- Email: [email protected]
IMPROVED APPROXIMATE FORMULATIONS FOR ANISOTROPIC HEAT CONDUCTION
Inr. C-.
Pergamon
Heat Mass Tran#er, Vol. 24, No. 6,pp. 869-S78, 1997 Copyright O 1997 ElsevierScienceLtd
PrintedintireUSA.Allrightsresenwd 0735-1933/97$17.00+ .00
PII S0735-1933(97)00073-O
IMPROVEDAPPROXIMATE FORMULATIONSFOR ANISOTROPIC HEAT CONDUCTION
F.M.L. Traiano, R.M. Cotta, and H.R.B. Orlande Mechanical EngineeringDepartment EE/COPPE UniversidadeFederal do Rio de Janeiro CX.Postal 68503- Cidade UniversitAria Rio de Janeiro, W, Brazil 21945-970
(Communicated by J.P. Hartnett and W.J. Minkowycz)
ABSTRACT Approximate lumped-differentialformulations for heat conduction in anisotropic media are
presented,here illustrated for a fsn-typeexample. The proposed formulation is as simple as the classical lumped system analysis, but considerably more accurate, since temperature gradients in the lumped direction are more adequately approximated. Results obtained for an orthotropic firs are compared to the exact two-dimensional solution, as well as to the classical lumped system analysis, for a range of the governing parameters, aspect ratio, thermal conductivity ratios and Biot numbers. Illustrative results are reported for a general anisotropic material. Q 1997ElsevierScienceLtd
Introduction
Several engineeringmaterials such as laminated metal sheets and composites cannot be considered as isotropic, and the dependenceof thermal conductivitywith direction has to be taken into account in the modelingof heat transfer problems [1], Compositesare of special interest to the aerospace industry because of its strength and reduced weight. Similarly, anisotropic materials are now in commonuse in the electronic industry, where the current miniaturization requires a reduction in weight and size and an increasing ability to remove the high heat density generated by the components. Exact solutions obtained by using different analytical methods such as the Classical Integral Transform Technique [1], are available in the literature for heat conductionproblems in orthotropic media. Althoughexact, such solutions may require the computation of many eigenvahresand eigenfonctionsin order to obtain the convergenceof the series solutions within a 869
870
F.M.L. Traiano, R.M. Cotta and H.R.B. Orlande
Vol. 24, No. 6
prescribed degree of accuracy. Analytical solutions are also available for some limited situations of heat conduction in general anisotropic media [1,2]. However, problems involving general aniaotropic ilnite regions are specially difficult to be solved analytically. Dit&ent numerical methods have been used fir the solution of either orthotropic or anisotropic materials such as finite-differences [3,4] and finite elements [5]. In this note, we apply the Coupled Jntegral Equations Approach (CIEA) of refs. [6-7] to obtain improved lumped-differentialfbnmdations for the heat conduction problem in orthotropic aud anisotropic two-dimensional media, here illustrated fix a typical fm problem. In such approach, one of the space variables is eliminatedby integration of the energy equation in the direction where the temperature gradient is expected to be less steep. Hermite’s integration formulae are utilized to approximate the integrals involvingthe temperature and its derivative in the direction where the lumping is performed. The present approach is found to be much simpler and computationdy faster than other more involved approaches, while maintaininga reasonable degree of accuracy, sufficient for most practical purposes.
Analysis The steady-state two-dimensional heat conduction problem in longitudinal anisotropic fins of rectangular profile can he described in dimensionlessfirm by:
$fyx,y,)+C: $O(x>y)~~; $e~x,y~ .0, in0
●
a(l,r)
O(O,Y)=1,
d72a
●9(1,Y)—
+ BilO(l,Y) =O
(1.a)
(1.b,c)
(1.d,e)
where the fm geometry is depicted in Figure 1, and the followingdimensionlessvariables are detined:
x .5; a
Y.;;
E12=~; kll
e=~;
K =: ; C; =K2E22; C: = 2KE12;
(2.a-fJ
E22=~;
Bil =w; kll
Bi2 =~ k22
(3.ad)
~–Tm
kll
where a and b are the fin dimensionsin the x and y directions, respectively, Tz is the ambient temperature and Tb is the uniformtemperature at the fm base. ‘Ihe heat transfer coefficients at the lateral surface and at the tip of the fin are h~ and h2, respectively, and the thermal conductivitytensor is given as in [1].
Vol. 24,No. 6
ANISOTROPICHEAT CONDUCTION
‘l--r%
n
b
871
At M
\\\\\\\\\\\\’m
x
FIG. 1 Geometry and coordinates system for anisotropic heat conductionm a fin-type application.
