained is integrated using the finite integral transform(ll),and,asin(15),weget
df’m-,,i (7) +/LfTm,i(T)= 0. dr
(36)
584
M. D. MIKHAILOV
Taking into account expression
(25), the equation (36) can be written in the form
(37) After substituting (37) in the inversion formula (12), Therefore the pseudo-steady m-order solution can the system (34)-(35). The solution (20), valid at second-order boundary parts the time integral contained q-times, can be written T(M,T)
=
l
s
w(WdV
{IwWlf,(W
d’V+
v
f[f 0
s
we come to the expression (33). be obtained by solving directly condition, after integrating by in the form:
k(N)fCNT T,d&s A(N)
V
1I
P(M,T)
+
dV dr + i
V
X
Tg(M,r)
+i
m=o
w(M)$~(ll4)fo(M)
G&i(M)
exp (-/&)
i=l
dI’-
i
O”
k(N) aY(N, 0) &(N) a
arm
V +
IV JI,t”)
1 a*+!fW,7) h(N) [I 1I
A(N)
awm 0) dV ap
k(N)
0
a++'
s
ds
A(N)
aq+iP(MTT) dv & a+ V
(381
is the pseudo-steady solution of order m at second-order boundary conditions. It can be shown, as for (37), that the expression (39) is a solution to the ditferentia equation w(M)
aTiLl (MT4 = div [k(M) grad T$(M, T)] a7
(401
at the conditions
aT7nwY7) = an
0
’
J
w(M) T, (M, T) dI’ = 0.
V
(41:
585
General solutions of the heat equation in finite regions
Therefore the pseudo-steady solution T:(M, 7) can be found by solving directly equation (40) at the conditions (4 1). The solution (32) and (38) are particularly convenient whenf(N, 7) and P(M, T)can be presented as q-order polynomials of the time. 5. EXPONENTIAL
SOURCE
FUNCTIONS
f(N, T) AND
P(M,T)
In this paragraph is studied the case
fW, 7) =fW) exp (-44 Introducing TM,
these expressions
P(it4,~)= P(M) exp(-&).
and
(42)
into the solution (19), we get
7) = Tf(M) exp (--d17) + T,(M)
exp (-&)
+ i
G&(M)
exp (-_cL~T)
i=l
(43)
and Tp (W = i
G&f (W &
1
i=l
2
j- $i (WP
W) dV.
(45)
V
It can easily be shown that the expression (44) is a solution to the differential equation div [k(M)grad T,(M)]+d,w(M)T,(M) = 0 (46) at the boundary condition (47) Analogically
the expression
(45) represents
the solution to the differential equation
div[k(M)grad
T,(M)]+d,w(M)T,(M)+P(M)=
0
(48)
at the boundary condition
A(Np$p
+B(N)T,(N)=O.
(49)
Therefore T,(M) and T,(M) canbe obtained by solving directly the systems (46)(47) and (48)-(49) respectively.
586
M. D. MIKHAILOV
Introducing the expressions boundary condition, we get:
(42) into the solution
T* (M, T) = Tf* (M) exp (-&-)
+ Tjf (M) exp (-&-)
1
+ s 1’
X
dI’+’
w(M)&(M)
1
dz
j-P(M) v
I s
AtW
dv}
G9dW exp (-&)
+ i
w(M) dV
k(N)fod,S
-expd(-dl’)
c
+ 1 - exp (-L&T)
(20), valid at second-order
{
i=l
1wUW+dMl.fW)
dV
V
(50) where
and
TXW =
5 Ci6iW)& j”WWYW i
i=l
It can be shown that the expression
k(N)A
(52)
dV.
V
flN)dS (IV)
(5 1) is a solution to the differential equation
= div [k(M)
grad Tf (M)] +d,w(M)TT(M)
(53)
at the boundary conditions
aT,*W) =- f(N) an A(N)
and
I
w(M)TF(M)dV=O.
(54)
V
Analogically
w(M)
I
w(M)
dV
the expression
(52) represents
the solution to the differential equation
s
P(M)dV=div[k(M)gradT$(M)]+d,w(M)T,*(M)+P(M)
(55)
,,
W
at the boundary conditions
amw = an
0
and
I
w(M)T$(M)
dV=
0.
(56)
V
Therefore T;” (M) and Tg (M) (53)-(54) and (55)-56) respectively.
can be obtained
by solving directly
the systems
587
General solutionsof the heat equationin finiteregions 6. ONE-DIMENSIONAL
SOLUTION AT f(N, T) AND OF THE TIME
P(M, T) INDEPENDENT
As an application of the general theory consider the one-dimensional by the differential equation
$[k(x)aTt;“I +P(x),
aT(x, 7) = ar
w(x)
s
x0
x
s
case, described
730
Xl,
(57)
at the conditions T(x, 0) =fo(x) A
aT(xo,T)+BoT(xo,T) = bo; ax
aTbl,T) +BlT(xl,
A
0
(58)
1
7) = bl.
