Mass Dispersion in Liquid Flow through Packed Beds - Industrial

Chemical Engineering Science 2008 63 (16), 4203-4217 ... A.A. Iordanidis , M. van Sint Annaland , A.E. Kronberg , J.A.M. Kuipers. Chemical Engineering...
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Mass Dispersion in Liquid Flow through Packed Beds Arno H. Benneker,† Alexander E. Kronberg,* Ivo C. Lansbergen, and K. Roel Westerterp Chemical Reaction Engineering Laboratories, Department of Chemical Engineering, Twente University of Technology, P.O. Box 217, 7500 AE Enschede, The Netherlands

Liquid flow experiments, in which a tracer was introduced as a point source, were performed in a thin two-dimensional packed bed. These simple experiments demonstrated the essential features of hydrodynamic dispersion in packed beds. The standard Fickian-type model and the wave model were compared with respect to their ability to describe the tracer propagation. The wave model does so adequately, in contrast to the erroneous predictions by the Fickian-type model. Arguments are provided to show that the application of hyperbolic-type equations improves the description of dispersion in packed-bed reactors. Introduction In comprehensive investigations, we still meet with difficulties in the design of wall-cooled packed-bed reactors using model calculations and independently measured kinetic and transport parameters.1-3 Residence time distribution experiments for the determination of axial dispersion coefficients in packed beds have shown that the values of these coefficients can depend considerably on the distance from the injection plane.4-7 Moreover, the independence of the transverse heat dispersion coefficient of the chemical reaction rate has been doubted.1,8 The drawbacks of the conventionally used parabolictype dispersion models have been discussed by many authors.9-12 A striking example, showing the disadvantages of the use of Fick’s diffusion law for the description of hydrodynamic mass dispersion, was given by Hiby.4 A colored tracer was continuously injected from a point source into a liquid in upward flow through a packed bed and dispersed inside that bed. Although the twodimensional standard dispersion model (SDM), where axial as well as radial dispersion are taken into account, predicts the presence of tracer also in the upstream parts of the packed bed, backmixing was not found relative to the system at rest, as shown in Figure 3 of Hiby.4 Moreover, the tracer dispersion in the radial direction can only be described reasonably well by the SDM at sufficiently large distances away from the injection point. Molecular diffusion can be neglected in the rapid liquid flow with Re0 ) 140 as applied by Hiby. Also, the experiments of Stephenson and Stewart13 demonstrated the shortcomings of the SDM. They measured the axial and lateral displacements of marker bubbles in liquid flow through a randomly packed bed. The introduced bubbles of air were sufficiently small and followed the fluid motion well. They found that the SDM fits the measured distributions poorly in both the axial and lateral directions. The SDM underestimates the sharpness of the peaks and cannot describe the * Author to whom correspondence should be addressed. Phone: 31 (0)53 4891088. Fax: 31 (0)53 4893663. E-mail: [email protected]. † Current address: DSM Research BV, P.O. Box 18, 6160 MD Geleen, The Netherlands.

bimodal distributions of the bubble travel distances in the axial direction, as always observed for Re0 < 100. Furthermore, their experiments showed that backward fluid motions are rare and that such displacements are consistently small compared to the packing dimensions; this in accordance with the experimental findings of Hiby.4 In this article, we present some simple experiments, similar to the experiments by Hiby,4 to demonstrate the essential features of dispersion in packed beds and perform point-source injections in a thin two-dimensional transparent packed bed. The colored tracer was injected close to the wall, as in Hiby’s work, as well as inside the packing. The experimental results are compared with the predictions of the two-dimensional SDM and the wave model. Transverse concentration profiles at different longitudinal positions for a salt solution injection were also measured by electrolytic conductivity experiments. Experiments were carried out with different particle sizes and at various flow rates; also the effect of a density difference between tracer and carrier was studied. For negligible density differences between tracer and carrier, the results of our visualization experiments are consistent with the results of Hiby4 and Stephenson and Stewart:13 no tracer was ever observed upstream from the injection point for the whole range of experimental conditions. These experimental results will be compared with the predictions of the conventional SDM and the two-dimensional hyperbolic wave model derived recently by Kronberg and Westerterp.14 The fundamental differences between the hyperbolic- and parabolic-type models will be demonstrated. The wave model predicts no backmixing and can adequately describe the colored plume. The SDM, in contrast, predicts strong backmixing. At downstream distances that are large compared to the particle diameter, the predictions of wave model and SDM approach each other. Experiments with the injection of different amounts of tracer confirm the strength of the wave model over the SDM: an increase in the tracer amount does not change the situation near the injection point. This is in accordance with the wave model predictions that the tracer concentration beyond a characteristic surface is zero for any injection rate. The observation is in contrast to the SDM predictions,

