Jump condition at the boundary between a porous catalyst and a homogeneous fluid

Transcription

1 From the SelectedWork of Francico J. Valde-Parada 2005 Jump condition at the boundary between a porou catalyt and a homogeneou fluid Francico J. Valde-Parada J. Alberto Ochoa-Tapia Available at:

2 Proceeding of 4 th ICCHMT May 17 20, 2005, Pari-Cachan, FRANCE ICCHMT' JUMP CONDITION AT THE BOUNDARY BETWEEN A POROUS CATALYST AND A HOMOGENEOUS FLUID F.J. Valdé Parada and J.A. Ochoa Tapia* Univeridad Autónoma Metropolitana-Iztapalapa; Av. San Rafael Atlixco 186; Col. Vicentina; México D.F., C.P ; MEXICO *Correpondence author: Tel +(52) (55) ; Fax +(52) (55) jaot@xanum.uam.mx ABSTRACT The method of volume averaging (Whitaker, 1999) i ued to derive the jump condition between a porou catalyt and a homogeneou fluid. To implify the analyi a firt order kinetic i ued and the convective tranport i neglected. The tatement for the derivation i in term of the patially moothed equation obtained by Wood et al. (2000). We have followed a imilar procedure to the one ued by thoe author to derive, beide a jump boundary condition that couple the two media tranport, a cloure problem to predict the ective reaction rate coicient at the interregion dividing urface a a function of the volume fraction and the Thiele modulu. Preliminary etimation of the jump coicient obtained by an approximated olution of the cloure problem are preented. KEYWORDS Cloure problem; Effective reaction rate; Jump coicient; Jump condition; Micropore-macorpore; Volume averaging. NOMENCLATURE a a v av ω length of the bae of the rectangle of the unit cell for the inter-region, m. interfacial area per unit volume of the porou 1 medium everywhere m. interfacial area per unit volume of the porou 1 medium in the ω region m. molar concentration of pecie A in the phae, mol / m D diffuivity tenor valid everywhere, D Knuden diffuivity tenor, m 2 / D mixture molecular diffuivity, 3 m 2 / m 2 / K ective, reaction rate coicient for the interregion, m / k ective, reaction rate coicient for the micropore-region, m/ k patial deviation of the ective, reaction rate coicient for the micropore-region, m/ k area average of the ective reaction rate coicient in the micropore-region, m/ l characteritic length of the unit cell in the homogeneou region, m. n unit normal vector directed from the η toward the ω region ε volume fraction of the phae everywhere INTRODUCTION The ma tranfer between a porou catalyt and a homogeneou fluid ha been a ubject of interet for a long time. However when modeling thi proce many aumption are often made and not tated; for example, it i neceary to derive rigorouly the tranport equation together with jump condition for thi kind of problem. Whitaker (1983) ha already derived the average tranport equation for the porou medium and a cloure to predict the

