98
Ind. Eng. Chem. Res. 1999, 38, 98-107
Effect of Internal Filtration on Slurry Reactor Performance Piet Huizenga, Johannes A. M. Kuipers,* and Wim P. M. van Swaaij Department of Chemical Engineering, Twente University of Technology, P.O. Box 217, 7500 AE Enschede, The Netherlands
In slurry bubble column reactors, generally small particles (99% pure AMS of a single batch derived from Merck. AMS was pretreated by leading it through a packed bed of alumina beads before it was fed to the reactor.1 In the experiments sieved fractions of 0.5 and 5 wt % Pd/Al2O3 powdered catalyst obtained from Engelhard Drijfhout were employed. Size distributions of the applied particles can be found in Figure 2, while an overview of relevant properties is given in Table 2. The temperature of the reactor contents was measured using a type J thermocouple submerged in the slurry, while a Druck DPI 262 pressure indicator was applied to determine the column pressure. Just above the top of the column the liquid circulation loop bent from the vertical to the horizontal direction. At this position a second pressure indicator of the same type was mounted horizontally to measure the pressure on the permeate side. Because the bend was also the highest point of the liquid circulation loop, a deaeration valve was installed there as well. Liquid composition
Figure 2. Particle size probability density functions of applied batches of catalyst: (A) 0.5 wt % Pd/Al2O3, dp,v ) 47 × 10-6 m; (B) 5 wt % Pd/Al2O3, dp,v ) 47 × 10-6 m.
100 Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999 Table 3. Relationships for Gas Holdup, kla and ks g )
(
kla )
kladisp 1 - g
(
) () )
ηsl,Darton Fsl
1 ln 1 + 1.6g-7/24 8 ηsl,Darton ) η1e36.15s,b2.5 Fsl ) s,bFs + (1 - s,b)Fl
-1/6
ω Fsl
-1/8
Ug
Darton4
O ¨ ztu¨rk et al.8
DA,l kladisp ) Shgl 2 dB
()
Shgl ) 0.62Sc0.50Bo0.33Ga0.29Fr0.68
Fg Fl
0.04
DA,l ks ) Shls dp,a
Sano et al.12 dp,a d′p
measurement of resistance against filtration, the system was allowed to stabilize (5-10 min). Reaction was initiated by switching the gas flow from nitrogen to hydrogen. For 1 h, every 5 min the resistance against filtration was recorded and a liquid sample of approximately 2 × 10-6 m3 was withdrawn from the system. After completion of the reaction, pure nitrogen was again fed to the reactor, and the maximum resistance against filtration was recorded by setting the gas velocity to 0.008 m/s and increasing the liquid flow rate to 23.0 × 10-6 m3/s. Then the hydrostatic pressure difference between the indicators at these conditions was recorded after shutting down the pump. Finally the hydrostatic pressure difference during reaction time was determined once more after readjusting the gas flow to its value in this period.
Shls ) (2.0 + 0.4Re1/4Sc1/3)
3. Theory was analyzed using a Carl Zeiss type A refractometer calibrated at 24 °C using AMS from the batch applied in the conversion experiments and >99% pure cumene derived from Merck. At the beginning of every experiment involving chemical conversion measurements, pretreated AMS was fed to the storage vessel using the funnel present on its top. Subsequently, nitrogen flow through the column was initiated at a superficial gas velocity of 0.012 m/s. The column pressure was set at approximately 2 × 104 kg‚m-1‚s-2 using the water locks. Liquid was fed to the column, and the circulation loop was deaerated. The desired suspension level (0.23 or 0.30 m) was reached by feeding additional liquid to the column. To determine the corresponding volumes of liquid in the reactor, corrections need to be made for gas holdup and the immersed filter volume. The former was calculated from the equation4 given in Table 3. It was found that the liquid volume amounted to 0.918 × 10-3 and 1.202 × 10-3 m3 for levels of 0.23 and 0.30 m, respectively. Having fed the desired amount of liquid to the reactor, the gas velocity was adjusted to its desired value (0.025 or 0.042 m/s) and the pump was stopped. The difference in hydrostatic pressure between the two pressure indicators (∆pstat) was recorded. Subsequently, the filter resistance was checked at the flow rate (operated at 6.0 × 10-6 or 18.4 × 10-6 m3/s) by measuring pressure difference between the pressure indicators (∆p) as well as the operating temperature. Measured pressure differences ranged from 3 × 102 to 2 × 104 kg‚m-1‚s-2. Resistance against filtration was calculated from the measured pressure drops using the following relation:
