Particle Capture and Plugging in Packed-Bed Reactors - Industrial

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Ind. Eng. Chem. Res. 1997, 36, 4620-4627

Particle Capture and Plugging in Packed-Bed Reactors Rashmi Narayan,† Jose´ R. Coury,‡ Jacob H. Masliyah,† and Murray R. Gray*,† Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, T6G 2G6 Canada, and DEQ/UFSCAR, 13565-905 Sao Carlos (SP), Brazil

Fine particles in the liquid feed to packed-bed reactors can be trapped in the catalyst bed, which eventually leads to excessive pressure drop. The fine particles can include coke, corrosion products, clays, and other minerals. The catalyst bed functions as a granular filter to remove particles much smaller than the size of the pores between the catalyst pellets. The efficiency for trapping the particles in the packed bed depends on the flow fields and the attractive forces between the packing and the fine particles. In order to understand the capture of fine particles from nonaqueous media, we studied a model system of carbon black in kerosene. Columns packed with glass beads and a catalyst were operated over a range of flow velocities to Reynolds numbers from 0.1 to 2.3, on the basis of the diameter of the packing in the bed. Flow was in the upward and in the downward direction. The filter coefficient and efficiency were sensitive to liquid velocity. Trapping was slightly more efficient with downward flow at low velocity. The pressure drop increased along the entire length of the packed bed, but the extent of increase at a given amount of deposit depended on the liquid velocity. Microscopy showed that the particles tended to deposit onto other particles, rather than smoothly coating the bed packing. At low velocities, more particles were deposited in the pores between the packing, giving a larger increase in pressure drop than that at high velocity. A model is presented for calculating pressure drop due to this type of deposition. Introduction Packed-bed reactors are widely used to process a wide range of liquid feeds. Operating in the trickle-bed or pulsed-flow regime, they are also widely used for hydrotreating and hydrocracking of petroleum fractions. The size of the catalyst pellets is selected on the basis of considerations of catalyst effectiveness, mechanical strength, and pressure drop. When the liquid feed contains fine particles, the interstices between the catalyst pellets can become filled with deposits, leading to a significant increase in pressure drop across the reactor. Several different types of particles may occur in hydrocarbon liquid feeds. The nondistillable fraction of some crude oils can contain iron sulfides (Koyama et al., 1995b), which can accumulate in the catalyst bed and increase pressure drop to the point that the reactor must be shut down before the catalyst activity has been fully utilized. Athabasca bitumen from mining operations contains clays, as a result of the extraction process. Some of these clays are entrained into the gas oil fraction during distillation, and subsequently enter downstream hydrotreaters. Similarly, coke solids can be entrained into gas oils from cokers and enter downstream hydrotreaters. Coke can also be formed in situ as a byproduct of cracking reactions (Koyama et al., 1995a). Large particles can be removed by filtration, but particles smaller than 20 µm cannot be easily filtered and so enter the reactors. Even a small concentration of solids can accumulate significantly over a period of months and can cause eventual plugging by deposition throughout the catalyst bed. The size range of the suspended particles depends on their origin and the use of upstream filtration. The bed of the hydrotreating catalyst is comprised of pellets of diameter 1-2 * Author to whom correspondence should be addressed. Phone: (403) 432-7965. Fax: (403) 432-2882. E-mail: [email protected]. † University of Alberta. ‡ DEQ/UFSCAR. S0888-5885(97)00101-2 CCC: $14.00

mm; therefore, particles of order 100 µm or larger will be strained out at the top of the bed. Particles smaller than approximately 20 µm will enter the catalyst bed and possibly undergo deposition. The removal of these particles by the catalyst packing in hydrotreating reactors is analogous to deep-bed or granular filtration, wherein liquid flows through a granular bed and deposits particles on the bed packing (Tien, 1989). This process has been extensively studied for aqueous suspensions, in part due to its significance in water purification. Studies on the deposition of fine particles from nonaqueous liquids are rare. Chowdiah et al. (1982) studied filtration of carbon black from tetralin by a bed of silica sand. The streaming potential across the filter bed was used to monitor the initial deposition of particles. The initial capture efficiency of the bed was very high, so that a distinct breakthrough curve was observed in particle concentration. The initial deposition was attributed to electrostatic attraction between the particles and the packing. Once the charge was neutralized, continued deposition at much lower efficiency was attributed to other mechanisms such as straining and interception of particles by the packing. Placing a high voltage across the packed bed can enhance the capture of particles, especially when the bed packing has a much higher dielectric than that of the liquid (Byers and Amarnath, 1995). Deposition of particles from a liquid suspension onto a collector surface can occur by several mechanisms, including interception, Brownian diffusion, gravitation, and electrostatic forces (Tien, 1989). Models of the forces on a moving particle, and its resulting trajectory during flow past a collector, have been developed by Spielman and FitzPatrick (1973) and Rajagopalan and Tien (1976). Unfortunately, these models are suitable for analyzing the initial deposition of particles on a clean surface. The increase in pressure drop in industrial reactors, however, requires much more extensive deposits of particles than the initial layer onto the clean surface. Most of the deposition must occur long after © 1997 American Chemical Society

Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997 4621

the initial transient removal observed by Chowdiah et al. (1982) has been completed. The literature on granular filtration of aqueous suspensions suggests that the efficiency of particle trapping will change as particles continue to be deposited in the bed. The most common case is an increase in trapping as the bed “ripens” (e.g., Deb, 1969; Wnek et al., 1975; Tian and Guthrie, 1995). Some studies observe decreasing efficiency with an increase in deposit (Vaidyanathan and Tien, 1989; Gimbel, 1989), similar to the observations of Chowdiah et al. (1982) in a nonaqueous medium. In all cases, pressure drop increases with deposit, but with various intensities (Rajagopalan and Tien, 1979). Modeling these complex phenomena is difficult partly because of the difficulty in verifying the models. Choo and Tien (1995a,b) recently presented a model for describing the variation in trapping efficiency and permeability with the amount of deposition of particles. They assumed that the deposit is formed in three phases: first, the trapped particles cover the collector surface smoothly; second, they form a multilayer particle deposit; and third, the particles clog the pores in the bed and straining (cake filtration) becomes the dominant mechanism. The authors argue that the three phases can act simultaneously according to the stage of particle buildup. The simulated results for each phase acting separately show bed efficiency decreasing with an increase in specific deposit in the first phase, and then increasing in the second and third phases. When acting together, the simulation showed an overall increase in the efficiency of trapping with an increasing amount of deposit. The bed permeability decreases with the specific deposit in all cases, as expected. Although very useful for providing an insight into the bed behavior, the models depend on parameters such as deposit distribution, porosity, permeability, and other factors. Most of these parameters have a clear physical significance and must be present in any representation of the packed bed. Nevertheless, the experimental tools presently available are not capable of measuring them. Deposition of particles in hydrotreating reactors occurs at extreme conditions of temperature and pressure. The chemistry of the liquid and possibly the particles would be altered by the hydrogenation reactions promoted by the catalyst. Unlike the liquid flow regime of granular filtration, both liquid and gas flow through the catalyst bed. If the reactor is operated in the pulsedflow regime, the local flows may be turbulent and highly variable with time. The present study was conducted under simplified conditions in order to understand the deposition of fine particles from nonaqueous suspensions under controlled conditions. Deposition and pressure drop due in a packed bed with liquid-phase flow only were studied using a model suspension of carbon black particles in kerosene. This work provides an understanding of particle behavior under well-defined conditions, to serve as a basis for subsequent studies of the role of other factors such as two-phase flow and chemical changes during hydrotreating. Materials and Methods Materials. The model system selected was carbon black particles (8 µm mean diameter) suspended in kerosene. Kerosene was selected due to its chemical similarity to the hydrocarbon streams of interest, its stability, and its low vapor pressure. Carbon black has some chemical similarity to the coke deposits which

Table 1. General Characteristics of the Material Utilized material

characteristics

carbon black

density: 1768 kg/m3 diameter: 5-10 µm; 8 µm average

kerosene

viscosity: 2.14 × 10-3 kg/m‚s density: 784.3 kg/m3

glass spheres

density: 2487 kg/m3 diameter: 0.925 mm (+0.85-1.00) porosity: 0.37 As: 45.95

catalyst pellets

density: 2487 kg/m3 diameter: 1.0 mm length: 4-10 mm; 7 mm average porosity: 0.42 As: 37.98 estimated Hamaker constant for glasscarbon-kerosenea ) 3.04 × 10-20 J

a

Narayan (1996).