An exact solution to the problem given by equations (1) is not readily available. To obtain an approximate soluticm,we use the fact that transversal temperature gradients are in general much smaller than longitudinaltemperature gradients for fin applications. We then integrate eq.(1.a) in the Indirection:
Y=l d2%(x)
~&
dx2
Y=l 2% +C2—
a’ y=~
~
(4)
=0 y=o
where r9m(X) denotesthe transversally averaged temperature. Substitutingeqs.(1.b, c) into w.(4):
2 1c; ae(x,o)1c; atyx,l)–~i2*(x~, =o d2r9m —+ Cp[e(x,l) –e(x,o)]+c1–~—––————— > ~2 2 Cj z 1 [ 2 c1 ix
(5a)
The boundary conditionsto solve equation (5a) are obtained from the integrationof eqs.(1.dj e) in 1’:
em(o). I;
●
de (1) ~[e(I,l)* 1
e(I,o)]+ Bil%(l} =0
(5.b,c)
In the classical lumpingapproach i% the reformulation of equations (5), the temperature gradients in the 1’direction are considered sufficiently smooth so that:
e(x,l) z WX80)z em(x); Then, equations (5) reduceto:
ae(x,l) 6wx,0)
—————— =— a’
6X
(6.a,b)
872
F.M.L. Traiano, R.M. Cotta and H.R.B. Orlande
d2Qm(x) -c~Bi2em(x)=0‘ ~2
em(o)=1;
V
0
Vol. 24, No. 6
1
(7a)
+BilQm(l) .0
(7.b,c)
The solutionof equations(7)canbe readily obtained as
em(x) = cosh(c,@x)-
C,@sinh(Cl@)
+Bilcosh(Cl~)
C1&cosh(C’l&)
+Bilsinh(C1fi)
sinh(C1@X)
(8)
The dimensionlessheat transtlx rate at the fin base is then obtained from:
We note that the dimensionless variable C2, defined by equation (2.fJ which gives the anisctropic characteristic of the fm, does not appear m the classical lumped formulation given by equations (7). In fact such forrmdatioo is identical to the me obtained with the classical hnnping approach as applied to orthotropic fins (C2=O).Therefore, it is inappropriate to be used for general anisotrcpic fins. In the approximate formulations here proposed, the transversal gradients are taken into account more accurately by using Hermite’s two-point integmtion formulae to evaluate the average temperature and derivative in the 1’ direction, following the Coupled Integral Equations Approach of [6-7]. Using Hermite’s formula of zeroth-orderto approximate such averaged quantities, the followingtwo coupled equations result:
Jo
k(X,Y)dY = em(x)=#e(x31) +e(x,o)l--+(xw
l~(x, y)dy=Q(X,l)– Wx.o)=-j ix ● iiwx,o) 1 %(X,1) JOa
[
(% 1
+ +7’’(X, q)
(11.a)
(11.b)
where primes denote derivatives in the Ydirection and ~ and q are located at some point between 0 and 1, in the error terms above. By neglectingthe error terms above and by subatitutiog boundary conditions (lb, c) into equations (11), we obtain, after some manipulations:
vol. 24, No. 6
ANISOTROPICHEAT CONDUCTION
$(X,1) = -+
Dtyx,l) 1 —.— ix +
-+$~:x) 4c~
em(x)
1++ [ dt&(X) ——. [ dx
873
(12.a)
1
C; d2QJX) 4C; dx2 1
a(x,o). 2dOw(X) tyx,o). 2ew(x)-O(x,I), ~ ax-z
(12,b)
%(X,1)
(12.c,d)
Equations (12) are substituted into equations (5) to obtain the improvedapproximate formulation:
(13.a)
em(o)=1,
(13.b,c)
(14.a,b)
The reader should note that the improved formulation given by equations (13) is very similar to the classical fin solution given by equations (7). Therefore, the solution of the proposed formulation is obtained from equation (8) by substituting Bil and Bi2 by Bi~ and Bi~, respectively. Similarly, the heat transfer rate at the fin base is obtained from equation (10) by usrng the same substitution. Such expressions for the average temperature and heat transfer rate maintain the simplicity of the classical tin solution but are considerablymore accurate, since the transversal gradients are approximated more appropriately than in the classical lumping approach. Furthermore, the proposed formulation takes into account the anisotropic effects throughthe modifiedBiot numbers Bi~ and Bi~, which contain the dimensionlessgroup C2, which is not present in the classical lumped formulation. The above approximate formulation is also valid for the special case of orthotropic fins, where the correspondentmodifiedBiot numbers are obtained from eqs. (14) by making C2=0. The formulation given by eqs. (13), where Hermite’s zeroth-order formula is used to approximate the average temperature, as well as the average temperature gradient in the Ydirection, is here denotedthe Ho,o/Ho,o approximation. Further improved formulations can also be obtained by using more accurate approximationsto the average temperature and gradient along the Ydirection. Such an approach is illustrated here for the case involvingan orthotropic fin (C2=O), by using the first-order Herrnite’sformula (corrected trapezoidal rule) to approximate the integral involvingthe temperature, while using the zeroth-
874
F.M.L. Traiano, R.M. Cotta and H.R.B. Orlande
Vol. 24, No. 6
order Hermite’s formula (trapezoidal rule) to approximate the integral involvingthe temperature gradient. Such approximationstake the form:
J:e(x,Y)dY =em(x)= ;[e(x,l)+e(x,o)]+ ; [7 I&(x,
I
o
a’
y)& .
