Single component and competitive adsorption of propane, carbon dioxide and butane on Vycor glass

Chemical Engineering Science 63 (2008) 1586–1601www.elsevier.com/locate/ces

Single component and competitive adsorption of propane, carbon dioxideand butane on Vycor glass

Jirina Reznícková Cermákováa, Ana Markovicb,∗, Petr Uchytila, Andreas Seidel-Morgensternb,c,1

aInstitute of Chemical Process Fundamentals, Academy of Sciences of the Czech Republic, 16502 Prague, Czech RepublicbMax-Planck Institute for Dynamics of Complex Technical Systems, D-39106 Magdeburg, Germany

cOtto-von-Guericke-University Magdeburg, Faculty for Process and Systems Engineering, D-39016 Magdeburg, Germany

Received 6 July 2007; received in revised form 23 October 2007; accepted 14 November 2007Available online 19 November 2007

Abstract

Equilibrium of gas phase adsorption on Vycor glass has been investigated. Adsorption isotherms for propane, carbon dioxide and butaneas pure gases, binary mixtures and ternary mixtures were determined experimentally as a function of temperature using a volumetric method.The single-component isotherms were described with the Langmuir and Freundlich equations. Additionally, a second order isotherm based onstatistical thermodynamics and an isotherm equation based on vacancy solution theory taking into account real phase behavior were used forfitting single-component equilibrium data. In order to describe the measured partial isotherms for binary mixtures, at first simple extensions ofthe single-component isotherm models were used, i.e., the conventional competitive Langmuir model and a multi-Freundlich equation based onthe ideal adsorbed solution theory (IAS). Since these two simple isotherm models failed to represent the unusual competitive behavior observed,three model extensions using additional mixture parameters were applied, i.e., two modified multi-Langmuir equations based on: (a) statisticalthermodynamics and (b) vacancy solution theory and a modified multi-Freundlich IAS model correcting spreading pressure uncertainties. Thesethree model equations were found to be capable to describe the observed behavior better. Finally, the measured partial adsorption equilibriumdata of the ternary system were correlated based on the extended equations using the determined additional binary parameters. The resultsobtained reveal the difficulty to predict accurately multi-component adsorption equilibria.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Porous media; Gases; Adsorption; Single-component isotherms; Multi-component isotherms; Parameter estimation

1. Introduction

In the past decade an increasing number of papers dealingwith microporous glass membranes has been published (e.g. inFOA7, 2001; Enke et al., 2003, in FOA8, 2004). This interest isdue to the fact that these membranes show good thermal stabil-ity, chemical resistance and considerable selectivity. Addition-ally, these materials possess significant adsorption capacities.

The porous glass named Vycor (Elmer, 1992) has frequentlybeen an object for conducting instructive studies on gas

∗ Corresponding author.E-mail addresses: [email protected] (A. Markovic),

[email protected](A. Seidel-Morgenstern).

1 Also for correspondence. Max-Planck Institute for Dynamics of ComplexTechnical Systems, D-39106 Magdeburg, Germany.

0009-2509/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2007.11.021

diffusion (Uchytil et al., 2005; Schlünder et al., 2006; Kazuhiroet al., 2006). The preparation of porous glass membranes wasfirst described by Hood and Nordberg (1938). The glass com-position and structure is a result of the removal of an ini-tially present soluble phase by acid leaching. Various averagepore diameters are accessible ranging from 2 to 200 nm. Thus,various transport mechanisms contribute to the overall trans-port, e.g. Knudsen diffusion, molecular diffusion, viscous flow,capillary condensation and surface diffusion. In most casestransport phenomena are simultaneously accompanied by gasadsorption.

Therefore, to study and understand the transport in thesematerials, it is a necessary prerequisite to quantify the equilibriaof adsorbing species.

Adsorption isotherms of several gases (e.g. CO2, H2O,C2H4, CH4, N2, C2H5OH and CH2Cl2) were already investi-gated for porous glasses (Bhandarkar et al., 1992). Often, the

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J.R. Cermáková et al. / Chemical Engineering Science 63 (2008) 1586–1601 1587

adsorption isotherm parameters of the Dubinin–Radushkevichequation and the Langmuir model were determined. Adsorp-tion isotherms of ethylene, propylene and isobutene weredetermined at 298 K on Vycor glass cylindrical pellet with anaverage pore diameter of 8.9 nm (Gilliland, 1972) and quan-tified with the Langmuir isotherm model. Adsorption effectswere observed even for so called inert gases (Hwang andKammermeyer, 1975; Shindo et al., 1983; Chang andLee, 2004). Experimental data for carbon dioxide, propaneand butane adsorption on Vycor glass were given forsingle-components and for binary mixtures C3H8.CO2 andC3H8.C4H10 at 298 and 313 K (Tuchlenski et al., 1998; Yanget al., 2005). The measured single-component isotherms werewell described by the Langmuir equation applying componentspecific maximum loading capacities. However, it was foundthat classical, multi-component Langmuir isotherms couldnot properly describe the competitive adsorption behavior. Adual adsorption model was suggested as possible concept todescribe more complex situations (Kapteijn et al., 2000).

In this work adsorption equilibria of three different singlegases and their binary and ternary mixtures were determined ex-perimentally on a Vycor glass sample, characterized by a meanpore diameter of 4 nm. Different adsorption isotherm modelssuggested in the literature were tested in order to representadsorption equilibria. Before describing the experiments, themain equations and theories are introduced briefly in the nextsection.

2. Theory

2.1. Single-component adsorption isotherm models

To quantify single-component adsorption equilibria, fouroften used adsorption equilibrium models are considered inthis work. The applied isotherm equations are the Langmuirisotherm (L1), a Quadratic isotherm (Q1—which is a sec-ond order truncation of an isotherm predicted by statisticalthermodynamics; Hill, 1962), an adsorption isotherm basedon Vacancy solution theory capable to describe real phasebehavior (L1,real), (Suwanayuen and Danner, 1980a) and theclassical Freundlich isotherm (F1).

