WSEAS TRANSACTIONS on HEAT and MASS TRANSFER

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1 Numerical Simulation o Mass Transer Aiming at Food Processes and Bioprocesses in Fixed-Bed Equipment: a Comparison Between Finite- Dierences and Lattice-Bolztmann Method JOSÉ A. RABI Faculty o Animal Science and Food Engineering (FZEA), University o São Paulo (USP) Av. Duque de Caxias Norte 5, , Pirassununga, SP BRAZIL jrabi@usp.br Abstract: - Lattice-Boltzmann method (LBM) has been alternatively used to numerically simulate mass transer in porous media. Supercritical luid extraction (SFE) and biospeciic ainity chromatography (BAC) are ixedbed processes that may beneit rom LBM as their model ramewors are very similar. By allowing the species concentration in the luid phase to vary both in time and along the bed axis, this wor has simpliied SFE and BAC models in order to obtain an identical governing partial dierential uation (PDE). Cast in dimensionless orm, such mutual PDE was numerically solved through either LBM or inite-dierences method (FDM). As ar as alse diusion is concerned, numerical simulations were compared and results encourage the use o LBM as a simulation tool to investigate either SFE or BAC processes. Key-Words: - mass transer, ixed bed, porous media, numerical simulation, inite-dierences method, lattice- Boltzmann method Introduction One may rely on distinct modeling approaches in order to simulate transport phenomena, with models varying between the ollowing extremes []: rom physics-based to data-driven, rom deterministic to stochastic, rom macroscopic to microscopic, and rom analytical to numerical. Combinations are allowed as one may propose, or instance, physicsbased models in either macroscopic, mesoscopic or microscopic scales []. In eect, one may use any scale to simulate transport phenomena within porous media [3], which is the case o those involving ood or bioproducts. Although comprehensive models or ood or bioprocesses may result complex [4], their numerical simulation has continuously increased as computational tools have been developed [5] while their importance have been recognized [6]. With widespread use or ood or bioprocesses, macroscopic modeling leads to partial dierential uations (PDEs) or observable properties (e.g. species concentration and temperature). As systems contain a huge number o particles, such properties are interpreted as average values at a point in the continuum. On the opposite end, the microscopic approach considers each single constituent particle, identiied together with inter-particle interactions to be used in Newton s law o motion. Reerred to as molecular dynamics (MD), the resulting simulation ruires large computational eort [3]. Lying between the two later modeling levels, the mesoscopic approach deals with global eects o particle via a so-called distribution unction, which attempts to depict the behavior o a relatively small particle collection. Based on such unction, lattice- Boltzmann method (LBM) is implemented so that one may distinguish it rom both MD simulation and classical macroscale discretization methods such as inite-dierences (FDM), inite-volumes (FVM) and inite-elements (FEM) [7]. In LBM, bul medium is treated as a collection o constituent particles occupying a discrete space (lattice sites). During a discrete time lag, particles travel between sites along pre-deined directions (lattice lins). When they arrive at adjacent sites, particles mutually collide and are rearranged. Such dynamics is described by two LBM steps reerred to as streaming and collision. By assuming that such dynamics obeys basic conservation principles while being isotropic, LBM may properly describe and simulate macroscopic medium behavior [8]. Being a discrete approach or the inetic theory, LBM has been applied to ood and bioprocesses and simulated systems have included (but not restricted to) lows in porous media, colloidal suspensions, polymer solutions and emulsions [9]. The present wor aims at LBM simulation o biospeciic ainity chromatography (BAC) as well o supercritical luid extraction (SFE) in ixed bed uipment. Even when E-ISSN: Issue 3, Volume 7, July 0