-@&
1
+J#vx>c)
1+-
(7(X,])– @(xvo)=-j a 1 —— 69(X,1) ● C9tyx,o) +“(x, q) n
[
(15.a)
(15.b)
and is here denotedthe Hl,l/HoO approximation. We solve equations (15) fir 8(X,Z) by neglecting the error terms and by using boundary conditions(1.b, c) with C2=0. After some manipulations, we obtain:
(16)
O(X,l) == +
By substitutingequation (16) into equations (5) aod making C2=0, we obtain the following higher order improvedformulation for orthotropic fins:
#em(x) &2
em(o)= I,
-
c~Bj~*@m(x) =0‘
in O
(17.a)
d@m(l)+Bjl e.,(l) ‘0 dx
(17.b,c) (18)
The solution of problem (17) is also obtained from the classical tin solution, equation (8), by substituting B12for Bij; . The use of the more accurate fuat-order Hermite’sformula to approximate both integrals involvingthe average temperature and gradient in general anisctropic fins, yields more involved approximate formulations. Althoughmore accurate, such formulations loose the simplicity inherent to those presented above, where the solution can be obtained directly from the classical fin solution, equation (8), by substituting the modified Biot numbers. Therefore, such formulations are avoided within the present scope. Table 1 summarizes the Biot numbers to be used in equations (8) and (10), in order to obtain the average temperature and the heat transfkr rate at the fin base, fkomthe dif%rentapproximate fh-rnulationsabove.
vol. 24, No. 6
ANISOTROPICHEAT CONDUCTION
875
TABLE 1 Definitionof Biot numbers to be used in the computationof the average temperature fkomequation (8) and the heat transfer rate from equatior Substitute Bi2 for Substitute Bil for Formulation Classical
t-- Lumped Analysis a
k
Bi2 - C~Bi2 — +Bil 1+— () 4 4
H@HO,O b
Hl,[email protected]
Bi2
Bil
~+ Bi2 d 4 4cj Bi2 Bi]
c
l+%
a. Theclassicalformulation forgeneralanisotropic andorthotropic fmsareidentical. Thus,theuseofthis formulation forgenerslsniwtropic fmsisinappropriate. b. Validforbothortbotmuic andgeneralanisotrouic furs(C9=0intheBiotnurnberexpressions fororthotropic fins). c. Derived hereforo~otropicks only. “ -
The exact solutionto the heat conductionproblemin orthotropicfins, which is given by equations(1) by making C2 = 0, is readily obtained by the classical integral transform methodartd is given by:
sinh$mcosh(&Y) fl~ +Bi~ O(X,Y) =22 ~ *=1[ pm+Bi2(Bi2+1) pm ~mC1sinh(&C1) +Bil cosh(ACx) sinh(pmclx) cOsh(pmclx) Bil sinh($mC1) +$mClcosh($mC1) )] (
(19)
where the eigenvalues & are the positive roots of & tan/3m= Bi2. The exact average temperature in the Ydirection is then obtained from the integration of equation (19), which gives:
fl~+Bi~ sinh2fim e.(x)=2f ~ — *=1flm [ +Bi~+Bi2 /3~ ~osh(flmclx)[
&Clsinh(%CI)+Bil cosh(p#l)sinh(pmclx) Bil sinh(&C1) +&C1cosh(~mC1) 1]
(20)
The exact heat transfer rate at the fm base is obtainedby substituting equation (20) into equation (9):
Cl sinh2/lm&C1sinh(/lmC1)+Bilcosh(AjCl) ~~ +Bi~ ‘0‘~~1p~+Bi2(Bi2 +]) & Bil sinh(/3mC1) +~mC1cosh(~mel)1 [
(21)
F.M.L. Traiano, R.M. Cotta and H.R.B. Orlande
Vol. 24, No. 6
l’he exact average temperature given by equation (20) and the exact heat transfer rate given by equation (21) for orthotropic tins, are next used in the comparison widr the approximate formulations developedpreviously irrthis section.