10.1021/ie000904t CCC: $22.00 © 2002 American Chemical Society Published on Web 03/09/2002

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which are very sensitive to the tracer quantity because of the infinitely fast signal propagation. The same experiments but with a point-source injection of a salt solution and tracer concentration measurements at different locations in the bed result in a large scatter of data points. This is caused by the physical mechanism of hydrodynamic dispersion: mixing in the longitudinal and transverse directions is due to the velocity field, which is irregular in time and space. The magnitude of this scatter can be considered to be a measure of the dispersion flux. The relaxation time for this dispersion effect is on the order of the time needed for a fluid element to pass a particle. Our analyses in previous work15-17 showed that the SDM does not give adequate results for systems with characteristic times on the order of the relaxation time. As a consequence, the SDM cannot be used for describing dispersion phenomena over a distance of a few particle diameters, in contrast to the wave model. The experimental results underline the necessity of replacing the parabolic-type equations by hyperbolictype equations. Particularly in cooled tubular reactors, where significant temperature variations are present over a distance of a few particles, the application of the traditionally used dispersion models is highly questionable. The available dispersion coefficients, as determined experimentally for packed beds and evaluated with conventional models, can also be applied in the hyperbolic models. Methods for estimating the value of the relaxation time in the hyperbolic model for packed beds are given. Solutions of the Two-Dimensional Standard Dispersion Model and Wave Model For tracer propagation in a two-dimensional bed with fluid flow in the z direction, the SDM in dimensionless form can be written as

∂C 1 ∂2C 1 ∂2 C ) 2 ∂Z Boz ∂Z Boy ∂Y 2

(1)

where Z ) z/dp, Y ) y/dp, C ) c/c0, Boz ) uzdp/Dzz, and Boy ) uzdp/Dyy. Recently, Westerterp et al.15-17 developed a onedimensional hyperbolic model, called the wave model, for the description of hydrodynamic dispersion in chemical reactors and contactors. Kronberg and Westerterp14 also derived a two-dimensional wave model of heat and mass transfer in packed beds. The derived mass conservation equation for a nonreactive and nonadsorptive solute in a fluid flowing through a packed bed is

∂2C 1 ∂2 C ∂C + Bn 2 ) ∂Z Boy ∂Y 2 ∂Z

(2)

where

Bn )

τuz2 - Dzz uzdp

(3)

Here, a possible asymmetry of the dispersion process with respect to any plane moving with the average velocity is neglected, because no experimental data are available to judge whether this effect is important. Bn can be called the backmixing number, and its sign

Figure 1. Definition of the velocities vz, wz, and uz for an arbitrary velocity profile in a tube. vz ) fluctuation (local) velocity, wz ) average fluctuation velocity (“wave” velocity), uz ) average interstitial velocity.

indicates whether real backmixing is present (negative value) or not (positive value). The difference between the SDM and wave model is obvious. The sign of the coefficient of the second derivative with respect to the longitudinal coordinate Z in eq 1 is negative; in the wave model equation, eq 2, it can be negative or positive, depending on the value of the model parameters. In exceptional cases, the coefficient can even be zero, resulting in the disappearance of the second derivative with respect to Z in the wave model equation. Note that this does not mean that axial dispersion is absent. Equation 2 transforms into eq 1 if Dzz . uz2τ; this corresponds to wz . uz, where wz is the average fluctuation velocity in the longitudinal direction (see Figure 1), which is physically unrealistic in the case of hydrodynamic dispersion in packed beds. However, in case of molecular diffusion, eq 2 can be safely replaced by eq 1 in most practical situations, because the fluid velocity uz is usually much lower than the molecular speed or the speed of sound. The solution of the SDM, eq 1, for a continuous injection of tracer at point Z ) 0, Y ) 0, with a flux per unit depth of the channel equal to 2buzc0 is18