2 model simpliications are evoed, inherent hurdles still justiy the use o numerical methods to simulate either BAC or SFE processes. Theory. Models or BAC and SFE processes Fixed-beds considered in this wor are cylinders o inner radius R and length L, either horizontally or vertically oriented as shown in Fig.. Bed inlet is at z = 0 while outlet is at z = L, with respect to luid low. In BAC the later is the percolating solution whereas in SFE it reers to the supercritical luid. Porosity ε is allegedly uniorm throughout the bed. BAC models have evoed uniorm luid low, sorption-desorption inetics and species transport by convection and/or diusion []-[7]. Those models have been o irst-order with respect to the spatial dependence so that species concentrations may vary along a given coordinate z (i.e., ixed-bed axis), apart rom depending on time t. Such concentrations have then been modeled as φ = φ(z, and θ = θ(z, in luid and solid phases, respectively, and related governing PDEs have been expressed as: φ φ φ ε + vz = Dz r& ε () θ = r& = φ( θmax θ) θ () where D z is species axial diusivity, and are respectively sorption and desorption coeicients and θ max is the maximum adsorption capacity o the chromatographic column. Giving the rate at which species are adsorbed rom luid to solid phase, term r& behaves as a sin in Eq. () and as a source in Eq. (). As initial conditions, one may prescribe: Fig.. Cylindrical coordinates (r-z) or ixed-beds considered in the present wor. I the volumetric low rate V & remains constant, the interstitial velocity v z = V & /( επr ) is uniorm. In model uations, one may use the so-called seepage velocity ~ = V & /( πr ) instead [0]. The idea when v z assessing velocities v ~ z and v z is depicted in Fig., being A = πr bed total cross-sectional area while A = εa and A s = ( ε) A are those occupied by luid and solid phases, respectively. at t = 0: φ = 0 and θ = 0 (3) while boundary conditions might be: φ at z = 0: φ = φ in ; at z = L: = 0 (4) being φ in 0 a nown inlet concentration. SFE models have ually evoed uniorm low, sorption-desorption inetics and species transport by convection and/or diusion. Although the later has been neglected [8]-[], diusion can indeed be inluential inside large uipments, thus justiying its modeling [3]. Lie time-dependent -D BAC models, species concentrations in SFE have been assumed as φ = φ(z, and θ = θ(z, respectively in luid and solid phases, so that governing PDEs have been expressed as: (a) (b) φ φ φ ε + vz = Dz r& ε (5) Fig.. Evaluation o (a) seepage velocity v ~ z = V & / A and (b) interstitial velocity = V & / A = V& /( εa). v z θ φ l = r& =, ti = µ t t θ i (6) P Di While v z, D z and ε have the same meaning as in BAC uations, P is a partition coeicient and t i is p E-ISSN: Issue 3, Volume 7, July 0

3 an intra-particle diusion reerence time evaluated rom particle eatures, namely, shape coeicient µ, characteristic length l p and intra-particle diusivity D i. I θ max is bed maximum extraction capacity, one may impose the ollowing initial conditions: at t = 0: φ = 0 and θ = θ max (7) while boundary conditions can be: φ at z = 0: φ = 0 ; at z = L: = 0 (8) While BAC and SFE are distinct processes, their model ramewors are quite similar to each other. As one may realize, governing PDEs or luid-phase concentration φ are basically the same, apart rom the act that the sorption-desorption term r& reers to dierent PDEs or the solid-phase concentration θ. Aiming