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CFD modelling and experimental validation of pressure drop and flow profile in a novel structured catalytic reactor packing
Chemical Engineering Science 56 (2001) 1713}1720
CFD modelling and experimental validation of pressure drop and #ow pro"le in a novel structured catalytic reactor packing H. P. A. Calis *, J. Nijenhuis , B. C. Paikert, F. M. Dautzenberg, C. M. van den Bleek Delft University of Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft, Netherlands ABB Corporate Research Center//Alstom Power, Segelhof, CH 5405 Baden-Da( ttwil, Switzerland ABB Lummus Global Inc., Technology Development Center, 1515 Broad Street, Bloomxeld, NJ, USA
Abstract Packed beds of catalyst particles are normally described using models that contain a number of empirical parameters. The development of computer technology and CFD models makes it tempting to try to (1) fully simulate the #ow in packed beds to obtain a more detailed understanding of the physical phenomena that take place in the bed, and (2) to use the CFD solutions to derive &simple' correlations suitable for design purposes. In this paper it is shown that a commercial CFD code (CFX-5.3) can be used to predict, with an average error of about 10%, the pressure drop characteristics of packed beds of spheres that have a tube-to-particlediameter ratio of 1.00 to 2.00. Packed beds with these unusually low tube-to-particle-diameter ratios can be used as unit cells in a novel type of structured catalytic reactor packing, proposed in this paper, that has very favorable pressure drop characteristics. The error of 10% in the pressure drop prediction by CFD is acceptable for design purposes. The CFD model is also able to predict local velocity pro"les that were measured with LDA. The CFD results have been used to "t a simple two-parameter model that describes the experimental pressure drop data with an average error of about 20%. For a grid-independent CFD solution of laminar #ow in a packed bed containing only 16 particles, already three million cells are required. However, it is anticipated that within "ve years from now the simulation of a packed bed containing a few hundred particles will be considered a &standard' problem in terms of memory and calculation time requirements. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Packed bed; Structured packing; Simulation; Fluid mechanics; Momentum transfer; Pressure drop
1. Introduction Randomly packed beds of catalyst particles have been the most important embodiment of heterogeneous catalytic reactors for more than 70 years. In spite of the increasing popularity of structured catalysts and reactors (Cybulski & Moulijn, 1998), randomly packed beds will most probably remain the &default' catalytic reactor for at least a few more decades, mainly because of their low cost compared to structured systems. The high economic value represented by packed beds has driven an almost incredible number of studies investigating the chemical reactor engineering characteristics of packed beds, such as pressure drop, mass and heat transfer and dispersion, and continues to do so.
* Corresponding author. Fax: #31-15-278-4452. E-mail address: [email protected] (H. P. A. Calis).
The modelling of the hydrodynamics of packed beds is mostly done with models that are based on a number of physical principles and contain a number of parameters that are determined from experiments in the lab. An example is the well-known Ergun correlation for pressure drop across a random bed of particles, which in its original form contains two empirically determined parameters c and c (Ergun, 1952): 1 u ¸ p"4f , (1) 2 d F c u /d with Re, F . (2) 4f" #c Re
In the remaining part of this paper, the Reynolds number is de"ned as in Eq. (2). With the parameters (c , c ) equal to (133,2.33), Eqs. (1) and (2) describe the pressure drop across randomly packed beds of spheres with a packing voidage of about 40%. Also the pressure drop across tube bundles and
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various structured packings is described by this correlation, using di!erent values for (c , c ). Even rough ducts can be described with this correlation, e.g., (c , c )"(64,0.65) for circular tubes with a dimensionless wall roughness of 0.05; however, the pressure drop around the transition from laminar to turbulent #ow is not adequately described by this correlation, because in an empty tube the transition takes place fairly suddenly instead of over an extended range of Re. Often the desire for more accurate models tuned to more speci"c or better-de"ned systems is an incentive for elaborating on the original models, introducing more model parameters that should be determined experimentally. For example, the friction factor correlation (Eq. (2)) is applicable to randomly packed beds of which the ratio of bed diameter to particle diameter is larger than, e.g., 15; for smaller ratios, the ratio can be introduced as an additional variable accompanied by one or more additional parameters (e.g., Winterberg & Tsotsas, 2000). The developments in computer technology and computational #uid dynamics (CFD) techniques will one day allow a radically di!erent approach, in which the packed bed is fully simulated using a CFD model, i.e., the #ow through the space between the particles is calculated by solving a.o. the Navies}Stokes equations. This yields a very detailed solution containing the local values of all relevant variables, such as pressure, velocities, temperature, viscosity, shear stress and so on. Such detailed solutions can be of great importance in the understanding of the phenomena that occur in the packed bed. Of course, for the design of packed-bed reactors fairly simple algebraic correlations remain indispensable. However, analysis of the results of CFD simulations can yield such correlations, as an alternative to experiments in the lab. Before we can rely on CFD models to study or design packed beds, we need to establish whether the model yields valid results. Even though CFD technology has developed tremendously over the past few decades, its use and further development still rely on experimental validation. The goal of this paper is to show that commercially available CFD software can be used to predict the single-phase pressure drop characteristics of packed beds of catalyst particles, and that analysis of the CFD solutions allows the derivation of simple algebraic pressure drop correlations (Eqs. (1) and (2)) that are in agreement with experimental data. For this purpose, packed beds with a very low tubeto-particle-diameter ratio, between 1.0 and 2.0, will be used. Such low values are seldomly encountered in practice; however, these extremely narrow packed beds may be viewed as unit cells of a novel type of structured catalytic reactor packing called composite structured packing (CSP; Strangio, Dautzenberg, Calis, & Gupta, 1998). The CSP concept is based on the well-known principle that in a randomly packed bed of particles, the
Fig. 1. Concept of the Composite Structured Packing (CSP).