C)

( )[ (

)]

axBozBoy Boz Boz 2 Boy exp Z K0 Z + Y2 π 2 2 Boz

1/2

(4)

where a ) b/dp and K0(x) is the zeroth-order modified Bessel function of the second kind. Equation 4 does not take into account the physical boundaries of a packed bed. Because we are particularly interested in the region close to the injection point and not in the regions near the physical boundaries of the experimental equipment, the influence of these boundaries on the tracer distribution is negligible in the region of our interest; therefore, we have used eq 4 for a comparison with the experimental results. The solution of eq 2 for (τuz2 - Dzz) > 0, corresponding to no backmixing in accordance with many experimental studies, with the boundary conditions

Z ) 0: C ) 2aδ(Y),

∂C )0 ∂Z

(5)

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is not equal to zero in the area surrounded by a surface |Y| ) Z/λ

C ) aλe-Z/(2Bn)

[

]

I0(P) Z I (P) + δ(Z - λY) + 2Bn 4Bn2P 1

(6)

where I0 and I1 are the zeroth- and first-order modified Bessel functions of the first kind, respectively, and

λ ) xBnBoy )

x

uz2 - wz2 xZ 2 - λ2Y 2 ; P ) 2Bn wy2

(7)

Equations 4 and 6 give essentially different concentration distributions near the injection point: the SDM predicts a wider spread of the tracer than the wave model. For large Z, provided that Z and Y are bounded, eqs 4 and 6 both develop in the Gaussian distribution

C)

a

x

Z π Boy

exp

(

)

-Y 2Boy 4Z

(8)

As Dzz no longer appears in eq 8, longitudinal dispersion has no influence on the concentration distribution far from the injection point. This analysis can explain the difficulties encountered in the determination of longitudinal dispersion coefficients in steady-state point-source injection measurements, such as in the work of Ziolkowski and Szustek,19 who found a large and random scatter in the longitudinal dispersion coefficients. Normally, the SDM solution is fitted to transverse concentration distributions, as measured in steady-state experiments at different longitudinal positions. This results in correct values for the transverse Bodenstein number because the available experimental data at larger distances can be handled well with the SDM. The determination of a longitudinal dispersion coefficient is more problematic: at large distances from the injection point, longitudinal dispersion does not significantly influence the model predictions, whereas at short distances, the SDM predictions are incorrect, as shown by Hiby4 and also in this article in Figure 4. The values for the longitudinal dispersion coefficient, determined in steady-state experiments by curve fitting with the SDM, are much smaller than those obtained in transient tracer experiments. This is demonstrated in Figure 2, where transverse concentration distributions at a distance of 5dp downstream from the injection point are plotted. The SDM predictions with Boz ) 0.5 and Boy ) 14, values measured for moderate Reynolds numbers, do not match the more realistic wave model predictions. The concentration distribution as predicted by the SDM can approach the distribution predicted by the wave model only if unrealistic large values of Boz, such as Boz ) 100, are used, as shown in the figure. Such large Bodenstein numbers in the SDM are also needed to describe adequately the Hiby experiment4 or the experiment shown in Figure 4 of this article. Experimental Setup Point-source injections were studied in a two-dimensional transparent bed. A visible tracer was injected as a point source, and its spreading was recorded photographically with a 50-mm camera. Tracer concentration

Figure 2. Transverse concentration profiles predicted by the twodimensional SDM and wave model at a distance of five particles downstream from the injection point (Z ) 5). Model parameters: λ ) 2, Boy ) 14, a ) 0.1.

