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An incremental analytical solution for gas-solid reactions, application to the grain model
Pergamon
Chemical Engineerin 9 Science, Vol. 51, No. 18, pp. 4253-4257, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved P I I : S0009-2509(96)00257-6 0009 2509/96 $15.00 + 0.00
AN INCREMENTAL ANALYTICAL SOLUTION FOR GAS-SOLID REACTIONS, APPLICATION TO THE GRAIN MODEL E. JAMSHIDI* and H. ALE E B R A H I M Department of Chemical Engineering, Amir-Kabir University (Tehran Polytechnic), Tehran, 15875-4413, Iran (First received 9 January 1995; revised manuscript received and accepted 27 February 1996) Abstract--Gas-solid reactions are very important in chemical and metallurgical process industries. The grain model is one of the most important models describing some of these reactions. This model gives two coupled partial differential equations. The governing equations of this model must be solved numerically. There are some analytical solutions that can be applied for very special and simplified cases. For rapid determination of conversion-time behavior, some approximate methods have also been proposed. In this work a new mathematical technique is applied for solution of the grain model equations. This technique makes it possible to solve the coupled partial differential equations for all physical shape of grains. The results of this technique are compared with existing numerical and approximate solutions. This technique is simple and rapid and fits numerical solution with reasonable accuracy. This solution also gives a mathematical expression which cannot be found in numerical solutions. Copyright @, 1996 Elsevier Science Ltd
We have the following assumptions:
INTRODUCTION
Most of the important gas-solid reactions in chemical and metallurgical industries consist of solid pellets produced from small particles or grains. These pellets are often initially porous, or become porous in the reaction conditions. For example, in order to enhance the production rate, solid particles are mixed with some vaporizing material before being pelletized (Ale Ebrahim and Jamshidi, 1990). As temperature increases in the furnace, the vaporizing material leaves the pellet. Therefore, some open channels have been produced, which makes it possible for gas to diffuse into interior of the pellet. There are several mathematical models describing noncatalytic gas solid reactions. The shrinking core model for nonporous pellets, has been replaced by volume reaction model (Ishida and Wen, 1968; Wen, 1968), and grain model (Szekely and Evans, 1970, 1971a, b; Sohn and Szekely, 1972; Szekely et al., 1976) for porous pellets. In this work the grain model has been considered. This model assumes that the overall reaction consists of intra pellet diffusion, and subsequent chemical reaction on the surface of each grain.
(1) Pseudo-steady-state approximation is valid. (2) Diffusion within the pellet is equimolar counter-diffusion. (3) The system is isothermal. (4) The solid structure is unaffected by the reaction. (5) The solid product around each grain is highly porous. Thus, the resistance of this product layer is negligible. (6) The reaction is irreversible and first order with respect to the gaseous reactant. (7) The pellet is slab or sphere, and the grains can be at any physical shape. The dimensionless governing equations in the general form are: 6~2a Fp - 10a 03,2 + - -y c~y
a 2 r *Fg ' a
Or* 30
-
a.
A mathematical model for such system has been studied before (Sohn and Szekely, 1972; Szekely et al., 1976). Consider a reaction:
(3)
With initial and boundary conditions: 0 = 0: r* = 1
MATHEMATICAL MODELING
(2)
(4)
y = 0: (~a
-~y - =o
(5)
c3-y= N~h(1 -- a).
(6)
y = 1: A(g) + vBB(s) --* C(g) + vDD(s).
* Corresponding author.
(1)
(3a
Where the dimensionless parameters are defined in notations. This model consists of two stages. In the 4253
4254
E. JAMSHIDI and H. ALE EBRAHIM
first stage (0 < 0~), diffusion of gas "A" and reaction between gas "A" and solid "B" are happening simultaneously. At the time 0 = 0~ all solid of the outer layer of the pellet has been reacted. Then second stage is started when 0 > 0c. In this stage the gas diffuses through the completely reacted outer layer of the pellet, in order to reach the diffusion-reaction zone. The partial differential equations of model were solved numerically (Szekely and Evans, 1971a). In the special case for Fg = 1 (slab grains), analytical solution is possible (Ishida and Wen, 1968; Szekely et al., 1976). Sohn and Szekely (1972), Szekely et al. (1973) solved these general equations for two limiting cases of reaction control or diffusion control regimes. Then at intermediate regime where both chemical reaction and diffusion are important, their approximate analytical solution is based on addition of two reaction times obtained from limiting cases. Evans and Ranade (1980) used the CDC 6400 computer for least-squares method to fit the above approximate solution to exact (numerical) solution. INCREMENTAL SOLUTION TECHNIQUE In this work the mathematical solution is applied for spherical or slab pellets, with spherical or cylindrical grains. Spherical pellets First stage. Coupled partial differential equations
describing the spherical pellets are eqs (2) and (3) with Fp = 3. In this mathematical method, it is assumed that r* is constant between two small time increments in eq. (2). This means that we put r* (j - 1, i) instead of r* (j, i) in eq. (2), where j and i are the time and position indexes, respectively. Thus, we can define a modified and variable Thiele modulus as --
2
M ( j , i) - x / a r
,v~-i
.
