Design of Entrained-Flow and Moving-, Packed-, and Fluidized-Bed

1024

Ind. Eng. Chem. Res. 1996, 35, 1024-1043

Design of Entrained-Flow and Moving-, Packed-, and Fluidized-Bed Sorption Systems: Grain-Model Kinetics for Hot Coal-Gas Desulfurization with Limestone Laurent A. Fenouil† and Scott Lynn*,‡ Department of Chemical Engineering and Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720

Concentration profiles in several configurations of sorption systems (moving, packed, and fluidized beds and entrained-flow) were determined using the grain and the unreacted shrinking core model for the description of the gas-solid reaction kinetics. General equations for the design and analysis of all these reactor configurations are given, and simple analytical solutions are proposed. The behavior of these reactors can be expressed as a function of only four parameters: the size of the sorbent pellet and the Pećlet, Sherwood (or Biot), and Damko¨hler numbers. These general equations were then applied to the lime(stone)/H2S systems. Using limestone particles with diameters on the order of 100 µm for entrained-flow and with diameters on the order of 1 mm for the other configurations, it was found that H2S could be removed from the hot coal gas to near its equilibrium value with more than 75% sorbent utilization and with reasonable bed depth (about 1 m, or 1-s contact time for entrained-flow) when the reactor temperature is maintained 25-50 °C above the calcination temperature of the calcium carbonate. Introduction In this paper we model the behavior of a sorbing bed of lime(stone) under typical coal-gas conditions. We will examine the characteristics and performances of moving (in both cocurrent and countercurrent configurations), fixed, and fluidized beds of lime(stone) particles as well as entrained-flow (or transport) reactor configurations. However, the kinetic models will be developed under very general assumptions and will be applicable to a large variety of gas-solid reactions and reactor configurations. It has been established by Fenouil and Lynn (1995) that lime (CaO), or limestone (CaCO3) under specific conditions, can be a suitable high-temperature sorbent for H2S in coal gas through the following reactions:

CaO + H2S h CaS + H2O

(1a)

CaCO3 + H2S h CaS + H2O + CO2

(1b)

We will make use of the physical and kinetic data obtained by Fenouil and Lynn (1995) as well as those from Borgwardt et al. (1984) to estimate the desulfurization performances of these calcium-based sorbents. Most of our attention will be focused on reaction (1a), since even millimeter-sized particles of CaO can be completely and rapidly converted to CaS. Nonetheless, data pertaining to CaCO3 will also be included since micron-sized limestone particles can also be successfully converted to CaS (Borgwardt and Roache, 1984). We will consider H2S as the only sulfur-containing species in the coal gas. Generally, H2S comprises more than 95% of the sulfur-containing species, the remainder †

Current address: Chemical Development Department, Shell Chemical, 3333 Highway 6 South, Houston, TX 77082. ‡ It is a pleasure to dedicate this paper to Jud King, who, for nearly 30 years, has been an esteemed hiking companion, research collaborator, coteacher of advanced plant design, and faculty colleague in many capacities.

0888-5885/96/2635-1024$12.00/0

being primarily COS (MERC, 1978; Yang and Chen, 1979; Wang et al., 1990), which was shown to react with CaO at a rate comparable to that of H2S (Borgwardt et al., 1984). Furthermore, we will assume that H2S is the only species reacting with CaO and that all other gaseous compounds are inert toward the sorbent. Hence, coal gas can be treated as a pseudobinary mixture of H2S and inerts. These two assumptions combined with the stoichiometry of reaction (1a), which guarantees equimolecular counterdiffusion inside (and outside) the pellets, will somewhat simplify the mathematical analysis.

Thermodynamic Equilibria for H2S Sorption Since reaction (1a) is endothermic while (1b) is exothermic, it follows that the minimum mole fraction of H2S at equilibrium is found at the calcination temperature of CaCO3 for a given coal-gas composition and total pressure. Figure 1 shows how the temperature, H2O and CO2 levels, at values typical of coal gas, affect the equilibrium H2S level that can be achieved using CaO or CaCO3 as sorbent. It is highly beneficial to operate near but somewhat above the calcination temperature (Fenouil and Lynn, 1995). For a given pressure and CO2 and H2O content in the gas, the equilibrium concentration of H2S at the calcination temperature of CaCO3 is given by:

yH2S ) 1.75 × 10-3yH2OyCO20.364P0.364

(2)

where P is expressed in bar. Equation 2 results from the expressions for the equilibrium constants for the limestone sulfidation and limestone calcination (Fenouil, 1995). For a coal gas containing 6.5% CO2 and 8% H2O, with yeq equal to 180 ppm H2S at 30 bar, a value of yout of 200 ppm H2S would be a practical design specification. This corresponds to 98% sulfur removal for a © 1996 American Chemical Society

Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996 1025

Figure 2. Diagramatic representation of the solid pellet. Figure 1. Effect of temperature, CO2, and H2O on H2S sorption from coal gas.

typical value for yin of 1% H2S. For the sorbent to be completely converted to CaS, limestone particles must first be calcined into CaO (Fenouil and Lynn, 1995). However, the rate of calcination of CaCO3 does not limit the kinetics of the reaction between CaO and H2S when the reaction temperature is maintained 30-50 °C above the calcination temperature of the CaCO3 (Fenouil and Lynn, 1995). The increased temperature does not raise the minimum level of H2S that can be achieved appreciably since the enthalpy of reaction (1a) is only about -65 kJ/mol.

Choice of a Kinetic Model for Gas-Solid Reactions A great number of models have been developed in recent years to describe the kinetics of gas-solid reactions. Some of them, such as the “unreacted shrinking core model” (Levenspiel, 1972), do not require specific knowledge of the internal structure of the reacting/ reacted solid. Other slightly more sophisticated models, such as the “grain model” (Szekely et al., 1976), require the knowledge of some physical parameters characterizing the internal structure of the solid (the specific surface area or the average pore size). Finally, the most sophisticated models include such considerations as pore-size distribution, change of porosity during the reaction, and pore plugging in the course of the reaction (Bhatia and Perlmutter, 1981; Yu and Sotirchos, 1987; Sotirchos and Zarkanatis, 1989; Froment and Bischoff, 1991, p. 127) and demand more extensive and precise information about the structure of the reacting solid. In the case of large particles of calcium-based sorbents for H2S sorption, CaO must first be produced in situ by calcination of CaCO3 prior to being reacted with H2S. Information such as pore-size distribution in the freshly formed CaO is difficult to predict with great precision, since slight variations in the calcination procedure can greatly influence the values of the specific surface area, porosity, and pore-size distribution in the final CaO product (Boynton, 1980; Borgwardt, 1989a,b; Rubiera et al., 1991; Fuertes et al., 1991, 1993). Fortunately, the shape of the breakthrough curve (in the case of a packed-bed configuration) was shown to be weakly influenced by the details built into the pellet model

(Efthimiadis and Sotirchos, 1993). Under realistic operating conditions, gas flow and mass-transfer considerations are often more significant that any slight modification in the local description of the reaction kinetics. Finally, another weakness of the more sophisticated kinetic models mentioned earlier is the impossibility of analytical solutions: they all require numerical solutions to obtain information on the reaction kinetics. In this paper, we will focus our attention on analytical solutions as often as possible because they provide a more useful, practical, and general description of the hot coal-gas clean-up unit and can readily be used for scale-up purposes. It has been shown that reaction (1a) proceeds via an unreacted shrinking core mechanism for large lime particles (diameter on the order of 1 mm) at temperatures above 700 °C (Fenouil and Lynn, 1995). However, below 700 °C and for small particle size (about 1 µm), experimental results published by Borgwardt et al. (1984) suggest that the grain model gives a more realistic description of the reaction kinetics (the time for the total reaction is a function of the initial specific surface area of CaO, regardless of the overall particle diameter). Consequently, the grain model was chosen for the description of the kinetics of reaction (1a) at the pellet level since it includes all four potential resistances to the reaction kinetics (mass transfer from the bulk gas to the solid surface, diffusion through the reacted layer of the pellet, diffusion through the reacted layer of the grain, and the chemical reaction itself). Nonetheless, since the simpler unreacted shrinking core model adequately describes the reaction kinetics for many cases, design equations based on the unreacted shrinking core model are also provided. A schematic description of a typical lime(stone) pellet is given in Figure 2, including some of the notation used in the derivation of the kinetic equations. The pellets are assumed to be spherical (radius R and porosity v) and composed of an agglomeration of smaller spherical particles (subsequently referred to as grains) of radius rg. It is also assumed that the pellets (and grains) do not change size in the course of the reaction and that the chemical reaction is reversible and first-order with respect to the gas-phase reactant (for both the forward and backward reactions) and is proportional to the surface of the unreacted core of the grain (or pellet for the unreacted shrinking core model). The overall kinetics of the reaction can then be found from a set of five ordinary first-order differential equations representing

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Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996

five material balances around the pellet or the grain:

-4πFsRc2

dRc ) 4πkgR2(C - Cs) dt

(3)

dRc dC ) 4πRc2De dt dRc

(4)

-4πFsRc2

drc dC -4πFsrc ) (1 - v)4πrc2Dg dt drc 2

(under the same pseudo-steady-state assumption for the solids) yields:

t ) τDP[1 - 3(1 - X)2/3 + 2(1 - X)] + τMT[X] + τR,SC[1 - (1 - X)1/3] (9) (unreacted shrinking core model), with:

τDP ) (5) τMT )

dRc ) 4πRc2ks(Ci - Ceq) dt

(6)

drc ) (1 - v)4πrc2ks(Cig - Ceq) dt

(7)

-4πFsRc2 -4πFsrc2

( (

(10)

Fs R 3(C - Ceq) kg

(11)

(

(12)

τR,SC )

τDG ) Equation 3 accounts for the diffusion of the reacting gas through the gas film surrounding the pellet (external mass-transfer resistance), eq 4 accounts for the diffusion of the reacting gas through the pellet’s product layer, eq 5 accounts for the diffusion through the grain’s product layer, eq 6 accounts for the chemical reaction at the interface between the completely reacted layer of the pellet and the partially (or completely) unreacted core of the pellet, and eq 7 accounts for the chemical reaction at the interface between the completely reacted product layer of the grain and the unreacted core of the grain. On the one hand, if the diffusivity of the gaseous reactants (or products) in the core of the reacting pellet is not significantly lower than that of the completely (or partially) reacted ash layer, eq 7 ought to be applied to describe the chemical reaction: the gases have the potential to reach the center of the pellet even if only a thin outside layer of the pellet is reacted. Equations 3-5 and 7 form the basic set that describe the grain model (Szekely et al., 1976). On the other hand, if the diffusivity in the core of the pellet is very significantly lower than that of the reacted layer, eq 5 must be applied instead of eq 7; virtually no gaseous reactant can reach the unreacted core. The kinetics described by eqs 3, 4, and 6 corresponds to the unreacted shrinking core model (Levenspiel, 1972). Finally, it should be noted that the intermediate case, in which the diffusivity of the gaseous reactants (or products) in the partially unreacted core is noticeably lower than that in the ash layer but not negligibly small, has been treated by Wen (1968) under a series of simplifying assumptions. His model is an extension of the unreacted shrinking core model in which a finite value of the diffusivity in the unreacted core is added. However, this model suffers from not including any description of the reaction kinetics at the grain level and will not be further considered. The set constituted by eqs 3-5 and 7 can be solved, assuming pseudo steady state for the solid, to give the time necessary to reach a given conversion of the solid:

t ) (τDP + τDG)[1 - 3(1 - X)2/3 + 2(1 - X)] + τMT[X] + τR[1 - (1 - X)1/3] (8) (grain model) and the set constituted by eqs 3, 4, and 6

τR )

(

)( ) )( ) )( )

Fs R2 6(C - Ceq) De

Fs R (C - Ceq) ks

)( )

Fs rg2 6(1 - v)(C - Ceq) Dg

(13)

(

(14)

)( )

