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On heat conduction problem in a semi-infinite periodically laminated layer
International Communications in Heat and Mass Transfer 32 (2005) 123 – 132 www.elsevier.com/locate/ichmt
On heat conduction problem in a semi-infinite periodically laminated layerB Roman Kulchytsky-Zhyhailo, Stanisyaw J. Matysiak* Faculty of Mechanical Engineering, Biay ystok University of Technology, 15-351 Biay ystok, ul. Wiejska 45 C, Poland
Abstract The paper deals with the heat conduction problem of a semi-infinite periodically stratified layer heated by a constant heat flux directed according to the layering, normal to the boundary being a cross-section of the composite components. The free heat exchange with surroundings is considered on the remaining parts of the boundary. The body is assumed to be composed of n periodically repeated two-layered, perfect bonding lamine. The problem is solved on two ways: (1) directly as heat conduction problem or (2) by using the homogenized model with microlocal parameters [R. Kulchytsky-Zhyhailo, S.J. Matysiak, On some heat conduction problem in a periodically two-layered body. Comparative results, Int. Commun. Heat Mass Transf., in press]. The obtained results are compared and presented in the form of figures. D 2004 Elsevier Ltd. All rights reserved. Keywords: Laminated composite; Temperature; Heat flux; Semi-infinite layer; Homogenized model
1. Introduction The subject of our present investigation is very important in various branches of modern engineering. Many engineering structures are made by bonding together periodically materials with different thermal properties. The heat conduction problems of composite materials can be solved by using variety methods: exact, approximate, and purely numerical. The special attention was devoted to construction of some homogenized models for heat conduction problems [1,2,4]. One of them, which were derived by B
Communicated by J.P. Hartnett and W.J. Minkowycz. * Corresponding author. Tel.: +48 22 5540507; fax: +48 22 5540001. E-mail address: [email protected] (S.J. Matysiak).
0735-1933/$ - see front matter D 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2004.08.023
124 R. Kulchytsky-Zhyhailo, S.J. Matysiak / International Communications in Heat and Mass Transfer 32 (2005) 123–132
using the nonstandard analysis method [5–7], is called the homogenized model with microlocal parameters. The paper stands for a continuation of our previous study [3] connected with comparison of some results obtained within the framework of two approaches of solutions of heat conduction problems for periodic composites: (1) formulation as some boundary value problem of heat conduction, (2) by using the homogenized model with microlocal parameters [5,8]. The heat conduction problems in laminated body with boundary in one of components of the composite have been solved in many papers (see for references [6]). However, solutions of boundary value problems for laminated composites with boundaries being a cross-section of periodically repeated materials are not presented within the framework of the homogenized model with microlocal parameters. In this paper we consider a semi-infinite layer with one part of its boundary being a cross-section of the layering and heated by a constant normal heat flux. On the remaining parts of layer boundary, the free heat exchange condition with surroundings is considered. The problem is solved by using the classical heat conduction description and by application of the homogenized model with microlocal parameters [5,8]. The first of approach leads to some system of algebraic equations which are solved numerically. The homogenized model presents an approximate solution of the problem. The obtained results are compared and discussed in details.
2. Formulations and solutions of the heat conduction problem Consider a rigid, semi-infinite layer composed of n periodically repeated two-layered, perfect joint lamine, Fig. 1. Let d 1, d 2 be the dimensionless thicknesses of the subsequent layers and d=d 1+d 2, (here
Fig. 1. Cross-section of the composite layer.
R. Kulchytsky-Zhyhailo, S.J. Matysiak / International Communications in Heat and Mass Transfer 32 (2005) 123–132 125
d=l/a, d 1=l 1/a, d 2=l 2/a, and l 1, l 2 are the thicknesses of the lamine, l=l 1+l 2, a is the thickness of the layer). Let (x, y, z) be the dimensionless coordinates related to the thickness of layer a. So, the laminated body occupies the region 0bxb1, yN0, lbzbl. The cross-section of the body, y=0 is assumed to be heated by a constant heat flux with intensity q 0, normal to the boundary. On the surfaces x=0 and x=1 we assume the conditions of free exchange of heat with surroundings. Let K 1, K 2 denote the coefficients of heat conductivities of the subsequent layers of composite.
