Liquid hold-up correlations for trickle beds without gas flow

Transcription

1 Chemical Engineering and Processing 43 (2004) 85/90 Short communication Liquid hold-up correlations for trickle beds without gas flow F. Larachi *, L. Belfares, I. Iliuta, B.P.A. Grandjean Department of Chemical Engineering, Center for Research on the Properties of Interfaces and Catalysis, Laval University, Québec, Canada G1K 7P4 Received 30 September 2002; accepted 6 December 2002 Abstract Dynamic and total liquid hold-up data required for trickle bed reactors without gas flow were gathered from the literature to generate an exhaustive database including over 1100 measurements. The slit model was satisfactorily representational of the evolution of the total liquid hold-up and yielded an average absolute relative error (AARE) of 21.3% for the gravity-driven liquid flow depicted in the database. Similarly, an alternate approach that combined artificial neural networks with dimensional analysis enabled derivation of an empirical dynamic liquid hold-up correlation h D /f(fr L, We L, Re L, Eo L ) yielding an AARE/14%. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Liquid hold-up; Trickle bed; Slit model; Neural network correlation 1. Introduction Aside from the traditional realm of petroleum industry, trickle-bed reactors are legion in the areas of biochemical processing, wastewater and gas scrubbing biological treatments [1 /3]. Trickle beds usually refer to downwardly fed gas /liquid flows through a porous layer of randomly dumped packing (or catalysts). In some instances, the limiting case of stagnant gas (v G /0) and gravity-driven liquid trickling constitutes a possible reactor operation where the liquid hold-up becomes the governing hydrodynamic parameter while the reactor experiences no pressure gradient (DP/H /0). The liquid texture that prevails in this case is mainly film-like and theory predicts that, besides the bed Ergun constants, liquid hold-up depends primarily on the liquid/solid wetting efficiency, and the liquid Reynolds and Galileo numbers [4,5]. Though in the broad sense, liquid holdup is a measure of the volume of liquid normalized with respect to the containing-vessel volume, two types are of chief significance in the design of trickle beds, namely the total and dynamic liquid hold-up components, respectively, h T and h D. Excluding the intra-particle contribution for porous particles, h T represents the total * Corresponding author. Tel.: / ; fax: / address: faical.larachi@gch.ulaval.ca (F. Larachi). external liquid volume occupying the extra-granular interstitial domain, whereas h D, usually measured after feed cut-off and bed drainage, represents an estimate of the portion of liquid in permanent renewal for a bed in operation. The difference between total and dynamic hold-ups yields the static, or residual, liquid hold-up, which is an upper bound measure of the passive liquid volume fraction. The present contribution continues a series of papers devoted to the database building, correlations development and analyses of the macroscopic transport parameters in trickle beds [6,7]. The endeavor is envisioned from the perspective of the inherited body of knowledge acquired over the past several decades in the area of multiphase reactors. The long term objective is to formulate efficient design tools that are validated over broad data repositories and that are reliable for the estimation of the transport parameters. Hence, the present work aims at providing two very accurate correlations, one for the dynamic liquid hold-up and another for the total liquid hold-up in trickle beds without gas flow. The study first traces from the literature the majority of hold-up data for v G /0 published over the last 60 years. Secondly, it evaluates the ability of the slit model to correlate h T. It further proposes a new correlation for h D based on the combination of artificial neural network (ANN) computing and dimensional analysis /03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved. doi: /s (03)

