CRITICAL COMPARISON OF MASS TRANSFER MODELS FOR INTERNALLY CONTROLLED SCF EXTRACTION OF SOLID SUBSTRATES

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1 CRITICAL COMPARISON OF MASS TRANSFER MODELS FOR INTERNALLY CONTROLLED SCF EXTRACTION OF SOLID SUBSTRATES Joé Manul dl Vall,* & Edgar Uquich Pont. Univ. Católica d Chil, Avda. Vicuña Macknna 486, Macul, Santiago, Chil. Univ. d La Frontra, Avda. Francico Salazar 45, Tmuco, Chil. * Fax: ( dlvall@ing.uc.cl. Thr modl or th urcritical luid (SCF xtraction o olid with dirnt intrnal ma tranr mchanim wr critically comard in thi work. Intrnal ma tranr hyothi includd: tranint diuion; linar driving orc (LDF; and dortiondiolution-diuion (DDD. A nitivity analyi wa rormd on th bai o Biot numbr (Bi ratio btwn intrnal and xtrnal ma tranr ritanc and charactritic xtrnal xtraction tim (τ ratio btwn th xtrnal ma tranr ritanc and ridnc tim o th SCF in th xtractor. Th ngativ ct o a -ordr o magnitud incra in Bi ( in dcraing xtraction rat wa quivalnt to that o a on-ordr o magnitud incra in τ (.. Th LDF aroximation could b ud or th two othr modl undr analyi i th total comoundd oroity o th bd (ε and articl (ε wa conidrd, a modl-dndnt dinition o Bi wa utilizd (Bik R/D K or Fickan and LDF modl, Bi k R/D or DDD modl, and th valu o Bi wr <. Th LDF modl wa alid to litratur data on ntial oil xtraction rom lavndr lowr and nnyroyal lav with urcritical carbon dioxid (SC-CO at bar and 5 C. Analyi o intrtitial olvnt vlocity ct uggtd that th convctiv ma tranr coicint in th SCF i mallr than rdictd by dimnionl corrlation or ackd bd orating with SCF. INTRODUCTION Ma tranr aramtr drivd rom data gnratd in a laboratory or ilot lant unit can aid in th caling-u and dign o indutrial SCF xtraction roc or olid ubtrat []. Paramtr valuation, in turn, dnd on th imlmntation o aroriat ma tranr modl or ackd bd. Unortunatly, inc modl with dirnt hyothi about th limiting ma tranr mchanim can dcrib tyical cumulativ xtraction lot (rcovrd olut vru xtraction tim or botanical ubtrat tratd with SCF, it i diicult to dicriminat btwn modl bad on thir itting caabiliti or xrimntal data []. In thi work, w xandd a rviou contribution by conidring altrnativ intrnal ma tranr mchanim rood in cializd litratur. Hyoth includd Fickan [] or arabolic concntration roil o ridual olut in th olid matrix [], and a dortiondiolution-diuion mchanim [3]. MATHEMATICAL MODELS Fickan modl. Thi corrond to th gnral modl o dl Vall t al. []. A dirntial ma balanc quation wa writtn or th SCF urrounding hrical articl o olid ubtrat in a ackd bd (qn.. Th lux o olut tranrrd rom th olid to th SCF (J wa timatd uing quation, which aum a contant artition coicint o udo-olut (KC / C btwn th olid matrix and SCF. Equation 3 rrnt olut *

2 diuion within th olid articl, and inally, quation 4a- rrnt th initial and boundary condition o th ytm. ε + u J z ε ( 3 k C r R, J C R K ( C D + r (3 C (4a C z, t (4b C r, Co (4c r, D r R, k C r R, K C LDF modl. Ma balanc quation ali in thi ca alo. Howvr, whn th concntration roil o ridual olut in th olid matrix i aumd to b arabolic, dinition and dirntial quation 3 can b rlacd by quation 5 and 6, rctivly []. In thi ca only avrag olut concntration in th olid matrix ( C ar o intrt. Initial condition 4a and boundary condition 4b wr maintaind in thi ca, and initial/boundary condition 4c- wr rlacd by quation 7. 5 k DK C J C k R + 5DKR K (5 J (6 C Co (7 DDD modl. Thi modl wa dcribd by in dtail by Goto t al. [3]. Ma balanc quation ali, but a ditinction i mad in thi ca btwn th olut bound to th olid matrix (C and in it or (C, which ar rlatd by dortion kintic. Howvr, it wa aumd that quilibrium i tablihd intantanouly in th or du to rlativly at * dortion, which can b charactrizd by a contant artition coicint o olut (KC / C btwn th olid matrix and luid ha within th or. Undr th aumtion, dinition and dirntial quation 3 wr rlacd by quation 8 and 9, rctivly. Initial condition 4a and boundary condition 4b wr alo maintaind in thi ca, but initial/boundary condition 4c- wr rlacd by quation a-c. 3 k J ( C C (8 r R, R (4d (4

