Normal and Parallel Permeability of Preform Composite Materials used in Liquid Molding Processes: Analytical Solution

Document Type: Research Paper

Authors

Mechanical Engineering, Shahrood University of Technology

Abstract

The permeability of the preform composite materials used in liquid molding processes such as resin transfer molding and structural reaction injection molding is a complex function of weave pattern and packing characteristics. The development of tools for predicting permeability as a function of these parameters is of great industrial importance. Such capability would speed process design and optimization and provide a step towards establishing processing-performance relations. In this study, both normal and parallel permeability of fibrous media comprised of ordered arrays of elliptical cylinders is studied analytically. A novel scale analysis technique is employed for determining the normal permeability of arrays of elliptical fibers. In this technique, the permeability is related to the geometrical parameters such as porosity, elliptical fiber diameters, and the tortuosity of the medium. Following a unit cell approach, compact relationships are proposed for the first time for the normal permeability of the studied geometries. A comprehensive analysis is also performed to determine the permeability of ordered arrays of elliptical fibers over a wide range of porosity and fiber diameters. The developed compact relationship is successfully verified through comparison with the present results. As a result of assuming an elliptical cross section for the fibers in this analytical analysis, an extra parameter comes to play; therefore, the present analytical solution will be more complicated than those developed for circular fiber type in the literature.

