Znd. Eng. Chem. Res. 1995,34,1114-1125
1114
The Sharp Interface Model: Zero-Order Reaction with Volume Change Mostafa Maalmi, b i n d Varma,* and William C. Strieder" Department of Chemical Engineering, University of Notre Dame, Notre Dame, Indiana 46556
The sharp interface model is examined for isothermal gas- solid reactions following zero-order kinetics with respect to the gas phase and first-order kinetics with respect to the solid phase for spherical particles either shrinking or growing during reaction. It is shown that depending on the relative magnitudes of the governing parameters which include the volume expansion factor, Damkohler number, and mass Biot number, it is possible to have up to three successive transitions from one controlling regime (external mass transfer, product layer diffusion, or chemical kinetics) to another as the reaction evolves with time. This behavior gives rise to various gas concentration and solid conversion profiles. A classification of all the possible situations is made. 1. Introduction
Noncatalytic gas-solid reactions are of substantial practical interest, and a variety of models have been developed since the pioneering work of Yagi and Kunii (1955) and Cannon and Denbigh (1957). The sharp interface model (sometimes called the shrinking core model) is commonly used, especially for nonporous solids. It has been studied widely in the literature for systems with or without particle size change during reaction for various reaction kinetics: first-order with respect to both solid and gas (Beveridge and Goldie, 1968;Wen, 1968;Ishida and Wen, 1968;Wen and Wang, 1970; Calvelo and Smith, 1970; Wen and Wei, 1971; Levenspiel, 1972; Szekely et al., 1976; Rehmat and Saxena, 1976), Langmuir-Hinshelwood type kinetics with respect to the gas and first-order with respect to the solid (Ramachandran, 1982; Erk and Dudukovic, 1984; Jayaraman et al., 1987; Cao et al., 19931, and second-, fractional-, and zero-order with respect to the gas and first-order with respect to the solid (Wen and Wei, 1971; Szekely et al., 1976). Also, various features of reactions following arbitrary monotone increasing kinetics, including dynamics of the interfacial reaction front, have recently been presented (Cao et al., 1995). The case of a zero-order reaction with respect t o the gas and first-order with respect to the solid, which applies for a strongly adsorbed gas on the solid (Wen, 1968; Dudukovic and Lamba, 1978), has received relatively little attention in the literature. An example of this type is the reaction between acetylene and Cu2O studied by Tamhankar et al. (1981). The gasification of solid carbon by steam (Wen et al., 1967) is also approximately zero-order with respect to the steam, and the CVD of silicon from a dichlorosilane-N2-H2 mixture is also approximately zero-order with respect t o Hz at low temperatures (Claassen and Bloem, 1980). Simonsson (1979) used the sharp interface model for the reduction of fluoride in wastewater by limestone, which has a zero-order dependence on the fluoride concentration in the fluid phase. This study was limited to the case of reaction with no structural change or external mass transfer resistance, and a transition from kinetic to diffision control during the course of the reaction was observed. However, as will be shown in the present
* Correspondence concerning this paper should be addressed to either A. Varma (E-mail:
[email protected])or W. C. Strieder (E-mail: william.c.strieder.