(j -- 1, i).
(7)
Inserting eq. (7) into eq. (2) yields 02a
2 Oa
(8)
+ -;Fy = M a"
By using this approximation, eq. (8) with boundary conditions (5) and (6) can be integrated as a
1 sinh (My) 0c y sinh (M)'
--
Equation (11) seems to be an implicit one. But in our approximation it is assumed that r* in the M is constant between two small time increments, therefore we can replace the r* in the M by r* which has been calculated before. End time for the first stage can be found by letting r* = 0 when y = 1 in eq. (11), thus we have 0 = 0c for end of the first stage. Then the second stage starts from 0 > 0c. Second stage. In this stage there are two zones. In zone 1 there is only diffusion, because the solid reactant is completely reacted. In the zone 2 (or interior J~ layer), the gas goes under diffusion and reaction simultaneously. Let us define al, r* and az, r~ for diffusion zone, and for diffusion-reaction zone, respectively. By applying the method of Ishida and Wen (1968), which has been applied for the slab grains and spherical pellets only, we have
al = am
y~ 1 - y + y/Nsh y 1 --y,,, +y,,,/N,h
+
1 --y,,,/y 1 --y,~ +ym/Nsh
y , sinh (My) a2 = am ysinh(My~) a,. =
1
1 +(1 - y ,
+ ym/N~h) [ M y m c o t h ( M y m ) - 1]
(12)
(13) (14)
M2 0 = 1 +--~- (1 _y.,)2 (1 + 2y,.) + (1 - y , . ) [ m y m c o t h ( M y m ) -- 1] 1 M 2
+~-~--~- (1 _ y 3 ) + ~lva[ M y , coth(My=) - 1] (15) Ymsinh ( M y ) r'~ = 1
ysinh(Mym)"
(16)
As M is a variable Thiele modulus, it is noticed that using a (the original Thiele modulus) in eqs (10) and (15) gives better fitting. Thus from eqs (15) and (16), we can compute r* at each time and position. Then from eq. (7) we can compute the new M from r* at the same position but at past time increment, and continue the procedure. The conversion of solid at each time can be obtained by numerical integration of the following equation:
(9)
X(O) = 1 -- 3
y2r*"(O, y) dy.
(17)
where Slab pellets 1
0c = 1 + ~ h [ M c o t h ( M ) -
1].
(10)
By using the same method, final equations for slab pellets and for negligible external mass transfer resistance are:
Integration of eq. (3) with initial condition (4) gives First stage
1 sinh(My) 0 r* = 1
Ocysinh(M)
= 1 -aO.
(11)
a
cosh(My) cosh (M)
(18)
Incremental analytical solution for gas-solid reactions cosh(My) cosh(M) 0 = 1 - aO
r*=l
0c = 1.
(19)
figurethan approximateit is clear thatsolution.OUr results have better accuracy
(20)
(4) Comparison of our results for Fp = 1, Fg = 2 with experimental data (reduction of nickel oxide with hydrogen), and approximate solution of Szekely et al.
Second stage M ( y -- 1)
al=l+
a2 =
(21)
coth (Myra) - M ( y , . - 1)
-
1.a 1
cosh(My) cosh(Mym) - M ( y m
(22)
1) sinh (My,,)
0.8 ~a
M 2
0 = 1 +--~-(1 - y m ) 2 + M ( 1 - y m ) t a n h ( M y m )
(23) 0.4
cosh ( M y ) cosh(My,.)
r~ = 1
4255
0.2 //
(24)
X(O)=l-flr*~'(O,y)dy.
//
/ i
L
i
i
i
i
0.5
1
1.5
2
2.5
3
Dimensionless
3.5
Time
(25) - - Num Sol
The conversion of solid can be calculated by eq. (25), and we use tr instead of M in eq. (23).
• Our Results + Approx Sol
Fig. 2. Comparison of our results with numerical and approximate solutions of Evans and Ranade (1980) for spherical pellets and cylindrical grains.
