1470
Ind. Eng. Chem. Res. 1997, 36, 1470-1479
Spatiotemporal Evolution of Conversion and Selectivity for Simultaneous Noncatalytic Gas-Solid Reactions in a Compact of Particles Mostafa Maalmi, William C. Strieder,*,† and Arvind Varma*,‡ Department of Chemical Engineering, University of Notre Dame, Notre Dame, Indiana 46556
A simple model for simultaneous isothermal noncatalytic gas-solid reactions, occurring in spherical particles either shrinking or growing with reaction, is developed. The particles are assumed to be packed following the hexagonal close-packed model and allowed to overlap. The model predicts how the surface area and porosity of the compact vary with reaction and describes the spatiotemporal evolution of solid reactant conversion as well as product selectivity. The influence of physicochemical variables on product uniformity is studied. Introduction A realistic model for noncatalytic gas-solid reactions without structural change is the so-called particle-pellet or grain model (Calvelo and Smith, 1970; Szekely and Evans, 1971a,b). In this model, the pellet is formed by an agglomeration of nonporous particles of spherical shape, and the reaction occurs by diffusion of gaseous reactant into the pellet to the reactive surface of individual particles. As time progresses, a layer of porous solid product builds up around the particles, so the sharp interface model (SIM) applies to the particles but not to the pellet as a whole. In the above studies, the pseudo-steady-state approximation (PSSA) was used, along with constant diffusivity values for diffusion in both the micropores of the product layer surrounding the individual solid particles as well as in the macropores of the pellet. A variant of this model was proposed by Szekely and Propster (1975) to account for particle-size distribution. Ramachandran and Smith (1977) presented a similar model to account for structural change due to both reaction and sintering for a spherical pellet packed with spherical particles. They studied the case of a firstorder reaction with respect to the gaseous reactant and also used the PSSA for both types of diffusion discussed above. The effect of porosity on the effective diffusivity was included, but the term involving its spatial derivative in the mass balance equation for diffusion in the macropores was neglected. They also included variation of the effective diffusivity, following the random pore model of Wakao and Smith (1962). Prasannan et al. (1985) used the same assumptions as Ramachandran and Smith (1977) to describe a first-order reaction with respect to the gaseous reactant with structural change, in the presence of inert solids. However, the variation of Knudsen diffusivity owing to evolution of the pores with time was not considered. Improvements of the above models to account for the overlapping of individual particles, which occurs in the case of particles expanding with reaction, have been proposed by several workers. Lindner and Simonsson (1981) described a partially sintered spheres model (PSSM) in which the initial solid structure is considered to consist of aggregates of spheres in an initial stage of * Authors to whom correspondence concerning this paper should be addressed. † E-mail:
[email protected]. ‡ E-mail:
[email protected]. S0888-5885(96)00467-8 CCC: $14.00
sintering. Assuming that each particle is, on the average, in contact with n other particles of the same initial radius, the dependence of the surface area on porosity was obtained from geometrical considerations. In the diffusion equation, the problem of overlapping and decrease of surface area with reaction was solved by taking the surface area to be the average of the external surface (pore surface) and the reaction surface, and the gas concentration gradient in the product layer was assumed to be linear. Bhatia and Perlmutter (1983) and Sotirchos and Yu (1988) have presented a general overlapping model for gas-solid systems that exhibit first-order kinetics with respect to the gaseous reactant concentration. This model is derived from random pore structure models, developed for the case where the system is initially formed of an unreacted solid matrix with randomly distributed overlapping cylindrical pores (Bhatia and Perlmutter, 1980, 