738
Ind. Eng. Chem. Res. 1990,29, 738-748
A Novel Model for Reaction in Trickle Beds with Flow Maldistribution Gregory A. Funk, Michael P. Harold, and Ka M. Ng* Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 01003
The effect of liquid maldistribution on reaction in a trickle-bed reactor is examined by using a discrete model. It is based on a computer-generated, two-dimensional bed packed with equal-sized spherical catalyst pellets. The configuration of the liquid inlet distributor is included in the model. The flow on and intraparticle diffusion and reaction in each and every catalyst pellet are sequentially analyzed, leading to predictions of the overall reactor behavior. T h e interplay between flow patterns, wetting on individual pellets, and effectiveness enhancement is demonstrated. T h e present model offers advantages over other conventional models in terms of overall reaction rate predictions, the ability in the description of irregular morphological features, the absence of the artificial backmixing of materials, and others. Gas-liquid cocurrent downflow trickle-bed reactors have been widely used in hydrogenation, hydrodesulfurization, and other hydrotreating processes. For this reason, modeling the performance of this reactor type has received considerable attention. Progress has been rather gradual, however. The primary source of difficulty is due to the complex gas-liquid flow in the bed (Ng and Chu, 1987). Depending on the gas and liquid flow rates, the reactor may operate in either the trickling, spray, pulsing, or bubble flow regime. In the trickling flow regime, the liquid flows down the bed primarily on the surfaces of the packings while the gas flows in the remaining void space. Spray flow occurs at a high gas-to-liquid flow rate ratio. The liquid phase, in the form of droplets, is entrained in the continuous gas phase. In contrast, in the bubble flow regime, the bed is filled with liquid and the gas passes through the bed as dispersed bubbles. In pulsing flow, gas pulses of relatively high gas content can be observed passing down the bed. These different flow patterns can significantly affect the mass-transfer rate of gas and liquid reactants into the catalyst pellets and thus the overall reaction rate of mass-transfer-controlled reactions. The difficulty in describing these flows in a reactor model is compounded by the fact that finer features are present in any flow regime. For instance, a wetted catalytic particle in the incomplete wetting trickling regime can be completely or partially covered by liquid. Indeed, the liquid holdup is comprised of films, rivulets, filaments, pendular structures, and liquid pockets. Films and rivulets are liquid streams that flow on the surface of a catalyst pellet, while filaments are liquid streanis that flow in pore channels among the particles. Pendular structures and liquid pockets are stationary. The former resides at particle-particle contact points while the latter occupies several contiguous pore chambers. Obviously, a realistic reactor model should account for the major effects of these flow features on reaction. This paper reports such an attempt to model reaction in trickle beds in the trickling regime. Trickling flow is selected because it is the most common flow regime in industrial practice and is known to develop severe liquid maldistribution. For reasons such as poor liquid distributor design, entire regions of the reactor might be bypassed by the liquid phase, resulting in unexpectedly low conversion and/or formation of hot spots. A better understanding of the effect of liquid maldistribution on reaction is of considerable interest in the design and operation of a trickle-bed reactor. Before proceeding further, we describe qualitatively below in more detail the physical picture of the incomplete wetting trickling regime at the particle level. This would help to show clearly the motivations behind o u r model.
Figure 1 shows schematically how liquid wets the packings in such a flow regime. The particles are, of course, in contact but are represented as isolated circles in this two-dimensional schematic diagram. There are two liquid streams, depicted as shaded regions, that correspond to the two liquid inlets at the top of the packed bed. The liquid phase in each stream passes from pellet to pellet down the reactor. The wetting efficiency of a given pellet, i.e., the fraction of the external surface area of a pellet covered by liquid, depends on the liquid flow rate and the direction from which the liquid impinges on the particle. Each liquid stream spreads out somewhat at the top of the bed but then maintains a relatively constant width. The gas reactants pass down the bed in the remaining interstitial void space. Pellets can be bypassed by liquid streams and are nonwetted. It should be pointed out that, even if a porous catalyst pellet is externally nonwetted, it is generally internally liquid-filled because of capillarity. Internal dewetting, however, can occur if the reaction is highly exothermic (Bhatia, 1988; Harold, 1988). Furthermore, when the reactor temperature is sufficiently high, a fraction of the volatile liquid reactants can be present in the gas phase because of vaporization (Collins et al., 1985; Ruecker and Akgerman, 1987). In this paper, we focus on the relatively simple situation of isothermal reaction in wetted catalysts, which are assumed to be internally liquid-filled. The liquid from the pellets upstream brings with it the nonvolatile liquid reactants, which can then diffuse into the pellet. The gas reactants enter the pellet either through the nonwetted part of the catalyst surface or through the wetted part after first dissolving in the liquid film. Thus, if the liquid and gas reactant concentrations external to a pellet are known, the conversion taking place within the pellet can be determined by analyzing the problem of external mass transport and intraparticle diffusion and reaction. And if the liquid flow pattern is known, the change in liquid reactant concentration can be followed from particle to particle for each catalyst pellet in the bed. This is essentially the approach taken in our model. After a brief review of the literature, we discuss the model in detail and then present the simulations. Review of the Literature A review of existing trickle-bed models indicates three stages of development. In the earliest studies, flow regimes and the various flow features were not considered, and mass-transport effects were not taken into account. Satterfield (1975) treated the gas and liquid phases as a single homogeneous phase, and a pseudo rate constant was used for reactor design. Henry and Gilbert (1973) suggested that the measured reaction rate was the product of an
0 1990 American Chemical Society
Ind. Eng. Chem. Res., Vol. 29, No. 5,1990 739
n
0 Nonwetted Particle 0 0 Figure 1. Schematic diagram of trickling flow in a trickle-bed reactor. The liquid phase entering through two inlets flows down the packed bed as two isolated streams. The catalyst particles covered by either stream can be either partially or completely wetted. Bypassed by the streams, some particles are totally nonwetted.
