Mass transprt with varying diffusin- and slubility cefficient thrugh a catalytic membrane layer Prceedings f Eurpean Cngress f Chemical Engineering (ECCE-6) Cpenhagen, 6-0 September 007 Mass transprt with varying diffusin- and slubility cefficient thrugh a catalytic membrane layer E. Nagy, a a Research Institute f Chemical and Prcess Engineering, University f Pannnia, H-800 Veszprém,Hungary Abstract The mass transprt, accmpanied by chemical reactin thrugh membrane reactr has been investigated in the case f varying diffusin cefficient and slubility cefficient. In the reality, bth parameters might depend n the cncentratin and/r n the inhmgeneity f the membrane layer. General mathematical mdels were develped t describe the mass transprt, taking int accunt the external mass transfer resistances as well, when the slubility cefficient can vary e.g. accrding t the Langmuir-Hinschelwd adsrptin thery r when the value f diffusin cefficient depends n the cncentratin/inhmgeneity in the membrane. A general slutin has been given that can be applied mst f the mathematical functins f the parameters mentined. The cncentratin distributin and the mass transfer rate will be given in clsed mathematical frms. The value f the mass transfer rates culd be strngly altered by the varying diffusin- and/r slubility cefficient. The mathematical mdel and the effect f the varying parameters has been discussed in this paper. Keywrds: catalytic membrane layer, dispersed catalyst particles, variable diffusin cefficient, variable slubility cefficient, nnlinear mass transfer;. Intrductin As a catalytic membrane reactr ne can use an intrinsically catalytic membrane ( e.g. zelite r metallic membranes) r a membrane that has been made catalytic by dispersin, impregnatin, etc. f catalytically active particles, as metallic cmplexes, metallic clusters r activated carbn, zelite particles, etc., thrughut the dense, plymeric- r inrganic membrane layers (Marcan and Tstsis, 00; Saracc et al., 999). A very imprtant task is t describe the mass transfer rate in rder t predict the cncentratin prfile in the feed phases r fr planning the measure f a membrane reactr. Recently, Nagy (007) develped mathematical mdels which
define the cncentratin distributin and the mass transfer rate in membrane layer with dispersed catalyst particles. Depending n the catalyst particle size, hetergeneus- (fr the case f micr-sized particles) and pseud-hmgeneus mdels (fr sub-micrn particles) were recmmended. Fr first-rder reactin it will be develped an analytical slutin fr the mass transprt. This mdel assumes cnstant diffusin in bth membrane layer and catalyst particles and cnstant slubility cefficient f the reactant in the catalyst particles. In the reality, bth parameters might depend n the cncentratin and/r n the inhmgeneity f the membrane layer. E.g., in the case f zelite catalyst particles r membrane, the diffusin f a single rganic cmpund r mixtures f hydrcarbns culd strngly vary with the cncentratin f cmpnent(s). The mass transprt culd be well described by the Maxwell-Stefan apprach cmbinatin with the Langmuir- Hinschelwd adsrptin thery (Krishna and Wesselingh, 997). The mass transprt f barely sluble liquid cmpnents can ften be described by the Flry-Huggins thery (Meuleman et al., 999) that gives als a strng cncentratin dependency f the diffusin cefficient. The slubility depends n the cncentratin especially in the case f inrganic membrane. Accrding t a Langmuir-istherm, the slubility cefficient gradually increases with the cncentratin up t a limiting value. These facts prve that the develpment f mathematical mdels shuld be imprtant fr the predictin f the effect f the varying parameters n the mass transprt. The mathematical mdels develped give the cncentratin distributins and the mass transfer rates in a membrane reactr in clsed mathematical frms. Tw imprtant cases are discussed here, separately: the effect f the varying slubility and that f the varying diffusin cefficient. The equatins develped give general slutin, they can be applied fr mntne functins f the value f slubility r any functin f the value f diffusin cefficient. E. Nagy. Theretical part The mass transprt thrugh