ax
Equations (4) and (5), defining the eigenvalues and the eigenfunctions, _g
[,,x)!%g]
AoWii(X,)
~+aoJlf(xo)
+PfW(X)+i(x)
A
= 0;
=
(59)
get the form
(60)
O
d&Cd
l~+BlJIi(xl)
=
O-
(61)
Different methods are described in [ 121 and [ 131 for solving the one-dimensional Sturm-Liouville problem. An effective method is given in [ 141. The expression (14) for the case considered will be
The solution (2 1) for the one-dimensional
case becomes 11
T(x, 7) = To(x) + 5 G&(x)
exp (--I.&)
i=l
w(X)~i~~>_h(X) dX
{I
3%
[WibI)/~~l -$,k(xA$4(X*) -AI+&
_ bdccxoj 9iCxO) -
[a++ boml
Ao+Bo
II
+
I
$h(x)p(x)
zo
where the pseudo-steady
I
d-.X
(63)
solution of zero-order $[k(x)
y]
To(x) is obtained by solving directly +2’(x)
=
o
(64)
at the conditions A
dTo(xo)
ok+
BoTo(xo)= bo;
A,-+
BITo(xl)
= bl
(65)
M. D. MlKHAILOV
588
obtaining
The solution (26), valid for B0 = B1 = 0, for the one-dimensional
case becomes
Xl
T" (x,
7) = T:(x)
+
1
w(x)fo(x)
xl
I
W(X) dx
(I
d.x+r
k(x,)$kk)~
33
X0
+
j!
P(X)
h]}+C
G&i(x)
exp
C-/&l{
7
i=l
X0
w(xI~~~(xMI(x)
b-
X0 Xl
-;
.[
k(x,M, y-
where the pseudo-steady by solving directly Xl
w(x)
s w(x)
k(x,)b,y+
1
I XII
solution of zero-order
k(xJ +- l
k(xo)++
d.x
0
+,(x)P(x)
dx
(67)
boundary conditions
j: P(x)
T$ (x)
dx] = $[k(x)F]
is obtained
+P(x)
(68)
x0
at the conditions (69) obtained TJW=,,
{[k(x+k(xo)2+ji
’ I 4x1
XB
dx
P(x) X0
b]
j&(
f w(x) to
d--+x
General solutions of the heat equation in finite regions
589
7. CONCLUSIONS
The problem treated by &er[2,3] becomes a special case of the general problem studied here. To obtain the Olcer’s solutions it is sufficient to let: cp(7) = 1, w(M) = 1, k(M) = 1, P(T) = 0, p(M) = 0, M = P, T = kt, P(M, T) = [Q(P, t)/k]. It is evident, that by use of the results obtained here, more complicated, compared with conductive heat transfer cases, problems can be resolved. It may be mentioned, in this connection, that the one-dimensional problem treated in [ 15- 191 are very special cases of the solutions (66) or (67).
NOMENCLATURE
P(& 7) s T(M, 7) T* (M, 7) To(M, 7)
boundary coefficient functions defined on S constant boundary coefficients on x = x0 constant boundary coefficients on x = x1 constant coefficients in equations (42) initial temperature distribution function in V one-dimensional initial temperature distribution function inx source function on S source function on x = x0 source function on x = x1 coefficient defined in equation (14) or (62) 1,2,3 ,..., ~0 0, 1,2, . . :, q a prescribed function defined in I/ a prescribed function defined in x point in V point on S outward normal of S internal heat source function per unit time per unit volume OfV internal heat source function per unit time per unit volume of one-dimensional region boundary of V unsteady temperature distributions defined in equations (l), (2) and (3) unsteady temperature distributions defined in equation (20) pseudo-steady temperature distributions of order zero, defined in equations (23) and (24)
590
M. D. MIKHAILOV
T8 (M, 7) pseudo-steady
(
),
Sw(M)+i(W(
temperature distributions of order zero, defined in equations (30), (3 1) and (29) pseudo-steady temperature distributions of order -m, defined in equations (34) and (3 5) T: CM,7) pseudo-steady temperature distributions of order m, defined in equations (40) and (4 1) T,(M) temperature distributions defined in equations (46) and (47) T,(M) temperature distributions defined in equations (48) and (49) T,(M) temperature distributions defined in equations (53) and (54) T,(M) temperature distributions defined in equations (55) and (56) X co-ordinate 7 time variable w(M) a prescribed function defined in V w(x) a prescribed function defined in x P(T) a prescribed function defined in T a prescribed function defined in I/ P(M) div ( ) divergence operator in M - space gradient vector in M - space grad ( ) finite integral transform of ( ) ) dv eigenfunctions in M - space 9i (M) eigenfunctions in x - space h(X) eigenvalues Pi