Results and Discussion
Heat conduction in odmtropic materials is considered first, since the exact twodimensiooal solution is readily available for comparison purposes, equations (19-21). Table 2 lists six test-cases selected for the analysis, with different combinations of the governingparameters, E22, K, Bil and Bi2, withinrangesof practical interest.Figures 2a-c show the transversallyaveragedtemperature distributions, O@(X), akmg
the longitudinal coordinate, X, as computed from the exact two-dimensional solution and from the three luwed~rential
formulationsdeveloped, namely, the
classical lumped system analysis,
Ho,&o.oad
Hl,l/HoO approximations. For cases 1 and 2 (Fig. 2a), as expected, due to the relmively smooth transversal temperature gradients, all three approximate fixrmrlations offer a reasonably good agreement with the filly differential formulation. As the Biot numbers are increased in cases 3 rmd 4 (Fig. 2.b), the classical lumping approach results start deviating from the exact solution, while the improved lumpeddifferential formulations remain quite reasonable. It carr also be observed that case 4, with larger value of E22 and, therefore, predominance of the transversal thermal conductivity coefficient, allows fix a better behavior of the classical lumped system analysis, with respect to case 3. Cases 5 and 6 (Fig. 2.c) combine the influenceof increasingthe thermal conductivitycoefficients ratio and Biot numbers. Specially in case 6, though with Biot number values not typical of fin-type applications, one can observe the marked deviations of the classical lumped solution from the exact one, but a still reasonably accurate prediction by the Hl,l~,O
formulation,which retains the same degree of mathematical simplicity as the classical approach.
TABLE 2 Sel
erial
Figures 3,a, b present the heat transfer rates at the tin’s base (X = O),as a frmction of E22, and for different values of Biot numbers (Fig. 3a), or aspect ratio (Fig. 3.b). Fig. 3.a indicates the increase in heat transfix rates for increasing values of E22. As the transversal thermal conductivity coefficient predominates, E22 > 1, a heat transfer enhancement is achievable with respect to the isctropic situation
Vol. 24, No. 6
877
ANISOTROPICHEAT CONDUCTION
(E22 = 1). Fig. 3.b shows the saturation of heat transfer enhancementas the aspect ratio is increa~ specially for materials with E22>1, when Khas little ef%ct on the heat transfer rates.
1.0
m WE
0.8
0,6
ass
0.4
2
1
IRk___J 0.0
0.2
0.4
x
0.6
0.8
1.0
FIG. 2.a Averagedtemperature distributions fix orthotropic material (cases 1 and 2, table 2).
FIGS. 2.b. C Same as above for (b)cases 3 and 4, table 2;and (c)cases 5 and 6, table 2.
?o~
m
- 0.1
1-
00
10
0.1 t
FIGS. 3a, b Heat transfer rate versus thermal conductivityratio (orthotropic); (a)difTereotBiz, (b)difl%mt K.
and
878
F.M.L. Traiano, R.M. Cotta and H.R.B. Orlande
Vol. 24, No. 6
Table 3 lists the fbur cases selected of general anisotropic heat conducticm,for differeut values of the governingparameters, namely, E22, E12, K, Bil and Bi2. Cases 1 and 2 (Fig. 4a) recover the orthotropic situation, horn the very low value of E12 considered,with similar conclusionas above. Cases 3 and 4 (Fig. 4.b) introduct the general anisotropy ei%ct, and dif%rent Biot numbers. It should be noticed that for both cases, the two approximate formulations are markedly differe@ which represents the importance of the transversal temperature gradients, now also directly affected by the longitudinalgradients, that composethe transversal heat flux componentexpression, not accounted for in the classical lumped system analysis.
Case
TABLE 3 Selectedcases for heat conductionin general anisctropic material. Bil K E22 E12 10-10 10-10
1
3 4
~ 0. 4
0.6 0.6
6.0
6.0 6.0 6.0
4 0.2 4 2. 0.24 2.4
Bi2
0.10 1.0 0.10
1.0
FIGS. 4a, b Temperature distributions for general anisotropic material (a)cases 1 and 2 (b)cases 3 and 4, table 3.
1. M.N. Ozisik, “HeatConduction”, John Wiley, New York (1980).
2. S.C. Huang and Y.P. Chang,.1 Heat Transfer,V. 106,646-648 (1984). 3. M.I. Flik and C.L. Tim, J HeatTransfer,V. 112, 10-15(1990). 4. J.C. McWhorter and M.H. Sad~ J Heat Transfer,V. 102,308-311 (1980). 5. A. Aziz,, Heat TransferEng., V. 14,63-70 (1993). 6. J.B. Aparecidoand R.M. Cotta,Heat Tran.f Eng., V. 11, no. 1,49-59 (1989). 7. F. Swt%noNetoaud R.M. Cotta, J HeatTramfer,V. 115,921-927 (1993).