distributions were obtained by measuring the electrolytic conductivity of a salt solution tracer at several transverse and longitudinal positions. In Figure 3, a simplified flow diagram of the experimental installation is presented. The flat Plexiglas box, 500 mm long, 300 mm wide, and 30 mm deep, was filled with 3.8-, 7.1-, or 9.7-mm-diameter glass spheres. The liquid carrier was stored in two stirred 60-L supply vessels under a constant pressure of about 0.3 MPa. The liquid flow through the bed was controlled by rotameters, the maximum flow rate being 14 L/min. The liquid entered the flat box at the bottom through four inlet tubes. Furthermore, the distributor consisted of a 5-mm Plexiglas plate with 2-mm-diameter perforations at a distance of 5 mm from each other. Horizontal baffles above the inlet tubes and a 50-mm-high calming section packed with 3.8-mm spheres were installed to avoid a preferential flow above the four inlet tubes. Several visualization tests verified that the macroscopic velocity was uniform in the transverse direction. For the visualization experiments, tap water or a NaCl aqueous solution (5.6 wt %) containing about 0.5 ppm of phenolphthalein on a weight basis was fed into the flat box. In the case of conductivity experiments, tap water was used as the carrier. The physical properties of the solutions used in the experimental program are shown in Table 1. Tracer was supplied from a 1-L vessel under a pressure of 0.15 MPa. The tracer was injected through a small capillary. For the visualization experiments, a continuous flow of a solution of caustic soda of 1 M NaOH was introduced into the bed through a capillary of 3-mm o.d. and 2-mm i.d. The tracer injection point was placed in the middle of the packing 30 mm above the calming section. To investigate possible wall effects, some experiments were performed with injection at the front wall or the side wall of the box. Phenolphthalein was added to the liquid in the supply vessel; it is colorless in the acid phase and changes to violet at around 10-6 M NaOH or at pH ) 8 or lower. In our experimental setup, the minimum observable concentration was around 10-4 M NaOH as determined via calibration. In most of the experiments, the tracer flow was adjusted to approximately the same velocity as the interstitial velocity through the packed bed. Electrolytic conductivity measurements were performed during the injection of a 5 wt % KBr solution through a 4-mm-i.d. capillary. The carrier was tap water; density adjust-

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Figure 3. Simplified flow diagram of experimental installation: (1) tracer supply vessel, (2) rotameter, (3) two-dimensional packed bed, (4) carrier supply vessel. Table 1. Physical Properties of the Aqueous Solutions Used in the Experimental Study liquid

density (kg/m3)

viscosity (10-3 Pa s)

tap water 5.6 wt % NaCl solution 1.0 M NaOH solution 5.0 wt % KBr solution

998 1039 1039 1035

1.002 1.098 1.235 1.079

ments by sugar addition did not help because of foam and large viscosity differences. Therefore, these experiments were carried out only under conditions where buoyancy effects were negligible. The detector, a circular platinum cell, was placed just above the packing at different transverse positions and at 50, 100, 150, and 300 mm above the injection point. Thus, after each set of transverse concentration distribution determinations, the bed height was increased with 50 or 150 mm of packing. The detection cell was calibrated, and the measured signals were converted to tracer concentrations. The detector had a circular diameter of 10 mm, so the signal represented the tracer concentration averaged over a cross-sectional area of 79 mm2. With this detector size, it was possible to measure so-called point concentrations of individual streams in the 9.7-mm packing as well as average concentrations in the 3.8-mm packing. The measured concentration distributions must be considered as only qualitatively as the detectors’ cross-sectional area was relatively large compared to the area of the tracer cloud. The observed values of the transverse Bodenstein numbers, for example, Boy ) 9 at Re0 ) 250, agreed reasonably with data in the literature. Visualization Experiments The experimental results with 3.8-, 7.1-, and 9.7-mm particles and at Reynolds numbers between 1 and 250 showed that the tracer material remained in a parabolashaped cloud downstream from the injection point. Throughout the entire range of experimental conditions, no backmixing relative to the packing occurred, that is,

Figure 4. Point-source injection in a two-dimensional packed bed compared to the predictions of the SDM (solid line) and the wave model (dotted line). No tracer is observed upstream from the injection point, as predicted by the wave model. The SDM predicts strong backmixing. Conditions: Re0 ) 150, dp ) 7.1 mm. Model parameters: Boz ) 1.0, Boy ) 12.3, λ ) 2.

