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Abstract

Adsorption equilibria of the pure component and binary gaseous mixtures of water vapor and organic compound (toluene or n-butanol) on two selected activated carbons: Sorbonorit 4 and BPL 4×6 were studied. A mathematical model of the adsorption equilibrium for binary systems with the virial mixture coefficients has been developed based on the two-dimensional virial equation of state. In the modeling, the experimental adsorption equilibrium data of pure components were correlated by the Toth isotherm for organic compounds, and Talu–Meunier or Qi–LeVan models for water vapor. Additionally, multitemperature isotherms for pure component were calculated to determine virial mixture coefficient of the two-dimensional virial equation of state depending on temperature. Adsorption isotherms determined for two-component systems were compared with multicomponent adsorption equilibrium data obtained experimentally by a dynamic method at relative humidities of the carrier gas set at 25, 50, 75, and 90%. A fairly good agreement between simulations and experimental results was obtained using expansion of the two-dimensional virial equation of state through D terms.

Keywords

Multicomponent adsorption equilibrium multitemperature adsorption equilibrium activated carbon water vapor organic compounds

Introduction

Purification of waste industrial gases is an important problem both from the economics and environment protection points of view. Water vapor is usually present in waste gases and ambient air directed to fixed adsorption beds for separation, purification, and recovery. The water vapor considerably affects both equilibrium and kinetics of adsorption processes used in volatile organic compounds (VOCs) removal from polluted gas streams. Unfortunately, research on pure water adsorption equilibrium is limited, in contrary to adsorption equilibrium of pure organic compounds (Chiang et al., 2001a, 2001b; Qi et al., 2006; Gironi and Piemonte, 2011; Lashaki et al., 2012; Nastaj and Chybowska, 2004; Ramirez et al., 2005; Wang et al., 2004, 2005). In literature, there are not many adsorption equilibrium data, especially on commonly used adsorbents such as various kinds of activated carbons (Kim et al., 2005; Nastaj and Aleksandrzak, 2013; Qi and LeVan, 2005a,b; Qi et al., 2006). There are several models describing pure water adsorption equilibrium (Do, 1998; Ladshaw et al., 2015; Qi and LeVan, 2005a,b; Ribeiro et al., 2008; Talu and Meunier, 1996; Tien, 1994). However, the accuracy of existing models needs to be improved. Thus, a generalized model (or graphical method) is still desired, which would be relatively simple and have only several parameters. It should be based on single adsorption isotherms but grant accurate calculations over a wide range of temperatures.

In general, the polluted gas stream directed to the adsorption installation for separation or purification contains more than one component. Multicomponent adsorption equilibrium is area of increasing research interest (Fletcher et al., 2002; Foo and Hameed, 2010; Monneyron et al., 2003; Myers and Prausnitz, 1965; Smith, 1991; Wang et al., 2012; Wood, 2002; Yun et al., 1999). However, reliable experimental data are scarce, especially for low concentration of the gas streams. Despite the fact that water has been known to have strong effects on organic components adsorption, multicomponent adsorption equilibrium data, or organic mixtures with water are also rarely reported in topical literature (Appel, 1998; Appel et al., 1998; Chybowska, 2007; Doong and Yang, 1987; Gun'ko et al., 2008; Linders et al., 2001; Manes, 1984; Nastaj and Chybowska, 2006; Nastaj et al., 2006; Okazaki et al., 1978; Qi, 2003; Qi and LeVan, 2005a,b; Qi et al., 2000; Taqvi et al., 1999).

The published models for multicomponent adsorption equilibrium are usually empirical and some of them lack a solid thermodynamic basis (Appel, 1998; Qi, 2003). They may describe some specific systems with fairly good results, but not in general applications. Thermodynamically consistent models for multicomponent adsorption equilibrium, accurate for both nonideal and ideal systems, simple in mathematical form and consisting of few parameters are necessary. A few approaches have been proposed for more accurate prediction of multicomponent adsorption equilibrium of organic–water vapor systems.

The approaches developed for estimation of the multicomponent adsorption equilibria on the basis of well-known models such as ideal adsorbed solution theory (Myers and Prausnitz, 1965) or potential theory (Grant and Manes, 1966) fail when water vapor is present in the adsorbed mixture. Although both models are thermodynamically consistent, they do not take into account the mechanism of capillary condensation during water vapor adsorption. Therefore, they cannot properly describe adsorption equilibria of organics–water mixtures. A number of models have been developed to predict multicomponent adsorption equilibria for systems containing water vapor.

Doong and Yang (1987) formulated a simple thermodynamic model for determination of the adsorption equilibrium for water–organic vapor mixtures on microporous adsorbents such as activated carbon and zeolites. They applied modified Dubinin–Radushkevich equation (Do, 1998) with the aid of the concept of maximum available pore volume. The model was positively verified by experiments for binary systems consisted of selected organic compound (methanol, acetone, benzene, and toluene) and water vapor.

Okazaki et al. (1978) proposed the model of binary adsorption equilibrium for organic compound–water vapor system on microporous adsorbents. They assumed that total concentration of adsorbed organic component results from three contributions. The first one is the amount adsorbed in not wetted pores in which no water vapor condensation takes place. The second contribution is a liquid-phase adsorption of organic within wetted pores in which capillary condensation occurs. The final contribution results from dissolution of organic vapor in the condensed water. Application of the model for binary system requires knowledge of isotherms for organic compound in gas and liquid state, isothermal liquid–vapor equilibrium, pore volume, and specific surface area of activated carbon. The model gave inferior predictions for selected hydrocarbon-water systems in comparison with Doong–Yang formulation (Doong and Yang, 1987).

On the basis of the potential theory, Manes (1984) developed a graphical method for determination of humidity influence on adsorption of organic compounds not soluble in water. He considered a gas mixture saturated with water vapor and assumed that molar volumes of adsorbed components are the same as in their saturated liquid state. Manes stated that if adsorbed volume of pure organic component (at its vapor pressure in the mixture) is equal or greater than the volume of pure adsorbed water, then water will not interfere with the adsorption of organic vapor in mixtures. In the opposite case, the adsorption of organic vapor will be reduced by volume of adsorbed pure water. For mixtures in which gas is not saturated with water vapor, Manes introduced the modification to his method. Qi et al. (2000) proposed simple alternative numerical solution to the method of Manes. The Manes method was satisfactorily verified experimentally for selected water–immiscible organic component–water systems.

Another approach to the modeling of multicomponent adsorption equilibrium is the virial mixture coefficient (VMC) method (Appel, 1998). It uses pure component two-dimensional virial equation of state (2D-VEOS) to develop multicomponent adsorption isotherms. Mixture virial terms of these equations represent molecular interactions between adsorbed components and reflect adjustment to the adsorption contributions from the pure components. The accuracy of the model depends on a number of VMCs, which constitute a polynomial expansion. The sufficient number of these coefficients should therefore be able to describe any adsorption mixture including highly nonideal system such as organic component–water vapor. The model gave good agreement with experimental data (Taqvi et al., 1999; Taqvi and LeVan, 1997) also in a multitemperature form (Appel et al., 1998).

