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Ternary adsorption kinetics of gases in activated carbon
Ternary adsorption kinetics of gases in activated carbon Xijun
Hu, Bradley
King and Duong
D. Do”
Department of Chemical Engineering, University of Queensland, Brisbane, OLD 4072, Australia Ternary adsorption kinetic experiments of ethane (light species), propane (intermediate species) and n-butane in activated carbon are collected under various concentration combinations, temperatures and particle sizes. The effects of these parameters on the ternary adsorption dynamics are investigated. All the experimental data are compared with the predictions by a multicomponent heterogeneous macropore, surface and micropore diffusion (HMSMD) model recently proposed by Hu and Do (A/CM J (1993) 39 1628) using only single-component equilibrium and mass transfer parameters. The model can accurately predict the adsorption rates of ethane, propane and n-butane, but a small error in the calculation of the adsorbed amount of propane at ternary equilibrium is observed. Keywords: activated carbon; adsorption kinetics; pore and surface diffusion; multicomponent
energy
Nomenclature a
b
bo Cb cim
CP cpi
E F
Ratio of the surface activation energy to the heat of adsorption Isotherm parameter (m3 kmol- ‘) Isotherm parameter (m” kmoll ‘) Adsorbate concentration in the bulk (kmol mm3) Imaginary adsorbate concentration inside the microparticle (kmol mp3) Adsorbate concentration in the macropore (kmol me3) Initial adsorbate concentration in the macropore (kmol m- “) Adsorbed concentration in the particle (kmol m- 3, Observed adsorbed concentration in the particle (kmol m- 3, Maximum adsorption capacity (kmol m - “) Macropore diffusivity (m’ s- ‘) Surface diffusivity (m’ SC’) Surface diffusivity at zero-energy level (m2 s - ‘) Adsorbate-adsorbent interaction energy (kJ mall ‘) Energy distribution function
Introduction
HMSMD
JP J,
km NC I
So T t
* Author to whom all correspondence should be addressed
Ltd
micro-
Gas chromatograph Heterogeneous macropore, surface and micropore diffusion Flux through the macropore (kmol mW2SC’) Flux through the solid (kmol rnp2 s- ‘) Film mass transfer coefficient (m s- ‘) Number of components Particle radial position (m) Microparticle radial position (m) Particle radius (m) Gas constant (kJ kmol- ’ K- ‘) Diffusion length of the microparticle (m) Geometric factor ( = 0, 1, 2, for slab, cylinder and sphere, respectively) Geometric factor of the microparticle Temperature (K) Time (s)
Greek letters
Ratio of the zero-coverage surface diffusivity in the microparticle coordinate to that in the particle coordinate Particle macropore porosity
theoretical studies with exnerimental sunoort on the multicomponent diffusion of gases in solid adsorbent are scarce and are restricted to binary systems’-‘. Any urouosed theorv should be further tested with exnerimental results of higher order systems such as ternary ones for its confidence as a predicting tool. Therefore, it I
In most practical adsorption processes there are more than two components present in the system. However,
0950-4214/94/030175-12 @ 1994 Butterworth-Heinemann
GC
distribution;
II
Gas Separation 81 Purification 1994 Vol 8 No 3
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Ternary adsorption
kinetics of gases in activated
carbon: X. Hu et al.
is the purposes of this paper to present some ternary experimental results of ethane, propane and n-butane adsorption kinetics in activated carbon under various temperatures, particle sizes and concentration combinations. The data are collected using a differential adsorption bed (DAB) apparatus. A recently proposed multicomponent heterogeneous macropore, surface and micropore diffusion (HMSMD) model* is used to predict the ternary adsorption dynamics using only singlecomponent equilibrium isotherm and mass transfer parameters. This model uses the chemical potential gradient as the driving force so that the concentration dependency of the diffusivity of the adsorbed species can be explained. The surface energetic heterogeneity is also taken into account in this model for both adsorption equilibrium and diffusion of the adsorbed species. The local diffusion flux of the adsorbed species inside the microparticle is calculated with the aid of an imaginary gas-phase concentration concept’. The predictions from the model are compared with the experimental results. Theory Physical system and assumptions
A bidispersed microporous particle is exposed to a bulk environment containing NC adsorbate species. During the adsorption process any heat generated from adsorption can be rapidly dissipated by a high flow of gas so that the system remains isothermal. The bulk concentrations are assumed constant. This is justified because of the very high flow rate of the adsorbate mixture used in the experiments. The free species diffuse in the macropores and the adsorbed species diffuse along the directions of both the particle and microparticle coordinates because the particle is bidispersed. The particle surface is considered heterogeneous, giving rise to a distribution of interaction energies between the gases and the solid. The diffusion of the adsorbed species is assumed to follow a parallel path for a specified energy level”. Local adsorption
isotherm and energy distribution
First a local extended Langmuir adsorption is assumed for an adsorption site with an adsorption energy E(k) for the species k:
C,CkWI = C,,(k) 1+
43, -WlC,(k)
“c” KL &W,(j)
j=l
where C,(k) is the gas-phase concentration, C,[k, E(k)] and b[k, E(k)] are the adsorbed-phase concentration and the affinity for adsorbate k for a given site of energy E(k), respectively, the maximum adsorbed-phase concentration is C,,(k), and the affinity parameter b[k, E(k)] is correlated to the interaction energy by the following equation : b[k, E(k)] = b,(k) exp
with b. being the affinity at the zero-energy level or at the infinite temperature.
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Gas Separation
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Let F[k, E(k)] denote the energy distribution in the adsorbed phase; then the macroscopically observed isotherm at a given set of gas-phase concentrations (C,(k); k = 1, NC) is
sm E
bCk-WI C,(k)
C,,,,,(k)= Cpdk)
’ 1+
bC.L WW,(j)
j=l
x FCk,WI
Wk)
(3)
In this article a uniform energy distribution is used: 1
f’k -WI
=
forEmin
< E < -kAk)
E,,,(k) - Emi” (4)
and F[k, E(k)] = 0 elsewhere. If the ordering of the energy sites from low to high is assumed to be the same for different adsorbates and the cumulative energy distribution function is also considered to be the same for all species in the mixture 11,the correlation of energies of different species for a uniform energy distribution is I2
E(j) - Li,(j) E(i)- Edi) E,,,(i) - Edi) = 4,,,,(j) - -kin(i)
(5)
Because the extended Langmuir equation has been chosen to describe the local adsorption isotherm (Equation (1)) it is necessary for different species to have the same maximum adsorption capacities in order to satisfy the thermodynamic consistency. The local diffusion
flux of the adsorbed species
Because the adsorbed species inside the microparticle is not in equilibrium with the gas species, the calculation of the local diffusion flux of the adsorbed species is achieved with the aid of the imaginary gas-phase concentration concept’. For a given set of adsorbed concentrations in the micropore, the imaginary gasphase concentrations which are in equilibrium with that set can be evaluated from any multicomponent equilibrium theory. These imaginary gas-phase concentrations are then used to compute the chemical potential inside the microparticle. Knowing this, the local diffusion fluxes of the adsorbed species k along the particle and microparticle coordinates J,[k, E(k)]r and J,Ck, E(W, , are evaluated from
CpCkE(k)1acid
J&t EWI, = - D,Ck E(k)1 J,[k, E(k)],&, = -jP(k)D,[k,
E(k)] c,,,,,,
(6)
ar
C;,(k) Irn
2@)
P
(7)
where r and rlc are the coordinates of the particle and microparticle, respectively, with their diffusion lengths being R and R,, Ci, is the imaginary gas-phase concentration, and D,[k, E(k)] is the zero-coverage diffusivity of the adsorbed species k, which has the
Ternary adsorption kinetics of gases in activated carbon: X. Hu et al.
following temperature
written:
dependency:
D,[k, E(k)] = D,,(k)exp(
- y)
%4
where a is the ratio of the surface activation energy to the heat of adsorption, and D,,(k) is the zero-coverage diffusivity of adsorbed species k at the zero-energy level in the particle coordinate. Besides the coordinates I and r@, Equation (7) differs from (6) by a factor of p’. The reason for this difference is that the zero-coverage surface diffusivity with zero energy, D,,, might be different in the grain coordinate from that in the particle coordinate because of the different tortuosities in different directions. Since different species with different molecular sizes can access the micropores with different degrees due to the micropore distributions, the tortuosity parameter B2 can be different for different components’.
qsk) at
(1 - %d
+
C,Ck,EtWCk,
J
E(k)1dE(k) I$ dr,
$ dr,
0
-
eY f
$ [r-V,(k)]
- (1 - Q,,); $
J,Ck JYW’Ck E(k)1 dE(k) c dr,
s
(13)
&
e dr, 1
0
Mass balance equations
with its boundary conditions as:
We can now write the mass balance equation in the microparticle using the diffusion flux of the adsorbed species obtained in the last section and the parallel-path model (PPM) of Kapoor and Yang”:
i,
a
J,Ck (sm
qtar, e
W)l,,,FCk Et41 Wk)
>
(9)
where s,, is the geometric factor of the microparticle ( = 0, 1 and 2 for slab, cylinder and sphere, respectively). The local adsorbed concentrations inside the microparticle are written in terms of the imaginary gas-phase concentration:
bCkE(k)1 Cimtk) 1+
0
(14)
R;
R,,
0
C,Ck WI = C,,(k)
r =
%(k) _ ar
+J,tk) + (1 - s,I)
J C,t-k W)lFCk, E(k)1Wk)
=-__i
r = 0;
F Nj, -WW&)
X
m
I (s 0
0
J,Ck EtWTk E(k)1d.W) e dr, >
s RN
rS” w dr P
0
=
k(k)CC,(k)
-
W41
(15)
The initial conditions of the system are: t = 0; C,(k) = CPi(k);
(10)
bCkEtk)lCpitk)
C,CkEt41= C,,(k) 1+
j=l
(16)
“c” KA EtjW,itj)
j=l
One of the boundary conditions of Equation (9) is that the flux at the centre of the microparticle is zero: rp = 0;
aC,t-kWI ar, =
o
(11) Experimental
Another boundary condition is the equilibrium relationship at the exterior surface of the microparticle: r,, = R,; m C,Ck, s0
=
s
WW’Ck WI Wk)
O” C,,(k) 0
1+
W, -WI C,(k)
F Ni j= 1
The above model equations are solved numerically after casting into appropriate non-dimensional form8*r3.
