Development of correlations for effective thermal conductivity of a tetrakaidecahedra structure in presence of combined conduction and radiation heat transfer

International Journal of Heat and Mass Transfer 127 (2018) 843–856

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Development of correlations for effective thermal conductivity of a tetrakaidecahedra structure in presence of combined conduction and radiation heat transfer Vipul M. Patel a, Miguel A.A. Mendes b, Prabal Talukdar a,⇑, Subhashis Ray c,⇑ a b c

Department of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India LAETA, IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal Institute of Thermal Engineering, Technische Universität Bergakademie Freiberg, Germany

a r t i c l e

i n f o

Article history: Received 12 April 2018 Received in revised form 21 June 2018 Accepted 8 July 2018

Keywords: Effective thermal conductivity Radiative properties Correlations Porous media Tetrakaidecahedra structure

a b s t r a c t The variations in the total effective thermal conductivity (keff ;t ) of a tetrakaidecahedra unit cell structure as functions of porosity (/), thermal conductivity of the solid phase (ks ) and the average temperature of the medium (T av g ), in the presence of combined conduction and radiation heat transfer, are presented in this article. For this purpose, the governing energy conservation equation is numerically solved using the blocked-off region approach based on the finite volume method. In addition, the variations in the radiative properties of the structure as functions of surface reflectivity (qs ), pore density (PPC) and / are investigated, for which, a pure radiation heat transfer based numerical model is developed and used. From the detailed numerical simulations, three different correlations for keff ;t are proposed. Correlation 1 is developed by fitting the raw simulated data, although its form does not respect some of the limiting conditions. Particularly for ks < 5 W=mK and in the absence of thermal radiation, it under-predicts the effective thermal conductivity due to pure heat conduction (keff ;PC ). Correlation 2, on the other hand, satisfies all possible limiting conditions, although it requires one additional simulation or correlation for keff ;PC . Finally, correlation 3 is obtained by superposing the effective thermal conductivities due to pure radiation (keff ;R ) and keff ;PC , while introducing an adjustable coefficient in order to account for the coupling between them. From the investigation on radiative properties, it is observed that the extinction coefficient increases with the decrease in / and with the increase in PPC as well as qs and hence keff ;t as well as keff ;R is expected to decrease for these conditions. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction The improved thermal-hydraulic properties of open cell foams, such as low overall density, high specific surface area, moderately high effective thermal conductivity (ETC, or keff ), considerably high tortuous flow path, allowing flow with comparatively lower pressure drop, attracts researchers to investigate their performance for various applications. Beforehand knowledge of the effective thermos-mechanical properties of porous foams allows the designer to improve the performance of systems that use them for different purposes. Some of the important applications of open cell porous foams may be identified as compact heat exchanger [1], fire retardants [2], radiant burner [3], convection to radiation converter (C-R-C) [4,5], porous media combustion [6], solar radiation ⇑ Corresponding authors. E-mail addresses: [email protected], [email protected] (P. Talukdar), [email protected], [email protected] (S. Ray). https://doi.org/10.1016/j.ijheatmasstransfer.2018.07.048 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.

absorber [7], etc. The study of fluid flow and heat transfer in porous media may be carried out using the simplified homogeneous medium approach (HMA), see e.g., [8,9,10]. However, the accuracy of the HMA extensively depends on the specified effective properties that appear in the considered averaged equations. For example, while analysing the convective heat transfer through porous media with the assumption of local thermal non equilibrium condition, an adequate knowledge of the interfacial/volumetric heat transfer coefficient is essential and the temperature distribution within the fluid-saturated porous media can be predicted with reasonable accuracy if the ETC is accurately known. For the high-temperature applications, the ETC of the porous media mainly depends on (i) thermal conductivity of the solid and the fluid phases, (ii) porosity, (iii) morphology of the porous structure, (iv) optical property of the solid surface, and (v) radiative properties of the involved phases. The existing theoretical and empirical models for predicting the ETC of porous foams with various degrees of complexity have been

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Nomenclature a A b B c ^e G I k L ! q, q ! r ^s T x, y, z

parameters used for fitting A coefficients in Eqs. (10) and (20); exponent in Eq. (16) parameters used for fitting B exponent in Eq. (10) parameters used for fitting b unit vector incident radiation (W=m2 ) radiation intensity (W=m2 sr) thermal conductivity (W=mK) length of the computational domain (m) heat flux (W=m2 ) position vector (m) direction vector for radiation intensity temperature (K) Cartesian coordinates (m)

Greek symbols a absorptivity b extinction coefficient (m1 ) DT temperature difference (K) e emissivity j absorption coefficient, (m1 ) q reflectance; reflectivity r Stefan Boltzmann constant (¼ 5:67  108 W=m2 K4 ); scattering coefficient, (m1) s transmittance / porosity

thoroughly reviewed by Coquard and Baillis [11], Coquard et al. [12] and Ranut [13] and hence they are briefly summarised here for the sake of completeness. Based on these reviews, the available models may be categorised into three groups, namely, (i) asymptotic solutions, (ii) empirical correlations and (iii) unit cell approach [13]. Asymptotic solutions: In this approach, all models, forming two limiting bounds for the ETC, can be put together in the asymptotic expressions. For any pure heat conduction case, the solutions, based on the parallel and the series resistance models, most often form the upper and the lower bounds of the ETC, respectively. These bounds are also referred to as the Weiner bounds [14]. Quite obviously, the actual keff lies within these two limiting values. In comparison to these bounds, slightly restrictive bounds were proposed by Hashin and Shtrikman [15]. The expressions of these bounds may be mathematically shown to be equivalent to the Maxwell-Eucken models [16] which were derived in order to estimate the effective electrical conductivity of spheres, dispersed in a continuous phase. However, for the case of two-phase system, as considered by Maxwell [16], the dispersed phase never forms any continuous pathways and hence the estimated effective properties were found to be more biased towards the property of the continuous phase [17]. Carson et al. [17] used the effective medium theory (EMT) in order to calculate keff of the two-phase system, where the involved phases are randomly dispersed. The calculated keff was found to be ‘unbiased’ towards the thermal conductivities of the involved phases. Empirical correlations: The simplified models, whose coefficients are determined by fitting the experimental data, are placed under the category of empirical correlations. For example, Calmidi and Mahajan [18] proposed a modified form of the parallel arrangement model in order to express keff of the metal foam. From

U

x X

scattering phase function scattering albedo solid angle, (sr)

