Ind. Eng. Chem. Res. 1994,33, 1380-1390
1380
Determination of Intraparticle Diffusivities Using a Single Pellet String Fixed Bed and a Shallow-Bed Diffusion Cell Weiruo Sun Laboratbrio de Saude Ptiblica, SSM Macau
Carlos A. V. Costa*and Alfrio E. Rodrigues Department of Chemical Engineering, University of Porto, 4099 Porto Codex, Portugal
Two experimental techniques and the corresponding theoretical models were used for determining gas adsorption equilibria and intraparticle mass transport parameters. These methods include a diffusion cell adapted t o be used with small particles (shallow-bed diffusion cell) and the chromatographic method using a single pellet string fixed bed. The experimental systems used were oxygen, nitrogen, and argon in helium over the RhBne Poulene 4A and 5A zeolite and LaRoche alumina at various temperatures. The conditions where other mass transport resistances can be neglected (film diffusion in the chromatographic method) or independently obtained (bulk diffusion in the cell and axial dispersion in the fixed bed) are discussed. Also, a sensitivity analysis is used which shows that it is not possible to estimate the microparticle diffusivities in 5A zeolite and LaRoche alumina and also the macropore diffusitivites in 4A zeolite. In the last case we assume that the macropore diffusivities are the same as in 5A because both zeolites come from the same manufacturer and so possibly only differ in the type and number of cations. The temperature dependency of the microparticle diffusivities follows an Arrhenius-like equation. The consistency of the results obtained by both methods confirms that the experimental techniques and the corresponding models are reliable. Also, the shallow-bed diffusion cell presents some advantages over the chromatographic method, such as simplicity,easy temperature control, no need for evaluation of flow patterns, and smaller amounts of materials. Introduction Adsorption equilibrium constants and intraparticle diffusivities are important fundamental data for process development and design of adsorption separations. The theoretical prediction of gas adsorption equilibria in zeolite crystals using molecular engineering principles is possible (Karavias and Myers, 1991). Nevertheless, due to the complexity of the prediction methods and to the differences between single zeolite crystals and pelletized adsorbents, the experimental determination is still the main tool for the evaluation of adsorption equilibria. The complexity of the macropore structure including pore shape, interconnectivity, and pore size distribution and the limited knowledge of effective diffusivities prevent a reliable prediction of intraparticle mass transport parameters. There is clearly a need for general, reliable, and reproducible experimental methods and models to obtain the adsorption equilibria and intraparticle effective diffusivitiesrequired by engineeringdevelopmentand design. The most commonly used method for the determination of adsorption equilibria and intraparticle effective diffusivities over a wide range of experimental conditions is the chromatographic technique (Haynes 1975; Chiang et al., 1984, I and II; Hyun and Danner, 1985). The principle of the chromatographic method is that the diffusion rate can be related to the measurements of the dynamic response of an adsorption column subjected to a perturbation in the feed adsorbate concentration. In the chromatographic method, the nondiffusive rate effects such as external mass transfer resistance and/or heat dissipation can be diminished by the choice of a suitable bulk phase flow rate. Usually, the column is packed with small particles and there is a bulk phase flow; the axial dispersion effect is unavoidable and thus will also change
* To whom correspondence should be addressed.
the transient response curve shape and give rise to difficulties in the interpretation of the experimental data. Theoretically, the importance of the axial dispersion can be decreased by increasing the bulk phase velocity. But the increase of bulk phase velocity will increase the column pressure drop, and thus the bulk phase density becomes a function of column position, which will greatly increase the complexity of the models. The use of the single pellet string fixed bed allows the increase of the flow rate while the bed pressure drop is kept negligible, due to the void space between particles and the column wall (Cresswell and Orr, 1982). The diffusion cell is another widely used technique for determining the intraparticle diffusivities (Suzuki and Smith, 1972; Biswas et al., 1987, I and 11). Traditionally, the diffusion cell experimental technique was essentially applied for single pellets. When the pellet dimensions are small, experimental measurements and cell design are limited by the system dead volume accurate calibration and by the observability of the response signals. In this case and in order to use the advantages of the diffusion cell technique, we used a shallow-bed diffusion cell (Sun et al., 1993) where the pressure difference across the bed was kept equal to zero. For porous solids with bidispersed pores, the steady-state exit concentration responses at both sides of the diffusion cell can only be used to calculate the macropore effective diffusivity since micropore diffusion does not make any contribution to the steady-state concentration responses. The experimental transient responses of both sides of the diffusion cell are used to estimate the mass transport parameters. The advantages of the diffusion cell technique are the experimental simplicity and negligible flow pattern effects. Using various gas-solid pairs that we think representative of many systems with theoretical and practical interest, the objectives of this work are the following: to
0888-5885/94/2633-1380$04.50/00 1994 American Chemical Society
compare the methods and improve the precision of the determination of intraparticle effective diffusivities using a shallow-bed diffusion cell or a single pellet string fixed bed; to develop experimental setups for both methods that are reliable, simple to operate, and allow for high precision; this implies not only averaging of experimental results but also minimization of the total number of parameters to beevaluated (avoidanceof entry effects,axial dispersion, film diffusion, heat effects, etc.) and particularly of those to be evaluated simultaneously, e.g., not independently; to determine both method’s weak points using sensitivity analysis; and to obtain the intraparticle effective diffusivities for air gases (Ar, 02,and N2) in zeolites 4A and 5A at various temperatures. The gas-solid experimental systems were selected not only due to a particular interest in the determination of their intraparticle transport parameters but also in order to cover a wide range of situations like micropore control, macropore control, both resistances, spherical and cylindrical particles, and bidispersed and monodispersed pores. The systems used are air gases (Ar, 0 2 , and N2) diffusing and adsorbing in zeolites 4A and 5A and alumina.
Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994 1381 Table 1. Mass Balance Equations for Bulk Phase and Chambers Single Pellet String Fixed Bed Bulk Phase
ac+ac ----1 a2c+ 3 @ b c f l a ac dy=, = ae az
p e a ~
IC: C(Z,O) = 0
Shallow-Bed Diffusion Cell Bulk Phase (3) IC: C(Z,O) = 0
BC:
ac -(o,e) az
(4)
Nf = -[c(o,e)
- c,cni
(5)
Nb
Top Chamber Equation
Models Tables 1 and 2 give the dimensionless mass balance equations where we assumed linear adsorption isotherms, spherical particles, negligible heat effects, and step input. For the single pellet string fixed bed (SPSFB)we further assumed negligible pressure drop and for the shallow-bed diffusion cell (SBDC)chambers of equal volume and perfectly mixed, equal flow rate to and from the chambers and no convection. The dimensionless variables and parameters are defined as
IC: CJO) = 0
(7)
Bottom Chamber Equation
IC: Cb(0) = 0
(10)
Table 2. Mass Balance Equations inside Macropores and Microparticles Macropore Equation
c, (Y,O) = 0 BC: ac, (OJ) = 0 and C,(l,O) = C(Z,B) (For SBDC) IC:
ay
where A is the pellet section area, ct, Cb, ce, and c are the concentrations of tracer in top and bottom chambers and entrance and bulk phases, D,,, D,, and Di are, respectively, the diffusivities in bulk phase, macropores, and microparticles, D , is the axial dispersion coefficient, F is the flow rate, kf is the film diffusion coefficient, K is the dimensionless adsorption constant, L is the thickness of the pellet or the length of the column, ro and Ro are, respectively, the microparticle and particle radii, u is the interstitial velocity, V, is the chamber volume, E, and Eb are, respectively, macropore and bulk porosities, and t , r, R, and z are the time and coordinate variables, corresponding, respectively, to the microparticle, particle, and shallow-bed or fixed-bed axis. The conditions for neglecting heat effects and for considering the interstitial velocity constant and the isotherm linear require that the experiments be conducted at low solute concentration. Equal volume chambers and flow rates are easily satisfied by carefully designing the chamber geometry and controlling the flow rates. The perfect mixing in both chambers is also attained by properly designing the chamber geometry and especially the inlet and outlet port distribution. The use of step or
ac, (o,e) = o and c, (z,i,e)= c(z,e)ay N, ac
dy=l
(For SPSFB)
(12)
(13a)
(13b)
Microparticle Equation
IC: Ci(X,Y,Z,O)= 0 (O,Y,Z,O) = 0 and Q(l,Y,Z,O)= KC,(Y,Z,B)
(15) (16)
pulse inputs is mainly dependent on the experimental technique. Experimentally it is usually more difficult to form an ideal Dirac impulse than a step. All these parameters and variables except the mass transport parameters (N,, Ni, Nf,Nb, and Pe) are available for a given diffusion cell or fixed bed, pellet, and
1382 Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994
-
Table 3. Pore Structure and Particle Prowrties samDle no. RP5A La A1 diameter, mm 1.6 3.8 length, mm 3.5 microparticlediameter, pma 1.84 bulk density, g/mL 0.720 0.734 apparent density, g/mL 1.13 1.2591 skeletal density, g/mL 1.86 1.6612 pore volume, mL/g 0.351 0.1922 48.0 22.532 specific area, A, m2/g average pore diameter (4V/A), wm 0.0258 0.0341 0.363 0.417 ebb 0.393 0.242 eaC a This value was measured from scanning electronic microscope photographs. Calculated by €b = (1 - packing weight/apparent density)/cell volume. CCalculated by ta = 1 - apparent density/ skeletal density. ~
1 gas tank
3 5 7 9
sampling valve ihermccouple heating tape oven
2 mass flow iontroller 4 dccurnte differential pressure sensor 6 needle valve
8 thenndl conduitiviry detector
Figure 1. Experimental setup diagram of the single pellet string fixed bed.
experimental operating conditions. For the special case where the velocities of gas in the column are sufficiently high and thus the film mass transport resistance can be neglected (kfand Nftend to infinity), eq 13b simplifies to
All these equations and the boundary conditions are linear. They can be solved either analytically or numerically using techniques such as the method of lines with orthogonal collocation (Villadsen and Michelsen, 1978). All the mass transport parameters existing in eqs 1-16 can be obtained by matching the theoretical and experimental transient or steady-state responses.
