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Dynamic modelling of periodically wetted catalyst particles
Chemical Engineering Science 60 (2005) 6254 – 6261 www.elsevier.com/locate/ces
Dynamic modelling of periodically wetted catalyst particles Wulf Dietrich∗ , Marcus Grünewald, David W. Agar Chair of Reaction Engineering, Department of Biochemical and Chemical Engineering,University of Dortmund, Emil-Figge-Str. 66, D-44227 Dortmund, Germany Received 24 November 2004; accepted 18 March 2005 Available online 15 June 2005
Abstract The design of periodically operated trickle-bed reactors requires extensive knowledge of the complex underlying processes occurring at various scales. The development of dynamic models offers detailed insights into system behaviour and allows quantitative investigations over a wide range of operating conditions without the expense of comprehensive experimental studies. The description of the dynamic behaviour of periodically wetted catalyst particles is an important element in the overall multi-scale model architecture. Unfortunately, very little information is available on the influence of various model assumptions to aid selection of appropriate unsteady-state models. A systematic evaluation of dynamic particle-scale modelling approaches based on various intra-particle concentration profiles and periodic wetting of the external particle surface is presented. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Reaction engineering; Multiphase reactors; Modelling; Dynamic simulation; Trickle-bed reactors; Periodic operation
1. Introduction The periodic operation of trickle-bed reactors offers considerable potential for process intensification and has, as a consequence, received considerable attention in both industrial and scientific research work over the last years (Haure et al., 1989; Lange et al., 1994). In order to predict the quantitative improvement in reactor performance, comprehensive models must be developed, since direct experimentation without the assistance of modelling offers little chance of successfully exploiting complex dynamic operation. The accurate description of the dynamic behaviour of catalyst particles with dynamically varying surface exposures to gas and liquid is a decisive building block in a multi-scale model architecture. However, to date no systematic evaluation of various dynamic modelling approaches for this task is available, and transient effects impose additional criteria ∗ Corresponding author. Tel.: +49 231 755 4347; fax: +49 231 755 2698.
E-mail addresses: [email protected] (W. Dietrich), [email protected] (D.W. Agar). 0009-2509/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2005.03.054
that mean a comparison of the corresponding steady-state models is invalid. Due to the industrial significance of trickle bed reactors, extensive research activities have been devoted to understanding the complex interactions between hydrodynamics, transport processes and chemical reaction, resulting in a number of basic models permitting a relatively good description of steady-state behaviour (Dudukovic et al., 2002). Several comparative studies on the impact of various model assumptions and numerical solution methods on effective reaction rates have been published over the last 20 years. Approximate expressions (Mears, 1974) and numerical solutions (Dudukovic and Mills, 1978) for effectiveness factors have been developed and compared for an infinite catalyst slab with partial external surface wetting and found to be adequate under very strong internal mass transfer limitations. Similar results were obtained for catalyst cubes (Herskowitz et al., 1979) and various geometries (Tan, 1988). The impact of evaporation taking place due to strongly exothermic reactions, which leads to incomplete internal pore wetting and multiple steady states, has also been investigated
W. Dietrich et al. / Chemical Engineering Science 60 (2005) 6254 – 6261
(Harold, 1988; Watson and Harold, 1994). The number and size of liquid rivulets covering the external particle surface was found to influence the effective reaction rate under steady-state conditions strongly (Ring and Missen, 1986; Zhu and Hofmann, 1997), except when extreme mass transfer limitations are present. The more uniformly the liquid is spread over the particle surface, the higher the effectiveness factor. Thus a broad selection of quantitative information is available to serve as basis for the selection of steady-state partially wetted catalyst particle models. An unsteady-state operating strategy leads to a further increase in the complexity of the physical and chemical processes determining reactor behaviour and therefore necessitates a more refined modelling approach, with special attention being given to the correct accounting for material accumulation within the particle pores. For dynamic operating conditions several models have been proposed, which primarily treat the phenomena occurring at the reactor scale (Gabarain et al., 1997; Stegasov et al., 1994). A one-dimensional dynamic particle model was proposed by Boelhouwer (2001) which distinguishes between continuously and periodically wetted zones. However, comparative studies examining the effect of different levels of model detail on the accuracy of simulation results, similar to those available for steady state conditions, have not yet been published. In this work, an evaluation of three dynamic particle-scale modelling approaches based on different assumptions with respect to intra-particle concentration profiles and a periodically fluctuating wetting of the external particle surface is presented. A single partially wetted particle exposed to periodic variations of the wetted fraction of external surface area is modelled as an essential module for use in multi-scale trickle-bed reactor models. The primary focus lies in the degree of symmetry assumed in concentration profiles and the corresponding modelling detail that results. The comparison of the various models is based on a selection of benchmark scenarios representing dynamic reaction conditions occurring over the whole length of a trickle-bed reactor. Special attention was paid to reaction conditions that have been found to cause significant deviations for approximate models under steady-state conditions, i.e., moderate Thiele moduli and strong external mass transfer limitations with a limiting gas phase reactant.
2. Modelling The behaviour of a partially wetted catalyst particle is modelled for a slab geometry under isothermal reaction conditions. It is assumed that internal wetting is complete due to capillary forces and that the wetting efficiency of the external particle surface is dependent on the liquid flow rate according to the correlation proposed by El-Hisnawi et al. (1982). No influence of the wetting geometry is considered, as the wetted surface area is averaged at the particle-scale
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resulting in a single wetted and dry fraction. Mass transfer between gas and liquid phases and the catalyst surface is described using film theory and Henry’s law and film theory, respectively. A general reaction equation (Eq. (1)) is formulated for the reaction of a non-volatile liquid with a sparingly soluble gaseous reactant obeying reaction kinetics which are first order with respect to both reactants: L1 (l) + G(g) → L2 (l),
(1)
r = kcL1 cG .
