Single Nonisothermal Noncatalytic Solid-Gas Reaction. Effect of Changing Particle Size A. Rehmat' and S. C. Saxena" Depadment of Energy Engineering, University of Illinois, Chicago, lllinois 60680
The conversion-time relationship for a noncatalytic nonisothermal solid-gas reaction is formulated employing a shrinking core model for a single spherical pellet in which for simplicity the solid-gas reaction is approximated by a first-order irreversible reaction. The formulation takes into account the changing size of the particle during reaction and also includes the influence of intermolecular gas film diffusion, intraparticle ash layer diffusion, and chemical reaction rate. Numerical calculations reveal that the influence of changing size on conversion-time relationship is significant whenever the reaction is controlled either by intermolecular and/or intraparticle diffusion. The effect of heat transfer coefficient, while found to be negligible for a growing particle, has a marked influence on a shrinking particle. This formulation and the conclusions drawn from the numerical calculations are directly applicable in modeling fluidized bed reactors.
Introduction In the analysis of a chemical reactor involving heterogeneous gas-solid reactions, a comprehensive study of the primary single nonisothermal noncatalytic reaction not only helps in arriving a t important conclusions but can also assist in the substantial simplification of the overall formulation. For simplicity, and as an initial instance, we consider a single spherical pellet. The single pellet results can then be used to predict the behavior of a fixed bed (Bowen and Lacey, 1969) or to analyze the operation of a fluidized bed (Yagi and Kunii, 1955). Heterogeneous gas-solid reactions involving a moving boundary occur frequently in chemical and metallurgical industrial processes including combustion and gasification of solid fuels, reduction of metallic oxides, roasting of sulfides, and the regeneration of carbon-deposited catalysts and hence have been extensively studied by many investigators, viz., Aris (1965); Cannon and Denbigh (1957); Ishida and Shirai (1969); Ishida and Wen (1968); Lemcoff and Cunningham (1973); Levenspiel (1962); Mcllvried and Massoth (1973); Shen and Smith (1965); Szekely and Evans (1970); Weisz and Goodwin (1963); Wen and Wei (1971); White and Carberry (1965); Williams et al. (1972); Yagi and Kunii (1955) and Yoshida and Kunii (1969). A gas-solid reaction usually involves heat and mass transfer processes and chemical kinetics. These processes are further complicated by the fluid flow, size, shape, and the configuration of the solid particles. In view of these complications many authors (Lemcoff and Cunningham, 1973; Levenspiel, 1962; Mcllvried and Massoth, 1973; Weisz and Goodwin, 1963; Yagi and Kunii, 1955; Yoshida and Kunii, 1969) have attempted to analyze the problem after neglecting some of the reaction resistances, or by using a simple form for the reaction rate expression or by restricting the system to isothermal conditions only. On the other hand, many works (Aris, 1967; Ishida and Shirai, 1969; Ishida and Wen, 1968; Shen and Smith, 1965; Wen and Wei, 1971; White and Carberry, 1965) have accounted for all the three resistances simultaneously and included expressions for the rate of reaction in terms of the parameters defining the interphase mass transfer, intraparticle diffusion, and chemical reaction rate constants. Different authors have employed varying forms for the resistance parameter. Though the majority of the workers have considInstitute of Gas Technology, Chicago, Ill. 60616.