[email protected]). 0888-588519512634-1114$09.00/0
Table 1. Volume Expansion Factor, z , for Some Gas-Solid Reactions reaction 2 reference charcoal + 0 2 HzO CO + Hz 0.10 Rehmat and Saxena, 1980 0.29 Wenand MnSOd(s) CO(g) MnOW + Wang, 1970 3coz(g) + soz(g) FezOs(s) + 3CO(g) 2 Fe(s) 3coz(g) 0.47 Wen, 1968 ZnS(s) 302(g) ZnO(s) + 2SOz(g) 0.59 Cannon and Denbigh, 1957 FeSz(s) llOz(g) FezOds) + 8SOz(g) 0.62 Wen, 1968 cOo(S)+ sos(g) cOsoo(g) 1.19 Habashi, 1969 3Si(s) + 2Nz(g) Si3Nds) 1.22 Moulson, 1979 CaO(s) + HZS(g) CaS(s) + HzO(g) 1.53 Wen and Wang, 1970 Ni(s) l/ZOz(g) NiOW 1.70 Carter, 1961 2Fe(s) + Oz(g) 2FeO(s) 1.77 Wen, 1968 UOds) 4HFW UFds) + 2HzO(d 1.90 Calvelo and Smith, 1970 2.22 Dhupe et al., 1987
-
+
+
+ +
+ +
--
+
-. -
--
-
study, it is possible to have up to three successive transitions from one regime to another, depending on the magnitudes of three parameters (i.e. the Damkohler number, mass Biot number, and particle volume expansion factor). The case of smaller values of the solid product to reactant molar volume ratio (expansionfactor z 1)has also not been examined previously in the literature. Table 1 shows that many important reactions fall in this category. We will establish that such reactions have certain features which are not present at higher volume expansion factors (z > 1). The objective of this study is to examine all possible combinations of chemical kinetics, internal diffusion, and external mass transfer rates for the case of a zeroorder reaction with respect t o the gas and first-order with respect to the solid. The analysis is carried out using the sharp interface model for a spherical particle undergoing size change during reaction, for the entire range of the three governing parameters cited above, and provides a comprehensive picture of the gas reactant concentration profile as well as the solid conversion with time. While we limit the analysis to isothermal particles, this assumption is valid for the majority of cases where the solid reactant is relatively nonporous and the solid product has little porosity, even when the reaction is highly exothermic (Mazet, 1992). 2. The Model Equations
Consider a noncatalytic gas-solid reaction of general
0 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34, No. 4,1995
1115
External boundary layer
/
I I I \ \
I I
.--LA. I
concentraI
I
rO
0
li: ro rs
radial distance
Figure 1. Schematic representation of the shrinking core model. “his illustrates how the unreacted core of a particle shrinks and the product layer grows with time.
stoichiometry
above are
occurring between the gaseous reactant A and solid reactant S, leading to gaseous and solid products B and P, respectively. The reaction is assumed to occur a t the sharp interface between the exhausted outer shell and the unreacted core of the solid (see Figure 1). In addition, we assume that the pseudo-steady state approximation applies, that the solid particle has a spherical shape, that the concentration of the solid reactant is uniform, that the reaction is zero-order with respect to the gas and first-order with respect to the solid, and that the process is isothermal. 2.1. Mass Balances in the External Boundary and Product Layers. The mass balance for diffusion of gaseous reactant A in the product layer is given by the following dimensionless equation
where kgA, DeA, and DA denote the mass transfer coefficient of the reactant gas A in the gas film surrounding the particle, the effective difisivity through the product Layer, and the molecular diffusivity of A, respectively. R(CUA~) is the dimensionless reaction rate, and for the ~ 0. case of a zero-order reaction, it equals 1for C U A > From the definitions of Biot number Bi and Sherwood number Sh, it is readily seen that the mass transfer coefficient kgA, and hence the Biot number, depends on the external radius of the particle as follows:
where the initial Biot number is the constant: along with the boundary conditions (BCs)
(7)
(3) From eqs 2-4, we have
and
I
(4)
The dimensionless quantities involved in the equations
and the concentration of reactant gas C U Aas ~ a function
1116 Ind. Eng. Chem. Res., Vol. 34, No. 4,1995 of the position of the reacting interface
5, is given by
transfer through the gas film surrounding the particle or by diffision through the product layer. In either case, the BC (eq 4) is replaced by
(=tc
w A = 0 at where a is a constant given by
a = 1 - l/Bio
The solution of eq 2 with BCs 3 and 17 is given by (10)
There is a relationship which arises from the reaction stoichiometry and the densities of the solid reactant S and product P and leads to the following relation between the dimensionless unreacted core and external particle radii:
693 = z + (1- 2)tc3
(18) The equation describing the movement of the interface is now given by equating the rate of consumption of the solid with the rate of diffusion of the reactant gas through the product layer or the gas film, i.e.
(11)
where z is the volume expansion factor, defined as the ratio of stoichiometrically equivalent volumes of solid product to solid reactant. The reaction conversion is defined as that of the solid:
X=l-5,3
and
(12)
2.2. Equations Describing Movement of the Unreacted Core Interface. The equation describing the movement of the unreacted core interface is given by the relation between the rate of consumption of solid reactant and, when mass transfer is not limiting (Le. wAc> 01, the rate of consumption of the gas by reaction. For the solid we have
(13)
and combining eq 18 and eq 19a or 19b leads to
(20) The integration of eq 20 gives
o=o*Da 1-2
while from the stoichiometry of the reaction dii, - vs dfiA --3g d0 V A d0
(14)
where iis and f i are ~ the dimensionless number of moles of the solid and gas reactants defined relative to the initial number of moles of the solid reactant and 0 is the dimensionless time. Equations 13 and 14 lead to the following equation describing the movement of the unreacted core (15)
(i)Constant Interfacial Velocity (CIV) Condition In the case of WAC > 0, the zero-order reaction rate is constant and is not limited by mass transfer. Under these conditions, the interface velocity given by eq 15 is also constant and upon integration gives a linear variation of the unreacted core radius with time:
and
where O*, &*, and &* are the time, reaction interface position, and external particle radius, respectively, at the onset of the mass transfer limiting regime. The mass transfer limiting regime has two asymptotic cases. The first occurs for Bio -, so that reactant gas diffusion through the product layer controls. Then from either BC (eq 3) or eq 8,
-
(OA, > 0).