CONCLUSION
In this work a new analytical solution has been developed for predicting the conversion-time behavior of grain model. This method is rapid and also gives some mathematical expression for gas and solid concentrations. The accuracy of this method is as follows: (1) Figure 1 is a comparison of our prediction with numerical and approximate solutions of Sohn and Szekely (1972), for Fp = Fg = 3. (2) Figure 2 shows conversion for Fp = 3, Fg = 2. There is very good agreement between our results, numerical solution and approximate solution of Evans and Ranade (1980). (3) In the case of appreciable external mass transfer resistance, comparison of our solution with numerical and approximate solutions of Sohn and Szekely (1972), for Fp = Fg = 3, is presented in Fig. 3. In this
1.2 ^ =
1
,
=
0.8
~
0.6
0.4 ~. = 1 , N s h = 1
0.2
O, 0
0.2
0.4
0.6
0.8
1
1.2
1.4
Dimensionless
- - Num Sol
1.6
1.8
2
i
i
2.2
2.4
Time
• Our Results + Approx Sol
Fig. 3. Comparison of our results with numerical and approximate solutions of Sohn and Szekely (1972) when external mass transfer resistance is important, for spherical pellets and spherical grains.
1.2 1.2
o"=1 1
~+,
•
+
I ~ u n 13
0.8
= 0.8 o
m 0.8 t..)
•
e_>0.6
8
0.4 0.2
0.4 0.2
0.5
1
1,5
2
2.5
Dimensionless
- - Num Sol
3
3.5
4
4.5
5
50
t
Fig. 1. Comparison of our results with numerical and approximate solutions of Sohn and Szekely (1972) for spherical pellets and spherical grains.
Rdn !2
i
i
i
=
i
i
i
i
100
150
200
250
300
350
400
450
500
Time (sec)
Time
• Our Results + Approx Sol
4-
]/ i
0
"
Exp D=a
I
O~, Results + ApproxSol/
J
Fig. 4. Comparison of our results with experimental data and approximate solution of Szekely et al. (1973) for slab pellets and cylindrical grains (Runs 12 and 13).
4256
E. JAMSHIDI and H. ALE EBRAHIM
(1973) is presented in Figs 4-6. We apply the same
NOTATION
i n d u c t i o n t i m e as this reference a p p l i e d , a n d u s e t h e ideal g a s l a w for c o m p u t i n g t h e h y d r o g e n c o n c e n t r a tion. O t h e r p a r a m e t e r s f r o m T a b l e 4 o f this reference c a n be o b t a i n e d . A s t h e s e figures s h o w t h e r e is a v e r y g o o d a g r e e ment between our predictions and existing solutions specially for s m a l l m o d u l u s ( r e a c t i o n c o n t r o l regime). T h e a c c u r a c y o f o u r m e t h o d w i t h r e s p e c t to n u m e r i c a l
a = Ca/Cag Ca Cag De
s o l u t i o n s is p r e s e n t e d in T a b l e s 1 - 4 for this c o n d i t i o n . T h u s , this m e t h o d c a n be a p p l i e d for r a p i d e s t i m a t i o n o f kinetic p a r a m e t e r s f r o m e x p e r i m e n t a l d a t a .
Fp i j
1.2
dimensionless gas concentration gas concentration in the pellet bulk gas concentration effective diffusivity of gas A in the pellet shape factor of grains = 1, 2, 3 for slab, cylinder, and sphere, respectively shape factor of the pellet position index time index
Fg
1.2 Run 9
1
+
Run 10
1
•
.~ 0.8
¢~ 0.8 Run 11
~_
~ 0.6
c~0.6
8 0.4
+
8 0.4
0.2
0.2 I
I
I
50
1O0
150
I
I
I
200
250
300
i 350
00
400
200
400
600
Time (sec) I
Exp Data
I
S00
1,000
1,200
1,400
Time (sec)
Our Results ÷ Approx Sol
Fig. 5. Comparison of our results with experimental data and approximate solution of Szekely et al. (1973) for slab pellets and cylindrical grains (Runs 9 and 11).
I--ExpData
" OurResults + ApproxSol I
Fig. 6. Comparison of our results with experimental data and approximate solution of Szekely et al. (1973) for slab pellets and cylindrical grains (Run 10).
Table 1. Error comparison for N,h = 1, of this work and approximate solution of Sohn and Szekely (1972), with respect to numerical solution (Nsh = 1, Fp = Fg = 3, 8 = 1) 05
Num. sol.
This work
Approx. sol.
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.093 0.153 0.214 0.288 0.364 0.424 0.499 0.558 0.616 0.702
0.075 0.148 0.218 0.286 0.352 0.415 0.475 0.534 0.589 0.642
0.085 0.166 0.245 0.320 0.393 0.462 0.528 0.590 0.649 0.704
% Error of this work - 19.4 - 3.3 1.9 - 0.7 - 3.3 - 2.1 - 4.8 - 4.3 - 4.4 - 8.5
% Error of approx, sol. - 8.6 8.5 14.5 11.1 8.0 9.0 5.8 5.7 5.4 0.3
Table 2. Error comparison for N,~ = 3, of this work and approximate solution of Sohn and Szekely (1972), with respect to numerical solution (N~h = 3, Fp = Fg = 3, 8 = 1) 0~
Num. sol.
This work
Approx. sol.
