Effective thermal conductivity of three-component composites containing spherical capsules

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Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/authorsrights Author's personal copy International Journal of Heat and Mass Transfer 73 (2014) 177–185 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt Effective thermal conductivity of three-component composites containing spherical capsules Alexander M. Thiele a, Aditya Kumar b, Gaurav Sant b,c, Laurent Pilon a,⇑ a Mechanical and Aerospace Engineering Department, Henry Samueli School of Engineering and Applied Science, University of California, Los Angeles, USA Civil and Environmental Engineering Department, Laboratory for the Chemistry of Construction Materials (LC2), Henry Samueli School of Engineering and Applied Science, University of California, Los Angeles, CA 90095, USA c California Nanosystems Institute (CNSI), Los Angeles, CA 90095, USA b a r t i c l e i n f o Article history: Received 27 November 2013 Received in revised form 25 January 2014 Accepted 1 February 2014 Available online 3 March 2014 Keywords: Effective medium approximation Composite materials Three-phase media Phase change materials Microballoons Composite spheres a b s t r a c t This paper presents detailed numerical simulations predicting the effective thermal conductivity of spherical monodisperse and polydisperse core–shell particles ordered or randomly distributed in a continuous matrix. First, the effective thermal conductivity of this three-component composite material was found to be independent of the capsule spatial distribution and size distribution. In fact, the study established that the effective thermal conductivity depended only on the core and shell volume fractions and on the core, shell, and matrix thermal conductivities. Second, the effective medium approximation reported by Felske (2004) [21] was in very good agreement with numerical predictions for any arbitrary combination of the above-mentioned parameters. These results can be used to design energy efficient composites, such as microencapsulated phase change materials in concrete and/or insulation materials for energy efficient buildings. Ó 2014 Elsevier Ltd. All rights reserved. 1. Introduction In 2010, building operations accounted for 41% of total US primary energy resource consumption [1]. Approximately, half of this energy was consumed for heating, ventilation, and air conditioning (HVAC) [1]. A common strategy to improve building energy efficiency is to use materials with a large thermal mass, e.g., concrete or brick [2,3]. While these materials can store large amounts of energy per unit mass, they operate passively, demonstrating only a sensible heat response [2,3]. To add an active or temperature sensitive dimension to the thermal behavior of building materials, there is interest in embedding phase change materials (PCMs) in building elements [4–7]. By reversibly undergoing solid–liquid phase transitions in relation to the temperature of their local environment, PCMs are able to actively and adaptively absorb and release latent heat required to induce phase transitions. These actions further enhance the thermal inertia of building systems. As such, if properly implemented, PCMs embedded in building materials can limit thermal exchange through exterior walls, reducing the need and cost for HVAC operations, and thus improving building energy efficiency. ⇑ Corresponding author. Tel.: +1 (310) 206 5598; fax: +1 (310) 206 2302. E-mail address: [email protected] (L. Pilon). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.02.002 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved. The incorporation of PCMs (e.g., paraffin waxes, hydrated salts, or fatty acids) in building composites is facilitated by encapsulating the PCMs in a polymeric shell [6,5,4,7]. This serves to isolate the PCM from high pH chemical environments common to building materials, thus enhancing durability and limiting contamination [4–7]. When PCMs are embedded in a cementitious material, the resultant composite consists of three distinct components in the form of matrix (often cement-based), shell (often polymer-based), and PCM (often organic in nature). Clearly, this is a complex threecomponent composite material whose effective thermal properties must be predicted accurately to estimate heat transfer across composite building walls. This study aims (1) to rigorously predict the effective thermal conductivity of three-component core–shell composite materials (2) to identify the controlling design parameters and (3) to derive design rules for composite walls. The results of this study could also be applicable to other multicomponent composites including self-healing microcapsule-doped polymers [8] and hollow glass microsphere-embedded syntactic foams [9], to name a few. 2. Background