Subscripts average av g b blackbody B, T pertaining to ‘Bottom’ and ‘Top’ boundary c unit cell e, eff effective E, W pertaining to ‘East’ and ‘West’ boundary f fluid face solid-fluid interface i incident n number of layers PC due to pure conduction R due to pure radiation s solid; surface t total Abbreviations CV control volume GA genetic algorithm HPM homogeneous participating medium FVM finite volume method PPC pores per centimetre RTE radiative transfer equation

the experimental data fit, the constant ‘A’ was found to be 0:181 and 0:195 for air and water as the working fluid, respectively. Bhattacharya et al. [19] used a combination of the series and the parallel arrangement models in order to estimate keff of the metal foam. The constant A ¼ 0:35, obtained by fitting the experimental data, satisfied all data points for both air and water as the fluid medium. Singh and Kasana [20] used the experimental data of Bhattacharya et al. [19] and proposed a separate correlation by combining the series and the parallel arrangement models. The exponent ‘F’, appeared in the resultant expression, highlights the fraction of material oriented in the direction of heat flow. ‘F’ was further expressed as functions of the porosity / and the ratio of thermal conductivities of the solid and the fluid phases (ks =kf ). Dietrich et al. [21] proposed a serial combination of the parallel and the series bounds of keff . For this purpose, an empirical constant was evaluated by fitting the experimental measurements, carried out for alumina, Mullite, and OBSiC foams. They observed that for the same solid material, keff of the foam could be approximately five times higher than that observed for the packed bed. The experimental approaches, used in order to determine keff of the open cell porous foams, may appear to be simple, but they are definitely not straightforward, as discussed in some of the previous studies [22,23,24]. For example, the contact area between the sensor and the foam sample, selection of the sensor diameter and the contact pressures are some of the key parameters which significantly affect the measurements of keff . Unit cell models: The representative unit cell based geometric or numerical models can be placed in this category. In the first category, i.e., in the unit cell based geometric models, in order to express the ETC of the porous foams as functions of the material properties and the porosity, different unit cell based geometric models were developed. For example, Calmidi and Mahajan [18]

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used a two-dimensional (2D) hexagonal unit cell with a square lump at the strut junction, whereas Bhattacharya et al. [19] replaced the square lump of Calmidi and Mahajan’s unit cell with a circular lump. Boomsma and Poulikakos [25], on the other hand, used a more realistic three-dimensional (3D) tetrakaidecahedra structure with cubic lumps and cylindrical struts. They observed that for Aluminium foams in the presence of common working fluids like air, water, or ethylene glycol, the variation in kf has a marginal effect on keff . The non-dimensional area ratios ‘e’ (the ratio of strut size to the lump size), appeared in the aforementioned model [18,19,25], were determined using the experimental data [18,19]. Yang et al. [26] pointed out that a constant value for parameter ‘e’ in the model of Boomsma and Poulikakos [25] led to geometrically impossible results, which they successfully eliminated by expressing the parameter ‘e’ as a third order polynomial function of the porosity. Yao et al. [27], on the other hand, proposed a new predictive model for keff where the struts of the tetrakaidecahedra structure were represented using concave tri-prism shaped ligaments. Unlike the earlier models [18,19,25], the model of Yao et al. [27] does not require any empirical parameter. In the second category, i.e., in the unit cell based numerical models, Mendes et al. [28] proposed an extremely simplified model by utilising the upper and the lower bounds of keff , requiring only one adjustable parameter ‘b’. This parameter is evaluated using the dimensionless keff (scaled with respect to the thermal conductivity of the solid phase), obtained under the vacuum condition from the numerical simulations performed only once for the representative unit cells, e.g., cubic or tetrakaidecahedra structures. The lower and the upper Hashin-Shtrikman bounds [15] were found to be the best in predicting the simulated keff data for any combination of the solid and the fluid phases. An extended version of this model, with two adjustable parameters, was later proposed by Mendes et al. [29]. Although these numerical models are based on pure heat conduction and are proved to be extremely accurate in the absence of thermal radiation, for the high-temperature applications, the radiation heat transfer is inevitable and hence its contribution to the combined keff requires to be appropriately addressed. For this purpose, Talukdar et al. [30] developed a combined conduction-radiation based numerical method for high temperature applications in order to determine the total ETC (keff ;t ) of the representative unit cell based structures. The extensive parametric study, carried out by Talukdar et al. [30], carefully examined the effects of various thermo-optical properties of the structural elements on keff ;t . This model was later employed in order to investigate the combined ETC of real foam structures, whose geometries were reconstructed by binarizing the grey-scaled voxel based information, obtained from the 3D computed tomography (CT) [31] scan images. It has been adequately demonstrated in the literature that the tetrakaidecahedra configuration of the unit cell is an appropriate structure, which can closely represent a real foam with reasonable accuracy. Therefore, it is possible to analyse the heat transfer behaviour of this structure by varying its porosity, pore density, thermal conductivities of the involved phases, etc., and hence systematic parametric studies can be carried out in contrast to a random real foam, where the spatial variations most often introduce uncertainties into the parametric analysis. These systematic parametric studies eventually help one to understand the behaviour of a real random foam, associated with their morphological structure. In this context, it is desirable to obtain comprehensive correlations for the calculation of effective properties, which, as will be shortly apparent, is still not readily available in the literature. Additionally, recent developments of the 3D-printing technology make it possible to produce ‘non-random’ foams (e.g. periodic) based on advanced-materials, which could not be manufactured earlier by

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Fig. 1. Structure of Tetrakaidecahedra unit cell.

the conventional methods [32]. This, therefore, potentiates the direct use of effective property correlations, obtained for the well-defined cell structures, like the tetrakaidecahedra, for a possible top-down optimization procedure and the subsequent fabrication of idealized porous structures for specific applications. In the present investigation, the total ETC keff ;t of a tetrakaidecahedra structure, as demonstrated in Fig. 1, are determined employing the numerical method, described by Talukdar et al. [30]. A parametric study is carried out in order to investigate and quantify the effects of boundary temperatures, thermal conductivity of the solid phase and porosity of the structures on keff ;t and these effects are represented by three different correlations. Although the employed numerical method of Talukdar et al. [30] has been already tested for random foam structures by Mendes et al. [31], in the present study, only a representative tetrakaidecahedra unit cell structure is considered not only because it offers a greater flexibility for the parametric study to be conducted with less computational effort, but also since the associated results can be directly used for the optimisation and the fabrication of non-random foams, as mentioned earlier. 2. Problem formulation and solution methodology 2.1. Modelling the geometry The geometry of a tetrakaidecahedra unit cell is shown in Fig. 1. The porosity of this structure can be varied by modifying the crosssectional area of struts. The blocked-off region approach, based on the Cartesian coordinates, as adequately explained by Talukdar et al. [30], is employed in order to obtain the simulated data, where the information about the locations of solid and fluid phases are transferred by assigning specific values to the identifiers, corresponding to each phase. For this purpose, the solid voxels are marked as ‘0’, whereas the fluid voxels are designated as ‘1’. The thermo-physical properties of the solid and the fluid phases are assumed to remain constant over the entire temperature range, considered in the present investigation. 2.2. Numerical methodology As mentioned earlier, in order to determine keff ;t of the tetrakaidecahedra unit cell structure, the numerical method, suggested by Talukdar et al. [30], is adopted. A brief description of the method is, however, provided here for the sake of completeness. The tetrakaidecahedra unit cell structure is assumed to be enclosed in a bounding cubic enclosure. The void space is assumed