REFERENCES [ 1] J. FOURIER, Theo& analityque de la chaleur, Paris (1822). 121 N. Y. GLCER, Int. .I. Heat Mass Transfer 7,307 (1964). [3] N. Y. GLCER, Int. J. Heat Muss Transfer 8,529 (1965). [4] N. Y. QLCER, Q. appl. Math. 24,380 (1967). [5] N. Y. OLCER, Br. J. appl. Phys. 18,89 (1967). [6] N. Y. GLCERand J. E. SUNDERLAND, Nuclr. EngngDes. 8,201(1968). [7] N. Y. OLCER,J. Heat Transfer, 45 (1969). [8] M. D. MIKHAILOV, Forsch. Ing.-Wes. 36,5 (1970). [9] M. D. MIKHAILOV, Problem of Heat and Mass Transfer, pp. 101-107. Energiya, Moscow (1970). [lOI N. YE:,KOCHIN, Vector Calculus and Principles of Tensor Calculus, p. 178. Moscow (1961). [I I] N. Y. OLCER, D. MILLER and J. E. SUNDERLAND, Chemical Engineering Progress Symposium Series, Number 59 61, 12 (1965). [ 121 E. KAMKE, Difierentialgleichungen, Leipzig: Akadem. Verl.-Ges. Geest & Portig (1959). [ 131 L. KOLLATZ, Eigenwertaufgaben mit Technischen Anwendungen, 2. Au8 Leipzig: Akadem. Verl.Ges. Geest & Portig (1963). [14] V.J. BERRYANDC.R.PRIMA,J.~~~~. Phys.23,195(1952). [I51 A. P. HATTON and A. QUARMBY, Int. J. Heat Mass Transfer 5,973 (1962). [ 161 F. P. FORABOSCHI and I. D. FEDERICO, Int. J. Heat Mass Transfer 7,3 15 (1964). cl71 I. MICHIYOSCHI and R. MATSUMOTO, Int. J. Heat Muss Transfer’l, 101 (1964). [ 181 J. SESTAK and F. RIEGER, Inr. J. Heat Mass Transfer 12,7 1 (1969). [19] SUN-NAN HONG and J. C. MATTHEWS, Int. J. Heat Muss Transfer 12,1699 (1969). (Received
1 September
197 1)
RCsumC-On obtient une solution analytique pour un grand nombre de problbmes de transfer& de chaleur en utilisant une transformation indgrale finie. La solution se presente sous la forme de series infinies et contient des termes.quasi constants et transitoires. 11est demontre que la solution de transfert de chaleur par conduction selon Olcer[2,3] constitue un cas special de la solution g&&ale obtenue. On peut aussi appliquer cette demiere lorsque Ton Ctudie les transfer& de chaleur darts les ecoulements laminaires et turbulents de fluides Newtoniens et non-Newtoniens dans les tuyaux et les conduites ainsi que les developpements thermiques a l’entree des canaux MHD.
General solutions of the heat equation in finite regions
591
Zusammenfamung - Eine analytische Liisung ftir eine grosse Klasse von Warmeiibertragungsproblemen wird unter Beniitzung eines endlichen Integraltransforms gefunden. Diese Losung wird in der Fop unendlicher Reihen erhalten und enth%lt quasistationiire und unstationtie Glieder Es wird gezeigt, dass Olcer’s W&meleitungslSsung[2,3] ein Spezialfall der erhahenen alggemeinen Losung ist. Die le.jztere kann such angeReihen erhahen und enWilt quasistetige und fliichtige Terme. Es wird gezeigt, dass Olcer’s leitende Warmeiibertragungsliisung [2,3] ein Spezialfall der erhaltenen allgemeinen Losung ist. Die letztere kann such angewandt werden wenn Wanneiibertragung in laminaren und turbulenten Stromungen Newton’scher und nicht-Newton’scher Fliissigkeiten in Rohren und Leitungen untersucht werden, Temperatureentwicklung im Eintrittsbereich von MHD-Kanalen et al. Sommario- Impiegando una transformazione integrale finita si scopre una soluzione analitica per una grande classe di problemi di trasferimento termico. La soluzione si ottiene in forma di serie infinite e contiene termini quasi uniformi e transienti. Si dimostra the la soluzione de1 transferimento termico conduttivo di Olcer [2,3] P un case speciale delta soluzione generale ottenuta. Quest’ultima pub anche venire applicata quando si studia il trasferimento termico in flusso laminare e turbolento di fluidi newtonianai e non newtoniani in tubi e condutture, lo sviluppo della temperatura nella zona d’ingresso dei canali MHD et al. A~CT~ICT - npwh4eHeHwe aso6pameHur no KOHeYHOMy HHTerpaJly AaeT aHaJlHTWIecKoe pemeHue Ilnfl 6onbmero Knacca npo6neM Tennonepenavw. ~OAyYeHO pt%leHHeBBHAe 6eCKOHe'iHbIX pSIAOB,COAep~alue'2 KBa3WCTaUHOHapHble
'IT0 OJIbCepOBCKOe PeIIIeHHe AJIffTeIIAOIIpOBOApemesria: BOSMO)I(HO A npwMeHwb o6uee ~IlleHHe lTpH W3yWHWH TellJfOIIepeAa'Ui B JlaMHHapHblX Ii Typ6yJIeHTHblX IlOTOKaX HbH)TOHOl?CKUX U H‘?Hb,OTOHOBCKHX lKHAKOCT& B Tpy6ax U KaHaJlaX, npri II3y'ieHHU pa3BHTWi TeMtlepaTypbI B BBOAHOfi o6nacTH MaI'HeTOTUApOAWHaMB‘iCCKHXKaHaJIOB HT. A.