Fs rg (1 - v)(C - Ceq) ks

where τDP is the characteristic time for diffusion through the pellet’s product layer, τMT the characteristic time for external mass transfer from the bulk gas to the surface of the pellet (film diffusion), τR,SC the characteristic time for chemical reaction at the interface between the unreacted core of the pellet and the reacted product layer, τDG the characteristic time for diffusion through the grain, and τR the characteristic time for chemical reaction at the interface between the unreacted core and the reacted layer of the grain. A derivation of the kinetic equations governing these two models can be found elsewhere (Levenspiel, 1972; Szekely et al., 1976). Here, we assume that the chemical reaction between CaO and H2S is reversible. This assumption has little impact on the reaction kinetics as long as the concentration of H2S in the coal gas is much larger than the equilibrium concentration (i.e., stays above about 1000 ppm) but becomes increasingly important as the level of H2S approaches its equilibrium value. Accounting for the reversibility of the reaction, combined with the assumption that the CaO/CaS interface is at thermodynamic equilibrium and that both the forward and backward reactions are first-order with respect to the partial pressure of H2S and H2O, respectively, leads to the conclusion that the overall reaction rate is proportional to PH2S interface - PH2S equilibrium rather than to PH2S interface alone. This explains the presence of the CH2S - CH2S equilibrium terms, rather than simply CH2S, in the expressions of the characteristic time constants defined in eqs 10-14. Intrinsic Kinetic Data for the Lime(stone)/H2S System Table 1 summarizes the necessary kinetic parameters needed to estimate the overall reaction rate for lime(stone) particles having a diameter of 1 mm. These data were primarily obtained from the experiments presented in Fenouil and Lynn (1995) and from Borgwardt et al. (1984). The values of rg are based on a specific surface area of 10 m2/g (high-temperature reactions) to 50 m2/g (low-temperature reactions) for lime (Fuertes et al., 1991; Rubiera et al., 1991) and a specific surface area

Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996 1027 Table 1. Summary of the Characteristics and Kinetic Parameters for the Reaction between H2S and Lime or Limestone (See Text for More Details) rg (m) De (m2/s) Dg (m2/s) ks (m/s) Fsorbent,pellet (mol/m3 pellet) Fsorbent,grain (mol/m3 grain)

lime (CaO)

limestone (CaCO3)

0.2 to 1 × 10-7 ∼5 × 10-6 ∼10 exp(-31170/T) .1010 exp(-31170/T) 25 000 59 300

4.1 × 10-6 ∼5 × 10-6 ∼0.1 exp(-31170/T) 5.2 × 103 exp(-19700/T) 25 000 27 100

Table 2. Predicted Time for Complete Conversion for the Lime(stone) Particles to CaS, Assuming That the Reaction Kinetics Is Solely Controlled by Pore Diffusion through the Entire Pellet (τDP), by Diffusion through the Grain (τDG), or by the Chemical Kinetics (τR)a lime (CaO)

limestone (CaCO3)

600 °C

900 °C

600 °C

900 °C

∼103 ∼103 ,103

∼103 ∼1 ,1

∼103 ∼1010 ∼105

∼103 ∼106 ∼103

τDP (s) τDG (s) τR (s) a

These estimates are based on the data presented in Table 1 for about 2% H2S in the gas phase (1 bar total pressure) and for a sorbent radius of 0.5 mm. See text for more details.

m2/g

Table 3. Predicted Time to Reach 5% Conversion to CaS for the Limestone Particles Assuming That the Reaction Kinetics Is Solely Controlled by Pore Diffusion through the Entire Pellet, by Diffusion through the Grain, or by the Chemical Kineticsa

of 0.25 for limestone (Fenouil et al., 1994). The value of De in Table 1, which holds for the 600-900 °C range, was determined experimentally for CaO (Fenouil and Lynn, 1995) and calculated for CaCO3 using the following equation: De ) (v/τp)(D-1 + DK-1)-1, assuming 8% porosity (v) (Fenouil et al., 1994) and 1.73 tortuosity factor (τp) (Froment and Bischoff, 1990, p 146) and using the Fuller et al. (Reid et al., 1987) correlation to determine the molecular diffusivity (D) of H2S in the gas mixture. The value of the Knudsen diffusivity, DK, is given by Fenouil and Lynn (1995). The value of Dg relies on experimental results published by Borgwardt et al. (1984; see their Figures 7 and 9) for the diffusion of H2S into CaS. The values of Dg are very sensitive to the composition of the gas phase as well as the presence of impurities in the CaS layer. Referring to the kinetic model developed earlier for the reaction between limestone and H2S under noncalcining conditions (Fenouil and Lynn, 1995), one expects Dg to have the same activation energy but a lower preexponential factor for the reaction between limestone and H2S than for the reaction between lime and H2S; the reaction is completed more rapidly with lime, thus reducing the time of exposure of CaS to hot coal gas and consequently its loss of surface area. Hence, the solidstate diffusion character of Dg (relative to its Knudsen diffusion part) is stronger in the case of the reaction between CaCO3 and H2S than with the reaction between CaO and H2S, resulting in an increase of the preexponential factor for Dg for the reaction with limestone. For CaO, the value of ks is supported by experimental results published by Borgwardt et al. (1984), who found no limitation due to chemical kinetics. For CaCO3, this value has been calculated from the experimental reaction rate between 560 and 660 °C (Fenouil and Lynn, 1995). Finally, the values of the molar densities are derived by assuming an 8% initial porosity for the limestone, with negligible shrinkage upon calcination, and by taking the values for the true density of 3.32 g/cm3 for CaO and 2.71 g/cm3 for CaCO3 (Borgwardt et al., 1984). Table 2 provides estimates for the characteristic times τDP, τDG, and τR for the kinetics of reactions (1a) and (1b). These characteristic times allow us to identify the rate-limiting step for the overall reaction kinetics and

pellet diffusion control (s) grain diffusion control (s) chemical reaction control (s)

600 °C

900 °C

∼1 e103 ∼103

∼1 ∼103 ∼10

a These estimates are based on the data presented in Table 1 for about 2% H2S in the gas phase (1 bar total pressure) and for a sorbent radius of 0.5 mm. See text for more details.

to estimate the time required for complete conversion of the sorbent particles (1 mm diameter) at 600 and 900 °C. For instance, it is predicted that the kinetics of the reaction between CaO and H2S is primarily controlled by the diffusion of H2S (or H2O) through the pores of the particle since τDP is significantly larger than τDG and τR at 900 °C. This was experimentally confirmed by the presence of a sharp interface between the unreacted CaO core and the CaS product layer observed in SEM photographs of millimeter-size particles of lime reacted with simulated coal gas at 900 °C (Fenouil and Lynn, 1995). Table 2 also predicts that the diffusion through the individual grains of the lime particle becomes ratelimiting for the reaction kinetics when the temperature is lowered to the vicinity of 600 °C. That fact was also experimentally confirmed: the sharpness of the CaO/ CaS interface diminishes as the temperature is brought below about 650 °C and disappears when the reaction temperature is further lowered. The characteristic time for pore diffusion is of the same order of magnitude for both reactions. However, τDP is smaller than both τDG and τR in the case of the reaction involving limestone. Thus, even though pore diffusion is rate-limiting for the CaO/H2S reaction, such is not the case for the CaCO3/H2S reaction. Instead, grain diffusion is rate-limiting at all temperatures above 660 °C, as well as very early in the reaction (below 5% conversion), and a period measured in days is required for complete conversion to CaS. The key difference between CaO and CaCO3 is the average size of the grains. Limestone particles have low specific surface areas (less than 0.5 m2/g) and typical grain sizes, rg, of the order of 5 µm. In contrast, lime particles obtained by calcining limestone have specific surface areas ranging from 10 to 50 m2/g depending on the calcination conditions and average grain sizes ranging from only 0.02 to 0.1 µm. This substantial difference in the value of rg explains the large difference in the values of τDG for CaO and CaCO3, since this value is proportional to rg2. It may be assumed that the net effect of the calcination of the limestone particles is simply to decrease the average grain size of the sorbent sufficiently so that grain diffusion does not limit the reaction rate at high temperatures (600 °C and above). Table 3 focuses on the initial reaction rate of the limestone with H2S. The value of the time for graindiffusion control at 600 °C shows that, below 660 °C,

1028


CaS does not recrystallize and the metastable, flat CaS layer (in which S2- ions have replaced CO32- ions) is sufficiently permeable to introduce no limitation to the overall reaction kinetics up to 5% conversion to CaS (Fenouil and Lynn, 1995). The first line in Table 3 suggests that pore diffusion throughout the particle is not rate-limiting. This was experimentally confirmed by the even distribution of sulfur ions throughout the entire limestone pellets exposed to simulated coal gas (Fenouil and Lynn, 1995). Table 3 also predicts that the chemical kinetics is rate-limiting at 600 °C at the beginning of the reaction (up to 5% conversion to CaS) but that the rate-limiting step switches to grain diffusion at higher temperatures (>900 °C). These two predictions have been confirmed by our experiments (Fenouil and Lynn, 1995). Moreover, the right orders of magnitude for the time necessary to reach 5% conversion are correctly predicted at 600 and 900 °C. Estimation of Physical Parameters for Sorption Systems Viscosity. The viscosity of a mixture of n gases, ηm, can be estimated using Wilke’s method, which was shown to be quite accurate for CO2/H2 systems (Gururaja et al., cited in Reid et al., 1987, p 410) combined with the Herning and Zipperer approximation (Reid et al., 1987, pp 407-410): i)n

ηm )

yiηiMi0.5 ∑ i)1 i)n

(15)

yiMi ∑ i)1

0.5

in which ηm is the viscosity of the mixture (N s m-2), ηi is the viscosity of pure component i (N s/m2), yi is the mole fraction of component i, and Mi is the molecular weight of component i (g/mol). The pure-component viscosity is given by the Chapman-Enskog equation (Reid et al., 1987, pp 392-395):

ηi )

8.44 × 10-11(MiT)0.5 σi2Ωv,i

(16)

with:

Table 4. Data Used in the Determination of Several Physical Properties of the Gasa H2S H2 N2 CO CO2 H2O CH4

(∑ ) j)n

Dim )

yj

j)1,j*iDij

-1

(18)

Pc (bar)

(Σv)i

i/k (K)

σi (Å)

373.2 33.2 126.2 132.9 304.1 647.3 190.4

89.4 13.0 33.9 35.0 73.8 221.2 46.0

27.52 6.12 18.5 18.0 26.9 13.1 25.14

301.1 59.7 71.4 91.7 195.2 809.1 148.6

3.623 2.827 3.798 3.690 3.941 2.641 3.758

a References: P and T data from Reid et al., 1987, pp 656c c 732. (Σv)i data from Reid et al., 1987, p 588. i/k and σi data from Reid et al., 1987, pp 733-734.

