CFD as a Design Tool for Fixed-Bed Reactors - American Chemical

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CFD as a Design Tool for Fixed-Bed Reactors Anthony G. Dixon* and Michiel Nijemeisland Department of Chemical Engineering, Worcester Polytechnic Institute, Worcester, Massachusetts 01609

Computational fluid dynamics (CFD) is a tool that is becoming more realistic for use in the description of the detailed flow fields within fixed beds of low tube-to-particle diameter ratio (N). The motivation for the use of CFD is presented by reviewing the current state of fixed-bed reactor modeling, with an emphasis on the treatment of the description of fluid flow within the bed. Challenges in the use of CFD for fixed beds of particles are treated here, selected results are presented for N ) 2 and N ) 4, and potential uses of the simulation information in design models for fixed-bed reactors are discussed. Introduction Computational fluid dynamics is a well-established tool for several areas of reaction engineering. However, for gas-solid reactors such as fixed beds, geometric complexity has so far prevented detailed modeling of their hydrodynamics. The usual approach to modeling fixed-bed reactors assumes plug flow and effective transport mechanisms. Recent work suggests that improved predictions of reactor performance can be obtained for slim tubes if the radial variation of the axial flow component is included, but effective parameters must still be retained, and good predictions of measured velocity profiles can only be obtained if an effective viscosity is also introduced. Modern CFD codes and the exponential growth of computer power are bringing realistic fixed-bed flow simulations into the realm of possibility. It is now feasible to obtain detailed flow fields in fixed beds of low tube-to-particle diameter ratio (N). The flow field is especially interesting in the near-wall region, where flow features can differ from those in the bed center. This information can be used directly in detailed threedimensional reactor simulations or to provide information for inclusion in process design models. This contribution presents the motivation and framework for the use of CFD calculations in deriving improved fixed-bed models for low N. Challenges in fixed-bed CFD are discussed, including the generation of representative bed geometries and formulation of the model for laminar and turbulent flows. Some examples of detailed calculations are presented that can be used to develop simplified models of fluid flow, heat transfer, and chemical reaction. Future needs are identified, including the extraction of useful knowledge from the large data sets that result from simulations. Background Modeling and simulation are essential tools in the analysis and scale-up of reactors, and as demands on reactor performance increase, model performance needs to be able to give the spatial distribution of reactants, catalysts, inerts, and products in detail at all times.1 Current heterogeneous reactor models have been based on fairly radical simplifying assumptions: homogeneity, * Corresponding author. Tel.: (508) 831-5350. Fax: (508) 831-5853. E-mail: [email protected] (A. G. Dixon).

effective transport parameters, pellet effectiveness factors, etc. These simplifications have been driven by a (fast-disappearing) need for computational simplicity and by the difficulties of the complex structure of random packed tubes. These idealized models have led, however, to problems. Even the most advanced models today cannot quantitatively represent reactor behavior if independently determined kinetics and transport parameters are used.2,3 Nowhere have the simplifications in reactor modeling been so sweeping as in representing the fluid flow through the reactor. Despite the realization that the global behavior of a flow or transport system depends directly on local flow structures,4 in most cases, the hydrodynamic modeling of fixed-bed reactors is still based on plug flow. The most recent developments have only gone as far as extending simple uniform onedimensional plug flow with a single constant velocity component to a single velocity component with variation perpendicular to flow.5 The simplified models of the past have been mainly intuitive or empirical responses to the need to decide which factors are crucial in a reactor model and which are of minor importance. The phenomena that must be accurately represented are those that have strong effects on rates of reaction or rates of heat and mass transfer. The traditional approach to this problem of modeling transport rates in fixed-bed reactors has been to introduce radial, and axial when needed, effective thermal conductivities and an apparent wall heat-transfer coefficient.6-10 The parameters lump together all of the physical phenomena contributing to the heat-transfer picture. These parameters are then estimated from experimental data by regression analysis of appropriate models,11 which usually assume the fluid flowing through the tube to be in unidirectional axial “plug” flow. Despite nearly 50 years of such efforts, a large degree of disagreement exists between the data from various workers,12-14 especially for the dimensionless heattransfer coefficient or wall Nusselt number. In addition, this approach yields no insight into the physical mechanisms at work or possible methods of improving the heat-transfer rates. The main limitation in our understanding is the lack of resolution of the detailed flow picture in these beds.15 In reality, thermal energy is transported by strong radial convective flows as fluid is displaced around the packing elements. Regions of stagnant and reverse flow