3 ective diffuivity for thi problem. However, in hi development the reaction rate at the interregion wa aumed negligible compared with the reaction inide the porou medium, thu he impoed condition of continuity of concentration and flux. Thu, it i deirable to have more general jump condition, imilar to thoe found for the tranport between homogeneou phae (Whitaker, 1992), which in turn may be implified for a pecific cae. Recently, Wood et al. (2000) derived the jump condition for the proce of diffuion and reaction at a catalytic urface in contact with a homogeneou fluid. Alo, a cloure problem wa developed and olved to obtain value of the urface ective reaction rate coicient for a firt order irreverible reaction. In thi work we follow a imilar approach. METHODOLOGY The tarting point for the derivation i the tranport problem in term of the average equation and jump condition derived by Wood et al. (2000). The micropore homogeneou region i identified a the ω region and the homogeneou part of the macropore fluid i referred a the η region. By homogeneou ω and η region we mean thoe portion of the region that are not influenced by the rapid change in tructure that occur in the interregion neighborhood. The ω region i made up κ phae and the fluid of olid material ( ) ( phae) ; the fluid i the ame exiting in the macropore part of the ytem. If the averaging volume atifie the uual characteritic length contraint, the fluid fraction in the bulk of the micropore region ε ω change very lowly with poition. The initial tatement of the tranport problem i in term of the equation developed by Wood et al. (2000). Uing the method of volume averaging (Whitaker, 1999), it i poible to obtain a generalized ma tranfer equation without length-cale contraint and that i, a conequence, valid everywhere (i.e. in the ω and η region a well a in the interregion neighborhood), ε ( r) = i( ε ( r) D( r) i ) t k ( r ) (1) In eq. (1), r i the poition vector ued to indicate that the ective coicient are poition-dependent. In addition, it ha been hown elewhere (Pérez- Córdova et al., 1995) that, for thi type of ytem, the geometric propertie uch a the poroity and the uperficial area are mooth function at the interregion and can be approximated by the polynomial expreion of the poition relative to the dividing urface. Jump condition From eq. (1), it i poible to obtain average equation for the homogeneou region. At the interregion, the mimatch of length cale could become a problem and alo the length-cale contraint ued to obtain the average equation for the homogeneou region may fail, thu jump condition are needed. Baed on the continuity of the averaged concentration field it i poible to impoe the following condition, =, at the interregion. (2) η ω On the other hand, following a development imilar to that of Valencia-López et al. (2003) for the problem of diffuion and adorption in an analogou ytem, it ha been obtained that: ( c D A ) ( ε c η A ω) n i + n i D i = K, at the interregion. (3) ω Until thi point, the analyi i almot the repetition of the work preented by Valencia López et al. (2003): the jump condition coicient K would be an adjutable parameter. However, following Wood et al. (2000) a cloure problem to predict it ha been derived. The cloure and the calculation procedure are decribed in the Appendix. RESULTS The approximate problem wa olved uing a finite difference cheme to obtain the field. Then, the area average in eq. (5) were computed to obtain the reaction rate ective coicient, K. The procedure wa repeated until convergence on the ective coicient wa achieved taking a tolerance 5 of 110 x. In order to tet the ect of the approximation given by eq. (10) on the K reult, the complete problem wa alo olved and it wa found that the

4 difference wa le than 1%. Such mall difference mot probably i reult of the integration in eq. (11). In Figure 1, it i hown the normalized ective reaction rate coicient (jump coicient) a a function of the ize of the unit cell and the parameter φ, which i analogou to the macrocopic Thiele modulu, and defined by, 2 k av ωl φ =. (4) D The dependence of the reult on thi parameter a well a on the apect ratio i analogou to that exhibited by the ective reaction rate coicient k obtained by Wood et al. (2000), for the ( ) proce of diffuion and reaction at a catalytic urface. A explained in the appendix, the problem wa olved uing a rectangular repreentation of the porou medium (dahed line) a well a a periodic-quare repreentation (continuou line); the reult obtained with both repreentation are hown in Figure 1; note that even though the behavior i analogou, the error between the coicient value may be uperior to 50% for a given Thiele modulu. Thi ugget the neceity of uing unit cell with more quare and without the rectangle approximation.(creo que e major decir: Thi ugget that the rectangle approximation i only valid for apect ratio value uperior to 0.5 ) K k φ Figure 1: Effective coicient a a function of φ and the apect ratio a/l. CONCLUSIONS The jump condition for the proce of diffuion and reaction at the micropore-fluid interregion i derived in term of an ective coicient which i a function of a cloure variable, for whoe prediction a boundary value problem wa derived. The jump condition K coicient, ha been etimated uing an approximate olution of the cloure variable. K how a trong dependence on the geometry of the interregion, the Thiele modulu and on the characteritic of the unit cell choen. ACKNOWLEGMENT The author whih to thank CONACyT for the financial upport provided (U45160-Y). REFERENCES 1. Gray, W.G., A derivation of the equation for multiphae tranport. Chem. Eng. Sci., 30, pp Pérez Córdova H.; Ochoa Tapia J.A., 1995, Cálculo de variable y propiedade promedio en la región entre un fluido y un medio poroo a partir de valore puntuale, Av. en Ing. Quím., 5, pp Prat M., 1989, On the boundary condition at the macrocopic level, Tranp. in Porou Media, 4, pp Valencia-López J.J.; Epinoa-Parede G.; Ochoa- Tapia J.A., Ma tranfer jump condition at the boundary between a porou medium and a homogeneou fluid. J. of Porou Media, 6, 1, pp Whitaker S., 1983, Diffuion and reaction in a micropore-macropore model of a porou medium, Lat. Am. J. Chem. Eng. Appl. Chem., 13, pp Whitaker S., 1992, The pecie ma jump condition at a ingular urface, Chem. Eng. Sci. 47, pp Whitaker S., 1999, The method of volume averaging, Kluwer academic publiher. 8. Wood, B.D, Quintard M., Whitaker S., 2000, Jump condition at non-uniform boundarie: the catalytic urface, Chem. Eng. Sci., 55, pp Appendix Cloure problem In eq. (3), it wa introduced the following ective reaction rate coicient definition for the averaging cale

