Authors
^{1} Department of Chemical Engineering, College of Engineering, University of Isfahan, Isfahan, Iran
^{2} Department of Chemical Engineering, College of Engineering, University of Isfahan, Isfahan, Iran Department of Chemical Engineering, School of Chemical, Petroleum and Gas Engineering, Shiraz University, Shiraz, Iran
Abstract
Keywords
Introduction
The prediction of adsorption equilibrium plays a crucial part in an adsorption bed design. One of the most important approaches to calculate gas adsorption equilibrium is using twodimensional equations of state. Many attempts have been made to develop twodimensional equations of state and apply them to perform gas adsorption equilibrium calculations. Hill (1946) described a systematic thermodynamic treatment for mobile and monolayer adsorption of gas mixtures on homogeneous surfaces. They defined a surface fugacity for each component and they have utilized Van der Waals (VDW) EOS to predict adsorption equilibrium for binary gas mixtures [1]. Payne et al. (1968) derived a twodimensional equation of state from the HirschfelderEyring modification of VDW and applied it to predict high pressure adsorption equilibrium of pure and mixed hydrocarbons on charcoal [2]. Friederich and Mullins (1972) have used VDW to predict adsorption equilibrium of a mixture of hydrocarbons on carbon black [3]. Patrekiejew et al. (1977) have developed the twodimensional forms of Eyring, RedlichKwong (RK) and PengRobinson (PR) EOSs using statistical mechanics [4]. Konno et al. (1985) have obtained an isotherm equation by modifying PR EOS [5]. Zhou (1994) introduced a general twodimensional equation of state (ZGR) and derived the fugacity equations to describe the adsorbed phase for adsorption from gas mixtures [6]. The results of ZGR has been compared with those of multicomponent Langmuir model and ideal solution theory and found out that ZGR produced better results in multicomponent gas adsorption systems. Zheng et al. (1998) proposed a modified twodimensional VDW EOS model for the prediction of multicomponent gas adsorption isotherms from corresponding singlecomponent adsorption equilibrium data. Then, they employed this equation to predict adsorption isotherms of CO–CO_{2} mixtures and CO_{2}–N_{2} mixtures on Cu(I)–NaY zeolite [7]. Pan (2004) applied 2D EOS of ZGR and PR using new mixing rules to predict the adsorption of pure gas and several binary mixtures of gases on a carbon bed utilizing. A new computational algorithm has been developed for this purpose [8]. Fitzgerald et al. (2005) employed 2DEOS of ZGR to predict the adsorption of a mixture of gases on wet Tiffany coal [9]. In that work, a modified form of Van der Waals equation was developed by adding a new term. The four key parameters of this equation were obtained based on fitting to adsorption equilibrium data of pure gases using nonlinear regression method. In addition, the results of this equation are compared with those of VDW, Eyring, ZGR, SRK and PR for adsorption equilibrium of several pure and binary systems. The main objective of the present study is to compare the results obtained by twodimensional EOS of VDW, Eyring, ZGR, SRK and PR in predicting equilibrium adsorption of pure gases and the mixtures of gases. The sets of experimental data on pure and mixture gas adsorption found in literature were introduced in tables 2 and 3. Also, it was intended to develop a powerful 2DEOS which is capable of producing more accurate results for predicting equilibrium adsorption of highly nonideal gas mixtures.
Development of RT 2D EOS
The following general 2D EOS can be developed from threedimensional equations of state (3D EOS) [10]:

(1) 
Where π, A, T, R and w are spreading pressure, surface area of the adsorbent, absolute temperature, universal gas constant and the material amount adsorbed per mass of adsorbent. α and β are two parameters of this equations. Applying different values of U, W and m, which are mentioned in table 1, one can generate different types of 2DEOSs.
Table 1. The values of W, U and m for generating different types of EOSs.
w 
U 
m 
2DEOS 
0 
0 
0 
VDW 
0 
0 
Eyring 

0 
0 
ZGR 

0 
1 
1 
SRK 
2 
1 
1 
PR 
VW equation can be obtained by using the following equation for :
(2) 
In the righthand side of the above equation, the first and second terms show repulsive and attractive energy of adsorption. By investigating equilibrium adsorption of different systems employing VDW, It was revealed that repulsive term is underpredicted and the attractive term is overpredicted overall. Thus, in order to modify these terms, the following corrections were added to the righthand side of equation 32:

(3) 
(4) 
Where λ is an additional parameter of the new 2D EOS. Using these two terms, the best results were obtained amongst the numerous different terms tested. As a result, the following equation was obtained for the term :

(5) 
The main difference between RT and VDW is the addition of the third parameter of λ.
Mixing rule
For RT equation the following common mixing rule were applied [11]:
(6) 

(7) 

(8) 

(9) 

(10) 
Where are mole fractions of the components i and j, and are the binary correction factors which can be determined by fitting the results into mixture experimental data. However, in this study these parameters were set equal to zero. Also α, β and λ are the mixture parameters and α_{i}, β_{i}, and λ_{i} are the pure parameters of the 2D EOS.
Fugacity coefficients in adsorbed phase
The fugacity coefficient relation can be derived using the following thermodynamic equation [11]:

(11) 
Where Z_{a} is the compressibility factor for twodimensional adsorbate phase which is defined as follows:
(12) 
Where a is defined as:
(13) 
Using the above equations the following general relations for mixture fugacity coefficient of the new EOS were found:
(14) 
Where Q_{1}, Q_{2} and δ are:
(15) 

(16) 

(17) 
The term w_{j} in the above equations denotes the mass of component j in the adsorbatephase mixture. It should be differentiated from w_{jj} which is the mass of component j in the pure adsorbate phase. The fugacity coefficient relations for pure components were obtained by taking i=j=1 in the equations, as follows:

(18) 
Where and w_{ii }are Fugacity coefficient and the mass of component i in the pure adsorbed phase.
For :

(19) 

(20) 
(21) 

For :

(22) 
(23) 
Equilibrium calculation for purecomponent system
The equilibrium relation for pure component is in the following form:
(24) 
Where isfugacity of component i in the pure gas phase. Also, k_{i} is the henry’s constant for singlecomponent adsorption which is the ratio of at very low pressures (P→0).
Taking logarithm from both sides of equation 24 yields:

(25) 
Substituting the relation for pure component fugacity coefficient in adsorbed phase gives:
(26) 
Knowing the values of temperature, pressure and also the parameters of α_{i}, β_{i}, k_{i} and λ_{i}, one can determine the total amount of the adsorbed phase (w_{ii}). The fugacity of pure component i in gas phase, , was calculated using 3D PR equation of state.
Determination of optimum parameters
In order to determine singlecomponent parameters,α_{i}, β_{i}, k_{i} and λ_{i}, the adsorption equilibrium several sets of experimental data found in literature for pure component adsoprtion on a particular adsorbents were used as shown in tables 2 and 3. These parameters were calculated by fitting w_{ii }determined by the 2D EOS to the experimental data for the pure systems. Then, using the mixing rules one can find the EOS parameters for a multicomponent system with a given composition.
Equilibrium calculation for a multicomponent system
The equilibrium relations used in equilibrium adsorption calculations of a multicomponent system are:
(27) 
Where N_{C} is the number of components. Knowing temperature, pressure and composition one can determine gasphase composition {y_{i}} and adsorbed –phase composition {w_{i}}. Determining the multicomponent parameters of α, β and λ using the mixing rules one can calculate (using equation 14) and , which is the fugacity of component i in the gasphase mixture (using 3D PR). Substituting these values into the set of equilibrium relations (equation 27), the following system of Nc equations for N_{C} variables which are the amount of each component in adsorbed mixture (w_{i}) was obtained:
(28) 
Solving the above using the iterative Newton numerical method equations one can determine the mass of each component in the adsorbed mixture (w_{i}).
Result and Discussion
The adsorption equilibrium data predicted by new 2D EOS (RT) were compared to the ones collected from literature. In order to conduct equilibrium calculations for a multicomponent system, the parameters of single component adsorption of all respective constituents are required. It means that a set of adsorption equilibrium data for a multicomponent system on a specific adsorbent cannot be employed until the singlecomponent equilibrium data of all components on that particular adsorbent are available. These data are necessary to find the singlecomponent parameters. This fact created a huge restriction on the adsorption equilibrium data released in literature. In this regard, the appropriate sets of data were collected by an extensive literature survey. In this study, 24 sets of purecomponent data containing 441 equilibrium points and 15 sets of binary data comprising 175 equilibrium points were used to compare the performances of different 2D EOSs. Table 4 compares the predictions of various 2D EOSs with each other and reported their respective relative average deviations (%RAD) which are defined as:
(29) 
Where and are the calculated and experimental amount of adsorbed phase of pure component i respectively.
Table 2. The comparison of relative average deviations in predicting singlecomponent adsorption equilibrium data using various 2D EOSs.
Ref 
T(K) 
Adsorbate 
Adsorbent 
%RAD 
NP 
System 

