Ind. Eng. Chem. Process Des. Dev. 1980, 19, 237-242
are zinc oxide reactant made visible after a highly brittle, thin surface layer was chipped away. Figure 12B shows the surface of the brittle exterior a t a magnification of 1260X. This layer looks to be much less porous than the original reactant, and dendritic growth, a clear sign of deposition from the vapor phase, is apparent. Nomenclature U A = stoichiometric coefficient of gaseous reactant us = stoichiometric coefficient of solid reactant A , = specific surface area of reactant, 12/m B = constant in Froessling equation (17) CA = concentration of gaseous reactant in porous pellet, mol/l3 C, = concentration of gaseous reactant in bulk gas, mol/13 C, = concentration of gaseous reactant at pellet surface, m01/13 C, = initial concentration of solid reactant in pellet, mol/13 C,' = initial concentration of solid reactant in grain, mol l3 DA = diffusion coefficient of gaseous reactant in pellet, 1 i t DA' = diffusion coefficient of gaseous reactant in grain product layer, P/t De, = effective diffusivity of gaseous reactant in pellet, 12/t D K = Knudsen diffusion coefficient, 12/t DM = molecular diffusivity, 12/t E = activation energy, energyimol C = mass flux of gas past pellet, m/Pt k M A= mass transfer coefficient of gaseous reactant, l / t ko = frequency factor, 14/molt k,' = first-order rate constant, 14/mol t K = constant in Froessling equation (17) m = constant in Froessling equation (17) MA = molecular weight of gaseous reactant, m/mol Ms = molecular weight of solid reactant, m/mol n = constant in Froessling equation (17) NRe = Reynolds number, dimensionless, 2rsC/kg Nsc = Schmidt number, dimensionless, p g / p g D M N S h = Sherwood number, dimensionless, 2kMArs/DM
1
237
r = radial coordinate, 1 rc/ = unreacted core radius of a single grain, 1 rs = radius of the spherical pellet, 1 rs/ = grain radius, 1 R = gas constant t = time, t T = temperature
W , = initial pellet mass, m x = local extent of reaction, dimensionless 4 = overall extent of reaction, dimensionless
Greek Letters = pellet porosity, dimensionless gg = bulk gas viscosity, milt pg = bulk gas density, m/13 po,= pellet density, m/13 ps = density of solid reactant in grain, m/13 L i t e r a t u r e Cited to
Beveridge, G. S.G., "Agglomeration International Symposium", Interscience, New York, 1962. Bresler, S.A,, Ireland, J. D., Chern. Eng., 79, 23 (1972). Gibson, J. B.,Ph.D. Thesis, Louisiana State University, 1977. Hughmark, G. A,, AIChE J . , 13, 1219 (1967). Phillipson,J. J., "Desulfurization",in "Catalyst Handbook", Wolf, London, 1970. Reid, R. C.,Prausnitz, J. M., Sherwocd, T. K., "The Properties of Gases and Liquids", McGraw-Hill, New York, 1977. Satterfield. C. N., "Mass Transfer in Heterogeneous Catalysis", M.I.T. Press, Cambridge, Mass., 1970. Smith, J. M., "Chemical Engineering Kinetics", McGraw-Hill, New York, 1970. Szekely, J., Evans, J. W., Sohn, H. Y., "Gas-Sold Reactions", Academic Ress, New York, 1976. Wakao, N., Smith, J. M., Chern. Eng. Sci., 17, 825 (1962). Westmoreland, P. R., Harrison, D. P., Environ. Sci. Techno/.,10,659 (1976). Westmoreland, P. R., Gibson, J. B., Harrison, D. P., Environ. Sci. Techno/.,11, 488 (1977).
Received for review February 5, 1979 Accepted November 20, 1979
This study was supported, in part, by a grant from the U.S. Environmental Protection Agency.
