Diffusion of Binary Gas Mixtures in Zeolite X Pellets Yi Hua Ma* and Ting Yueh Lee Chemical Engineering Department, Worcester Polytechnic Institute. Worcester, Massachusetts 0 1609
A mathematical model is presented to describe the diffusion of gas mixtures in zeolite pellets. The model is used to determine the intracrystalline diffusion coefficients of each component in a constant volume and well-stirred system for two mixtures: n-butane-isobutane and isobutane- 1-butene. The diffusion coefficients of the mixture are determined by the experimental mixture rate curves which are in the order of cm2/s, which is approximately one order of magnitude smaller than those of pure components. In the case of the isobutane-1-butene mixture, displacement of isobutane by I-butene was experimentally observed. This displacement phenomenon is explained by the fact that the total number of sites available for sorption is finite and the fluxes of 1-butene and isobutane are opposite in direction. One distinct feature of the current model is that only the intracrystalline diffusion coefficients determined during the period of no displacement are needed to describe the displacement phenomenon. This explanation is verified by the good agreement between the experimental data and the theoretical curve.
Due to potential applications in sorption processes and in catalysis, much work has been done recently to study the rate of sorption of gases in molecular sieves. Essentially all of the work is concentrated on the study of rates of sorption of single pure gases in zeolitic materials. However, multicomponent sorption is involved in practical applications. Although substantial information on multicomponent sorption equilibria is available (Barrer and Robins, 1953; Glessner and Myers, 1969; Joubert and Zwiebel, 19711, there is little information available in the literature concerning the rate of sorption of gaseous mixtures in molecular sieves. Recent studies on diffusion in binary sorbate systems either considered one of the components to be “inert” or assumed counter diffusion to occur as recently summarized by Ma and Roux (1973). Work by Kokoszka (1970) on the rate of sorption of propane and butane from helium by 5A molecular sieves showed that the sorption rates of propane and butane in helium were lower than their respective rates of sorption as pure components. Round et al. (1966) presented a numerical solution to the equation describing the diffusion of a gaseous mixture. Chemical potential was used as the driving force for diffusion in their analysis. Their calculations were applied to the experimental results of the diffusion of mixtures of nitrogen and methane in 4A molecular sieve reported by Habgood (1958). They were able to use their model to predict qualitatively the maximum experimentally observed in the sorption rate curve of nitrogen. Ma and Roux (1973) investigated the diffusion of mixtures of sulfur dioxide and carbon dioxide in sodium mordenite. They found that when COThad been preadsorbed on sodium mordenite, the introduction of SO2 resulted in an almost complete desorption of Cog. They also observed a maximum in the sorption rate curve of COS when a mixture of CO? and SO. was simultaneously adsorbed on the mordenite. They presented a simplified mathematical model to describe the observed phenomenon qualitatively. Most recently, Kargen and Bulow (1975) studied the mixture sorption of benzene and n-heptane on NaX zeolite. The irreversible thermodynamics and chemical potential were used to provide a satisfactory description on the part of the displaced curve of the weakly sorbed component. Values of the diffusion coefficients were not given. In all the work described above, either powder material was used or intracrystalline diffusion was assumed to be the controlling step. Moreover, only qualitative discussions were presented. In particular, no attempt was made to evaluate the diffusion coefficients of the individual sorbates. 44
Ind. Eng. Chem., Fundam., Vol. 16, No. 1, 1977
The present work utilizes the concept of a well-stirred reactor to study the rate of sorption of gaseous mixtures in spherical pellets of CaX(Na) zeolite. The objective was to investigate the nature of the competitive sorption of a gaseous mixture in zeolite and to develop models which can be used for the determination of the diffusion coefficients of individual sorbate gases in a mixture. Apparatus and Procedure In most experimental systems, the rate of sorption has been related either to the decrease of the pressure within the sorption chamber or to the weight increase of the sorbent. However, such techniques are inadequate for mixture studies as it is impossible to distinguish the sorption of different components. Consequently, a well-stirred constant volume sorption system was employed. The spherical sorbent pellets (about 1g) were placed in two baskets which were rotated a t 3300 rev/min in a constant volume sorption chamber, which is shown in Figure 1. Equal amounts of each sorbate gas were premixed thoroughly and a measured quantity of the mixture was injected into the sorption chamber. The amounts of each component adsorbed as a function of time were followed by withdrawing gas samples which were analyzed by a gas chromatograph. A schematic diagram of the sorption system is shown in Figure 2. The system was assumed to reach equilibrium when the composition in the gas phase remained constant. Detailed description of the procedure can be found in (Ma and Lee, 1976).
