Rahimi, A., Talaie, M. (2017). Introduction of a Novel Two-Dimensional Equation of State to Predict Gas Equilibrium Adsorption in Highly-Nonideal Systems. Gas Processing, 5(2), 1-12. doi: 10.22108/gpj.2018.100147.1004

Abdolazim Rahimi; Mohamad Reza Talaie. "Introduction of a Novel Two-Dimensional Equation of State to Predict Gas Equilibrium Adsorption in Highly-Nonideal Systems". Gas Processing, 5, 2, 2017, 1-12. doi: 10.22108/gpj.2018.100147.1004

Rahimi, A., Talaie, M. (2017). 'Introduction of a Novel Two-Dimensional Equation of State to Predict Gas Equilibrium Adsorption in Highly-Nonideal Systems', Gas Processing, 5(2), pp. 1-12. doi: 10.22108/gpj.2018.100147.1004

Rahimi, A., Talaie, M. Introduction of a Novel Two-Dimensional Equation of State to Predict Gas Equilibrium Adsorption in Highly-Nonideal Systems. Gas Processing, 2017; 5(2): 1-12. doi: 10.22108/gpj.2018.100147.1004

Introduction of a Novel Two-Dimensional Equation of State to Predict Gas Equilibrium Adsorption in Highly-Nonideal Systems

^{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

Abstract The accurate calculations of adsorption equilibrium for multicomponent gas systems are of great importance in many applications. In this paper, five two-dimensional equations of state 2D-EOS, i.e. Van der Waals, Eyring, Zhou-Ghasem-Robinson, Soave-Redlich-Kwong and Peng-Robinson, were examined to find out their abilities to predict adsorption equilibrium for pure and multi-component gas adsorption systems. Also, a new 2D-EOS named Rahimi-Talaie (RT) was developed for accurately predicting adsorption equilibrium of the gas mixtures having highly non-ideal behavior. The pure parameters of all these equations were obtained by fitting 2D-EOS into pure gas-adsorption equilibrium data, and then the mixture parameters were calculated by recommended mixing rules. It was concluded that all equations were capable of accurately predicting pure adsorption equilibrium. However, among the six above-mentioned 2D-EOSs, RT was more successful to provide more accurate prediction of gas-mixture adsorption equilibrium, especially for the mixture showing azeotrope behavior.

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 two-dimensional equations of state. Many attempts have been made to develop two-dimensional 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 two-dimensional equation of state from the Hirschfelder-Eyring 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 two-dimensional forms of Eyring, Redlich-Kwong (RK) and Peng-Robinson (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 two-dimensional 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 two-dimensional VDW EOS model for the prediction of multicomponent gas adsorption isotherms from corresponding single-component 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 2D-EOS 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 two-dimensional 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 2D-EOS which is capable of producing more accurate results for predicting equilibrium adsorption of highly non-ideal gas mixtures.

Development of RT 2D EOS

The following general 2D EOS can be developed from three-dimensional 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 2D-EOSs.

Table 1. The values of W, U and m for generating different types of EOSs.

w

U

m

2D-EOS

0

0

0

VDW

0

0

Eyring

0

0

ZGR

0

1

1

SRK

-2

1

1

P-R

VW equation can be obtained by using the following equation for :

(2)

In the right-hand 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 under-predicted and the attractive term is over-predicted overall. Thus, in order to modify these terms, the following corrections were added to the right-hand 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 two-dimensional 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 adsorbate-phase 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 pure-component 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 single-component 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 single-component 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 multi-component system

The equilibrium relations used in equilibrium adsorption calculations of a multi-component system are:

(27)

Where N_{C} is the number of components. Knowing temperature, pressure and composition one can determine gas-phase composition {y_{i}} and adsorbed –phase composition {w_{i}}. Determining the multi-component 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 gas-phase mixture (using 3-D 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 2-D EOS (RT) were compared to the ones collected from literature. In order to conduct equilibrium calculations for a multi-component system, the parameters of single component adsorption of all respective constituents are required. It means that a set of adsorption equilibrium data for a multi-component system on a specific adsorbent cannot be employed until the single-component equilibrium data of all components on that particular adsorbent are available. These data are necessary to find the single-component 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 pure-component data containing 441 equilibrium points and 15 sets of binary data comprising 175 equilibrium points were used to compare the performances of different 2-D EOSs. Table 4 compares the predictions of various 2-D 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 single-component adsorption equilibrium data using various 2-D EOSs.

Ref

T(K)

Adsorbate

Adsorbent

%RAD

NP

System

R-T

P-R

SRK

ZGR

EYRING

VDW

[12]

298.15

i-C_{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

i-C_{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 multi-component adsorption equilibrium data using various 2-D EOSs.

