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Evaluation of effective thermal conductivity of porous foams in presence of arbitrary working fluid
International Journal of Thermal Sciences 79 (2014) 260e265
Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Evaluation of effective thermal conductivity of porous foams in presence of arbitrary working fluid Miguel A.A. Mendes*, Subhashis Ray*, Dimosthenis Trimis Institute of Thermal Engineering, Technische Universität Bergakademie Freiberg, Gustav-Zeuner-Strasse 7, D-09596 Freiberg, Germany
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 July 2013 Received in revised form 5 November 2013 Accepted 8 January 2014 Available online 22 February 2014
The Effective Thermal Conductivity (ETC) of open-cell porous foams can be either measured using experimental techniques or predicted from the detailed numerical simulation, considering the complex foam structure obtained from three-dimensional (3D) Computed Tomography (CT)-scan images. An alternative approach could be to consider simplified models for a quick and accurate estimation of the ETC. A model for ETC of open-cell porous foams, using an extremely simplified approach, has been proposed recently by Mendes et al. [1] and it provides an expression for the ETC with one adjustable parameter. It relies upon a single numerical prediction of the dimensionless ETC under vacuum condition, based on the detailed foam structure, obtained from 3D CT-scan information. Using experimental techniques, however, the vacuum condition is difficult to achieve. Therefore, it would be more suitable to conduct the measurement of ETC in presence of a commonly available fluid, like air or water, in order to determine the model parameter. From present result it can be concluded that lower thermal conductivity working fluid, like air, is the most suitable for evaluating the model parameter. Nevertheless, higher thermal conductivity working fluid, like water, also yields an accurate estimation of the model parameter if the thermal conductivity of the solid matrix is also sufficiently high. Ó 2014 Elsevier Masson SAS. All rights reserved.
Keywords: Open-cell porous foams Effective thermal conductivity Simplified model Experimental technique Optimal working fluid
1. Introduction Most research efforts on modeling of practical heat and mass transfer applications involving porous media are largely based upon the popular homogenization approach, where the transport processes are analyzed using macroscopic models, thereby effectively neglecting the detailed micro-scale effects of the porous structure [2]. Nevertheless, for obtaining accurate predictions, these macroscopic models require reliable information about effective thermo-physical properties of porous media those essentially reflect micro-scale effects in an implicit manner. In this respect, the effective thermal conductivity (ETC) of porous media is an essential macroscopic parameter. The most common approach for modeling macroscopic heat conduction through a porous medium is to treat it as a homogeneous medium with an ETC that accounts for the contribution of thermal conductivities of both solid matrix and fluid phase. This assumption is valid whenever the characteristic dimension of the physical problem is much larger than the characteristic dimension
* Corresponding authors. E-mail addresses: [email protected] (M.A.A. Mendes), ray@ iwtt.tu-freiberg.de (S. Ray). 1290-0729/$ e see front matter Ó 2014 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.ijthermalsci.2014.01.009
of the porous structure, which is typically represented by its pore size. The ETC of porous media can be either measured with experimental techniques [3e6] or numerically predicted from simulations, considering the detailed morphology of porous media. Owing to the very high porosity of foams, the total heat transfer can take place by both conduction as well as thermal radiation, if the operating temperature as well as the applied temperature difference are relatively high. The structural information, required for numerical simulations, nowadays can be easily obtained from high resolution, three-dimensional (3D) Computed Tomography (CT) scan images for both conduction [1,7,8] and radiation [9e13], although the present investigation is focused only on the conduction heat transfer. Nevertheless, even in presence of combined mode of heat transfer, evaluation of the ETC due to pure heat conduction would find its importance [14,15]. However, such detailed approach could prove to be quite time consuming, since the ETC of a specific structure for a given working fluid is obtained individually on a case to case basis. Therefore, in order to achieve a reasonable compromise between accuracy and measuring (or computational) effort, an alternative approach could be to obtain simplified models for quick and accurate evaluation of the ETC, ideally presenting a wide range of applicability [13].
M.A.A. Mendes et al. / International Journal of Thermal Sciences 79 (2014) 260e265
Nomenclature b dp k Lr
structure parameters characteristic pore dimension [m] thermal conductivity [W/m K] representative size of porous medium domain [m]
Greek symbols ɸ macroscopic porosity Superscript e dimensionless variable Subscript eff effective f fluid HS, lower lower HashineShtrikman bound for ETC HS, upper upper HashineShtrikman bound for ETC max maximum generic bound for ETC min minimum generic bound for ETC s solid
261
~ ði:e:; k eff;s ¼ keff;s =ks Þ is independent of the thermal conductivity of solid phase. Second, the dimensionless ETC under vacuum condition implicitly includes the quantitative morphological information of porous media that is relevant for solving any further heat conduction problem. Using experimental techniques, however, the vacuum condition is very difficult to achieve, if not impossible, although it is quite simple to assume the same during a numerical simulation. Therefore, from the experimental perspective, it is always better to conduct the measurement of ETC in presence of a commonly available fluid, like air or water. It is evident that using the suggested experimental approach, the CT-scan data of foam structures would not be required. The main objective of the present study is to propose a suitable method for the experimental determination of adjustable parameter required in the simplified model of Mendes et al. [1]. As will be shortly apparent, during the present investigation, detailed numerical simulations are performed in order to mimic the real-life experiments. Merits and demerits of this simplified approach for evaluating the ETC of open-cell porous foams using a single point measurement are thoroughly discussed in this paper. Therefore, the outcome of this study is expected to help experimentalists in evaluating the ETC in presence of any working fluid from a single measurement, using a commonly available fluid that would suit the purpose. 2. Theoretical considerations
