Ind. Eng. Chem. Fundam. 1980, 19, 11-17
11
ARTICLES
An Evaluation of a Method for Investigating Sorption and Diffusion in Porous Solids James F. Kelly and 0. Maynard Fuller” Department of Chemical Engineering, McGili University, Montreal, Quebec H3A 2A 7, Canada
Transient response curves are obtained by perturbing the steady-state concentration of the input stream to a “gradient-free’’sorption vessel. The response curves are interpreted by means of a mathematical model to give values of the slope of the sorption isotherm and the effective diffusivi. The method was evaluated using the system: zeolite NaX-propylene-nitrogen. The assumptions of the mathematical model are well supported except for the assumption of negligible truncation error in the values of the sorption parameter. The sorption isotherms and heats of sorption agree reasonably well with other results where comparisons can be made. The concentration dependence of the effective diffusivity is explained by the mathematical model and the nonlinearity of the sorption isotherm.
Introduction For both scientific and technological purposes, accurate, quantitative expressions are needed for the laws of chemical reactions in porous, catalytic solids and for the physicochemical phenomena that are associated with these reactions. Rate laws of the Langmuir-Hinshelwood type assume the hypothesis: heats of sorption are independent of surface coverages. Calorimetric studies of sorption by Gravelle (19731, for example, have shown that this hypothesis is usually false. Consequently, Langmuir-Hinshelwood rate laws for reactions at low surface coverages may not give accurate predictions at high coverages and vice versa. This situation points to the need for an alternate method of determining sorption parameters under reaction conditions. Tamaru (1964) suggested that methods that are based on interpretations of the transient responses to pulsed inputs offer alternatives to the conventional LangmuirHinshelwood approach to determining sorption and other parameters in heterogeneous catalysis. Considerable progress has been made in this direction. There is now available a variety of methods for determining a sorption parameter and a diffusion parameter in situations where they are both independent of concentration. As examples, chromatographic methods have been presented by Suzuki and Smith (1971), Gangwal et al. (1971,1978),Hashimoto and Smith (1973, 1974), and Haynes and Sarma (1973), and nonchromatographic methods have been presented by Villermaux and Matras (1973) and Dogu and Smith (1976). A chromatographic method for obtaining approximate nonlinear sorption isotherms was proposed by de Vault (1943) and developed by Cremer and Huber (1961) and Huber and Keulemans (1963). Kawazoe et al. (1974) presented an improved chromatographic method that takes into account the effects of dispersion, gas-particle mass transfer and intraparticle diffusion. This method permits the determination of nonlinear sorption isotherms, concentration-dependent diffusivities, and parameters for 0019-7874/80/1019-0011$01.00/0
the other phenomena as well. Shah and Ruthven (1977) compared sorption isotherms and intracrystalline zeolitic diffusivities obtained by chromatography with corresponding results from gravimetric methods. They found reasonable agreement for both sorption and diffusion. In this paper we evaluate a nonchromatographicmethod of obtaining nonlinear sorption isotherms and diffusivities. This method was proposed by Kelly and Fuller (1972) and has been applied by Kelly (1975). A related method was reported by Frost (1978). The physical arrangement of the apparatus is essentially the same as that of the “gradient-free’’ reactor developed by Berty (1974). A similar mathematical model was used in a gravimetric sorption method by Ruthven and Derrah (1972) and Youngquist et al. (1971). In this paper we confine our evaluation to the method of moments, an established procedure for interpreting transient response curves. Mathematical Model The sorption vessel is initially in a steady state where there is a steady inflow and outflow of gas composed of the sorbate and a nonsorbed, diluent gas. The concentrations of sorbate in the streams entering and leaving the vessel are equal and constant. Then, a step function in composition in the inflow is caused by the switching of an initial, inflowing gas stream with another one of different composition. The response of the sorbate concentration in the outflow is continuously measured and recorded. The purpose of the mathematical model is to permit the interpretation of this response in terms of values of a sorption parameter and a diffusion parameter for the sorbent-sorbate system. The following assumptions have been used to simplify the mathematical model: (1)The composition of the bulk gas phase in the sorption vessel is uniform (gradientless). (2) The sorption process is isothermal. (3) The change in total flow rate accompanying a change in composition of inflowing gas is negligible. (4) The concentration of the sorbate in the gas space of the macropores is in equilibrium 0 1980 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980