no tracer was observed upstream from the injection point. The standard dispersion model would predict the propagation of alkali upstream from the injection point. Similar results were obtained with the injection point near the box wall. To avoid confusion, we explicitly note here that, in our equipment, upstream means in the downward direction, and in like manner, downstream is in the upward direction. In Figure 4, a typical result is presented for the 7.1mm particles and Re0 ) 150. The predictions of the SDM and of the wave model, as calculated with eqs 4 and 6, respectively, are also given in the figure; the lines bound the region where the tracer should be visible and correspond to a concentration of 5 × 10-4 M or pH ) 8.7. The values for the longitudinal and transverse

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Bodenstein numbers were estimated with the correlations of Gunn20 and found to be Boz ) 1.0 and Boy ) 12.3. Furthermore, a ) b/dp ) 0.1, and c0 ) πb2/(2bd)c/0 ) 0.0524 M, where d denotes the depth of the twodimensional bed and c/0 is the concentration of the tracer in the liquid supplied through the capillary. The SDM yields acceptable results only if the longitudinal and transverse dispersion coefficients vary with the axial position and decrease considerably near the injection point (large Bodenstein numbers). In that case dispersion coefficients are no longer universal constants and no longer have any physical meaning. Because no backmixing was observed, a minimal relaxation time for the wave model can be calculated if the Bodenstein number for longitudinal dispersion is known. In this case, Bn > 0, or uz2τ > Dzz ) wz2τ. Physically, this means that the average fluctuations in the longitudinal velocity do not exceed the interstitial average velocity or that wz < uz. Now, τ > Dzz/uz2 ) (1/ Boz)(dp/uz) ) dp/uz, if we use the value of Boz ) 1.0 calculated with the correlation of Gunn.20 The slope of the tracer plume boundary near the injection point, characterized by dZ/dY, equals λ as defined in eq 7. Note that this slope, dZ/dY, of the plume boundary for the case of a variation of the relaxation time from 1.0dp/uz to 3.0dp/uz varies from 0 to 5. The value of λ in the wave model in Figure 4 was chosen to be 2, which corresponds to τ ) 1.4dp/uz. Thus, the characteristic time constant for exchange of mass between different streams is on the order of the time that a fluid element needs to pass a particle. This corresponds with the physical concept of interweaving streamlines that mix continuously in the voids between the particles; the time necessary to travel from one void to the next is approximately dp/uz. Experiments with the injection of different amounts of tracer confirm the strength of the wave model over the SDM for the description of tracer propagation. Increasing the amount of tracer by a factor of 3 did not result in different shapes of the visible cloud. Particularly near the injection point, no influence of the injection quantity on the spreading was seen, in accordance with the predictions of the wave model. In contrast, the predictions of the SDM for the boundary of the visible tracer plume are very sensitive to the tracer quantity: the cloud predicted by the SDM should become considerably larger in the upstream direction when the amount of tracer is increased. During point-source injections with density differences of 4% between the tracer and the bulk liquid, buoyancy forces evidently affected the fluid flow. The effect was more pronounced in experiments with low Reynolds numbers and large particle diameters, as was found previously by Benneker et al.21 for gas-phase dispersion in packed beds. Even extensive backmixing of the denser tracer fluid in the upward liquid flow was observed: tracer was detected more than 3 particle diameters, that is, 30 mm, upstream from the injection point. Measurements of Concentration Profiles Injections of a 5 wt % KBr solution were performed at Reynolds numbers between 10 and 250 and with 3.8-, 7.1-, and 9.7-mm-diameter particles. The measurements with the 9.7-mm particles revealed a large scatter in time and space of the concentration distribution near the injection point. The detector signal continuously

Figure 5. Transverse concentration profiles at different longitudinal positions. (a) Re0 ) 150, dp ) 9.7 mm; (b) Re0 ) 70, dp ) 3.8 mm.