Numerous models have been derived to predict the effective thermal conductivity of two-component composites as reviewed by Progelhof et al. [10], for example. Comparatively, few models Author's personal copy 178 A.M. Thiele et al. / International Journal of Heat and Mass Transfer 73 (2014) 177–185 Nomenclature A parameter in Eq. (4) Ac cross-sectional area, m2 B parameter in Eq. (4) CD centroidal distance between two proximal capsules, lm D diameter, lm k thermal conductivity, W/m K L unit cell length, lm N number of unit cells n normal unit vector p number of spherical capsules in a unit cell r radius, lm q00x ; q00y ; q00z heat flux along the x-, y-, and z-directions, W/m2 00x q area-averaged heat flux along the x-direction, W/m2 ts thickness of capsule shell, i:e. t s ¼ ðDs  Dc Þ=2; lm T temperature, K To; TL temperature at x ¼ 0 and x ¼ L, K ratio of shell diameter to core diameter, d ¼ Ds =Dc volume fraction of phase ‘‘i’’ in the composite structure volume fraction of core in the capsule, /c=s ¼ /c =ð/c þ /s Þ volume fraction of capsules in the composite structure, /cþs ¼ /c þ /s volume fraction of closely-packed capsules numerator and denominator of the Felske model (Eq. (3)) d /i /c=s /cþs /max HN ; HD Subscripts c refers to core cþs refers to core–shell particle cr refers to the critical thermal conductivity ratios eff refers to effective properties m refers to matrix s refers to shell Greek symbols b parameter in Eq. (9) Dx minimum mesh size, lm exist for three-component composites [11–21]. Several models were developed for liquid and gas phases in a porous solid matrix such as building materials or soil [17,18]. Other models require prior knowledge of the temperature gradients in each component of the composite to determine the effective thermal conductivity [13,14]. The most practical models provide explicit analytical expressions for the effective thermal conductivity of three-component composites based on the constituent thermal conductivities and on the geometric parameters of the composite structure such as core and shell diameters and/or volume fractions. Lichtenecker [20] proposed an ad hoc expression for the electrical permittivity of a composite consisting of any number of randomly mixed components [22]. Woodside and Messmer [22], among others, have applied this model to the effective thermal conductivity of three-component composites expressed as [20,22,23], / / / keff ¼ kc c ks s kmm ð1Þ where kc ; ks , and km are the thermal conductivities of the core, shell, and matrix materials, respectively. Similarly, /c ; /s , and /m ¼ 1  /c  /s , are the volume fractions of the core, shell, and matrix materials, respectively. Woodside and Messmer [22] referred to Eq. (1) as a ‘‘geometric mean’’ and noted that it corresponds to the arithmetic mean of the logarithms of the constituent thermal conductivities. Zakri et al. [23] analytically derived Lichtenecker’s [20] model (Eq. (1)) for the effective electrical permittivity of three-component composites. They concluded that Eq. (1) is ‘‘physically founded,’’ despite criticism from Reynolds and Hough [24] who suggested that the model ‘‘lacked a theoretical basis.’’ Note that Eq. (1) predicts that keff vanishes if the thermal conductivity of either the core or the shell vanishes. This is obviously not the case since heat conduction could still take place through the continuous matrix material. Brailsford and Major [19] developed a model for the effective thermal conductivity of monodisperse homogeneous particles randomly distributed in a continuous matrix. This two-component model was equivalent to the Maxwell–Garnett model for electrical conductivity [25]. Brailsford and Major [19] extended the twocomponent model to account for monodisperse homogeneous particles made of two different materials randomly distributed in a continuous matrix. Then, the effective thermal conductivity of three-component media was given by [19], keff ¼ m m km /m þ kc /c ð2k3k þ ks /s ð2k3k m þkc Þ m þks Þ ð2Þ m m /m þ /c ð2k3k þ /s ð2k3k m þkc Þ m þks Þ Model predictions for two-component media agreed well with experimental data for the effective thermal conductivity of solid glass spheres surrounded by air or water [19]. However, experimental validation was not reported for three-component composite materials. Felske [21] derived a model, using the self-consistent field approximation [26], to predict the effective thermal conductivity of monodisperse spherical capsules randomly distributed in a continuous matrix. This effort was motivated by the need to estimate the effective thermal conductivity of syntactic foam insulation. The geometry considered in the derivation consisted of a spherical volume of matrix material containing a concentric core–shell particle with volume fractions representative of the overall composite. The model accounted for contact resistance at the shell-matrix interface. An exact series solution of the heat conduction equation was obtained for the temperature distribution in each phase. In absence of contact resistance, the model can be expressed as [21], keff ¼ HN km HD ð3Þ Here, the numerator HN and denominator HD are expressed as [21], HN ¼ 2 1  /cþs A þ 1 þ 2/cþs B and