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.168 0.311 0.433 0.554 0.662 0.771 0.856 0.914 0.957 0.986
0.151 0.291 0.422 0.542 0.652 0.751 0.838 0.912 0.966 0.993
0.184 0.339 0.471 0.583 0.678 0.758 0.825 0.880 0.923 0.957
% Error of this work - 10.1 - 6.4 - 2.5 - 2.2 - 1.5 - 2.6 - 2.1 - 0.2 0.9 0.7
% Error of approx, sol.
---
9.5 9.0 8.8 5.2 2.4 1.7 3.6 3.7 3.6 2.9
4257
Incremental analytical solution for gas-solid reactions Table 3. Error comparison for N~h = 0% of this work and approximate solution of Sohn and Szekely (1972), with respect to numerical solution (Ns, = 0% Fp = Fg = 3, ~ = 1) 05
Num. sol.
This work
Approx. sol.
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
0.300 0.536 0.723 0.847 0.915 0.974 0.994 1.000
0.297 0.550 0.754 0.901 0.977 0.996 1.000 1.000
0.373 0.574 0.708 0.805 0.876 0.927 0.962 0.985
% Error of this work
% Error of approx, sol.
- 1.0 2.6 4.3 6.4 6.8 2.3 0.6 0.0
24.3 7.1 - 2.1 - 5.0 - 4.3 - 4.8 - 3.2 - 1.5
Table 4. Error comparison for N~h = oo, of this work and approximate solution of Evans and Ranade (1980), with respect to numerical solution (Nsh = co, Fp = 3, F, = 2, ~ = 1) 0~
Num. sol.
This work
Approx. sol.
0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.255 0.438 0.614 0.749 0.880 0.934 0.972
0.236 0.451 0.642 0.806 0.928 0.967 0.987
0.256 0.449 0.618 0.749 0.861 0.918 0.955
km k~ M Me Nsh = kmR/De r G rgo r* = rg/rgo R t
x(o) y = r/R Ym
external mass transfer coefficient surface rate c o n s t a n t modified Thiele modulus defined by eq. (7) molecular weight of reactant solid Sherwood number distance from the center of the pellet radius of unreacted core in the grain initial grain radius dimensionless unreacted radius in the grain characteristic pellet length time solid conversion at time 0 dimensionless position in the pellet dimensionless position of moving b o u n d a r y between the reaction zone a n d p r o d u c t layer in the pellet
Greek letters pellet porosity 0 = ks CA. M e t / p . rgo dimensionless time dimensionless time for com0c plete reaction of first stage, defined by eq. (10) stoichiometric coefficient of N ~N density of solid r e a c t a n t Pe
a = R ~1
Dergo
reaction m o d u l u s
% Error of this work
% Error of approx, sol.
- 7.5 3.0 4.6 7.6 5.5 3.5 1.5
O"
0.4 2.5 0.7 0.0 - 2.2 - 1.7 - 1.7
generalized reaction m o d u l u s
REFERENCES
Ale Ebrahim, H. and Jamshidi, E., 1990, Reduction of zinc oxide concentrate by petroleum materials. M. S. Thesis, Tehran Polytechnic, Amir-Kabir University. Evans, J. W. and Ranade, M. G., 1980, The grain model for reaction between a gas and a porous solid--a refined approximate solution to the equations. Chem. Engng Sci. 35, 1261-1262. Ishida, M. and Wen, C. Y., 1968, Comparison of kinetic and diffusional models for solid-gas reactions. A.LCh.E.J. 14, 311-317. Sohn, H. Y. and Szekely, J., 1972, A structural model for gas-solid reactions with a moving boundary--III. A general dimensionless representation of the irreversible reaction between a porous solid and a reactant gas. Chem. Engn9 Sci. 27, 763-778. Szekely, J. and Evans, J. W., 1970, A structural model for gas-solid reactions with a moving boundary. Chem. Engng Sci. 25, 1091-1107. Szekely, J. and Evans, J. W., 1971a, Studies in gas-solid reactions. Part I. A structural model for the reaction of porous oxides with a reducing gas. Met. Trans. 2, 16911698. Szekely, J. and Evans, J. W., 1971b, A structural model for gas-solid reactions with a moving boundary--II. The effect of grain size, porosity and temperature on the reaction of porous pellet. Chem. Engng Sci. 26, 1901-1913. Szekely, J., Evans, J. W. and Sohn, H. Y., 1976, Gas-Solid Reactions. Academic Press, New York. Szekely, J., Lin, C. I. and Sohn, H. Y., 1973, A structural model for gas-solid reactions with a moving boundary-V. An experimental study of the reduction of porous nickel-oxide pellets with hydrogen. Chem. Engng Sci. 28, 1975-1989. Wen, C. Y., 1968, Non-catalytic heterogeneous solid-fluid reaction models. Ind. Engng Chem. 60, 34-54.