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have been identified by nuclear magnetic resonance (NMR) imaging experiments,16-19 which are limited to liquid flows at very low flow rates. These flow features are thought to be strongly connected to poor heattransfer performance near the wall. To understand them, and to quantify them for gas flows at the large flow rates of industrial practice, realistic three-dimensional simulation of the fluid flow in a fixed bed is necessary. The design approach discussed here is the development of gas-solid reactor models based on the systematic analysis of the flow fields within them. Computational fluid dynamics (CFD) can provide details of flow patterns in the interstices of a fixed-bed reactor, identifying regions that could lead to enhanced or poor transport. The connection to process- or macroscopiclevel models for reactor design and analysis is the main challenge. Fixed-Bed Transport Models Classical Approaches. The earliest approaches6,7 to modeling transport in low-N fixed beds assumed the fluid to be in uniform, one-dimensional axial flow (plug flow), lumped all mechanisms for radial transport into an effective thermal conductivity kr or diffusivity Dr, and represented the observed increase in resistance to heat transfer near the containing wall by an apparent wall heat-transfer coefficient hw. These parameters were incorporated into models that viewed the bed as a single-phase continuum (pseudohomogeneous models). Axial transport was usually regarded as negligible under the high flow rates of industrial conditions. The 50 years of research that have followed the initial introduction of effective transport parameters have seen the parameters correlated against fluid flow rates, fluid properties, bed properties, and catalyst particle shapes and sizes.8,20,21 Theories for the prediction of the parameters have been developed, on the basis of more fundamental mechanisms that could be isolated and measured, at least in theory.9,22 Much discussion in the literature has taken place over how to measure temperature data in fixed beds,23 how to eliminate unwanted effects such as bed length,24 how to obtain unbiased estimates of the effective parameters,25 whether the parameters depend on reaction rates in the catalyst pellets,26 and many more issues. Continuum models have been developed that discard the pseudohomogeneous assumption27 and regard the bed as being composed of two phases, each with its own set of effective parameters as well as interphase coefficients. Approaches from the literature of porous media have been used28-30 to try to put the derivation of the continuum equations and the parameters in them on a more rigorous basis. The result of all of this effort is a plethora of correlations in the literature, which are usually in very good agreement with the data of the authors who developed them and in less good agreement with the data of other workers. There is a well-documented inability to combine independently measured kinetics with heat and mass transport to model reactor performance without further adjustment of the parameters. There is a flourishing debate as to what the effective parameters really represent9,22,26 and, in some cases, as to whether they should be used at all.31 The inescapable conclusion is that a designer of a fixed-bed multitubular

reactor would not be able to predict a priori with any certainty the behavior of the reactor. Recent Approaches. The deficiencies in our ability to model fixed-bed reactors might stem from the values of the transport parameters, the limited ranges of the models used, the inadequacy of the kinetics, or a combination of all three and maybe other sources. Several recent attempts have been made to radically change the approach to fixed-bed reactor modeling. A revival of the cell model approach has been tried,32,33 which is likely to meet the same problems as the original, as it must rely on idealized pictures of mixing in the interstices of the packing, and as it is extended to accommodate both heat and mass transfer only with difficulty. The approach to the dispersion of mass and heat through Fick’s and Fourier’s laws has been challenged, and a wave model first developed over 30 years ago34 is now being extended and revised.35 A data-based approach has been proposed36 that dispenses with the need to measure and correlate effective parameters. Microscopically valid transport equations are constructed for a segment of the bed. They allow the theoretical use of a Green’s function to write the solution at one space location and time (x, t) in terms of a known solution at another (x′, t′). The inlet feed, reaction, and wall heat-transfer terms then become known sources in the equations. Although a large amount of work has been done on bed structure and velocity profiles, this work has focused on the usually empirical relation of the radial variation of the axial velocity component, vz(r), to the radial variation of the bed porosity, (r), and has been extensively reviewed.37 Recently, there has been considerable work by the groups of Vortmeyer38 and Eigenberger39 using the extended Brinkman-Forchheimer-Darcy equation to obtain vz(r). This equation is already wellknown in the literature of flow in porous media,28,29 where it has been derived through the volume-averaging method and gives vz(r) in terms of (r)