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to be filled up with the participating media, which is considered as gas in the present study. The ‘East’ and the ‘West’ boundaries of the enclosure are maintained at isothermal conditions, but at different temperatures, namely, T E and T W , respectively, with T E > T W , such that heat is transferred from the ‘East’ boundary to the ‘West’. Both ‘East’ and ‘West’ boundaries are considered as black surfaces, i.e., their emissivity e is assumed to be exactly equal to unity. In addition, all other transverse boundaries, that are parallel to the applied temperature gradient, are subjected to the adiabatic boundary condition and hence the integrated heat transfer through any surface, parallel and equal to the ‘East’ or the ‘West’ boundaries, remain identical and depends only on the imposed temperature gradient, the geometry of the unit cell and the thermal conductivities of the solid and the fluid phases. In order to reduce the influence of these boundaries on the predicted ETC, their emissivity is considered to be zero (e ¼ 0). The solid matrix of the tetrakaidecahedra structure is characterised by the thermal conductivity ks and the surface emissivity e. The thermal conductivity and the absorption coefficient of the entrapped, stagnant gas are denoted as kf and j, respectively. In the absence of convection, which is assumed for the present study, the heat transfer takes place only by conduction through both solid and fluid phases and by radiation through the fluid phase. The steady-state energy equation, governing the combined conduction-radiation heat transfer through the computational domain may be written as:

r  ðkrTÞ  r  ! qR ¼ 0

ð1Þ

where the local thermal conductivity k is assigned the value for the solid phase ks or that of the fluid phase kf , depending on the voxel information, as described before. At the solid-fluid interface, the thermal conductivity k is modelled using the weighted harmonic mean of ks and kf . For uniform grid, which is adopted for the present investigation owing to the available voxel based information, this is given as:

kface ¼

2kf ks ðkf þ ks Þ

ð2Þ

The diffusive term in Eq. (1), including the temperature gradient

rT, is discretised using the central difference scheme for every control volume (CV). These CVs coincide with either the solid or the fluid voxels. The set of discretised equations is solved in order to obtain the resultant temperature field using the Dirichlet boundary conditions at the ‘East’ and the ‘West’ boundaries, while applying homogeneous Neumann condition at all other faces that are parallel to the imposed temperature gradient. The divergence ! of radiation heat flux vector r  q R is treated as a source term in Eq. (1) and is implicitly upgraded after solving the standard radiative transfer equation (RTE). In the present investigation, the finite volume method (FVM), proposed by Chai and Patankar [33] and applied further by Talukdar et al. [30,31], is used in order to solve the standard RTE. Once, the distribution of radiation intensity is ! obtained, r  q R is calculated from the following relation:

h

! ! ! r! q R ¼ jð r Þ 4pIb ð r Þ  Gð r Þ

i

ð3Þ

where j is the absorption coefficient of the participating gas and the black-body radiation intensity Ib is given as:

rT ! Ib ð r Þ ¼

4

ð4Þ

p

! The irradiation term Gð r Þ, appearing in Eq. (3), is evaluated as:

! Gð r Þ ¼

Z

4p

! Ið r ; ^sÞdX

ð5Þ

Finally, from the converged temperature and radiation fields, the total ETC of the unit cell may then calculated as:

keff ;t ¼

qav g;E DT EW =L

ð6Þ

where qav g;E is the average heat flux due to conduction and radiation on ‘East’ boundary, and L is the distance between the ‘East’ and the ‘West’ boundary surfaces. A comprehensive parametric study is carried out in order to investigate the effects of the average temperature of the computational domain T av g ¼ ðT E þ T W Þ=2, the thermal conductivity of the solid matrix ks , and the porosity / of the unit cell on keff ;t of the considered structure. The temperatures of both ‘East’ and ‘West’ boundaries are varied for quantifying the effect of T av g . For this purpose, the temperature difference DT EW ¼ T E  T W is always kept fixed at 100 K for all tested cases so that the assumption of constant thermal property still remains valid. Thus, different values of T E , considered for the present investigation, are: 800 K, 1000 K, 1200 K, 1500 K and 2000 K. Since T W ¼ T E  100 K is always specified, the corresponding average temperatures are recorded as T av g ¼ 750 K, 950 K, 1150 K, 1450 K and 1950 K, respectively. The parametric study is also conducted for different ks , ranging from 0:5 W=mK to 200 W=mK, namely, 0:5, 1, 5, 7, 10, 20, 50, 100, 150 and 200, with all units being defined in W=mK, in order to investigate the effect of ks on keff ;t . The thermal conductivity of the stagnant air kf inside the structure is taken as 0:029 W=mK for the present investigation, which is typical for air at high temperature. Therefore, the ratio thermal conductivities of the solid and the fluid phases ks =kf is varied from approximately 17 to 7  103 (17:24 to 6; 896:55, to be precise). The previous study, conducted by Talukdar et al. [30], suggested that the absorption coefficient of the stagnant air j and the emissivity of the strut surface e have minimal influence on keff ;t . Therefore, in the present investigation, the emissivity of the strut surfaces is kept fixed at 0:7, whereas the absorption coefficient of the participating air is taken as 0:1 m1 . As mentioned earlier, the porosity of the tetrakaidecahedra structure is varied by modifying the cross-sectional area of the struts. Six different porosities are considered in the present study, namely, 0:944, 0:919, 0:87, 0:83, 0:77 and 0:72. 2.3. Determination of radiative properties of porous media The unit cell based numerical model, developed by Patel and Talukdar [34], is modified and extended for the present investigation in order to determine the radiative properties of the tetrakaidecahedra structure. The employed model can be divided into three essential tasks: (i) numerical estimation of the effective reflectance qe;c and the transmittance se;c of a single unit cell, (ii) determination of the effective reflectance qe;n and the transmission se;n of n-layered unit cell structures by substituting qe;c and se;c in recurrence relationships, described later and (iii) prediction of the radiative properties of an equivalent homogeneous participating media (HPM) by inverting qe;n and se;n into the direct solution of standard RTE, integrated with the genetic algorithm (GA). The assumptions invoked in the proposed model may be summarised as: (i) the characteristic dimensions of the unit cell are much greater than the wavelength of the thermal radiation, (ii) the solid matrix is always considered to be cold and hence no thermal radiation can be emitted by the solid surface, (iii) the considered unit cell is surrounded by (or, attached to) a large number of tetrakaidecahedra cells in the transverse directions, i.e., in the directions parallel to the imposed temperature gradient or heat flux, which allows one to implement the symmetry boundary

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847

condition at the transverse boundaries, (iv) the reflected radiation from the solid surface is considered to be diffuse in nature, (v) the thermo-physical properties of both solid and fluid phases remain constant over the range of temperature, encountered in the present investigation.