Chemical Engineerin 9 Science, Vol. 51, No. 18, pp. 4253-4257, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved P I I : S0009-2509(96)00257-6 0009 2509/96 $15.00 + 0.00
AN INCREMENTAL ANALYTICAL SOLUTION FOR GAS-SOLID REACTIONS, APPLICATION TO THE GRAIN MODEL E. JAMSHIDI* and H. ALE E B R A H I M Department of Chemical Engineering, Amir-Kabir University (Tehran Polytechnic), Tehran, 15875-4413, Iran (First received 9 January 1995; revised manuscript received and accepted 27 February 1996) Abstract--Gas-solid reactions are very important in chemical and metallurgical process industries. The grain model is one of the most important models describing some of these reactions. This model gives two coupled partial differential equations. The governing equations of this model must be solved numerically. There are some analytical solutions that can be applied for very special and simplified cases. For rapid determination of conversion-time behavior, some approximate methods have also been proposed. In this work a new mathematical technique is applied for solution of the grain model equations. This technique makes it possible to solve the coupled partial differential equations for all physical shape of grains. The results of this technique are compared with existing numerical and approximate solutions. This technique is simple and rapid and fits numerical solution with reasonable accuracy. This solution also gives a mathematical expression which cannot be found in numerical solutions. Copyright @, 1996 Elsevier Science Ltd
We have the following assumptions:
INTRODUCTION
Most of the important gas-solid reactions in chemical and metallurgical industries consist of solid pellets produced from small particles or grains. These pellets are often initially porous, or become porous in the reaction conditions. For example, in order to enhance the production rate, solid particles are mixed with some vaporizing material before being pelletized (Ale Ebrahim and Jamshidi, 1990). As temperature increases in the furnace, the vaporizing material leaves the pellet. Therefore, some open channels have been produced, which makes it possible for gas to diffuse into interior of the pellet. There are several mathematical models describing noncatalytic gas solid reactions. The shrinking core model for nonporous pellets, has been replaced by volume reaction model (Ishida and Wen, 1968; Wen, 1968), and grain model (Szekely and Evans, 1970, 1971a, b; Sohn and Szekely, 1972; Szekely et al., 1976) for porous pellets. In this work the grain model has been considered. This model assumes that the overall reaction consists of intra pellet diffusion, and subsequent chemical reaction on the surface of each grain.
(1) Pseudo-steady-state approximation is valid. (2) Diffusion within the pellet is equimolar counter-diffusion. (3) The system is isothermal. (4) The solid structure is unaffected by the reaction. (5) The solid product around each grain is highly porous. Thus, the resistance of this product layer is negligible. (6) The reaction is irreversible and first order with respect to the gaseous reactant. (7) The pellet is slab or sphere, and the grains can be at any physical shape. The dimensionless governing equations in the general form are: 6~2a Fp - 10a 03,2 + - -y c~y
a 2 r *Fg ' a
Or* 30
-
a.
A mathematical model for such system has been studied before (Sohn and Szekely, 1972; Szekely et al., 1976). Consider a reaction:
(3)
With initial and boundary conditions: 0 = 0: r* = 1
MATHEMATICAL MODELING
(2)
(4)
y = 0: (~a
-~y - =o
(5)
c3-y= N~h(1 -- a).
(6)
y = 1: A(g) + vBB(s) --* C(g) + vDD(s).
* Corresponding author.
(1)
(3a
Where the dimensionless parameters are defined in notations. This model consists of two stages. In the 4253
4254
E. JAMSHIDI and H. ALE EBRAHIM
first stage (0 < 0~), diffusion of gas "A" and reaction between gas "A" and solid "B" are happening simultaneously. At the time 0 = 0~ all solid of the outer layer of the pellet has been reacted. Then second stage is started when 0 > 0c. In this stage the gas diffuses through the completely reacted outer layer of the pellet, in order to reach the diffusion-reaction zone. The partial differential equations of model were solved numerically (Szekely and Evans, 1971a). In the special case for Fg = 1 (slab grains), analytical solution is possible (Ishida and Wen, 1968; Szekely et al., 1976). Sohn and Szekely (1972), Szekely et al. (1973) solved these general equations for two limiting cases of reaction control or diffusion control regimes. Then at intermediate regime where both chemical reaction and diffusion are important, their approximate analytical solution is based on addition of two reaction times obtained from limiting cases. Evans and Ranade (1980) used the CDC 6400 computer for least-squares method to fit the above approximate solution to exact (numerical) solution. INCREMENTAL SOLUTION TECHNIQUE In this work the mathematical solution is applied for spherical or slab pellets, with spherical or cylindrical grains. Spherical pellets First stage. Coupled partial differential equations
describing the spherical pellets are eqs (2) and (3) with Fp = 3. In this mathematical method, it is assumed that r* is constant between two small time increments in eq. (2). This means that we put r* (j - 1, i) instead of r* (j, i) in eq. (2), where j and i are the time and position indexes, respectively. Thus, we can define a modified and variable Thiele modulus as --
2
M ( j , i) - x / a r
,v~-i
.