Different types of simplified models can be found in the literature [1,13,16e23] for the prediction of the ETC of porous foams. Most of these models provide analytical expressions or closed form correlations for the ETC, based on either experimental data or theoretical considerations, that grossly simplify either the material morphology or the solution method. In some of these simplified models [16e18], porosity was considered as the only structural parameter. However, the direct influence of porous media morphology on the magnitude of ETC was neglected by these models, although its influence on the conduction heat transfer is known to be relevant, see e.g., [8]. In order to take into account the morphology of open-cell porous foams, several other authors [19e23] proposed simplified models for the ETC by approximating the geometry of porous media as a cluster of elementary cells with simplified geometry. Complete heat conduction problem in such cases is typically modeled as a network of thermal resistances and in some particular situations, further simplifications are also made with respect to the solution method. Alternatively, simplified models for the ETC of open-cell porous foams can also be given in terms of correlations, those are obtained uniquely from numerical simulations of conduction heat transfer using the detailed morphology of porous media, where no further simplifications or assumptions are introduced for computations [1,13]. Recently, a model for estimating the ETC of open-cell porous foams using a simplified approach has been proposed by Mendes et al. [1]. This methodology provides an explicit expression for the ETC, with only one adjustable parameter. It relies upon a single numerical prediction of the dimensionless ETC under vacuum condition, based on the detailed geometry of porous media, which is obtained from 3D CT-scan images. Development of a correlation for the ETC of porous media, in the form proposed by Mendes et al. [1], is useful since it applies to any fluidesolid combination and hence the result could be extended even in presence of any hazardous fluid for which experiments may be impossible to conduct. Two main advantages of this approach are the following. First, it requires significantly reduced computational time for calculating the dimensionless ETC under vacuum condition, which scales to the order of the porosity of foams and the result of computation
Let us consider a porous medium, formed by a solid and a fluid phase, with different thermal conductivities ks and kf, respectively. The simplified ETC model, in dimensionless form, proposed by Mendes et al. [1] is given as:
~ ~ ~ k eff ¼ bkmin þ ð1 bÞkmax
(1)
where all the thermal conductivities are made dimensionless with ~ ¼ respected to the thermal conductivity of the solid phase ks i.e., k ~ ~ k=ks : Further, kmin and kmax are the generic minimum and maximum dimensionless bounds for ~ keff and b is an adjustable parameter to be determined either from a single measurement or from an equivalent detailed numerical prediction of the ETC at a ~ ; acselected reference condition. It may be noted here that k eff ~ ~ cording to Eq. (1), is a linear combination of kmin and kmax with weighting factors b and 1 b, respectively. However, it will be shortly evident that the generic lower and upper bounds of ETC are, in general, nonlinear functions of the dimensionless fluid conduc~ ; for a particular porous medium. Consequently, Eq. (1) tivity k f ~ as function of k ~: represents a complex nonlinear behavior of k eff f The previous study of Mendes et al. [1] showed that in general, the HashineShtrikman bounds serve the best for all the investi~ ~ gated open-cell porous foams i.e., when k ¼ k and min
HS;lower
~ ~ kmax ¼ k HS;upper are selected. These bounds are given as [24]:
~ ~ k min ¼ kHS;lower
h i ~ 2k ~ þ12 k ~ 1 ð1 fÞ k f f f ¼ kf 1 ð1 fÞ 2~ kf þ 1 þ ~
~ ~ k max ¼ kHS;upper ¼
~ 2 1k ~ f 2þk f f ~ þ 1k ~ f 2þk f
(2a)
(2b)
f
where is the macroscopic porosity of the medium. It must be noted here that both lower and upper bounds of the dimensionless ETC ~ ¼ k =ks and f can be uniquely determined from known values of k f f in a straightforward manner from Eq. (2).
262
M.A.A. Mendes et al. / International Journal of Thermal Sciences 79 (2014) 260e265
~ Considering the reference ETC k eff;ref evaluated for a known ~ reference thermal conductivity ratio of fluid to solid k f;ref ; a generic expression for b can be obtained from Eq. (1) as:
b ¼
~ ~ k max;ref keff;ref ~ ~ kmax;ref kmin;ref
(3)
~ ~ ~ ~ where k min;ref and kmax;ref are respective values for kmin and kmax at ~ the reference condition kf;ref ; those are directly evaluated from Eq. (2). It may be recognized here that the expression for b, according to Eq. (3), behaves similar to that of an interpolation factor. If the numerical simulation or the pseudo-experiment is carried ~ out under vacuum condition, i.e., for k f;ref ¼ 0; lower and upper ~ ~ HashineShtrikman bounds for k min and kmax ; respectively, are obtained according to Eq. (2) as:
~ k min ~
~ ¼ k HS;lower ~
¼ 0
~max k ~
~ ¼ k HS;upper ~
¼
kf;ref ¼0
kf ;ref ¼0
kf;ref ¼0
kf;ref ¼0
(4a)
2ð1 fÞ ð2 þ fÞ
(4b)
Denoting the dimensionless ETC of porous medium under vac~ uum condition as k eff;s and using Eqs. (3) and (4), the following expression for b is readily obtained [1]:
bj~k
f;ref ¼0
¼ 1
~ k eff;s ð2 þ fÞ 2 ð1 fÞ
(5)
As demonstrated by Mendes et al. [1] through several case ~ studies, k eff;s implicitly contains the morphological information of the porous medium in a quantitative manner that is relevant for heat transfer due to pure conduction. Additionally, its calculation is performed only once for a particular porous medium since the dimensionless results are independent of the thermal conductivity of solid phase. To that extent, ~ keff;s can be recognized as a geometric factor that solely depends on the internal structure of porous media. It is important to note that the evaluation of b according to Eq. ~ (5) would be most meaningful when k ¼ ~ k is evaluated eff;ref
eff;s
based on a detailed numerical simulation, because only in this situation the vacuum condition can be accurately imposed. How~ is evaluated in presence of a commonly available ever, if k eff;ref
~ fluid, the value of k f ;ref would lie in the intermediate range between zero and unity since the thermal conductivity of the fluid is expected to be less than that of the solid. Therefore, instead of using Eq. (5), as originally proposed by Mendes et al. [1], Eq. (3) should be invoked for the evaluation of b. This alternative approach can be ~ regarded more practical for experimentally obtaining k and eff;ref
hence the value of b to be used in Eq. (1) for any other working fluid. It is very important to note here that an experimental approach does not require an explicit information about the foam structure, obtained for example, from the 3D CT-scan data. The major issue, therefore, boils down to finding out a suitable reference fluid for could be obtained based on a single which bzb~ kf ;ref ¼0