f’Ck E(k)1Wk) (12)
W)lC,(j)
Let s denote the particle shape factor, with a value of 0, 1 or 2 for slab, cylinder or sphere, respectively, Ed the macropore porosity, and Jp the pore diffusion flux, the total mass balance equation for the particle can be
Equilibrium analysis of a single-component
gas
Activated carbon Type 967 supplied by Ajax Chemical Co., Australia as 1.8 mm diameter cylindrical extrudate was used as the absorbent. The structural properties and the pore size distribution of Ajax activated carbon can be found in ref. 14. The absorbates used in this study were pure ethane, propane and n-butane in gas cylinders supplied by Commonwealth Industrial Gases in Brisbane. A high-accuracy volumetric adsorption isotherm rig (Figure I) was used to measure the single-component equilibrium of gases in Ajax activated carbon. The isotherm measurement rig has two separate chambers. One has a supply bomb and the other contains an adsorption cell. Each chamber has a separate MKS transducer (maximum range 1000 mmHg) to record the
Gas Separation
& Purification
1994 Vol 8 No 3
177
Ternary adsorption
kinetics of gases in activated
carbon: X. Hu et al.
150 cc Supply Bomb
Needle Valves Toggle Valve
Alcatel Molecular Drag Pump
HBBiee female
-
female coupliq
male to NPT VCR tee
L
VCR cross
+ Water Bath
1
ewaqlok tee
7
J
El-
I t
Activated Carbon
Figure 1
Schematic
diagram of volumetric
adsorption
measurement
pressure. l/4” stainless steel tubes and VCR fittings supplied by CAJON Co., Macedonia, Ohio, USA are used in the system. A l/2” glass tube, which has a round bottom, is used as the adsorption cell and is connected to the system via a CAJON reducing adaptor (l/2” to l/4”). The supply bomb section has a total volume of 159.0 cm3 and the sample cell section has a total volume of 38.4 cm3. A K-type thermocouple is mounted inside the supply bomb to measure its temperature. Another K-type thermocouple is installed in the glass sample cell and is surrounded by activated carbon to record the temperature of the adsorbent. Pure gas cylinders are connected via needle valves, and CAJON toggle valves are used to isolate the adsorbate. Before an equilibrium experiment starts, an Alcatel molecular drag pump is used to evacuate the system to 10d3 mmHg. The operation procedure is described below. Ajax activated carbon was crushed to about 0.1 mm and dried in an oven at 200°C for three hours to remove excess moisture. The carbon was then weighed (0.603 g was used) and loaded into the adsorption cell. The whole system was first evacuated to vacuum using an Alcatel molecular drag pump and the activated carbon particles in the adsorption cell were heated up to 300°C and kept at this temperature and under vacuum overnight to degas any possible adsorbed species. Pure gas (ethane, propane or n-butane) was then dosed into the supply bomb and the pressure and temperature were recorded when constant readings were reached.
178
Gas Separation
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Vol 8 No 3
Thermal couple
rig
A small amount of pure gas was dosed from the supply bomb into the adsorption cell which was kept isothermal by using a water bath. After the pressure in the adsorption cell becomes constant (typically about one to two hours), the pressures and temperatures in both supply bomb and adsorption cell were recorded. The amount adsorbed which was in equilibrium with the pressure in the gas phase of the cell was calculated from the amount supplied from the supply bomb via the ideal P-V-T relationship. Steps 4 and 5 were repeated to get the equilibrium isotherm data at higher pressures until a full isotherm curve was obtained. Kinetics measurement A DAB apparatus was used to collect the dynamic responses of the adsorption process of the gases in activated carbon. As seen in Figure 2, the DAB rig has four major elements: gas mixing system, adsorption bed, sample bomb and gas analysis section. Rotameters (R-l to R-6) supplied by Porter Instrument Company are used to control the flow. The size of the glass tube and the type of float inside the tube are chosen according to the flow rate desired. The flow rate of a rotameter is calibrated by using a bubble soap meter. Needle valves (V-l to V-6) are used to provide the on/off function. Porter pressure regulators (PR-1 to PR-6) are mounted before the rotameters to keep a constant head pressure of 100 kPa. The gases are mixed after the rotameters and the concentrations of adsorbates are calculated from the flow
Ternary adsorption
kinetics of gases in activated
carbon: X. Hu et al.
GAS MIXING
Water / Bath
-l
I I
I -- 4-J
I
ADSFEFON
I
I
1
s
NEEDLE VALVE rzl
GAS CHROYATOGRAPH AND INTEGRATOR
1 PR-1
( PR-2
( PR-3
( PR-4
( PR-5
TWV-4
NEEDLE VALVE
OVEN AND SAMPLING VALVE
1 PR-6
TWV-3
VACUUY
I
\
GAS ANALYSIS
NitrogenNitrugen Ethane Figure
2
Schematic
Propane n-Ehtaoe
SECTION
diagram of differential adsorption bed
rates and their concentrations in the supply cylinders. A four-way valve (FWV-1) is used to send either the mixed adsorbates or the purging gas (nitrogen) to a three-way valve (TWV-1). This three-way valve controls the connection either to the adsorber for adsorption or to the GC for calibration. l/8” stainless steel tubes are used in all gas lines to minimize the dead space. The adsorber section consists of a four-way valve (FWV-2), the adsorber and a three-way valve (TWV-2). The four-way valve serves for the isolation of the adsorber from the adsorbate gas mixture. To ensure good mixing of the gases, the mixed gas is bypassed to vent for a few minutes before it is switched to the adsorber. A water bath is used to keep the temperature of the adsorber constant. The isothermality of the system is maintained by using a high flow rate of the adsorbates. This is confirmed by inserting a thermocouple inside the adsorber and monitoring its temperature reading. The adsorber is constructed by modifying a 314” stainless steel CAJON plug, of which the interior is machined out to create an open chamber to host the adsorbent. A 100 pm mesh is built into a stainless steel gasket and is used to seal the adsorber before a CAJON cap is screwed onto the plug. l/S” stainless steel tubes are silver-soldered to the plug and cap as inlet and outlet gas lines which are then connected to the four-way and three-way valves (FWV-2 and TWV-2) via l/4” VCO o-ring fittings. The sample bomb is built from a 316 stainless steel hollow bar which has a depth of 40 mm and an insider
diameter of 55 mm. The hollow bar is sealed with stainless steel plate at the bottom and flanged with a viton o-ring at the top. The sample bomb section has an effective volume of 108.1 cm3. A barksdale series 302 pressure transducer is used to measure the bomb pressure. The millivolt output of the barksdale transducer is calibrated against an MKS transducer and the response of the barksdale transducer is found to be linear. The temperature of the sample bomb is measured by mounting a l/8” K-type thermocouple inside the bomb. The sample bomb is then connected to either the sample system of the GC or the vacuum pump through a three-way valve (TWV-3). The contents of the gas sample are analysed using a Shimadzu FID gas chromatograph (type GC-8A). A l/16” stainless steel tube is used to connect the gas sample and the six-port stainless steel Valco sampling valve attached to the GC. The gas sample from either the sample bomb or the calibration line is directed to the sample loop via a three-way valve (TWV-4). The gas from the outlet of the sample valve is vented to the atmosphere via a water trap, which ensures a constant pressure in the sample loop at zero flow. A DURAPAK column packed in l/8” tubing is used to separate ethane, propane and n-butane, with nitrogen as the carrier gas. The GC is operated isothermally at 100°C and the detector responses are integrated using a Hewlett Packard 3394A integrator. Ajax activated carbon particles of different sizes and shapes were prepared before the kinetic experiments. The
Gas Separation
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Ternary adsorption
kinetics of gases in activated
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1.8 mm diameter cylinders were obtained by simply sealing the end surfaces of the supplied extrudates with epoxy resin, hence only the cylindrical surface of the particle was exposed to gases. For slab particles, the cylindrical surface was coated with epoxy leaving only the end surfaces exposed to gases for adsorption. The end surfaces of the slab particles were carefully trimmed to obtained a desired length. The operation procedure of the DAB is stated below. Slab or cylindrical particles of Ajax activated carbon were loaded to the absorber. About 0.1 g of particles was used in the dynamic studies. The adsorption bed was initially heated up to a high temperature (200°C) under vacuum overnight and then degassed by purging the heated bed with a small flow of pure nitrogen for 15 min. The bed was brought to the adsorption temperature using a water bath and the adsorbates were flushed through the bed at a flow rate of 50@1500 cm3 min-’ for a predetermined adsorption time before the adsorber was isolated. The sample was desorbed to a pre-evacuated bomb by heating up the adsorption bed to 200°C and remaining at this temperature for 40 min. At the same time the four-way valve (FWV-1) was switched to rotameter 2 and pure nitrogen was flushed through the gas line between FWV-1 and FWV-2 to vent in order to clean any adsorbates remaining in the gas line. After the gas line was cleaned, FWV-2 was switched to the adsorber and pure nitrogen was slowly flushed through the adsorber (around 10 cm3 min-‘) for 1615 min. The pressure and temperature of the bomb were measured and recorded and then the adsorbate concentrations in the bomb were analysed by gas chromatography. The total amount in the sample bomb was calculated from its temperature, volume and pressure via the ideal P-V-T relationship. This total amount includes the adsorbed amount inside the particles and those adsorbates trapped in the voids of the adsorber at isolation (section between FWV-2 and TWV-2). The adsorbed amount was the result of the product of the total amount in the bomb and the adsorbate mole fraction obtained from GC analysis minus the dead volume contribution. By repeating steps 3-6 for different exposure times an uptake curve can be obtained for the specified experimental conditions. In order to accurately measure the dead volume contribution of the adsorbates trapped in the voids of the adsorber at isolation (section between FWV-2 and TWV-2) an empty cell having the same internal dimensions was used to substitute the adsorber. The mixed adsorbates were then flushed through the empty cell under the same operation conditions (concentration, temperature and flow rate) for some time and isolated. The amounts of different species inside the empty cell were then passed to the pre-evacuated sample bomb and further flushed by purging pure nitrogen to bring the pressure of the sample bomb to the same value as in the adsorption kinetics. The gas mole fractions were analysed by the GC. The dead volume contribution was then calculated as the product of the total amount in the sample bomb (via the P-V-T relationship) and the molar fraction.
