Ind. Eng. Chem. Res. 1989,28, 1550-1553
1550
COMMUNICATIONS Reversible, Noncatalytic Reactions between Gases and Solids in Fixed Beds Noncatalytic reactions between gases and solids are normally assumed to be irreversible because product gases are eventually swept away by a flow of reactant gas. If diffusion from the point of reaction is slow, however, any potential for reversibility will slow down the overall rate of reaction and should be taken into account. A model previously developed for the irreversible case, in which reaction is confined to a narrow zone between product layer and unreacted core, has been extended to account for reversibility. A multivariable optimization technique has been used with experimental data obtained from the reaction of discrete pellets, to find the equilibrium and rate constants that characterized the reaction. Those were then used to predict the performance of fixed beds of the pellets. Families of curves are presented for various values of the reaction parameters. Noncatalytic reactions between gases and solids are usually assumed to occur irreversibly when being analyzed in connection with the design of large-scale chemical reactions. The assumption may be justified for highly porous solids reacting with a pure gaseous reactant. In those circumstances, products diffuse rapidly from the point of reaction to the exterior surface and are swept away. When, however, diffusion of gases through the solid is difficult, product gases accumulate in contact with solid product, so the possibility of a reverse reaction must be considered. Even when conditions favor a reverse reaction, true reversibility is unlikely to be achieved since the physical form of the reformed solid is unlikely to be identical with its form before reaction. When the reactor takes the form of a fixed bed, all solid downstream of the points of reaction within the bed will be exposed to a mixture of product and reactant gases. Thus, the conditions exist for the reverse element of the reaction to occur so that the rate of overall conversion is decreased. If the gaseous feed contains some product gases, the total conversion of the solid will be limited by equilibrium considerations. If, however, the fixed bed is fed with pure reactant gas, the solid reactant will eventually be completely converted whatever the interim degree of reversibility. Szekely and Evans (1971) and Evans and h a d e (1979, 1980), have considered reversibility in gas-solid noncatalytic reactions when using the grain model. Their analysis was restricted to reactions where 1mol of gaseous reactant yields 1 mol of gaseous product.
ters are chosen so as to maximize the agreement between them. The model used in this paper assumes that an unreacted core of solid is surrounded by a layer of solid product, the two being separated by a narrow zone of reaction. It differs from the classical shrinking core model in that the change from product to reactant occurs not instantaneously at a sharp interface within the solid but over a narrow, though finite, zone of reaction. Described as a diffuse interface model, it was developed originally for irreversible reaction of the type A(g) + bB(s) products (2) It was shown to give more realistic values of energies of activation and of orders of reaction than the sharp interface model (Bowen and Cheng, 1969; Mutasher, 1987; Bowen et al., 1989). In the present paper, equations for a reversible reaction will be restricted to certain orders of reaction and solutions obtained as a guide to more complex systems. Initially, it will be assumed that the reaction shown in eq 1is first order with respect to A and mth and nth order, respectively, for D and E. It should be noted that m and n will not necessarily be equal to d and e in a complex reaction. Orders of reaction will be taken as zero with respect to solid, in both directions. At an incremental surface of thickness dr, within the reaction zone, the net forward reaction can be written as
-
The Model Consideration will be given to the design of a fixed bed reactor in which the following reaction is occurring:
Single-Pellet Equation It is typical to obtain kinetic data for gas-solid reactions by exposing discrete pellets of solid to the reactant gas in conditions similar to those experienced by pellets at the inlet to the fixed bed. Experimental plots of conversion against time are compared with the predictions of a theoretical model of the reacting pellet, and kinetic parame-
--W A '
dt
-
1
dmen
OSS8-58S5/89/262S-1550$01.50/0 0 1989 American Chemical Society
(5)
Ind. Eng. Chem. Res., Vol. 28, No. 10, 1989 1551 For a reversible reaction, conditions at the interface between the reaction zone and the unreacted core will correspond to chemical equilibrium. Hence, dmen -CA*)~+~ K= (6) (d + e)m+n CA* For the particular case of m + n = 1, eq 5 and 6 combine to give
(c,
--CWA' - 4 d k f [ (1 + dmen )(CA - CA*)] dr K(e + d )
dt