Experiments Figure 1 shows the experimental setup used for the chromatographic method. The experimental procedure for the single pellet string fixed bed begins by flowing an inert carrier gas through the column packed with the porous solid at the desired temperature until the response curve at the column exit becomes a steady straight line. Detectors located at the inlet and outlet of the column record the gas stream composition, that is, the concentration curves of the adsorbate as a function of time. At time zero the carrier stream is turned to the gas mixture stream with a known solute concentration. The column influent and effluent concentrations are monitored continuously. The choice of the column diameter should satisfy the requirements of the single pellet string fixed bed (Scott et al., 1974); that is, the ratio of the column internal diametertotheparticlediametershould beabout 1.1-1.4. According to the solid samples, we used a column with dimensions 2.0 X 1500 mm (internal diameter X length) for R h h e Poulenc 4A and 5A zeolites and 4.5 X 1500 mm (internal diameter X length) for LaRoche alumina. The column length was chosen in order to have a reasonable Peclet number and residence time. In the shallow-bed diffusion cell we used an experimental setup similar to the one used for the single pellet diffusion cell described by Sun et al. (1993). The choice of the cell geometry should allow for the maximum sensitivity of the
response to the mass transport parameters. The cell dimension used is 2.8 X 2.0 cm (diameter X thickness), and the corresponding geometric parameters, N,, are 17.88, 17.88, and 20.54 for Rh6ne Poulene 4A and 5A and LaRoche alumina, respectively. Experiments were carried out by applying positive concentration steps at the inlet of the top chamber. Inlet tracer concentrations are from 3.5 to 20% in volume and are accurately controlled by the mass flow controllers. The pressure difference through the cell is kept negligible by using the same tubing and connectors and a self-made device. The concentrations of the outlet streams are measured by two microvolume thermal conductivity detectors, and the output signals from these detectors are automatically recorded by the data acquisition system which is composed of a computer and an A/D conversion board. Using a blocked cell, it was also verified experimentally that the chambers are perfectly mixed (Sun et al., 1993). Table 3 shows the solid sample geometry and pore structure information. Rhbne Poulenc 4A and 5A zeolites are cylindrical pellets and LaFtoche alumina are spheres. Figure 2a,b shows the detailed differential pore diameter distributions of Rh6ne Poulenc 5A and LaRoche alumina measured by mercury porosimetry. In these figures, V is the pore volume (m3/kg) and d is the pore diameter (m). The pore network structure of zeolite particles shows the typical bidisperse pore distribution. The micropore dimensions are completely determined by the crystal structure and have uniform size. Because the 5A zeolites are usually obtained from the corresponding 4A zeolites by ion exchange, the pore structure characteristics measured by the mercury porosimetry should be the same when both are produced by the same source. In the range of 0.1-1 pm there is a distribution peak. These peaks are expected since the zeolite macropores are composed of the voids among crystals and binding material. The crystal size and the binding material particle dimension are practically the same (1pm), and thus the void dimensions should be close to that dimension. The LaRoche alumina pore size distribution seems continuous, and possibly,there is a distribution peak in the limit of mercury porosimetry detection. In our mass transport models, one of the interesting pore structure parameters is the macropore porosity that can be directly obtained from the mercury porosimetry measurements. The micropore structure characteristics are less important since in our mass transport models there are no variables or parameters explicitly linked to the micropore structure properties, and the effectivemicropore diffusivities and adsorption equilibrium constants were expressed in microparticle dimension basis.
Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994 1383
Combining eqs 17 and 18 gives a relationship that links Nf and N b : 0.001
0.1
10
1000
Pore dlameter ( ~rm ) 0.10
0.08
~
3 g! 0.06
B s 01 ii
0.04
e
0 0.02
0.00
0.001
0.01
0.1
1
10
100
Pore diameter ( Ir m )
Figure 2. Differential pore size distribution obtained by mercury porosimetry: (a, top) R h h e Poulenc 5 4 (b, bottom) LaFZoche alumina.
Results and Discussion I. Adsorption Equilibria. The adsorption equilibrium constants can be obtained by using the global mass balance between input and output for each experimental run (Sun et al., 1993). Table 4 shows the average adsorption equilibrium constants of nitrogen, oxygen, and argon in RhGne Poulenc 4A and 5A zeolites obtained by both experimental techniques at different temperatures and using several tracer concentrations. The LaRoche alumina adsorbed amount is too small to be accurately measured; nevertheless the results obtained by both methods fairly agree. The adsorbed amount of permanent gases in 4A and 5A zeolites at ambient temperature and pressure is small, and again the possible experimental errors in concentration and flow rate measurements accumulate in the mass balance error. Although some of the absolute values obtained by both methods are different, the variation with temperature is consistent. The adsorption equilibrium constant for nitrogen in RhGne Poulenc 5A at 40 "C compares reasonably well with the results obtained by Garcin (1991) using a gravimetric technique as shown by Sun et al. (1993). In the range of the tracer gas concentrations used for the determination of the mass transport rate parameters, the adsorption isotherms are linear, which confirms the validity of the assumption of a linear equilibrium relationship. It should be noted that the determination of the adsorption equilibria by these methods is not limited to the linear region because the adsorbed amount of gases in the solid is calculated by a global mass balance that is independent of the mass transport rate model. 11. Bulk Phase, Film Diffusion and Inlet Effects. (a) Shallow-Bed Diffusion Cell. In the shallow-bed diffusion cell model there are four mass transport parameters that need to be estimated from experimental data: N b , Nf,N,, and N i . The steady-state responses of both chambers can be easily obtained
So among these four parameters only three need to be estimated from the experimental transient-state responses. If we manage to evaluate separately one of them, N i , N,, and N b (or Nf),then only two parameters need to be estimated, which improves the estimation convergency and precision. The molecular diffusionshould be the only contribution to the bulk phase mass transport when there is not a pressure difference across the cell. Due to the increase of the diffusional path length and the variation of the path geometry, the effective bulk phase diffusivity, Dm,, must be smaller than the corresponding molecular diffusivity, Dm,and the relationship between them can be expressed by means of a tortuosity, fbdk Db