(

)

µf ∂2vz 1 ∂vz dP FF 2 + µeff 2 + 0)- vz vz (1) dz r ∂r K ∂r xK where

K)

3(r)dp2 150(1 - (r))2

F)

;

1.75

x1503(r)

(2)

The solution of this equation gives vz(r), which is then incorporated into the energy balance and mass balance equations for the reactor, along with reaction terms. Either homogeneous or heterogeneous bed-scale models could be used. For a homogeneous model, for example, the energy balance gives

Fcpvz(r)

(

)

∂2T 1 ∂T ∂T ) keff 2 + ∂z r ∂r ∂r

(3)

with a boundary condition at r ) R that idealizes the extra resistance near the wall to a resistance at the wall by introducing the wall heat-transfer coefficient via the equation

-keff

dT ) hw(T(R) - Tw) dz

(4)

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The authors38,39 have claimed improved predictions of reactor behavior at low N, confirming that the next generation of models needs to include more information about the flow field in the reactor. One drawback to this particular approach is that the effective radial conductivity, keff, and apparent wall heat-transfer coefficient, hw, are retained and, indeed, represent different phenomena than they did in the previous plug-flow models, so that new correlations must be developed for them.40 Because the use of vz(r) is again an approximation, although a better one than plug flow, we can expect that the disagreement and confusion regarding the lumped parameters is likely to be repeated. In addition, neither group could obtain reasonable results for the velocity profile without introducing an effective viscosity, µeff, into the Brinkman term in the momentum equation above. This is a new lumped parameter that must be correlated. These approaches do not bring us any closer to understanding and representing the phenomena in fixed beds at a fundamental level. What is needed are experimental and computational techniques that allow us to understand and model fixedbed phenomena on the particle or subparticle level. Experimental techniques to do this have improved markedly in recent years, such as laser-Doppler velocimetry (LDV),41 nuclear magnetic resonance (NMR) imaging,42 and particle tracking methods.43,44 These methods all allow observations of flow inside the fixed bed without disturbing the bed structure and can thus further our understanding, but each is subject to severe limitations. LDV requires windows for optical access and is restricted to beds of very low N where such voids occur naturally. It also requires the fluid to be refractive-index-matched with the transparent material of the column. NMR methods are also restricted to liquids, and usually, the flow rate must be very low for the signal strength to be high enough for detection. Particle tracking methods require observation and counting of the markers, and problems with choice of fluid similar to those with LDV are found. Computational techniques for fluid flow at the pore scale in fixed beds have so far been few. Network models have been used by several investigators.45,46 The lattice Boltzmann method (LBM) has also enjoyed some prominence recently.47-49 Solution of the full Navier-Stokes equations to obtain the microscopic flow field around particles in a fixed bed, for realistic flow rates, has usually been considered to be unachievable as a result of computational limitations and geometrical complexities, which so far have been beyond computational fluid dynamics (CFD) codes.50 Early computations were made for creeping flow in a cubic array of spheres51 and for simplified geometries such as two spheres near a wall.52 Flow around single particles53,54 and several particles in a row55 has also been studied, in which the axisymmetric nature of the flow allowed a two-dimensional simulation. Full three-dimensional simulations of beds of small numbers of spheres, including wall effects, have been started by our group.56-59 Most recently, a complete bed of 44 spheres with N ) 2 has been solved and validated by comparison to experimental data.60 Both experimental and computational methods for fixed beds at the pore (interstice) scale have revealed a very complex picture, especially near the containing wall, of strong radial flow components, regions of reverse flow (back flow), stagnant eddies, and so on. These hydrodynamic features can be expected to play impor-