1981; Gavalas, 1980; Sotirchos and Yu, 1985). They considered that the initial porous structure can be represented by a population of particles of a given geometry in a certain size range, randomly distributed in space. Each grain gives rise to two concentric grains, the unreacted core and the solid product, that are allowed to overlap with other grains. The evolution of the surface areas of the grains is determined by Avrami’s model (1940) which states that on the average the increment in the volume enclosed by the actual (overlapped) system is only a certain fraction of the growth in the nonoverlapped system. The problem of reactant concentration was solved by assuming uniform concentration over surfaces of the same curvature in the product layer surrounding a certain grain and taking into account the decrease in surface area due to the overlapping of adjacent particles in the mass balance equation of gas diffusion in the individual grains. In all prior works, only single reactions were considered. In this paper, we propose a simple model for parallel noncatalytic gas-solid reactions, using the hexagonal close-packed model (HCPM) as a basis to estimate the evolution of the structure of the system. The initial state of the system can be arbitrary; i.e., the particles can be initially nonoverlapping (free powder) or partially sintered (pressed pellets). A new way to solve the mass balance equation for reactant gas diffusion in the case of overlapped particles is proposed. The evolution of the solid reactant conversion as well as product selectivity is examined, where both are functions of time and position in the compact. The use of © 1997 American Chemical Society
Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 1471 Table 1. Some Examples of Simultaneous Gas-Solid Reactions, along with Their Volume Expansion Factors, z
a
reaction
za
MnSO4(s) + CO(g) f MnO(s) + CO2(g) + SO2(g) MnSO4(s) + 4CO(g) f MnS(s) + 4CO2(g)
0.29 0.47
Wen and Wei, 1970
CuS(s) + 2O2(g) f CuSO4(s) 2CuS(s) + O2(g) f Cu2S(s) + SO2(g)
2.13 0.68
Mukherjee and Gangily, 1965
3FeO(s) + CO2(g) f Fe3O4(s) + CO(g) 2FeO(s) + CO2(g) f Fe2O3(s) + CO(g)
1.18 1.21
Habashi, 1969
3Fe2O3(s) + CO(g) f 2Fe3O4(s) + CO2(g) Fe2O3(s) + CO(g) f 2FeO(s) + CO2(g) Fe2O3(s) + 3CO(g) f 2Fe(s) + 3CO2(g)
0.97 0.83 0.47
Habashi, 1969
3Si(s) + 2N2(g) f R-Si3N4(s) 3Si(s) + 2N2(g) f β-Si3N4(s)
1.22 1.21
Moulson, 1979
2PbS(s) + 3O2(g) f 2PbO(s) + 2SO2(g) 2PbS(s) + 4O2(g) f 2PbSO4(s)
0.75 1.53
Szekeley et al., 1976
3UO2(s) + O2(g) f U3O8(s) 3UO2(s) + 1/2O2(g) f U3O7(s)
1.56b 0.98b
Habashi, 1969
Ti(s) + 1/2O2(g) f TiO(s) 4Ti(s) + 3O2(g) f 2Ti2O3(s)
1.22 1.47
ref
Density values taken from Weast (1986), neglecting thermal volume expansion. b Density values taken from Aronson et al. (1957).
the exhausted outer shell and the unreacted core of each solid particle. In addition, we assume that the particles are packed following the HCPM (Van Vlack, 1980) and that they are spherical in shape with the same initial radius, r0. The initial structure of the compact is defined by the parameter d, the distance between the centers of two adjacent particles, as shown in Figure 1. It is also assumed that diffusion in the macropores (i.e., interstices between particles) is unidirectional and that the process is isothermal. Diffusion in Individual Particles. For the spatially overlapped region, the mass balance equation for diffusion of gaseous reactant A in the product layer of each individual solid particle, assuming that the PSSA applies and diffusion occurs radially, is given by the dimensionless equation
(
)
dωA1 DeA d 2 λξ ) 0; d h - ξs < ξ < ξs dξ ξ2 dξ Figure 1. Schematic representation of the hexagonal close-packed model (HCPM).
the HCPM, although an idealization, may be appropriate for a variety of systems, especially those involving fine particles with a narrow size distribution, e.g., silicon nitride synthesis (Pigeon and Varma, 1993), reduction of manganese oxide with hydrogen (De Bruijn et al., 1980), and nickel oxide reduction with hydrogen (Szekely and Evans, 1971b). Some examples of simultaneous gas-solid reactions, along with their volume expansion factor z, are given in Table 1. The latter is defined as the ratio of stoichiometrically equivalent solid reactant to solid product volumes.