2 86 F. Larachi et al. / Chemical Engineering and Processing 43 (2004) 85/90 2. Brief description of database In view of its broad coverage of the literature, the present study brings together a vast majority of data published over the past 60 years on the total and dynamic liquid hold-ups in trickle bed reactors operated under stagnant gas conditions. An expanded database including over 1100 measurements taken from 24 references [8/31] between 1938 and 2000 was used to evaluate the goodness of representation of the proposed correlations. Table 1 displays the breadth of operating variables characterizing liquid hold-up. It also exhibits results on 26 packing varieties. The hold-ups were mostly measured through residence time distribution tests, and bed draining and weighing. They were categorized as dynamic or total liquid hold-ups. When the latter was not known, a static liquid hold-up correction by means of the Saez and Carbonell [4] correlation was used to convert the measured dynamic hold-ups into total hold-ups. The laminar and turbulent Ergun constants required in the slit model must be known a priori. They have been estimated according to three procedures:. Taken from the original sources when provided therein.. When not available in the original sources, they were estimated from the dry pressure drop versus gas superficial velocity data for the same packing and column size. For Ergun constants identification, the Table 1 Intervals of operating conditions for the liquid dynamic and total holdups in trickling flow at v G /0 Category Properties Ranges Operating conditions Packing & bed properties Liquid properties Pressure (P)/10 6 (Pa) 0.1: 6 Temperature (T) (K) 279: 300 Superficial velocity (v L )/10 2 (m/s) 3.24E/3: 5.6 Particle diameter (d e )/10 3 (m) 1: 25.4 Bed porosity (o) 0.34: 0.93 Bed specific surface area (a t ) (per m) 100: 3600 Sphericity factor (F) 0.157: 1 Packed bed height (H) (m) 0.2: 2.76 Column diameter (d C )/10 2 (m) 3: 45.7 Density (r L ) (kg/m 3 ) 663: 1299 Viscosity (m L )/10 3 (Pa s) 0.3: 18.5 Surface tension (s L )/10 3 (N/m) 25: 80 Liquids: H 2 O, H 2 O/ETG (40%), H 2 O/Sodium octadecil sulfate (0.5 g/l), H 2 O/sorbitol, H 2 O/Dupont petrowet agent, organic mixture, H 2 O/isopropanol (80 vol.%.), H 2 O/sucrose (40%), H 2 O/ terginol, H 2 O/NaCl (12.4/45.6%), benzene, ethanol, kerosene, n - hexane, methanol. Packing shape: sphere, bead, extrudate, Raschig ring, Intalox, Berl saddle. Packing material: glass, ceramic, porous alumina, carbon, stainless steel, polyethylene, porcelain, siliconecoated glass. slit model simplified to the case of single-phase flow was used in these circumstances: DP 1 E 1 Re G Re2 G E 2 (1) Hgr G f 2 Ga G Ga G f. When neither the Ergun constants nor the /DP/H versus v G data were accessible, E 1 and E 2 were estimated using literature correlations [32]. 3. Theory 3.1. Slit model The slit model approximates the trickle bed hydrodynamics by assuming geometrical, kinematic and dynamic similarities between a hypothetical slit flow in a well defined geometry and the actual bed-scale flow taking place in a random porous medium [5,33]. The slit model is suitable when the liquid evolves in a film-like structure, which is typical of the low interaction regime (i.e. trickle flow regime) or when the liquid trickles down without gas flow. Besides the liquid properties and gas density, the variables describing the film flow at the bed scale are, the bed porosity, o, the laminar and turbulent Ergun constants, E 1, E 2, the equivalent particle diameter, d v, the sphericity factor, f, the wetting efficiency (or the percentage of packing specific area wetted by liquid) and the total liquid hold-up, h T. Under gas stagnant conditions and assuming partially wetted beds, the slit model simplifies to the following implicit expression [5]: h 3 T rg r L 1 h 2 T oh r G e o 3 r L E1 h 2 e f 2 Re L Re2 L E 2 h e Ga L Ga L f 0 (2) When the gas-to-liquid density ratio is very small which is the case for most of the experiments summarized in the database, Eq. (2) turns to the following simplified explicit form: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E h T o 1 h 2 3 e Re L Re2 L E 2 h e (3) f 2 Ga L Ga L f The wetting efficiency, h e, was evaluated using a recent comprehensive empirical correlation from reference [34] ANN correlation Alternatively to the slit model, an empirical ANN correlation was also developed for the dynamic liquid hold-up using dimensionless groups as the independent variables. The method surrounding the application of ANNs and the identification of the best set of dimen-

