Microsoft Word - 35fakandu.docx CHEMICAL ENGINEERING TRANSACTIONS VOL. 53, 2016 A publication of The Italian Association of Chemical Engineering Online at www.aidic.it/cet Guest Editors: Valerio Cozzani, Eddy De Rademaeker, Davide Manca Copyright © 2016, AIDIC Servizi S.r.l., ISBN 978-88-95608-44-0; ISSN 2283-9216 On the Mesoscopic Model of Mass Transfer in Microporous Membranes Gulmira Sakhmetova, Arnold Brener*, Marat Satayev, Korlan Korabayeva State University of South Kazakhstan, Tauke Khan, 5, Shymkent, 160012, Kazakhstan amb_52@mail.ru The paper deals with the simplified mesoscopic model for describing the membrane purifying process. The problem of working out the reliable models and the appropriate calculating methods is very important both for extracting valuable components and for purifying gases from hazardous substances. The main supposition of the model is that mass transfer through the microporous crystalline films is determined by a superposition of two streams. The first stream is the diffusion stream described by the Fick law, and the second stream is conditioned by the random walk of molecules about active centres of the grid. It is supposed that characteristic time delay of diffusing substance inside the membrane is proportional to the probability of its capturing by active centres into the membrane. On the base of such consideration the asymptotic analysis of the submitted concept has been carried out. 1. Introduction The complex problem of gas purifying from various toxic or ballast ingredients has always been and remains to this day very relevant (Makaruk et al., 2010). Nowadays this problem gets of particular importance in the connection with the development of alternative energetic. Power and chemical production based on biogas technology becomes nowadays an important and promising component in long-term economic planning (Li Sun et al., 2014). The contribution of biogas energetic into the power production shows steady growth during the last decade (McKendry, 2002). Intensity of biomass production in chemical industry also increases (Di Donato et al., 2009). Many techniques to solve this problem have been offered (Ryckebosch et al., 2011). At the same time the complete ingredients content for many processes is not given, or it can vary into wide range (Teplyakov et al., 1996). Therefore for optimal designing the power and chemical production devices with allowance for the scaling problem the engineering practise needs the appropriate scientific well-founded methods. One of the main methods of gas, and of biogas purification especially, is the membrane technology (Esteves et al., 2008). Creation of engineering methods for calculating mass transfer through thin microporous films or membranes is a relevant science problem (Basu et al., 2010). Microscopic approach to calculating the mass transport inside membranes is accompanied however with great computing difficulties (Giacomin & Lebowitz, 1997). The mesoscopic approach under a correct employment calls for using stochastic integro- differential equations taking into account the nonequilibrium transport phenomena (Katsoulakis & Viachos, 2000). Such equations can be obtained in the long-rage order limit for intermolecular forces (Katsoulakis & Viachos, 2004). Standard practice in permeation modeling has employed the partial equilibrium assumption that enforces fixed boundary loadings. The more rigorous approach entails use of Robin boundary conditions at each side of the membrane. It can be shown that non-linearity of equation leads to the certain delay of concentration wave regarding the initial moment of the process (Viachos & Katsoulakis, 2000). But the known mesoscopic models in strict approach also entail the principal mathematical difficulties (Snyder et al., 2003). In this paper we try to offer and submit the simplified engineering model for describing that process. Previous results of our computer experiments with models of relaxation transfer cores showed that we can extend the mesoscopic approach for describing heat and mass transfer over less space scales as diffusion or heat DOI: 10.3303/CET1653009 Please cite this article as: Sakhmetova G., Brener A., Satayev M., Korabayeva K., 2016, On the mesoscopic model of mass transfer in microporous membranes, Chemical Engineering Transactions, 53, 49-54 DOI: 10.3303/CET1653009 49 boundary layers (Brener, 2006).The main supposition of our model is that mass transfer through the microporous crystalline films is determined by a superposition of two streams. The first stream is the diffusion stream described by the usual Fick law, and the second stream is conditioned by the random walk of molecules about active centers of the lattice (Brener et al., 2009). In our opinion such supposition corresponds to the mesoscopic models (Lam et al., 2002). 