fluctuated, and the signal averaged over time did not exhibit a smooth normal distribution as a function of the transverse coordinate. We found that, with increasing distance from the injection point, the transverse concentration profiles transformed to a normal distribution and the scatter in time diminished. The experiments with the small 3.8-mm particles did not show significant fluctuations or spatial scatter in the transverse concentration profiles at all. Typical transverse concentration profiles for both packings are presented in Figure 5. Scatter similar to that shown Figure 5b was also observed by Stephenson and Stewart13 and Ziolkowski and Szustek.19 The frequency distributions of the lateral displacements of small marker bubbles determined by Stephenson and Stewart did not exhibit a smooth normal distribution. Ziolkowski and Szustek found a large scatter for different beds of the values of the local tracer concentrations at the same position in the bed when the bed was repacked after each concentration distribution measurement. Furthermore, the results coincided qualitatively with temperature variations in point measurements of the temperature in packed beds.22,23 The observations mentioned previously are characteristic for the mechanism responsible for hydrodynamic dispersion in packed beds. The longitudinal and transverse dispersion originates from the tortuous flow of the liquid in the form of blending and separating streams in the voids between the particles. A dispersion flux can arise only if the concentrations of streams that move in an opposite direction with respect to a plane moving

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with the average interstitial velocity are different. If the size of the probe is comparable to or smaller than the particle size, the point measurements show a scatter of data, as demonstrated for the 9.7-mm particles; this scatter can be considered as a measure of the dispersion flux.14 In turbulent flow, the continuously changing flow pattern causes fluctuations in time in the detector signal. Evidently, the branching effect of the solid packing generating a network of interwoven streamlines is a dominant factor in the dispersion of the point-source tracer. The relaxation time for this process is on the order of dp/uz, i.e., the time covered by a fluid element in travelling before mixing with another fluid element. Because characteristic times in cooled tubular reactors, for instance, of radial heat dispersion and/or chemical reaction, are on the order of (a few) dp/uz, previous analyses16,17 of the SDM and wave model taught us that the latter is superior in describing the dispersion phenomena in those systems. If the size of the probe is larger than the particle size, the detector signal gives the average of several streams with different concentrations. In that case, the detector is no longer a point probe, and scatter and fluctuations level out. This is confirmed in our experiments with the 3.8-mm particles. Discussion In the current article, simple point-source experiments in a two-dimensional packed bed have been presented. Emphasis has been placed on the dispersion of the tracer at distances of a few particles from the injection point. Considering such a small region around the injection point might seem of no practical importance. However, Klingman and Lee24 have stated that an acceptable dispersion model at the very least should adequately describe such a simple problem as injecting a tracer into a stream through a packed bed. Furthermore, an adequate description of mass and heat transfer over distances of a few particles frequently is of major importance for the reliable design and a safe operation of wall-cooled tubular reactors, in which the local hot spot is confined to a few particle layers in the tube. In these reactors, where tube-to-particle-diameter ratios of 4-10 are typical for highly exothermic reactions, significant differences in the temperature over a distance of a few particles exist, and a good knowledge of the heat transfer in radial direction to the wall is highly desirable. This was clearly demonstrated by Schouten et al.3 Apparently, the traditionally used SDM is not adequate for describing the dispersion processes at distances of a few particle diameters, as confirmed in this article. The wave model does not include the physical shortcomings of the SDM and appears to be a suitable model for hydrodynamic fluid dispersion. Only for slowly varying concentration and temperature fields do the predictions of the SDM coincide with the wave model predictions. To apply the results obtained to gas-phase systems, one should realize that the role of molecular diffusion is much different for gases than for liquids. In the case of high gas flow rates, as commonly encountered in practice, molecular diffusion does not play a role, so the results obtained in this study can equally well be applied to gases flowing through packed beds. The available data on longitudinal and transverse dispersion obtained by transient experiments can be