ered first-order reactions, Williams et al. (1972) have based their formulation on an n t h order chemical reaction. Lemcoff and Cunningham (1973) and Wen and Wei (1971) considered first-order multiple reactions. Most of these workers have limited their models by neglecting the structural changes as reaction proceeds. Such changes are likely to occur in a number of situations due to sintering, agglomeration, growing or shrinking particle size. Shen and Smith (1965) and White and Carberry (1965) have incorporated the concept of changing size into their analysis but restricted their investigations to isothermal cases only. Most gas-solid reactions have been analyzed by the shrinking core model assuming that the particles are nonporous and the size remains invariant during the reaction. In gas-solid reactions, the overall reaction rate is controlled by a series of three events which may occur simultaneously. These are: (1) diffusive transport of gaseous reactants from the bulk stream through the gas film surrounding the particle to the external surface of the solid particle (interphase transport), (2) diffusion of reactant through a product layer (a blanket of ash) a t the surface of the unreacted core, the thickness of which changes with time (intraparticle diffusion), and (3) chemical reactions between gaseous reactant and the unreacted solid a t the product-reactant boundary (reaction-front or surface) which is again time dependent. The system is assumed to be a t quasi-steady state, an assumption analyzed and justified by Bischoff (1963). Many authors (Lemcoff and Cunningham, 1973; Szekely and Evans, 1970; Yoshida and Kunii, 1969), in an attempt to improve upon this shrinking core model, have given treatments in which solid is regarded as porous and the chemical reactions are assumed to take place in the pores rather than a t the interface. Both of these procedures have provided a satisfactory overall interpretation of the limited available experimental data. The purpose of the present analysis is to incorporate the change in size of the solid during a nonisothermal gas-solid reaction taking into consideration all the essential parameters. The ultimate aim of this work is to broadly define the physical criteria that influence gas-solid reactivity and to define the optimum conditions for carrying out such reactions in fixed and fluidized-bed reactors. For this purpose the shrinking core model is employed. The description of the model is readily available in the literature (Levenspiel, 1962; Shen and Smith, 1965; Wen and Wei, 1971; Yagi and Kunni, 1955) and the following assumptions are implicit in Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 2, 1976
343
Y
and
' Y
,ASH
LAYER
4%
H&e
NSho = N s ~ ( R o / ~ R ) ( D A / D ~ A )
(6)
The solution of eq 1 in conjunction with the boundary conditions of eq 4 and 5 after lengthy manipulations finally leads to the following relation for the concentration of gas A a t the reaction surface
R
R,
r
0
rc
rc
r
R,
1 -=1+
R
Figure 1. Gas-solid reaction of a growing particle: the concentra-
UcNShOts'
WAC
+
tion and temperature profiles.
the present work: (a) the pellet is spherical, (b) the pellet is in a gas stream of constant composition and temperature, (c) the reaction a t the surface of the solid is first order with respect to the concentrations of the gaseous and solid reactants and is irreversible, (d) the system is a t quasi-steady state, and (e) the reacted and unreacted regions are separated by a sharp interface constituting a plane a t which the reaction occurs.
Theoretical Formulation Let a spherical pellet of initial radius Ro a t time t = 0 be exposed to a gas mixture containing a reactant A whose concentration CAO in the bulk phase remains constant. The reactant gas diffuses to the surface of the pellet through the gas film and initially reacts with the solid a t its outer surface; but, subsequently, the reaction will take place a t a front which moves inward. Depending upon the density of the solid product formed, the particle will either grow or shrink as the reaction proceeds. The process will then require the radial diffusion of reactant gas and the gaseous reaction products through the solid product layer. At any given instant let the outer radius of either a growing or a shrinking particle be R. Figure 1 shows the instantaneous concentration of A and the temperature a t various locations in the pellet. The following single reaction is considered in the present analysis. aA(g) + S(s)
ki
eE(g) + jSl(s)
Based on an unreacted shrinking core model for a nonisothermal system under the quasi-steady-state assumption, a material balance for a reactant gas component A leads to the relation
Many simplifying assumptions are implicit in the derivation of this relationship. The effective diffusivity, de^, is taken to be proportional to T1.O over the entire diffusion regime as a compromise of its dependence as T1.5-2 in the molecular diffusion regime and as in the Knudsen diffusion regime. The gases behave ideally and the total pressure remains constant throughout the reaction. The temperature dependence of the rate constant is taken to be of Arrhenius type. Consequently
