STUDY OF LIQUID DISTRIBUTION IN A TRICKLE BED REACTOR

Transcription

1 STUDY OF LIQUID DISTRIBUTION IN A TRICKLE BED REACTOR Manuel Martínez Supervisors: Dr. Jordi Pallarès Dr. Francesc Xavier Grau June 2007 Master in Chemical and Process Engineering

2 Contents 1 Introduction and background 1 2 Formulation of the problem to investigate Model validation Numerical simulation of the trickle-bed reactor Objectives 6 4 Methodology Mathematical model Porous media model Granular phase model Model validation Numerical simulation of the trickle bed reactor Numerical model Results and discussion Model validation Porous media model Granular phase model Numerical simulation of the trickle bed reactor Central region Near-wall region Conclusions 21 7 Future work 22

3 List of Figures 2.1 Sketch of the reactor Liquid volume fraction contours at the inlet of the first bed Isosurface of liquid volume fraction of Computational domain for the central region simulation Computational domain for the near-wall region simulation Contours of liquid volume fraction of 0.12 in a r-z plane of the cylindrical bed. Porous media model validation Contours of liquid volume fraction of 0.12 in a r-z plane of the cylindrical bed. Effect of capillary term Contours of liquid volume fraction of 0.12 in a r-z plane of the cylindrical bed. Granular phase model validation Liquid distribution in the zone surrounding one chimney in the central region of the reactor. Isosurface of volume fraction of Contours of liquid volume fraction on horizontal slices of the four different beds. The position of the slices is indicated in Figure Liquid distribution in the zone surrounding one chimney in the near-wall region of the reactor. Isosurface of volume fraction of Contours of liquid volume fraction on horizontal slices of the four different beds. The position of the slices is indicated in Figure

4 List of Tables 2.1 Bed porosities and lengths Physical properties Stages in the development of the code

5 Nomenclature A, coefficient for fluid-particle interaction, m 2 B, coefficient for fluid-particle interaction, m 1 C 2, inertial resistance coefficient, m 1 d e, equivalent diameter, m d min, minimum equivalent 3 diameter of the area between three spheres in contact, m, where d min = 0.5 d π p d p, particle diameter, m f, wetting efficiency, dimensionless F, force per unit volume, N m 3 F D, drag force per unit volume, N m 3 F ρg ρ l, pressure factor, dimensionless g, gravity acceleration, m s 2 p, capillary pressure, Pa S m, source term, kg s 1 m 3 u, interstitial velocity, m s 1 U, superficial velocity, m s 1 v, velocity, m/s t, time, s X, exchange coefficient, kg m 3 s 1 Greek letters α, permeability, m 2 ɛ, porosity, dimensionless µ, viscosity, Pa s θ, volume fraction, dimensionless ρ, density, kg m 3 σ s, surface tension, N m 1 τ, stress, N m 2

6 Subscripts l, liquid phase g, gas phase s, solid phase k, fluid

7 Acknowledgements I would like to thank My supervisors, Dr. Pallarès and Dr. Grau, for their help and support. Everyone at ECoMMFiT. Repsol-YPF for the financial support.

8 Abstract Numerical simulations of isothermal non-reactive two-phase flow distribution in the packed beds of a reactor used for fuel hydrodesulfuration are reported. The reactor considered has, above the beds, a distribution tray equipped with chimneys. The boundary conditions at the inlet of the reactor beds were obtained from simulations of the flow at the exit of a chimney [Martínez et al. [1]]. An Eulerian three-phase model that considers the particles of catalyst as a granular static phase has been used following the Holub single slit model for particle fluid interaction to compute the liquid-solid and gas-solid drag coefficients. A numerical simulation of the dispersion of waterair flow in a column of glass beads using this model was initially carried out and results were found to be in reasonable agreement with numerical and experimental data available in the literature. The simulations consider the flow dispersion in the central region of the reactor beds as well as in the region close to the cylindrical lateral wall of the reactor. In both cases the distributions of the liquid concentration do not change significatively as the depth of the bed is increased except at the top part of the first bed, in the third bed and at the interface of two beds with different porosity. The liquid in the near-wall region tends to attach the wall of the reactor as the depth of the bed is increased.