12
with the sorbed concentration on adjacent surfaces. (5) At the dividing surface between the bulk gas phase and the porous sorbent, the deviation from equilibrium is negligible. This is equivalent to the assumption of a negligible mass transfer resistance. (6) The effect of intracrystalline (for zeolites) or micropore (for nonzeolite) diffusion on the mass transport through the porous sorbent is negligible compared to macropore diffusion. These assumptions lead to a mass balance for the sorbate in the bulk gas phase of the sorption vessel
Here we have introduced another assumption: (7) In the experiments that are to be interpreted by this model, the deviations from the steady state will be small enough to allow us to neglect the errors caused by the terms omitted from eq 4,5, and 6. With the aid of the linearizationsshown above and a new variable, w = E,U, eq 1 and 2 may now be written in terms of the deviation variables yi, yo, and w
where the initial condition is
c,
C,(O) = Ob) A mass balance for the sorbate in the porous sorbent yields
a w = -Da2w at L2 az2
a c, 8% t + em 7=
aw -(t,O) az and the boundary conditions are
=0
and
c, = c,
C,(O,Z) =
(2b)
C,(O,Z)
(24
c,
where
a - (t,O) = 0 az
+=
a C, - (t,O) = 0
(
$)c:
and
az
and
C,(t,l) = G[C,(t,Ul = G[C,(t)l
(2f)
where eq 2f represents the equilibrium sorption isotherm. In this representation, we envision that the sorbate capacity of the intracrystalline volume is large compared to the capacity of the macropore volume. A t the same time, mass transport occurs primarily by gaseous diffusion through the macropores and secondarily by surface diffusion along the boundary of macropores. The factor X2 represents the combined effect of a tortuous path and a varying area for mass transport. The values of h2 for gaseous and surface diffusion have been assumed to be equal for the sake of simplicity. In what follows, we will use the following deviations from the initial steady states y&) = Ci(t) Y O W
=
ei
c o w - e,
- e, = C,(t,z) - c,
(3a)
Note that $ and D are not cons_tantsbut are functions of the steady-state concentration C,. The parameter ($ em) that appears in eq 7a and 10 is the derivative of the total sorbate (sorbed material plus gas in macropores) per unit pellet volume with respect to gaseous sorbate concentration, while $ is the slope of the sorption isotherm. The parameters D and $ are related to the moments of the response to a step input. An expression for the response in the Laplace domain is obtained after transforming eq 7 and 8. Then, we use the well-known relation between moments in the time domain and derivatives in the Laplace domain and Jeffreson's (1970) relation between the moments of the responses to an impulse and a step function to obtain
+
(3b)
u(t,z) = C,(t,z)
(34
u(t,z)
(34
and
The nonlinear terms in eq la, 2a, and 2f are now to be linearized. The terms D,(C,) and G(C,) are expanded in Taylor series about their steady-state values and the steady-state equations are subtracted from the original equations to give the following approximations where
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980
(13)
13
FITTED CURVES
30
and y,(t) is the response to a step input. All of the quantities on the right-hand sides of eq 11 and 1 2 are obtained from experimental measurements. Experimental Section Apparatus, Materials, and Procedure. The chemical system that was chosen for use in evaluating the method consisted of zeolite NaX as the sorbent, propylene as the sorbate, and nitrogen as the diluent gas. This choice was made with the intention of making a study of the catalytic isomerization of cyclopropane, work that will be reported later. We found that propylene in contact with zeolite NaX reacts a t temperatures around 300 "C to give condensable products. Spectroscopic evidence suggested that the main constituents of this condensate were propylene dimer and trimer, as might be expected. By lowering the temperature below 255 " C , we reduced the rate of this reaction and avoided the interference of the products with the sorption and the concentration measurement. The maximum propylene concentration at the highest experimental temperature was limited for the same reason. The sorption vessel, the signal generator, and the detector were the principal units of apparatus. The sorption vessel contained six evenly spaced, 2 in. diameter cylindrical cavities in which powdered solid sorbent was pressed to give compact pellets that were sealed at the sides and bottoms. The pellet depth could be adjusted between 0.1 and 3.0 cm by partially filling the cavities with salt. The impeller in the bulk gas phase over the pellets was driven