The binary diffusivities were calculated using the Fuller et al. method (Reid et al., 1987, p 588), which was shown to be one of the most accurate correlations for predicting binary diffusivities over a large range of conditions:

Dij )

4.52 × 10-4T1.75 P[2(Mi-1 + Mj-1)-1]0.5[(Σv)i1/3 + (Σv)j1/3]2

(19)

where Dij is the binary diffusion coefficient of i and j (m2/s), P is the total pressure (Pa), and (Σv)i is the molecular diffusion volume of species i (see Table 4). For large values of the total pressure, eq 19 needs to be corrected since Dij is not exactly proportional to the inverse of the total pressure. However, Takahashi’s correlation (Reid et al., 1987, p 592) predicts that no significant correction (i.e., less than 5%) need be applied when the reduced temperature TR (T/Tcrit) is larger than 3 while the reduced pressure PR (P/Pcrit) is lower than 3. In the 1050-1400 K, 1-30 bar range of interest all the species present in coal gas meet this constraint, except H2O for which TR is close to 2. However, since PR is well under 3 for H2O, no correction is needed. Mass-Transfer Coefficient. (i) Packed Bed. Numerous correlations for mass transfer in packed beds have been suggested in recent years (Froment and Bischoff, 1990, pp 128-129; Szekely et al., 1976, pp 266-267). Most of these correlations provide comparable predictions when the Reynolds number varies from 1 to 100, the likely range to be encountered in the coal-gas clean-up unit. A practical correlation, valid for packed beds with identical spheres and a void fraction of 0.37, has been developed by Hougen and Yoshida et al. (Froment and Bischoff, 1990, p 128) and can be conveniently expressed as follows:

Ωv,i ) 1.16145(Ti*)-0.14874 + 0.52487 × exp(-0.77320Ti*) + 2.16178 exp(-2.43787Ti*) (17) for Ti* ) (kT)/i between 0.3 and 100, and where Ωv,i is the collision integral for species i, k is Boltzmann’s constant (1.38048 × 10-23 J K-1), σi is the hard-sphere diameter of the gas molecule (m), i is a characteristic Lennard-Jones energy of species i (see Table 4) (J), and T is the absolute temperature (K). Diffusivity. The diffusivity of H2S (i) in the gas mixture, Dim, was estimated using Blanc’s law (Reid et al., 1987, p 597), which is the asymptotic form of the Stephan-Maxwell equation for dilute systems (since the concentration of H2S in coal gas is only of the order of 1%):

Tc (K)

Sh ) 1.17Pe0.49Sc-0.16

if Re < 95

Sh ) 0.74Pe0.59Sc-0.26

if Re > 95

(20)

uGR uGR kgR ν ; Pe ) ; Sc ) ; Re ) D D D ν

(21)

where:

Sh )

and where ν is it the viscosity of the coal-gas mixture and D represents the value of the molecular diffusivity of H2S in the gas mixture. The preceding expression is of practical importance since it is expressed in terms of the Pećlet number, Pe, which can be easily determined. Further, the Schmidt number, Sc, will always be close to unity, and its influence on the Sherwood number, Sh, will be negligible. (ii) Moving Bed. The exact nature of the masstransfer correlations in moving beds is strongly influ-


enced by the nature of the flow of the gas and solid phases. We will use, as a first approximation, the same correlations as for packed beds. It should, however, be kept in mind that flow maldistributions that could be the result of the pellet movements may affect the overall mass-transfer coefficient. In fact, it has long been observed that gas/solid heat-transfer coefficients are much smaller in moving beds than those predicted by standard packed-bed correlations, sometimes by as much as a factor of 15 (Paterson et al., 1992). It has recently been verified that gas channeling, caused by slight inhomogeneities in the structure of the moving bed (particularly the fluctuations in the bed voidage), can account for the lower values of the heat-transfer coefficients (Crawshaw et al., 1993). These inhomogeneities should affect the values of the gas/solid masstransfer coefficients in a similar way. (iii) Entrained Bed. The loading of the solids in the gas will be sufficiently small (less than 1 part solid for 1000 parts of gas by volume) that the Froessling correlation for a single sphere moving through an infinite volume of fluid can be used to estimate the Sherwood number (valid for low concentrations of the reacting gas):

( )

uoR Sh ) 1 + 0.424 D

0.5

Sc

-0.17

) 1 + 0.424Pe00.5Sc-0.17 (22)

where u0 is the relative velocity between the fluid and the solid. (iv) Fluidized Bed. As we will see in the FluidizedBed Design section, we do not have to worry about masstransfer limitations from the bubble phase to the emulsion phase when the solid particles have a diameter larger than 0.5 mm. In the emulsion phase, the actual value of the Sherwood number was found to be bounded by those found for packed beds and for a single particle in an infinite fluid medium, respectively described by eqs 20 and 22 (see Kunii and Levenspiel, 1991, pp 258260). More precise estimates of Sh will require a better knowledge of the specifics of the gas/solid system under consideration. Under the expected operating conditions of the high-temperature coal-gas cleanup, Sh will stay close to unity in all of the four reactor configurations we consider. Pressure Drop. (i) Moving and Packed Beds. The pressure drop across the solid bed will be estimated using the well-accepted Ergun equation (Szekely et al., 1976):

[

]( )( )

75(1 - )ν FguG ∆P ) 1.75 + L RuG 2R

2

1- 3

particles (3-mm glass spheres) is actually smaller than in an equivalent packed bed (Paterson et al., 1992). The somewhat higher value of, and lesser spatial regularity of, the voidage in a moving bed (compared to a packed bed) has been suggested as the cause for the difference in the pressure drops. (ii) Fluidized Bed. The pressure drop per unit length of bed can be estimated with eq 23; replace uG by the value of the minimum fluidization velocity, umf, given by eq 24 (below). Minimum Fluidization Velocity. The superficial velocity of the gas must be below the minimum fluidization velocity, umf, of the solids if the desulfurization unit is to be operated as a moving bed of particles with upward flow of gas. Assuming that the Ergun equation appropriately describes the pressure drop across the bed, umf is given by:

( )[(

umf ) 21.43(1 - )

( ) )( )) ]

1 + 2.489 × 10-3

(

Fp - Fg gR Fg ν2

3

3 × (1 - )2

0.5

- 1 (24)

where umf is the minimum fluidization velocity (m/s), Fp is the density of the solid (kg/m3), and g is the standard acceleration of gravity (about 9.81 m/s2). Other Physical Data. Some simplifications can be made without significant loss in the generality of the analysis of high-temperature H2S removal with calciumbased sorbents: (i) is generally close to 0.4 for fixed or moving beds. (ii) For coarse particles in fluidized beds, mf is generally very close to 0.5, very slightly higher at higher pressures (1-4%), but is not affected by the temperature (Kunii and Levenspiel, 1991, p 74). (iii) Fp is approximately equal to 1500 kg/m3 and Fs(1 - v) to 25 000 mol/m3 for lime derived from large limestone particles. (iv) The calcination temperature for calcium carbonate, at which H2S removal is optimal, will range between 800 and 1000 °C for most coal gases. Hence, estimating the value of the physical characteristics of the gas phase at an average temperature of 900 °C, as is done in the remaining portion of this paper, will not introduce serious error. (v) The average molecular weight of a typical coal gas (Mm) does not differ much from 20 g/mol. It follows that Fg can be approximated by:

Fg ≈ (23)

where ∆P is the frictional pressure drop across the bed (Pa), L is the bed length (m), ν is the kinematic viscosity of the gas mixture and is equal to ηm/Fg (m2/s), Fg is the density of the gas mixture (kg/m3), and is the void fraction in the bed. The Ergun equation is valid for spatially uniform beds in the absence of large variations of pressure and temperature so that the fluid can be considered incompressible. These conditions generally hold for H2S sorption. Equation 23 was also shown to be accurate even at high temperatures and pressures (Kunii and Levenspiel, 1991, p 74). Recently, it has been clearly demonstrated that the pressure drop in countercurrent gas flow (0.14-0.84 m/s) through a moving bed of

ν R

MmP P ≈ 0.0024 ≈ 2.0 × 10-6P RgT T

()

(25)

(vi) Approximating the value of ν by 50 P-1 results in the following expression for the minimum fluidization velocity:

( )[(

umf ≈ 10.7

ν R

)

]

R3T 0.5 1 + 7610 2 -1 ≈ Pν 535 [(1 + 3571PR3)0.5 - 1] (26) RP

Moving-Bed Design. Using the kinetic expressions obtained in eqs 3-7, it is possible to model the performance of a moving bed of lime or limestone with the help of some simplifying assumptions. Assumptions: (1) isothermal moving bed; (2) steady state, N and uG are constant; (3) plug-flow conditions

1030


for the gas phase and the solid phase with no axial dispersion; i.e., one-dimensional model along the z-axis (no radial dispersion); (4) the bed void fraction, , is constant along the bed; (5) the grain model or the unreacted shrinking core model for spherical particles (and grains) describes the sulfidation kinetics at the particle level; (6) the chemical reaction rate is first-order with respect to the gaseous reactant concentration; (7) the solid particles all have the same diameter; (8) the gas is ideal. The calcination of CaCO3 is endothermic (∆H ≈ 100 kJ/mol at 900 °C) and the sulfidation of the lime is exothermic (∆H ≈ -65 kJ/mol at 900 °C). However, the ratio between the heat released by these reactions and the sensible heat of the gas is small throughout the bed because H2S is only about 1% of the coal gas, and no significant axial temperature gradient results from the reactions. The design of the sulfur removal unit must preclude severe heat losses to avoid a sizable radial temperature gradient. Since the kinetics of the reaction between large pellets of lime and hydrogen sulfide is controlled by a combination of external mass transfer and internal pore diffusion, both of which are relatively insensitive to temperature, we may neglect any temperature effect on the overall reaction kinetics (assumption 1). Assumption 8 is only used to relate partial pressures, concentrations, and mole fractions. What may be considered the most critical assumption is the flow patterns of both the gas and solid phases in the reactor (assumption 3). These patterns will be affected not only by the shape of the reactor but also by random maldistribution of the gas and solids within the bed. A priori predictions of these nonidealities cannot be made without prior knowledge of the exact form and flow configuration of the reactor, and even with this information they remain quite difficult to compute. At this stage of the design, we account for these uncertainties through an “efficiency” factor in the mass-transfer correlations. Axial dispersion is ignored for the following reasons: first, the convective maldistribution mentioned earlier will probably have a much larger impact on the overall concentration profile than will axial diffusion. Second, a large number of experimental studies on packed beds consistently show that the influence of axial dispersion remains negligible if the total length of the bed is at least 50 times larger than the size of the individual solid pellets, which will be the case for the configurations that will be further considered (Froment and Bischoff, 1990, p 447). Design Equations for Moving Beds. A differential mass balance on the sulfur atoms yields:

uG

∂C ∂X + (1 - )Fs )0 ∂z ∂t

(27)

One can then define the following dimensionless numbers:

[ (

Sh′′ ) Sh′ 1 +

D 2(τ + τ )( ) (D )] ) τ

rg 1 1 - v R

Sh′ )

e

Pe′ )

( )

rg 1 1 - v R

2

De Dg

in the case of a cocurrent configuration.

6(τDP + τDG) τF

( )( )

(33)

(34)

][ ( )( ) ( )]

rg 2 De R 1 (1 - v) 1 + ) rg 1 - v R Dg 6(τDP + τDG) (35) τR Da′ )

ksR 6τDP ) De τR,SC

(36)

where τF is a characteristic flow time in the moving bed defined as:

τF )

( )[

]

R (1 - )FS uG C - Ceq

(37)

Sh′ (Sh′′), Pe′ (Pe′′), and Da′ (Da′′) are respectively modified Sherwood (also called Biot), Pećlet, and Damko¨hler numbers. The combination of eq 8 or 9 and eq 27 yields:

(Y - Yeq) dY + )0 dZ Pe′′ 1 1 + (Y-1/3 - 1) + (Y-2/3) 3 Sh′′ Da′′

[

(38)

]

for the grain model (for the unreacted shrinking core model Pe′′, Sh′′, and Da′′ are replaced by Pe′, Sh′, and Da′, respectively), where:

Z ) z/R

(39)

(

)

(

)

y - yin yout - yin for countercurrent flow (40)

Y ) 1 - X ) (1 - Xout) + (Xout - Xin)

y - yin yout - yin for cocurrent flow (41)

(28)

Y ) 1 - X ) (1 - Xin) - (Xout - Xin)

(29)

with the initial condition:

in the case of a countercurrent configuration, and:

N(X - Xin) ) uG(Cin - C)

)

(32)

6τDP uGR D 1 ) Pe ) De 1 - τF (1 - )De

or, at a given z in the bed:

N(Xout - X) ) uG(Cin - C)

(31)

MT

2τDP kgR D ) Sh ) De De τMT

[( )

Da′′ ) Da′

DG)

DP

g

[ ( )( ) ( )]

Pe′′ ) Pe′ 1 +

An overall mass-balance over the total bed length gives:

N(Xout - Xin) ) uG(Cin - Cout)

2

Y ) Yin ) 1 - Xout

at Z ) 0 for countercurrent flow (42)

(30) Y ) Yin ) 1 - Xin

at Z ) 0 for cocurrent flow (43)


Results for Moving Beds. Equation 38 can be solved analytically to give (see Appendix):

Z)

[(

)

uGR uGrg rg 1 + I(Y) + Dg R(1 - v) 3(1 - ) De uG uG rg J(Y) + K(Y) (44) kg ks R(1 - v)

( )(

( )

( )

) ]

( ) ]

Pe′′ 1 1 I(Y) + J(Y) + K(Y) (45) 3 Sh′′ Da′′ 2(τDP + τDG) τMT τR Z) I(Y) + J(Y) + K(Y) (46) τF τF 3τF

[

Z)

[

]

[ ]

[ ]

(grain model), or

Z)

[( )

( )

( ) ]

uGR uG uG 1 I(Y) + J(Y) + K(Y) kg ks 3(1 - ) De

(47)

( )

(48)

( ) ]

Pe′ 1 1 I(Y) + J(Y) + K(Y) 3 Sh′ Da′ 2τDP τMT τR,SC Z) I(Y) + J(Y) + K(Y) τF τF 3τF Z)

[

[ ]

[ ]

[ ]

Figure 3. Gas-phase concentration profile of H2S in a countercurrent moving bed of calcined limestone particles (eqs 40 and 44).

