Ind. Eng. Chem. Res. 1997, 36, 5125-5132
5125
A Model for the Oxidation of Sulfur Dioxide in a Trickle-Bed Reactor Potnuru V. Ravindra, D. Prahlada Rao, and Musti S. Rao* Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208016, India
A reaction rate model based on complete wetting of the catalyst particles in a trickle-bed reactor is presented. It is considered that the pendular rings are formed at the contact points of the particles and the liquid flows down as thin films on the remaining surface between the pendular rings. Assuming that the flow in the films is laminar, the liquid in the pendular rings is wellmixed, and a fraction of the liquid flowing in the films bypasses the pendular rings, the reaction rate was found. The data obtained for oxidation of sulfur dioxide on activated carbon were used to test the model predictions of the dependence of reaction rate on liquid velocity. The reaction rates in nonprewetted beds were found to be lower than those in prewetted beds. The proposed model correlated the observed reaction rates satisfactorily, with the fractional thickness of the film that bypasses the pendular ring as an adjustable parameter. The model also predicted the reaction rate trends, including the minimum of the oxidation rate reported by Mata and Smith (Chem. Eng. J. 1981, 22, 229-235). Introduction Trickle-bed reactors are fixed-bed catalytic reactors in which gas and liquid flow concurrently downward. They are used for carrying out a variety of multiphase reactions such as hydrogenation and oxidation, in which the gaseous reactant is sparingly soluble in the liquid phase. The primary source of difficulty in modeling trickle-bed reactor performance is the complex nature of gas-liquid flow in the bed. The rational design of these reactors is still out of reach. Several models have been proposed in the literature to predict the performance of trickle-bed reactors. In the early studies, mass transport effects were neglected, and the gas and liquid phases were treated as a single homogeneous phase (pseudohomogeneous models). The catalyst particles were assumed to be partially wet and the reaction was assumed to occur only in the wetted zones of bed in order to have a simple relationship between the apparent reaction rate and the extent of contact between liquid and solid. The apparent rate was assumed to be proportional to the liquid holdup (Henry and Gilbert, 1973) and to the fractional coverage of the catalyst surface with the flowing liquid, known as wetting efficiency (Mears, 1974). These pseudohomogeneous models are able to predict the behavior of trickle-bed reactors for liquid-limiting reactions such as hydrodesulfurization of heavy distillates (Paraskos et al., 1975), hydrodenitrogenation of hydrotreated oils (Van Klinken and Van Dongen, 1980), and decomposition of hydrogen peroxide over activated carbon (Koros, 1976). However, when the limiting reactant is present in the gas phase, a minimum in the reaction rate was observed for hydrogenation of R-methylstyrene (Herskowitz et al., 1979) and oxidation of sulfur dioxide (Mata and Smith, 1981). The pseudohomogeneous models fail to predict the observed reaction rate trends. Heterogeneous models accounting for external mass-transfer resistance have been proposed to explain the minimum in the reaction rate. In these models, both the wetted and unwetted zones of the catalyst particles were assumed to contribute to the reaction rate. The gaseous reactant * Author to whom correspondence should be addressed. S0888-5885(97)00340-0 CCC: $14.00
must overcome both mass-transfer resistances due to flowing liquid film to reach the wetted catalyst pellet, whereas on the nonwetted surface, the gaseous reactant is in direct contact with the liquid in the pores. Some investigators assumed that the nonwetted surface of the catalyst is covered with a stagnant liquid film and offers a significant resistance to the transport of gaseous reactant (Mills and Dudukovic, 1984), whereas others assumed that the dry surface is devoid of liquid film (Herskowitz et al., 1979; Capra et al., 1982; among others). The recent models of reaction rates in trickle-bed reactors are based on the assumption that the catalyst particles are partially wetted with the flowing liquid (Beaudry et al., 1987; Harold and Ng, 1987; among others). A comprehensive review on modeling of reaction rates is given by Harold (1993). A recent study on liquid flow texture employing the dye adsorption technique (Ravindra et al., 1997) suggests that the particles are likely to be completely covered by the flowing liquid. The objective of this study is to develop a reaction rate model based on complete wetting of the particles and to validate the model using the rate data of oxidation of sulfur dioxide in the presence of activated carbon as a catalyst. Reactor Model Consider the bed to be made up of spherical particles of uniform size, packed in a cubic array. The bed can be visualized as identical strings, with the neighboring particles touching at their equators. With uniform inlet liquid distribution, each string of spheres receives an equal amount of liquid. Since the strings are identical in nature, it can be assumed that the flow does not take place across the strings. In such a packing, each particle is in contact with six other particlessone at the top and another at the bottom of the particle and four along its equator. Let us now consider the liquid flow over a string of particles. The side view of pendular rings formed at the contact points of a sphere is shown in Figure 1a. The five shaded regions represent the pendular rings (the sixth one is hidden behind the central one). The unshaded region represents the film flow region. The liquid in between the streamlines, not © 1997 American Chemical Society
5126 Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997
and
δvθ/δx ) 0
at x ) 0
(3)
Equation 1 can be solved to obtain
vθ )
Fg sin θ 2 δ (1 - x2/δ2) 2µ
for β e θ e π - β (4)
The film thickness, δ, can be shown to be Figure 1. Flow over a spherical particle in a cubic packing: (a) particle with pendular rings; (b) a quadrant of the particle surface; (c) mixing in the pendular ring.