HD ¼ 2 þ /cþs A þ 1  /cþs B

ð4Þ where the parameters A and B are given by [21], ! ! 1 kc /c=s ks A¼ 1þ 2 /c=s B¼ 2þ ! ! 1 kc 1 ks 2 1 /c=s km /c=s km  1 and ð5Þ Here, /cþs is the volume fraction of the composite occupied by the capsule and /c=s is the volume fraction of the core with respect to the capsule. 3 They are expressed as /cþs ¼ ðDs =Dm Þ3 and /c=s ¼ ðDc =Ds Þ where Dc ; Ds , and Dm are the diameters of the core, shell, and matrix domains, respectively. The volume fraction of core–shell capsules /cþs can be written as /cþs ¼ /c þ /s . Pal [12] Author's personal copy 179 A.M. Thiele et al. / International Journal of Heat and Mass Transfer 73 (2014) 177–185 noted that the Felske model [21] ‘‘generally describes thermal conductivity data well when the core–shell volume fraction /cþs is less than about 0.2,’’ but no evidence was provided to demonstrate this claim. Park et al. [11] also developed a model predicting the effective thermal conductivity of monodisperse spherical capsules randomly distributed in a continuous matrix based on a two-step approach. First, the effective thermal conductivity of the twocomponent core–shell capsule denoted by kcþs was modeled based on the core and shell thermal conductivities and on the volume fraction of core with respect to the core–shell composite /c=s . It was based on a modified Eshelby effective medium approximation (EMA) [27,28] and expressed as [11],     2 1  /c=s ks þ 1 þ 2/c=s kc    kcþs ¼  ks 2 þ /c=s ks þ 1  /c=s kc 3. Analysis 3.1. Schematics ð6Þ The effective thermal conductivity keff of the three-component composite was then expressed based on the core–shell effective thermal conductivity kcþs , the matrix thermal conductivity km , and the core– shell volume fraction /cþs as [11],

2 1  /cþs km þ 1 þ 2/cþs kcþs

keff ¼ km 2 þ /cþs km þ 1  /cþs kcþs The aim of this study is to predict and to identify the dominant parameters controlling the effective thermal conductivity of threecomponent composite materials. To do so, detailed ‘‘numerical experiments’’ were performed to investigate the effects of (1) core and shell dimensions and volume fractions, (2) spatial distribution of the capsules, (3) size distribution of the capsules, and (4) core, shell, and matrix thermal conductivities. The results were compared with the previously reviewed EMAs to identify the most appropriate one and its range of validity. The present study examined various composite representative volumes consisting of different packing arrangements of monodisperse and polydisperse spherical capsules distributed in a continuous matrix. Fig. 1 shows three-component unit cells with q y '' ð7Þ ð8Þ T1 z T2 y x qx ''

2 þ d3 kkmc  2 1  d3 kkms

1 þ 2d3  1  d3 kkcs L q y '' 0 0 (b) where /cþs;max is the maximum capsule volume fraction for a given packing arrangement and b was expressed as [12], b¼ 0 0 qx '' After careful consideration, combining Eqs. (6) and (7) led to the Felske model [21] given by Eqs. (3)–(5). Pal [12] developed an implicit model to predict the effective thermal conductivity of three-component composites of monodisperse spherical capsules randomly distributed in a continuous matrix. This model was derived using the differential effective medium approach [29]. The resulting model was an implicit function of the volume fraction of capsules expressed as [12],  1=3    /cþs;max keff b1 /cþs ¼ 1 b  keff =km km /cþs;max qz '' (a) q y '' qz '' 0 qz '' 0 0 qx '' T1 z T2 ð9Þ where d is the shell to core diameter ratio, i.e., d ¼ Ds =Dc or d3 ¼ /1 c=s . The model accounted for the upper limit of the capsule volume fraction /cþs;max corresponding to close packing. Predictions by Eqs. (8) and (9) were reported to agree well with experimental data for thirteen different samples of two-phase media for ‘‘reasonable values’’ of /cþs;max [12]. However, /cþs;max was taken as 0.7, 0.85, or 1 which seems arbitrary and large. Indeed, the maximum volume fraction reaches 0.74 for face-centered cubic packing and 0.6 for randomly distributed monodisperse solid spheres [30]. Note that in the case of composite building materials with PCM, the capsule volume fraction is typically much smaller than the packing limit, as large volume fractions could compromise the mechanical strength of the wall [7]. Overall, several EMAs have been proposed in the literature for the effective thermal conductivity of three-component composite materials consisting of monodisperse capsules in a continuous matrix. However, these models are significantly different from one another and their validation against experimental