and
k1(T) = k1° exp(-El/RT)
Similarly, the heat balance equation under the assumption of quasi-steady state is given by
with the boundary conditions
RTo
-
dU
(x) (,>
= 4101
(y)
= r/Ro
N N ~=ON N , ( R o / ~ R( k)l k d Equations 11, 12, and 13 give
(2) Eliminating
(3) The boundary conditions are
344
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 2, 1976
(1 -
&)]
where
The dimensionless parameters E and W A are defined as
t
(10)
uc- 1 =
WAC
from eq 7 and 15 we get
(14)
Since the solid product layer does not have the sanie volume as the solid reactant consumed, a change in particle size will occur. This can be accounted for by introducing the following parameter volume of S1 formed - j p s M ( S 1 ) z=-volume of S consumed MSp(S1)
(17) Substituting eq 27 and 28 into eq 26 we get
T h u s Z = 1 for a pellet whose size remains invariant as the reaction proceeds. Z > 1 for an expanding and Z < 1 for a shrinking particle. From the stoichiometry of the reaction, we have
Combination of eq 17 and 18 gives ts3 =
z+ (1 - z)tc3
In dimensionless form eq 29 is
We now define a characteristic time, T, as (19)
This expression when .introduced for Es in eq 16 accounts partially for the effect of varying pellet size. A similar efO they fect must also be considered on Nsho and N N ~since also depend upon the overall size of the particle. The experimental correlations giving the gas film mass transfer in terms of Sherwood number are generally expressed as functions of Schmidt and Reynolds number and the functional form of the correlation depends upon the flow regime (Rowe et al., 1965). The experimental data for the laminar regime suggests the following correlation (Ranz and Marshall, 1952)
Numerical estimates of K1 and K2 have been given by Ranz and Marshall (1952). A similar expression is also available (Ranz and Marshall, 1952) for the convective heat t,ransfer coefficient in terms of Nusselt number. In the laminar regime it is expressed as a function of Reynolds number as follows
NN" = K1
The reaction rate is assumed to be first order with respect to the concentrations of the gaseous and solid reactants and having the following form
+ K2(NPr)1/3(NRe)1/2
Using eq 6 and 14, we obtain the following and NNUO
NShO
7 has significance inasmuch as it represents the time required for the complete reaction of the pellet a t the temperature of the bulk gas and when all the diffusion resistances are negligibly small. Further let us introduce a dimensionless time 6 such that
6 = t/r
(32)
Equation 30 will now simplify to dtc do
--WAC
(33)
U,
with the boundary condition that
i, = 1 a t 0 = 0
(34)
Equations 33, 34, 7 , and 16 completely describe a system of gas-solid nonisothermal first-order reaction. Since eq 16 does not represent U , as an explicit function of E,, numerical approach is essential for the solution of the differential eq 33. Such results are described in the next section. In case the temperature of the system remains constant, the results of the above analysis become much simpler and such a case of isothermal combustion will be discussed now. The expression for the surface reaction and hence WAC are easily obtained by substituting U , = 1 in eq 7 viz.
and
1
-WAC =
[,+-
41Ec2 NShOEs 2
+ 41Ec (1 -
k) ]
(35)
Similarly, eq 33 for such a case simplifies to dEC - WAC dB
The relations of eq 22-25 and 19 in conjunction with eq 7 and 16 enable the determination of WAC and U,, respectively, with & as the only parameter in each case for any given system characterized by such parameters as Kl', K3, K4, Cbl,Z, PI, and EIIRTo. Next we relate the rate of disappearance of the solid reactant with the rate of surface reaction. From the stoichiometry of the chemical reaction we have
~ N A dNs -adt dt The rate of disappearance of solid S is given by
(36)
The boundary conditions remain the same as above for the general case of nonisothermal reaction. If eq 19, 22, and 35 are substituted in eq 36, the resulting differential equation has the following analytical solution
o = (1 - E,)
41
+-(I 2
- & 2 ) - ___ 41 (1 - p2/3) + 2(1 - 2 )
where p = (,3(1 -
z)+ z
(38)
This result has been derived earlier by Shen and Smith (1965) and their eq 29 is found to be equivalent to the above eq 37. Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 2, 1976
345
Finally, we relate the conversion of solid reactant S to the parameter of the system E,. Since the spherical particle is used, the relation is simple, viz.
x = 1 - Ec3
(39)
Here X is defined as the ratio of the moles of solid reacted to the initial moles of solid in the particle. Finally, for the case of a single reaction that we have considered, the effectiveness factor, 11, following the definition of Ishida and Wen (1968), is given by 11=
4.rrrC2akI(T~)CSOCAC 4rrc2akI( To)CSOCAO
0
(X)vs:reduced time (0) for an isothermal reaction, changing particle size ( Z ) ,small and large mass transfer coefficients ( K 3 ) ,and a low value of the intraparticle diffusion resistance (& = 0.139). K1’ = 2.0.