3. Approach 1. Formulation within the framework of classical equations of the heat conduction Let the subsequent layers be numbered by numbers 1, . . . , 2n starting from the left side of the layer (from x=0 to x=1), Fig. 1, and Ti , i=1, . . . , 2n be the temperature in the i-th layer. Thus, we consider the equation of heat conduction for stationary case in the i-th layer B2 Ti B2 Ti þ ¼ 0; Bx2 By2
i ¼ 1; N ; 2n;
ð1Þ
the conditions of free thermal exchange with surroundings BT1 Bi1 T1 ¼ 0 ; Bx BT2n þ Bi2 T2n ¼ 0; Bx
for x ¼ 0;
for x ¼ 1;
ð2aÞ
yN0;
ð2bÞ
yN0;
where Bi1 ¼ a a=K1 ;
Bi2 ¼ a a=K2 :
ð3Þ
Here a is the coefficient of heat exchange. The following continuity conditions on interfaces T2i1 ¼ T2i ;
K1 BT2i1 =Bx ¼ K2 BT2i =Bx;
for x ¼ ði 1Þd þ d1 ;
T2i ¼ T2iþ1 ;
K2 BT2i =Bx ¼ K1 BT2iþ1 =Bx;
for x ¼ id;
i ¼ 1; 2; N ; n
ð4Þ
and i ¼ 1; 2; N ; n 1;
ð5Þ
are assumed. Moreover, the conditions on the heated boundary K1 BT2i1 =By ¼ aq0 ; K2 BT2i =By ¼ aq0 ;
for y ¼ 0; for y ¼ 0;
ði 1Þd V x V ði 1Þd þ d1 ; ði 1Þd þ d1 V x V id;
i ¼ 1; 2; N ; n;
i ¼ 1; 2; N ; n
ð6aÞ ð6bÞ
and the regularity conditions in infinity Ti Y0;
for yYl;
are taken into account.
i ¼ 1; 2; N ; 2n
ð7Þ
126 R. Kulchytsky-Zhyhailo, S.J. Matysiak / International Communications in Heat and Mass Transfer 32 (2005) 123–132
The solution of the above-presented problem will be found by using the method of Fourier cosine transformation. Denoting the cosine transform of function f(x,y) by rffiffiffiffi Z l ˜f ðcÞ ð x; sÞ ¼ F f ð x; yÞ; yYs ¼ 2 f ð x; yÞcosðsyÞdy; ð8Þ p 0 the solution of the considered problem can be written in the form rffiffiffiffi 2 q0 a 2 ð c Þ T˜ 2i1 ð x; sÞ ¼ s þ C4i3 ðsÞsinhðsðði 1Þd þ d1 xÞÞ p K1 i ¼ 1; 2; N ; n; þ C4i2 ðsÞcoshðsðði 1Þd þ d1 xÞÞ;
ðcÞ T˜ 2i ð x;sÞ
rffiffiffiffi 2 q0 a 2 ¼ s þ C4i1 ðsÞsinhðsðid xÞÞ þ C4i ðsÞcoshðsðid xÞÞ; p K2
ð9Þ
i ¼ 1; 2; N ; n; ð10Þ
where the functions C i (s),i=1,2, . . . ,4n should be determined from the following system of 4n linear algebraic equations rffiffiffiffi 2 q0 aBi1 2 s ; C1 ðsÞðs coshðsd1 Þ þ Bi1 sinhðsd1 ÞÞ þ C2 ðsÞðs sinhðsd1 Þ þ Bi1 coshðsd1 ÞÞ ¼ p K1 ð11aÞ rffiffiffiffi 2 q 0 a K1 1 s2 ; C4i2 ðsÞ C4i1 ðsÞsinhðd2 sÞ C4i ðsÞcoshðd2 sÞ ¼ p K1 K2
i ¼ 1; 2; N ; n; ð11bÞ
K1 C4i3 ðsÞ C4i1 ðsÞcoshðd2 sÞ C4i ðsÞsinhðd2 sÞ ¼ 0; K2
i ¼ 1; 2; N ; n;
ð11cÞ
rffiffiffiffi 2 q0 a K1 2 C4i ðsÞ C4iþ1 ðsÞsinhðd1 sÞ C4iþ2 ðsÞcoshðd1 sÞ ¼ 1 s ; i ¼ 1; 2; N ; n 1; p K1 K2 ð11dÞ K2 C4i1 ðsÞ C4iþ1 ðsÞcoshðd1 sÞ C4iþ2 ðsÞsinhðd1 sÞ ¼ 0; K1 rffiffiffiffi 2 q0 aBi2 2 sC4n1 ðsÞ Bi2 C4n ðsÞ ¼ s : p K2 The system of Eqs. (11a)–(11f) will be solved numerically.