The classical Langmuir isotherm (L1) describing the adsorp-tion equilibrium of a component i can be presented in followingforms:

qi = bHipi

1 + bipi

or pi = 1

bi

�i

1 − �i

with

bHi = qsat,ibi and �i = qi

qsat,i. (1)

This equation correlates at constant temperature the surfaceconcentrations or loadings qi in equilibrium with the gas phasepressures pi . It contains a temperature dependent parameter bi

and the limiting saturation loadings qsat,i as free parameters.For simplification, the saturation loading are often considerednot to depend on temperature.

Applying statistical thermodynamics Hill (1962) suggesteda more general equilibrium isotherm as a ratio between poly-nomial functions of same degree n:

qi = qsat,ipiP′(pi)

P (pi)

= qsat,ipi(ai + 2bipi + · · · + nzipn−1i )

1 + aipi + bip2i + · · · + zip

ni

, (2)

where P ′(pi) it the first derivative of the polynomial P(pi).The Langmuir model (Eq. (1)) corresponds to the polynomial

of the first degree. A Quadratic isotherm (Q1) corresponds tothe polynomial of the second degree:

qi = qsat,ipi(ai + 2bipi)

1 + aipi + bip2i

. (3)

This equation contains three instead of two parameters in theLangmuir isotherm. The bi consider additional interactions ofmolecules adsorbed above the monolayer.

An isotherm model (L1,real) taking into account the realphase behavior of the adsorbed phase is based on the va-cancy solution theory developed by Suwanayuen and Danner(1980a). In the single-component model the adsorbed and thegas phases are treated as two vacancy solutions. All interac-tions between adsorbate–vacancy caused by nonidealities of theadsorbed phase are accounted for by activity coefficients.

The general form of the L1,real model for adsorption of purecomponents is given with Eq. (4).

pi =[qsat,i

bHi

�i

1 − �i

]fi(�i ). (4)

The �i are dimensionless loadings (Eq. (1)). The term fi(�i )

represents the non-ideality of the system in the term of theactivity coefficient, as defined e.g. by the Wilson equation(Eq. (5)) (Suwanayuen and Danner, 1980a).

fi(�i ) = �iV

1 − (1 − �V i)�i

�iV + (1 − �iV )�i

× exp

(−�V i(1 − �V i)�i

1 − (1 − �V i)�i

− (1 − �iV )�i

�iV + (1 − �iV )�i

).

(5)

The free parameters �iV and �V i account for adsorbate–vacancyinteractions. The activity coefficients depend on compositionand temperature.

For fi = 1 Eq. (4) is equivalent to the Langmuir isotherm(Eq. 1).

Finally, the Freundlich isotherm (F1) can be described as:

qi = kipni

i or pi =(

qi

ki

)1/ni

, (6)

where ki, ni are generally temperature dependent free pa-rameters. The Freundlich equation is often used to describeadsorption on heterogeneous surfaces when interactions be-tween adsorbent and adsorbate are not equal for each site(Do, 1998). For low partial pressures Eq. (6) does not reduce toHenry’s law.

1588 J.R. Cermáková et al. / Chemical Engineering Science 63 (2008) 1586–1601

2.2. Competitive adsorption in multi-component systems

Below are summarized at first simple extensions of single-component isotherms. Then, several more sophisticated com-petitive isotherm models are given, which contain additionalmixture parameters.

2.2.1. Models based exclusively on single-componentparameters

(a) Multi-Langmuir equation (Lmix,0 model): The well-known extension of the L1 model (Eq. (1)) to mixtures is

qi = qsat,ibipi

1 +∑Nj=1pjbj

or pi = �i/bi(1 −∑N

j=1�j

) with

�i = qi

qsat,ifor i = 1, . . . , N . (7)

Thermodynamic consistency requires all qsat,i to be identical(Broughton, 1948).

(b) Multi-Freundlich equation based on IAS theory (Fmix,0model): The IAS theory (Myers and Prausnitz, 1965) is a widelyused general framework to derive mixture adsorption isothermsfrom available single-component adsorption isotherms, e.g. L1,Q1, F1.

Hereby, the calculation of equilibrium loadings from par-tial pressures usually requires an iteration procedure. Regard-ing this aspect an exception is e.g. the case of single Freudlichisotherms (F1) for which the following analytical solution ex-pressing the equilibrium pressures as a function of the corre-sponding equilibrium loadings holds (Seidel and Gelbin, 1988):

pi = qi∑Nj=1qj

⎛⎝ N∑

j=1

ni

nj

qj

⎞⎠

1/ni

k−1/ni

i for i = 1, . . . , N .

(8)

2.2.2. Models using additional mixture parameters(a) Modified multi-Langmuir isotherm based on statistical

thermodynamics (Lmix,1 model): The isotherm models basedon statistical thermodynamics can be extended to competitiveadsorption (Hill, 1962; Lin et al., 1989; Guiochon et al., 1994;Mihlbachler et al., 2002). The quadratic mixture isotherm fortwo components i and j are based on six single-componentisotherms parameters: qsat,i , qsat,j , ai , aj , bi , bj . In their binarymixture a cross term bij has to be measured with the binarymixtures quantifying interactions between the two components.

qi = qsat,ipi(ai + 2bipi + bijpj )

1 + aipi + ajpj + bip2i + bjp

2j + bijpipj

,

i, j = 1, 2, . . . , N . (9)

The corresponding competitive isotherm for a ternary mixture,based solely on binary interaction parameters is:

qi = qsat,ipi(ai + 2bipi + bijpj + bikpk)

1 + aipi + ajpj + akpk + bip2i + bjp

2j + bkp

2k + bijpipj + bikpipk + bjkpjpk

, i, j, k = 1, 2, 3, . . . , N . (10)

Thus, neglecting ternary interactions (i.e., cijk = 0) the pa-rameters derived from single components and binary mixturesare the only ones needed for the prediction of ternary mixtureisotherms.

(b) Vacancy solution theory (Lmix,2 model): Predictions ofmulti-component equilibrium data from single-component datais also possible using an extension of the vacancy solutiontheory (Suwanayuen and Danner, 1980b).