180
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9 The GC responses were calibrated once daily. In the
calibrations, nitrogen and adsorbate (ethane, propane or n-butane) were mixed at different flow rates via rotameters R-3 and R-4/R-5/R-6 and then passed to the sampling valve of the GC for analysis. The mole fraction of gases was calculated from their relative flow rates. The mole fractions were then plotted versus the integrated area of the detector responses. A linear relationship was observed experimentally. Results
and discussion
First, the single-component equilibrium of the gases in Ajax activated carbon is analysed. Because the gas concentrations used in our dynamic studies were always less than 30 kPa, only those equilibrium data having a gas pressure less than 35 kPa were used in the determination of isotherm parameters. All experimental data for three temperatures (10, 30 and 60°C) were used simultaneously in the fitting with the Langmuir-Uniform distribution equation (Unilan). A least squares method was employed in the fitting process. The maximum adsorption capacity (C,,) was set to be the same for all three species in order for the local multicomponent Langmuir equation to satisfy the thermodynamic consistency. This parameter, however, is assumed to be temperature dependent. The other parameters in the isotherm equation, b,, Emin and E,,,, are temperature independent but depend upon the adsorbate. The experimental results (symbols) and the model fittings (solid lines) of the adsorption equilibria of ethane, propane and n-butane in Ajax activated carbon are shown in Figure 3. The fitting between the model and the experimental data is excellent in all cases. The extracted equilibrium parameters are tabulated in Table I.
Second, the single-component adsorption kinetic experiments were carried out and the uptake data were used to extract the dynamic parameters (D,, and R,//?) for the single component. The experimental results for ethane and propane in Ajax activated carbon are available in ref. 13 for different temperatures, particle sizes and bulk concentration combinations. The flow rate used for ethane and propane was 500 cm3 min- ‘, which was found to be adequate to provide a differential condition in the cell containing about 0.2 g activated carbon having a slab geometry with a full length of 2.4 mm. The adsorption dynamics of n-butane in Ajax activated carbon were measured by a similar procedure but a higher flow rate of 1000 cm3 min-’ was required to maintain the differential condition. The reason for this is that the adsorption equilibrium isotherm of butane is highly favourable (Figure 3c), implying that more gas will be adsorbed from the bulk environment. Hence a higher molar rate of supply of n-butane is required to maintain the differential condition. For a 1.8 mm diameter cylindrical particle, the adsorption kinetics are very fast; hence the experiments with ethane and propane in such a particle were carried out using a higher gas flow rate of 1500 cm3 min - 1 to dissipate any heat generated from adsorption. The single-component experimental adsorption data (symbols) of ethane, propane and n-butane in Ajax activated carbon are plotted in Figure 4 for different bulk concentrations and particle sizes and shapes. The adsorption kinetics were carried out at a constant temperature of 30°C and the total pressure was 1 atm.
Ternary adsorption kinetics of gases in activated carbon: X. Hu et al. I
I
I
1
-l
a v
3o”c 60°C
20
I
I
I
Figure 3 Adsorption equilibrium isotherm of gases in Ajax activated carbon: (a) ethane; (b) propane; (c) n-butane
‘able 1
These experimental results were then used to extract the dynamic parameters of the HMSMD model. The ratio of the surface activation energy to the heat of adsorption, a, was taken as 0.5 for all three species. The pore diffusivity was computed from the combined molecular and Knudsen diffusivitiesi5 and a tortuosity of three 13. The shape of the microparticle was assumed to be spherical (sll = 2) although other geometries can also be used. Therefore, the fitting parameters in the HMSMD model are the zero-coverage surface diffusivity at zero-energy level (D,c) and the effective diffusion path length in the microparticle (RJB). The model fittings are shown in Rgure 4 as solid lines. It is seen in this figure that the fitting between the model and the experimental data is very good for all cases tested. The kinetic parameters are tabulated in Table 2. The equilibrium and kinetic parameters obtained for the single-component systems were then used in the HMSMD model to predict the ternary adsorption kinetics of ethane, propane and n-butane in Ajax activated carbon under various conditions. The gas flow rate through the DAB was 1500 cm3 min-’ in all ternary kinetic experiments. Figure 5 shows the adsorption kinetics of 10% ethane, 10% propane and 10% n-butane in Ajax activated carbon of 2.4 mm full-length slabs at 30°C and 1 atm. Because n-butane is the slowest moving and most strongly adsorbing species, the other two species, ethane and propane, exhibit an overshoot phenomenon in their fractional uptake before their final equilibrium is reached. The degree of overshoot is about 5.5 for ethane and 2.5 for propane. The HMSMD model predicts well the uptake and fractional curves of ethane, including its degree of overshoot and the time at which the overshoot occurs. Because the adsorption affinity of ethane in Ajax activated carbon is much weaker compared with that of propane and n-butane, the adsorbed amount of ethane at final equilibrium is so small that its amount in the void space of the adsorber and the tubing system (between FWV-2 and TWV-2 in Figure 2) can make a significant contribution to the measurement of the total amount of ethane in the sample bomb. Therefore, a small error in the measurement of the contribution from the void space might generate a large error in the fractional uptake of ethane. In Figure 5, the experimental data of ethane have an error of about 10% in the maximum fractional uptake, which corresponds to a 3% error in the void space concentration measurement. In the ternary adsorption dynamic studies, the measured amount of ethane in the void space of the adsorber is about three times the final adsorbed amount of ethane inside the activated carbon.
Isotherm parameters of ethane, propane and n-butane in Ajax activated carbon
Species
FC)
$noI
Ethane
10 30 60 10 30 60 10 30 60
4.75 4.47 4.11 4.75 4.47 4.11 4.75 4.47 4.11
Propane
n-Butane
mm3)
b, (kPa-‘)
Emi, (kJ mol-‘)
6.11 x lo-’
16.7
27.1
4.17 x 1O-6
21 .3
34.6
5.60 x 1O-5
15.8
34.9
Gas Separation
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1994 Vol 8 No 3
181
Ternary adsorption
kinetics of gases in activated
carbon: X. Hu et al.
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2 4.4
mm
2.6
slab
0.0
mm
slab
0.0
1.0 3
0.8
0.8
2
0.6
0.6
0.4
0.4
2
0.2 4.4
mm
slab
0.0
k
O
2000
1000
0
400
0
100
800
1.0 0.8 0.6 0.4 0.2 0.0
0
500
1000 1500
Time Figure 4
(seconds)
Adsorption dynamics of gases in Ajax activated carbon
That is, the ethane adsorbed concentration inside the particle is only about 25% of the total amount of ethane in the sample bomb. However, the adsorbed amount of ethane is so small that a 10% error of the maximum fractional uptake of ethane becomes negligible when it is plotted as the real adsorbed amount. In Figure 5 it is seen that the agreement between the model predictions and the experimental data is excellent for n-butane uptake and fractional uptake. Some deviations are observed for propane but the model describes its kinetic
182
200
Gas Separation
81 Purification 1994 Vol 8 No 3
behaviour well. The error in the overshoot degree of propane is due to the incorrect prediction of ternary equilibrium for propane. The heterogeneous extended Langmuir equation gives a higher value for the amount of propane adsorbed than the experimental data, even at the final steady state, which in turn results in a prediction of lower uptake overshoot for propane. This raises the importance of accurate measurement and calculation of the single- and multicomponent adsorption equilibrium isotherms.