i ¼ 1; 2; N ; n 1;
ð11eÞ
ð11f Þ
R. Kulchytsky-Zhyhailo, S.J. Matysiak / International Communications in Heat and Mass Transfer 32 (2005) 123–132 127
4. Approach 2. Formulation within the framework of the homogenized model with microlocal parameter The homogenized model with microlocal parameters for the stationary two-dimensional case of heat conduction problem for periodically two-layered composite is given by the following relations [5,8] T ð x;yÞ ¼ hð x;yÞ þ hð xÞcð x;yÞ; ð12Þ P where h is an unknown function interpreted as the macrotemperature, c is unknown function called the thermal microlocal parameter, and h is given d-periodic function taken in the form ( x 0:5d1 ; for 0 V xV d1 ; gx d1 hð xÞ ¼ ð13Þ 0:5d1 þ ; for d1 V x V d; 1g 1g hð x þ dÞ ¼ hð xÞ and g ¼ d1 =d:
ð14Þ
Since |h(x)|bd for every x, then for small d the underlined term in Eq. (12) will be neglected, but the derivative hV(x) is not small, so we have BT Bh c þ hVð xÞc; Bx Bx
T ch;
BT Bh c : By By
The homogenized model of heat conduction is described by equations [3] 2 B h B2 h Bc Bh ˜ K ¼ 0; Kˆ c ¼ ½ K ; þ 2 þ ½K 2 Bx By Bx Bx where g2 K˜ ¼ gK1 þ ð1 gÞK2 ; ½K ¼ gðK1 K2 Þ; Kˆ ¼ gK1 þ K2 : 1g
ð15Þ
ð16Þ
ð17Þ
From Eq. (16), it follows that B2 h B2 h K˜ 1 K T 2 þ 2 ¼ 0; Bx By
ð18Þ
where K4 ¼
K1 K2 : ð1 gÞK1 þ gK2
ð19Þ
The heat flux vector in a layer of the i-th kind, i=1, 2 is given by Bh Bh ðiÞ q ð x;yÞ ¼ K4 ; Ki ;0 ; i ¼ 1; 2; Bx By
ð20Þ
where K4uK1
½K 1 Kˆ
¼ K2 1 þ
g ½K : 1 g Kˆ
ð21Þ
128 R. Kulchytsky-Zhyhailo, S.J. Matysiak / International Communications in Heat and Mass Transfer 32 (2005) 123–132
The problem considered in Approach 1 can be described by using the homogenized model with microlocal parameter by Eq. (18), the conditions of free thermal exchange with surroundings Bh Bih ¼ 0; Bx
for x ¼ 0;
Bh þ Bih ¼ 0; Bx
for x ¼ 1;
ð22aÞ
yN0;
ð22bÞ
yN0;
where Bi ¼ aa=KT;
ð23Þ
the condition in infinity ð24Þ
hY0; for yYl; and the boundary condition on the heated part of boundary in the form: Bh K˜ ¼ aq0 ; By
y ¼ 0;
ð25Þ
0 V x V 1:
The boundary condition (25) stands for the condition of averaging heat flux on the cross-section of the layering. The solution of the boundary value problem can be obtained by using the cosine Fourier transform method. So, we obtain rffiffiffiffi 2 q0 a ð c Þ h˜ ð x;sÞ ¼ ð26Þ f1 þ BiðAðS ÞsinhðSxÞ BðS ÞcoshðSxÞÞg; p K4S 2 where AðS Þ ¼ BðS Þ ¼
S sinhðS Þ þ Bi coshðS Þ Bi ; ðS 2 þ Bi2 ÞsinhðS Þ þ 2BiS coshðS Þ ðS 2
S coshðS Þ þ Bi sinhðS Þ þ S ; þ Bi2 ÞsinhðS Þ þ 2BiS coshðS Þ
sffiffiffiffiffiffiffi K˜ : S¼s KT
ð27Þ
The inverse Fourier transform of the function h˜(c) presented in (26) will be calculated numerically and results will be shown in the form of graphs.