The Effect of Bulk Flow Due to Volume Change in the Gas Phase on Gas-Solid Reactions: Initially Nonporous Solids Hong Yong Sohn" and Hun-Joon Sohn DepaHment of Metallurgy and Meta//urgica/ Engineering, University of Utah, Salt Lake City, Utah 84 112
The effect of bulk flow due to volume change in the gas phase on the rate of noncatalytic gas-solid reactions has been studied systematically. The model has been formulated in general terms so as to allow the incorporation of specific details of an actual system. The computed results show that the effect of bulk flow can be quite large. This effect increases with the importance of diffusion through the product layer in determining the overall rate of reaction. The law of additive reaction times previously proposed for reactions without volume change has been applied and found to yield a useful approximate solution also for this type of reaction systems. As a result, an approximate analytical equation for the conversion vs. time relationship incorporating chemical reaction, product layer diffusion, and external mass transfer has been derived.
Introduction Heterogeneous noncatalytic gas-solid reactions play an important role in many extractive metallurgical processes such as the reduction of oxide ores or the roasting of sulfide ores. Extensive research has been conducted for this type of reactions because of their wide commercial applications. Recent advances due to the development of more sophisticated mathematical models and experimental techniques have contributed t o our understanding of reaction 0196-4305/80/1119-0237$01.00/0
mechanisms and reactor design problems. Since Yagi and Kunii (1955) introduced the so-called shrinking unreacted core model, a great deal of effort has been directed toward the development of alternatives to this model for heterogeneous noncatalytic gas-solid reactions. These are described in some detail in a recent monograph and articles (Sohn, 1976a,b, 1978,1979; Szekely et al., 1976). However, most of the work has been restricted to the systems in which bulk flow within the solid 0 1980 American Chemical Society
238
Ind. Eng. Chem. Process
Des. Dev., Vol. 19, No. 2, 1980
due to the volume change of gas is negligible. Weekman and Gorring (1965) noted that the effect of bulk flow on the overall rate of catalytic reactions in porous catalysts could be very large. Szekely (1967) also discussed the effect of bulk flow on mass transfer for a homogeneous chemical reaction. Among heterogeneous noncatalytic gas-solid reactions, the roasting of sulfide minerals is a typical example in which the volume of the reactant gas is different from that of the product gas. Other examples of this type of reactions are the Boudouard reaction C ( S ) + COdg) = 2CO(g) the carbonylation of nickel Ni(s) + 4CO(g) = Ni(CO),(g)
Substituting eq 4 in eq 3 and rearranging, we get -DeCT dxA/dr NA = 1 - (1 - Y)X* Combining eq 2 and 5 and rewriting the resulting equation in a dimensionless form, we obtain
in which
and the reduction of halides FeCl,(s) + H2(g) = Fe(s) + 2HCl(g)
8
Haung and Bartlett (1976) reported the oxidation kinetics of a lime-copper concentrate pellet. They mentioned the bulk flow due to volume change in deriving the mass balance equation, but assumed equimolar counterdiffusion to analyze the experimental data. Natesan and Philbrook (1969) studied the oxidation kinetics of sphalerite. The rate of this reaction is controlled by the diffusion through the product layer and was analyzed by considering the bulk flow due to volume change. The mathematical model developed was specific for their system and not in a generalized form which could be applied easily to other systems. Furthermore, their model applied only to a reaction the overall rate of which is controlled by the product layer diffusion. An investigation has been carried out to determine systematically the effect of bulk flow on the rate of a gas-solid reaction which is controlled in general by both intrinsic kinetics and mass transfer processes. In this paper, we present the results for the reaction of an initially nonporous solid according to the shrinking core scheme. Mathematical Formulation Let us consider a gassolid reaction which can