through a magnetic coupling at speeds up to 2000 rpm. The concentration pulses and steps were generated by switching between inlet streams that had different compositions. The switching time was less than 35 ms. The time delays and dispersions in the inlet and outlet lines were negligible for the normal experimental conditions. The measured first moments of experimental square waves deviated by about 1% from those of ideal square waves a t the flow rates used in our experiments. The deviations of the measured second moments of experimental square waves from those of ideal square waves were negligible. A modified version of the Wilks Miran miniature infrared analyzer was used to detect the concentration of the sorbate in the effluent gas stream from the sorption vessel. An electronic filter was added to improve the signal-tonoise ratio and the combination of instrument and filter had a time constant of about 40 ms. The propylene absorption band a t 6.1 ym was used because it offered good sensitivity to changes in propylene concentration with a 3-cm pathlength sample cell. The sample cell volume was 0.3 cm3. This is well below the limiting volume of 27 cm3 that was calculated from Schmayler and Hoelscher's (1971) criterion for negligible sample cell effects. There was a small but noticeable deviation from the Beer-Lambert law a t higher concentrations, so second-order polynomial calibration curves were used. There was some variation in the calibration, so the instrument was recalibrated for each experiment. The propylene purity was 99% and the nitrogen purity was 99.99%. The NaX zeolite sorbent was commercial Linde 13X Molecular Sieve, supplied in the form of pellets. The original pellets were crushed and screened to give a +30 - 80 (Canadian Standard Sieves) fraction that was pressed in the sorption vessel to form the 2-in. pellets. A wet pressing technique was used to provide a more even pressure distribution and, thus, a more uniform texture. The pellets were pressed at 23 000 psi and acetone was used as the liquid pressing aid. The sorbent was conditioned
d
EXPERIMENTAL DATA
+ 233 O°C
.+
I231 5OC
05
0 x
10
MOLE FRACTION PROPYLENE
+
Figure 1. Values of + t, vs. x o at 231.5 and 233.0 "C. The dimensions of + t, are g-mol (total)/cm3pellet X cm3 macropore/ g-mol (gas). is the slope of the sorption isotherm.
+ +
204 5OC
2330'C
0
05
x
10
MOLE FRACTION PROPYLEIUE
Figure 2. Sorption isotherms. The dimensions of C, are g-mol/cm3 micropore volume X lo4. To convert C, to mmol/g, multiply the numbers on the graph by 0.0296.
by evaporating the acetone, heating at 350 "C for 6 h, followed by slow cooling to 250 "C over a 6-h period and then cooling in a vacuum oven to 50 "C overnight. Before use, the vessel containing the pellets was heated to the sorption temperature and flushed with dry nitrogen for 2 h. The procedure followed for sorption experiments was to establish a steady state of flow, temperature, and concentration and then to perturb this steady state with a step change in inlet concentration. The response was recorded by an oscillographic recorder. After a new steady state was attained by the system, another step change was made in the inlet concentration. Kelly (1975) has presented additional details of apparatus design and procedure. Experimental Results The sorption results are presented in Figures 1 and 2. A least-squares procedure was used to fit second-degree polynomials to the graphs of # + cm vs. xo as shown in Figure 1. After the subtraction of cm, these polynomials were divided by c,, then integrated, to give the sorption isotherms of Figure 2. The reproducibility of the sorption results may be judged by comparing the results of experiments at temperatures of 231.5 and 233.0 "C in Figures 1and 2. The pellet depths for the two experiments were 0.427 and 0.132 cm, respectively. The different values of
14
Ind. Eng. Chem. Fundam., Vol. 19, No. 1 , 1980
Table I. Diffusivities at 108.5 "C"
X ~-
0.0 0.049 0.10 0.20 0.31 0.40 0.50 0.60 0.70 0.81 0.90
d
Table 111. Estimated Bounds of $ '/$ -a/(1
M, mole fraction
b
$
0.048 0.054 0.092 0.114 0.092 0.093 0.099 0.099 0.113 0.083 0.101
227.0 189.6 84.8 42.1 22.8 15.5 11.4 7.2 6.0 2.7 1.3
C
0.96 1.6 4.6 8.0 14 12 12
0.85 1.02 2.3 4.5 8.3 12 16
10
30
d
temp, "C
mean
maximum
0.05 0.03
108.5 156.5 204.5 231.5 252.5
0.242 0.164 0.158 0.085 0.082
0.580 0.251 0.597 0.136 0.147
of D is large. No values of D were calculated when this difference was less than 10% of the term containing pis. Molecular diffusion was the main mode of transport for propylene in the macropores of the zeolite pellets. Values of DpNfor propylene and nitrogen were estimated by the method of Fuller et al. (1966). The values of D, were obtained by correcting for Knudsen diffusion by 1 -1= - 1 (15)
"-T = 108.5 'C; L = 0.30 cm; D, = 0.18 cm*/s;h 2 = 4. D x lo4 cm2/s. [ D /($ + e m ) h 2 ] X lo4 cm2/s. [D,$ /($ + E m ) A z ] x $0Fcm2/s.
