Skip to main content
SpringerLink
Log in

Coupled Water and Salt Transport in Porous Materials: Rapid Determination of a Varying Diffusion Coefficient from Experimental Data

  • Published:
Transport in Porous Media Aims and scope Submit manuscript

Abstract

A direct method for an accurate and rapid evaluation of a varying salt diffusion coefficient, \(D\), from experimental data is proposed for a coupled water and salt transport in porous materials. The evaluation uses data on the moisture and salt concentration profiles and is based on a formula obtained from the Boltzmann-Matano method. The coupled transport is described by the diffusion-advection model of Bear and Bachmat. A simple expression for \(D\) in the center of the concentration interval is deduced from the formula to provide a rapid estimate on \(D\). Possible extensions of this analytical approach are pointed out, suggesting that it can serve as a convenient general tool in engineering calculations. The theoretical results are applied to a laboratory experiment in which a coupled moisture and chloride transport had been investigated in a lime plaster, and the chloride diffusion coefficient had been obtained numerically in dependence on the chloride concentration. The agreement with the numerical results is shown to be rather good, except at low concentrations where our analytical results should be more reliable. It is also shown that the unusually high value of the calculated chloride diffusion coefficient—about three orders of magnitude higher than for free chloride ions in water—cannot be explained by possible inaccuracies in the measurements and/or numerical calculations. The reason is that changes in the measured profiles’ data could cause a change in \(D\) of just the same order of magnitude. This shows that, besides diffusion and advection, additional mechanisms take part in the considered chloride transport.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Abbreviations

\(a_i\) :

Slope of moisture or concentration profile (\(\hbox {m}^{-1}\, \hbox {s}^{1/2}\))

\(C\) :

Total concentration (\(\hbox {kg} \, \hbox {m}^{-3}\))

\(\mathsf {C}\) :

Model profile for total concentration (\(\hbox {kg} \, \hbox {m}^{-3}\))

\(C_\pm \) :

Limiting values of total concentration (\(\hbox {kg} \, \hbox {m}^{-3}\))

\(C_b\) :

Concentration of a bound salt (\(\hbox {kg}\, \hbox {m}^{-3}\))

\(C_f\) :

Concentration of a free salt (\(\hbox {kg} \, \hbox {m}^{-3}\))

\(\mathsf {C}_f\) :

Model profile for free salt concentration (\(\hbox {kg} \, \hbox {m}^{-3}\))

\(D\) :

Salt diffusion coefficient (\(\hbox {kg} \, \hbox {m}^{-3}\))

\(D^*\) :

Salt diffusion coefficient at concentration profile center (\(\hbox {kg} \, \hbox {m}^{-3}\))

\(\mathcal {D}^*\) :

Salt diffusion coefficient at free concentration profile center (\(\hbox {kg} \, \hbox {m}^{-3}\))

\(h_i\) :

Half-height of moisture profile (\(-\)) or concentration profile (\(\hbox {kg} \, \hbox {m}^{-3}\))

\(K_H\) :

Henry constant (\(-\))

\(\pmb {v}\) :

Darcy velocity (\(\hbox {m}\, \hbox {s}^{-1}\))

\(w\) :

Volumetric moisture content (\(-\))

\(\mathsf {w}\) :

Model profile for volumetric moisture content (\(-\))

\(w_\pm \) :

Limiting values of volumetric moisture content (\(-\))

\(\eta \) :

Boltzmann variable \(x/\sqrt{t}\) (\(\hbox {m}\, \hbox {s}^{-1/2}\))

\(\eta _i\) :

Position of moisture or concentration profile (\(\hbox {m}\, \hbox {s}^{-1/2}\))

\(\kappa \) :

Moisture diffusivity (\(\hbox {m}^{2}\, \hbox {s}^{-1}\))

\(\kappa ^*\) :

Moisture diffusivity at moisture profile center (\(\hbox {m}^{2}\, \hbox {s}^{-1}\))

References

  • Bear, J., Bachmat, Y.: Introduction to Modelling of Transport Phenomena in Porous Media, vol. 4. Kluwer, Dordrecht (1990)

    Book  Google Scholar 

  • Boltzmann, L.: Zur integration des diffusiongleichung bei variabeln diffusions coefficienten. Wiedemmans Ann. Phys. 53, 959 (1894)

    Article  Google Scholar 

  • Bruining, H., Darwish, M., Rijnks, A.: Computation of the longitudinal and transverse dispersion coefficient in an adsorbing porous medium using homogenization. Transp. Porous. Med. 91, 833 (2012)

    Article  Google Scholar 

  • Carmeliet, J., Hens, H., Adan, S.R., Brocken, H., Cerny, R., Pavlik, Z., Hall, C., Kumaran, K., Pel, L.: Determination of the liquid water diffusivity from transient moisture transfer experiments. J. Therm. Environ. Bldg. Sci. 27, 277 (2004)

    Google Scholar 

  • Carslaw, H.S., Jaeger, J.C.: Conduction of Heat in Solids. Clarendon Press, Oxford (1959)

    Google Scholar 

  • Crank, J.: The Mathematics of Diffusion. Clarendon Press, Oxford (1975)