Ternary adsorption kinetics of gases in activated carbon: X. Hu et al. Table 2 Kinetic parameters of ethane, propane and n-butane in Ajax activated carbon
Species
10%
0
5%
v
5% c,
model
Ethane
10 30 60 10 30 60 10 30 60
Propane
n-Butane
4.03 4.48 5.23 3.20 3.47 4.16 2.75 3.04 3.57
c, c,
0
1 .oo
0.863
3.36
10.2
8.42
5.75
B a = 0.5 bsr=2
%
6
-
_
2.0
OE 1
1.5
; ti -
model
1.0
0 0
500
Time
1500
1000
1000
500
1500
(seconds)
Figure 6 Ternary adsorption kinetics of ethane, propane and n-butane in Ajax activated carbon of 2.4 mm full-length slabs at 30°C and 1 atm
G-E \ z
2.0 0
0 v
1.5 -
E 2
10% c, 10% c, 10% c, model
1.0 0
5%
c,
a 5%c, v 5%c, __ model 0
500
Time
lob0
1500
(seconds)
Figure 5 Ternary adsorption kinetics of ethane, propane and n-outane m Ajax activated carbon of 2.4 mm full-length slabs at 30°C and 1 atm
Another experiment with a different bulk concentration combination of 10% ethane, 5% propane and 5% n-butane is shown in Figure 6. Since the concentrations of propane and n-butane here are lower than those in Figure 5, the faster-adsorbed ethane displaced by the later coming propane and n-butane in the final equilibrium is less, so that we have a lower overshoot degree of ethane uptake of about 4.3. The HMSMD
0
600
200
400
Time
(seconds)
800
Figure 7 Ternary adsorption kinetics of ethane, propane and n-butane in Ajax activated carbon of 2.4 mm full-length slabs at 30°C and 1 atm
Gas Separation
& Purification 1994 Vol 8 No 3
183
Ternary adsorption
kinetics
of gases in activated
-
0
200
model
400
Time
carbon: X. Hu et al.
600
(seconds)
Figure 8 Ternary adsorption kinetics of ethane, propane and n-butane in Ajax activated carbon of 2.4 mm full-length slabs at 30°C and 1 atm
-
500
0
-
model
1500
1000
10%
0
5% c,
-
Time
1000
1500
Gas Separation
& Purification
1994
c,
0
10%
0
5% c,
v
5% c,
-
0
200
Time
(seconds)
Figure 9 Ternary adsorption kinetics of ethane, propane and n-butane in Ajax activated carbon of 2.4 mm full-length slabs at 60°C and 1 atm
184
300
model
_
-.500
model
200
-
model
-
100
0
c,
0
05%C,
0
model predictions are again in very good agreement with the experimental data for ethane and n-butane. The model reasonably predicts the uptake rate of propane with some deviations in the degree of its overshoot. This underprediction of the overshoot degree is again due to the overprediction of the final adsorbed amount of propane at equilibrium. The ternary adsorption fractional uptakes in Ajax activated carbon of 2.4 mm full-length slabs at 30°C are measured for two other bulk concentration combinations, 5% for all species (Figure 7) and 5% ethane, 10% propane, 5% n-butane (Figure 8). Again the agreement between the model predictions and the data is within experimental error for ethane and n-butane, with some deviations for propane uptake. Comparing Figures 6 and 7, one comes to the conclusion that a decrease in ethane bulk concentration will yield a higher overshoot degree of ethane fractional uptake. The reason for this is that the percentage amount of adsorbed ethane displaced by the late penetrating propane and n-butane at final equilibrium is higher when the ethane bulk concentration is lower. The ternary adsorption kinetics of 10% ethane, 5% propane and 5% n-butane at a higher temperature, 60°C is shown in Figure 9. The same particle (2.4 mm slab) and total pressure (1 atm) as in Figures 5-8 are used here. Since the pore diffusivity is calculated from combined Knudsen and molecular diffusivities and the
Vol 8 No 3
400
600
800
1000
(seconds)
Figure 10 Ternary adsorption kinetics of ethane, propane and n-butane in Ajax activated carbon of 1.8 mm diameter cylinders at 30°C and 1 atm
Ternary
E
1.0
-
model
~
model
adsorption
kinetics
of gases
in activated
carbon:
-
b-4
X. t-/u et al.
model
_
-
0
200
Time
400
600
800
1000
0
(seconds)
Figure 11 Ternary adsorption kinetics of ethane, propane and n-butane in Ajax activated carbon of 1.8 mm diameter cylinders at 60°C and 1 atm
zero-coverage surface diffusivity at zero-energy level (Dko) is independent of temperature, the HMSMD model can predict the dynamics at different temperatures without any extra fitting parameter. The agreement between the model and the experimental data, including the final equilibrium, is reasonably good, close to the experimental error. The overshoot degree of ethane fractional uptake is lower at 60°C than at 30°C (Figure 6). This is expected because the adsorption energy of propane and n-butane is higher than that of ethane (Table I). Therefore, the adsorption affinity and surface diffusivity of propane and n-butane are more sensitive to temperature than those of ethane. As a result the difference in the adsorption affinity and surface diffusivity between different species is narrowed when the temperature is higher. Having studied the ternary adsorption kinetics for different concentration combinations and temperatures, we now explore the ternary adsorption dynamics in Ajax activated carbon of different shapes and sizes. The adsorption kinetics of 10% ethane, 5% propane and 5% n-butane in 1.8 mm diameter cylindrical Ajax activated carbon are shown in Figure 10 for an adsorption temperature of 30°C and in Figure II for 60°C. The model predictions are generally in good agreement with the experimental data, but the isotherm equation gives a higher prediction of the adsorbed amount of propane. Again the overshoot degree of ethane fractional uptake is found to be lower when the temperature is higher.
1000
2000
3000
Time
(seconds)
4000
Figure 12 Ternary adsorption kinetics of ethane, propane and n-butane in Ajax activated carbon of 4.2 mm full-length slabs at 30°C and 1 atm
Finally, the ternary adsorption kinetics in a larger slab of Ajax activated carbon having a full length of 4.2 mm is shown in Figure 12. The concentration is 10% for all three species and the adsorption process is carried out at a temperature of 30°C and a total pressure of 1 atm. The agreement between the model and the experimental data is reasonable for ethane and n-butane. The discrepancy for propane uptake is still due to the improper prediction of the propane adsorbed amount.
Conclusions Ternary adsorption kinetics data for ethane, propane and n-butane in activated carbon are collected via a DAB under different operating conditions, such as bulk concentration combinations, temperatures, particle sizes and shapes. Because n-butane is the slowest diffusing and most strongly adsorbed species, both ethane and propane exhibit a maximum in their uptakes during the simultaneous adsorption process. The overshoot degree of ethane uptake is found to decrease with an increase in ethane bulk concentration, or with a decrease in propane and n-butane concentrations. A higher adsorption temperature is also found to reduce the overshoot degree of ethane uptake. The HMSMD model proposed by Hu and Do’ is used to predict the ternary adsorption kinetics by using only the information from singlecomponent mass transfer and adsorption equilibrium
Gas Separation
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Ternary adsorption kinetics of gases in activated carbon: X. Hu et al.
parameters. The agreement between the model and the experimental data is found to be good in most cases studied for ethane and n-butane uptake. The model can also predict well the adsorption rate of propane but fails to predict its overshoot degree accurately because of the slightly higher prediction of the propane equilibrium adsorbed amount. This demonstrates the potential of the HMSMD model as a mathematical tool in the prediction of multicomponent adsorption kinetics of gases in activated carbon.
7
8
9
Acknowledgement
10
Financial support from the Australian Research Council and the REGS of the University of Queensland is gratefully acknowledged.