measurement. It may, therefore, be reiterated here that the experimental methodology, proposed in this article for the evaluation of b from Eq. (3), does not necessarily require to consider the structure of ~ foams in an explicit manner. However, as in the case of k eff;s ; ~ numerically calculated based on the CT-scan data, keff;ref obtained from the experiment directly depends on the considered foam structure. Therefore, the respective value of b also takes into
account the morphology of the specific foam in an implicit manner. In general, many correlations available in the literature do not take into account the morphology of a particular foam, since adjustable parameters of those correlations are evaluated from sets of measurements, including geometry of foams arising out of different structures [16e18]. 3. Results and discussions The ETC is predicted for real open-cell foam structures in the ~ < 1: Raw data from complete k =ks range lower than unity, i.e., k f
f
detailed numerical simulations obtained by Mendes et al. [1] for four different samples of ceramic and metal foams (named here as Foam 1 to Foam 4) are considered for analysis in the present study and the predicted ETCs are used as reference for the comparison with simplified model. These detailed predictions are obtained using an in-house code, which is enabled with excellent parallel computation facility and is based on finite volume method 1. Structures of these foams are retrieved from 3D CT-scan images [25,26] and are shown in Fig. 1, whereas the relevant geometric information about these foam samples is presented in Table 1. Mesh resolutions for detailed ETC predictions are the same as those used ~ in order to obtain the CT-scan data 1. In addition, the k predicted eff;s
from detailed simulations under vacuum condition, i.e., by neglecting the contribution of the fluid phase, along with the values ; obtained from Eq. (5), are also presented in the same of b~ kf ;ref ¼0
table for all the four foam samples. Fig. 2 presents the comparison between detailed predictions ~ and respective estimations from the simplified model (1) of k eff
for different open-cell porous foams, using values of b obtained from Eq. (5), i.e., assuming vacuum condition. It can be observed that in general, estimated values are in excellent agreement with simulated values, indicating the validity of the model for a wide range of conductivity ratio ð0 < ~ k 1Þ: Respective relative errors f
for the estimations of ~ keff are presented in Fig. 2(b). The figure clearly shows that for all the foam samples, model (1) with b evaluated under vacuum condition, yields a maximum relative ~ : It may error that is consistently lower than 5% for any value of k f
be further noted that relative errors tend to vanish in the limits ~ /0 and ~ as k k /1; indicating model (1) respects the limiting f
f
values of ~ keff when the adjustable parameter b is evaluated from Eq. (5) under vacuum condition. Nevertheless, the overall results indicate that this simplified approach for estimating the ETC based on the CT-scan information of open-cell porous foams is quite adequate for practical purposes and could alternatively be applied based on a single measurement of ETC for a selected reference fluid as long as the required condition bzb~ is kf;ref ¼0
satisfied. As discussed earlier, the experimental evaluation of ~ ~ ~ keff;ref ¼ k eff;s ; by setting the vacuum condition, i.e., kf ;ref ¼ 0; is not practically feasible. Therefore, a different reference condition, where ~ kf ;ref depends on the working fluid, would be preferred for ~ experimental condition in order to obtain k eff;ref and hence b can be estimated using Eq. (3). ~ The choice of optimal reference condition k ; for evaluating f ;ref
the parameter b in the simplified model (1), requires a closer look. Fig. 3 presents the value of b, calculated according to Eq. (3), as ~ ; for different open-cell porous foams. One can function of k f;ref
easily observe from the figure that, as reported by Mendes et al. [1], the value of b strongly depends on the foam structure. Nevertheless, the most important observation is that for each foam
M.A.A. Mendes et al. / International Journal of Thermal Sciences 79 (2014) 260e265
263
Fig. 1. Samples of open-cell porous foams obtained from 3D CT-scan images: (a) Foam 1 and (b) Foam 2 (Al2O3eC ceramic foams); (c) Foam 3 (Al2O3 ceramic foam); (d) Foam 4 (FeCrAl-alloy metal foam).
structure, the value of b remains almost constant for lower values of ~ k ; whereas it changes rapidly with ~ k for higher values. The f ;ref
f;ref
reason for such a behavior can be explained as follows. In the limit ~ ~ as ~ kf /1; the rate of change in k eff with respect to kf , i.e., ~ ; for a given foam can be shown to equal the porosity d~ keff =dk f ~ kf ¼1
f of that particular foam. This can be readily observed from Fig. 2(a) for Foams 2 and 3, those have similar porosities of 0.74 and 0.79, respectively (see Table 1 for details). In other words, this means that ~ tends to become insensitive to the chosen value of b in the k eff
vicinity of ~ kf ¼ 1: As a result, any small error incurred by model (1)
Table 1 Geometric information about foam samples presented in Fig. 1. Sample
Foam 1
Pores per inch (ppi) Lr (mm) CT resolution (mm) dp (mm)
10 18 65 3 0.57 23.98 0.27
f
2 ~ k eff ;s 10 bjk~ ¼0 ; Eq. (5) f;ref
Foam 2
Foam 3
Foam 4
18 65 4 0.74 11.04 0.42
30 18 65 2.8 0.79 9.43 0.38
9 17 2.8 0.88 3.15 0.62
~ ~ in predicting k eff;ref for higher values of kf ;ref is compensated by a large change in the required value of b and hence the adjustable ~ for its higher values. parameter b appears to be too sensitive to k f ;ref
It may be recognized here that such a compensation of b for higher ~ although satisfies perfectly the reference condition, values of k f ;ref
~ (i.e., the desired ETC) for other leads to erroneous prediction of k eff ~ which is clearly values (particularly the lower values) of k f
demonstrated in Fig. 3(b). ~ The influence of k f ;ref on the maximum absolute value of the relative error for the estimation of ETC, with b obtained from Eq. (3), is shown in Fig. 3(b) for different open-cell porous foams. One can ~ clearly conclude from the figure that lower values of k f;ref ; for which b is already proven to be less sensitive, yield a lower relative error for the estimations of ETC according to the simplified model (1). ~ Typical ranges of k f ;ref for different combinations of commonly available working fluids (air or water) and foam materials are also presented in Fig. 3. It can be authentically concluded from Fig. 3(b) that the evaluation of b according to Eq. (3) is more accurate if lower thermal conductivity fluid, e.g. air, is considered as the working fluid. Nevertheless, working fluids with higher thermal conductivity, like water, could be also suitable only if the thermal conductivity of the solid matrix is also high enough e.g., for metal foams.