(49)

(unreacted shrinking core mechanism), with:

I(Y) ) F1(Y) + 3Yeq-1/3[F2(Y) + F3(Y)]

(50)

J(Y) ) -F1(Y)

(51)

K(Y) ) 3Yeq-2/3[F2(Y) + F4(Y)]

(52)

and with:

[ [

F1(Y) ) ln

(53)

(Ain3 - 1)A3

]

(54)

2Ain + 1 1 2A + 1 tan-1 - tan-1 x3 x3 x3

(55)

F2(Y) ) -

F3(Y) )

]

(A3 - 1)Ain3

F4(Y) ) -

(1 - A)3(1 - Ain3) 1 ln 6 (1 - A3)(1 - Ain)3

[ ( ) ( )] [ ( ) ( )] ( ) ( )

2Ain - 1 1 2A - 1 tan-1 - tan-1 x3 x3 x3 A)

Yeq Y

1/3

and Ain )

Yeq Yin

(56)

1/3

(57)

The length of the bed (and the H2S profile) can be expressed as a function of only five parameters: Sh′′, Pe′′, Da′′, yeq, and R. It is interesting to note that, for a given Sh′′ and Da′′ number as well as yeq (which is fixed by the thermodynamics of the lime/H2S reaction), the bed length is directly proportional to the product R × Pe′′. Xin and Cin (or yin) are fixed by the process conditions, i.e., by the concentration of H2S in the coal gas to be treated and by the degree of conversion specified for the sorbent entering the bed. We are free to chose Xout (which will determine the degree of sorbent utilization) and yout (which will determine the degree of H2S removal from the coal gas), provided that yout is larger than yeq.

Figure 4. Gas-phase concentration profile of H2S in a cocurrent moving bed of calcined limestone particles (eqs 41 and 44).

The maximum value of the superficial velocity uG is the fluidization velocity of the bed if the gas is flowing vertically upward. The solid flow rate is then fixed by the overall mass balance (eq 29). Equations 44-57 can then be used to obtain the H2S profile within the moving bed. The total bed length can be obtained by inserting Yout for Y in eqs 44-46. Examples of Moving Beds. Figures 3 and 4 show two examples of calculations using reasonable values of the various physical parameters of CaO/H2S in the coal-gas system. The Sherwood number is used as a free parameter to demonstrate the influence that external mass transfer can have on the system. Table 5a provides ranges of possible values for Sh′′, Pe′′, and Da′′ for the CaO/H2S system in high-temperature coal gas (600-1000 °C) for the range of superficial velocities given in Table 5b. The lower and upper bounds of these estimates have been conservatively estimated, and most actual values should be close to the middle of the proposed range. Table 5a shows that Da′′ remains so large, especially when the radius of the sorbent is larger than 0.5 mm, that it hardly influences the overall reaction kinetics. A relatively conservative value of Da′′ ) 103 was applied for Figures 3 and 4. As expected, the required length is always larger with cocurrent flow of gas and solid than in the countercurrent configuration. This difference is more pronounced when the reaction is controlled by the chemical kinetics, or by internal diffusion through the sorbent, since the reaction rate is influenced not only by the concentration

1032


Table 5. Ranges of Values as a Function of the Pellet Radius and the Total Pressure for the Moving- and Packed-Bed Configurations R ) 10-4 m 1 bar 30 bar

1 bar 30 bar a

R ) 10-3 m

R ) 10-2 m

(a) Values For Pe′′, Sh′′, and Da′′ for Reaction (1a)a 10-1 < Pe′′ < 1 10 < Pe′′ < 103 1 < Sh′′ < 102 1 < Sh′′ < 104 10 < Da′′ 103 < Da′′ 1 < Pe′′ < 10 5 × 10 < Pe′′ < 5 × 103 5 < Sh′′ < 5 × 102 1 < Sh′′ < 104 102 < Da′′ 104 < Da′′

5 × 102 < Pe′′ < 5 × 104 10 < Sh′′ < 105 105 < Da′′ 103 < Pe′′ < 105 10 < Sh′′ < 105 106 < Da′′

(b) Design Values for the Superficial Velocity, uG (m/s) 0.001-0.02 0.1-1 0.001-0.02 0.03-0.3

0.5-5 0.1-1

These estimates are based on the values of uG given in Table 5b.

of H2S in the coal gas but also by the degree of conversion of the solid. Conversely, the profile of H2S in the bed becomes identical for the two configurations as the Sherwood number tends toward zero. In this case, the degree of conversion of the solid has no effect on the overall reaction rate because gas-phase diffusion is the limiting factor. Using the values presented in parts a and b of Table 5, the required length of the bed appears to be somewhere between a few centimeters to a few meters, depending chiefly on the radius of the sorbent pellet and the superficial velocity of the gas. Most of the sorption occurs within the first half of the bed, the second half being used essentially to “polish” the gas. Packed-Bed Design. Assumptions: (1) isothermal packed bed; (2) uG is constant; (3-8) same as for the Moving-Bed Design section. Design Equations for Packed Beds. A sulfur mass balance on a differential cross-section of the bed yields:

uG

∂C ∂C ∂X + (1 - )Fs + )0 ∂z ∂t ∂t

(58)

It is necessary to keep the third term in the left-hand side of eq 58 since steady state is never reached in a packed bed. This term is small compared to the other two and could be neglected as a first approximation (Szekely et al., 1976). However, the elimination of this term does not really simplify the calculations for the determination of the breakthrough curve so we kept it. At the pellet level the reaction kinetics can be described by:

E′(X) )

[

[ ][ 3De

FsR

2

∂X ) E′(X) [C - Ceq] ∂t

1+

( ) (DD )]

rg 1 (1 - v) R

2

e

(59)

(63)

in eq 58 to replace z and t. The quantity v0 is the velocity of the constant pattern and is given by an overall sulfur mass balance over the bed:

v0 )

uG(Cin - Ceq) (1 - )Fs

(64)

if one assumes that the solid (CaO) can be completely converted to CaS. With this change of variable, eq 58 then becomes:

[(

1-

uG v0

)]

dX dC + [Fs(1 - )] )0 dτ dτ

(65)

(since ∂X/∂t ) dX/dτ), with the following boundary conditions:

for τ ) τDP + τMT + C ) Cin and X ) 1 τR,SC (unreacted shrinking core model) (66) C ) Cin and X ) 1

for τ ) τDP + τDG + τMT + τR (grain model) (67)

Integration of eq 65 gives:

×

g

]

-1

(60)

(grain model) or by:

E(X) ) 3De 1 1 + (1 - X)-1/3 - 1 + (1 - X)-2/3 2 Sh′ Da′ Fs R

[(

1-

)]

uG [C - Cin] + [Fs(1 - )][X - 1] ) 0 v0

or, using eq 64:

∂X ) E(X) [C - Ceq] ∂t

(unreacted shrinking core model).

τ ) t - z/v0

-1

1 1 + (1 - X)-1/3 - 1 + (1 - X)-2/3 Sh′′ Da′′

[ ][

Equations 58 and 59 (or eqs 58 and 61) constitute a system of hyperbolic partial differential equations that must be solved numerically. Several resolution methods have been proposed (Park et al., 1984). However, the equation of the breakthrough curve can be determined analytically if a “constant pattern” is reached in the packed bed. In this case, it is possible to perform the following change of variable:

(61)

]

-1

(62)

[ ]

v0 uG C - Ceq ) (Cin - Ceq) v0 1uG

(68)

X-

(69)

which gives the relationship between C and X at any point of the breakthrough curve.


The combination of eqs 59 and 65 yields:

(Y* - Y* dY* eq) + )0 dZ* Pe′′ 1 1 + (Y*-1/3 - 1) + (Y*-2/3) 3 Sh′′ Da′′ (70)

[

]

(grain model), and

(Y* - Y* eq) dY* + )0 dZ* Pe′′ 1 1 + (Y*-1/3 - 1) + (Y*-2/3) 3 Sh′ Da′ (71)

[

]

partial pressure of H2S in the coal gas. Moreover, if the hypotheses of no-slip and negligible axial diffusion of H2S hold throughout the reaction zone, the entrainedflow reactor can be modeled as a small batch reactor moving at the plug-flow gas velocity (i.e., uG/). The first design equation, mass balance on the sulfur atoms, can be written:

( (

)( )(

) )

v0t v0 1R uG

-1

≈Z-

(Cin - Cout) ) Fs(1 - )(Xout - Xin) v0t R

(72)

v0 y - yin y - yin Y* ) 1 - X ) 1 ≈ uG yeq - yin yeq - yin

at Z* ) 0

(74)

It can be shown that a constant pattern will be established as soon as the pellet at the very front of the packed bed is completely converted. Complete conversion of these front pellets is obtained after τDP + τDG + τMT + τR seconds (grain model) or τDP + τMT + τR,SC seconds (unreacted shrinking core model), in which C is taken at Cin in the expression of the τ’s. Results for Packed Beds. Equation 70 (or eq 71) can then be solved analytically (see the appendix):

( )

( )

1 1 Pe′′ I(Y*) + J(Y*) + K(Y*) Z* ) 3 Sh′′ Da′′

[

]

Z* )

( )

( )

1 1 Pe′ I(Y*) + J(Y*) + K(Y*) 3 Sh′ Da′

[

(

)

Cin - Cout dX dC + )0 dt Xout - Xin dt

(Y - Yeq) dY + )0 Z 1 Pe′′ 1 d + (Y-1/3 - 1) + (Y-2/3) 3 Sh′′ Da′′

()

[

]

(76)

(unreacted shrinking core model), with I(Y*), J(Y*), and K(Y*) given by eqs 50-52. As anticipated on purely physical grounds, the shape of the breakthrough curve is very similar to the concentration profile obtained for a countercurrent moving bed of sorbent pellets for which Xin is equal to zero, Xout is 1, and yout is yeq. Figure 3 can then be used for an approximate breakthrough curve. Finally, eq 73 can be used to obtain the conversion profile for the sorbent throughout the packed bed once a constant pattern has been established. Entrained-Flow (or Transport) Reactor Design. Assumptions: (1) isothermal entrained flow; (2) steady state, uG and N are constant; (3) no-slip reactor conditions (i.e., the solid particles move at the same velocity as the gas phase); (4) no axial dispersion, i.e., onedimensional model along the z-axis (no radial dispersion); (5-8) same as for the Moving-Bed Design section. Design Equations for Entrained-Flow Reactors. It should be recognized at the outset that in such a system the solid loading (i.e., the volume fraction of the solid within the reactor) is extremely small because the molar density of the solid reactant is so much higher than that of the H2S in the gas phase; the solid loading will vary between 10-5 and 10-3 v/v depending on the

(80)

if the grain model is used in the description of the reaction kinetics, and:

(Y - Yeq) dY + )0 Z 1 Pe′ 1 d + (Y-1/3 - 1) + (Y-2/3) 3 Sh′ Da′

()

[

(81)

]

if the unreacted shrinking core model is used, with:

Y ) (1 - Xin) - (Xout - Xin)

]

(79)

The second design equation describing the kinetics of the reaction between H2S and the solid sorbent is given by eqs 59-62. The combination of eqs 59 and 79 yields:

(75)

(grain model), or:

(78)

Combining eqs 77 and 78, one obtains:

(73)

(since uG is significantly larger than vo) with the following boundary condition:

Y* ) Y* in ) 0

(77)

in which is fixed by an overall mass balance on the batch reactor:

(unreacted shrinking core model), where:

Z* ) Z -

dC dX + (1 - )Fs )0 dt dt

(

)

y - yin )1-X yout - yin

(

Yeq ) (1 - Xin) - (Xout - Xin)

)

yeq - yin yout - yin

(82)

(83)

and with the initial condition:

Y ) Yin ) 1 - Xin

at Z ) 0

(84)