This article presents the application of VMC method to study coadsorption of water and organic components in binary (water–n-butanol and water–toluene) systems for both single temperature and multitemperature cases.

VMC approach in multicomponent adsorption equilibria modeling

2D-VEOS

The VMC approach utilizes equation of state (EOS) to estimate the multicomponent adsorption isotherms. The 2D-VEOS for multicomponent systems is defined as (Taqvi and LeVan, 1997):

\frac{π S}{R T} = q + \frac{1}{S} \sum_{i} \sum_{j} q i q j B ij + \frac{1}{S 2} \sum_{i} \sum_{j} \sum_{k} q i q j q k C ijk + \dots

(1)

where π is the spreading pressure of adsorbed mixture at temperature T, S is the specific surface area of the adsorbent, and q_i is the concentration of component i in the solid phase. Virial coefficients

B ij, C ijk

, etc., account for molecular interactions between adsorbed components. Interactions between adsorbed particles and adsorbent are described by Henry's law constant. These interactions are taken into account by pure component isotherms. In equation (1), virial coefficients with the same subscripts (

B 11, C 333

…) represent interactions between particles of the same component. Interactions between particles of different components are described by VMCs such as

B 12, C 123

, etc. The sequence of subscripts in virial coefficients is of no importance therefore, for example,

B 21 = B 12

. For the case of binary adsorption equilibrium, equation (1) can be rearranged by grouping together terms concerning pure components and mixture:

\frac{π S}{R T} = q 1 + \frac{1}{S} B 11 q_{1}^{2} + \frac{1}{S 2} C 111 q_{1}^{3} + \dots + q 2 + \frac{1}{S} B 22 q_{2}^{2} + \frac{1}{S 2} C 222 q_{2}^{3} + \dots + \frac{2}{S} B 12 q 1 q 2 + \frac{3}{S 2} C 112 q_{1}^{2} q 2 + \frac{3}{S 2} C 122 q 1 q_{2}^{2} + \dots

(2)

Equation (2) can be rewritten to the following form:

\frac{π S}{R T} = \frac{π S}{R T} | pure 1 + \frac{π S}{R T} | pure 2 + \frac{π S}{R T} | mixture

(3)

where the individual fractions on the right side of this equation stand for the first pure component, the second pure component and mixture, respectively. The obtained relation illustrates clearly that mixture term is an adjustment to the adsorption contributions from all the pure components. If only one component is adsorbed, equation (3) reduces to the pure component EOS.

Multicomponent adsorption isotherm using VMC approach

To derive the multicomponent adsorption isotherm from the 2-D VEOS, Van Ness (1969) proposed the classical thermodynamic approach. He provided the following relation between spreading pressure of adsorbed components and their fugacities derived from Helmholtz free energy:

\ln \frac{f i}{f_{i}^{*}} = \int_{S}^{\infty} [(\frac{\partial (π S / π S R T R T)}{\partial q i}) T, S, q j - 1] \frac{d S}{S}

(4)

where

f i

is the fugacity of a component i in a gas mixture and

f_{i}^{*} = q i / q i (K i S) (K i S)

is the fugacity at reference state, which corresponds to ideal gas at surface for

S \to \infty

. Substitution of equation (3) into (4) yields:

\ln \frac{f i}{f_{i}^{*}} = \int_{S}^{\infty} [(\frac{\partial (π S / π S R T | R T | purei)}{\partial q i}) T, S - 1] \frac{d S}{S} + \int_{S}^{\infty} (\frac{\partial (π S / π S R T | R T | mixture)}{\partial q i}) T, S, q j \frac{d S}{S}

(5)

First integral in equation (5) is, according to equation (4), equaled to $\ln (f purei / f purei f_{i}^{*} f_{i}^{*})$ for a pure component i. Thus,

\ln f i = \ln f purei + \int_{S}^{\infty} (\frac{\partial (π S / π S R T | R T | mixture)}{\partial q i}) T, S, q j \frac{d S}{S}

(6)

Equation (6) shows that, in multicomponent systems, the adsorption equilibrium of a component i is a sum of adsorption of pure component at concentration q_i in a mixture and a value of adjustment term resulted from interactions with other components.

Assuming ideal gas behavior, the fugacity f_i of a component i in equation (6) can be replaced by partial pressure p_i. Substituting the virial 2-D VEOS mixture terms from equation (1) in place of $π S / π S R T | R T | mixture$ into equation (6) gives after integration and rearrangement, the final form of the multicomponent adsorption isotherm:

\ln p i = \ln p purei + \frac{2}{S} \sum_{j} B ij q i + \frac{3}{2 S 2} \sum_{j} \sum_{k} C ijk q j q k + \dots

(7)

The derived equilibrium isotherm for multicomponent adsorption can be extended to multitemperature form by including a temperature dependence of the virial coefficients VC:

VC = V C 0 + \frac{V C 1}{T}

(8)

where VC = B_ij, C_ijk, D_ijkl, etc. The pure component adsorption terms in equation (7) are in this case described by multitemperature isotherms. Thus, the general form of the multitemperature adsorption isotherm derived according to the VMC approach is as follows:

\ln p i (T) = \ln p purei (T) + \frac{2}{S} \sum_{j} (B 0 ij + \frac{B 1 ij}{T}) q i + \frac{3}{2 S 2} \sum_{j} \sum_{k} (C 0 ijk + \frac{C 1 ijk}{T}) q j q k + \dots

(9)

To calculate multicomponent adsorption equilibrium using the VMC approach, the isotherm of each pure component must be known. Additionally, values of interaction of the VMCs (B₁₂, C₁₂₂…) should be determined. Typically, they are calculated using arithmetic or geometric mean for mixing rules (Appel, 1998; Appel et al., 1998; Steele, 1974). However, due to sufficient amount of available experimental data, Appel's, LeVan's and Finn's (Appel et al., 1998) approach is used here. It relies on determination of interaction of the VMCs directly as approximation parameters of equations (7) and (9) by minimizing the following objective function:

f min = \sum_{i = 1}^{n} \sum_{j = 1}^{N} (\ln p_{i j}^{cal} - \ln p_{i j}^{\exp})

(10)

for N data points of calculated and measured partial pressures of component i, and n numbers of components.

Investigations of adsorption equilibria

Materials

Two commercial activated carbons were used in experimental studies of adsorption equilibria, namely Sorbonorit 4 (Norit, The Netherlands) and BPL (Calgon Carbon Corporation, USA). The basic properties of these adsorbents are presented in Table 1 (Chiang et al., 1998, 2001a, 2001b; Reucroft et al., 1971; Zumkeller and Bart, 1997).

Table 1.

Basic physical properties of adsorbents.

Property	Sorbonorit 4	BPL 4×6
S_p (BET) (m²/g)	1400	1100
ρ_b (kg/m³)	400	440
ρ_p (kg/m³)	680	700
ɛ (m³/m³)	0.411	0.371
ɛ_p (m³/m³)	0.707	0.620
r_e (nm)	1.524	1.455

BET: Brunauer, Emmett and Teller.

The isotherms were obtained for one-component systems composed of activated carbon and pure component (water, n-butanol or toluene) and two-component systems (organic compound–water–activated carbon). Basic physical properties of adsorbed components are shown in Table 2.