in general be expressed by A(g) + b*B(s)---* v.C(g) + d*D(s) (1) In the following formulation we shall consider an isothermal, first-order irreversible reaction of a nonporous solid producing a porous product layer according to the shrinking-core scheme. We shall also consider only the case where the product layer is sufficiently permeable so that the total pressure is constant throughout the system. It will further be assumed that the pseudo-steady-state approximation for gas-phase mass balance is valid, and the solid retains its original overall shape and size during the course of reaction. The mathematical description of such a system is obtained by writing a mass balance for the reactant species within the product layer. For the three basic geometries-slabs, long cylinders and spheres, we have
3
(v
-
1)X&
(9)
We see that the effect of bulk flow is represented by a single parameter 6. The boundary conditions are atv=l, y = l (10) and
where
Boundary condition 10 assumes that external mass transfer offers a negligible resistance. The effect of external mass transfer can be incorporated by writing an appropriate equation in lieu of eq 10. Since our major purpose in this work is to show the effect of bulk flow and external mass transfer will not materially affect the overail conclusion, we neglect its effect at this point. A separate discussion on the effect of external mass transfer will be presented in a subsequent section. Boundary condition 11 is obtained by equating NA at the interface between the unreacted core and the product layer with the rate of surface reaction per unit area. (See eq 13 below.) The solution of eq 6 will give the concentration profile and, from which, the flux NAwhen the reaction front is at qc. The rate of consumption of the solid can be equated to the rate of chemical reaction or the flux of gas A as follows
which, combined with eq 12, can be rewritten in the following dimensionless form
where where the shape factor F, has the value of 1, 2, or 3 for each of the three geometries, respectively. The molar flux NAis given by NA = (NA + NC)XA- D e C ~ V x A (3) and from stoichiometry Nc = -UNA
(15) Integration of eq 13 will yield rc, and thus conversion
X, as a function of time recognizing the following relationship
(4)
x = 1- 0,FP
(16)
This completes the statement of the problem. Upon integrating eq 6 and applying boundary condition 10, we obtain
0.8
-
X
z- 0.6 .
By substituting eq 17 in eq 11, we get 1-1)= I
Clf3q,'-F~ (18)
Evaluating y c from eq 17 a t q = qc and making use of eq 16 and 18, eq 14 can be rewritten as
Integration of eq 19 will yield the X vs. t* relationship. Since the integration constant C1cannot be obtained explicitely, a third-order Richmond iteration technique (Lapidus, 1962) was used to solve eq 18 for given values of F,, os2, and X . Equation 19 was integrated by the Adams method of stepwise integration. Results and Discussion Asymptotic Behavior. (i) When os approaches zero, i.e., when the overall reaction is controlled by chemical reaction, y = 1 throughout and from eq 14 we have qc = 1 - t* (20) from which we obtain t* = 1 - (1 - X)'/FP 3 g F p ( X )
(21)
In this case volume change has no effect on the overall rate. (ii) When 8 = -1, which is the case when v = 0 and XAb = 1, again y = 1 throughout from eq 17 and 18, and eq 20 and 21 apply, regardless of the relative importance of chemical reaction and diffusion. This result can also be expected from physical reasoning that, when there is no gaseous product and the bulk gas is pure reactant, only the reactant gas exists in the system under a constant total pressure as we have assumed. (iii) When us approaches infinity, i.e., when chemical reaction is very fast compared with diffusion, the overall rate of reaction is controlled by product layer diffusion and the concentration of' gas A will be zero a t the reaction interface. In this case, boundary condition 11 can be replaced by a t q = vc, y =0 (22)
qr
Applying eq 22 to eq 17 we obtain c1
=
-1n (1 + 0) 0
(
)
- F, -___ 2-Fp - 1
-In (1 0
+ 0)
=
2-Fp (1- X)'2-Fp'/J-p- 1
(23)
0:2
0:4
016
d.8
-dX =-
CI
dt+
2
l:o
IJZ
to
Figure 1. The effect of % on the conversion vs. time relationship for a small value of u:.