Figure 1. Schematic view of the apparatus: (a) setup for upward flow tests; (b) setup for downward flow tests; (c) detail of the bed with pressure measurement planes.

form in some hydrotreaters. The carbon-black suspension was stabilized by Aerosol OT (sodium bis(ethylhexyl)sulfosuccinate). A 6 mmol/L solution of AOT in cyclohexane was added to kerosene to form a 0.3 mmol/L solution of AOT. The carbon black was added to this solution and homogenized in a high-speed blender. The carbon black concentration was determined by spectrophotometry. Two materials were used for the packing of the bed, also called the collector: glass beads and actual hydrotreating catalyst pellets (Ni and Mo on γ-alumina). The glass spheres were washed and sieved between 0.71 and 0.85 mm mesh screens. The general characteristics of the materials are listed in Table 1. Experimental Apparatus. Figure 1 shows a schematic view of the experimental apparatus used in this work. It consisted of a fixed bed contained in a cylindrical column of Plexiglas of 2.54 cm in diameter and 30 cm in length, supported by a wire mesh. The column was connected to a differential pressure transducer. The column was also connected to a metering pump by 1/4 in. stainless steel tubing in such a way that was possible to direct the flow upward or downward through the bed (see Figure 1a and 1b). The detail of the bed with the pressure measurement planes is depicted in Figure 1c. The feed tank, containing 8 L of carbon-black suspension, was mixed with a mechanical stirrer throughout the experiment and was kept recirculating through the bed for the time stipulated for each test.

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The apparatus was conditioned before each experiment by pumping suspension through the empty tube for 24 h. The pump was then stopped and the column removed and filled with packing. This procedure allowed the system to reach equilibrium in respect to particle trapping in the tubing, bends, valves, etc. The pump was then switched on and the initial pressure drop measured in the five planes. The initial particle concentration was measured by sampling in the tank. The measurements of the tank concentration and the pressure drops were repeated at selected time intervals. A typical experiment lasted 27 h. The static pressure was measured at five equidistant points along the column, using a differential pressure (DP) cell (Rosemount Instruments). These five points defined the sections 1-4, shown in Figure 1c, in which the pressure drop was recorded throughout the study. The particle concentration was measured on 5 mL samples of the suspension using a spectrophotometer (Shimadzu UV-160), previously calibrated for the concentration range under study. The measured absorbances were in the range 0.25-1.75. The amount of deposit in the bed was expressed in terms of σ, the specific deposit, defined as mass of particles trapped per unit of empty bed volume. For an experiment with recirculation of the suspension, a mass balance on the tank gives

VL σ ) [C(t0) - C(t)] Vb

(1)

The basic filter equation relates the concentration gradient along the filter column, along the x-ordinate, to the local concentration C:

dC ) -λ′(x)C dx

(2)

where λ′(x) is the local filter coefficient at position x in the bed. Integrating over the length of the filter gives an average filter coefficient in terms of the inlet and outlet concentrations:

λ)

1 ln(Cinlet/Coutlet) L

(3)

where L is the length of the bed. Another measure of the efficiency of the trapping of the particles is the bed efficiency. The rate of particles trapped by the bed (dmpr/dt) can be related to the bed efficiency as follows:

dmpr ) E(t) ωp(t) dt

(4)

where ωp(t), the mass flow rate of particles through the bed, is related to the suspension volumetric flow rate, Q, as

ωp(t) ) C(t)Q

(5)

The specific deposit, σ, is defined as

σ)

mpr Vb

(6)

Replacing eqs 5 and 6 in eq 4 and rearranging gives

E)-

Vb 1 dσ Q C(t) dt

(7)

Substituting the expression for σ, given in eq 1, the expression for E becomes

E)-

VL 1 dC(t) Q C(t) dt

(8)

The two measures of the filtration process, E and λ, can be related by first recognizing the definition of bed efficiency implicit in eq 4:

E)

Cinlet - Coutlet Cinlet

(9)

Rearranging eq 3 and substituting into eq 9 gives a relationship between E and λ:

E ) 1 - e-λL

(10)

The Reynolds number was calculated on the basis of the diameter of the packing in the bed:

Re ) dcU/ν

(11)

where dc was the diameter of the glass beads or the equivalent diameter of the catalyst pellets, U was the superficial liquid velocity, and ν was the kinematic viscosity. Results and Discussion Particle Removal and Pressure Increase as a Function of Concentration. The first series of experiments varied the initial concentration of the carbon black in suspension in the range 96-200 mg/L and operated the bed in two modes. Four experiments used recirculation of the liquid, with constantly decreasing liquid concentration, while one used constant concentration with a single pass through the bed. The initial outlet concentrations of carbon black were g17 mg/L and then the particle concentration in the outlet increased with time. Final outlet concentrations were ca. 40 mg/L in experiments with recirculation of the suspension and 115 mg/L in the experiment with a single pass through the bed. The results for the filter coefficient are illustrated in Figure 2, as a function of specific deposit. The corresponding pressure drop data are shown in Figure 3, where the pressure drop at each time is normalized by dividing by the initial pressure drop. These data clearly show that the specific deposit serves to normalize experiments conducted at different concentrations, and that the filter coefficient is not dependent on the concentration of particles in the suspension. With the exception of three outlying points, the data of Figure 3 show that the increase in pressure drop is also scaled with the value of σ. In a dilute suspension, the filter coefficient should be independent of concentration. Under these conditions particles pass through the bed independently, and the probability of a particle being deposited on the packing depends on the state of the bed (i.e., σ) and not on the other particles in solution. Particle-particle interactions are not significant in a stable suspension at such low concentrations of solids. The particle removal is, therefore, a true first-order process. The data of Figures 2 and 3 show that the concentration can be selected for convenience at an arbitrary level, and that recirculating and single-pass operation give equivalent results. In every case, high filter coefficients were observed until 0.5-1 mg/mL of specific deposit had accumulated

Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997 4623

Figure 2. Filter coefficient as a function of specific deposit for different initial concentrations of carbon black particles (Re ) 0.5; upward flow with glass beads). Filter coefficients were calculated from eq 3.

and then the filter coefficient remained constant over an extended period of time. On the basis of the observations of Chowdiah et al. (1982), the initial high value of the filter coefficient can be attributed to electrostatic interactions between opposite charges on the carbon black and the glass beads. Once the charge on the beads was neutralized, the filter coefficient remained at a lower steady-state value for an extended time. This hypothesis is consistent with electrokinetic studies. The glass beads and the catalyst were both positively charged, with ζ-potentials of 16 and 31 mV, respectively, as measured by streaming potential (Narayan, 1996). Carbon black in hydrocarbons with AOT has a ζ-potential ranging from -40 to -80 mV (Chowdiah et al., 1982; Kitahara et al., 1971). While the calculated values of the filter coefficient at small values of σ were consistent between replicate experiments (Figure 2), the values were not accurate for several reasons. The 30 cm column gave significant averaging of the concentration of particles in the bed at short times, and hence averaging of the filter coefficient. Time lags between the outlet sample point and the bed gave additional uncertainty. An accurate estimate of the true filter coefficient of the clean bed would require the use of a much shallower bed. In any case, these errors were insignificant once the steadystate filter coefficient was established. Visual observation showed that the carbon black was deposited down the entire length of the packed bed. This observation was confirmed by measuring the pressure gradient along the bed. As illustrated in Figure 4, the pressure drop was slightly higher in the inlet section of the bed, but the pressure drop definitely increased over each section of the bed. The carbon particles were, therefore, deposited along the length of the bed and not strained out in the first few centimeters. Particle Removal as a Function of Reynolds Number. The data of Figure 5 illustrate the variation of the bed-trapping efficiency (E) with specific deposit for upward flow at four different Re. The efficiencies were calculated from eq 8 using the Euler approximation to calculate dC(t)/dt at each value of σ. The rate

Figure 3. Increase in pressure drop as a function of specific deposit at different initial particle concentrations.

Figure 4. Axial pressure drop profile in the fixed bed, at three different particle specific deposits, for upward flow at Re ) 0.199 with glass beads (see reference plane in Figure 1c).

of change of concentration in the reservoir was g5 mg/ (mL‚h); therefore, the changes in concentration of carbon black were well within the accuracy of the analytical method. The initial efficiency decreased with Re, which was in qualitative agreement with the theoretical analyses of Spielman and FitzPatrick (1973). The initial efficiencies were in reasonable agreement with the correlation of Rajagopalan and Tien (1976), which predicted E ) 0.75 at Re ) 0.2 and E ) 0.45 at Re ) 2. The change in Re in these experiments was due only to changes in the liquid velocity. An increase in superficial velocity will decrease deposition due to gravitational forces and London-van der Waals forces and increase deposition due to the diffusion of particles. When the contributions of the mechanisms are added, both theoretical models suggest a decrease in the initial bed efficiency with increasing superficial velocity.