264
M.A.A. Mendes et al. / International Journal of Thermal Sciences 79 (2014) 260e265
Fig. 2. Comparison between detailed numerical predictions and estimations from the ~ ; for simplified model (1), with b evaluated under vacuum condition, as functions of k f different foam samples: (a) dimensionless ETC; (b) estimated relative errors for the ETC.
4. Conclusions In the present study, a recently proposed simplified model for the ETC is considered in order to investigate the possible experimental determination of the adjustable parameter in the model, using a commonly available fluid (i.e., either air or water). Detailed numerical simulations of the ETC, for real open-cell porous foams, are considered in order to mimic the real life experiments. The structure of these foams is obtained from 3D CT-scan images. Predictions, obtained from the simplified model, are compared against results from detailed simulations for the complete range of thermal conductivity ratio of fluid to solid phases lower than unity. Based on this comparison, the choice of the optimal thermal conductivity ~ ratio of the reference fluid to solid phase k f;ref for evaluating the model parameter and hence the working fluid for a single point measurement is investigated. From the present results it can be safely concluded that lower thermal conductivity working fluid, like air, is the most suitable for evaluating the adjustable model parameter b. Nevertheless, relatively higher thermal conductivity working fluids, like water, also could yield accurate estimations for the model parameter b only when the thermal conductivity of the solid matrix is also reasonably high.
Fig. 3. Influence of reference condition ~kf;ref on the evaluation of b from Eq. (3) for different foam samples: (a) effect on estimated value of b; (b) maximum absolute relative error for the estimation of ETC.
Finally, following the recommendation of this paper, along with the conclusions drawn by Mendes et al. [1], it would now be possible to design simple experiment in order to determine the ETC of porous foams in presence of any hazardous working fluids (e.g. in presence of highly inflammable gas, like hydrogen, products of partial oxidation, etc., or liquid metals), where performing an experiment would be nearly impossible. Acknowledgments The authors would like to thank the German Research Foundation (DFG) for supporting the investigation in the subproject B02, which is part of the Collaborative Research Center CRC 920. References [1] M.A.A. Mendes, S. Ray, D. Trimis, A simple and efficient method for the evaluation of effective thermal conductivity of open-cell foam-like structures, Int. J. Heat. Mass Transf. 66 (2013) 412e422. [2] M. Kaviany, Principles of Heat Transfer in Porous Media, in: Mechanical Engineering Series, Springer-Verlag, 1995. [3] C. Tseng, M. Yamaguchi, T. Ohmori, Thermal conductivity of polyurethane foams from room temperature to 20 k, Cryogenics 37 (1997) 305e312. [4] V. Calmidi, R. Mahajan, The effective thermal conductivity of high porosity fibrous metal foams, J. Heat. Transf. 121 (1999) 466e471. [5] A. Bhattacharya, V. Calmidi, R. Mahajan, An analytical-experimental study for the determination of the effective thermal conductivity of high porosity
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fibrous foams, in: R. Sullivan (Ed.), Application of Porous Media Methods for Engineered Materials, vol. 233ASME, 1999, pp. 13e20. A. Sullins, K. Daryabeigi, Effective thermal conductivity of high porosity open cell nickel foam, in: 35th AIAA Thermophysics Conference, Anaheim, CA, 2001. J. Petrasch, B. Schrader, P. Wyss, A. Steinfeld, Tomography-based determination of the effective thermal conductivity of fluid-saturated reticulate porous ceramics, J. Heat. Transf. 130 (2008) 032602e032610. R. Coquard, D. Baillis, Numerical investigation of conductive heat transfer in high-porosity foams, Acta Mater. 57 (2009) 5466e5479. J. Petrasch, P. Wyss, A. Steinfeld, Tomography-based monte carlo determination of radiative properties of reticulate porous ceramics, J. Quant. Spectrosc. Radiat. Transf. 105 (2007) 180e197. M. Loretz, E. Maire, D. Baillis, Analytical modelling of the radiative properties of metallic foams: contribution of X-ray tomography, Adv. Eng. Mater. 10 (2008) 352e360. S. Haussener, P. Coray, W. Lipiski, P. Wyss, A. Steinfeld, Tomography-based heat and mass transfer characterization of reticulate porous ceramics for high-temperature processing, J. Heat. Transf. 132 (2009) 023305. R. Coquard, D. Baillis, E. Maire, Numerical investigation of the radiative properties of polymeric foams from tomographic images, J. Thermophys. Heat. Transf. 24 (2010) 647e658. R. Coquard, D. Rochais, D. Baillis, Conductive and radiative heat transfer in ceramic and metal foams at fire temperatures, Fire Technol. 48 (2012) 699e 732. P. Talukdar, M.A.A. Mendes, R.K. Parida, D. Trimis, S. Ray, Modelling of conduction radiation in a porous medium with blocked-off region approach, Int. J. Therm. Sci. 72 (2013) 102e114. M.A.A. Mendes, P. Talukdar, S. Ray, D. Trimis, Detailed and simplified models for evaluation of effective thermal conductivity of open-cell porous foams at high temperatures in presence of thermal radiation, Int. J. Heat. Mass Transf. 68 (2014) 612e624.