= Dm/Tbdk
(20)
Schneider and Smith (1968) presented the above equation as a parallel pore model of the bed, and thus Tbdk should be independent of the solute and of the temperature. Because the responses at steady state are independent of the intraparticle pore network structure, the bulk phase tortuosity should be also independent of the solid pore structure and only dependent on the packing way and particle geometry. Experimental values of Tbdk for silica gel are 1.45-1.59 (Schneider and Smith, 1968). The theoretical prediction of fbdk for a bed randomly packed with uniform spherical particles is &' (Wheeler, 1955). To evaluate the effective bulk phase molecular diffusivity, we used eq 20 with the theoretical value for the tortuosity (Tbdk = A)and molecular diffusivity calculated by the Chapman-Enskog equation (Reid et al., 1988). In order to confirm such a procedure, we calculated the diffusivities using three-parameter regression from the dynamic responses of both chambers for nitrogen in RhGne Poulenc 5A a t 40 and 95 "C. The effective bulk phase diffusivities respectively estimated by regression and by eq 20 with fbulk = d iare 0.5337 and 0.5556 at 40 "c and 0.6767 and 0.7269 at 95 "C. These results show that the values obtained by optimization are slightly smaller than the estimated ones but the difference between them is less than 5 % . Such small error is within the uncertainty range of the experimental determination and parameter regression, so we adopted this theoretical estimation method for the calculation of the effective bulk phase diffusivities. This procedure should apply at least for shallow beds packed with spherical or almost spherical particles; this is the shape of the majority of commercial adsorbents. When the effective bulk phase diffusivities can be evaluated, the film mass transport coefficients can be calculated using the steady-state responses of both chambers and this calculation is independent of the dynamic responses. These will be used to estimate the remaining two parameters. The results of this methodology are shown in Table 5
1384 Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994 Table 4. Average Dimensionless Adsorption Equilibrium Constants Obtained Using the Shallow-Bed Diffusion Cell and the Single Pellet String Fixed Bed RP 4A R P 5A LaRoche alumina sample SBDC SPSFB A%a SBDC SPSFB A% SBDC SPSFB A% Ar 18.5 40 OC 2.7 2.2 2.6 2.9 11.5 1.8 1.5 16.7 70 OC 1.8 2.1 2.5 19.0 10.0 2.0 1.4 1.6 12.5 95 o c 1.5 1.6 1.8 16.7 6.7 1.5 1.4 14.3 1.2
N2
40 OC 70 OC 95 o c
5.3 3.3 2.7
5.3 3.0 2.3
0.0 9.1 14.8
10.5 6.9 4.4
9.3 6.3 4.6
11.4 8.7 4.5
1.4 1.3 1.3
1.4 1.3 1.2
0.0 0.0 7.7
40 "C 70 "C 95 o c
3.1 2.1 1.9
3.5 2.3 2.1
12.9 9.5 10.5
3.6 2.5 1.8
3.3 2.6 1.9
8.3 4.0 5.6
1.3
1.3 1.2 1.2
0.0 9.1 9.1
0 2
1.1 1.1
SBDC - SPSFBl SBDC
"A% =J
Table 5. Experimental Conditions and Corresponding Estimated Film Mass Transport Coefficients in the Shallow-Bed Diffusion Cell sample RP4A RP5A LaAl
lo 0 . 8 0 -
Ar
40 OC 70 OC
flow rate, mL/s
kr, cm/s
0.126 11.52 0.138 13.21 0.148 14.34
0.126 10.63 0.138 13.30 0.148 13.63
0.126 11.45 0.138 12.78 0.148 14.17
flow rate, mL/s kr, cm/s flow rate, mL/s kt, cm/s flow rate, mL/s kr, cm/s
0.126 11.27 0.138 12.30 0.148 14.13
0.126 11.26 0.138 13.19 0.148 14.98
0.126 11.79 0.138 13.19 0.148 13.55
flow rate, mL/s kr,cm/s flow rate, mL/s kr, cm/s flow rate, mL/s kr, cm/s
0.126 13.04 0.138 12.96 0.148 15.59
0.126 12.29 0.138 13.67 0.148 14.09
0.126 12.59 0.138 13.18 0.148 15.68
kr,cm/s
flow rate, mL/s
kr, cm/s 95 OC
flow rate, mL/s
N2 40 OC 70 OC 95 OC
70 OC
95 OC
where the experimental film diffusion coefficients for argon, nitrogen, and oxygen at 40,70, and 95 OC in RhBne Poulenc 4A and 5A and LaFtoche alumina are displayed. As expected, the difference among film diffusion coefficients at the same temperature for the various gasaolid pairs is small. (b)Single Pellet String Fixed Bed. In the SPSFB, there is a concentration detector at the inlet and outlet of the column and the same tubing, connectors, and tubing length were used for the connection between the column and detectors in order to maintain the same delays and effects on concentration responses. In our mass transport model we assumed that the inlet perturbation is an ideal concentration step. In practice the input step dispersion is unavoidable. The experimental input can be approximately expressed using the following equation
c = 1 - exp( -
t)
where Bo is the dimensionless time at dimensionless concentration C = 0.6321. The input is independent of the solid sample, and for different gases the difference should be small. In our experiments the range of flow rates is small, and thus we used eq 21 with different measured Bo to simulate the real input steps. The influence of nonideal step input in the response can be shown by
0.0747
Pe=3M Na = 4.072 Ni = 76.11
1
0 2
40 OC
e,,=
2
3 4 5 6 Dlmrnrionlrrr time e
7
Figure 3. Simulated breakthrough curves with ideal (dashed line) and real (solid line) input steps for oxygen in LaRoche alumina at 70 OC.
comparing the corresponding breakthrough curves. As an illustration Figure 3 compares the calculated breakthrough curves for an ideal step and for a real input for oxygen in LaFtoche alumina at 70 OC. For this case the standard deviation is less than 0.05, that is, of the same order of magnitude as the experimental measurement error for the concentration responses. The concentration detector for the input stream is fixed in an oven, and thus the input concentration dispersion measured by the detector includes the tubing and detector dispersions. The true input should be somewhere between the ideal and measured inputs, and thus we can use the response to the ideal step to calculate the model parameters. One of the main advantages of the single pellet string fixed bed is that there is a large space between the pellets and the column wall and thus a high gas velocity can be used without a visible pressure drop. In order to show the effects of varying the velocity on the concentration response curves, we use the Ranz-Marshall empirical correlation (Yang, 1987)as a rough estimation of the film mass transport coefficient:
-2kfRo - 2.0 + 0.6 (L)1'3 Dm PDm
2R,G ( u)(22) 112
where G is the superficial mass flux based on the unit cross sectional area of the empty bed and p is the gas density. This correlation shows that the film mass transport coefficient is proportional to the square root of the flow rate. The particle radius for cylindrical geometry was calculated by the equivalent hydraulic radius defined as
R, = 3V0/S0
(23)
Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994 1385 1.00
I
c
c "
0.60:
1
0.60;
0.60;
u
f 0.40; -s g 0.20:
PP
Pe = 181 Na=6.819 Ni = 241.7 Nf = 48 and
E
0 --s
-
E
a
- 00 . 0 0
0.80j
10
20
30 Dimensionless time
e
40
50
0.40Nf = 1.98
Na = 3.93 Ni = 54.6 Pe -81 and Zoo0
0.20: 0.0040
....,....,....,.