3 Table 2 Inputs, output and connectivity weights of the ANN correlation for dynamic hold-up at v G /0 Normalized output Normalized inputs 1 log(fr G j P5 (4) V 1 L =(7: )) 1 exp j1 v 5:9829 ijv i 1 C 1 exp P v 2 L (5) Fr 9 j1 v L gd jg j e We L r L v2 L d e s L Domain of applicability V 2 log(re L=0:17) 2:65445 Re L r L v L d e m L (1 o) V 3 log(we L=(4: )) 7:1308 Eo L r L gd2 e f2 o 2 s L (1 o) 2 V 4 log(eo=0:06) 1:39162 C log(h D=(4: )) 7.14/10 8 5/Fr i5j58 G 9 1 L 5/ /Re L 5/ /10 8 5/We L 5/ /Eo 5/ :9497 NN connectivity weights v ij / / / / / / / / / / / / / / / / / / / / / / / v j / / / / A user-friendly spreadsheet of the neural correlation is accessible at: V 5 1 F. Larachi et al. / Chemical Engineering and Processing 43 (2004) 85/90 87

4 88 F. Larachi et al. / Chemical Engineering and Processing 43 (2004) 85/90 Fig. 1. Measured vs. predicted parity plots of (a) total liquid hold-up using the simplified slit model (Eq. (3)), (b) dynamic liquid hold-up using the ANN correlation. sionless groups to be involved in the correlations were addressed elsewhere [35] and will be skipped here. Briefly, several dimensionless groups are generated by clustering the dimensional variables having potential impact on h D to form the smallest set of dimensionless combinations best correlated to the output. The structure of the correlation is described by Eqn. 4 and 5 from Table 2. In these equations, V and G define the input and hidden layer vectors and G 9 and V 5 are bias constants set equal to one, v ij and v j are the connectivity weights. Equations 4 and 5 correlate the neural network normalized output, C, to the normalized input vector V. The weights are adjusted by minimizing, via the quasi-newton/broyden /Fletcher/ Goldfarb/Shanno algorithm, a quadratic training error on a learning set, i.e. 70% of the h D values from the database. A good measure for robustness of a welltrained neural network correlation is decided based on the generalization error calculated using the 30% remaining portion of data. This error must be close to the learning error for inputs/output instances dissimulated during learning. Table 2 lists the optimized weights of the h D correlation. 4. Results and discussion Fig. 1a is a parity plot of the measured versus calculated total liquid hold-up values. The calculated h T values were made based on the simplified slit model form given by Eq. (3). The overall average absolute relative error (AARE) for the 714 data was 21.3%. More than 90% of the h T data was predicted to fall within the 9/2AARE envelopes. The ANN correlation for the dynamic liquid hold-up (746 data) was prepared based on the forces appearing in the dimensionless groups of the slit model (Eq. (3)). Besides the gravitational, inertial and viscous forces, accounting for the capillary force was shown to improve the fit quality in comparison with the slit model where this force was ignored. The following dimensionless numbers (Fr L, We L, Re L, Eo L ) represent the liquid phase forces relationships which were found to be the most effectively correlated with the dynamic liquid hold-up. The best neural network correlation contained thus four nodes in the input layer and the bias (Fr L, We L, Re L, Eo L, 1), as well as nine nodes in the hidden layer. Fig. 1b illustrates how well the predictions of the ANN h D -correlation, summarized in Table 2, represent the measured h D from the database. The 523 data, i.e. 70% of the training set (black circles in Fig. 1b) yielded an AARE equal 13.7%. The remaining 223 data from the test set (empty circles in Fig. 1b) gave an AARE of 14.8%. The overall AARE for the 746 data was 14%. As seen in Fig. 1b, more than 90% of the 746h D data was predicted to fall within the 9/2AARE envelopes. Note that the Froude and Reynolds numbers here accomplish the same function as the Galileo and the Reynolds numbers in the slit model. Introducing two additional dimensionless groups to account for the effect of capillary forces via Weber and Eötvös numbers appeared to improve noticeably the hold-up correlations, 14% for the ANN correlation versus 21.3% for the slit model Eq. (3). 5. Concluding remarks An exhaustive database of dynamic and total liquid hold-ups in trickle beds without gas flow was set using the open literature information traced back to The database was composed of more than 1100 data and was used to validate two estimation methods. The first one was the slit model simplified to the case with zero gas velocity and gravity-driven liquid flow through the trickle bed. It was used to estimate the total liquid hold-up and yielded an average error of 21.3%. The second was a correlation based on ANNs and dimensionless analysis. It was used for dynamic liquid hold-up representation and yielded an average error of 14%.

