Cake ltration analysis the eect of the relationship between the pore liquid pressure and the cake compressive stress

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1 Chemical Engineering Science 56 (21) Cake ltration analyi the eect of the relationhip between the pore liquid preure and the cake compreive tre C. Tien, S. K. Teoh, R. B. H. Tan Department of Chemical and Environmental Engineering, National Univerity of Singapore, 1 Kent Ridge Crecent, Singapore 11926, Singapore Received 27 February 21; received in revied form 18 May 21; accepted 28 May 21 Abtract The relationhip between the pore liquid preure (p L ) and the cake compreive tre (p ) i required in analyzing cake ltration. Thi relationhip (for the one-dimenional cae) i commonly aumed to be dp L +dp = although imilar but dierent one may alo be advanced from the multiphae ow theory. The eect of thee dierent expreion i tudied by examining the correpondence between cake ltration data and the compreion permeation cell meaurement of two pecic ytem.? 21 Elevier Science Ltd. All right reerved. Keyword: Cake ltration; Contitutive relationhip; Compreive tage; Pore liquid preure; Specic cake reitance; Slurry 1. Introduction In cake ltration operation, liquid olid upenion ow through media which allow the paage of the upending liquid but retain particle at the uptream ide of the media. In mot cae, lter cake exhibit compreive behavior, namely, they become more compact a the cake compreive tre increae. On the other hand, liquid ow rate through lter cake depend upon liquid preure gradient in addition to other variable. Analyi of cake ltration therefore require the knowledge between the liquid preure within the cake and the cake compreive tre. Thi information i alo needed in applying cake propertie obtained from the compreion permeability (C P) cell meaurement for the prediction of cake ltration performance. The origin of the cake compreive tre may be explained a follow (Walker, Lewi, McAdam, & Gilliland, 1937): The ow of liquid through a lter cake impart uid drag on particle contituting the cake. Since thee particle are contiguou, the drag force experienced by individual particle are tranmitted and accumulated along the direction of the liquid ow, giving Correponding author. Tel.: ; fax: addre: chetanbh@nu.edu.g (R. B. H. Tan). rie to a compreive tre in the cake phae. Baed on intuitive reaoning, Ruth (1935) and later Tiller (1953) propoed for the one-dimenional cae, dp L +dp =; (1) where p L and p are the pore liquid preure and the compreive tre. According to Eq. (1), the um of p L and p i contant acro a cake. Thi relationhip ha been widely ued in ltration analyi for the lat three decade. The relationhip between p L and p can be etablihed, in principle, by applying the multi-phae ow theory and carrying out volume averaging the continuity equation (momentum) of the liquid and particle phae. Depending upon the aumption and procedure ued in volume averaging, the following relationhip between p L and p have been obtained (Willi, Shen, & Gray, 1974; Rietema, 1982). (1 )dp L +dp =; (2) (1 )dp L + dp =; (3) d[(1 )p L ]+d[ p ]=: (4) For convenience in ubequent dicuion, Eq. (1), (2), (3) and (4) will be referred to a cae (1), (2), (3) and (4), repectively, in the following ection /1/$ - ee front matter? 21 Elevier Science Ltd. All right reerved. PII: S 9-259(1)263-9

2 5362 C. Tien et al. / Chemical Engineering Science 56 (21) Fig. 1. Repreentation of one-dimenional cake ltration. A natural quetion ariing from thi multiplicity of the relationhip i it eect on the analyi and prediction of cake ltration. One may ak which one of the expreion hould be ued in applying C P cell reult to ltration calculation? Such a quetion wa raied a quarter of a century ago (Willi et al., 1974). It i omewhat a urprie that thi line of inquiry ha not been further purued ince. The purpoe of the preent work i to examine the eect of thee dierent expreion on cake ltration analyi through comparion of reult of ltration experiment and C P cell meaurement of two pecic ytem. For the purpoe of interpreting the ltration data, the conventional ltration equation will be re-derived in a manner o that the eect of the p L p relationhip i manifeted explicitly Explicit expreion of p L We will rt derive expreion of p L a a function of p for the one-dimenional cae which i commonly encountered in cake ltration. A chematic repreentation i given in Fig. 1. The x-coordinate tart from the medium urface and x = L(t) correpond to the cake upenion interface, and L, the cake thickne. The upenion i at a preure of P and the preure at the downtream ide of the medium i aumed to be zero. If the preure drop acro the cake and medium are p c and p m one ha p c +p m = P : (5) At the cake upenion interface, the cake i free of compreion and p = at x = L(t); (6) p reache a maximum at the cake media interface, or p = p m at x =: (7) From Eq. (1) (7), p L i found to be Cae (1) (i.e. Eq. (1)): p L = P p ; (7a) p m = P p m = P if p m i inignicant: (7b) Cae (2) (i.e. Eq. (2)): p L = P pm p dp 1 ; (8a) dp 1 = P p m = P if p m i inignicant: Cae (3) (i.e. Eq. (3)): p L = P pm p (8b) 1 dp ; (9a) 1 dp = P p m = P if p m i inignicant: Cae (4) (i.e. Eq. (4)): (9b) p L = 1 1 P 1 p ; (1a)