o = o * - (Ec - 5,")
(17)
OAs
also from eq 10, a by
-
-.
(22)
1. From eq 18, OA is then given
(16)
where O* and &* are the time and reaction interface position at the onset of the CIV regime. (ii) Mass Transfer Limiting Regimes (OA, = 0). In the case of a zero-order reaction in a porous catalyst pellet, above a critical Thiele modulus, a dead zone develops when the fluid reactant concentration drops to 0 ( h i s , 1975). Here, when the reactant gas concentration at the reaction interface becomes 0, the process continues but is controlled either by external mass
(23)
and the movement of the interface is given by eq 21 by substituting a = 1. In the other asymptotic case, Bio is small and mass transfer through the external film surrounding the particle controls the rate. In this case, w h 0 from eq 18 and, hence, the reactant gas concentration, W A 0
- -
everywhere within the particle. Also, a the a terms dominate in eq 21.
-- 1 B i o
Ind. Eng. Chem. Res., Vol. 34, No. 4, 1995 1117 so that
1.5
3. The General Behavior of Function figc) 1
We note from eqs 8 and 9 that the gas reactant concentration profile depends on the shape of the function A&), the ratio BidDa,and the evolution of 6, with time. Hence, a careful study of A&) is necessary prior to any computation (Cao et al., 1993 and 1995). In this section, the possible shapes of A&) are investigated for all possible values of z and Bio. From eq 9 for f and eq 11 for &, we have
/
,
,
,
.
I
.o
h
0.5
0.0
2.0
and note that
for all values of Bio and z. The first and second derivatives of A&) are.given as
1 .o
0.0
0
0.5
1
5, Note that f '(0) = Bio is always positive, while f '(1)= 1 - z (Bio - 1)changes sign from positive to negative as Bio increases across
for any positive volume expansion factor z . Since for all z > 0.5, the second derivative f "(&I is always positive below and negative above the unit value of Bio, the plot of A&) versus the reaction interface coordinate Ec can be constructed easily. The curves in Figure 2a are labeled from the bottom to the top for increasing Bio. For Bio = 1, A&) is a straight line of unit slope. For Bio < Bil, the slope ofA&) is positive a t both ends, and from either f "(&) > 0 for 0 < Bio 1orff'(&) < 0 for 1 < Bio < Bil, no extremum in Atc)can exist. For Bio > Bil, the slope off at eC= 1becomes negative and f"(&) 0, so a single maximum in f ( & ) above the value of unity must occur. In the Bio-z plane (Figure 3), the single maximum region is labeled I11 and the region with no extremum is labeled I below and I1 above Bio = 1. For 0 < z < 0.5, from eq 26, f " ( & ) has a single 0 at the inflection point z
Figure 2. (a) Shapes of the function fl&) defined by eq 24 for z > 0.5 and ascending values of the Biot number: I, Bio < 1; 11, 1 < Bi, < Bil; 111,Bio > Bil. (b) Shapes of the function f(&) defined by eq 24 for z < 0.5 and ascending values of the Biot number: Iv, Bi, < 1;v, 1 < Bio < Biz; VI,Biz < Bio < Bi3; VII, Bi3 < Bio < Bi1; VIII, Bio > Bil.
0.01, 0
I
IV I
,
I
,
.
0.25
1
I
I
;
.
I
,
0.5
.
,
,
0.75
,
.
,
, 1
,
,
,
,
1
1.25
z Figure 3. Classification of the Bio-z plane into regions giving different shapes for f(&).