tant roles in heat transfer, dispersion, and chemical reaction. To date, no studies have investigated this connection at the level of the phenomena occurring in the interstices between particles. Computational Fluid Dynamics Modeling of Fixed Beds In CFD simulations, the Navier-Stokes mass and momentum conservation balances are solved for a quantity of mesh volumes; additional balances for heat transfer and turbulence modeling parameters can be added. The basic equations and background of these balances are stated in standard references.61 In this section, we present a brief summary of some of the main points regarding the use of CFD for fixed-bed modeling. We also present some results from two studies of fixed beds with N ) 2 and N ) 4 to show the kind of information that can be obtained from CFD modeling. The study with N ) 2 was conducted as a benchmarking case to validate the use of CFD in fixed-bed modeling.60 Only a summary of the main points is presented here. The first step in CFD simulation is the creation of a representative geometric model of the desired flow situation. After a geometric model has been created, a volume mesh for the numerical simulation is needed. The mesh must differentiate between the solid and fluid parts within the geometry and have a proper control volume density so that it shows all flow features without increasing computational intensity unnecessarily. Especially for turbulent conditions, the mesh needs to be fine in constricted flow areas, i.e., near sphere-sphere contact points and sphere-wall contact points. Another point for the numerical simulation of flow is that all mesh elements need a finite dimension on all edges, which does not allow actual contact points between solid parts in the geometry. Thus, a gap must be introduced between solid surfaces in the model. A sensitivity study to the introduced gap was performed60 to allow for an appropriate flow solution. When a sufficiently accurate mesh has been generated, the numerical simulation can be performed. Boundary and initial conditions need to be specified. Boundary conditions include wall temperatures and flow inlet and outlet conditions; initial conditions define the first step in the numerical simulation and the iteration process and concern the fluid flow field and the initial temperatures of both the fluid and the bed internals. The geometric model of the flow situation needs to be identical to the experimental setup to perform a validation, which necessitates an accurate modeling technique. For the direct validation study, a structured packed bed with N ) 2 was chosen; this specific ratio allows for a very structured bed that can be modeled with mathematical accuracy. Beds with low tube-toparticle diameter ratios show structured packings, which is an advantage in creating a simulation geometry. The N ) 2 bed packs regularly in the experimental setup, and an identical geometric model was created (see Figure 1). The mesh used in this validation model was an unstructured tetrahedral mesh, also shown in Figure 1. The complexity of the packed-bed geometry necessitated an unstructured mesh. The minimal density of the mesh was determined after a sensitivity study was performed on a single-sphere model. Specific care was taken in meshing the areas in the flow-constricted areas near sphere-sphere and sphere-wall contact areas; the


Figure 2. Velocity vector plot for N ) 2 for a four-layer section over the entire bed diameter in the y ) x plane at Re ) 1922; legend shows velocity magnitude in meters per second.

Figure 1. Geometry and mesh for fixed bed with N ) 2.

mesh was slightly refined in these areas to allow for less distorted mesh volumes. The resulting mesh contained 430 000 control volumes and required approximately 11 h of CPU time on a 500-MHz DEC Alpha workstation for a turbulent solution and approximately 5.5 h for a laminar solution. Numerical simulations were performed under conditions similar to those under which the experiments were performed. In a range of Reynolds numbers (373-1922), radial temperature profiles above the bed were established, and these values were directly compared with experimentally obtained radial temperature profiles. Both simulation and experimental temperature profiles were acquired under steady-state conditions. A comparison of the temperature profiles showed that CFD could qualitatively and quantitatively predict the experimental results accurately. A selection of the flow fields and temperature contours available from the CFD simulations is shown in Figures 2-5. In Figure 2, the flow field shows strong axially directed bypass flow down the near-wall voids present in the N ) 2 packing (refer to Figure 1) and weaker flow components passing the particle contact points. The arrows representing the velocity vectors point in the direction of the local flow, and their lengths are proportional to the magnitude of the velocity at each

Figure 3. Velocity vector plot for N ) 2 for a four-layer section over the entire bed diameter in the x ) 0 plane at Re ) 1922; legend shows velocity magnitude in meters per second.

location. Open spaces correspond to solid volumes that have been removed from the picture for clarity. The velocity field shown in Figure 3 corresponds to a bed section passing through the particle centers in the y-z plane. This section clearly shows the circulatory

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Figure 4. Velocity vector field for N ) 2 at a sphere-sphere contact point in the x ) 0 plane at Re ) 1922; legend shows velocity magnitude in meters per second.