Figure 2. Conversion
Simplifying eq 40 and substituting eq 7 and 10 we get 1 11=
(41) For the isothermal case ( U , = l),the effectiveness factor has a maximum value of 1 which is obtained a t the complete reaction (Fc = 0). Results and Discussion In this section we examine the consequences of the above theory by considering a number of illustrations each referring to a particular condition specified by a set of parameters. We first consider the isothermal case because this will provide an initial base to analyze the nonisothermal reactions. In each case, the results are presented in terms of the conversion-time relationship since it is a good measure of the progress of the reaction. The conversion for a nonisothermal reaction is a function of the parameters 8, Kl’, K3, Kq, 41, 2, (3, and EllRTo, while for an isothermal case the controlling parameters are 8, Kl‘, K3, $1, and 2 only. The various characteristics discussed below examine the variation of X with 8 for the two general cases (isothermal and nonisothermal) for a range of specific numerical values for the remaining parameters. For these numerical calculations, K1’ has been arbitrarily assigned a value of 2.0. In this treatment, we have assumed that the reaction rate is much faster than the diffusion rates corresponding to the gas film and ash layer. As a result, the diffusion processes will control the rate. This assumption is plausible because if the reaction is the rate-controlling step, the changes in size of the particle will have no effect on its conversion time. For such a case, the parameter & effectively represents a quantity which can be regarded as the measure of intraparticle diffusion resistance. Figures 2 and 3 give conversion characteristic for a spherical particle under isothermal conditions corresponding to low and high values of intraparticle diffusion resistance (41), respectively, and for a range of mass transfer coefficient (K3).In both the cases, the change in the particle size as the reaction proceeds has a marked effect on the conversion-time relationship. For the case of low mass transfer rate across the film (K3 = O.l), irrespective of the value of d1, the increase in the size of the particle always helps to decrease the reaction time for a specified conversion. This can be explained in the following way. As the particle grows, the surface area through which the gas can diffuse also increases, and consequently, more gaseous reactant is made available at the core for reaction. The reaction in such a case is controlled by the gas film diffusion rate. It should be noted, however, that when the intraparticle diffusion resistance (41) is large, the influence on the reaction time due to increase in size of the particle is consid346
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 2, 1976
Figure 3. Conversion (X)vs. reduced time (0) for an isothermal reaction, changing particle size ( Z ) ,small and large mass transfer coefficients ( K 3 ) ,and a high value of intraparticle diffusion resistance (61 = 10.0). K1’ = 2.0.
O,‘
01
02
03
04
05
d6
0
Figure 4. Conversion (X) vs. reduced time (0) for a nonisothermal reaction, changing particle size ( Z ) ,low mass ( K 3 ) and heat (K4 = 1.0) transfer coefficients, and a low value of the intr9particle diffusion resistance ($1 = 0.139). K1‘ = 2.0; @ = 0.02; E1/RTo = 25.
erably reduced as seen in Figure 3. On the other hand, when the mass transfer rate through the gas film is large (K3 = 50), the influence of the changing size on conversion vs. time relationship is reversed. T o accomplish the same conversion now, it takes longer for the growing particle than for the shrinking particle. The time difference will depend upon the value of &. This can be inferred by comparing the curves corresponding to K3 = 50 in Figures 2 and 3. When the mass transfer across the film is large, the resistance due to intraparticle diffusion plays a relatively more important role. For a growing particle, it would take longer
0 Figure 5 . Conversion (X) vs. reduced time ( 8 ) for a nonisothermal reaction and high intraparticle diffusion resistance: effect of particle size (2).K3 = 0.1; Kd = 1.0; = 6.0; K1' = 2.0; p = 0.02; E l / RTo = 25.
1 01
05
1
l=l "
5
10
5
Figure 6. Conversion (x) vs. reduced time (0) for a nonisothermal reaction: effect of intraparticle diffusion resistance ($1). K3 = 0.1; K j = 1.0; Z = 10; K1' = 2.0; 81 = 0.02; El/RTo = 25.
