Thermally induced diffusion of chemicals under steady-state heat transfer in saturated porous media

doi:10.1016/j.ijheatmasstransfer.2020.119664

International Journal of Heat and Mass Transfer

Volume 153, June 2020, 119664

https://doi.org/10.1016/j.ijheatmasstransfer.2020.119664 Get rights and content

Highlights

•
Analytical models to the one-dimensional coupled molecular and thermal diffusions of chemicals in saturated porous media is presented.
•
The generalized integral transform technique is successfully adopted to derive the model under Neumann and Dirichlet outflow boundaries and steady state heat transfer.
•
We present a comparative assessment of the importance of Soret effect and temperature dependent diffusion coefficient on non-isothermal diffusion.

Abstract

Thermally induced diffusion of chemicals is observed in many natural phenomena and engineering processes. We present analytical solutions to the non-isothermal diffusion of chemicals under combined effects of molecular and thermal diffusion processes and under variable diffusion coefficient with temperature. The generalized integral transform technique (GITT) is adopted to successfully derive the analytical solutions to the one-dimensional boundary problem under Neumann and Dirichlet outflow conditions. The accuracy of the analytical model developed is examined against three benchmarks including two experimental datasets. The analytical models are applied to develop an understanding of how the chemical transfer under non-isothermal conditions is governed by a combination of the Soret effect and temperature dependency of diffusion coefficient. We demonstrate that the mass transfer components due to thermal diffusion (or Soret effect) and the temperature dependency of diffusion coefficients are equally important to develop an accurate assessment of non-isothermal diffusion problem. Under the conditions of the case study considered, we show that a threshold for the Soret coefficient exists at which the temperature dependency of chemical diffusion coefficient can induce equally or larger influences on the chemical compared to the influence of thermal diffusion. The analytical solutions proposed also provide modelling benchmarks to examine the accuracy of alternative multiphysics numerical models for studying the non-isothermal diffusion problems in porous media.

Introduction

Studies of coupled heat and mass transfer phenomena in porous media have a wide applications in various engineering applications and natural phenomena. It has been shown that temperature gradient can induce mass transfer (chemicals) in fluid and porous systems which is known as thermal diffusion or Soret effect (or Soret-Ludwig effect) [30,[46]]. Similarly, the heat transfer can be induced as the result of mass flow which is described as Dufour effect ([10]; Ahmed et al., 2019). Thermal diffusion (or Soret effect) and Dufour effects provide theoretical descriptions of coupling between fluxes (mass and heat) and forces (temperature and concentration gradients) which have been extended based on thermodynamics of irreversible processes [39].

The focus of this paper is on thermal diffusion or Soret effects which has been overserved and reported in many natural phenomena and engineering processes such as uranium enrichment [32], petrology [9,19], petroleum reservoirs [5], solar ponds [3], radioactive waste and fusion reactors [29], aerosols [20], thermo-optical trapping of gas bubbles [51], shock waves [7], swimmers and engines [31], granular media [18], clays [40], double diffusive convection [33,45], polymer fractionation [13], biopolymers and prebiotic evolution [23,36], thermo-taxis [4] and Maxwell liquid [1,2].

Thermal diffusion is a second-order transport process compared to the molecular diffusion. However, the importance of Soret effect on the non-isothermal diffusion is well recognized as a non-negligible governing mechanism of mass flow which is evident from experimental and analytical research (e.g. [5,29,32,41]). The magnitude of the thermal diffusion effect on the overall non-isothermal diffusion of chemicals in an aqueous solution or porous system is controlled by the Soret coefficient, concentration, composition and temperature gradient in the system [6,11,25,26]. Thermal diffusion is a direct driving mechanism for the chemicals to diffuse under a temperature gradient. However, it is well established that the diffusion coefficient of chemicals in aqueous solutions is also a temperature dependent parameter (e.g. [25]). Therefore and in addition to the direct influence by the thermal diffusion, temperature dependency of the diffusion coefficients can indirectly affect the non-isothermal mass diffusion.