particles close to the wall are arranged in a speci"c way (because they cannot protrude through the wall), accompanied by a higher local bed voidage and a lower tortuosity. As a result, at a given pressure drop across a packed bed, the #uid velocity close to the wall is signi"cantly higher than far away from the wall (e.g., Schwartz & Smith, 1953, Winterberg & Tsotsas, 2000). To put it di!erently, the pressure gradient required to obtain a certain #uid velocity close to the wall is signi"cantly smaller than the pressure gradient needed to obtain the very same #uid velocity far away from the wall. This principle can be exploited by "lling a multichannel framework, such as a monolith, with ordinary catalyst particles, as shown in Fig. 1. Depending on the design (channel and particle shapes, channel-to-particle-diameter ratio, etc.) a CSP creates a very signi"cantly lower pressure drop (up to a factor of 15) per unit catalyst mass than a randomly packed bed containing the same catalyst particles, due to the high voidage and the alignment of the particles. Alternatively, the CSP concept allows the use of smaller particles compared to a traditional random bed with the same pressure drop, favoring catalyst e$ciency and selectivity. The following approach is followed to reach the stated goal. First we de"ne a number of narrow packed beds of particles, or, to put it di!erently, a number of CSP con"gurations, by "xing the channel-to-particle-diameter ratio as well as the positions of all particles. Next, a commercial CFD code (CFX-5.3) is used to simulate single-phase #ow through these packed beds. The results are expressed as friction factors versus Reynolds numbers. In parallel, lab-scale packed beds are constructed with exactly the same geometry as the packings used in the simulations. The lab-scale packings are used to experimentally determine pressure drop and local #uid velocities (using laser-Doppler anemometry, LDA). Following, the simulated results are compared to the experimental values and "nally the simple algebraic pressure drop correlation of Eqs. (1) and (2) is "tted to the CFD results and compared with the experimental pressure drop data. In a second paper on this topic (Calis, Romkes, Dautzenberg, & Van den Bleek, 2001) we will
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follow a similar approach to study particle-to-#uid heat and mass transfer in packed beds.
2. De5nition of the packed beds Keeping in mind the concept of the CSP, in which monoliths are used to provide the structural framework to contain catalyst particles, channels with a square cross-section are a logical choice for the current study. Spherical particles were chosen to represent the catalyst particles, as these were anticipated to be more challenging from a CFD point of view than e.g. cylinders, even though cylinders will generally give higher pressure drop reductions compared to spheres. A number of channels with square cross-sections and "lled with spherical particles (i.e., unit cells of a CSP) are shown in Fig. 2. For any channel-to-particle-diameter ratio, both regular (repeating) and irregular (random) packings are possible. The random packings at N"2.42 and 4.00 shown in Fig. 2 were de"ned using a simulation algorithm, in which particles were dropped in a channel one by one, and each subsequent particle would "nd the lowest position available in the bed at that moment, i.e., the position with the lowest energy. This algorithm yields the densest random packing, as opposed to the loosest random packing (Jaeger & Nagel, 1992; Sloane, 1984; Eppstein, 2000). Without proof it is stated here that for any channel-to-particle-diameter ratio, the densest packing that can be obtained is always a regular packing. Note, however, that at a certain channel-to-particle-diameter ratio more than one regular packings may be
Fig. 3. Dependence of the voidage of a packing of uniform spheres in channels with square cross-sections, as a function of the channel-toparticle-diameter ratio (N). Solid line marked with &d': voidage of the densest possible packing (i.e., a regular packing) at a given N; solid line marked with &r': voidage of the &densest random packing' at a given N. (䉱) voidage of &loosest random packing' of an in"nitely wide bed; (䊏) voidage of &densest random packing' of an in"nitely wide bed; (䢇) voidage of &absolutely densest packing' of an in"nitely wide bed; all for uniform spheres.
possible; see for example the two regular packings at N"2.00 in Fig. 2. For the range of 1.00)N)4.00, the positions of the particles in both regular and irregular packed beds containing up to 157 uniform spheres were calculated, and used to calculate the voidage as a function of the channel-to-particle-diameter ratio. The result is presented in Fig. 3. It is obvious that if one wants to simulate a packed bed using CFD techniques, the amount of required memory and calculation time will increase with the number of particles in the bed. At the current state of development, we consider a number of grid cells of a few million to be a reasonable maximum. Higher numbers are certainly possible, but are considered excessive at this moment. As we will show, this limits the number of particles in the simulated packed bed to about 30 (laminar #ow, without modelling heat transfer from particle to #uid). From the con"gurations shown in Fig. 2, the "rst "ve were selected for CFD simulation and experimental pressure drop and LDA measurements. The packing with N"1.00 ("0.48) was chosen because it is the simplest packed bed that is possible; N"1.15 ("0.60) to study the e!ect of a small deviation of N on pressure drop, to verify that the tolerance in the experimental packed beds was su$ciently small; N"1.47 ("0.68) because it has a very high voidage; N"2.00 because two regular particle arrangements are possible, A ("0.48) and B ("0.54), with signi"cantly di!erent voidages.
3. Simulation Fig. 2. Arrangement of uniform spheres in channels with a square cross-section, for various ratios of channel-to-particle-diameter (N). Top row: front views; bottom row: isometric views.
The most important aspects of the CFD model are: E The model is three-dimensional.
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E To limit the computational demands, in each case only 8}16 particles were used. E An unstructured grid was used, which is unavoidable because of the complicated geometry of the #ow domain (i.e., the space between the particles). On the walls of the solids, a number of layers of prismatic grid cells were used, the remaining space was "lled with tetrahedral cells. E The #uid was de"ned to be incompressible. This is allowed because the #uid #ow in packed beds has (in practice) very low Mach numbers, and because pressure drop characteristics obtained with a constant density are easily translated for situations with varying density. E The #ow was assumed to be stationary, even though it is not yet clear if transient phenomena (compare vortex shedding with cylinders in cross-#ow) might in#uence the calculated #ow "eld and pressure drop. E Both laminar and turbulent #ow were studied; the Reynolds number (based on interstitial velocity and hydraulic packing diameter) was varied from 1.0;10\ to 5.0;10, to cover the full range from fully laminar (where the friction factor is inversely proportional to Re) to fully turbulent #ow (where the friction factor is independent of Re). E Two turbulence models were used: a k}-model (because it is the most widely used turbulence model) and a Reynolds-Stress Model (RSM; because isotropic turbulence, assumed in the k}-model, may not be valid in the considered geometries). A zero-equation turbulence model was only used to facilitate convergence of a more elaborate turbulence model. E Standard wall functions were used in the case of turbulent #ow. E In almost all cases the "nal solution was obtained with a second-order discretization scheme. E The model was implemented in the commercial CFD package CFX-5.3 (AEA Technology). E The geometry of the packed beds was built using CFX-5 Build and the meshing was done using the CFX-5 mesh generator. To avoid the generation of mesh cells of poor quality at the points of contact between particles themselves and between particles and channel wall, all particles were shrunk by 1% of the diameter after the initial de"nition of the bed. In other words, the positions of all particles in the packed bed were calculated based on a certain particle diameter d , but the particles actually de"ned in the N CFD domain had a diameter of only 0.99d . As a result, N the packings contained no points of contact. The values of the packing voidage and hydraulic diameter to be used in the calculation of the friction factor from the simulated pressure drop were corrected for the smaller particle diameter. To check whether this correction is adequate, an extra case was simulated in which a factor of 0.98
Fig. 4. Impression of the CFD mesh in a region between a spherical particle and the channel wall.