(j -- 1, i).
(7)
Inserting eq. (7) into eq. (2) yields 02a
2 Oa
(8)
+ -;Fy = M a"
By using this approximation, eq. (8) with boundary conditions (5) and (6) can be integrated as a
1 sinh (My) 0c y sinh (M)'
--
Equation (11) seems to be an implicit one. But in our approximation it is assumed that r* in the M is constant between two small time increments, therefore we can replace the r* in the M by r* which has been calculated before. End time for the first stage can be found by letting r* = 0 when y = 1 in eq. (11), thus we have 0 = 0c for end of the first stage. Then the second stage starts from 0 > 0c. Second stage. In this stage there are two zones. In zone 1 there is only diffusion, because the solid reactant is completely reacted. In the zone 2 (or interior J~ layer), the gas goes under diffusion and reaction simultaneously. Let us define al, r* and az, r~ for diffusion zone, and for diffusion-reaction zone, respectively. By applying the method of Ishida and Wen (1968), which has been applied for the slab grains and spherical pellets only, we have
al = am
y~ 1 - y + y/Nsh y 1 --y,,, +y,,,/N,h
+
1 --y,,,/y 1 --y,~ +ym/Nsh
y , sinh (My) a2 = am ysinh(My~) a,. =
1
1 +(1 - y ,
+ ym/N~h) [ M y m c o t h ( M y m ) - 1]
(12)
(13) (14)
M2 0 = 1 +--~- (1 _y.,)2 (1 + 2y,.) + (1 - y , . ) [ m y m c o t h ( M y m ) -- 1] 1 M 2
+~-~--~- (1 _ y 3 ) + ~lva[ M y , coth(My=) - 1] (15) Ymsinh ( M y ) r'~ = 1
ysinh(Mym)"
(16)
As M is a variable Thiele modulus, it is noticed that using a (the original Thiele modulus) in eqs (10) and (15) gives better fitting. Thus from eqs (15) and (16), we can compute r* at each time and position. Then from eq. (7) we can compute the new M from r* at the same position but at past time increment, and continue the procedure. The conversion of solid at each time can be obtained by numerical integration of the following equation:
(9)
X(O) = 1 -- 3
y2r*"(O, y) dy.
(17)
where Slab pellets 1
0c = 1 + ~ h [ M c o t h ( M ) -
1].
(10)
By using the same method, final equations for slab pellets and for negligible external mass transfer resistance are:
Integration of eq. (3) with initial condition (4) gives First stage
1 sinh(My) 0 r* = 1
Ocysinh(M)
= 1 -aO.
(11)
a
cosh(My) cosh (M)
(18)
Incremental analytical solution for gas-solid reactions cosh(My) cosh(M) 0 = 1 - aO
r*=l
0c = 1.
(19)
figurethan approximateit is clear thatsolution.OUr results have better accuracy
(20)
(4) Comparison of our results for Fp = 1, Fg = 2 with experimental data (reduction of nickel oxide with hydrogen), and approximate solution of Szekely et al.
Second stage M ( y -- 1)
al=l+
a2 =
(21)
coth (Myra) - M ( y , . - 1)
-
1.a 1
cosh(My) cosh(Mym) - M ( y m
(22)
1) sinh (My,,)
0.8 ~a
M 2
0 = 1 +--~-(1 - y m ) 2 + M ( 1 - y m ) t a n h ( M y m )
(23) 0.4
cosh ( M y ) cosh(My,.)
r~ = 1
4255
0.2 //
(24)
X(O)=l-flr*~'(O,y)dy.
//
/ i
L
i
i
i
i
0.5
1
1.5
2
2.5
3
Dimensionless
3.5
Time
(25) - - Num Sol
The conversion of solid can be calculated by eq. (25), and we use tr instead of M in eq. (23).
• Our Results + Approx Sol
Fig. 2. Comparison of our results with numerical and approximate solutions of Evans and Ranade (1980) for spherical pellets and cylindrical grains.