used as parameters for the wave model, even though they were determined using the Fickian-type dispersion model, because the measurements usually are made with the sensing elements located at quite a distance from the tracer injection point. In that case, relaxation effects are not important, and the concentration profiles can also be described with the conventional parabolictype equations. Exceptions have been found during the determination of axial dispersion in liquid flow through packed beds. Several authors have recognized that the axial dispersion coefficient in a packed bed depends on the column length and found that the dispersion coefficient increases with the distance from the injection plane. This indicates that relaxation effects are still important.4-7 Also, theoretical considerations of dispersion in porous media have led to dispersion coefficients that are much larger than ever measured in porous media.25 It should be realized that longitudinal dispersion originates from several mechanisms, each characterized by its own relaxation time. This dispersion can be governed by (1) the branching effect in a packing generating a network of interwoven streamlines, (2) the mass exchange between flowing and stagnant zones in the vicinity of pellet contact points, and (3) channeling and wall effects. In the laminar and transition regimes, where stagnant zones comprise a significant part of the liquid hold-up, the relaxation time for the exchange between the flowing and stagnant zones is about dp2/ Dmol. The ratio of the relaxation time to the average residence time is RepScdp/L, whereas for liquids, Sc is between 102 and 103. In many dispersion experiments in liquids, the relaxation time of this mechanism is comparable to or larger than the average residence time. If we further approximate the average fluctuation velocity to be on the order of uz, then Dzz ∼ uz2τ, and Boz ∼ 1/RepSc. Such low Bodenstein numbers are not measured, as in the usual experimental setups, the column lengths are too small for the mechanism discussed to be noticeable. The axial Bodenstein numbers, as experimentally observed, decrease with the distance in such cases; the above-mentioned low values of Boz ∼ 1/RepSc have not been observed. At higher Reynolds numbers, the flow becomes fully turbulent, the volume of the stagnant zones diminishes, and the mechanism governed by molecular diffusion becomes unimportant. For a first estimate of the relaxation time, the minimum, which follows from the no-backmixing requirement, can be used. In this case, wz,min ) uz, resulting in τmin ) Dzz/uz2 ) dp/uzBoz. More precise values can be obtained experimentally by measuring the slope of tracer boundaries near an introduced disturbance, as has been demonstrated in this article. At this point, we remark that the relaxation time must always be known to justify the applicability of the SDM. Conclusions The applicability of hyperbolic-type equations for the description of hydrodynamic dispersion in packed beds was investigated. A comparison with measurements of tracer propagation in a liquid flowing through a packed bed demonstrated that the wave model could adequately describe the experimental results with universal model parameters. The SDM gave acceptable results only if the model parameters were taken to be spatially dependent. These results support the concept that the

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traditionally used parabolic equations are not valid for describing rapid processes, as they occur above all in cooled tubular reactors. Because the wave model can be considered as more general than the SDM and because it uses rather similar model parameters, this model can be applied to great advantage in reactor modeling to improve the description of mass and heat transfer profiles in packed beds. Acknowledgment We owe a great deal to M. J. Lensselink, who took part in the experimental program. We further express our appreciation to G. H. Banis and F. ter Borg for their technical assistance. Nomenclature a ) dimensionless half-width of the injection source, b/dp b ) half-width of the injection source Bn ) Backmixing number, (uz2τ - Dzz)/uzdp Bo ) Bodenstein number, udp/D c ) tracer concentration c0 ) tracer inlet concentration in the two-dimensional bed, c/0πb2/2bd / c0 ) tracer concentration in the capillary C ) dimensionless concentration, c/c0 d ) depth of the two-dimensional bed D ) dispersion coefficient dp ) packing diameter j ) dispersion flux Rep ) Reynolds number based on interstitial velocity, Fuzdp/η Re0 ) Reynolds number, Fu0dp/η Sc ) Schmidt number, η/FDmol uz ) interstitial velocity u0 ) superficial velocity v ) fluctuation velocity w ) average fluctuation velocity y ) transverse coordinate Y ) dimensionless transverse coordinate, y/dp z ) longitudinal coordinate Z ) dimensionless longitudinal coordinate, z/dp Greek Letters  ) porosity λ ) constant defined in eq 10, [(uz2 - wz2)/wy2]1/2 η ) viscosity of the fluid F ) density τ ) relaxation time Subscripts y ) transverse direction z ) longitudinal direction

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Received for review October 19, 2000 Revised manuscript received December 11, 2001 Accepted December 12, 2001 IE000904T