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[16] R. Singh, H.S. Kasana, Computational aspects of effective thermal conductivity of highly porous metal foams, Appl. Therm. Eng. 24 (2004) 1841e1849. [17] E. Solórzano, J.A. Reglero, M.A. Rodríguez-Pérez, D. Lehmhus, M. Wichmann, J.A. de Saja, An experimental study on the thermal conductivity of aluminium foams by using the transient plane source method, Int. J. Heat. Mass Transf. 51 (2008) 6259e6267. [18] B. Dietrich, G. Schell, E.C. Bucharsky, R. Oberacker, M.J. Hoffmann, W. Schabel, M. Kind, H. Martin, Determination of the thermal properties of ceramic sponges, Int. J. Heat. Mass Transf. 53 (2010) 198e205. [19] L.R. Glicksmann, M.A. Schuetz, Heat transfer in foams, in: N. Hilyard, A. Cunningham (Eds.), Low Density Cellular Plastics: Physical Basis of Behaviour, vol. 248Chapman & Hall, London, 1994, pp. 104e152. [20] X. Fu, R. Viskanta, J.P. Gore, Prediction of effective thermal conductivity of cellular ceramics, Int. Commun. Heat. Mass Transf. 25 (1998) 151e160. [21] K. Boomsma, D. Poulikakos, On the effective thermal conductivity of a threedimensionally structured fluid-saturated metal foam, Int. J. Heat. Mass Transf. 44 (2001) 827e836. [22] A. Ahern, G. Verbist, D. Weaire, R. Phelan, H. Fleurent, The conductivity of foams: a generalisation of the electrical to the thermal case, Colloids Surf. A: Physicochem. Eng. Aspects 263 (1e3) (2005) 275e279. [23] J.F. Wang, J.K. Carson, J. Willix, M.F. North, D.J. Cleland, A symmetric and interconnected skeleton structural (siss) model for predicting thermal and electrical conductivity and youngs modulus of porous foams, Acta Mater. 56 (2008) 5138e5146. [24] Z. Hashin, S. Shtrikman, A variational approach to the theory of the effective magnetic permeability of multiphase materials, J. Appl. Phys. 33 (1962) 3125e 3131. [25] M. Emmel, C.G. Aneziris, Development of novel carbon bonded filter compositions for steel melt filtration, Ceram. Int. 38 (2012) 5165e5173. [26] V. Skibina, Bestimmung der wärmeleitfähigkeit von nichtdurchströmten zellularen metallen (Ph.D. thesis), TU Bergakademie Freiberg, Germany, 2012.
Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Evaluation of effective thermal conductivity of porous foams in presence of arbitrary working fluid Miguel A.A. Mendes*, Subhashis Ray*, Dimosthenis Trimis Institute of Thermal Engineering, Technische Universität Bergakademie Freiberg, Gustav-Zeuner-Strasse 7, D-09596 Freiberg, Germany
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 July 2013 Received in revised form 5 November 2013 Accepted 8 January 2014 Available online 22 February 2014
The Effective Thermal Conductivity (ETC) of open-cell porous foams can be either measured using experimental techniques or predicted from the detailed numerical simulation, considering the complex foam structure obtained from three-dimensional (3D) Computed Tomography (CT)-scan images. An alternative approach could be to consider simplified models for a quick and accurate estimation of the ETC. A model for ETC of open-cell porous foams, using an extremely simplified approach, has been proposed recently by Mendes et al. [1] and it provides an expression for the ETC with one adjustable parameter. It relies upon a single numerical prediction of the dimensionless ETC under vacuum condition, based on the detailed foam structure, obtained from 3D CT-scan information. Using experimental techniques, however, the vacuum condition is difficult to achieve. Therefore, it would be more suitable to conduct the measurement of ETC in presence of a commonly available fluid, like air or water, in order to determine the model parameter. From present result it can be concluded that lower thermal conductivity working fluid, like air, is the most suitable for evaluating the model parameter. Nevertheless, higher thermal conductivity working fluid, like water, also yields an accurate estimation of the model parameter if the thermal conductivity of the solid matrix is also sufficiently high. Ó 2014 Elsevier Masson SAS. All rights reserved.
Keywords: Open-cell porous foams Effective thermal conductivity Simplified model Experimental technique Optimal working fluid
1. Introduction Most research efforts on modeling of practical heat and mass transfer applications involving porous media are largely based upon the popular homogenization approach, where the transport processes are analyzed using macroscopic models, thereby effectively neglecting the detailed micro-scale effects of the porous structure [2]. Nevertheless, for obtaining accurate predictions, these macroscopic models require reliable information about effective thermo-physical properties of porous media those essentially reflect micro-scale effects in an implicit manner. In this respect, the effective thermal conductivity (ETC) of porous media is an essential macroscopic parameter. The most common approach for modeling macroscopic heat conduction through a porous medium is to treat it as a homogeneous medium with an ETC that accounts for the contribution of thermal conductivities of both solid matrix and fluid phase. This assumption is valid whenever the characteristic dimension of the physical problem is much larger than the characteristic dimension
* Corresponding authors. E-mail addresses: [email protected] (M.A.A. Mendes), ray@ iwtt.tu-freiberg.de (S. Ray). 1290-0729/$ e see front matter Ó 2014 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.ijthermalsci.2014.01.009
of the porous structure, which is typically represented by its pore size. The ETC of porous media can be either measured with experimental techniques [3e6] or numerically predicted from simulations, considering the detailed morphology of porous media. Owing to the very high porosity of foams, the total heat transfer can take place by both conduction as well as thermal radiation, if the operating temperature as well as the applied temperature difference are relatively high. The structural information, required for numerical simulations, nowadays can be easily obtained from high resolution, three-dimensional (3D) Computed Tomography (CT) scan images for both conduction [1,7,8] and radiation [9e13], although the present investigation is focused only on the conduction heat transfer. Nevertheless, even in presence of combined mode of heat transfer, evaluation of the ETC due to pure heat conduction would find its importance [14,15]. However, such detailed approach could prove to be quite time consuming, since the ETC of a specific structure for a given working fluid is obtained individually on a case to case basis. Therefore, in order to achieve a reasonable compromise between accuracy and measuring (or computational) effort, an alternative approach could be to obtain simplified models for quick and accurate evaluation of the ETC, ideally presenting a wide range of applicability [13].