2.3.2. Calculation of reflectance qe;n and transmittance se;n using recurrence relations The results of qe;c and se;c , obtained for a single unit cell, may now be extended for the multi-layered porous structure using the following recurrence relations [34]:

2.3.1. Numerical estimation of effective reflectance qe;c and transmittance se;c The numerical model, proposed by Patel and Talukdar [34], is modified and the modifications are implemented in order to determine the effective reflectance qe;c and transmittance se;c of a unit cell structure. Similar to the case of energy conservation equation, according to this model, the tetrakaidecahedra unit cell is assumed to be enclosed within a cubic computational domain. A radiation heat flux of unit magnitude (qi ¼ 1 W=m2 ) is assumed to enter through the ‘Bottom’ boundary. The solid matrix is characterised by the surface reflectivity qs . Therefore, qs fraction of the incident radiation heat flux, intercepted by the solid matrix, is reflected back in a diffuse manner, whereas 1  qs fraction of the incident radiation heat flux is absorbed by the solid matrix. Since the solid is assumed to be maintained at cold condition, no energy could be emitted back from the solid matrix. The ‘Top’ boundary of the computational domain is also considered at the cold condition. As explained before, the infinite cell assumption in the transverse direction permits one to implement the symmetry boundary condition at all other boundaries, that are parallel to the direction of the imposed heat flux. In the numerical method of Patel and Talukdar [34], a diffuse reflection approach was used in order to implement the symmetry boundary condition, according to which, the radiation intensities arriving the void CV at the transverse boundaries are reflected back to the unit cell in a diffuse manner. The magnitude of radiation intensities, reflected from the void CV, are calculated using the average radiation heat flux intercepted by that particular CV. In this manner, one would expect that the reflectance and the transmittance, obtained using the unit cell model would be identical to that when multiple unit cells are placed side by side in the transverse direction. The earlier model [34], however, fails to replicate such desired behaviour. Moreover, the aforementioned diffuse reflection approach, considered in the previous investigation, leads to the over-prediction of both extinction coefficient b and scattering albedo x. In order to overcome the aforementioned difficulty, in the present investigation, the assumption of diffuse reflection is replaced by the specular reflection. For a ray tracing method, like FVM, the specular reflection condition may be implemented in either of the two ways: (i) reflect the incident intensity in the direction of its corresponding reflection angle or (ii) use the periodic boundary condition for the radiation intensities. In the latter approach, the intensity, leaving from any of the transverse boundaries, is allowed to enter the computational domain through the corresponding opposite boundary. Since this latter approach is relatively easy and straightforward to implement in FVM, it is selected for the present investigation. Once the solution is converged, the radiation heat flux, reflected back to the ‘Bottom’ boundary qB and the radiation heat flux, transmitted to the ‘Top’ boundary qT are calculated using the following equations:

qe;n ¼ qe;n1 þ

Z

Z ! ! ! ! qB ¼ ! Ið r ; ^sÞð r :^ez ÞdX qT ¼ ! Ið r ; ^sÞð r :^ez ÞdX r :^ez <0 r :^ez>0

ð7Þ

The quantities qB , qT and qi are used in order to evaluate the effective reflectance qe;c and transmittance se;c of the unit cell structure as follows:

qe;c ¼

qB qi

se;c ¼

qT qi

ð8Þ

se;n ¼

qe;c s2e;n1 1  qe;c qe;n1

se;c se;n1

1  qe;c qe;n1

ð9aÞ

ð9bÞ

where the subscript ‘n  1’ stands for the ðn  1Þth layer of the unit cells and the appropriate value for the layer n is selected based on the pore density of the structure of interest. 2.3.3. Determination of radiative properties of HPM In the previous step, an equivalent homogeneous participating media (HPM) is considered in order to represent the multilayered structure of the unit cell, described in Section 2.3. An optimization procedure, developed by coupling the GA with the FVM for solving the RTE [34] is then used in order to predict the extinction coefficient b and the scattering albedo x of the HPM. The values of qe;n and se;n , obtained using the aforementioned recurrence relations, are then inverted in the combined FVM-GA model. The GA appropriately adjusts the radiative properties of the HPM such that the reflectance qHPM and the transmittance sHPM match reasonably well with qe;n and se;n of the structure, as obtained before. Since the pore density is defined in terms of the pores per centimetre (PPC), the dimension of the HPM is also considered as 1 cm. In Section 2.3.1, the reflection of the radiation intensities, striking the solid matrix, is assumed to be diffuse and hence the scattering phase function in the RTE for the HPM is also considered to be equal to unity (i.e., U ¼ 1), i.e., the scattering from the solid matrix is also assumed to be isotropic. 3. Results and discussion Based on the carefully conducted grid independence study, which is not presented here for the sake of brevity, it is observed that 53  53  53 CVs for the computational domain of size 1  1  1 cm3 are sufficient enough in order to ensure that the solution is free from any effect of the grid size. In the FVM for solving radiation, the polar (h) and the azimuthal (/) angels are discretized into 10  8 uniform angles, respectively. The numerical model, employed in order to estimate keff ;t has been already validated by Talukdar et al. [30] and hence it is not repeated here. As mentioned earlier, during the present study, extensive numerical simulations are carried out by varying the thermal conductivity of the solid phase ks , the boundary temperatures T E and T W (and hence T av g ) and the porosity / of the HPM in order to explore their effects on the total ETC keff ;t . In this section, the importance and the contribution of radiation heat transfer on keff ;t is first presented and the detailed correlations for keff ;t are then developed from the simulated data. 3.1. Contribution of radiation and conduction modes of heat transfer to keff ;t Since the numerical method, used for evaluating keff ;t , considers both conduction and radiation heat transfer, in order to identify the relative contributions from each of these modes, separate simulations are also performed by employing the pure heat conduction based numerical model, developed by Mendes et al. [28], i.e., ! by neglecting the radiation source term r  q R in Eq. (1). The