ReceivedJanuary 28, 1997
Pergamon
Heat Mass Tran#er, Vol. 24, No. 6,pp. 869-S78, 1997 Copyright O 1997 ElsevierScienceLtd
PrintedintireUSA.Allrightsresenwd 0735-1933/97$17.00+ .00
PII S0735-1933(97)00073-O
IMPROVEDAPPROXIMATE FORMULATIONSFOR ANISOTROPIC HEAT CONDUCTION
F.M.L. Traiano, R.M. Cotta, and H.R.B. Orlande Mechanical EngineeringDepartment EE/COPPE UniversidadeFederal do Rio de Janeiro CX.Postal 68503- Cidade UniversitAria Rio de Janeiro, W, Brazil 21945-970
(Communicated by J.P. Hartnett and W.J. Minkowycz)
ABSTRACT Approximate lumped-differentialformulations for heat conduction in anisotropic media are
presented,here illustrated for a fsn-typeexample. The proposed formulation is as simple as the classical lumped system analysis, but considerably more accurate, since temperature gradients in the lumped direction are more adequately approximated. Results obtained for an orthotropic firs are compared to the exact two-dimensional solution, as well as to the classical lumped system analysis, for a range of the governing parameters, aspect ratio, thermal conductivity ratios and Biot numbers. Illustrative results are reported for a general anisotropic material. Q 1997ElsevierScienceLtd
Introduction
Several engineeringmaterials such as laminated metal sheets and composites cannot be considered as isotropic, and the dependenceof thermal conductivitywith direction has to be taken into account in the modelingof heat transfer problems [1], Compositesare of special interest to the aerospace industry because of its strength and reduced weight. Similarly, anisotropic materials are now in commonuse in the electronic industry, where the current miniaturization requires a reduction in weight and size and an increasing ability to remove the high heat density generated by the components. Exact solutions obtained by using different analytical methods such as the Classical Integral Transform Technique [1], are available in the literature for heat conductionproblems in orthotropic media. Althoughexact, such solutions may require the computation of many eigenvahresand eigenfonctionsin order to obtain the convergenceof the series solutions within a 869
870
F.M.L. Traiano, R.M. Cotta and H.R.B. Orlande
Vol. 24, No. 6
prescribed degree of accuracy. Analytical solutions are also available for some limited situations of heat conduction in general anisotropic media [1,2]. However, problems involving general aniaotropic ilnite regions are specially difficult to be solved analytically. Dit&ent numerical methods have been used fir the solution of either orthotropic or anisotropic materials such as finite-differences [3,4] and finite elements [5]. In this note, we apply the Coupled Jntegral Equations Approach (CIEA) of refs. [6-7] to obtain improved lumped-differentialfbnmdations for the heat conduction problem in orthotropic aud anisotropic two-dimensional media, here illustrated fix a typical fm problem. In such approach, one of the space variables is eliminatedby integration of the energy equation in the direction where the temperature gradient is expected to be less steep. Hermite’s integration formulae are utilized to approximate the integrals involvingthe temperature and its derivative in the direction where the lumping is performed. The present approach is found to be much simpler and computationdy faster than other more involved approaches, while maintaininga reasonable degree of accuracy, sufficient for most practical purposes.
Analysis The steady-state two-dimensional heat conduction problem in longitudinal anisotropic fins of rectangular profile can he described in dimensionlessfirm by:
$fyx,y,)+C: $O(x>y)~~; $e~x,y~ .0, in0
●
a(l,r)
O(O,Y)=1,
d72a
●9(1,Y)—
+ BilO(l,Y) =O
(1.a)
(1.b,c)
(1.d,e)
where the fm geometry is depicted in Figure 1, and the followingdimensionlessvariables are detined:
x .5; a
Y.;;
E12=~; kll
e=~;
K =: ; C; =K2E22; C: = 2KE12;
(2.a-fJ
E22=~;
Bil =w; kll
Bi2 =~ k22
(3.ad)
~–Tm
kll
where a and b are the fin dimensionsin the x and y directions, respectively, Tz is the ambient temperature and Tb is the uniformtemperature at the fm base. ‘Ihe heat transfer coefficients at the lateral surface and at the tip of the fin are h~ and h2, respectively, and the thermal conductivitytensor is given as in [1].
Vol. 24,No. 6
ANISOTROPICHEAT CONDUCTION
‘l--r%
n
b
871
At M
\\\\\\\\\\\\’m
x
FIG. 1 Geometry and coordinates system for anisotropic heat conductionm a fin-type application.