3 C. Tien et al. / Chemical Engineering Science 56 (21) p m = 1 m P 1 m p m 1 = P m 1 m if p m i inignicant; (1b) where i the cake olidoity at the zero tre tate and m i the value of at p = p m. From thee expreion, with the knowledge of v. p (i.e. the contitutive relationhip), the relationhip of p L v. p acro a cake can be readily determined Parabolic law of contant-preure ltration The conventional cake ltration theory i baed on the following aumption: (1) The olid (particle) velocity i negligible. (2) The moving boundary eect of the upenion cake interface can be ignored. (3) Darcy law can be ued to etimate liquid ow rate. (4) The relevant cake propertie, i.e. the olidoity and permeability are function of the compreive tre only. The ma continuity equation of the liquid and olid phae are ( l = (11a) (11b) where q l and q denote the liquid and olid upercial velocitie. i the cake poroity and equal to 1. The um of the above two expreion ( + )= and q l + q =(q l ) m = contant; (12) where (q l ) m i the liquid permeation velocity through the medium. With aumption (1), q l i contant acro the cake and equal to the rate of permeation (i.e. ltration rate). If v i the cumulative of ltrate per unit medium area, on account of aumption (1), one ha q l = dv dt = : (13) The above expreion may be rearranged to give dv dt dx = k dp L : (14) Integrating the above equation over the entire cake thickne and noting that (dv=dt) i contant acro the cake and and k being function of p only, one ha dv L pm dx = (k )( f )dp ; (15) dt where f = dp L (16) dp and i the particle denity. The rt integral of Eq. (15) give the ma of cake olid per unit medium area, w. By denition L w = dx = v 1 m ; (17) where i the particle ma fraction of the upenion and m i the overall ma ratio of wet cake to dry cake, which can be expreed a L m =1+ (1 ) dx L : (18) dx Subtituting Eq. (17) into Eq. (15), one ha dv dt = 1 m pm (k )( f )dp : (19) v For the ow of liquid through the medium, one ha dv dt = p m ; (2) R m where R m i the medium reitance. From Eq. (19) and (2) and uing the relation of Eq. (5), one ha P =p c +p m = v dv dt 1 m p c dv pm + R (k )( f m )dp dt : (21) If one dene an average pecic cake reitance, av over compreive tre ranging from to P m to be [p c =p m ] [ av ] pm = [ p m : (22) (k )( f )dp ]=p m Eq. (21) may be rewritten a ( ) dv P = v dt [1 m] [ dv av] pm + R m dt : (23) Further integration with time give v(t) P t = [1 m] [ av] pm v dv + R m v = { [av ] pm 2 1 m } v 2 + R m v; (24)