(2)
Three different modelling approaches have been considered, which mainly differ in the symmetry and thus spatial resolution of concentration profiles within the particle. A schematic comparison of the model concepts is depicted in Fig. 1. In the first case (see Fig. 1a), two-dimensional concentration profiles are calculated whilst diffusional mass transfer inside the particle is allowed in both x- and y-directions. This leads to the particle mass balances as proposed by Dudukovic and Mills (1978), which has been extended to account for dynamic conditions. jcp,L1 j2 cp,L1 j2 cp,L1 = De,L1 − kcp,L1 cp,G , (3) + jt jx 2 jy 2 jcp,G j2 cp,G j2 cp,G − kcp,L1 cp,G . = De,G + jt jx 2 jy 2
(4)
The partial wetting of the particle surface, which changes with time depending on the liquid flow rate, is considered in the surface boundary conditions. For the liquid reactant L1 mass transfer can only occur at the wetted surface (Eq. (5)):
jcp,L1 = Bi l,L1 cl,L1 − cp,L1 |x=0 jx x=0
jcp,L1 = 0 for w (t) < y < 1. jx x=0
for 0 < y < w (t),
(5) (6)
Depending on the y-coordinate at the particle surface and the time variable wetting fraction w (t) the boundary conditions for component G account for mass transfer from either the liquid (Eq. (7)) or gas (Eq. (8)) phases. Elsewhere flat profiles were assumed at the domain boundaries (Eq. (9)): jcp,G jx x=0 = Bi l,G (cl,G − cp,G |x=0 ) for 0 < y < w (t), (7) jcp,G jx x=0 cg,G for w (t) < y < 1, (8) = Bi g,G − cp,G x=0 HG jcp,i jcp,i =0, =0, with i={L1 , G}. (9) jx x=dp jy y={0,lp }
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W. Dietrich et al. / Chemical Engineering Science 60 (2005) 6254 – 6261
(g)
pw
(g)
ηw (t)
ηw (t)
ηw,min
ηw,min (I)
pw
(g) ηw (t)
x cw
(I)
ηw,min
cw
(I)
y x
x
(a)
Case 1 (2D)
(b)
Case 2 (1D)
(c)
Case 3 (0D)
Fig. 1. Schematic representation of the model strategies and assumptions.
A more simplified model, as suggested by Boelhouwer (2001), is represented by case 2 (see Fig. 1b), in which the domain is divided into a permanently wetted zone and a zone with periodically changing wetting. The relative portions of the total particle volume assigned to each zone are determined according to the minimum and maximum wetting efficiency during the periodic wetting process. These values have to be determined a priori by evaluating the wetting correlation for the upper and lower boundaries of the intended flow rate modulation. The concentration profiles are assumed to be symmetrical in each zone without any lateral mass transfer between different zones, which results in one-dimensional mass balances for each zone: jccw,L1 j2 ccw,L1 w,min =w,min De,L1 − kccw,L1 ccw,G , jt jx 2 (10) (w,max −w,min )
jcpw,L1 jt
j2 cpw,L1 =(w,max −w,min ) De,L1 −kcpw,L1 cpw,G , jx 2 (11) 2 jccw,G j ccw,G w,min −kccw,L1 ccw,G , =w,min De,G jt jx 2 (12) (w,max − w,min )
jcpw,G jt
j2 cpw,G =(w,max −w,min ) De,G −kcpw,L1 cpw,G . jx 2 (13)
For the permanently wetted zone only liquid–solid mass transfer has to be considered in the boundary conditions: jccw,L1 w,min = w,min Bi l,L1 (ccw,L1 |x=0 − cl,L1 ), jx x=0 (14) jccw,G w,min = w,min Bi l (ccw,G |x=0 − cl,G ). (15) jx x=0
For the periodically wetted zone both gas–solid (only component G) and liquid–solid mass transfer are permitted and weighted according to the wetted particle surface fraction: (w,max − w,min )
jcpw,L1 jx x=0
= (w − w,min )Bi l,L1 (ccw,L1 |x=0 − cl,L1 ), jcpw,G (w,max − w,min ) jx x=0 cg,G = (w,max − w )Bi g,G cpw,G |x=0 − H + (w − w,min )Bi l,G (ccw,G |x=0 − cl,G ).
(16)
(17)
The third case is a further simplification of case 2, which can be derived if intra-particle profiles are neglected and an average particle concentration (see Fig. 1c) is employed. Mass transfer inside the particle is described using an effective internal mass transfer coefficient based on the assumption of linear driving forces: ke,i =
2De,i . dp
(18)
As in case 2 for the continuously wetted zone, only mass transfer from the liquid phase is considered (Eqs. (19) and (20)), while in the periodically wetted zone mass transfer of G from gas can take place as well (Eqs. (21) and (22)):
w,min
jccw,L1 1 1 −1 =w,min + (cl,L1 − ccw,L1 ) jt kls,L1 ke,L1 − w,min kccw,L1 ccw,G , (19)
jccw,G 1 1 −1 = w,min w,min + (cl,G − ccw,G ) jt kls,G ke,G − w,min kccw,L1 ccw,G , (20)
W. Dietrich et al. / Chemical Engineering Science 60 (2005) 6254 – 6261
jcpw,L1 jt −1 1 1 = (w − w,min ) + (cl,L1 − cpw,L1 ) kls,L1 ke,L1 − (w,max − w,min )kcpw,L1 cpw,G , (21)
(w,max − w,min )
jcpw,G (w,max − w,min ) jt 1 1 −1 + (cl,G − cpw,G ) = (w − w,min ) kls,G ke,G 1 1 −1 cg,G + − cpw,G + (w,max − w ) kgs,G ke,G H − (w,max − w,min )kcpw,L1 cpw,G . (22) The three different particle models were each incorporated into a differential axial element of a fixed-bed reactor. The corresponding mass balance equations are given as follows1
jcl,L jcl,L = −ul −(w,max −w )kls,L ap (cl,L −cpw,L |x=0 ) jt jz (23) −(w − w,min )kls,L ap (cl,L − ccw,L |x=0 ), c jcl,G jcl,G g,G = − ul + ka gl,G − cl,G jt jz H − (w − w,min )kls,G ap (cl,G − cpw,G |x=0 ) − w,min kls,G ap (cl,G − ccw,G |x=0 ),
jcg,G jcg,G = − ug − ka gl,G jt jz
c
g,G
H c
− (w,max − w )kgs,G ap
− cl,G g,G
H
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Table 1 Simulation parameters and periodic operating conditions Catalyst characteristics Pore depth Particle length Effective diffusivity Bed porosity
dp lp De,i
Correlated parameters Wetting efficiency w Liquid holdup l Liquid–solid mass transfer coeff.a kls Gas–solid mass transfer coeff.a Gas–liquid mass transfer coeff.a Periodic operation parameters Period length Period split Superficial gas velocity Superficial liquid velocity (min) Superficial liquid velocity (max)
kgs kgl
Value 10−4 m 10−3 m De,i = 0.1 · Di 0.4 Reference (El-Hisnawi et al., 1982) (Larachi et al., 1991) (Dharwadkar and Sylvester, 1977) (Schlünder, 1994) (Midoux et al., 1984)
Value 400 s 0.5 (BASE–PEAK) uG 0.2 m/s ul,min 0.001 m/s ul,max 0.01 m/s
a Mass transfer coefficients have been scaled to yield the desired Bi numbers.