interesected by the pendular rings, flows into the bottom pendular ring (see Figure 1b). Thus, the entire liquid that flows over the particle passes through the top as well as the bottom pendular rings. On the basis of geometrical considerations (see Figure 1c), it was assumed that the liquid flowing in the inner part of the film enters the bottom pendular ring and is well-mixed (the hatched portion) and the remaining upper part bypasses the pendular ring as depicted in Figure 1c. In contrast, it was considered that all the liquid that is intercepted by the middle pendular ring flows into it and is well-mixed. The gas flowing cocurrently with the liquid is in contact with the liquid film and the surfaces of the pendular rings. In view of the geometrical symmetry, the liquid flow over a quadrant is shown in Figure 1b. In region I, liquid flows as a film between the top and bottom pendular rings. In region II, there is a pendular ring in the middle. Mass-transfer and reaction rates for a string can be estimated from the rates on individual spheres. Satterfield et al. (1969) and Hanika et al. (1971) obtained equations to determine mass-transfer and reaction rates for a string of catalyst particles with flow. Their method has been adapted to model the reaction rates in a trickle-bed reactor. Flow in Liquid Film. First, we consider that there are no middle pendular rings. Later on, the effect of middle pendular rings on the film flow is accounted for. We make the following assumptions: (a) Flow in the liquid film is steady, fully developed, laminar, and free from ripples. (b) The velocity component perpendicular to the particle surface is negligible. (c) Gas exerts negligible drag on the gas-liquid interface. (d) The film thickness is much smaller in comparison to the radius of the particle. Under these assumptions, the equations of motion reduce to
d2vθ
Fg sin θ ) 2 µ dx
for β e θ e π - β
(1)
where θ is the angle measured from the vertical axis of the sphere, x is the perpendicular distance into the film from the gas-liquid interface, F and µ are the density and viscosity of the liquid, respectively, g is the acceleration due to gravity, β is the filling angle (see Figure 1c), and vθ is the θ-component of velocity in the film. Employing the boundary conditions
vθ ) 0
at x ) δ
(2)
δ)
[
3µQ 2πRFg sin2 θ
]
1/3
for β e θ e π - β
(5)
where Q is the liquid flow rate per string and R is the radius of the particle. Now consider the regions shown in Figure 1b. The film flow in region I is similar to the flow over a single sphere. In region II, there is a pendular ring in the middle. In the region above the pendular ring, the film flow would be the same as that over a sphere and eqs 4 and 5 are applicable. The nature of the outflow of the pendular ring is not known. The flow rate could be more at the lower part of the pendular ring. As the nature of the flow is not known, we considered the flow below the pendular ring to be a mirror image of the flow over the ring in region II. Mass Transfer. The gaseous reactant diffuses through the liquid film to reach the catalyst surface. Its concentration profile and the transfer rate through the film can be obtained by solving the differential mass balance equations with appropriate boundary conditions
D
∂2C vθ ∂C ) R ∂θ ∂x2
(6)
where D and C are the diffusivity and the concentration of the limiting reactant in the liquid, respectively. The boundary conditions are as follows: (i) Considering that equilibrium prevails at the gasliquid interface, we have
C ) C*
at x ) 0
(7)
(ii) As the amount of the gaseous reactant that reaches the catalyst surface through the liquid film is equal to the rate of reaction, for a first-order reaction, we get
-D
∂C φCsDe 1 1 ) ∂x R tanh φ φ
[
]
at x ) δ
(8)
and
φ ) R(kv/De)1/2
(9)
where φ is the Thiele modulus, Cs is the concentration of the gaseous reactant on the solid surface, and De is the effective diffusivity of the gaseous reactant. (iii) Considering that the inlet liquid stream has a uniform concentration of the dissolved gaseous reactant, we have
C ) Ci
at θ ) β
(10)
The inlet liquid concentration for the subsequent spheres can be determined as described in the following section. Pendular Rings. Considering the liquid in the pendular ring to be well-mixed, the rate of disappear-
Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997 5127
rpr ) AmksC ho
(11)
fractional volumetric flow rate of the liquid that flows into the pendular ring from region I. This is dictated by the filling angle and is given as
Am ) 2πR2(1 - cos β)
(12)
f ) 1 - 4β/π
ance of gaseous reactant in the pendular ring is
(20)