Table 11. Diffusivities at 252.5 "CY
-
Dg
X
M, mole fraction
$
b
C
0.0 0.051 0.098 0.15 0.19 0.25 0.30 0.34 0.39 0.48
0.052 0.047 0.048 0.047 0.054 0.048 0.045 0.048 0.097 0.103
26.4 23.7 18.5 15.5 15.0 13.5 12.2 8.4 9.0 6.1
1.4 1.7 3.8 2.9 2.7 2.2 2.5
1.2 1.4 2.1 2.2 2.4 2.6
3.2
3.5
"_T = 252.5 'C; L D x l o 3 cm2/s.
=
a
+ a)
1.8
0.427 cm; D , = 0 . 3 1 cm2/s;h 2 = 4 . + e m ) h 2 ] x l o 3 cm2/s.
[D,E,/($
pellet depth were used deliberately to show the sensitivity of the sorption results to changes in this variable. A comparison of the curves for these two experiments in Figure 2 indicates that the reproducibility of the sorption isotherms obtained by this method would be satisfactory for most purposes. We asked whether the Langmuir and Freundlich sorption isotherms agree with our data. The extent of agreement of the Langmuir equation was judged from the linearity of the data when graphed in coordinates of $-lI2 vs. x,. Typically, the relation was approximately linear for x , < 0.5 and diverged from linearity for x , > 0.5. Although linearity was approached more closely at higher temperatures, none of our data sets gave linear relations over the entire interval of 0 I x , I 1. A similar test of the Freundlich equation showed that it gave a poorer fit to our data than the Langmuir equation. The isosteric heats of sorption, calculated from the experiments at 108.5 and 156.5 "C and 0 Ix , < 0.2, increased from 9.8 to 10.2 ked/ mol with increasing concentration. The small change in the heat of sorption explains the approximate fit of the Langmuir equation to the data at low coverage. Typical diffusion results_are presented in Tables I and 11. At every temperature D increases with increasing x,. If the capacity for sorbate is too small, as in the experiment with thin (0.132 cm) pellets at 233.0 "C,th_efirst moment is small and the errors in the calculation of D increase. The reason for this can be seen by rewriting eq 12 in the form
When the term containing pls is small, the difference between the terms in brackets is small and the variability
+-
DPN
DK
The contribution of Knudsen diffusion to D, was relatively small because the macropores of the zeolite pellets were relatively large and the total pressure was approximately 1 atm for all experiments. Values of D, at 108.5 and 156 " C for small values of C, were estimated using the heat of sorption and Sladek's (1974) correlation. As shown in Table I, these estimates indicate that the contribution of surface diffusion is small compared to molecular diffusion in our experiments. The density of the porous pellets was measured and was found to be 1.155 g/cm3. According to Breck (1974), the content of clay binder in commercial NaX is 2070, the crystal density of NaX is 1.43 g/cm3, the micropore volume of NaX is 0.352 cm3/g, and the micropore volume accessible to propylene is 0.296 cm3/g. These data give (accessible) micropore and macropore porosities of 0.213 and 0.429, respectively, for our pellets. Evaluation. In evaluating this method we will consider both the assumptions of the mathematical model and the results. First of all, the sorbate content in the macropores was less than 2% of the total in the experiments at lower temperatures, so the term involving t, in eq 2a could have been neglected as was done by Youngquist et al. (1971) and Ruthven and Derrah (1972). However, at higher temperatures and partial pressures of propylene, the sorbate content of the macropores approached 4% of the total. There could be a noticeable loss in accuracy if the term involving t, were omitted for experiments at even higher temperatures and partial pressures. Second, the concentration dependency of D appears to be satisfactorily explained by eq 10 and the nonlineazity of the sorption isotherm. The mean of the ratio of D to D,c,/h2($ E,) was 1.10 f 0.15 (95% confidence limits) for X2 = 4. This agrees with earlier work reported by Youngquist et al. (1971) and Ruthven and Derrah (1972) on Knudsen diffusion of light hydrocarbons in beads of zeolite CaA. Assumptions 1, 2, and 3 were tested directly. The gas phase mixing was tested by comparing the vessel volumes, computed from pulse response tests, to the static volume, measured by filling the vessel with water. Excellent agreements (around 1% deviation) were found for impeller speeds between 560 and 1100 rpm. The temperature of the sorbent and the bulk gas phase were measured by thermocouples and the difference between them was less than 0.5 "C for all sorption experiments. This was about the same as the precision of the temperature control of the sorption vessel, so the departure from isothermal sorption