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Figure 5. Bed-trapping efficiency as a function of specific deposit, with four different Re, for upward flow with glass beads. Efficiency was calculated from eq 8.

Figure 5 also shows a decrease in E with specific deposit in all cases. This behavior has been reported before (Vaidyanathan and Tien, 1989; Gimbel, 1989) and was attributed to the mode of deposition: in these first stages, wherein the trapped particles would deposit as a smooth layer on the collector surface, without altering the geometry of the packing. An alternate explanation would be blocking of deposition by particles already on the surface (Johnson and Elimelech, 1995); however, the ionic interactions normally responsible for such phenomena are unlikely in a nonaqueous medium. After approximately 2-3 mg/mL of deposit had formed, the filter coefficient reached a constant value, as illustrated in Figure 2. Following the efficiency in Figure 5, this steady-state filter coefficient decreased with increasing Re (Figure 6). More trapping was consistently observed with downward flow as compared to upward flow, although the difference was only of the order of 30% on average. In downward flow, due to the density difference between the carbon black and the kerosene, gravity would serve to pull the particles closer to the collector, whereas in upward flow gravity would tend to act against deposition. The same filter coefficients were observed, within experimental error, for catalyst pellets (Figure 7). Therefore, the chemistry of the collector surface was not important. The variability of the data were greater with the catalyst due to its size and internal porosity. The pellets were too long to pack reproducibly in the apparatus, and the catalyst internal pore volume tended to release minute gas bubbles which disrupted the liquid flow. Pressure Drop across the Bed. The increase in pressure drop across the packing was a function of the liquid velocity. As illustrated in Figure 8, the increase in pressure drop was much more pronounced when the superficial velocity was low, i.e., low Re. The same trend was observed with flow in the upward direction and in the downward direction. The increase in pressure drop was more variable with a catalyst, particularly when a bubble formed. The maximum value of ∆P/∆P0 for a catalyst was 1.64 at a specific deposit of 3.5 mg/mL (Re ) 0.23), equivalent to the range of values observed for

Figure 6. Steady-state filter coefficients as a function of superficial velocity for glass beads. Error bars show standard deviation for experiments repeated in triplicate. The lines were calculated by linear regression of the data for upward and downward flow.

Figure 7. Steady-state filter coefficients as a function of superficial velocity for the catalyst. The regression line was calculated using all of the data. The slopes for data from upward and downward flows were not significantly different.

the glass beads. The increase in pressure drop as a function of σ at Re ) 0.1 was intermediate between the smooth-coating mode and the pore-blocking mode of particle deposition (Pendse et al., 1978; Rajagopalan and Tien, 1979). The even pressure drop along the length of the column also suggested that pore blocking was not a reasonable description of the deposition morphology. The change in ∆P/∆P0 as a function of Re suggested that the structure of the deposits in the packed bed depends on the flow conditions. High superficial velocity gave a deposit that caused less pressure drop at a given specific deposit. A microscope was used to observe the packing directly, by turning off the pump during an experiment in the downward flow direction and allowing the liquid to drain to the top of the packing. The top of the apparatus was then removed and the

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Figure 8. Increase in pressure drop with specific deposit as a function of Re (upward glass beads).

images from the microscope were recorded. Figure 9a shows a typical deposit at low Re in a pore between catalyst pellets. The deposit was characterized by particle-on-particle deposition to form a loose floc in a location to give maximal increase in ∆P/∆P0. At high Re, on the other hand, the deposits formed as small domains scattered over the entire surface of the pellets (Figure 9b). Similar deposits formed on the surfaces of the glass beads. These observations confirmed that the nature of the deposits depended on the liquid velocity, and they also showed that deposition on the collector was highly nonuniform. Interpreting filtration efficiency and pressure drop on the basis of an assumption that the packing is evenly covered is clearly incorrect. These deposits were not consistent with types of unfavorable surface interactions considered by Johnson and Elimelech (1995) or Bai and Tien (1996), wherein ionic interactions opposed either the achievement of monolayer coverage or the initial deposition of particles. In this case, the particle-particle interactions were more favorable than the deposition on the clean collector surface. The pressure drop in packed beds is commonly calculated using the Ergun equation:

Figure 9. Photographs of the structure of deposit on the catalyst. (a) at Re ) 0.23. Scale: 1 cm ) 0.33 mm. (b) at Re ) 2.3. Scale: 1 cm ) 0.33 mm.