11
12
13
References Ma, Y.H. and Roux, A.J. Multicomponent rates of sorption of SO, and CO, in sodium mordenite AIChE J (1973) 19 1055-1059 Ma, Y.H. and Lee, T.Y. Diffusion of binary gas mixtures in zeolite X pellets Ind Eng Chem Fundam (1977) 16 4448 Carlson, N.W. and Dranoff, J.S. Competitive adsorption of methane and ethane on 4A zeolite Fundamentals ofAdsorption (Ed A.L. Liapis) Engineering Foundation, New York (1987)
186
Gas Separation
& Purification 1994 Vol 8 NO 3
14
15
Qureshi, W.R. and Wei, J. One- and two-component diffusion in zeolite ZSM-5, I. Theory J Catal (1990) 126 126-146 Qureshi, W.R. and Wei, J. One- and two-component diffusion in zeolite ZSM-5, II. Experimental J Catal (1990) 126 147-172 Yang, R.T., Chen, Y.D. and Yeh, Y.T. Prediction of cross-term coefficients in binary diffusion: diffusion in zeolite Chem Eng Sci (1991) 46 3089-3099 Hu, X., Rao, G.N. and Do, D.D. Multicomponent sorption kinetics of ethane and propane in activated carbon: simultaneous adsorntion Gas Sea Purif (1993) 7 3945 Hu, -X. and D6, DID: Role of energy distribution in multicomponent sorption kinetics in bidispersed solids AIChE .I (1993) 39 1628-1640 Hu, X. and Do, D.D. Multicomponent adsorption kinetics of hydrocarbons onto activated carbon: contribution of micropore resistance Chem Eng Sci (1993) 48 1317-1323 Kapoor, A. and Yang, R.T. Surface diffusion on energetically heterogeneous surfaces AIChE .I (1989) 35 1735-1738 Valenzuela, D.P., Myers, A.L., Tab, 0. and Zwiebel, 1. Adsorption of gas mixtures: effect of energetic heterogeneity ArChi J (1988) -M 397402 Kauoor, A., Ritter, J.A. and Yang. R.T. An extended Lanumuir _ model for adsorption of gas mixtures on heterogeneous surfaces Lungmuir (1990) 6 66@664 Hu, X., Rao, G.N. and Do, D.D. Effect of energy distribution on sorption kinetics in bidispersed particles AIChE J (1993) 39 249-261 Gray, P.G. and Do, D.D. Adsorption and desorption of gaseous sorbates on a bidispersed particle with Freundlich isotherm. II. Experimental study of sulphur dioxide sorption on activated carbon particles Gas Sep Purif(1989) 3 201-208 Reid, R.C., Prausnitz, J.M. and Polling, B.E. The Properties of Gases and Liquids 4th Edn, McGraw-Hill, New York (1987) I
I
_,
Hu, Bradley
King and Duong
D. Do”
Department of Chemical Engineering, University of Queensland, Brisbane, OLD 4072, Australia Ternary adsorption kinetic experiments of ethane (light species), propane (intermediate species) and n-butane in activated carbon are collected under various concentration combinations, temperatures and particle sizes. The effects of these parameters on the ternary adsorption dynamics are investigated. All the experimental data are compared with the predictions by a multicomponent heterogeneous macropore, surface and micropore diffusion (HMSMD) model recently proposed by Hu and Do (A/CM J (1993) 39 1628) using only single-component equilibrium and mass transfer parameters. The model can accurately predict the adsorption rates of ethane, propane and n-butane, but a small error in the calculation of the adsorbed amount of propane at ternary equilibrium is observed. Keywords: activated carbon; adsorption kinetics; pore and surface diffusion; multicomponent
energy
Nomenclature a
b
bo Cb cim
CP cpi
E F
Ratio of the surface activation energy to the heat of adsorption Isotherm parameter (m3 kmol- ‘) Isotherm parameter (m” kmoll ‘) Adsorbate concentration in the bulk (kmol mm3) Imaginary adsorbate concentration inside the microparticle (kmol mp3) Adsorbate concentration in the macropore (kmol me3) Initial adsorbate concentration in the macropore (kmol m- “) Adsorbed concentration in the particle (kmol m- 3, Observed adsorbed concentration in the particle (kmol m- 3, Maximum adsorption capacity (kmol m - “) Macropore diffusivity (m’ s- ‘) Surface diffusivity (m’ SC’) Surface diffusivity at zero-energy level (m2 s - ‘) Adsorbate-adsorbent interaction energy (kJ mall ‘) Energy distribution function
Introduction
HMSMD
JP J,
km NC I
So T t
* Author to whom all correspondence should be addressed
Ltd
micro-
Gas chromatograph Heterogeneous macropore, surface and micropore diffusion Flux through the macropore (kmol mW2SC’) Flux through the solid (kmol rnp2 s- ‘) Film mass transfer coefficient (m s- ‘) Number of components Particle radial position (m) Microparticle radial position (m) Particle radius (m) Gas constant (kJ kmol- ’ K- ‘) Diffusion length of the microparticle (m) Geometric factor ( = 0, 1, 2, for slab, cylinder and sphere, respectively) Geometric factor of the microparticle Temperature (K) Time (s)
Greek letters
Ratio of the zero-coverage surface diffusivity in the microparticle coordinate to that in the particle coordinate Particle macropore porosity
theoretical studies with exnerimental sunoort on the multicomponent diffusion of gases in solid adsorbent are scarce and are restricted to binary systems’-‘. Any urouosed theorv should be further tested with exnerimental results of higher order systems such as ternary ones for its confidence as a predicting tool. Therefore, it I
In most practical adsorption processes there are more than two components present in the system. However,
0950-4214/94/030175-12 @ 1994 Butterworth-Heinemann
GC
distribution;
II
Gas Separation 81 Purification 1994 Vol 8 No 3
175
Ternary adsorption
kinetics of gases in activated
carbon: X. Hu et al.
is the purposes of this paper to present some ternary experimental results of ethane, propane and n-butane adsorption kinetics in activated carbon under various temperatures, particle sizes and concentration combinations. The data are collected using a differential adsorption bed (DAB) apparatus. A recently proposed multicomponent heterogeneous macropore, surface and micropore diffusion (HMSMD) model* is used to predict the ternary adsorption dynamics using only singlecomponent equilibrium isotherm and mass transfer parameters. This model uses the chemical potential gradient as the driving force so that the concentration dependency of the diffusivity of the adsorbed species can be explained. The surface energetic heterogeneity is also taken into account in this model for both adsorption equilibrium and diffusion of the adsorbed species. The local diffusion flux of the adsorbed species inside the microparticle is calculated with the aid of an imaginary gas-phase concentration concept’. The predictions from the model are compared with the experimental results. Theory Physical system and assumptions
A bidispersed microporous particle is exposed to a bulk environment containing NC adsorbate species. During the adsorption process any heat generated from adsorption can be rapidly dissipated by a high flow of gas so that the system remains isothermal. The bulk concentrations are assumed constant. This is justified because of the very high flow rate of the adsorbate mixture used in the experiments. The free species diffuse in the macropores and the adsorbed species diffuse along the directions of both the particle and microparticle coordinates because the particle is bidispersed. The particle surface is considered heterogeneous, giving rise to a distribution of interaction energies between the gases and the solid. The diffusion of the adsorbed species is assumed to follow a parallel path for a specified energy level”. Local adsorption
isotherm and energy distribution
First a local extended Langmuir adsorption is assumed for an adsorption site with an adsorption energy E(k) for the species k:
C,CkWI = C,,(k) 1+
43, -WlC,(k)
“c” KL &W,(j)
j=l
where C,(k) is the gas-phase concentration, C,[k, E(k)] and b[k, E(k)] are the adsorbed-phase concentration and the affinity for adsorbate k for a given site of energy E(k), respectively, the maximum adsorbed-phase concentration is C,,(k), and the affinity parameter b[k, E(k)] is correlated to the interaction energy by the following equation : b[k, E(k)] = b,(k) exp
with b. being the affinity at the zero-energy level or at the infinite temperature.
176
Gas Separation
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1994 Vol 8 No 3
Let F[k, E(k)] denote the energy distribution in the adsorbed phase; then the macroscopically observed isotherm at a given set of gas-phase concentrations (C,(k); k = 1, NC) is
sm E
bCk-WI C,(k)
C,,,,,(k)= Cpdk)
’ 1+
bC.L WW,(j)
j=l
x FCk,WI
Wk)
(3)
In this article a uniform energy distribution is used: 1
f’k -WI
=
forEmin
< E < -kAk)
E,,,(k) - Emi” (4)
and F[k, E(k)] = 0 elsewhere. If the ordering of the energy sites from low to high is assumed to be the same for different adsorbates and the cumulative energy distribution function is also considered to be the same for all species in the mixture 11,the correlation of energies of different species for a uniform energy distribution is I2
E(j) - Li,(j) E(i)- Edi) E,,,(i) - Edi) = 4,,,,(j) - -kin(i)
(5)
Because the extended Langmuir equation has been chosen to describe the local adsorption isotherm (Equation (1)) it is necessary for different species to have the same maximum adsorption capacities in order to satisfy the thermodynamic consistency. The local diffusion
flux of the adsorbed species
Because the adsorbed species inside the microparticle is not in equilibrium with the gas species, the calculation of the local diffusion flux of the adsorbed species is achieved with the aid of the imaginary gas-phase concentration concept’. For a given set of adsorbed concentrations in the micropore, the imaginary gasphase concentrations which are in equilibrium with that set can be evaluated from any multicomponent equilibrium theory. These imaginary gas-phase concentrations are then used to compute the chemical potential inside the microparticle. Knowing this, the local diffusion fluxes of the adsorbed species k along the particle and microparticle coordinates J,[k, E(k)]r and J,Ck, E(W, , are evaluated from
CpCkE(k)1acid
J&t EWI, = - D,Ck E(k)1 J,[k, E(k)],&, = -jP(k)D,[k,
E(k)] c,,,,,,
(6)
ar
C;,(k) Irn
2@)
P
(7)
where r and rlc are the coordinates of the particle and microparticle, respectively, with their diffusion lengths being R and R,, Ci, is the imaginary gas-phase concentration, and D,[k, E(k)] is the zero-coverage diffusivity of the adsorbed species k, which has the
Ternary adsorption kinetics of gases in activated carbon: X. Hu et al.
following temperature
written:
dependency:
D,[k, E(k)] = D,,(k)exp(
- y)
%4
where a is the ratio of the surface activation energy to the heat of adsorption, and D,,(k) is the zero-coverage diffusivity of adsorbed species k at the zero-energy level in the particle coordinate. Besides the coordinates I and r@, Equation (7) differs from (6) by a factor of p’. The reason for this difference is that the zero-coverage surface diffusivity with zero energy, D,,, might be different in the grain coordinate from that in the particle coordinate because of the different tortuosities in different directions. Since different species with different molecular sizes can access the micropores with different degrees due to the micropore distributions, the tortuosity parameter B2 can be different for different components’.
qsk) at
(1 - %d
+
C,Ck,EtWCk,
J
E(k)1dE(k) I$ dr,
$ dr,
0
-
eY f
$ [r-V,(k)]
- (1 - Q,,); $
J,Ck JYW’Ck E(k)1 dE(k) c dr,
s
(13)
&
e dr, 1
0
Mass balance equations
with its boundary conditions as:
We can now write the mass balance equation in the microparticle using the diffusion flux of the adsorbed species obtained in the last section and the parallel-path model (PPM) of Kapoor and Yang”:
i,
a
J,Ck (sm
qtar, e
W)l,,,FCk Et41 Wk)
>
(9)
where s,, is the geometric factor of the microparticle ( = 0, 1 and 2 for slab, cylinder and sphere, respectively). The local adsorbed concentrations inside the microparticle are written in terms of the imaginary gas-phase concentration:
bCkE(k)1 Cimtk) 1+
0
(14)
R;
R,,
0
C,Ck WI = C,,(k)
r =
%(k) _ ar
+J,tk) + (1 - s,I)
J C,t-k W)lFCk, E(k)1Wk)
=-__i
r = 0;
F Nj, -WW&)
X
m
I (s 0
0
J,Ck EtWTk E(k)1d.W) e dr, >
s RN
rS” w dr P
0
=
k(k)CC,(k)
-
W41
(15)
The initial conditions of the system are: t = 0; C,(k) = CPi(k);
(10)
bCkEtk)lCpitk)
C,CkEt41= C,,(k) 1+
j=l
(16)
“c” KA EtjW,itj)
j=l
One of the boundary conditions of Equation (9) is that the flux at the centre of the microparticle is zero: rp = 0;
aC,t-kWI ar, =
o
(11) Experimental
Another boundary condition is the equilibrium relationship at the exterior surface of the microparticle: r,, = R,; m C,Ck, s0
=
s
WW’Ck WI Wk)
O” C,,(k) 0
1+
W, -WI C,(k)
F Ni j= 1
The above model equations are solved numerically after casting into appropriate non-dimensional form8*r3.