3. Results (24)
− cpw,G |x=0 . (25)
To account for the attenuation characteristics of pulses travelling along a periodically operated trickle-bed reactor, different gradients and amplitudes of liquid pulses were imposed, corresponding to the whole spectrum of conditions occurring along the length of the reactor. The liquid flow modulation affects the model results through the correlations that have been implemented (see Table 1) to calculate the wetting fraction on the one hand and mass transfer coefficients from liquid to solid on the other. The partial differential equations were solved based on the method of lines (Schiesser, 1991) using finite differences. In order to resolve the steep gradients arising under more severe reaction conditions ( 5, Bi l 1) a locally refined grid was used in the spatial domain for cases 1 and 2. Initial conditions for all variables and cases were set to correspond to the steady-state solution for the average flow rate of periodic operation. 1 In particular these equations apply for case 2. For case 1 mass transfer fluxes would have to be integrated over the particle length. For case 3 internal mass transfer coefficient would have to be taken into account as in Eqs. (19–22).
Simulations were carried out for parameters covering a wide range of typical reaction conditions observed in industrial trickle bed reactors. Mass transfer coefficients and kinetic constants were varied to give Thiele moduli from 0.1 to 10 and Biot numbers Bi l,G from 0.1 to 100 with respect to the dissolved gaseous component. The Biot number for direct gas–solid mass transfer Bi g,G was set at a constant value of 100 for all cases examined. The two different scenarios of liquid flow rate variation under the cyclic conditions indicated in Table 1 were investigated, namely a symmetric rectangular profile representing conditions at the reactor inlet and a continuous asymmetric S-shaped profile typically found further downstream in a periodically operated trickle-bed reactor (see Fig. 2). The shape of the liquid flow rate profile was taken from reactor scale simulations using a non-linear liquid hold-up correlation that have been presented elsewhere (Lange et al., 2004). Under reaction conditions of practical relevance, e.g. = 1 . . . 5 and Bil,G = 1, the simulations predict conversion enhancements through periodic operation of c. 30% for a rectangular flow rate modulation profile. Fig. 3 shows values for two different sets of reaction conditions corresponding to a differential reactor element with a length of 0.01 m. If these values are extrapolated up to catalyst bed lengths of 0.2 . . . 0.5 m they are qualitatively consistent with the results of experimental studies reported in the literature (Khadilkar et al., 1999; Lange et al., 1994; Turco et al., 2001). Under steady-state conditions as well as for periodic operation according to scenario 1 the models of cases 1 and 2
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Fig. 4 depicts the results for the first scenario. Excellent agreement can be observed between the 1D and 2D-models while the simplified case 3 exhibits strong deviations from the other two except at low Thiele moduli. With decreasing external mass transfer limitations, i.e., increasing Bi l , an even more pronounced underestimation of the average reaction rate is obtained in case 3. In order to analyse these findings, intra-particle reaction rate profiles at a given time shortly after reducing the liquid flow rate and wetting efficiency, respectively, are illustrated in Fig. 5. Under these conditions the dry portion of the particle surface is exposed to high concentrations of G in the gas phase, while the liquid reactant inside the pores is not yet depleted. The resulting strong driving forces should emphasise the consequences of differences in the underlying model assumptions. It becomes apparent that an averaging of mass balances is only appropriate along the particle length dimension (see Fig. 5a), where the profiles predicted by case 1 (2D) and case 2 (1D) are in close agreement. The assumption of a homogenous concentration throughout the whole pore length in case 3 gives rise to a buffer effect damping the steep intra-particle gradients, especially in the periodically wetted zone (see Fig. 5b). At high Thiele moduli in particular this causes a considerable diminishing of the reaction driving forces in comparison to the other two models. The comparison of the different models for the second scenario is illustrated in Fig. 6 . In this instance the onedimensional model (case 2) shows more obvious deviations from the two-dimensional model (case 1) than in the first scenario, in particular with increasing mass transfer limitations, be they external or internal. Based on the reaction rate profiles shown in Fig. 7 two explanations for the discrepancies between cases 1 and 2 may be identified. Due to the slower propagation of the wetting front from minimum to maximum wetting conditions, the dry and wetted fractions of the particle change continuously
VG . VL
Scenario 1
t . VL
Scenario 2
t
Fig. 2. Pulse attenuation along reactor length showing the two benchmark scenarios investigated.
predict almost the same conversion while case 3 yields values which are about 20% lower. However, for the second scenario a very strong increase in conversion is predicted by case 2 in contrast to a decrease for case 1 and a moderate increase for case 3. A detailed performance comparison of the three models was carried out based on a modified catalyst effectiveness factor , which was defined by the ratio of time-averaged periodic reaction rates to the corresponding steady-state reaction rates without mass transfer limitations.
Case 1 (2D) Case 2 (1D) Case 3 (0D) = 1, Bil,G = 1
2.0%
= 5, Bil,G = 1
4.0%
+18% conversion [ - ]
conversion [ - ]
+17% 1.5%
+29% 1.0% 0.5% 0.0%
3.5%
+28% 1.0% 0.5% 0.0%
steady state
scenario 1
scenario 2
steady state
scenario 1
scenario 2
Fig. 3. Conversion enhancement by periodic operation for = 1 . . . 5, Bi l = 1. Comparison of scenarios 1 and 2 (see Fig. 2).
W. Dietrich et al. / Chemical Engineering Science 60 (2005) 6254 – 6261 1
1 = 0.1
= 0.1 0.1
0.1
=1
=1
0.01
η′
η′
6259
0.001
Case 1 (2D) Case 2 (1D) Case 3 (0D)
= 10
0.0001 0.1
10
1
100
Bil
0.01 = 10
Case 1 (2D) Case 2 (1D) Case 3 (0D)
0.001
0.0001 0.1
1
10
100
Bil
Fig. 4. Simulation results for reactor inlet section.
Fig. 6. Simulation results for downstream trickle-bed scenario with asymmetric s-shaped pulses.
with time. However, in the two-zone models of cases 2 and 3 the zones that are used for averaging the mass balances are constant. As illustrated in Fig. 7a, the same concentrations and reaction rates are assumed to prevail throughout the entire periodically wetted zone. This results in an inadequate weighting of reaction rates compared to the two-dimensional mass balances of case 1. Furthermore, at the surface of the periodically wetted zone mass transfer from the gas and liquid phases is weighted according to the wetting efficiency. When the wetting efficiency is between maximum and minimum values the partially wetted surface fraction in case 2 is exposed to both phases at once, resulting in an enhancement of the mass transfer from the gas phase whilst still maintaining the supply of liquid reactant. This results in higher concentrations of both reactants in the partially wetted zone and thus an overestimate of the reaction rate in comparison to case 1, which becomes even more pronounced with increasing external mass transfer limitations. In the first scenario and in general for case 1, no simultaneous mass
transfer from gas and liquid phases to a single pore can take place. The model in case 3 exhibits a combination of the phenomena described above for case 2 together with the damping of concentration fluctuations found for the first scenario. If external mass transfer limitations are very strong the overestimation of mass transfer due to the two-zone approach and the underestimation of the reaction rate caused by neglecting intra-particle profiles cancel one another out. Since the extent of the individual contributions of these opposing effects cannot be estimated a priori and varies with time, it has to be concluded that the assumption of an average pore concentration instead of allowing for intra-particle concentration profiles is unsuitable for quantifying accumulation effects within the particle pores. If the simulation results are compared with the behaviour of periodically operated trickle-bed reactors that has been found experimentally, the two-dimensional model of case 1 seems to provide more realistic results than the two-zone
Fig. 5. Intra-particle reaction rate profiles for reactor inlet section (t = 210 s, = 1, Bi l = 1).