Hereby, a mixture of N components is treated as anN + 1 component vacancy solution. All interactions betweenadsorbate–vacancy and additionally adsorbate–adsorbate in-teractions caused by nonideality of the adsorbed phase areaccounted for by activity coefficients. The following final equi-librium isotherm for mixtures was suggested (Suwanayuen andDanner, 1980b):

pi = �iziqT

qsat,i

qsat,T

�iV

bHi

exp(�V i − 1)

× exp

(�ai

RT

), i = 1, . . . , N (11)

with:

�ai

RT= −

(1 + qsat,T − qsat,i

qT

)× ln(�V xV ), i = 1, . . . , N , (12)

qsat,T =N∑

i=1

xiqsat,i , qT =N∑

i=1

qi ,

zi = qi

qT

, xi = zi

qT

qsat,T,

xV = 1 −N∑

i=1

xi, i = 1, . . . , N , (13)

ln(�i ) = 1 − ln

⎡⎣N+1∑

j=1

xj�ij

⎤⎦−

N+1∑k=1

[xk�ki∑N+1

j=1 xj�kj

],

i = 1, . . . , N, N + 1 with N + 1 = V . (14)

In the above equation � is the spreading pressure of adsorbedmixture, ai is partial molar surface area, zi is the mole fractionof component i in the vacancy free adsorbed phase, xi is themole fraction of component i in the adsorbed phase vacancysolution. Eq. (14) is the Wilson equation.

Besides the single-component parameters qsat,i , �iV , �V i

for mixtures of N components appear additionally all possiblebinary interactions between adsorbed components, e.g. for a bi-nary mixture �ij and �ji . These parameters must be estimatedfrom experimentally mixture data. If they are equal to 1, thereare no interactions between the adsobates. That means multi-component behavior can be estimated based only on single-component isotherms.


(c) Multi-Freundlich IAS model correcting spreading pres-sure uncertainties (Fmix1 model): As the Freundlich isothermdoes not comprise Henry’s law behavior at low pressure, Seideland Gelbin (1988) and Seidel (1989) suggested a possible wayto treat this deficit in predicting competitive isotherms. Theidea is to combine a linear isotherm model valid in an initialrange up to a certain critical pressure pi,min with the Freundlichisotherm above that pressure. This approach allows with theIAS-theory to estimate for each component a spreading pres-sure correction ��i as follows:

��i =∫ pimin

0

q0truei

p0i

dp0i −

∫ pimin

0

q0Freundlichi

p0i

dp0i ,

i = 1, . . . , N . (15)

In mixtures an arbitrary component can be selected as a ref-erence component (below component 1) with respect to whichthe values of the other components will be shifted. It followsnow at a certain partial pressure p0

i , for the spreading pressurea modified value �∗

i (p0i ) which is used within the IAS calcu-

lation procedure:

�∗i (p

0i ) =

∫ p0i

0

q0i,Freudlich

p0i

dc0i + ��i or

�∗i − ��1 =

∫ p0i

0

q0i,Freundlich

p0i

dc0i + ��i1 (16)

for i = 1, . . . , N with ��i1 = ��i − ��1. (17)

For mixtures of N components there are N−1 parameters ��i1which must be determined from mixture experiments and whichmust satisfy the condition of rotation of indices i.e., ��ij =��i1−��j1. These additional parameters should be applicableto predict the adsorption isotherms of binary mixtures whichdo not contain the reference component, 1.

Based on this concept the corrected competitive FreundlichIAS isotherm (Fmix1) takes finally the following form:

p1 = q1∑Nj=1qi

(q0

1

k1

)1/n1

for the reference component 1, (18)

pi = qi∑Nj=1qj

(ni

n1q0

1 − ��i1ni

)1/ni

k−1/ni

i

for all other components, i �= 1. (19)

The hypothetical loading of the adsorbed reference componentq0

1 has to be determined from:

q1

q01

+N∑

i=2

qi

ni(q01/n1 − ��i1)

= 1. (20)

To apply the model in case of N-component mixture, an equa-tion of Nth order must be solved.

In this paper the corrected multi-Freundlich equation wasapplied for binary and ternary mixtures. Then, the solution of

Eq. (20) requires solving a quadratic or cubic equation, respec-tively. For a binary mixture holds:

q01 = C + (C2 − ��21n1q1)

1/2 where

C = 1

2

(q1 + n1

n2q2 + ��21n1

). (21)

In the case of a ternary mixture holds:

q03

1 − q02

1

(q1 + q2

n1

n2+ q3

n1

n3+ (��31 + ��21)n1

)

+ q01

((��31 + ��21)q1n1 + n2

1

(��31

q2

n2

+ ��21q3

n3+ ��21��31

))

− q1n21��21��31 = 0. (22)

The Newton–Ralphson method is suggested as an effective toolfor determining q0

1 from higher order equations. Finally, thecorresponding equilibrium pressures p1 and pi can be calcu-lated explicitly using Eqs. (18) and (19).

If all correction parameters ��ji are zero, the predictionsbecome equal to the predictions of the general competitive Fre-undlich isotherms based on IAS theory (Eq. (8)).

3. Experiments

3.1. Vycor glass as a sorption material

The characteristics of the Vycor glass used in this studyas supplied in tubular form by Corning Inc. (USA) are listedup in Table 1. This glass is a “moisture adsorbing material”,because water rapidly infiltrates into the pores. For that reason,to remove impurities being adsorbed at the surface, the sampleswere treated initially with 30% H2O2 or concentrated HNO3up to 373 K until the brown color disappeared, then washedin water (3 min was enough for filling all pores with water).Finally, the membrane was dried at 433 K in an inert atmospherefor 3 h. Later, the glass sample was activated by heating undervacuum at 393 K for 3 h, what was sufficient to fully desorb themoisture.

The Vycor glass applied has a narrow pore size distribu-tion with a mean pore diameter of about 4 nm (as measuredby N2 adsorption). In the adsorption equilibrium experimentsthis material was exposed to the pure components propane,

Table 1Structural data of Vycor glass membrane (Corning, code 7930)

Average pore diameter, dp 4 nmApparent density, �ap 1.49 g cm−3

Skeletal density, �s 2.02 g cm−3

Porosity, � 0.284Specific pore volume, Vs 0.17 cm3 g−1

BET specific surface area, As 170 m2 g−1

Thickness of the samplesa, � 1.20 mm

aTubular membranes were crashed for the adsorption equilibrium study.