intrinsic rate and the liquid holdup in the bed. Mears (1974) argued that this assumption was not physically justifiable and proposed to replace liquid holdup with wetting efficiency. These earlier homogeneous models were found to be suitable for relatively slow liquid reactant limited reactions such as decomposition of hydrogen peroxide over activated carbon and hydrodenitrogenation of hydrotreated oil (van Klinken and van Dongen, ,1980). When a limiting reactant is present in the gas phase and/or the reaction is not slow, heterogeneous models accounting for external mass-transfer resistance are essential. The gas and liquid phases are assumed to be in plug flow (El-Hisnawi and Dudukovic, 1982), or axial dispersion is accounted for (Hofmann, 1977). Considerable research took place to study the corresponding heterogeneous model parameters-wetting efficiency (Schwartz et al., 1976), dispersion coefficients (Michell and Furzer, 1972), and mass-transfer coefficients (Goto and Smith, 1975; Sylvester and Pitayagulsarn, 1975). These heterogeneous models have gained acceptance and have been confirmed with experimental data. However, an implicit assumption in all the heterogeneous models is that each model parameter is of a constant value throughout the reactor. For example, for a bed with an overall wetting efficiency of 0.5, it is assumed that each and every pellet is half-wetted. Although uniform flow distribution can be achieved in laboratory experiments, liquid maldistribution can be a serious problem in commerical reactors. This has led to the third stage of development. Flow patterns are explicitly taken into account in what can be called flow and reaction models. Stanek et al. (1981) assumed that radial liquid spreading was governed by a diffusion-like model. The driving force for radial flow was the radial gradient of the axial liquid flow rate per unit area. Liquid distributors of different configurations were readily simulated with this model. On the basis of the assumption that the rate of reaction was a constant as long as the catalyst pellet was wetted to any degree, conversion was calculated for any flow pattern generated. This model represents a major innovation in modeling the reaction in trickle-bed reactors but has a couple of weak points. For
Particle Wetted on Left Particle Wetted on Right
8
Particle Wetted on Both Sides
@
Completely Wetted Particle
Figure 2. Computer-generated two-dimensional bed of equal-sized spheres serving as the porous medium. Liquid enters the bed a t a single inlet tube and flows from particle to particle down the bed.
a diffusion-type model, the liquid propagates indefinitely. This is not realistic because radial spreading is expected to teas\? when the axial flow rate is sufficiently low. As mentioned previously, the rate of reaction in a catalyst pellet depends on whether it is completely or partially wetted. As radial liquid spreading takes place and the wetting efficiency decreases down the bed, a constant rate of reaction is unlikely. Flow and reaction models based on percolation theory represent another contribution to this area (Crine et al., 1980; Crine and L’Homme, 1982 and other subsequent papers). The approach is basically a combination of simulation and a phenomenological description of flow and reaction in a trickle-bed reactor. Since the relevant transport processes are not analyzed in detail, the potential of these models has not yet been fully realized. The model described in this paper can be regarded as an addition to the flow and reaction models. Predictions of liquid flow and distribution are based on physical grounds and provide more details than existing models. In fact, the wetting efficiency of each individual catalyst pellet is accounted for. External mass transfer, intraparticle diffusion, and reaction of each particle are also included.
Model Development The model can be conveniently divided into four parts: (1)A porous medium model is needed to account for the interactions between the fluids and the solid matrix. (2) The liquid flow pattern is generated through a flow simulation. (3) Intraparticle diffusion and reaction analysis provides the rate of reaction in a single pellet. (4) Finally, the determination of the model parameters is discussed. Porous Medium Model. A two-dimensional computer-generated bed packed with equal-sized spheres serves as the porous medium model (Figure 2). It is formed by dropping the spheres one by one along the top of an initially empty column, which has two imaginary walls and a floor. The column can have any height and width but
740 Ind. Eng. Chem. Res., Vol. 29. No. 5. 1990
has a thickness equal to one sphere diameter. Each sphere free falls down the column. All collisions are assumed to be inelastic. If a sphere hits the floor, it stays at the position where it lands. If it collides with any previously dropped spheres, it rolls until a stable position is reached. Thus, a sphere can rest on the floor or on top of two previously dropped spheres. Periodic conditions are applied along the two imaginary walls. If a sphere rolls out of the bed at the left wall, it reenters at the right wall. Thus, the bed in Figure 2 can be repeated indefinitely in the horizontal direction, and any content propagating down this model is not hindered by any walls, as if it were in the middle of a large reactor. This porous medium model is a simpler version of a three-dimensional computer-generated packed bed for which geometrical characteristics of the packing as well as the pore space have been determined (Chan and Ng, 1986, 1988). Flow Distribution. The liquid enters the sphere pack from inlet tubes positioned at prescribed locations and spreads out from sphere to sphere as it flows down the bed. The flow rate from each inlet tube can be prespecified. Figure 2 shows a column with one single inlet tube. Details of how the liquid spreads have been described elsewhere (Zimmerrnan and Ng, 1986; Zimmerman et ai., 1987) and are not repeated here. Briefly, the liquid flows down the bed from contact point to contact point. If the flow rate impinging on a sphere is sufficiently high, the liquid completely wets the sphere; otherwise, the sphere is only partially wetted, The wetting efficiency of a pellet, E,, can be calculated as a ratio of the liquid flow rate onto a single pellet, yp, to the critical liquid flow rate, nd,rg, for complete wetting (Hartley and Murgatrovd. 1964: Ng. 1986):
and statistical analysis (Fox, 1987). Wetting efficiencies of individual pellets, however, were not determined in these previous models. Diffusion a n d Reaction i n a Single Pellet. The conversion provided by each individual catalyst pellet is estimated by using a two-dimensional diffusion and reaction model. Since a more detailed description of this single-pellet model is available (Funk et al., 1988, 1989), only a brief summary of the important aspects is presented here. An infinitely long pellet with a square cross section and a width equal to S is assumed to be wetted symmetrically at the corners by four identical liquid films. An infinite square slab can be used to approximate the reaction within a spherically shaped catalyst as long as the ratio of the pellet volume to the external surface area is the same. This implies that S should be set equal to two-thirds of the diameter of a spherical catalyst pellet (Funk et al. 1989). Each of the four liquid films contains an equal concentration of a nonvolatile liquid reactant, CBp Due to capillarity, the pores of the catalyst particle are assumed to be completely liquid-filled. A gaseous environment, with a constant partial pressure of the gas reactant (A), completely surrounds the wetted pellet. A t this partial pressure, the equilibrium solubility of species A is equal to CAe. An isothermal, irreversible reaction of the general type A,,,
+ uB(,,
-
products(l,
(2)
occurs between the dissolved gas reactant and the nonvolatile reactant within the uniformly active pellet. The intrinsic reaction rate, r, is assumed to he first order with respect to A and B; i.e., r = h,CACB
(3)
Species A and B respectively satisfy the following dimensionless diffusion-reaction equations: where d, is the pellet diameter, I?, is the critical mass flow rate per unit length for complete wetting, is a proportionality constant, a is the surface tension, 1.1 and p are the fluid viscosity and density, respectively, and g is the acceleration due to gravity. The value of [ for solid, nonporous surfaces is estimated to be about 0.1 (Ng, 19861, and unless stated otherwise, this value is used in all the simulations in this paper. The value of f for a porous catalyst surface is difficult to estimate. However, it is expected to be less than 0.1 because, due to capillarity, it is easier to wet a porous surface. As illustrated in Figure 2, a completely wetted pellet is indicated by a darkened circle, while a nonwetted pellet is unfilled. Also shown are the symbols used for a partially wetted sphere that is wetted on the left, on the right, or on both left and right, depending on the way liquid comes from above. The shaded areai fraction of each circle corresponds to the wetting efficiency. Note that, since the bed is periodic, if any liquid flows out of one of the imaginary walls, it reenters the bed a t the other wall. The gas flow rate is assumed to have no effect on the liquid distribution. Flow patterns thus generated have been found to be sufficiently realistic for predicting dispersion in trickle beds, which depends on liquid distribution (Zimmerman et al., 1987) and resemble those observed in a CAT (Computer Assisted Tomography) scanner (Lutran et al., 1990). As mentioned earlier, flow patterns can also be generated by diffusion-type models and percolation theory. In addition, flow distribution has also been studied through Monte Carlo simulation (Ahtchi-Mi and Pedersen, 1986)
d2uA/as12 + a2UA/as22 =
(4)
~~uB+AuB
a2LlB/ds12+ d2UB/ds2' = $2muAuB
(5)
The dimensionless variables in eqs 4 and 5 are defined as UA
CA = -
C.A,
CB ug = CBt
x1
=
s/4
x2
sp = __ s/4
where C , and De, are the concentration and effective diffusivity of species i, respectively, and x1 and x2 represent distances in the two Cartesian-coordinate directions. Due to the symmetric wetting configuration, zero-flux boundary conditions can he used along s1 = 2 and s2 = 2 ; Le..