The outlet liquid concentration can be found from eq 14 and the rate of reaction in the pendular ring from
and
ks ) Rkvη/3
(13)
where Am is the surface area of the sphere encompassed by the pendular ring, ks is the apparent reaction rate constant, and C h o is the average concentration of the outlet liquid. From the mass balance over the pendular ring, we get
hi QmC C ho ) Qm + Amks
(14)
∫βπ-βksCs(θ) 2πR2 sin θ dθ
(15)
The rate of reaction in the film flow of region II between the top and middle pendular rings, rII,t, is
rII,t )
∫βπ/2-βksCs(θ) 2πR2 sin θ dθ
(16)
We consider the effect of gravity on the volume of the pendular ring to be negligible. The rate of reaction in the middle pendular ring, rm, is
ho rm ) ksAmC
(17)
The rate of reaction in the film flow of region II between the middle and bottom pendular rings, rII,b, is
rII,b )
π-β ksCs(θ) 2πR2 sin θ dθ ∫π/2+β
(18)
The bottom pendular ring receives liquid from regions I and II. The compositions of liquid films flowing from regions I and II into the pendular ring are different. Recall that a fraction of these two liquid films mixes in the bottom pendular ring (see Figure 1c), and the remaining fraction bypasses to regions I and II of the next sphere. Let the liquid film thickness be δ (measured normal to the flow). We consider that the liquid flowing adjacent to the solid of fδδ thickness flows into the pendular ring and the rest bypasses the pendular ring. The average inlet concentration of the liquid film flowing in the pendular ring, C h i, is
∫0f δvθCII,i dx ∫f δvθCI,i dx +f C h i ) (1 - f) ∫0f δvθ dx ∫fδδvθ dx δ
δ
(21)
The rate of reaction in the pendular ring, rpr, is apportioned equally between the bottom pendular ring of the nth sphere and top pendular ring of the (n + 1)th sphere. Thus, we have
1 (n) ) r(n+1) ) rpr rpb pt 2
(22)
The rate of reaction on the ith sphere is
where Qm is the volumetric flow rate and C h i is the average concentration of the inlet liquid. Overall Rate of Reaction. The overall rate of reaction was obtained from the reaction rates of individual spheres. For any sphere in the string, the reaction rate is the sum of reaction rates in region I, region II, and top and bottom pendular rings (see Figure 1b). The rate of reaction in region I, rI, is
rI )
rpr ) ksAmC ho
δ
δ
(19)
δ
where CI,i and CII,i are the concentrations of the liquid films of regions I and II, respectively, and f is the
ri ) rpt,i + frI,i + rpb,i + (1 - f)(rII,t + rpr + rII,b)
(23)
The overall rate of reaction is N
Ra )
ri ∑ i)1 NVp
(24)
where N is the total number of particles in a string and Vp is the volume of a particle. Method of Solution. The concentration profiles in the film regions were obtained by solving eq 6 with the appropriate initial and boundary conditions using an explicit difference scheme. To minimize the computational effort, only one quadrant was considered in view of symmetry. Numerical integration was used to obtain the reaction rates and the average exit concentrations of the liquid films of different regions. The outlet liquid concentration and reaction rates were obtained from eqs 14 and 11, respectively, for the middle pendular rings. Equation 22 was used for reaction rates in the case of top or bottom pendular rings. Recall that it was considered that the inner part of the film, fδδ, enters the bottom pendular ring whereas the outer part, (1 - fδ)δ, bypasses it. Therefore, the lower part of the film entering the second sphere has uniform concentration along thickness fδδ and throughout the periphery. In contrast, concentration varies with x in the upper part of the film. The exit concentration profiles of the film of the first sphere in regions I and II were taken as the inlet profiles for the corresponding regions of the second sphere. With these initial conditions, the calculations were performed for the second and subsequent spheres. The details are available elsewhere (Ravindra, 1995). Experimental Apparatus and Procedure We selected the liquid-phase oxidation of sulfur dioxide
1 SO2 + O2 + H2O f H2SO4 2
(25)
as the test reaction for the following reasons. First, the reaction kinetics has already been studied (Komiyama and Smith, 1975) and is first order in oxygen and zerothorder in water and sulfur dioxide. Second, the reaction
5128 Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997
Figure 2. Schematic diagram of the experimental setup: (A) liquid distributor; (B) test section; (C) gas-liquid separator; (1) deionized water tank; (2) liquid rotameter; (3) needle valves; (4) trickle-bed reactor; (5) water saturation tank; (6) pump; (7) liquid sample port; (8) mixing chamber; (9) gas rotameter; (10) soap bubble flow meter; (11) absorber.