~0C~li
W IIepCXOAHble WIeHbl. nOKa3aH0,
[2, 31 aanaeTca YacTHbrMcnyvaebt nonyreHHor0 o6qero
Sci., 1972, Vol. 10, pp. 577-591.
GENERAL
Pergamon
Press.
Printed
inGreat Britain
SOLUTIONS OF THE HEAT IN FINITE REGIONS
EQUATION
M. D. MIKHAILOV Institute for Mechanical and Electrotechnical Engineering, Sofia, Bulgaria Abstract- Using a finite integral transform an analytical solution is found for a large class of heat transfer problems. This solu$on is obtained in form of infinite series and contains quasi-steady and transient terms. It is shown that the Olcer’s conductive heat transfer solution[2,3] is a special case of the general solution obtained. The latter can also be applied when studying heat transfer in laminar and turbulent flows of Newtonian and non-Newtonian fluids in pipes and ducts, temperature development in the entrance region of MHD channels and elsewhere. 1. INTRODUCTION
the famous fundamental Fourier’s work [ 11, an immense number of papers and books appeared on the analytical heat transfer theory. Digital computers made these complicated solutions accessible for technological applications. The information flow in this field is growing lately more and more. It is sometimes easier to solve and tabulate a problem as a new one rather than to find out whether its solution is published. Therefore it seems quite useful to find a general solution, which could easily permit to obtain numerous special cases with practical application. In [2,3] Olcer presented such general solutions of the unsteady heat conduction in a finite homogeneous region of arbitrary geometry and initial conditions with general boundary conditions, including the boundary conditions of the first, second and third kind (prescribed surface temperature, prescribed heat flux and Newtonian convections) or any combination of these three. This solution was reported again in[4] and used by Olcer to solve a number of particular problems [5,6,7]. Recently the analytical heat transfer theory is concentrating on more complicated, compared with conductive heat transfer cases, problems, namely: heat transfer in laminar and turbulent flows of Newtonian or non-Newtonian fluids in pipes, ducts; heat transfer by Hartmenn’s flow in thermal entrance region etc. In [8] and [9] the author presented a one-dimensional and three-dimensional solution of a broad class of heat transfer problems, including the above-mentioned as special cases. The present paper is a further development of these results in a form, which contains in itself as a special case the results obtained by &er. FOLLOWING
2. STATEMENT
AND
SOLUTION
OF THE PROBLEM
Subject of the present study will be the partial differential equation cp(T)W(M)
aT(g,7,=
div [(k(M) grad T(A4, T)] +
rP(~)ww) -PW)lTef,
7) +pwf,7)
(1)
with the initial condition
TW, 0) =f,(W 577
1J.E.S.
Vol.
IO No. 7-A
(2)
578
M. D. MIKHAILOV
and the general boundary condition (3) The solution of the Sturm-Liouville
problem
div [k(M) grad JI(ZW1-t [P’w(M) A(N)
-_p(M)]JI(M)
a+(N) ,,+Bw)w)
= 0
(4)
=0
is granted for known. Let $$(M) and t&(M) be two eigenfunctions, values pi and pj
corresponding
to two different eigen-
I-&(W+CI,(W
= P(W&(W
-div
[k(M) grad$i(Wl
/-+(WJI,(W
=p(M)ljrj(W
-div
[k(M)
(61
gradWW1
(7:
where Am
+B(N)$i(N)
=O
and
A(fv)T
+B(N)JII(N)
= 0.
(8:
Multiplying (6) by I,!+(M), subtracting from the result the multiplied with e*(M) equation (7) and integrating the result obtained, after taking into account the well. known Gauss Theorem, which transforms the volume integral into a surface one [lo] one gets
From the boundary conditions (8) follows VJfW
GiCN>
&I,(N)
T
aJij(N)
=
O
(10
an
i.e. the eigenfunctions, corresponding to different eigenvalues (pi # pj) are orthogona with a weight w(M) . To solve equation (1) at the conditions (2) and (3) the finite integral transform Fi(T) = J w(M)&(M)T(M,r)
dV
(11
V
is to be used. It follows, from the orthogonal&y of the eigenfunctions expanded into a series as T(M, 7) = I:1 W&W%(W
I,IJ~ (M) , that T (M, 7) can bc (12
General solutions of the heat equation in finite regions
579
where Gil=
j- w(M)g(M)
dV
(13)
V
and (12) is sometimes called the inversion formula for (11). Let us carry out the transition to boundary pI + pj in equations (9). For (13) the following expression is obtained:
G;’ =
& * 1k(N)
(14) JIiow
8
-
an
Multiplying (1) by I&(M), subtracting from the result the multiplied with T(M, T) equation (4) and integrating the result obtained, one gets, as in (9),
(P(T)v+[pf-P(T)]Fi(T) =I k(N) s
+
I $iOW’W,
7) a’.