3 ε r, C C D r, D + K( ε o r R, k C + r ( C C r R, (9 (a (b (c SENSITIVITY ANALYSIS Th Fickan and LDF modl wr r-writtn in trm o a dimnionl tim [ (t u/h ], axial oition [ ξ (z/h ], radial oition [ δ (r/r ], and olut concntration in th SCF [ Y (K C /C o ] and olid ha [ X (C /C o ; X ( C /C o ]. On th on othr hand, dimnionl concntration or th DDD modl wr r-dind a: YC /C o, or th SCF ha; Y C /C o, or th luid ha within th or; and XC /C o, or th olid ha, whr C o K C o /K and K ε +K(-ε. Tabl ummariz th dimnionl dirntial ma balanc quation, initial condition, and boundary condition or th two ha and th thr modl. Clo xamination uggt that th olution o th dirntial quation in Tabl dnd on th artition o olut btwn th ha (K, K, bd and articl oroity (ε, ε and two dimnionl aramtr, namly: i Biot numbr (Bik R/D K or Fickan and LDF Bi k R/D or DDD modl, which rrnt th ratio btwn intrnal and xtrnal modl, ma tranr ritanc; and, ii charactritic xtrnal xtraction tim (τ u/k a H, whr a 3/R, which rrnt th ratio btwn th xtrnal ritanc to ma tranr and th ridnc tim o th SCF in th xtractor. A nitivity analyi wa rormd on th bai o Bi and τ, which i ummarizd in Figur or th LDF modl (th ba ca wa: K, ε.6, Bi, and τ.. Extraction rat incrad a a rult o a dcra in ithr Bi or τ, but th ct o a on-ordr o magnitud chang in τ (.- wa imilar to that o a -ordr o magnitud chang in Bi (-. dl Vall t al. [] tudid th ct o variation in K and ε by man o anothr dimnionl aramtr (Γε/(-εK that i rlatd to th artition o olut btwn th SCF and olid ha undr quilibrium condition, and concludd that a Γ incra (and th olut i mor tightly hld by th olid matrix, th amount o olut carrid out by th SCF dcra, thu incraing xtraction tim.

4 Tabl. Dimnionl dirntial ma balanc quation, initial condition, and boundary condition or th SCF ha and olid matrix ha or th Fickan, LDF and DDD modl. Modl SCF ha Solid ha / Por within olid ha ξ ε ε τ Fickan + ( X Y Y, ξ Y, ξ X + 3 K τbi δ δ δ X δ, ξ, δ δ δ, ξ, δ, ξ, Bi ( X Y δ, ξ, ξ, ξ ε 5 ε τ Bi + 5 LDF + ( X Y Y, ξ Y, ξ ε DDD + ( Y Y ξ Y, ξ Y, ξ ε τ τ X, ξ 5 Bi + 5 ( X Y Y τ 3 K Bi δ K X δ, ξ, K Y δ, ξ, K δ δ, ξ, Bi + δ δ ( Y Y δ, ξ, ξ, Figur comar rdiction o th Fickan, LDF and DDD modl or two combination o Bi and τ. Solut artition btwn th ha (K and total oroity (ε T ε+ε (-ε.6 wr kt contant in all ca. Two valu o articl oroity wr alo comard or th DDD modl (ε. and.375 that rultd in dirnt valu o bd oroity (ε.5 and.375, rctivly. Prdictd cumulativ xtraction lot wr virtually th am or th thr modl undr analyi or at xtraction (Fig. A, and mall dirnc wr obrvd or low xtraction (Fig. B. Th LDF aroximation wa inaroriat or <6. Thi i in agrmnt with Do & Ric [4], who howd that ridual radial olut concntration roil can b aumd to b arabolic in ha only whn /Biτ 3 (/Biτ 6 in Fig. B. On th othr hand, Goto t al. [3] uggtd that th LDF i aroriat only whn Bi<. Figur B alo uggt that xtraction rat imrov lightly a a rult o an incra in ε or long xtraction tim. It can b concludd that th LDF aroximation can b alid or th two othr modl undr analyi rovidd that th total oroity o th bd and articl (ε T i conidrd, that a modl-dndnt