R)
∆p - ∆pstat ηlJ
(1)
After the first 10 × 10-6 m3 was purged into a waste flask, a liquid sample of approximately 2 × 10-6 m3 was taken. The desired load of catalyst particles (0.007 or 0.0138 kg) was subsequently added to the system through a hole in the lid. After the lid was closed again, the pump was temporarily shut down to allow for an additional measurement of hydrostatic pressure difference between the two pressure indicators. The gas velocity was increased to 0.047 m/s to remove any filter cake that might have formed. The pump was switched on at the desired flow rate, and the gas velocity was gradually reduced to its desired value. Prior to the
In this section, first the relevant results of previous hydrodynamic studies are highlighted. Then a model is presented which describes mass transfer and chemical reaction in series in a slurry reactor while accounting for the influence of a filter cake. 3.1. Internal Filtration in Slurry Reactors. In previous studies5 the simultaneous effects of gas velocity, permeate flux, kinematic liquid viscosity, particle diameter, gas density, and load of solids on cake buildup in filtering slurry bubble columns were thoroughly investigated. Two models describing solid-phase hydrodynamics in the vicinity of the filter were developed. The main conclusion of these models is that solids concentration in the vicinity of the filter is uniquely determined by a single dimensionless number, that is
( )
/s,b ) f
Jxυ1
dpxUgg
(2)
This conclusion was validated by measuring solids concentration indirectly. For this purpose, the cake resistance against filtration for a system operating in batch with respect to solids was determined for a broad set of experimental conditions. Subsequently, the cake resistance was scaled by its maximum value for the batch of solids present in the system to arrive at a quantity called cake ratio. This maximum cake resistance occurs when all particles are present in the cake. The cake ratio can be considered an indirect measure for solids concentration in the vicinity of the filter provided that two conditions are fulfilled. The first is that cake resistance should monotonically increase with cake volume, while the second states that solids volume in the bulk should monotonically increase with solids concentration in the vicinity of the filter. Since measured cake ratios obtained for several loads of solids could be well described by the dimensionless number appearing in eq 2 on the one hand and theoretical calculations showed that the cake ratio was a good measure for solids concentration in the vicinity of the filter on the other hand, it was concluded that the theories developed were valid. The crucial implication of these theories for reactor operation is that as long as solids concentration in the vicinity of the filter remains below its equilibrium value (eq 2) no cake will be formed. If this value is, however,
Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999 101
surface, obeying (1, 0) kinetics:
NA,b ) ksas(cA,l - cA,s)Vl
(5)
NA,b ) krwcA,sVl
(6)
Our kinetic description is in accord with the ones reported in the literature for the hydrogenation of AMS indicating a first order with respect to hydrogen and a zero order with respect to the liquid component.1 In eq 6 kr represents an effective rate constant incorporating possible effects of internal diffusion limitation, which is defined per unit mass of solids. The equation describing liquid-solid mass transfer (eq 5) can be related to solids loading with the following expression:
6 w ) k′sw ksas ) ks dpFp
(7)
When eqs 5-7 are combined, it follows that the contribution of the bulk to the chemical conversion rate can be written as
NA,b ) K′swcA,lVl
(8)
1 1 1 ) + K′s k′s kr
(9)
where
Figure 3. Mass transfer and chemical reaction in series according to the film model: (A) chemical conversion due to suspended catalyst particles; (B) chemical conversion taking place in a filter cake.