auA/as, = auB/as, = 0 at
S,
=2
= 1, 2
I
(7)
'rhus, eqs 4 and 5 only have to be solved over one quadrant of the pellet. Since the liquid reactant is nonvolatile, there is no transport of B from the gas phase: Le., duRjds, = 0
at s, = 0 (nonwetted portion) 1
= 1. 2 (8)
Film theory was used to obtain the following boundary relations for the transport of the liquid reactant across the
Ind. Eng. Chem. Res., Vol. 29, No. 5, 1990 741
"
n
I
Table 11. Parameter Values Used To Stimulate the Data of Herskowitz et al. (1979) CAe= 2.84 X lo4 mol/cm3 T = 40 "C CBf = 7.55 x iow3mol/cm3 P = 1 atm DeA = 9.30 X cm2/s S = 0.108 cm de^ = 7.00 X IO4 cm2/s d, = 0.162 cm p = 7.17 X g/cm/s p p = 1.5 g/cm3 p = 0.893 g/cm3 c, = 0.5 cm3 void/cm3 pellet u = 32 dyn/cm c, = 0.5 cm3 void/cm3 reactor g = 981 cm/s2 k , = 7437 cm3/mol/s u = l
nil
07
ii,
q
io:
i i i
L1L\
Figure 3. Dependence of single catalyst particle effectiveness on liquid flow rate with dimensionless liquid reactant concentration as a parameter. Calculated by using the diffusion-reaction model of Funk et al. (1988). Table I. Dimensionless Reaction Model Parameter Values Used in Simulations BiA,w= 6.81-9.49 @ = 0.4069 BiB,w = 110.6-142.8 m = 13.3 BiA,n = 1000.0
wetted portion and the gas reactant across both the wetted and the nonwetted portions of the external catalyst surface: - d ~ B / d ) ~=i BiB,,(1 - uB) a t si = 0 (wetted portion) -aUA/asi
i = 1, 2 (9)
= BiA,,(l - u,) a t si = 0 (nonwetted portion)
i = 1, 2 (10)
BiA,,( 1 - U A ) a t si = 0 (wetted portion)
i = 1, 2 (11)
-aU~/dSi=
where the dimensionless Biot numbers are defined as
A numerical solution to the coupled, nonlinear partial differential equations, eqs 4 and 5, subject to the boundary conditions, eqs 7-11, is obtained by the method of finite differences. A detailed discussion of the numerical method is presented elsewhere (Funk, 1990). Predictions obtained from this single-pellet model are found to be in agreement with experimental data (Funk et al., 1989). Some typical results are presented in Figure 3, which shows the dependence of the catalyst effectiveness, v, on the liquid flow rate over the pellet for different values of the dimensionless liquid reactant concentration in the film, UBp The effectiveness is defined as actual reaction rate ll= krCAeCBf
Table I presents a list of the relevant dimensionless model parameters used for Figure 3, which are derived from a more basic set of model variables to be described in Table 11. The value selected for the Thiele modulus, 4, corresponds to a rapid, mass-transport-limited reaction. In addition, the high value of m, which represents the ratio of the gaseous reactant effective diffusivity to that of the liquid reactant, implies that the gaseous reactant diffuses more rapidly than the liquid reactant within the support. As discussed below, the external mass transfer of reactants
A and B through the wetted part of external surface is dependent on the liquid flow rate. Therefore, a range of values are used for both wetted-part Biot numbers, BiA,w and BiB,,. Also notice that the gas reactant is more easily transferred across the nonwetted portion than the wetted portion of the external surface, Le., BiA,n > BiA,,. Figure 3 shows that the effectiveness exhibits a maximum and a local minimum for a fixed value of the dimensionless liquid reactant concentration. The maximum, which occurs in the partial wetting regime, has been shown in several previous studies to result from the system undergoing a liquid-reactant-limited to gas-reactant-limited transition. This phenomenon is known as the effectiveness enhancement by partial wetting (Harold and Ng, 1987; Yentekakis and Vayenas, 1987; Funk et al., 1988, 1989). As the liquid flow rate approaches zero, the rate must also approach zero since this corresponds to a complete absence of the liquid reactant. Increases in the liquid flow rate increase the degree of liquid coverage. This leads to an increase in the supply of the liquid reactant as well as the overall reaction rate. However, at an intermediate wetting efficiency, the gas reactant supply becomes the rate-limiting process. Since the gas reactant is ordinarily less easily transferred across a wetted surface due to the added resistance of the liquid film, further increases in the liquid flow rate cause a decrease in the overall reaction rate. Once the pellet is completely wetted, the effectiveness again increases with increasing liquid flow rate as a result of enhanced external mass transport of the reactants. Since the numerical solution of the single-pellet reaction model requires a substantial amount of computer time, it is impractical to repeat the effectiveness calculations for all the pellets in a bed that is sufficiently large to allow for a realistic treatment of liquid maldistribution. Instead, we solved the single-pellet model for different values of the local conditions, i.e., wetting efficiencies (E,) and liquid reactant concentrations ( UBf). Interpolation with bicubic splines was then used to estimate the effectiveness values for any intermediate conditions. Except where specified otherwise, we assumed that the partial pressure of the volatile reactant does not change for a given bed. This should be a reasonable assumption for a moderately sized bed for most reaction systems. In fact, we can view the bed as a window of observation of a certain location within a large reactor. Determination of Model Parameters. A host of model parameters are needed. For a given system, most physical properties such as density and viscosity can be found in a standard reference handbook. Information specific to the system, such as kinetics data, has to be determined with experiments. Since external mass transport has been shown to depend on the liquid flow rate under complete wetting conditions in previous experimental studies, we include the variation of the masstransfer coefficients with liquid flow for the completely wetted catalyst. A correlation for the mass transport of a sparingly soluble gas into a liquid flowing down a flat surface ( k g J is provided by Sherwood et al. (1975):
742 Ind. Eng. Chem. Res., Vol. 29, No. 5, 1990