4 5364 C. Tien et al. / Chemical Engineering Science 56 (21) where { } ( [av ] v pm = 2 1 m )/ [ av ] pm [1 m] v dv v 2 : (25) It hould be noted that a m and av correpond to the average value of the ma ratio of wet to dry cake and the pecic cake reitance dened by Eq. (22), thu {[ av ] pm =(1 m)} i the average of a quantity, which, in turn, i compoed of average quantitie according to Eq. (18), (22) and (25). Intuitively, one may aume that { } [av ] pm = [ av] pm (26) 1 m (1 m) with the value of the right hand ide evaluated according to the nal cake thickne. With thi approximity, Eq. (24) become P t = [ av ] pm v 2 (1 m) 2 + R mv: (27) Furthermore, for cae (1), and with negligible media reitance, dp L +dp =;f = 1; p m = P, Eq. (24) become P t = [ av] P 2(1 m) v2 (28) and Eq. (22) become P [ av ] P = P : (29) (k )dp Eq. (28) and (29) contitute the o-called parabolic law of contant preure ltration according to the conventional cake ltration theory. Eq. (28) give the ltration volume v. time relationhip. It alo etablihe the linear t=v v. v plot, commonly ued in determining the contitutive relationhip, i.e. the pecic cake reitance a a function of the cake compreive tre. The imple derivation given above how that there exit a general expreion between v and t irrepective of the relationhip between p L and p. On the other hand, Eq. (22) alo reveal that the relationhip between p L and p i required if the reult obtained from the C P cell meaurement are to be applied for predicting cake ltration performance. The p L p relationhip i eential to the development of the contitutive relationhip from ltration experimental reult. Thi latter point will be dicued in the following ection. 2. Experiment 2.1. Apparatu and material The experimental reult ued in examining the eect of the relationhip between p L and p are ltration data (v v. t and L v. t) and C P cell data ( v. p and v. p ). Thee data were obtained uing our newly developed multifunctional tet cell which, in eence, conit of a cylindrical chamber attached to a computer-controlled preciion univeral tet machine (Shimadzu Autograph AGS Serie) and ha the capability of functioning a both a variable-volume ltration chamber and a C P cell. A chematic diagram of the multifunctional cell i hown in Fig. 2. The internal diameter of thi cell i 7:5 cm. Detailed information about the cell can be found elewhere (Tan, He, & Tien, 1999; Teoh, Tan, He, & Tien, 21). Two type of upenion; CaCO 3 H 2 O and Kaolin H 2 O were ued in the experimental work. Kaolin and CaCO 3 powder were obtained from Merck, Singapore and Fiher Scientic, Singapore, repectively. The tet upenion were prepared by uing de-ionized water. Table 1lit the condition under which the experiment were conducted. 3. Reult 3.1. C P cell reult In C P cell meaurement, uniformly packed cake were formed under pecied load. The olidoity of the cake were calculated from the amount of the olid particle ued to form the cake and the cake thickne. The cake permeability wa then determined by carrying out the claical ow rate-preure drop meaurement. A typical et of reult of a particular cake (CaCO 3 ), i.e. =1 v. p and [=( k ) 1 ] v. p i hown in Fig. 3. The reult of v. p and v. p are often expreed by the power-law equation (Carman, 1938; Shirato, Murae, Kato, & Shigeo, 197; Tiller & Leu, 198) or [ = 1+ p ] ; (3a) P a [ =[ k ] 1 = 1+ p ] n ; (3b) P a where ;P a ;;nand are empirical contant although and can be interpreted a the value of and at the zero compreion tate. With and o expreed, the cake permeability become k = k (1 + p =P a ) with k =( ) 1 and = + n. Both the CaCO 3 and Kaolin reult of and were tted with the power-law equation. The repective value of the tting parameter are given in Table 2. The degree of tting can be een from Fig Filtration reult The ltration experiment were conducted under the condition of contant preure. The reult obtained are v

5 C. Tien et al. / Chemical Engineering Science 56 (21) Fig. 2. Schematic diagram of the newly developed multifunctional tet cell: (a) general arrangement, (b) a a C P cell, (c) a a variable chamber ltration cell.