M.A.A. Mendes et al. / International Journal of Thermal Sciences 79 (2014) 260e265
Nomenclature b dp k Lr
structure parameters characteristic pore dimension [m] thermal conductivity [W/m K] representative size of porous medium domain [m]
Greek symbols ɸ macroscopic porosity Superscript e dimensionless variable Subscript eff effective f fluid HS, lower lower HashineShtrikman bound for ETC HS, upper upper HashineShtrikman bound for ETC max maximum generic bound for ETC min minimum generic bound for ETC s solid
261
~ ði:e:; k eff;s ¼ keff;s =ks Þ is independent of the thermal conductivity of solid phase. Second, the dimensionless ETC under vacuum condition implicitly includes the quantitative morphological information of porous media that is relevant for solving any further heat conduction problem. Using experimental techniques, however, the vacuum condition is very difficult to achieve, if not impossible, although it is quite simple to assume the same during a numerical simulation. Therefore, from the experimental perspective, it is always better to conduct the measurement of ETC in presence of a commonly available fluid, like air or water. It is evident that using the suggested experimental approach, the CT-scan data of foam structures would not be required. The main objective of the present study is to propose a suitable method for the experimental determination of adjustable parameter required in the simplified model of Mendes et al. [1]. As will be shortly apparent, during the present investigation, detailed numerical simulations are performed in order to mimic the real-life experiments. Merits and demerits of this simplified approach for evaluating the ETC of open-cell porous foams using a single point measurement are thoroughly discussed in this paper. Therefore, the outcome of this study is expected to help experimentalists in evaluating the ETC in presence of any working fluid from a single measurement, using a commonly available fluid that would suit the purpose. 2. Theoretical considerations
Different types of simplified models can be found in the literature [1,13,16e23] for the prediction of the ETC of porous foams. Most of these models provide analytical expressions or closed form correlations for the ETC, based on either experimental data or theoretical considerations, that grossly simplify either the material morphology or the solution method. In some of these simplified models [16e18], porosity was considered as the only structural parameter. However, the direct influence of porous media morphology on the magnitude of ETC was neglected by these models, although its influence on the conduction heat transfer is known to be relevant, see e.g., [8]. In order to take into account the morphology of open-cell porous foams, several other authors [19e23] proposed simplified models for the ETC by approximating the geometry of porous media as a cluster of elementary cells with simplified geometry. Complete heat conduction problem in such cases is typically modeled as a network of thermal resistances and in some particular situations, further simplifications are also made with respect to the solution method. Alternatively, simplified models for the ETC of open-cell porous foams can also be given in terms of correlations, those are obtained uniquely from numerical simulations of conduction heat transfer using the detailed morphology of porous media, where no further simplifications or assumptions are introduced for computations [1,13]. Recently, a model for estimating the ETC of open-cell porous foams using a simplified approach has been proposed by Mendes et al. [1]. This methodology provides an explicit expression for the ETC, with only one adjustable parameter. It relies upon a single numerical prediction of the dimensionless ETC under vacuum condition, based on the detailed geometry of porous media, which is obtained from 3D CT-scan images. Development of a correlation for the ETC of porous media, in the form proposed by Mendes et al. [1], is useful since it applies to any fluidesolid combination and hence the result could be extended even in presence of any hazardous fluid for which experiments may be impossible to conduct. Two main advantages of this approach are the following. First, it requires significantly reduced computational time for calculating the dimensionless ETC under vacuum condition, which scales to the order of the porosity of foams and the result of computation
Let us consider a porous medium, formed by a solid and a fluid phase, with different thermal conductivities ks and kf, respectively. The simplified ETC model, in dimensionless form, proposed by Mendes et al. [1] is given as:
~ ~ ~ k eff ¼ bkmin þ ð1 bÞkmax
(1)
where all the thermal conductivities are made dimensionless with ~ ¼ respected to the thermal conductivity of the solid phase ks i.e., k ~ ~ k=ks : Further, kmin and kmax are the generic minimum and maximum dimensionless bounds for ~ keff and b is an adjustable parameter to be determined either from a single measurement or from an equivalent detailed numerical prediction of the ETC at a ~ ; acselected reference condition. It may be noted here that k eff ~ ~ cording to Eq. (1), is a linear combination of kmin and kmax with weighting factors b and 1 b, respectively. However, it will be shortly evident that the generic lower and upper bounds of ETC are, in general, nonlinear functions of the dimensionless fluid conduc~ ; for a particular porous medium. Consequently, Eq. (1) tivity k f ~ as function of k ~: represents a complex nonlinear behavior of k eff f The previous study of Mendes et al. [1] showed that in general, the HashineShtrikman bounds serve the best for all the investi~ ~ gated open-cell porous foams i.e., when k ¼ k and min
HS;lower
~ ~ kmax ¼ k HS;upper are selected. These bounds are given as [24]:
~ ~ k min ¼ kHS;lower
h i ~ 2k ~ þ12 k ~ 1 ð1 fÞ k f f f ¼ kf 1 ð1 fÞ 2~ kf þ 1 þ ~
~ ~ k max ¼ kHS;upper ¼
~ 2 1k ~ f 2þk f f ~ þ 1k ~ f 2þk f
(2a)
(2b)
f
where is the macroscopic porosity of the medium. It must be noted here that both lower and upper bounds of the dimensionless ETC ~ ¼ k =ks and f can be uniquely determined from known values of k f f in a straightforward manner from Eq. (2).