2. Theoretical details 2.1 Main concept We submit an approach for accounting molecular hopping drift inside the membrane based on simple stochastic model. Namely, let us suppose that characteristic time delay of diffusing substance inside the membrane is proportional to the probability p of its capturing by active centres into the membrane. Thus, this probability can be supposed as proportional to a product of two probabilities: 21 ppp = (1) The first probability 1p is proportional to the admixture concentration, since the more concentration the more a probability of molecules capturing (Horntrop, 2010). The second probability 2p is proportional to the specific number of free active centres inside the membrane. This number is proportional, in turn, to the specific number of active centres except of the number of centres occupied by captured molecules of an admixture. And the last number can be supposed as proportional to the admixture concentration, but only if this concentration is not so great that it would limit capturing potential of active centres (Figure 1). Figure 1: Molecular hopping drift inside the membranes Thus, the probability p can be written as follows )( 21 cknckp −= (2) Here c is the admixture concentration, 1k , 2k are the coefficients of proportionality, n is the complete specific number of active centres inside the membrane. Using the well-known Einstein’s formula for the random walk diffusivity we obtain x c pJ m ∂ ∂ −= τ δ 2 (3) Here mJ is the substance flux induced by the molecular hopping drift inside the membrane (Horntrop & Majda, 1994), δ is the typical lattice space scale, τ is the typical relaxation time, x is the coordinate along the diffusion direction. Thus, the approximation reflecting the qualitative peculiarities of the offered model can be written as follows ( ) 01 =    −− ∂ ∂ ∂ ∂ − ∂ ∂ ccD x c D xt c m χ , (4) where D is the usual diffusion coefficient, nk2=χ and mD is a kind of diffusion coefficient describing the random walk about the adjacent active centres of the lattice. The appropriate expression for mD reads x h δ 50 τ δ 2 1nkDm = . (5) In the case of sufficiently large concentration of the admixture, i.e. when the condition noted before deriving formula (2) may be not correct and the parameter 1→χ , Eq(4) acquires a look of (Brener et al., 2009) ( ) 01 =    −− ∂ ∂ ∂ ∂ − ∂ ∂ ccD x c D xt c m (6) The further analysis of Eq(4) will be carried out on the likely approximation of constD = , γDDm = , 1<<γ , (7) where γ is a special migration factor. 2.2 Asymptotic analysis From Eqs (5) and (7) it follows ( )[ ]cc x D x c D t c χγ − ∂ ∂ = ∂ ∂ − ∂ ∂ 12 2 (8) Let us look for the solution of Eq(8) in the form of asymptotic series  ∞ = = 0j j j cc γ (9) Under the homogeneous boundary conditions the zero-order term can be represented in the form of Fourier series       −     =  ∞ = Dt h i h xi Aс i i 2 22 1 00 expsin ππ (10) Here iA0 is the amplitude of the −i mode, and h is the typical depth. However, in reality the boundary conditions can be inhomogeneous: )(0 xfc = at 0=t ; )(0 tgc = at 0=x ; )(0 tuc = at hx = ; (11) In that case the first-order solution can be obtained on the base of special methods (Horntrop et al., 2001) as following ( )             −=  ∞ = h xi h tDi tM h c i i ππ sinexp 2 2 22 1 0 (12) Here ( ) dttu h tDi h Di dttg h tDi h Di dx h xi xftM t i th i )(exp)1( expsin)()( 0 2 22 0 2 22 0         −−       +     = ππ πππ (13) Both under the homogeneous and under the inhomogeneous conditions the subsequent approximations will be linearized step by step. 51 Particularly, the first-order approximation reads ( )[ ]0021 2 1 1 cc x D x c D t c χγ − ∂ ∂ = ∂ ∂ − ∂ ∂ (14) As it was above noted the case of the small concentration of an admixture is more interesting. So let us consider this case more detail: Cc ε= , 1<<ε (15) Eq(8) transforms to the following form x C D x C D x C D t C ∂ ∂ = ∂ ∂ + ∂ ∂ − ∂ ∂ )( 2 2 2 εχγγ (16) By using asymptotic approximation methods regarding the small parameter ε the asymptotic series and the zero-order equation read j j jCC ε ∞ = = 0 (17) 002 0 2 0 = ∂ ∂ + ∂ ∂ − ∂ ∂ x C D x C D t C γ (18) Let us look for the solution of Eq(18) in the form of running waves:             −= t D xKCC 2 exp ~ 00 γ . (19) ( ) 02020 2 0 ~ 2 ~ 2 ~~ C K KD x C KD x C D t C       −+ ∂ ∂ −+ ∂ ∂ = ∂ ∂ γ γ (20) From Eq(20) it immediately follows that at some conditions, for example at 2γ=K , the concentration running waves along the membrane depth can be arisen. The appropriate concentrate wave velocity and length are 22 mDDW == γ (21) mD D22 ==Λ γ (22) The equations for the first-order reads x C D x C D x C D t C ∂ ∂ = ∂ ∂ + ∂ ∂ − ∂ ∂ )( 201 2 1 2 1 χγγ (23) Since while solving Eq(23) the homogeneous boundary conditions can be used the solution looks as the Duhamel form ( )dssstxZC t  −= 0 1 ;, (24) The function Z can be found from the following auxiliary problem 52 02 2 = ∂ ∂ − ∂ ∂ − ∂ ∂ x Z D x Z D t Z γ (25) Moreover, it can be used the homogeneous boundary conditions and the obvious initial condition ( ) x C DZ ∂ ∂ −= 2 0γχ at 0=t (26) The main interesting conclusion which