Table 2.

Basic physical properties of adsorbed components (Yaws, 1999).

Property	Toluene	n-Butanol	Water
M (g/mol)	92.141	74.123	18.015
T_b (K)	383.78	390.81	373.15
μ (D)	0.36	1.66	1.85
p_s (Pa)
293.15 K	2914.69	656.61	2339.53
313.15 K	7881.06	2538.45	7386.97
333.15 K	18,515.49	8055.85	19,942.17
353.15 K	38,819.63	21,757.48	47,372.91

Water adsorption equilibria

Adsorption equilibria of water on activated carbons Sorbonorit 4 and BPL were obtained experimentally by the static gravimetric method (system IGA-002, Hiden Isochema, UK).

An adsorbent sample, weighing approximately 75 mg, was placed in the stainless steel sieve, which was suspended to the microbalance. All these components of the apparatus are located in the stainless steel vessel. The gas temperature near the sample was measured with a platinum resistance thermometer (Pt100). The pressure in the reactor was measured using the manometer with resolution of 0.1 Pa. The repeatability of isotherms is better than ±1%.

Before each measurement, the sample was degassed “in situ” at 453.15 K under vacuum (10⁻⁶ Pa) for 2 hours. This ultra-high vacuum was achieved using a system of two vacuum pumps: the diaphragm pump MD1 (Vacuumbrand, Germany) and turbomolecular pump TMU071P (Pfiffer Vacuum Technology, Germany). After degassing, measurement was made for selected value of temperature and various pressures. Adsorbed compound in the liquid phase was used as source of vapor, and was placed in a thermostatic vessel at 328.15 K. For the each set point of pressure and temperature, the real-time processor (RTP) enabled automatic recording of the changes in sample mass due to adsorption. The IGA (Intelligent Gravimetric Analyser) system based on measurements of mass and built-in algorithm of calculations enabled prediction of the asymptotic adsorption value. After reaching 99% of the estimated mass value, the IGA system passed to the next point of isotherm (next set point of pressure). Each measurement was carried out until all the pressure set points were examined. Then, the sample was degassed and measurements were made for next temperature. Details of the system IGA are described in our previous work (Nastaj and Aleksandrzak, 2013).

For the system water–Sorbonorit 4, the adsorption isotherms were obtained at temperatures 293.15, 303.15, 313.15, and 323.15 K. The system water–BPL was investigated at temperatures 293.15, 313.15, and 323.15 K.

Adsorption equilibria of water on Sorbonorit 4 and BPL activated carbons are described by two models: Talu–Meunier (T-M) and Qi–LeVan (Q-LV) isotherm. The T-M model was derived based on thermodynamic analysis and theory of self-associating water molecules in micropores of activated carbon. This model was considered as BDDT (Brunauer et al., 1940) type V isotherm and is therefore often used for prediction of adsorption equilibrium of water. The T-M isotherm equation is given by Talu and Meunier (1996):

p = \frac{H Ψ}{1 + K Ψ} \exp (\frac{Ψ}{q m})

(11)

where H is the Henry's law constant, K indicates the association equilibrium constant, and q_m is the water maximum adsorption capacity. Parameter Ψ is defined as:

Ψ = \frac{- 1 + \sqrt{1 + 4 K ζ}}{2 K}

(12)

where

ζ = q m q / q m q (q m - q) (q m - q)

(13)

The other model applied for adsorption equilibrium of water–activated carbons system is the Q–LV isotherm (Qi and LeVan, 2005a,b):

p = \frac{q *}{ξ 0 + ξ 1 q * + ξ 2 q * 2 + ξ 3 q * 3 + \dots + ξ n q * n}

(14)

where

ξ 0

ξ 1

ξ 3

, and

ξ n

are model parameters. The Q–LV model can describe accurately the entire range of water adsorption on activated carbons using a small number (usually four) of parameters. Parameters of both models (equations (11) and (14)) were determined using Statistica (Statsoft) by nonlinear least-squares regression of experimental data. The average relative error (ARE) was defined to qualify data fitting as:

δ = \frac{1}{N} \sum_{i = 1}^{N} | \frac{q \exp - q calc}{q \exp} | \cdot 100

(15)

Obtained isotherm parameters and the corresponding AREs are given in Table 3. Fittings of the T-M and Q-LV models to the experimental adsorption equilibrium data for the systems: water–Sorbonorit 4 activated carbon and water–BPL activated carbon at various temperatures are presented in Figure 1. Rapid increase in adsorbed phase concentration caused by water vapor capillary condensation is observed at relative pressure of about 0.6 at 293.15 K up to about 0.75 at 323.15 K (water saturation pressures are presented in Table 2). Additionally, at higher temperatures, the slope of isotherms becomes less steep. Water uptake on BPL activated carbon is greater in comparison with Sorbonorit 4 at the high values of the water vapor pressures. It could be attributed to slightly higher fraction of micropores in particles of the BPL activated carbon.

Table 3.

Parameters and AREs of T-M and Q-LV isotherms for adsorption of water on Sorbonorit 4 and BPL activated carbons.

System	Water–Sorbonorit 4				Water–BPL
Model	T-M isotherm
T (K)	293.15	303.15	313.15	323.15	293.15	313.15
q_m (mol/kg)	23.1269	23.1952	22.5808	29.6110	28.2368	27.7811
H (kPa)	1.2933	2.3765	4.5191	5.8863	1.4121	4.9278
K (kg/mol)	0.9467	0.9563	1.0674	0.7243	0.9079	1.0088
δ_W (%)	4.20	4.85	4.78	5.70	9.68	5.77
Model	Q-LV isotherm
T (K)	293.15	303.15	313.15	323.15	293.15	313.15
$ξ 0$ (mol/(kg kPa))	1.6785	0.9311	0.5020	0.2591	1.2159	0.2747
$ξ 1$ (1/kPa)	0.5227	0.2767	0.1631	0.1247	0.5674	0.2335
$ξ 2$ (kg/mol)	0.0280	0.0168	0.0105	0.0013	0.0138	−0.0006
$ξ 3$ (kg/(mol kPa))	−0.00173	−0.00099	−0.00062	−0.00016	−0.00088	−0.00016
δ_W (%)	4.00	3.89	4.46	3.82	3.43	4.72

ARE: average relative error; T-M: Talu–Meunier; Q-LV: Qi–LeVan.

Figure 1.

Experimental and predicted T-M and Q-LV isotherms for water adsorption onto Sorbonorit 4 and BPL activated carbons, at various temperatures.

Figure 2 presents selected experimental data points for relative humidities: RH = 25%, RH = 50%, RH = 75%, and RH = 90% for both the Sorbonorit 4 and BPL activated carbons. The water adsorption capacities practically overlap in this case. These RH values of carrier air were arbitrary chosen in the experimental measurements and multicomponent equilibria modeling.

Figure 2.

Experimental points of adsorption equilibrium capacities for Water–Sorbonorit 4 carbon and Water–BPL carbon systems at various relative humidities: RH = 25%, RH = 50%, RH = 75%, RH = 90%.