Keywords

[1] Tomadakis M. M., Robertson T. J. , 2005, Viscous permeability of random fiber structures: comparison of electrical and diffusional estimates with experimental and analytical results, Journal of Composite Materials 39(2): 163-188.
[2] Gostick J. T., Fowler M. W., Pritzker M. D., Ioannidis M. A., Behra L. M. , 2006, In-plane and through-plane gas permeability of carbon fiber electrode backing layers, Journal of Power Sources 162(1): 228-238.
[3] Ismail M. S., Hughes K. J., Ingham D. B., Ma L., Pourkashanian M., 2012, Effects of anisotropic permeability and electrical conductivity of gas diffusion layers on the performance of proton exchange membrane fuel cells, Applied Energy 95: 50-63.
[4] Tamayol A., Hooman K., 2011, Thermal assessment of forced convection through metal foam heat exchangers, Journal of Heat Transfer 133(11): 111801-111808.
[5] Tamayol A., McGregor F., Bahrami M., 2012, Single phase through-plane permeability of carbon paper gas diffusion layers, Journal of Power Sources 204: 94-99.
[6] Kaviany M., 1995, Principles of Heat Transfer in Porous Media, Springer-Verlag, New York.
[7] Yazdchi K., Luding S., 2012, Towards unified drag laws for inertial flow through fibrous materials, Chemical Engineering Journal 207–208: 35-48.
[8] Jackson G. W., James D. F., 1986, The permeability of fibrous porous media, The Canadian Journal of Chemical Engineering 64(3): 364-374.
[9] Tamayol A., Bahrami M., 2011, Transverse permeability of fibrous porous media, Physical Review E 83(4): 046314.
[10] Mattern K. J., Deen W. M. , 2008, Mixing rules for estimating the hydraulic permeability of fiber mixtures, AIChE Journal 54(1): 32-41.
[11] Happel J., 1959, Viscous flow relative to arrays of cylinders, AIChE Journal 5(2): 174-177.
[12] Carman P. C. , 1938, The determination of the specific surface of powders, Journal of the Chemical Society, Transactions 57: 225-234.
[13] Sullivan R. R., 1942, Specific surface measurements on compact bundles of parallel fibers, Journal of Applied Physics 13(11): 725-730.
[14] Sparrow E. M., Loeffler A. L., 1959, Longitudinal laminar flow between cylinders arranged in regular array, AIChE Journal 5(3): 325-330.
[15] Hasimoto H., 1959, On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres, Journal of Fluid Mechanics 5(02): 317-328.
[16] Kuwabara S., 1959, The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds numbers, Journal of the Physical Society of Japan 14: 527-532.
[17] Sangani A. S., Acrivos A., 1982, Slow flow past periodic arrays of cylinders with application to heat transfer, International Journal of Multiphase Flow 8(3): 193-206.
[18] Drummond J. E., Tahir M. I., 1984, Laminar viscous flow through regular arrays of parallel solid cylinders, International Journal of Multiphase Flow 10(5): 515-540.
[19] Hellou M., Martinez J., El Yazidi M., 2004, Stokes flow through microstructural model of fibrous media, Mechanics Research Communications 31(1): 97-103.
[20] Tamayol A., Bahrami M., 2009, Analytical determination of viscous permeability of fibrous porous media, International Journal of Heat and Mass Transfer 52(9): 2407-2414.
[21] Gebart B. R., 1992, Permeability of unidirectional reinforcements for RTM, Journal of Composite Materials 26(8): 1100-1133.
[22] Van der Westhuizen J., Prieur Du Plessis J., 1996, An attempt to quantify fibre bed permeability utilizing the phase average Navier-Stokes equation, Composites Part A: Applied Science and Manufacturing 27(4): 263-269.
[23] Sahraoui M., Kaviany M., 1992, Slip and no-slip velocity boundary conditions at interface of porous, plain media, International Journal of Heat and Mass Transfer 35(4): 927-943.
[24] Sobera M. P., Kleijn C. R., 2006, Hydraulic permeability of ordered and disordered single-layer arrays of cylinders, Physical Review E 74(3): 036301-036311.
[25] Clague D. S., Phillips R. J., 1997, A numerical calculation of the hydraulic permeability of three-dimensional disordered fibrous media, Physics of Fluids 9: 1562-1572.
[26] Nabovati A., Llewellin E. W., Sousa A., 2009, A general model for the permeability of fibrous porous media based on fluid flow simulations using the lattice Boltzmann method, Composites Part A: Applied Science and Manufacturing 40(6): 860-869.
[27] Higdon J. J. L., Ford G. D., 1996, Permeability of three-dimensional models of fibrous porous media, Journal of Fluid Mechanics 308: 341-361.
[28] Dahua Sh., Lin Y., Youhong T., Jintu F., Feng D. , 2013, Transverse permeability determination of dual-scale fibrous materials, International Journal of Heat and Mass Transfer 58(1–2): 532-539.
[29] Dahua Sh., Lin Y., Jintu F., 2014, On the longitudinal permeability of aligned fiber arrays, Journal of Composite Materials 0021998314540192.
[30] Xiaohu Y.,Tian Jian L., Tongbeum K., 2014, An analytical model for permeability of isotropic porous media, Physics Letters A 378(30–31): 2308-2311.
[31] Dahua Sh., Lin Y., Jintu F., 2015, Longitudinal permeability determination of dual-scale fibrous materials, Composites Part A: Applied Science and Manufacturing 68: 42-46.
[32] White F.M.,2003, Fluid Mechanics, McGraw-Hill Higher Education.
[33] Archie G. E., 1942, The electrical resistivity log as an aid in determining some reservoir characteristics, Transactions of the AIME 146(99): 54-62.
[34] Shen L., Chen Z., 2007, Critical review of the impact of tortuosity on diffusion, Chemical Engineering Science 62(14): 3748-3755.
[35] Versteeg H. K., Malalasekera W., 1995, An Introduction to Computational Fluid Dynamics, Longman Scientific and Technical, Essex, UK.
[36] Bergelin O. P., Brown G. A., Hull H. L., Sullivan F. W., 1950, Heat transfer and fluid friction during viscous flow across banks of tubes–III. A study of tube spacing and tube size, Transactions of the ASME 72: 881-888.
[37] Chmielewski C., Jayaraman K., 1992, The effect of polymer extensibility on crossflow of polymer solutions through cylinder arrays, Journal of Rheology 36: 1105-1126.
[38] Khomami B., Moreno L. D., 1997, Stability of viscoelastic flow around periodic arrays of cylinders, Rheologica Acta 36(4): 367-383.
[39] Kirsch A. A., Fuchs N. A., 1967, Studies on fibrous aerosol filters—II. Pressure drops in systems of parallel cylinders, Annals of Occupational Hygiene 10(1): 23-30.
[40] Sadiq T. A. K., Advani S. G., Parnas R. S., 1995, Experimental investigation of transverse flow through aligned cylinders, International Journal of Multiphase Flow 21(5): 755-774.
[41] Zhong W. H., Currie I. G., James D. F., 2006, Creeping flow through a model fibrous porous medium, Experiments in Fluids 40(1): 119-126.
[42] Skartsis L., Kardos J.L., 1992, The newtonian permeability and consolidation of oriented carbon fiber beds, Proceedings of American Society of Composites Technical Conference 5 :548-556.
[43] Sangani A. S., Yao C. , 1988, Transport processes in random arrays of cylinders: II-viscous flow, Physics of Fluids 31: 2435-2444.
[44] Tamayol A., Bahrami M., 2010, Parallel flow through ordered: An analytical approach, Journal of Fluids Engineering 132: 114502.