flow typical near particle-particle contact points, the strong radially directed flow components immediately before and immediately following a particle, and regions of reverse flow and near-stagnant flow next to the walls. Thus, in some parts of the bed, flow is rapidly bypassing the packing in the wall regions, whereas in other parts downstream of wall-particle contact points, the flow is slow, and even backflow is taking place. Accelerated flow can be seen in the narrow interstices between particles. A closer view of these features is presented in Figure 4. The region of near-stagnant flow corresponds to the contact point of two of the particles that have been removed for clarity. The last picture for N ) 2 shows temperature contour plots for the entire tube of 44 spheres for three Re values in Figure 5. A comparison of the three tubes shows the development of the temperature profile occurring further downstream with increasing Re, as would be expected. The regions of low temperature at the bottoms of the figures correspond to the unheated flow-calming section of the apparatus. The higher-temperature contours within the outlines of the particles demonstrate the effects of the higher thermal conductivity of the solids. The development of an interrupted boundary layer near the wall is also evident, with temperature intrusions into the bed through the solid. These pictures of near-wall temperature fields demonstrate the complexity of the flow and heat-transfer phenomena near the wall of a fixed bed. Following the validation study, a geometric model of a larger N ) 4 packed bed was developed. When the tube-to-particle diameter ratio is increased the number of particles in the packing increases substantially. Because the mesh density needs to be similar to the density used for the N ) 2 bed, the mesh size increases dramatically with the increase in N. To keep the mesh size limited and the numerical simulation reasonable,

Figure 5. Temperature contour plot for N ) 2 for section of entire bed in the x ) 0 plane for (a) Re ) 373, (b) Re ) 986, (c) Re ) 1922; legend shows temperature in Kelvin.

translational periodic boundaries on the flow inlet and outlet are necessary. Periodic boundaries can be enforced on a packing by defining particle positions in a plane and forcing identical particle positions on the top and bottom of the packing, thus creating identical boundaries. During close examination of the N ) 4 packing, it was found that several packing structures developed, from the imposed planar particle organization at the bottom through a transition region to a repeating 3D structure. The repeating structure could be isolated and modeled with translational periodic boundary conditions as a representative part of a continuing structure in a regular N ) 4 bed. In low-N beds, the wall-induced packing structure dominates. The N ) 4 geometry showed axially repeating alternating layers of nine spheres along the wall. In the central void created by the wall layer, we observed a spiraling three-sphere arrangement. The spiraling occurred to obtain identical layer spacing in the central and wall structures. The wall layer supported the central spiral and imposed its layer spacing thereupon, resulting in an axially repetitive structure. The N ) 4 geometry was created with a coarse mesh for a laminar flow prestudy, which is shown in Figure 6. The mesh for this model was too coarse to find an accurate flow field, but it can give an indication of the large-scale flow structures, such as bypassing, reverse


Figure 8. Path lines for N ) 4 for a six-layer section over the entire bed diameter in the x ) 0 plane at Re ) 180; legend shows axial velocity component in meters per second.

Figure 6. Geometry and mesh for fixed bed with N ) 4.

bypassing features near the wall and regions of backflow downstream of the particles. These features are further reinforced in Figure 8, in which pathlines colored by axial velocity components are shown; thus negative values can be seen, showing regions of backflow. Discussion

Figure 7. Velocity vector plot for N ) 4 for a six-layer section over the entire bed diameter in the x ) 0 plane at Re ) 180; legend shows velocity magnitude in meters per second.

flow, and radial flow components. These are shown in Figure 7 for the flow field in a bed section parallel to the overall axial flow direction for Re ) 180. The tortuous nature of the flow as it moves past particleparticle contact points is clearly seen, as well as the

The CFD results of the previous section show that this approach can rapidly generate a great deal of information. Even a steady-state CFD analysis without heat transfer yields a database of results that comprises the spatial coordinates (r, θ, and z) and the associated velocity components vr, vθ, and vz, for up to 500 000 control volumes in the fluid. The addition of conduction through particles and temperature for each of the volumes complicates the analysis of the data still further. As computer speed and memory increase and the size of CFD models increases along with them, the problem of obtaining useful knowledge from a large database of information will only get worse. For fixed-bed modeling and design, we can consider different uses of the data. (i) The velocity components could be used directly in a three-dimensional heterogeneous model of the fixed bed. This would be specific to the particular packing used in the CFD simulation and would be of little help in new reactor designs or in assessing variability from one reactor tube to another. It would also result in an extremely computationally complex reactor model, in which the level of detail of the transport was incompatible with the level of knowledge of the reaction kinetics. (ii) The minimum use of the CFD information would be to average the axial velocity component over a suitable representative averaging volume (or area) to obtain vz(r) for insertion into eq 3. This use of the data would have the advantage of providing information that