where r is the mass flow rate per unit length, p is the liquid density, x is the distance in the direction of flow, 6 is the liquid film thickness, D is the molecular diffusivity of the dissolved gas in the liquid, and t is the time a surface fluid element is exposed to the gas over the distance x. The liquid film thickness and exposure time can be estimated easily based on an analysis of film flow over a sphere (Lynn et al., 1955). The distance x is approximated as ~ d , / 2 . Similarly, a correlation proposed by Norris and Streid (1940) and used to estimate the liquid-to-solid masstransfer coefficient (kl,):
Note that kgl and kl, are independent of the liquid flow rate, q p under partial wetting conditions. This is based on the fact that an increase in the liquid flow rate results in an increase in the width of the flowing film but the volumetric flow rate per unit length equals to a constant, rCp(eq 1). With a constant flow rate per unit length, the film thickness as well as the mass-transport resistance does not change.
Flow and Reaction Simulation Algorithm The simulation procedure takes advantage of the fact that liquid flows from particle to particle down the bed, and thus the flow of information concerning the pellets is also from top to bottom of the reactor. This absence of backmixing allows the examination of flow and reaction of each pellet, one at a time, until all the pellets in the bed are considered. There are nine steps: 1. Specify the locations, sizes, and liquid flow rates of each inlet tube. 2. Determine the amount of liquid that a pellet receives, if any, from the liquid distributor. Of course, the input liquid primarily goes to those pellets on top of the packed bed. More details will be given below. 3. Starting with the highest sphere in the bed, sum all the liquid flow rates from higher spheres that are in contact with the current sphere and/or from the liquid inlet distributor. 4. Determine the concentration of the liquid reactant flowing onto the current sphere by averaging over the input flows. 5. On the basis of the total liquid flow rate over the current sphere, estimate the wetting efficiency by using eq 1. 6. Calculate the conversion of the liquid reactant given by the current sphere. 7. Identify the spheres below and in contact with the current sphere. 8. Determine the liquid flow rates at the various outlets of the current sphere. 9. Return to step 3 for the next highest sphere until all spheres have been considered. Unless specified otherwise, all the results presented below are based on a bed with 1000 particles. It has a width of 15.7 pellet diameters and a height of approximately 60 pellet diameters. As pointed out earlier, because of this relatively small size, we are in essence considering reaction at a certain location, or window, within a much larger reactor. This size was not selected because we were limited by our computing facilities. It is simply sufficiently large to account for the impact of the flow features on
.
0
.-
I u fcn'ii
Figure 4. Comparison of sphere pack model predictions of the reaction rate as a function of the superficial liquid velocity to the experimental data of Herskowitz et al. (1979).
reaction. In fact, by "scrolling" the window, little additional computer memory is needed to simulate reaction in a bed of any height.
Results Comparison with Experimental Data. The predictive capability of the model is first confirmed with the experimental data of Herskowitz et al. (1979). They studied the hydrogenation of a-methylstyrene over a Pd/Al,O, catalyst in a differential trickle-bed reactor. This particular reaction system has been studied in numerous experimental investigations (Babcock et al., 1957; Satterfield et al., 1969; Turek and Lange, 1981; among others). The experimental conditions and model parameters are listed in Table 11. Two points should be mentioned. First, Herskowitz et al. assumed that the reaction obeyed pseudo-first-order kinetics, i.e., first order in hydrogen and zeroth order in a-methylstyrene. This is reasonable only if the intraparticle concentration of a-methylstyrene is sufficiently higher than that of hydrogen. However, this approximation must certainly break down if the wetting efficiency is so low that the supply of the liquid reactant is severely limited. To remove this approximation, we assume that the reaction obeys bimolecular first-order kinetics. The rate constant in eq 3 is equal to the pseudo-first-order rate constant determined by Herskowitz et al. divided by the pure amethylstyrene molar concentration. Second, as mentioned previously, the value of the proportionality constant, 6 (eq 1),is not known for a porous catalytic surface. For this reason, we treated it as an adjustable parameter in matching the experimental data. The prediction in Figure 4 is based on a 5 value of 0.03. It should be emphasized that 5 has essentially no effect on the predicted overall reaction rate. Adjusting 6 is nothing more than specifying that complete wetting occurs at u = 0.5 cm/s, as indicated by the experimental data. Note that since the cross-sectional area of the actual reactor is not the same as that of the model, superficial velocity is used to ensure that comparison is based on the same liquid flow rate per unit area. Two different sets of reaction rate vs superficial velocity data are included. The filled circles are for equilibrium feed conditions; Le., the a-methylstyrene is presaturated with hydrogen. Unsaturated liquid gives the results shown as unfilled circles. Both sets of experimental data show the same general trends. As the liquid superficial velocity increases from around 0.5 cm/s, the observed reaction rate gradually increases due to an enhanced external masstransport rate. If the liquid flow rate is decreased from the same point of 0.5 cm/s, the reaction rate increases
Ind. Eng. Chem. Res., Vol. 29, No. 5, 1990 743 4.0
I
I
0.0
0.2
0.4
0.6
0.8
1.0
q, (mL/s)
Figure 5. Dependence of reaction rate on liquid flow rate for uniform, three-tube, and single-tube inlet distributions of the liquid. For each inlet configuration, the calculations were repeated over three different randomly packed beds.