1 0 5 0 Figure 7. Conversion ( X i vs. reduced time ( 8 ) for a nonisothermal reaction: effect of heat (K4) and mass (K3) transfer coefficients and changing particle size ( Z ) . = 0.139; K1' = 2.0; pi = 0.02; 001
01
EllRTo = 25.
for the gas to reach the reaction front than for the case of a shrinking particle, and it is directly reflected in the corresponding reaction times. In such a case, t h e reaction is controlled by the intraparticle diffusion rate. T h e resistance due to intraparticle diffusion is also present in the case of low mass transfer rate across the film, but is of less consequence as long as the reaction is controlled by the film diffusion. Figures 4 t o 9 illustrate the influence of various parameters on the conversion-time relationship for the case of a nonisothermal reaction. We have considered only those parameters which are directly related to the heat and mass transfer process, viz., K1, K3, and $1, and which play an im-
Figure 8. Conversion (X) vs. reduced time (8) for a nonisothermal reaction and a high intraparticle diffusionresistance: effect of particle growth ( 2 )and mass transfer-coefficient (K3).& = 6.0; K4 = 0.1 to 100; Ki' = 2.0; pi = 0.02; Ei/RTo = 25.
0 Figure 9. Conversion (XIvs. reduced time ( 8 ) for a nonisothermal reaction of a &inking (2 = 0.75) and high intraparticle diffusion resistance: effect of ( K ~an!) heat ( K ~transfer ) coefficients. $1 = 6.0; KI' = 2.0; 01 = 0.02; E1/RTo= 25.
portant role when the size of the particle changes during the reaction. In these calculations, /3 and E&TO are assigned constant values of 0.02 and 25, respectively, following Wen and Wei (1971). In Figure 4 the intraparticle diffusion resistance ($1) and the heat transfer coefficient ( K 4 ) are kept small t o investigate the effect of K3. When K3 is small ( K 3 = O . l ) , the initial trend for the conversion vs. time for a growing and a shrinking particle is the same as that for the isothermal case, viz., it takes longer for a shrinking particle t o achieve the same conversion than for a growing particle. However, as the reaction proceeds and the temperature of the particle rises due t o the heat of reaction and low heat transfer coefficient, the conversion depends upon the nature of the change in the particle size. For a shrinking particle, because of the smaller volume and smaller surface area for the heat transfer, the temperature of the particle rises more rapidly than for a growing particle. Since the intraparticle diffusion resistance 41 is inversely proportional to the temperature, the rate of the mass transport through the ash layer is greatly enhanced for the shrinking particle resulting in the reversal of the trend. Now the shrinking particle is consumed more rapidly than the growing particle. Notice that this reversal occurs earlier if K3 is increased. From this observation, it is evident that in the nonisothermal case, the resistance due t o the intraparticle diffusion has a considerable influence on the reaction time due to its dependence on the temperature. If the resistance due t o the intraparticle diffusion is increased ($1 = 6.0) as shown in Figure 5 , its Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 2, 1976
347
effect on the conversion-time relationship is felt from the start of the reaction, namely larger reaction times for the particles which exhibit larger growth. The effect is more pronounced as the reaction progresses due to the faster increase in the temperature of the shrinking particle. The comparison of Figures 4 and 5 also illustrates the increased reaction time due to the increased intraparticle diffusion resistance when other parameters are kept constant. The latter effect is shown independently in Figure 6 in which the conversion-time relationship is plotted for a given system with increasing intraparticle diffusion resistance. Figure 7 illustrates the effect of the heat transfer coefficient (K4) on the growing and shrinking particle. For the growing particle (2 = loo), due to the large surface area for the heat transfer, the change in the heat 'transfer coefficient has negligible effect on the conversion-time relationship as shown by the narrow band when K4 is changed from 0.1 to 100. In contrast, the heat transfer coefficient has a significant effect on the shrinking particle. When the heat transfer coefficient is small (K4 = 0.1) due to more heat retention and resulting high temperature, the reaction proceeds faster and consequently the particle gets consumed much faster than a corresponding growing particle. A similar trend is also apparent in Figure 4. However, when Kq is increased substantially, the behavior corresponds to the isothermal case due to the rapid heat transfer. In this case, the conversion-time for a shrinking particle exceeds the conversion-time for the growing particle as explained in connection with Figure 2. T o highlight the parametric dependence of X ,the results of Figure 7 are further examined in detail in Figures 8 and 9 for growing and shrinking particles