5 F. Larachi et al. / Chemical Engineering and Processing 43 (2004) 85/90 89 Appendix A: Nomenclature AARE average absolute relative error, AARE / N 1 aj1/y calc,i /y exp,i j a c external surface packing area per unit reactor volume (per m) d e grain effective diameter, 6(1/o)/a c (m) d v grain equivalent diameter, diameter of sphere having same volume as grain, d v /d e /f (m) Eo L Eötvös number, Eo L /r L gd 2 e f 2 o 2 s 1 L (1/o) 2 E 1, E 2 Ergun constants Fr L Froude number, Fr L /v 2 L d 1 e g 1 Ga a a-phase Galileo number, Ga a /d 3 v gr 2 a o 3 (1/ o) 3 2 m a h D dynamic liquid hold-up h T total (external) liquid hold-up Re a a-phase Reynolds number, Re a /v a d e r a (1/ o) 1 m 1 a in ANN correlation Re a a-phase Reynolds number, Re a /v a d v r a (1/ o) 1 m 1 a in slit model v a a-phase superficial velocity (m/s) We L Weber number, We L /r L v 2 L d e s L Greek symbol DP/H pressure drop (Pa/m) o bed porosity G hidden-layer vector f sphericity factor h e wetting efficiency r a a-phase density (kg/m 3 ) s L surface tension (N/m) v neural network connectivity weights V i neural network normalized input variables C neural network output Subscripts a gas or liquid calc calculated exp experimental G gas phase L liquid phase References [1] R. Lobo, S. Revah, T. Viveros-Garcia, An analysis of a tricklebed bioreactor: carbon disulfide removal, Biotechnol. Bioeng. 63 (1999) 98/109. [2] W.J.H. Okkerse, S.P.P. Ottengraf, B. Osinga-Kuipers, M. Okkerse, Biomass accumulation and clogging in biotrickling filters for waste gas treatment. Evaluation of a dynamic model using dichloromethane as a model pollutant, Biotechnol. Bioeng. 63 (1999) 418/430. [3] C. Benthack, B. Srinivasan, D. Bonvin, An optimal operating strategy for fixed-bed bioreactors used in wastewater treatment, Biotechnol. Bioeng. 72 (2001) 34/40. [4] A.E. Saez, R.G. Carbonell, Hydrodynamic parameters for gas/ liquid cocurrent flow in packed beds, AIChE J. 31 (1985) 52/62. [5] I. Iliuta, F. Larachi, The generalized slit model: pressure gradient, liquid holdup and wetting efficiency in gas/liquid trickle flow, Chem. Eng. Sci. 54 (1999) 5039/5045. [6] I. Iliuta, A. Ortiz-Arroyo, F. Larachi, B.P.A. Grandjean, G. Wild, Hydrodynamics and mass transfer in trickle-bed reactors: an overview, Chem. Eng. Sci. 54 (1999) 5329/5337. [7] I. Iliuta, F. Larachi, B.P.A. Grandjean, G. Wild, Gas/liquid interfacial mass transfer in trickle-bed reactors: state-of-the-art correlations, Chem. Eng. Sci. 54 (1999) 5633/5645. [8] W.S. Dodds, L.F. Stutzman, B.J. Sollami, R.J. McCarter, Pressure drop and liquid hold-up in concurrent gas absorption, AIChE J. 6 (1960) 390/393. [9] W.S. Dodds, L.F. Stutzman, B.J. Sollami, R.J. McCarter, Concurrent absorption mass transfer, AIChE J. 6 (1960) 197/201. [10] A. Burghardt, G. Bartelmus, M. Jaroszynski, A. Kolodziej, Hydrodynamics and mass transfer in a three-phase fixed-bed reactor with cocurrent gas/liquid downflow, Chem. Eng. J. 58 (1995) 83/99. [11] W.J.A. Wammes, Hydrodynamics in a cocurrent gas/liquid trickle-bed reactor at elevated pressures, Ph.D. thesis, Twente University, Enschede, The Netherlands, [12] W.J.A. Wammes, K.R. Westerterp, The influence of the reactor pressure on the hydrodynamics in a cocurrent gas/liquid tricklebed reactor, Chem. Eng. Sci. 45 (1990) 2247/2254. [13] W.J.A. Wammes, S.J. Mechielsen, K.R. Westerterp, The transition between trickle flow and pulse flow in a cocurrent gas/liquid trickle-bed reactor at elevated pressures, Chem. Eng. Sci. 45 (1990) 3149/3158. [14] W.J.A. Wammes, K.R. Westerterp, Hydrodynamics in a pressurized co-current gas/liquid trickle-bed reactor, Chem. Eng. Technol. 