5. Numerical results The main aim of the considerations is to compare the results for temperature and heat flux distributions in the composite body obtained within the framework of the exact (classical) formulation (Approach 1) and the approximate formulation by using the homogenized model with microlocal parameters (Approach 2). Moreover, it seems to be very interesting the analysis of influence of increasing of number 2n of the layers being the composite components in Approach 1 on the solution in comparison with the results obtained in Approach 2.
R. Kulchytsky-Zhyhailo, S.J. Matysiak / International Communications in Heat and Mass Transfer 32 (2005) 123–132 129
Fig. 2. The dimensional temperature in the heated region.
The dimensionless temperature TK*/q 0a in the heated region is shown in Fig. 2. The waved curves present the solution given in Approach 1 and the thickly marked curves present the solutions given in the framework of homogenized model. The composite is assumed to be composed of 20 layers (10 basic units) in the case shown in Fig. 2a. Fig. 2b shows the dimensionless temperature for three cases of numbers of repeated layers: 20 (n=10)-waved curve 1, 40 (n=20)-waved curve 2, and 80 (n=40)-waved curve 3. From the presented results in Fig. 2a and b, it follows that the results obtained within the framework of the homogenized model with microlocal parameters stand for a good approximation of the results obtained in Approach 1. The accuracy of the approximation increases with an increasing of number n of lamine. Figs. 3 and 4 present the dimensionless component of heat flux q y /q 0 along the boundary (normal to the boundary y=0) on the different depths from boundary y=0 for two values of ratio K 1/K 2=4 and K 1/
130 R. Kulchytsky-Zhyhailo, S.J. Matysiak / International Communications in Heat and Mass Transfer 32 (2005) 123–132
Fig. 3. Dimensional heat flux parallel to the layering at the points on different depths of the heated region and K 1/K 2=8.
R. Kulchytsky-Zhyhailo, S.J. Matysiak / International Communications in Heat and Mass Transfer 32 (2005) 123–132 131
Fig. 4. Dimensional heat flux parallel to the layering at the points on different depths for the heated region for K 1/K 2=4.
132 R. Kulchytsky-Zhyhailo, S.J. Matysiak / International Communications in Heat and Mass Transfer 32 (2005) 123–132
K 2=8, respectively. The black curves present the component of heat flux obtained within the framework of the Approach 1, the gray curves present the solutions for the homogenized model (Approach 2) and the upper curves shown the component of heat flux q y /q 0 in layers of the first kind (with the coefficient of heat conductivity K 1), the lower curves in layers of the second kind (with the coefficient of heat conductivity K 2). These curves are extended to both kinds of layers for a better visibility.
6. Final remarks and conclusions The obtained results stand for a some comparison of solutions for the stationary heat conduction problem in periodically two-layered semi-infinite layer obtained for two formulations given in Approaches 1 and 2. From Fig. 2a and b it follows that the temperature distribution in the heated region obtained within the homogenized model stands for a good approximation for the temperature given for, bexact formulationQ (Approach 1). The accuracy of the approximation increases with decreasing of the thickness of basic layer d (or increasing of number n of layers). For n=40, what it corresponds with the thickness of basic unit d=0.025, from Fig. 2b (see the solid curve and the curve denoted by 3) we observe a well-fitting of the temperature distribution obtained by the homogenized model. Figs. 3 and 4 show the dimensionless component of heat flux q y /q 0 for two cases of ratio K 1/K 2=8 and K 1/K 2=4 calculated for different distances from the heated boundary for the case of 20 basic layers (d=0.05). We observe that greater differences between the results obtained within the Approaches 1 and 2 are for small depths from the boundary. When distances from the heated boundary are greater than 0.5 d, the results obtained for the homogenized model stand for a very good approximation for the heat flux. The above analysis confirms the conclusions presented in our previous studies [3]. The homogenized model of heat conduction with microlocal parameters can be applied for stationary problems of heat conduction in periodically layered composites.
Acknowledgements The investigation described in this paper is a part of the research No. W/WM/1/04 sponsored by the Polish State Committee for Scientific Research and realized in Bialystok University of Technology. References [1] [2] [3] [4] [5] [6] [7] [8]
J.L. Aurialt, Int. J. Heat Mass Transfer 26 (1983) 861. H. Bufler, Arch. Appl. Mech. 70 (2000) 103. R. Kulchytsky-Zhyhailo, S.J. Matysiak, Int. Commun. Heat Mass Transf., in review. A. Maewal, ASME J. Appl. Mech. 46 (1979) 945. S.J. Matysiak, Cz. Woz´niak, Acta Mech. 63 (1986) 223. S.J. Matysiak, J. Theor. Appl. Mech. 2 (33) (1995) 481. I.N. Sneddon, The Use of Integral Transforms, McGraw-Hill Book, London, 1972. Cz. Woz´niak, Int. J. Eng. Sci. 25 (1987) 483.