    Google Scholar 

  • de Groot, S.R., Mazur, P.: Non-Equilibrium Thermodynamics. Dover Publications, New York (1984)

    Google Scholar 

  • Freeze, R.A., Cherry, J.A.: Groundwater. Prentice Hall, Englewood Cliffs (1979)

    Google Scholar 

  • Hall, L.D.: An analytical method of calculating variable diffusion coefficients. J. Chem. Phys. 21, 87 (1953)

    Article  Google Scholar 

  • Kailasam, S.K., Lacombe, J.C., Glicksman, M.E.: Evaluation of the methods for calculating the concentration-dependent diffusivity in binary systems. Metall. Mater. Trans. A 30, 2605 (1998)

    Article  Google Scholar 

  • Korecký, T., Keppert, M., Maděra, J., Černý, R.: Water transport parameters of autoclaved aerated concrete: Experimental assessment of different modeling approaches. J. Build. Phys. (2014 In press), doi:10.1177/1744259114535727

  • Kumar, A., Jaiswal, D.K., Kumar, N.: Analytical solutions of one-dimensional advection-diffusion equation with variable coefficients in a finite domain. J. Earth. Syst. Sci. 118, 539 (2009)

    Article  Google Scholar 

  • Matano, C.: On the relation between the diffusion-coefficients and concentrations of solid metals (the nickel-copper system). Jpn. J. Phys. 8, 109 (1933)

    Google Scholar 

  • Medved’, I., Černý, R.: Surface diffusion in porous media: A critical review. Micropor. Mesopor. Mat. 142, 405 (2011)

    Article  Google Scholar 

  • Medved’, I., Černý, R.: Osmosis in porous media: A review of recent studies. Micropor. Mesopor. Mat. 170, 299 (2013)

    Article  Google Scholar 

  • Nikitin, A.G., Spichak, S.V., Vedula, Y.S., Naumovets, A.G.: Symmetries and modelling functions for diffusion processes. J. Phys. D 42(055), 301 (2009)

    Google Scholar 

  • Pavlík, Z., Michálek, P., Pavlíková, M., Kopecká, I., Maxová, I., Černý, R.: Water and salt transport and storage properties of mšené sandstone. Constr. Build. Mater. 22, 1736 (2008)

    Article  Google Scholar 

  • Pavlík, Z., Fiala, L., Maděra, J., Pavlíková, M., Černý, R.: Computational modelling of coupled water and salt transport in porous materials using diffusion-advection model. J. Frankl. Inst. 348, 1574 (2011)

    Article  Google Scholar 

  • Pel, L., Kopinga, K., Kaasschieter, E.F.: Saline absorption in calcium-silicate brick observed by nmr scanning. J. Phys. D 33, 1380 (2000)

    Article  Google Scholar 

  • Philip, J.R., Vries, D.A.D.: Moisture movement in porous materials under temperature gradients. Trans. Amer. Geophys. Union 38, 222 (1957)

    Article  Google Scholar 

  • Rolle, M., Hochstetler, D.L., Chiogna, G., Kitanidis, P.K., Grathwohl, P.: Experimental investigation and pore-scale modeling interpretation of compound-specific transverse dispersion in porous media. Transp. Porous Med. 93, 347 (2012)

    Article  Google Scholar 

  • Rolle, M., Chiogna, G., Hochstetlera, D.L., Kitanidis, P.K.: On the importance of diffusion and compound-specific mixing for groundwater transport: An investigation from pore to field scale. J. Contam. Hydrol. 153, 51 (2013)

    Article  Google Scholar 

  • Samson, E., Marchand, J.: Modeling the transport of ions in usaturated cement-based materials. Comput. Struct. 85, 1740 (2007)

    Article  Google Scholar 

  • Šimunek, J., van Genuchten, M., Šejna, M.: The HYDRUS-1D Software Package for Simulating the One-Dimensional Movement of Water, Heat, and Multiple Solutes in a Variably-Saturated Media (Version 3.0). University of California Riverside, Riverside (2005)

    Google Scholar 

  • Vach, M., Svojtka, M.: Evaluation of molar volume effect for calculation of diffusion in binary systems. Metall. Mater. Trans. B 43, 1446 (2012)

    Article  Google Scholar 

  • Voronina, V., Pel, L., Kopinga, K.: Effect of osmotic pressure on salt extraction by a poultice. Constr. Build Mater. 53, 432 (2014)

    Article  Google Scholar 

Download references

Acknowledgments

This research was supported by the Czech Science Foundation, Project No. P105/12/G059.

Author information

Authors and Affiliations

  1. Department of Materials Engineering and Chemistry, Faculty of Civil Engineering, Czech Technical University in Prague, Thakurova 7, 16629, Prague, Czech Republic

    Igor Medved’ & Robert Černý

  2. Department of Physics, Constantine The Philosopher University, 94974, Nitra, Slovakia

    Igor Medved’

Authors
  1. Igor Medved’
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Robert Černý
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Igor Medved’.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Medved’, I., Černý, R. Coupled Water and Salt Transport in Porous Materials: Rapid Determination of a Varying Diffusion Coefficient from Experimental Data. Transp Porous Med 105, 597–610 (2014). https://doi.org/10.1007/s11242-014-0386-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-014-0386-4

Keywords

Search

Navigation