14 (1991) 406/413. [15] W.J.A. Wammes, S.J. Mechielsen, K.R. Westerterp, Influence of pressure on the liquid hold-up in a cocurrent gas/liquid tricklebed reactor operating at low gas velocities, Chem. Eng. Sci. 46 (1991) 409/417. [16] W.J.A. Wammes, J. Middelkamp, W.J. Huisman, C.M. debaas, K.R. Westerterp, Hydrodynamics in a cocurrent gas/liquid trickle bed at elevated pressures, AIChE J. 37 (1991) 1849/1862. [17] J. Levec, A.E. Saez, R.G. Carbonell, The hydrodynamics of trickling flow in packed beds: II-Experimental observations, AIChE J. 32 (1986) 369/380. [18] A.E. Saez, R.G. Carbonell, J. Levec, The hydrodynamics of trickling flow in packed beds: I-Conduit models, AIChE J. 32 (1986) 353/368. [19] H.L. Shulman, C.F. Ullrich, N. Wells, Performance of packed columns: III-Hold-up for aqueous and non-aqueous systems, AIChE J. 1 (1955) 259/264. [20] H.L. Shulman, C.F. Ullrich, N. Wells, Performance of packed columns: I-Total, static and operating holdups, AIChE J. 1 (1955) 247/253. [21] M. Kohler, W. Richarz, Investigation of liquid hold-up in tricklebed reactors, German Chem. Eng. 8 (1985) 295/300. [22] J.C. Charpentier, Hydrodynamique des colonnes à garnissage avec écoulements de gaz et de liquide à co-courant et contrecourant: aspects statique et dynamique de la texture du liquide, Ph.D. thesis, Faculté des Sciences, Université de Nancy, Nancy, France, [23] J.C. Charpentier, C. Prost, W. Van Swaaij, P. Le Goff, tude de la rétention de liquide dans une colonne à garnissage arrosé à cocourant et à contre-courant de gaz/liquide, Chim. Ind. Génie Chimique 99 (1968) 803/826. [24] J.C. Charpentier, C. Prost, P. Le Goff, Chute de pression pour des écoulements à co-courant dans les colonnes à garnissage arrosé: comparaison avec le garnissage noyé, Chem. Eng. Sci. 24 (1969) 1777/1794.

6 90 F. Larachi et al. / Chemical Engineering and Processing 43 (2004) 85/90 [25] V. Tukac, J. Hanika, Influence of catalyst particles orientation on the pressure drop and the liquid dispersion in the trickle-bed reactor, Chem. Eng. Sci. 47 (1992) 2227/2232. [26] B. Jesser, J. Elgin, Studies of liquid holdup in packed towers (I. Relations of operating holdup on various packing materials to liquor velocity and physical properties), Trans. AIChE J. 39 (1948) 277/298. [27] C.W. Simmons, H.B. Osborn, Tower absorption coefficients (V. Determination and effect of free volume), Ind. Eng. Chem. 26 (1934) 529/531. [28] T. Otake, K. Okada, Liquid holdup in packed towers*/operating holdup without gas flow, Kagaku Kogaku 17 (1953) 176/184. [29] H.J. Pant, A.K. Saroha, K.D.P. Nigam, Measurement of liquid holdup and axial dispersion in trickle bed reactor using radiotracer technique, Nukleonika 45 (2000) 235/241. [30] J.B. Wijffels, J. Verloops, F.J. Zuiderweg, Wetting of catalyst particles under trickle flow conditions, ACS Monograph Series, Chem. Reaction Eng.-II 133 (1974) 151/163. [31] S. Ushida, S. Fujita, Liquid holdup in packed towers, J. Soc. Chem. Ind. Jpn. 41 (1938) 275. [32] I. Iliuta, F. Larachi, B.P.A. Grandjean, Pressure drop and liquid holdup in trickle flow reactors: improved Ergun constants and slip correlations for the slit model, Ind. Eng. Chem. Res. 37 (1998) 4542/4550. [33] R.A. Holub, M.P. Dudukovic, P.A. Ramachandran, Pressure drop, liquid holdup and flow regime transition in trickle flow, AIChE J. 39 (1993) 302/321. [34] F. Larachi, L. Belfares, B.P.A. Grandjean, Prediction of liquid/ solid wetting efficiency in trickle flow reactors, Int. Commun. Heat Mass Transfer 28 (2001) 595/600. [35] F. Larachi, Z. Bensetiti, B.P.A. Grandjean, G. Wild, Two-phase frictional pressure drop in flooded-bed reactors: a state-of-the-art correlation, Chem. Eng. Technol. 21 (1998) 887/893.