On heat conduction problem in a semi-infinite periodically laminated layerB Roman Kulchytsky-Zhyhailo, Stanisyaw J. Matysiak* Faculty of Mechanical Engineering, Biay ystok University of Technology, 15-351 Biay ystok, ul. Wiejska 45 C, Poland
Abstract The paper deals with the heat conduction problem of a semi-infinite periodically stratified layer heated by a constant heat flux directed according to the layering, normal to the boundary being a cross-section of the composite components. The free heat exchange with surroundings is considered on the remaining parts of the boundary. The body is assumed to be composed of n periodically repeated two-layered, perfect bonding lamine. The problem is solved on two ways: (1) directly as heat conduction problem or (2) by using the homogenized model with microlocal parameters [R. Kulchytsky-Zhyhailo, S.J. Matysiak, On some heat conduction problem in a periodically two-layered body. Comparative results, Int. Commun. Heat Mass Transf., in press]. The obtained results are compared and presented in the form of figures. D 2004 Elsevier Ltd. All rights reserved. Keywords: Laminated composite; Temperature; Heat flux; Semi-infinite layer; Homogenized model
1. Introduction The subject of our present investigation is very important in various branches of modern engineering. Many engineering structures are made by bonding together periodically materials with different thermal properties. The heat conduction problems of composite materials can be solved by using variety methods: exact, approximate, and purely numerical. The special attention was devoted to construction of some homogenized models for heat conduction problems [1,2,4]. One of them, which were derived by B
Communicated by J.P. Hartnett and W.J. Minkowycz. * Corresponding author. Tel.: +48 22 5540507; fax: +48 22 5540001. E-mail address: [email protected] (S.J. Matysiak).
0735-1933/$ - see front matter D 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2004.08.023
124 R. Kulchytsky-Zhyhailo, S.J. Matysiak / International Communications in Heat and Mass Transfer 32 (2005) 123–132
using the nonstandard analysis method [5–7], is called the homogenized model with microlocal parameters. The paper stands for a continuation of our previous study [3] connected with comparison of some results obtained within the framework of two approaches of solutions of heat conduction problems for periodic composites: (1) formulation as some boundary value problem of heat conduction, (2) by using the homogenized model with microlocal parameters [5,8]. The heat conduction problems in laminated body with boundary in one of components of the composite have been solved in many papers (see for references [6]). However, solutions of boundary value problems for laminated composites with boundaries being a cross-section of periodically repeated materials are not presented within the framework of the homogenized model with microlocal parameters. In this paper we consider a semi-infinite layer with one part of its boundary being a cross-section of the layering and heated by a constant normal heat flux. On the remaining parts of layer boundary, the free heat exchange condition with surroundings is considered. The problem is solved by using the classical heat conduction description and by application of the homogenized model with microlocal parameters [5,8]. The first of approach leads to some system of algebraic equations which are solved numerically. The homogenized model presents an approximate solution of the problem. The obtained results are compared and discussed in details.
2. Formulations and solutions of the heat conduction problem Consider a rigid, semi-infinite layer composed of n periodically repeated two-layered, perfect joint lamine, Fig. 1. Let d 1, d 2 be the dimensionless thicknesses of the subsequent layers and d=d 1+d 2, (here
Fig. 1. Cross-section of the composite layer.
R. Kulchytsky-Zhyhailo, S.J. Matysiak / International Communications in Heat and Mass Transfer 32 (2005) 123–132 125
d=l/a, d 1=l 1/a, d 2=l 2/a, and l 1, l 2 are the thicknesses of the lamine, l=l 1+l 2, a is the thickness of the layer). Let (x, y, z) be the dimensionless coordinates related to the thickness of layer a. So, the laminated body occupies the region 0bxb1, yN0, lbzbl. The cross-section of the body, y=0 is assumed to be heated by a constant heat flux with intensity q 0, normal to the boundary. On the surfaces x=0 and x=1 we assume the conditions of free exchange of heat with surroundings. Let K 1, K 2 denote the coefficients of heat conductivities of the subsequent layers of composite.