Pure water multitemperature isotherm

The multitemperature T-M isotherm for water–activated carbon systems is given by:

p = \frac{H (T) Ψ}{1 + K (T) Ψ} \exp (\frac{Ψ}{q m})

(16)

where H and K are temperature-dependent parameters defined as:

H = \exp (H 0 + \frac{H 1}{T})

(17)

K = \exp (K 0 + \frac{K 1}{T})

(18)

and Ψ is given by equation (10).

The multitemperature Q–LV isotherm describing adsorption equilibrium of water on activated carbons is given by the following relationship (Qi and LeVan, 2005a,b):

p A = P y A = p A ref (q_{A}^{*}, T ref) \exp [\frac{δ 0 + δ 1 q_{A}^{*}}{R} (\frac{1}{T} - \frac{1}{T ref}) + \frac{δ 2}{2 R} (\frac{1}{T 2} - \frac{1}{T_{ref}^{2}})]

(19)

where the reference adsorption isotherm at a particular temperature T_ref is defined as:

p A ref (q_{A}^{*}, T ref) = \frac{q_{A}^{*}}{ξ 0 + ξ 1 q_{A}^{*} + ξ 2 q_{A}^{* 2} + ξ 3 q_{A}^{* 3} + \dots + ξ n q A * n}

(20)

The parameters of the multitemperature isotherms in equations (16) through (20) were determined by nonlinear least-squares regression of experimental data. Only the water–Sorbonorit 4 equilibrium was analyzed in this case because isotherms for this system were measured in a sufficiently wide temperature range. Parameters of the multitemperature Q-LV isotherm were obtained for reference temperature T_ref = 313.15 and are shown in Table 4 together with multitemperature T-M isotherm parameters. Experimental and correlated multitemperature isotherms for water–Sorbonorit 4 system are depicted in Figure 3. The fittings obtained by Q-LV model are slightly better than those using T-M isotherm, especially at higher temperatures.

Table 4.

Parameters and AREs of multitemperature T-M and multitemperature Q-LV isotherms for adsorption of pure water on Sorbonorit 4 activated carbons.

System	Water–Sorbonorit 4
Model	Multitemperature T-M
q_m (mol/kg)	22.8146
H₀ ( $kPa \cdot kg / mol$ )	18.7807
H₁ ( $kPa \cdot kg \cdot K / mol$ )	−5414.4941
K₀ (kg/mol)	0.5485
K₁ ( $kg \cdot K / mol$ )	−159.1277
δ_W (%)	5.40
Model	Multitemperature Q-LV
δ₀ (J/mol)	−9748.0489
δ₁ (J/kg)	74.0393
δ₂ (J K/mol)	−10869768.1807
δ_W (%)	4.57

ARE: average relative error; T-M: Talu–Meunier; Q-LV: Qi–LeVan.

Figure 3.

Experimental and predicted multitemperature T-M and Q-LV isotherms for water adsorption onto Sorbonorit 4 activated carbon.

Pure organic component adsorption equilibria

The pure component adsorption equilibria of toluene and n-butanol were determined by a gravimetric method using the apparatus shown schematically in Figure 4. An adsorbent sample (about 6 mg) was placed in a glass column kept at constant temperature. A stream of dry air was mixed at required ratio with second stream saturated with organic component and delivered to the adsorption column. Adsorption uptake was determined on the basis of periodical measurements of a sample weight. The measurements were carried out until no increment of sample weight was observed. Adsorption isotherm of toluene–Sorbonorit 4, toluene–BPL, n-butanol–Sorbonorit 4, and n-butanol–BPL systems were obtained at temperatures 293.15, 313.15, 333.15, and 353.15 K (Chybowska, 2007; Nastaj and Chybowska, 2004).

Figure 4.

Scheme of apparatus to study adsorption equilibrium of pure components by gravimetric method: (1a,1b) Needle valves, (2a,2b) rotameters, (3) saturator with liquid component to be adsorbed, (4a,4b) thermostats, (5) mixer, (6) adsorbent, (7) analytical balance and (8) protection column.

The adsorption equilibria for these systems are described using the Toth model:

q = \frac{q m p}{(b + p n) 1 / 1 n n}

(21)

where constants q_m, b, and n were determined by nonlinear least-squares regression of experimental data. The ARE δ of the Toth model fitting to experimental data was calculated using equation (15). All isotherm constants and the corresponding AREs (δ) are given in Table 5. The experimental and correlated isotherms for both systems are depicted in Figures 5 and 6.

Table 5.

Parameters and AREs of Toth isotherm for adsorption of toluene and n-butanol on Sorbonorit 4 and BPL activated carbons.

Model	Toth isotherm
System	Toluene–Sorbonorit 4				Toluene–BPL
T (K)	293.15	313.15	333.15	353.15	293.15	313.15	333.15	353.15
q_m (mol/kg)	6.2762	5.5140	5.5766	6.3979	5.5450	4.7539	4.6798	5.5303
b (Paⁿ)	0.0644	0.0837	0.1222	0.1671	0.0611	0.0749	0.1034	0.1540
n (–)	0.2450	0.3470	0.3102	0.2550	0.1859	0.2684	0.2794	0.2328
δ_T, %	0.38	0.33	0.55	0.56	0.21	0.32	0.66	0.86
System	n-Butanol–Sorbonorit 4				n-Butanol–BPL
T (K)	293.15	313.15	333.15	353.15	293.15	313.15	333.15	353.15
q_m (mol/kg)	8.6635	6.3286	5.9974	6.1029	7.3057	5.0937	5.0061	5.7551
b (Paⁿ)	0.0670	0.0719	0.1095	0.1812	0.0588	0.0652	0.1190	0.2130
n (–)	0.1875	0.4112	0.4457	0.4753	0.1484	0.3762	0.4304	0.3868
δ_B (%)	0.85	1.13	1.58	1.57	0.85	0.57	0.68	0.61

ARE: average relative error.

Figure 5.

Experimental and predicted Toth isotherms for toluene adsorption onto Sorbonorit 4 and BPL activated carbons, at various temperatures.

Figure 6.

Experimental and predicted Toth isotherms for n-butanol adsorption onto Sorbonorit 4 and BPL activated carbons, at various temperatures.

Isotherms for both organic compounds can be identified according to the BDDT classification (Brunauer et al., 1940) as type I, which is typical for organics adsorption onto microporous adsorbents. Adsorption uptake of n-butanol is greater for both Sorbonorit 4 and BPL activated carbons than that of toluene. Additionally, adsorption uptakes of both organic compounds, i.e. toluene and n-butanol, are greater for Sorbonorit 4 compared with BPL, by about 10 to 20%. The obtained experimental results show that water adsorption uptake on the BPL is slightly larger in comparison with Sorbonorit 4, which could be explained by higher water affinity to the BPL activated carbon.

Pure organic component multitemperature isotherms.

The pure component adsorption equilibria of toluene and n-butanol are described by the multitemperature Toth isotherm:

q = \frac{q m p}{(b (T) + p n) 1 / 1 n n}

(22)

The parameter b(T) in equation (22) depends on the temperature T according to the following relation:

b (T) = b 0 \exp [- \frac{n Δ H}{R T}]

(23)

where

Δ H

is the isosteric heat of adsorption, parameters

b 0

and n are specific for given adsorbate–adsorbent system.