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effective thermal conductivity due to pure heat conduction, determined in this manner, is termed as keff ;PC , which can also be evaluated using the simplified model, proposed by Mendes et al. [28], where the adjustable parameter appearing in the predictive model is obtained from the unit cell based simulation, carried out under vacuum condition. Fig. 2 presents the ratio of effective thermal conductivities keff ;t =keff ;PC for a unit cell with porosities of / ¼ 0:72 and 0:944 as functions of ks for the lowest and the highest average temperatures considered for the present investigation, T av g ¼ 750 K and 1950 K, respectively. It is evident from the figure that for higher values of ks and particularly for lower porosity, the influence of conduction heat transfer is more prominent and consequently the ratio keff ;t =keff ;PC approaches to unity in this limit for both cases of T av g . As expected, with the decrease in ks , the contribution from radiation heat transfer to keff ;t increases for both / ¼ 0:72 and 0:944, where the higher ratio of effective thermal conductivities for lower values of ks clearly indicates a comparatively higher contribution from the thermal radiation. The contribution of radiative mode of heat transfer also increases further with the increase in both / and T av g , as may be observed by comparing the results for / ¼ 0:72 and 0:944 with T av g ¼ 750 K and 1950 K in Fig. 2. The effects of porosity / on keff ;t =keff ;PC are presented for two different values of ks , while keeping T av g fixed at 750 K in Fig. 3(a) and 1950 K in Fig. 3(b). For the highest ks ¼ 200 W=mK, the conduction mode of heat transfer dominates considerably over the radiation mode and hence the effects of porosity on keff ;t =keff ;PC is observed to be almost insignificant for T av g ¼ 750 K and less prominent for T av g ¼ 1950 K. For the lowest ks ¼ 0:5 W=mK, however, keff ;t =keff ;PC

is observed to be considerably (approximately one order of magnitude) higher for T av g ¼ 1950 K than that for T av g ¼ 750 K. Further, the contribution from radiation heat transfer for lower values of ks is substantial and as expected, its contribution further increases with the increase in porosity, i.e., with the increase in the fraction of fluid phase within the unit cell through which the thermal radiation can be transferred. 3.2. Correlations for keff ;t In the present investigation, considerable effort is made in order to express the effects of ks , T av g , and / on keff ;t in the form of three different generalised correlations, that are named as correlations 1, 2 and 3. The functional form of correlation 1 may be recognised as a modified version of the parallel resistance model, where the layers of the involved phases are aligned parallel to the direction of heat flow. The performance of correlation 1 is carefully analysed for limiting conditions in order to extend the range of applicability, especially for lower values of ks 6 5 W=mK. With this effort, an alternative correlation 2 is developed, where ks =kf , appearing in correlation 1, is replaced by keff ;PC =kf . The behaviour of correlation 2 and its accuracy under limiting conditions are then discussed. As will be shortly apparent, the coefficients and exponents, appearing in the aforementioned correlations 1 and 2, are obtained by fitting the simulated data and they are expressed as functions of / and T av g . Correlation 3, on the other hand, articulates keff ;t as an algebraic sum of the effective thermal conductivities due to pure radiation keff ;R , expressed using the Rosseland mean extinction coefficient, and pure heat conduction keff ;PC .

Fig. 2. Effects of ks on keff ;t =keff ;PC for (a) / ¼ 0:72 and (b) / ¼ 0:944.

Fig. 3. Effects of / on keff ;t =keff ;PC for (a) T av g ¼ 750 K and (b) T av g ¼ 1950 K.

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Fig. 4. Variations of keff ;t =kf as functions of ks =kf for (a) T av g ¼ 750 K and (b) T av g ¼ 1950 K.

3.2.1. Development of correlation 1 For porosities / ¼ 0:72 and 0:944, the numerically predicted keff ;t =kf as functions of ks =kf are presented in Fig. 4(a) and (b) for average temperatures T av g ¼ 750 K and 1950 K, respectively. It may be noted that in the present investigation, an increase in ks =kf essentially implies an increase in ks alone since kf for air (i.e., the fluid phase) is kept fixed at 0:029 W=mK for all investigated cases. Irrespective of / and T av g , the observed increasing trend of keff ;t =kf with the increase in ks =kf in these figures could be mainly attributed to the increase in conduction heat transfer through the solid struts of the porous medium. The figure also shows that keff ;t =kf is higher for T av g ¼ 1950 K than that for T av g ¼ 750 K, owing to the higher contribution of radiation heat transfer. It is also interesting to note that irrespective of T av g and particularly for lower values of ks =kf , keff ;t =kf is higher for higher porosity. This trend, however, is reversed beyond a certain value of ks =kf that strongly depends on both T av g and /. Such behaviour is expected and can be explained in view of the relative contributions of conduction and ratidation heat transfer by noting the following: (i) thermal radiation does not explicitly depend on either ks or kf (and hence on ks =kf ) and its contribution towards the total heat transfer increases rapidly with the increase in T av g and moderately with the increase in /; (ii) since the thermo-physical properties of both solid and fluid phases are assumed to be constants, conduction heat transfer does not explicitly depend on T av g (although it depends on DT, but it is kept fixed at 100 K) and its contribution increases substantially with the increase in ks (and hence ks =kf since kf is kept fixed at 0:029 W=mK) and significantly with the decrease in /, i.e., with the increase in fraction of solid phase within the unit cell. Nevertheless, the simulated data could be represented by correlation 1, whose functional form may be considered as follows:

  keff ;t ks ¼ A/ þ ð1  /ÞB kf kf

in Fig. 5 for / ¼ 0:72 and 0:944, which may be functionally expressed as:

A ¼ a1 T aa2v g

ð11Þ

Employing the case-specific values of a1 and a2 , evaluated using the LSM, R2  1 is obtained for both porosities. The variations in the coefficients a1 and a2 , appearing in Eq. (11), are presented in Fig. 6 that may be further expressed as functions of / in Eqs. (12a) and (12b), respectively:

a1 ¼ a11 expða12 /Þ

ð12aÞ

a2 ¼ a21 þ a22 /

ð12bÞ 2

The coefficients of determination are obtained as R ¼ 0:988 and 0:995 for a1 and a2 , respectively. For a given porosity, the exponent B in Eq. (10) is found to be almost independent of T av g for the considered ranges of temperature. Therefore, the average value of B for a given /, which is independent of T av g , is considered for further analysis. The variation of B as a function of / is presented in Fig. 7, which may be correlated as:

B ¼ b1  b2 /

ð13Þ 2

Using LSM for evaluating b1 and b2 , R ¼ 0:975 is obtained. Substituting the expressions for A and B from Eqs. (11) and (13), respectively, while using Eq. (12) for expressing a1 and a2 , correlation 1 for keff ;t =kf in Eq. (10) may be expressed as:

  i keff ;t h ks ¼ a11 expða12 /ÞT aðav21g þa22 /Þ / þ ð1  /Þðb1 b2 /Þ kf kf

ð14Þ

where the constants are evaluated as: a11 ¼ 5  106 , a12 ¼ 4:6, a21 ¼ 2:2, a22 ¼ 0:75, b1 ¼ 2:52 and b2 ¼ 1:14. Fig. 8 shows the performance of the final form of correlation 1 in Eq. (14) as comparison

ð10Þ

The constant A and the exponent B in Eq. (10) are evaluated by the least square method (LSM) i.e., by minimising the sum of the square of errors (or, maximising the R2 values). The performance of correlation 1 for the aforementioned values of / and T av g , obtained using the case specific values of A and B is also presented in Fig. 4, which shows an excellent agreement between the simulated data and the correlated values of keff ;t =kf with R2min  1 and 0:959 for T av g ¼ 750 K and 1950 K, respectively. In order to determine the functional dependence of A on / and T av g , its variations are now critically examined. The variations in A as a function of T av g , ranging from 750 K to 1950 K, are illustrated

Fig. 5. Variations of A in Eq. (10) as functions of T av g for / ¼ 0:72 and 0:944.