rapidly, indicating the onset of partial wetting. It is easier for hydrogen to enter the nonwetted part of a partially wetted catalyst pellet, thus resulting in a higher rate of reaction. Although not demonstrated by this set of data, the reaction rate must possess a local maximum a t a sufficiently low liquid flow rate. As the supply of a-methylstyrene disappears, the reaction rate must approach zero. The model prediction is shown as a solid line, and its general trends are in agreement with the experimental data. At low liquid flow rates, the predicted reaction rates are expectedly higher than the experimental data. Actual liquid distributors never provide perfectly uniform liquid input. In the simulations for Figure 4, the liquid inlet distributor is assumed and can be made to be perfectly uniform. This creates a large number of partially wetted catalyst pellets, and the phenomenon of effectiveness enhancement is fully expressed, resulting in a higher predicted rate of reaction. This point will be considered in more detail below. Another possible reason for the overprediction of the reaction rate is related to the presence of more complicated flow features. Liquid pockets and pendular rings covering an active catalyst pellet block off a portion of the catalyst surface and decrease the external mass transport of gas-phase hydrogen into the affected pellets, thereby decreasing the overall reaction rate. In its present form, the sphere-pack model does not account for these factors. For the complete wetting region, the predicted rate is less than the experimental data. This difference is related to the way the external mass transfer is modeled. We calculated the effectiveness of a single pellet based on the assumption that the liquid flow and concentration profiles within the liquid film are fully developed. In the actual experiments, entrance effects may not be insignificant. Thus, the average hydrogen concentration in the liquid film in the model is lower than that in the experiments with equilibrium feed, resulting in an underprediction of the reaction rate. The model predictions are lower than, but close to, the nonequilibrium feed data. In the experimental apparatus, there was an inlet section in the reactor where a-methylstyrene was in contact with hydrogen. As pointed out by the original authors, the nonequilibrium feed was not completely devoid of hydrogen and consequently the rather close agreement with our predictions. Effect of Liquid Distributor. The effect of the inlet liquid distribution is shown in Figure 5. The reaction system and operating conditions for this simulation as well as for all other simulations in the remainder of this paper,
Figure 6. Fractional liquid coverage of each particle in a packed bed for a uniform inlet distribution and six different flowrates. The areal fraction of each circle shaded black represents the degree of liquid coverage.
other than a few stated exceptions, are the same as those listed in Table 11. Three different liquid inlet configurations are considered: a perfectly uniform distributor, three equally spaced inlet tubes, and a single inlet tube. For each inlet configuration, we calculate the reaction rate as a function of liquid flow rate over three different computer-generated, randomly packed beds in order to investigate the impact of the packing configuration. Note that individual data points, with a different symbol for each of the three randomly packed beds, are used for the single inlet case because they cannot be easily represented by smooth curves. Typical flow patterns for the uniform inlet case are shown in parts a-f of Figure 6, which correspond to the flow patterns a t a flow rate of 0.04,0.1,0.2, 0.3, 0.4, and 0.7 mL/s, respectively. For clarity, only the top 400 pellets of the 1000-pellet bed are shown. The uniform liquid distributor is actually represented by 1000 evenly spaced inlet tubes, although a small number of tubes are shown in Figure 6. A sphere receives an amount of liquid that is equal to 1/1000th of the total liquid flow rate into the bed if a vertical line, unobstructed by any other spheres, can be drawn to connect the inlet tube and the sphere under consideration. Obviously, spheres a t the top of the bed would receive the majority of the input liquid. We assume that, for spheres receiving input liquid from the distributor, the liquid flows on both sides of the sphere if it is not completely wetted. A t low liquid flow rates such as shown in Figure 6a, a large number of very thin rivulets flow from particle to particle via the contact points. For simplicity, we start shading at the left or right extremes of the circle. For this reason, although these thin rivulets appear to be disconnected, they are obviously continuous in reality. Visual observation of Figure 6 indicates that the degree of wetting of all the pellets in each window is not significantly different. Since most of the pellets experience a nearly identical wetting condition, the functional dependence of the reaction rate on the liquid flow rate in a bed with a uniform inlet distribution (Figure 5) is qualitatively similar to the reaction in a single catalyst pellet (Figure 3). Also,
744
Ind. Eng. Chem. Res., Vol. 29, No. 5, 1990
( a ) q, = 0.04 mL/s
1 1 1
(d) q, = 0 3 mL/s
(b) q, = 0.1 mL/s I
I
I
( e ) q, = 0 4 mL/s
( c ) q, = 0.2 mL/s !
I
I
(0 q , =
I1
I :
0 7 mL/s
1 1 1
Figure 7. Fractional liquid coverage of each particle in a packed bed for a three-tube inlet distribution and six different flow rates. The areal fraction of each circle shaded black represents the degree of liquid coverage.
the three curves for uniform input in Figure 5 are close to each other, demonstrating that packing configuration does not play a major role when the liquid is injected uniformly. This is not true for nonuniform inlets, as illustrated next. As shown in Figure 5 for three-tube and single-tube distributors, changes in the packing geometry affect both the overall rate of reaction and the functional dependence of the rate on the total liquid flow rate. Let us consider the case of three inlet tubes. The overall reaction rate is substantially lower than the case of uniform inlet and can display two local maxima. Examination of the flow patterns in Figure 7 for the corresponding liquid flow rates reveals the underlying causes. When liquid is injected from several discrete locations a t the top of the column, it spreads out gradually from the points of injection from particle to particle. Spreading ceases if the local flow rate is not sufficiently high, resulting in three isolated streams. As a result, many of the pellets in Figure 7a-e are not wetted and therefore do not contribute to reaction. With a significant fraction of the pellets not involved in reaction, the overall reaction rate is considerably lower. Before proceeding further, we should mention that each of the three depicted inlet tubes is actually made up of 100 evenly spaced inlet tubes in the computer simulation, each carrying 300th of the total input liquid flow rate to the reactor. The curves with a double maximum are also a result of the flow pattern. Let us consider the curve with a slightly higher rate at 0.1 mL/s (Figure 5). For liquid flow rate ranging from 0 to 0.2 mL/s, a maximum is present because of the effectiveness enhancement of the partially wetted pellets. As the liquid flow rate is increased from 0.2 mL/s (Figure 7c) to 0.3 mL/s (Figure 7d), it so happens that in an averaged sense the three liquid streams widen just enough to wet more nonwetted pellets, and the phenomenon of effectiveness enhancement is again encountered. That is, the rate attains a maximum a t an intermediate degree of wetting for this new group of partially wetted spheres. Whether or not a second maximum is observed depends on the packing configuration. As a stream widens, it can increase the wetting efficiency of a few already
(d) qr = 0.3 mL/s
(c) q, = 0.2 mL/s
(b) q, = 0.1 mL/s
( a ) q, = 0.04 mL/s
- =
I
(c) qr = 0.4 mL/s
I
I
(f)
q, = 0.7 mL/s
Figure 8. Fractional liquid coverage of each particle in a packed bed for a single-tube inlet distribution and six different flow rates. The areal fraction of each circle shaded black represents the degree of liquid coverage.