at preliminary LBM simulations or BAC and SFE processes, this wor attempts to proit rom the aoresaid similarity between model uations.. LBM undamentals The underlying concept o LBM is to replace the nowledge about each constituent particle (in terms o position and velocity) by a suitable description o their overall eect through a distribution unction r r = (, c,. Obeying Boltzmann s uation, such unction gives the probability o inding, at time t, particles about position r with speeds within c r and r r c + dc. Once becomes nown, one may then assess macroscopic properties o interest [9]. In the absence o external orces, Boltzmann s uation is written as the ollowing advection uation with a source (or sin) term: r r + c = Ω( ) = τ ( ) (9) where the collision operator Ω = Ω() gives the variation rate o unction due to collisions between particles. In the above uation, the so-called BGK (Bhatnagar-Gross-Kroo) approach was evoed so as to linearize such operator as Ω ( ) = ( )/ τ, where τ is reerred to as relaxation time and is the local uilibrium distribution unction [4]. In LBM, Eq. (9) is discretized along pre-deined lattice lins (directions). Distances z separating adjacent lattice sites are discrete (subscript reers to a given lin) and so is time, so that t is a discrete advancing time step. Lattice arrays are identiied as DnQm, where n is the problem dimension (e.g., n = or -D problems) while m is the number o lattice lins (= number o distribution unctions to be solved). Entailing a central site and two neighbors, Fig. 3 setches a -D lattice array nown as DQ3, which is similar to DQ. Through the lattice lins connecting the central site to its neighbors, particles may then stream with either orward or bacward r r velocities, c = + c zˆ or c = c zˆ (c = z/ t = lattice speed, ẑ = unit vector), as Fig. 3 suggests. Central velocity is null, i.e., c 0 = 0. Fig. 3. Either DQ or DQ3 lattice arrays or onedimensional (-D) LBM simulations. Two usual -D lattice arrays are shown in Fig. 4. Including the null velocity at the central site, DQ5 array comprises 5 lattice speeds c r ( = 0,,, 3, 4) but it cannot be used or low simulations [3] so that DQ9 array should be used instead, which entails 5 lattice speeds c r ( = 0,,..., 7, 8). (a) (b) Fig. 4. (a) DQ5 and (b) DQ9 lattice arrays or two-dimensional (-D) LBM simulations. With solid lines representing lattice lins, Fig. 5 shows two common 3-D arrays: D3Q5 and D3Q9. There are 5 lattice speeds c r ( = 0,,..., 3, 4) in the ormer and 9 lattice speeds c r ( = 0,,..., 7, 8) in the later. E-ISSN: Issue 3, Volume 7, July 0

4 ( z + z, t + = t + (4) (a) (b) Fig. 5. (a) D3Q5 and (b) D3Q9 lattice arrays or three-dimensional (3-D) LBM simulations. By writing Eq. (9) or a direction at a position z and time t, one then obtains -D lattice-boltzmann uation under BGK approach, namely: t + c = ) (0) τ where c = z / t ( z = ± z, depending on the streaming direction). For DQ3 arrays, Eq. (0) is written or = 0, and (i.e., or 0, and ) while or DQ arrays it is only written or = and as unction 0 ( = 0) is disregarded. Space-time discretization o Eq. (0) renders the ollowing algebraic uation (the order o let-hand side terms has been changed or aesthetic purposes): c ( z + z, t + z t + = t t + + τ () By introducing the so-called relaxation parameter ω = t/τ, the previous uations is rewritten as: ( z + z, t + = [ ω] + ω () whose evolution is carried out in two steps [3]. In the collision step (= time evolution), distribution unctions or each direction are updated at each lattice site rom instant t to t + t as: t + = [ ω] + ω (3) In the streaming step (= spatial evolution), collision results are transported to adjacent sites according to: In LBM, distinct systems can be simulated by suitably handling the relaxation parameter ω and the uilibrium