Determined parameters of multitemperature Toth isotherms for toluene–Sorbonorit 4 and n-butanol–Sorbonorit 4 systems are shown in Table 6, whereas experimental and correlated isotherms are depicted in Figures 7 and 8. Values of AREs are slightly higher than those for single temperature isotherms, but in general the fitting quality is good.

Table 6.

Parameters and AREs of multitemperature Toth isotherm for adsorption of pure toluene and n-butanol on Sorbonorit 4 activated carbon.

System	Toluene–Sorbonorit 4	n-Butanol–Sorbonorit 4
q_m (mol/kg)	6.2884	6.6913
b₀ (kPa)ⁿ	12.5596	283.1080
$Δ H$ (J/mol)	50,724.2327	55,891.2409
n (–)	0.2510	0.3817
δ_T, δ_B (%)	1.28	2.23

ARE: average relative error.

Figure 7.

Experimental and predicted multitemperature Toth isotherms for toluene adsorption onto Sorbonorit 4 activated carbon.

Figure 8.

Experimental and predicted multitemperature Toth isotherms for n-butanol adsorption onto Sorbonorit 4 activated carbon.

Multicomponent adsorption equilibria

The binary adsorption equilibria were studied experimentally by a dynamic method using the experimental set-up shown in Figure 9. Concentrations of particular gas components were adjusted by means of a mass flow controller (Aalborg, USA) and syringe pumps (kdScientific, USA), which were dosing liquid components into the static mixer. Adsorption took place in a column filled with about 5 g of an adsorbent and kept inside the thermostat. Temperature and humidity of outlet gas mixture were measured by HygroClip SC05 probe (Rotronic, Switzerland). Concentration of organic component in outlet gas was determined using the gas chromatograph GC07 (Labio, Czech Republic) with FID detector. The process was terminated when organic component concentration in outlet stream exceeded 95% of its concentration in inlet stream. On the basis of obtained concentration breakthrough curves and mass balance of the process, the equilibrium concentrations of components were calculated.

Figure 9.

Scheme of apparatus to study multicomponent adsorption equilibria: (1) mass flow controller (MFC), (2a,2b) syringe pumps, (3) mixer, (4a,4b) coil, (5) adsorption column, (6) thermostat, (7) rotameter, (8) needle valve, (9) heat exchanger, (10) concentration measurement, (11) HygroClip probe, (12) computer, and (13) outlet gas purification column.

The binary adsorption equilibria data were obtained by this method for water–toluene–Sorbonorit 4 and water–toluene–BPL systems at temperature 293.15 K and relative humidity (RH) of 50 and 90%. Another measurements were made for water–toluene–Sorbonorit 4 system at temperature 313.15 K and RH set at 25, 50, 75, and 90% as well as for water–n-butanol–Sorbonorit 4 system at temperature 313.15 K and RH of 50 and 90% (Chybowska, 2007). Obtained isotherms were used to validate the VMC model for binary systems, in both single temperature and multitemperature form, and are presented in the following sections.

Results and discussion

The VMC model was validated for binary systems composed of organic components (toluene or n-butanol) and water vapor mixtures adsorbed on Sorbonorit 4 and BPL activated carbons. Both single-temperature and multitemperature VMC models were analyzed, using parameters of pure component isotherms in single-temperature and multitemperature form, respectively. VMCs were determined by optimization of the objective function equation (10) using Maple software (Maplesoft) by means of nonlinear programming based on the modified Newton method. The simulated VMC isotherms obtained for different values of humidity and temperature levels were compared with experimental results and fitting quality was then evaluated using equation (15).

Validation of single-temperature VMC model for water vapor–organic component binary systems

Utilizing the VMC approach, the binary adsorption of two components: (1) organic compound (toluene or n-butanol) and (2) water vapor is considered. The multicomponent adsorption isotherm defined by equation (9) is written now as a set of two equations. The first equation is composed of the pure isotherm term given by Toth model for organic component and the mixture term derived from 2-D VEOS. For the sake of VMC approach, the Toth isotherm (equation (22)) is rearranged into the form:

p = (\frac{b}{(q / q m) - n - 1}) 1 / n

(24)

In the second equation of the set, the pure isotherm for water adsorption is used—either T-M or Q-LV model. Thus, the resulting set of equations for binary system: organic component–water vapor is composed of an equation for the first component (toluene or n-butanol):

\ln (p 1) = \frac{1}{n 1} \ln (\frac{b 1}{(q 1 / q 1 s) - n 1 - 1}) + \frac{2}{S} B 12 q 2 + \frac{3}{S 2} C 112 q 1 q 2 + \frac{3}{2 S 2} C 122 q_{2}^{2} + + \frac{4}{S 3} D 1122 q 1 q_{2}^{2} + \frac{4}{S 3} D 1112 q_{1}^{2} q 2 + \frac{4}{3 S 3} D 1222 q_{2}^{3} + \dots

(25a)

and for water, with either T–M isotherm:

\ln (p 2) = \ln (\frac{H Ψ}{1 + K Ψ}) + \frac{Ψ}{q m} + \frac{2}{S} B 12 q 1 + \frac{3}{2 S 2} C 112 q_{1}^{2} + \frac{3}{S 2} C 122 q 1 q 2 + + \frac{4}{S 3} D 1222 q 1 q_{2}^{2} + \frac{4}{S 3} D 1122 q_{1}^{2} q 2 + \frac{4}{3 S 3} D 1112 q_{1}^{3} + \dots

(25b)

where Ψ is defined by equation (12), or using the Q–LV isotherm:

\ln (p 2) = \ln (\frac{q 2}{ξ 0 + ξ 1 q 2 + ξ 2 q_{2}^{2} + ξ 3 q_{2}^{3} + \dots + ξ n q_{2}^{n}}) + \dots + \frac{2}{S} B 12 q 1 + \frac{3}{2 S 2} C 112 q_{1}^{2} + \frac{3}{S 2} C 122 q 1 q 2 + + \frac{4}{S 3} D 1222 q 1 q_{2}^{2} + \frac{4}{S 3} D 1122 q_{1}^{2} q 2 + \frac{4}{3 S 3} D 1112 q_{1}^{3} + \dots

(25c)

The data were correlated using expansion of the EOS equation through D terms, since expansion through E terms did not improve significantly the model fitting. The binary system was described using six VMCs, three parameters for the Toth isotherm, and three parameters for the T-M or four for Q-LV isotherm. VMCs were determined as approximation parameters of equations (25a) through (25c) by minimizing the objective function given by equation (10). It should be noted, that calculated earlier parameters of the Toth, T-M and Q-LV isotherms for single components were introduced into equations (25a to 25c) as constants. Values of the determined VMCs and the corresponding AREs are given in the Tables 7 through 9. Exemplary experimental and correlated coadsorption isotherms for toluene–water–Sorbonorit 4, toluene– water–BPL, n-butanol–water–Sorbonorit 4, and n-butanol–water–BPL systems are depicted in Figures 10 through 13.