9 Chapter 1 Introduction and background In the trickle bed reactors used in the petrochemical industry for fuel hydrodesulfuration processes the gas and the liquid flow co-currently downward through a packed bed of catalytic solid particles. The uniformity of the twophase flow in the bed is an important parameter for the correct operation of the catalyst. For example, a poor distribution of the liquid-gas mixture reduces the effective volume of the reactor in which the hydrodesulfuration takes place and this can produce hot spots that may decrease the life of the catalyst. The interest to obtain a uniform well mixed two-phase flow in catalytic beds has motivated most of the studies carried out to determine the liquid spreading in packed beds. Cihla and Schmidt [2] proposed a series of analytical solutions to an advection-diffusion equation to predict the liquid spreading in a column, for different types of inlet distributors. The mathematical model assumes a radial diffusion term with an adjustable diffusion coefficient and an advection term along the axial direction. Baker et al. [3] carried out experimental studies in columns of different sizes and filled with particles of different shapes and sizes to evaluate the radial liquid spreading. These authors found that initial uniform distribution is essential, since the flow from a single stream needs a length equivalent to four or five column diameters to become uniform. Jiang et al. [4, 5] simulated the macroscale multiphase flow in packed beds and compared the results with available experimental data. The CFDLIB package, based in the Eulerian k-fluid model, was used in their simulations. The Holub (Holub et al. [6]) model for particle-fluid interactions and the Attou (Attou and Ferschneider [7]) model for gas-liquid interaction were implemented in the model. The comparison showed that the k-fluid model predicted reasonably well the pressure gradient and the global liquid saturation in the trickle flow regime for liquid upflow in a cylindrical packed bed. Boyer et al. [8] studied liquid spreading from a point source in a 1

10 column through gamma-ray tomography and CFD simulation. CFDLIB was also used in their simulations. The comparison showed that the prediction of the radial liquid spreading was strongly affected by the capillary pressure term. Several models for this term were tested and a modification of this capillary pressure term was proposed to fit experimental data better. 2

11 Chapter 2 Formulation of the problem to investigate The prediction of the liquid and gas distributions in a porous bed of a catalytic reactor by means of numerical simulations is the main objective of this study. In order to achieve this, it is convenient to validate the different models available in the commercial CFD code Fluent,which has been used in this study, before carrying out the simulation of the flow in the reactor. Thus, two problems have been investigated: the model validation using data available in the literature and the numerical simulation of the isothermal non-reactive two-phase flow dispersion in the beds of the reactor. 2.1 Model validation Boyer et al. [8] studied numerically and experimentally the two phase flow dispersion in a vertical cylindrical column of spherical glass particles of 1.99 mm of diameter. The column had an inner diameter of 0.4 m and a bed height of 1.80 m. Although the axial porosity distribution is reported, an average porosity of has been considered for the present simulation. The fluids used were water and air at ambient temperature. The liquid was introduced through a tube of 8 mm of inner diameter located in the center of the top of the bed while the air was injected all around the liquid tube over the whole cross-section of the bed. The liquid and gas flow rates were m 3 /h and 45 m 3 /h, respectively. 3

12 2.2 Numerical simulation of the trickle-bed reactor The reactor considered has a distribution tray with chimneys above the catalyst bed to produce a uniform two-phase flow distribution at the inlet of the bed. The bed is divided in four sections with different particle sizes and porosity. A sketch of the reactor is shown in Figure 2.1. The beds porosities and lengths are shown in Table 2.1. The liquid has an initial distribution before entering the beds, due to the effect of the distribution tray. The relevant physical properties of the liquid and the gas, which are assumed to be constant, are shown in Table 2.2. It is desired to know the liquid distribution along the reactor. Once the most adequate model has been chosen in the first part of the study (model validation), numerical simulations of the reactor are performed. As a first step, this model has been used to simulate the isothermal three-dimensional two-phase flow dispersion in the reactor bed without considering the chemical reactions and the heat transfer processes that occur in a real reactor. The implementation of the chemical reactions and the heat transfer is left for future studies. Table 2.1: Bed porosities and lengths Bed Porosity Bed length (mm) Table 2.2: Physical properties Density (kg m 3 ) Viscosity (kg m 1 s 1 ) Liquid Gas Two different cases have been studied: the first one considers the flow dispersion in the bed underneath a group of chimneys installed in a central region of the reactor, far from the cylindrical lateral wall of the reactor. The second case considers the dispersion near the walls of the reactor. 4