RT 
PR 
SRK 
ZGR 
EYRING 
VDW 

[12] 
298.15 
iC_{4}H_{10} 
13X 
2.88 
2.92 
2.94 
2.84 
2.86 
2.90 
16 
1 
[12] 
298.15 
C_{2}H_{4} 
13X 
3.61 
4.69 
4.75 
4.08 
4.55 
5.04 
30 
2 
[12] 
323.15 
iC_{4}H_{10} 
13X 
1.48 
1.47 
1.48 
1.67 
1.59 
1.48 
16 
3 
[12] 
323.15 
C_{2}H_{4} 
13X 
2.77 
2.67 
2.66 
2.99 
2.77 
2.70 
25 
4 
[12] 
298.15 
C_{2}H_{6} 
13X 
1.97 
3.46 
3.32 
4.61 
2.32 
2.92 
31 
5 
[12] 
323.15 
C_{2}H_{6} 
13X 
2.63 
2.77 
2.67 
3.00 
1.75 
2.47 
20 
6 
[12] 
298.15 
CO_{2} 
13X 
2.69 
4.84 
5.06 
4.11 
4.87 
5.44 
20 
7 
[12] 
323.15 
CO_{2} 
13X 
2.69 
2.56 
2.56 
3.00 
2.77 
2.65 
17 
8 
[13] 
300 
CH_{4} 
13X 
0.54 
0.52 
0.53 
4.07 
1.95 
0.54 
18 
9 
[13] 
300 
C_{3}H_{8} 
13X 
1.83 
1.86 
1.74 
2.36 
1.74 
2.57 
15 
10 
[13] 
300 
CH_{4} 
5A 
4.00 
3.94 
3.95 
4.63 
3.94 
3.95 
18 
11 
[13] 
300 
C_{2}H_{6} 
5A 
0.82 
1.23 
1.18 
2.26 
1.21 
0.91 
15 
12 
[13] 
300 
C_{3}H_{8} 
5A 
5.58 
5.74 
5.65 
5.23 
5.34 
6.13 
12 
13 
[14] 
293 
CO_{2} 
13X 
2.11 
1.98 
2.44 
2.20 
2.59 
3.81 
38 
14 
[14] 
293 
C_{2}H_{6} 
13X 
0.55 
1.64 
1.43 
2.90 
1.28 
0.76 
30 
15 
[15] 
423 
C_{3}H_{8} 
4A 
2.37 
1.99 
2.15 
6.67 
4.35 
2.44 
5 
16 
[15] 
423 
C_{3}H_{6} 
4A 
2.99 
3.01 
3 
3.21 
3.06 
2.99 
20 
17 
[15] 
473 
C_{3}H_{8} 
4A 
2.25 
2.24 
2.24 
2.50 
2.12 
2.25 
5 
18 
[15] 
473 
C_{3}H_{6} 
4A 
2.70 
2.78 
2.76 
3.68 
3.06 
2.71 
18 
19 
[16] 
298 
O_{2} 
10X 
0.51 
0.50 
0.50 
0.59 
0.53 
0.50 
19 
20 
[16] 
298 
N_{2} 
10X 
0.26 
0.33 
0.34 
0.25 
0.30 
0.35 
19 
21 
[17] 
298 
CH_{4} 
Norit R1 
.49 
1.64 
1.82 
3.12 
0.60 
2.04 
12 
22 
[17] 
298 
N_{2} 
Norit R1 
.69 
0.76 
0.79 
5.18 
2.03 
.84 
10 
23 
[17] 
298 
CO_{2} 
Norit R1 
1.63 
2.87 
3.07 
3.28 
1.58 
3.36 
12 
24 
 
 
 