Table 7.

VMCs and AREs for water–toluene–Sorbonorit 4 binary systems.

Coefficient	T = 293.15 K, RH = 50%		T = 293.15 K, RH = 90%		T = 313.15 K, RH = 25%
Coefficient	T–M	Q–LV	T–M	Q–LV	T–M	Q–LV
B₁₂/S	0.31871	0.40350	0.10347	0.09110	−1.50768	−1.31857
C₁₁₂/S²	−0.04737	−0.07129	0.03444	0.04203	1.16596	1.12475
C₁₂₂/S²	−0.16657	−0.19205	0.00862	0.00932	0.96995	0.88554
D₁₁₁₂/S³	−0.00180	0.00006	−0.00208	−0.00279	−0.16908	−0.16715
D₁₁₂₂/S³	0.01690	0.01886	−0.00320	−0.00349	−0.22566	−0.22286
D₁₂₂₂/S³	0.01348	0.01522	−0.00015	−0.00013	−0.09056	−0.07606
δ_T (%)	1.55	1.55	0.81	1.46	0.60	0.58
δ_W (%)	4.15	4.09	4.35	7.35	1.47	9.63
Coefficient	T = 313.15 K, RH = 50%		T = 313.15 K, RH = 75%		T = 313.15 K, RH = 90%
Coefficient	T–M	Q–LV	T–M	Q–LV	T–M	Q–LV
B₁₂/S	−1.32193	−1.12443	0.55926	0.62898	−0.13950	−0.17056
C₁₁₂/S²	0.24302	0.17055	−0.27859	−0.28666	0.21339	0.24155
C₁₂₂/S²	1.42729	1.35600	−0.03200	−0.05661	0.07028	0.07067
D₁₁₁₂/S³	0.00257	0.01003	0.04477	0.04405	−0.02879	−0.03323
D₁₁₂₂/S³	−0.10620	−0.09918	0.00164	0.00285	−0.01641	−0.01716
D₁₂₂₂/S³	−0.31588	−0.30957	0.00183	0.00368	−0.00201	−0.00190
δ_T (%)	2.54	2.62	1.53	0.98	5.33	3.17
δ_W (%)	3.95	4.07	20.25	7.69	6.17	6.66

VMC: virial mixture coefficient; ARE: average relative error; RH: relative humidity; T-M: Talu–Meunier; Q-LV: Qi–LeVan.

Figure 10.

Experimental and predicted VMC coadsorption isotherms for toluene–water binary mixtures adsorption onto Sorbonorit 4 and BPL activated carbon at 293.15 K, for various humidity levels.

Figure 11.

Experimental and predicted VMC coadsorption isotherms for toluene-water binary mixtures adsorption onto Sorbonorit 4 activated carbon at 313.15 K, for various humidity levels.

Figure 12.

Experimental and predicted VMC coadsorption isotherms for n-butanol–water binary mixtures adsorption onto Sorbonorit 4 and BPL activated carbon at 293.15 K, for various humidity levels.

Figure 13.

Experimental and predicted VMC coadsorption isotherms for n-butanol–water binary mixtures adsorption onto Sorbonorit 4 activated carbon at 313.15 K, for various humidity levels.

Values of adsorption uptake q for each component were calculated using Mathcad's (PTC Inc.) procedure for solving nonlinear equations system of equations (25a) through (25c) by Levenberg–Marquardt method. It caused generation of an additional error and decreased the total accuracy of used approach. Also, it should be noted that pure component isotherms replace the pure component virial terms. Therefore, the error of pure component isotherms correlation significantly contributes to the final accuracy of the VMC method.

For water–toluene–Sorbonorit 4 binary systems, the AREs values of water adsorption vary from 1.47 to 20.25% (Table 7) and are generally higher than those for toluene. The fitting quality for toluene is much better, with AREs in the range of 0.58–5.33%. For the water–toluene–BPL binary systems, AREs for water range from 4.20 to 5.71%, whereas the AREs for toluene are in the range 0.96–15.13% (Table 9). Water–n-butanol–Sorbonorit 4 binary systems give AREs values for water spanning from 1.28 to 13.99%, and from 0.86 to 5.93% for n-butanol (Table 8). The best fitting is obtained in this case at RH = 90%. On the BPL activated carbon, AREs values of water are contained in the range of 4.60–8.75%, whereas AREs for n-butanol are within 2.80–22.22% (Table 9).

Table 8.

VMCs and AREs for water–n-butanol–Sorbonorit 4 binary systems.

Coefficient	T = 293.15 K, RH = 50%		T = 293.15 K, RH = 90%		T = 313.15 K, RH = 50%		T = 313.15 K, RH = 90%
Coefficient	T–M	Q–LV	T–M	Q–LV	T–M	Q–LV	T–M	Q–LV
B₁₂/S	0.66679	0.70792	0.25075	0.26478	−3.10210	−2.88370	0.01260	−0.02076
C₁₁₂/S²	−0.13528	−0.14310	0.01868	0.01388	2.05673	2.00020	0.01623	0.02667
C₁₂₂/S²	−0.19392	−0.20654	−0.01642	−0.01881	2.26394	2.14961	0.03002	0.03076
D₁₁₁₂/S³	0.01134	0.01176	−0.00240	−0.00195	−0.26223	−0.25799	0.00192	0.00097
D₁₁₂₂/S³	0.00961	0.01030	−0.00158	−0.00142	−0.42927	−0.42333	−0.00417	−0.00424
D₁₂₂₂/S³	0.01569	0.01650	0.00067	0.00076	−0.25913	−0.23578	−0.00095	−0.00089
δ_B (%)	1.86	1.87	5.55	5.93	0.92	0.96	1.04	0.86
δ_W (%)	8.18	8.10	13.99	9.52	4.19	4.87	1.70	1.28

VMC: virial mixture coefficient; ARE: average relative error; RH: relative humidity; T-M: Talu–Meunier; Q-LV: Qi–LeVan.

Table 9.

VMCs and AREs for water–toluene–BPL and water–n-butanol–BPL binary systems.

Organic component	Toluene				n-Butanol
Coefficient	T = 293.15 K, RH = 50%		T = 293.15 K, RH = 90%		T = 293.15 K, RH = 50%		T = 293.15 K, RH = 90%
Coefficient	T–M	Q–LV	T–M	Q–LV	T–M	Q–LV	T–M	Q–LV
B₁₂/S	−9.94428	−9.88067	0.14024	0.14294	−5.85463	−5.85067	−0.18769	−0.18717
C₁₁₂/S²	3.85415	3.83921	0.34582	0.34876	2.55363	2.56021	0.40824	0.41299
C₁₂₂/S²	2.18989	2.16510	0.00555	−0.00145	1.75790	1.74934	0.03267	0.02775
D₁₁₁₂/S³	−0.42209	−0.42129	−0.05334	−0.05427	−0.25688	−0.25806	−0.05089	−0.05167
D₁₁₂₂/S³	−0.19256	−0.19070	−0.02048	−0.01979	−0.20609	−0.20585	−0.01810	−0.01791
D₁₂₂₂/S³	−0.14881	−0.14697	0.00114	0.00153	−0.13147	−0.13079	0.00051	0.00081
δ_T, δ_B (%)	0.96	0.97	14.93	15.13	2.80	2.83	21.56	22.22
δ_W (%)	4.98	4.20	4.81	5.71	4.61	4.60	8.60	8.75

VMC: virial mixture coefficient; ARE: average relative error; RH: relative humidity; T-M: Talu–Meunier; Q-LV: Qi–LeVan.