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W. Dietrich et al. / Chemical Engineering Science 60 (2005) 6254 – 6261
Fig. 7. Intra-particle reaction rate profiles for downstream trickle-bed scenario with asymmetric s-shaped pulses (t = 210 s, = 1, Bi l = 1).
approaches. On the one hand it would be expected that the conversion enhancement decreases asymptotically towards the steady-state value with increasing pulse attenuation. While case 1 shows a decrease in conversion enhancement for s-shaped profiles (scenario 1) in comparison to step profiles (scenario 2), case 3 predicts a moderate increase and case 2 a considerable increase. On the other hand it can be assumed that the existence of large numbers of “semi-wetted” pores as implied by the two-zone models is somewhat unrealistic and causes an overestimation of mass transfer rates. A stepwise motion of the wetting line due to film rupture and rivulet formation seems physically more credible (Khanna and Nigam, 2002). The experimental data required for a quantitative discrimination of the particle models will have to be obtained from reaction and mass transfer rate measurements employing a differential reactor in order to establish distinct flow rate profiles and eliminate the attenuation effects, which would occur in a full length reactor.
4. Conclusions Based on the comparison of three different modelling strategies for a partially wetted slab key aspects in the selection of an appropriate dynamic model for a partially wetted catalyst particle have been demonstrated. Not only the extent of the mass transfer limitations but also the time dependency of the surface wetting is an important parameter that has to be taken into consideration. When integrated into a reactor scale model, a two-dimensional particle model requires a very large number of grid points if a reasonably high spatial resolution is to be achieved and thus rapidly becomes numerically exorbitant. A compromise between accuracy and computational effort could involve a multiple-zone approach. The less steep the profiles the greater the number of zones that are required. Thus, the number of zones would have to be increased over the length of the reactor according to the attenuation characteristics of the liquid pulses. Addi-
tionally, an adjustment of the boundaries between continuously and periodically wetted zones is necessary along the length of the reactor, as the amplitude of pulses decreases from top to bottom. This necessitates preliminary simulations of the hydrodynamics in order to provide the minimum and maximum wetting efficiency values. Considering the strong influence of the wetting geometry on catalyst performance reported under steady state conditions, the detailed investigation of the wetting morphology for various particle geometries for periodic variations of the surface wetting would be an important step in the analysis of the dynamic trickle-bed reactor behaviour. The rigorous free surface simulation of multiphase flows in randomly packed beds of arbitrarily shaped porous particles is likely to remain an insoluble problem even in the medium term. Thus, the development of models for periodically operated multiphase reactors will have to rely on approximate particle models for some time to come. This work represents a tentative first step towards acquiring a broad basis of experience to aid the selection of dynamic models for the description of periodically wetted catalyst particle systems.
Notation ap Bi g,i Bi l,i cg,i cl,i ccw,i cpw,i
specific particle surface, m2 /m3 k dp Biot number for dry surface, Bi g,i = gs,i De,i , dimensionless k dp Biot number for wetted surface, Bi l,i = ls,i De,i , dimensionless bulk gas phase concentration, mol/m3 bulk liquid phase concentration, mol/m3 concentration in continuously wetted zone, mol/m3 concentration in periodically wetted zone, mol/m3
W. Dietrich et al. / Chemical Engineering Science 60 (2005) 6254 – 6261
cpw,i cp,i De,i dp H ka gl k ke,i kls kgs lp ug ul l
w w,max w,min cw pw
concentration in periodically wetted zone, mol/m3 concentration in particle pores (2D case), mol/m3 effective diffusivity, m2 /s 1lpore length, m Henry’s constant of component G, dimensionless gas–liquid mass transfer coefficient, 1/s reaction rate constant, m3 /mol s effective mass transfer coefficient, m/s liquid–solid mass transfer coefficient, m/s gas–solid mass transfer coefficient, m/s particle size, m superficial gas velocity, m superficial liquid velocity, m bed voidage, dimensionless liquid hold-up, dimensionless
kc
c
l,L1 Thiele modulus, =dp Dl,G , dimensionless e,G cl,G modified particle effectiveness factor, dimensionless wetting efficiency, dimensionless maximum wetting efficiency arising over cycle, dimensionless minimum wetting efficiency arising over cycle, dimensionless continuously wetted domain partially wetted domain
References Boelhouwer, J.G., 2001. Nonsteady operation of trickle-bed reactors—hydrodynamics, mass and heat transfer. Dissertation, Technische Universiteit Eindhoven. Dharwadkar, A., Sylvester, N.D., 1977. Liquid-solid mass transfer in trickle beds. A.I.Ch.E. Journal 23, 276–378. Dudukovic, M.P., Mills, P.L., 1978. Catalyst effectiveness factor in tricklebed reactors. ACS Symposium Series 65, 387–399. Dudukovic, M.P., et al., 2002. Multiphase catalytic reactors: a perspective on current knowledge and future trends. Catalysis Reviews 44, 123–246. El-Hisnawi, A.A., et al., 1982. Trickle-bed reactors: dynamic tracer tests, reaction studies, and modeling of reactor performance. ACS Symposium Series 196, 421–440.