Results for Entrained-Flow Reactors. Equations 80-84 are very similar to those for the description of a cocurrent moving bed of sorbent. They can be solved analytically in the same manner to give (see the appendix):

( )

( ) ]

(85)

( ) ]

(86)

Z Pe′′ 1 1 ) I(Y) + J(Y) + K(Y) 3 Sh′′ Da′′

[

(for the grain model), or:

( )

1 1 Z Pe′ ) I(Y) + J(Y) + K(Y) 3 Sh′ Da′

[

(for the unreacted shrinking core model) with I(Y), J(Y), and K(Y) given in eqs 50-52. Figure 4 can then be used to describe the reaction profile within the reactor, provided Z/ is substituted for Z. However, as we mentioned earlier, is here in excess of 0.99, and Figure 4, neglecting the 1/ correction on Z, can be used directly. It is also easy to translate Z (spatial variable) into a “contact time” by using the

1034


following expression:

tcontact )

z ZR ZR ) ≈ uG uG uG

(87)

Even if we have the same mathematical structure in the description of a cocurrent moving bed and an entrained bed reactor, fundamental physical differences exist between the two reactor configurations. First, the mass-transfer correlations used to determine the values of Sh′ and Sh′′ will differ. Most importantly, the value of 1 - appearing in the definition of Pe′ and Pe′′ is of the order of 10-6-10-4 for an entrained-flow reactor compared to 0.5-0.6 for a moving-bed reactor. More precisely, the molar density of the lime(stone) particles is about 25 000 mol/m3, whereas in a coal gas containing 1% H2S, the molar density of H2S is about 0.10 mol/m3 at 1 bar and 3.0 mol/m3 at 30 bar. From the 1:1 Cato-S ratio required by the stoichiometry of reaction (1), it then follows that 1 - is about 4.0 × 10-6 at 1 bar and 1.2 × 10-4 at 30 bar, which corresponds to a solid loading of respectively 4 and 120 ppmv (or 20 and 600 ppmw, assuming 20 g/mol for a typical coal gas as recommended in Other Physical Properties and 100 g/mol for lime(stone)). Thus, for similar conditions in the gas and solid phases, the length of the reacting zone, z, must be about 103-105 times larger for the entrainedflow configuration if limestone with the same particle size is used. Alternatively, by making the radius of the sorbent particles 1-3 orders of magnitude smaller for the entrained-flow configuration, one maintains a similar length for the sorption zone. Fluidized-Bed Design. One of the key difficulties in the description of fluidized beds is the complex nature of the hydrodynamics. It is crucial to determine clearly the contacting regime (i.e., bubbling vs nonbubbling beds, and slow bubbling, turbulent, or fast fluidization) under which the bed is operating before one can attempt meaningful modeling. Scores of models accounting for various intricate bed dynamics have been published (Kunii and Levenspiel, 1991). Comparisons between model predictions and experimental data show that simplistic approaches, which assume that the gas moves upward in a plug flow, or completely mixed, fashion, are not adequate descriptions for fluidized-bed reactors. Even the more sophisticated two-region model proposed by Toomey and Johnstone in 1952, which represents the first serious attempt to incorporate the nonhomogeneities of the bed, produced poor descriptions of the bed’s performances (Kunii and Levenspiel, 1991, p 155). More realistic hydrodynamic models are required for accurate predictions of the reaction kinetics. Fortunately, significant progress has been made in recent years in devising reasonable models that do not require the use of advanced numerical techniques and yet correlate well with experimental data (Kunii and Levenspiel, 1990). The hydrodynamics of fluidized beds depends heavily on such variables as the total bed height, average particle size, particle size distribution, physical properties of the solids and of the fluids, superficial velocity of the fluid phase. Rather than investigating all the potential cases, we narrow our focus by clearly defining the expected range of these important variables. Assumptions. (i) Particle Characteristics. As we did for the other reactor configurations, we treat particles with diameters larger than 0.5 mm. The solid particles then fall in the standard Geldart D class for every temperature (700-1100 °C) and pressure (1-35

bar) relevant to high-temperature coal-gas cleanup (Kunii and Levenspiel, 1991, p 189). We also assume that all the particles have the same size and neglect any variation in the overall particle size or density as the reaction proceeds. (ii) Range of Superficial Velocities of the Gas Phase. The superficial velocity of the gas phase, uG, must be larger than the minimum fluidization velocity, umf, and less than the elutriation velocity, uM. We assume that the Ergun equation describes the pressure drop across the bed; umf is then given by eq 24. The elutriation velocity is, as a first approximation, the terminal settling velocity of the solids, ut, obtained when the weight of the particle equates the drag force of the fluid acting on the particle (Kunii and Levenspiel, 1991, pp 80-81):

ut* )

[

]

4.5 0.418 + R*2 R*0.5

-1

(88)

where:

[ [

ut* ) ut

R* ) R

] ]

Fg2 ηm(Fp - Fg)g Fg(Fp - Fg)g ηm

2

1/3

(89)

1/3

Note that the ratio between umf and ut, while being as large as 100 for fine particles, is only of the order of 10 for large ones. There are several reasons for also choosing uG as close from umf as possible without impairing the bed operability. First, we want to minimize the pressure drop by minimizing the height of the bed while maintaining a reasonable value for the contact time between the gas and the solids. We also desire to minimize the quantity of bypass of the “bubble phase” (the discontinuous phase formed by the ensemble of the quasi-solid free bubbles moving upward) with respect to the “emulsion phase” (the continuous phase comprising the totality of the fluidized bed minus the bubbles). As we will see in more detail in the Bed Hydrodynamics section, this is achieved by reducing the relative fraction of the bubble phase. Since this fraction is proportional to uG - umf, keeping uG close to umf reduces the quantity of gas that bypasses the emulsion phase. We also want to maintain uG below 2-2.5umf to prevent the appearance of channeling, slugging, and explosive bubbling (Kunii and Levenspiel, 1990). Finally, keeping uG close to umf also minimizes elutriation of solids. For large particles, umf is sufficiently large to permit an acceptable gas throughput. (iii) Behavior of the Solid in the Emulsion Phase. It is reasonable to assume that the solids are well mixed in a fluidized bed with a height-to-diameter ratio not exceeding 3:1 (Szekely et al., 1976, p 229). We will overlook the elutriation of fines that creates a carryover stream in addition to the bed overflow. This assumption, combined with the disregard of any additional reaction occurring in the freeboard zone located above the top of the fluidized bed, results in a somewhat conservative estimate of both the average solid- and gasphase conversions since the reaction rate of fines is larger than that of larger particles (the reaction rate is proportional to the first or second power of the diameter).


We also neglect the effect of the change in the solid density during the course of the reaction. This will also lead to a slightly conservative prediction of the solidand gas-phase conversion: lime particles are not as dense as limestone or calcium sulfide particles, which creates some inhomogeneity in the bed. However, the lighter particles (i.e., those containing a large fraction of unreacted lime) will tend to concentrate on the top of the bed, introducing a slight “countercurrent” component to the otherwise well-mixed particles. The presence of this countercurrent component will actually be beneficial, especially if close-to-equilibrium removal of H2S is required (see section on Moving-Bed Design). (iv) Bed Hydrodynamics. For fluidized gas/solid systems, bubbles appear as soon as fluidization begins, especially when the diameter of the solid particles is larger than 0.2 mm (Kunii and Levenspiel, 1991, see their Figure 7, p 74). Under the limitations described in the preceding sections, we are under the “slowbubbling fluidized bed of large particles” regime described in detail by Kunii and Levenspiel (1990, 1991). The flow through the emulsion is faster than the rise velocity of the bubbles. Consequently, it can be assumed that the fluidizing gas uses the bubbles as shortcuts through the bed. It is then reasonable to assume that the gas moves up the bed in plug flow with occasional shortcuts through the bubbles. One can envision a gas element entering the bed being rapidly exchanged between the emulsion and bubble phase. Hence, ignoring any conversion in the bubble phase (since very little solid is present in the bubbles) is equivalent to assuming that the gas in the emulsion phase rises by plug flow. The fraction of the gas in the bubble phase (which has the same composition as that in the emulsion phase at any height of the bed) simply bypasses the emulsion phase, i.e., does not “see” the solid reactant. It should then be noted that, as with the other reactor configurations, we are dealing with a one-dimensional problem. Fortunately, the hydrodynamics of beds of “large particles” (i.e., of diameter larger than 1 mm) is somewhat simpler to describe than that of fine particles. The principal results concerning this regime are summarized below (for more details refer to Kunii and Levenspiel, 1990, 1991):

db0 ) 0.823lor0.8(uG - umf)0.4 for db0 < lor (square array of holes) (94) for db0 < lor db0 ) 0.777lor0.8(uG - umf)0.4 (equilateral triangle array of holes) where lor is the spacing between the holes of the distributor.

fraction of the gas passing through the emulsion phase, i.e., contacting the solids (Frac): umf uG

Frac ) (1 - δ)

(95)

void fraction in the fluidized bed (f): f ) 1 - (1 - δ)(1 - mf)

(96)

(v) Bed Dimensions. We should aim for a bed having a diameter-to-height ratio close to 1 because of the mechanical constraints for large, high-pressure vessels. Moreover, it has been observed that the bubbles formed at the distributor may grow to the bed diameter to form slugs in beds with greater height-todiameter ratios. For large particles (Geldart D type), this generally results in so-called “flat slugs”: the bed separates into rising slices of emulsion separated by gas, in which solid particles continuously rain from slice to slice (Kunii and Levenspiel, 1991, p 132). Careful measurements by Baeyens and Geldart (1974) and Steward and Davidson (cited in Kunii and Levenspiel, 1991, p 132) showed that slugging will occur if:

ub > umf + 0.22xdt

(97)

and will begin at the height zs given by:

zs ) 1.34dt0.175

(98)

fraction of bed in the bubble phase (δ): uG - umf δ) ub + 2umf

(90)

bubble phase rise velocity (ub): ub ) (uG - umf) + 2.22xdb

(91)

bubble diameter (db):

(

db ) dbM - (dbM - db0) exp -

)

0.3z dt

(92)

theoretical maximum bubble size (dbM): dbM ) 1.48(uG - umf)0.4dt0.8 minimum (or entry) bubble size (db0): db0 ) 0.28(uG - umf)2

if db0 > lor

(dt and zs being expressed in m and ub and umf being expressed in m/s). (vi) Other Assumptions. As with the other designs, we will assume that the fluidized bed is isothermal, that the gas is ideal, that the grain model for spherical particles (and grains) describes the sulfidation kinetics at the particle level, and that the chemical reaction rate is first-order with respect to the gaseous reactant concentration. Design Equations for Fluidized Beds. A differential mass balance on the sulfur atoms between the height z and z + dz yields:

uG (93)

〈 〉

∂C ∂X + (1 - f)Fs Frac ∂z ∂t

at z

)0

(99)

where Frac represents the fraction of the gas which is actually in contact with the solid (i.e., the fraction of the gas in the emulsion phase) and where a quantity surrounded by the brackets 〈 〉 is taken as its average value.