Table 1 Typical mesh quantities used in the CFD simulations Turbulence model Edge length of the tetrahedral cells (mm) Number of layers of prismatic cells (dimensionless) Thickness of the "rst layer of prismatic cells (mm) Expansion rate of the prismatic cells (dimensionless)
Laminar 0.4 5
k} 1.0 0
0.052
n.a.
1.2
n.a.
instead of 0.99 was used. The resulting friction factors di!ered less than 0.5%, which justi"es the approach. The mesh used for a certain packed bed can be characterized by the edge length of the tetrahedral cells, the thickness of the "rst layer of prismatic cells on the surface of the solids (sphere or channel wall), the expansion rate of the prismatic cells and the number of layers of prismatic cells; see Fig. 4. In most simulations a sphere diameter of 12.7 mm was used, in combination with the mesh quantities listed in Table 1. An edge length of 0.4 mm on a 12.7 mm diameter sphere corresponds to a number of about 7300 triangles in the surface mesh of the sphere, whereas an edge length of 1.0 mm corresponds to about 1200 triangles. For laminar #ow and a case with N"2.00 A with 16 spheres, a total of about three million grid cells was needed when using a tetrahedral mesh edge length 0.4 mm in combination with "ve layers of prismatic cells on the solid surfaces. Extrapolating the rate of development of computer and CFD technology, these "gures suggest that within "ve years the simulation of a packed bed containing a few hundred particles will be considered a &standard' problem in terms of memory and calculation time requirements.
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Fig. 5. Dependence of the simulated pressure drop across a packed bed as a function of the (inverse) thickness of the wall cells. (䊏) cases where no layers of prismatic cells were used, i.e., the wall cells were tetrahedral; (䢇) cases where layers of prismatic cells were used. The arrow indicates the wall cell thickness used in most laminar simulations.
For laminar #ow the grid independence of the solution was checked by varying the grid type and grid density near the walls. In Fig. 5 the resulting pressure drops across the packed bed are plotted as a function of the thickness of the wall cells. This "gure indicates that wall cells of 0.052 mm thickness are adequate in this situation, as a further reduction of the thickness by a factor 2.7 results in only a 2.5% higher simulated pressure drop. For turbulent #ow the thickness of the cell walls is expressed in the value of y>, which in the case of standard wall functions should preferably be between 30 and 300 or at least between 10 and 1000. However, in packed beds, large deviations of the local velocity near the solid surfaces exist because of more or less stagnant zones near the points of contact between spheres. This results in a variation of the y> values over the sphere surface of a factor 40 when a homogeneous surface mesh is used, making it di$cult to meet the y> criterion everywhere on the sphere surface. More important, the condition that the #ow should be well developed in order for the standard wall model to be valid, is most likely not fully met in the case of a packed bed. These factors will introduce a systematic error in the simulated pressure drop characteristics that is hard to assess beforehand. The average y> value on the sphere surfaces was about 4.4;10\ Re, i.e., ranging from about 4.4 at Re"1.0;10 to about 440 at Re"1.0;10. These values can be improved by using a non-uniform surface mesh on the spheres and by adapting the mesh density to the Reynolds number. However, this was beyond the scope of the present study. The use of the RSM turbulence model as an alternative to the k} yielded only slightly di!erent pressure drop results (less than 10% discrepancy). This di!erence was considered too small to justify the extra computational demand of the RSM model.
Fig. 6. Schematic of the experimental setup. Right: detail.
4. Experimental Experimental data were obtained using channels with walls made of PMMA, with square cross-sections of 12.7}26.0 mm, and a length of 700 mm. Highprecision polyethylene spheres (~"12.7 mm for N"1.00, 1.15, 2.00A and 2.00 B, versus 9.5 mm for N"1.47) were supported by a wire mesh screen. The inlet region of the channels contained a 40 mm high bed of 2 mm spheres followed by a 40 mm section of 400 cells/inch monolith, to obtain a #at inlet velocity pro"le. The channels contained pressure taps every 100 mm. See Fig. 6. For the pressure drop experiments, dried air at ambient conditions was used. Experimental Reynolds numbers ranged from about 100 to about 6000. Pressure drops were measured across a bed length of about 500 mm. Local velocity pro"les were measured with laser-Doppler anemometry (LDA), using water at ambient conditions, seeded with 0.3 m TiO particles. Both horizontal and vertical cross-sections through the bed were measured for beds with channel-to-particle-diameter ratios of 1.47 (at Re"486 and 1.11;10) and 2.00B (at Re"575 and 7.84;10).
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Fig. 7. Pressure pro"le obtained through CFD simulation of a packed bed with N"1.00. The area-averaged pressure was calculated on planes perpendicular to the longitudinal axis of the tube, midway between subsequent particles.
5. Results and discussion A very important question in the simulation of a random bed with a certain channel-to-particle-diameter ratio is what bed length (or what number of particles) is needed to obtain results that are representative for &in"nitely' long beds. In the present study the number of particles in the simulations ranged from 8 (N"1.00) to 16 (N"2.00 A). Fig. 7 shows the simulated pressure pro"le in a packed bed with N"1.00. The constant pressure gradient indicates the absence of a signi"cant entrance region in the packed bed * or, to put it di!erently, each subsequent particle represents a new entrance region, and in this the "rst particle is not di!erent from the subsequent particles. The #ow appears to be periodical right from the "rst particle onwards, at least in terms of pressure drop.