13 Figure 2.1: Sketch of the reactor 5

14 Chapter 3 Objectives The specific objectives are: 1. To validate the different models available in the commercial CFD code Fluent to predict the two-phase flow dispersion in porous media. 2. To obtain the liquid distribution along the reactor simulating the packed beds with the most adequate model according to Objective 1. 6

15 Chapter 4 Methodology The procedure to carry out the simulations has been the following: 1. Grid generation using the commercial software Gambit. 2. Numerical simulation using the commercial software Fluent. (a) Choice of discretization schemes for the equations to be solved. (b) Set the boundary conditions. 3. In the case of Objective 1, comparison with experimental data available in the literature. 4.1 Mathematical model Two models for the simulation of two-phase flow in porous media have been tested to predict the experimental and numerical results of Boyer et al. [8]. The models considered are the porous media model and the granular phase model. Both models consider the continuity (Eq. 4.1) and momentum (Eq. 4.2) equations for each phase as starting point. t (θρ) + (θρ v) = S m (4.1) t (θρ v) + (θρ v v) = θ P + (τ) + θρ g + F (4.2) 7

16 4.1.1 Porous media model In this model a two-phase flow (gas and liquid) is solved, and the solid particles are modeled as a porous media. The mathematical treatment of this media is carried out through the addition of a pressure drop term (Eq. 4.3) based on the Ergun equation to the momentum equations. This term consists of a viscous term, proportional to the viscosity (µ) of the fluid phase, and an inertial term, proportional to the square of the velocity (U). P = µ α U i + C 2 ρ 2 U i U i (4.3) In Eq. 4.3 the viscous and inertial resistance coefficients, α and C 2, can be computed using Eqs. 4.4 and 4.5, respectively: 1 α α = d2 p Granular phase model ɛ 3 (1 ɛ) 2 (4.4) C 2 = 3.5 d p (1 ɛ) ɛ 3 (4.5) In this model particles are defined as a static granular phase with the corresponding volume fraction. The phase interaction is modeled through drag coefficients, and in this study, they have been modeled according to the single-slit model proposed by Holub et al. [6]. In this model the drag force, that has to be included in the corresponding momentum equations for each phase, can be expressed as, F Dks = θ k θ s X ks (u k u s ) (4.6) where the exchange or drag coefficient X k s is defined as and X ks = ( A ks µ k U k + B ks ρ k U 2 k (1 ɛ)2 A ks = 180 θ 3 d 2 p ) 1 (1 ɛ) u k (4.7) (4.8) (1 ɛ) B ks = 1.8 (4.9) θ 3 d p 8

17 The particular forms of the drag forces used for the simulation are given in Eqs and have been included in the simulation code through userdefined functions Model validation The two-dimensional axisymmetric computational domain was divided into rectangular finite volumes using a uniform grid distribution similar to that used by Boyer et al. [8], with 1 cm spacing in axial direction and 0.4 cm in radial direction. As reported by Jiang et al. [9] and also according to preliminary simulations using the k-ɛ turbulence model, the Reynolds stress term does not influence the behavior of the two phase flow in the bed for the flow conditions considered. Consequently, the simulations were carried out without any turbulence model Numerical simulation of the trickle bed reactor Central region The computational domain for this case consists in a prism of rhombic cross section. The dimensions of this cross section, which is centered in the projection of one chimney, were selected according to the symmetry elements of the distribution of chimneys in the tray. The velocity and volume fraction distributions imposed on the inlet top surface of the computational domain were taken from a previous simulation of the flow in a chimney reported in Martínez et al. [1] and shown in Figure 4.1. Figure 4.2 shows an isosurface of volume fraction 0.3 of the flow inside a chimney. The liquid, as can be seen, enters the chimney through the lateral holes, while the gas enters through the top opening.as can be seen in both figures, two main regions are formed. Figure 4.3 shows the computational domain for this case. The projection of the chimney is indicated in Figure 4.3 on the top of the bed. The four different beds are also indicated in this figure. Periodic boundary conditions are imposed on the lateral faces of the domain to model the effect of the surrounding chimneys. The height of the fourth bed has been considered to be 450 mm, because preliminary simulations carried out predict fully developed flow conditions for bed depths larger than 70 mm. A hexahedral regular mesh of cells has been used in the numerical simulation corresponding to this case. 9

