Effect of double-diffusive heat transfer on thermal conductivity of porous sintered ceramics: Macrotransport analysis

International Journal of Heat and Mass Transfer 54 (2011) 4844–4855

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Effect of double-diffusive heat transfer on thermal conductivity of porous sintered ceramics: Macrotransport analysis T. Gambaryan-Roisman a, M. Shapiro b,⇑, A. Shavit b a b

Institute for Technical Thermodynamics and Center of Smart Interfaces, Technische Universität Darmstadt, Petersenstr. 32, 64287 Darmstadt, Germany Laboratory of Transport Processes in Porous Materials, Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel

a r t i c l e

i n f o

Article history: Available online 28 July 2011 Keywords: Effective thermal conductivity Species segregation Composite materials Pores Nonstationary diffusion Conduction

a b s t r a c t Heat conduction in composite materials has been traditionally considered as occurring via conductance through the solid phase and also in pores containing gas. In this paper these mechanisms are considered together with thermal effects associated with motion of mass species – impurities, or lattice defects, normally present in all sintered ceramic materials. Diffusion and reversible segregation and desegregation of these species (from solid bulk to pore surfaces and back), is induced by an externally applied temperature gradient and accompanied by heat release and absorption on pore surface. A general method is developed which allows calculation of the effective thermal conductivity of ceramic materials with complicated microstructure subject to the segregation–diffusion processes. The model is based on the macrotransport analysis of heat transfer in composite materials, generalized to include the thermal effect associated with reversible segregation and diffusion in the grain boundary region. The method is illustrated by calculation of the effective thermal conductivity of ceramics containing chains of pores in the grain boundaries. The physical circumstances, at which segregation–diffusion processes significantly affect thermal conductivity dependence on temperature measured in vacuum, are outlined. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Thermal conductivity is an important material property, required in many applications. For porous ceramic materials this property determines insulating capacity and heat transfer regimes of furnaces, high-temperature installations, etc. [1–4]. The effective thermal conductivity tensor of a composite material, keff, is defined via the Fourier law

 ¼ keff  rT; q

ð1Þ

 is an average heat flux and rT is an average temperature where q gradient. This expression is employed in measurements of keff by the stationary method [5]. keff depends on the material constituents and microstructure. Porosity as one microstructural parameter may be used for calculation of keff only when the microstructural geometry is specified [6–9]. Several methods are known for calculation of the effective thermal conductivity of heterogeneous materials. Most of them pertain to arrays of spheres, cylinders or disks [10–12]. These studies are inapplicable to sintered porous ceramics because of their very special irregular microstructure, such as grain boundaries containing pores of different sizes, etc. [13]. When the microstructure is ⇑ Corresponding author. Fax: +972 4 829 3185. E-mail address: [email protected] (M. Shapiro). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.06.039

known, homogenization methods [14–18] can be used to calculate thermal diffusivity and conductivity and other transport properties in terms of appropriate microstructure parameters. This is possible when a hierarchy of length scales can be identified: one a microscale l, associated with the material interstitial structure, and the other – macro-scale, L  l, e.g. the one associated with the whole sample. Further, one can employ multiple time scales or multiple length scales analysis, envisioned in by a specific homogenization scheme chosen [14,15], or the volume averaging method [16] to calculate the effective material properties. Macrotransport theory [18] is a variant of the homogenization methods for calculating the materials effective macroscale properties based on the assumption that the material microstructure can be approximated by a spatial periodic geometry. Rather than employing the Fourier definition (1), this method considers the nonstationary heat diffusion resulting from a heat pulse instantaneously released within the material originally prevailing in equilibrium. Modeling of this process allows calculation of the material effective thermal diffusivity aeff, which is related to keff via the material density q and specific heat cp:

keff ¼ aeff ðqcp Þeff :

ð2Þ

Macrotransport theory had been generalized to include coupled transport of multi-component and reactive species in porous media [19,20] and in microfluidic networks [21] and for the systems

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Nomenclature a B e B

thermal diffusivity tensor field defined by Eq. (A23) nondimensional B-field b field defined by Eq. (A24) ~ b nondimensional b-field C concentration of impurities b C mole fraction of impurities c excess bulk concentration function defined by Eq. (10) cm mth local moment of the function c cp specific heat at constant pressure long-time zero-order moment of the function c c0,1 D diffusivity of impurities Ds0 pre-exponential factor of calculations of Ds d surface layer thickness activation energy for surface diffusion of impurities Eact KC, Kg, KT parameters defined by Eq. (18) k thermal conductivity tensor k thermal conductivity scalar L distance between the pore chains, grain size l distance between pores within a chain of pores l1, l2, l3 basic lattice vectors M average molar mass of the bulk solid Mm mth total energy moment ~x -fields introe x -and b mC, mT parameters for calculating the B duced in Eqs. (32) and (33) Nm mth total concentration moment n1, n2, n3 integer numbers defining a unit cell Q heat release of the segregation process R global radius-vector R0 global point of heat impulse release r radial coordinate rp pore radius

involving coupling between diffusion of heat and mass, known as thermo-diffusion and diffusional thermal effects [22,23]. Different from the above is the process of heat transfer associated with reversible segregation and diffusion of impurities on surfaces of pores [24,25]. The impurities may be foreign atoms or lattice defects, present in most ceramic materials. These species diffuse in the bulk solid phase and along the pore surfaces. They tend to concentrate (segregate) and diffuse along the surfaces of small pores existing in the region of contact between the crystalline grains, composing the ceramic materials. At equilibrium, the surface concentration of segregated species, Cs, is determind by the temperature, and bulk concentration of species, C. Normally, Cs increases with increasing C and decreasing T [26]. If an external macroscopic temperature gradient is imposed on the material, the surface concentration of the impurities on the ‘‘cooler’’ side of a pore becomes higher than that at the ‘‘hotter’’ side of the same pore. The impurities are segregated from the bulk onto the cooler pore sites. This process is accompanied by heat release and species desegregation from the hotter pore sites, resulting in heat absorption. This process of heat release and absorption (analogous to heat pipe mechanism), adds to the heat transport across the pores filled with gas. Steady distributions of the impurities concentrations in the bulk and on the pores surfaces are maintained by the impurities bulk diffusion towards the cold sites of each pore. As such, heat in porous solids flows along a double-diffusive pass, wherein some portion of heat transported due to the motion of the mass species contributes to the effective material thermal conductivity and thermal diffusivity. The effect of impurities segregation and surface diffusion on the effective thermal conductivity of materials with small volume frac-

r s T t Z z

local radius-vector surface area temperature time complex function, defined by Eq. (5.87) complex variable, z = x + iy