data has been limited mainly to two-component media. Therefore, it remains unclear which one of these models is the most appropriate and accurate. In addition, to the best of our knowledge, no study has rigorously investigated the effects of the capsules’ spatial and size distributions on the effective thermal conductivity of three-component composites. y x qx '' L q y '' 0 0 (c) q y '' qz '' 0 qz '' 0 0 qx '' T1 z T2 y x qx '' L q y '' 0 0 qz '' 0 Fig. 1. Schematic and computational domain of a single unit cell consisting of capsules distributed in a continuous matrix with (a) simple, (b) body-centered, and (c) face-centered cubic packing arrangement. Core and shell diameters and unit cell length corresponding to core and shell volume fractions /c and /s were denoted by Dc ; Ds , and L, respectively. Author's personal copy 180 A.M. Thiele et al. / International Journal of Heat and Mass Transfer 73 (2014) 177–185 (a) simple, (b) body-centered, and (c) face-centered cubic packing arrangements along with the associated Cartesian coordinate system. The inner core and outer shell diameters were given by Dc and Ds , respectively with shell thickness ts ¼ ðDs  Dc Þ=2, and the length of the unit cell was denoted by L. For any packing arrangement of monodisperse capsules, the core and shell volume fractions /c and /s were expressed as,   /c ¼ ppD3c and /s ¼ 6L3 pp D3s  D3c 3.3. Governing equations and boundary conditions ð10Þ 6L3 where p is the number of spherical capsules per unit cell. It was equal to 1, 2, and 4 for simple, body-centered, or face-centered cubic arrangements, respectively. To study the effects of the capsule’s size and spatial distributions in detail, a microstructural stochastic packing algorithm was implemented [31]. This algorithm considered a size distribution corresponding to an average outer shell diameter Ds of 18 lm with 10th and 95th percentile diameters equal to 9 lm and 33 lm, respectively, and a shell thickness t s of 1 lm. It placed spherical capsules in a 3D representative volume of arbitrary size until the desired core phase volume fraction was achieved. Microstructural generation and packing was performed such that the minimum centroidal distance C D between two proximal capsules was always greater than the sum of their radii r 1 and r 2 ; i.e., C D > r 1 þ r 2 . The packing algorithm placed capsules at random locations in the volume in accordance with two packing rules: (1) the size and number of capsules maintained the desired size distribution and (2) the desired core phase volume fraction was achieved within 0.5%. Fig. 2 shows examples of computational volumes consisting of 38 to 61 monodisperse or polydisperse capsules randomly distributed in a continuous matrix. Fig. 2(a)–(d) correspond to cases 3, 6, 9, and 10 summarized in Table 1, respectively. 3.2. Assumptions To make the problem mathematically tractable, the following assumptions were made: (1) steady-state heat conduction (a) prevailed. (2) All materials were isotropic and had constant properties. (3) There was no heat generation. (4) Interfacial contact resistance was neglected, and (5) phase change and natural convection in the core phase were absent. This last assumption stemmed from the fact that even if microcapsules were filled with liquid (e.g. molten PCM) the Rayleigh number would be small. (b) Under the above assumptions, the local temperatures in the core, shell, and matrix denoted by T c ; T s , and T m were governed by the steady-state heat diffusion equation in each domain, given by, r2 T c ¼ 0; r2 T s ¼ 0; and r2 T m ¼ 0 ð11Þ These equations were coupled through the boundary conditions. Heat conduction took place mainly in the x-direction of the unit cell or representative cube (Figs. 1 or 2) by imposing the temperature on the faces of the cube located at x ¼ 0 and x ¼ L such that for 0 6 y 6 L and 0 6 z 6 L, T ð0; y; zÞ ¼ T o and T ðL; y; zÞ ¼ T L ð12Þ By virtue of symmetry, the heat flux through the four lateral faces vanished, i.e., q00y ðx; 0; zÞ ¼ q00y ðx; L; zÞ ¼ 0 and q00z ðx; y; 0Þ ¼ q00z ðx; y; LÞ ¼ 0 ð13Þ where q00y ðx; y; zÞ and q00z ðx; y; zÞ are the heat fluxes along the y- and zaxes, respectively. They are given by Fourier’s law, i.e., q00y ¼ k@T=@y and q00z ¼ k@T=@z. The boundary temperatures on the faces x ¼ 0 and x ¼ L were taken as T o ¼ 294 K and T L ¼ 292 K. Coupling between the temperatures of the different domains was achieved by imposing continuous heat flux across their interfaces, i.e., km @T m @T s ¼ ks @n m=s @n m=s and  ks @T s @T c ¼ kc @n s=c @n s=c ð14Þ where n is the unit normal vector at any given point on the matrix/ shell and shell/core interfaces, designated by subscript m=s and s=c, respectively. 