2

(1 - ) µU FU2 (1 - ) -∆P ) 150 + 1.75 (12) L dc 3 dc2 3 If the particles coat the packing evenly, then the diameter of the packing increases and the porosity decreases. The increase in pressure drop is then given by (Tian and Guthrie, 1995)

[

∆P/∆P0 ) 1 +

][

]

σ(t) σ(t) 1(1 - 0)(1 - d) 0(1 - d)

-3

(13)

where the deposit is uniformly distributed in the bed, with a porosity of d. Like other models for smooth deposition, this equation does not match the actual observations of heterogeneous deposition illustrated in Figure 9b. It also fails to match the observed increase

Figure 10. Schematic diagram of surface area available for liquid contact for calculation of hydraulic diameter (Equation 14).

in pressure drop, with a prediction of ∆P/∆P0 ) 1.05, regardless of Re. From the observed deposition behavior, illustrated in Figure 9, pressure drop increases partly because the pores become constricted and partly because the attached particles increase the hydraulic radius of the packing. As illustrated in Figure 10, the particles will alter the local flows and increase drag more than if the coating is assumed to be smooth. If the hydraulic

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The pressure drop increased rapidly with relatively small amounts of deposit in the bed. The pressure increase was higher in the inlet section of the bed, but pressure drop increased throughout the entire bed. Low Re gave a more pronounced increase in pressure drop because the deposits accumulated in the flow channels between the beads or pellets in the bed. The Ergun equation was extended to model the pressure drop by allowing for the change in the hydraulic radius of the packing due to particle deposition. Acknowledgment The authors acknowledge the generous support of the Natural Sciences and Engineering Research Council of Canada, through a Strategic Grant, and the support and assistance of Syncrude Canada Ltd. J. R. Coury is indebted to CAPES-Brazil for the financial support given (proc. no. 0657/96-10). Nomenclature Figure 11. Increase in pressure drop with specific deposit, showing range of predicted pressure drops from the Ergun equation (Equation 12) using the modified collector diameter (Equation 14).

diameter of the collector is defined as the ratio of surface to volume, then the deposition of particles gives

dc(t) ) 6

Vc + Vd Sc - RSd + (1 - R)Sd

(14)

where Vc and Sc are the initial volume and surface area of the collectors (i.e., bed packing). The adjustable parameter R is the fraction of the surface area of the deposited particles that is occluded at the point of contact with the collector or with other particles. The predicted increase in pressure drop depends on the value of R as illustrated in Figure 11. Although this model is consistent with the observed structure of the deposits at higher values of Re, its major limitation to this model is that the value of empirical parameter, R, cannot be predicted a priori. Significance for Reactor Operation. This initial study showed that the composition of the granules in the bed was unimportant for particle trapping, because glass beads and the catalyst gave similar behavior. Once the initial particles had attached to the packing, subsequent deposits were dominated by particleparticle interactions. Particle deposition at room temperature and pressure may, therefore, provide insights into particle behavior at reactor conditions. This study also suggests that while the theory for granular filtration may be applicable to particle deposition in nonaqueous media, the structure of the deposits should be examined in order to model pressure drop behavior. The next stage of our work will determine the mechanisms of particle deposition in gas-liquid flow through packed beds. Conclusions The efficiency of trapping of fine particles decreased with the amount of deposit in the bed and decreased with increasing Re. Flow in the downward direction gave slightly more trapping than in the upward direction. Equivalent results were obtained with glass beads and with a hydrotreating catalyst.