wetted particles. If the wetting efficiency is greater than the wetting efficiency that gives a maximum effectiveness for a single catalyst, the reaction rate actually decreases. Apparently, these opposing effects cancel out for one of the three packing configurations, which exhibits a single maximum. The case of a single liquid inlet is even more complicated. Again, the rate must approach zero as the liquid flow rate approaches zero. However, the reaction rate is a very complex function of the liquid flow rate and the packing configuration within the bed. Figure 8 shows the flow patterns at different flow rates. With one single liquid inlet, this system is limited to a single stream, which widens with increasing liquid flow rate. As one would expect, comparison of Figures 6 and 7 to Figure 8 shows that there are more completely nonwetted spheres for the latter a t any given flow rate, indicating that a single tube distributor is inadequate for wetting the bed under consideration. There is another interesting feature in Figure 5. Despite inadequate wetting, the overall reaction rate of the single tube distributor reactor is slightly higher than that of the three-tube inlet and uniform inlet reactors at sufficiently high liquid flow rates. Examination of Figures 6f, 7f, and 8f reveals the underlying cause. At a liquid flow rate of 0.7 mL/s, the uniform inlet completely wets all the catalyst pellets; the three-tube inlet leaves some pellets totally unwetted but partially wets a number of pellets. In the single-tube inlet reactor, there are even more totally unwetted and partially wetted pellets. Recall that a partially wetted catalyst pellet has a higher reaction rate due to the effectiveness enhancement. Apparently, the higher reactivity of the partially wetted pellets overcompensates the loss in reactivity of the totally nonwetted pellets in the single inlet reactor. The overall wetting efficiency &,, i.e., the average wetting efficiency for all the particles in the bed, for each distributor as a function of liquid flow rate is shown in Figure 9. For liquid flow rates less than 0.2 mL/s, the wetting efficiency is nearly identical for all three distributors. The difference becomes considerably larger a t higher liquid flow rates. The three-tube distributor pro-
Ind. Eng. Chem. Res., Vol. 29, NO. 5, 1990 745 1.0
Uniform
'
0.0 0.0
0.2
0.4
0.6
0.8
1.0
004 00
'
Figure 9. Dependence of overall wetting efficiency on liquid flow rate for uniform, three-tube, and single-tube liquid inlet distributions. 4.0 1
I
-
0.0
0.2
0.4 0.6 q T(mL/s)
04
06
OS
10
q I (mL/c)
q, (mL/s)
0.0
4
'
02
0.8
1.0
Figure 10. Dependence of overall reaction rate on liquid flow rate with the inlet liquid reactant concentration as a parameter.
vides a wetting efficiency close to that of a perfect distributor, while the single tube distributor has a much lower value. It is now the opportune time to reexamine Figure 4. Comparison of Figures 6a and 7a shows that many more catalyst pellets are wetted when a perfect liquid distributor is available a t the top of the bed even though the overall wetting efficiencies in the two beds are nearly identical (Figure 9). This illustrates that a difference in flow pattern, amplified by the phenomenon of effectiveness enhancement, can have a significant impact on overall reaction rate. Of course, a perfect liquid distributor is not expected in an actual experiment. Therefore, the experimental reaction rate of Herskowitz et al. is closer to the predicted rate with three inlet tubes and lower than that with a perfect distributor. Effect of Inlet Liquid Concentration. Figure 10 shows the predictions of the model for changing inlet concentrations of the liquid reactant by adding an inert solvent. From another point of view, these results also demonstrate how the reaction rate changes down a reactor as the liquid reactant is depleted. At any liquid flow rate, a pure liquid reactant feed ( C B = ~ C B P ) has the highest reaction rate. Thus, as the concentration of the liquid reactant decreases down the bed, the overall reaction rate decreases for the bimolecular first-order kinetics used in the analysis. It should be pointed out that this monotonic behavior may not occur for systems that obey more complicated Langmuir-Hinshelwood-type rate expressions (Funk et al., 1989). Note that the flow rate at which the maximum reaction rate occurs increases with decreasing liquid reactant concentration. As the system becomes increasingly liquid reactant limited, a higher liquid flow rate is required to supply the necessary amount of liquid reactant.
Figure 11. Dependence of reaction rate on liquid flow rate for loo-, 1000-, and 2000-particle beds, which show increasing effects of pressure drop on reaction.
Effect of Pressure Drop. We have so far assumed that the pressure within a window of observation is a constant. Of course, pressure actually drops for two reasons, after neglecting pressure drop due to acceleration and gravity. One is due to frictional loss; the other is due to consumption of the gaseous reactant. We assume that the latter contributes little to pressure drop. This is a reasonable assumption for most systems. For instance, it can be shown readily that less than 1% of the incoming hydrogen was consumed in the reaction system of Herskowitz et al. Three beds with different heights but the same width of 10 pellet diameters are considered. The inlet pressure is 1.6 atm in each case. There are 100, 1000, and 2000 pellets, respectively. The liquid inlet distributor is perfectly uniform. The two-phase pressure drop due to friction loss, APlg,can be estimated with a correlation proposed by Sat0 et al. (1973): (APIJAPI)'.~ = 1.30 + 1.85(AP,/APg)-0.425 (16) The single-phase flow pressure drops, AI',and APg,can be estimated with a correlation by Tallmadge (1970): 150
+ 4.2
[
pd,u ~
F(1- C r I ]
5'6
dp2t,3 = AL put1 - t r ) 2
(17)
The impact of pressure drop on the overall reaction rate is displayed in Figure 11. A decrease in the pressure reduces the amount of dissolved gaseous reactant in the liquid, which leads to a reduction in the reaction rate for a pellet. Thus, the overall reaction rate decreases with an increasing number of particles for any given liquid flow rate. Now, let us focus on the region for qr larger than 0.4 mL/s. Recall that an increase in reaction rate with increasing liquid flow rate under complete wetting conditions is caused by an increase in the external mass-transport rate of the gaseous reactant through the liquid film. This is again observed in the 100-particle bed. However, an increase in the liquid flow rate increases the frictional pressure drop, which in turn lowers the reaction rate. As the height of the bed is increased, the negative influence of pressure on reaction becomes more apparent. For the beds with lo00 and 2000 pellets, the reaction rate no longer increases with increasing liquid flow rate but, instead, shows a steady decline. Clearly, this effect should be accounted for in large-scale trickle-bed reactors.