18 Figure 4.1: Liquid volume fraction contours at the inlet of the first bed Near-wall region Figure 4.2: Isosurface of liquid volume fraction of 0.3 The computational domain for the near-wall region, shown in Figure 4.4, consists in a prism of rectangular cross section. The projections of the chim- 10

19 neys contained in the rectangle representing the top of the bed are indicated in Figure 4.4. As shown in Figure 4.4, the top surface of the computational domain contains the circular projection of one chimney located near the wall and three halves of the projection of one chimney distributed along the perimeter of the top surface. Specific distributions of volume fraction and inlet velocity are imposed on the top of the bed as inlet conditions, according to numerical simulations of the flow at the exit of one chimney, reported in Martínez et al. [1] and previously shown in Figure 4.1. The no slip condition is imposed at the wall and symmetry boundary conditions are imposed at the lateral faces of the domain, which are perpendicular to the y-direction. At the boundaries opposed to the wall, the distributions of velocity and volume fraction imposed were taken from a simulation of the flow in a central region of the bed. A hexahedral regular mesh of cells has been used in this simulation. Figure 4.3: Computational domain for the central region simulation 11

20 Figure 4.4: Computational domain for the near-wall region simulation 4.2 Numerical model In all cases, pressure-velocity coupling has been solved through Phase-coupled SIMPLE algorithm ([10]), while momentum and volume fraction equations have been discretized with a first order Upwind scheme ([10]). 12

21 Chapter 5 Results and discussion The results have been divided in two parts according to the previously defined objectives: 5.1 Model validation Porous media model Figure 5.1 shows the comparison of the experimental and numerical results of Boyer et al [8] and the present numerical results obtained with the porous media model. The lines plotted in this figure correspond to a value of liquid volume fraction of It should be noted that the liquid is introduced at the center (r<4 mm) of the top of the bed (z=1.8 m). As can be seen in Figure 5.1, the liquid spreading for the conditions considered is clearly underpredicted by the porous media model. A simulation carried out with a porosity of 0.339, corresponding to the minimum value of the distribution reported by Boyer et al. [8], shows no significant differences with that presented in Figure 5.1 using a porosity value of As shown in Figure 5.1, the porous media model is not capable to reproduce the data reported by Boyer et al. [8]. As suggested by Jiang et al. [4, 5] the inclusion of a capillary pressure term Eq. 5.1 in the momentum equations increases the flow dispersion. θ l (p l p g ) (5.1) Two cases with different capillary pressure correlations have been simulated, one with the correlation given by Attou and Ferschneider [7] (Eq. 5.2) and another with the correlation proposed by Grosser et al. [11] (Eq. 5.3). 13

22 Figure 5.1: Contours of liquid volume fraction of 0.12 in a r-z plane of the cylindrical bed. Porous media model validation p l = p g 2σ s (1 f) [ θ s 1 θ g ] 1 [ ] F d p d min ( ) ρg [ ( )] 180θs 1 θs θ l p l = p g (1 f) σ s ln (1 θ s ) d e θ l ρ l (5.2) (5.3) In Eq. 5.2 the pressure factor, F, can be computed as: F ( ) ρg ρ l = ρ g ρ l (5.4) and the wetting efficiency, f, is given in Eq. 5.5 according to [12]: f = ( θl 1 θ s ) (5.5) 14

23 It should be noted that the computation of capillary pressure term has been added to the simulation code using a user-defined function. The effect of the addition of the capillary pressure term using Eq. 5.2 on the liquid distribution is shown in Figure 5.2. The results obtained using Eq. 5.3 are not significantly different in comparison with those shown in Figure 5.2 and consequently have been omitted. As can be seen, the model with the capillary term predicts approximately the same liquid spreading in comparison with the default porous media model and it does not provide a noticeable improvement of the radial spreading. Figure 5.2: Contours of liquid volume fraction of 0.12 in a r-z plane of the cylindrical bed. Effect of capillary term Granular phase model Figure 5.3 shows contours of liquid volume fraction for the granular phase model and their comparison with the experimental and numerical simulation results of Boyer et al. [8]. As it can be seen in Figure 5.3.d, the granular 15