respectively. As stated while discussing Figure 7 , the effect of heat transfer coefficient is negligible on a growing particle. Consequently, the behavior of the growing particle when considered alone should resemble that of an isothermal case. In Figure 8 two specific cases of the particle growth are presented. When 2 is kept constant and mass transfer coefficient ( K 3 ) through the gas film is increased, the result is the decrease in the reaction time for the same conversion. Also, for the larger value of K3 (K3 = lo), if the size of the particle is increased, it takes longer for a larger particle t o result in the same conversion due to the increased path for the intraparticle diffusion. Both of these effects are similar t o those found in Figure 2 for the isothermal case. In contrast, Figure 9 illustrates the effect of the heat transfer coefficient ( K 4 ) on a shrinking particle. As explained while discussing Figure 7, the smaller the heat transfer coefficient (K4),the shorter is the reaction time a t a given mass transfer rate. This is borne out by the results of Figure 9 for the two cases, viz., K3 = 0.1 and K3 = 10 when K4 is varied from 0.1 to 100. The plots of this figure also emphasize the expected result that for a given value of the heat transfer coefficient, it takes shorter time to accomplish a given conversion as the mass transfer coefficient is increased. For the purpose of illustrating the usefulness of the effectiveness factor, the curves of 9 vs. X are plotted in Figure 10 for certain selected parameters. I t has been indicated earlier in the text that for a growing particle (2 = 10) the diffusion primarily controls the rate of reaction. This effect is also reflected in Figure 10 (curves 1 and 2 by high values of 9. However, for a shrinking particle (curve 3) the reaction is initially chemical reaction controlled as evident by low values of 7 and a transition in rate controlling step occurs a t X = 0.66 to diffusion control. This is consistent with the observations made while discussing the results of Figure 4. The possibility of geometrical instability exists in 348
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 2, 1976
10.00
c I 1 I =1 R y 1 2 5 , +1-0.139,p, 10.02,for all curvi 2-10, K, -10, K, -1.0
I
I
I
I
I I
I
I
I
I X
Figure 10. Effectiveness factor cases.
( q ) vs.
conversion (X) for selected
the later stages of conversion for this case as can be deduced by the steep slope of the 9 vs. X curve. The above investigations have attempted to highlight the effect of changing particle size on the reaction rate for a nonisothermal gas-solid reaction. In the past, the contribution of changing particle size for nonisothermal gas-solid reactions has been ignored probably in the belief that it was of little consequence. The present work stresses the need to include this feature in the formulation where diffusion is the controlling resistance. The general results obtained here will find ample use in the formulation, description, and design of fixed and fluidized-bed reactors. G e n e r a l Conclusions The effect of change in size of the particle on the conversion-time relationship has been examined in this work for isothermal as well as nonisothermal gas-solid reactions employing a shrinking core model. The following observations are explicit from the solutions of the generalized formulations derived and discussed above. For the isothermal case, both the intermolecular diffusion and the intraparticle diffusion resistances have significant influence over the conversion-time relationship. When only the intermolecular diffusion is prominent, the increase in the size of the particle helps to decrease the reaction time due to increase in the flux of the reactant through the increased surface area. On the other hand, when the reaction is governed by intraparticle diffusion only, increase in size of the particle increases the reaction time due to increased path for the reactant to reach the core. When both of these resistances are present simulta-
neously, they tend to balance the effect of t h e particle growth on conversion-time characteristics. For the nonisothermal case when the heat transfer coefficient is small, the increase in particle temperature increases the magnitude of the intraparticle diffusion coefficient significantly and thereby decreasing the intraparticle diffusion resistance. This effect is more pronounced in a shrinking particle than in a growing particle. Consequently, when the intermolecular diffusion and the intraparticle diffusion resistances are present simultaneously, the shrinking particle requires less time to yield a particular conversion than the growing particle. This trend is opposite t o what is found for the case of isothermal reaction and highlights the importance of considering the nonisothermal nature of the overall particle. However, large heat transfer coefficient yields the results which are similar t o the isothermal case, namely, the growing particle results in a faster conversion than the shrinking