distribution unction. While the later governs the transport phenomenon (i.e., o mass, momentum or energy), the ormer dictates the related coeicient (i.e., mass diusivity, inematic viscosity or thermal diusivity). For mass transport, diusivity D z depends on ω as well as on the so-called lattice sound speed c s as: Dz = + Dz = c t s ω c t ω s (5) where the so-called lattice sound speed is assessed as c s = c = z/ t or both DQ and DQ3 arrays [3]. For either DQ4 or DQ5 arrays it becomes c s = c / and or DQ9, D3Q5 or D3Q9 arrays it is c s = c / 3. For luid low problems, expression or inematic viscosity υ is similar to the previous uation by simply replacing D z or υ whereas or heat transer one must then replace D z by thermal diusivity α [3],[9]. As cited, the uilibrium distribution unction is deined according to the transport phenomenon to be simulated. Being φ the transported quantity, the ollowing expression applies or relatively low luid low velocities v r [3],[9]: r r r r ( c v) ( c v) φ cs cs = w r r ( v v) (6) cs w being weighting actors satisying the condition w =. In the DQ array, or example, w 0 = 0 reers to the central site while w = w = ½ reers to each streaming direction (Fig. 3). Last but not least, at any position z and instant t, one may retrieve the transported quantity φ = φ(z, rom the distribution unctions as [3],[9]: 3 Numerical method φ( z, = (7) 3. Dimensionless ormulation One may beneit rom the similarity between BAC and SFE model uations to implement preliminary LBM simulators. In view o that, this wor applies E-ISSN: Issue 3, Volume 7, July 0

5 LBM only to the species concentration φ in the luid phase so that simpliied orms o Eqs. () and (5) are considered by ignoring the source or sin term r&. Hence, an identical governing PDE or φ is then obtained or both BAC and SFE, namely: θ φ φ φ r& = = 0 + vz = Dz (8) Aiming at a dimensionless ormulation o such mutual uation, dimensionless variables or the luid-phase species concentration φ, time t and axial coordinate z are introduced as: φ t Φ =, T =, φ t re Z = z z (9) where φ re 0 can be identiied to any reerence concentration while t and z were introduced in the previous section. By recalling that c s = c = z/ t and v z = v or -D problems, Eq. (8) can then be cast into the ollowing dimensionless orm: Φ Φ Φ + Ma = T Z Pe m Z (0) where Ma and Pe m are lattice-based Mach and masstranser Péclet numbers, respectively deined as: v v t c z ( z) Ma = =, Pem = = () c z D t s z D z In order to solve Eq. (0), one may impose the ollowing initial condition on Φ: at T = 0: Φ = 0 () as well as the ollowing boundary conditions: at Z = 0 (inle: Φ = (3) Φ at Z = N z = L/ z (outle: = 0 Z (4) where N z + is the number o lattice sites, end points included. In Eq. (3), reerence concentration was identiied to the non-null inlet concentration (φ re = φ in ) in line with BAC. Without loss o generality, one may instead identiy it to the non-null maximum concentration (φ re = φ max ) in line with SFE. 3. Preliminary LBM or BAC and SFE In order to implement LBM simulators or BAC and SFE, one actually needs two distinct distribution unctions (z, and s (z,, sharing the same lattice and respectively reerring to species concentration in luid and solid phases. Given the underlying physics o governing PDEs, Eqs. (), (), (5) and (6) (namely, diusion-convection in luid phase and stationary solid medium), the ollowing uilibrium distribution unctions and s can be adopted [3]: = w φ( z, ± s [ v c] z (5) = w φ( z, (6) where the sign o v z /c depends on the streaming direction. In line with the lattice array (but always ulilling the condition w = ), weighting actors w are the same or and s [3]. Yet, relaxation actors ω and ω s are dierent or each phase: luid phase (D z 0): ω D = z + (7) c z solid phase (D z = 0): ω = s (8) As a irst step