along with the boundary conditions (BCs)
dωA1 dξ λ
)
dξ
)
dωA2 ; ξ)d h - ξs dξ
(3)
(4)
where
{
h /2 (before overlapping) 1 for ξs e d λ ) 3d h h /2 (after overlapping) - 5 for ξs g d ξs In these equations,
Consider the system of noncatalytic gas-solid reactions
Bi0 )
occurring between the gaseous reactant A and solid reactant S, leading to solid products Pi and gaseous products Bi (i ) 1, 2, ..., n). The n reactions are assumed to occur simultaneously at the sharp interface between
Bi0 (ω - ωAs); ξ ) ξs ξs Ap
dωA1
Model Equations
νAiA(g) + νSiS(s) f νPiPi(s) + νBiBi(g), i ) 1, 2, ..., n (1)
(2)
(5)
kgAr0 CA rs r d , ωA ) , ξ ) , ξs ) , d h) (6) DeA CA0 r0 r0 r0
where kgA and DeA denote the mass transfer coefficient of reactant gas A in the gas film surrounding the particle and the effective diffusivity through the product layer, respectively. Further, d h and ξs are the dimensionless distance between the centers of adjacent particles and dimensionless external particle radius.
1472 Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997
As seen in eqs 2 and 4, it is assumed that after the overlapping of adjacent particles there is a reduction in the external surface area available for reactant gas flux by a factor λ (ratio of external surface area of the particle at a given time to the surface area of the particle if no overlapping had occurred). This reduction is accounted for in the region from the external surface (radius rs) to the point where there is no overlapping (d - rs), as shown in Figure 1. Since only radial diffusion is assumed, in order to keep the problem mathematically tractable, the amount of gas diffused is redistributed uniformly at radial position (d - rs), and the diffusion process toward the unreacted core continues as if there were no overlapping. This model appears to be closer to reality than considering that the diffusion is hindered both in the overlapped and the nonoverlapped regions, leading to use of the same diffusion equation in both regions (Lindner and Simonsson, 1981; Sotirchos and Yu, 1988). For diffusion in the nonoverlapped region, we have
(
)
DeA d 2 dωA2 ξ ) 0; ξc < ξ < d h - ξs dξ ξ2 dξ
(7)
Maalmi et al., 1995)
ξ3s ) z + (1 - z)ξ3c
where z is the overall volume expansion factor, which can be written in terms of vi, the volume fraction of product Pi, as follows
FPi|νSi|MS
n
1 ) z
vi ∑ F ν i)1
dωA2
n
νA DaiR h i(ωAc); ∑ i)1
) -(
i
dξ
ξ ) ξc
(9)
g(ωAc) ≡ -Bi0
∫01X(y,θ) dy
(10)
νA DaiR h i(ωAc) ∑ i)1 ξc
]
Bi0ξc 1 - (1 - 1/Bi0) ≡ f(ξc) (11) ξs The equation describing the movement of the unreacted core interface of each particle is given by
dξc
n
)dθ
∑ i)1
[
νSi Dai
νAi Da1
h i(ωAc) Dai R Daj R h j(ωAc)
(17)
Of more practical interest is the global (or integral) selectivity, defined as the ratio of the total amounts of products Pi and Pj formed during the course of the reaction up to time θ
Sij(y,θ) )
nPi nPj
n j Pi )
n j Pj
(18)
while the average selectivity value within the compact is
) n
[
(16)
The differential (i.e. instantaneous) selectivity of solid product Pi relative to solid product Pj is defined as the ratio of the reaction rates, at the sharp interface, of the two parallel reactions leading to the formation of these two products:
λωAp - ωAc
i
(15)
while its average value within the compact is
σij(y,θ) )
where Dai and R h i(ωAc) represent the Damkohler number and the dimensionless reaction rate for reaction i, respectively. From eqs 2-9, ωAc can be obtained by solving the following equation (cf. Cao et al., 1995).