Fig. 8 shows simulated streamlines for a packed bed with N"1.47. Also the streamlines around the "rst particle suggest that the #ow at the "rst part of the packed bed is identical to the #ow more downstream. The conclusion of these two "gures and similar "gures for other channel-to-particle-diameter ratios, is that already a short bed of about 6 particle diameters is representative of a much longer bed. Fig. 9 shows a comparison between a.o. the experimentally determined and the simulated friction factors as a function of the Reynolds number. It is seen that, in general, the agreement between the experimental and simulated friction factor is satisfactory * a typical deviation between the two is about 10%, which is certainly acceptable in the design of a packed-bed reactor. Because the simple correlation of Eqs. (1) and (2) contains only two parameters, and because in general the correlation is seen to even describe the friction factor in the transition range of Re quite satisfactory, it is tempting to try to "t the correlation using only two simulated data points. If this approach works well, one would only need to do two simulations for any given packed bed, one in the fully laminar range (to determine the parameter c ) and one in the fully turbulent range (to determine c ). This approach was followed, and the resulting parameter values are listed in Table 2. The resulting correlations are shown in Fig. 9 as well. For N"1.00, 2.00A and 2.00B, the comparison with the experimentally determined friction factors is very good: the average deviation is only about 5%. However, for N"1.15 and 1.47, the deviation is much larger, with an average of about 30%. The larger deviation is attributed to the higher voidage; note that the friction factor of an empty duct ("1.00) in the transition range of Reynolds cannot be adequately described with Eqs. (1) and (2). Also in Fig. 9 the prediction of the friction factor based on the original Ergun correlation, using (c , c )"(133,2.33), is shown. The agreement with the
Fig. 8. Streamlines from CFD simulation of the #ow in a packed bed with N"1.47.
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Fig. 9. Comparison of friction factors obtained from experiments (䊏), CFD simulation, both laminar () and turbulent (*), Eq. (2) "tted to two CFD data points for each packing (solid line marked with &C'), and the original Ergun correlation, Eq. (2) with (c , c )"(133,2.33) (solid line marked with &E').
Table 2 Results of "tting Eqs. (1) and (2) to two CFD data points per packing N (dimensionless)
c (dimensionless)
c (dimensionless)
1.00 1.15 1.47 2.00 A 2.00 B
1.48;10 1.14;10 7.28;10 1.12;10 2.14;10
5.03;10\ 3.59;10\ 3.67;10\ 6.51;10\ 1.04
experimental friction factors is poor: the Ergun overpredicts the experimental friction factors by an average of 80%. Comparison of the Ergun correlation with the results of the CFD simulations shows that the deviation is mainly due to poor prediction of the turbulent contribution to the friction factor. This is an interesting observation, because for empty ducts or tubes (as opposed to packed beds) the laminar contribution to the friction factor of a duct or channel is strongly dependent on the shape of channel (in other words: the geometry), whereas the turbulent contribution is not, and can normally be described in terms of hydraulic diameter only. In the case of the packed beds of the present study, the opposite is the case. Fig. 10 presents a comparison between experimental #ow pro"les measured with LDA and simulated #ow pro"les obtained with the CFD model. The agreement between experiments and simulation is satisfactory, and was found to be the same for other Reynolds numbers
Fig. 10. Comparison of #ow pro"les in a horizontal cross-section through the bed (i.e., perpendicular to the longitudinal axis of the bed), obtained with LDA (left) and CFD (right), for channel-to-particlediameter ratios of 1.47 (top) and 2.00B (bottom). The horizontal bars in the drawings on the right indicate the locations of the sample planes.
and for vertical cross-sections instead of the horizontal cross-sections shown in Fig. 10. These results indicate that the CFD model is not only able to predict an integral performance characteristic like the pressure drop across the packed bed, but also local velocities. Consequently, we anticipate that CFD simulation of packed beds can be of signi"cant help in obtaining a detailed understanding of the physical phenomena that take place in packed beds.
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6. Conclusions It has been shown that a commercial CFD code can be used to predict the pressure drop characteristics of packed beds of spheres that have a channel-to-particlediameter ratio of 1.00 to 2.00, with an average error of about 10%. This error is acceptable for design purposes. The local velocity pro"le on a cross-section of the packed bed, predicted by the CFD model, agrees well with the pro"le measured with LDA. The pressure drop characteristics over the full relevant range of Reynolds numbers can be modelled with a simple two-parameter model, of which the two parameters are obtained from only two CFD simulations, one at fully laminar #ow and one at fully turbulent #ow. Even though the geometry of the packed bed of spheres contains fairly strong curvature, the k} turbulence model appears to be adequate, and yields approximately (within about 10%) the same results as the RSM. The size of the grid cells needed for a grid-independent solution is so small that about three million cells are required for the simulation of a packed bed containing 16 particles. It is anticipated that within "ve years from now the simulation of a packed bed containing a few hundred particles will be considered a &standard' problem in terms of memory and calculation time requirements.
Notation c , c model parameters, dimensionless d hydraulic packing or channel diameter, including F the surface of the particles and the channel wall, m
d N 4f ¸ N Re u y>
particle diameter, m friction factor, dimensionless packing bed length, m ratio of channel diameter to particle diameter, dimensionless Reynolds number, as de"ned in Eq. (2), dimensionless super"cial #uid velocity, m/s dimensionless thickness of wall cells in CFD mesh, dimensionless
Greek letters
packing voidage, m /m dynamic #uid viscosity, Pa s #uid density, kg/m
References Calis, H. P. A., Romkes, S., Dautzenberg, F. M., & Van den Bleek, C. M. (2001). First International Conference on Structured Catalysts and Reactors, Delft, The Netherlands, accepted for presentation. Cybulski, A., & Moulijn, J. A. (1998). Structured catalysts and reactors. New York: Marcel Dekker Inc. Eppstein, D. (2000). http://www.ics.uci.edu/&eppstein/junkyard/ cover.html. Ergun, S. (1952). Chemical Engineering Progress, 48, 89. Jaeger, H. M., & Nagel, S. R. (1992). Science, 255, 1523. Sloane, N. J. A. (1984). Scientixc American, 250, 92. Strangio, V. A., Dautzenberg, F. M., Calis, H. P. A., & Gupta, A. (1998). Fixed Catalytic Bed Reactor; International patent application PCT/US99/06242, priority date March 23, 1998. Schwartz, C. E., & Smith, J. M. (1953). Industrial Engineering Chemistry, 45, 1209. Winterberg, M., & Tsotsas, E. (2000). American Institute of Chemical Engineering Journal, 46(5), 1084.