towards LBM simulations o BAC and SFE, LBM was applied only or the luid-phase concentration, in view o the common governing PDE, Eq. (0). DQ array was employed so that weighting actors are w = w = ½. With respect to the uilibrium distribution unction and to the dimensionless species concentration, the ollowing expressions hold (or = or ): [ ± Ma] ( Z, T ) = w Φ( Z, T ) (9) Φ( Z, T ) = ( Z, T ) (30) With the help o Eqs. () and (7), one may write the relaxation actor as: ω = Pe m + (3) With respect to the initial condition (at T = 0), one may evoe Eq. () so as to impose Z,0) = w Φ(,0) (3) ( Z E-ISSN: Issue 3, Volume 7, July 0

6 ( Z Z,0) = w Φ(,0) (33) At bed inlet, one obtains the boundary condition or (0,T) via streaming rom the adjacent site (,T), thus (0,T) remains the only unnown. By imposing Φ(0,T) = as given by Eq. (3), one may use Eq. (30) to yield the ollowing condition or at Z = 0: 0, T ) = (0, ) (34) ( T Approximating Φ/ Z = 0 in Eq. (4) via irst-order inite dierences, the ollowing boundary conditions are obtained or and at bed outlet (Z = N z ): ( N z z T, T ) = ( N, ) (35) ( N z z T, T ) = ( N, ) (36) 4 Results and discussion Relying on BGK-DQ approach, the present wor programmed LBM in Fortran (standard 90/95) so as to simulate time-dependent -D species transer as governed by Eq. (0) subjected to Eqs. (), (3), (4). Resulting rom the numerical implementation o streaming and collision steps, LBM codes ollow those encountered in [4] or transport phenomena o similar nature. The proposal here is to chec out the proper implementation o LBM codes as regards to species concentration in luid phase. Parameter N z was set to yield 5 sites (end points included). For the sae o comparison, a inite-dierences method (FDM) code was ually implemented with 5 grid points (end points included). With respect to how Eq. (0) is discretized in such FDM code, it is worth remaring that: Time derivative Φ/ T was discretized via irstorder orward dierences while explicit ormulation was used or the remaining terms. Instabilities were indeed observed or T > 0.3 so that such scheme should be replaced by either a ully or partially implicit scheme [5],[6]. In contrast, no restrictions on the advancing time step DT must be imposed or LBM simulations. Convective term Φ/ Z was discretized by means o upwind scheme, which may yield alse (numerical) diusion. Particularly noted ahead or lower D z, such eect may explain why gradients in FDM proiles are smoother when compared to LBM counterparts. Proiles or Φ simulated at T = 600 via LBM and FDM are compared in Fig. 6 or some distinct Pe m values with Ma = 0.. Bearing in mind, or instance, a bed length L = m and an interstitial velocity v z = m/s (i.e., laboratory scale), such Pe m and Ma values then yield z = m and t = 0.5 s (thus, T = 600 renders t = 300 s) while they may reer to the ollowing values or species diusivity D z and relaxation actor ω : Pe m = 0.8 D z = m /s and ω = 4/7; Pe m = 4.0 D z = m /s and ω = 4/3; Pe m = 8.0 D z = m /s and ω = 8/5. (Many other combinations are obviously easible by assuming distinct values or L and v z ). Φ (dimensionless) Ma = 0., no source / sin LBM, Pe = 0.8 LBM, Pe = 4.0 LBM, Pe = 8.0 FDM, Pe = 0.8 FDM, Pe = 4.0 FDM, Pe = Z (dimensionless) Fig. 6. LBM and FDM simulations o Φ proiles at T = 600 using Ma = 0. (ixed) and Pe m = 0.8, 4 or 8: diusive-convective transport. Despite LBM and FDM simulators were able to reproduce expected Φ proiles, dierences between numerical results become apparent as Pe m increases or, in view o Eq. (), as v z (convection) increases with regard to D z (diusion). One might assign such dierences to alse diusion in FDM. In eect, urther comparisons between LBM and FDM simulations were carried out by neglecting the convective term in Eq. (0), i.e., by setting Ma = 0. A diusion-dominant PDE or the dimensionless concentration Φ in the luid phase is obtained: Φ Φ = T Pe m Z (37) Moreover, by imposing v z = 0 in Eq. (9), luidphase uilibrium distribution unctions become analogous to solid-phase counterparts, Eq. (30). One may then write (in dimensionless orm): ( Z, T ) = w Φ( Z, T ), =, (38) E-ISSN: Issue 3, Volume 7, July 0