T . . , . . . .
1
2
3 4 5 6 Dimensionless time 8
7
8
Figure 4. Comparisonamong simulatedbreakthroughcurves using different film mass transport coefficients for NdHe in LaRoche alumina at 40 "C.
Figure 5. Comparison amongsimulated breakthroughcurves using different axial dispersioncoefficientsfor NdHe in LaRochealumina at 40 O C .
where Va and Sa are, respectively, the volume and surface area of the cylindrical particle. The Ranz-Marshall empirical correlation for the calculation of the mass transport coefficient was developed for a column packed with small particles; that is, the ratio of column to the particle diameter is greater than 20 so that the wall effects are negligible, which, when applied to the single pellet string column, does not give precise results. But we just use this correlation as an order of magnitude estimation. Figure 4 shows the theoretical concentration response curves of nitrogen at 40 "C in RhGne Poulenc 5A with different film mass transport coefficients ranging from 10 times smaller than the estimatedvalue by eq 22 to infinity. Within this large range, which certainly includes the estimation error, the response curves are coincident. Because the film mass transport resistance is in series with other diffusional resistances inside the particle, their relative importance can be illustrated by considering the ratios of the dimensionless mass transport parameters. For this system where macropore diffusion is the rate controlling step, the ratio of Nf/Nais of the order of 100 and thus the film resistance can be safely neglected. For the case of gases in 4A zeolite, the main diffusional resistance is the micropore diffusion and the contribution of film mass transport resistance to the total resistance will be even less important. The differences among the film mass transport coefficients of argon, nitrogen, and oxygen are small. The temperature dependency of the film mass transport coefficient is similar to that of the macropore diffusivity. As a conclusion, for our experimental operating conditions and systems the film resistance effects on the concentration responses can be neglected. The conclusions for these gases in LaRoche alumina are similar. As it is shown later, the macropore resistance of the LaRoche alumina particles is larger than in 5A zeolite, and thus the film diffusion resistance can also be neglected. Similarly, the axial dispersion coefficients or Peclet numbers can be related to the flow rate and the gas physical properties by the followingempirical equation (Yang,1987)
In the range studied the response curves are coincident, meaning that in this range the axial dispersion coefficient does not cause an observable error in the simulated response curves. In the case of 5A zeolite, the macropore diffusion resistance is of the same order of magnitude as for the alumina and thus the axial dispersion effect on the responses should be similar. Because of the relatively large microparticle diffusional resistance, the axial dispersion effect on the response variance is even less important in 4A zeolite. 111. Intraparticle Diffusivities. (a) Shallow-Bed Diffusion Cell. In our SBDC mass transport model the mass transport resistances for bulk phase, macropores, and micropores were taken in series and thus when one of them is much larger than the others this resistance becomes the rate-controlling step. The relative importance between any two diffusion resistances linked in series can be illustrated by considering the ratios of the time constants, rab = N.$& and r& = Na/Ni. In d l our cases the variation of Nb and Na is small compared to the variation of Ni for different solid samples, gases, and temperatures. rab is in the range of 1.3-5.5, and thus both bulk and macropore diffusion resistances make similar contributions to the total mass transport resistance. The variation of r i covers a larger range. When r i >> 1, the micropore diffusion becomes the rate-controlling step. When r i 11.3 ,0259 3.6 x 10-7 0.0133 .610 >13.2 .0267 4.6 x 10-7 0.0137 .612 >14.7 .0287 5.4 x 10-7 0.0071
.649 .00363 .0259 1.1 x 10-10 0.0117 .573 .00467 .0229 1.6 X 10-lo 0.0116 ,581 .00554 .0254 2.0 x 10-10 0.0063
.649 >11.3 .0259 3.6 x 10-7 0.0128 .573 >22.8 .0229 7.2 x 10-7 0.0125 .581 >26.7 .0254 9.2 x 10-7 0.0122
.556 .234 .0222 7.4x 10-9 0.0094 ,583 .468 .0255 1.6 X 10-6 0.0095 .586 0.762 .0275 2.8 X 10-8 0.0023
.556 >20.5 .0222 6.5 x 10-7 0.0066 .583 >26.1 .0255 9.0 x 10-7 0.0044 .586 >30.8 ,0275 1.1 x 10-6 0.0047
.190 >2.20 .0288
70 "C
95 "C
/---
0
-5 0.20
Ni r 0.w489 Na = 0.610 Nf=5.76 Ns= 17.81
(P
0.0121 .188 >2.65 .0312 0.0086 .179 3.00 .0319 0.0113
sE 0.10
i5
0.00 0
0.0089 .265 >5.53 ,0441 0.0085 ,179 >3.00 .0319 0.0097 .263 >3.58 .0399 0.0075 .243 >4.40 ,0405 0.0099 .254 >5.06 .0480 0.0104
and macropore diffusion is mainly molecular diffusion. Thus in a small temperature range, the temperature dependency of the ratio between the bulk phase and macropore diffusion resistances should be small. Table 6 shows the results obtained using the shallowbed diffusion cell for the effective macropore and micropore diffusivities of argon, nitrogen, and oxygen at 40, 70, and 95 "C in RhBne Poulenc 4A and 5A and LaRoche alumina samples. For alumina samples the microparticle diameters are unknown, and thus only the dimensionless mass transport rate parameters are shown. For RhBne Poulenc 5A the microparticle diffusion parameters listed in Table 6 are possibly the lower limit values. Due to the similar macropore network structures of 4A and 5A zeolites obtained from the same source, the macropore diffusivities for 4A zeolite were taken to be the same as for 5A zeolite. As an illustration, Figure 10 shows the experimental and simulated response curves of both chambers for argon in RhGne Poulenc 4A at 70 OC. The dashed lines are simulatedvalues, and the solid lines are experimental data. The standard deviations, S (square root of the average square of the differences between simulated and average experimental top and bottom transient response curves),