Greek letters bC, bT coefficients introduced in Eq. (11) C parameter defined by Eq. (15) e material porosity or void fraction s volume s0 volume of unit cell ~x -fields introe x - and b parameters for calculating the B kC, kT duced in Eqs. (32) and (33) m unit normal vector r excess surface concentration function; or parameter defined by Eq. (10) rm mth local moment of the function r long-time zero-order moment of the function r r0,1 h excess temperature function defined by Eq. (10) hm mth local moment of the function h h0,1 long-time zero-order moment of the function h / angular coordinate W, w parameters defined by Eq. (43) Subscripts b bulk solid eff effective g gas p pore s surface

tion of isolated pores can be calculated using the Maxwell approach [29]. In [27,28] this method has been used to assess the relative influence of the segregation–diffusion heat transfer mechanism in model ceramic materials and its importance with respect to the bulk and pore-phase conductivities. It has been shown that the contribution of the segregation–diffusion mechanism to the pore-phase conductivity increases with increasing temperature and becomes comparable with the thermal conductivity of air at atmospheric pressure. Applying the concept of pore-phase conductivity to materials containing micro-cracks, the behavior of effective thermal conductivity of such materials in vacuum had been predicted and reasonable agreement with experimental results had been achieved [7]. Yet when pores are uniformly distributed within the material bulk, the effect of species surface-segregation and diffusion is small [7]. This is no longer true in situations when the pores are concentrated in specific regions, thereby impeding heat transfer. In such cases porosity contributes significantly to the material overall thermal resistance and its effective thermal properties. One example of such materials is metal oxide ceramics, wherein chains of pores exist in the grain boundary region between poorly sintered grains [13]. At room temperature surface diffusivity of the segregated impurities is low, and the contribution of the heat transmitted by surface segregation–desegregation mechanism is negligible. However with increasing temperature the bulk thermal conductivity kbs decreases in accordance with Eucken law [6]: kbs = A/T (with parameter A dependent on the material structure). In contrast, the surface diffusivity of the segregated species is amplified when T increases. This together with the heat transferred through the grain boundaries due to segregation and desegregation of species on pore surfaces becomes dominant.

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In such circumstances even for pure materials, such as yttrrium oxide, keff can grow with increasing T. This is especially dramatically manifested in low gas pressure conditions, where conduction across pores volume is suppressed. It had been shown [6,27] that even a small amount of these defects can qualitatively change the temperature dependence of keff via their segregation and diffusion along surfaces of small pores (having size of about 1 lm) prevailing in the grain boundary region. This conjecture had received additional support by considering the kinetics of these species transport [28] (see discussion on this matter in Section 3). No model has so far been developed, which allows describing the influence of segregation–diffusion mechanism on the effective thermal conductivity of materials with arbitrary complex porous microstructure. This paper is aimed at investigation of contribution of segregation and diffusion processes to the effective thermal conductivity of materials with arbitrary microstructure. Towards this goal the macrotransport theory is extended to physical circumstances where heat transfer is associated both with transport of bulk thermal energy and also energy associated with diffusing and segregating species. The developed theory is used to evaluate the effect of the pores proximity in the grain boundary region on thermal conductivity of sintered ceramic materials in circumstances where the segregation–diffusion mechanism is important. 2. Effective thermal conductivity Consider dense homogeneous porous materials, for example sintered oxide ceramics without additives or binders wherein heat conduction both in the solid and gas phases occurs in the Fourier regime. These ceramic materials may contain micro-pores in the grain boundary region (see Fig. 1) with sizes however well above the phonon mean-free path, which is normally in order of 200– 400 Å [30]. Therefore, the effect of phonon scattering at the pore surfaces [30] on the effective thermal conductivity of material is negligible. Normally heat in the solid phase is transferred by conductance and radiation. The material porosity is considered to be low so that the effect of radiation (photon scattering) on keff is neg-

ligible. Phonon scattering on grain boundaries or other structural defects may diminish kbs. This is important for materials having significant amount of crystal lattice point defects (oxygen vacancies or impurities), as for example ceramics stabilized by foreign additives [31]. The amount of such defects in pure materials here considered is insignificant, so this process does not contribute to keff. Other processes, associated with ‘‘hopping’’ of oxygen vacancies also have a small effect on keff and in all cases they are negligible at temperatures exceeding 100 °C [31]. If necessary the effect of this mechanism may be incorporated in the T-dependence of the material solid-phase thermal conductivity kbs. The technique used here for calculation of thermal conductivity is related to a nonstationary method of thermal conductivity measurement. In contrast to the stationary methods it is aimed at measurement of thermal diffusivity aeff and subsequent determination of thermal conductivity via (2) using the heat capacity (qcp)eff. Such methods include heating at constant rate [32], hot wire [33,34], or laser flash method [5,33,35,36]. In the monotonous heating method thermal diffusivity is determined by monitoring the temperature difference between the lateral sides of the specimen and its center. In the flash method it is determined by monitoring of temperature at the back surface of the specimen. All these nonstationary processes may be viewed as a superposition of an infinite sequence of infinitesimally short heat pulses distributed within the specimen volume. The temperature field resulting from one pulse of unit total heat in a small domain near a point R0 of the specimen may be calculated by the basic conduction heat transfer equations. This fundamental unsteady solution T(R, tjR0 ) may be used as the Green function to construct any temperature field prevailing in conditions of a given nonstationary method. Therefore, the information about keff and hence, aeff is contained within this basic solution T(R, tjR0 ), representing conditional probability density of locating the so-called thermal energy tracer, which moves stochastically (diffuses) across the material. aeff may indeed be extracted from this solution through the application of the macrotransport theory via the Einstein-type definition [18]:

lim ðDR  DRÞ2 ¼ 2taeff ;

ð3Þ

t!1

where DR = R  R0 is the tracer displacement with respect to its initial position. For materials with internal energy determined by temperature only, the mean (marked with overbar) displacement and square displacement appearing above are [18]

Z

3

ðR  R0 ÞTðR; tjR0 Þd R; Z 3 ðDR  DRÞ2 ¼ ðR  R0  DRÞ2 TðR; tjR0 Þd R;

DR ¼

ð4aÞ

V

ð4bÞ

V

Fig. 1. Schematic of a spatially-periodic porous material.

where V denotes the total (infinite) volume of the material. As it will be shown below, for the materials here considered these definitions should be modified to include the internal energy transport associated with bulk and surface transport of impurities taking place within the material. In the presence of these segregation–diffusion processes the temperature field is coupled with the bulk (C, mol/m3) and surface (Cs, mol/m2) concentrations of impurities within the material. Calculation of effective thermal conductivity requires determination of temporal evolution of C and Cs fields due to the unit heat pulse. Since the segregation-desegregation processes are accompanied by the heat release and absorption, the internal energy is defined not only by the material temperature, and Eqs. (4a), (4b) should be modified accordingly. The heat impulse released at a moment t = 0 at a local point R0 , within the effectively infinite spatially-periodic porous material produces three nonuniform evolving fields: normalized temperature field T(R, tjR0 ), and the concomitant normalized concentration

T. Gambaryan-Roisman et al. / International Journal of Heat and Mass Transfer 54 (2011) 4844–4855

fields C(R, tjR0 ), and Cs(R, tjR0 ). The latter two result from the coupling between the heat and mass transport processes. Neglecting thermal diffusion and the diffusional thermal (Dufour) effect, and assuming the diffusion to be governed by the Fick law, these processes can be described by the following set of differential equations:

@T  r  ðk  rTÞ ¼ dðR  R0 ÞdðtÞ @t in the bulk phase ðsolid and gasÞ :

qcp

r 2 s0 ;

@C  r  ðD  rCÞ ¼ 0 in the solid phase : r 2 sbs ; @t @C s ¼ rs  ðDs  rs C s Þ þ m  D  rC; @t

on pores surfaces :

ð5Þ

r 2 sp ; ð7Þ

to be solved subject to the condition of continuity of T across the pore surface sp and the condition for the heat flux on sp:

ðm  k  rTÞbs  ðm  k  rTÞg þ Q m  D  rC ¼ 0; on pores surfaces :

r 2 sp :