f’Ck E(k)1Wk) (12)
W)lC,(j)
Let s denote the particle shape factor, with a value of 0, 1 or 2 for slab, cylinder or sphere, respectively, Ed the macropore porosity, and Jp the pore diffusion flux, the total mass balance equation for the particle can be
Equilibrium analysis of a single-component
gas
Activated carbon Type 967 supplied by Ajax Chemical Co., Australia as 1.8 mm diameter cylindrical extrudate was used as the absorbent. The structural properties and the pore size distribution of Ajax activated carbon can be found in ref. 14. The absorbates used in this study were pure ethane, propane and n-butane in gas cylinders supplied by Commonwealth Industrial Gases in Brisbane. A high-accuracy volumetric adsorption isotherm rig (Figure I) was used to measure the single-component equilibrium of gases in Ajax activated carbon. The isotherm measurement rig has two separate chambers. One has a supply bomb and the other contains an adsorption cell. Each chamber has a separate MKS transducer (maximum range 1000 mmHg) to record the
Gas Separation
& Purification
1994 Vol 8 No 3
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Ternary adsorption
kinetics of gases in activated
carbon: X. Hu et al.
150 cc Supply Bomb
Needle Valves Toggle Valve
Alcatel Molecular Drag Pump
HBBiee female
-
female coupliq
male to NPT VCR tee
L
VCR cross
+ Water Bath
1
ewaqlok tee
7
J
El-
I t
Activated Carbon
Figure 1
Schematic
diagram of volumetric
adsorption
measurement
pressure. l/4” stainless steel tubes and VCR fittings supplied by CAJON Co., Macedonia, Ohio, USA are used in the system. A l/2” glass tube, which has a round bottom, is used as the adsorption cell and is connected to the system via a CAJON reducing adaptor (l/2” to l/4”). The supply bomb section has a total volume of 159.0 cm3 and the sample cell section has a total volume of 38.4 cm3. A K-type thermocouple is mounted inside the supply bomb to measure its temperature. Another K-type thermocouple is installed in the glass sample cell and is surrounded by activated carbon to record the temperature of the adsorbent. Pure gas cylinders are connected via needle valves, and CAJON toggle valves are used to isolate the adsorbate. Before an equilibrium experiment starts, an Alcatel molecular drag pump is used to evacuate the system to 10d3 mmHg. The operation procedure is described below. Ajax activated carbon was crushed to about 0.1 mm and dried in an oven at 200°C for three hours to remove excess moisture. The carbon was then weighed (0.603 g was used) and loaded into the adsorption cell. The whole system was first evacuated to vacuum using an Alcatel molecular drag pump and the activated carbon particles in the adsorption cell were heated up to 300°C and kept at this temperature and under vacuum overnight to degas any possible adsorbed species. Pure gas (ethane, propane or n-butane) was then dosed into the supply bomb and the pressure and temperature were recorded when constant readings were reached.
178
Gas Separation
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Vol 8 No 3
Thermal couple
rig
A small amount of pure gas was dosed from the supply bomb into the adsorption cell which was kept isothermal by using a water bath. After the pressure in the adsorption cell becomes constant (typically about one to two hours), the pressures and temperatures in both supply bomb and adsorption cell were recorded. The amount adsorbed which was in equilibrium with the pressure in the gas phase of the cell was calculated from the amount supplied from the supply bomb via the ideal P-V-T relationship. Steps 4 and 5 were repeated to get the equilibrium isotherm data at higher pressures until a full isotherm curve was obtained. Kinetics measurement A DAB apparatus was used to collect the dynamic responses of the adsorption process of the gases in activated carbon. As seen in Figure 2, the DAB rig has four major elements: gas mixing system, adsorption bed, sample bomb and gas analysis section. Rotameters (R-l to R-6) supplied by Porter Instrument Company are used to control the flow. The size of the glass tube and the type of float inside the tube are chosen according to the flow rate desired. The flow rate of a rotameter is calibrated by using a bubble soap meter. Needle valves (V-l to V-6) are used to provide the on/off function. Porter pressure regulators (PR-1 to PR-6) are mounted before the rotameters to keep a constant head pressure of 100 kPa. The gases are mixed after the rotameters and the concentrations of adsorbates are calculated from the flow
Ternary adsorption
kinetics of gases in activated
carbon: X. Hu et al.
GAS MIXING
Water / Bath
-l
I I
I -- 4-J
I
ADSFEFON
I
I
1
s
NEEDLE VALVE rzl
GAS CHROYATOGRAPH AND INTEGRATOR
1 PR-1
( PR-2
( PR-3
( PR-4
( PR-5
TWV-4
NEEDLE VALVE
OVEN AND SAMPLING VALVE
1 PR-6
TWV-3
VACUUY
I
\
GAS ANALYSIS
NitrogenNitrugen Ethane Figure
2
Schematic
Propane n-Ehtaoe
SECTION
diagram of differential adsorption bed
rates and their concentrations in the supply cylinders. A four-way valve (FWV-1) is used to send either the mixed adsorbates or the purging gas (nitrogen) to a three-way valve (TWV-1). This three-way valve controls the connection either to the adsorber for adsorption or to the GC for calibration. l/8” stainless steel tubes are used in all gas lines to minimize the dead space. The adsorber section consists of a four-way valve (FWV-2), the adsorber and a three-way valve (TWV-2). The four-way valve serves for the isolation of the adsorber from the adsorbate gas mixture. To ensure good mixing of the gases, the mixed gas is bypassed to vent for a few minutes before it is switched to the adsorber. A water bath is used to keep the temperature of the adsorber constant. The isothermality of the system is maintained by using a high flow rate of the adsorbates. This is confirmed by inserting a thermocouple inside the adsorber and monitoring its temperature reading. The adsorber is constructed by modifying a 314” stainless steel CAJON plug, of which the interior is machined out to create an open chamber to host the adsorbent. A 100 pm mesh is built into a stainless steel gasket and is used to seal the adsorber before a CAJON cap is screwed onto the plug. l/S” stainless steel tubes are silver-soldered to the plug and cap as inlet and outlet gas lines which are then connected to the four-way and three-way valves (FWV-2 and TWV-2) via l/4” VCO o-ring fittings. The sample bomb is built from a 316 stainless steel hollow bar which has a depth of 40 mm and an insider
diameter of 55 mm. The hollow bar is sealed with stainless steel plate at the bottom and flanged with a viton o-ring at the top. The sample bomb section has an effective volume of 108.1 cm3. A barksdale series 302 pressure transducer is used to measure the bomb pressure. The millivolt output of the barksdale transducer is calibrated against an MKS transducer and the response of the barksdale transducer is found to be linear. The temperature of the sample bomb is measured by mounting a l/8” K-type thermocouple inside the bomb. The sample bomb is then connected to either the sample system of the GC or the vacuum pump through a three-way valve (TWV-3). The contents of the gas sample are analysed using a Shimadzu FID gas chromatograph (type GC-8A). A l/16” stainless steel tube is used to connect the gas sample and the six-port stainless steel Valco sampling valve attached to the GC. The gas sample from either the sample bomb or the calibration line is directed to the sample loop via a three-way valve (TWV-4). The gas from the outlet of the sample valve is vented to the atmosphere via a water trap, which ensures a constant pressure in the sample loop at zero flow. A DURAPAK column packed in l/8” tubing is used to separate ethane, propane and n-butane, with nitrogen as the carrier gas. The GC is operated isothermally at 100°C and the detector responses are integrated using a Hewlett Packard 3394A integrator. Ajax activated carbon particles of different sizes and shapes were prepared before the kinetic experiments. The
Gas Separation
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1994 Vol 8 No 3
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Ternary adsorption
kinetics of gases in activated
carbon: X. Hu et al.
