Theoretical analyses of solute concentration-gradient driven water flow and solute concentration variation in solution-saturated semi-permeable materials
Source abstract
Abstract
Based on the commonly-used definition of the semi-permeable material (SPM), in which the solvent (such as water) can pass through but the solute cannot move freely, a novel constitutive equation is presented, in this paper, to theoretically express the intrinsic constitutive relationship between the porosity variation rate with time and the solute concentration variation rate with time. From this theoretical finding, a mathematical model is established to describe the solute concentration-gradient driven water flow and solute concentration variation problem in solution-saturated semi-permeable materials (SSSPMs). In particular, the solute concentration variation in the considered problem can be mathematically described as a linear homogeneous second-order partial differential equation with a variable coefficient in the front of the time term. A special mathematical transform is presented and used to solve this equation, so that the analytical solution for the solute concentration variation in the considered problem has been derived in a purely mathematical manner. The derived analytical solution is then used to provide some theoretical understanding of solute concentration variations in the SSSPM layer.
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Research context
OpenAlex topics: Membrane-based Ion Separation Techniques · Membrane Separation Technologies · Nanopore and Nanochannel Transport Studies
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References (31)
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1. Barbour S L, Fredlund D G. Mechanisms of osmotic flow and volume change in clay soils. Can Geotech J, 1989, 26: 551–56210.1139/t89-068 ↗
2. Kaczmarek M, Hueckel T. Chemo-mechanical consolidation of clays: Analytical solutions for a linearized one-dimensional problem. Transp Porous Media, 1998, 32: 49–7410.1023/A:1006530405361 ↗
3. Peters G P, Smith D W. The influence of advective transport on coupled chemical and mechanical consolidation of clays. Mech Mater, 2004, 36: 467–48610.1016/S0167-6636(03)00072-3 ↗
4. Zheng X, Wang L, Xu Y. Analytical solutions of 1-D chemo-hydromechanical coupled model of saturated soil considering osmotic efficiency. Num Anal Meth GeoMech, 2021, 45: 2522–254010.1002/nag.3275 ↗
5. Zheng X, Li X, Xu Y, et al. Influence of coupling effect in the chemohydro-mechanical consolidation of saturated soil layer. Num Anal Meth GeoMech, 2022, 46: 2446–246010.1002/nag.3413 ↗
6. Zhang Z, Masum S A, Thomas H R, et al. Modeling fully coupled hydraulic-mechanical-chemical processes in a natural clay liner under mechanical and chemico-osmotic consolidation. Environ Sci Pollut Res, 2018, 25: 36173–3618310.1007/s11356-018-3532-7 ↗
7. Han L, Zhang Z. Analytical solutions for a fully coupled hydraulic-mechanical-chemical model with non-zero initial concentration of contaminant. Num Anal Meth GeoMech, 2023, 47: 2513–253010.1002/nag.3589 ↗
8. Zhang Z, Masum S A, Tian G, et al. Modelling non-isothermal volume change and solute transport behaviours of a semi-permeable clay soil under the combined influence of mechanical loading, chemical-osmosis, and thermo-osmosis. Eng Geol, 2021, 293: 10627110.1016/j.enggeo.2021.106271 ↗
9. Biesheuvel P M, Rutten S B, Ryzhkov I I, et al. Theory for salt transport in charged reverse osmosis membranes: Novel analytical equations for desalination performance and experimental validation. Desalination, 2023, 557: 11658010.1016/j.desal.2023.116580 ↗
10. Biesheuvel P M, Porada S, Elimelech M, et al. Tutorial review of reverse osmosis and electrodialysis. J Membrane Sci, 2022, 647: 12022110.1016/j.memsci.2021.120221 ↗