ð8Þ

Here k is thermal conductivity either of the solid phase, kbs or the gas phase, kg = Ikg, where I is an identity tensor, D and Ds are bulk and surface species diffusivities; d (R  R0 ) and d (t) denote the Dirac delta-functions. m is the unit normal vector directed into the bulk, rs = (I  mm)  r is the surface gradient operator and Q is the segregation energy, measured in J/mol. The subscript ‘‘bs’’ denotes the quantities pertaining to the bulk solid, and the subscript ‘‘g’’ denotes the gas filling the pores A local equilibrium is assumed to prevail between the bulk and surface concentration of impurities at any temperature T, which implies that C and Cs are related by a segregation isotherm [27,28] (for example, the Arrhenius, Langmuir–McLean or Bragg– Williams isotherms [26]):

C s ¼ f ðC; TÞ;

r 2 sp ;

ð9Þ

Define the excess temperature, bulk and surface concentration functions:

h ¼ T  TI;

c ¼ C  CI ;

r ¼ C s  C sI ;

ð10Þ

with respect to the initial respective values TI, CI and CsI, and linearize the segregation equation in the form

r ¼ bC c  bT h; r 2 sp ;

ð11Þ

where

bC ¼ @f =@C

and bT ¼ @f =@T

ð12Þ

are both calculated at TI. Defined as above, the microscale functions h, c, and r and their gradients tend to zero as jR  R0 j ?1. The above equations form a basis for calculation of the effective thermal diffusivity according to Eqs. (2)–(4). This is done by generalization of the method of moments used in the simpler pure thermal or pure mass transport problems. The material microstructure is characterized by a certain representative region (unit cell) of volume s0, reproduced infinitely in space (spatially-periodic structure). The treatment of the problem (5)–(9) follows the general macrotransport paradigm [18] and is summarized in Appendix A. The solution may be expressed via two intracellular vector functions B(r) and b(r) obeying equations (A26)–(A31) and three scalar fields h0,1, c0,1, r0,1, respectively obeying the stationary forms of Eqs. (5)–(7). The effective thermal conductivity tensor is given by the following equation (see (A37)):

keff ¼

1 qcp h0;1 2

Z sp

½m  kbs  rðBBÞbs  kg m  rðBBÞg dsp ;

where qcp is the appropriately unit cell averaged product qcp. Eq. (13) will be analyzed in the following section. 3. Discussion To establish the main parameters governing keff in the presence of segregation–diffusion processes, introduce dimensionless variables:

~r ¼ ð6Þ

ð13Þ

4847

r ; rp

e ¼ B; B rp

~¼ b ; b rp C

ð14Þ

where rp is a characteristic pore size in the unit cell, and parameter C is

C¼1þ

sbs

ð15Þ

sp bC

Normally sbs/sp is of order of pore size rp (usually several microns), and bC is of order of surface segregation layer thickness d (several Angstroms). Hence C is a very large number (about 104) for typical values of the micro-geometrical parameters and the commonly used segregation isotherms. The dimensionless form of Eq. (A37) is

keff 1 1 ¼ ~0 kbs 2 s

Z ~0 @s

e ~s; m  K  rð Be BÞd

ð16Þ

where K = k/kbs and kbs is the characteristic value of the bulk solid conductivity. Furthermore,

s~0 ¼

s0 r 3p

;

~sp ¼

sp : r2p

ð17Þ

For uniform microscale material properties (see Eqs. (A15)– (A18)) the set of differential equations and boundary conditions ~ fields accept the e and b for determination of the dimensionless B forms (A38)–(A43), wherein in view of the relationship C  1, the third boundary condition in (A43) may be substituted by con~ ~ dition of periodicity of function bð r Þ. As such, the dimensionless ~ ~ e ~ fields Bð r Þ and bð r Þ depend on the following parameters:

Kg ¼

kg ; kbs

KT ¼

DS Q bT ; kbs r p

KC ¼

Ds bC : D rp

ð18Þ

Parameter KT characterizes the effect of heat transfer rate induced by the segregated and desegregated species across the pores with respect to the conductive heat transfer in their vicinity [27,28]. If KT  1, the former heat transfer rate is negligible. In this case o e ~ Bð r Þ combined with Eq. (16) yield thermal conductivity keff of a porous material due to solely heat conduction in the continuous phases. It will be shown in the following sections that keff increases with increasing KT. On the other hand, parameter KC determines the diffusion mass transfer rate in the surface layer relative to the bulk diffusion mass transfer around the pore [27,28]. The first describes the surface diffusion of the impurities from the cold to the hot side of the pore. The latter is responsible for the return of the impurities species to the cold side of the pore via the near-pore region to maintain the steady-state heat and mass transfer regime. Parameter KC characterizes the limiting role of the bulk diffusion in the process of impurities transfer into the surface layer. Therefore, keff decreases with increasing of KC. In the limit of very large values of KC the problem reduces to classical thermal conductivity in porous material. Expressions (13) and (16) for the effective thermal conductivity tensor are valid for arbitrary unit cell structure. It may contain grains of different compositions, pores of different sizes and shapes and distances between each other, micro-and macrocracks. Expression (16) is tested in the dilute limit e  1, that is, for a material

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with uniformly distributed cylindrical and spherical pores, separated from each other by large distances. In this case the volumetric fraction of pores is small, Towards this goal consider first a homogeneous material of constant thermal conductivity of the bulk solid, kbs, with square array of cylindrical pores with radius a. In this dilute limit the square unit cell, in which the set of equations and boundary conditions (A38)–(A43) is to be solved, can be substituted by the equivalent area circle with external radius pffiffiffi rp = e. The solution is sought in the form

e ~rÞ ¼ er Bð e ~rÞ; Bð

~ ~r Þ; ~ r~Þ ¼ er bð bð

ð19Þ

e ~rÞ where er is a unit vector in cylindrical coordinate system, and Bð ~ ~rÞ are scalar functions of radius ~r measured from the pore and bð center. This solution form implies that the boundary conditions (A43) can be replaced by

e ¼ 1= B

pffiffiffi

pffiffiffi

e; b~ ¼ 0; ~r ¼ 1= e:

ð20Þ

Solving Eqs. (A38)–(A42), (20) and neglecting the terms of order e2 e ~rÞ: one obtains the following expressions for Bð

   pffiffiffi e ~rÞ ¼ 1 þ e j  1 ~r þ j  1 1 ; 1 < ~r 6 1= e; Bð ~ jþ1 jþ1 r   j 1 2~r e Bð~rÞ ¼  1 þ e ; 0 < ~r 6 1; jþ1 jþ1

ð21Þ ð22Þ

where

j¼

kg KT þ : kbs K C þ 1

ð23Þ

Substituting the expressions (19), (21), (23) into Eq. (16), one obtains the following expression for the effective thermal conductivity:

  j1 keff ¼ kbs 1 þ 2e I: jþ1

ð24Þ

In a similar manner one can calculate the effective thermal conductivity of a homogeneous material with a cubical array of spherical pores and e  1:

  j1  1 I; keff ¼ kbs 1 þ 3e j1 þ 2

ð25aÞ

where

j1 ¼

kg 2K T þ : kbs K C þ 1

ð25bÞ

Eqs. (24) and (25) are identical to the expressions for keff, derived for e  1 invoking the intuitive Maxwell approach [27–29]. Analysis of these equations shows that keff increases with increasing parameter KT and decreases with increasing KC, which behavior accords with the physical interpretation of these quantities discussed at the beginning of this section. When KT  1, or KT  KC, the contribution of the species segregation and diffusion to keff is negligible. Alternatively, this mechanism dominates heat transfer in circumstances where KT  1 and KT  KC. Referring to Eq. (18) one can quantify this situation by the following inequalities