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could be directly used in reactor simulations and that would be compatible with existing approaches. The vz(r) expression in eqs 1 and 2 could be used to estimate µeff, i.e., the CFD calculations could be used to provide a constitutive relation for an averaged tube-scale description. This use of the data disregards a great deal of the fine structure of the flow field, as two of the three velocity components are discarded as being averaged out. This would not be our preferred approach, as we believe that heat transfer and reactor performance are strongly affected by the particle-scale features of the flow field. (iii) An attractive approach to fixed-bed modeling is to move forward, away from the use of effective parameters, but still retaining a general, structure-based approach to transport that would be useful for reactor design. For example, Dixon et al.62 postulated that radial transport of heat takes place by solid conduction and the radial displacement of flow around particles. They constructed a model of fixed-bed heat transfer that employed the true fluid and solid thermal conductivities of the phases and a two-dimensional velocity field composed of the components vz(r) and vr(r,z). The original work used a computer-generated packing, which was then tesselated and transformed into a network model. A pressure gradient was applied, and flows through the network branches were calculated. These flows were then averaged to obtain the velocity components. The resulting model represented heat transfer through the bed center quite well but was not accurate near the wall. This result was explained by the fact that the flow channels parallel to the wall could not be included in the network and also by the fact that, during the averaging process, a “cancellation effect” was observed, in which strong radial flows to and from the wall were added to give no net flow, although the net heat transfer could have been significant. Nonetheless, the development of a model completely without effective parameters was a significant step toward more physically based fixed-bed models. CFD simulations could be used in conjunction with the approach of Dixon et al.62 to replace the need for a network model and computed branch flows. One could anticipate improved modeling of flows near the wall, using CFD. However, the choice of which velocity components to retain and determination of their dependence on spatial coordinates was purely intuitive. A more systematic approach is desired, in which important features of the flow could be identified and linked to structural features of the bed, such as particle size, shape, and N. Searching through the myriad of flow pictures that could be generated by CFD postprocessing is becoming beyond human capability as models of larger-N beds are created, especially as more features of the flows are internal to the bed. The efficient performance of such searches will require tools from the realm of information sciences, such as pattern recognition, classification, and feature extraction. The incorporation of prior knowledge will be necessary to guide these steps. The coupling of CFD to information technology tools has the potential of making a strong impact on reactor design. Conclusions An approach to fluid flows in low tube-to-particle diameter fixed beds is needed that is general enough to be applicable for reactor design purposes, but that

retains the detailed flow features that contribute to transport of heat and mass and to strong local gradients that could influence reaction kinetics. This approach should provide a link to bed structure, so that, if a packing or packing characteristics are known, flow features, transport rates, and reactor performance can be rapidly assessed. Computational fluid dynamics simulations of flow through packed-bed structures can provide highly detailed and reliable information about the temperature and flow fields. The challenge for the future is to use the information that will be available to gain knowledge and understanding that will allow us to develop reduced models that are detailed enough for design purposes, but still intuitively understandable and computationally tractable. Some of the recent tools of information technology will be of use in this endeavor. Acknowledgment The authors thank the DuPont educational fund for their financial support. Also, Fluent, Inc., is acknowledged for the university license of their CFD software. Notation cp ) fluid heat capacity [J/(kg K)] dp ) particle diameter (m) F, K ) constants defined in eq 2 hw ) apparent wall heat-transfer coefficient [W/(m2 K)] keff ) effective radial thermal conductivity [W/(m K)] N ) tube-to-particle diameter ratio (dt/dp) P ) pressure (Pa) r ) radial coordinate (m) R ) tube radius (m) Re ) Reynolds number (Fvzdp/µ) T, Tw ) temperature and wall temperature, respectively (K) vz ) axial gas velocity (m/s) V, Vf ) averaging volume and volume in fluid phase, respectively (m3) x, y, z ) coordinates (m) Greek Symbols ) bed voidage F ) fluid density (kg/m3) µf ) fluid viscosity (N s/m2) µeff ) effective bed viscosity (N s/m2) θ ) angular coordinate

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Received for review December 7, 2000 Revised manuscript received June 6, 2001 Accepted June 6, 2001 IE001035A

CFD as a Design Tool for Fixed-Bed Reactors - American Chemical

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