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~ Considering the reference ETC k eff;ref evaluated for a known ~ reference thermal conductivity ratio of fluid to solid k f;ref ; a generic expression for b can be obtained from Eq. (1) as:
b ¼
~ ~ k max;ref keff;ref ~ ~ kmax;ref kmin;ref
(3)
~ ~ ~ ~ where k min;ref and kmax;ref are respective values for kmin and kmax at ~ the reference condition kf;ref ; those are directly evaluated from Eq. (2). It may be recognized here that the expression for b, according to Eq. (3), behaves similar to that of an interpolation factor. If the numerical simulation or the pseudo-experiment is carried ~ out under vacuum condition, i.e., for k f;ref ¼ 0; lower and upper ~ ~ HashineShtrikman bounds for k min and kmax ; respectively, are obtained according to Eq. (2) as:
~ k min ~
~ ¼ k HS;lower ~
¼ 0
~max k ~
~ ¼ k HS;upper ~
¼
kf;ref ¼0
kf ;ref ¼0
kf;ref ¼0
kf;ref ¼0
(4a)
2ð1 fÞ ð2 þ fÞ
(4b)
Denoting the dimensionless ETC of porous medium under vac~ uum condition as k eff;s and using Eqs. (3) and (4), the following expression for b is readily obtained [1]:
bj~k
f;ref ¼0
¼ 1
~ k eff;s ð2 þ fÞ 2 ð1 fÞ
(5)
As demonstrated by Mendes et al. [1] through several case ~ studies, k eff;s implicitly contains the morphological information of the porous medium in a quantitative manner that is relevant for heat transfer due to pure conduction. Additionally, its calculation is performed only once for a particular porous medium since the dimensionless results are independent of the thermal conductivity of solid phase. To that extent, ~ keff;s can be recognized as a geometric factor that solely depends on the internal structure of porous media. It is important to note that the evaluation of b according to Eq. ~ (5) would be most meaningful when k ¼ ~ k is evaluated eff;ref
eff;s
based on a detailed numerical simulation, because only in this situation the vacuum condition can be accurately imposed. How~ is evaluated in presence of a commonly available ever, if k eff;ref
~ fluid, the value of k f ;ref would lie in the intermediate range between zero and unity since the thermal conductivity of the fluid is expected to be less than that of the solid. Therefore, instead of using Eq. (5), as originally proposed by Mendes et al. [1], Eq. (3) should be invoked for the evaluation of b. This alternative approach can be ~ regarded more practical for experimentally obtaining k and eff;ref
hence the value of b to be used in Eq. (1) for any other working fluid. It is very important to note here that an experimental approach does not require an explicit information about the foam structure, obtained for example, from the 3D CT-scan data. The major issue, therefore, boils down to finding out a suitable reference fluid for could be obtained based on a single which bzb~ kf ;ref ¼0
measurement. It may, therefore, be reiterated here that the experimental methodology, proposed in this article for the evaluation of b from Eq. (3), does not necessarily require to consider the structure of ~ foams in an explicit manner. However, as in the case of k eff;s ; ~ numerically calculated based on the CT-scan data, keff;ref obtained from the experiment directly depends on the considered foam structure. Therefore, the respective value of b also takes into
account the morphology of the specific foam in an implicit manner. In general, many correlations available in the literature do not take into account the morphology of a particular foam, since adjustable parameters of those correlations are evaluated from sets of measurements, including geometry of foams arising out of different structures [16e18]. 3. Results and discussions The ETC is predicted for real open-cell foam structures in the ~ < 1: Raw data from complete k =ks range lower than unity, i.e., k f
f
detailed numerical simulations obtained by Mendes et al. [1] for four different samples of ceramic and metal foams (named here as Foam 1 to Foam 4) are considered for analysis in the present study and the predicted ETCs are used as reference for the comparison with simplified model. These detailed predictions are obtained using an in-house code, which is enabled with excellent parallel computation facility and is based on finite volume method 1. Structures of these foams are retrieved from 3D CT-scan images [25,26] and are shown in Fig. 1, whereas the relevant geometric information about these foam samples is presented in Table 1. Mesh resolutions for detailed ETC predictions are the same as those used ~ in order to obtain the CT-scan data 1. In addition, the k predicted eff;s
from detailed simulations under vacuum condition, i.e., by neglecting the contribution of the fluid phase, along with the values ; obtained from Eq. (5), are also presented in the same of b~ kf ;ref ¼0
table for all the four foam samples. Fig. 2 presents the comparison between detailed predictions ~ and respective estimations from the simplified model (1) of k eff
for different open-cell porous foams, using values of b obtained from Eq. (5), i.e., assuming vacuum condition. It can be observed that in general, estimated values are in excellent agreement with simulated values, indicating the validity of the model for a wide range of conductivity ratio ð0 < ~ k 1Þ: Respective relative errors f
for the estimations of ~ keff are presented in Fig. 2(b). The figure clearly shows that for all the foam samples, model (1) with b evaluated under vacuum condition, yields a maximum relative ~ : It may error that is consistently lower than 5% for any value of k f
be further noted that relative errors tend to vanish in the limits ~ /0 and ~ as k k /1; indicating model (1) respects the limiting f
f
values of ~ keff when the adjustable parameter b is evaluated from Eq. (5) under vacuum condition. Nevertheless, the overall results indicate that this simplified approach for estimating the ETC based on the CT-scan information of open-cell porous foams is quite adequate for practical purposes and could alternatively be applied based on a single measurement of ETC for a selected reference fluid as long as the required condition bzb~ is kf;ref ¼0
satisfied. As discussed earlier, the experimental evaluation of ~ ~ ~ keff;ref ¼ k eff;s ; by setting the vacuum condition, i.e., kf ;ref ¼ 0; is not practically feasible. Therefore, a different reference condition, where ~ kf ;ref depends on the working fluid, would be preferred for ~ experimental condition in order to obtain k eff;ref and hence b can be estimated using Eq. (3). ~ The choice of optimal reference condition k ; for evaluating f ;ref
the parameter b in the simplified model (1), requires a closer look. Fig. 3 presents the value of b, calculated according to Eq. (3), as ~ ; for different open-cell porous foams. One can function of k f;ref
easily observe from the figure that, as reported by Mendes et al. [1], the value of b strongly depends on the foam structure. Nevertheless, the most important observation is that for each foam
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263
Fig. 1. Samples of open-cell porous foams obtained from 3D CT-scan images: (a) Foam 1 and (b) Foam 2 (Al2O3eC ceramic foams); (c) Foam 3 (Al2O3 ceramic foam); (d) Foam 4 (FeCrAl-alloy metal foam).