+
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980
was not significant. The constancy of the total flow rate was checked by Bonsu (1975) in similar experiments. He found changes of as much as 2%, but this was not significant because it was about the same as the precision of the flow measurements. The mass transfer coefficients from pellets in the sorption vessel were calculated from a correlation obtained from experiments with naphthalene. Typical mass transfer coefficients were around 4 cm/s for an impeller speed of 700 rpm. If, contrary to assumption 5, we assume that there is a difference between sorbate concentrations in the bulk of the gas and the layer adjacent to the porous sorbent, then a mass flux balance on the sorbate gives
This may be combined with eq 7a to give
or Yo - Y s
Yi
-YO
< -F
Ak
because (dy,/dt)/(yi -yo) was always positive in our experiments. The largest value of F / A k was around 0.025 and the fractional difference (Yo - ys)/(yi - yo) near the beginning of a step change was usually around 0.01, according to eq 17. These results confirm assumption 5. In parallel research, Oberoi (1978) found that the characteristic time, L2/D,of the intracrystalline diffusivity for the system NaX-propylene-helium at 125 "C is around 100 s. At similar concentrations of propylene and 108.5 "C we obtained characteristic times that were larger by about an order of magnitude. At higher concentrations, both characteristic times were smaller, but the one for intracrystalline diffusivity was always smaller than characteristic time of our experiment by an order of magnitude. This difference in characteristic times justifies assumption 6. The infrared detector was not entirely satisfactory for our purpose. The signal-to-noise ratio restricted the magnitude of the step changes in concentration to no less than about 0.05 mole fraction propylene. Smaller step changes would have been in closer accord with assumption 7 of the model, but the signal-to-noise ratio would have deteriorated. As a consequence of the magnitude of the step change and the truncation error of eq 6, there is an error in the sorption parameter. Let the error in $ be represented by
p=+t-JB
(19)
and let
where M is the difference between the final and initial steady states of C,. An estimate of the upper bound of is given by -P< - -
JB-
a
l + a
The means and maxima of this estimate are given in Table
15
111. The results of this table give a rough indication of the inaccuracy caused by the truncation error and concentration changes of the magnitudes that were used by Kelly. As one might expect, the largest errors are associated with the isotherms that have the largest second derivatives, that is, the isotherms at lower temperatures. Although the actual errors are not known, the bounds are undesirably large in these cases. One solution to this problem could be to use an improved detector that would give an acceptable signal-to-noise ratio for values of M of 0.005 to 0.01, that is, smaller than the actual values of M by a factor of about 10. Oberoi et al. (1979) suggest that the problem may also be solved by averaging the two values of $ that are obtained from the responses to one positive and one negative perturbation from the same steady state. The two contributions of the second derivative of G to the error in the average value of $ are of opposing signs, so the net error would be greatly reduced without reducing M. Some of the experimental results can be compared with previous results. A rough check of the sorption capacity at 108.5 "C can be made because this is near the critical temperature, 91.9 "C. The ratio of the accessible volume in NaX to the moles sorbed a t 1 atm should be close to the critical volume for propylene, but larger because of the higher temperature. The experimental molar volume was 212 cm3/mol while the critical molar volume is 181 cm3/mol (Kobe and Lynn, 1953). This is only a semiquantitative comparison and should not obscure the possibility of inaccuracy of the sorption isotherm, which has been discussed above. Sorption in zeolites is often expected to agree with the Langmuir equation. However, Loughlin and Ruthven (1972) and Schirmer et al. (1968) have reported failures of the Langmuir equation to fit sorption data for systems similar to ours. Our heat of sorption of 9.8 to 10.2 kcal/mol agrees reasonably well with Habgood's (1964) value of 10.9 f 1kcal/mol. For a system similar to