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Gabarain, L., et al., 1997. Analysis of rate enhancement in a periodically operated trickle-bed reactor. A.I.Ch.E. Journal 43, 166–172. Harold, M.P., 1988. Steady-state behavior of the nonisothermally partially wetted and filled catalyst. Chemical Engineering Science 43, 3197–3216. Haure, P.M., et al., 1989. Periodic operation of a trickle-bed Reactor. A.I.Ch.E. Journal 35, 1437–1444. Herskowitz, M., et al., 1979. Effectiveness factors and mass transfer in trickle-bed reactors. A.I.Ch.E. Journal 25, 272–283. Khadilkar, M.R., et al., 1999. Parametric study of unsteady-state flow modulation in trickle-bed reactors. Chemical Engineering Science 54, 2585–2595. Khanna, R., Nigam, K.D.P., 2002. Partial wetting in porous catalysts: wettability and wetting efficiency. Chemical Engineering Science 57, 3401–3405. Lange, R., et al., 1994. Investigations of periodic operated trickle-bed reactors. Chemical Engineering Science 49, 5615–5621. Lange, R., et al., 2004. Unsteady operation of trickle bed reactors. Chemical Engineering Science 59, 5355–5361. Larachi, F., et al., 1991. Experimental study of a trickle-bed reactor operating at high pressure: two-phase pressure drop and liquid saturation. Chemical Engineering Science 46, 1233–1246. Mears, D.E., 1974. The role of liquid hold up and effective wetting in the performance of trickle-bed reactors. ACS Symposium Series 133, 218–277. Midoux, N., et al., 1984. Interfacial area and liquid side mass transfer coefficient in trickle bed reactors operating with organic liquids. Chemical Engineering Science 39, 781–794. Ring, Z.E., Missen, R.W., 1986. Trickle bed reactors: effect of wetting geometry on overall effectiveness factor. Canadian Journal of Chemical Engineering 64, 117. Schiesser, W.E., 1991. The Numerical Method of Lines. Academic Press, London. Schlünder, E.U., 1994. VDI-Wärmeatlas. VDI-Verlag, Düsseldorf. Stegasov, A.N., et al., 1994. Modelling of catalytic SO2 oxidation for continuous and periodic liquid flow through a trickle bed. Chemical Engineering Science 49, 3699–3710. Tan, C.-S., 1988. Effectiveness factors of n-th order reactions for incomplete wetting particles in trickle-bed reactors. ChemicaI Engineering Science 43, 1281–1286. Turco, F., et al., 2001. Investigation of periodic operation of a trickle-bed reactor. Chemical Engineering Science 56, 1429–1434. Watson, P.C., Harold, M.P., 1994. Rate enhancement and multiplicity in a partially wetted and filled pellet: experimental study. A.I.Ch.E. Journal 40, 97–111. Zhu, X., Hofmann, H., 1997. Effect of wetting geometry on overall effectiveness factors in trickle beds. Chemical Engineering Science 52, 4511–4524.
Dynamic modelling of periodically wetted catalyst particles Wulf Dietrich∗ , Marcus Grünewald, David W. Agar Chair of Reaction Engineering, Department of Biochemical and Chemical Engineering,University of Dortmund, Emil-Figge-Str. 66, D-44227 Dortmund, Germany Received 24 November 2004; accepted 18 March 2005 Available online 15 June 2005
Abstract The design of periodically operated trickle-bed reactors requires extensive knowledge of the complex underlying processes occurring at various scales. The development of dynamic models offers detailed insights into system behaviour and allows quantitative investigations over a wide range of operating conditions without the expense of comprehensive experimental studies. The description of the dynamic behaviour of periodically wetted catalyst particles is an important element in the overall multi-scale model architecture. Unfortunately, very little information is available on the influence of various model assumptions to aid selection of appropriate unsteady-state models. A systematic evaluation of dynamic particle-scale modelling approaches based on various intra-particle concentration profiles and periodic wetting of the external particle surface is presented. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Reaction engineering; Multiphase reactors; Modelling; Dynamic simulation; Trickle-bed reactors; Periodic operation
1. Introduction The periodic operation of trickle-bed reactors offers considerable potential for process intensification and has, as a consequence, received considerable attention in both industrial and scientific research work over the last years (Haure et al., 1989; Lange et al., 1994). In order to predict the quantitative improvement in reactor performance, comprehensive models must be developed, since direct experimentation without the assistance of modelling offers little chance of successfully exploiting complex dynamic operation. The accurate description of the dynamic behaviour of catalyst particles with dynamically varying surface exposures to gas and liquid is a decisive building block in a multi-scale model architecture. However, to date no systematic evaluation of various dynamic modelling approaches for this task is available, and transient effects impose additional criteria ∗ Corresponding author. Tel.: +49 231 755 4347; fax: +49 231 755 2698.
E-mail addresses: [email protected] (W. Dietrich), [email protected] (D.W. Agar). 0009-2509/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2005.03.054
that mean a comparison of the corresponding steady-state models is invalid. Due to the industrial significance of trickle bed reactors, extensive research activities have been devoted to understanding the complex interactions between hydrodynamics, transport processes and chemical reaction, resulting in a number of basic models permitting a relatively good description of steady-state behaviour (Dudukovic et al., 2002). Several comparative studies on the impact of various model assumptions and numerical solution methods on effective reaction rates have been published over the last 20 years. Approximate expressions (Mears, 1974) and numerical solutions (Dudukovic and Mills, 1978) for effectiveness factors have been developed and compared for an infinite catalyst slab with partial external surface wetting and found to be adequate under very strong internal mass transfer limitations. Similar results were obtained for catalyst cubes (Herskowitz et al., 1979) and various geometries (Tan, 1988). The impact of evaporation taking place due to strongly exothermic reactions, which leads to incomplete internal pore wetting and multiple steady states, has also been investigated
W. Dietrich et al. / Chemical Engineering Science 60 (2005) 6254 – 6261
(Harold, 1988; Watson and Harold, 1994). The number and size of liquid rivulets covering the external particle surface was found to influence the effective reaction rate under steady-state conditions strongly (Ring and Missen, 1986; Zhu and Hofmann, 1997), except when extreme mass transfer limitations are present. The more uniformly the liquid is spread over the particle surface, the higher the effectiveness factor. Thus a broad selection of quantitative information is available to serve as basis for the selection of steady-state partially wetted catalyst particle models. An unsteady-state operating strategy leads to a further increase in the complexity of the physical and chemical processes determining reactor behaviour and therefore necessitates a more refined modelling approach, with special attention being given to the correct accounting for material accumulation within the particle pores. For dynamic operating conditions several models have been proposed, which primarily treat the phenomena occurring at the reactor scale (Gabarain et al., 1997; Stegasov et al., 1994). A one-dimensional dynamic particle model was proposed by Boelhouwer (2001) which distinguishes between continuously and periodically wetted zones. However, comparative studies examining the effect of different levels of model detail on the accuracy of simulation results, similar to those available for steady state conditions, have not yet been published. In this work, an evaluation of three dynamic particle-scale modelling approaches based on different assumptions with respect to intra-particle concentration profiles and a periodically fluctuating wetting of the external particle surface is presented. A single partially wetted particle exposed to periodic variations of the wetted fraction of external surface area is modelled as an essential module for use in multi-scale trickle-bed reactor models. The primary focus lies in the degree of symmetry assumed in concentration profiles and the corresponding modelling detail that results. The comparison of the various models is based on a selection of benchmark scenarios representing dynamic reaction conditions occurring over the whole length of a trickle-bed reactor. Special attention was paid to reaction conditions that have been found to cause significant deviations for approximate models under steady-state conditions, i.e., moderate Thiele moduli and strong external mass transfer limitations with a limiting gas phase reactant.