Application of the general theory developed here for calculation of keff of materials with more complicated pore geometry, including pore chains and cracks, normally requires numerical solutions of boundary value problems in complicated spatial regions. The central point in the implementation of the general theory developed above is determination of the material representative volume (unit cell) in which the boundary value problems for h0,1, c0,1, r0,1, B and b should be solved. This should adequately reflect the material microstructure in terms of appropriate statistical characteristics, such as pores and grains distributions. One approach to this task is porous medium computer reconstruction, which has been employed for calculation of the effective permeability of geological materials (sandstones) [37]. Existence of the effective thermophysical properties of a composite material is associated with the certain time tcell which should pass after the initial heat pulse release. In circumstances outlined above, where the segregation–diffusion processes contribute negligibly to the effective thermal conductivity, tcell is governed by the intracellular heat diffusion, i.e., tcell;T ¼ s2=3 0 =a0 , where a0 is the characteristic value of the material bulk thermal diffusivity (normally lowermost of the solid and gas thermal diffusivity). In materials characterized by Ds  a0 (heat transfer is controlled by surface segregation and diffusion) tcell is dominated by the surface diffusivity, which constitutes the slowest link in the heat transfer path. The characteristic time scale of this double-diffusive heat 2=3 transfer is clearly t cell;C ¼ s0 =Ds . In view of the relationship Ds  a0 one can see that tcell,C  tcell,T of the measurement time needed for achievement of the true (stationary) thermal conductivity value relative to the measurements performed in conditions where the segregation–diffusion processes are suppressed and do not affect the heat transfer. The above considerations are confirmed experimentally. Thermal conductivity of a Y2O3 specimen measured by the stationary method at the gas pressure p = 102 Pa has slowly drifted during the first 10–15 h of measurement until the stationary value of keff has been reached [7,28]. In contrast for atmospheric gas pressure, when the intrapore heat transferred due to the segregated species is negligible compared to the comparable heat flow through the gas filling the pores, this stationary value is already established after the first 2–3 h of measurement. The temporal variation of the apparent thermal conductivity has been estimated using the Maxwell approach by assuming that the temperature field is quasi-stationary so that the bulk and surface concentrations of impurities slowly change with time [28]. This treatment may be generalized using the macrotransport approach, where the governing Eqs. (5)–(9) are solved by the multiple time scale analysis for tcell,T  t < tcell,C, where the limit t ? 1 in Eq. (3) will have the meaning t  tcell,T. One of the important structural parameters affecting keff is pore chains existing in the grain boundary region [13]. In the next section this specific problem is addressed for materials where segregation and diffusion of impurities significantly affect the heat transfer (such as sintered Y2O3).

ð26Þ

4. Effective thermal conductivity of materials containing chains of pores

That is, the contribution of segregation and surface diffusion to keff is significant in materials with low kbs, high bulk solid species diffusivity D, high surface diffusivity of impurities Ds, high segregation energy Q, small pore sizes and strong temperature dependence of the segregated species concentration on pores. Further insight into the dependence of the criteria (26) upon the physical parameters may be gained by selecting an appropriate surface segregation isotherm (see Section 4).

Here the application of the general model developed in the previous section is illustrated by calculating keff of a material having chains of cylindrical pores of radii rp on the grain boundaries (see Fig. 1). This structure is typical for sintered Y2O3 ceramics [25]. A simplified model of such material structure is sketched in Fig. 2, where the grains are slabs of thickness L and infinite in two other dimensions. If the grain size L is much larger than the distance l between the centers of the pores, interaction between the adjacent

KT ¼

Ds QbT  1; kbs r p

DQbT  1: kbs bC

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4849

Fig. 2. Schematic of a porous material with cylindrical pores chains.

pore chains can be neglected, and the problem of calculation of keff can be reduced to calculating the fields of temperature and bulk and surface concentrations in infinite medium with a single chain of pores. Here keff is calculated in the case where the pores are evacuated so the heat and mass transfer are calculated only in the solid phase and on the pore surfaces. The solution involves systematic application of the general scheme outlined in Appendix A (Eqs. (A38)–(A43)) to the material geometry shown in Fig. 2 under the general assumption L  l. The mathematical details of the solution are summarized in Appendix B. As a result, the x-component of the effective thermal conductivity tensor has the following form (see B23):

ðkeff Þxx ¼ kbs

4pmT r 2p 1þ Ll

!1 ¼ kbs ð1 þ 4emT Þ1 ;

ð27Þ

where e ¼ pr2p =ðLlÞ is the material porosity and the parameter mT is determined from the solution of the system of equations (B10)– (B16). As in the case where the pores are homogeneously distributed, one can see that the effect of the surface segregation heat transfer is lumped together with porosity which contains the parameter ~l ¼ l=r p , characterizing the proximity of the pores. If mT < 0, then, according to Eq. (27), the effective thermal conductivity increases with increasing porosity due to strong contribution of segregation-surface diffusion mechanism into the overall heat transfer. In the limit of small e mT is given by Eq. (B20) where one can see that condition mT < 0 is equivalent to KT > KC + 1. The heat transferred by the segregation–diffusion mechanism comes mainly from the diffusion of species in the vicinity of the pore chains in the grain boundary region. Therefore values of bulk diffusivity and bulk concentration of impurities should be taken from this region [27]. Figs. 3 and 4 depict keff calculated for dense Y2O3 with the pertinent thermophysical data taken from [13,24,27]. The effective thermal conductivity is calculated for the Langmuir–McLean segregation isotherm [13]:

bs ¼ C

ð30ÞbT ¼

(a)

b expðQ =RTÞ C ; b expðQ=RTÞ 1þC

ð28Þ

b ¼ CM=q and C b s ¼ C s M are measured in where the concentrations C qd mole fractions, and T is the physical temperature (K). Herein M and q are the molar mass and the density of the material, and d is a surface layer thickness. For Y2O3 the molar mass of the bulk solid is M = 0.226 kg/mol, q = 5.02  103 kg/m3 and the segregation layer b s paramthickness d is taken 10 nm [24]. With the above function C eters bC and bT introduced in Eq. (12) accept the form

@C s d expðQ=RTÞ bC ¼ ; ¼ b expðQ =RTÞ2 @C ½1 þ C

ð29Þ

(b)

Fig. 3. keff/kbs ratio as a function of pores density within the pore chain. (a) KC = 10; b ¼ 0:01; rp =L ¼ 0:05; Ds ¼ Ds0 expðEact =RTÞ; Ds0 ¼ (b) KC = 100. Q ¼ 75 kJ=mol; C 0:0606 m2 =s; Eact ¼ 41:0 kJ=mol. 1 – 1100 K, 2 – 900 K, 3 – 700 K, 4 – in the absence of segregation–diffusion processes.