ours, NaX-propane, Dubinin et al. (1968) also report calorimetric heats of sorption that increase as the propane concentration is increased. Koubek et al. (1975) have given a theoretical explanation for an increase in the heat of sorption with an increase in sorbate concentration. The effective diffusivities in our system are about 4 times greater than those found by Derrah (1972) at the same concentration of propylene, but this is to be expected because of the difference between the dominant modes of diffusion in their experiments and ours and the difference in the macropore porosities. The tortuosity factors calculated by Ruthven and Derrah ranged from 4.5 to 6.8, while our X2 is approximately4. The difference is probably due to differences in pore structure. In sum, all of the assumptions of the model, except assumption 7, are reasonably well supported. The satisfaction of assumption 7 would require smaller step sizes and an improved detector or the averaging of positive and negative pulses. The results are internally consistent and in reasonable agreement with what is known about the sorption and diffusion behavior of hydrocarbon-zeolite systems. Compared to equilibrium methods for determining sorption isotherms and steady-state methods for determining diffusivities, the method has the advantages of relatively short experiments, and relative ease of application at high temperatures. The linearized mathematical model of eq 7-10 permits the determination of nonlinear sorption isotherms and concentration dependent diffusivities. On the other hand, a truncation error in the $ values may be introduced by the use of large step sizes.
16
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980
This type of error could cause greater errors in sorption isotherms than those of the volumetric and gravimetric equilibrium methods. For porous sorbents having a subsJantial proportion of dead-end macropores, the value of D will not agree with the value found by a steady-state method. However, this is not so much a question of error as it is a question of the consistency of the method with the interests of the experimenter. The interpretation of response curves is a common problem for all transient methods. The merits and demerits of an interpretation by means of moments of the response curves have been pointed out by Rony and Funk (1971) and hderssen and White (1970) among others. An interpretation by moments does not require a lot of computing, but it does amplify errors due to detector drift and noise in prolonged response curves and this amplification is worse for higher moments. We attribute much of the scatter in the D values to interpretation by moments and the amplification of relative errors by taking the difference of two quantities in eq 14. There are a variety of transient methods for sorption and diffusion in porous solids, but we will restrict our comparison to the class of methods based on gas-solid chromatography. The replacement of a chromatographic column by a sorption vessel having a mechanically agitated gas phase can eliminate the need to estimate dispersion and mass transport parameters and the need to make corrections for these effects. There is no need to correct for the variation in volumetric flow rate from the entry to the exit of the column, because there is no analogue to this effect in an agitated vessel. On the other hand, the determination of micropore diffusivities by this method does not appear to be practical because it is difficult to press strong pellets that are thin enough to eliminate the effect of diffusion in macropores. Effective diffusivities in commercial pellets would not be obtained using the vessel described here. The difference in pressing techniques would cause differences in the textures of the porous solids that would cause difference in diffusivities. However, we believe that a vessel of the type described by Berty (1974) would allow the determination of effective diffusivities for commercial pellets. Opportunities for improvement of the method presented here consist mainly in alternatives to the interpretations of response curves by moments and a more sensitive and stable detector that is still usable over the full range of sorbate concentrations. Nomenclature A = area of dividing surface between agitated gas phase and pellets of porous sorbent, cm2 Ci, C, = concentrations of sorbate in the streams flowing into and out of the sorption vessel, g-mol/cm3 C, = concentration of sorbate in the gas of the macropores of the porous sorbent, g-mol/cm3 