structure, the value of b remains almost constant for lower values of ~ k ; whereas it changes rapidly with ~ k for higher values. The f ;ref
f;ref
reason for such a behavior can be explained as follows. In the limit ~ ~ as ~ kf /1; the rate of change in k eff with respect to kf , i.e., ~ ; for a given foam can be shown to equal the porosity d~ keff =dk f ~ kf ¼1
f of that particular foam. This can be readily observed from Fig. 2(a) for Foams 2 and 3, those have similar porosities of 0.74 and 0.79, respectively (see Table 1 for details). In other words, this means that ~ tends to become insensitive to the chosen value of b in the k eff
vicinity of ~ kf ¼ 1: As a result, any small error incurred by model (1)
Table 1 Geometric information about foam samples presented in Fig. 1. Sample
Foam 1
Pores per inch (ppi) Lr (mm) CT resolution (mm) dp (mm)
10 18 65 3 0.57 23.98 0.27
f
2 ~ k eff ;s 10 bjk~ ¼0 ; Eq. (5) f;ref
Foam 2
Foam 3
Foam 4
18 65 4 0.74 11.04 0.42
30 18 65 2.8 0.79 9.43 0.38
9 17 2.8 0.88 3.15 0.62
~ ~ in predicting k eff;ref for higher values of kf ;ref is compensated by a large change in the required value of b and hence the adjustable ~ for its higher values. parameter b appears to be too sensitive to k f ;ref
It may be recognized here that such a compensation of b for higher ~ although satisfies perfectly the reference condition, values of k f ;ref
~ (i.e., the desired ETC) for other leads to erroneous prediction of k eff ~ which is clearly values (particularly the lower values) of k f
demonstrated in Fig. 3(b). ~ The influence of k f ;ref on the maximum absolute value of the relative error for the estimation of ETC, with b obtained from Eq. (3), is shown in Fig. 3(b) for different open-cell porous foams. One can ~ clearly conclude from the figure that lower values of k f;ref ; for which b is already proven to be less sensitive, yield a lower relative error for the estimations of ETC according to the simplified model (1). ~ Typical ranges of k f ;ref for different combinations of commonly available working fluids (air or water) and foam materials are also presented in Fig. 3. It can be authentically concluded from Fig. 3(b) that the evaluation of b according to Eq. (3) is more accurate if lower thermal conductivity fluid, e.g. air, is considered as the working fluid. Nevertheless, working fluids with higher thermal conductivity, like water, could be also suitable only if the thermal conductivity of the solid matrix is also high enough e.g., for metal foams.
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M.A.A. Mendes et al. / International Journal of Thermal Sciences 79 (2014) 260e265
Fig. 2. Comparison between detailed numerical predictions and estimations from the ~ ; for simplified model (1), with b evaluated under vacuum condition, as functions of k f different foam samples: (a) dimensionless ETC; (b) estimated relative errors for the ETC.
4. Conclusions In the present study, a recently proposed simplified model for the ETC is considered in order to investigate the possible experimental determination of the adjustable parameter in the model, using a commonly available fluid (i.e., either air or water). Detailed numerical simulations of the ETC, for real open-cell porous foams, are considered in order to mimic the real life experiments. The structure of these foams is obtained from 3D CT-scan images. Predictions, obtained from the simplified model, are compared against results from detailed simulations for the complete range of thermal conductivity ratio of fluid to solid phases lower than unity. Based on this comparison, the choice of the optimal thermal conductivity ~ ratio of the reference fluid to solid phase k f;ref for evaluating the model parameter and hence the working fluid for a single point measurement is investigated. From the present results it can be safely concluded that lower thermal conductivity working fluid, like air, is the most suitable for evaluating the adjustable model parameter b. Nevertheless, relatively higher thermal conductivity working fluids, like water, also could yield accurate estimations for the model parameter b only when the thermal conductivity of the solid matrix is also reasonably high.
Fig. 3. Influence of reference condition ~kf;ref on the evaluation of b from Eq. (3) for different foam samples: (a) effect on estimated value of b; (b) maximum absolute relative error for the estimation of ETC.
Finally, following the recommendation of this paper, along with the conclusions drawn by Mendes et al. [1], it would now be possible to design simple experiment in order to determine the ETC of porous foams in presence of any hazardous working fluids (e.g. in presence of highly inflammable gas, like hydrogen, products of partial oxidation, etc., or liquid metals), where performing an experiment would be nearly impossible. Acknowledgments The authors would like to thank the German Research Foundation (DFG) for supporting the investigation in the subproject B02, which is part of the Collaborative Research Center CRC 920. References [1] M.A.A. Mendes, S. Ray, D. Trimis, A simple and efficient method for the evaluation of effective thermal conductivity of open-cell foam-like structures, Int. J. Heat. Mass Transf. 66 (2013) 412e422. [2] M. Kaviany, Principles of Heat Transfer in Porous Media, in: Mechanical Engineering Series, Springer-Verlag, 1995. [3] C. Tseng, M. Yamaguchi, T. Ohmori, Thermal conductivity of polyurethane foams from room temperature to 20 k, Cryogenics 37 (1997) 305e312. [4] V. Calmidi, R. Mahajan, The effective thermal conductivity of high porosity fibrous metal foams, J. Heat. Transf. 121 (1999) 466e471. [5] A. Bhattacharya, V. Calmidi, R. Mahajan, An analytical-experimental study for the determination of the effective thermal conductivity of high porosity
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