CONCLUSION
In this work a new analytical solution has been developed for predicting the conversion-time behavior of grain model. This method is rapid and also gives some mathematical expression for gas and solid concentrations. The accuracy of this method is as follows: (1) Figure 1 is a comparison of our prediction with numerical and approximate solutions of Sohn and Szekely (1972), for Fp = Fg = 3. (2) Figure 2 shows conversion for Fp = 3, Fg = 2. There is very good agreement between our results, numerical solution and approximate solution of Evans and Ranade (1980). (3) In the case of appreciable external mass transfer resistance, comparison of our solution with numerical and approximate solutions of Sohn and Szekely (1972), for Fp = Fg = 3, is presented in Fig. 3. In this
1.2 ^ =
1
,
=
0.8
~
0.6
0.4 ~. = 1 , N s h = 1
0.2
O, 0
0.2
0.4
0.6
0.8
1
1.2
1.4
Dimensionless
- - Num Sol
1.6
1.8
2
i
i
2.2
2.4
Time
• Our Results + Approx Sol
Fig. 3. Comparison of our results with numerical and approximate solutions of Sohn and Szekely (1972) when external mass transfer resistance is important, for spherical pellets and spherical grains.
1.2 1.2
o"=1 1
~+,
•
+
I ~ u n 13
0.8
= 0.8 o
m 0.8 t..)
•
e_>0.6
8
0.4 0.2
0.4 0.2
0.5
1
1,5
2
2.5
Dimensionless
- - Num Sol
3
3.5
4
4.5
5
50
t
Fig. 1. Comparison of our results with numerical and approximate solutions of Sohn and Szekely (1972) for spherical pellets and spherical grains.
Rdn !2
i
i
i
=
i
i
i
i
100
150
200
250
300
350
400
450
500
Time (sec)
Time
• Our Results + Approx Sol
4-
]/ i
0
"
Exp D=a
I
O~, Results + ApproxSol/
J
Fig. 4. Comparison of our results with experimental data and approximate solution of Szekely et al. (1973) for slab pellets and cylindrical grains (Runs 12 and 13).
4256
E. JAMSHIDI and H. ALE EBRAHIM
(1973) is presented in Figs 4-6. We apply the same
NOTATION
i n d u c t i o n t i m e as this reference a p p l i e d , a n d u s e t h e ideal g a s l a w for c o m p u t i n g t h e h y d r o g e n c o n c e n t r a tion. O t h e r p a r a m e t e r s f r o m T a b l e 4 o f this reference c a n be o b t a i n e d . A s t h e s e figures s h o w t h e r e is a v e r y g o o d a g r e e ment between our predictions and existing solutions specially for s m a l l m o d u l u s ( r e a c t i o n c o n t r o l regime). T h e a c c u r a c y o f o u r m e t h o d w i t h r e s p e c t to n u m e r i c a l
a = Ca/Cag Ca Cag De
s o l u t i o n s is p r e s e n t e d in T a b l e s 1 - 4 for this c o n d i t i o n . T h u s , this m e t h o d c a n be a p p l i e d for r a p i d e s t i m a t i o n o f kinetic p a r a m e t e r s f r o m e x p e r i m e n t a l d a t a .
Fp i j
1.2
dimensionless gas concentration gas concentration in the pellet bulk gas concentration effective diffusivity of gas A in the pellet shape factor of grains = 1, 2, 3 for slab, cylinder, and sphere, respectively shape factor of the pellet position index time index
Fg
1.2 Run 9
1
+
Run 10
1
•
.~ 0.8
¢~ 0.8 Run 11
~_
~ 0.6
c~0.6
8 0.4
+
8 0.4
0.2
0.2 I
I
I
50
1O0
150
I
I
I
200
250
300
i 350
00
400
200
400
600
Time (sec) I
Exp Data
I
S00
1,000
1,200
1,400
Time (sec)
Our Results ÷ Approx Sol
Fig. 5. Comparison of our results with experimental data and approximate solution of Szekely et al. (1973) for slab pellets and cylindrical grains (Runs 9 and 11).
I--ExpData
" OurResults + ApproxSol I
Fig. 6. Comparison of our results with experimental data and approximate solution of Szekely et al. (1973) for slab pellets and cylindrical grains (Run 10).
Table 1. Error comparison for N,h = 1, of this work and approximate solution of Sohn and Szekely (1972), with respect to numerical solution (Nsh = 1, Fp = Fg = 3, 8 = 1) 05
Num. sol.
This work
Approx. sol.
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.093 0.153 0.214 0.288 0.364 0.424 0.499 0.558 0.616 0.702
0.075 0.148 0.218 0.286 0.352 0.415 0.475 0.534 0.589 0.642
0.085 0.166 0.245 0.320 0.393 0.462 0.528 0.590 0.649 0.704
% Error of this work - 19.4 - 3.3 1.9 - 0.7 - 3.3 - 2.1 - 4.8 - 4.3 - 4.4 - 8.5
% Error of approx, sol. - 8.6 8.5 14.5 11.1 8.0 9.0 5.8 5.7 5.4 0.3
Table 2. Error comparison for N,~ = 3, of this work and approximate solution of Sohn and Szekely (1972), with respect to numerical solution (N~h = 3, Fp = Fg = 3, 8 = 1) 0~
Num. sol.
This work
Approx. sol.