Fig. 1. Scheme of setup applied for volumetric technique, whereV1—adsorption chamber within Vycor glass sample, V2—storage chamber,V3—additional chamber, P—pressure sensor.

carbon dioxide and butane. Then multi-component adsorptionwas investigated. The adsorption of all three binary mixtureswas measured for different mixture compositions at differenttemperatures. Finally, a few experiments were carried out withternary mixture, at one temperature (343 K). The measurementswere repeated several times, in order to get reproducible data.

3.2. Measurement technique

The measurement of the adsorption isotherms was based onthe well-known volumetric method. To perform the adsorptionmeasurements, one tubular glass membrane (wall thickness:1.2 mm) was crashed into small fragments.

The sorption isotherms were measured using the setupdepicted in Fig. 1. The temperature controlled apparatus con-sisted of three chambers (V1, V2 and V3), connected by ex-pansion valves. These chambers could be evacuated and filledwith gases individually (gas purity: 5.0). The Vycor glasspieces were placed in the adsorption chamber V1. At the be-ginning of the experiment the whole apparatus was evacuated,and then the chamber V1 was closed, while the volume storagechamber V2 was filled with a gas mixture of a certain compo-sition at a certain pressure. Then, the chambers V1, V2 wereconnected. After reaching equilibrium the common pressurewas measured. The time needed to achieve equilibrium wasaround 30 min. For sorption measurements at higher pressure,the volume V2 was enlarged by connection with an additionalchamber V3. In the chamber V3, a magnetic impeller wasplaced to ensure a homogeneous mixture composition in bothvolumes V2 and V3. The start and final composition of themixtures were measured by a gas chromatographic methodwith thermal conductivity detector (TCD). The concentrationswere calculated from the areas of the chromatographic peaksand the related calibration curves.

Binary and ternary mixtures were prepared separately bysubsequentially filling the storage chamber with the respective

pure gases. The desired concentrations were controlled by mea-suring the pressure in the chamber during the filling process.

The amounts of gas adsorbed were calculated from the ma-terial balance as the difference between the initial amount ofgas and the amount of gas at equilibrium. Finally, the adsorbedamounts were related to the solid volume to determine the cor-responding adsorbed phase concentrations q in mol cm−3:

q = nads

V= nads�ap

m(1 − �), (23)

where m is the mass of the Vycor glass sample, �ap is theapparent density of Vycor glass and � is its porosity.

3.3. Experimental program and procedure of estimatingisotherm model parameters

The four isotherm models L1, Q1, L1,real and F1 were exam-ined first with pure gas adsorption data for the three differentgases (CO2, C3H8, C4H10) at 298, 313, 323 and 343 K.

The evaluations were carried out by fitting the free param-eters of the isotherm equations to the experimental data. Allparameters were found by minimization of the standard devi-ation (Eq. (24)) between experimental and theoretical values.The subscript x denotes here pressures (x = p in Eqs. (4) and(5)), or loadings (x = q in Eqs. (1), (3), and (6)).

x = 100%

√√√√ 1

Nd − Np

Nd∑k=1

(xk exp − xk calc

xk exp

)2

,

x = p, q. (24)

For comparison between applied models, the standard devia-tions of loadings were recalculated also for Eqs. (4) and (5) byiterative procedure. The study of multi-component isothermswas carried out for binary and ternary gas mixtures.

Three binary mixtures C3H8:CO2, C3H8:C4H10 andCO2:C4H10 with various mole compositions in gas phase(0.90:0.10; 0.50:0.50; 0.35:0.65; 0.10:0.90) at 298 and 343 Kwere investigated. Finally, five selected ternary mixtureswith the initial molar composition of C3H8:CO2:C4H10 =0.377:0.285:0.338 were equilibrated for different total pres-sures with the Vycor sample at 343 K.

Competitive gas adsorption based on the conventional multi-Langmuir and the multi-Freundlich models (Lmix0, Fmix0), us-ing only single-component parameters have been evaluated asa first attempt. Then the more complex models with additionalparameters introduced above were analyzed.

It should be mentioned that in case of binary or ternarymixtures, the objective function has been taken as a sum ofsquares of the differences between calculated and experimentalpartial values for each component in mixture (Eq. (25)).

x=100%

√√√√ 1

Nd − Np

Nd∑k=1

(N∑

i=1

(xi,k exp−xi,kcalc

xi,k exp

)2

+ · · ·)

,

xi = pi, qi . (25)

Again xi corresponds to the specific model (in Eqs. (7), (9),and (10) xi=qi and in Eqs. (8), (11), (18), and (19) xi=pi). The


subscript i denotes the component in mixture. For comparisonbetween applied models, also in the mixture case, the standarddeviations of loadings were recalculated (for Eqs. (8), (11),(18), and (19)) using an iterative procedure.

4. Results and discussion

All equilibrium data determined experimentally are shownin Figs. 2–5 and 14. Upon request the row data can be obtainedfrom the authors.

4.1. Single isotherms

The results of the adsorption measurements of propane (com-ponent 1, triangles), carbon dioxide (component 2, circles) andbutane (component 3, squares) at four different temperaturesare presented in Figs. 2a–d in the whole pressure range cov-ered (0–2.5 bar). Subsequently, the experimental results wereanalyzed theoretically for all gases only up to 1.40 bar. Abovethis pressure capillary condensation of butane occurs at roomtemperature in pores of 4 nm size (pc ≈ 1.40 bar at T =298 K,according to Kelvin equation) and a comparison of all gases is

Fig. 2. Single gas adsorption isotherms of: (a) propane, (b) carbon dioxide, (c) butane at four different temperatures (in K) and (d) comparison between thesethree gases at 298 K.

more difficult. Also the applied simplified models introducedabove might be not applicable in a wider pressure range.

Figs. 2a–c show the decrease of adsorption capacity with in-creasing temperature. Butane has at 298 K the strongest affinityto the Vycor surface compared to the other two gases (Fig. 2d).Carbon dioxide is stronger adsorbed than propane. At highertemperatures gave analogous trends hold.