The behavior of fiec)depends on the sign of the slope at the inflection point. From eqs 25 and 28,
113
E,'= (K) and f "(&I goes from positive to negative for Bio < 1and negative to positive for Bio 1,as EC varies from 0 t o 1. As before, the case Bio = 1is a straight line of unit slope.
and with increasing Biot number, f '(52) crosses from
1118 Ind. Eng. Chem. Res., Vol. 34,No. 4, 1995 I'
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1 ............... 1 ............................... .
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f m a x ................................................
1 4...................
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8
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.......................
mm
fmin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.I
13
10 , .
1-
0.5
Volume expansion factor, z Figure 4. Summary of the various cases that arise from different values of z and the ratio BidDa.
1 " ' " " " ' " " " " ' t I i n ..........
v,
0'
,
,
,
,
,
,
,
,
,
,
,
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,
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0.5
L,
k c Or O A c
Y 0' kC3
d m a x
5c2
5cl
0.5
kc or
1
WAC
Figure 5. Shapes of the functions fitc) and g ( o h )given by eq 9 versus tCand W A ~respectively. , This figure serves as an illustration of how the shape of function f for case 111-5of Figure 4 affects the solution of eq 9.
Figure 6. Shapes of the functions Atc) and g ( o A c ) given by eq 9 versus & and ~ hrespectively. , This figure serves as an illustration of how the shape of function f for case VII-8of Figure 4 affects the solution of eq 9.
positive to negative at
on either side of ti. The curve for which fmax = 1 is represented by Bi3. For Bio > Bil as for z > 0.5, we encounter the single maximum region (region VIII).
1
BZ0 = 1 -
24/3(1- z)1/3]-1
= Bi, (30) 3 In Figure 2b, for the lowest three curves, we have 0 < Bio < Biz, all three derivatives f ' ( O ) , f'(l),and f'(6c') are positive, and no relative extremum exists. For Bio > Biz, the inflection point derivative changes sign, f ' ( t 2 ) < 0, a local maximum fmax must exist between 0 and 52, and as f '( 1)remains positive, a local minimum fmin occurs to the right of the inflection point 6 : < &< 1. While the local maximum value fmax is initially less than 1, as Bio increases, fmax also increases and crosses unity at Bi3. Finally, in the uppermost curve of Figure 2b, both f'(1) and f'(e2) are negative and a global maximum in A&) above unity is observed. In the Bio-z plane (Figure 3), for 0 < z < 0.5, the two lowest regions IV and V are below Biz, so there is no extremum. For Biz < Bio < Bil in regions VI and VII, two local extrema, a maximum and a minimum, exist
4. The Variety of Gas Reactant Concentration
and Solid Conversion Profiles Depending on the shape of function A&) and the ratio BidDa, a variety of gas concentration and solid reactant conversion profiles arise. They can be classified as shown in Figure 4, where each case exhibits unique characteristic features. For brevity, in the following, we present details of only two cases, the first corresponding to case 111-5 and the second to case VII-8 of Figure 4. In all cases, we are interested in describing how oh and & (hence solid conversion,X)evolve in time. Case 111-5. This corresponds to the case where Bio > Bil, z > 0.5, and BidDa < 1. Using Figure 5 , we note from eq 9 that since A&) = g ( W A c ) for a solution to exist, there is no solution from & = 1 to a specific value &I
Ind. Eng. Chem. Res., Vol. 34,No. 4,1995 1119
Time, 8
Time, 8
Time, 8
Time, 8
t
/
Time, 0 Figure 7. Evolution of the interfacial reactant gas concentration W A ~ unreacted , core radius &, and solid conversion X versus time for case 111-5.
that is numerically determined from eq 9, with W A set ~ equal to 0. During this stage, there is a mass transfer limitation and the reactant gas concentration a t the reaction interface oh remains equal to 0. In the second stage (from to 01,external and internal mass transfer as well as reaction are all significant, so o~~is given by the solution of eq 9 and the CIV condition prevails. Thus, for & = 1 to tC= &I, we have
Time, 8 Figure 8. Evolution of the interfacial reactant gas concentration O A ~ ,unreacted core radius &, and solid conversion X versus time for case VII-8.