3. Approach 1. Formulation within the framework of classical equations of the heat conduction Let the subsequent layers be numbered by numbers 1, . . . , 2n starting from the left side of the layer (from x=0 to x=1), Fig. 1, and Ti , i=1, . . . , 2n be the temperature in the i-th layer. Thus, we consider the equation of heat conduction for stationary case in the i-th layer B2 Ti B2 Ti þ ¼ 0; Bx2 By2
i ¼ 1; N ; 2n;
ð1Þ
the conditions of free thermal exchange with surroundings BT1 Bi1 T1 ¼ 0 ; Bx BT2n þ Bi2 T2n ¼ 0; Bx
for x ¼ 0;
for x ¼ 1;
ð2aÞ
yN0;
ð2bÞ
yN0;
where Bi1 ¼ a a=K1 ;
Bi2 ¼ a a=K2 :
ð3Þ
Here a is the coefficient of heat exchange. The following continuity conditions on interfaces T2i1 ¼ T2i ;
K1 BT2i1 =Bx ¼ K2 BT2i =Bx;
for x ¼ ði 1Þd þ d1 ;
T2i ¼ T2iþ1 ;
K2 BT2i =Bx ¼ K1 BT2iþ1 =Bx;
for x ¼ id;
i ¼ 1; 2; N ; n
ð4Þ
and i ¼ 1; 2; N ; n 1;
ð5Þ
are assumed. Moreover, the conditions on the heated boundary K1 BT2i1 =By ¼ aq0 ; K2 BT2i =By ¼ aq0 ;
for y ¼ 0; for y ¼ 0;
ði 1Þd V x V ði 1Þd þ d1 ; ði 1Þd þ d1 V x V id;
i ¼ 1; 2; N ; n;
i ¼ 1; 2; N ; n
ð6aÞ ð6bÞ
and the regularity conditions in infinity Ti Y0;
for yYl;
are taken into account.
i ¼ 1; 2; N ; 2n
ð7Þ
126 R. Kulchytsky-Zhyhailo, S.J. Matysiak / International Communications in Heat and Mass Transfer 32 (2005) 123–132
The solution of the above-presented problem will be found by using the method of Fourier cosine transformation. Denoting the cosine transform of function f(x,y) by rffiffiffiffi Z l ˜f ðcÞ ð x; sÞ ¼ F f ð x; yÞ; yYs ¼ 2 f ð x; yÞcosðsyÞdy; ð8Þ p 0 the solution of the considered problem can be written in the form rffiffiffiffi 2 q0 a 2 ð c Þ T˜ 2i1 ð x; sÞ ¼ s þ C4i3 ðsÞsinhðsðði 1Þd þ d1 xÞÞ p K1 i ¼ 1; 2; N ; n; þ C4i2 ðsÞcoshðsðði 1Þd þ d1 xÞÞ;
ðcÞ T˜ 2i ð x;sÞ
rffiffiffiffi 2 q0 a 2 ¼ s þ C4i1 ðsÞsinhðsðid xÞÞ þ C4i ðsÞcoshðsðid xÞÞ; p K2
ð9Þ
i ¼ 1; 2; N ; n; ð10Þ
where the functions C i (s),i=1,2, . . . ,4n should be determined from the following system of 4n linear algebraic equations rffiffiffiffi 2 q0 aBi1 2 s ; C1 ðsÞðs coshðsd1 Þ þ Bi1 sinhðsd1 ÞÞ þ C2 ðsÞðs sinhðsd1 Þ þ Bi1 coshðsd1 ÞÞ ¼ p K1 ð11aÞ rffiffiffiffi 2 q 0 a K1 1 s2 ; C4i2 ðsÞ C4i1 ðsÞsinhðd2 sÞ C4i ðsÞcoshðd2 sÞ ¼ p K1 K2
i ¼ 1; 2; N ; n; ð11bÞ
K1 C4i3 ðsÞ C4i1 ðsÞcoshðd2 sÞ C4i ðsÞsinhðd2 sÞ ¼ 0; K2
i ¼ 1; 2; N ; n;
ð11cÞ
rffiffiffiffi 2 q0 a K1 2 C4i ðsÞ C4iþ1 ðsÞsinhðd1 sÞ C4iþ2 ðsÞcoshðd1 sÞ ¼ 1 s ; i ¼ 1; 2; N ; n 1; p K1 K2 ð11dÞ K2 C4i1 ðsÞ C4iþ1 ðsÞcoshðd1 sÞ C4iþ2 ðsÞsinhðd1 sÞ ¼ 0; K1 rffiffiffiffi 2 q0 aBi2 2 sC4n1 ðsÞ Bi2 C4n ðsÞ ¼ s : p K2 The system of Eqs. (11a)–(11f) will be solved numerically.