The Langmuir isotherm (Eq. (1)), the Quadratic isothermwith additionally second order parameters (Eq. (3)), theisotherm based on Vacancy solution theory (Eqs. (4) and (5))and the Freundlich isotherm (Eq. (6)) were used for describingthe experimental data. In the fitting process it turned out thatno satisfactory agreement between experimental and theoret-ical loadings could be achieved assuming a single saturationcapacity value valid for all components. Also no effect of tem-perature on qsat,i was detectable. Thus, for each componentin all models a single specific value was estimated. The fittedparameters for all four equations are listed in Table 2.

The Langmuir isotherm model includes two parameters, thelimiting adsorption amount qsat,i and the temperature depen-dent parameter bi . Relative small deviations between L1-modeland experiments were obtained for all three gases, (q ≈ 9%).

1592 J.R. Cermáková et al. / Chemical Engineering Science 63 (2008) 1586–1601Ta

ble

2Pa

ram

eter

sof

sing

le-c

ompo

nent

isot

herm

sm

odel

sa

Com

pone

ntL

1-m

odel

(Eq.

(1))

Q1-m

odel

(Eq.

(3))

L1,

real

-mod

el(E

qs.

(4)

and

(5))

F 1-m

odel

(Eq.

(6))

Nd

T,K

qsa

t,i

bi

qi

(%)

qsa

t,i

ai

bi

qi

(%)

qsa

t,i

bH

i�

iV/�

Vi

pi/ q

i(%

)bk

n q

i(%

)(m

olcm

−3)

(bar

−1)

(mol

cm−3

)(b

ar−1

)(b

ar−2

)(m

olcm

−3)

(mol

cm−3

bar−1

)(m

olcm

−3ba

r−n)

Prop

ane

298

1.04

e−

30.

639

5.66

6.24

e−

41.

040.

232

5.02

8.38

e−

46.

79e

−4

0.23

1/4.

274

4.96

/5.

783.

93e

−4

0.77

44.

8915

313

0.43

87.

830.

712

0.10

68.

804.

64e

−4

0.24

1/4.

217

5.7/

16.3

2.78

e−

40.

712

8.07

732

30.

336

4.77

0.57

90.

050

5.10

3.76

e−

40.

226/

4.48

77.

00/5.

672.

33e

−4

0.71

94.

846

342

0.21

810

.50

0.42

1−1

.2e

−3

6.90

2.80

e−

40.

326/

4.98

12.1

/15

.21.

78e

−4

0.73

75.

368

Car

bon

diox

ide

298

1.30

e−

31.

025

6.08

8.68

e−

41.

510

0.34

34.

231.

55e

−3

1.32

e−

30.

651/

1.53

46.

77/8.

326.

24e

−4

0.73

95.

6412

313

0.67

77.

440.

973

0.16

94.

248.

26e

−4

2.18

3/1.

623

6.98

/8.

214.

80e

−4

0.67

44.

137

323

0.52

33.

330.

801

0.08

02.

917.

02e

−4

0.58

4/1.

714

6.41

/8.

674.

10e

−4

0.65

23.

167

342

0.29

34.

690.

451

0.02

54.

543.

91e

−4

0.93

2/1.

241

7.53

/9.

812.

64e

−4

0.76

17.

358

But

ane

298

4.60

e−

30.

301

6.90

1.05

e−

31.

250

0.92

39.

855.

86e

−3

1.40

e−

30.

339/

2.95

8.78

/10

.11.

09e

−3

0.96

16.

4114

313

0.18

413

.80

0.87

40.

266

15.9

01.

01e

−3

0.25

3/3.

941

7.58

/11

.17.

42e

−4

0.89

88.

678

323

0.14

813

.80

0.76

30.

119

13.1

08.

47e

−4

0.21

7/4.

618

6.51

/8.

435.

60e

−4

0.81

24.

856

342

0.09

718

.60

0.56

60.

0034

8.69

6.36

e−

40.

644/

4.95

110

.3/12

.13.

73e

−4

0.75

24.

126 L1-parameters for butane gave larger deviations for the loadings

at higher temperatures (around 18%).In the Quadratic isotherm (Eq. (3)) both parameters ai and

bi were estimated as a function of temperature. The standarddeviation is only slightly smaller in comparison with L1-model.The bi parameters are positive for all three components andsmaller in comparison with first order parameters ai .

The Vacancy solution theory (Eqs. (4) and (5)) quantifiesreal behavior of adsorbed phase with Wilson parameters. TheL1,real-model is a corrected L1-model describing the real phasebehavior. Thus, the isotherm contains four independent param-eters qsat,i , bHi , �iV and �V i . The standard deviations of thevacancy solution theory are very close to L1-model. The differ-ence between corresponding values of qsat,i for L1,real-modeland L1-model are due to the relation of Henry constant bHi asqsat,ibi for L1-model (Eq. (1)). The qsat,i for L1-model appearto be more reliable. To be more precise correlation betweenthe four parameters of the L1,real-model reduces the confidenceinto the estimated parameters. The corresponding activity co-efficients �i for pure components are in the range of 1–1.65.Thus, the behavior seems to be not so far from ideal.

In case of Freundlich isotherm, the parameters k and n de-pend on temperature in a quite complex way. For butane bothparameters decrease with temperature. For carbon dioxide andpropane the parameter n is almost constant.

The results of a statistical analysis performed indicate thatF1-model exhibits the best fitting results: an average standarddeviation q = 6.7%, for all 12 isotherms (3 components at4 temperatures). For comparison: (L1-model: q = 8.62%,Q1-model: q = 7.44% and L1,real-model: q = 9.14%).

All four models can be directly extended to predict gas mix-ture adsorption equilibria based on pure component parametersas discussed in the next section.

4.2. Binary competitive isotherms

General trends of the experimental results measured for thethree binary gas mixtures were evaluated quantitatively in orderto see general trends before applying the theoretical models.

Figs. 3–5 represent the equilibria behavior at 298 K. Athigher temperatures the tendencies are similar, however, loweradsorbed amount are found.

The results for mixtures of propane–carbon dioxide are pre-sented in Fig. 3.

For the same conditions more carbon dioxide is adsorbedon Vycor glass than propane. Pure component data are shownin addition to indicate changes when the two gases competeto occupy the available active sites. In Fig. 3a can be seenthat the adsorbed amount of propane in the mixture can beeven larger than for pure propane (e.g. for 0.90:0.10 mixture,�). This unusual behavior could be explained by adsorption ofpropane on the surface already covered by carbon-dioxide. Incase of carbon dioxide in mixture with propane (Fig. 3b), theobserved adsorbed amounts in the mixture are always lowerthan the observed for pure carbon dioxide. This illustrates themore typical competition of two components within mixtures.