(14)
i
X(y,θ) ) 1 - ξ3c
and
r0Ri(CA0) Dai ) DeACA0
vi
∑ i)1 z
In the analysis below, z is assumed to be constant during the course of the reaction. This assumption is generally valid, as discussed in detail elsewhere (Maalmi, 1996). The local conversion of the solid at position y is defined as
Xav(θ) ) (8)
n
)
S PiMPi
with BCs
h - ξs λωA1 ) ωA2; ξ ) d
(13)
]
R h i(ωAc)
Sav(θ) )
dθ
(12)
Product Selectivity and Volume Expansion The dimensionless unreacted core and external particle radii, ξc and ξs, are related by the equation (cf.
(19)
In eq 18, n j Pi is the dimensionless number of moles of product Pi, given by the stoichiometry of the reaction and eq 12 as follows:
dn j Pi
We consider here only the case ωAc > 0, since the possibility of ωAc ) 0 arises only when a reaction is of zero order. For the former, as discussed elsewhere (Maalmi et al., 1995), the reaction rate is not limited by mass transfer and the interfacial velocity is obtained by numerical integration of eq 12.
∫01Sij(y,θ) dy
) -3
νPi Dai R h (ω ) ξ2 νAi Da1 i Ac c
(20)
Mass Transfer in the Compact. The equation for diffusion of reactant gas in the macropores of the compact, assuming a slab geometry, can be written in dimensionless form as
∂ωAp (y,θ)
1 )
∂θ
(
)
∂ Deff(y,θ) ∂ωAp
+ c2ψ2 ∂y Deff(1,0) ∂y n c1 Dai S h a(y,θ) νAi R h i(ωAc) (21) c2 Da1 i)1
∑
Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 1473
(22)
where 0 and p denote, respectively, the initial porosity of the compact and the residual porosity (i.e., percolation threshold), and Dm is the combined molecular-Knudsen diffusivity
(23)
1 1 1 ) + Dm DA DK
along with the boundary conditions (BCs)
∂ωAp Deff(1,0) (1 - ωAp); y ) 1, θ > 0 ) Bie ∂y Deff(1,θ) ∂ωAp ) 0; y ) 0, θ > 0 ∂y
with
and the initial conditon (IC)
ωAp ) 1; 0 e y e 1, θ ) 0
MSCA0 4x2π ; c2 ) 3 FS d h
(25)
Bie is the mass Biot number for the compact and ψ is a modified Thiele modulus defined by
x
ψ)L
R1(CA0)
(26)
Deff(1,0) CA0r0
ωAc and ωAp are the dimensionless reactant gas concentrations at the unreacted core interface of the solid particles and in the pores of the compact, respectively, while (y,θ) is the compact porosity at postion y and time θ, obtained from geometrical considerations.
{
[]
4x2π ξs 3 ; 3 d h ) 3 ξs 20x2π ξs 1+ - 6x2π 3 d h d h 1-
[]
[]
ξs e
2
+
d h 2
h π d d h ; e ξs e x2 2 x3 (27)
The void area Av between adjacent particles, shown in Figure 1, is given by
Av r20
( )
) 3ξs2 arccos
πξs2 d h + 2ξs 2 d h2 [x3 - 3{(2ξs/d h )2 - 1}1/2] (28) 4
This expression can be simplified, using a Taylor expansion, to yield
Av r20
≈
d h (d h - x3ξs)2 ξs
(29)
which gives a simplifed expression for the macropore radius
reff )
( ) d h πξs
1/2
(d h - x3ξs)r0
(30)
The effective diffusion coefficient, Deff, is assumed to be given by a modified Archie’s law (1942), developed by Tomadakis and Sotirchos (1991) for random structures
[
x
(24)
In eq 21, c1 and c2 are constants which represent geometrical and chemical properties of the system and are defined by
c1 )
(32)
]
0 - p (y,θ) Deff(y,θ) ) Dm η(θ)0) (y,θ) - p
-γ
(31)
DK ) 9700reff
T MA
(33)
Results and Discussion Method of Solution. The partial differential eq 21 was discretized in space following an orthogonal collocation scheme (Villadsen and Michelsen, 1978; Finlayson, 1980). The resulting ODEs, along with the other ODEs describing the movement of the unreacted solid interface (eq 12) and the amount of solid products formed during reaction (eq 20) at each collocation point, were integrated in time numerically using marching techniques [Gear and Runge-Kutta (IMSL, 1989)]. The algebraic equations that yield the interfacial gas concentration, ωAc, at each collocation point (eq 11) were solved by a modified Newton-Raphson algorithm [Muller’s method and Powell’s method (IMSL, 1989)]. When the PSSA is used for diffusion in the macropores of the compact, eq 21 becomes an ODE that is discretized in the same way as without the PSSA, but the result is a system of nonlinear equations solved by Powell’s method (IMSL, 1989). For the single particle analysis, eq 21 was ignored. The numerical methods employed can be summarized as follows method