As ) Happel parameter C ) particle concentration, kg/m3 d ) diameter, m E ) bed-trapping efficiency mpr ) mass of particles trapped in the bed, kg P ) pressure, Pa Q ) volumetric flow rate, m3/s Re ) collector Reynolds number ()FUdc/µ) S ) surface area, m2 t ) time, t U ) superficial velocity, m/s V ) volume, m3 Greek Letters R ) adjustable parameter in eq 14  ) bed porosity µ ) liquid viscosity, kg/ms F ) density, kg/m3 σ ) mass of deposited particles per unit bed volume, mg/ mL ωp ) particle mass flow rate, kg/s Subscripts 0 ) initial condition b ) packed bed c ) collector or packing d ) deposit of particles i0 ) initial carbon black concentration in the tank inlet ) inlet conditions for packed bed L ) liquid suspension outlet ) outlet conditions for packed bed p ) particle

Literature Cited Bai, R.; Tien, C. A new correlation for the initial filter coefficient under unfavorable surface interactions. J. Colloid. Interface Sci. 1996, 179, 631-634. Byers, C. H.; Amarnath, A. Understand the potential of electroseparations. Chem. Eng. Prog. 1995, 91 (2), 63-69. Choo, C.; Tien, C. Analysis of the Transient Behavior of DeepBed Filtration. J. Colloid Interface Sci. 1995a, 169, 13-33. Choo, C.; Tien, C. Simulation of Hydrosol Deposition in Granular Media. AIChE J. 1995b, 41, 1426-1442. Chowdiah, P.; Wasan, D. T.; Gidaspow, D. Electrokinetic phenomena in the filtration of colloidal particles suspended in nonaqueous media. AIChE J. 1982, 27, 975-984.

Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997 4627 Deb, A. K. Theory of Sand Filtration. J. Sanit. Eng. Div. NY, Am. Soc. Civ. Eng. 1969, 95, 399-422. Gimbel, R. Theoretical Approach to Deep Bed Filtration. In Water, Wastewater and Sludge Filtration; Vigneswaran, S., Ben Aim, R., Eds.; CRC Press, Boca Raton, FL, 1989; pp 17-56. Johnson, P. R.; Elimelech, M. Dynamics of colloid deposition in porous media: Blocking based on random sequential adsorption. Langmuir 1995, 11, 801-812. Kitahara, A.; Fujii, T.; Katano, S. Dependence of ζ-potential upon particle size and capillary radius at streaming potential study in non-aqueous media. Bull. Chem. Soc. Jpn. 1971, 44, 32423245. Koyama, H.; Nagai, E.; Torii, H.; Kumagai, H. Simple changes reduce catalyst deactivation, pressure-drop buildup. Oil Gas J. 1995a, 93 (47), 68-71. Koyama, H.; Nagai, E.; Torii, H.; Kumagai, H. Japanese refiner solves problems in resid desulfurization unit. Oil Gas J. 1995b, 93 (46), 82-90. Narayan, R. Particle Capture from Non-Aqueous Media on Packed Beds. MSc. Dissertation, University of Alberta, 100 p, 1996. Pendse, H.; Tien, C.; Rajagopalan, R.; Turian, R. M. Dispersion measurement in clogged filter beds: A diagnostic study on the morphology of particle deposits. AIChE J. 1978, 24, 473-485. Rajagopalan. R.; Tien, C. Trajectory Analysis of Deep-Bed Filtration with the Sphere-in-cell Porous Media Model. AIChE J. 1976, 22, 523-533.

Rajagopalan. R.; Tien, C. The theory of deep bed filtration. In Progress in Filtration and Separation; Wakeman, R. J., Ed.; Elsevier: Amsterdam, 1979; Vol. 1, pp 179-270. Spielman, L. A.; FitzPatrick, J. A. Theory for Particle Collection under London and Gravity Forces. J. Colloid Interface Sci. 1973, 42, 607-623. Tian, C.; Guthrie, R. I. L. The Dynamic Process of Liquid Metal Filtration. Light Metals 1995, 1263-1272. Tien, C. Granular Filtration of Aerosols and Hydrosols; Butterworths: Boston, MA, 1989. Vaidyanathan, R.; Tien, C. Hydrosol Deposition in Granular BedssAn Experimental Study. Chem. Eng. Commun. 1989, 81, 123-144. Wnek, W. J.; Gidaspow, D.; Wasan, D. T. The Role of Colloid Chemistry in Modelling Deep-Bed Liquid Filtration. Chem. Eng. Sci. 1975, 30, 1035.

Received for review February 3, 1997 Revised manuscript received August 15, 1997 Accepted August 24, 1997X IE970101E

X Abstract published in Advance ACS Abstracts, October 1, 1997.