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.168 0.311 0.433 0.554 0.662 0.771 0.856 0.914 0.957 0.986
0.151 0.291 0.422 0.542 0.652 0.751 0.838 0.912 0.966 0.993
0.184 0.339 0.471 0.583 0.678 0.758 0.825 0.880 0.923 0.957
% Error of this work - 10.1 - 6.4 - 2.5 - 2.2 - 1.5 - 2.6 - 2.1 - 0.2 0.9 0.7
% Error of approx, sol.
---
9.5 9.0 8.8 5.2 2.4 1.7 3.6 3.7 3.6 2.9
4257
Incremental analytical solution for gas-solid reactions Table 3. Error comparison for N~h = 0% of this work and approximate solution of Sohn and Szekely (1972), with respect to numerical solution (Ns, = 0% Fp = Fg = 3, ~ = 1) 05
Num. sol.
This work
Approx. sol.
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
0.300 0.536 0.723 0.847 0.915 0.974 0.994 1.000
0.297 0.550 0.754 0.901 0.977 0.996 1.000 1.000
0.373 0.574 0.708 0.805 0.876 0.927 0.962 0.985
% Error of this work
% Error of approx, sol.
- 1.0 2.6 4.3 6.4 6.8 2.3 0.6 0.0
24.3 7.1 - 2.1 - 5.0 - 4.3 - 4.8 - 3.2 - 1.5
Table 4. Error comparison for N~h = oo, of this work and approximate solution of Evans and Ranade (1980), with respect to numerical solution (Nsh = co, Fp = 3, F, = 2, ~ = 1) 0~
Num. sol.
This work
Approx. sol.
0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.255 0.438 0.614 0.749 0.880 0.934 0.972
0.236 0.451 0.642 0.806 0.928 0.967 0.987
0.256 0.449 0.618 0.749 0.861 0.918 0.955
km k~ M Me Nsh = kmR/De r G rgo r* = rg/rgo R t
x(o) y = r/R Ym
external mass transfer coefficient surface rate c o n s t a n t modified Thiele modulus defined by eq. (7) molecular weight of reactant solid Sherwood number distance from the center of the pellet radius of unreacted core in the grain initial grain radius dimensionless unreacted radius in the grain characteristic pellet length time solid conversion at time 0 dimensionless position in the pellet dimensionless position of moving b o u n d a r y between the reaction zone a n d p r o d u c t layer in the pellet
Greek letters pellet porosity 0 = ks CA. M e t / p . rgo dimensionless time dimensionless time for com0c plete reaction of first stage, defined by eq. (10) stoichiometric coefficient of N ~N density of solid r e a c t a n t Pe
a = R ~1
Dergo
reaction m o d u l u s
% Error of this work
% Error of approx, sol.
- 7.5 3.0 4.6 7.6 5.5 3.5 1.5
O"
0.4 2.5 0.7 0.0 - 2.2 - 1.7 - 1.7
generalized reaction m o d u l u s
REFERENCES
Ale Ebrahim, H. and Jamshidi, E., 1990, Reduction of zinc oxide concentrate by petroleum materials. M. S. Thesis, Tehran Polytechnic, Amir-Kabir University. Evans, J. W. and Ranade, M. G., 1980, The grain model for reaction between a gas and a porous solid--a refined approximate solution to the equations. Chem. Engng Sci. 35, 1261-1262. Ishida, M. and Wen, C. Y., 1968, Comparison of kinetic and diffusional models for solid-gas reactions. A.LCh.E.J. 14, 311-317. Sohn, H. Y. and Szekely, J., 1972, A structural model for gas-solid reactions with a moving boundary--III. A general dimensionless representation of the irreversible reaction between a porous solid and a reactant gas. Chem. Engn9 Sci. 27, 763-778. Szekely, J. and Evans, J. W., 1970, A structural model for gas-solid reactions with a moving boundary. Chem. Engng Sci. 25, 1091-1107. Szekely, J. and Evans, J. W., 1971a, Studies in gas-solid reactions. Part I. A structural model for the reaction of porous oxides with a reducing gas. Met. Trans. 2, 16911698. Szekely, J. and Evans, J. W., 1971b, A structural model for gas-solid reactions with a moving boundary--II. The effect of grain size, porosity and temperature on the reaction of porous pellet. Chem. Engng Sci. 26, 1901-1913. Szekely, J., Evans, J. W. and Sohn, H. Y., 1976, Gas-Solid Reactions. Academic Press, New York. Szekely, J., Lin, C. I. and Sohn, H. Y., 1973, A structural model for gas-solid reactions with a moving boundary-V. An experimental study of the reduction of porous nickel-oxide pellets with hydrogen. Chem. Engng Sci. 28, 1975-1989. Wen, C. Y., 1968, Non-catalytic heterogeneous solid-fluid reaction models. Ind. Engng Chem. 60, 34-54.