7 Also subjected to boundary and initial conditions as given by Eqs. (3), (33), (34), (35) and (36), Eq. (37) was solved via LBM and FDM. Resulting Φ proiles simulated at T = 600 are compared in Fig. 7. In the absence o the convective term (and o alse diusion), it is worth observing that LBM and FDM proiles are practically coincident or each Pe m. Φ (dimensionless) Ma = 0, no source / sin LBM, Pe = 0.8 LBM, Pe = 4.0 LBM, Pe = 8.0 FDM, Pe = 0.8 FDM, Pe = 4.0 FDM, Pe = Z (dimensionless) Fig. 7. LBM and FDM simulations o Φ proiles at T = 600 using Ma = 0 (ixed) and Pe m = 0.8, 4 or 8: diusion-only (convective transport neglected). Additional comparisons between LBM and FDM simulations were accomplished by adding a constant term R & into the right-hand side o Eq. (37) as: and FDM results are practically coincident or each Pe m. While it is worth recalling that Eq. (39) lacs a convective term (thus alse diusion is absen, one veriies that both simulators were able to reproduce the expected inluence o the source term. Φ (dimensionless) Ma = 0, source: R = 0.0 LBM, Pe = 0.8 LBM, Pe = 4.0 LBM, Pe = 8.0 FDM, Pe = 0.8 FDM, Pe = 4.0 FDM, Pe = Z (dimensionless) Fig. 8. LBM and FDM simulations o Φ proiles at T = 600 using Ma = 0 (ixed) and Pe m = 0.8, 4 or 8: diusion-only with source term. In order to examine prospective alse diusion eects together with those rom the source term, urther comparisons between LBM and FDM were carried out by solving the ollowing PDE or the dimensionless luid-phase concentration: Φ = T Pe m Φ r t + R&, R& & = (39) Z φ re Φ Φ Φ + Ma = + R& (4) T Z Pe m Z In view o Eqs. () and (5), one may interpret it as an attempt to account or the presence o the solid phase. From the luid phase standpoint, such new term in Eq. (39) behaves as a source in SFE ( R & > 0 ) or as a sin in BAC ( R & < 0 ). In LBM, one inserts sources or sins in the right-hand side o Eq. (9) so that the collision step is extended to [3]: ( Z, T + T ) = [ ω ] ( Z, T ) + + ω ( Z, T ) + w R& T (40) where weighting actors are again w = w = ½. Also subjected to boundary and initial conditions as imposed by Eqs. (3), (33), (34), (35) and (36), with R & = 0. 0 as a source term, Eq. (39) was solved via LBM and FDM. Advancing time steps were T =.0 or LBM and T = 0.3 or FDM so as to avoid numerical instabilities. Simulated Φ proiles at T = 600 are shown in Fig. 8 where one notes that LBM Such previous PDE becomes a dimensionless orm o either Eq. () or (5) provided that the source or sin term R & is suitably modeled in line with BAC or SFE processes (which evoes the solution o the related PDE or the solid-phase concentration). Also subjected to the same boundary and initial conditions, Eqs. (3), (33), (34), (35) and (36), Eq. (4) was solved with R & = 0. 0 as a source term and T =.0 and T = 0.3 as advancing time steps or LBM and FDM, respectively. Simulated Φ proiles at T = 600 are shown in Fig. 9, where alse diusion eects in FDM results can be once again identiied, yet relatively to minor extent i compared to those in Fig. 6. As expected, alse diusion eects become more evident or higher Pe m (higher convection in relation to diusion), together with the sweeping eect o the percolating luid low, when compared to counterpart proiles in Fig. 8. E-ISSN: Issue 3, Volume 7, July 0