1.8 mm diameter cylinders were obtained by simply sealing the end surfaces of the supplied extrudates with epoxy resin, hence only the cylindrical surface of the particle was exposed to gases. For slab particles, the cylindrical surface was coated with epoxy leaving only the end surfaces exposed to gases for adsorption. The end surfaces of the slab particles were carefully trimmed to obtained a desired length. The operation procedure of the DAB is stated below. Slab or cylindrical particles of Ajax activated carbon were loaded to the absorber. About 0.1 g of particles was used in the dynamic studies. The adsorption bed was initially heated up to a high temperature (200°C) under vacuum overnight and then degassed by purging the heated bed with a small flow of pure nitrogen for 15 min. The bed was brought to the adsorption temperature using a water bath and the adsorbates were flushed through the bed at a flow rate of 50@1500 cm3 min-’ for a predetermined adsorption time before the adsorber was isolated. The sample was desorbed to a pre-evacuated bomb by heating up the adsorption bed to 200°C and remaining at this temperature for 40 min. At the same time the four-way valve (FWV-1) was switched to rotameter 2 and pure nitrogen was flushed through the gas line between FWV-1 and FWV-2 to vent in order to clean any adsorbates remaining in the gas line. After the gas line was cleaned, FWV-2 was switched to the adsorber and pure nitrogen was slowly flushed through the adsorber (around 10 cm3 min-‘) for 1615 min. The pressure and temperature of the bomb were measured and recorded and then the adsorbate concentrations in the bomb were analysed by gas chromatography. The total amount in the sample bomb was calculated from its temperature, volume and pressure via the ideal P-V-T relationship. This total amount includes the adsorbed amount inside the particles and those adsorbates trapped in the voids of the adsorber at isolation (section between FWV-2 and TWV-2). The adsorbed amount was the result of the product of the total amount in the bomb and the adsorbate mole fraction obtained from GC analysis minus the dead volume contribution. By repeating steps 3-6 for different exposure times an uptake curve can be obtained for the specified experimental conditions. In order to accurately measure the dead volume contribution of the adsorbates trapped in the voids of the adsorber at isolation (section between FWV-2 and TWV-2) an empty cell having the same internal dimensions was used to substitute the adsorber. The mixed adsorbates were then flushed through the empty cell under the same operation conditions (concentration, temperature and flow rate) for some time and isolated. The amounts of different species inside the empty cell were then passed to the pre-evacuated sample bomb and further flushed by purging pure nitrogen to bring the pressure of the sample bomb to the same value as in the adsorption kinetics. The gas mole fractions were analysed by the GC. The dead volume contribution was then calculated as the product of the total amount in the sample bomb (via the P-V-T relationship) and the molar fraction.
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9 The GC responses were calibrated once daily. In the
calibrations, nitrogen and adsorbate (ethane, propane or n-butane) were mixed at different flow rates via rotameters R-3 and R-4/R-5/R-6 and then passed to the sampling valve of the GC for analysis. The mole fraction of gases was calculated from their relative flow rates. The mole fractions were then plotted versus the integrated area of the detector responses. A linear relationship was observed experimentally. Results
and discussion
First, the single-component equilibrium of the gases in Ajax activated carbon is analysed. Because the gas concentrations used in our dynamic studies were always less than 30 kPa, only those equilibrium data having a gas pressure less than 35 kPa were used in the determination of isotherm parameters. All experimental data for three temperatures (10, 30 and 60°C) were used simultaneously in the fitting with the Langmuir-Uniform distribution equation (Unilan). A least squares method was employed in the fitting process. The maximum adsorption capacity (C,,) was set to be the same for all three species in order for the local multicomponent Langmuir equation to satisfy the thermodynamic consistency. This parameter, however, is assumed to be temperature dependent. The other parameters in the isotherm equation, b,, Emin and E,,,, are temperature independent but depend upon the adsorbate. The experimental results (symbols) and the model fittings (solid lines) of the adsorption equilibria of ethane, propane and n-butane in Ajax activated carbon are shown in Figure 3. The fitting between the model and the experimental data is excellent in all cases. The extracted equilibrium parameters are tabulated in Table I.
Second, the single-component adsorption kinetic experiments were carried out and the uptake data were used to extract the dynamic parameters (D,, and R,//?) for the single component. The experimental results for ethane and propane in Ajax activated carbon are available in ref. 13 for different temperatures, particle sizes and bulk concentration combinations. The flow rate used for ethane and propane was 500 cm3 min- ‘, which was found to be adequate to provide a differential condition in the cell containing about 0.2 g activated carbon having a slab geometry with a full length of 2.4 mm. The adsorption dynamics of n-butane in Ajax activated carbon were measured by a similar procedure but a higher flow rate of 1000 cm3 min-’ was required to maintain the differential condition. The reason for this is that the adsorption equilibrium isotherm of butane is highly favourable (Figure 3c), implying that more gas will be adsorbed from the bulk environment. Hence a higher molar rate of supply of n-butane is required to maintain the differential condition. For a 1.8 mm diameter cylindrical particle, the adsorption kinetics are very fast; hence the experiments with ethane and propane in such a particle were carried out using a higher gas flow rate of 1500 cm3 min - 1 to dissipate any heat generated from adsorption. The single-component experimental adsorption data (symbols) of ethane, propane and n-butane in Ajax activated carbon are plotted in Figure 4 for different bulk concentrations and particle sizes and shapes. The adsorption kinetics were carried out at a constant temperature of 30°C and the total pressure was 1 atm.
Ternary adsorption kinetics of gases in activated carbon: X. Hu et al. I
I
I
1
-l
a v
3o”c 60°C
20
I
I
I
Figure 3 Adsorption equilibrium isotherm of gases in Ajax activated carbon: (a) ethane; (b) propane; (c) n-butane
‘able 1
These experimental results were then used to extract the dynamic parameters of the HMSMD model. The ratio of the surface activation energy to the heat of adsorption, a, was taken as 0.5 for all three species. The pore diffusivity was computed from the combined molecular and Knudsen diffusivitiesi5 and a tortuosity of three 13. The shape of the microparticle was assumed to be spherical (sll = 2) although other geometries can also be used. Therefore, the fitting parameters in the HMSMD model are the zero-coverage surface diffusivity at zero-energy level (D,c) and the effective diffusion path length in the microparticle (RJB). The model fittings are shown in Rgure 4 as solid lines. It is seen in this figure that the fitting between the model and the experimental data is very good for all cases tested. The kinetic parameters are tabulated in Table 2. The equilibrium and kinetic parameters obtained for the single-component systems were then used in the HMSMD model to predict the ternary adsorption kinetics of ethane, propane and n-butane in Ajax activated carbon under various conditions. The gas flow rate through the DAB was 1500 cm3 min-’ in all ternary kinetic experiments. Figure 5 shows the adsorption kinetics of 10% ethane, 10% propane and 10% n-butane in Ajax activated carbon of 2.4 mm full-length slabs at 30°C and 1 atm. Because n-butane is the slowest moving and most strongly adsorbing species, the other two species, ethane and propane, exhibit an overshoot phenomenon in their fractional uptake before their final equilibrium is reached. The degree of overshoot is about 5.5 for ethane and 2.5 for propane. The HMSMD model predicts well the uptake and fractional curves of ethane, including its degree of overshoot and the time at which the overshoot occurs. Because the adsorption affinity of ethane in Ajax activated carbon is much weaker compared with that of propane and n-butane, the adsorbed amount of ethane at final equilibrium is so small that its amount in the void space of the adsorber and the tubing system (between FWV-2 and TWV-2 in Figure 2) can make a significant contribution to the measurement of the total amount of ethane in the sample bomb. Therefore, a small error in the measurement of the contribution from the void space might generate a large error in the fractional uptake of ethane. In Figure 5, the experimental data of ethane have an error of about 10% in the maximum fractional uptake, which corresponds to a 3% error in the void space concentration measurement. In the ternary adsorption dynamic studies, the measured amount of ethane in the void space of the adsorber is about three times the final adsorbed amount of ethane inside the activated carbon.
Isotherm parameters of ethane, propane and n-butane in Ajax activated carbon
Species
FC)
$noI
Ethane
10 30 60 10 30 60 10 30 60
4.75 4.47 4.11 4.75 4.47 4.11 4.75 4.47 4.11
Propane
n-Butane
mm3)
b, (kPa-‘)
Emi, (kJ mol-‘)
6.11 x lo-’
16.7
27.1
4.17 x 1O-6
21 .3
34.6
5.60 x 1O-5
15.8
34.9
Gas Separation
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1994 Vol 8 No 3
181
Ternary adsorption
kinetics of gases in activated
carbon: X. Hu et al.
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2 4.4
mm
2.6
slab
0.0
mm
slab
0.0
1.0 3
0.8
0.8
2
0.6
0.6
0.4
0.4
2
0.2 4.4
mm
slab
0.0
k
O
2000
1000
0
400
0
100
800
1.0 0.8 0.6 0.4 0.2 0.0
0
500
1000 1500
Time Figure 4
(seconds)
Adsorption dynamics of gases in Ajax activated carbon
That is, the ethane adsorbed concentration inside the particle is only about 25% of the total amount of ethane in the sample bomb. However, the adsorbed amount of ethane is so small that a 10% error of the maximum fractional uptake of ethane becomes negligible when it is plotted as the real adsorbed amount. In Figure 5 it is seen that the agreement between the model predictions and the experimental data is excellent for n-butane uptake and fractional uptake. Some deviations are observed for propane but the model describes its kinetic
182
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behaviour well. The error in the overshoot degree of propane is due to the incorrect prediction of ternary equilibrium for propane. The heterogeneous extended Langmuir equation gives a higher value for the amount of propane adsorbed than the experimental data, even at the final steady state, which in turn results in a prediction of lower uptake overshoot for propane. This raises the importance of accurate measurement and calculation of the single- and multicomponent adsorption equilibrium isotherms.