Results and Discussion The Stefan-Maxwell equation should normally be used in formulating a multicomponent diffusion problem. However, due to the complexity involved in the solution of such a problem, it is generally possible to utilize the concept of an effective diffusion coefficient in the mixture. In so doing, the problem can be considerably simplified and becomes tractable. The diffusion coefficient defined in this manner is normally a function of concentration. However, if the concentrations involved are low, it is possible to assume a constant diffusion coefficient and the model previously developed by Ma and Lee (1976) can then be applied to the diffusion of gas mixtures in zeolites. The partial differential equations and the boundary conditions describing the system are
CARRIER DRIVE MOTOR
QAS INJECTION
SAMPLE TO SAMPLE LOOP
COOLANT OUT To P Vu am c upu m U B
fiF
~ ~ - C h r o r n o t o g r a p h
C, H y d r o c a r b o n s
COOLANT IN
Figure 1. Constant volume sorption chamber.
where D,, and D,, are respectively the effective binary intracrystalline and intercrystalline diffusion coefficients of component j with respect to the mixture. Using the Laplace Transform Technique, the solution of eq 1, 2, and 3 with boundary conditions (4) is (Ma and Lee, 1976) Y,(%,S) =
[S + p,K, (d:coth d: -''])I x cosh [S + @,K, coth dz - l)]"'
V,
X
coth
He
Figure 2. Schematic diagram of sorption apparatus: A, vacuum line; B, mercury trap; a, sorbate inlet; b, sample outlet; D, MacLeod vacuum gauge; E, gas cylinder; F, moisture trap; G, 100-cm gas buret; H, 100-crn3gas buret; I, sorption chamber; Q, carrier gas valve.
d:
The validity of assuming a linear equilibrium isotherm has been a subject of discussion for a long time and thus requires some additional brief comments. It should be emphasized that in the present model, the equilibrium is assumed to exist between the gas phase in the macropores and the surface of the zeolite crystals (eq 4a). As the size of the macropores is relatively small, it can probably be expected that the sorbate concentration in the macropores is small although the initial concentration in the bulk gas phase may be large. The average gas phase concentration in the macropores is probably considerably less than that of the bulk gas phase for most part of the experiment. Thus, only near the end of the experiment may the assumption of a linear isotherm be somewhat questionable. The model was used t o calculate the intracrystalline diffusion coefficients of individual sorbate gases in binary mixtures. The mixtures were n-butane-isobutane and isobutane-1-butene. T h e main reason for selecting these two mixtures was that n-butane and isobutane have approximately the same affinity for the adsorbent CaX(Na) while isobutane and 1-butene have considerably different affinities for CaX(Na) due, in part, to the presence of a double bond in 1-butene. The diffusion coefficients of two binary sorbate systems, n -butme-isobutane and isobutme-1 -butene, calculated from the model a t 35 "C, are shown in Tables I and I1 together with the data employed in the calculations. Also shown in the tables are the diffusion coefficients of pure n-butane, isobutane, and 1-butene. The macropore diffusion coefficients in the mixtures are estimated from the following expression.
(7) The molecular diffusion coefficient D,, in the mixture was averaged from values of D,, calculated from
- 1)]1'1
(5)
where j = 1,2. Inversion of eq 5 can be obtained by using Bellman's method of numerical inversion to give
which is the normalized form of the change of sorbate concentration in the bulk phase and is experimentally measurable. The assumptions involved in the derivation of eq 1,2, and 3 can be found in the work of Ma and Lee (1976).
I - Y_ J Dim = _
gL
(j=1,2)
(8)
I # J D], with initial and final concentrations of component J in the gas phase. A tortuosity factor of 4.5 (Ma and Lee, 1976) was used in the calculation. I t should be pointed out that only 0, was varied in the determination of the diffusion coefficient. Other parameters were either experimentally determined or estimated independently. Comparisons between the experimental data and theoretical curves are shown in Figures 3 and 4, which show relatively good agreement. Deviations a t the final
Ind. Eng. Chem., Fundam., Vol. 16, No. 1 , 1977
45
.-0 L
10
10
08
0.0
0 .-e
06
0
0 6
0
E
LL
0 C
1
.-
0
c
I\.
I
I
04
I --
C
Theo.
al 0 0 C
V
1 01
,
- -4 The0
Exp
a Iso-butane
A N-butane
1
130-butane
Displacement O f Iso-butane
A
!