where Rb is the rate of reaction per unit volume of reactor. It can be related to the rate per pellet, R,: 3(1 - e ) (for spherical pellets) Rb = R, 4?rr,3
(7)
The model assumes linear concentration gradients to exist through a narrow reaction zone. Hence,
Integrating across the zone from rc to rc + z gives a total reaction rate:
NBis the amount of solid reactant in a pellet at time t:
where f is the fraction of the diffuse interface reaction zone that has been reacted, between the original concentration qa and the equilibrium concentration q*. Hence,
where
+=- I+--+2l(
;($) -
If accumulation in the gas phase is negligible, the rate at which molecules of gaseous reactant diffuse through the product layer and into the reaction zone is equal to the rate of reaction. Hence, the overall rate is
1--
From eq 14 and 18, an expression for drcfdt can be obtained:
--dr, dt
r,
Also Zone thickness, z , can be obtained from eq 9 and 11: where D,is some effective diffusion coefficient in the zone. If, from the point of view of diffusion, the zone is perceived as consisting of two halves, in the first, D, applies, and in the second, D,, then
From (10) and (ll),it can be shown that
so that eq 9 becomes A = --m
dt
r
1
Fixed Bed Equation Excluding diffusional effects, the mass conservation equation can be written as
22
=
'*
dmen
+
K(e + d )
I
(20)
Results and Discussion The mathematical model is defined by eq 14-20. Most of the coefficients are functions of temperature and could be included as such in the solution, together with a statement of energy balance. For the purposes of illustrating an application of the model, it will be assumed that any heat of reaction can be dispersed sufficiently quickly to prevent an appreciable change in temperature of the pellet or of the bed. A bed of length 100 mm will be chosen, operating at 673 K with a gas velocity u = 2.62 X lo9 ms-'. Typical values chosen for the coefficients are as follows: D p = 1.67 X lo4 m2 s-l, D, = 1.48 X lo4 m2 s-l, K = 1, and kf = 333.3 s-l. Multiples of some of these values are also used, as indicated. The model is solved with a finite difference technique (Carnhan et al., 1969) and a modified Runge-KuttaMerson method (Hall and Watt, 1976). The following boundary and initial conditions were used: C, = C, for L = Oand t > 0, C, = OforL > 0 and t = 0, and r, = rs for 0 < L < LT and t = 0. Figure 1 gives a diagrammatic
1552 Ind. Eng. Chem. Res., Vol. 28, No. 10, 1989
Reaction progress
01
WLt h
t Lme
Figure 1. Concentration profile during the progression of the reaction zone.
___
,________ breakthrough curves 1
0 9
J ~
-
~
__ + n
r-
Figure 3. Fractional conversion as a function of equilibrium constant and bed length (t = 2400 s).
reaction time (sec)
Figure 2. Dimensionless concentration of reactant gas leaving a bed, as a function of equilibrium constant and time. Table I. Effect of System Properties on Reaction Zone Thickness 108D,, m2 108D,,m2 kf, s-l K z,mm 167 148 333.33 0.1 0.0333 1.0 0.0734 167 148 333.33 10 0.0936 167 148 333.33 0.0967 333.33 100 167 148 0.0971 333.33 lozo 167 148 L, mm 10aDo,, m2 s-l 108D,,m2 s-l k , s-l K =1 K = 100 0.0132 333.33 0.0101 167 1.48 167 14.8 333.33 0.0306 0.0403 167 1480 333.33 0.1014 0.1336 1.67 148 333.33 0.0106 0.014 0.0422 148 333.33 0.0321 16.7 0.0966 0.1274 1670 148 333.33
representation of the progression of the reaction within a single pellet. Within the core, the gaseous concentrations are at their equilibrium values, so no net reaction occurs there. The effect of reversibility is most evident in Figures 2 and 3 for different values of K. Following a short period during which nitrogen is swept from the bed, the effluent concentration reaches a plateau corresponding to equilibrium conditions. These persist for a while, but the equilibrium is eventually destroyed along the length of the bed as fresh gaseous reactant enters. Eventually a second breakthrough will occur when the effluent concentration rises to the inlet value.
L
-r C - r
- -
~
_ -
?
,
_
_
_
- - -
I _
Figure 4. Effect of product diffusivity on fractional conversion along a bed (t = 600 9).
Table I shows the effect of diffusivities and chemical constants on the reaction zone thickness. The model will be less accurate as z/rc increases. Pellets of 6-mm diameter have been used as a basis for the tabulated values. The effects of changing D, and D,are shown in Figures 4 and 5. In the former figure, D,is kept constant and four values are assumed for D,, each at two levels of K. D, has a dominant influence on the progress of the reaction, which is negligible as D, approaches zero. In Figure 5 , D, is kept constant and D,varied at two levels of K. Again D, has an important effect on the extent of the reaction, though it is not as dominant as D,. Many other models of gassolid noncatalytic reaction take no account of the diffusion characteristics of the core. The rate at which the core can
Ind. Eng. Chem. Res., Vol. 28, No. 10, 1989 1553
CT = total concentrationof reactant and product gases, kmol
i ti
2C
d?