100
200 300 400 Dlmenslonlerr tlme 8
500
Figure 10. Simulated (dashed lines) and experimental (solid lines) shallow-bed diffusion cell responses for argon in Rhbne Poulenc 4A at 70 O C . Table 7. Experimental Activation Energies of Micropore Diffusivities Obtained Using the Shallow-Bed Diffusion Cell
.270 >4.43 .0410
02
40 "C
-I
activation energy, kJ/mol
Laporte 4A Ar N2 02 25.14 21.02 22.46
Rh6ne Poulenc 4A Ar NZ 02 29.56 18.78 23.21
that can be used to assess the precision of the simulated results are also listed in Table 6. The simulated results for RhBne Poulenc 5A were generally poorer than those obtained for the corresponding 4A, since for the 5A zeolite the microparticle diffusional resistance is negligible and so the whole mass transport process of gases in 5A is faster and 5A zeolites have larger adsorption capacities than 4A. These factors result in more uncertainty in the experimental responses. The results in Table 6 also show that the microparticle diffusion resistance in LaRoche alumina is negligible when compared with the macropore diffusion resistance. The pore volume of LaRoche alumina is concentrated in the region of diameters smaller than 5 pm, and no significant distribution peak exists. The micropores are made of small A1203 crystals, and their dimension should be larger than the eight-membered-ring aperture of zeolite A. Also, there is not a defined boundary between macro- and micropores. When we use the bidispersed pore model to represent the mass transport inside such alumina particles, we do not know the microparticle dimension and thus we cannot distinguish a microparticle diffusivity. Also, we can say that the precision of the macropore diffusivities obtained by the fitting technique is the same as for the 5A zeolite and that the microparticle diffusion parameters cannot be accurately obtained. The theoretical study of permanent gas diffusion in blocked zeolite micropores showed that such diffusion is an activated process (Kiirger and Ruthven, 1989)and that the temperature dependency of diffusivities can be expressed by the Arrhenius equation. The experimental activation energies are listed in Table 7. The temperature range of the experimental data is not sufficiently wide, and the presence of possible experimental and parameter regression errors implies that the absolute values of activation energies for these systems should be considered as approximate. Nevertheless the tendency is qualitatively good. (b) Single Pellet String Fixed Bed. Similar to the case for the shallow-bed diffusion cell, the precision of the parameter estimation using the single pellet string fixed bed is dependent on the sensitivity of the concentration response curves to the parameters to be estimated. Only the diffusivity of the mass transport rate-controlling step can be determined accurately in the diffusion resistances
1388 Ind. Eng. Chem. Res., Vol. 33, No. 5,1994 Table 8. Comparison among the Various Mass Transport Resistances in the Sinale Pellet String Fixed Bed Rt R. Ri R, ~~
the three gases at different temperatures where the various resistances are defined as
R, = D, -
~
AI
40 "C 70 "C 95 "C
Nz 40 "C 70 "C 95 "C 02
40 "C 70 "C 95 "C
0.0022 0.0024 0.0398 0.0019 0.0021 0.0304 0.0017 0.0018 0.0283
0.156 O.ooOo27
0.000 20 O.OO0 98 O.OO0 14 O.OO0 17 O.OO0 89 0.000 12 O.OO0 15 O.OO0 81
0.0026 0.0029 0.0207 0.0023 0.0028 0.0203 0.0020 0.0025 0.0184
0.0601 0.0000062
O.OO0 15 O.OO0 15 O.OO0 88 o.OO0 12 O.OO0 13 O.OO0 81 o.OO0 11 o.OO0 11 o.OO0 74
0.0022 0.0025 0.0200 0.0018 0.0019 0.0218 0.0018 0.0017 0.0199
0.00118 O.ooOo13
RP 4A RP 5A La A1 RP 4A RP 5A LaAl RP 4A RP 5A La A1
0.0057 0.0057 0.0026 0.0061 0.0061 0.0027 0.0064 0.0064 0.0028
O.OO0 13 O.OO0 14 O.OO0 98 o.OO0 11 O.OO0 13 O.OO0 89 o.OO0 10 o.OO0 11
RP 4A
0.0055 0.0055 0.0025 0.0059 0.0059 0.0026 0.0061 0.0061 0.0027
O.OO0 18
RP 5A La A1 RP 4A RP 5A La A1 RP 4A RP 5A La A1 RP 4A RP 5A La A1 RP 4A RP 5A La A1 RP 4A RP 5A La A1
0.0059 0.0059 0.0026 0.0063 0.0063 0.0028 0.0066 0.0066 0.0029
0.000 79
0.0773 O.ooOo24 0.0701 O.ooOo24
axial dispersion resistance
UL
film diffusion resistance
R,=---
L 1- 'b 3Kf
I"
(1- e,)R
macropore diffusion resistance
micropore diffusion resistance
0.0657 0.0000072 0.0618 0.0000083
0.00071 o.ooOo12 O.OOO48 0.m11
in series model. The relative importance of axial dispersion, film mass transport resistance, and macropore and micropore diffusional resistance contributions to the total mass transport resistance can be studied using moment analysis. Ruthven (1984) gives the variance for a fixed bed with axial dispersion, film mass transport resistance, and macropore and micropore diffusional resistances:
where R = t a + (1- c,)K, $1 is the first-order noncentral moment, and u2 is the second-order central moment. The various terms in eq 25 are equivalent to the corresponding mass transport resistances. It is evident that the contributions of axial dispersion and of the various mass transport resistances are linearly additive. Because of the additivity of the various mass transport resistances, we can examine them by comparing the relative sizes of the various terms. Again let us make use of our experimental results to illustrate this procedure. Table 8 shows the relative sizes of axial dispersion, film mass transport, and macropore and micropore diffusional resistances at the experimental conditions. The axial dispersion and film mass transport coefficients were estimated by eqs 23 and 24. It should be mentioned that eq 25 was developed with the assumption that the boundary condition C(z = -) is limited, while in our model we considered the boundary condition at the column outlet as aC/aZ = 0. For long columns the difference between the response concentrations using these two different boundary conditions is small. In our experimental system the column is quite long and we just used these tools to compare the relative importance of the various mass transport resistances. Table 8 shows the various mass transport resistances in singlet pellet string fixed bed for