Comparison with Predictions of Simplified Models In this section, we compare the predictions of our model
746
Ind. Eng. Chem. Res., Vol. 29, No. 5, 1990 4
.
0
1
3
004
00
"
02
"
.
"
0 4 Oh q~(l1lL's)
08
IO
Figure 12. Comparison of the sphere pack model (solid lines and circles) and average wetting efficiency model (dashed lines) for uniform, three-tube, and single-tube inlet distributions of liquid.
to those of two approximate models that have been proposed in previous studies. The objective is to demonstrate the advantages that our more detailed approach has to offer in modeling trickle-bed reactors. The first approximate method uses an average wetting efficiency to characterize the state of wetting over the entire bed. The reaction rate data for one of the three randomly packed beds in Figure 5 are reproduced in Figure 12. As in Figure 5, the uniform and three-tube distributor cases are represented as solid lines, while the single tube inlet is represented by individual solid circles. The reaction rates based on an overall wetting efficiency are shown as dashed lines. These are calculated as follows. For a given liquid distributor and a specified liquid flow rate, the wetting efficiency averaged over the entire bed can be obtained from Figure 9. On the basis of this average value, we can use the same single-pellet diffusion and reaction model described earlier to estimate the effectiveness and then the overall reaction rate. For the case of a uniform liquid inlet distributor, the two models agree fairly well over the entire range of liquid flow rates considered. This is hardly surprising because, as illustrated in Figure 6, the wetting condition of each particle is approximately the same. The differences between the models become much more pronounced as the level of liquid maldistribution increases. For the case of the three-tube distributor, the predictions of the average wetting efficiency model are higher than that of the current model. Averaging the wetting efficiency is equivalent to assuming that some of the completely nonwetted pellets in the bed also take part in the reaction. Thus, higher reaction rates are obtained, particularly a t low liquid flow rates. Similar overprediction can be seen in the single-tube distributor case. However, there is an additional interesting feature. The average wetting efficiency model predicts significantly higher rates for the single-tube inlet case even at high liquid flow rates. This is again due to the inability of a single parameter, the average wetting efficiency, to describe the state of wetting in a randomly packed bed. As Figure 9 shows, for the single tube inlet, the average wetting efficiency can never equal unity since many particles a t the top of the bed would not be wetted at any liquid flow rate (Figure 8). As a result, the reaction rate based on an average wetting efficiency corresponds to that of a partially wetted pellet even a t a high liquid flow rate. For a gas reactant-limited reaction, the rate is overestimated. Figure 13 compares the predictions of our model to those using a weighting factor approach. For instance, Beaudry et al. (1987) suggested a quadratic expression to describe the dependence of the overall reactor effectiveness, vr, on
0.0
0.2
0.4
0.6
0.8
1.0
q W/s)
Figure 13. Comparison of overall reaction rate as a function of liquid flow rate calculated from the sphere pack and weighting factor models.
the bed-averaged wetting efficiency, E,, and the catalyst effectiveness at three discrete wetting levels; i.e., = (1 - E w ) 2 ~ o+ (1 - Ew)Ew~l,z +
(18)
where vr is the overall average effectiveness within the reactor and to,v1/2, and v1 are the effectiveness values of a single nonwetted, half-wetted, and completely wetted catalyst pellet, respectively. In order to compare the weighting factor model to our model, a relation is needed between the overall wetting efficiency and the total liquid flow rate. We assume that it varies linearly with the total liquid flow rate to the reactor, reflecting a relation similar to eq 1; i.e., Ew
= Qr/Qrc
(19)
In the case of a perfectly uniform inlet, the value of the flow rate a t which complete wetting occurs, qro can be estimated from Figure 9 to be 0.65 mL/s. The catalyst effectiveness of a nonwetted pellet is obviously zero; the values for half and complete wetting are estimated from our single pellet diffusion and reaction model. As shown in Figure 13, the maximum reaction rate predicted by the weighting factor approach is lower and lies to the right of the sphere-pack model maximum. We can see in Figure 3 that the maximum effectiveness occurs at a wetting efficiency close to zero. In using only the effectiveness at half-wetting in eq 18, it is equivalent to placing the maximum in the vicinity of 50% overall bed wetting efficiency, while ignoring the actual maximum effectiveness of a single catalyst. Another drawback of the weighting factor approach is that, with a constant q1 in eq 8, it cannot account for the slight increase in reaction rate with increasing liquid flow rate under complete wetting conditions (qr > 0.65 mL/s).
Concluding Remarks We have proposed a novel model to investigate reaction in a trickle-bed reactor with significant liquid maldistribution. The novelity lies in the fact that a passage is established to link particle level physics to macroscopic reactor performance. By following the flow on and reaction in each catalyst particle, pellet by pellet, the flow pattern and the extent of reaction of the entire reactor are determined. Conventional reactor models based on differential equations are ill-suited for problems where irregular morphological features are present. For these cases, it is generally extremely difficult to specify the appropriate boundary conditions. In contrast, this discrete model handles the flow pattern in a trickle-bed reactor in a rather natural manner.