5 ud, and that valu o Bi ar not too larg. To illutrat th ct o th dinition o Bi, an additional imulatd xtraction lot i includd in Figur A or th DDD modl and K, ε., ε.5, τ., and Bi (corronding to Bik R/D K. FITTING OF LITERATURE DATA Th u o th LDF modl or itting xrimntal cumulativ xtraction lot i illutratd in Figur 3 or lctd litratur data on ntial oil xtraction with SC-CO. Data corrond to tudi on th ct o olvnt ratio or th xtraction o camhor and nchon rom lavndr (Lavandula tocha ubci C. Boi lowr [5], and o ntial oil rom nnyroyal (Mntha ulgium L. lav [6]. Both t o xrimnt wr rormd with SC-CO at bar and 5 ºC. Partition aramtr (K wr timatd by lotting th ntial oil yild vru ciic olvnt conumtion or ach on o th two xrimntal t, and calculating th lo o th initial traight ortion [7]. Valu o K wr 7.6 or lavndr, and 6.6 or nnyroyal. W rocdd to timat bt-it valu o k, 5k D K/(k R+5D K, or ach condition. Dimnionl corrlation or th convctiv ma tranr coicint in th SCF ha (k hav th gnral orm: n.33 N a (N (N ( Sh R Sc whr N Sh (k R/D i th dimnionl Shrwood numbr, N R (ρur/µ, th dimnionl Rynold numbr, and N Sc (µ/ρd, th dimnionl Schmidt numbr. Th hyical rorti o th loadd SCF ha (ρ, µ, D wr timatd uing th rocdur rood by dl Vall t al. [8] uing PM885.4 g/mol and V c 3 cm 3 /mol or a tyical olut in lant ntial oil [7]. Whn th olvnt condition rmain unchangd, quation rduc to: n n- k a U R ( In a cond tag, bt-it valu o k or ach xrimnt wr ud to dtrmin bt it valu o a (.89, n (.8, and ubtrat-dndnt D (.5x -9 m / or lavndr, 3x -9 m / or nnyroyal. Valu o n in dimnionl corrlation or ma tranr coicint in ackd bd rang rom.6 [9] and.83 []. Bt-it valu o k timatd uing th aormntiond rocdur rangd x 6 m/, which ar about tim mallr than rdictd uing th corrlation o Tan t al. [], which ha bn uggtd or

6 th xtraction o vgtabl ubtrat with SC-CO in a ackd bd [8]. Modl itting wa obviouly wort or th data o Akgün t al. [5] than that o Ri-Vaco t al. [6] (c. Fig. 3. Th valu o Bi rangd rom.3 to. or lavndr lowr and rom.7 to.5 or nnyroyal lav, or which th LDF aroximation i adquat rgardl o th intrnal ma tranr mchanim. Acknowldgmnt Funding by Fondcyt (rojct rom Chil i gratly acknowldgd. REFERENCES [] DEL VALLE, J.M., NAPOLITANO, P., FUENTES, N., Ind. Eng. Chm. R., 39,,. 47 [] PEKER, H., SRINIVASAN, M.P., SMITH, J.M., McCOY, B.J., AIChE J., 38, 99,. 76 [3] GOTO, M., ROY, B.C., KODAMA, A., HIROSE, T., J. Chm. Eng. Jaan, 3, 998,. 7 [4] DO, D.D., RICE, R.G., AIChE J., 3, 986,. 49 [5] AKGÜN, M., AKGÜN, N.A., DINÇER, S., Ind. Eng. Chm. R. 39,,. 473 [6] REIS-VASCO, E.M.C., COELHO, J.A.P., PALAVRA, A.M.F., MARRONE, C., REVERCHON, E., Chm. Eng. Sci. 55,,. 97 [7] REVERCHON, E., MARRONE, C., Chm. Eng. Sci. 5, 997,. 34 [8] DEL VALLE, J.M., RIVERA, O., MATTEA, M., RUETSCH, L., DAGHERO, J., FLORES, A., J. Surcrit. Fluid (ubmittd [9] WAKAO, N., KAGUEI, S., Hat and Ma Tranr in Packd Bd, Gordon and Brach: Nw York, 98 [] TAN, C.-S., LIANG, S.-K., LIOU, D.-C., Chm. Eng. J. 38, 988,. 7