2. Modelling The behaviour of a partially wetted catalyst particle is modelled for a slab geometry under isothermal reaction conditions. It is assumed that internal wetting is complete due to capillary forces and that the wetting efficiency of the external particle surface is dependent on the liquid flow rate according to the correlation proposed by El-Hisnawi et al. (1982). No influence of the wetting geometry is considered, as the wetted surface area is averaged at the particle-scale
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resulting in a single wetted and dry fraction. Mass transfer between gas and liquid phases and the catalyst surface is described using film theory and Henry’s law and film theory, respectively. A general reaction equation (Eq. (1)) is formulated for the reaction of a non-volatile liquid with a sparingly soluble gaseous reactant obeying reaction kinetics which are first order with respect to both reactants: L1 (l) + G(g) → L2 (l),
(1)
r = kcL1 cG .
(2)
Three different modelling approaches have been considered, which mainly differ in the symmetry and thus spatial resolution of concentration profiles within the particle. A schematic comparison of the model concepts is depicted in Fig. 1. In the first case (see Fig. 1a), two-dimensional concentration profiles are calculated whilst diffusional mass transfer inside the particle is allowed in both x- and y-directions. This leads to the particle mass balances as proposed by Dudukovic and Mills (1978), which has been extended to account for dynamic conditions. jcp,L1 j2 cp,L1 j2 cp,L1 = De,L1 − kcp,L1 cp,G , (3) + jt jx 2 jy 2 jcp,G j2 cp,G j2 cp,G − kcp,L1 cp,G . = De,G + jt jx 2 jy 2
(4)
The partial wetting of the particle surface, which changes with time depending on the liquid flow rate, is considered in the surface boundary conditions. For the liquid reactant L1 mass transfer can only occur at the wetted surface (Eq. (5)):
jcp,L1 = Bi l,L1 cl,L1 − cp,L1 |x=0 jx x=0
jcp,L1 = 0 for w (t) < y < 1. jx x=0
for 0 < y < w (t),
(5) (6)
Depending on the y-coordinate at the particle surface and the time variable wetting fraction w (t) the boundary conditions for component G account for mass transfer from either the liquid (Eq. (7)) or gas (Eq. (8)) phases. Elsewhere flat profiles were assumed at the domain boundaries (Eq. (9)): jcp,G jx x=0 = Bi l,G (cl,G − cp,G |x=0 ) for 0 < y < w (t), (7) jcp,G jx x=0 cg,G for w (t) < y < 1, (8) = Bi g,G − cp,G x=0 HG jcp,i jcp,i =0, =0, with i={L1 , G}. (9) jx x=dp jy y={0,lp }
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W. Dietrich et al. / Chemical Engineering Science 60 (2005) 6254 – 6261
(g)
pw
(g)
ηw (t)
ηw (t)
ηw,min
ηw,min (I)
pw
(g) ηw (t)
x cw
(I)
ηw,min
cw
(I)
y x
x
(a)
Case 1 (2D)
(b)
Case 2 (1D)
(c)
Case 3 (0D)
Fig. 1. Schematic representation of the model strategies and assumptions.
A more simplified model, as suggested by Boelhouwer (2001), is represented by case 2 (see Fig. 1b), in which the domain is divided into a permanently wetted zone and a zone with periodically changing wetting. The relative portions of the total particle volume assigned to each zone are determined according to the minimum and maximum wetting efficiency during the periodic wetting process. These values have to be determined a priori by evaluating the wetting correlation for the upper and lower boundaries of the intended flow rate modulation. The concentration profiles are assumed to be symmetrical in each zone without any lateral mass transfer between different zones, which results in one-dimensional mass balances for each zone: jccw,L1 j2 ccw,L1 w,min =w,min De,L1 − kccw,L1 ccw,G , jt jx 2 (10) (w,max −w,min )
jcpw,L1 jt
j2 cpw,L1 =(w,max −w,min ) De,L1 −kcpw,L1 cpw,G , jx 2 (11) 2 jccw,G j ccw,G w,min −kccw,L1 ccw,G , =w,min De,G jt jx 2 (12) (w,max − w,min )
jcpw,G jt
j2 cpw,G =(w,max −w,min ) De,G −kcpw,L1 cpw,G . jx 2 (13)
For the permanently wetted zone only liquid–solid mass transfer has to be considered in the boundary conditions: jccw,L1 w,min = w,min Bi l,L1 (ccw,L1 |x=0 − cl,L1 ), jx x=0 (14) jccw,G w,min = w,min Bi l (ccw,G |x=0 − cl,G ). (15) jx x=0
For the periodically wetted zone both gas–solid (only component G) and liquid–solid mass transfer are permitted and weighted according to the wetted particle surface fraction: (w,max − w,min )
jcpw,L1 jx x=0
= (w − w,min )Bi l,L1 (ccw,L1 |x=0 − cl,L1 ), jcpw,G (w,max − w,min ) jx x=0 cg,G = (w,max − w )Bi g,G cpw,G |x=0 − H + (w − w,min )Bi l,G (ccw,G |x=0 − cl,G ).
(16)
(17)
The third case is a further simplification of case 2, which can be derived if intra-particle profiles are neglected and an average particle concentration (see Fig. 1c) is employed. Mass transfer inside the particle is described using an effective internal mass transfer coefficient based on the assumption of linear driving forces: ke,i =
2De,i . dp
(18)
As in case 2 for the continuously wetted zone, only mass transfer from the liquid phase is considered (Eqs. (19) and (20)), while in the periodically wetted zone mass transfer of G from gas can take place as well (Eqs. (21) and (22)):
w,min
jccw,L1 1 1 −1 =w,min + (cl,L1 − ccw,L1 ) jt kls,L1 ke,L1 − w,min kccw,L1 ccw,G , (19)
jccw,G 1 1 −1 = w,min w,min + (cl,G − ccw,G ) jt kls,G ke,G − w,min kccw,L1 ccw,G , (20)
W. Dietrich et al. / Chemical Engineering Science 60 (2005) 6254 – 6261
jcpw,L1 jt −1 1 1 = (w − w,min ) + (cl,L1 − cpw,L1 ) kls,L1 ke,L1 − (w,max − w,min )kcpw,L1 cpw,G , (21)
(w,max − w,min )
jcpw,G (w,max − w,min ) jt 1 1 −1 + (cl,G − cpw,G ) = (w − w,min ) kls,G ke,G 1 1 −1 cg,G + − cpw,G + (w,max − w ) kgs,G ke,G H − (w,max − w,min )kcpw,L1 cpw,G . (22) The three different particle models were each incorporated into a differential axial element of a fixed-bed reactor. The corresponding mass balance equations are given as follows1
jcl,L jcl,L = −ul −(w,max −w )kls,L ap (cl,L −cpw,L |x=0 ) jt jz (23) −(w − w,min )kls,L ap (cl,L − ccw,L |x=0 ), c jcl,G jcl,G g,G = − ul + ka gl,G − cl,G jt jz H − (w − w,min )kls,G ap (cl,G − cpw,G |x=0 ) − w,min kls,G ap (cl,G − ccw,G |x=0 ),
jcg,G jcg,G = − ug − ka gl,G jt jz
c
g,G
H c
− (w,max − w )kgs,G ap
− cl,G g,G
H
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Table 1 Simulation parameters and periodic operating conditions Catalyst characteristics Pore depth Particle length Effective diffusivity Bed porosity
dp lp De,i
Correlated parameters Wetting efficiency w Liquid holdup l Liquid–solid mass transfer coeff.a kls Gas–solid mass transfer coeff.a Gas–liquid mass transfer coeff.a Periodic operation parameters Period length Period split Superficial gas velocity Superficial liquid velocity (min) Superficial liquid velocity (max)
kgs kgl
Value 10−4 m 10−3 m De,i = 0.1 · Di 0.4 Reference (El-Hisnawi et al., 1982) (Larachi et al., 1991) (Dharwadkar and Sylvester, 1977) (Schlünder, 1994) (Midoux et al., 1984)
Value 400 s 0.5 (BASE–PEAK) uG 0.2 m/s ul,min 0.001 m/s ul,max 0.01 m/s
a Mass transfer coefficients have been scaled to yield the desired Bi numbers.
