Alshawabkeh, A. N.

Johannesson, Björn

Ottosen, Lisbeth M.

Paz-Garcia, Juan Manuel

Ribeiro, A. B.

Rodríguez-Maroto, J. M.

Online Research Database In Technology

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Downloaded from orbit.dtu.dk on: Dec 19, 2017Coupling the transport of water and aqueous species in finite element modeling ofelectrokineticsPaz Garcia, Juan Manuel; Johannesson, Björn ; Ottosen, Lisbeth M.; Alshawabkeh, A. N.; Ribeiro, A. B.;Rodríguez-Maroto, J. M.Publication date:2011Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Paz-Garcia, J. M., Johannesson, B., Ottosen, L. M., Alshawabkeh, A. N., Ribeiro, A. B., & Rodríguez-Maroto, J.M. (2011). Coupling the transport of water and aqueous species in finite element modeling of electrokinetics.Poster session presented at 12th Mediterranean Congress of Chemical Engineering, Barcelona, Spain.COUPLING THE TRANSPORT OF WATER AND AQUEOUS SPECIES IN FINITE ELEMENT MODELING OF ELECTROKINETICSJ.M. Paz-García1, B. Johannesson1, L.M. Ottosen1, A.N. Alshawabkeh2, A.B. Ribeiro3 and J.M. Rodríguez-Maroto41 Department of Civil Engineering, Technical University of Denmark; 2Department of Civil and Environmental Engineering, Northeastern University, Boston, USA; 3Department of Environmental Sciences and Engineering, New University of Lisbon, Portugal; 4Department of Chemical Engineering, University of Málaga, SpainIntroductionThe electrokinetic transport of ionic and non-ionic species through a porous media under the effect of an external electric field can be mathematically  described by  the Nernst-Planck-Poisson (NPP) system of equations. Mass balance equation for the transport of water: Ap ∂θ∂t= −∇Jθ +Gθ⎛⎝⎜⎞⎠⎟; Jw = −Dθ∇θMoisturetransport − kh∇hHydraulictransport −θke∇φElectroomosis Nernst-Planck equation: Apθ∂ni∂t= −∇J i +Gi⎛⎝⎜⎞⎠⎟; J i = vwniAdvection −Dieff∇niDiffusion −Uieff ni∇φElectromigration  Poisson’s equation:  ε∇2φ = F nizii=1M∑where: • AdvectionvwniMovement of the species in the solution due to the fluid’s bulk motion. • Diffusion (Fick’s law)−Dieff∇niMovement of species due to a gradient of chemical potential.• Electromigration−Uieff ni∇φMovement of charged species in an electric field.• Total Moisture transport (Richards’ equation)−Dθ∇θMotion of water in unsaturated porous media. • Hydraulic flow (Darcy’s law)−kh∇hMovement of water in saturated porous media, due to gradient of hydraulic gradients. • Electroosmotic flow −θke∇φMotion of the liquid induced by an applied electric potential.Figure 1: Electrokinetic transport phenomena1. Electromigration is the main mechanism affecting ionic transport. Electroosmotic advection is important for the transport of non-charged particles.2. Richards’ equation describes the movement of both vapor and liquid water on the basis of a single variable, the moisture content. Capillary  forces are assumed the main transport mechanism. 3. The moisture diffusivity  is assumed to be non-linear function of the water content and the nature of the porous material. 4. Fully  saturated conditions are rare. Hydraulic flow may  be negligible with respect to the other transport contributors. Unsaturated conditions are assumed in the proposed numerical model.TerminologyTable 1: Terminology for the mathematical description of the modelTerm:θ kgwmpore−3( ) Moisture contentni moli kgw−1( ) Molal concentrationφ V( ) Electrical potential Ionic charge Hydraulic headCross-sectional areaPorosityIndex for the ith speciesM Total number of aqueous chemical speciesFlux termGeneration termWater velocityEffective diffusivityEffective ion mobilityTotal moisture diffusivityElectroosmotic permeability coefficientHydraulic conductivityFaraday constantTable 2: Terminology for the matrices in the Finite Element Formulation.FEM Matrix State variable  parameter ApDθ Aθ pke ApDθni Aθ pkeni Aθ pDieff Aθ pUieffε Fzi ApΔt ApθΔtChemical GenerationWater part icipates in several chemical reactions along the domain of the treated body  as, for example, p rec ip i ta t ion , d isso lu t ion and hydration of salts. In electrokinetics, electrolysis reactions may occur at the electrodes as a consequence of the applied current. And so, water is consume at the vicinities of the electrodes. H2O12O2 + 2H+ + 2e−2H2O + 2e− H2+2OH−Somewhere in the middle of the treated body, these H+ and OH- fronts can meet each other producing water.Numerical ModelThe strongly  coupled non-linear NPP system can be satisfactory  solved by  finite elements integration. It can also be combined to additional physicochemical aspects such as the electrode processes and the chemical interaction between the species in the system for modeling of electrokinetic transport processes. Let the vector a be the discrete form at the state variables at the nodal points, and f the load vectors for the state variables indicated with the corresponding subscript. Generation terms are implicitly  included in the load vectors. The weak formula for the extended NPP system in the finite element method is: Mass balance equation for the transport of water:Nernst-Planck equation:( ) ( ) 0t t t mta t dif t eoa em ti i i i i i i iiθ φ−Δ− + + + + + =C a a K a K a K K a fPoisson’s equation: 10Mt ti iiφ φ φ=+ + =∑E a K a f !A global matrix system of equations is obtained by: Where: C =Cθ 0   00 C1   0       CM 00 0  0 0⎡⎣⎢⎢⎢⎢⎢⎢⎢⎤⎦⎥⎥⎥⎥⎥⎥⎥; a t =aθta1ta Mtaφt⎡⎣⎢⎢⎢⎢⎢⎢⎢⎤⎦⎥⎥⎥⎥⎥⎥⎥; f =fθf1fMfφ⎡⎣⎢⎢⎢⎢⎢⎢⎢⎤⎦⎥⎥⎥⎥⎥⎥⎥; K =K mt 0   K eoK1mta K1dif 0  K1eoa +K1em( ) 0   K Mmta   K Mdif K Meoa +K Mem( )0 E1  EM Kφ⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥The proposed system can be used for simulating reactive transport of water and aqueous species in porous material under the effect of an externally  applied electric field, as well as for free diffusion problems. ConclusionsPetrov-Galerkin (PG) or Galerkin least square (PLS) are assumed standard for convection dominated problems, but they  required a complicated formulation which usually implies high computation time. By means of the proposed coupled system, including the advective term directly  in the Nernst-Planck equation of each aqueous species apart from in the water transport equation itself, the convenience Galerkin method of weighted residuals can be used for the solution of the non-linear coupled transport of water and aqueous species in porous media. AcknowledgmentsRodriguez-Maroto acknowledges the financial support of the Ministerio de Ciencia e Innovación of the Spanish G o v e r n m e n t t h r o u g h t h e p r o j e c t E R H M E S , CTM2010-16824. Paz-García acknowledges the Danish Agency  of Science Technology  and Innovation for the economic support through the project 274-08-0386 “Fundamentals of Electrokinetics in in-homogeneous matrices”.