1036


By inspection of eq 99, it appears that the average reaction rate ∂x/∂t can be rewritten as follows:

〈 〉 ∂X ∂t

) at z

〈 〉 ∂X ∂t

(C - Ceq)z

where 〈C - Ceq〉 stands for the average of the difference between the actual concentration and the equilibrium concentration over the entire fluidized bed and, therefore, is not a function of z. Moreover, 〈∂X/∂t〉〈C-Ceq〉 also does not depend on z, since we assume that the solid is well mixed. Equation 100 can thus be integrated to give:

-ln

[

] [

][

][ ]

〉

dX dt

]∫

〈C-Ceq〉

z

[

0

[

without introducing an error larger than 5% for bed diameters larger or equal to 5 m; this number would rise to about 10% for beds having a diameter of only 2 m. It should also be noted that the value of db* is always larger than that of db, which results again in a slight underestimation of the bed performances. Replacing db by db* in the expression for δ given by eq 90 and changing the integration variable from z to V, one obtains:

]

2(V - V0) (108)

(1 - δ)2 dz] (101)

where δ is given by eq 90. Defining a characteristic flow time (for the gas) τF* as follows:

τ F* )

]( )

where:

β1 )

Fs(1 - mf) R 〈C - Ceq〉 uG

(102)

(

[

] [ ][ ]

β2 )

β3 )

[∫

dz (1 - δ)2 0 R

]

z

(103) (τF* is the analogue to τF defined earlier for moving beds and τB is the average residence time for the sorbent in the fluidized bed.) We want to keep δ significantly lower than 1 to minimize the potential bypass of the gas. In this case, the following approximation can be made:

∫0z(1 - δ)2 dz ≈ ∫0z dz ) z

( [ ][ ]

)

y - yeq τF* umf ≈ exp [〈X〉〈C-Ceq〉]Z yin - yeq τB uG

(105)

(where Z ) z/R). From the approximately exponential concentration profile given by eq 105, it follows that 〈C - Ceq〉 is given approximately by:

(Cin - Cout) ≈ Cin - Ceq ln Cout - Ceq yin - yout P (106) RgT yin - yeq ln yout - yeq

∫0L(C - Ceq) dz ≈

1 L

[

( )

]

[

V)

]

where L is the total height of the fluidized bed. A more rigorous solution for eq 103 should account for the finite value of δ given by eq 90. For values of z below the value of zs, eq 92, an expression giving the

(

3uG - umf umf uG2 - umf2

(109)

umf2

)

2.22xdb* uG + umf + umf umf

V0 )

(

)

2.22xd0 uG + umf + umf umf

Average Conversion of the Sorbent. The average conversion of the solid in the fluidized bed is given by:

〈X〉〈C-Ceq〉 )

(104)

and eq 103 becomes:

〈C - Ceq〉 )

)

dt 1.35 u (u - umf) R dbM - db0 mf G

and anticipating the simple relationship between the average reaction rate for the sorbent and its average conversion presented below, we finally obtain:

y - yeq τF* umf -ln ) (〈X〉〈C-Ceq〉) yin - yeq τ B uG

(107)

V ) Z + β1[β2 ln( ) + β3(V-1 - V0-1) ∫0z(1 - δ)2 dz R V0

y - yeq Fs 1 - mf umf ) × yin - yeq uG uG 〈C - Ceq〉

[〈

(dbM - db0) RZ dt

db* ) db0 + 0.3

(100)

〈C - Ceq〉

for 〈C-Ceq〉

diameter of the bubbles can be approximated by:

t)∞ E(t) X(t)at 〈C-C ∫t)0

eq〉

dt

(110)

where E(t) is the residence time distribution function for the solids in the fluidized bed. The values of X(t) (or rather t(X)) is given by eq 8, for the grain model, and eq 9, for the unreacted shrinking core model. Assuming that the solids are well mixed in the fluidized bed, E(t) is given by:

E(t) )

( )

1 t exp τB τB

(111)

Combining eqs 110 and 111, we obtain: Φ)1 exp(-θΦ)X(Φ) dΦ + ∫Φ)0 Φ)∞ θ∫Φ)1 exp(-θΦ) dΦ

〈X〉 ) θ

(112)

where θ stands for (τDP + τDG + τMT + τR)/τB, θ1,4 for (τDP + τDG)/τB, θ2 for τMT/τB, θ5 for τR/τB, and Φ for t/(τDP + τDG + τMT + τR) for the grain model (for the unreacted shrinking core model, θ stands for (τDP + τMT + τR,SC)/ τB, θ1,4 for τDP/τB, θ2 for τMT/τB, θ5 for τR,SC/τB, and Φ for t/(τDP + τMT + τR,SC)).


(

Table 6. Numerical Values of Gn for n ) 0-9 Gn

n

Gn

0 1 2 3 4

0.333 0.267 0.115 3.43 × 10-2 7.86 × 10-3

5 6 7 8 9

1.46 × 10-3 2.28 × 10-4 3.08 × 10-5 3.66 × 10-6 3.89 × 10-7

Making the change of variable u ) (1 can be rewritten as follows:

∫u)0 exp[θ5u + 3θ1,4u u)1

〈X〉 ) 3 exp(-θ)

X)1/3,

2

+

Equation 113 cannot readily be solved in closed form in the most general case where all potential resistances to the overall reaction rate have to be considered. However, several cases of interest can be solved in closed form.

n)∞

[

(-6)n

θ5 ∑ n)0 (n + 3)!

5

〈X〉 )

n

]

(-1)

n)∞

〈X〉 )

n

1

1

where

γ′′ )

Sh′′ - 1 Sh′′

Then, performing an infinite series development of the exponential and inverting the integral and summation signs (which is legitimate since the integral defined in eq 113 converges), one finally obtains:

( (

2

〈X〉 ) 3 exp -θ1,4 1 +

)) ∑

n)∞

[

Sh′′

Gn*θ1,4n]

n)0

where

(-1)n Sh′′ 2n+3 n! Sh′′ - 1

(

∫01-(1/Sh′′)u2n+2(2u - 3)n du

)

(121)

or

(ii) mass-transfer control (i.e., θ1,4 ) θ5 ) 0) 1 [1 - exp(-θ2)] θ2

(120)

(114)

1 1 1 3 θ5 + ... ≈ 1 - θ5 + θ52 4 20 120 (115)

〈X〉 )

y)γ′′ exp[-θ1,4γ′′2y2(2y - 3)]y2 dy ∫y)0

Gn* )

(i) chemical reaction control (i.e., θ1,4 ) θ2 ) 0) 2 2 3 1 - + 2(1 - exp(-θ5)) θ5 θ5 θ

))

2 γ′′-3 × Sh′′

eq 112

(θ2 - 2θ1,4)u3]u2 du (113)

〈X〉 )

(

〈X〉 ) 3 exp -θ1,4 1 +

n

n)∞

Gn* ) 3n

1

( )( p

3

)

Sh′′ - 1 Sh′′

1

p

(2n + p + 3)(n - p)!p!

When Sh′′ is equal to 1, u can be replaced by y ) (3θ1,4)0.5u in eq 113 to yield:

〈X〉 ) 3 exp(-3θ1,4)

(116)

θ2n ≈ 1 - θ2 + θ22 - θ23 + ... ∑ 2 6 24 n)0 (n + 1)!

∑ p)0

2

-

y)x3θ ∫y)0

exp(y2)y2 dy (122)

1,4

and, after performing a similar series development for the exponential, 〈X〉 is finally given by eq 123: n)∞

〈X〉 ) 3 exp(-3θ1,4)

(117)

3n

θ1,4n ∑ (2n + 3)n! n)0

(123)

(iii) internal diffusion control (i.e., θ2 ) θ5 ) 0) n)∞

〈X〉 ) 3 exp(-θ1,4)[

Gnθ1,4n] ∑ n)0

(118)

In both cases the series expansion of 〈X〉 (valid for small values of θ1,4 and large values of Sh′′) is given by eq 124:

〈X〉 ≈ 1 - R1θ1,4 + R2θ1,42 - R3θ1,43 + ... (124)

where

(-1)n Gn )

n!

with:

∫

1 2n+2 u (2u 0

- 3) du )

n)∞

3n

n

( ) 2

∑ -3 p)0

R1 )

p

1

(2n + p + 3)(n - p)!p!

19 41 1 θ 2θ 3 + ... 〈X〉 ≈ 1 - θ1,4 + 5 420 1,4 4620 1,4

R2 )

(119)

(see case iv, below, for more details). Numerical values of G0 to G9 are given in Table 6.

(iv) external mass-transfer and internal diffusion control (i.e., θ5 ) 0) This is actually the most relevant case for the kinetics of the reaction between H2S and CaO at high temperatures. When Sh′′ is different from 1, one can replace u by y ) γ′′u in eq 113 to obtain:

R3 )

1 1 + 5 Sh′′

19 2 19 + + 420 60Sh′′ 3Sh′′2

(125)

123 29 41 1 + + + 4620 1540Sh′′ 110Sh′′2 3Sh′′3

We chose to perform the development as a function of θ1,4 rather than θ2 because under most operating conditions of the lime(stone) fluidized bed θ1,4 will be significantly larger than θ2, since Sh′′ will remain larger than unity (see Table 5a). In the absence of any internal diffusion resistance, the value of 〈X〉 can be precisely calculated without the need of an infinite series expansion. However, when internal diffusion has to be accounted for in the overall

1038


Figure 5. Average sorbent conversion in a fluidized bed as a function of θ ) θ1,4 for internal diffusion control, θ ) θ2 for masstransfer control, and θ ) θ5 for chemical reaction control (obtained by numerical integration of eq 113).

Figure 6. Comparison of the exact solution of eq 118 (obtained by numerical integration) to its infinite-series expansion truncated after the second or third term (eq 119).

reaction kinetic expression, we need to either solve eq 113 numerically for different values of θ1,4 and θ2 or use a truncated “version” of the infinite series expansion proposed in eq 121. Figure 5 gives the average solid conversion as a function of θ for the three limiting cases presented in i-iii. It appears that θ needs to be less than 2 if a sorbent utilization larger than 75% is to be required. Within this range of values for θ, Figure 6 shows that the error introduced by truncating the infinite series defined in eq 118 after the third terms stays below 3%. Finally, Figure 7 provides values of 〈X〉 when both external mass transfer and internal diffusion influence the overall reaction kinetics using numerical integration of eq 113 (case iv). Average Reactivity of the Sorbent. The average reactivity of the solids is given by the following expression:

Figure 7. Average sorbent conversion in a fluidized bed as a function of θ1,4 and Sh′′ (obtained from numerical integration of eq 113).

〈∂X∂t 〉

〈C-Ceq〉

)

〈dXdt 〉

〈C-Ceq〉

)

t)∞ E(t) [ ∫t)0

]

dX(t) dt

at 〈C-Ceq〉

dt (126)

where E(t), the residence time distribution function for the solids in the fluidized bed, is again given by eq 111 (the first equality results from the fact that the average reactivity does not depend on z, since the solid particles are considered well mixed). An integration by parts yields a very simple relationship between the average reactivity and the average conversion:

1 ) [〈X〉 - X(t)0)] 〈dX dt 〉 τ

(127)

B

or more simply:

) 〈dX dt 〉 τ

〈X〉

(128)

Application to Calcium-Based Sorbents for HighTemperature Coal-Gas Desulfurization in Fluidized Beds. (i) uG ) umf. Equation 101 can be simplified to eq 129 when the bed is operated at the incipient fluidization velocity, umf:

([ ]

)

y - yeq τF* ) exp [〈X〉〈C-Ceq〉]Z yin - yeq τB

(129)

None of the specific properties of the gas or the sorbent explicitly appears in the equation governing the concentration profile of H2S throughout the bed, other than in Sh′′, Pe′′, and Da′′. It has been established that the chemical kinetics does not affect the overall reaction kinetics for the lime/H2S system. Hence, the average sorbent conversion can be described by the equations developed in section iv, above, beginning with eq 120. Furthermore, if more than 85% conversion is desired for the solid, Figure 6 shows that 〈X〉 can be accurately estimated by:

B

if fresh (unreacted) sorbent is fed to the bed.

〈X〉 ≈ 1 - R1θ1,4 + R2θ1,42

(130)


([ ]

)

τF* y - yeq ) exp [〈X〉〈C-Ceq〉]Zζ yin - yeq τB

(132)

where

Z′ ) ζZ )

Figure 8. H2S concentration profile in a fluidized bed of calcined limestone particles at uG ) umf, for 95% average conversion of the sorbent (obtained from eqs 129 and 131).