3.4. Data processing y Based on Fourier’s law, the effective thermal conductivity of the core–shell composite medium was computed from the imposed temperature difference along the x-direction, the domain length 00x along the x-direction accordL, and the area-averaged heat flux q ing to, z y x (c) z x (d) 00x L q TL  To 00x ðxÞ ¼ where q 1 Ac ZZ q00x ðx; y; zÞdy dz ð15Þ Here, Ac is the cross-sectional area of the computational domain perpendicular to the x-axis. Due to the heterogeneous nature of the composite medium the heat flux was not uniform over a given cross-section perpendicular to the x-axis. However, it was system00x ðxÞ was the same atically verified that the area-averaged heat flux q at any cross-section between x ¼ 0 and x ¼ L. 3.5. Method of solution z y keff ¼  z x y Fig. 2. Computational cells containing monodisperse p ¼ 39; L ¼ 75 lm, /c ¼ 0:198, and /s ¼ 0:041, and (c) /c ¼ 0:105, and /s ¼ 0:045, as well as polydisperse p ¼ 38; L ¼ 75 lm, /c ¼ 0:197, and /s ¼ 0:075, and (d) /c ¼ 0:095, and /s ¼ 0:035. x capsules with (a) p ¼ 49; L ¼ 100 lm, capsules with (b) p ¼ 61; L ¼ 100 lm, The governing equation (11) along with the boundary conditions given by Eqs. (12)–(14) were solved using finite element methods. The numerical convergence criteria was defined such that the maximum relative difference in the predicted local area00x ðxÞ was less than 0.5% when reducing the averaged heat flux q mesh size by a factor of 2. Converged solutions were obtained by Author's personal copy 181 A.M. Thiele et al. / International Journal of Heat and Mass Transfer 73 (2014) 177–185 Table 1 Numerical and analytical predictions of the effective thermal conductivity of composites consisting of monodisperse or polydisperse capsules randomly distributed in a continuous matrix. The average outer diameter and thickness of the shell are Ds;av g ¼ 18 lm and ts ¼ 1 lm, respectively for all cases. Monodisperse Polydisperse Monodisperse Monodisperse Monodisperse Polydisperse Polydisperse Polydisperse Monodisperse Polydisperse Numerical Eq. (17) p L (lm) /c /s kc (W/m K) ks (W/m K) km (W/m K) keff (W/m K) keff (W/m K) 19 22 39 39 39 38 38 38 49 61 75 75 75 75 75 75 75 75 100 100 0.097 0.095 0.198 0.198 0.198 0.197 0.197 0.197 0.105 0.095 0.041 0.041 0.084 0.084 0.084 0.075 0.075 0.075 0.045 0.035 0.21 0.21 0.21 10 100 0.21 10 100 10 50 1.3 1.3 1.3 100 10 1.3 100 10 20 10 0.4 0.4 0.4 30 30 0.4 30 30 50 20 0.41 0.41 0.42 30.17 33.41 0.41 29.68 33.85 42.83 21.30 0.41 0.41 0.42 30.17 33.42 0.41 29.72 33.78 42.89 21.28 imposing the minimum mesh size to be Dx ¼ ðDs  Dc Þ=4 and the maximum growth rate to be 1.5. The number of finite elements needed to obtain a converged solution ranged from 12,873 to 1,451,237 depending on the size of the computational cell and on the core and shell dimensions. In order to validate the computational tool, a unit cell containing capsules with face-centered cubic packing arrangement was simulated with the same boundary conditions given by Eqs. (12)–(14) but assuming ks ¼ kc ¼ km . As expected, the predicted area-averaged heat flux at x ¼ L fell within 0.5% of Fourier’s law gi00x ðxÞ ¼ km ðT o  T L Þ=L for L ¼ 20:3 lm, km ¼ 0:4 W/m K, ven by q T o ¼ 294 K, and T L ¼ 292 K. Note also that the area-averaged heat 00x ðxÞ was the same at any cross-section along the x-direction. flux q (a) 4. Results and discussion 4.1. Effect of capsule dimensions and packing arrangement Fig. 3 shows the effective thermal conductivity keff for domains comprised of 1 to 20 stacked unit cells for simple, body-centered, and face-centered cubic packing arrangements (Fig. 1). Two sets of volume fractions were considered: (i) /c ¼ 0:25 and /s ¼ 0:1 and (ii) /c ¼ 0:05 and /s ¼ 0:0165. The diameters Dc and Ds and the unit cell length L were adjusted with each packing arrangement to achieve the desired volume fractions. The core, shell, and matrix thermal conductivities were taken to be kc ¼ 0:21 W/m K (b) Fig. 3. Effective thermal conductivity for linear arrays of N unit cells with two different combinations of volume fractions /c and /s and packing arrangements SC, BCC, and FCC. Core, shell, and matrix thermal conductivities were kc ¼ 0:21 W/m K, ks ¼ 1:3 W/m K, and km ¼ 0:4 W/m K, respectively. % Difference 0.02 0.02 0.01 0.00 0.03 0.11 0.14 0.22 0.14 0.10 [32], ks ¼ 1:3 W/m K [33], and km ¼ 0:4 W/m K [34], respectively. These values correspond to paraffin wax PCM in silica shells embedded in cement. Fig. 3 establishes that keff was independent (i) of the number of stacked unit cells, as expected from symmetry considerations, and (ii) of the choice of packing arrangement. The same conclusions were reached for different volume fractions. Therefore, a single unit cell with a face-centered cubic packing arrangement will be considered in the remainder of this study as Effective thermal conductivity, keff (W/m.K) 1 2 3 4 5 6 7 8 9 10 Input Effective thermal conductivity, keff (W/m.K) Size distribution 0.42 Dc fixed L fixed 0.40 0.38 0.36 0.34 0.32 0.30 s 0.28 0.0 = 0.025 0.1 0.2 0.3 0.4 Volume fraction of core, c 0.5 0.80 0.75 