Coupling the transport of water and aqueous species in finite element modeling of electrokinetics

                       General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.  • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal   If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.     Downloaded from orbit.dtu.dk on: Dec 19, 2017Coupling the transport of water and aqueous species in finite element modeling ofelectrokineticsPaz Garcia, Juan Manuel; Ottosen, Lisbeth M.; Johannesson, Björn ; Alshawabkeh, A. N.; Ribeiro, A. B.;Rodríguez-Maroto, J. M.Published in:Abstracts. 12th Mediterranean Congress of Chemical EngineeringPublication date:2011Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Paz-Garcia, J. M., Ottosen, L. M., Johannesson, B., Alshawabkeh, A. N., Ribeiro, A. B., & Rodríguez-Maroto, J.M. (2011). Coupling the transport of water and aqueous species in finite element modeling of electrokinetics. InAbstracts. 12th Mediterranean Congress of Chemical Engineering SEQUI.12th Mediterranean Congress of Chemical Engineering 2.Applied Chemical Engineering 2.2 Environmental Technology   COUPLING THE TRANSPORT OF WATER AND AQUEOUS SPECIES IN FINITE ELEMENT MODELING OF ELECTROKINETICS  J.M. Paz-García1, L.M. Ottosen1, B. Johannesson1, A.N. Alshawabkeh2, A.B. Ribeiro3 and J.M. Rodríguez-Maroto4  1 Department of Civil Engineering, Technical University of Denmark, jugra@byg.dtu.dk 2Department of Civil and Environmental Engineering, Northeastern University, USA 3Department of Environmental Sciences and Engineering, New University of Lisbon, Portugal 4Department of Chemical Engineering, University of Málaga, Spain  Scientific topics: Environmental Technology  The electrokinetic transport of ionic and non-ionic species through a porous media under the effect of an external electric field can be mathematically described by the Nernst-Planck-Poisson (NPP) system of equations. The Nernst-Planck mass balance equation describes the transport of aqueous species by a combination of the diffusion and electromigration terms. Since the ions themselves contribute to the local electric potential, Poisson’s equation of electrostatics, that relates the electrostatic potential to local ion concentrations, is solved simultaneously to describe this effect. This strongly coupled non-linear system has been satisfactory solved by finite element integration, and combined to additional physicochemical aspects such as the electrode processes and the chemical interaction between the species in the system for modeling of electrokinetic transport processes.  Proper models of the electrokinetic phenomena have to take into account the transport of the pore solution itself. Water is suggested to electroosmotic flow under electric fields. Electroosmotic advection is of mayor importance when considering the transport of non-charged particles such as organic contaminants from soil. The transport of water may dry the material at the anode surroundings, what may interrupt the ionic current and stop the process. Capillary forces and hydraulic gradient, between others, can also affects to the transport of water in the porous media. In the present work, the NPP system has been extended in order to include the water transport equation, and to calculate the advective flow contribution on the ionic transport. In this model, the transport of water is assumed to be a combination of hydraulic gradient, capillary suction forces and electroosmotic flow. In the model presented here, the advective term has been implicitly included in the Nernst-Planck equation of each individual aqueous species. Thus, this coupling strategy between the water and the ionic transport together with a numerical procedure for solving the non-linearity character, allows a simplified finite element analysis.  References J.M. Paz-García, B. Johannesson, L.M. Ottosen, A.B. Ribeiro, and J.M. Rodríguez-Maroto, “Modeling of electrokinetic processes by finite element integration of the Nernst-Planck-Poisson system of equations”. Separation and Purification Technology, 2011, 79, 183-192.   

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Coupling the transport of water and aqueous species in finite element modeling of electrokinetics

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