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Fig. 6. Variations of (a) a1 and (b) a2 in Eq. (11) as functions of /.

Fig. 7. Variation of B in Eq. (10) as function of /.

T av g . Since this term does not contain any thermal conductivity of either the solid or the fluid phase, it is expected that this term represents the contribution only from radiation heat transfer. Similarly, the second term on the right-hand side of Eq. (14), which does not contain T av g , is expected to account only for the contribution from conduction heat transfer. Nevertheless, in order to functionally represent the physical situation, the proposed correlation requires satisfying at least the following limiting conditions: (i) as the porosity tends to zero (i.e., when the medium is filled only with the solid phase), the correlation should recover the thermal conductivity of the solid phase, i.e., keff ;t  ks ; (ii) as T av g tends to zero, the contribution of radiation heat transfer should not exist and hence the correlation should recover the effective thermal conductivity under pure heat conduction situation, i.e., keff ;t  keff ;PC ; (iii) as / tends to unity (i.e., without any solid phase) and T av g tends to zero (i.e., without thermal radiation), the thermal conductivity of the fluid phase should be obtained from the correlation, i.e., keff ;t  kf . It may be easily verified that correlation 1 satisfies the first limiting condition. For the second limiting condition, on the other hand, the reduced form of Eq. (14) may be written as:

  keff ;PC ks ¼ ð1  /Þðb1 b2 /Þ kf kf

ð15Þ

It is evident from Eq. (15) that the third limiting condition as / ! 1 is not respected by the proposed correlation and hence it is not expected to perform well for higher porosity. Nevertheless, as mentioned earlier in Section 3.1, keff ;PC for different combinations of / and ks are also evaluated using the numerical model, proposed by Mendes et al. [28]. Fig. 9 shows the comparison between

Fig. 8. Performance of correlation 1.

between the simulated and the correlated values of keff ;t that are observed to be in reasonably good agreement with each other. The coefficient of determination is evaluated as R2 ¼ 0:997. The average absolute difference between the values, predicted using correlation 1 and the simulation data, is found to be only 5:45%, whereas the maximum absolute difference is observed as 29:43%. Quite clearly, the error in prediction increases with the decrease in ks and increase in T av g , which is also evident according to the variations presented in Fig. 4. In addition, it is also observed that almost 69% and 83% of the data lie within 5% and 10%, respectively, of that predicted by Eq. (14). After fitting the data and obtaining the functional forms for A and B, it is now evident that the first term on the right hand side of Eq. (14) contains the porosity / and the average temperature

Fig. 9. Comparison of correlation 1 in Eq. (15) with numerically calculated keff ;PC =kf in the pure heat conduction limit as functions of / for ks ¼ 0:5 W=mK, 5 W=mK and 10 W=mK.

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the simulated keff ;PC =kf and the corresponding values, obtained from the reduced form of correlation 1 in Eq. (15), for ks ¼ 0:5 W=mK, 5 W=mK and 10 W=mK. Although the overall agreement appears to be reasonably good with R2min ¼ 0:998, a careful inspection reveals that for lower values of ks and particularly for higher porosity, Eq. (15) underpredicts the simulated data. The difference between the results decreases as ks increases and for ks P 5 W=mK, both results almost coincide with each other, although the comparisons for other values of ks are not presented in Fig. 9 for the sake of brevity. Nevertheless, since for most of the open cell porous foams, employed in different applications, ks is generally higher than 5 W=mK, correlation 1 in Eq. (14) may be safely recommended for the evaluation of keff ;t in presence of combined conduction and radiation heat transfer. 3.2.2. Development of correlation 2 It is already shown in the previous section that for i.e., ks 6 5 W=mK and particularly for higher porosity, correlation 1 fails to satisfy the second limiting condition as T av g ! 0. It also does not satisfy the third limiting condition as / ! 1 along with T av g ! 0. Therefore, in order to extend the applicability of correlation 1 for lower values of ks , the second term on the right hand side of Eq. (10) is now replaced by keff ;PC =kf for correlation 2. In addition, the first term on the right hand side of correlation 1 is also modified as T aAv g / and hence the resulting form of correlation 2 is expressed as:

keff ;t keff ;PC ¼ T aAv g / þ kf kf

ð16Þ

For a given combination of / and T av g , the exponent A can be evaluated from the variations of keff ;t =kf as functions of keff ;PC =kf using the LSM. Typical variations in keff ;t =kf for T av g ¼ 750 K and 1950 K are presented in Fig. 10(a) and (b), respectively, for two different porosities / ¼ 0:72 and 0:944. The figure also shows the performance of correlation 2 in Eq. (16), obtained using the case specific values of A, which, like correlation 1, indicates reasonably good agreement between the simulated and the correlated data for

Fig. 11. Variations of A in Eq. (16) as functions of T av g for / ¼ 0:72 and 0.944.

specific values of a1 and a2 . The coefficients of determination are obtained as R2 ¼ 0:997 and 0:998 for / ¼ 0:72 and 0:944, respectively. From the evaluated data, since a1 is observed to be an extremely weak function of /, its value, averaged for all porosities, a1 ¼ a1;av g ¼ 1:816 is recommended for further use. The other coefficient a2 may be further expressed as a linear function of porosity as follows:

a2 ¼ a21 þ a22 /

The variation of a2 in Eq. (17) and the performance of Eq. (18) are shown in Fig. 12. The coefficient of determination is obtained as R2 ¼ 0:890. Using Eqs. (17) and (18), the final form of correlation 2 in Eq. (16) may be written as:

keff ;t =kf with R2min  1 and 0:964, for T av g ¼ 750 K and 1950 K, respectively. For a given porosity, the exponent A can be correlated as a function of T av g as:

A ¼ a1 þ a2 logðT av g Þ

ð17Þ

For the considered range of porosity, a1 and a2 are evaluated by employing the least square method. The variations in A, evaluated in this manner, for / ¼ 0:72 and 0:944 are presented in Fig. 11, along with that predicted by Eq. (17), obtained using the case

ð18Þ

Fig. 12. Variation of a2 as a function of porosity /.

Fig. 10. Variations of keff ;t =kf as functions of keff ;PC =kf for (a) T av g ¼ 750 K and (b) 1950 K.