particle for a given intermolecular diffusion resistance. For a given set of conditions and a particular growth, it is observed that an increase in either the intermolecular diffusion resistance or the intraparticle diffusion resistance increases the conversion-time as expected. In the calculations we have considered a wide range of values of the heat and mass transfer coefficients and typical values for the intraparticle diffusion resistance and other constants. Various specific situations are expected to lie in these ranges and the particular set of values will depend very much on the nature of the particle. We plan to discuss such cases in a future publication dealing with the processing of coal in a fluidized bed. Acknowledgments The authors are thankful to S. Szepe and T. P. Chen for a critical reading of the manuscript and for making a number of interesting comments. Nomenclature
a = stoichiometric coefficient of the gaseous component A A = gasreactant C = total concentration of gases, mol/ft3 C A = concentration of the gas component A, mol/ft3 CAO = concentration of A in the bulk phase, mol/ft3 CAS = concentration of A at the surface of the particle, mol/ft3 CAC = concentration of A a t the core of the particle, mol/ ft3 C s o = initial concentration of the solid reactant, mol/ft3 C , = specific heat of the bulk gas a t constant pressure, Btu/(lb)(OR) D A = molecular diffusivity of the component A in the bulk gas phase, ft2/h de^ = effective diffusivity of the component A in the ash layer, fP/h e = stoichiometric coefficient for the gaseous component
E E1 = activation energy, Btu/mol g = signifies the gaseous state h = overall convective and linearized radiative heat transfer coefficient, Btu/(h)(ft2)(OR) AH1 = heat of reaction per mole of reactant, Btu/mol j = stoichiometric coefficient of the solid component SI k = thermal conductivity of the bulk gas, Btu/(h)(ft)(OR) k , = effective thermal conductivity of ash layer, Btu/ (h)(ft)(OR) k l = reaction rate constant, ft4/(mol)(h) k1° = frequency factor based on surface area, ft4/(mol)(h) k , ~= mass transfer coefficient for the component A across the gas film, ft/h
K1 = a numerical constant which occurs in the correlation of Sherwood number K1' = KJ2, dimensionless K2 = a numerical constant which occurs in the correlation of Sherwood number K2' = K2/2, dimensionless K3 = defined by eq 24 K4 = defined by eq 25 M s = molecular weight of the solid S M(S1) = molecular weight of the solid S1 N A = number of moles of A, mol N s = number of moles of S, mol N N " = Nusselt number = (2Rh/k), dimensionless N N ~=O ",(Ro/2R)(k/ke), dimensionless Np, = Prandtl number = (C,w)/k, dimensionless N R = ~ Reynolds number = 2uRp/w, dimensionless NR~= O NR,(Ro/R), dimensionless Ns, = Schmidt number = (w/pD~),dimensionless N S h = Sherwood number = ( ~ R ~ , A / D A dimensionless ), Nsho = N ~ ~ ( R ~ / ~ R ) ( D Adimensionless /D~A), p = defined by eq 38, dimensionless r = radial distance from the center of the spherical particle, f t rc = radius of the unreacted core, f t R = pellet or particle radius, f t 40 = initial pellet or particle radius, f t R = gas constant, Btu/(mol)(OR) s = signifies the solid state S = solid reactant S1 = solid product t = time, h T = temperature, OR To = temperature of the bulk gas phase, OR T , = temperature of the unreacted core surface, OR T , = temperature of the outer surface of the particle, OR u = flow velocity of bulk gas, ft/h U = reduced temperature, T/To, dimensionless U , = reduced core surface temperature, T,/To, dimensionless U s = reduced particle surface temperature, T,/To, dimensionless X A = mole fraction of component A, dimensionless X A O = value O f X A in the bulk gas phase, dimensionless Z A S = value of X A at the outer surface of the particle, dimensionless X A C = value of X A a t the unreacted core surface, dimensionless X = conversion of the solid reactant S defined by eq 39, dimensionless 2 = A parameter to characterize the growth or shrinkage of the particle, defined by eq 17, dimensionless Greek Letters = [ CA&,A( To)( - M l ) R ]/k$1, dimensionless 8 = reduced time defined by eq 32, dimensionless 7 = effectiveness factor defined by eq 40, dimensionless w = viscosity of the bulk gas, lb/(ft)(h) E = reduced distance, r/Ro, dimensionless E, = reduced core length of the particle, rc/Ro, dimensionless ts = reduced size of the particle, R/Ro, dimensionless p = density of the bulk gas, lb/ft3 ps = density of the solid S, lb/ft3 p(S1) = density of the solid S I , lb/ft3 T = characteristic time defined by eq 31, hr 41 = a parameter to characterize the ratio of intraparticle diffusion resistance to the reaction resistance and is equal to aRok 1(To)CSO/DeA(To),dimensionless W A = reduced value of x ~ , x A ] x ~dimensionless o, WAC = reduced value of X A C , X A ~ X A O ,dimensionless WAS = reduced value of X A S , X A ~ X A O dimensionless ,
Literature Cited Aris, R., Ind. Eng. Chern., Fundarn., 4,293 (1965). Bischoff, K. B., Chern. Eng. Sci., 18, 711 (1963).