The data in Table 8 show that the microparticle diffusional resistances for argon and nitrogen in Rh8ne Poulenc 4A always dominate the various resistances. Whereas for all gases in 5A zeolite, the microparticle diffusion resistance is negligible when compared with the macropore diffusion resistance. The axial dispersion effects on the outlet concentration response variance are of the same order of magnitude as the macropore diffusion resistances. For LaRoche alumina the macropore diffusion resistance is large due to its larger particle dimensions. The film mass transport resistance is negligible when compared to other resistances. For 5A zeolite and alumina the main mass transport resistance is macropore diffusion and thus the comparison between the film and macropore diffusion resistances can explain their relative importance. In our experimental conditions, the ratio between the macropore and the film diffusion resistances is around 15-20; that is, if we neglect the film mass transport resistance, the error will be around 5 % . If we further consider the axial dispersion term, the error will decrease since the axial dispersion has also an important contribution to the response variance. For argon and nitrogen in 4A zeolite, neglecting the film mass transport resistance will be safer because the microparticle diffusional resistance is much larger than all other resistances. Even for oxygen in 4A zeolite the existence of the microparticle diffusion resistance will reduce the relative importance of film mass transport resistance. The relative importance between macropore and microparticle diffusion resistances of argon and nitrogen in 4A and 5A zeolite shows that only the controlling resistance and the corresponding mass transport coefficients can be determined by our experimental and parameter regression techniques. For oxygen in 4A zeolite, the macropore and microparticle diffusion resistances are of the same order of magnitude and, although both can be calculated simultaneously by the fitting technique, it is a better policy to determine one of them independently as we did for the SBDC. Experimental results are also listed in Table 9 for SPSFB. Figure 11 illustrates the experimental and simulated breakthrough curves for oxygen in Rh8ne Poulenc 5A at 70 "C. This plot shows that the simulated results are quite good. The standard deviations for all the experiments are also listed in that table.
Conclusion Two experimental techniques and the corresponding theoretical models were used for determining gas adsorption equilibria and mass transport parameters. These techniques include a diffusion cell adapted to be used with
Ind. Eng. Chem. Res., Vol. 33, No. 5,1994 1389 1.00 I
s 0.80-
'i:
E
0.60:
3 0.40: g 0.20:
1-
Na = 7.603 Ni =271.3
E
a
0.00-r.. 0
.
I
.. . . , .. . .
2
4
6 8 1 0 1 2 1 4 DltneSiOnleS8 t h e
e
Figure 11. Simulated (dashed line) and experimental (solid line) breakthrough curves for oxygen in Rh8ne Poulenc 5A at 70 O C . Table 9. Effective Macropore and Micropore Mass Transport Rate Parameters Obtained with the Single Pellet String Fixed Bed
Ar 4OoC 7OoC 95OC
Nz 40°C 7OoC 95OC
02 4OoC 7OoC 95OC
RP 4A RP5A La Al RP 4A RP5A La Al RP 4A RP5A La Al RP4A RP5A La Al RP 4A RP5A La Al RP 4A RP5A La Al RP4A RP5A La A1 RP 4A RP5A La Al RP 4A RP5A La A1
u, cm/s
N.
Ni
9.12 9.12 7.81 10.0 10.0 8.56 10.7 10.7 9.18 9.12 9.12 7.81 10.0 10.0 8.56 10.7 10.7 9.18 9.12 9.12 7.81 10.0 10.0 8.56 10.7 10.7 9.18
6.264 6.264 2.882 6.615 6.615 2.676 7.033 7.033 2.687 6.819 6.819 3.930 6.557 6.557 3.861 6.867 6.867 4.072 7.314 7.314 3.908 7.603 7.603 3.494 7.598 7.598 3.694
.0225 >114.2 >26.2 .0492 >133.7 >30.5 .0912 >156.7 >36.5 .0367 >241.7 >54.6 .0466 >270.5 >64.8 .0541 >293.0 >76.11 2.436 >218.6 >43.5 4.879 >273.1 >53.9 7.424 >316.7 >59.3
D., cm% .014 48 .014 48 ,021 66 .016 77 .016 77 .022 05 .019 08 .019 08 .023 74 .015 76 .015 76 .029 55 .016 62 .016 62 .03182 .Ol8 63 .Ol8 63 .035 99 .016 91 .016 91 .029 39 .019 27 ,019 27 .028 79 .020 61 .020 61 ,032 65
Di,crnZ/s S 4.1 X 10-11 0.021 >2.1 x 10-7 0.0062 0.077 9.8 X 10-l1 0.016 >2.7 X lk7 0.0055 0.073 1.2 X 10-'0 0.013 3.3 X le7 0.0045 0.062 6.7 X 10-" 0.023 >4.4 x 10-7 0.012 0.051 9.3 x lo-" 0.018 >5.5 X 10-7 0.0093 0.064 1.2 X 0.017 6.3 X le7 0.0077 0.039 4.4 X 0.0048 >4.0 X l W 7 0.0073 0.048 9.7 X 0.0044 >5.4 X 10-7 0.0063 0.057 1.6 X 1 P 0.0044 >6.8 X lP7 0.0052 0.033
small particles, the shallow-bed diffusion cell, and the chromatographic method using a single pellet string fixed bed. They were used to determined adsorption equilibrium constants and mass transport parameters for nitrogen, oxygen, and argon in helium over Rhdne Poulenc 4A and 5A zeolites and LaRoche alumina at 40,70, and 95 OC. The adsorption equilibrium constants were calculated from the mass balance for each experimental run. When we compare the results obtained by both methods, the errors are less than 20%, possibly due to the experimental errors that also influence the global mass balance. The temperature dependency of the adsorption equilibrium constant is in the right direction. The shallow-bed diffusion cell was designed and operated under conditions where the chambers are perfectly mixed and secondary effects such as heat effects and convectiveflow across the cell are avoided. The estimation of the intraparticle diffusivities assumed that other diffusion resistances can be neglected or determined without using the transient responses. This is the case for bulk phase diffusion and film mass transport. The bulk phase effective diffusivity was determined by calculating the molecular diffusivity by using the Chapman-Enskog equation and the theoretical value of h for the tortuosity. It was verified that this procedure can be used in the case of our experiments and extended to other cases.