exceeded, a cake is built up on the filter until eq 2 is satisfied. 3.2. Mass Transfer Accompanied by Chemical Reaction and Cake Formation. The description of mass transfer and reaction in series is well-known in the literature.6 In this subsection a similar description is developed in which the effect of cake formation is incorporated (Figure 3). In our derivation gas-side mass-transfer resistance is neglected since in our experiments pure hydrogen was used and the applied liquid has a low vapor pressure (300-500 kg‚m-1‚s-2). The mass-transfer rate of hydrogen (denoted by A) from gas-liquid interface to liquid bulk is given by
NA ) kla(c/A,l - cA,l)Vl
(3)
Note that NA is a molar flow rate and not a molar flux in the above equation. Further note that kla is based on liquid volume instead of gassed slurry volume. All hydrogen transferred to the bulk at pseudo steady state is converted either in the bulk (Figure 3A) or in the cake (Figure 3B):
NA ) NA,b + NA,c
(4)
In the bulk hydrogen is transported to the liquid-solid interface where it subsequently reacts at the catalyst
Bearing in mind that unconverted hydrogen flows back to the reactor through the circulation loop, the contribution of the cake to the chemical conversion rate can be written as
NA,c ) φv,lcA,lζA,c )
ζA,c c V τl A,l l
(10)
For calculation of the hydrogen conversion in the cake (ζA,c) the model described in appendix A can be used. This model comprises the extent to which chemical conversion at the catalyst surface is hindered by masstransfer limitations, and calculation of the chemical conversion in the cake itself. When eqs 3, 4, 8, and 10 are combined, the following equation can be derived describing the hydrogen conversion rate in a filtering slurry system:
NA )
1 + kla
c/A,lVl 1 K′sw +
(11) ζA,c τl
The validity of the above equation for the description of the chemical conversion rate in a filtering slurry bubble column is tested in section 4. Widely accepted correlations for calculation of the volumetric mass-transfer coefficient from the gas-liquid interface to liquid bulk (kla) and the liquid-solid masstransfer coefficient (ks) can be found in Table 3. Values calculated from these correlations are compared to their experimental counterparts in section 4. To allow calculation of kla, a value for gas holdup is calculated from the equation by Akita and Yoshida7 as suggested by
102 Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999
Figure 4. History dependency of filtration resistance: gas ) nitrogen, liquid ) AMS, 0.007 kg of 0.5 wt % Pd catalyst, ug ) 0.027 m/s, level ) 0.30 m.
O ¨ ztu¨rk et al.8 An explicit form of this equation corrected for slurry properties4 is applied however. 4. Results Hydrodynamics. Initially some experiments where performed with water as the liquid phase, glass beads as the solid phase, and nitrogen as a gas. Contrary to the experiments described in previous work,5 cake ratio dependencies for these experiments did not show unique combinations of orders with respect to gas velocity, particle diameter, and liquid viscosity. The order in gas velocity, for example, significantly depended on liquid viscosity. These anomalous results can be explained by axial nonuniformities in the bulk partly due to the inhomogeneous gas injection, causing solids volume in the bulk to not monotonically increase with solids concentration in the vicinity of the filter. This effect makes the cake ratio an inappropriate measure for the latter quantity (subsection 3.1). Calculations indicate that for the system under consideration the minimum gas velocity required to suspend the particles9 is within the range of gas velocities studied. Furthermore, for nonuniform gas injections such as the one applied in this work, even a higher minimum gas velocity is required to suspend the particles as compared to uniform distributors.10 Therefore, axial nonuniformities in the bulk can be expected. Because the purpose of the present work is to show the influence of cake formation on reactor performance and the cake resistance is always measured separately, the system as it functioned was still considered acceptable. Then experiments were performed with AMS as the liquid phase, 0.5 wt % Pd/Al2O3 catalyst as the solid
