구형 및 회전타원체 입자의 격자배열에서의 유효열전도도

김 재 영, 윤 병 준

포항공과대학교 화학공학과

Effective Thermal Conductivity

of Periodic Arrays of Spheres and Spheroids

Jae Young Kim and Byung Jun Yoon

Department of Chemical Engineering, POSTECH

Introduction

Determination of bulk or effective properties of two-phase system is one of the most important problems in the theory of transport phenomena.  In many chemical engineering processes two phase systems such as packed beds, composite materials, porous media, and colloidal dispersions are frequently encountered.  In these systems the determination of effective properties is important in order to understand and utilize their properties properly.

In this study we calculate the effective thermal conductivity of composites which contain perfectly conducting inclusions.  Inclusions are arranged in simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC) lattices.  Inclusions of spherical and spheroidal shapes are considered.  A boundary element method (BEM) is used.  Numerical results are compared with available analytical results for spheres at various particle concentrations.  Anisotropy of the effective thermal conductivity for the systems with spheroidal inclusions are also discussed.

Theory

The temperature fields inside and outside inclusion satisfy Laplace's equation.  Assuming that the applied heat flux is along the x-axis, the appropriate boundary conditions are given by

for BCC and FCC arrays.  Here the superscripts in and out denote the surfaces of flux-in and flux-out, respectively.  The symbol Sp is the particle surface, Sx the surfaces of x=[eqn: +-]1 for SC (x=0,1 for BCC and FCC), and Sn the other 4 surfaces of the unit cell.

Using Green's identity temperature at any point can be represented by the following integral equation;

[eqn: -4 pi beta T({bold x}) = INT _{ S_T} {partialG}over{partialn}({bold x,y})T_0 dS({bold y})- INT _{ S_n}G({bold x,y}){partialT}over{partialn}({bold y})dS({bold y})]

[eqn: + INT _{ S_n}{partialG}over{partialn}({bold x,y})T({bold y})dS({bold y}),]

[eqn: G({bold x,y}) = 1/|{bold x}-{bold y}|.]

Here, ST and Sn are the surfaces where  T and [eqn: {partial T}/{partial n}] are given as the boundary conditions, respectively.  We solve the equation numerically at the boundary to determine the unknowns [1].  We discretize the boundary surface into flat triangular elements as shown in Fig. 1.  When the concentration of inclusion is very high, the unknowns [eqn: {partial T}/{partial n}] on Sn varies rapidly in the vicinity of nearly contacting point, for which we further refine the boundary elements.

From Fourier's law of heat conduction the effective thermal conductivity ke can be calculated using

[eqn: k_e = { INT _{ S}LEFT ( {partialT}over{partialn} RIGHT ) dS} over { TRIANGLE T CDOT TRIANGLE A/TRIANGLE l}]

Results and Discussion

Fig. 2 shows the numerical results for spherical inclusions.  Analytic results are also shown [2].  At critical volume fraction [eqn: phi _c] neighboring particles touch each other, so [eqn: k_e] diverges.  Our numerical results agree fairly well with the analytic solution even at very high volume fractions.

Fig. 3 shows the results for prolate spheroids (a) and oblate spheroids (b) in SC arrays.  The symbol[eqn: k_e^||] ([eqn: k_e^BOT]) is the effective thermal conductivity when the symmetric axis of spheroid is parallel (perpendicular) to the direction of the applied heat flux.  The aspect ratio [eqn: alpha] is defined as the ratio of the minor axis to the major axis. The symbol [eqn: R_max] is the ratio of the longest dimension of spheroid to the length of unit cell.  Depending on the aspect ratio and volume fraction of spheroidal inclusions [eqn: k_e^||] and [eqn: k_e^BOT] can differ significantly.

References

[1].  Beskos, D. E. : "Boundary Element Methods in Mechanics", North-Holland, Amsterdam(1987).

[2].  McPhedran, R. C. and Mckenzie, D. R. : Proc. R. Soc. Lond. A, 359, 45(1978).