Then for the interval
&I > i$
>
0, from eq 9,
(33) and from eq 15, with &* = &I at O* = 81 given by eq 32,
e = el - (tC- tCl) and with &* = 1 at O* = 0, eq 21a yields
(34
Case VII-8. This corresponds to the case where z < 0.5, Bi3 < Bio < Bil, and &in(&) -= BidDa 1. From
1120 Ind. Eng. Chem. Res., Vol. 34,No. 4,1995
; 3 c- 01 3
1 :
.d m
E
3
E e, 0 E
ii
C 0
'1. Time, 8 Figure 9. Variety of possible profiles for the gas concentration at the unreacted core interface Table 2. Interfacial Gas Concentration (ah)and Unreacted Core Radius (&) Profiles for Each Region as Classified in Figure 4O VI-10 1-1 VI-11 1-2 VI-12 11-1 VI-13 11-2 VII-6 111-3 VII-7 111-4 VII-8 111-5 VII-9 IV-1 VIII-3 Iv-2 VIII-4 v-1 VIII-5 V-2
(UJA~)versus
dimensionless time 0.
where
Third stage:
e=e,-
The first letter (lowercase) refers to the O J A ~profile shown in Figure 9, while the second letter (uppercase) denotes the tCprofile shown in Figure 10.
Figure 6, we see that eq 9 has no solution from 5, = 1 to a given value &I, during this stage there is mass transfer limitation, and O A ~is constantly equal to 0. In the second stage (from to gc2), OA, is given by eq 33 and CIV condition prevails. During this period, it may be seen from Figure 6 that LOA, first increases from 0 t o WA,," along the straight line g ( O A c ) and then returns to 0 again. In the third stage (from 5,2 t o &3), there is again mass transfer limitation and OA, remains equal to 0. In the last stage (from 5,3 to O), WA, increases monotonically from 0 to 1and the CIV condition prevails again. Finally, note that &I, 5,2, and &3 are all determined numerically from eq 9, with OA, set equal to 0. In the four stages, expressions for the reactant gas interfacial concentration W A ~the , interfacial position &, and time 6 are given as follows:
where
Fourth stage:
where
(41) In Figures 7 and 8 are shown the evolution of reactant gas concentration at the reaction interface (oAJ, the unreacted core radius (&I, and the solid conversion (XI with time ( 6 ) for the two cases discussed above. In Figure 7,we note that WA, remains equal to 0 until because of mass transfer control. Toward the end, owing to the loss of solid reactant surface area, the reaction slows down sufficiently, allowing the gas concentration to start building up. In Figure 7b, the change in slope at may be noted. The four stages described above are observed in Figure 8. In Figure 8b,
Ind. Eng. Chem. Res., Vol. 34, No. 4, 1995 1121
Time, 8 Figure 10. Four distinct profiles (A-D) of unreacted core radius & showing all possible combinations of operating regimes: CIV, constant interfacial velocity regime; MT, mass transfer control regime.
the decrease of the unreacted core radius occurs with four different rates: The first corresponds to mass transfer control from eC= 1to &I, the second linear slope corresponds to the CIV regime from &I to &2, when the gas is completely depleted by the reaction there is mass transfer (both external and internal diffusion) control once again from Ec2 to &3, while in the last stage ( t C 3 to 01, the gas concentration rises again and the slope is identical to the second one and linear. Summary of Possible Reaction Behavior. The main parameters of interest in this study are the BiotDamkohler number ratio (BidDa) and the expansion factor (z). For gas-solid reactions in general, the ratio BidDa (which is equal to ShDACAd2r&CAO)) can have large values (10-103) as in the case of a low reaction rate, small particle sizes, or a high initial concentration of the gas reactant. This ratio can also be small (10-l1)when CAOis small, when R ( C A ~is)high, or when DA is small as in the case of diffusion of reacting gas through a liquid phase t o reach the solid surface (Simonson, 1979; Tamhankar et al., 1981) and also in the case of large particles. The volume expansion factor, z can also vary, as shown in Table 1. From Figure 4, we note that there are a total of 22 possible cases for the case of a zero-order reaction depending on the Biot-Damkohler number ratio (Bid Da) and the expansion factor ( 2 ) . As shown in Figure 9 and Table 2, some of these cases give rise to the same shape of the O A ~versus 0 curve, leading to 12 different profiles. Nevertheless, there are only four distinct physical situations (A-D, see Figure 10) regardless of the shape of the W A ~ - O curve, and they reflect how the unreacted core radius & (hence solid conversion, X)
