An effective thermal conductivity model for fractal porous media with rough surfaces

Abstract

Quantitative evaluation of the effective thermal conductivity of porous media has received wide attention in science and engineering since it is a key thermophysical parameter in characterizing heat transfer properties. Based on fractal characters of tortuous capillary tubes and rough surfaces in micro-pores, we proposed a theoretical model of the effective thermal conductivity in porous media with rough surfaces. This model considers the geometrical parameters of porous media, including porosity, micro-pore fractal dimension, tortuosity fractal dimension, and relative roughness. The calculated normalized effective thermal conductivity was then validated against published experimental data. The results show good agreement between them. The influence of geometrical factors, porosity and relative surface roughness, on the effective thermal conductivity in porous media with rough surfaces are discussed and analyzed extensively.

Cited as: Qin, X., Zhou, Y., Sasmito, A.P. An effective thermal conductivity model for fractal porous media with rough surfaces. Advances in Geo-Energy Research, 2019, 3(2): 149-155, doi: 10.26804/ager.2019.02.04

References

Askari, R., Hejazi, S.H., Sahimi, M. Effect of deformation on the thermal conductivity of granular porous media with rough grain surface. Geophys. Res. Lett. 2017, 44(16): 8285-8293.

Behrang, A., Taheri, S., Kantzas, A. A hybrid approach on predicting the effective thermal conductivity of porous and nanoporous media. Int. J. Heat Mass Transf. 2016, 98: 52-59.

Cai, J., Hu, X., Standnes, D.C., et al. An analytical model for spontaneous imbibition in fractal porous media including gravity. Colloid Surf. A-Physicochem. Eng. 2012, 414: 228-233.

Cai, J., Perfect, E., Cheng, C., et al. Generalized modeling of spontaneous imbibition based on Hagen-Poiseuille flow in tortuous capillaries with variably shaped apertures. Langmuir 2014, 30(18): 5142-5151.

Cai, J., Yu, B., Zou, M., et al. Fractal analysis of surface roughness of particles in porous media. Chin. Phys. Lett. 2010, 27(2): 157-160.

Carson, J.K., Lovatt, S.J., Tanner, D.J., et al. Thermal conductivity bounds for isotropic, porous materials. Int. J. Heat Mass Transf. 2005, 48(11): 2150-2158.

Chen, Y., Zhang, C. Role of surface roughness on thermal conductance at liquid-solid interfaces. Int. J. Heat Mass Transf. 2014, 78: 624-629.

Clarkson, C.R., Solano, N., Bustin, R.M., et al. Pore structure characterization of North American shale gas reservoirs using USANS/SANS, gas adsorption, and mercury intrusion. Fuel 2013, 103(1): 606-616.

Clauser, C. Heat transport processes in the Earth’s crust. Surv. Geophys. 2009, 30(3): 163-191.

Dai, S., Cha, J.H., Rosenbaum, E.J., et al. Thermal con-ductivity measurements in unsaturated hydrate-bearing sediments. Geophys. Res. Lett. 2015, 42(15): 6295-6305.

Feng, Y., Yu, B., Zou, M., et al. A generalized model for the effective thermal conductivity of unsaturated porous media based on self-similarity. J. Porous Media 2007, 10(6): 551-567.

Ghanbarian, B., Daigle, H. Thermal conductivity in porous media: Percolation-based effective-medium approxima-tion. Water Resour. Res. 2016, 52(1): 295-314.

Guo, L., Xu, H., Gong, L. Influence of wall roughness models on fluid flow and heat transfer in microchannels. Appl. Therm. Eng. 2015, 84: 399-408.

Jougnot, D., Revil, A. Thermal conductivity of unsaturated clay-rocks. Hydrol. Earth Syst. Sci. 2010, 14(1): 91-98.

Kaviany, M. Conduction Heat Transfer in Principles of Heat Transfer in Porous Media. New York, USA, Springer Science & Business Media, 2012.

Kooi, H. Spatial variability in subsurface warming over the last three decades; insight from repeated borehole temperature measurements in The Netherlands. Earth Planet. Sci. Lett. 2008, 270(1): 86-94.

Kou, J., Wu, F., Lu, H., et al. The effective thermal conductivity of porous media based on statistical self-similarity. Phys. Lett. A 2009, 374(1): 62-65.

Li, B., Xu, W., Tong, F. Measuring thermal conductivity of soils based on least squares finite element method. Int. J. Heat Mass Transf. 2017, 115(Part B): 833-841.

Li, Z., Du, D., Guo, Z. Experimental study on flow characteristics of liquid in circular microtubes. Microsc. Therm. Eng. 2003, 7(3): 253-265.

Li, Z., He, Y., Tang, G., et al. Experimental and numerical studies of liquid flow and heat transfer in microtubes. Int. J. Heat Mass Transf. 2007, 50(17): 3447-3460.