CFD modelling and experimental validation of pressure drop and #ow pro"le in a novel structured catalytic reactor packing H. P. A. Calis *, J. Nijenhuis , B. C. Paikert, F. M. Dautzenberg, C. M. van den Bleek Delft University of Technology, DelftChemTech, Julianalaan 136, 2628 BL Delft, Netherlands ABB Corporate Research Center//Alstom Power, Segelhof, CH 5405 Baden-Da( ttwil, Switzerland ABB Lummus Global Inc., Technology Development Center, 1515 Broad Street, Bloomxeld, NJ, USA
Abstract Packed beds of catalyst particles are normally described using models that contain a number of empirical parameters. The development of computer technology and CFD models makes it tempting to try to (1) fully simulate the #ow in packed beds to obtain a more detailed understanding of the physical phenomena that take place in the bed, and (2) to use the CFD solutions to derive &simple' correlations suitable for design purposes. In this paper it is shown that a commercial CFD code (CFX-5.3) can be used to predict, with an average error of about 10%, the pressure drop characteristics of packed beds of spheres that have a tube-to-particlediameter ratio of 1.00 to 2.00. Packed beds with these unusually low tube-to-particle-diameter ratios can be used as unit cells in a novel type of structured catalytic reactor packing, proposed in this paper, that has very favorable pressure drop characteristics. The error of 10% in the pressure drop prediction by CFD is acceptable for design purposes. The CFD model is also able to predict local velocity pro"les that were measured with LDA. The CFD results have been used to "t a simple two-parameter model that describes the experimental pressure drop data with an average error of about 20%. For a grid-independent CFD solution of laminar #ow in a packed bed containing only 16 particles, already three million cells are required. However, it is anticipated that within "ve years from now the simulation of a packed bed containing a few hundred particles will be considered a &standard' problem in terms of memory and calculation time requirements. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Packed bed; Structured packing; Simulation; Fluid mechanics; Momentum transfer; Pressure drop
1. Introduction Randomly packed beds of catalyst particles have been the most important embodiment of heterogeneous catalytic reactors for more than 70 years. In spite of the increasing popularity of structured catalysts and reactors (Cybulski & Moulijn, 1998), randomly packed beds will most probably remain the &default' catalytic reactor for at least a few more decades, mainly because of their low cost compared to structured systems. The high economic value represented by packed beds has driven an almost incredible number of studies investigating the chemical reactor engineering characteristics of packed beds, such as pressure drop, mass and heat transfer and dispersion, and continues to do so.
* Corresponding author. Fax: #31-15-278-4452. E-mail address: [email protected] (H. P. A. Calis).
The modelling of the hydrodynamics of packed beds is mostly done with models that are based on a number of physical principles and contain a number of parameters that are determined from experiments in the lab. An example is the well-known Ergun correlation for pressure drop across a random bed of particles, which in its original form contains two empirically determined parameters c and c (Ergun, 1952): 1 u ¸ p"4f , (1) 2 d F c u /d with Re, F . (2) 4f" #c Re
In the remaining part of this paper, the Reynolds number is de"ned as in Eq. (2). With the parameters (c , c ) equal to (133,2.33), Eqs. (1) and (2) describe the pressure drop across randomly packed beds of spheres with a packing voidage of about 40%. Also the pressure drop across tube bundles and
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various structured packings is described by this correlation, using di!erent values for (c , c ). Even rough ducts can be described with this correlation, e.g., (c , c )"(64,0.65) for circular tubes with a dimensionless wall roughness of 0.05; however, the pressure drop around the transition from laminar to turbulent #ow is not adequately described by this correlation, because in an empty tube the transition takes place fairly suddenly instead of over an extended range of Re. Often the desire for more accurate models tuned to more speci"c or better-de"ned systems is an incentive for elaborating on the original models, introducing more model parameters that should be determined experimentally. For example, the friction factor correlation (Eq. (2)) is applicable to randomly packed beds of which the ratio of bed diameter to particle diameter is larger than, e.g., 15; for smaller ratios, the ratio can be introduced as an additional variable accompanied by one or more additional parameters (e.g., Winterberg & Tsotsas, 2000). The developments in computer technology and computational #uid dynamics (CFD) techniques will one day allow a radically di!erent approach, in which the packed bed is fully simulated using a CFD model, i.e., the #ow through the space between the particles is calculated by solving a.o. the Navies}Stokes equations. This yields a very detailed solution containing the local values of all relevant variables, such as pressure, velocities, temperature, viscosity, shear stress and so on. Such detailed solutions can be of great importance in the understanding of the phenomena that occur in the packed bed. Of course, for the design of packed-bed reactors fairly simple algebraic correlations remain indispensable. However, analysis of the results of CFD simulations can yield such correlations, as an alternative to experiments in the lab. Before we can rely on CFD models to study or design packed beds, we need to establish whether the model yields valid results. Even though CFD technology has developed tremendously over the past few decades, its use and further development still rely on experimental validation. The goal of this paper is to show that commercially available CFD software can be used to predict the single-phase pressure drop characteristics of packed beds of catalyst particles, and that analysis of the CFD solutions allows the derivation of simple algebraic pressure drop correlations (Eqs. (1) and (2)) that are in agreement with experimental data. For this purpose, packed beds with a very low tubeto-particle-diameter ratio, between 1.0 and 2.0, will be used. Such low values are seldomly encountered in practice; however, these extremely narrow packed beds may be viewed as unit cells of a novel type of structured catalytic reactor packing called composite structured packing (CSP; Strangio, Dautzenberg, Calis, & Gupta, 1998). The CSP concept is based on the well-known principle that in a randomly packed bed of particles, the
Fig. 1. Concept of the Composite Structured Packing (CSP).