evolves with time. (i) In A, there is no external nor internal mass transfer limitation and the unreacted core radius decreases linearly in time. Also, O A ~has the i.e. W A has ~ a maximum (minireverse shape of fitc), mum) if fc&) has a minimum (maximum) (1-1,11-1,IV1, V-1, 111-3,VIII-3, VII-6, and VI-10). (ii) In case B, there is initially mass transfer limitation and OJA~= 0 until a certain location where the reaction slows ~ to build up down, CIV condition prevails, and W A starts (1-2, 11-2, IV-2, V-2, 111-5, VIII-5, VII-9, VI-11, and VI13). For this case, the unreacted core radius versus time profile exhibits a positive or negative-positive curvature followed by a steeper straight line. (iii) In case C, initially the CIV condition holds up to a certain location &I where W A drops ~ to 0, and then mass transfer is limiting from to &2 with W A ~= 0, while toward the end, the reaction slows down and CIV condition again prevails (111-4, VIII-4, and VII-7). The unreacted core radius decreases with a linear slope under the CIV condition, while the mass transfer control region is characterized by a negative-positive curvature. (iv) The final case D occurs only for z < 0.5. Initially, mass ~ 0 until EC = transfer is the limiting process and W A = $1. Since Bi = BidE,, the mass transfer coefficient k g A increases as the process evolves, and a t tc= &I, it becomes large enough so that mass transfer is not limiting any more. Thus, from &I to &2, there is no mass transfer limitation. At Ec2, the gas is depleted by ~ to 0 again, and mass transfer the reaction, W A drops becomes the controlling step from &2 to &3. At &3, the reaction becomes slower and the interfacial gas concentration WAC builds up again to unity (VII-8 and VI-12).
1122 Ind. Eng. Chem. Res., Vol. 34,No. 4,1995 21.0
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Time, 8
Figure 11. Influence of the expansion factor z on the interfacial gas concentration, the unreacted core radius, and the solid conversion for an example of case A of Figure 10 (Bio = 10,Da = 2). This illustrates the fact that z has no influence on conversion in this case.
Figure 12. Influence of the expansion factor z on the interfacial gas concentration, the unreacted core radius, and the solid conversion for an example of case C of Figure 10 (Bio = 10,Da = 5 ) . This illustrates how z affects conversion due to mass transfer limitation.
Figures 11 and 12 show the influence of the volume expansion factor z on the interfacial gas concentration, unreacted core radius, and solid conversion profiles for cases A and C, respectively. In Figure l l a , note that all four curves start at the same concentration OJA~.The product layer thickness always increases with reaction, although the magnitude of the increase diminishes as z decreases. On the other hand, following eq 6, the external mass transfer coefficient increases (decreases) with reaction for z < 1 (> 1). The interplay of these two effects is exhibited in the curvature and slope of ~ In Figure llb,c, we observe no influthe W A profiles. ence of z because in case A, the movement of the sharp
interface is CIV-controlled and is given by eq 15 which is independent of z . However, when z increases in Figure 12 (case C),the reaction becomes slower (the solid conversion decreases) due to the mass transfer limitation, and W A remains ~ equal to 0 over a longer period of time. In Figure 12b, the slope is linear and independent of z in the first stage, the slope decreases in the second stage, and in the last stage where CIV controls again the slope is higher and identical to the first stage. The influence of Biot number on the reactiondiffision process for an example of case B and for specific values of z and Da is shown in Figure 13. As
Ind. Eng. Chem. Res., Vol. 34,No. 4, 1995 1123
.$
0.5
0
0
Time, 8
Time, 0
Time, 8
Time, 8
X
I
C
0
. I
v)
t
2 5 C
0
0
P
. I
0
v)
0.0
l-
0
1.875
o . o , . . , , 0
3.75
Time, 8 Figure 13. Influence of Bio on the interfacial gas concentration, the unreacted core radius, and the solid conversion for an example of case B of Figure 10 (z = 0.9,Da = 15).
Bio decreases, the period for which WAC remains 0 widens and hence, the conversion decreases which is expected owing to the additional resistance to mass transfer. The influence of Da, for an example of case C and for fixed values of z and Bio, is shown in Figure 14. As Da increases, the mass transfer control period becomes larger and the solid conversion decreases. This is understandable since as Da increases, the intrinsic reaction rate increases, accelerating the onset and duration of the second stage which is mass transfercontrolled. As can be Seen from Figures 5 and 6,ifg(0) is not finite (R(0)= 0), which is the case for a reaction that is not zero-order with respect t o the gas reactant, there is always a solution to eq 9. The transitions discussed
,
.
.