(R1)∫ (1 - δ) dZ Z

0

provided that the Pećlet number is defined as in the preceding case (i.e., R ) 1), that is, with being taken equal to mf. The main difficulty resides in the fact that ζ is an explicit function of the superficial velocity of the gas, of some physical properties of the gas and solid phases, and of the physical dimensions of the reactor (through dt) and gas distributor (through lor). However, as we mentioned earlier, the value of δ is bounded by its minimum value at the top of the fluidized bed (at least 0) and by its maximum value at the bottom of the bed, just above the gas distributors, where db is equal to db0 given by eq 94. Hence:

ζmax )

( )(

)

1 1 2+q > > ζ > ζmin ) R R 1+R+q 1 1.17R + 0.83 (133) R 2.17R - 0.17

( )(

)

where

q)

Figure 9. H2S concentration profile in a fluidized bed of calcined limestone particles at uG ) umf, for 85% average conversion of the sorbent (obtained from eqs 129 and 131).

from which we obtain:

()

τF* 6(1 - mf) 3(1 - mf) R1 ) × θ1,4 ) τB Pe′′(1 - f) Pe′′(1 - f) R2

[ [

R2 1 - 1 - 4(1 - 〈X〉) 2 R1

]] 0.5

(131)

making (y - yeq)/(yin - yeq) a function only of Z/Pe′′ and Sh′′ (since f ) mf for uG ) umf). It should, however, be remembered that this simple solution only applies when 〈X〉 is larger than 0.85 and Sh′′ larger than 10, which is expected for millimeter-sized lime(stone) particles. For lower sorbent utilization a numerical approach similar to that described in the following section should be applied. Figures 8 and 9 provide the conversion profiles in the gas phase as a function of 3Z/Pe′′, and using Sh′′ as the parameter, for two values of the average conversion of the solid (95% and 85%). These plots show the high sensitivity of the profile to 〈X〉: even if the bed dimensions remain reasonable, especially for 〈X〉 being equal to 0.85, lower average conversion of the sorbent decreases the bed height. (ii) uG ) rumf (2.5 > r > 1). The H2S concentration profile in now given by:

2.22xdb umf

(db being expressed in m and umf being expressed in m/s). Figure 10 displays these values of ζmin and ζmax as a function of R. It can be seen that a quick conservative estimate of the bed performances can be easily made by using the value of ζmin given by eq 133 and using Figures 8 and 9 after substituting Z′ for Z. This will not introduce severe error if R remains close to unity or if the value of umf is larger than that of the square root of dt (i.e., for larger values of R). For larger values of R, or lower values of umf, the quantity 1 - δ needs to be integrated over the bed’s length. This operation requires that the nature and physical properties of the coal gas as well as the physical characteristics of the fluidized bed reactor be known beforehand, so the entire set of constitutive equations described in the preceding sections should be used. The concentration profile of H2S through the fluidized bed, y(z), depends on a large number of parameters:

y ) y(yin,P,T,Mm,ν,yeq,Fp,τB,uG,dt,lor,Sh′′,θ,R,z) (134) However, some of these parameters do not influence the concentration profile very much, or their range of variation is so limited under realistic conditions that their values can be averaged (see the section Estimation of Physical Characteristics). The following steps must be taken to determine the H2S concentration profile through the fluidized bed. First, the physical properties of the gas/solid system and the reactor dimensions must be determined (yin, P, and Mm, the average density of the gas). A thermodynamic analysis of the lime(stone)/coal gas systems will reveal yeq and the optimum temperature at which to operate the bed and will allow us to calculate the kinematic viscosity of the gas, ν. Then, the degree of sulfur removal must be chosen, provided it does not exceed the thermodynamic predictions. This choice will determine

1040


tion and reasonable bed depth (about 1 m, or 1 s contact time for entrained flow) when the reactor temperature is maintained 25-50 °C above the calcination temperature of the calcium carbonate. Since no pilot-scale sorption systems using limestone have yet been built, the validity of the equations has not been tested. However, they should serve as a starting point for the design of such systems and as a basis for correlating the data obtained. Once validated, with modification as necessary, the equation should be of use in the scale-up of pilot-plant data. Nomenclature

Figure 10. Range of variation of ζ as a function of R (ζM and ζm are given by eq 133).

the value of 〈C - Ceq〉. Finally, one must choose an appropriate degree of utilization of the sorbent. This will determine the value of the average residence time of the sorbent, τB. The H2S concentration profile will then be given by eq 132 once R, whose value is mainly constrained by the value of umf, is chosen (the actual values of dt and lor do not strongly influence the final concentration profile). From the preceding analysis, it appears that only P, τB (or 〈X〉), R, R, and the intrinsic kinetic attributes of the sorbent (through Sh′′, Pe′′, and Da′′) emerge as the really important variables influencing the fluidized-bed reactor performances. Conclusions Design equations describing entrained-flow as well as moving-, packed-, and fluidized-bed systems using limestone particles for the high-temperature sorption H2S have been developed and solved analytically for steady-state operation (or for constant-pattern breakthrough in the case of a fixed bed) using the grain model for spherical grains and pellets to describe the kinetics of the noncatalytic gas/solid reaction between CaO and H2S. The concentration profile of H2S and CaS along the bed length as well as the degree of utilization of the sorbent, the degree of removal of H2S from the gas, and the required bed total length can be expressed as a function of only five parameters: (1) R, the radius of the limestone particles; (2) yeq, the thermodynamic equilibrium mole fraction of H2S in the gas under bed conditions; (3) Sh′′ ) 2(τDP + τDG)/τMT, a modified Sherwood number (or Biot number); (4) Da′′ ) 6(τDP + τDG)/τR, a modified Damko¨hler number; (5) Pe′′ ) 6(τDP + τDG)/τF, a modified Pećlet number. These equations, combined with the kinetic data we gather on the CaCO3/H2S and CaO/H2S systems, provide a practical way to design any of the reactor configurations considered in this work. Using limestone particles with diameters of the order of 100 µm for entrained flow, and with diameters of the order of 1 mm for the other configurations, it was found that H2S could be removed from the hot coal gas to near its equilibrium value with more than 75% sorbent utiliza-

A ) see eq 57 Ain ) see eq 57 C ) concentration of H2S in the bulk gas (mol of H2S/m3 of gas) Ceq ) equilibrium concentration of H2S at bed conditions (mol of H2S/m3 of gas) Ci ) H2S concentration at the core/reacted layer interface of the pellet (mol of H2S/m3 of gas) Cig ) H2S concentration at the core/reacted layer interface of the grain (mol of H2S/m3 of gas) Cs ) concentration of H2S at the surface of the pellet (mol of H2S/m3 of gas) Csg ) concentration of H2S at the surface of the grain (mol of H2S/m3 of gas) db ) bubble diameter in the fluidized bed (m) db* ) approximate value of the bubble diameter in the fluidized bed (m) dbM ) maximum theoretical bubble diameter in the fluidized bed (m) db0 ) minimum (or entry) bubble diameter in the fluidized bed (m) dt ) internal diameter of the fluidized bed reactor (m) D ) diffusivity of H2S in the gas mixture (m2/s) Da′ ) modified Damko¨hler number: (ksR)/(De) Da′′ ) modified Damko¨hler number (see eq 35) De ) effective diffusivity of H2S in the reacted layer of the solid (m2/s) Dg ) effective diffusivity of H2S in the grains of the solid (m2/s) Dij ) binary diffusivity of components i and j (m2/s) Dim ) diffusivity of component i in the gas mixture (m2/s) DK ) Knudsen diffusivity for H2S in the pores of the lime(stone) pellets (m2/s) E(t) ) residence time distribution function for the solids in the fluidized bed E(X) ) see eq 62 E′(X) ) see eq 60 F1(Y) ) see eq 53 F2(Y) ) see eq 54 F3(Y) ) see eq 55 F4(Y) ) see eq 56 Frac ) fraction of the gas phase passing through the emulsion phase in the fluidized bed g ) standard acceleration of gravity (about 9.81 m/s2) I(Y) ) see eq 50 J(Y) ) see eq 51 K(Y) ) see eq 52 k ) Boltzmann’s constant (1.38048 × 10-23 J‚K-1) kg ) mass-transfer coefficient in the gas film around the solid particle (m/s) ks ) rate coefficient for the chemical kinetics (m/s) L ) bed length (m) lor ) spacing of the holes in the distributor of the fluidized bed (m) Mi ) molecular weight of component i (kg/mol) Mm ) average molecular weight of the gas mixture (kg/ mol)

Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996 1041 N ) molar flow of active solid sorbent per unit area of bed (mol Ca‚m-2‚s-1) P ) total pressure (Pa) Pc ) critical pressure (Pa) Pe ) Pećlet number: uGR/D Pe′ ) modified Pećlet number: Pe(D/De)/(1 - ) Pe′′ ) modified Pećlet number (see eq 33) PR ) reduced pressure ()P/Pc) R ) effective average radius of the lime/limestone pellet (m) R* ) dimensionless effective radius of the lime/limestone pellet (m) Rc ) radius of the unreacted core of the pellet (m) rc ) radius of the unreacted core of the grain (m) rg ) effective grain radius of the grains of the lime/ limestone pellet (m) Rg ) ideal gas constant (8.31439 Pa m3 mol-1 K-1) Sc ) Schmidt number: v/D Sh ) Sherwood number: kgR/D Sh′ ) modified Sherwood number (Biot number): Sh(D/ De) Sh′′ ) modified Sherwood number (see eq 30) t ) time (s) T ) absolute temperature (K) Tc ) critical temperature (K) Ti* ) (kT)/i TR ) reduced temperature ()T/Tc) ub ) bubble phase rise velocity in the fluidized bed (m/s) uG ) superficial velocity of gas (volumetric flow rate per unit bed area) (m/s) uM ) elutriation velocity (m/s) umf ) minimum fluidization velocity (m/s) ut ) terminal velocity of the pellet in the gas phase (m/s) ut* ) dimensionless terminal velocity of the pellet in the gas phase (m/s) V ) see eq 109 V0 ) see eq 109 v0 ) constant pattern velocity (m/s) X ) fraction of lime converted to CaS at position z in bed (0 < X < 1) y ) mole fraction of H2S in the gas phase yi ) mole fraction of component i Y ) see eqs 40 and 41 Y* ) see eq 73 z ) length along the bed (m) Z ) dimensionless length along the bed: z/R Z* ) dimensionless length along the bed (see eq 72) zs ) height in the fluidized bed at which flat slugging appears (m)

θ5 ) τR/τB Φ ) t/(τDP + τDG + τMT + τR) Fg ) density of the gas mixture (kg/m3) Fp ) density of the solid, including its void volume (kg/m3) FS ) molar density of the solid, including its void volume (mol of solid per m3 of solid) σi ) hard-sphere diameter of the gas molecule (m) τ ) t - z/vo (s) τB ) average residence time of the sorbent in the fluidized bed (s) τp ) tortuosity factor for the sorbent pellet τDP ) characteristic particle diffusion time: FSR2/(6De(C Ceq)) (s) τMT ) characteristic mass-transfer time: FSR/(3kg(C - Ceq)) (s) τR,SC ) characteristic reaction time: FSR/(ks(C - Ceq)) (s) τDG ) characteristic grain diffusion time: FSrg2/[6(1 - v)Dg(C - Ceq)] (s) τR ) characteristic reaction time: FSrg/[ks(1 - v)(C - Ceq)] (s) τF ) characteristic flow time: ((1 - )FS/(C - Ceq))(R/uG) (s) τ*F ) characteristic flow time: ((1 - mf)FS/〈C - Ceq〉)(R/ uG) (s) ν ) kinematic viscosity of the gas mixture ()ηm/Fg) (m2/s) Ωv,i ) collision integral for species i (Σv)i ) molecular diffusion volume of species i (see Table 4) ζ ) see eq 132 ζmin ) see eq 132 ζmax ) see eq 132

Appendix (A) Derivation of the Design Equations for Moving Bed. The first part of this appendix provides a more detailed derivation of the design equations for the entrained-flow reactor configuration. The derivation is given for the case where the unreacted shrinking core model describes the reaction kinetics but will be similar in the case where the grain model is applied (Pe′, Sh′, and Da′ just have to be respectively replaced by Pe′′, Sh′′, and Da′′). A differential sulfur mass balance cross section of the bed between z and z + dz gives:

uG

Greek Letters R ) uG/umf R1 ) see eq 125 R2 ) see eq 125 R3 ) see eq 125 β1 ) see eq 109 β2 ) see eq 109 β3 ) see eq 109 δ ) fraction of the fluidized bed in the bubble phase ∆P ) pressure drop across the bed (Pa) ) bed porosity (bed void fraction) i ) characteristic Lennard-Jones energy of species i (see Table 4) (J) f ) void fraction in the fluidized bed mf ) void fraction in the emulsion phase at incipient fluidization (≈0.5) v ) porosity of the solid reactant ηi ) pure component i viscosity (N‚s‚m-2) ηm ) viscosity of the mixture (N‚s‚m-2) θ ) (τDP + τDG + τMT + τR)/τB θ1,4 ) (τDP + τDG)/τB θ2 ) τMT/τB