24 phase model fits reasonably well the experimental data near the exit of the bed, although in the initial section of the domain, the model over predicts the liquid spreading that reaches a developed condition flow for z<1.7 m. Figure 5.3: Contours of liquid volume fraction of 0.12 in a r-z plane of the cylindrical bed. Granular phase model validation 5.2 Numerical simulation of the trickle bed reactor The granular phase model, with the fluid-particle interaction model of Holub et al. [6] implemented in the code through user defined functions, has been chosen for the simulation of the trickle-bed reactor Central region The liquid distribution in a central region of the trickle-bed reactor can be seen in Figure 5.4, in terms of an isosurface of the liquid volume fraction 16

25 corresponding to the value The circular region on the top of the bed represents the projection of one chimney and the arrow indicates the inlet direction of gas into the chimney. It can be seen in Figure 5.4 that the liquid in the first bed spreads slightly to the outer part of the computational domain, while the gas occupies the center. In the second bed, the liquid concentrates again while in the third bed it spreads continuously until the interface with the fourth bed. In the fourth bed the distribution of the concentration of liquid does not change significatively as the depth of the bed is increased. The change of the liquid distribution at the interface of the beds agrees with the different porosity of the beds. For example, it can be seen in Figure 5.5 that the liquid is more dispersed in the first bed, with lower porosity (ɛ=0.33, Fig. 5.5.a), while it is more concentrated in the second bed, in which the porosity is 0.53 (Fig. 5.5.b). The same reasoning can be applied to Figures 5.5.c and 5.5.d. which show the liquid distribution on slices of the third (ɛ=0.45) and fourth (ɛ=0.40) beds. However, it should be noted that the re-concentration of liquid in the second bed seems to lack physical meaning, since it was expected, at least, to maintain the same distribution that the first bed. Due to that fact, another simulation is currently being carried out including the capillary term. In principle, this term was discarded since in the model validation it seemed not to affect significatively the liquid spreading. However, since the model validation and the reactor simulation deal with two different systems, for this particular case it could have some effect. The results obtained at the moment seem to point in that direction. 17

26 Figure 5.4: Liquid distribution in the zone surrounding one chimney in the central region of the reactor. Isosurface of volume fraction of 0.10 Figure 5.5: Contours of liquid volume fraction on horizontal slices of the four different beds. The position of the slices is indicated in Figure

27 5.2.2 Near-wall region The liquid distribution in the near-wall region is shown in Figure 5.6, in terms of an isosurface of liquid volume fraction (θ l =0.10). The circular regions indicated on the top of the bed represent the projections of the above chimneys, while the arrows indicate the inlet direction of gas into the chimney. As previously seen in the case of the flow underneath a chimney located in the central region of the reactor, the liquid spreads slightly in the first bed to the outer part of the computational domain, while the gas occupies the center, it concentrates again in the second bed and in the third bed it spreads continuously until the interface with the fourth bed. As in the central region case, there is no significantly liquid spreading in the fourth bed except for the chimney projection that is closest to the wall. Figure 5.6 shows that the liquid tends to attach the wall of the reactor as the depth of the bed is increased. This can be observed noting the inclination towards the wall of the surface depicted in Figure 5.6. As in Figure 5.4, corresponding to the dispersion of the flow underneath a central chimney, the liquid distribution in the bed does not change significatively except at the top of the first bed, in the third bed and at the interfaces of the different beds. The effect of the different porosity of the beds can be seen in Figure 5.7 that shows contours of liquid volume fraction in four horizontal slices of the beds indicated in Figure 5.6. It can be seen in Figure 5.7 that the beds of lower porosity show a more dispersed liquid than the beds of higher porosity. For example, the liquid is more dispersed in the first bed (Fig. 5.7.a, ɛ=0.33) than in the second bed (Fig. 5.7.b, ɛ=0.53). Figures 5.7.c and 5.7.d show liquid more dispersed due to the lower porosity of these beds, 0.45 and 0.40 respectively, in comparison with the second bed. As it has been previously exposed for the central region, the results obtained for the second bed are currently under investigation for the same reason that has been previously explained. 19

28 Figure 5.6: Liquid distribution in the zone surrounding one chimney in the near-wall region of the reactor. Isosurface of volume fraction of 0.10 Figure 5.7: Contours of liquid volume fraction on horizontal slices of the four different beds. The position of the slices is indicated in Figure