5 K k k = +, (5) Where the operator i repreent the area averaging, and i a cloure variable that can be obtained from the olution of the following boundary value problem defined in a unit cell characteritic of the interregion (ee Figure 1): 2 av ( r) D = k. (6) ε ( r) B.C.1 At A κ, nκ i =, (7) D B.C.2 At y = 2 h, = η, (8) B.C.3 At y = 0, = ω, (9) Alo periodicity boundary condition are impoed in the urface normal to the plane in which the dividing urface i contained. Eq. (6) i valid everywhere in the ytem and it i the bai to derive the cloure problem for the ω and η region can be developed. The interfacial boundary condition, eq. (7) i the reult of ubtituting Gray patial concentration decompoition (Gray, 1975) in the reaction boundary condition for the point concentration. The condition at the top and bottom urface of the unit cell, mut be found from the olution of the analogou problem for the unit cell totally contained in the homogeneou region. It ha been hown, that the olution of the cloure problem for the η region i η = 0. However, the ω field i not known a priori, thu it i neceary to derive it boundary value problem and olve it. The derivation i obtained from the equation for by fixing ε = εω ; the olution although poible reult very expenive in term of computer time. Thu, it i deirable an approximate problem for the interregion whoe olution i cloe to the one obtained with the exact ω field at y = 0. Approximate problem At thi point, the problem i not retricted to any particular geometry. In thi work, a periodic-quare model of the porou medium i choen (ee Figure 2) in order to make the problem more treatable. To etimate, it i neceary to chooe an adequate unit cell repreentative of the interregion. In Figure 2, k we how the ω region unit cell and everal model for the interregion unit cell; uch viualization i taken from Prat (1989). To reduce the computer time required to olve the exact unit cell we approximate the olid particle in the unit cell for the interregion a a rectangle intead of everal quare. The number of quare in the unit cell depend on how far from the interregion mut the homogeneou ω region be located. Thi iue will be expanded in the next ection. Unit cell for the inter-region Unit cell for the homogeneou ω-region Figure 2: Unit cell to olve the cloure problem. It i convenient to conider a change on the boundary condition given by eq. (9): The boundary identified at y = 0 eparate the homogeneou ω region from the interregion, whoe length ( 2h ) mut be large enough to account for the rapid change in propertie that do not occur in the homogeneou region, in thi ene, it i reaonable to think of continuity of the field. A a conequence of the above we hall impoe the continuity of the field derivative at y = 0. The cloure problem for the ω region ha been developed elewhere (Whitaker, 1999). We ue thoe reult to replace the boundary condition at y = 0, Eq. (9), by

6 At y = 0, = 0, (10) y Then, it can be hown that due to the ymmetry of the unit cell thi approximate problem can be olved in a half of the unit cell a the one hown in Figure 2. 4 t International Conference on Computational Heat and Ma Tranfer