eq 21
eq 11
eq 12
eq 20
without m ODEs (m + 1) alg. eq. (m + 1) ODEs n(m + 1) ODEs PSSA with m alg. eq. (m + 1) alg. eq. (m + 1) ODEs n(m + 1) ODEs PSSA
where n and m are the number of reactions and internal collocation points, respectively. In the discussion below, for simplicity, n was taken equal to 2. The results of this discussion are also valid for n > 2, when only two of the products are of interest. The percolation threshold, p, for the HCPM is ∼0.04 (the porosity level at which the macropore system percolates and there is pore close-off), and γ and η(θ)0) were taken equal to 1 and 2, respectively. In practice, p is always higher than 0.04 because the particles are not perfectly spherical in shape, and both p and γ have to be determined experimentally. Analysis for a Single Particle. For the case of a single particle, there is no compact effect involving diffusion in the macropores, and λ and ωAp are both equal to unity in eqs 2-11. From eq 11, it can be noted that the unreacted core interface gas concentration, ωAc, and hence the selectivity profiles, depend on the shape of the two functions f(ξc) and g(ωAc) and on the evolution of the reaction interface position with time (eq 12). A detailed study of the function f(ξc) has been presented elsewhere (Maalmi et al., 1995) and shows that depending on the values of Bi0 and z, f(ξc) has different shapes and features as shown in Figure 2. The curves Bi1, Bi2, and Bi3 that define the different regions I-VIII of Figure 3, and correspond to the labeled curves in Figure 2, are discussed in that study. For g(ωAc), it is readily seen that its first derivative is negative for reactions
1474 Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997
Figure 2. Shapes of the function f(ξc) defined by eq 11. (a) z > 0.5, (b) z < 0.5.
Figure 4. Variety of possible profiles for global selectivity, S12, versus dimensionless time, θ (a1 < a2).
Figure 3. Classification of the Bi0-z plane into regions giving different shapes for f(ξc).
following R h i(ωAc) > 0; hence, it is always monotonically decreasing. A comparison of the functions f and g leads to the interfacial concentration, ωAc (Cao et al., 1995). In Figures 4 and 5 are summarized the different possible shapes of the global and differential selectivity curves versus dimensionless time, for fixed Damkohler numbers and reaction orders, for the case of two parallel reactions (n ) 2). We note that for regions III and VIII, S12 has a maximum, for regions I, II, IV, and V it decreases monotonically, and for VI and VII S12 goes through a minimum then a maximum. The difference between cases VI and VII is that the minimum in VII is an absolute minimum, whereas in VI it is local. The differential selectivity follows the same trends as S12 with more pronounced extrema, especially in regions III and VIII. The different slopes in the differential selectivity profiles between cases III and VIII, and between V and I and II and IV, arise from the different characteristics of the function f(ξc) in each of these regions. The shapes of curves in Figures 4 and 5 are reversed if the sign of (a1 - a2) changes.
Figure 5. Variety of possible profiles for differential selectivity, σ12, versus dimensionless time, θ (a1 < a2).