occurs at room temperature and atmospheric pressure. Finally, the analytical determination of sulfuric acid by a titrimetric method is simple and accurate. Figure 2 shows the experimental setup, which is selfexplanatory. The trickle-bed consisted of a liquid distributor, a packed-bed section, and a gas-liquid separator. The bed was rectangular in cross section with inner dimensions of 6 × 8 cm. The height of the packed section was 20 cm. To distribute the liquid uniformly over the bed, 99 stainless steel capillary tubes of 7.0 cm length and 0.1 cm inner diameter were used. To introduce the liquid through a point inlet, a stainless steel tube of 0.4 cm inner diameter was employed in place of the capillary tubes. The test section was packed with granular activated carbon. Deionized water from a tank (1) was fed to the liquid distributor through a rotameter. Air from a compressor was fed to the mixing chamber (8) through a rotameter. Sulfur dioxide was passed through a soap bubble flow meter to the mixing chamber. The air and sulfur dioxide mixture was fed to the trickle-bed reactor through the gas chamber of the liquid distributor. Two different startup procedures were employed. In the first, deionized water was introduced in a bed of dry activated carbon at a low flow rate and the flow rate was gradually increased to the desired rate. Then the gas flow was set at a desired rate. The state of this bed is referred to as nonprewetted bed. In the other procedure, deionized water was introduced at a low flow rate and the bed was filled with water by closing the outlets. Once the water level reached the top of the packing, the water flow rate was set at the desired value and the outlets were gradually opened so as to drain the excess water while maintaining the liquid flow at the desired rate. Then the gas flow was initiated. The state of the bed, thus started, is referred to as prewetted bed. Experiments were carried out using both unsaturated and presaturated water with air. The details of the operating conditions employed are given in Table 1. The outlet liquid was drawn periodically to monitor the attainment of steady state. The sulfuric acid content of the liquid effluent was analyzed by the method described by Hartman and Coughlin (1972), in which sulfuric acid concentrations were obtained from total acidity and sulfurous acid determinations. Trip-
Figure 3. Transient behavior of reaction rates in nonprewetted and prewetted beds. Table 1. Operating Conditions for the Reaction Rate Studies temperature pressure liquid phase superficial velocity gas phase air flow SO2 flow rate total gas flow rate ySO2 yO2 yN2 solid phase particle diameter reactor dimensions bed porosity
25 °C 1 atm deionized water 1.0-8.0 mm/s air and SO2 mixture 79.62 cm3/s 1.38 cm3/s 81.00 cm3/s 0.017 0.206 0.777 granular activated carbon 1.89 mm 6 × 8 × 21 cm3 0.36
licate sampling was carried out at each flow rate. It was considered that the steady state was attained if two successive reaction rates were within 3% of one another. The mole fraction of oxygen in the gas phase is taken to be uniform throughout the test section as the consumption of oxygen is negligible compared to the inlet rate. The rate of reaction of the limiting reactant was calculated from the sulfuric acid concentration. A more detailed description of the experimental apparatus and procedure can be found elsewhere (Ravindra, 1995). Results and Discussion A series of experiments with different liquid velocities was carried out to determine the reaction rates in prewetted and nonprewetted beds for liquid feed saturated with air. The transient behavior of the reaction rates in prewetted and nonprewetted beds is shown in Figure 3. The prewetted beds took less time to reach steady state compared to the nonprewetted beds. For instance, at a liquid velocity of 1 kg/m2‚s, the nonprewetted bed took about 6 h, whereas the prewetted one attained steady state in only 3 h. The slower attainment of steady state in nonprewetted beds compared to prewetted beds could be because of the slow process of filling of the pores in the former since the particles were initially dry when packed. For prewetted beds, the trends of the reaction rates are similar to the ones reported by Haure et al. (1992). It is evident that the prewetted beds should be preferred over nonprewetted beds, for the former gave higher reaction rates and attained steady state faster.
Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997 5129 Table 2. Parameters Used in the Simulation of Reaction Rates D R C* µ F N β g ηkv b
Figure 4. Dependence of the reaction rate on liquid velocity in nonprewetted and prewetted beds (uniform liquid inlet).
Figure 5. Reaction rate vs liquid velocity for uniform and single liquid inlets (prewetted beds).
Figure 4 shows the dependence of steady-state reaction rate on liquid velocity in prewetted and nonprewetted beds. The reaction rates for the nonprewetted beds are lower than those for prewetted beds. The exit liquid distribution measured with 16 cells at the bed outlet (Ravindra et al., 1997) revealed that the liquid distribution in prewetted beds was more uniform compared to that in the nonprewetted beds. The higher flow rates in some of the cells in nonprewetted beds indicated that a part of the bed cross section had liquid-flooded regions and the rest had film flow of liquid over the particles. The contribution of the flooded region to the reaction rate is expected to be negligible since the particles inside the flooded region were not exposed to the gaseous reactant. Hence, lower reaction rates were observed in nonprewetted beds compared to prewetted beds. The effect of inlet liquid distribution on the reaction rate for prewetted beds is shown in Figure 5. It presents the reaction rates for a single liquid inlet with and without prepacking. Glass beads of 1.9 mm diameter, packed to a height of 6 cm, were used as prepacking. The corresponding rates with uniform liquid inlet measurements (Figure 4) are included for comparison. Marginally higher reaction rates were observed in the case of beds with prepacking than in those without prepacking for a single liquid inlet. However, the rates for a single inlet with prepacking were slightly lower than those for a uniform liquid inlet. This can be explained based on the visual observations of liquid texture with the dye-adsorption technique (Ravindra et al., 1997). With dye measurements it was noticed that
2.6 × 10-5 cm2/s 0.095 cm 2.56 × 10-7 (g mol)/cm3 1 × 10-2 g/cm‚s 1.0 g/cm3 110 30° 981 cm/s2 0.18 1/s 0.36
for the single liquid inlet the spread of liquid took place within 4 cm of the bed depth. Therefore, a fraction of the particles in the top layers does not participate in the reaction. As a result, the reaction rates were lower for the case of a single liquid inlet without prepacking. However, with a prepacking depth of 6 cm, all the catalyst particles below the prepacking zone had received liquid even with a single liquid inlet. Hence, the reaction rates were slightly higher compared to the ones without prepacking. The higher rates with uniform liquid inlet could be attributed to more uniform liquid distribution. Estimation of Reaction Rates from the Model. The reaction rates were estimated using intrinsic rate constant, diffusivities, filling angle, and fractional mixing thickness as given below. Komiyama and Smith (1975) studied the kinetics of oxidation of sulfur dioxide with activated carbon (BPL-Pittsburgh Calgon Corp.). Since the same type of catalyst was used in the present study, the kinetic rate constant and other parameters were taken from Komiyama and Smith (1975). Satterfield et al. (1969) measured the static filling angle of a pendular ring between spheres in a string of alumina particles of 8 mm diameter and found it to be 33°. However, in view of the lack of data on the dynamic filling angles, it was assumed to be 30°. Table 2 gives the parameters used in the calculations. Calculation of fδ. The number of particles that could be packed in a square array in the cross section of the packed section was found. From the inlet liquid flow rate to the bed, the flow rate for a string of spheres was found. The only unknown parameter was fδ. For a guess value of fδ, the rate of formation of sulfuric acid was found and compared with the experimental value. The regula-falsi iterative method was used to find fδ to attain the difference between the calculated and experimental values to a prescribed tolerance. Similarly, fδ was found for the data reported by Mata and Smith (1981). Figure 6 shows the effect of percent saturation on rA over the individual spheres at different liquid velocities. The spheres are numbered from the top. For the sake of convenience, the rates over the individual spheres are shown as a continuous curve, instead of discrete points. Depending on the percent saturation of the feed, there was a sharp change in rA over the top few spheres. Thereafter, rA attained a constant value. The variation in rA was solely due to the percent saturation of the inlet liquid. The constant value and the sphere number at which it was attained were independent of percent saturation. However, the sphere number at which the constant value was attained increased with liquid velocity. The effect of percent saturation on the concentration profile has been examined. Figure 7 presents typical profiles at the equator for fδ ) 0.3 at L ) 7 kg/m2‚s. It can be seen that the concentration profiles differ widely
5130 Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997
Figure 6. Effect of percent saturation of oxygen in the inlet liquid at different liquid velocities on the reaction rate.