macropore volume C, = concentration in the micropores (zeolite), g-mol/cm3 micropore volume. This may also be interpreted as a surface concentration for a nonzeolite. DpN = molecular diffusivity of propylene and nitrogen, cm2/s DK = Knudsen diffusivity of propylene in macropores, cm2/s D, = gaseous diffusivity of propylene in macropores, cmz/s D, = surface diffusivit of propylene on the boundary surfaces of macropores, cmP/s D = effective diffusivity defined by eq 10, cm2/s F = flow rate of gas into and out of sorption vessel, cm3/s G(C,) = representation of C, as a function of C,; sorption isotherm k = mass transfer coefficient, cm/s L = pellet depth, cm M = pulse magnitude, g-mol/cm3
t = time t f = the time required for a response to reach a new steady state after a step change in input, s u = deviation of C, from its initial steady-state value, g-mol/ cm3 micropore volume V = volume of the bulk gas phase in the sorption vessel, cm3 u = deviation of C, from its initial steady-state value, g-mol/ cm3 macropore volume w = tpu,g-mol/cm3 pellet volume xi, x , = mole fractions corresponding to yi, yo yi, yo = deviations of Ci and C, from their initial steady state y, = deviation of C, from its steady state at the dividing surface between the bulk gas and a pellet of sorbent, assuming equilibrium at the surface z = dimensionless distance from closed end of pellet Greek Letters a = a constant defined by eq 20 t, = macropore porosity of a pellet, cm3macropore volume/cm3 pellet volume tp = micropore porosity of a pellet, cm3micropore volume/cm3 pellet volume X2 = correction factor for tortuosity and diameter variations of macropores I,,*= nth moment of the response to a step function input, (g-mol)(s) n+l / cm3 T* = A L / F , s '7 = V / F , s $ = t,(aG/aC,)c,, (g-mol sorbed/pellet volume)/(g-mol in gas/gas volume) Subscripts i = inflow 0 = outflow Superscript B ( t ) = the initial steady-state value of the function B ( t ) ,for example a = approximate t = true e = error Literature Cited Anderssen, A. S., White, E. T., Chem. Eng. Sci., 25, 1015 (1970). Berty, J., Chem. Eng. Prog., 70, 78 (1974). Bonsu, A. K.. M.Eng. Thesis, McGili University, Montreal, 1975. Breck, D. W.,"Zeolite Molecular Sieves: Structure, Chemistry and Uses",p 429, Wiley. New York, N.Y., 1974. Cremer, E., Huber, H. F., Angew. Chem., 73, 461 (1961). de Vault, D., J . Am. Chem. Soc., 85, 532 (1943). Dogu, G., Smith, J. M., Chem. Eng. Sci., 31, 123 (1976). Dubinin, M. M., Islrikyan, A. A., Sarakhov, A. I., Serpinskii, V. V., In.Akad. Nauk SSSR, Ser. Khim., No. 8 , 1599 (1968). Frost, A. C., "Measurement of the Effective Diffuslvtty from Effluent Concentration of a Flow-Through Diffusion Cell", paper presented at the 65th National Meeting of the American Institute of Chemical Engineers, Philadelphia, Pa.. June 4-8, 1978. Fuller, E. N., Schecter, P. D., Glddings, J. C., Ind. Eng. Chem.,58, 19 (1966). Gangwal, S. K., Hudgins, R. R., Bryson, A. W., Silveston, P. L., Can. J . Chem. Eng., 49, 113 (1971). Gangwal, S.K., Hudgins, R. A., Silveston, P. L., Can. J . Chem. Eng., 58, 554 (1978). Gravelle, P. C., Adv. Catal., 22 (1973). Habgood, H. W.,Can. J . Chem., 42, 2340 (1964). Hashimoto, N., Smith, J. M., Ind. Eng. Chem. Fundam., 12, 353 (1973). Hashimoto, N., Smith, J. M., Ind. Eng. Chem. Fundam., 13, 115 (1974). Haynes, H. W.,Sarma, P. N., AIChE J., 19, 1043 (1973). Huber, J. F. K.. Keulemans, A. I.M., In "Gas Chromatography 1962", M. van Swaay Ed., p 26, Butterworths, London, 1963. Jeffreson, C. P., Chem. Eng. Sci., 25, 1319 (1970). Kawazoe, K., Suzuki, M., Chihara, K., J . Chem. Eng. Jpn., 7, 151 (1974). Kelly, J. F., Ph.D. Thesis, McGill University, Montreal, 1975, copies available from the National Library of Canada, Ottawa. Kelly, J. F., Fuller. 0. M.,Can. J . Chem. Eng., 5 0 , 534 (1972). Kobe, K. A., Lynn, R. E., Chem. Rev., 52, 117 (1953). Koubek, J., PaSek, J., Volf, J., J . Colloid Interface Sci., 51, 491 (1975). Loughlin, K. M., Ruthven, D. M., Chem. Eng. Sci., 27, 1401 (1972). Oberoi, A. S., McGill University, personal communication, April 1978. Oberoi, A. S., Fuller, 0. M., Kelly, J. F., Ind. Eng. Chem. Fundam., accompanying paper in this Issue, 1980. Rony, P. R., Funk, J. F., J . Chromatogr. Sci., 9, 215 (1971). Ruthven, D. M., Derrah, R. I., Can. J . Chem. Eng., 50, 743 (1972). Schlrmer, W., Fiedrich, G., G'ossman, A,, Stach, H., "Mdecuhr Sieves", p 276, The Society of Chemical Industry, London, 1968. Shah, D. E.,Ruthven, D. M., AIChE J . , 23, 804 (1977). Schmayler, D. K., Hoelscher, H. E., AIChE J., 17, 241 (1971).