0.70 0.65 Ds fixed L fixed Lichtenecker model Brailsford model Felske model 0.60 0.55 0.50 0.45 0.40 c 0.35 0.0 0.1 0.2 0.3 0.4 Volume fraction of shell, s = 0.05 0.5 Fig. 4. Effective thermal conductivity for (a) different values of /c with /s ¼ 0:025 and (b) different values of /s with /c ¼ 0:05. The volume fractions were varied by adjusting either the diameter or unit cell length. Here, kc ¼ 0:21 W/m K, ks ¼ 1:3 W/m K, and km ¼ 0:4 W/m K. Predictions by the Lichtenecker, Brailsford, and Felske models are also shown. Author's personal copy representative of any composite media consisting of ordered monodisperse capsules. Fig. 4 shows the effective thermal conductivity keff of a composite containing monodisperse capsules as a function of (a) the core volume fraction /c ranging from 0.0 to 0.55 for a constant shell volume fraction of /s ¼ 0:025 and (b) the shell volume fraction /s ranging from 0.0 to 0.55 for a constant core volume fraction of /c ¼ 0:05. The desired volume fractions were imposed by either adjusting the relevant diameter ðDc or Ds Þ while holding the unit cell length L constant or by adjusting the unit cell length L and holding the relevant diameter constant. Here also, the core, shell, and matrix thermal conductivities were taken as kc ¼ 0:21 W/ m K, ks ¼ 1:3 W/m K, and km ¼ 0:4 W/m K, respectively. Fig. 4(a) and (b) establish that keff depended only on /c and /s and not on the individual geometric parameters Dc ; Ds , and L. Overall, this section demonstrated that the effective thermal conductivity of a composite material containing monodisperse capsules was a function only of five parameters namely, the volume fractions /c and /s and the constituent material thermal conductivities kc ; ks , and km ; i:e., keff ¼ keff ð/c ; /s ; kc ; ks ; km Þ. 4.2. Effect of core and shell volume fractions For any packing arrangement of monodisperse spherical capsules the term /c=s used in Eqs. (5) and (6) can be written in terms of /c and /s so that, 1 / ¼1þ c /s /c=s ð16Þ Then, the Felske model [21], given by Eq. (3) can be written in terms of /c and /s as,   h  i 2km ð1  /c  /s Þ 3 þ 2 //s þ //s kkcs þ ð1 þ 2/c þ 2/s Þ 3 þ //s kc þ 2 //s ks c c c c   h   i keff ¼ ð2 þ /c þ /s Þ 3 þ 2 //cs þ //sc kkcs þ ð1  /c  /s Þ 3 þ //cs kkmc þ 2 //cskkms ð17Þ Similar operation can also be performed for the models proposed by Park [11] and Pal [12]. Thus, the EMAs previously reviewed satisfy the relationship keff ¼ keff ð/c ; /s ; kc ; ks ; km Þ. However, it remains unclear which one accurately predicts the effective thermal conductivity retrieved from detailed numerical simulations based on Eq. (15). Fig. 4 compares the effective thermal conductivity keff of a composite containing monodisperse capsules retrieved numerically with that predicted by the Lichtenecker [20], Brailsford [19], and Felske [21] models given respectively by Eqs. (1), (2), and (17) as a function of (a) the core volume fraction /c for /s ¼ 0:025 and (b) the shell volume fraction /s for /c ¼ 0:05. Here also, kc ¼ 0:21 W/m K, ks ¼ 1:3 W/m K, and km ¼ 0:4 W/m K, respectively. Fig. 4(a) and (b) indicate that keff decreased as /c increased and increased as /s increased, for the values of kc ; ks , and km considered. More importantly, they indicate that predictions by the Felske model (Eq. (17)) fell within 0.5% of numerical predictions, i.e., within numerical uncertainty. The other models underpredicted keff by 2.4% to 5.4% for the values of /c ; /s ; kc ; ks , and km considered. These relative errors are expected to increase as the thermal conductivity mismatch between the three phases increases. 4.3. Effect of constituent thermal conductivities Fig. 5 plots the effective thermal conductivity keff of a composite material containing monodisperse capsules as a function of matrix thermal conductivity km ranging from 1 to 50 W/m K for volume fractions /c ¼ 0:2 and /s ¼ 0:145 and two combinations of core and shell thermal conductivities, namely, (i) kc ¼ 5 W/m K and Effective thermal conductivity, keff (W/m.K) A.M. Thiele et al. / International Journal of Heat and Mass Transfer 73 (2014) 177–185 40 Lichtnecker model, Eq. (1) Brailsford model, Eq. (2) Felske model, Eq. (17) 35 30 25 20 kc = 10 W/m.K ks = 30 W/m.K 15 10 kc = 5 W/m.K ks = 10 W/m.K 5 0 0 5 10 15 20 25 30 35 40 45 50 Thermal conductivity of matrix, km (W/m.K) Fig. 5. Effective thermal conductivity keff of core–shell composite as a function of the thermal conductivity of the continuous phase km obtained numerically and predicted by the Lichtenecker, Brailsford, and Felske models given by Eqs. (1), (2) and (17), respectively. The volume fractions of core and shell were /c ¼ 0:2 and /s ¼ 0:145. ks ¼ 10 W/m K and (ii) kc ¼ 10 W/m K and ks ¼ 30 W/m K. This study demonstrated that predictions of keff by the Felske model (Eq. (17)) fell within 0.3% of the numerical predictions for km up to 500 W/m K (see