catalytic membrane layer, with dispersed catalyst particles, can be given by means f hmgeneus and/r hetergeneus mdels. When the catalyst particles are enugh small, d p <<δ (d p <-3 µm, d p - particle size, δ- thickness f the membrane layer) then the pseud-hmgeneus mdel is recmmended. In this paper the hmgeneus mdel is applied t predict the mass transfer rate in the case f variable diffusin cefficient and slubility as well as accmpanied by first-rder, irreversible chemical reactin... Variable slubility cefficient Details f the pseud-hmgeneus mdel applied, is given in Nagy s paper (Nagy, 007), in the case f cnstant slubility and diffusivity. The cncave slubility (adsrptin r absrptin) istherm is apprached by means f tw linear istherms (Nagy, 999), fr the sake f an analytical slutin (Fig. A). The istherms have H and H slubility cnstant. Thus the cncentratin bundary
layer in the membrane is divided int tw sectins (0 t X and X t), namely, the sectin f 0 X X where the dimensinless cncentratin falls between and A as well as the sectin f X <X with cncentratin regime frm A t 0 (Fig. B). Thus the linear istherms can be generally expressed as fllws: A d H i (A A i ) + C i with i, () The slpes f the linear segments (H i ) and the pint f intersectin (C i ) between the tw linear segments characterize the mathematical equatin f the linear istherm. Thus, fr the secnd zne C i is equal t zer, C 0. The differential mass balance equatin fr the tw sectins in the membrane can be given as fllws: where d A 6ωε D i β p Ai Ci dx ε ( Hi[A ] + ) 0 D p Ha β p p R tanh ( ) Ha p and with i, () The value f β p gives the mass transfer cefficient int the catalyst particles (fr details see Nagy, 007), while Ha p is the dimensinless reactin rate cnstant (Rradius f particles, D p diffusin cefficient in the particle, k- reactin rate cnstant, ε- catalyst phase hldup). The bundary cnditins: If X0 then A Aa/a *, Xx/δ If X then A0 if XX then A X X A X X + Ha p k R D p E. Nagy if XX then da D dx X X da D dx X X + The slutin f the abve equatin system gives: A F i X G i X i e λ + i e λ + K i (3) After the slutin (details are nt given here), the mass transfer rate can be given as fllws: Dλ J δ ( K)( tanhλ tanhλ +λ / λ) + tanhλ +λ /( λ λ ) tanh K / csh λ 3
C K + ( ) K ( ) tanh λ tanhλ tanhλ λ + K tanhλ cshλ λ tanhλ +λ /( λ tanh λ) (4) (5) λ X 6β pεδ H ( ε) d p D λ H ( X) λ H X λ C K A X H Applying Eqs. (4) and (5), the mass transfer rate can be predicted as a functin f the reactin rate as well as with the different linear appraches f the riginal, curvature istherm. Fig A. Illustratin f the apprach f the curved istherm (cntinuus line) by means f tw linear istherms (dtted lines); the cncentratin distributin in the membrane layer is pltted in Fig. B... Variable diffusin cefficient The diffusin cefficient might vary as a functin f the cncentratin [DD(a)] and/r space crdinate [DD(x)]. Fr an analytical slutin f these systems, the membrane layer is divided int M sub-layers (M value must be high enugh due t the assumptin f cnstant diffusivity in these sub-layers, see Nagy, 007) with cnstant diffusivity. Thus, yu can get rdinary differential equatin system with cnstant parameters. This equatin system can be slved analytically. Fr the case f the cncentratin dependent diffusivity the cncentratin distributin in the membrane layer must be knwn in rder t determine the crrect values f D(a). This calculatin prcess needs 3-4 steps f iteratin. When the diffusin cefficient 4
depends n the space crdinate, the mass transfer rate at the membrane interface can be expressed withut knwing the cncentratin distributin in the catalytic membrane layer. The cncentratin distributin and the mass transfer rates are expressed by clsed, explicit mathematical equatins fr pseud first-rder, irreversible chemical reactin. E. Nagy Fig. B. Illustratin f the cncentratin distributin in the tw znes f the membrane layer Fr ne-dimensinal transprt in rectangular crdinates the general mlar species cntinuity equatin is: d dx D da dx Q 0 (6) In general case the diffusin cefficient and the cnvective velcity can depend n the space crdinate, thus DD(x), DD(a) r DD(a,x). As a surce term, Q we cnsider first-rder kinetics, Qk a (very easy t cnsider the zer rder reactin as well, but nt discussed here). In the bundary cnditins the external mass transfer resistance shuld be taken int accunt. The external bundary cnditins are as fllws: if x0 then β * af af D da dx x 0 + (7) 5