17
Ind. Eng. Chem. Fundam. 1980, 19, 17-21 Sladek, K. J., Gilliland, E. R., Baddour, R. F., I d . Eng. Chem. Fundam., 13, 100 (1974). Suzuki, M., Smith, J. M., Chem. €ng. Sci., 26, 221 (1971). Tamaru, K., Adv. Catal., 15, 65 (1964). Villermaux, J., Matras, D., Can. J . Chem. €ng., 51, 636 (1973). . prod. Res. BV., Youngquist, G. R., Allen, J. L., Elsenberg, J., rnd. ~ n gchem. 10, 308 (1971).
Received f o r review October 27, 1978 Accepted September 13, 1979 The authors gratefully acknowledge the support of the National Research Council of Canada and the J. W. McConnell Foundation.
Methods of Interpreting Transient Response Curves from Dynamic Sorption Experiments Agyapal S. Oberol, 0. Maynard Fuller,’ and James F. Kelly Department of Chemical Engineering, McGill University, Montreal, Quebec H3A 2 A 7, Canada
Two new methods for obtaining sorption and diffusion parameters are based on an integrodifferential form of the conservation equation for the sorbate in the bulk gas phase of a “gradient-free’’ sorption vessel. Both methods and the method of moments are evaluated with respect to the agreement of theoretical and experimental response curves, the variability within sets of parameter values, linearizationerrors, and computing effort. They are about equal in providing agreement between theoretical and experimental curves, but all three suffer from significant linearizationerrors when used for sorption isotherms that include a pronounced “knee”. When programming and computing effort are important, the method of moments is preferred: if not, direct fitting of the response curve is preferred.
Introduction Several methods of interpreting the response curves that are obtained from transient sorption experiments have recently been reviewed in this journal by Ramachandran and Smith (1978). This paper is a report on two new methods and a slight variant of an old one. We are particularly concerned with the reproducibility of the original transient response curves, the variability of the results, the handling of linearization errors, and computing effort. All of the methods discussed here will be illustrated using Kelly’s (1975) experimental data and the mathematical model of Kelly and Fuller (1980). The Representation of Response Curves by Empirical Equations We found that Kelly’s experimental curves could be represented with reasonable accuracy by functions of the form 7 = -aiexp(-bit) ti-1
5t
e
ti; i = 1, 2,
(1)
...
where 1 + 7 is the dimensionless response to a step input. The constants ai and bi for each segment were found by linear regression of In (-7) on t. Most of the curves could be represented by two line segments and none required more than three. Examples of the approximations that were combined for typical experimental response curves are presented in Figure 1. An empirical representation such as eq 1 removes the noise in an experimental curve. This permits the curve to be differentiated without amplifying the noise and it permits the curve to be extrapolated to very large values of time, for which the signal-to-noise ratio is poor. Both of these features help in interpreting the experimental curves. 0019-7874/80/1019-0017$01 .OO/O
Methods DTF and ITF. For method DTF, one of the two differential equations in the model of Kelly and Fuller is solved analytically (see Appendix) to give
-d?l + - -*+ +7 dt (2)
and 7(0+) = -1. This integro-differential equation is then solved numerically to give a function 7=
f(4;D , J,)
(3)
that is made to agree closely with an experimental response curve by varying the parameters D and J,. When L 2 / Dis small, the solution is insensitive to changes in the value of D and the integral in eq 2 may be approximated to permit an analytical solution 7 = -exp[-t/[~+
+ 7*($ + tm)]]
(4)
in which J, is varied to fit the experimental response data. For method ITF, eq 2 is first solved for J, + t,
By differentiating with respect to time, we obtain
7’7’’) = 0 (6) 0 1980 American Chemical Society