(1%
V
From (3) and (5) A (N) and B(N) obtained, one gets
are determined.
h,(N) -
After summing up the results
CWiW)/anl
A(N)+B(N)
*
(16)
Substituting the result in (15), for Ti (7) is obtained in ordinary, first order, linear differential equation, which is easily solved, using the transformed according to (11) initial condition (2). Using this solution in the inversion formula (12), the desired solution of the problem is obtained as follows:
+
J
&(M)P(M,T)
b
If p(M) = 0 and B(N)
dl’ d7 . I
I
(17)
= 0 (second order boundary conditions), then p = 0 and and eigenfunction of the Sturm-Liuville problem (4)-
I,IJ~ = const, are also eigenvalue
M. D. MIKHAILOV
580
(5). Therefore, in this case an ~ditional term, co~esponding to the zero-eigenvalue, appears in the solution and (17) takes the form
~li(M)f’(M,
+
7) dV
(18)
s V
Consider further the special cases ~(7) = 1 and p(r) = 0, where equations (17) and ( 18) take the form:
(19)
These expressions (19) and (20), although representing exact solutions, are not in a convenient form for practical purposes. In the case where the time dependent source functions f(N, 7) and P(M, 7) are given by their Taylor series expansion in time or by exponential lotions, the solutions should be booed into sums of quasi steady and transient terms readily applicable for the purpose of nail evaluations.
581
General solutions of the heat equation in finite regions 3. ZERO-ORDER
SOLUTIONS
For faster convergence of the series (19) and (20) the time integral contained integrated by parts with respect to time. Then the solution ( 19) becomes
WWW,
0) dv
is
1 (21)
where
(22) is called according to 6lcer [2] pseudo-steady zero-order solution. It is easy to show, that the expression (22) is a solution to the differential equation div [k(M)gradT,(M,
0
T)]--p(M)To(M,7)+P(M,7)=
(23)
at the boundary condition
(24) And indeed, after multiplying (23) by $i (M) , and subtracting from it equation (4) multiplied by To (M, T) , the result obtained is integrated using the finite integral transform (1 l), and, as in (15), one gets:
Toi = f
s
k(Nlf(N,
TpW) - hWWan1 A WI +Bw)
ds+
I
V
$i(M)P(M,
1
7) dV . (25)
Substituting (25) in the inversion formula (12) we come to the expression (22). Therefore, the pseudo-steady zero-order solution can be obtained by solving directly equation (23) at the boundary condition (24). The solution (20), valid at second-order boundary condition, after integrating by
582
M. D. MIKHAILOV
parts the time integral contained, can be written, as in (2 l), in the form:
T* CM,7) = T,*CM,T) +
2 G&(M)
exp (-p?r){
1 w(M)rj~~(M)fo(M) dV
i=l
V
1
+
s w(WdI’
{I v
w(Wf,(W
k(N)f(N, r) d&Y A(N)
dV+ 0
s
V
+
I
dV
f’(M,T)
V
I
dr
(26
1
where I,*
= 5 GRi(M)$[ i=l
s
(27
V
is a pseudo-steady zero-order solution. For the case discussed p(M) = 0, B(N)
I
dV]
f k(Nlf(N,r)~dS+IP(M,~)~i(M)
w(M)&(M)
dI’=-5
V
= 0 i.e. [a$i (N)/an]
.
1 k(N)y ”
= 0, therefore we ge
dS.
(28
s
Taking into account (28), after multiplying both parts of the expression (27) b w(M) and integrating with the volume, we get the following condition, which is to b met by the pseudo-steady zero-order solution at second-order boundary conditions IV w(M)T$(M,7)
dV=
It can be shown, as in (22), that the expression equation k(N)
f(N, mdS+ 7)
s V
P(M,7)dV
0.
(29
(27) is a solution to the differentia
1
=div[k(M)gradT,*(M,~)l
+f’(M, 7)
(30
at the conditions (29) and (31
Therefore the pseudo-steady zero-order solution TZ (M, T) can be obtained solving directly equation (30) at the conditions (29) and (3 1).
b
583
General solutions of the heat equation in finite regions
The zero-order solutions obtained (21) and (26) are particularly convenient the source functionsf(iV, 7) and P(M, 7) are not functions of the time. 4. HIGHER-ORDER
when
SOLUTIONS
An even faster convergence of the series (19) and (20), the time integral contained is q-times integrated by parts with respect to time. Then (19) can be written in the form
+
I
&CM)
V
dV ““p:T~~ O) 1
(-l)q+l Texp I
+ /$c9+1,
k(N) aq+YV, 7) $f W) -
(&)
[I
0
+
$*(M)
V
a++1
s
A (W
[W*,(Whl a +Bw)
1I
aq+lp(~, 7) dV dr aTq+l
(32)
where
(33) is called according to &er [ 111 pseudo-steady solution of order m. It can easily be shown, that the expression (33) is a solution (when m # 0) of the differential equation w(M)
aTm--1 WV7) = div [k(M) grad T,(M, T)] -p(M)T,(M, a7
T)
(34)
at the boundary condition A
(N)
dTmW97) +B(N)T,(N, an
7) = 0.