Analysis for a Compact of Particles. For the case of compacts, only the top layer of particles is expected to follow the classification discussed above for a single particle. Numerical calculations for a compact of particles were made following the procedure described above. In these calculations, while the other parameters were as defined in the captions for figures, the values
Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 1475
Figure 6. Effect of Thiele modulus, ψ, on the average solid conversion, Xav, in the compact and validity of the PSSA. Da1 ) Da2 ) 2.5; z ) 1.2; 0 ) 0.34; c2 ) 1.2 × 10-4; a1 ) a2 ) 1.
for Bi0 and Bie were maintained fixed at 1000 and 10, respectively. In Figure 6, the influence of the modified Thiele modulus, ψ, on the evolution of the average solid conversion in the compact, Xav, with and without use of the PSSA, is shown. It can be seen that the average solid conversion increases as ψ decreases, due to decrease in resistance to macropore diffusion within the compact. It is also seen that the difference in conversion obtained with and without PSSA for diffusion in the macropores becomes important as ψ increases above unity. For example, in Figure 6, the relative error in conversion introduced by using the PSSA at θ ) 0.8 is less that 1% for ψ ) 1, 6.7% for ψ ) 2, and 9% for ψ ) 4. This can be understood readily if analyzed in terms of characteristic times for reaction and diffusion in the compact. The characteristic time for compact diffusion is τc ) L2/Deff, while that for reaction is τr ) CA0r0/ R1(CA0), and their ratio τc/τr equals ψ2. Thus for ψ < 1, the reaction is slower than diffusion in the compact and the PSSA applies, while the converse is true for ψ > 1. In several studies reported in the literature, the PSSA
was used for diffusion in compacts without discussing its validity. The evolution of the solid conversion at various axial positions in the compact, for different volume expansion factors z, for the case of two reactions is shown in Figure 7. It is readily seen that, as expected, the average solid conversion in the compact decreases as z increases. It is also seen that the difference in conversion between the top and bottom layers of the compact becomes larger as z increases, due to the hindrance to diffusion in the compact caused by the expansion of particles. For z ) 2 (Figure 7d), the reaction ceases (θ ) 4.23) due to percolation before full conversion is reached, leaving the compact with a highly reacted outer surface and poorly reacted inner core. This type of behavior was observed experimentally in the case of reaction-bonded silicon nitride synthesis (Pigeon and Varma, 1993) and was described as a “watermelon skin” formation at the outer edge of the compact, although in this specific case the reaction between solid silicon and nitrogen gas does not cease completely but is dramatically slowed down. However, in the case of CaO-SO2 (Hartman and Coughin, 1976; Simons and Rawlins, 1980), the reaction ceases completely before full conversion is reached owing to the high volume expansion factor value (z ) 3.1). In Figures 8 and 9 are represented the evolution of the global selectivity S12 in the compact, for shrinking and expanding particles, for the cases a1 > a2 and a1 < a2, respectively (all other parameters are the same). In Figure 8, for both shrinking (z ) 0.75) and expanding particles (z ) 2), the difference in selectivity at various positions within the compact becomes smaller and the average selectivity decreases as the reaction proceeds. However, the corresponding unreacted core interface and macropore gas concentrations, which are represented in Figures 10 and 11, have different evolution for the two cases. For the case z ) 0.75, as particles shrink with reaction, more space is created within the
Figure 7. Effect of volume expansion factor, z, on the distribution of solid conversion, X, in a compact. Da1 ) 20; Da2 ) 10; 0 ) 0.34; ψ ) 2; c2 ) 1.2 × 10-4; a1 ) 2; a2 ) 1.
1476 Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997
Figure 8. Effect of volume expansion factor, z, on the distribution of selectivity, S12, in a compact (a1 > a2). Da1 ) 20; Da2 ) 10; 0 ) 0.34; ψ ) 2; c2 ) 1.2 × 10-4; a1 ) 2; a2 ) 1.
Figure 9. Effect of volume expansion factor, z, on the distribution of selectivity, S12, in a compact (a1 < a2). Da1 ) 20; Da2 ) 10; 0 ) 0.34; ψ ) 2; c2 ) 1.2 × 10-4; a1 ) 1; a2 ) 3/2.
compact and the unreacted core interface and macropore gas concentrations tend to uniformity (see Figure 10a,b), which makes all layers react more or less uniformly. For the case z ) 2, as the particles expand with reaction, the porosity becomes smaller, the resistance to diffusion becomes important, and the gas concentration is different from one layer to another, as
Figure 10. Evolution of reactant gas concentrations, ωAc and ωAp, along the compact with time, for a case of shrinking particles (z ) 0.75). Da1 ) 20; Da2 ) 10; 0 ) 0.34; ψ ) 2; c2 ) 1.2 × 10-4; a1 ) 2; a2 ) 1.