11. Biesheuvel P M, Zhao R, Porada S, et al. Theory of membrane capacitive deionization including the effect ofthe electrode pore space. J Colloid Interface Sci, 2011, 360: 239–24810.1016/j.jcis.2011.04.049 ↗
12. Suss M E, Porada S, Sun X, et al. Water desalination via capacitive deionization: What is it and what can we expect from it? Energy Environ Sci, 2015, 8: 2296–231910.1039/C5EE00519A ↗
13. Gamaethiralalage J G, Singh K, Sahin S, et al. Recent advances in ion selectivity with capacitive deionization. Energy Environ Sci, 2021, 14: 1095–112010.1039/D0EE03145C ↗
14. Nield D A, Bejan A. Convection in Porous Media. New York: Springer-Verlag, 199210.1007/978-1-4757-2175-1 ↗
15. Phillips O M. Flow and Reactions in Permeable Rocks. Cambridge: Cambridge University Press, 1991
16. Zhao C, Hobbs B E, Mühlhaus H B. Finite element modelling of reactive mass transport problems in fluid-saturated porous media. Commun Numer Meth Engng, 1999, 15: 501–51310.1002/(SICI)1099-0887(199907)15:7<501::AID-CNM264>3.0.CO;2-T ↗
17. Zhao C, Hobbs B E, Walshe J L, et al. Finite element modeling of fluid-rock interaction problems in pore-fluid saturated hydrothermal/sedimentary basins. Comput Methods Appl Mech Eng, 2001, 190: 2277–229310.1016/S0045-7825(00)00304-2 ↗
18. Zhao C, Reid L B, Regenauer-Lieb K. Some fundamental issues in computational hydrodynamics of mineralization: A review. J Geo-Chem Explor, 2012, 112: 21–3410.1016/j.gexplo.2011.10.005 ↗
19. Bear J. Dynamics of fluids in porous media. Soil Sci, 1975, 120: 162–16310.1097/00010694-197508000-00022 ↗
20. Chadam J, Hoff D, Merino E, et al. Reactive infiltration instabilities. IMA J Appl Math, 1986, 36: 207–22110.1093/imamat/36.3.207 ↗
21. Lai K H, Chen J S, Liu C W, et al. Effect of medium permeability anisotropy on the morphological evolution of two non-uniformities in a geochemical dissolution system. J Hydrol, 2016, 533: 224–23310.1016/j.jhydrol.2015.11.039 ↗
22. Zhao C, Schaubs P, Hobbs B E. Computational simulation of seepage instability problems in fluid-saturated porous rocks: Potential dynamic mechanisms for controlling mineralisation patterns. Ore Geol Rev, 2016, 79: 180–18810.1016/j.oregeorev.2016.05.002 ↗
23. Chen J S, Liu C W. Numerical simulation of the evolution of aquifer porosity and species concentrations during reactive transport. Comput Geoscis, 2002, 28: 485–49910.1016/S0098-3004(01)00084-X ↗
24. Chen J S, Liu C W. Interaction of reactive fronts during transport in a homogeneous porous medium with initial small non-uniformity. J Contam Hydrol, 2004, 72: 47–6610.1016/j.jconhyd.2003.10.006 ↗
25. Imhoff P T, Mann A S, Mercer M, et al. Scaling DNAPL migration from the laboratory to the field. J Contam Hydrol, 2003, 64: 73–9210.1016/S0169-7722(02)00106-7 ↗
26. Chen J S, Liu C W, Lai G X, et al. Effects of mechanical dispersion on the morphological evolution of a chemical dissolution front in a fluid-saturated porous medium. J Hydrol, 2009, 373: 96–10210.1016/j.jhydrol.2009.04.014 ↗
27. Schmidt Mumm A, Brugger J, Zhao C, et al. Fluids in geological processes—The present state and future outlook. J GeoChem Explor, 2010, 106: 1–710.1016/j.gexplo.2010.01.005 ↗
28. Zhao C, Hobbs B E, Ord A. Effects of mathematical transforms on theoretical analysis and computational simulation of chemical dissolution-front instability within fluid-saturated porous media. J Hydrol, 2021, 600: 12653110.1016/j.jhydrol.2021.126531 ↗
29. Carslaw H S, Jaeger J C. Conduction of Heat in Solids. Oxford: Clarendon Press, 1959
30. Terzaghi K. Theoretical Soil Mechanics. New York: John Wiley and Sons, 194310.1002/9780470172766 ↗
31. Zhao C, Liu Q. Investigating effects of structural deformation regimes on mineralization distributions in fluid-saturated rocks: Computational simulation approach through generic models. Minerals, 2023, 13: 66410.3390/min13050664 ↗