An exact solution to the problem given by equations (1) is not readily available. To obtain an approximate soluticm,we use the fact that transversal temperature gradients are in general much smaller than longitudinaltemperature gradients for fin applications. We then integrate eq.(1.a) in the Indirection:
Y=l d2%(x)
~&
dx2
Y=l 2% +C2—
a’ y=~
~
(4)
=0 y=o
where r9m(X) denotesthe transversally averaged temperature. Substitutingeqs.(1.b, c) into w.(4):
2 1c; ae(x,o)1c; atyx,l)–~i2*(x~, =o d2r9m —+ Cp[e(x,l) –e(x,o)]+c1–~—––————— > ~2 2 Cj z 1 [ 2 c1 ix
(5a)
The boundary conditionsto solve equation (5a) are obtained from the integrationof eqs.(1.dj e) in 1’:
em(o). I;
●
de (1) ~[e(I,l)* 1
e(I,o)]+ Bil%(l} =0
(5.b,c)
In the classical lumpingapproach i% the reformulation of equations (5), the temperature gradients in the 1’direction are considered sufficiently smooth so that:
e(x,l) z WX80)z em(x); Then, equations (5) reduceto:
ae(x,l) 6wx,0)
—————— =— a’
6X
(6.a,b)
872
F.M.L. Traiano, R.M. Cotta and H.R.B. Orlande
d2Qm(x) -c~Bi2em(x)=0‘ ~2
em(o)=1;
V
0
Vol. 24, No. 6
1
(7a)
+BilQm(l) .0
(7.b,c)
The solutionof equations(7)canbe readily obtained as
em(x) = cosh(c,@x)-
C,@sinh(Cl@)
+Bilcosh(Cl~)
C1&cosh(C’l&)
+Bilsinh(C1fi)
sinh(C1@X)
(8)
The dimensionlessheat transtlx rate at the fin base is then obtained from:
We note that the dimensionless variable C2, defined by equation (2.fJ which gives the anisctropic characteristic of the fm, does not appear m the classical lumped formulation given by equations (7). In fact such forrmdatioo is identical to the me obtained with the classical hnnping approach as applied to orthotropic fins (C2=O).Therefore, it is inappropriate to be used for general anisotrcpic fins. In the approximate formulations here proposed, the transversal gradients are taken into account more accurately by using Hermite’s two-point integmtion formulae to evaluate the average temperature and derivative in the 1’ direction, following the Coupled Integral Equations Approach of [6-7]. Using Hermite’s formula of zeroth-orderto approximate such averaged quantities, the followingtwo coupled equations result:
Jo
k(X,Y)dY = em(x)=#e(x31) +e(x,o)l--+(xw
l~(x, y)dy=Q(X,l)– Wx.o)=-j ix ● iiwx,o) 1 %(X,1) JOa
[
(% 1
+ +7’’(X, q)
(11.a)
(11.b)
where primes denote derivatives in the Ydirection and ~ and q are located at some point between 0 and 1, in the error terms above. By neglectingthe error terms above and by subatitutiog boundary conditions (lb, c) into equations (11), we obtain, after some manipulations:
vol. 24, No. 6
ANISOTROPICHEAT CONDUCTION
$(X,1) = -+
Dtyx,l) 1 —.— ix +
-+$~:x) 4c~
em(x)
1++ [ dt&(X) ——. [ dx
873
(12.a)
1
C; d2QJX) 4C; dx2 1
a(x,o). 2dOw(X) tyx,o). 2ew(x)-O(x,I), ~ ax-z
(12,b)
%(X,1)
(12.c,d)
Equations (12) are substituted into equations (5) to obtain the improvedapproximate formulation:
(13.a)
em(o)=1,
(13.b,c)
(14.a,b)
The reader should note that the improved formulation given by equations (13) is very similar to the classical fin solution given by equations (7). Therefore, the solution of the proposed formulation is obtained from equation (8) by substituting Bil and Bi2 by Bi~ and Bi~, respectively. Similarly, the heat transfer rate at the fin base is obtained from equation (10) by usrng the same substitution. Such expressions for the average temperature and heat transfer rate maintain the simplicity of the classical tin solution but are considerablymore accurate, since the transversal gradients are approximated more appropriately than in the classical lumping approach. Furthermore, the proposed formulation takes into account the anisotropic effects throughthe modifiedBiot numbers Bi~ and Bi~, which contain the dimensionlessgroup C2, which is not present in the classical lumped formulation. The above approximate formulation is also valid for the special case of orthotropic fins, where the correspondentmodifiedBiot numbers are obtained from eqs. (14) by making C2=0. The formulation given by eqs. (13), where Hermite’s zeroth-order formula is used to approximate the average temperature, as well as the average temperature gradient in the Ydirection, is here denotedthe Ho,o/Ho,o approximation. Further improved formulations can also be obtained by using more accurate approximationsto the average temperature and gradient along the Ydirection. Such an approach is illustrated here for the case involvingan orthotropic fin (C2=O), by using the first-order Herrnite’sformula (corrected trapezoidal rule) to approximate the integral involvingthe temperature, while using the zeroth-
874
F.M.L. Traiano, R.M. Cotta and H.R.B. Orlande
Vol. 24, No. 6
order Hermite’s formula (trapezoidal rule) to approximate the integral involvingthe temperature gradient. Such approximationstake the form:
J:e(x,Y)dY =em(x)= ;[e(x,l)+e(x,o)]+ ; [7 I&(x,
I
o
a’
y)& .