Figure 8. fδ vs Reynolds number for saturated and unsaturated liquid feeds.
Figure 7. Concentration profiles in the liquid film at the equator of the particle for fδ ) 0.3 and L ) 7 kg/m2‚s.
Figure 9. Estimated reaction rate vs liquid velocity for oxidation of sulfur dioxide.
with the percent saturation for the top few spheres. But, they gradually converged and finally merged. The sphere number at which the merging took place corresponds to the sphere at which rA attained a constant value (see Figure 6). There is no interface transport of the gaseous reactant even at the fifth sphere for the saturated feed. The percent saturation of the inlet liquid may have significant influence on the reaction rates in laboratory reactors in which the bed heights are small. But, this may have negligible effect in industrial reactors. Correlation of the Data. The fδ values were determined from the rate data of oxidation of sulfur dioxide reported by Mata and Smith (1981) and those obtained in the present work. Figure 8 shows the variation of fδ with the Reynolds number for saturated and unsaturated liquid feeds. fδ increased with liquid flow rate. The liquid motion in the pendular ring is a result of the momentum imparted by the incoming liquid. As the momentum influx increases, the intensity of the fluid motion within the pendular ring also increases, which, in turn, increases the erosion of the bypassing film. Therefore, fδ is expected to increase with liquid flow rate. The degree of saturation of gaseous reactant may be assumed to have negligible or no effect on the film flow and on the pendular ring volume. Therefore, fδ should be independent of the degree of saturation. The close agreement between the estimated fδ for both the saturated and unsaturated liquid feeds for the rate data supports the validity of the model. It can be seen that
there is a considerable deviation in the values of fδ obtained using the data for a saturated feed of Mata and Smith (1981) and from those calculated in the present work. This could be attributed to the variation in the characteristics of the activated carbon from one batch of production to another and of the pendular rings, which depend on the particle size. On the other hand, for an unsaturated feed, the fδ’s for the data of Mata and Smith (1981) were in close agreement with those in the present work. However, it may be pointed that there is an uncertainity in the analysis of oxygen concentration for an unsaturated feed, as reported by Mata and Smith (1981). The estimated fδ values have been correlated with the corresponding film Reynolds number for the data in the present work, using a quadratic polynomial given as
fδ ) 0.259 + 0.00789Ref + 0.00136Ref2
(26)
To examine the ability of the model to estimate the rates, the reaction rates were computed using the relation for fδ (eq 21). The estimated reaction rates are shown in Figure 9. The model predicted the reaction rate trends, including the minimum. However, the deviations are larger in the case of the saturated feed of Mata and Smith (1981). As discussed earlier, this could be because of different activities and wetting characteristics of the activated carbon. To ascertain the cause for the minimum, the reaction rate contributions from film flow and pendular rings were analyzed. Table 3 gives the individual contribu-
Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997 5131 Table 3. Individual Contributions of Film Flow and Pendular Rings to the Overall Reaction Rate L (kg/m2‚s)
reaction rate in the film flow region × 108 ((g mol)/cm3‚s)
reaction rate in pendular rings × 108 ((g mol)/cm3‚s)
1 2 3 4 5 6 7
1.868 1.461 1.235 1.106 0.989 0.940 0.886
0.920 0.934 0.971 1.020 1.072 1.148 1.276