The highest AREs were obtained for n-butanol–water–BPL system at 293.15 K and RH = 90% (Table 9). In one case, the VMC model failed to give stable results when the Q-LV isotherm for water was used, which can be seen in Figure 12. Generally, the used approach was proved to be useful in binary adsorption equilibrium modeling, providing reliable calculated values of component loadings q. There is not unambiguous advantage of the Q-LV or T-M model use.

An analysis of adsorption isotherms at 293.15 K shown in Figure 10 indicates a slight decrease of water adsorption capacity at RH = 50% with increasing toluene partial pressure. However, at RH = 90%, the sharp decrease of water adsorption capacity takes place for very low concentrations of organic component. In this case, the capillary condensation phenomenon is observed both for toluene and n-butanol adsorption on Sorbonorit 4 and BPL activated carbons (Figures 10 through 13).

For all examined binary systems, in the presence of water vapor, the loadings of toluene or n-butanol have always lower values than those for pure component isotherms at RH = 0. This decrease is proportional to RH; however, this phenomenon is not so significant. Water does not interact strongly with carbonaceous adsorbents, and organic compound fills much of the pore volume in binary coadsorption systems. At high RH, adsorbed water molecules act as secondary sites for further water adsorption. Yet, the phenomena of water clusters formation and capillary condensation depend on how many hydrogen bonds between water molecules and adsorbent surface was formed initially.

From analysis of Figures 10 and 12, it can be concluded that n-butanol adsorption capacity is slightly higher in comparison with that of toluene on BPL activated carbon at temperature of 293.15 K. This fact can be explained by higher equilibrium adsorption capacity of pure n-butanol than that of toluene. Furthermore, the reason may be that toluene is recognized as only slightly (0.53 g/dm³ at 25℃) water-soluble component (Sanemasa et al., 1982), contrary to n-butanol, which is partly (68 g/dm³ at 25℃) water-soluble (Yalkowsky et al., 2010).

Considerable decreasing of the water adsorption capacities for toluene–water–Sorbonorit 4 system is observed at temperature 313.15 K in comparison with 293.15 K (see Figures 11 and 10, respectively). Similar behavior exists for n-butanol–water–Sorbonorit 4 system in the same temperatures (see Figures 13 and 12, respectively).

A general conclusion can be drawn, that equilibrium adsorption capacities of both organic components and water on the BPL activated carbon are somewhat lower than those on the Sorbonorit 4 activated carbon. It can be explained by differences in the texture of these adsorbents. The differences can mainly be attributed to specific surface area, total porosity, particle porosity, and mean pore radius.

Validation of multitemperature VMC model for water vapor–organic component binary systems

In the isotherm equations for binary multitemperature adsorption, the multitemperature isotherms for pure component systems are used: multitemperature Toth isotherm for toluene/n-butanol–activated carbon system and multitemperature T-M or multitemperature Q-LV isotherm for water–activated carbon systems. The resulting equations for multitemperature adsorption isotherms derived from 2-D VEOS by VMC approach for organic compound and water systems are as follows:

\ln (p 1) = \frac{1}{n 1} \ln (\frac{b 01 \exp [- \frac{n 1 Δ H 1}{R T}]}{(q 1 / q 1 s) - n 1 - 1}) + \frac{2}{S} (B 120 + \frac{B 121}{T}) q 2 + + \frac{3}{S 2} (C 1120 + \frac{C 1121}{T}) q 1 q 2 + \frac{3}{2 S 2} (C 1220 + \frac{C 1221}{T}) q_{2}^{2} + + \frac{4}{S 3} (D 11220 + \frac{D 11221}{T}) q 1 q_{2}^{2} + \frac{4}{S 3} (D 11120 + \frac{D 11121}{T}) q_{1}^{2} q 2 + \frac{4}{3 S 3} (D 12220 + \frac{D 12221}{T}) q_{2}^{3} + \dots

(26a)

\ln (p 2) = \ln (\frac{H (T) Ψ}{1 + K (T) Ψ}) + \frac{Ψ}{q m} + \frac{2}{S} (B 120 + \frac{B 121}{T}) q 1 + + \frac{3}{S 2} (C 1220 + \frac{C 1221}{T}) q 1 q 2 + \frac{3}{2 S 2} (C 1120 + \frac{C 1121}{T}) q_{1}^{2} + + \frac{4}{S 3} (D 12220 + \frac{D 12221}{T}) q 1 q_{2}^{2} + \frac{4}{S 3} (D 11220 + \frac{D 11221}{T}) q_{1}^{2} q 2 + \frac{4}{3 S 3} (D 11120 + \frac{D 11121}{T}) q_{1}^{3} + \dots

(26b)

\ln (p 2) = \ln (p 2 ref (q 2, T ref) \exp [\frac{δ 0 + δ 1 q_{A}^{*}}{R} (\frac{1}{T} - \frac{1}{T ref}) + \frac{δ 2}{2 R} (\frac{1}{T 2} - \frac{1}{T_{ref}^{2}})]) + + \frac{2}{S} (B 120 + \frac{B 121}{T}) q 1 + \frac{3}{S 2} (C 1220 + \frac{C 1221}{T}) q 1 q 2 + \frac{3}{2 S 2} (C 1120 + \frac{C 1121}{T}) q_{1}^{2} + + \frac{4}{S 3} (D 12220 + \frac{D 12221}{T}) q 1 q_{2}^{2} + \frac{4}{S 3} (D 11220 + \frac{D 11221}{T}) q_{1}^{2} q 2 + \frac{4}{3 S 3} (D 11120 + \frac{D 11121}{T}) q_{1}^{3} + \dots

(26c)

The binary adsorption equilibria data of the systems: water–toluene–Sorbonorit 4, water–toluene–BPL, water–n-butanol–Sorbonorit 4, and water–n-butanol–BPL were correlated using expansion of the EOS equation through D terms. System of equation (26) is a multitemperature version of equations set (25a), with 12 VMCs, 4 parameters for multitemperature Toth isotherm (Table 6), and 5 parameters for T-M (Table 4) or 7 for Q-LV isotherm with reference temperature T_ref = 313.15 (Tables 3 and 4). Determined virial coefficients of analyzed binary adsorption systems and the corresponding AREs are given in Tables 10 and 11. Exemplary experimental and correlated multitemperature coadsorption isotherms for toluene–water–Sorbonorit 4 and n-butanol–water–Sorbonorit 4 systems are depicted in Figures 14 through 16.

Figure 14.

Experimental and predicted multitemperature VMC isotherms for toluene-water and n-butanol–water mixtures adsorption onto Sorbonorit 4 activated carbon at 293.15 K, for various humidity levels.

Figure 15.