Substituting C1 in terms of X given in eq 23 into eq 24 and integrating, we obtain F,(1 - X ) 2 / F-~2(1 - X) In (1 + 8 ) t+=1E PFp(x) 0 Fp- 2 (26) The conversion function P F p ( Xin ) eq 26 can be rewritten for different values of F, in the following familiar forms: for Fp = 1 (slabs) In (1 + 8 ) t+ = p l ( X ) = x2 (27) 8 for Fp = 2 (long cylinders), by applying L'HGpital's rule In (1 + 8 ) t+ = P 2 ( X ) X (1 - X)In (1 - X ) (28) 0 for Fp = 3 (spheres) In (1 0) -~ t+ = P3(X) = 1 - 3(1 - X ) 2 ' 3+ 2(1 - X ) (29) 0 The dimensionless time t+ defined by eq 25 and the conversion functions P F p ( Xhave ) previously been used in the conversion-vs.-time relationships for the diffusion controlled gas-solid reactions without volume change (Sohn, 1976a,b, 1979; Szekely et al., 1976). The relationship for such a system is given by
+
+
t+ = P F P ( X )
(30)
Comparison of eq 26 and 30 indicates that, for diffusioncontrolled gas-solid reactions, the net effect of volume changes is to change the time required to attain a certain conversion by a factor of O/ln (1 + 0). If the effective diffusivity in the product layer is determined from plots of equations such as eq 30 without accounting for the bulk flow term, a serious error can result. The General Case. In the general case, eq 19 must be integrated with the aid of eq 16 and 18. Figures 1-3 show conversion vs. time for spheres for various 0 and .0: The solution for the system of 8 = 0 is given by (Sohn, 1976a, 1979; Szekely et al., 1976) t* = gFp(X) + .,2.PFP(X)
Equation 19 is rewritten as
where
]
Ob
(31)
For a small value of:u (say less than 0.1) in which case chemical reaction is predominantly rate-controlling, the effect of volume change is seen to be negligible as shown in Figure 1. As T(: increases, however, the effect becomes quite large as can be seen in Figures 2 and 3. For a slablike solid (F, = 1)the effect is seen to be larger than for a sphere by comparing Figures 3 and 4. In Figure 5 the
240
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 2, 1980 1.o
~
a
o.8 x 20.6
.-,,_ - -
t
0:. 1.0
e = -0.8
e. - 0 . 5 e = 0.0 e = 0.5 e. 1.0
2 In K
2
.-
--+
0.4
0
u
1.0
I
0.5
3.0
2.0
t'
1.0 ttE
Figure 2. The effect of 0 on the conversion vs. time relationship for an intermediate value of u,2.
1.5
t*/a2
Figure 5. The effect of 0 on the PFp(X)vs. time relationship for large values of 6:. 1.5
J
P
0.2 "'
01 0
2.0
6.0
4.0
8.0
10
12
14
16
0.5.
0.8
.0.4
1'
Figure 3. The effect of 0 on the conversion vs. time relationship for a large value of u,2 (F,= 3). 1.o
I
0.8
-
20.6
-
X
P v)
K W
0.4
0
-
/
' x
0
0.4
0.8
0
//,/ .
_.'
/ '
/ '
,'
q?.10
e:
-0.8 e.-O.5 0.0
e=
e- 0.5 e. 1.0
u
Figure 4. The effect of 0 on the conversion vs. time relationship for a large value of u: (F,= 1).
conversion vs. time relationship for large values of u? is plotted using PFp(X)and t+. As in the case of 8 = 0, it is seen for other values of 8 that the diffusion-controlled asymptote given by eq 26 is approached when (r: is greater than 10. It is noted that when the volume of the product gas is greater than that of the reactant gas, Le., 8 > 0, the outward bulk flow makes it more difficult for the reactant gas to diffuse into the interior. This results in a lower rate of reaction. When the volume of the product is smaller (8 < 0), the inward bulk flow speeds up the diffusion thus increasing the rate. Figure 6 shows the effect of 8 on the time for complete conversion. When there is no volume change in the gas phase (8 = 0), the value of t*x=,/(l + us2)is unity regardless of us2as can be seen from eq 31. Furthermore, when the overall rate is completely controlled
Figure 6. The effect of 0 on the time for complete conversion.
by chemical reaction (u: = 0), t*x,l/(l + u,2) is again unity regardless of 8 from eq 21. For other values of u: and 8 Figure 6 gives the effect of 8 on t*x=l. The effect of 8 is greatest when usis infinity and from eq 26 this effect can be expressed by the following analytical relationship t*x=1 t+x=l = 2,,
e In (1
+ 8)
Application of the Law of Additive Reaction Times Recently one of the authors (Sohn, 1978) proposed the law of additive reaction times for gassolid reactions which can be applied in either an integral or a differential form. This law states that the time required to attain a certain conversion can be approximated by the sum of the time required to reach the same conversion in the absence of resistance due to the intrapellet diffusion of fluid reactant and the time required to reach the same conversion under the control of the intrapellet diffusion. The law was also shown to be useful for the reaction of a porous solid following nucleation kinetics. It would be extremely useful to have a closed form solution for the system under consideration in which the effect of volume change is significant. Therefore, the law of additive reaction times was tested. The mathematical representation of the law is given by (Sohn, 1978) (33) t ( X ) = t(x)ln,40 + t(X)lbl-D) Substituting eq 21 and 26 together with eq 25 into eq 33, we obtain
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 2, 1980
241
-EXACT SOLUTION _ _ _ _ _ APPROXIMATE SOLUTION
1.0
0
2.0
3.0
1.