3. Results (24)
− cpw,G |x=0 . (25)
To account for the attenuation characteristics of pulses travelling along a periodically operated trickle-bed reactor, different gradients and amplitudes of liquid pulses were imposed, corresponding to the whole spectrum of conditions occurring along the length of the reactor. The liquid flow modulation affects the model results through the correlations that have been implemented (see Table 1) to calculate the wetting fraction on the one hand and mass transfer coefficients from liquid to solid on the other. The partial differential equations were solved based on the method of lines (Schiesser, 1991) using finite differences. In order to resolve the steep gradients arising under more severe reaction conditions ( 5, Bi l 1) a locally refined grid was used in the spatial domain for cases 1 and 2. Initial conditions for all variables and cases were set to correspond to the steady-state solution for the average flow rate of periodic operation. 1 In particular these equations apply for case 2. For case 1 mass transfer fluxes would have to be integrated over the particle length. For case 3 internal mass transfer coefficient would have to be taken into account as in Eqs. (19–22).
Simulations were carried out for parameters covering a wide range of typical reaction conditions observed in industrial trickle bed reactors. Mass transfer coefficients and kinetic constants were varied to give Thiele moduli from 0.1 to 10 and Biot numbers Bi l,G from 0.1 to 100 with respect to the dissolved gaseous component. The Biot number for direct gas–solid mass transfer Bi g,G was set at a constant value of 100 for all cases examined. The two different scenarios of liquid flow rate variation under the cyclic conditions indicated in Table 1 were investigated, namely a symmetric rectangular profile representing conditions at the reactor inlet and a continuous asymmetric S-shaped profile typically found further downstream in a periodically operated trickle-bed reactor (see Fig. 2). The shape of the liquid flow rate profile was taken from reactor scale simulations using a non-linear liquid hold-up correlation that have been presented elsewhere (Lange et al., 2004). Under reaction conditions of practical relevance, e.g. = 1 . . . 5 and Bil,G = 1, the simulations predict conversion enhancements through periodic operation of c. 30% for a rectangular flow rate modulation profile. Fig. 3 shows values for two different sets of reaction conditions corresponding to a differential reactor element with a length of 0.01 m. If these values are extrapolated up to catalyst bed lengths of 0.2 . . . 0.5 m they are qualitatively consistent with the results of experimental studies reported in the literature (Khadilkar et al., 1999; Lange et al., 1994; Turco et al., 2001). Under steady-state conditions as well as for periodic operation according to scenario 1 the models of cases 1 and 2
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W. Dietrich et al. / Chemical Engineering Science 60 (2005) 6254 – 6261 VL
Fig. 4 depicts the results for the first scenario. Excellent agreement can be observed between the 1D and 2D-models while the simplified case 3 exhibits strong deviations from the other two except at low Thiele moduli. With decreasing external mass transfer limitations, i.e., increasing Bi l , an even more pronounced underestimation of the average reaction rate is obtained in case 3. In order to analyse these findings, intra-particle reaction rate profiles at a given time shortly after reducing the liquid flow rate and wetting efficiency, respectively, are illustrated in Fig. 5. Under these conditions the dry portion of the particle surface is exposed to high concentrations of G in the gas phase, while the liquid reactant inside the pores is not yet depleted. The resulting strong driving forces should emphasise the consequences of differences in the underlying model assumptions. It becomes apparent that an averaging of mass balances is only appropriate along the particle length dimension (see Fig. 5a), where the profiles predicted by case 1 (2D) and case 2 (1D) are in close agreement. The assumption of a homogenous concentration throughout the whole pore length in case 3 gives rise to a buffer effect damping the steep intra-particle gradients, especially in the periodically wetted zone (see Fig. 5b). At high Thiele moduli in particular this causes a considerable diminishing of the reaction driving forces in comparison to the other two models. The comparison of the different models for the second scenario is illustrated in Fig. 6 . In this instance the onedimensional model (case 2) shows more obvious deviations from the two-dimensional model (case 1) than in the first scenario, in particular with increasing mass transfer limitations, be they external or internal. Based on the reaction rate profiles shown in Fig. 7 two explanations for the discrepancies between cases 1 and 2 may be identified. Due to the slower propagation of the wetting front from minimum to maximum wetting conditions, the dry and wetted fractions of the particle change continuously
VG . VL
Scenario 1
t . VL
Scenario 2
t
Fig. 2. Pulse attenuation along reactor length showing the two benchmark scenarios investigated.
predict almost the same conversion while case 3 yields values which are about 20% lower. However, for the second scenario a very strong increase in conversion is predicted by case 2 in contrast to a decrease for case 1 and a moderate increase for case 3. A detailed performance comparison of the three models was carried out based on a modified catalyst effectiveness factor , which was defined by the ratio of time-averaged periodic reaction rates to the corresponding steady-state reaction rates without mass transfer limitations.
Case 1 (2D) Case 2 (1D) Case 3 (0D) = 1, Bil,G = 1
2.0%
= 5, Bil,G = 1
4.0%
+18% conversion [ - ]
conversion [ - ]
+17% 1.5%
+29% 1.0% 0.5% 0.0%
3.5%
+28% 1.0% 0.5% 0.0%
steady state
scenario 1
scenario 2
steady state
scenario 1
scenario 2
Fig. 3. Conversion enhancement by periodic operation for = 1 . . . 5, Bi l = 1. Comparison of scenarios 1 and 2 (see Fig. 2).