Majumdar, A., Bhushan, B. Role of fractal geometry in roughness characterization and contact mechanics of surfaces. J Tribol. 1990, 112(2): 205-216.

Miao, T., Cheng, S., Chen, A., et al. Analysis of axial thermal conductivity of dual-porosity fractal porous media with random fractures. Int. J. Heat Mass Transf. 2016, 102: 884-890.

Pia, G., Corcione, C.E., Striani, R., et al. Thermal conductivity of porous stones treated with UV light-cured hybrid organic-inorganic methacrylic-based coating. Experimen-tal and fractal modeling procedure. Prog. Org. Coat. 2016, 94: 105-115.

Pia, G., Sanna, U. An intermingled fractal units model to evaluate pore size distribution influence on thermal conductivity values in porous materials. Appl. Therm. Eng. 2014, 65(1-2): 330-336.

Poljacek, S.M., Risovic, D., Furic, K., et al. Comparison of fractal and profilometric methods for surface topography characterization. Appl. Surf. Sci. 2008, 254(11): 3449-3458.

Prasad, V., Kladias, N., Bandyopadhaya, A., et al. Evaluation of correlations for stagnant thermal conductivity of liquid-saturated porous beds of spheres. Int. J. Heat Mass Transf. 1989, 32(9): 1793-1796.

Qin, X., Cai, J., Xu, P., et al. A fractal model of effective thermal conductivity for porous media with various liquid saturation. Int. J. Heat Mass Transf. 2019, 128: 1149-1156.

Ranut, P. On the effective thermal conductivity of aluminum metal foams: Review and improvement of the available empirical and analytical models. Appl. Therm. Eng. 2016, 101: 496-524.

Revil, A. Thermal conductivity of unconsolidated sediments with geophysical applications. J. Geophys. Res. 2000, 105(B7): 16749-16768.

Sadeghi, M., Ghanbarian, B., Horton, R. Derivation of an explicit form of the percolation-based effective-medium approximation for thermal conductivity of partially saturated soils. Water Resour. Res. 2018, 54(2): 1389-1399.

Sahimi, M. Flow and Transport in Porous Media and Fractured Rock: From Classical Methods to Modern Approaches. New Jersey, USA, John Wiley & Sons, 2011.

Shen, H., Ye, Q., Meng, G. Anisotropic fractal model for the effective thermal conductivity of random metal fiber porous media with high porosity. Phys. Lett. A 2017, 381(37): 3193-3196.

Tang, G., Li, Z., He, Y., et al. Experimental study of compressibility, roughness and rarefaction influences on microchannel flow. Int. J. Heat Mass Transf. 2007, 50(11): 2282-2295.

Wang, F., Li, X. The stagnant thermal conductivity of porous media predicted by the random walk theory. Int. J. Heat Mass Transf. 2017, 107: 520-533.

Wang, J., Carson, J.K., North, M.F., et al. A new approach to modelling the effective thermal conductivity of heterogeneous materials. Int. J. Heat Mass Transf. 2006, 49(17-18): 3075-3083.

Wang, M., Wang, J., Pan, N., et al. Mesoscopic predictions of the effective thermal conductivity for microscale random porous media. Phys. Rev. E 2007, 75(3): 036702.

Wei, W., Cai, J., Hu, X., et al. An electrical conductivity model for fractal porous media. Geophys. Res. Lett. 2015, 42(12): 4833-4840.

Woodside, W., Messmer, J.H. Thermal conductivity of porous media. I. Unconsolidated sands. J. Appl. Phys. 1961, 32(9): 1688-1706.

Xu, P., Yu, B., Yun, M., et al. Heat conduction in fractal tree-like branched networks. Int. J. Heat Mass Transf. 2006, 49(19-20): 3746-3751.

Yang, S., Liang, M., Yu, B., et al. Permeability model for fractal porous media with rough surfaces. Microfluid. Nanofluidics 2015, 18(5): 1085-1093.

Yang, S., Yu, B., Zou, M., et al. A fractal analysis of laminar flow resistance in roughened microchannels. Int. J. Heat Mass Transf. 2014, 77: 208-217.

Yu, B., Cai, J., Zou, M. On the physical properties of apparent two-phase fractal porous media. Vadose Zone J. 2009, 8(1): 177-186.

Yu, B., Cheng, P. Fractal models for the effective thermal conductivity of bidispersed porous media. J. Thermophys Heat Trans. 2002a, 16(1): 22-29.

Yu, B., Cheng, P. A fractal permeability model for bi-dispersed porous media. Int. J. Heat Mass Transf. 2002b, 45(14): 2983-2993.

Zou, M., Yu, B., Cai, J., et al. Fractal model for thermal contact conductance. J. Heat Transfer 2008, 130(10): 101301.