supplementary material). On the other hand, predictions by the Brailsford model (Eq. (2)) and the Lichtenecker model (Eq. (1)) underpredicted keff by up to 3% and 60%, respectively. The discrepancies between these model’s predictions and numerical simulations increased with increasing km . Fig. 6 plots the effective thermal conductivity keff of a composite containing monodisperse capsules as a function of the core thermal conductivity kc ranging from 1 to 500 W/m K for volume fractions /c ¼ 0:2 and /s ¼ 0:145 and two combinations of matrix and shell thermal conductivities, namely, (i) km ¼ 5 W/m K and ks ¼ 10 W/m K and (ii) km ¼ 10 W/m K and ks ¼ 30 W/m K. Fig. 6 demonstrates that predictions by the Felske model (Eq. (17)) fell within 0.2% of the numerical predictions while the other models deviated by more than 10% for the values of km and ks considered. Fig. 6 also indicates that keff asymptotically reached a plateau as kc increased. This can be attributed to the fact that as kc becomes Effective thermal conductivity, keff (W/m.K) 182 30 ks = 30 W/m.K km = 10 W/m.K 25 20 ks = 10 W/m.K km = 5 W/m.K 15 10 Lichtnecker model, Eq. (1) Brailsford model, Eq. (2) Felske model, Eq. (17) 5 0 0 100 200 300 400 500 Thermal conductivity of core, kc (W/m.K) Fig. 6. Effective thermal conductivity keff of core–shell composite as a function of the thermal conductivity of the core phase kc obtained numerically and predicted by the Lichtenecker, Brailsford, and Felske models given by Eqs. (1), (2) and (17), respectively. The volume fractions of core and shell were /c ¼ 0:2 and /s ¼ 0:145. Author's personal copy A.M. Thiele et al. / International Journal of Heat and Mass Transfer 73 (2014) 177–185 much greater than ks and km , the temperature gradient throughout the core material vanishes. Then, the core provides negligible thermal resistance to heat conduction through the composite medium and thus does not affect keff . From a mathematical point of view, for kc  ks and kc  km , Eq. (17) simplifies to   2ð1  /c  /s Þ //sckkms þ ð1 þ 2/c þ 2/s Þ 3 þ //cs   keff ¼ ð2 þ /c þ /s Þ //c ks s þ ð1  /c  /s Þ 3 þ //cs k1m ð18Þ Similarly, Fig. 7 plots the effective thermal conductivity keff of a composite material containing monodisperse capsules as a function of shell thermal conductivity ks ranging from 1 to 500 W/m K for volume fractions /c ¼ 0:2 and /s ¼ 0:145 and two combinations of core and matrix thermal conductivities, namely, (i) kc ¼ 5 W/m K and km ¼ 10 W/m K and (ii) kc ¼ 10 W/m K and km ¼ 30 W/m K. Here also, the Felske model [Eq. (17)] agreed very well with the numerical predictions while the other models deviated by more than 33% for the values of kc and km considered. For ks  kc and ks  km ; keff asymptotically converged to a function independent not only of ks but also of kc given by, keff ¼ ð1 þ 2/c þ 2/s Þ km ð1  /c  /s Þ ð19Þ In this case, ks did not contribute to keff because the shell thermal resistance was negligible compared with that of the matrix. In addition, heat can be transferred through the capsule via two paths: through the shell and the core, or along the shell around the core. When ks  kc and ks  km , the latter path provided the least resistance to heat transfer. Then, the highly conducting shell thermally ‘‘short-circuited’’ the core and kc did not affect keff . As a result, keff was only a function of km . Finally, Figs. 5–7 show that the Felske model (Eq. (17)) predicted the effective thermal conductivity of composites containing monodisperse capsules within the estimated numerical uncertainty for all volume fractions /c and /s considered and for a wide range of thermal conductivities ks ; kc , and km . It remains to be shown whether this model is also valid for polydisperse and/or randomly distributed capsules. 4.4. Effect of capsule spatial and size distributions Effective thermal conductivity, keff (W/m.K) Ten composite structures consisting of monodisperse and polydisperse microcapsules randomly distributed in a continuous 183 matrix were generated as described previously. The number of capsules p in the computational domain ranged from 19 to 61 and the thickness of the shell was taken as t s ¼ 1 lm. Table 1 summarizes the different values of p; L; /c ; /s ; kc ; ks , and km considered in each case. It also compares the numerically predicted effective thermal conductivity keff of these composite microstructures to that predicted by the Felske model (Eq. (17)). Cases 1 and 2 indicate that the numerically predicted keff was the same for composites with monodisperse or polydisperse capsules for the same values of /c ; /s ; kc ; ks , and km . Table 1 also shows that keff predicted by the Felske model (Eq. (17)) fell within 0.25% of numerical predictions for a wide range of constituent thermal conductivities kc ; ks , and km and volume fractions /c and /s . In summary, these results established that the effective thermal conductivity of three-component composites consisting of capsules distributed in a continuous matrix was independent of capsule size distribution and of their spatial distribution. In all cases, the Felske model (Eq. (17)) predicted the effective thermal conductivity within numerical uncertainty. 