8 Φ (dimensionless) Ma = 0., source: R = 0.0 LBM, Pe = 0.8 LBM, Pe = 4.0 LBM, Pe = 8.0 FDM, Pe = 0.8 FDM, Pe = 4.0 FDM, Pe = Z (dimensionless) Fig. 9. LBM and FDM simulations o Φ proiles at T = 600 using Ma = 0. (ixed) and Pe m = 0.8, 4 or 8: diusion-convection with source. 5 Concluding remars As ar as species concentration in the luid phase is concerned, time-dependent -D uations or BAC or SFE processes in ixed beds are so similar that one is able to arrive at the same governing PDE by neglecting each related sorption-desorption term. Proiting rom such lieness, this wor implemented preliminary LBM simulators or either BAC or SFE, by starting rom such mutual uation and casting it into dimensionless orm. FDM simulators were also implemented or comparison purposes in terms o numerical validity and perormance. Dimensionless concentration proiles simulated through LBM and FDM were practically coincident as regards to the common governing PDE as well as with respect to variations o it, namely, diusiondominant scenario (i.e., convection neglected) and inclusion o a source term. It is then believed that dierences noted between LBM and FDM proiles can be assigned to alse (i.e., numerical) diusion eects attributable to the upwind (i.e., irst-order) discretization o convective term in FDM. Besides, LBM simulations proved to be exempt rom the well-nown numerical stability criteria that must be obeyed by FDM simulations when the later rely on an explicit scheme with respect to time. Those previous results are encouraging having in mind the ability to deal with more comprehensive simulations o both BAC and SFE processes in ixed beds. A primary objective is to widen each model ramewor in order to include the PDE or species concentration in the solid phase. Other extensions include -D (or 3-D) domains and the inclusion o additional transport phenomena (e.g., heat transer and/or bed hydrodynamics). Acnowledgement The author thans FAPESP (São Paulo Research Foundation, Brazil) or their inancial support to the research project (No. 05/0538-) rom which the present wor derives. Nomenclature A cross-sectional area (m ) c lattice speed (m s ) D species (mass) diusivity (m s ) distribution unctions (dimensionless) P partition coeicient (dimensionless) sorption constant (suitable units) desorption constant (suitable units) L length o the ixed bed (m) l P particle characteristic length (m) Ma Mach number (dimensionless) N z last site/grid point index (dimensionless) Pe m (mass) Péclet number (dimensionless) R inner radius o the ixed bed (m) R & dimensionless source or sin term r lattice position (m) r& source or sin term (suitable units) s distribution unction (dimensionless) T dimensionless time t time (s) V & volumetric low rate (m 3 s ) v luid velocity (m s ) w weighting actors (dimensionless) Z dimensionless axial coordinate z axial coordinate o the ixed bed (m) Gree symbols ε ixed bed porosity (dimensionless) Φ dimensionless luid-phase concentration φ luid-phase concentration (suitable units) µ shape coeicient (dimensionless) θ solid-phase concentration (suitable units) τ relaxation time (s) Ω collision operator (s ) ω relaxation parameter (dimensionless) Subscripts and superscripts uilibrium distribution unction luid phase i intra-particle diusion coeicient/time in bed inlet lattice direction max maximum adsorption/extraction capacity re non-null reerence concentration s solid phase or lattice sound speed z axial coordinate o the ixed bed 0 central lattice site E-ISSN: Issue 3, Volume 7, July 0