phase, and 99.9% pure nitrogen as a gas. Alternatingly, the liquid flow rate was gradually increased and decreased in a set of subsequent experiments. In Figure 4 results are shown for experiments in which all data points have been taken after 15 min of process time. It can be seen from this figure that in the first set of experiments in which flow rate was decreased, resistance exceeds its previously measured value at low flow rates. When the liquid flow rate is increased subsequently, all measured resistances are consistently higher than the previously obtained values. If the liquid flow rate is decreased again, the process repeats itself. In additional experiments it was shown that the measured resistance significantly increased in time at low permeate flux, while it remained approximately constant at high permeate flux. For the low permeate flux, however, no decoloring of the slurry was observed visually, indicating that the volume of solids in the slurry did not change significantly. This suggests that the cake ratio is not a unique measure for volume of solids in the filter cake for this system. The observed behavior could then be attributed to the nonsphericity of the catalyst particles (Figure 5), leading to increased packing efficiency with time at low permeate fluxes. Thermal Effects in a Filter Cake. In the experiments involving chemical conversion, typically a temperature of 297 K is applied at a pressure of 1.2 × 105 kg‚m-1‚s-2, leading to an AMS conversion of typically 10% in 1 h. At these conditions the adiabatic temperature rise of the liquid passing through the cake can be estimated as 0.25 K, while 0.06 K is found to be the maximum temperature difference between solid catalyst and permeating liquid. For more detailed information concerning these calculations, the reader is referred to the thesis of Huizenga.11 On the basis of these calculations, it can be concluded that in the system under consideration heat effects are virtually absent in the filter cake. For the vast majority of industrially applied systems, physical properties are in the same order of magnitude as those for the present system. Therefore, in practice heat effects will not play a role in filter cakes in a filtering slurry reactor unless the system runs at extremely high hydrogen partial pressures or a highly soluble gaseous component is involved. Influence of Cake Buildup on Chemical Conversion Rate. Initially conversion experiments were performed at a level of 0.30 m with a duration of 6 h. In these experiments after 2 h the liquid flow rate was changed from 6.0 × 10-6 m3‚s-1 (J ) 2.2 × 10-3 m‚s-1) to 18.4 × 10-6 m3‚s-1 (J ) 6.9 × 10-3 m‚s-1) and
Figure 5. Microscopic pictures of applied particles: (A) glass spheres, dp,v ) 57 × 10-6 m; (B) 0.5 wt % Pd/Al2O3, dp,v ) 47 × 10-6 m.
Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999 103
Figure 6. Time-dependent behavior in a typical experiment at a low permeate flux: 0.007 kg of 0.5 wt % Pd catalyst, ug ) 0.025 m/s, level ) 0.23 m, J ) 2.2 × 10-3 m/s. (A) Liquid composition. (B) Filtration resistance.
switched back again after an additional 2 h. The chemical conversion rate dropped after switching to the higher flow rate and increased again when the setup was subsequently run at the lower flow rate. These results indicate that at the higher permeate flux additional particles move into the filter cake, leading to a reduced bulk reactivity and an associated drop in the conversion rate. After we switch back to the low flow rate, some particles are resuspended into the slurry bulk, thereby again increasing the conversion rate. In some of these experiments, however, particles tended to stick to the wall of the column and subsequently showed a considerable extent of sedimentation. The reason for this behavior was not clear. Therefore, it was decided to run subsequent experiments for only 1 h at a level of 0.23 m, thereby circumventing these practical problems sufficiently. Typical results for an experiment with a duration of 1 h are shown in Figures 6 and 7 for low and high permeate fluxes, respectively. From Figures 6A and 7A it can be observed that the AMS content of the liquid decreases in a linear fashion with time, indicating that the volumes of solid catalyst present in cake and bulk and the activity of the catalyst do not vary much. From Figures 6B and 7B it can be seen that the measured resistance shows a certain degree of variation. The initial increase of cake resistance observed at low
Figure 7. Time-dependent behavior in a typical experiment at a high permeate flux: 0.007 kg of 5 wt % Pd catalyst, ug ) 0.025 m/s, level ) 0.23 cm, J ) 6.9 × 10-3 m/s. (A) Liquid composition. (B) Filtration resistance.