?nc
6L1'
8:'
13:
I ~ l ~ i" 7l
Figure 5. Effect of core diffusivity on fractional conversion along a bed ( t = 600 s).
be penetrated by reactant gas is an important reaction parameter.
Conclusions In gas-solid, noncatalytic reactions, the effects of reversibility may be important. If the reactant gas is contaminated with product gas, both the rate of reaction and the ultimate conversion will be affected. If the reactant gas is pure but the diffusion of gases to and from the reaction site is hindered, accumulation of products at the site will slow down the reaction but not affect the ultimate conversion. The effect of the degree of reversibility is most easily seen when the concentration of reactant gas in the effluent from a fixed bed is plotted as a function of time. As the equilibrium constant decreases and the reaction becomes more reversible, the concentration of the reactant gas in the effluent emerges at a constant value for an increasing part of the reaction time. The magnitude of the constant value becomes progressively higher with reversibility. The effects of changing the other reaction parameters have been highlighted, particularly values of diffusivity in the product layer and the unreacted core. The latter is normally ignored in gas-solid, topochemical models but is clearly important insofar as the reactions can only proceed if the core can be penetrated. In its present form, the model is confined to certain orders of reaction, such that m + n = 1. It can be extended easily to m n = 2, but higher values would prove more intractable.
+
Nomenclature A = gaseous reactant b = stoichiometric number B = solid reactant CA = concentration of gaseous reactant, kmol m-3 CA* = equilibrium concentration of gaseous reactant at any point within the pellet, kmol m-3 C, = concentration of gaseous reactant at outer surface of the pellet, kmol m-3 C, = concentration of gaseous reactant at the inlet to the fixed bed, kmol m-3
m-3 d = stoichiometric number D = gaseous product D,, D,, D, = diffusion coefficient in the core, product, and reaction zone, m2 s-l e = stoichiometric number E = gaseous product f = fractional conversion of the reactant solid in the reaction zone g = gaseous phase j = stoichiometric number J = solid product k f = forward reaction velocity constant for diffuse interface model k , = backward reaction velocity constant for diffuse interface model K = equilibrium constant in diffuse interface model L = length of reaction bed, mm LT = total length of reaction bed, mm m , n = orders of reaction with respect to gaseous products NA, NB = number of kilomoles of gaseous/solid reactant = number of kilomoles of gaseous/solid reactant in a zonal increment dr P = total pressure of the system, bar qa = concentration of solid reactant initially, kmol m-3 q* = equilibrium concentration of solid reactant, kmol m-3 r, = radius of unreacted core, mm rs = outside radius of the spherical pellet, mm R = gas constant (8.314 kJ/(kmol K)) Rb = rate per unit volume of bed at which moles of component A are reacted, kmol m-3 s-l R, = rate at which single pellet reacts, kmol s-' s = solid phase t = time, s T = reaction temperature, K u = superficial reactant gas velocity, m s-l z = thickness of the reaction zone, mm Greek Symbols
+ = function of order of reaction in eq 12 e =
voidage of bed
Literature Cited Bowen, J. H.; Cheng, C. K. Chem. Eng. Sci. 1969,24, 1829. Bowen, J. H.; Khan, A. R.; Mutasher, E. I. Chem. Eng. Res. Des. 1989, 67, 58.
Carnhan, S.; Luther, H. A,; Wilkes, J. 0. Applied numerical methods; Wiley: New York, 1969. Evans, J. W.; Ranade, M. Chem. Eng. Sci. 1979,35, 1261. Evans, J. W.; Ranade, M. Ind. Eng. Chem. Process Des. Dev. 1980, 19, 118. Hall, G.; Watt, J. M. Modern numerical methods for ordinary differential equations; Clarendon Press: Oxford, 1976. Mutasher, E. I. Ph,D. Thesis, University of Wales, University College of Swansea, Swansea, Wales, 1987. Szekely, J.; Evans, J. W. Chem. Eng. Sci. 1971, 26, 1901. *Present address: Department of Chemical & Process Engineering, University of Dar Es Salaam, P.O. Box 35131, Dar Es Salaam, Tanzania. t Present address: School of Chemical Engineering, Tel-Mohammed University of Technology, Baghdad, Iraq. Present address: Department of Chemical Engineering, University of Technology, Loughborough, Leicestershire LE11 3TU.
*
E. I. Mutasher,' A. R. Khan,' J. H. Bowen* Department of Chemical Engineering University of Wales University College of Swansea Swansea, SA2 8PP Wales Received for review March 11, 1988 Revised manuscript received May 10, 1989 Accepted May 23, 1989