I
0.8
I
1.6
Time, 8 Figure 14. Influence of Da on the interfacial gas concentration, the unreacted core radius, and the solid conversion for an example of case C of Figure 10 (z = 0.75, Bio = 10).
above do not occur then, and the evolution of WAC follows a pattern similar to case A but with a time-varying interfacial velocity given by
in Place ofeq 15. In this case, from eq 9, the function is no longer a straight line, although it retains its character as a monotone decreasing function for > 0. Also note from normal reactions following R’(wA~) eq 9 that the function f l & ) is independent of reaction kinetics.
&?(WAC)
1124 Ind. Eng. Chem. Res., Vol. 3 4 , No. 4, 1995
5. Concluding Remarks
Greek Letters
Following the sharp interface model, a spherical particle undergoing product layer expansion or shrinkage during a n isothermal reaction with a zero-order dependence on the gas phase and first-order dependence on the solid phase exhibits a variety of interfacial gas concentration and solid conversion profiles. However, when these are translated into how the reaction interface shrinks L e . solid conversion increases) with time, then there are only a total of four different types of behavior possible. Each type is characterized by a specific shift between the constant interfacial velocity (CIV) and mass transfer (MT) control regimes and occurs for a different range of the three governing physicochemical parameters: the volume expansion factor z , the Damkohler number Da,and the mass Biot number Bio. It is shown that up to three successive transitions from one regime to another can occur during the reaction-mass transfer process and that the CIV regime is approached asymptotically in all situations. It is also shown that the shape of the function /I& holds) the key to understanding the physical behavior of the process. When is monotone increasing, then there is a possibility of only one transition from the MT to the CIV regime; when figc) exhibits a maximum, then two successive regime transitions can occur, from the CIV to the MT and then again to the CIV regime. has two extrema, then three succesFinally, when sive regime transitions a r e possible from the model, from MT to CIV twice (see Figure 10). One may expect that t h e behavior of expanding (z > 1)or shrinking particles ( z < 1)would be different, but as shown in this work, the critical value of the volume expansion factor z is 0.5.
a = constant, (1 - 1/Bi0)
fccc)
A&)
Acknowledgment We gratefully acknowledge the donors of the Petroleum Research Fund, administered by t h e American Chemical Society, for partial support of this research.
Nomenclature
Bi
= mass Biot number, defined by eq 5
C = concentration D = molecular diffusivity
Da = Damkohler number, defined by eq 5 De = effective diffusivity through the product layer f = function defined by eq 9 g = function defined by eq 9 ko = reaction rate constant k, = mass transfer coefficient M = molecular weight n, = number of moles of species i ii, = dimensionless number of moles of species i defined relative to the initial number of moles of the solid reactant, 3 M s n , l ( 4 z r o 3 ~ s ) R = reaction rate OCA' = ko) R = dimensionless reaction rate, R(CA)/R(CAO) = 1 for CA 0 r = particle radius at time t rc = unreacted core radius ro = initial particle radius rs = external particle radius t = time X = solid conversion, defined by eq 1 2 z = volume expansion factor, V P M P Q ~ V & S Q P
’
v = stoichiometric coefficient radius, defined by eq 5 o = dimensionless gas concentration, defined by eq 5 0 = dimensionless time, v & f & o t / v ~ s r o Q = density
5 = dimensionless
Subscripts A = gas reactant B = gas product c = unreacted core surface max = maximum min = minimum P = solid product s = external particle surface S = solid reactant 0 = initial value Superscript i = inflection point