6 5366 C. Tien et al. / Chemical Engineering Science 56 (21) Table 1 Experimental condition ued in ltration experiment Sytem Variable CaCO 3 Kaolin P (Pa) 1: 1 5 7:7 1 5 : : 1 5 (kg=m 3 ) (kg=kg).2.5 Filtration time () Up to 253 Up to 37,23 Final cake thickne.2.2 (m) Fig. 4. Plot of t=v v. v of CaCO 3. Reult hown in Fig. 4. (o) ltration data; ( ) t line 1; ( ) t line 2. Fig. 3. Contitutive relationhip obtained from C P cell meaurement for CaCO 3 cake. 1 v. p ; ( ) C P data; ( ) tting with Eq. (3a), (3b) v. p ; (o) C P data; ( ) tting with Eq. (3b). Table 2 Value of the parameter of the contitutive equation of Eq. (3a) and (3b) from C P cell meaurement Quantity CaCO 3 cake Kaolin cake (dimenionle) n (dimenionle) P a (Pa) 4: :7 1 4 (dimenionle).2.34 (m=kg) 3: : v. t and the time when the cake thickne reached a particular value. Thi time wa determined in the following manner. When the multifunction cell wa ued for ltration experiment, the poition of the upper piton (ee Fig. 2c) wa xed, thereby encloing it and the lower piton, a chamber of known volume. The tet upenion entered into the chamber through an orice at the center of the piton (ee Fig. 2c). Initially, there wa ucient pace between the upper piton and the cake urface for the entering upenion to be ditributed over the entire cake urface. However, a the cake thickne approache the chamber height, upenion ditribution are hindered and the reduction of the eective ltration area lead to a udden reduction of the ltration rate (Murae, Iritani, Cho, Nakanomori, & Shirato, 1987). From the occurrence of thi udden reduction in ltration, the time wa determined. To illutrate the procedure, the reult of t=v v. v, of one typical experiment (ltration of CaCO 3 upenion at P =7:7 1 5 Pa) are hown in Fig. 4. A in contant preure ltration, v decreae with t but a more ignificant reduction occur at t 7. A hown in Fig. 4, the reult are repreented by two linear egment. The interecting point of thee two egment give the value of the time when L equal the chamber height Aeing the eect of the p L p relationhip Reult of p L p Once the information of v. p i known, the relationhip between p L and p and the maximum value of p m (p at the cake medium interface) can be readily determined from the variou equation given before [i.e. Eq. (7a) (1b)]. The reult are hown in Fig. 5 and 6. In Fig. 5, the relationhip of p L v. p for P equal to and Pa of CaCO 3 cake and Kaolin cake are hown and Fig. 6 give the reult of p m v. P.Iti clear that the numerical value of p m can be either greater or le than the preure drop acro the cake, depending upon the p L p relationhip ued Correpondence between the pecic cake reitance determined from the ltration data and the C P-cell meaurement reult. A a general expreion relating v and t, Eq. (27) can be ued to determine [ av ] from ltration experimental data following the conventional procedure baed on the t=v v. v plot. On the other hand, if the contitutive relationhip of v. p and v. p are known, [ av ] can be etimated according to Eq. (22). By comparing thee [ av ] value, the more appropriate relationhip between p L and p, applicable to cake ltration analyi may be identied.

7 C. Tien et al. / Chemical Engineering Science 56 (21) Fig. 5. Reult of p L v. p of lter cake according to ( ) Eq. (1); ( ) Eq. (2); ( ) Eq. (3); ( ) Eq. (4). (a) CaCO 3 cake, (b) Kaolin cake, at (o): P =2 1 5 Pa and () Pa. Such comparion are hown in Fig. 7a (CaCO 3 cake) and b (Kaolin cake). It i clear that for the ame contitutive relationhip, the calculated [ av ] varie depending upon the p L p relationhip (i.e., f ) ued. In Fig. 7, [ av ] obtained from contant-preure ltration data of CaCO 3 cake wa found to agree with the calculated [ av ] with a p L p relationhip of Cae 2. On the other hand, for Kaolin cake, the bet agreement wa found if the p L p relationhip i of Cae 3. Becaue of the relatively mall amount of data we have preently, the quetion concerning a proper election of the p L p relationhip to be applied in cake ltration analyi cannot be anwered at thi time. Neverthele, ome peculation may be oered if one conider that the cake compreive tre originate from the drag force exerted on individual particle contituting a cake due to the relative motion between the cake olid and pore liquid, the cake tructure may play a role in the magnitude of p. On the other hand, a hown in Fig. 5a and b, the relationhip between p L and p i approximately linear ince the variation of acro a cake i not, even for compreible cake, very ignicant. Thu a an approximation, one may aume dp L dp = f = A: Fig. 6. p m v. P according to ( ) Eq. (1); ( ) Eq. (2); ( ) Eq. (3); ( ) Eq. (4). (a) CaCO 3 cake, (b) Kaolin cake. A may be determined on the bai of bet agreement between the calculated [ av ] according to Eq. (22) and [ av ] determined from ltration data and Eq. (28). Once the value of A i known for a ucient number of dierent type of cake, attempt can then be made to correlate A with certain cake characteritic uch a the cake compreibility or poroity, which may be repreented, repectively, by the exponent n of Eq. (3b) or of Eq. (3a). It i our plan to purue uch a tudy in the near future. 4. Concluion The knowledge of the relationhip between the pore liquid preure, p L and the cake phae compreive tre, p i eential to the analyi of cake ltration and in applying C P cell meaurement for the prediction of cake ltration performance. Unfortunately, thi relationhip i not known preently although everal poible one have been advanced. The conventional parabolic law of contant preure ltration wa derived on the bai of a particular type of p L p relationhip, namely, p L + p = contant or Eq. (1). In the preent, we demontrate that thi parabolic law can be extended to other type of p L p relationhip by dening a generalized pecic