Figure 7. Comparison between the exact and the approximate solutions for u,2 = 1.
Figure 9. Comparison between the exact and the approximate solutions for u,2 = 10.
transfer rate than that defined otherwise (Bird et al., 1960). Again from stoichiometry Ncs = - v N ~ (36) From eq 35 and 36, we get
o’2
0
_ _ - APPROXIMATE
t 1/
L
0
0.2
0.4
0.6
0.8
SOLUTION
1.0
1.2
1
1
1.4
1‘
Figure 8. Comparison between the exact and the approximate solutions for us2= 0.1.
Figure 7 shows the comparison between the exact numerical solution and the approximate solution given by eq 34. It is seen that the approximate solution gives a reasonable representation of the exact solution (maximum 10% error). The comparison was made for :u = 1 for which the difference between the exact and the approximate solutions is largest. As CT becomes : smaller or larger, the agreement becomes better, as can be seen in Figures 8 and 9 for a small and a large value of us2, respectively. In fact as :u approaches either zero or infinity, eq 34 becomes asymptotically exact. It has thus been verified that the law of additive reaction times is valid even for the system in which there is volume change in the gas phase. The advantages of such a closed-form solution, especially in the analysis and design of multiparticle systems in which a particle size distribution and/or bulk concentration and temperature gradients may exist, have been discussed elsewhere (Sohn and Szekely, 1972; Sohn, 1978).
Effect of External Mass Transfer The incorporation of external mass transfer into the conversion vs. time relationship for the case of negligible volume change is relatively straightforward and has been discussed previously (Sohn and Szekely, 1972; Szekely et al., 1976). When the effect of bulk flow due to volume change is significant, the problem becomes rather complex. Bird et al. (1960) recommended the following equation for external mass transfer accompanied by bulk flow - XAb(hrAs
Ncs) =
.IZ,(Xh
- XAb)
(35)
The mass transfer coefficient defined by this equation is reported to be less dependent on concentration and mass
The overall rate can be expressed in terms of the rate of conversion of the solid or the rate of external mass transfer of the fluid reactant. Thus
I
pBVp d X / d t = bA,.(-NAs)
(38)
Under the pseudo-steady-state approximation, the term d X / d t can be obtained from eq 34 by replacing XAb in the definition oft* with the mole fraction at the surface X A
~
dt
where prime designates derivative with respect to X. Substituting eq 37 and 39 into eq 38, and rearranging, we get
where
Finally, on substituting eq 40 in eq 39 and integrating, we obtain
The availability of a closed-form approximate solution given by eq 34 has enabled us to incorporate the effect of external mass transfer analytically. The validity of the above procedure for incorporating the effect of external mass transfer has been verified previously (Sohn and Szekely, 1972) and thus will not be repeated here.