-@&
1
+J#vx>c)
1+-
(7(X,])– @(xvo)=-j a 1 —— 69(X,1) ● C9tyx,o) +“(x, q) n
[
(15.a)
(15.b)
and is here denotedthe Hl,l/HoO approximation. We solve equations (15) fir 8(X,Z) by neglecting the error terms and by using boundary conditions(1.b, c) with C2=0. After some manipulations, we obtain:
(16)
O(X,l) == +
By substitutingequation (16) into equations (5) aod making C2=0, we obtain the following higher order improvedformulation for orthotropic fins:
#em(x) &2
em(o)= I,
-
c~Bj~*@m(x) =0‘
in O
(17.a)
d@m(l)+Bjl e.,(l) ‘0 dx
(17.b,c) (18)
The solution of problem (17) is also obtained from the classical tin solution, equation (8), by substituting B12for Bij; . The use of the more accurate fuat-order Hermite’sformula to approximate both integrals involvingthe average temperature and gradient in general anisctropic fins, yields more involved approximate formulations. Althoughmore accurate, such formulations loose the simplicity inherent to those presented above, where the solution can be obtained directly from the classical fin solution, equation (8), by substituting the modified Biot numbers. Therefore, such formulations are avoided within the present scope. Table 1 summarizes the Biot numbers to be used in equations (8) and (10), in order to obtain the average temperature and the heat transfkr rate at the fin base, fkomthe dif%rentapproximate fh-rnulationsabove.
vol. 24, No. 6
ANISOTROPICHEAT CONDUCTION
875
TABLE 1 Definitionof Biot numbers to be used in the computationof the average temperature fkomequation (8) and the heat transfer rate from equatior Substitute Bi2 for Substitute Bil for Formulation Classical
t-- Lumped Analysis a
k
Bi2 - C~Bi2 — +Bil 1+— () 4 4
H@HO,O b
Hl,[email protected]
Bi2
Bil
~+ Bi2 d 4 4cj Bi2 Bi]
c
l+%
a. Theclassicalformulation forgeneralanisotropic andorthotropic fmsareidentical. Thus,theuseofthis formulation forgenerslsniwtropic fmsisinappropriate. b. Validforbothortbotmuic andgeneralanisotrouic furs(C9=0intheBiotnurnberexpressions fororthotropic fins). c. Derived hereforo~otropicks only. “ -
The exact solutionto the heat conductionproblemin orthotropicfins, which is given by equations(1) by making C2 = 0, is readily obtained by the classical integral transform methodartd is given by:
sinh$mcosh(&Y) fl~ +Bi~ O(X,Y) =22 ~ *=1[ pm+Bi2(Bi2+1) pm ~mC1sinh(&C1) +Bil cosh(ACx) sinh(pmclx) cOsh(pmclx) Bil sinh($mC1) +$mClcosh($mC1) )] (
(19)
where the eigenvalues & are the positive roots of & tan/3m= Bi2. The exact average temperature in the Ydirection is then obtained from the integration of equation (19), which gives:
fl~+Bi~ sinh2fim e.(x)=2f ~ — *=1flm [ +Bi~+Bi2 /3~ ~osh(flmclx)[
&Clsinh(%CI)+Bil cosh(p#l)sinh(pmclx) Bil sinh(&C1) +&C1cosh(~mC1) 1]
(20)
The exact heat transfer rate at the fm base is obtainedby substituting equation (20) into equation (9):
Cl sinh2/lm&C1sinh(/lmC1)+Bilcosh(AjCl) ~~ +Bi~ ‘0‘~~1p~+Bi2(Bi2 +]) & Bil sinh(/3mC1) +~mC1cosh(~mel)1 [
(21)
F.M.L. Traiano, R.M. Cotta and H.R.B. Orlande
Vol. 24, No. 6
l’he exact average temperature given by equation (20) and the exact heat transfer rate given by equation (21) for orthotropic tins, are next used in the comparison widr the approximate formulations developedpreviously irrthis section.