permeate fluxes (Figure 6B) can be attributed to the nonsphericity of the particles as explained earlier. The slightly decreasing trend in cake resistance observed toward the end of an experiment (Figures 6B and 7B) can partly be explained by a certain extent of sticking of the catalyst to the column wall which could not be completely prevented. Particles sticking to the wall are withdrawn from the bulk, leading to additional particles being resuspended from the cake. The time average of the resistance obtained from a 1-h experiment is used to arrive at a measure of volume of solids in the filter cake. It is assumed that all remaining particles are present in the slurry bulk of the reactor. The maximum resistance was measured separately at a high permeate flux at the end of the runs. For 0.007 kg of particles its value was found to be 1.73 × 109 m-1 for the 0.5 wt %Pd catalyst, while for the 5 wt % Pd catalyst, a value of 1.50 × 109 m-1 was observed. For filter resistances measured at the start of every experiment, an average value of 1.68 × 108 m-1 was observed. On the basis of the average resistances measured for every experiment, the cake ratio can now be calculated according to
CR )
Rtot - Rfilter Rtot,max - Rfilter
(12)
104 Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999 Table 4. Overview of Experimental Results (Identifiers of Experiments Correspond to Those of Table 5) experiment A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
catalyst Pd content [wt %] 0.5 0.5 0.5 5.0 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 5.0 5.0 5.0 amount of catalyst [10-3 kg] 7.03 6.99 7.00 7.01 7.00 7.00 7.00 7.00 6.99 7.00 13.82 13.81 7.01 7.00 7.01 level [10-2 m] 30 30 30 30 23 23 23 23 23 23 23 23 23 23 23 Ug [10-2 m‚s-1] 2.5 2.5 2.5 2.5 4.2 4.2 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 -6 3 -1 φl [10 m ‚s ] 6.2 6.0 6.0 6.2 6.0 6.0 18.4 18.4 6.0 6.0 6.0 6.0 6.0 6.0 18.4 CRsingle experiment 0.17 0.23 0.37 0.47 0.20 0.06 0.83 0.95 0.50 0.60 0.47 0.51 0.42 0.54 0.90 CRaverage 0.26 0.26 0.26 0.47 0.13 0.13 0.89 0.89 0.55 0.55 0.49 0.49 0.48 0.48 0.90 NA,obs [10-4 mol‚s-1] 2.75 2.73 3.08 2.57 3.19 3.58 1.02 1.31 2.40 2.41 3.37 3.64 2.62 2.57 1.38
Figure 8. Overall resistance against chemical conversion as a function of reciprocal weight of catalyst present in the bulk of the reactor.
Especially for the early experiments in which less measurement points were taken, cake ratios vary considerably between comparable series (Table 4). The reproducibility of the observed chemical conversion rates is however high, indicating that the volume of solids present in the cake should not vary much between these experiments. Therefore, cake ratios of experiments with the same load and type of solids, permeate flux, gas velocity, and liquid level are averaged, thereby leveling out effects of hysteresis in cake buildup (Table 4). It is now assumed that the volume of solids present within the cake is proportional to these averaged cake ratios. In Table 4 an overview of the measured chemical conversion rates for the various combinations of operating conditions is given. It can be seen from this table that the chemical conversion rate (NA,obs) drops with an increase of the liquid flow rate (φl), a decrease of the gas velocity (Ug), a decrease of the amount of catalyst, and a decrease of the palladium content of the catalyst. The first two trends can be explained by increased cake buildup, while the latter two can be explained by decreased quantity and activity of the catalyst, respectively. Increased cake buildup implies increased solids withdrawal from the bulk, leading to reduction of bulk conversion rate (w decreases in eq 11). In Figure 8 all measured conversion rates are given in the form of the well-known plot of cA,l*Vl/NA versus 1/w, where w follows from
w ) wtot(1 - CR)
(13)
Values for kla and K′s can now be determined from experimental results obtained at constant palladium content, level, and gas velocity. The largest set of experiments meeting this requirement are those obtained for the 0.5 wt % Pd catalyst, a level of 0.23 m, and a gas velocity of 0.025 m/s (experiments G-L).