Quantities

P (KPa)

T (K)

Adsorbent

Adsorbate

NAME

wi

137.8

298.15

13X Zeolite

i-C_{4}H_{10}(1)-C_{2}H_{4}(2)

A

w

wi

137.8

323.15

13X Zeolite

i-C_{4}H_{10}(1)-C_{2}H_{4}(2)

B

w

wi

137.8

298.15

13X Zeolite

i-C_{4}H_{10}(1)-C_{2}H_{6}(2)

C

w

wi

137.8

323.15

13X Zeolite

i-C_{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

84.61-98.41

423

4A Zeolite

C_{3}H_{6}(1)-C_{3}H_{8}(2)

K

w

wi

102.93-118.99

473

4A Zeolite

C_{3}H_{6}(1)-C_{3}H_{8}(2)

L

w

wi

6.38-28.65

298

10X Zeolite

O_{2}(1)-N_{2}(2)

M

w

w_{i}

151-5975

298

Activated carbon

CH_{4}(1)-N_{2}(2)

N

w

w_{i}

98-5916

298

Activated carbon

CH_{4}(1)-CO_{2}(2)

O

w

wi

OVERALL

w

ref

NP

R-T

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 pure-component 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 single-component 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 2-D 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 highly-nonideal 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 highly-nonideal behavior were selected to be compared with the predictions of different EOSs. For each case the single-component 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 isobutene-ethylene on zeolite 13X

Table 4. The single-component parameters and %RAD for various EOSs

Adsorbate

2-D-EOS

NP

10^{-4 }α_{i}

β_{i}

Ln k_{i}

λ_{i}

%AAD

P (KPa)

T (K)

Ref

Iso-butane

VDW

16

-2.286

0.4074

8.554

---

2.90

24-137.84

298.15

[12]

Eyring

16

-0.0452

0.3680

8.953

---

2.86

24-137.84

298.15

ZGR

16

1.829

0.3439

9.993

----

2.84

24-137.84

298.15

SRK

16

-3.782

0.4135

9.069

----

2.94

24-137.84

298.15

P-R

16

-4.416

0.4104

9.126

----

2.92

24-137.84

298.15

R-T

16

-1.872

0.3912

8.405

-0.3912

2.88

24-137.84

298.15

Ethylene

VDW

30

-0.4643

0.2362

5.065

----

5.04

24-137.84

298.15

[12]

Eyring

30

0.6790

0.2093

5.612

----

4.55

24-137.84

298.15

ZGR

30

1.802

0.1973

6.627

----

4.08

24-137.84

298.15

SRK

30

-0.6323

0.2405

5.129

----

4.75

24-137.84

298.15

P-R

30

-0.7564

0.2396

5.157

----

4.69

24-137.84

298.15

R-T

30

0.3001

0.2559

6.131

2.406

3.61

24-137.84

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 dioxide-ethylene on zeolite 13X

Table 5. The single-component parameters and %RAD for various EOSs

Adsorbate

2-D-EOS

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

67-137.87

323.15

[12]

Eyring

25

6.527

0.1968

4.375

---

2.77

67-137.87

323.15

ZGR

25

20.170

0.1942

5.335

----

2.99

67-137.87

323.15

SRK

25

-7.511

0.2236

3.926

----

2.66

67-137.87

323.15

P-R

25

-8.822

0.2233

3.951

----

2.67

67-137.87

323.15

R-T

25

1.497

0.2479

4.210

1.419

2.77

67-137.87

323.15

Carbon dioxide

VDW

17

-8.398

0.0966

4.0477

----

2.65

0.44-137.84

323.15

[12]

Eyring

17

1.120

0.1180

4.4618

----

2.77

0.44-137.84

323.15

ZGR

17

11.470

0.1265

5.3763

----

3

0.44-137.84

323.15

SRK

17

-8.481

0.1333

4.0770

----

2.56

0.44-137.84

323.15

P-R

17

-9.670

0.1356

4.1082

----

2.56

0.44-137.84

323.15

R-T

17

-2.571

0.1505

4.2643

1.080

2.69

0.44-137.84

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 Nitrogen-oxygen on zeolite 10X

Table 6. The single-component parameters and %RAD for various EOSs

Adsorbate

2-D-EOS

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

92.9-212.39

273.15

[16]

Eyring

25

67.6354

0.8062

-1.368

---

0.537

92.9-212.39

273.15

ZGR

25

128.2308

0.9011

-0.3409

----

0.592

92.9-212.39

273.15

SRK

25

11.0082

0.4634

-1.855

----

0.504

92.9-212.39

273.15

P-R

25

10.7484

0.4289

-1.858

----

0.503

92.9-212.39

273.15

R-T

25

32.0132

0.7882

-1.831

1.244

0.510

92.9-212.39

273.15

Nitrogen

VDW

17

-10.8017

5.00E-05

-0.3447

----

0.351

99.5-212.4

273.15

[16]

Eyring

17

0.9828

0.1212

0.0285

----

0.307

99.5-212.4

273.15

ZGR

17

12.3812

0.1370

0.9070

----

0.253

99.5-212.4

273.15

SRK

17

-9.2520

0.0917

-0.3313

----

0.342

99.5-212.4

273.15

P-R

17

-9.7411

0.1022

-0.3143

----

0.333

99.5-212.4

273.15

R-T

17

18.4780

0.4298

0.8526

3.199

0.262

99.5-212.4

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.

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