I
I
1
TsI
I
0
Figure 3. Comparisonof experimental and theoretical sorption curves for n-butane-isobutane mixture on CaX(Na) pellets.
Figure 4.Comparison of experimental and theoretical sorption curves for isobutane-1-butene mixture on CaX(Na) pellets.
Table I. Isobutane and n-Butane Mixture Diffusion n-Butane Temperature, O C Initial amount, mmol S, mmol/g Henry's law constant, K R,, cm R,, cm D,, cm2/s Dlrn cm2/s D K , cm2/s D,, cm2/s
Isobutane
35 1.59
35 1.49
1.505 31 500
1.772 18 000
0.2302 1.08 X 0.046 (0.0517)a 0.441 0.402 5.11 x 10-13 (2.226 X 10-13)R c8 (porosity) 0.5 Tortuosity factor 4.5
0.2302 1.08 X 0.043 (0.0472)a 0.368 0.402 8.101 X (2.298 X 10-lL)a 0.5 4.5
0 Pure component diffusion coefficients have been revised from reference (Ma and Lee, 1976).
stage of sorption shown in Figure 3 may be caused by a slight displacement of isobutane by n-butane. From Table I, it can be seen that the magnitude of the intracrystalline diffusion coefficients for the mixture are approximately one order of magnitude smaller than those of pure components. This indicates that the presence of a second sorbate gas will affect the diffusion properties of the other gas although the two gases have about the same affinity for the zeolite. Table I1 shows that the intracrystalline diffusion coefficient of isobutane in the mixture is about one order of magnitude smaller than that of the pure component. This is due to the dominating strong interaction between 1-butene and the zeolite. This is consistent with the results previously reported (Ma and Roux, 1973). I t should be noted that isobutane is displaced by 1-butene at a later stage of the diffusional process as shown in Figure 4.The treatment of this system is thus somewhat complicated. For time t < t,, where t , is the time when 1-butene starts t o displace isobutane, there are enough sites available for the sorption of both types of molecules. Thus, e q 1,2, and 3 can be used for the determination of the intracrystalline diffusion coefficients. The values tabulated in Table I1 were determined in this manner. Once all the available sites are filled, compe46
Ind. Eng. Chem., Fundam., Vol. 16,No. 1, 1977
Table 11. Isobutane and l-Butene Mixture Diffusion Isobutane Initial mmol Temperature, "C
1.52 35 1.772 18 000
S, Henry's law constant, K R,, cm R!, cm D,,, cm'ls D,,, cmlls D K , cm'ls D , , cm2/s
1-Butene 1.64 35 2.212 81 900
0.2302 1.08 X 10-4 0.0427 (0.O472ln 0.367 0.402 5.168 X (2.298 X lo-") t d (porosity) 0.5 Tortuosity factor 4.5 1
0.2302 1.08 x 10-4 0.0452 (0.0496)" 0.400 0.410 5.017 X (9.704 X 0.5 4.5
a Pure component diffusion coefficients have been revised from reference (Ma and Lee, 1976).
tition for sorption between the two types of molecules develops. At this stage the affinities of the species toward the zeolitic structure play an important role. The less favorable molecules (low affinity) are displaced by the more favorable (high affinity) molecules. Consequently, a n increase in concentrations of the species with low affinity is observed in the gas phase as shown in Figure 4. For the convenience of discussion, component 2 (1-butene) is assumed to have a higher affinity toward the solid than that of component 1 (isobutane). At the beginning, the total number of molecules diffusing into the sieve is where N,,, is the maximum total capacity for both species. For t < t,, N l ( t )and N*(t)increase with time. At t = t,, all the available sites are occupied and
N l ( t ) + N z ( t ) = N ( t ) = N,,, Thus, for time t
> t, aN1 + aN 2 = 0 at
at
(10)
Table 111. Separation Factors of Mixtures
__
n-Butane(1)-Isobutane(2) 35 1.75 1.81
5
1.41 0.79
60 1.43 1.29
~~~
Isobutane( l)-l-Butene(2) 5 1.55 15.17
35 4.55 16.00
60 5.92 22.24
Table IV. Diffusion of Mixtures in CaX(Na) Pellets T . "C 5
35
Isobutane Initial mmol k Do 0
D,
n-Butane
1.357 60 900 0.038 1 1.259 x 1.055 x
1.807 85 700 0.0427 1.49 X 1.040 X
Isobutane Initial mmol k Do