∂C ∂X + (1 - )Fs )0 ∂z ∂t

(A.1)

We then need to introduce the following kinetic expression (shrinking core model for spherical particles in the case of a reaction controlled by the diffusion through the sulfide layer and the mass transfer in the gas film around the solid particle) to eliminate time as a variable in eq A.1:

t ) τ1[1 - 3(1 - X)2/3 + 2(1 - X)] + τ2[X] + τ3[1 - (1 - X)1/3] (A.2) where:

τ1 )

FsR2 6De(C - Ceq)

(A.3)

τ2 )

F sR 3kg(C - Ceq)

(A.4)

1042


τ3 )

F sR ks(C - Ceq)

Da′ )

(A.5)

( )[

]

(A.6)

[

]

(

)

(

)

y - yin yout - yin for countercurrent flow (A.16a) y - yin yout - yin for cocurrent flow (A.16b)

Y ) (1 - Xin) - (Xout - Xin)

-1

(A.7)

Using the expression of dX/dt given by eq A.5 in eq A.1 gives the following time-independent equation: uG

(A.15)

Y ) (1 - Xout) + (Xout - Xin)

Differentiation of eq A.2 yields:

τ3 dX ) 2τ1[(1 - X)-1/3 - 1] + τ2 + (1 - X)-2/3 dt 3

(A.14)

Z ) z/R

A characteristic flow time can also be defined as follows:

R (1 - )Fs τ6 ) uG C - Ceq

ksR 6τ1 ) De τ3

to transform eq A.6 into:

Y - Yeq dY + )0 dZ Pe′ 1 1 + (Y-1/3 - 1) + (Y-2/3) 3 Sh′ Da′

[

d(C - Ceq) + dz

with the initial condition:

C - Ceq

(1 - )

R2 R R + [(1 - X)-1/3 - 1] + [1 - X]-2/3 3kg 3De 3ks

)0

(A.8) Now we can use an overall mass balance over the total bed length to obtain the value of the molar flow rate of solid, N, as a function of the superficial velocity of the gas, uG:

Cin - Cout N ) uG Xout - Xin

(A.9)

and an overall mass balance between the bed entrance (z ) 0) and any point located at height z:

N(Xout - X) ) uG(Cin - C)

(A.10b)

in the case of a cocurrent configuration, to obtain a relationship between X and C and, thus, eliminate C as a variable in eq A.6:

(

)

Xout - Xin X ) Xout + (C - Cin) Cin - Cout

Y ) Yin ) 1 - Xout

)

(

Xout - Xin (C - Cin) Cin - Cout

A ) (Yeq/Y)1/3

( )

( )( )

uGR D 1 ) Pe Pe′ ) De 1 - (1 - )De

(A.20)

in eq A.6 finally results in the following expression:

dZ ) Pe′

[(

)(

)

Sh′ - 1 A2 1 3 Sh′ A -1 A Yeq-2/3 1 A Yeq-1/3 3 3 Da′ A -1 A -1

(

)

)] dA (A.21)

(

which, using the following boundary condition (valid for both counter- and cocurrent configurations):

A ) Ain ) (Yeq/Yin)1/3

at Z ) 0

(A.22)

( ) ]

(A.23)

can be easily integrated to give:

Z)

(A.11a)

( )

Pe′ 1 1 I(Y) + J(Y) + K(Y) 3 Sh′ Da′

[

where:

I(Y) ) F1(Y) + 3Yeq-1/3[F2(Y) + F3(Y)] (A.24) (A.11b)

for a cocurrent configuration. One can then define the following nondimensional variables:

2τ1 kgR D ) Sh ) Sh′ ) De De τ2

(A.19)

The following change of variable:

for a countercurrent configuration, and:

X ) Xin -

at Z ) 0 for countercurrent flow (A.18)

Y ) Yin ) 1 - Xin at Z ) 0 for cocurrent flow

(A.10a)

in the case of a countercurrent configuration, and:

N(Xin - X) ) uG(C - Cin)

]

(A.17)

(A.12)

(A.13)

J(Y) ) -F1(Y)

(A.25)

K(Y) ) 3Yeq-2/3[F2(Y) + F4(Y)]

(A.26)

with:

[ [

F1(Y) ) ln

F2(Y) ) -

]

(A3 - 1)Ain3

(Ain3 - 1)A3

(A.27)

]

(1 - A)3(1 - Ain3) 1 ln 6 (1 - A3)(1 - Ain)3

(A.28)

F3(Y) ) F4(Y) ) -


[ (

)

(

)]

(A.29)

[ (

)

(

)]

(A.30)

2Ain + 1 1 2A + 1 tan-1 - tan-1 x3 x3 x3

2Ain - 1 1 2A - 1 tan-1 - tan-1 x3 x3 x3

(B) Inversion of Equation 8 (When τMT ) τR ) 0). The purpose of the second part of this appendix is to show the possibility of inverting the expression between time and conversion for the diffusion-limited case of the grain or shrinking core model for spherical geometry:

τ ) 1 - 3(1 - X)2/3 + 2(1 - X)

(B.1)

Using the new variable u ) (1 - X)1/3, eq B.1 becomes:

3 1-τ )0 u3 - u2 + 2 2

(B.2)

which can be solved for u for any value of τ (Perry and Green, 1984). The quantity R ) [(1 - 2τ)2 - 1]/64 being negative (since τ lies between 0 and 1), eq B.2 has three unequal real roots. The root of interest (the one between 0 and 1) is given by:

u)

[

]

cos-1(2τ - 1) - 2π 1 + cos 2 3

(B.3)

where the argument of cos-1(2τ - 1) is taken between 0 and π. X is finally given by:

X)1-

[

[

]]

cos-1(2τ - 1) - 2π 1 + cos 2 3

3

(B.4)

where, once again, the argument of cos-1(2τ - 1) is taken between 0 and π. Literature Cited Baeyens, J.; Geldart, D. An Investigation Into Slugging Fluidized Beds. Chem. Eng. Sci. 1974, 29, 255-265. Bhatia, S. K.; Perlmutter, D. D. A Random Pore Model for FluidSolid Reactions. II. Diffusion and Transport Effects. AIChE J. 1981, 27 (2), 247-254. Borgwardt, R. H. Sintering of Nascent Calcium Oxide. Chem. Eng. Sci. 1989a, 44 (1), 53-60. Borgwardt, R. H. Calcium Oxide Sintering in Atmospheres Containing Water and Carbon Dioxide. Ind. Eng. Chem. Res. 1989b, 28, 493-500. Borgwardt, R. H.; Roache, N. F. Reaction of H2S and Sulfur with Limestone Particles. Ind. Eng. Chem. Process Des. Dev. 1984, 23, 742-748. Borgwardt, R. H.; Roache, N. F.; Bruce, K. R. Surface Area of Calcium Oxide and Kinetics of Calcium Sulfide Formation. Environ. Prog. 1984, 3 (2), 129-135. Boynton, R. S. Chemistry and Technology of Lime and Limestone, 2nd ed.; John Wiley & Sons: New York, 1980; Chapters 6 and 7. Crawshaw, J. P.; Paterson, W. R.; Scott, D. M. Gas Residence Time Distribution Studies of Fixed, Moving and Frozen Beds of Spheres. Chem. Eng. Res. Des.sTrans. Inst. Chem. Eng. 1993, 71 (A), 643-648. Efthimiadis, E. A.; Sotirchos, S. V. Sulfidation of LimestoneDerived Calcines. Ind. Eng. Chem. Res. 1992, 31, 2311-2321.

Efthimiadis, E. A.; Sotirchos, S. V. Experimental Validation of a Mathematical Model for Fixed-Bed Desulfurization. AIChE J. 1993, 39 (1), 99-110. Fenouil, L. A. Kinetic and Structural Studies of the Sulfidation of Large Particles of Lime and Limestone Under Simulated CoalGas. Ph.D. Dissertation, University of California at Berkeley, Berkeley, CA, 1995. Fenouil, L. A.; Lynn, S. Study of Calcium-Based Sorbents for HighTemperature H2S Removal. Parts 1, 2, and 3. Ind. Eng. Chem. Res. 1995, 34, 2334-2348. Fenouil, L. A.; Towler, G. P.; Lynn, S. Removal of H2S from CoalGas Using Limestone: Kinetic Considerations. Ind. Eng. Chem. Res. 1994, 33 (2), 265-272. Froment, G. F.; Bischoff, K. B. Chemical Reactor Analysis and Design, 2nd ed.; John Wiley & Sons: New York, 1990. Fuertes, A. B.; Alvarez, D.; Rubiera, F.; Pis, J. J.; Marban, G. Surface Area and Pore Size Changes During Sintering of Calcium Oxide Particles. Chem. Eng. Commun. 1991, 109, 7388. Fuertes, A. B.; Alvarez, D.; Rubiera, F.; Pis, J. J.; Marban, G. Simultaneous Calcination and Sintering Model for Limestone Particles Decomposition. Chem. Eng. Res. Des.sTrans. Inst. Chem. Eng., Part A 1993, 71 (A), 69-76. Kunii, D.; Levenspiel, O. Fluidized Reactor Models. 1. For Bubbling Beds of Fine, Intermediate, and Large Particles. 2. For the Lean Phase: Freeboard and Fast Fluidization. Ind. Eng. Chem. Res. 1990, 29, 1226-1234. Kunii, D.; Levenspiel, O. Fluidization Engineering, 2nd ed.; Butterworths-Heinemann Series in Chemical Engineering; Butterworths-Heinemann: Stoneham, MA, 1991. Levenspiel, O. Chemical Reaction Engineering, 2nd ed.; John Wiley & Sons: New York, 1972; Chapter 12. MERC Hot Gas Cleanup Task Force. Chemistry of Hot Gas Cleanup in Coal Gasification and Combustion, February 1978; MERC/SP-7812. Park, H. C.; Kimura, S.; Sakai, Y.; Tone, S,; Otake, T. An Unsteady State Analysis of Packed Bed Reactors for Gas-Solid Reactions. J. Chem. Eng. Jpn. 1984, 17 (3), 269-274. Paterson, W. R.; Crawshaw, J. P.; Hart, G.; Parker, S. R.; Scott, D. M.; Young, J. P. Pressure Drop in Countercurrent Gas Flow Through Moving Particulate Beds. Chem. Eng. Res. Des.sTrans. Inst. Chem. Eng. 1992, 70 (A), 252-254. Perry, R. H.; Green, D. W. Perry’s Chemical Engineers’ Handbook, 6th ed.; McGraw-Hill Book Co.: New York, 1984; p 2.15. Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill Co.: New York, 1987. Rubiera, F.; Fuertes, A. B.; Pis, J. J.; Artos, V.; Marban, G. Changes in Textural Properties of Limestone and Dolomite During Calcination. Thermochim. Acta 1991, 179, 125-134. Sotirchos, S. V.; Zarkanatis, S. Pellet-Model Effects on Simulation Models for Fixed-Bed Desulfurization Reactors. AIChE J. 1989, 35 (7), 1137-1147. Szekely, J.; Evans, J. W.; Sohn, H. Y. Gas Solid Reactions; Academic Press: New York, 1976. Wang, J. C. P.; Groves, F. R.; Harrison, D. P. Modeling High Temperature Desulfurization in a Fixed-Bed Reactor. Chem. Eng. Sci. 1990, 45 (7), 1693-1701. Wen, C. Y. Noncatalytic Heterogeneous Solid Fluid Reaction Models. Ind. Eng. Chem. 1968, 60 (9), 34-54. Yang, R. T.; Chen, J. M. Kinetics of Desulfurization of Hot Fuel Gases with Calcium Oxide. Reaction Between Carbonyl Sulfide and Calcium Oxide. Environ. Sci. Technol. 1979, 13, 549-553. Yu, H. C.; Sotirchos, S. V. A Generalized Pore Model for GasSolid Reactions Exhibiting Pore Closure. AIChE J. 1987, 33 (3), 382-393.

Received for review April 14, 1995 Revised manuscript received July 19, 1995 Accepted July 25, 1995X IE950245Y X Abstract published in Advance ACS Abstracts, February 15, 1996.

Design of Entrained-Flow and Moving-, Packed-, and Fluidized-Bed

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