Using the latter relationship, one can re-write criteria (26) in the following form

b qb Ds Q 2 C C 2

kbs RT Mr p

 1;

^q DQ 2 C kbs RT 2 M

 1;

ð31Þ

@C s @T

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(a)

(b)

Fig. 4. Thermal conductivity of material with cylindrical pores chains on grain b ¼ 0:01; r p =L ¼ 0:05; Ds ¼ boundaries. (a) KC = 10; (b) KC = 100. Q ¼ 75 kJ=kg; C Ds0 expðEact =RTÞ; Ds0 ¼ 0:0606 m2 =s; Eact ¼ 41:0 kJ=mol. 1: 2rp/l = 0.5 (e = 3.93  102); 2: 2rp/l = 0.7 (e = 5.50  102); 3: 2rp/l = 0.9 (e = 7.07  102).

allowing quantifying physical circumstances, when the segregationsurface diffusion of species controls heat transfer in porous solids. One can see that in addition to the conclusion drawn on the bases of Eq. (26), the effect of surface segregation is directly proportional b , present in the bulk to the absolute amount of diffusing species C material. The results are presented in Figs. 3 and 4. The following parameters have been used for calculations: the segregation energy Q ¼ 75 kJ=mol; r p =L ¼ 0:05, the average bulk concentration of b ¼ 0:01. The temperature dependence for the surface impurities C diffusion coefficient has been taken in the form Ds = Ds0exp (-Eact/ RT), with Ds0 = 0.0606 m2/s, Eact = 41.0 kJ/mol, as for diffusion of oxygen vacancies in Y2O3[24]. The bulk solid conductivity of Y2O3 is taken as kbs = 6720/T W/(m.K) [27]. The average pore size in the grain boundary region is taken 1 lm [6]. The values of factor KC have been taken equal to 10 [(Fig. 4(a)] and 100 [Fig. 4(b)]. This factor depends on the surface diffusivity, Ds, and bulk diffusivity in the grain boundary region, D (see Eq. (18)). These quantities for diffusing oxygen vacancies within Y2O3 can be specified only to within several orders of magnitude [13,27]. Using literature data together with typical relationships between surface and grain boundary diffusivities, one can approximately evaluate that (dDs)/(rpD) = O(1) [13,27]. This yields the values of KC ranging between zero and about 200. The trend of temperature dependence of KC depends on relation between the activation energies of Ds and D. In the present work the influence of KC on the effective ther-

mal conductivity of the material is analyzed without referring to the dependence of this parameter upon temperature. Fig. 3 shows the effective thermal conductivity in the range 0.4 < 2rp/l < 0.9 for which the material porosity range is 3.14  102 < e < 7.07  102. In spite of small values of porosity, the material thermal conductivity significantly differs from kbs, especially at relatively large values of 2rp/l. This significant influence of transport processes in the pore region on keff is attributed to the chain-type pores arrangement (Fig. 2) which constitute a significant resistance to heat flow. One can see that for a dilute pores arrangement where 2rp/l < 0.6 (e < 4.71  102), the influence of the pore chains on keff is insignificant (keff does not differ substantially from kbs). Clearly the effect of the segregation-surface diffusion transfer is manifested via the material microstructure, embodied within 2rp/l and the material temperature. For low temperatures the surface diffusivity is small and the concomitant parameter KT (see Eqs. (12) and (18)) is also small. This describes the situation where the segregation–diffusion mechanism is suppressed. As a result the thermal conductivity diminution is entirely determined by the materials microstructrure. In this situation with increasing pore density (increasing 2rp/l) less solid material remains that conducts heat. Therefore keff decreases with 2rp/l. Ds increases with T, thereby triggering the segregation–diffusion heat transfer. This is manifested by keff increasing with T. Furthermore, the effect of surface segregation–diffusion is proportional to the relative surface area, embodied in the parameter 2rp/l. The heat thus transferred along pore surface offsets the shortage of bulk solid area (acting to diminish keff). For sufficiently large temperatures this mechanism becomes dominant, causing keff to grow with increasing T to the extent that at 1100 K it exceeds kbs. This trend, however takes place only for sufficiently low KC (see Fig. 3a), when sufficient amount of species can be supplied by bulk diffusion to the pore surface area. In the limit of large KC (see Fig. 3b) slow bulk diffusion acts as a bottle neck, diminishing the contribution of heat transfer along pore surface. In this situation keff decreases with 2rp/l for all temperatures. The above trends are illustrated in Fig. 4a,b depicting the dependence of keff upon temperature. The material is assumed to possess the microstructure as schematically depicted in Fig. 2 with rp/ L = 0.05. The 2rp/l ratio, which determines the density of the pores arrangement within the chain, takes the values 0.5 (the pores are relatively far from each other), 0.7 and 0.9 (very dense arrangement). The bulk solid thermal conductivity kbs decreases with T, according to the Eucken law, and keff also follows this trend. At higher temperatures the contribution of the heat transfer along pore surfaces is significant and may lead to increasing of keff with temperature due to increase of KT (see Fig. 4a, curve 3). When Ds reaches its maximal value (at about 800 °C), characteristic of the Arrhenius-like temperature dependence, further increase of T does not incur any increase of KT – a parameter controlling the effect of segregation-surface diffusion on keff. Accordingly for higher temperatures keff again decreases with increasing T. When the bulk diffusivity in the near-boundary region is low (KC is large, Fig. 4b) the effect of segregation–diffusion mechanism on keff is less pronounced, and does not incur nonmonotonic dependence of keff on temperature. The above described increase of keff with T is consistent with the experimentally collected trend of keff(T) reported in [25,27] for Y2O3 – based ceramic with the total porosity of 7  102. This porosity value corresponds to our factor 2rp/l = 0.9 if all the material porosity is confined in the grain boundary area. Increase of the measured keff with temperature is clearly observed at temperatures 500–600 °C. This is somewhat lower than the growth of keff shown in Fig. 4a for KC = 10. The data currently available on the segregation – surface diffusion parameters do not enable sufficiently

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accurate evaluation of this parameter, which affects both absolute values of keff and its T-dependence. In order to directly compare the results of the present model with the experimental results, a much more elaborate characterization of material is required. The knowledge of the material porosity which is normally provided as a characteristic value of material microstructure is by far insufficient for predicting effective thermal conductivity. For example, referring to materials with pore chains at the grain boundary shown in Fig. 4a, one can see that change of porosity from e = 3.93  102 to e = 7.07  102 leads to a more than 33% decrease of keff at 200°C. In contrast, in a material with spherical pores uniformly distributed within the material this increase in porosity leads only to a 4.7% reduction of keff. Therefore, the present paper stresses the importance of adequate material characterization, where the porosity is described not only by the total volume of pores but also their distribution, especially on grain boundaries. Also our research shows the importance of the reliable data on the impurities’ concentration for adequate modelling of keff. More accurate experimental data are needed to fully assess the predictive capacity of the model developed. However, the general scheme proposed in this work and the governing physical parameters identified allow to properly design the requisite experiments and to rationalize the data to be collected. Material porosity can also affect keff by means of radiation. This mechanism can affect keff beginning from temperatures exceeding several hundred degrees competing with and even prevailing over the segregation–diffusion mechanism [7]. Other heat transfer mechanisms responsible for leveling off the diminution of keff with T in dense ceramics are photon scattering due to oxygen vacancies, provided these are present in significant amounts in the material bulk, as for example in yttria stabilized zirconia [31]. For low porosity case e L; ~l  1 (dilute square array of cylinders) expression (27) reduces to (see Eq. (B18))

 1 K T þ K C þ 1 keff ¼ kbs 1 þ 2e KT þ KC þ 1   K T þ K C þ 1  kbs 1  2e ; KT þ KC þ 1

ð32Þ

which is identical to expression (23) and (24) upon substituting in Eq. (23) kg = 0. This agrees with the Maxwell type expression for keff derived in [27,28]. For multiphase material with randomly distributed pores with complicated shapes and variety of sizes the calculation of effective thermal conductivity requires adequate material characterization, including determination of statistical properties of the pores and phases distributions, and numerical solution of the boundary value problem formulated in this paper.