if xδ then β * af af DM dam dx xδ Fr the slutin the membrane shuld be divided M sub-layer, in the directin f the mass transprt, that is perpendicular t the membrane interface, with thickness f δ ( δδ/m). In each sectin the parameters D m, can be regarded as cnstant parameter (m dentes the m th sub-layer perpendicular t the interface). Fr the m th sub-layer f the membrane layer, using dimensinless quantities, it can be given (the value f λ is equal t the well knwn Hatta-number, λ Ha ) : where d A m dx λ m A m 0 (8) λ m k δ D m The slutin f Eq. (8) is as fllws: A m ( λ ) ( λ ) T m X P m X m e + m e with X m- < X < X m (9) T m and P m are parameters can be determined by means f the bundary cnditins fr the m th sub-layer (with m M). The bundary cnditins at the internal interfaces f the sub-layers ( m M-; X m m X) can be btained by the fllwing, well knwn equatins: da m Dm+ dam+ dx Dm dx at XX m (0a) H m A m A m+ at XX m (0b) The mass transfer rate n the upstream side f the membrane can be given: with J ( T P ) λ D () 6
F r F M r X M 0 e λ + δ Q * M n E M H * H n E T M F ( Θ+ tanh( λ X) ) M + Θ tanh( λ X) + n E M H * and Θ H * 0 λsh ( +Θ) csh () ( Θ) T P r0 (3) +Θ with r 0 AfH * 0 where Ei Fi Fi tanh( M i X) i E λ + + fr i,3,,m n M + i * HM+ i tanh Fi M i i E + + fr i,3,,m- n M + i * HM+ i with and as well as where with E tanh F M ShMλ + M tanh H * M ShMλ + M H * M M Q Ei i Bi + Ci fr i,3,,m- Fi Ci Bi rmi Qi sinh M+ i fr i,3,,m- rmi + Qi csh M+ i fr i,3,,m- 7
C B r M rδ sinh r M + rδ csh M M with r δ A f H * M D i λ n i i ; D i + λi+ H * i H i (H M+ H δ ) fr i,,3,,m- H i + Applying the abve equatins, the mass transfer rate can be btained easily using Eq. (). The cncentratin distributin can be simply btained frm the bundary cnditins (0a) and (0b) in the knwledge f the T and P values. Abut the slutin details will be given elsewhere. The mass transfer rate shuld be replaced int the differential mass balance equatin given fr the tube (r shell) side f the membrane layer and the effect f the cncentratin dependency f the parameters n the utlet cncentratin can be simulated. 3. Results and discussin 3.. Variable slubility cefficient Fig. cntains tw nnlinear slubility curves (curves and ), each f them were apprached by tw linear istherms given by dtted lines. The expressins f the linear istherms e. g. fr the mre curvature curve are as fllws: A d 0.64(A- A )+0.77 fr the regime f X X as well as A d 5.A fr 0 X X. The value f A in this case is equal t 0.5. The essential f the calculatin methdlgy, that yu shuld vary the value f X until the Eq. (5) is fulfilled in rder t get the value f X. Then ne can calculate the mass transfer rate, J, by means f Eq. (4). Typical curves illustrate the effect f the nnlinearity n the mass transfer rate in Fig. 3. The slubility curve with higher curvature has much higher value f the enhancement in the mass transfer (curve in Fig. 3). (E gives the mass transfer rate related t that f physical mass transfer). The curve dented by lin in Fig. 3. gives the enhancement with linear slubility istherm (straight line, curve 3, in Fig. ). Using a single linear slubility fr the curve 3 in Fig., ne can simply get that H0.9 (HA d /A at X). With increasing reactin rate the E value increases mre significantly than the E lin value. These results prve that the curvature f the slubility istherms culd have significant influence n the mass transfer rate int the catalytic membrane layer. It will be demnstrated the effect f the curvature n the mass transfer rate f hydrcarbns int zelite membrane as a functin f the reactin rate. 8