(35)
And indeed, after multiplying (34) by I&((M) and subtracting from it equation (4) multiplied by T, (M, T) the result obtained is integrated using the finite integral transform(ll),and,asin(15),weget
df’m-,,i (7) +/LfTm,i(T)= 0. dr
(36)
584
M. D. MIKHAILOV
Taking into account expression
(25), the equation (36) can be written in the form
(37) After substituting (37) in the inversion formula (12), Therefore the pseudo-steady m-order solution can the system (34)-(35). The solution (20), valid at second-order boundary parts the time integral contained q-times, can be written T(M,T)
=
l
s
w(WdV
{IwWlf,(W
d’V+
v
f[f 0
s
we come to the expression (33). be obtained by solving directly condition, after integrating by in the form:
k(N)fCNT T,d&s A(N)
V
1I
P(M,T)
+
dV dr + i
V
X
Tg(M,r)
+i
m=o
w(M)$~(ll4)fo(M)
G&i(M)
exp (-/&)
i=l
dI’-
i
O”
k(N) aY(N, 0) &(N) a
arm
V +
IV JI,t”)
1 a*+!fW,7) h(N) [I 1I
A(N)
awm 0) dV ap
k(N)
0
a++'
s
ds
A(N)
aq+iP(MTT) dv & a+ V
(381
is the pseudo-steady solution of order m at second-order boundary conditions. It can be shown, as for (37), that the expression (39) is a solution to the ditferentia equation w(M)
aTiLl (MT4 = div [k(M) grad T$(M, T)] a7
(401
at the conditions
aT7nwY7) = an
0
’
J
w(M) T, (M, T) dI’ = 0.
V
(41:
585
General solutions of the heat equation in finite regions
Therefore the pseudo-steady solution T:(M, 7) can be found by solving directly equation (40) at the conditions (4 1). The solution (32) and (38) are particularly convenient whenf(N, 7) and P(M, T)can be presented as q-order polynomials of the time. 5. EXPONENTIAL
SOURCE
FUNCTIONS
f(N, T) AND
P(M,T)
In this paragraph is studied the case
fW, 7) =fW) exp (-44 Introducing TM,
these expressions
P(it4,~)= P(M) exp(-&).
and
(42)
into the solution (19), we get
7) = Tf(M) exp (--d17) + T,(M)
exp (-&)
+ i
G&(M)
exp (-_cL~T)
i=l
(43)
and Tp (W = i
G&f (W &
1
i=l
2
j- $i (WP
W) dV.
(45)
V
It can easily be shown that the expression (44) is a solution to the differential equation div [k(M)grad T,(M)]+d,w(M)T,(M) = 0 (46) at the boundary condition (47) Analogically
the expression
(45) represents
the solution to the differential equation
div[k(M)grad
T,(M)]+d,w(M)T,(M)+P(M)=
0
(48)
at the boundary condition
A(Np$p
+B(N)T,(N)=O.
(49)
Therefore T,(M) and T,(M) canbe obtained by solving directly the systems (46)(47) and (48)-(49) respectively.
586
M. D. MIKHAILOV
Introducing the expressions boundary condition, we get:
(42) into the solution
T* (M, T) = Tf* (M) exp (-&-)
+ Tjf (M) exp (-&-)
1
+ s 1’
X
dI’+’
w(M)&(M)
1
dz
j-P(M) v
I s
AtW
dv}
G9dW exp (-&)
+ i
w(M) dV
k(N)fod,S
-expd(-dl’)
c
+ 1 - exp (-L&T)
(20), valid at second-order
{
i=l
1wUW+dMl.fW)
dV
V
(50) where
and
TXW =
5 Ci6iW)& j”WWYW i
i=l
It can be shown that the expression
k(N)A
(52)
dV.
V
flN)dS (IV)
(5 1) is a solution to the differential equation
= div [k(M)
grad Tf (M)] +d,w(M)TT(M)
(53)
at the boundary conditions
aT,*W) =- f(N) an A(N)
and
I
w(M)TF(M)dV=O.
(54)
V
Analogically
w(M)
I
w(M)
dV
the expression
(52) represents
the solution to the differential equation
s
P(M)dV=div[k(M)gradT$(M)]+d,w(M)T,*(M)+P(M)
(55)
,,
W
at the boundary conditions
amw = an
0
and
I
w(M)T$(M)
dV=
0.
(56)
V
Therefore T;” (M) and Tg (M) (53)-(54) and (55)-56) respectively.
can be obtained
by solving directly
the systems
587
General solutionsof the heat equationin finiteregions 6. ONE-DIMENSIONAL
SOLUTION AT f(N, T) AND OF THE TIME
P(M, T) INDEPENDENT
As an application of the general theory consider the one-dimensional by the differential equation
$[k(x)aTt;“I +P(x),
aT(x, 7) = ar
w(x)
s
x0
x
s
case, described
730
Xl,
(57)
at the conditions T(x, 0) =fo(x) A
aT(xo,T)+BoT(xo,T) = bo; ax
aTbl,T) +BlT(xl,
A
0
(58)
1
7) = bl.