i ¼ 1; 2; N ; n 1;
ð11eÞ
ð11f Þ
R. Kulchytsky-Zhyhailo, S.J. Matysiak / International Communications in Heat and Mass Transfer 32 (2005) 123–132 127
4. Approach 2. Formulation within the framework of the homogenized model with microlocal parameter The homogenized model with microlocal parameters for the stationary two-dimensional case of heat conduction problem for periodically two-layered composite is given by the following relations [5,8] T ð x;yÞ ¼ hð x;yÞ þ hð xÞcð x;yÞ; ð12Þ P where h is an unknown function interpreted as the macrotemperature, c is unknown function called the thermal microlocal parameter, and h is given d-periodic function taken in the form ( x 0:5d1 ; for 0 V xV d1 ; gx d1 hð xÞ ¼ ð13Þ 0:5d1 þ ; for d1 V x V d; 1g 1g hð x þ dÞ ¼ hð xÞ and g ¼ d1 =d:
ð14Þ
Since |h(x)|bd for every x, then for small d the underlined term in Eq. (12) will be neglected, but the derivative hV(x) is not small, so we have BT Bh c þ hVð xÞc; Bx Bx
T ch;
BT Bh c : By By
The homogenized model of heat conduction is described by equations [3] 2 B h B2 h Bc Bh ˜ K ¼ 0; Kˆ c ¼ ½ K ; þ 2 þ ½K 2 Bx By Bx Bx where g2 K˜ ¼ gK1 þ ð1 gÞK2 ; ½K ¼ gðK1 K2 Þ; Kˆ ¼ gK1 þ K2 : 1g
ð15Þ
ð16Þ
ð17Þ
From Eq. (16), it follows that B2 h B2 h K˜ 1 K T 2 þ 2 ¼ 0; Bx By
ð18Þ
where K4 ¼
K1 K2 : ð1 gÞK1 þ gK2
ð19Þ
The heat flux vector in a layer of the i-th kind, i=1, 2 is given by Bh Bh ðiÞ q ð x;yÞ ¼ K4 ; Ki ;0 ; i ¼ 1; 2; Bx By
ð20Þ
where K4uK1
½K 1 Kˆ
¼ K2 1 þ
g ½K : 1 g Kˆ
ð21Þ
128 R. Kulchytsky-Zhyhailo, S.J. Matysiak / International Communications in Heat and Mass Transfer 32 (2005) 123–132
The problem considered in Approach 1 can be described by using the homogenized model with microlocal parameter by Eq. (18), the conditions of free thermal exchange with surroundings Bh Bih ¼ 0; Bx
for x ¼ 0;
Bh þ Bih ¼ 0; Bx
for x ¼ 1;
ð22aÞ
yN0;
ð22bÞ
yN0;
where Bi ¼ aa=KT;
ð23Þ
the condition in infinity ð24Þ
hY0; for yYl; and the boundary condition on the heated part of boundary in the form: Bh K˜ ¼ aq0 ; By
y ¼ 0;
ð25Þ
0 V x V 1:
The boundary condition (25) stands for the condition of averaging heat flux on the cross-section of the layering. The solution of the boundary value problem can be obtained by using the cosine Fourier transform method. So, we obtain rffiffiffiffi 2 q0 a ð c Þ h˜ ð x;sÞ ¼ ð26Þ f1 þ BiðAðS ÞsinhðSxÞ BðS ÞcoshðSxÞÞg; p K4S 2 where AðS Þ ¼ BðS Þ ¼
S sinhðS Þ þ Bi coshðS Þ Bi ; ðS 2 þ Bi2 ÞsinhðS Þ þ 2BiS coshðS Þ ðS 2
S coshðS Þ þ Bi sinhðS Þ þ S ; þ Bi2 ÞsinhðS Þ þ 2BiS coshðS Þ
sffiffiffiffiffiffiffi K˜ : S¼s KT
ð27Þ
The inverse Fourier transform of the function h˜(c) presented in (26) will be calculated numerically and results will be shown in the form of graphs.