follows from condition (26) is that the non-linearity of diffusion equation with allowance for the migration factor leads to the non-locality of perturbations propagation. It is manifested as the delay of concentration waves relative to the initial moment: 2 4 mD D t χ ε≈Δ (27) At the low migration factor the small parameter ε can be evaluated as Λ ≈ h ε (28) The small parameter ε begin compete with the parameter χ that can deform the concentration wave shape at zero order. The appropriate zero-order solution after some rearrangements can be written as following ( )      −−      −= tx D t D x D DDA C mm ν νχχ ν 22 exp0 , (29) where ν is the phase velocity. The expression for the deformed concentration wave shape with account to the migration diffusion factor reads       −= ∗∗ t xDA C ω λν exp0 (30) Here the typical wave length is νχ λ 2− =∗ mD D (31) The typical wave frequency reads D Dm 4 4 22 ν ω − =∗ (32) 3. Conclusions The mesoscopic approach can be applied to describing heat and mass transfer in microporous membranes over less space scales as diffusion or heat boundary layers. As it follows from the analysis of submitted mesoscopic model the role of migration diffusion can be appreciable, and this phenomenon with appropriate amendments should be accounted while engineering calculations of the effectiveness of the membrane gas purification process. It is also shown that non-linearity of the master equation leads to the phase delay of the concentration wave relative the initial moment. As the result it can be concluded that submitted approach allows describing intricate phenomena accompanying the admixture diffusion through the microporous membranes. It is likely to be useful for engineering practice in design of processes and apparatuses for membrane technology with allowance for the scaling phenomena. 53 Reference Basu S. et al., 2010, Membrane-based technologies for biogas separations, Chemical Society Reviews, 39, No 2, 750-768. Brener A. M., 2006, Nonlocal equations of the heat and mass transfer in technological processes, Theoretical Foundations of Chemical Engineering, 40, No 6, 564-572. Brener A., Muratov A., Golubev V., 2009, Modeling of mass transport phenomena in microporous layers, Proceedings of the 9th International Conference, Mathematics and Computers in Science and Enginering, No5, 419-423. Di Donato P. et al., 2014, Biomass and Biopolymer Production Using Vegetable Wastes as Cheap Substrates for Extremophiles, Chemical Engineering Transactions, 38, 163-169, DOI: 10.3303/CET1438028. Esteves I. A. A. C. et al., 2008, Adsorption of natural gas and biogas components on activated carbon, Separation and Purification Technology, 62, No 2, 281-296. Giacomin G., Lebowitz J. L., 1997, Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits, Journal of statistical Physics, 87, No 1-2, 37-61. Horntrop D. J., Majda A.J., 1994, Subtle statistical behavior in simple models for random advection-diffusion, J. Math. Sci. Univ. Tokyo, 1, 23-70. Horntrop D. J., Katsoulakis M. A., Vlachos D. G., 2001, Spectral methods for mesoscopic models of pattern formation, Journal of Computational Physics, 173, No 1, 364-390. Horntrop D. J., 2010, Concentration effects in mesoscopic simulation of coarsening, Mathematics and Computers in Simulation, 80, No 6, 1082-1088. Katsoulakis M. A., Vlachos D. G., 2000, From microscopic interactions to macroscopic laws of cluster evolution, Physical review letters, 84, No 7, 1511. Katsoulakis M. A., Vlachos D. G., 2004, Mesoscopic modeling of surface processes, Dispersive Transport Equations and Multiscale Models. Springer, New York, 179-198. Lam R., Vlachos D. G., Katsoulakis M. A. Homogenization of mesoscopic theories: Effective properties of model membranes //AIChE journal. – 2002. – Т. 48. – №. 5. – С. 1083-1092. Li Sun, Bin Xu, Smith R., 2014, Power and Chemical Production Analysis Based on Biomass Gasification Processes, Chemical Engineering Transactions, 38, 61-67, DOI: 10.3303/CET1438011. Makaruk A., Miltner M., Harasek M., 2010, Membrane biogas upgrading processes for the production of natural gas substitute, Separation and Purification Technology, 74, No1, 83-92. McKendry P., 2002, Energy production from biomass (part 1): overview of biomass, Bioresource technology, 83, No1, 37-46. Ryckebosch E., Drouillon M., Vervaeren H., 2011, Techniques for transformation of biogas to biomethane, Biomass and bioenergy, 35, No 5, 1633-1645. Snyder M. A., Vlachos D. G., Katsoulakis M. A., 2003, Mesoscopic modeling of transport and reaction in microporous crystalline membranes, Chemical engineering science, 58, No 3, 895-901. Teplyakov V. et al., 1996, Integrated membrane systems for gas separation in biotechnology: potential and prospects, World Journal of Microbiology and Biotechnology, 12, No 5, 477-485. Vlachos D. G., Katsoulakis M. A., 2000, Derivation and validation of mesoscopic theories for diffusion of interacting molecules, Physical review letters, 85, No 18, 3898. 54