ax
Equations (4) and (5), defining the eigenvalues and the eigenfunctions, _g
[,,x)!%g]
AoWii(X,)
~+aoJlf(xo)
+PfW(X)+i(x)
A
= 0;
=
(59)
get the form
(60)
O
d&Cd
l~+BlJIi(xl)
=
O-
(61)
Different methods are described in [ 121 and [ 131 for solving the one-dimensional Sturm-Liouville problem. An effective method is given in [ 141. The expression (14) for the case considered will be
The solution (2 1) for the one-dimensional
case becomes 11
T(x, 7) = To(x) + 5 G&(x)
exp (--I.&)
i=l
w(X)~i~~>_h(X) dX
{I
3%
[WibI)/~~l -$,k(xA$4(X*) -AI+&
_ bdccxoj 9iCxO) -
[a++ boml
Ao+Bo
II
+
I
$h(x)p(x)
zo
where the pseudo-steady
I
d-.X
(63)
solution of zero-order $[k(x)
y]
To(x) is obtained by solving directly +2’(x)
=
o
(64)
at the conditions A
dTo(xo)
ok+
BoTo(xo)= bo;
A,-+
BITo(xl)
= bl
(65)
M. D. MlKHAILOV
588
obtaining
The solution (26), valid for B0 = B1 = 0, for the one-dimensional
case becomes
Xl
T" (x,
7) = T:(x)
+
1
w(x)fo(x)
xl
I
W(X) dx
(I
d.x+r
k(x,)$kk)~
33
X0
+
j!
P(X)
h]}+C
G&i(x)
exp
C-/&l{
7
i=l
X0
w(xI~~~(xMI(x)
b-
X0 Xl
-;
.[
k(x,M, y-
where the pseudo-steady by solving directly Xl
w(x)
s w(x)
k(x,)b,y+
1
I XII
solution of zero-order
k(xJ +- l
k(xo)++
d.x
0
+,(x)P(x)
dx
(67)
boundary conditions
j: P(x)
T$ (x)
dx] = $[k(x)F]
is obtained
+P(x)
(68)
x0
at the conditions (69) obtained TJW=,,
{[k(x+k(xo)2+ji
’ I 4x1
XB
dx
P(x) X0
b]
j&(
f w(x) to
d--+x
General solutions of the heat equation in finite regions
589
7. CONCLUSIONS
The problem treated by &er[2,3] becomes a special case of the general problem studied here. To obtain the Olcer’s solutions it is sufficient to let: cp(7) = 1, w(M) = 1, k(M) = 1, P(T) = 0, p(M) = 0, M = P, T = kt, P(M, T) = [Q(P, t)/k]. It is evident, that by use of the results obtained here, more complicated, compared with conductive heat transfer cases, problems can be resolved. It may be mentioned, in this connection, that the one-dimensional problem treated in [ 15- 191 are very special cases of the solutions (66) or (67).
NOMENCLATURE
P(& 7) s T(M, 7) T* (M, 7) To(M, 7)
boundary coefficient functions defined on S constant boundary coefficients on x = x0 constant boundary coefficients on x = x1 constant coefficients in equations (42) initial temperature distribution function in V one-dimensional initial temperature distribution function inx source function on S source function on x = x0 source function on x = x1 coefficient defined in equation (14) or (62) 1,2,3 ,..., ~0 0, 1,2, . . :, q a prescribed function defined in I/ a prescribed function defined in x point in V point on S outward normal of S internal heat source function per unit time per unit volume OfV internal heat source function per unit time per unit volume of one-dimensional region boundary of V unsteady temperature distributions defined in equations (l), (2) and (3) unsteady temperature distributions defined in equation (20) pseudo-steady temperature distributions of order zero, defined in equations (23) and (24)
590
M. D. MIKHAILOV
T8 (M, 7) pseudo-steady
(
),
Sw(M)+i(W(
temperature distributions of order zero, defined in equations (30), (3 1) and (29) pseudo-steady temperature distributions of order -m, defined in equations (34) and (3 5) T: CM,7) pseudo-steady temperature distributions of order m, defined in equations (40) and (4 1) T,(M) temperature distributions defined in equations (46) and (47) T,(M) temperature distributions defined in equations (48) and (49) T,(M) temperature distributions defined in equations (53) and (54) T,(M) temperature distributions defined in equations (55) and (56) X co-ordinate 7 time variable w(M) a prescribed function defined in V w(x) a prescribed function defined in x P(T) a prescribed function defined in T a prescribed function defined in I/ P(M) div ( ) divergence operator in M - space gradient vector in M - space grad ( ) finite integral transform of ( ) ) dv eigenfunctions in M - space 9i (M) eigenfunctions in x - space h(X) eigenvalues Pi