8 5368 C. Tien et al. / Chemical Engineering Science 56 (21) P p m upenion preure olid compreive tre at the cake medium interface q l liquid upercial velocity q olid upercial velocity (q l ) m liquid permeation velocity through the medium R m medium reitance olid particle ma fraction in the upenion t time v cumulative ltrate volume per unit medium area w ma of cake olid per unit medium area x ditance from medium urface Greek letter Fig. 7. Comparion of experimental determined av and etimate value baed on dierent p L p relationhip. (a) CaCO 3 cake, (b) Kaolin cake. (o) ltration data; ( ) cae 1; ( ) cae 2; ( ) cae 3; ( ) cae 4. cake reitance. Thu, the commonly ued cale-up procedure baed on the parabolic law remain valid provided it limitation are properly recognized. On the other hand, knowledge of the p L p relationhip i required in the etimation of the average pecic cake reitance from the cake contitutive equation. For two type of cake, the degree of agreement between [ av ] calculated from the cake contitutive equation and thoe determined from cake ltration experiment vary ignificantly with the p L p relationhip ued. The data we have at the preent time are not ucient to render any concluion. A poible approach of olving thi uncertainty i uggeted. Notation k k L m p L p P a ;n cake permeability cake permeability at zero tre cake thickne overall ma ratio of wet cake to dry cake pore liquid preure olid compreive tre empirical contant cake poroity cake olidoity cake olidoity at zero tre m cake olidoity at cake media interface cake pecic reitance cake pecic reitance at zero tre av average cake pecic reitance [ av ] average quantity of average cake pecic reitance empirical contant liquid vicoity olid particle denity liquid denity p c preure drop acro cake p m preure drop acro medium Acknowledgement The author would like to thank The National Science & Technology Board (NSTB) for funding thi tudy. Reference Carman, P. C. (1938). Fundamental principle of indutrial ltration (a critical review of preent knowledge). Tranaction-Intitution of Chemical Engineer, 16, Murae, T., Iritani, E., Cho, J. H., Nakanomori, S., & Shirato, M. (1987). Determination of ltration characteritic due to udden reduction in ltration area of lter cake urface. Journal of Chemical Engineering Japan, 2(3), Rietema, K. (1982). Science and technology of dipered two-phae ytem I and II. Chemical Engineering Science, 37, Ruth, B. F. (1935). Derivation of general ltration equation. Indutrial Engineering Chemitry, 27, Shirato, M., Murae, T., Kato, H., & Shigeo, F. (197). Fundamental analyi for expreion under contant preure. Filtration & Separation, 7, 277. Tan, R. B. H., He, D., & Tien, C. (1999). Multifunction tet cell for cake ltration tudie. Singapore Patent No

9 C. Tien et al. / Chemical Engineering Science 56 (21) Teoh, S. K., Tan, R. B. H., He, D., & Tien, C. (21). A multifunction tet cell for cake ltration tudie. Tranaction of the Filtration Society, 1, Tiller, F. M. (1953). The role of poroity in ltration I: numerical method. Chemical Engineering Progre, 49, Tiller, F. M., & Leu, W. F. (198). Baic data tting in ltration. Journal of Chinee Intitute of Chemical Engineer, 11, Walker, W. H., Lewi, W. K., McAdam, W. H., & Gilliland, E. R. (1937). Principle of Chemical Engineering (3rd ed.). New York: McGraw-Hill. Willi, M. S., Shen, M., & Gray, K. J. (1974). Invetigation of the fundamental aumption relating compreion permeability data with ltration. Canadian Journal of Chemical Engineering, 52,