Predictive capability for a quantitative assessment of the effects of thermally induced mass flow in porous media is required for the analysis of non-isothermal chemical transport, especially in cases where long term impacts of combined effects of molecular diffusion and thermal diffusion are of interest. In this paper, we present a generic analytical solution to the problem of one-dimensional non-isothermal diffusion of chemicals under combined effects of molecular diffusion, thermal diffusion and variable diffusion coefficient under steady state heat transfer in porous media.

Existing mathematical formulations present sound theoretical descriptions for the non-isothermal diffusion of chemicals in porous media by which the contributions of thermal diffusion have (e.g. [42,47,50]). The developments of analytical solutions for such non-linear and coupled mathematical problems are rather very limited. Analytical solutions for chemical diffusion in porous media [12,16,27,44,49] consider the molecular diffusion, advection, degradation and adsorption of chemical. However, the analytical solutions presented above for chemical diffusion neglect the effect of a thermally coupled process. Xie et al. [48] presented an analytical solution for the chemical diffusion under a thermal gradient in semi-infinite porous media which considers the Soret effects. However, the solution proposed assumes a constant diffusion coefficient. Temperature effects on the diffusion coefficients are not negligible. As an example, the experimental results of chemical diffusion coefficients of ionic species in a clay soil at elevated temperature indicated a strong influence on enhancing the chemical transport process as the result of temperature-dependency of the diffusion coefficient [34,40]. To the knowledge of authors, no analytical solution has been presented which considers the combined processes of molecular diffusion and thermal diffusion under variable diffusion coefficient. We aim to address the gap in knowledge, especially in relation to extent of the direct effect (Soret effects) and indirect effect (temperature dependency of diffusion coefficient) on the overall non-isothermal mass diffusion in porous media.

This paper presents new analytical solutions for thermally coupled solute transport in a finite porous media. The method of generalized integral transform technique (GITT) is adopted to derive a solution to the thermally coupled diffusion for a finite domain, due to a steady state heat transfer in the domain. The model is tested against a set of bench marks. An example simulation of chemical transport under a temperature gradient is also provided to demonstrate the effects of thermal diffusion and temperature dependency of the diffusion coefficient on the overall transport of chemical in porous media.

Section snippets

Non-isothermal diffusion of chemicals in porous media

Theoretical formulations of heat conduction and chemical diffusion in saturated porous media are presented which are solved by the analytical solutions proposed in this paper. The general governing equation for multicomponent chemical diffusion in porous media under coupled electrochemical and thermal potentials (e.g. [43,47]) can be simplified for the one-dimensional problem of diffusion of a single chemical species as: $R_{d} \frac{\partial C}{\partial t} = \frac{\partial}{\partial x} [τ D (T) \frac{\partial C}{\partial x}] + \frac{\partial}{\partial x} [τ S_{T} D (T) C \frac{\partial T}{\partial x}]$ where, C represents the

Analytical solution (Dirichlet boundary condition)

In order to derive the analytical solution to the mathematical formulation of diffusion of chemicals in saturated porous media under steady-state heat transfer and Dirichlet boundary condition, the inhomogeneous inflow boundary condition was changed into homogeneous conditions by adopting the approach proposed by Deng et al. [8]: $U (x, t) = C (x, t) + C_{0} (\frac{x}{L} - 1)$

Substituting Eq. (10) into Eq. (6) yields the following equation: $\begin{matrix} R_{d} \frac{\partial U}{\partial t} & = & D (T) \frac{\partial^{2} U}{\partial x^{2}} + A (S_{T} D (T) + M D_{0}) (\frac{\partial U}{\partial x} - \frac{C_{0}}{L}) \\ + A^{2} S_{T} M D_{0} (U + C_{0} (1 - \frac{x}{L})) \end{matrix}$

By combining Eq. (9)

Analytical solution (Neumann boundary condition)