Formulae allowing calculation of keff for a fairly general material microstructure are developed and applied to materials having small amounts of isolated pores and pore chains. The effect of increasing grain boundary porosity is shown to diminish keff for weak segregation and surface diffusion processes (small KT and/ or large KC). On the other hand, for strong segregation (large value of KT and small value of KC) a reverse trend is predicted, namely keff increases with growing porosity. The theoretically calculated non-monotonic temperature dependence of keff is consistent with the experimental data collected for Y2O3 – based ceramic with the total porosity of 7  102 [25,27]. The currently available knowledge of the transport properties of the segregation species does not enable sufficiently accurate evaluation of the model parameters, which affects both absolute values of keff and its T-dependence. Accurate experimental data are needed to fully assess the predictive capacity of the model developed. However, the proposed general scheme and the identified governing physical parameters allow to properly design the requisite experiments and to rationalize the data to be collected. Appendix A. Derivation of expression for effective thermal conductivity The material microstructure is modeled by spatially-periodic geometry. This implies that the material is composed of identical elementary cells (see Fig. 1). The position vector of each point relative to a fixed origin, R, can be represented in the form R = Rn + r = n1l1 + n2l2 + n3l3 + r, where the set (l1, l2, l3) of three non-complanar vectors is termed basic lattice vectors, and r denotes the local position vector of any interstitial point within the cell (n1, n2, n3). One of such possible simplified geometric models is shown in Fig. 1. The elementary cell volume s0 consists of the bulk solid volume, sbs, and the pore volume, sp, separated by the pore surface of total area sp i.e. sbs + sp = s0. As such the material porosity is e = sp/s0. The thermal conductivities of the bulk solid material and the gas filling the pores are both described by a function k(r). Similarly, D(r) and Ds(r), are the bulk and surface diffusivities of the segregated species. These properties are normally scalars (isotropic tensors). All these functions, as well as the mass density q(r), the specific heat, cp(r), are spatially periodic. In most important cases these functions may be assumed piecewise constant, as, for example, in multiphase materials, wherein each phase has uniform properties. Then the thermal conductivity of the bulk solid is denoted by kbs, and the thermal conductivity of the gas filling the pores is denoted by kg. A.1. Local moments

5. Summary and conclusions The developed macrotransport model of heat transfer in porous composite materials provides a generic scheme for calculating the effective thermal conductivity keff in circumstances where the heat transfer is affected by segregation and diffusion of impurities. This situation is quantified in terms of parameters KT and KC, containing the species diffusivities in the bulk and along pore surfaces. For the Langmuir–McLean segregation isotherms the condition when pore surface segregation and diffusion mechanism is significant and governed by inequalities (31), is formulated in terms of the material thermodynamic and heat-mass transfer parameters. The contribution of segregation and surface diffusion to keff is significant in materials with low kbs, high bulk solid species diffusivity D, high surface diffusivity of impurities Ds, small pores and significant amount of impurities available within the bulk material. This contribution is however independent of the pore size.

Define the mth local moments of the functions h, c, and r within a unit cell as

hm ðr; tjr 0 Þ ¼

X

Rn  R0n

m   h Rn  R0n ; r; tjr0 ;

m

n

¼ 0; 1; 2; . . .

ðA1Þ

with similar definitions for cm and rm. We assume that the microscale Green function h, c and r tend to zero when jRn  R0n j ! 1 sufficiently fast so that the sums in the definitions of hm, cm and rm converge. Combining Eq. (A1) and Eq. (5), we have the following differential equations governing the local moments hm, m = 0, 1, 2, . . .:

qcp

@hm ¼ r  ðk  rhm Þ ¼ dmo dðr  r 0 ÞdðtÞ; @t

r 2 so ;

ðA2Þ

In addition to this the functions hm, cm, and rm should satisfy Eqs. (6), (7), (8), (11) with relevant substitutions.

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To formulate boundary conditions for hm, cm, and rm on the cell boundary @ s, consider an arbitrary local tensor field A(r). Define a ‘‘jump’’ sAt in the value of this function at geometrically equivalent points, lying on the opposite ith faces of the surface of the unit cell as

In particular,

sAti ¼ Aðr þ li Þ  AðrÞ;

dN0 ¼ 0 ) N 0  const ¼ 0: dt

i ¼ 1; 2; 3:

ðA3Þ

In terms of this jump function the following boundary conditions should be satisfied [10]:

srh0 ti ¼ 0;

sh0 ti ¼ 0;

sh1 ti ¼ srh0 ti ;

ðA4Þ

srh1 ti ¼ srðrh0 Þti ;   srh2 ti ¼ sr h1 h1 h1 ti ; 0

sh1 h1 h1 0 ti ; 1

sh2 ti ¼

ðA5Þ ðA6Þ

which can be derived from the continuity of the Green functions and their gradients across the unit cell faces. Similar conditions should be satisfied by the zero-, first-and the second-order local moments of the bulk and surface concentrations fields. A.2. Total moments

Mm ¼

qcp hm d3 r  Q

so

Z

rm dsp ;

ðA7Þ

sp

and the total concentration moments:

Nm ¼

Z

3

qc m d r þ

so

Z

rm dsp :



0; t < 0

ðA8Þ

ðA11Þ

:

1; t P 0

ðA12Þ

The constant in Eq. (A12) is identically equal to zero because the value of the integral N0, as calculated from Eq. (A8) for m = 0, is equal to zero for t < 0 and does not change with time for t P 0. Equation (A11) reflects the energy conservation for the system. Equation (A12) expresses the conservation of matter within the system. A.3. Zero-order moments In the following treatment we shall extensively use the longtime zero-order moments, h0,1(r), c0,1(r), r0,1(r), which can be easily shown to be stationary fields. The zero-order moments satisfy the differential equation

r 2 so ;

r  ðk  rh0;1 ðrÞÞ ¼ 0;

Define the total energy moments:

Z

dM 0 ¼ dðtÞ ) M0 ¼ dt

ðA13Þ

the set (6), (7), (8), (11) and boundary conditions (A4) with relevant substitutions. This set should be complemented by the normalization condition:

M0;1 ¼

Z

qcp h0;1 ðrÞd3 r  Q

so

Z

r0;1 ðrÞdsp ¼ 1;

ðA14Þ

sp

sp

The zero-order energy moment, M0, is equal to the total excess internal energy that the material gains as a result of heat release (see below Eq. (A14)). The first integral on the right hand side of Eq. (A7) describes the excess energy of the bulk and the gas phase, and the second integral describes the surface energy. The ‘‘minus’’ sign before the second integral reflects the fact that the chemical potential of the segregated substance on the surface is lower than in the bulk (which is a cause of segregation). The first energy moment M1 is associated with energy displacement from the origin, R0n . The second energy moment is associated with the energy dispersion (see below Eq. (A33)) [14]. The zero-order concentration moment, N0, is equal to the total excess quantity of the impurities in the material, including the impurities within the bulk (first integral) and on the pores surface (the second integral). Clearly, N0  0, which is a consequence of the mass conservation law. N1 and N2 are associated with displacement and dispersion of species, respectively. The time derivatives of the energy moments can be calculated using Eqs. (A1), (A2), (A7), (7), and (8), and the Gauss theorem:

Z

Z

dM m d d qcp hm d3 r  Q rm dsp ¼ dt s0 dt sp dt Z Z 3 ¼ r  ðk  rhm Þd3 r þ dm0 dðr  r0 ÞdðtÞd r s0 s0 Z Q m  D  rcm dsp sp Z Z ¼ m  k  rhm ds  ½m  kbs  rðhm Þbs @ s0 sp Z  kg m  rðhm Þg dsp  Q m  D  rcm dsp þ dm0 dðtÞ sp Z X m  k  srhm tds þ dm0 dðtÞ: ¼ i¼1;2;3

which follows from Eqs. (A7), (A11). This represents the energy conservation condition, where the first term in the right hand side of Eq. (A14) is the internal energy associated with the bulk, and the second term is the internal energy associated with the surface. Consider the case where the material properties are homogeneous in the bulk and within the pore:

kðrÞ ¼



kg I; r 2 sp ; 

qðrÞcp ðrÞ ¼ DðrÞ  D;

ðA15Þ

r 2 sbs ;

kbs ;

qg cpg ; r 2 sp ; qbs cpbs ; r 2 sbs ;

ðA16Þ

r 2 sbs ;

ðA17Þ

r 2 sp ;

ðA18Þ

Ds ðrÞ  Ds ;

where kg, kbs, qg, qbs, cpg, cpbs, D and Ds are constant values. The long-time zero-order local moments are

h0;1 ¼ c0;1 ¼

1

s0 q c p

ðA19Þ

;

bT sp

1

ðA20Þ

;

sbs þ bC sp s0 qcp b s 1 r0;1 ¼  T bs ; sbs þ bC sp s0 qcp

ðA21Þ

where

qcp ¼ eqg cpg þ ð1  eÞqbcbp þ Q

ðA9Þ

bT sp ð1  eÞ : sbs þ bC sp

ðA22Þ

Usually, the third term in right-hand side of Eq. (A22), describing the contribution of segregation into the effective specific heat of the solid, is very small.

si

A.4. First-order moments

The time derivatives of the concentration moments are:

X Z dN m m  k  srcm tds ¼ dt i¼1;2;3 si

ðA10Þ

The local first-order moments are defined by the system of equations (5), (6), (7), (8), (11), (A5). The steady state solution of this system can be written in the form

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h1;1 ¼ h0;1 ðrÞBðrÞ;

ðA23Þ

c 1;1 ¼ c0;1 ðrÞBðrÞ;

ðA24Þ

r1;1 ¼ bC c0;1 ðrÞbðrÞ  bT h0;1 ðrÞBðrÞ;

ðA25Þ

where B(r) and b(r) are vector fields to be determined below. In the case of constant material properties (see Eqs. (A15)– (A18)), the vector fields B(r) and b(r) satisfy the following set of equations and boundary conditions:

r2 B ¼ 0;

r 2 s0 ;

ðA26Þ

r2 b ¼ 0;

r 2 sbs ;

ðA27Þ

2 S ðbC c 0;1 b

DS  r

 bT h0;1 BÞ þ c0;1 m  D  rb ¼ 0;

r 2 sp ;

ðA28Þ

h0;1 ðm  k  rBÞbs  h0;1 ðm  k  rBÞg þ c0;1 Q m  D  rb ¼ 0; r 2 sp ;

ðA29Þ

sBt ¼ srt;

srBt ¼ 0;

sbt ¼ srt;

srbt ¼ 0;

ðA30Þ B continuous on sp

ðA31Þ

The total first-order long-time energy and concentration moments, M1,1 and N1,1 are identically equal to zero, which can be easily shown by substituting Eqs. (A19)–(A21) into Eqs. (A9) and (A10) with m = 1. This means that no macroscopic translational motion of energy and concentration occurs within the material. A.5. Second-order moments

dM 2 ¼ dt i¼1;2;3

i¼1;2;3



m  k  sr

si

h1 h1 h1 0

 tds:

t!1

ðA32Þ

X Z dM 2 m  k  srh2;1 tds ¼ dt i¼1;2;3 si  X Z m  k  sr h1;1 h1;1 h1 ¼ 0;1 tds i¼1;2;3

¼ h0;1

m  k  srðBBÞtds:

ðA33Þ

si

A.6. Effective thermal conductivity The effective thermal diffusivity can be calculated from the definition (3). However, in the presence of segregation–diffusion phenomenon the expressions (4a), (4b) should be modified to account for the contribution of the concentration term to the internal energy:

Z

3

ðR  R0 ÞhðRn  R0n ; r; tjr0 Þd R Z   ðR  R0 Þr Rn  R0n ; r; tjr0 dSp ; Q V

ðA34Þ

Sp

ðDR  DRÞ2 ¼

Z

  3 ðR  R0  DRÞ2 h Rn  R0n ; r; tjr 0 d R V Z    Q ðR  R0  DRÞ2 r Rn  R0n ; r; tjr0 dsp :

ðA35Þ

Combining Eq. (4a) with (A34), (A35), one obtains the expression for the effective thermal diffusivity via the moments:

aeff ¼

~ 2B e ¼ 0; r

~r 2 s ~0 ;

ðA38Þ

~ ¼ 0; ~ 2b r

~r 2 s ~bs ;

ðA39Þ

e ; e ¼ ð BÞ ð BÞ bs g

~r 2 ~sp ;

ðA40Þ

~ ¼ 0; r~ 2 ~sp ; e  K g m  ðr e þ KT m  r ~ BÞ ~ BÞ ~b m  ðr bs g KC 1 ~ ~2e ~ ¼ 0; ~r 2 ~sp ; ~ 2b ~b r mr s  rs B þ KC ~ ¼ 0; ~ ¼ sr~=Ct; sr e ¼ s~r t; sr ~ Bt e ¼ 0; sbt ~ bt s Bt

ðA41Þ ðA42Þ ðA43Þ

1 dðR  RÞ2 1 dðM 2  M 1 M 1 Þ lim : ¼ lim 2 t!1 2 t!1 dt dt

Write down the x-component of the equations set (A38)–(A43) with C  1 and Kg = 0 for a unit cell having a shape of a rectangle ~ 6 ~l=2; e with length L and width l  L; s0 ¼ f~l=2 6 y L=2 6~ x6e L=2g, with a pore of radius rp at the center of the cell (see Fig. 2):

~ 2B e x ¼ 0; r

si

X Z

i¼1;2;3

DR ¼

Equation (A37) is a fundamental result of the macrotransport analysis of heat transfer in a porous material in the presence of segregation–diffusion processes. It gives a straightforward way of calculation of effective thermal conductivity tensor for any given geometry of the unit cell and any material properties. ~ rendered dimensionless as per e and b With the vector fields B Eq. (14), Eqs. (A26)–(A31) adopts the following form:

Appendix B. Calculation of keff for materials with chains of pores

The long-time moment takes the form

lim

ðA37Þ

m  k  srh2 tds

si

X Z

¼

X Z 1 qcp h0;1 m  k  srðBBÞtds 2 i¼1;2;3 si Z 1 ¼ qcp h0;1 ½m  kbs  rðBBÞbs  kg m  rðBBÞg dsp : 2 sp

keff ¼ qcp aeff ¼

where the constants Kg, KC and KT are given in Eq. (18), and the ~ are defined in Eq. (14). e and b dimensionless variables ~ r; B

Equation (A9) for m = 2 takes the form:

X Z

Using the fact that M1  0 and Eq. (A32), we have for the effective thermal conductivity:

ðA36Þ

~r 2 s0 ; ~r P 1; 2~ ~ r bx ¼ 0; ~r 2 s0 ; ~r P 1; ~x ~x K T @ b @B þ ¼ 0; ~r ¼ 1; ~ @ r K C @~r ~x @ 2 B ~x ex @2b 1 @b  þ ¼ 0; ~r ¼ 1; 2 2 K C @~r @/ @/ ~x t ¼ 0; ~B e x t ¼ 0; sb e x t ¼ s~xt; sr sB

ðB1Þ ðB2Þ ðB3Þ ðB4Þ ~x t ¼ 0; ~b sr

ðB5Þ

(see Eq. (A3) for the definition of the jump operator [j. . .j]), where all coordinates are normalized by rp and corresponding dimensionless quantities are marked by tilde, the subscript x denotes the x-component of the vector. Similar set of equations, with subscript x replaced by y and the e y t ¼ sy ~t, can be written first condition of (B5) replaced by s B ~ e down for the y-component of the B-and b-fields. The fields ~x ð~ e x ð~ ~Þ and b ~Þ, satisfying the set (B1)–(B5), should be even B x; y x; y ~y ð~ e y ð~ ~, and the fields B ~Þ and b ~Þ should be even functions of y x; y x; y functions of ~ x. Construct the solution of Eqs. (B1) and (B2) for e L  ~l in the form

 1 4pmT e x ð~x; y e ð0Þ  ½~x þ mT ReZð~x; y ~Þ ¼ B ~ B ; k Þ 1 þ ; T x ~le L 1

 4pmT ~ð0Þ  4pmC ~x þ mC ReZð~x; y ~x ð~x; y ~Þ ¼ b ~ ; kC Þ 1 þ ; b x ~le ~le L L

ðB6Þ ðB7Þ

~ð0Þ are arbitrary constants, Re denotes the real part e xð0Þ and b where B x of complex number, the function Z is given by

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( ) i sinh½pð~z  ikÞ=~l ~; kÞ ¼ log ; Zð~x; y k sinh½pð~z þ ikÞ=~l

ðB8Þ

~z ¼ ~ ~ is a complex variable, and the constants mT, mC, kT and kC x þ iy are real numbers to be determined below. Function Z is a superposition of dipoles of equal dipole moment, distributed over the inter~ ¼ n~l; n ¼ 0; 1; 2; : . . .. vals k 6 ~x 6 k; y It can be easily checked that the expression in the brackets appearing as argument of the logarithm function in (B8) approaches expði2pk=~lÞ at ~ x  ~l and expði2pk=~lÞ at ~ x  ~l. Therefore, the difference

~; kÞ  Zð~x  ~l; y ~; kÞ ¼ Zð~x  ~l; y

4p ~l

ðB9Þ

is constant. Solution (B6) and (B7) satisfies the Laplace equations (B1) and ~x and the boundary conditions (B5). This solution e x and b (B2) for B contains four arbitrary constants, to be determined from conditions (B3) and (B4). The form of solution (B6) and (B7) does not allow to satisfy the boundary conditions (B3) and (B4) exactly at each point on the cylinders surfaces, 0 6 / 6 p. We satisfy these conditions at the point / = 0 and in integral sense from / = 0 to / = p/2. The resulting expressions for determination of the constants mT and mC are

mT ¼

K T  K C wðkC Þ  1

kT

v ðkT Þ K T wðkT Þ þ K C wðkC Þ þ 1 þ w½K C  K T wðkT Þ  1

;

kC K C 1 þ wðkT Þ : mC ¼ v ðkC Þ K T wðkT Þ þ K C wðkC Þ þ 1 þ w½K C  K T wðkT Þ  1

ðB10Þ ðB11Þ

Parameters kT and kC have to be determined from the following transcendental equations:

1

K T  K C wðkC Þ  1

v ðkT Þ K T wðkT Þ þ K C wðkC Þ þ 1 þ w½K C  K T wðkT Þ  1 ¼

1 K T  K C WðkC Þ  1 ; VðkT Þ K T WðkT Þ þ K C WðkC Þ þ 1 þ W½K C  K T WðkT Þ  1

1

ðB12Þ

1 1 þ WðkT Þ ; VðkC Þ K T WðkT Þ þ K C WðkC Þ þ 1 þ W½K C  K T WðkT Þ  1

ðB13Þ

where

( ) 2p sinð2pk=~lÞ sin½ð1  kÞp=~l ; v ðkÞ ¼ ~ ; VðkÞ ¼ log l cosð2pk=~lÞ  coshð2p=~lÞ sin½ð1 þ kÞp=~l ðB14Þ wðkÞ ¼ 1 

WðkÞ ¼ w¼

2p sinhð2pk=~lÞ ; ~l cosð2pk=~lÞ  coshð2p=~lÞ

1 p ½cosð2p=~lÞ  cosð2pk=~lÞ sinð2pk=~lÞ ; VðkÞ 2~l sin2 ½ð1  kÞp=~l sin2 ½ð1 þ kÞp=~l

4pkC 4pkC ; W¼ : ~le ~le L mðkC Þ LVðkC Þ

ðB15Þ ðB16Þ

In the dilute (low porosity) limit ~l  1, one obtains from (B8) that

~; kÞ ¼ 2 ReZð~x; y

cos / ; ~r

1   ~x ð~x; y ~ð0Þ  cos / 4pmC ~r þ 2mC =~r 1 þ 4pmT ~Þ ¼ b b x ~le ~le L L ~ ~ 4 e r þ 2= r ð0Þ ~  cos /mC ¼b : x 1 þ 4emT

ðB18Þ

ðB19Þ

In this case the boundary conditions (B3) and (B4) can be satisfied on the entire pore surface. Introducing the expressions (B18) and (B19) into (B3) and (B4), we obtain two linear equations for mT and mC, which, accounting for the smallness of porosity, have the following solutions:

1 KC  KT þ 1 ; 2 KC þ KT þ 1 KC : mC ¼ KT þ KC þ 1 mT ¼

ðB20Þ ðB21Þ

The xx-component of the effective thermal conductivity is calculated by (cf. Eq. (16)):

ðkeff Þxx 1 1 ¼ ~0 2s kbs

Z ~ 0 @ tau



@

@



mx ~ ð Be x Be x Þ þ my ~ ð Be x Be y Þ d~s; @x @x

ðB22Þ

which after simple transformations reduces to the expression

2 3 ! ! Z ~ ex ex ðkeff Þxx 1 l=2 4 e @ B @B e 5dy ~ Bx ¼  Bx ~le kbs @ ~x @ ~x L ~l=2 e ~x¼e ~ x¼ L=2 L=2  1 4pmT ¼ 1þ ¼ ð1 þ 4emT Þ1 ; ~le L

ðB23Þ

valid for any value of ~l satisfying the condition e L  ~l. In the dilute limit upon substitution of the value of mT from Eq. (B20) into Eq. (B23) a Maxwell-type expression for the effective thermal conductivity is obtained [27]. References

1 þ wðkT Þ

v ðkC Þ K T wðkT Þ þ K C wðkC Þ þ 1 þ w½K C  K T wðkT Þ  1 ¼

 1 e x ð~x; y e ð0Þ  cos /ð~r þ 2mT =~rÞ 1 þ 4pmT ~Þ ¼ B B x ~le L ~ ~ = r r þ 2m T ð0Þ e ¼ B x  cos / 1 þ 4emT

ðB17Þ

which corresponds to a single dipole. In this limit the solutions (B6) and (B7) take the following form:

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