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Receiued for reuiew August 15, 1975 Accepted November 6 , 1975 This work is supported by the United States Energy Research and Development Administration, Washington, D.C., under Contract No. E(49-18)-1787. Computing services used in this research were provided by the Computer Center of the University of Illinois a t Chicago Circle.
Predictability of Reverse Osmosis Separations of Partially Dissociated Organic Acids in Dilute Aqueous Solutions Takeshi Matsuura, J. M. Dickson,' and S. Sourirajan' Division of Chemistry, National Research Council of Canada, Ottawa, Canada, K I A OR9
Data on free energy parameters for H+ and several RCOO- ions and the corresponding nonionized carboxylic acids have been generated with respect to cellulose acetate membranes from experimental reverse osmosis data. The necessary steric parameters have also been similarly generated. These parameters offer a means of predicting reverse osmosis separations of partially dissociated carboxylic acids in aqueous solutions from data on membrane specifications only, given in terms of pure water permeability constant and solute transport parameter for sodium chloride. This is illustrated in this paper with respect to several carboxylic acids (including monocarboxylic, hydroxycarboxylic, and dicarboxylic acids) in reverse osmosis systems involving cellulose acetate membranes and preferential sorption of water at the membrane-solution interface.
Introduction
Reverse osmosis separations of ionic solutes in aqueous solutions have been discussed in terms of the governing free energy parameters (Matsuura et al., 1 9 7 5 ~ )This . discussion has been extended to include reverse osmosis separations of nonionized polar organic solutes in aqueous solutions (Matsuura e t al., 1976). The practical utility of the above approach for predicting reverse osmosis separations of completely ionized or completely nonionized molecules has been illustrated (Matsuura et al., 1975a, 1975c, 1976). On the basis of the above work, it is reasonable to expect that it should also be possible to predict reverse osmosis separations of partially dissociated solutes in dilute aqueous solutions by appropriate combination of data on free energy parameters for the ionized and nonionized parts of the solute molecule. This possibility is illustrated in this paper. For the purpose of this work, the data on reverse osmosis separations of organic acids reported earlier (Matsuura and Sourirajan, 1973) were reanalyzed, along with data obtained in additional reverse osmosis experiments with cellulose acetate membranes carried out specifically for the purpose of this analysis. Thereby the necessary data on free energy parameters for a number of ionized and nonionized species were generated. These parameters, along with those already available in the literature (Matsuura et al., 1975c, 1976) are used to illustrate the predictability of reCo-op student, Department of Chemical Engineering, University of Waterloo, Waterloo, Ont., Canada. 350
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verse osmosis separations of partially dissociated organic acids in dilute aqueous solutions using porous cellulose acetate membranes, just from reverse osmosis data for the system NaCl-H20 for such membranes. Experimental Section
Laboratory-made 316(10/30)-type cellulose acetate membranes (films 6 to 11) (Pageau and Sourirajan, 1972), and dilute single-solute aqueous feed solutions were used in reverse osmosis experiments. Hydrochloric acid, nitric acid, and sodium salts of pivalic, isobutyric, valeric, propionic, acetic, benzoic, p-chlorobenzoic, m -nitrobenzoic, p -nitrobenzoic, o-chlorobenzoic, and o-nitrobenzoic acids (acids 33, 34, 36, 38, 39, 43, and 50 to 54 listed in Table I) were used as solutes in the concentration range 0.001 to 0.003 gmol/l. The sodium salts of acids 33, 34, 36, and 50 to 54 (Table I) were obtained by neutralization of the acid with equimolal quantities of sodium hydroxide; the other sodium salts were obtained commercially. Under the experimental conditions used, each of the above solutes existed essentially as completely ionized molecules in aqueous solution. An all-stainless steel apparatus was used for reverse osmosis experiments; other details of the experiment were the same as before (Matsuura and Sourirajan, 1971). The effective area of the film used was 13.2 cm2. The experiments were carried out a t 250 psig a t the laboratory temperature (23-25 "C). All data in this paper refer to 250 psig and 25 "C. In each experiment, the fraction solute separation f defined as