particles close to the wall are arranged in a speci"c way (because they cannot protrude through the wall), accompanied by a higher local bed voidage and a lower tortuosity. As a result, at a given pressure drop across a packed bed, the #uid velocity close to the wall is signi"cantly higher than far away from the wall (e.g., Schwartz & Smith, 1953, Winterberg & Tsotsas, 2000). To put it di!erently, the pressure gradient required to obtain a certain #uid velocity close to the wall is signi"cantly smaller than the pressure gradient needed to obtain the very same #uid velocity far away from the wall. This principle can be exploited by "lling a multichannel framework, such as a monolith, with ordinary catalyst particles, as shown in Fig. 1. Depending on the design (channel and particle shapes, channel-to-particle-diameter ratio, etc.) a CSP creates a very signi"cantly lower pressure drop (up to a factor of 15) per unit catalyst mass than a randomly packed bed containing the same catalyst particles, due to the high voidage and the alignment of the particles. Alternatively, the CSP concept allows the use of smaller particles compared to a traditional random bed with the same pressure drop, favoring catalyst e$ciency and selectivity. The following approach is followed to reach the stated goal. First we de"ne a number of narrow packed beds of particles, or, to put it di!erently, a number of CSP con"gurations, by "xing the channel-to-particle-diameter ratio as well as the positions of all particles. Next, a commercial CFD code (CFX-5.3) is used to simulate single-phase #ow through these packed beds. The results are expressed as friction factors versus Reynolds numbers. In parallel, lab-scale packed beds are constructed with exactly the same geometry as the packings used in the simulations. The lab-scale packings are used to experimentally determine pressure drop and local #uid velocities (using laser-Doppler anemometry, LDA). Following, the simulated results are compared to the experimental values and "nally the simple algebraic pressure drop correlation of Eqs. (1) and (2) is "tted to the CFD results and compared with the experimental pressure drop data. In a second paper on this topic (Calis, Romkes, Dautzenberg, & Van den Bleek, 2001) we will
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follow a similar approach to study particle-to-#uid heat and mass transfer in packed beds.
2. De5nition of the packed beds Keeping in mind the concept of the CSP, in which monoliths are used to provide the structural framework to contain catalyst particles, channels with a square cross-section are a logical choice for the current study. Spherical particles were chosen to represent the catalyst particles, as these were anticipated to be more challenging from a CFD point of view than e.g. cylinders, even though cylinders will generally give higher pressure drop reductions compared to spheres. A number of channels with square cross-sections and "lled with spherical particles (i.e., unit cells of a CSP) are shown in Fig. 2. For any channel-to-particle-diameter ratio, both regular (repeating) and irregular (random) packings are possible. The random packings at N"2.42 and 4.00 shown in Fig. 2 were de"ned using a simulation algorithm, in which particles were dropped in a channel one by one, and each subsequent particle would "nd the lowest position available in the bed at that moment, i.e., the position with the lowest energy. This algorithm yields the densest random packing, as opposed to the loosest random packing (Jaeger & Nagel, 1992; Sloane, 1984; Eppstein, 2000). Without proof it is stated here that for any channel-to-particle-diameter ratio, the densest packing that can be obtained is always a regular packing. Note, however, that at a certain channel-to-particle-diameter ratio more than one regular packings may be
Fig. 3. Dependence of the voidage of a packing of uniform spheres in channels with square cross-sections, as a function of the channel-toparticle-diameter ratio (N). Solid line marked with &d': voidage of the densest possible packing (i.e., a regular packing) at a given N; solid line marked with &r': voidage of the &densest random packing' at a given N. (䉱) voidage of &loosest random packing' of an in"nitely wide bed; (䊏) voidage of &densest random packing' of an in"nitely wide bed; (䢇) voidage of &absolutely densest packing' of an in"nitely wide bed; all for uniform spheres.
possible; see for example the two regular packings at N"2.00 in Fig. 2. For the range of 1.00)N)4.00, the positions of the particles in both regular and irregular packed beds containing up to 157 uniform spheres were calculated, and used to calculate the voidage as a function of the channel-to-particle-diameter ratio. The result is presented in Fig. 3. It is obvious that if one wants to simulate a packed bed using CFD techniques, the amount of required memory and calculation time will increase with the number of particles in the bed. At the current state of development, we consider a number of grid cells of a few million to be a reasonable maximum. Higher numbers are certainly possible, but are considered excessive at this moment. As we will show, this limits the number of particles in the simulated packed bed to about 30 (laminar #ow, without modelling heat transfer from particle to #uid). From the con"gurations shown in Fig. 2, the "rst "ve were selected for CFD simulation and experimental pressure drop and LDA measurements. The packing with N"1.00 ("0.48) was chosen because it is the simplest packed bed that is possible; N"1.15 ("0.60) to study the e!ect of a small deviation of N on pressure drop, to verify that the tolerance in the experimental packed beds was su$ciently small; N"1.47 ("0.68) because it has a very high voidage; N"2.00 because two regular particle arrangements are possible, A ("0.48) and B ("0.54), with signi"cantly di!erent voidages.
3. Simulation Fig. 2. Arrangement of uniform spheres in channels with a square cross-section, for various ratios of channel-to-particle-diameter (N). Top row: front views; bottom row: isometric views.
The most important aspects of the CFD model are: E The model is three-dimensional.
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E To limit the computational demands, in each case only 8}16 particles were used. E An unstructured grid was used, which is unavoidable because of the complicated geometry of the #ow domain (i.e., the space between the particles). On the walls of the solids, a number of layers of prismatic grid cells were used, the remaining space was "lled with tetrahedral cells. E The #uid was de"ned to be incompressible. This is allowed because the #uid #ow in packed beds has (in practice) very low Mach numbers, and because pressure drop characteristics obtained with a constant density are easily translated for situations with varying density. E The #ow was assumed to be stationary, even though it is not yet clear if transient phenomena (compare vortex shedding with cylinders in cross-#ow) might in#uence the calculated #ow "eld and pressure drop. E Both laminar and turbulent #ow were studied; the Reynolds number (based on interstitial velocity and hydraulic packing diameter) was varied from 1.0;10\ to 5.0;10, to cover the full range from fully laminar (where the friction factor is inversely proportional to Re) to fully turbulent #ow (where the friction factor is independent of Re). E Two turbulence models were used: a k}-model (because it is the most widely used turbulence model) and a Reynolds-Stress Model (RSM; because isotropic turbulence, assumed in the k}-model, may not be valid in the considered geometries). A zero-equation turbulence model was only used to facilitate convergence of a more elaborate turbulence model. E Standard wall functions were used in the case of turbulent #ow. E In almost all cases the "nal solution was obtained with a second-order discretization scheme. E The model was implemented in the commercial CFD package CFX-5.3 (AEA Technology). E The geometry of the packed beds was built using CFX-5 Build and the meshing was done using the CFX-5 mesh generator. To avoid the generation of mesh cells of poor quality at the points of contact between particles themselves and between particles and channel wall, all particles were shrunk by 1% of the diameter after the initial de"nition of the bed. In other words, the positions of all particles in the packed bed were calculated based on a certain particle diameter d , but the particles actually de"ned in the N CFD domain had a diameter of only 0.99d . As a result, N the packings contained no points of contact. The values of the packing voidage and hydraulic diameter to be used in the calculation of the friction factor from the simulated pressure drop were corrected for the smaller particle diameter. To check whether this correction is adequate, an extra case was simulated in which a factor of 0.98
Fig. 4. Impression of the CFD mesh in a region between a spherical particle and the channel wall.