Experimental and predicted multitemperature VMC isotherms for toluene-water mixtures adsorption onto Sorbonorit 4 activated carbon at 313.15 K, for various humidity levels.

Figure 16.

Experimental and predicted multitemperature VMC isotherms for n-butanol–water mixture adsorption onto Sorbonorit 4 activated carbon at 313.15 K, for various humidity levels.

Table 10.

Multitemperature VMCs for water–toluene–Sorbonorit 4 and water–n-butanol–Sorbonorit 4 binary systems.

Organic component	Toluene				n-Butanol
Coefficient	T = 293.15 K, RH = 50%		T = 293.15 K, RH = 90%		T = 293.15 K, RH = 50%		T = 293.15 K, RH = 90%
Coefficient	T–M	Q–LV	T–M	Q–LV	T–M	Q–LV	T–M	Q–LV
B₁₂/S	−9.94428	−9.88067	0.14024	0.14294	−5.85463	−5.85067	−0.18769	−0.18717
C₁₁₂/S²	3.85415	3.83921	0.34582	0.34876	2.55363	2.56021	0.40824	0.41299
C₁₂₂/S²	2.18989	2.16510	0.00555	−0.00145	1.75790	1.74934	0.03267	0.02775
D₁₁₁₂/S³	−0.42209	−0.42129	−0.05334	−0.05427	−0.25688	−0.25806	−0.05089	−0.05167
D₁₁₂₂/S³	−0.19256	−0.19070	−0.02048	−0.01979	−0.20609	−0.20585	−0.01810	−0.01791
D₁₂₂₂/S³	−0.14881	−0.14697	0.00114	0.00153	−0.13147	−0.13079	0.00051	0.00081
δ_T, δ_B (%)	0.96	0.97	14.93	15.13	2.80	2.83	21.56	22.22
δ_W (%)	4.98	4.20	4.81	5.71	4.61	4.60	8.60	8.75

VMC: virial mixture coefficient; RH: relative humidity; T-M: Talu–Meunier; Q-LV: Qi–LeVan.

Table 11.

Estimated values of AREs of multitemperature VMC isotherms for water–toluene–Sorbonorit 4 and water–n-butanol–Sorbonorit 4 binary systems.

ARE	T = 293.15 K, RH = 50%		T = 293.15 K, RH = 90%		T = 313.15 K, RH = 25%		T = 313.15 K, RH = 50%		T = 313.15 K, RH = 75%		T = 313.15 K, RH = 90%
ARE	T–M	Q–LV	T–M	Q–LV	T–M	Q–LV	T–M	Q–LV	T–M	Q–LV	T–M	Q–LV
System	Water–toluene–Sorbonorit 4
δ_T (%)	2.31	2.50	5.26	5.21	1.70	1.90	2.56	2.60	1.71	1.55	11.18	15.11
δ_W(%)	7.50	8.44	5.16	5.87	29.62	28.75	12.45	6.46	32.49	28.05	33.80	22.78
System	Water–n-butanol–Sorbonorit 4
δ_T (%)	1.79	1.64	15.86	15.54	–	–	1.39	1.81	–	–	11.71	9.98
δ_W (%)	11.36	9.93	17.83	16.75	–	–	44.87	47.86	–	–	33.59	18.37

ARE: average relative error; VMC: virial mixture coefficient; RH: relative humidity; T-M: Talu–Meunier; Q-LV: Qi–LeVan.

The comparison of fitting methods of the experimental binary organic-water mixture isotherms with that multitemperature approach leads to conclusion that multitemperature VMC model is nearly as accurate as single-temperature VMC model at low and moderate humidity levels (RH = 25%; RH = 50%). However, at higher humidity levels (RH = 75%; RH = 90%), the fittings errors (AREs) are considerably higher, especially for water.

Conclusions

In the article, modeling studies and validation of multicomponent and multitemperature adsorption equilibria of water vapor and organic compounds (n-butanol, toluene) binary mixtures onto activated carbons (Sorbonorit 4 and BPL) were performed. The results proved that thermodynamically consistent VMC model can successfully be used to predict the isotherms for such highly nonideal mixtures containing water. To get accurate prediction of the binary adsorption equilibria in such systems, the reliable pure component isotherms must be determined. In the work, good fittings to experimental pure components data were obtained using T-M or Q-LV models for water, and Toth model for organic components. The pure component isotherms were approximated in two ways, namely as single-temperature or multitemperature models, which were incorporated into single-temperature and multitemperature VMC model, respectively.

The number of parameters in the VMCs model equations is critical to attain the required accuracy of predictions for adsorption equilibria in the case of multicomponent systems. Namely, more parameters offers higher accuracy but the increased number of parameters should compromise higher number of experimental points to be generated during usually limited series of trials. This issue is of prime importance in the case of multitemperature models. In the modeling, when VMC approach is used, the reverse computation method of the nonlinear arithmetic equations system solution must be applied. However, simultaneously, the numerical stability problem arises concerning the capability of equations system solution to meet actual adsorption capacities of the system components.

In binary organic-water systems, the loadings of toluene or n-butanol have always lower values, in the presence of water vapor, than those for pure component isotherms (at RH = 0).

At humidity levels over 75%, the capillary condensation of water vapor can clearly be observed. Probably, at such high humidity, the adsorbed water molecules act as secondary sites for further water adsorption. Thus, the phenomena of water clusters formation, according to association theory, and capillary condensation depend on how many hydrogen bonds between water molecules and adsorbent surface were formed initially.

In binary systems, weakly water miscible organics (hydrophobic toluene) and water vapor show competitive coadsorption at all humidity values on Sorbonorit 4 and BPL activated carbons. Similarly, partly water miscible organics (hydrophilic n-butanol) and water vapor show competitive coadsorption, at humidity above 50%.

The presented approach to modeling of investigated systems gave fairly good prediction results for both the single-temperature and multitemperature VMC models. Despite of somewhat lower prediction accuracy, multitemperature VMC models exhibit useful advantage, namely enable the equilibria determination for unknown temperatures (interpolation and extrapolation). However, in multitemperature VMC models, the determined regressed constants lose the physical meaning.

Most precise results of fitting to the experimental data were obtained for binary systems: organic–water mixtures, especially when many data points were obtained. In these cases, the similar accuracy was observed no matter which model of pure water equilibrium in the VMC approach was exploited, i.e. either T-M or Q-LV isotherm.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Appendix 1

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Modeling of multicomponent and multitemperature adsorption equilibria of water vapor and organic compounds on activated carbons

Abstract

Keywords

Introduction

VMC approach in multicomponent adsorption equilibria modeling

2D-VEOS

Multicomponent adsorption isotherm using VMC approach

Investigations of adsorption equilibria

Materials

Water adsorption equilibria

Pure water multitemperature isotherm

Pure organic component adsorption equilibria

Pure organic component multitemperature isotherms.

Multicomponent adsorption equilibria

Results and discussion

Validation of single-temperature VMC model for water vapor–organic component binary systems

Validation of multitemperature VMC model for water vapor–organic component binary systems

Conclusions

Footnotes

Declaration of Conflicting Interests

Funding

Appendix 1

References