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keff ;t keff ;PC ¼ T a½av1gþða21 þa22 /Þ logðT av g Þ / þ kf kf

ð19Þ

where the coefficients, appearing in Eq. (19), are obtained as a1 ¼ 1:816, a21 ¼ 0:314, and a21 ¼ 0:024. The performance of correlation 2, presented Fig. 13 as a comparison between the numerically predicted results and the correlated values of keff ;t , exhibits a reasonably good agreement with R2 ¼ 0:998, which is marginally higher than that for correlation 1. However, the absolute maximum and average errors, associated with correlation 2, are obtained as 31:20% and 5:58%, respectively, that are marginally higher than correlation 1, although about 69% and 82% of the data are found to lie within 5% and 10%, respectively, of the predicted values. At this point, it is worthwhile to mention that since keff ;PC satisfies all possible limiting conditions in the absense of thermal radiation [28], it may also be easily verified that correlation 2 satisfies all three limiting conditions, mentioned earlier in Section 3.2.1. It is also evident that although correlation 1 is relatively more complex than correlation 2, as the former requires specifying six (in constrast to three for the latter) constants, this form, however, does not require any further simulation. On the other hand, although correlation 2 always requires one additional simulation for the specific geometry under vacuum condition in order to determine the ETC due to pure heat conduction keff ;PC , it may still be preferred due to its simplicity. 3.2.3. Development of correlation 3 It may be noted that the exponent of T av g in correlation 1, that represents the contribution of radiation heat transfer, varies from 2:72 for / ¼ 0:72 to 2:91 for / ¼ 0:944. Since this exponent is very close to 3 (see Eq. (12b) and Fig. 6b for clarification), the first term on the right hand side of Eq. (10) may be replaced by the ETC due to pure radiation, which may be obtained according to the simplified Rosseland approximation keff ;R ¼ 16rT 3av g =3b. The contribution of conduction heat transfer keff ;PC may be added separately, as adopted for correlation 2 in Eq. (16). Therefore, correlation 3 may be proposed as:

keff ;t ¼

16rT 3av g þ keff ;PC 3Ab

face reflectivity qs ¼ 1  es ¼ 0:3, obtained as a function of /, is then used in the first term on the right hand side of Eq. (20). For the sake of completeness, the detailed discussion on the variations of b for different surface reflectivity, porosity, and pore density is presented in Section 3.3. The variation in b as a function of / is presented in Fig. 14, which may be represented as:

b ¼ c1  c2 /

ð21Þ 2

where the coefficient of determination is obtained as R ¼ 0:998. Using the correlated values of b from Eq. (21), the adjustable parameter A in Eq. (20) is then evaluated for each combination of T av g and / using the least square method. The variations in keff ;t as functions of ks and the performance of correlation 3, obtained employing the case specific values of A, for T av g ¼ 750 K and 1950 K are presented in Fig. 15(a) and (b), respectively, where the results only for / ¼ 0:72 and 0:944 are shown for brevity. The figures clearly show that the simulated and the correlated data are in excellent agreement with each other for T av g ¼ 750 K with R2min  1 and in reasonably good agreement for T av g ¼ 1950 K with R2min ¼ 0:964. For a given porosity, A is found to be a weakly varying linear function of T av g and hence could be expressed as:

A ¼ a1 þ a2 T av g

ð22Þ

The variations of A, as functions of T av g for / ¼ 0:72 and 0:944 are presented in Fig. 16, along with that obtained according to Eq. (22) using the case specific values of a1 and a2 . The coefficients of determination are obtained as R2 ¼ 0:974 and R2  1 for / ¼ 0:72 and 0:944, respectively. The coefficients a1 and a2 are further correlated as functions of / using the quadratic and the linear functions, respectively as follow:

a1 ¼ a10  a11 / þ a12 /2

ð23aÞ

a2 ¼ a21 þ a21 /

ð23bÞ

The variations in a1 and a2 along with the performance of Eq.

ð20Þ

where b is the extinction coefficient for the tetrakaidecahedra unit cell structure and an additional coefficient A is deleberately introduced that takes care of the expected coupling between the contributions due to conduction and radiation modes of heat transfer. The extinction coefficient b for the considered structure with different porosities are first obtained using the modified numerical method of Patel and Talukdar [34], as briefly explained in Section 2.3. The values of b of a unit cell for a unit pore density (PPC ¼ 1) and a sur-

(23) are presented in Fig. 17, where R2 ¼ 0:989 and 0:889 are obtained for a1 and a2 , respectively. Using Eqs. (21)–(23), the resultant expression for correlation 3 in Eq. (20) may be written as:

keff ;t ¼

16rT 3av g 2

3½ða10  a11 / þ a12 / Þ þ ða21 þ a21 /ÞT av g ðc1  c2 /Þ

þ keff ;PC ð24Þ

where the constants, appearing in Eq. (24), are obtained as, a10 ¼ 22:26,

a11 ¼ 54:94,

a12 ¼ 37:24,

a21 ¼ 4:3  104 ,

4

a22 ¼

3  10 , c1 ¼ 722, c2 ¼ 672. Fig. 18 demonstrates the performance

Fig. 13. Performance of correlation 2.

Fig. 14. Variation of b with / for

es ¼ 0:7.

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Fig. 15. Variation of keff ;t as a function of ks for (a) T av g ¼ 750 K and (b) 1950 K.

Fig. 16. Variation of A with T av g for / ¼ 0:72 and 0:944.

Fig. 18. Performance of correlation 3.

of correlation 3, where the simulated and the correlated data for keff ;t are compared, and the agreement is found to be quite acceptable. Although the original form of correlation 3 in Eq. (20) is extremely simple, the absolute maximum and average errors, associated with this correlation are obtained as 29:56% and 5:42%, respectively. While the maximum error is almost identical to that obtained for correlation 1, and lower than correlation 2, the average error is the minimum amongst all three correlations. The coefficient

one additional simulation for a given geometry in order to determine keff ;PC . In comparison to Eq. (16), however, Eq. (20) shows that it is possible to superimpose the effects of conduction and radiation heat transfer in the forms of keff ;PC and keff ;R , respectively, provided an adjustable coefficient A, which is functions of / and T av g , is introduced in order account for the coupling between two modes of heat transfer. This may be regarded as a remarkable simplification of the outcome of an apparently complex heat transfer process.

of determination is obtained as R2 ¼ 0:998. In addition, it is also observed that approximately 68% (marginally lower than that for correlations 1 and 2) and 83% (same as that for correlation 1, and marginally higher than that for correlation 2) of the data lie within 5% and 10%, respectively, of that predicted by Eq. (24). Inspection of the original forms of correlations 2 and 3 in Eqs. (16) and (20), respectively, reveals that both of them requires

3.3. Effective radiative properties of the structure In this section, the effects of surface reflectivity qs , porosity /, and pore density (PPC) on the extinction coefficient b and scattering albedo x of the tetrakaidecahedra unit cell structure is pre-

Fig. 17. Variation of (a) a1 and (b) a2 as functions of / for

es ¼ 0:7.