 
2.09 
2.51 
2.54 
3.14 
2.47 
2.67 
441 
TOTALL 
Table 3. The comparison of relative average deviations in predicting multicomponent adsorption equilibrium data using various 2D EOSs.
Quantities 
P (KPa) 
T (K) 
Adsorbent 
Adsorbate 
NAME 

wi 
137.8 
298.15 
13X Zeolite 
iC_{4}H_{10}(1)C_{2}H_{4}(2) 
A 

w 

wi 
137.8 
323.15 
13X Zeolite 
iC_{4}H_{10}(1)C_{2}H_{4}(2) 
B 

w 

wi 
137.8 
298.15 
13X Zeolite 
iC_{4}H_{10}(1)C_{2}H_{6}(2) 
C 

w 

wi 
137.8 
323.15 
13X Zeolite 
iC_{4}H_{10}(1)C_{2}H_{6}(2) 
D 

w 

wi 
137.8 
298.15 
13X Zeolite 
C_{2}H_{4}(1)CO_{2}(2) 
E 

w 

wi 
137.8 
323.15 
13X Zeolite 
C_{2}H_{4}(1)CO_{2}(2) 
F 

w 

wi 
345 
300 
13X Zeolite 
CH_{4}(1)C_{3}H_{8}(2) 
G 

w 

wi 
345 
300 
5A Zeolite 
CH_{4}(1)C_{2}H_{6}(2) 
H 

w 

wi 
345 
300 
5A Zeolite 
C_{2}H_{6}(1)C_{3}H_{8}(2) 
I 

w 

wi 
101.3 
293 
13X Zeolite 
CO_{2} (1)C_{2}H_{6}(2) 
J 

w 

wi 
84.6198.41 
423 
4A Zeolite 
C_{3}H_{6}(1)C_{3}H_{8}(2) 
K 

w 

wi 
102.93118.99 
473 
4A Zeolite 
C_{3}H_{6}(1)C_{3}H_{8}(2) 
L 

w 

wi 
6.3828.65 
298 
10X Zeolite 
O_{2}(1)N_{2}(2) 
M 

w 

w_{i} 
1515975 
298 
Activated carbon 
CH_{4}(1)N_{2}(2) 
N 

w 

w_{i} 
985916 
298 
Activated carbon 
CH_{4}(1)CO_{2}(2) 
O 

w 

wi 
OVERALL 

w 
ref 
NP 
RT 
PR 
SRK 
ZGR 
Eyring 
VDW 
%RAD 

[12] 
10 
5.69 
19.77 
22.25 
22.70 
23.35 
24.65 
2.20 
3.21 
3.54 
3.50 
3.70 
3.86 

[12] 
8 
5.24 
9.80 
16.99 
10.30 
9.25 
8.28 
2.52 
4.32 
3.71 
4.29 
4.62 
3.97 

[12] 
10 
32.21 
49.51 
44.03 
38.16 
36.54 
37.43 
6.07 
6.69 
6.79 
6.77 
6.38 
6.38 

[12] 
7 
22.51 
34.24 
33.24 
30.04 
29.81 
31.57 
5.39 
4.03 
3.97 
4.86 
4.03 
4.25 

[12] 
6 
12.70 
20.57 
15.36 
9.67 
10.24 
11.51 
1.06 
8.03 
5.82 
3.18 
4.00 
5.72 

[12] 
5 
8.64 
17.13 
14.06 
18.84 
8.87 
9.04 
1.25 
5.75 
4.20 
1.24 
1.55 
3.00 

[13] 
8 
26.95 
50.97 
50.91 
18.45 
26.67 
43.94 
3.41 
1.90 
1.71 
3.88 
2.99 
1.68 

[13] 
27 
14 
14.77 
14.18 
13.45 
13.71 
13.90 
6.04 
8.35 
7.64 
6.06 
6.24 
6.62 

[13] 
16 
22.32 
19.88 
20.72 
21.40 
21.33 
20.21 
6.72 
7.22 
6.99 
6.94 
6.99 
6.90 

[14] 
6 
10.25 
49.31 
40.4 
7.23 
10.09 
9.90 
1.81 
14.21 
9.31 
4.19 
5.030 
5.09 

[15] 
7 
11.18 
11.52 
9.25 
14.13 
12.20 
11.37 
2.32 
4.06 
3.15 
2.77 
2.64 
2.88 