Table 1 Typical mesh quantities used in the CFD simulations Turbulence model Edge length of the tetrahedral cells (mm) Number of layers of prismatic cells (dimensionless) Thickness of the "rst layer of prismatic cells (mm) Expansion rate of the prismatic cells (dimensionless)
Laminar 0.4 5
k} 1.0 0
0.052
n.a.
1.2
n.a.
instead of 0.99 was used. The resulting friction factors di!ered less than 0.5%, which justi"es the approach. The mesh used for a certain packed bed can be characterized by the edge length of the tetrahedral cells, the thickness of the "rst layer of prismatic cells on the surface of the solids (sphere or channel wall), the expansion rate of the prismatic cells and the number of layers of prismatic cells; see Fig. 4. In most simulations a sphere diameter of 12.7 mm was used, in combination with the mesh quantities listed in Table 1. An edge length of 0.4 mm on a 12.7 mm diameter sphere corresponds to a number of about 7300 triangles in the surface mesh of the sphere, whereas an edge length of 1.0 mm corresponds to about 1200 triangles. For laminar #ow and a case with N"2.00 A with 16 spheres, a total of about three million grid cells was needed when using a tetrahedral mesh edge length 0.4 mm in combination with "ve layers of prismatic cells on the solid surfaces. Extrapolating the rate of development of computer and CFD technology, these "gures suggest that within "ve years the simulation of a packed bed containing a few hundred particles will be considered a &standard' problem in terms of memory and calculation time requirements.
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Fig. 5. Dependence of the simulated pressure drop across a packed bed as a function of the (inverse) thickness of the wall cells. (䊏) cases where no layers of prismatic cells were used, i.e., the wall cells were tetrahedral; (䢇) cases where layers of prismatic cells were used. The arrow indicates the wall cell thickness used in most laminar simulations.
For laminar #ow the grid independence of the solution was checked by varying the grid type and grid density near the walls. In Fig. 5 the resulting pressure drops across the packed bed are plotted as a function of the thickness of the wall cells. This "gure indicates that wall cells of 0.052 mm thickness are adequate in this situation, as a further reduction of the thickness by a factor 2.7 results in only a 2.5% higher simulated pressure drop. For turbulent #ow the thickness of the cell walls is expressed in the value of y>, which in the case of standard wall functions should preferably be between 30 and 300 or at least between 10 and 1000. However, in packed beds, large deviations of the local velocity near the solid surfaces exist because of more or less stagnant zones near the points of contact between spheres. This results in a variation of the y> values over the sphere surface of a factor 40 when a homogeneous surface mesh is used, making it di$cult to meet the y> criterion everywhere on the sphere surface. More important, the condition that the #ow should be well developed in order for the standard wall model to be valid, is most likely not fully met in the case of a packed bed. These factors will introduce a systematic error in the simulated pressure drop characteristics that is hard to assess beforehand. The average y> value on the sphere surfaces was about 4.4;10\ Re, i.e., ranging from about 4.4 at Re"1.0;10 to about 440 at Re"1.0;10. These values can be improved by using a non-uniform surface mesh on the spheres and by adapting the mesh density to the Reynolds number. However, this was beyond the scope of the present study. The use of the RSM turbulence model as an alternative to the k} yielded only slightly di!erent pressure drop results (less than 10% discrepancy). This di!erence was considered too small to justify the extra computational demand of the RSM model.
Fig. 6. Schematic of the experimental setup. Right: detail.
4. Experimental Experimental data were obtained using channels with walls made of PMMA, with square cross-sections of 12.7}26.0 mm, and a length of 700 mm. Highprecision polyethylene spheres (~"12.7 mm for N"1.00, 1.15, 2.00A and 2.00 B, versus 9.5 mm for N"1.47) were supported by a wire mesh screen. The inlet region of the channels contained a 40 mm high bed of 2 mm spheres followed by a 40 mm section of 400 cells/inch monolith, to obtain a #at inlet velocity pro"le. The channels contained pressure taps every 100 mm. See Fig. 6. For the pressure drop experiments, dried air at ambient conditions was used. Experimental Reynolds numbers ranged from about 100 to about 6000. Pressure drops were measured across a bed length of about 500 mm. Local velocity pro"les were measured with laser-Doppler anemometry (LDA), using water at ambient conditions, seeded with 0.3 m TiO particles. Both horizontal and vertical cross-sections through the bed were measured for beds with channel-to-particle-diameter ratios of 1.47 (at Re"486 and 1.11;10) and 2.00B (at Re"575 and 7.84;10).
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Fig. 7. Pressure pro"le obtained through CFD simulation of a packed bed with N"1.00. The area-averaged pressure was calculated on planes perpendicular to the longitudinal axis of the tube, midway between subsequent particles.
5. Results and discussion A very important question in the simulation of a random bed with a certain channel-to-particle-diameter ratio is what bed length (or what number of particles) is needed to obtain results that are representative for &in"nitely' long beds. In the present study the number of particles in the simulations ranged from 8 (N"1.00) to 16 (N"2.00 A). Fig. 7 shows the simulated pressure pro"le in a packed bed with N"1.00. The constant pressure gradient indicates the absence of a signi"cant entrance region in the packed bed * or, to put it di!erently, each subsequent particle represents a new entrance region, and in this the "rst particle is not di!erent from the subsequent particles. The #ow appears to be periodical right from the "rst particle onwards, at least in terms of pressure drop.