Ind. Eng. Chem. Res., Vol. 29, No. 5 , 1990 747 This more detailed approach has led to two interesting conclusions: (1)Liquid maldistribution can significantly lower the rate of reaction in a trickle-bed reactor. As shown in Figure 5 , the reaction rate with a single-tube distributor is about one-third of that with a uniform distributor. (2) Conventional models that assume a uniform wetting efficiency throughout the reactor can overpredict the reaction rate by the same margin (Figure 12). Furthermore, as pointed out by Sundaresan et al. (1980), conventional continuum models for catalytic fixed bed reactors predict unrealistic backmixing of materials and an infinite speed of signal propagation. This discrete model does not have these limitations. Recently, in order to avoid the same limitations, Schnitzlein and Hofmann (1987) suggested a network model of stirred tanks and plug flow reactors to represent the packings in a packed bed reactor. In this regard, our model is similar to theirs. Admittedly, the present model can use further refinements. For instance, the presence of pendular structures and liquid pockets is ignored (Figures 6-8). Hence, strictly speaking, the model represents a reactor that is not prewetted with liquid. However, stationary pendular structures and liquid pockets in regions of the reactor bypassed by flowing liquid do not receive fresh liquid reactant from the incoming streams. Without a continuous supply of liquid reactant, they do not participate in the reaction anyway and should not play a significant role in our predictions. Other simplifications are not as easy to justify. A twodimensional model is not exactly representative of an actual three-dimensional bed. The coordination number in a three-dimensional sphere pack, Le., the number of spheres in touch with a given sphere, is about six (Chan and Ng, 1986), whereas it is about four in this study. Also, a partially wetted pellet in our model cannot have more than two films or rivulets. This is probably not true in reality. As mentioned previously, catalyst effectiveness is calculated based on a two-dimensional pellet with four identical films. This number does not correspond to the one or two films in our flow model or a number probably ranging from one to three films expected in a three-dimensional bed. (This is because the most likely packing configuration in a sphere pack is that a sphere is in contact with three spheres from above and three below.) Therefore, although we believe that the predicted qualitative trends are valid, more work is needed to improve the model. Furthermore, the present study is still limited in scope. Only bimolecular first-order kinetics are considered (eq 3) although effectiveness values based on other kinetic rate expressions are available (Funk et al., 1989). The flow distribution model is not applicable for beds made up of very fine particles in which a more predominant capillary action is expected to retain more liquid in the bed. Also, the present study is limited to isothermal conditions, but many hydrogenation reactions are highly exothermic. It is not difficult to imagine that the model can be extended to account for evaporation from each pellet. It is now widely recognized that computer experiments can reveal phenomena that are difficult or impossible to observe in actual experiments. With the ever-increasing computing capabilities, our discrete model, with appropriate modifications, should provide a versatile alternative to the conventional models. Acknowledgment We express our appreciation to Mobil Research and Development and the National Science Foundation (Grant
CBT-8700554) for support of this research. The University of Massachusetts Engineering Computing Center provided the facilities for the simulations. Nomenclature A = gaseous reactant B = liquid reactant BiA,n = nonwetted-part Biot number for A = wetted-part Biot number for A BiB,, = wetted-part Biot number for B Ci = liquid-phase concentration of species i CAe= equilibrium solubility concentration of gaseous reactant CBf = bulk-liquid-reactant concentration d = diameter D = molecular diffusivity De = effective diffusivity E , = wetting efficiency E , = average wetting efficiency for the bed g = acceleration due to gravity kij = mass-transfer coefficient from phase i to phase j h, = rate constant L = length = VDeA/DeB P = pressure q = volumetric liquid flow rate qc = volumetric flow rate for complete wetting r = reaction rate sl, s2 = dimensionless coordinates S = width of square catalyst pellet t = exposure time of a surface fluid element T = temperature u = superficial velocity ui = dimensionless concentration of species i UBf =. CBP/ c A ~ x = distance in the direction of flow x l , x 2 = coordinates
Greek Symbols r = mass flow rate per unit length wetted
rc = critical
mass flow rate per unit length for complete wetting S’ = pressure drop 6 = liquid film thickness t = void fraction 7 = catalyst effectiveness = viscosity v = stoichiometric coefficient = constant in eq 1 p = density u = surface tension @ = Thiele modulus Subscripts A = gaseous reactant B = liquid reactant c = critical e = equilibrium or effective f = film g = gas 1 = liquid n = nonwetted p = pellet r = reactor s = solid w = wetted Superscript o = pure liquid reactant
Literature Cited Ahtchi-Ali, B.; Pedersen, H. Very Large Lattice Model of Liquid Mixing in Trickle Beds. Znd. Eng. Chem. Fundam. 1986,25, 108.
748 Ind, Eng. Chem. Res.. Voi. 29. No. 5. 1990 Babcock, B. D.; Mejdell, G. T.; Hougen, 0. A. Catalyzed Gas-Liquid Reactions in Trickle-Bed Reactors. AIChE J . 1957, 3, 366. Beaudry. E.; Dudukovic. M. P.; Mills, P. L.. Trickle-Bed Reactors: Liquid Diffusional Effects in a Gas-Limited Reaction. AIChE J . y State Multiplicity and Partial Internal Wetting
icles. AIChE J . 1988, 34, 969. Clian, S. K.: Ng, K. M. Geometrical Characteristics of a ComputerGenerated Three-Dimensiona! Packed Column of Equal and Unequal Sized Spheres, with Special Reference to Wall Effects. ('ummun. 1986, 48, 216. g, K. M. Geometrical Characteristics of the Pore Space n Packing of Equal Speres. Powder Technol. 1988,54. 147
Collins. G . M.; Hess, R. K.; Akgerman, A. Effect of Volatile Liquid Phase on Trickle Bed Reactor Performance, Chem. Eng. Commu::. 1985, 32. '281. Crine, M.;L'Homme. G. A. The Percolation Theory: A Powerful Approach ~I to the Design of Trickle-Bed Reactors Tool for a N C J V and Columns. Am. Chem. Soc., Symp. Ser. 1982, N o . 196, 407. Crine. 311.; Marchot, P.: L'Homme, G. A. A Phenomenological Description of the Trickle-Bed Reactors Application to the Hydrotreating of Petroleum Fractions. Chem. Eng. Sci. 1980, 35, 51. El-Hisnawi, A. A,: Dudukovic, M. P.; Mills, P. L. Trickle-Bed Reactors: Dvnamic Tracer Tests. React Studies, and Modeling r Performance. ,Im.P h ~ m . ., F x m p . S e r . 1982, "40. n the Liquid Flow Distribution in Trickle-Bed Reactors. I n d . En$. Chcm. Res. 1987. 26, 2413. Fuiik. G "A. Effect of \Vetting on Catalytic Gas-Liquid Reactions. 1'h.D. Thesis. University of Massachusetts, Amherst, 1990. Harold, M. P.; Ng, K. M. Effectiveness of a Partially atalvst for Bimolecular Reaction Kinetics. A I C h E J . 1988 'ii i , % l Funk. G A . Harold. M I' ., Ne. K. M Reactant Adsomtion Effects .,n Partially Wetiid Catalyst Performance. Chew. Ekg" Sei. 1989, ui
4 4 . 2509.
Goto, 5.; Smith, J. M. Trickle-Bed Reactor Performance. Part I. Holdup and Mass Transfer Effects. AICizE J . 1975, 21, 706. Harold, M. P. Steady-State Behavior of the Nonisothermal Partially Wetted and Filled Catalyst. ?hem. Eng. Sci. 1988, 4