5. Numerical results The main aim of the considerations is to compare the results for temperature and heat flux distributions in the composite body obtained within the framework of the exact (classical) formulation (Approach 1) and the approximate formulation by using the homogenized model with microlocal parameters (Approach 2). Moreover, it seems to be very interesting the analysis of influence of increasing of number 2n of the layers being the composite components in Approach 1 on the solution in comparison with the results obtained in Approach 2.
R. Kulchytsky-Zhyhailo, S.J. Matysiak / International Communications in Heat and Mass Transfer 32 (2005) 123–132 129
Fig. 2. The dimensional temperature in the heated region.
The dimensionless temperature TK*/q 0a in the heated region is shown in Fig. 2. The waved curves present the solution given in Approach 1 and the thickly marked curves present the solutions given in the framework of homogenized model. The composite is assumed to be composed of 20 layers (10 basic units) in the case shown in Fig. 2a. Fig. 2b shows the dimensionless temperature for three cases of numbers of repeated layers: 20 (n=10)-waved curve 1, 40 (n=20)-waved curve 2, and 80 (n=40)-waved curve 3. From the presented results in Fig. 2a and b, it follows that the results obtained within the framework of the homogenized model with microlocal parameters stand for a good approximation of the results obtained in Approach 1. The accuracy of the approximation increases with an increasing of number n of lamine. Figs. 3 and 4 present the dimensionless component of heat flux q y /q 0 along the boundary (normal to the boundary y=0) on the different depths from boundary y=0 for two values of ratio K 1/K 2=4 and K 1/
130 R. Kulchytsky-Zhyhailo, S.J. Matysiak / International Communications in Heat and Mass Transfer 32 (2005) 123–132
Fig. 3. Dimensional heat flux parallel to the layering at the points on different depths of the heated region and K 1/K 2=8.
R. Kulchytsky-Zhyhailo, S.J. Matysiak / International Communications in Heat and Mass Transfer 32 (2005) 123–132 131
Fig. 4. Dimensional heat flux parallel to the layering at the points on different depths for the heated region for K 1/K 2=4.
132 R. Kulchytsky-Zhyhailo, S.J. Matysiak / International Communications in Heat and Mass Transfer 32 (2005) 123–132
K 2=8, respectively. The black curves present the component of heat flux obtained within the framework of the Approach 1, the gray curves present the solutions for the homogenized model (Approach 2) and the upper curves shown the component of heat flux q y /q 0 in layers of the first kind (with the coefficient of heat conductivity K 1), the lower curves in layers of the second kind (with the coefficient of heat conductivity K 2). These curves are extended to both kinds of layers for a better visibility.
6. Final remarks and conclusions The obtained results stand for a some comparison of solutions for the stationary heat conduction problem in periodically two-layered semi-infinite layer obtained for two formulations given in Approaches 1 and 2. From Fig. 2a and b it follows that the temperature distribution in the heated region obtained within the homogenized model stands for a good approximation for the temperature given for, bexact formulationQ (Approach 1). The accuracy of the approximation increases with decreasing of the thickness of basic layer d (or increasing of number n of layers). For n=40, what it corresponds with the thickness of basic unit d=0.025, from Fig. 2b (see the solid curve and the curve denoted by 3) we observe a well-fitting of the temperature distribution obtained by the homogenized model. Figs. 3 and 4 show the dimensionless component of heat flux q y /q 0 for two cases of ratio K 1/K 2=8 and K 1/K 2=4 calculated for different distances from the heated boundary for the case of 20 basic layers (d=0.05). We observe that greater differences between the results obtained within the Approaches 1 and 2 are for small depths from the boundary. When distances from the heated boundary are greater than 0.5 d, the results obtained for the homogenized model stand for a very good approximation for the heat flux. The above analysis confirms the conclusions presented in our previous studies [3]. The homogenized model of heat conduction with microlocal parameters can be applied for stationary problems of heat conduction in periodically layered composites.
Acknowledgements The investigation described in this paper is a part of the research No. W/WM/1/04 sponsored by the Polish State Committee for Scientific Research and realized in Bialystok University of Technology. References [1] [2] [3] [4] [5] [6] [7] [8]
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