In order to derive the analytical solution to the mathematical formulation of diffusion of chemicals under steady-state heat transfer and Neumann boundary condition, the inhomogeneous inflow boundary condition was changed into homogeneous conditions by adopting: $U (x, t) = C (x, t) - C_{0}$

Substituting Eq. (30) into Eq. (6) yields the following equation: $\frac{R_{d}}{τ} \frac{\partial U}{\partial t} = D (T) \frac{\partial^{2} U}{\partial x^{2}} + A (S_{T} D (T) + M D_{0}) \frac{\partial U}{\partial x} + A^{2} S_{T} M D_{0} (U + C_{0})$

The initial condition of U(x, t) becomes: ${U (x, t) |}_{t = 0} = F (x) - C_{0}$

Similar to the derivation of analytical

Model applications and verification

In this section, the accuracy of the analytical solutions developed is examined against three bench marks including:

i)
pure thermal diffusion against an alternative analytical solution provided by Xie et al. [48]
ii)
diffusion of chemicals under variable temperature in clay which are tested against experimental data by Mon et al. [34] and
iii)
thermal diffusion in compacted clay which were developed based on experimental results of by Rosanne et al. [40].

The results of the proposed analytical solutions will

Model application for coupled thermal and chemical diffusion in porous media

In this section, thermally induced diffusion of chemicals in saturated porous media will be investigated. The main objective is to investigate that the extent of the effects of primary governing processes and parameters. The interest in particular is on to obtain an evaluation of the contributions of Soret effect and diffusion coefficient on the overall process. The effects of two key parameters including diffusion coefficient which is temperature dependent (D_T) and the Soret coefficient (S_T)

Conclusions

Analytical solutions to the one-dimensional coupled molecular and thermal diffusions of chemical in saturated porous media were presented which consider the combined effects of thermal diffusion and temperature dependency of diffusion coefficient under steady-state heat transfer. The solutions propose utilises a generalized integral transform technique (GITT) for two boundary problems including under Neumann and Dirichlet boundary conditions. The accuracy of analytical model was tested against

Author statement

Huaxiang Yan: Main developer of concept, theory and analytical solutions, verifications and analysis, writing up the original draft, editing and revision.

Majid Sedighi: Major contribution to the development of idea, concept and theory, analysis of results, co-writing up the original draft, editing and revision. Supervision of the research.

Haijian Xie: Major contributions to the development of concept and checking the solutions, verifications and analysis, writing up the original draft, editing

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

Haijian Xie acknowledges the financial support received from the National Key Research and Development Program of China (Grant No. 2018YFC1802303), from the National Natural Science Foundation of China (Grant Nos. 41672288, 41977223, and 41931289) and from the Zhejiang Provincial Natural Science Foundation (Gran No.LR20E080002).

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View full text

Thermally induced diffusion of chemicals under steady-state heat transfer in saturated porous media

Highlights

Abstract

Introduction

Section snippets

Non-isothermal diffusion of chemicals in porous media

Analytical solution (Dirichlet boundary condition)

Analytical solution (Neumann boundary condition)

Model applications and verification

Model application for coupled thermal and chemical diffusion in porous media

Conclusions

Author statement

Declaration of Competing Interest

Acknowledgment

Int. J. Heat and Mass Transf.

J. Hydrol.

J. Electroanal. Chem.

Sci. Total Environ.

Int. Commun. Heat and Mass Transf.

Adv. Water Resour.

J. Nuclear Mater.

Nuclear Instrum. Methods Phys. Res. Section A

Int. Commun. Heat and Mass Transf.

J. Colloid and Interface Sci.

Int. J. Heat and Mass Transf.

Comput. Geotech.

J. Colloid Interface Sci.

Significance of thermophoresis, thermal-diffusion and diffusion-thermo on the flow of Maxwell liquid film over a horizontal rotating disk

Phys. Scr.

Joule heating effects in thermally radiative swirling flow of Maxwell fluid over a porous rotating disk

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Thermotaxis of human sperm cells in extraordinarily shallow temperature gradients over a wide range

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The Soret effect and isotopic fractionation in high-temperature silicate melts

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