W. Dietrich et al. / Chemical Engineering Science 60 (2005) 6254 – 6261 1
1 = 0.1
= 0.1 0.1
0.1
=1
=1
0.01
η′
η′
6259
0.001
Case 1 (2D) Case 2 (1D) Case 3 (0D)
= 10
0.0001 0.1
10
1
100
Bil
0.01 = 10
Case 1 (2D) Case 2 (1D) Case 3 (0D)
0.001
0.0001 0.1
1
10
100
Bil
Fig. 4. Simulation results for reactor inlet section.
Fig. 6. Simulation results for downstream trickle-bed scenario with asymmetric s-shaped pulses.
with time. However, in the two-zone models of cases 2 and 3 the zones that are used for averaging the mass balances are constant. As illustrated in Fig. 7a, the same concentrations and reaction rates are assumed to prevail throughout the entire periodically wetted zone. This results in an inadequate weighting of reaction rates compared to the two-dimensional mass balances of case 1. Furthermore, at the surface of the periodically wetted zone mass transfer from the gas and liquid phases is weighted according to the wetting efficiency. When the wetting efficiency is between maximum and minimum values the partially wetted surface fraction in case 2 is exposed to both phases at once, resulting in an enhancement of the mass transfer from the gas phase whilst still maintaining the supply of liquid reactant. This results in higher concentrations of both reactants in the partially wetted zone and thus an overestimate of the reaction rate in comparison to case 1, which becomes even more pronounced with increasing external mass transfer limitations. In the first scenario and in general for case 1, no simultaneous mass
transfer from gas and liquid phases to a single pore can take place. The model in case 3 exhibits a combination of the phenomena described above for case 2 together with the damping of concentration fluctuations found for the first scenario. If external mass transfer limitations are very strong the overestimation of mass transfer due to the two-zone approach and the underestimation of the reaction rate caused by neglecting intra-particle profiles cancel one another out. Since the extent of the individual contributions of these opposing effects cannot be estimated a priori and varies with time, it has to be concluded that the assumption of an average pore concentration instead of allowing for intra-particle concentration profiles is unsuitable for quantifying accumulation effects within the particle pores. If the simulation results are compared with the behaviour of periodically operated trickle-bed reactors that has been found experimentally, the two-dimensional model of case 1 seems to provide more realistic results than the two-zone
Fig. 5. Intra-particle reaction rate profiles for reactor inlet section (t = 210 s, = 1, Bi l = 1).
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Fig. 7. Intra-particle reaction rate profiles for downstream trickle-bed scenario with asymmetric s-shaped pulses (t = 210 s, = 1, Bi l = 1).
approaches. On the one hand it would be expected that the conversion enhancement decreases asymptotically towards the steady-state value with increasing pulse attenuation. While case 1 shows a decrease in conversion enhancement for s-shaped profiles (scenario 1) in comparison to step profiles (scenario 2), case 3 predicts a moderate increase and case 2 a considerable increase. On the other hand it can be assumed that the existence of large numbers of “semi-wetted” pores as implied by the two-zone models is somewhat unrealistic and causes an overestimation of mass transfer rates. A stepwise motion of the wetting line due to film rupture and rivulet formation seems physically more credible (Khanna and Nigam, 2002). The experimental data required for a quantitative discrimination of the particle models will have to be obtained from reaction and mass transfer rate measurements employing a differential reactor in order to establish distinct flow rate profiles and eliminate the attenuation effects, which would occur in a full length reactor.
4. Conclusions Based on the comparison of three different modelling strategies for a partially wetted slab key aspects in the selection of an appropriate dynamic model for a partially wetted catalyst particle have been demonstrated. Not only the extent of the mass transfer limitations but also the time dependency of the surface wetting is an important parameter that has to be taken into consideration. When integrated into a reactor scale model, a two-dimensional particle model requires a very large number of grid points if a reasonably high spatial resolution is to be achieved and thus rapidly becomes numerically exorbitant. A compromise between accuracy and computational effort could involve a multiple-zone approach. The less steep the profiles the greater the number of zones that are required. Thus, the number of zones would have to be increased over the length of the reactor according to the attenuation characteristics of the liquid pulses. Addi-
tionally, an adjustment of the boundaries between continuously and periodically wetted zones is necessary along the length of the reactor, as the amplitude of pulses decreases from top to bottom. This necessitates preliminary simulations of the hydrodynamics in order to provide the minimum and maximum wetting efficiency values. Considering the strong influence of the wetting geometry on catalyst performance reported under steady state conditions, the detailed investigation of the wetting morphology for various particle geometries for periodic variations of the surface wetting would be an important step in the analysis of the dynamic trickle-bed reactor behaviour. The rigorous free surface simulation of multiphase flows in randomly packed beds of arbitrarily shaped porous particles is likely to remain an insoluble problem even in the medium term. Thus, the development of models for periodically operated multiphase reactors will have to rely on approximate particle models for some time to come. This work represents a tentative first step towards acquiring a broad basis of experience to aid the selection of dynamic models for the description of periodically wetted catalyst particle systems.
Notation ap Bi g,i Bi l,i cg,i cl,i ccw,i cpw,i
specific particle surface, m2 /m3 k dp Biot number for dry surface, Bi g,i = gs,i De,i , dimensionless k dp Biot number for wetted surface, Bi l,i = ls,i De,i , dimensionless bulk gas phase concentration, mol/m3 bulk liquid phase concentration, mol/m3 concentration in continuously wetted zone, mol/m3 concentration in periodically wetted zone, mol/m3
W. Dietrich et al. / Chemical Engineering Science 60 (2005) 6254 – 6261
cpw,i cp,i De,i dp H ka gl k ke,i kls kgs lp ug ul l
w w,max w,min cw pw
concentration in periodically wetted zone, mol/m3 concentration in particle pores (2D case), mol/m3 effective diffusivity, m2 /s 1lpore length, m Henry’s constant of component G, dimensionless gas–liquid mass transfer coefficient, 1/s reaction rate constant, m3 /mol s effective mass transfer coefficient, m/s liquid–solid mass transfer coefficient, m/s gas–solid mass transfer coefficient, m/s particle size, m superficial gas velocity, m superficial liquid velocity, m bed voidage, dimensionless liquid hold-up, dimensionless
kc
c
l,L1 Thiele modulus, =dp Dl,G , dimensionless e,G cl,G modified particle effectiveness factor, dimensionless wetting efficiency, dimensionless maximum wetting efficiency arising over cycle, dimensionless minimum wetting efficiency arising over cycle, dimensionless continuously wetted domain partially wetted domain
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