Results and Discussion
Heat conduction in odmtropic materials is considered first, since the exact twodimensiooal solution is readily available for comparison purposes, equations (19-21). Table 2 lists six test-cases selected for the analysis, with different combinations of the governingparameters, E22, K, Bil and Bi2, withinrangesof practical interest.Figures 2a-c show the transversallyaveragedtemperature distributions, O@(X), akmg
the longitudinal coordinate, X, as computed from the exact two-dimensional solution and from the three luwed~rential
formulationsdeveloped, namely, the
classical lumped system analysis,
Ho,&o.oad
Hl,l/HoO approximations. For cases 1 and 2 (Fig. 2a), as expected, due to the relmively smooth transversal temperature gradients, all three approximate fixrmrlations offer a reasonably good agreement with the filly differential formulation. As the Biot numbers are increased in cases 3 rmd 4 (Fig. 2.b), the classical lumping approach results start deviating from the exact solution, while the improved lumpeddifferential formulations remain quite reasonable. It carr also be observed that case 4, with larger value of E22 and, therefore, predominance of the transversal thermal conductivity coefficient, allows fix a better behavior of the classical lumped system analysis, with respect to case 3. Cases 5 and 6 (Fig. 2.c) combine the influenceof increasingthe thermal conductivitycoefficients ratio and Biot numbers. Specially in case 6, though with Biot number values not typical of fin-type applications, one can observe the marked deviations of the classical lumped solution from the exact one, but a still reasonably accurate prediction by the Hl,l~,O
formulation,which retains the same degree of mathematical simplicity as the classical approach.
TABLE 2 Sel
erial
Figures 3,a, b present the heat transfer rates at the tin’s base (X = O),as a frmction of E22, and for different values of Biot numbers (Fig. 3a), or aspect ratio (Fig. 3.b). Fig. 3.a indicates the increase in heat transfix rates for increasing values of E22. As the transversal thermal conductivity coefficient predominates, E22 > 1, a heat transfer enhancement is achievable with respect to the isctropic situation
Vol. 24, No. 6
877
ANISOTROPICHEAT CONDUCTION
(E22 = 1). Fig. 3.b shows the saturation of heat transfer enhancementas the aspect ratio is increa~ specially for materials with E22>1, when Khas little ef%ct on the heat transfer rates.
1.0
m WE
0.8
0,6
ass
0.4
2
1
IRk___J 0.0
0.2
0.4
x
0.6
0.8
1.0
FIG. 2.a Averagedtemperature distributions fix orthotropic material (cases 1 and 2, table 2).
FIGS. 2.b. C Same as above for (b)cases 3 and 4, table 2;and (c)cases 5 and 6, table 2.
?o~
m
- 0.1
1-
00
10
0.1 t
FIGS. 3a, b Heat transfer rate versus thermal conductivityratio (orthotropic); (a)difTereotBiz, (b)difl%mt K.
and
878
F.M.L. Traiano, R.M. Cotta and H.R.B. Orlande
Vol. 24, No. 6
Table 3 lists the fbur cases selected of general anisotropic heat conducticm,for differeut values of the governingparameters, namely, E22, E12, K, Bil and Bi2. Cases 1 and 2 (Fig. 4a) recover the orthotropic situation, horn the very low value of E12 considered,with similar conclusionas above. Cases 3 and 4 (Fig. 4.b) introduct the general anisotropy ei%ct, and dif%rent Biot numbers. It should be noticed that for both cases, the two approximate formulations are markedly differe@ which represents the importance of the transversal temperature gradients, now also directly affected by the longitudinalgradients, that composethe transversal heat flux componentexpression, not accounted for in the classical lumped system analysis.
Case
TABLE 3 Selectedcases for heat conductionin general anisctropic material. Bil K E22 E12 10-10 10-10
1
3 4
~ 0. 4
0.6 0.6
6.0
6.0 6.0 6.0
4 0.2 4 2. 0.24 2.4
Bi2
0.10 1.0 0.10
1.0
FIGS. 4a, b Temperature distributions for general anisotropic material (a)cases 1 and 2 (b)cases 3 and 4, table 3.
1. M.N. Ozisik, “HeatConduction”, John Wiley, New York (1980).
2. S.C. Huang and Y.P. Chang,.1 Heat Transfer,V. 106,646-648 (1984). 3. M.I. Flik and C.L. Tim, J HeatTransfer,V. 112, 10-15(1990). 4. J.C. McWhorter and M.H. Sad~ J Heat Transfer,V. 102,308-311 (1980). 5. A. Aziz,, Heat TransferEng., V. 14,63-70 (1993). 6. J.B. Aparecidoand R.M. Cotta,Heat Tran.f Eng., V. 11, no. 1,49-59 (1989). 7. F. Swt%noNetoaud R.M. Cotta, J HeatTramfer,V. 115,921-927 (1993).
ReceivedJanuary 28, 1997