Once the effective molecular diffusivity is known, the film mass transport coefficients can be readily calculated using the steady-state responses. Also, only the effective diffusivity corresponding to the rate-controlling intraparticular mechanism can be determined with accuracy. In this case the responses are sensible in a fairly good range of the parameter under estimation. When both intraparticle diffusion mechanisms are important, the simultaneous determination of the macro and micro effective diffusivities is possible but the results may not be accurate. The single pellet string fixed bed was designed and operated in order to allow high and constant interstitial velocity with negligible head loss and to avoid entry and heat effects. Axial dispersion and film diffusion can mitigate the estimation of the intraparticle diffusivities. High velocities were used, and it was verified that in this case film diffusion can be neglected. Axial dispersion cannot be neglected, and it was calculated by a correlation. It was verified that the accuracy of this procedure was sufficient because the sensitivity of the system to uncertainties in the value of the Peclet number is low. So with this method we also can manage to reduce the estimation procedure to the intraparticle diffusion parameters. The disadvantages already stated for the SBDC also hold for this case. The procedures were illustrated for various gas-solid systems with the following results. Because of the relatively large window aperture and relatively small crystal dimension of 5A zeolite, the diffusion of argon, nitrogen, and oxygen is macropore rate controlled. When compared with the macropore diffusion resistance, the microparticle diffusion resistance is negligible. In this case the effective macropore diffusivity is the only adjustable parameter in our model and the high sensitivity of both experimental technique responses allows accurate optimization results to be obtained. For LaRoche alumina the situation is similar and thus the dimensionless micropore diffusion coefficients cannot be accurately estimated. The diffusion of argon and nitrogen in R h h e Poulenc 4A is microparticle rate controlled since the window aperture is significantly reduced due to the presence of an exchangeable ion. The microparticle diffusion resistance dominates over the macropore and axial dispersion resistances, and thus the estimation of effectivemicroparticle diffusivities is accurate. This is not true for macropore diffusivities. Taking into account that the macropore structures of 4A and 5A zeolites produced by the same manufacturer are probably the same, the 5A zeolite effective macropore diffusivities were used for 4A zeolite. The effective microparticle diffusivity of oxygen in 4A zeolite is much larger than those of argon and nitrogen, and in this case micro- and macropore diffusion resistances are of the same order of magnitude. Both effective diffusivities can be obtained by the fitting technique although with lower accuracy. The consistent results obtained by both experimental techniques confirm that both methods and the corresponding models are reasonable and the experimental results can satisfy the requirements for engineering development and design. Traditionally the diffusion cell technique is used with a single pellet. The shallow-bed diffusion cell has some operational advantages over the chromatographic method, mainly simplicity, ease of temperature control when working at temperatures different from the ambient temperature, no need to deal with flow patterns, and use of a smaller amount of adsorbent and gases. The successful
1390 Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994
use of the diffusion cell for small particles broadens ita application field. Acknowledgment
W.S. acknowledges FundaGBo Oriente for a research grant. Nomenclature
A = cross section area, m2 C = dimensionless concentration Ct = dimensionless top chamber concentration cb = dimensionless bottom chamber concentration Cr = dimensionless steady-state top chamber concentration Ct = dimensionless steady-state bottom chamber concentration c = concentration, mol-ma ct = top chamber concentration, mol.m3 Cb = bottom chamber concentration, m ~ l - m - ~ c, = inlet concentration, mobm-3 D, = axial dispersion coefficient, m2w1 D, = effective intrapellet diffusivity, m 2 d D, = molecular diffusivity, m2.s-1 D,, = effective bulk phase diffusivity, m 2 d d = pore diameter, m F = flow rate, m3.s-1 G = mass flux, kg-rn-2d kf = film mass transport coefficient, ms-l K = dimensionless adsorption equilibrium constant L = pellet thickness or column length, m M = molecular mass, kgmol-l N , = dimensionless macropore diffusion parameter Nb = dimensionless bulk diffusion parameter Nf = dimensionless film diffusion parameter Ni = dimensionless microparticle diffusion parameter N , = dimensionless geometric parameter P = pressure, Pa q, = microparticle concentration, mol.m3 ro = microparticle radius, m Ro = particle radius, m R, = dimensionless macropore mass transport resistance 8, = dimensionlessaxial dispersion mass transport resistance Rf= dimensionless film mass transport resistance Ri = dimensionless microparticle mass transport resistance S = standard deviation t = time, s u = interstitial velocity, ms-1 V, = chamber volume, m3 V = specific pore volume, m3.kg-1 uo = intrapellet velocity, ms-1 Z = dimensionless axial variable z = axial variable, m t, = macropore porosity Cb = bulk phase porosity 4, = ratio between the microparticle and the macropore volumes &, = ratio between the particle and bulk phase volumes u = viscosity, kpm-1.5-2 v = residence time, s 0 = dimensionless time p = gas density, mobma pI = solid density, kg-m-3 7 = tortuosity
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Received for review January 12, 1994 Accepted January 28, 1994. e Abstract published in Advance ACS Abstracts, March 15, 1994.