Figure 11. Evolution of reactant gas concentrations, ωAc and ωAp, along the compact with time, for a case of expanding particles (z ) 2.0). Da1 ) 20; Da2 ) 10; 0 ) 0.34; ψ ) 2; c2 ) 1.2 × 10-4; a1 ) 2; a2 ) 1.
seen in Figure 11. In Figure 10a, the unreacted core gas concentration, ωAc, in the top layers decreases due to the resistance to diffusion in the product layer that forms around the particles. For the lower layers, however, ωAp is initially low, the reaction rate is low,
Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 1477
Figure 12. Effect of the volume expansion factor, z on average global selectivity, Sav, in the compact. Da1 ) 20; Da2 ) 10; 0 ) 0.34; ψ ) 2; c2 ) 1.2 × 10-4; a1 ) 2; a2 ) 1.
the compact, Sav, for the specified parameters. It is seen that the selectivity decreases with an increase in either the volume expansion factor or the Thiele modulus, because in both cases the resistance to diffusion in the macropores increases. In Figure 12, all the curves start at the same point, because the effect of z is not sensed initially by the process. However, in Figure 13, they start at different points because of the difference in the degree of resistance to diffusion immediately from the start of the reaction. It is interesting to note that for ψ ) 4, the global selectivity S12 remains essentially constant with time, because the resistance to diffusion in the compact is high, which makes the gas concentrations ωAc and ωAp relatively small from the start of the reaction. Consequently, the selectivity remains almost constant during reaction because in this case (a1 > a2) the instantaneous selectivity is proportional to ωAc. Note, however, that for high values of ψ, the overall rate is low (cf. Figure 6), so that there is a trade-off between high reaction rate and variation of properties within the compact. Concluding Remarks
Figure 13. Effect of the modified Thiele modulus, ψ, on average global selectivity, Sav, in the compact. z ) 1.2; Da1 ) 20; Da2 ) 10; 0 ) 0.26; c2 ) 1.2 × 10-4; a1 ) 2; a2 ) 1.
and with the increase of ωAp, due to the decrease of resistance to diffusion in the macropores, ωAc increases; i.e., the increase in ωAc due to macropore diffusion is more important than its decrease due to reaction. Toward the end of the reaction, ωAc and ωAp are the same in all layers and tend to unity. In Figure 11a, ωAc decreases in all layers, although it increases slightly initially in the lower layers. Ιn Figure 11b, ωAp increases initially as in Figure 10b, but as the process continues the particles swell, and contrary to Figure 10b, where the particles shrink, the macropore gas concentration decreases except for the top layer (i.e., the exposed boundary) where the gas accumulates, due to the increasing resistance in the compact, until it reaches unity. Since the differential selectivity σ12 varies proportionally to ωAca1-a2 for power-law kinetics, S12 which is an integrated form of σ12, has an evolution similar to ωAc in Figure 8; i.e., S12 first increases and then decreases if ωAc increases and decreases. For the case of Figure 9, where a1 < a2, the profiles for ωAc and ωAp are similar to those shown in Figures 10 and 11, but S12 varies in an opposite manner to ωAc. It is interesting to observe in Figure 9 that for z ) 0.75, the difference in selectivity between the top and bottom layers of the compact is relatively small, varying from 4.5 to 5.7. However, for z ) 2, this difference is more important and varies from 5.5 to 9, which will result in a nonuniform product. Figures 12 and 13 show respectively the influence of the volume expansion factor, z, and the modified Thiele modulus, ψ, on the evolution of average selectivity in