[15] 
6 
10.85 
8.98 
9.46 
14.82 
12.50 
10.89 
2.97 
3.80 
3.45 
3.99 
3.44 
3.26 

[16] 
11 
10.08 
10.79 
11.27 
6.92 
33.64 
11.19 
2.34 
4.01 
3.49 
9.43 
18.85 
1.74 

[17] 
24 
9.51 
10.60 
10.56 
8.37 
10.46 
16.64 
3.69 
6.16 
4.54 
5.78 
4.12 
3.72 

[17] 
24 
18.33 
20.44 
20.11 
18.72 
18.13 
22.66 
2.82 
6.14 
4.66 
3.55 
2.64 
3.50 


175 
15.04 
20.79 
20.22 
16.36 
18.22 
19.10 
3.87 
6.11 
5.17 
5.13 
5.37 
4.44 
As expected, the results obtained by all EOSs are in good agreement with the experimental data for purecomponent systems. Adsorption of propane on zeolite 5A using VDW having %RAD of 6.13 is the worst case. This table shows that RT EOS having the total relative average error of 2.09 demonstrates the best performance, while the highest total average error is associated with ZRG having the error of 3.14. Using the optimum values of the singlecomponent parameters (α_{i}, β_{i}, k_{i} and λ_{i}) obtained through the calculations related to table 4, the multicomponent parameters (α, β and λ) were determined. Table 5 shows the relative average deviations, %RAD (wi) and %RAD(w), of different 2D EOSs for binary systems. These relative errors were calculated using the following equations:
(30) 
(31) 
Where N_{C} is the total number of components and N_{p} is the total number of equilibrium points in a set of data. The superscript c and e stand for calculated and experimental respectively. w and w_{i} are total amount of adsorbed phase and the amount of component i in adsorbed phase respectively.
As it is apparent from Table 3, RT equation of state having %RAD (w_{i})=15.04 and %RAD(w)=3.87 reveals the best prediction among the others. PR is ranked as the worst having %RAD (w_{i})=20.79 and %RAD(w)=6.11. The good prediction of highlynonideal systems where shows azeotropic behavior is the primary advantage of the new EOS. In order to demonstrate this capability, three sets of experimental data showing highlynonideal behavior were selected to be compared with the predictions of different EOSs. For each case the singlecomponent parameters and %RAD for various EOSs are shown through the tables. Also, the variation of one component (w_{i}) and total (w) adsorbed amounts were plotted versus mole fraction of one component. Because of the limitation of the data released in literature on gas adsorption, the following three cases were considered.
Case 1: Adsorption of isobuteneethylene on zeolite 13X
Table 4. The singlecomponent parameters and %RAD for various EOSs
Adsorbate 
2DEOS 
NP 
10^{4 }α_{i} 
β_{i} 
Ln k_{i} 
λ_{i} 
%AAD 
P (KPa) 
T (K) 
Ref 
Isobutane 
VDW 
16 
2.286 
0.4074 
8.554 
 