Optimum values for predicting the observed chemical conversion rate in these experiments are kla ) 0.184 s-1 and K′s ) 3.63 × 10-2 m3‚kg-1‚s-1. These optima are calculated by minimizing the sum of the squared deviations between observed and calculated conversion rates for the set of experiments considered. For different level or gas velocity, optimum values were found by assuming K′s to remain constant, while kla is assumed to be independent of the palladium content of the catalyst. In Table 5 an overview of the calculation results is presented. In these calculations complete hydrogen conversion in the filter cake is assumed, which needs to be checked afterward. As can be witnessed from Table 5, the chemical conversion rate can be described rather well with the fitted values for the model constants. In addition, the fitted values of kla and K′s show trends that can be explained physically (Tables 4 and 5). The explanation for the increase of kla and K′s with the gas velocity and palladium content of the catalyst, respectively, is rather obvious, although increases larger than those observed would be expected. Furthermore, it is known that in the acceleration zone of a bubble column kla may be higher, leading to a decrease in kla with an increase of the liquid level. It can, therefore, be concluded that, despite the crude nature of the assumptions made, a consistent physical description is obtained. Values calculated for kla at gas velocities of 0.025 and 0.042 m/s are found to be 0.07 and 0.11 s-1 for the system considered, while corresponding values for calculated k′s both amount to 0.05 m3‚kg-1‚s-1. In the latter calculation the correction for nonsphericity12 was set to unity. The experimentally observed values for kla are in the same order of magnitude but are considerably higher, which may at least partly be attributed to the low aspect ratio (i.e., length/diameter ratio) of our column. This is confirmed by the experimentally observed strong decrease of kla with an increase of the slurry level. As should be expected (eq 9), k′s exceeds measured values of K′s. Chemical reaction rate constants kr separately measured in a stirred cell were found to be in the same order of magnitude as K′s but showed larger variations in rate between the different catalysts. Chemical Conversion in the Cake Itself. From the maximum cake resistances obtained at a solids loading of 0.007 kg using the 0.5 wt % Pd and 5 wt % Pd catalysts, cake solidities of 0.660 and 0.652, respectively, can be estimated from
Rc,max )
s,c (1 - s,c)
1 3
dp,a2
Ms,tot FsπdfilterHfilter
(14)
In the above equation cake buildup on the cylindrical filter is treated as if the filter area was flat, thereby
Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999 105 Table 5. Interpretation of Experimental Results (Identifiers of Experiments Correspond to Those of Table 4) experiment w [kg‚m-3] τl [s] K′s [10-2 m3‚kg-1‚s-1] kla [s-1] NA,obs [10-4 mol‚s-1] NA,calc [10-4 mol‚s-1]
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
4.33 194 3.63 0.118 2.75 2.86
4.31 200 3.63 0.118 2.73 2.86
4.31 200 3.63 0.118 3.08 2.83
3.09 194 4.14 0.118 2.57 2.57
6.63 153 3.63 0.194 3.19 3.36
6.63 153 3.63 0.194 3.58 3.41
0.84 50 3.63 0.184 1.02 1.24
0.84 50 3.63 0.184 1.31 1.23
3.43 153 3.63 0.184 2.40 2.36
3.43 153 3.63 0.184 2.41 2.37
7.68 153 3.63 0.184 3.37 3.50
7.67 153 3.63 0.184 3.64 3.53
3.97 153 3.62 0.184 2.62 2.66
3.97 153 3.62 0.184 2.57 2.60
0.76 50 3.62 0.184 1.38 1.20
Figure 9. Influence of conversion in the cake on the overall resistance against chemical conversion: 0.5 wt % Pd catalyst, ug ) 0.025 m/s, level ) 0.23 m.
neglecting the increase of permeable area with an increase of the cake thickness. Calculations revealed that this effect indeed has a minor influence. Since K′s incorporates (eq 9) the effect of both reaction (kr) and liquid-solid mass transfer in the slurry bulk (k′s), its value can serve as a lower boundary to kr. Assuming the Sherwood number to equal 3.89,13 it follows from eq A3 that, for the experimental conditions applied, conversion at the catalyst surface is typically limited by liquid-solid mass transfer in the cake to an extent of 70%. Therefore, the reaction rate constant k1 calculated from eq A2 needs to be corrected with an effectiveness factor, as suggested in the appendix (eq A4). For all experiments involved, the cake thickness can now be calculated using the cake ratio measured:
δc )
Ms,tot CR s,cFsπdfilterHfilter
(15)
It is found that for cake ratios exceeding 0.3 the condition δc/dp > 10 is met, ensuring the validity of eq A5. The Peclet number then follows from this equation, and subsequently the chemical conversion in the cake can be found from eq A6. Calculations show that conversion in the cake, except for low cake ratios ( 10), the Bodenstein number Bd varies between 0.5 and 2.0.6 A worst case estimate for the Peclet number can therefore be obtained from
Pe ) Bd
δc δc ) dp,a 2dp,a
(A5)
With the Peclet number known, the conversion in the cake can be calculated from the equation due to Danckwerts:6 ζA,c ) 1 4q Pe Pe 2 (1 + q ) exp - (1 - q) - (1 - q2) exp - (1 + q) 2 2
{
}
{
}
(A6)
Received for review May 11, 1998 Revised manuscript received September 3, 1998 Accepted September 21, 1998 IE980294L