REFERENCES [ 1] J. FOURIER, Theo& analityque de la chaleur, Paris (1822). 121 N. Y. GLCER, Int. .I. Heat Mass Transfer 7,307 (1964). [3] N. Y. GLCER, Int. J. Heat Muss Transfer 8,529 (1965). [4] N. Y. QLCER, Q. appl. Math. 24,380 (1967). [5] N. Y. OLCER, Br. J. appl. Phys. 18,89 (1967). [6] N. Y. GLCERand J. E. SUNDERLAND, Nuclr. EngngDes. 8,201(1968). [7] N. Y. OLCER,J. Heat Transfer, 45 (1969). [8] M. D. MIKHAILOV, Forsch. Ing.-Wes. 36,5 (1970). [9] M. D. MIKHAILOV, Problem of Heat and Mass Transfer, pp. 101-107. Energiya, Moscow (1970). [lOI N. YE:,KOCHIN, Vector Calculus and Principles of Tensor Calculus, p. 178. Moscow (1961). [I I] N. Y. OLCER, D. MILLER and J. E. SUNDERLAND, Chemical Engineering Progress Symposium Series, Number 59 61, 12 (1965). [ 121 E. KAMKE, Difierentialgleichungen, Leipzig: Akadem. Verl.-Ges. Geest & Portig (1959). [ 131 L. KOLLATZ, Eigenwertaufgaben mit Technischen Anwendungen, 2. Au8 Leipzig: Akadem. Verl.Ges. Geest & Portig (1963). [14] V.J. BERRYANDC.R.PRIMA,J.~~~~. Phys.23,195(1952). [I51 A. P. HATTON and A. QUARMBY, Int. J. Heat Mass Transfer 5,973 (1962). [ 161 F. P. FORABOSCHI and I. D. FEDERICO, Int. J. Heat Mass Transfer 7,3 15 (1964). cl71 I. MICHIYOSCHI and R. MATSUMOTO, Int. J. Heat Muss Transfer’l, 101 (1964). [ 181 J. SESTAK and F. RIEGER, Inr. J. Heat Mass Transfer 12,7 1 (1969). [19] SUN-NAN HONG and J. C. MATTHEWS, Int. J. Heat Muss Transfer 12,1699 (1969). (Received
1 September
197 1)
RCsumC-On obtient une solution analytique pour un grand nombre de problbmes de transfer& de chaleur en utilisant une transformation indgrale finie. La solution se presente sous la forme de series infinies et contient des termes.quasi constants et transitoires. 11est demontre que la solution de transfert de chaleur par conduction selon Olcer[2,3] constitue un cas special de la solution g&&ale obtenue. On peut aussi appliquer cette demiere lorsque Ton Ctudie les transfer& de chaleur darts les ecoulements laminaires et turbulents de fluides Newtoniens et non-Newtoniens dans les tuyaux et les conduites ainsi que les developpements thermiques a l’entree des canaux MHD.
General solutions of the heat equation in finite regions
591
Zusammenfamung - Eine analytische Liisung ftir eine grosse Klasse von Warmeiibertragungsproblemen wird unter Beniitzung eines endlichen Integraltransforms gefunden. Diese Losung wird in der Fop unendlicher Reihen erhalten und enth%lt quasistationiire und unstationtie Glieder Es wird gezeigt, dass Olcer’s W&meleitungslSsung[2,3] ein Spezialfall der erhahenen alggemeinen Losung ist. Die le.jztere kann such angeReihen erhahen und enWilt quasistetige und fliichtige Terme. Es wird gezeigt, dass Olcer’s leitende Warmeiibertragungsliisung [2,3] ein Spezialfall der erhaltenen allgemeinen Losung ist. Die letztere kann such angewandt werden wenn Wanneiibertragung in laminaren und turbulenten Stromungen Newton’scher und nicht-Newton’scher Fliissigkeiten in Rohren und Leitungen untersucht werden, Temperatureentwicklung im Eintrittsbereich von MHD-Kanalen et al. Sommario- Impiegando una transformazione integrale finita si scopre una soluzione analitica per una grande classe di problemi di trasferimento termico. La soluzione si ottiene in forma di serie infinite e contiene termini quasi uniformi e transienti. Si dimostra the la soluzione de1 transferimento termico conduttivo di Olcer [2,3] P un case speciale delta soluzione generale ottenuta. Quest’ultima pub anche venire applicata quando si studia il trasferimento termico in flusso laminare e turbolento di fluidi newtonianai e non newtoniani in tubi e condutture, lo sviluppo della temperatura nella zona d’ingresso dei canali MHD et al. A~CT~ICT - npwh4eHeHwe aso6pameHur no KOHeYHOMy HHTerpaJly AaeT aHaJlHTWIecKoe pemeHue Ilnfl 6onbmero Knacca npo6neM Tennonepenavw. ~OAyYeHO pt%leHHeBBHAe 6eCKOHe'iHbIX pSIAOB,COAep~alue'2 KBa3WCTaUHOHapHble
'IT0 OJIbCepOBCKOe PeIIIeHHe AJIffTeIIAOIIpOBOApemesria: BOSMO)I(HO A npwMeHwb o6uee ~IlleHHe lTpH W3yWHWH TellJfOIIepeAa'Ui B JlaMHHapHblX Ii Typ6yJIeHTHblX IlOTOKaX HbH)TOHOl?CKUX U H‘?Hb,OTOHOBCKHX lKHAKOCT& B Tpy6ax U KaHaJlaX, npri II3y'ieHHU pa3BHTWi TeMtlepaTypbI B BBOAHOfi o6nacTH MaI'HeTOTUApOAWHaMB‘iCCKHXKaHaJIOB HT. A.
~0C~li
W IIepCXOAHble WIeHbl. nOKa3aH0,
[2, 31 aanaeTca YacTHbrMcnyvaebt nonyreHHor0 o6qero