2.90 
24137.84 
298.15 
[12] 
Eyring 
16 
0.0452 
0.3680 
8.953 
 
2.86 
24137.84 
298.15 

ZGR 
16 
1.829 
0.3439 
9.993 
 
2.84 
24137.84 
298.15 

SRK 
16 
3.782 
0.4135 
9.069 
 
2.94 
24137.84 
298.15 

PR 
16 
4.416 
0.4104 
9.126 
 
2.92 
24137.84 
298.15 

RT 
16 
1.872 
0.3912 
8.405 
0.3912 
2.88 
24137.84 
298.15 

Ethylene 
VDW 
30 
0.4643 
0.2362 
5.065 
 
5.04 
24137.84 
298.15 
[12] 
Eyring 
30 
0.6790 
0.2093 
5.612 
 
4.55 
24137.84 
298.15 

ZGR 
30 
1.802 
0.1973 
6.627 
 
4.08 
24137.84 
298.15 

SRK 
30 
0.6323 
0.2405 
5.129 
 
4.75 
24137.84 
298.15 

PR 
30 
0.7564 
0.2396 
5.157 
 
4.69 
24137.84 
298.15 

RT 
30 
0.3001 
0.2559 
6.131 
2.406 
3.61 
24137.84 
298.15 
Figure 1. The variation of adsorbate mole fraction of isobutene versus gas mole fraction of isobutene.
Figure 2. the variation of total amount of adsorbate versus gas mole fraction of isobutene.
Case 2: Adsorption of carbon dioxideethylene on zeolite 13X
Table 5. The singlecomponent parameters and %RAD for various EOSs
Adsorbate 
2DEOS 
NP 
10^{4 }α_{i} 
β_{i} 
Ln k_{i} 
λ_{i} 
%AAD 
P (KPa) 
T (K) 
Ref 
Ethylene 
VDW 
25 
6.322 
0.2108 
3.887 
 
2.70 
67137.87 
323.15 
[12] 
Eyring 
25 
6.527 
0.1968 
4.375 
 
2.77 
67137.87 
323.15 

ZGR 
25 
20.170 
0.1942 
5.335 
 
2.99 
67137.87 
323.15 

SRK 
25 
7.511 
0.2236 
3.926 
 
2.66 
67137.87 
323.15 

PR 
25 
8.822 
0.2233 
3.951 
 
2.67 
67137.87 
323.15 

RT 
25 
1.497 
0.2479 
4.210 
1.419 
2.77 
67137.87 
323.15 

Carbon dioxide 
VDW 
17 
8.398 
0.0966 
4.0477 
 
2.65 
0.44137.84 
323.15 
[12] 
Eyring 
17 
1.120 
0.1180 
4.4618 
 
2.77 
0.44137.84 
323.15 

ZGR 
17 
11.470 
0.1265 
5.3763 
 
3 
0.44137.84 
323.15 

SRK 
17 
8.481 
0.1333 
4.0770 
 
2.56 
0.44137.84 
323.15 

PR 
17 
9.670 
0.1356 
4.1082 
 
2.56 
0.44137.84 
323.15 

RT 
17 
2.571 
0.1505 
4.2643 
1.080 
2.69 
0.44137.84 
323.15 
Figure 3. The variation of adsorbate mole fraction of ethylene versus gas mole fraction of ethylene.
Figure 4. The variation of total amount of adsorbate versus gas mole fraction of ethylene.
Case 3: Adsorption of Nitrogenoxygen on zeolite 10X
Table 6. The singlecomponent parameters and %RAD for various EOSs
Adsorbate 
2DEOS 
NP 
10^{4 }α_{i} 
β_{i} 
Ln k_{i} 
λ_{i} 
%AAD 
P (KPa) 
T (K) 
Ref 
Oxygen 
VDW 
25 
12.6552 
0.5593 
1.852 
 
0.505 
92.9212.39 
273.15 
[16] 
Eyring 
25 
67.6354 
0.8062 
1.368 
 
0.537 
92.9212.39 
273.15 

ZGR 
25 
128.2308 
0.9011 
0.3409 
 
0.592 
92.9212.39 
273.15 

SRK 
25 
11.0082 
0.4634 
1.855 
 
0.504 
92.9212.39 
273.15 

PR 
25 
10.7484 
0.4289 
1.858 
 
0.503 
92.9212.39 
273.15 

RT 
25 
32.0132 
0.7882 
1.831 
1.244 
0.510 
92.9212.39 
273.15 

Nitrogen 
VDW 
17 
10.8017 
5.00E05 
0.3447 
 
0.351 
99.5212.4 
273.15 
[16] 
Eyring 
17 
0.9828 
0.1212 
0.0285 
 
0.307 
99.5212.4 
273.15 

ZGR 
17 
12.3812 
0.1370 
0.9070 
 
0.253 
99.5212.4 
273.15 

SRK 
17 
9.2520 
0.0917 
0.3313 
 
0.342 
99.5212.4 
273.15 

PR 
17 
9.7411 
0.1022 
0.3143 
 
0.333 
99.5212.4 
273.15 

RT 
17 
18.4780 
0.4298 
0.8526 
3.199 
0.262 
99.5212.4 
273.15 
Figure 5. the variation of adsorbate mole fraction of oxygen versus gas mole fraction of oxygen.
Figure 6. The variation of total amount of adsorbate versus gas mole fraction of oxygen.
As it can be seen from figures 1 through 6 for all cases, especially for case 1, the performance of the RT EOS outreaches the rest significantly.