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sented and discussed for completeness. The variations in b, j and rs as functions of qs are presented in Fig. 19(a) and (b) for / ¼ 0:72 and 0:944, respectively, while keeping PPC ¼ 1. The increasing trend of b with the increase in qs may be explained by observing the effects of qs on j and rs of the structure. As qs increases, the thermal radiation, reflected from the strut surfaces, also increases and as a result, the radiation absorbed by the solid matrix decreases since as ¼ es ¼ 1  qs is considered in the present investigation. This phenomenon is clearly reflected by the increasing trend of scattering coefficient rs and the decreasing trend of absorption coefficient j with the increase in qs , as demonstrated in Fig. 19. The figure also particularly shows that as expected, j ! 0 as qs ! 1. However, since the increase in rs dominates over the decrease in j, the resultant b shows an increasing trend with the increase in qs , where b  rs is observed in the limit as qs ! 1. These effects are also found to be more prominent for lower values of /. Fig. 20(a) demonstrates the effects of PPC on b of the structure as functions of qs for / ¼ 0:944. It may be noted that as the pore density increases, the physical size of the unit cell reduces and hence the number of cells, participating in the radiation interaction, increases. This results in increase in both j and rs , which ultimately increases b of the HPM. The isotropic assumption, considered for the present investigation, marginalises the effect of PPC on x of the HPM and hence for a given qs , x is observed to be almost insensitive to PPC, as demonstrated in Fig. 20(b). It may also be noted that since the scattering albedo is defined as x ¼ rs =ðj þ rs Þ and since rs increases while j decreases with the increase in qs , x also increases with the increase in qs , as may be verified from Fig. 20(b). In addition, since j ! 0 as

Fig. 19. Variations of b,

qs ! 1 (refer to Fig. 19 for clarification), x tends to unity in this limit, irrespective of PPC. Fig. 21 shows the effects of / on b and x of the considered structure. As the porosity of the medium decreases, the thickness of the struts and hence their cross-sectional area increases. As a consequence, more fraction of radiation is then subjected to absorption and scattering. Therefore, b increases with the decrease in / and the increase in qs , although it is less sensitive to the change in qs for higher values of / and more sensitive to the change in / for higher values of qs . On the other hand, although x increases with the incease in qs and the decrease in /, it is more sensitive to the change in qs and less sensitive to the change in /. As mentioned before with respect to Fig. 20(b), x tends to unity in the limit as qs ! 1, irrespective of /. Therefore, x is observed to be more sensitive to the change in / for lower values of qs and its variation with / decreases drastically with the increase in qs , which is also evident from Fig. 21(b). In order to further explain these observations, the variations in b, j and rs as functions of / are presented for two extreme limits of surface reflectivity qs ¼ 0:3 and 0:99 in Fig. 22(a) and (b), respectively, for the unit cell structure with PPC ¼ 4. Owing to the high surface area to volume ratio, the scattering phenomena in this structure is observed to be more dominant. For qs ¼ 0:99, the absorption coefficient is almost zero (refer also to Fig. 19 for clarification), and most of the radiation energy is transported mainly due to scattering. As a result, b  rs is observed in Fig. 22(b) and the variation in j is not shown in this figure. This phenomenon is also reflected in Figs. 20(b) and 21(b), where x  1 is obtained for qs ¼ 0:99, irrespective of PPC and /. For qs ¼ 0:3, however, the radiation energy transport is dependent on both absorption

j and rs of the structure with PPC ¼ 1 as functions of qs for (a) / ¼ 0:72 and (b) / ¼ 0:944.

Fig. 20. Effects of PPC on (a) b and (b) x of the structure for / ¼ 0:944.

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Fig. 21. Effects of / on (a) b and (b) x of the structure for PPC ¼ 4.

Fig. 22. Variations of b,

j and rs as functions of / of the structure with PPC ¼ 4 for (a) qs ¼ 0:3 and (b) qs ¼ 0:99.

and scattering, which is evident from the results, presented in Fig. 22(b). In view of the radiative properties, presented in this section, it is now evident that since b increases with the decrease in / and the increase in PPC as well as qs , keff ;R decreases in these conditions (refer to Eq. (20) for clarification). On the other hand, although keff ;PC is independent of qs and PPC [28], it increases with the decrease in / (see Fig. 9 for clarification). In addition, the coefficient A, which is introduced in Eq. (20) in order to account for the coupling between the conduction and the radiation modes of heat transfer, is expected to be functions of both qs and PPC. Therefore, it may not be that straightforward to apriori predict the effects of qs and PPC on keff ;t , unless detailed investigation is performed. These issues will be taken up in near future.

4. Conclusions In the present investigation, a combined conduction and radiation heat transfer based numerical model is used in order to determine the total effective thermal conductivity of a tetrakaidecahedra structure. Extensive parametric study is conducted that explores the effects of average temperature, thermal conductivity of the solid phase and porosity on keff ;t of the considered structure. Stagnant air with kf ¼ 0:029 W=mK is considered as the working fluid for all considered cases. For quick evaluation of keff ;t in the presence of combined conduction and radiation heat transfer, three different correlations are proposed from a large set of simulated data. Based on the present study, the following major conclusions may be drawn:

1. The information about the porosity of the tetrakaidecahedra structure, the thermo-physical properties of the involved phases and the average temperature of the structure is sufficient in order to predict keff ;t using correlation 1. However, in the absence of thermal radiation and particularly for ks < 5 W=mK, correlation 1 fails to recover the effective thermal conductivity due to pure heat conduction keff ;PC . 2. Correlation 2 provides a fairly good prediction of keff ;t and satisfies all possible limiting conditions, although it requires a separate simulation (or correlation) for keff ;t . 3. Correlation 3 uses the superimposition principle in order to account for the contributions of both conduction and radiation modes of heat transfer, while introducing an adjustable coefficient that takes care of the coupling between two modes of heat transfer. However, this correlation requires the extinction coefficient b, which needs to be calculated separately or may be found from existing data base. 4. The extinction coefficient b increases with the increase in PPC as well as qs and with the decrease in /. Therefore, keff ;R and hence keff ;t is expected to decrease for these conditions. Conflict of interest The authors declared that there is no conflict of interest. Acknowledgement V.M. Patel and P. Talukdar would like to thank the Science and Education Research Board (SERB), Department of Science and Technology, Government of India, for the financial support in order to

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carry out the present research work. They would also like to acknowledge HPC facility of IIT Delhi for providing the computational resources. S. Ray would like to thank the German Research Foundation (DFG) for financially supporting the subproject B02 under the framework of Collaborative Research Centre SFB 920, granted to the Technische Universität Bergakademie Freiberg, Germany.

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