Fig. 3. Experimental results for binary mixture at 298 K presented in dependency of partial loadings vs. partial pressures: (a) propane(1) and (b) carbondioxide(2) for four different mixture compositions (defined in figure legend) in comparison with pure component behavior.

Fig. 4. Experimental results for binary mixture at 298 K presented in dependency of partial loadings vs. partial pressures: (a) propane(1) and (b) butane(3) forfour different mixture compositions (defined in figure legend) in comparison with pure component behavior.

Fig. 5. Experimental results for binary mixture at 298 K presented in dependency of partial loadings vs. partial pressures: (a) carbon dioxide(2) and (b)butane(3) for four different mixture compositions (defined in figure legend) in comparison with pure component behavior.


Fig. 6. Comparison of Lmix0- and Fmix0-isotherm predictions (no additionalparameters) with experimental results for binary system propane(1)–carbondioxide(2) for mixture composition 0.50:050 at 298 K.

Fig. 4 shows competitive adsorption equilibria between thecomponents propane and butane. Comparing with Fig. 3a, evena stronger unusual effect of propane adsorption was observedin competition with butane (Fig.4a). Propane in mixture (10%propane and 90%) butane is more adsorbed than pure propane.Fig.4b shows more typical results: with increasing compositionof propane, the adsorption of butane decreases.

In carbon dioxide–butane mixtures presented in Fig. 5, bu-tane is found to be more adsorbed on Vycor glass than car-bon dioxide. In the whole range of conditions investigated bothgases are less adsorbed in mixtures than as single components.This expected competition can be seen in Fig. 5.

4.2.1. Correlating the data based on single-componentparameters

The prediction of multi-component adsorption equilibriausing only single-component isotherm information is the mostattractive option. Therefore, two models exclusively basedon single-component parameters were applied first: multi-Langmuir isotherm (Lmix0) and multi-Freundlich isothermbased on IAS theory (Fmix0). Typical results are presentedin Fig. 6 for binary mixtures of propane and carbon dioxideat 298 K. As can be seen, both fitted isotherms present verysimilar general behavior (competition in mixtures decreasedthe adsorbed amount of both components compared to theirpure gas behavior). But, as discussed above experimental datashowed that the adsorbed amounts of propane in mixtureswith carbon dioxide are higher than predicted by the Lmix0-and Fmix0-models (adsorbed amount of carbon dioxide arelower than expected). For these two components, the mixtureisotherms even intersect with each other. However, the unusualadsorption behavior observed in this study (Figs. 3a and 4a)cannot be accounted correctly with classical Langmuir and Fre-undlich isotherms. The large standard deviations (q ≈ 40%)

between the Lmix0- and Fmix0-models and experimental results

can be seen in Tables 3 and 5, respectively. Similar unsatis-factory results of predicting component isotherms on Vycorglass with simple models were reported already by Tuchlenskiet al. (1998) and Yang et al. (2005). Thus, additional correctionparameters coming from mixture experiments are needed forbetter description.

4.2.2. Correlating the data based on models with additionalparameters

The following modified competitive isotherm models wereapplied:

(a) Modified multi-Langmuir isotherm based on statisticalthermodynamics (Lmix1-model), Eq. (9).

(b) Vacancy solution theory (Lmix2-model), Eqs. (11)–(14).(c) Multi-Freundlich IAS correcting spreading pressure uncer-

tainties (Fmix1-model), Eqs. (18)–(20).

(a) Modified multi-Langmuir isotherm based on statisticalthermodynamics (Lmix1-model): In addition to parameters takenfrom single-component isotherms (qsat,i , ai and bi given inTable 2) one further interaction parameters bij had to be deter-mined for each binary mixture, according to Eq. (9).

For illustrating the influence of fitted additional parameter,Figs. 7–9 present experimental results in comparison with themodified model (Lmix1) and the simpler conventional Langmuirmodel (Lmix0) at 298 K for mole fractions of 0.50 for eachcomponent in all three mixtures.

All parameter values bij are displayed for different temper-atures in Table 3. The second degree parameters bi and the in-teraction parameters bij reduce limitations of the Lmix0-modelsignificantly. The average standard deviations of Lmix1-model(q = 22.92%) are approximately two times smaller comparedto the classical Lmix0-model.

All estimated interaction parameters are positive for two bi-nary mixtures, except for the mixture propane–butane, whereonly a negative parameter is able to predict the unusual in-crease of propane in competition with butane. All parametersare decreasing with temperature.

(b) Vacancy solution theory (Lmix2-model): The Vacancy so-lution theory (Suwanayuen and Danner, 1980a) distinguishesbetween interactions of gas and adsorbed phase molecules andadsorbate–adsorbate interactions. In order to estimate these in-teractions in the mixture, adsorbate–vacancy interactions arequantified first from single-component adsorption data, given inTable 2 (qsati , bHi , �iV and �V i). For all mixtures additionallyadsorbate-adsorbate interaction parameters (�ij and �ji) werefitted from binary mixture data according to Eqs. (11)–(14).Estimated parameters with average standard deviations of pres-sure and loadings are presented in Table 4.

For the mixture carbon dioxide and butane the parame-ters �23 and �32 are almost equal to 1. This means thatadsorbate–adsorbate interactions between these two compo-nents may be neglected. However, for the mixture propane andbutane these parameters account to �13=−3.21 and �31=1.54and therefore it is suggested that nonidealities cannot be


Fig. 7. Comparison of Lmix0- and Lmix1-isotherms prediction with experimental results for binary system propane(1)–carbon dioxide(2) for mixture composition0.50:0.50 at 298 K: (a) propane and (b) carbon dioxide.

Fig. 8. Comparison of Lmix0- and Lmix1-isotherms prediction with experimental results for binary system propane(1)–butane(3) for mixture composition0.50:0.50 at 298 K: (a) propane and (b) butane.