242
Ind. Eng. Chem. Process Des. Dev. 1980, 19, 242-246
Concluding Remarks In this paper, the effect of bulk flow on the reaction of an initially nonporous solid with a gas has been examined. It has been shown that this effect can be quite large, increasing with the importance of the product-layer diffusion. When the overall rate is completely controlled by diffusion, an exact analytical solution including the effect of bulk flow has been obtained. For intermediate values of,:a the law of additive reaction times has been shown to provide an approximate closed-form solution. Thus, an equation for conversion vs. time has been derived which includes the effect of chemical reaction, product-layer diffusion, and external mass transfer. The results presented in this paper were obtained for constant temperature and bulk fluid concentration. It has been shown, for a gas-solid reaction without volume change, that the differential form of the approximate solution holds valid even when the temperature of the system or the bulk concentration varies with time, as long as the solid is isothermal at any given instant (Sohn, 1978). The same can be extended to the present case. This would be accomplished by writing eq 34 and 39 in dimensional form and incorporating the variation of temperature and bulk concentration with time. The resulting expressions are expected to yield a close approximation to the results of the exact numerical solution. The variation of temperature and bulk concentration may be spatial rather than temporal, but the development will be analogous. The differential form of the solution is very useful in analyzing multiparticle systems because it obviates the necessity to numerically solve the governing differential equation such as eq 6 at every time step and for different particle sizes. Nomenclature A, = external surface area of the particle b = stoichiometry coefficient C, = integration constant CT = total molar concentration of gas D = molecular diffusivity De = effective diffusivity F, = particle shape factor (equal to 1, 2, and 3 for slabs, long cylinders, and spheres, respectively) gFp(X)= conversion function defined in eq 21 k = reaction rate constant k , = mass transfer coefficient given in eq 35
N = flux of each species P p P ( X )= conversion function defined in eq 27, 28, and 29 r = distance coordinate Sh = Sherwood number Sh* = modified Sherwood number defined by eq 41 t = time t* = dimensionless time defined by eq 16 tf = dimensionless time defined by eq 25 xA = mole fraction of gas A X = fractional conversion of solid y = dimensionless concentration defined by eq 7 Greek Letters = dimensionless distance defined by eq 8 0 = volume change parameter defined by eq 9 Y = number of moles of gaseous product produced from one mole of gaseous reactant = number of moles of B per unit volume of the solid = dimensionless group defined by eq 1 2 Subscripts A = gas A b = bulk phase B = solid B c = value at reaction front C = gas C D = solid D s = value at external surface L i t e r a t u r e Cited Bird, R. B., Stewart, W. E., Lightfoot, E. N., "Transport Phenomena", Wiley, New York, 1960. Haung, H. H., Bartlett, R. W., Met. Trans. 8, 78, 369 (1976). Lapidus, L., "Digital Computation for Chemical Engineers", McGraw-Hill, New York, 1962. Natesan, K., Philbrook, W. O., Trans. TMS-AIM€, 245, 2243 (1969). Sohn, H. Y., Hwahak Konghak(J. Korean Insf. Chem. Eng.), 14, 3 (1976a). Sohn, H. Y., HwahakKongkhak(J. Korean Inst. Chem. Eng.), 14, 65 (1976b). Sohn, H. Y., Met. Trans. 8 , #E,89 (1978). Sohn, H. Y., in "Rate Processes of Extractive Metallurgy", p 1, H. Y. Sohn and M. E. Wadsworth, Ed., Plenum Press, New York, 1979. Sohn, H. Y., Szekely, J., Chem. Eng. Sci., 27, 763 (1972). Szekely, J., Chem. Eng. Sci., 22, 777 (1967). Szekeiy, J., Evans, J. W., Sohn, H.Y.. "Gas-Sold Reactions", Academic Press, New York, 1976. Weekman, V. W., Gorring, R. L., J. Cafal., 4, 260 (1965). Yagi, S., Kunni, D., Proceedings, 5th Symposium (International) on Combustion", p 231, Reinhold, New York, 1955.
Received for review February 12, 1979 Accepted January 7 , 1980
This work was supported in part by a research grant from the University of Utah Research Committee.
Simple Conversion Relationships for Noncatalytic Gas-Solid Reactions Hong H. Lee Department of Chemical Engineering, University of Florida, Gainesville, Florida 326 1 7
A simple conversion relationship is derived for noncatalytic gas-solid reactions in which the pore structure undergoes a change due to the reaction. Algebraic manipulation of the relationship yields the predicted conversion, given the physical properties of the solid reactant. An approximate conversion relationship is also developed for those reactions for which the physical properties are not known. This approximate relationship is a useful tool for correlating the conversion since the consistency of the correlation result can be checked to the extent the available information would allow. These results are applied to two reaction systems involving nickel oxide.
Various models have been developed for noncatalytic gas-solid reactions (Yagi and Kunii, 1955; Petersen, 1957; 0196-4305/80/1119-0242$01.00/0
Szekely and Evans, 1970; Chu, 1972; Ramachandran and Smith, 1977). Chu (1972) developed the parallel plate
0 1980 American Chemical Society