29 Chapter 6 Conclusions Numerical simulations of the isothermal two-phase flow dispersion in the bed of a hydrodesulfuration reactor have been reported. For the flow conditions considered, the porous media model is not adequate to simulate the flow dispersion in a catalytic bed, even the inclusion in the model of the capillary pressure effect. A granular phase model has been used for the two simulations carried out to predict the flow dispersion in the trickle bed. One simulation considers the flow underneath the central region of the reactor, far from the walls, and the other considers the flow in the bed near the cylindrical lateral wall of the reactor. In both cases, most of the liquid spreading takes place in the first and third beds. The distributions of the liquid concentration do not change significatively as the depth of the bed is increased except in the third bed and at the interface of two beds with different porosity. The simulations indicate that, for the flow conditions considered, a completely uniform liquid distribution is not achieved. The values of local liquid volume fractions at the bottom of the trickle-bed range between 0.05 and However, these results are not definitive and they should be taken with caution, since this study is being further carried out due to the apparent lack of meaning of the flow behaviour in the second bed of the reactor. 21

30 Chapter 7 Future work The work that has been presented in this report will serve as the basis for the development of a Doctoral Thesis to be carried out in the next three years. The same reactor will be the object of study in the thesis, but instead of using a commercial CFD code, the Thesis will consist in the development of a domestic three-dimensional code able to predict the flow, chemical species and temperature distributions in the reactor. Thus, not only continuity and momentum equations will need to be solved, but also chemical species transport and energy equations. The development of the code will take place in six different stages, as described in Table 7.1. Table 7.1: Stages in the development of the code Stage Dimension Chemical reactions Heat transfer 1 2D No No 2 2D Yes No 3 2D Yes Yes 4 3D No No 5 3D Yes No 6 3D Yes Yes This code will be a useful tool for the performance and design optimization of this type of reactors. If the goal is achieved, it will be an important contribution to the field of computational studies of trickle bed reactors, since the results published at the moment in the literature do not combine phases distribution, chemical reactions and heat transfer. 22

31 Bibliography [1] Martínez, M., López, J., Pallarès, J., López,A., Albertos, F., Grau, F.X. Numerical simulation of liquid distribution in a trickle-bed reactor. Part I: Study of the flow in a distribution chimney. In preparation for submission to Computers and Chemical Engineering [2] Cihla, Z., Schmidt, O. A study of the flow of liquid when freely trickling over the packing in a cylindrical tower. Collection Czechoslov. Chem. Commun. (1957) 22, pp [3] Baker, T., Chilton, T.H., Vernon, H.C. The course of liquor flow in packed towers. Presented at the Wilmington, Delaware, Meeting (1935). [4] Jiang, Y., Khadilkar, M.R., Al-Dahhan, M.H., Dudukovic, M.P. CFD of multiphase flow in packed-bed reactors: I. k-fluid modeling issues. AIChE Journal (2002) 48, pp [5] Jiang, Y., Khadilkar, M.R., Al-Dahhan, M.H., Dudukovic, M.P. CFD of multiphase flow in packed-bed reactors: II. Results and applications. AIChE Journal (2002) 48, pp [6] Holub, R.A., Dudukovic, M.P., Ramachadran, P.A. A phenomenological model for pressure drop, liquid holdup, and flow regime transition in gasliquid trickle flow. Chemical Engineering Science (1992) 47, pp [7] Attou, A., Ferschneider, G. A two-fluid hydrodynamic model for the transition between trickle and pulse flow in a cocurrent gas-liquid packed bed reactor. Chemical Engineering Science (2000) 55, pp [8] Boyer, C., Koudil, A., Chen, P., Dudukovic, M.P. Study of liquid spreading from a point source in a trickle bed via gamma-ray tomography and CFD simulation. Chemical Engineering Science (2005) 60, pp [9] Jiang, Y., Khadilkar, M.R., Al-Dahhan, M.H., Dudukovic, M.P. Single phase flow modeling in packed beds: discrete cell approach revisited. Chemical Engineering Science (2000) 55, pp

32 [10] Versteeg, H.K., Malalasekera, W. An introduction to computational fluid dynamics. The finite volume method. Pearson Education Limited, [11] Grosser, K.A., Carbonell, R.G., Sundaresan, S. Onset of pulsing in twophase cocurrent downflow through a packed bed. AIChE Journal (1988) 34, pp [12] El-Hisnawi, A.A. Tracer and reaction studies in trickle-bed reactors. D.Sc. Thesis, Washington University, St. Louis, USA (1981). [13] Fluent 6.1 Documentation, Fluent Inc., Lebanon, NH, USA 24