Fig. 9. Comparison of Lmix0- and Lmix1-isotherms prediction with experimental results for binary system carbon dioxide(2)–butane(3) for mixture composition0.50:0.50 at 298 K: (a) carbon dioxide and (b) butane.


neglected. The temperature dependence of these interactionswas not investigated.

The obtained average standard deviation for all 6 isotherms(3 mixtures at 2 temperatures) is quite satisfactory (q ≈ 25%).As an example a comparison between experimental results withapplied Lmix2-model is shown in Fig. 10 for all three mixturesat 298 K and molar fractions of 0.50 for all components.

(c) Multi-Freundlich IAS correcting spreading pressure un-certainties (Fmix1-model): As another alternative, the experi-mental binary data were correlated finally according to the IAStheory applied to the competitive Freundlich isotherm.

By minimizing the standard deviation between the predictedpressures defined by Eqs. (18), (19), and (21) and experimentaldata as (objective function Eq. (25)), the additional free param-eters ��i,ref were determined. They are presented in Table 5.With these parameters it was possible to decrease considerablystandard deviations obtained with general Fmix0-model (withfactor roughly about 2

3 ). The average standard deviation of ad-sorbed amount (q ≈ 16.63%, obtained iteratively) was foundto give the best description of the binary isotherms parameters.

A typical agreement between competitive and modified Fre-undlich isotherms, Fmix0 and Fmix1, respectively, can be alsoseen in Figs. 11–13 for all three mixtures at 298 K and for molefractions of 0.50 for all components.

Validation of the condition of the indices rotation for theinvestigated three components reveals the empirical characterof the ��i,ref . Thus, a binary mixture which not consists the

Table 3Fitting parameters of adsorption equilibrium for Lmix1-model, Eq. (9)

T (K) Mixture Lmix0 :q

a (%)Fittedparameters

Lmix1 :q (%)

298 Propane(1)–carbondioxide(2)

32.84 b12 = 0.536 16.32

343 49.06 b12 = 0.341 20.54

298 Propane(1)–butane(3)

50.36 b13 = −1.302 28.56

343 22.74 b13 = −0.358 31.30

298 Carbon dioxide(2)–butane (3)

45.05 b23 = 0.228 18.28

343 61.59 b23 = 0.178 22.53

aAverage standard deviations of Lmix0-isotherm, Eq. (7) using L1-parametersfrom Table 2.

Table 4Fitting parameters of adsorption equilibrium for the Lmix2 model, Eqs. (11)–(14)

T (K) Mixture Fitted parameters p/qa (%)

298 Propane(1)–carbon dioxide(2) �12 = −2.11; �21 = 1.12 21.11/26.8343 �12 = −1.44; �21 = 0.843 .24.76/25.31

298 Propane(1)–butane(3) �13 = −3.21; �31 = 1.54 26.62/28.1343 �13 = −1.84; �31 = 1.321 .28.20/29.4

298 Carbon dioxide(2)–butane(3) �23 = 1.145; �32 = 0.982 17.53/19.15343 �23 = 1.082; �32 = 1.01 17.86/21.01

aRecalculated iteratively.

reference component cannot be predicted directly from resultsfor the other two mixtures containing the reference component(e.g. at T = 298 K, ��31 − ��21 = ��32 leads to −5.27e −4 + 1.76e − 4 = −3.51e − 4, where in contrast the best fittedvalue is 2.89e − 5). The fitted ��i,ref parameters vary withtemperature only slightly.

In conclusion, correlating the data with additional pa-rameters, gave better statistical results in comparison withclassical models. The average standard deviations of binarymixtures data around 20% are quite satisfactory, since this er-ror is only two times bigger compared with the error of singlegas isotherms.

4.3. Ternary competitive isotherms (consistent with binarymixture parameters)

The five equilibrium states measured for the ternary systempropane(1)–carbon dioxide(1)–butane(3) were finally also an-alyzed. Data were collected for mixtures with the followinginitial composition: y1 = 0.377, y2 = 0.285 and y3 = 0.338 atT = 343 K. The experimental results are presented in Fig. 14aas the dependencies of the partial loadings on partial pressures.Based on estimated parameters of single components (Table 2)and binary mixtures (Tables 3–5), the three suggested modelsLmix1, Lmix2 and Fmix1 were applied for the prediction of theternary mixture isotherms without using any additional fittingparameters.

The agreement between experimental and calculated resultsis given in Table 6 as average standard deviations.

The Lmix1-model gave a systematic overestimation of partialloadings. The average standard deviation is q = 50.58%. Theagreement between experimental and predicted pressures basedon Lmix2-mode gave a standard deviation of q = 42.21% (ob-tained iteratively). The Fmix1-model was tested with propane asreference component 1 using the following ��i,ref parameters(Table 5): ��21 = −1.78e − 4 and ��31 = −2.43e − 4. Thismodel gave an average standard deviation q = 28.12%, whatis better in comparison with the other two models tested. Amore adequate analysis specifying eventually ternary interac-tion parameters was considered not to be reasonable with onlyfive data points.

In Fig. 14b is illustrated a relative ratio of estimated totaladsorbed amount using the Fmix1-model as the best fit of threeapplied models and experimentally observed amount vs. total


Fig. 10. Comparison of Lmix2-isotherm predictions with experimental results for binary systems: (a) propane(1)–carbon dioxide(2), (b) propane(1)–butane(3) and(c) carbon dioxide(2)–butane(3) at 298 K for mixture compositions 0.50:0.50 of all three binary mixtures (lines are just connections between measured points).

Table 5Corrected extrapolation error ��i,ref

a for Fmix1-model, Eqs. (18)–(21)

T (K) Mixture Fmix0 : pb (%) Corrected error (mol cm−3) Fmix1 : p/q

c (%)

298 Propane–carbon dioxide (ref = 1, i = 2) 33.08 ��21 = −1.76e − 4 17.08/16.53343 30.01 ��21 = −1.78e − 4 .21.64/17.16

298 Propane–butane (ref = 1, i = 3) 33.54 ��31 = −5.27e − 4 15.41/17.35343 25.10 ��31 = −2.43e − 4 .11.20/14.92

298 Carbon dioxide–butane (ref = 2, i = 3) 25.54 ��32 = 2.89e − 5 14.23/15.60343 29.50 ��32 = 2.18e − 5 15.43/18.25

aDerived from the lowest measured pressure pmin,i = 2.64 mbar (Eq. (15)) in the single system.bAverage standard deviations of Fmix0-isotherm, Eq. (8), using F1-parameters from Table 2.cRecalculated iteratively.

pressure. The systematic deviation is evident, experimentallyobserved total amounts adsorbed are higher than predicted bythe Fmix1-model.