Fig.. Different curved slubility curves and their appraching by tw linear istherms 4.E-0.E-0.E+00 Hatta-szám, Fig. 3 Enhancement f the mass transfer rate as a functin f the reactin rate applying the slubility curves given in Fig.. Parameter values used fr calculatin: DD p x0-0 m /s, δ00 µm, ε0., fr the linear istherm: H0.9) 3.. Variable diffusin cefficient The value f T m and P m were determined by means f the bundary cnditins btained fr the internal and external interfaces. Accrding t the Maxwell-Stefan, the Flry-Huggins theries r the Vignes equatins (Bitter, 99), there can be given several different mathematical functins between cncentratin and diffusin cefficient. As illustratins, it is shwn here sme typical examples hw the varying 9
diffusin cefficient can alter the cncentratin distributin and the mass transfer rate in the catalytic membrane layer. Fr the sake f simplicity a linear change f the D(a) value was assumed [DD (±Km/M)] in these examples. Tw typical figures illustrate the change f the cncentratin in the membrane layer when the diffusin cefficient increases with increasing cncentratin (Fig. 4a) r when it decreases with increasing cncentratin (Fig. 4b). The tendency f curves is well knwn, thus, their detailed discussin here is nt necessary. Fr physical mass transprt, the mass transfer rate and/r the cncentratin distributin can be calculated by means a simpler equatin, namely: A m Tmm X+ Pm with m,,,m (6) A T f m (7) D m M + + X D Sh D M Sh M j D j E. Nagy Pm T m + + T j X T m (m) X Sh (8) j.0 0.8 membrane thickness,- cncentratin,- 0.6 0.4 0. 0.0 0.0 0. 0.4 0.6 0.8.0 0
Fig. 4a. Cncentratin distributin in the membrane withut chemical reactin (Ha0, A 0, β β ) f.0 0.8 membrane thickness,- cncentratin,- 0.6 0.4 0. 0.0 0.0 0. 0.4 0.6 0.8.0 Fig. 4b Cncentratin distributin in the membrane withut chemical reactin (Ha0, A 0, β β ) f Hw the chemical reactin alters the enhancement is pltted in Fig. 5, in the case when the diffusin cefficient increases with the cncentratin. The value f K parameter changes between 0 and 0. The dtted line gives the mass transfer rate with cnstant diffusin cefficient. The actual mass transfer rate is related t that btained by D cnstant diffusin cefficient and withut chemical reactin. The mass transfer rate gradually increases with the Hatta-number. As can be seen the varying diffusin cefficient has strng effect n the mass transfer rate at even at higher value f Ha. enhancement in mass transfer,-.0 9.0 7.0 5.0 3.0.0 0.0.0.0 3.0 4.0 5.0 Hatta number,- Fig. 5. The effect f the chemical reactin rate n the mass transfer rate related t that withut chemical reactin and cnstant D diffusin cefficient in the catalytic
membrane layer. The value f diffusin cefficient increases with the cncentratin [DD (+Km/M), A 0, β β ] f 4. Cnclusin Relatively simple mathematical equatins are presented in rder t predict the mass transprt in membrane reactr in the case f nnlinear mass transfer. Clsed equatins were develped t calculate the effect f varying slubility and/r diffusin cefficient. Figures presented prve strng effect f the cncentratin dependency f the diffusin cefficient and slubility. Acknwledgement This wrk was supprted by the Hungarian research Fundatin under Grants OTKA 6365/006 Ntatin a cncentratin, ml/m 3 a f cncentratin in the bulk feed phase, ml/m 3 a * f cncentratin n the membrane interface n the feed side, ml/m 3 A dimensinless cncentratin, (a/a f) A d cncentratin in particles, ml/m 3 A f external bulk phase cncentratin in the feed side, ml/m 3 D diffusin cefficient, m /s D p diffusin cefficient in particles, m /s d p particle size, m H slubility cefficient, - H * m H m /H m+ k reactin rate cnstant, /sec M number f sub-layer with cnstant parameters in the membrane layer, m R particle radius, m Sh D /( δβ ) Sh M D M /( δβ ) Greek letters β ε mass transfer cefficient, m/s hldup
ω specific area f the catalyst particles in the membrane, m /m 3 Indices feed phase dwn stream phase References Bitter, J.G.A., Transprt mechanisms in membrane separatin prcesses, 99, Amsterdam Krishna, R., J.A. Wesselingh, 997, Chem. Eng. Sci., 5, 86-9. Markan, J.G.S., Tstsis, T.T., Catalytic Membranes and Membrane Reactrs, Wiley-VCH, (00). Meuleman, K., B.B. Bsch, M.H.V. Mulder, H. Stratmann, 999, AIChE J., 45, 53-60 Saracc, G. Nemagus, H.W.J.P., Versteeg, G.F., van Swaaij, W.P.M., (999), Chem. Eng. Sci. 54 997-07 Nagy E., (007), Industrial & Engineering Chemistry Research, 46, 95-306. Nagy E., (999), Chem. Eng. Jurnal, 7, 43-5 3