The simplified model presented in this paper, based on the hexagonal close-packed model (HCPM), can be a useful tool to analyze the behavior of many noncatalytic gas-solid reactions, especially for parallel reactions where selectivity is an important property of the system. Other packing models (such as FCC, BCC, etc.), if they represent a given system better, can also be used. In this case, only the expressions of , λ, and reff need to be modified. For the case of a single particle, it was shown that the eight different regions of the Bi0-z plane give rise to four basic shapes of the global (Sij) and differential selectivity (σij) profiles. Depending on the sign of (ai aj), Sij and σij can be monotonically decreasing (increasing), have a maximum (minimum), or have two extrema (a minimum and a maximum). It was shown that the PSSA for the compact is valid only when ψ < 1, i.e., when the characteristic time for reaction is larger than the characteristic time for diffusion in the macropores of the compact. It was also shown that the selectivity can be strongly affected by the volume expansion factor and that the solid conversion as well as the selectivity can evolve nonuniformly in the compact. This makes full conversion, as well as a final product with uniform properties, difficult to achieve in some situations. The results of this analysis can be used to identify operating conditions where these deleterious effects can be avoided. Acknowledgment We gratefully acknowledge the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. Nomenclature a ) reaction order Av ) void area, defined by eq 29 Bi0 ) initial mass Biot number for a single particle, ShDA/2DeA Bi ) mass Biot number for a single particle, Bi0/ξs Bie ) mass Biot number for the compact, ShDA/Deff(1,0) h3 c1 ) constant, 4x2π/d c2 ) constant, MSCA0/FS C ) concentration
1478 Ind. Eng. Chem. Res., Vol. 36, No. 5, 1997 d ) distance separating the centers of two adjacent particles d h ) d/r0 DA ) molecular diffusivity Da ) Damkohler number, (R(CA0) r0)/(DeACA0) De ) effective diffusivity through the product layer Deff ) effective diffusivity in the macropores of the compact, defined by eq 31 DK ) Knudsen diffusivity, defined by eq 33 Dm ) combined diffusivity, defined by eq 32 f ) function, defined by eq 11 g ) function, defined by eq 11 ki ) reaction constant of reaction i kg ) mass transfer coefficient around a single particle kge ) mass transfer coefficient through the gas film around the compact L ) length of the compact M ) molecular weight ni ) number of moles of species i n j i ) dimensionless number of moles of species i defined relative to the initial number of moles of the solid reactant, (3MSni)/(4πr30FS) r ) particle radius at time t rc ) unreacted core radius r0 ) initial particle radius rs ) external particle radius ai Ri ) rate of reaction i ()kiCAc ) R h i ) dimensionless reaction rate, Ri(CA)/Ri(CA0) S h a ) ratio of the total surface area of the unreacted cores per unit lattice volume at y and θ, to the initial value Sav ) average global selectivity in a compact, defined by eq 19 Sij ) global selectivity of two solid products Pi and Pj, defined by eq 18 Sh ) Sherwood number, 2Kgr/DA t ) time T ) temperature vi ) volume fraction of product i X ) solid conversion, 1 - ξ3c Xav ) average solid conversion in a compact, defined by eq 16 y ) dimensionless position in the compact, Y/L Y ) position in the compact zi ) volume expansion factor for reaction i, (νPiMPiFS)/ (|νSi|MSFPi) z ) average volume expansion factor, defined by eq 14 Greek Letters R ) constant, 1 - 1/Bi0 γ ) parameter, defined in eq 31 ()1) ) porosity or compact porosity η ) tortuosity λ ) ratio of the nonoverlapped surface of a particle to its total surface in the absence of overlapping ()1 before overlapping) ν ) stoichiometric coefficient (0 for products) θ ) dimensionless time, MSR1(CA0)t/FSr0 F ) density σ ) differential selectivity, defined by eq 17 ω ) dimensionless gas concentration, C/C0 ξ ) dimensionless radius, r/r0 ψ ) modified Thiele modulus, defined by eq 26 Subscripts av ) average value A ) gas reactant B ) gas product c ) unreacted core surface i ) species i or reaction i K ) Knudsen diffusion
p ) macropores P ) solid product s ) external particle surface S ) solid reactant 0 ) initial value
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Received for review July 31, 1996 Revised manuscript received December 4, 1996 Accepted December 19, 1996X IE960467B
X Abstract published in Advance ACS Abstracts, February 1, 1997.