Fig. 8 shows simulated streamlines for a packed bed with N"1.47. Also the streamlines around the "rst particle suggest that the #ow at the "rst part of the packed bed is identical to the #ow more downstream. The conclusion of these two "gures and similar "gures for other channel-to-particle-diameter ratios, is that already a short bed of about 6 particle diameters is representative of a much longer bed. Fig. 9 shows a comparison between a.o. the experimentally determined and the simulated friction factors as a function of the Reynolds number. It is seen that, in general, the agreement between the experimental and simulated friction factor is satisfactory * a typical deviation between the two is about 10%, which is certainly acceptable in the design of a packed-bed reactor. Because the simple correlation of Eqs. (1) and (2) contains only two parameters, and because in general the correlation is seen to even describe the friction factor in the transition range of Re quite satisfactory, it is tempting to try to "t the correlation using only two simulated data points. If this approach works well, one would only need to do two simulations for any given packed bed, one in the fully laminar range (to determine the parameter c ) and one in the fully turbulent range (to determine c ). This approach was followed, and the resulting parameter values are listed in Table 2. The resulting correlations are shown in Fig. 9 as well. For N"1.00, 2.00A and 2.00B, the comparison with the experimentally determined friction factors is very good: the average deviation is only about 5%. However, for N"1.15 and 1.47, the deviation is much larger, with an average of about 30%. The larger deviation is attributed to the higher voidage; note that the friction factor of an empty duct ("1.00) in the transition range of Reynolds cannot be adequately described with Eqs. (1) and (2). Also in Fig. 9 the prediction of the friction factor based on the original Ergun correlation, using (c , c )"(133,2.33), is shown. The agreement with the
Fig. 8. Streamlines from CFD simulation of the #ow in a packed bed with N"1.47.
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Fig. 9. Comparison of friction factors obtained from experiments (䊏), CFD simulation, both laminar () and turbulent (*), Eq. (2) "tted to two CFD data points for each packing (solid line marked with &C'), and the original Ergun correlation, Eq. (2) with (c , c )"(133,2.33) (solid line marked with &E').
Table 2 Results of "tting Eqs. (1) and (2) to two CFD data points per packing N (dimensionless)
c (dimensionless)
c (dimensionless)
1.00 1.15 1.47 2.00 A 2.00 B
1.48;10 1.14;10 7.28;10 1.12;10 2.14;10
5.03;10\ 3.59;10\ 3.67;10\ 6.51;10\ 1.04
experimental friction factors is poor: the Ergun overpredicts the experimental friction factors by an average of 80%. Comparison of the Ergun correlation with the results of the CFD simulations shows that the deviation is mainly due to poor prediction of the turbulent contribution to the friction factor. This is an interesting observation, because for empty ducts or tubes (as opposed to packed beds) the laminar contribution to the friction factor of a duct or channel is strongly dependent on the shape of channel (in other words: the geometry), whereas the turbulent contribution is not, and can normally be described in terms of hydraulic diameter only. In the case of the packed beds of the present study, the opposite is the case. Fig. 10 presents a comparison between experimental #ow pro"les measured with LDA and simulated #ow pro"les obtained with the CFD model. The agreement between experiments and simulation is satisfactory, and was found to be the same for other Reynolds numbers
Fig. 10. Comparison of #ow pro"les in a horizontal cross-section through the bed (i.e., perpendicular to the longitudinal axis of the bed), obtained with LDA (left) and CFD (right), for channel-to-particlediameter ratios of 1.47 (top) and 2.00B (bottom). The horizontal bars in the drawings on the right indicate the locations of the sample planes.
and for vertical cross-sections instead of the horizontal cross-sections shown in Fig. 10. These results indicate that the CFD model is not only able to predict an integral performance characteristic like the pressure drop across the packed bed, but also local velocities. Consequently, we anticipate that CFD simulation of packed beds can be of signi"cant help in obtaining a detailed understanding of the physical phenomena that take place in packed beds.
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6. Conclusions It has been shown that a commercial CFD code can be used to predict the pressure drop characteristics of packed beds of spheres that have a channel-to-particlediameter ratio of 1.00 to 2.00, with an average error of about 10%. This error is acceptable for design purposes. The local velocity pro"le on a cross-section of the packed bed, predicted by the CFD model, agrees well with the pro"le measured with LDA. The pressure drop characteristics over the full relevant range of Reynolds numbers can be modelled with a simple two-parameter model, of which the two parameters are obtained from only two CFD simulations, one at fully laminar #ow and one at fully turbulent #ow. Even though the geometry of the packed bed of spheres contains fairly strong curvature, the k} turbulence model appears to be adequate, and yields approximately (within about 10%) the same results as the RSM. The size of the grid cells needed for a grid-independent solution is so small that about three million cells are required for the simulation of a packed bed containing 16 particles. It is anticipated that within "ve years from now the simulation of a packed bed containing a few hundred particles will be considered a &standard' problem in terms of memory and calculation time requirements.
Notation c , c model parameters, dimensionless d hydraulic packing or channel diameter, including F the surface of the particles and the channel wall, m
d N 4f ¸ N Re u y>
particle diameter, m friction factor, dimensionless packing bed length, m ratio of channel diameter to particle diameter, dimensionless Reynolds number, as de"ned in Eq. (2), dimensionless super"cial #uid velocity, m/s dimensionless thickness of wall cells in CFD mesh, dimensionless
Greek letters
packing voidage, m /m dynamic #uid viscosity, Pa s #uid density, kg/m
References Calis, H. P. A., Romkes, S., Dautzenberg, F. M., & Van den Bleek, C. M. (2001). First International Conference on Structured Catalysts and Reactors, Delft, The Netherlands, accepted for presentation. Cybulski, A., & Moulijn, J. A. (1998). Structured catalysts and reactors. New York: Marcel Dekker Inc. Eppstein, D. (2000). http://www.ics.uci.edu/&eppstein/junkyard/ cover.html. Ergun, S. (1952). Chemical Engineering Progress, 48, 89. Jaeger, H. M., & Nagel, S. R. (1992). Science, 255, 1523. Sloane, N. J. A. (1984). Scientixc American, 250, 92. Strangio, V. A., Dautzenberg, F. M., Calis, H. P. A., & Gupta, A. (1998). Fixed Catalytic Bed Reactor; International patent application PCT/US99/06242, priority date March 23, 1998. Schwartz, C. E., & Smith, J. M. (1953). Industrial Engineering Chemistry, 45, 1209. Winterberg, M., & Tsotsas, E. (2000). American Institute of Chemical Engineering Journal, 46(5), 1084.