4.5. Critical condition for effective thermal conductivity As previously mentioned, encapsulated PCM can be used to reduce and delay the thermal load in concrete buildings. However, the addition of PCM microcapsules should not increase the effective thermal conductivity keff of the composite wall meant to provide not only large thermal mass but also act as thermal insulation [7]. Based on Eq. (17), the critical core to matrix thermal conductivity ratio above which keff becomes larger than km can be written as, kc km   cr ¼   2 kkms  1  3 //cs km  1  3 //cs ks ð20Þ This expression can be used as a thermal design rule for core–shell composite materials with monodisperse or polydisperse and ordered or randomly distributed capsules. Fig. 8 plots the critical core to matrix thermal conductivity ratio ðkc =km Þcr given by Eq. (20) as a function of the shell to matrix thermal conductivity ratio ks =km . Four combinations of core and shell volume fractions were used, namely, (i) /c ¼ 0:4 and /s ¼ 0:191, (ii) /c ¼ 0:2 and /s ¼ 0:124, (iii) /c ¼ 0:1 and /s ¼ 0:082, and (iv) /c ¼ 0:05 and /s ¼ 0:054. All curves passed through the same point ð1; 1Þ corresponding to kc ¼ ks ¼ km ¼ keff . Each design curve represents the ensemble of conditions for which keff ¼ km . The area 60 Lichtnecker model, Eq. (1) Brailsford model, Eq. (2) Felske model, Eq. (17) 50 40 kc = 10 W/m.K km = 30 W/m.K 30 20 10 kc = 5 W/m.K km = 10 W/m.K 0 0 100 200 300 400 500 Thermal conductivity of shell, ks (W/m.K) Fig. 7. Effective thermal conductivity keff of core–shell composite as a function of the thermal conductivity of the shell phase ks obtained numerically and predicted by the Lichtenecker, Brailsford, and Felske models given by Eqs. (1), (2) and (17), respectively. The volume fractions of core and shell were /c ¼ 0:2 and /s ¼ 0:145. Fig. 8. Critical core and shell conductivity ratios ðkc =km Þcr and ðks =km Þ (a) for different values of matrix thermal conductivity km with /c ¼ 0:4 and /s ¼ 0:191 and (b) for core and shell volume fractions /c and /s . Author's personal copy 184 A.M. Thiele et al. / International Journal of Heat and Mass Transfer 73 (2014) 177–185 under the curve correspond to the desirable conditions for which keff is smaller than km . 4.6. Comparison with experimental data Several studies have experimentally measured the effective thermal conductivity of three-component core–shell composite materials [35–40]. Liang and Li [35] measured the effective thermal conductivity of polydisperse hollow glass microspheres randomly distributed in a polypropylene matrix. These measurements were then compared with numerical predictions using finite element methods [36]. Surprisingly, the measured effective thermal conductivity was larger than that of the individual constituent materials which cast doubt on the data. Other studies reported the effective thermal conductivity of three-component composites but did not report the thermal conductivities of the constituents and/or the relevant geometric parameters such as the shell and/or the core volume fractions [37–40]. However, these parameters are necessary in order to accurately validate numerical predictions and effective medium approximations. In addition, contact resistance between the different phases may affect the experimental measurements. This effect was considered in the general case of the Felske model [21]. 5. Conclusion This study established that the effective thermal conductivity was independent of the capsules’ spatial distribution and size distribution. The effective thermal conductivity was found to depend solely on the core and shell volume fractions and on the core, shell, and matrix thermal conductivities. The Felske model (Eq. (17)) predicted the effective thermal conductivity of the composite material within numerical uncertainty for the wide range of parameters considered. This model was used to identify conditions under which the effective thermal conductivity keff of the composite materials remained smaller than that of the matrix material. This thermal design rule will be useful in developing PCM-composite materials for energy efficient buildings. Acknowledgments This report was prepared as a result of work sponsored by the California Energy Commission (Contract: PIR:-12-032), the National Science Foundation (CMMI: 1130028) and the University of California, Los Angeles (UCLA). It does not necessarily represent the views of the Energy Commission, its employees, the State of California, or the National Science Foundation. 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