Literature Cited Aris, R. The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts; Oxford University Press: Oxford, 1975; VOl. I. Beveridge, G. S. G.; Goldie, P. J. Effectiveness Factors and Instability in Non-Catalytic Gas-Solid Reactions: the Effect of Solid Heat Capacity. Chem. Eng. Sei. 1968,23,913-929. Calvelo, A.; Smith, J. M. Intrapellet Transport in Gas-Solid NonCatalytic Reactions. Chemeca ‘70,Conf 1970,No. 3,1-24. Cannon, K. J.; Denbigh, K. G. Studies on Gas-Solid Reactions I. Chem. Eng. Sei. 1957,6,145-154. Cao, G.; Varma, A.; Strieder, W. Approximate Solutions for Nonlinear Gas-Solid Noncatalytic Reactions. AZChE J. 1993, 39,913-917. Cao, G.; Strieder, W.; Varma, A. Analysis and Shape Inequalities for Nonlinear Gas-Solid Reactions with Changing Volume. AZChE J. 1995,41,324-336. Carter, R. E.Kinetic Model for Solid-state Reactions. J. Chem. Phys. 1961,34,2010-2015. Claassen, W. A. P.; Bloem, J. Rate-Determining Reactions and Surface Species in CVD of Silicon. J. Cryst. Growth 1980,50, 807-815. Dhupe, A. P.; Jayaraman, V. K.; Gokarn, A. N.; Doraiswamy, L. K. An Experimental Study of the Effect of Inerts on Gas-Solid Reactions. Chem. Eng. Sei. 1987,42,2285-2290. Dudukovic, M.P.; Lamba, H. S. A Zone Model for Reactions of Solid Particles with Strongly Adsorbing Species. Chem. Eng. Sei. 1978,33,471-478. Erk, H.F.; Dudukovic, M. P. Self-Inhibited Rate in Gas-Solid Noncatalytic Reactions. The Shrinking Core Model. Znd. Eng. Chem. Fundam. 1984,23,49-54. Habashi, F. Principles of Extractive Metallurgy; Gordon and Breach New Yo&, 1969;Vol. I. Ishida, M.; Wen, C. Y. Effectiveness Factors and Instability in Solid-Gas Reactions. Chem. Eng. Sei. 1968,23,125-137. Jayaraman, V. K.; Hlavacek, V.; Puszynski, J. An Expanding Core Model for a Heterogeneous, Noncatalytic, Gas-Solid Reaction. Znd. Eng. Chen. Res. 1987,26,1048-1050. Levenspiel, 0.Chemical Reaction Engineering; John Wiley & Sons: New York, 1972. Mazet, N. Modeling of Gas-Solid Reactions 1. Nonporous Solids. Znt. Chem. Eng. 1992,32,271-284. Moulson, A.J. Reaction-Bonded Silicon Nitride: Its Formation and Properties. J. Muter. Sei. 1979,14,1017-1051. Ramachandran, P. A. Analysis of Non-Catalytic Reactions Following Langmuir-Hinshelwood Kinetics. Chem. Eng. J . 1982, 23,223-226. Rehmat, A,; Saxena, S. C. Single Nonisothermal Noncatalytic Solid-Gas Reaction. Effect of Changing Particle Size. Znd. Eng. Chem. Process Des. Dev. 1976,15,343-350. Rehmat, A.; Saxena, S. C. Agglomeration of Ash in Fluidized-Bed Gasification of Coal by Steam-Oxygen(or air) Mixture. Znd. Eng. Chem. Process Des. Deu. 1980,19, 223-230.
Ind. Eng. Chem.Res.,Vol. 34,No. 4, 1995 1125 Simonsson, D. Reduction of Fluoride by Reaction with Limestone Particles in a Fixed Bed. Znd. Eng. Chem. Process Des. Dev. 1979,18,288-292. Szekely, J.; Evans, J. W.; Sohn, H. Y. Gas-Solid Reactions; Academic Press: New York, 1976. Tamhankar, S. S.; Gupte, S. P.; Chaudhari, R. V. Kinetics of a Non-Catalytic Slurry Reaction: Reaction of Acetylene with Cuprous Oxide Suspended in Water. Chem. Eng. J. 1981,222, 15-24. Wen, C. Y. Noncatalytic Heterogeneous Solid Fluid Reaction Models. Znd. Eng. Chem. 1968, 60, 34-54. Wen, C. Y.; Wang, S. C. Thermal and Diffusional Effects in SolidGas Reactions. Znd. Eng. Chem. 1970, 62, 30-51. Wen, C. Y.; Wei, L. Y. Simultaneous Nonisothermal Noncatalytic Solid-Gas Reactions. AZChE J. 1971, 17, 272-280.
Wen, C. Y.; Abraham, 0. C.; Talwalker, A. T. A Kinetic Study of the Reaction of Coal Char with Hydrogen-Steam Mixtures. Adv. Chem. Ser. 1967, 69, 253-271. Yagi, S.; Kunii, D. Studies on Fluidized-Solids Reactors for Particles with Decreasing Diameters. Chem. Erg., Jpn. 1955, 19,500-506.
Received for review June 23, 1994 Revised manuscript received January 10, 1995 Accepted January 31, 1995@ IE940392G Abstract published in Advance ACS Abstracts, March 15, 1995. @