Ternary adsorption kinetics of gases in activated carbon: X. Hu et al. Table 2 Kinetic parameters of ethane, propane and n-butane in Ajax activated carbon
Species
10%
0
5%
v
5% c,
model
Ethane
10 30 60 10 30 60 10 30 60
Propane
n-Butane
4.03 4.48 5.23 3.20 3.47 4.16 2.75 3.04 3.57
c, c,
0
1 .oo
0.863
3.36
10.2
8.42
5.75
B a = 0.5 bsr=2
%
6
-
_
2.0
OE 1
1.5
; ti -
model
1.0
0 0
500
Time
1500
1000
1000
500
1500
(seconds)
Figure 6 Ternary adsorption kinetics of ethane, propane and n-butane in Ajax activated carbon of 2.4 mm full-length slabs at 30°C and 1 atm
G-E \ z
2.0 0
0 v
1.5 -
E 2
10% c, 10% c, 10% c, model
1.0 0
5%
c,
a 5%c, v 5%c, __ model 0
500
Time
lob0
1500
(seconds)
Figure 5 Ternary adsorption kinetics of ethane, propane and n-outane m Ajax activated carbon of 2.4 mm full-length slabs at 30°C and 1 atm
Another experiment with a different bulk concentration combination of 10% ethane, 5% propane and 5% n-butane is shown in Figure 6. Since the concentrations of propane and n-butane here are lower than those in Figure 5, the faster-adsorbed ethane displaced by the later coming propane and n-butane in the final equilibrium is less, so that we have a lower overshoot degree of ethane uptake of about 4.3. The HMSMD
0
600
200
400
Time
(seconds)
800
Figure 7 Ternary adsorption kinetics of ethane, propane and n-butane in Ajax activated carbon of 2.4 mm full-length slabs at 30°C and 1 atm
Gas Separation
& Purification 1994 Vol 8 No 3
183
Ternary adsorption
kinetics
of gases in activated
-
0
200
model
400
Time
carbon: X. Hu et al.
600
(seconds)
Figure 8 Ternary adsorption kinetics of ethane, propane and n-butane in Ajax activated carbon of 2.4 mm full-length slabs at 30°C and 1 atm
-
500
0
-
model
1500
1000
10%
0
5% c,
-
Time
1000
1500
Gas Separation
& Purification
1994
c,
0
10%
0
5% c,
v
5% c,
-
0
200
Time
(seconds)
Figure 9 Ternary adsorption kinetics of ethane, propane and n-butane in Ajax activated carbon of 2.4 mm full-length slabs at 60°C and 1 atm
184
300
model
_
-.500
model
200
-
model
-
100
0
c,
0
05%C,
0
model predictions are again in very good agreement with the experimental data for ethane and n-butane. The model reasonably predicts the uptake rate of propane with some deviations in the degree of its overshoot. This underprediction of the overshoot degree is again due to the overprediction of the final adsorbed amount of propane at equilibrium. The ternary adsorption fractional uptakes in Ajax activated carbon of 2.4 mm full-length slabs at 30°C are measured for two other bulk concentration combinations, 5% for all species (Figure 7) and 5% ethane, 10% propane, 5% n-butane (Figure 8). Again the agreement between the model predictions and the data is within experimental error for ethane and n-butane, with some deviations for propane uptake. Comparing Figures 6 and 7, one comes to the conclusion that a decrease in ethane bulk concentration will yield a higher overshoot degree of ethane fractional uptake. The reason for this is that the percentage amount of adsorbed ethane displaced by the late penetrating propane and n-butane at final equilibrium is higher when the ethane bulk concentration is lower. The ternary adsorption kinetics of 10% ethane, 5% propane and 5% n-butane at a higher temperature, 60°C is shown in Figure 9. The same particle (2.4 mm slab) and total pressure (1 atm) as in Figures 5-8 are used here. Since the pore diffusivity is calculated from combined Knudsen and molecular diffusivities and the
Vol 8 No 3
400
600
800
1000
(seconds)
Figure 10 Ternary adsorption kinetics of ethane, propane and n-butane in Ajax activated carbon of 1.8 mm diameter cylinders at 30°C and 1 atm
Ternary
E
1.0
-
model
~
model
adsorption
kinetics
of gases
in activated
carbon:
-
b-4
X. t-/u et al.
model
_
-
0
200
Time
400
600
800
1000
0
(seconds)
Figure 11 Ternary adsorption kinetics of ethane, propane and n-butane in Ajax activated carbon of 1.8 mm diameter cylinders at 60°C and 1 atm
zero-coverage surface diffusivity at zero-energy level (Dko) is independent of temperature, the HMSMD model can predict the dynamics at different temperatures without any extra fitting parameter. The agreement between the model and the experimental data, including the final equilibrium, is reasonably good, close to the experimental error. The overshoot degree of ethane fractional uptake is lower at 60°C than at 30°C (Figure 6). This is expected because the adsorption energy of propane and n-butane is higher than that of ethane (Table I). Therefore, the adsorption affinity and surface diffusivity of propane and n-butane are more sensitive to temperature than those of ethane. As a result the difference in the adsorption affinity and surface diffusivity between different species is narrowed when the temperature is higher. Having studied the ternary adsorption kinetics for different concentration combinations and temperatures, we now explore the ternary adsorption dynamics in Ajax activated carbon of different shapes and sizes. The adsorption kinetics of 10% ethane, 5% propane and 5% n-butane in 1.8 mm diameter cylindrical Ajax activated carbon are shown in Figure 10 for an adsorption temperature of 30°C and in Figure II for 60°C. The model predictions are generally in good agreement with the experimental data, but the isotherm equation gives a higher prediction of the adsorbed amount of propane. Again the overshoot degree of ethane fractional uptake is found to be lower when the temperature is higher.
1000
2000
3000
Time
(seconds)
4000
Figure 12 Ternary adsorption kinetics of ethane, propane and n-butane in Ajax activated carbon of 4.2 mm full-length slabs at 30°C and 1 atm
Finally, the ternary adsorption kinetics in a larger slab of Ajax activated carbon having a full length of 4.2 mm is shown in Figure 12. The concentration is 10% for all three species and the adsorption process is carried out at a temperature of 30°C and a total pressure of 1 atm. The agreement between the model and the experimental data is reasonable for ethane and n-butane. The discrepancy for propane uptake is still due to the improper prediction of the propane adsorbed amount.
Conclusions Ternary adsorption kinetics data for ethane, propane and n-butane in activated carbon are collected via a DAB under different operating conditions, such as bulk concentration combinations, temperatures, particle sizes and shapes. Because n-butane is the slowest diffusing and most strongly adsorbed species, both ethane and propane exhibit a maximum in their uptakes during the simultaneous adsorption process. The overshoot degree of ethane uptake is found to decrease with an increase in ethane bulk concentration, or with a decrease in propane and n-butane concentrations. A higher adsorption temperature is also found to reduce the overshoot degree of ethane uptake. The HMSMD model proposed by Hu and Do’ is used to predict the ternary adsorption kinetics by using only the information from singlecomponent mass transfer and adsorption equilibrium
Gas Separation
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Ternary adsorption kinetics of gases in activated carbon: X. Hu et al.
parameters. The agreement between the model and the experimental data is found to be good in most cases studied for ethane and n-butane uptake. The model can also predict well the adsorption rate of propane but fails to predict its overshoot degree accurately because of the slightly higher prediction of the propane equilibrium adsorbed amount. This demonstrates the potential of the HMSMD model as a mathematical tool in the prediction of multicomponent adsorption kinetics of gases in activated carbon.
7
8
9
Acknowledgement
10
Financial support from the Australian Research Council and the REGS of the University of Queensland is gratefully acknowledged.
11
12
13
References Ma, Y.H. and Roux, A.J. Multicomponent rates of sorption of SO, and CO, in sodium mordenite AIChE J (1973) 19 1055-1059 Ma, Y.H. and Lee, T.Y. Diffusion of binary gas mixtures in zeolite X pellets Ind Eng Chem Fundam (1977) 16 4448 Carlson, N.W. and Dranoff, J.S. Competitive adsorption of methane and ethane on 4A zeolite Fundamentals ofAdsorption (Ed A.L. Liapis) Engineering Foundation, New York (1987)
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14
15
Qureshi, W.R. and Wei, J. One- and two-component diffusion in zeolite ZSM-5, I. Theory J Catal (1990) 126 126-146 Qureshi, W.R. and Wei, J. One- and two-component diffusion in zeolite ZSM-5, II. Experimental J Catal (1990) 126 147-172 Yang, R.T., Chen, Y.D. and Yeh, Y.T. Prediction of cross-term coefficients in binary diffusion: diffusion in zeolite Chem Eng Sci (1991) 46 3089-3099 Hu, X., Rao, G.N. and Do, D.D. Multicomponent sorption kinetics of ethane and propane in activated carbon: simultaneous adsorntion Gas Sea Purif (1993) 7 3945 Hu, -X. and D6, DID: Role of energy distribution in multicomponent sorption kinetics in bidispersed solids AIChE .I (1993) 39 1628-1640 Hu, X. and Do, D.D. Multicomponent adsorption kinetics of hydrocarbons onto activated carbon: contribution of micropore resistance Chem Eng Sci (1993) 48 1317-1323 Kapoor, A. and Yang, R.T. Surface diffusion on energetically heterogeneous surfaces AIChE .I (1989) 35 1735-1738 Valenzuela, D.P., Myers, A.L., Tab, 0. and Zwiebel, 1. Adsorption of gas mixtures: effect of energetic heterogeneity ArChi J (1988) -M 397402 Kauoor, A., Ritter, J.A. and Yang. R.T. An extended Lanumuir _ model for adsorption of gas mixtures on heterogeneous surfaces Lungmuir (1990) 6 66@664 Hu, X., Rao, G.N. and Do, D.D. Effect of energy distribution on sorption kinetics in bidispersed particles AIChE J (1993) 39 249-261 Gray, P.G. and Do, D.D. Adsorption and desorption of gaseous sorbates on a bidispersed particle with Freundlich isotherm. II. Experimental study of sulphur dioxide sorption on activated carbon particles Gas Sep Purif(1989) 3 201-208 Reid, R.C., Prausnitz, J.M. and Polling, B.E. The Properties of Gases and Liquids 4th Edn, McGraw-Hill, New York (1987) I
I
_,