tions of film flow and pendular rings to the overall reaction rate. Based on these results, a plausible explanation for the minimum is as follows. At low liquid flow rates, the film thickness is less and fδ is also less. The contribution of the film region to the overall reaction rate is higher compared to that of pendular rings. As the liquid flow rate increases, the film becomes thick. Therefore, the contribution of the film region decreases. On the other hand, the reaction rates in the pendular rings increase because of the increased flow into the pendular rings. Thus, the contribution to the reaction rates from the film flow and that due to the pendular rings may give rise to minima in the reaction rates. Conclusions Reaction rates in prewetted and nonprewetted beds were observed to be different. This was attributed to the difference in liquid distribution. Also, the time required to attain steady state in nonprewetted beds was higher than that in prewetted beds. In prewetted beds, the inlet liquid distribution was found to have a negligible effect on the reaction rates. A reaction rate model based on complete wetting of the catalyst particle was proposed in the present study. In this model, an adjustable parameter fδ was used in estimating the reaction rates. A functional relationship was derived between fδ and the film Reynolds number from the rate data of the present study. The model could predict the rate trends, including the minimum, reported by Mata and Smith (1981). The higher reaction rates at higher liquid flow rates were attributed to increased mixing in the pendular rings at these flow rates. The proposed model appears to be satisfactory. However, the model has to be extended to incorporate the flow distribution and random orientation of the contact points using computer-generated random sphere pack models (Funk et al., 1990). Further, fδ needs to be determined from experiments. The model may be extended to account for heat effects and evaluation for different kinetic schemes.
Ci ) concentration of limiting reactant in the inlet liquid stream, (g mol)/cm3 C h i ) average concentration of limiting reactant in liquid entering the pendular ring, (g mol)/cm3 C h o ) average concentration of limiting reactant in liquid leaving the pendular ring, (g mol)/cm3 Cs ) concentration of limiting reactant at the catalyst surface, (g mol)/cm3 CI,i ) concentration profile of the liquid film of region I CII,i ) concentration profile of the liquid film of region II D ) diffusivity of limiting reactant in the liquid, cm2/s De ) effective diffusivity of limiting reactant, cm2/s f ) fraction of liquid that flows into the pendular ring from region I fδ ) fractional thickness of film that flows into the pendular ring (see Figure 1c) g ) acceleration due to gravity, cm/s2 ks ) apparent reaction rate constant, ηkvR/3, cm/s kv ) intrinsic rate constant, 1/s L ) superficial mass flow rate of liquid, kg/m2‚s N ) total number of particles in a string Q ) volumetric flow rate of liquid per string, cm3/s Qi ) volumetric flow rate of liquid in the ith layer of the liquid film, cm3/s Qm ) volumetric flow rate of liquid into the pendular ring, cm3/s rI ) reaction rate in region I, (g mol)/s rII,t ) reaction rate in film flow in the upper part of region II, (g mol)/s rII,b ) reaction rate in film flow in the lower part of region II, (g mol)/s rA ) reaction rate per sphere, (g mol)/s ri ) reaction rate in the ith sphere, (g mol)/s rpr ) reaction rate in the pendular ring, (g mol)/s (n) ) reaction rate in the bottom pendular ring of the nth rpb sphere, (g mol)/s ) reaction rate in the top pendular ring of the (n + r(n+1) pt 1)th sphere, (g mol)/s R ) radius of a particle, cm Ra ) overall reaction rate, (g mol)/cm3‚s Ref ) film Reynolds number, 4QF/Wµ Vp ) volume of a particle, cm3 vθ ) θ-component of velocity in the film, cm/s W ) wetted perimeter at the equator, cm x ) perpendicular distance from the gas-liquid interface into the liquid film, cm y ) mole fraction Greek Letters β ) filling angle δ ) film thickness of the liquid on a particle, cm b ) bed void fraction η ) effectiveness factor θ ) angle measured from the vertical axis of the sphere µ ) viscosity of liquid, P F ) density of the liquid, g/cm3 φ ) Thiele modulus
Acknowledgment We gratefully acknowledge the financial support of the Department of Science and Technology, Government of India. Nomenclature Am ) surface area of the sphere encompassed by the pendular ring, cm2 C ) concentration of limiting reactant in the liquid, (g mol)/ cm3 C* ) equilibrium concentration of limiting reactant, (g mol)/cm3
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Received for review May 12, 1997 Revised manuscript received September 22, 1997 Accepted September 28, 1997X IE9703402
X Abstract published in Advance ACS Abstracts, November 1, 1997.