9 orward (-D) lattice direction ( ) bacward (-D) lattice direction ( ) ~ seepage velocity ^ unit vector Acronyms BAC Biospeciic ainity chromatography D*Q* Lattice arrangements (arrays) FDM Finite-dierence method LBM Lattice-Boltzmann method SFE Supercritical luid extraction Reerences: [] Gersheneld N., The Nature o Mathematical Modeling, Cambridge University Press, 999. [] Datta A.K., Sablani S.S., Mathematical modeling techniques in ood and bioprocess: an overview. In: Handboo o Food and Bioprocess Modeling Techniques, Sablani S.S., Rahman M.S., Datta A.K., Mujumdar A.R. (editors), CRC Press, 007, pp. -. [3] Mohamad A.A., Applied Lattice Boltzmann Method or Transport Phenomena, Momentum, Heat and Mass Transer, University o Calgary, 007. [4] Romano V.R., Foreword, Journal o Food Engineering, vol. 7, 005, pp [5] Norton T., Sun D.-W., An overview o CFD applications in the ood industry. In: Computational Fluid Dynamics in Food Processing, Sun D.-W. (editor), CRC Press, 007, pp. -4. [6] Wang L., Sun D.-W., Recent developments in numerical modelling o heating and cooling processes in the ood industry - a review. Trends in Food Science and Technology, vol. 4, 003, pp [7] Wol-Gladrow D.A., Lattice-Gas Cellular Automata and Lattice Boltzmann Models: an Introduction, Springer, 000. [8] Succi S., The Lattice Boltzmann Equation or Fluid Dynamics and Beyond, Oxord University Press, 00. [9] van der Sman R.G.M., Lattice Boltzmann Simulation o Microstructures, In: Handboo o Food and Bioprocess Modeling Techniques, Sablani S.S., Rahman M.S., Datta A.K., Mujumdar A.R. (editors), CRC Press, 007, pp [0] Nield D.A., Bejan A., Convection in Porous Medium, Springer-Verlag, 999. [] Chase H.A., Prediction o the perormance o preparative ainity chromatography, Journal o Chromatography, vol. 97, 984, pp [] Cowan G.H., Gosling I.S., Laws J.F., Sweetehham W.P., Physical and mathematical modeling to aid scale-up o liquid chromatography, Journal o Chromatography, vol. 363, 986, pp [3] Sridhar P., Sastri N.V.S., Moda J.M., Muherjee A.K., Mathematical simulation o bioseparation in an ainity paced column, Chemical Engineering Technology, vol. 7, 994, pp [4] Kempe H., Axelsson A., Nilsson B., Zacchi G., Simulation o chromatographic processes applied to separation o proteins, Journal o Chromatography A, vol. 846, 999, pp. -. [5] Leict L., Månsson A., Ohlson S., Prediction o ainity and inetics in biomolecular interactions by ainity chromatography, Analytical Biochemistry, vol. 9, 00, pp [6] Özdural A.R., Alan A., Kerho P.J.A.M., Modeling chromatographic columns: nonuilibrium paced-bed adsorption with nonlinear adsorption isotherm, Journal o Chromatography A, vol. 04, 004, pp [7] Yun J., Lin D.-Q., Yao S.-J., Predictive modeling o protein adsorption along the bed height by taing into account the axial nonuniorm liquid dispersion and particle classiication in expanded beds, Journal o Chromatography A, vol. 095, 005, pp [8] Sovová H., Rate o the vegetable oil extraction with supercritical CO - I. Modeling o extraction curves, Chemical Engineering Science, vol. 49, 994, pp [9] Reverchon E., Mathematical modeling o supercritical extraction o sage oil, AIChE Journal, vol. 4, 996, pp [0] França L.F., Meireles M.A.A., Modeling the extraction o carotene and lipids rom pressed palm oil (Elaes guineensis) ibers using supercritical CO, Journal o Supercritical Fluids, vol. 8, 000, pp [] Wu W., Hou Y., Mathematical modeling o extraction o egg yol oil with supercritical CO, Journal o Supercritical Fluids, vol. 9, 00, pp [] Lucas S., Calvo M.P., García-Serna J., Palencia C., Cocero M.J., Two-parameter model or mass transer processes between solid matrixes and supercritical luids: analytical solution, Journal o Supercritical Fluids, vol. 4, 007, pp [3] Gaspar F., Lu T., Santos R., Al-Duri B., Modelling the extraction o essential oils with compressed carbon dioxide, Journal o E-ISSN: Issue 3, Volume 7, July 0

10 Supercritical Fluids, vol. 5, 003, pp [4] Qian Y.H., D Humières D., Lallemand P., Lattice BGK models or Navier-Stoes uation, Europhysics Letters, vol. 7, 99, pp [5] Patanar S.V., Numerical Heat Transer and Fluid Flow, Hemisphere, 980. [6] Ferziger J.H., Peric M., Computational Methods or Fluid Dynamics, Springer-Verlag, 00. E-ISSN: Issue 3, Volume 7, July 0