The achieved standard deviation of describing the ternaryequilibria (30%) is three times more compared to errors fordescribing the single-component isotherms.

5. Conclusions

Subsequently, various models for the description of single-and multi-component adsorption phenomena were applied toquantify adsorption on Vycor glass. The single-componentisotherms were described using four different models: the


Fig. 11. Comparison of Fmix0- and Fmix1-isotherms prediction with experimental results for binary system propane(1)–carbon dioxide(2) for mixture composition0.50:0.50 at 298 K: (a) propane and (b) carbon dioxide.

Fig. 12. Comparison of Fmix0- and Fmix1-isotherms prediction with experimental results for binary system, propane(1)–butane(3) for mixture composition0.50:0.50 at 298 K: (a) propane and (b) butane.

Fig. 13. Comparison of Fmix0- and Fmix1-isotherms prediction with experimental results for binary system carbon dioxide(1)–butane(3) for mixture compositions0.50:0.50 at 298 K: (a) carbon dioxide and (b) butane.


Fig. 14. Ternary mixture propane (1)–carbon dioxide(1)–butane(3) at 343 K: (a) experimental data presented in dependency of partial loadings vs. partialpressures and (b) relative ratio between total loadings estimated from Fmix1-isotherm (as best fit) and experimental values vs. total pressures.

Table 6Average deviations between applied isotherm equations (Lmix1, Lmix2, Fmix1) and experimental data for ternary system based on binary fitted parameters at 343 K

Mixture T (K) Eq. (10)a Eqs. (11)–(14)a Eqs. (18)–(20), and 22a

Lmix1 : q (%) Lmix2 : q (%) Fmix1 : q (%)

Propane(1)–carbon dioxide(2)–butane(3) 343 50.58 42.21 28.12

aFitting parameters used from binary mixture estimation: Lmix1 (Table 3), Fmix1 (Table 5) and Lmix2 (Table 4) at 343 K.

Langmuir, Quadratic, Vacancy solution theory and Freudlichisotherms. The best results gave the Freundlich model (q =6.7%) with only two parameters.

The predictions of multi-component equilibrium data usingonly single-component isotherm information was not success-ful. In contrast, reasonable description of the measured partialisotherms in binary mixtures was possible with three differ-ent models Lmix1, Lmix2 and Fmix2 based on single-componentisotherms with additional parameters.

A few data for the ternary mixture could be described quan-titatively using all information available for the study of thebinary system.

However in general, it can be concluded that it is a diffi-cult task to describe and to choose reliable method of predict-ing the measured multi-component equilibrium which couldbe flexible enough to account for nonidealities of the complexmixture behavior.

Notation

ai partial molar surface area, m2 mol−1

ai first order constant of the statistical model ofsingle-adsorption isotherm, Eq. (3), bar−1

bi constant of the Langmuir adsorptionisotherm, Eq. (1), bar−1

bi quadratic constant of the statistical model ofsingle-adsorption isotherm, Eq. (1), bar−2

bij interacted constant of the statistical model ofbinary-adsorption isotherm, Eq. (11), bar−2

C parameter defined in Eq. (21), mol cm−3

k parameter of Freudlich Eq. (6),mol cm−3 bar−n

m weight of membrane sample in Eq. (23), gn exponential parameter of Freundlich Eq. (6)nads adsorbed moles of gases in Eq. (23), molNc number of components in mixtureNd number of experimental dataNp number of parametersp partial pressure,barp0 saturated vapor pressure, barqi adsorbed phase concentration of i

component, mol cm−3

qsat,i constant of the Langmuir adsorptionisotherm, Eq. (1), mol cm−3

q01 hypothetical adsorbed concentration of

reference component-1, Eq. (21), mol cm−3

R gas constant, kJ mol−1 K−1

T temperature, KV volume of the sample in Eq. (23), cm3

V1 adsorption chamber, cm3

V2 storage chamber, cm3

V3 additional chamber, cm3

xi mole fraction in adsorbed phase vacancysolution


zi mole fraction in the vacancy free adsorbedphase

zi nth order constant of the statistical model ofsingle-adsorption isotherm, Eq. (2), bar−n

Greek letters

�i activity coefficient of adsorbed phase� porosity of adsorbed phaseij binary correction parameter�ij , �ji adsorbated–adsorbated interaction

parameters from Wilson Eq. (14)�iV ,�V i adsorbate–vacancy interaction parameters

from Eq. (5)

�i modified spreading pressure(�i = �i a

RT

),

mol cm−3

�i spreading pressure in Eqs. (11) and (12),Nm−1

��i1 shift of extrapolation error of component ifrom reference component 1 defined byEq. (17), mol cm−3

�ap apparent density, g cm−3

x(x) standard deviation, defined by Eqs. (24)and (25), %

x(x) average standard deviation, %

Subscripts

0 initial valueexp experimental valuei, j, k componentmin minimal valuemix value of mixture

Superscript

0 single component

Abbreviations

F1 Freundlich isothermFmix0 multi-Freundlich isotherm based on IAS

theoryFmix1 multi-Freundlich isotherm corresponding

spreading pressure uncertaintiesL1 single-Langmuir isothermL1,real single isotherm based on vacancy solution

theoryLmix0 classical multi-Langmuir isothermLmix1 modified multi-Langmuir based on statistical

thermodynamicsLmix2 binary isotherm based on vacancy solution

theoryQ1 quadratic isotherm

Acknowledgement

The authors would like to thank Prof. E.-U. Schlünder(Karlsruhe University, Germany) for intensive discussions

and instructive suggestions. The financial supports of Fondder Chemischen Industrie and the Grant Agency of theCzech Academy of Science (Grant no. 1QS401250509 andIAA4072402) are also gratefully acknowledged.

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Single component and competitive adsorption of propane, carbon dioxide and butane on Vycor glass

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