Abstract
The present study provides a modified approach in determining gas adsorption equilibria based on two-dimensional equations of state (2-D EOS). The proposed model utilizes temperature-dependent parameters in the general form of the 2-D EOS. These parameters were considered similar to other well-known isotherm models, such as Langmuir as specifics function of temperature. The proposed model was examined against various experimental single- and multi-component adsorption isotherm data. In most of the investigated cases, the proposed model reduces the error of predictions compared with temperature-independent two-dimensional equations of state. Moreover, utilizing temperature-dependent two-dimensional equations of state, isosteric heat of adsorption was theoretically obtained and compared with experimental heats of adsorption for different homogeneous and heterogeneous adsorption systems. Applying temperature-dependent parameters within 2-D EOS enables us to describe the heterogeneity of considered adsorption systems quite well. Predicted isosteric heats are in good accordance with the experimental data.
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Abbreviations
- 2-D EOS:
-
Two-dimensional equation of state
- a :
-
Specific molar volume of adsorbent (\(\hbox {m}^{2}/\hbox {mol}\))
- A :
-
Surface area per mass of adsorbent (\(\hbox {m}^{2}/\hbox {kg}\))
- %AAD:
-
percentage of absolute average deviation
- f :
-
Fugacity (kPa)
- k :
-
Parameter of the TI 2-D EOS model (mol/kPa kg)
- \(k_{1}\) :
-
Parameter of the TD model (mol/kPa kg)
- \(k_{2}\) :
-
Parameter of the TD model parameter (\(\hbox {kPa\,m}^{3}/\)\(\hbox {mol}\))
- m :
-
2-D EOS constant (dimensionless)
- M :
-
Mass of adsorbent (kg)
- n :
-
Mole of components in adsorbed phase (mol)
- NDP:
-
Number of data points
- R :
-
Universal gas constant (\(\hbox {kPa}\,\hbox {m}^{3}/\hbox {mol}\,\hbox {K}\))
- \(S_{1},S_{2}\) :
-
Terms in the isosteric heat of adsorption equations
- T :
-
Temperature (K)
- \(T_{1},T_{2}\) :
-
Terms in the fugacity equations of adsorbed phase in the TI model
- \(\bar{{T}}_1 ,\bar{{T}}_2\) :
-
Terms in the fugacity equations of adsorbed phase in the TD model
- U :
-
2-D EOS constant (dimensionless)
- W :
-
2-D EOS constant (dimensionless)
- x :
-
Mole fraction in adsorbed phase (dimensionless)
- y :
-
Mole fraction in gas phase (dimensionless)
- Z :
-
Compressibility factor of adsorbed phase (dimensionless)
- \(\alpha \) :
-
Parameter of the TI 2-D EOS model (kPa, \(\hbox {m}^{3}\,\hbox {kg/mol}^{2}\))
- \(\alpha _{1}\) :
-
Parameter of the TD model (kPa, \(\hbox {m}^{3}\,\hbox {kg/mol}^{2}\))
- \(\alpha _{2}\) :
-
Parameter of the TD model (kPa, \(\hbox {m}^{3}\,\hbox {kg/K\,mol}^{2}\))
- \(\beta \) :
-
Parameter of the TI 2-D EOS model (kg/mol)
- \(\beta _{1}\) :
-
Parameter of the TD model (kg/mol)
- \(\beta _{2}\) :
-
Parameter of the TD model (kg/mol K)
- \(\theta \) :
-
Fractional loading in the TI model (dimensionless)
- \(\bar{{\theta }}\) :
-
Fractional loading in the TD model (dimensionless)
- \(\phi \) :
-
Fugacity coefficient (dimensionless)
- \(\pi \) :
-
Spreading pressure (kPa m)
- \(\varPsi \) :
-
Deviation from ideality of isotherms in the TI model (dimensionless)
- \({\varPsi }'\) :
-
Deviation from ideality of Langmuir isotherm (dimensionless)
- \(\bar{{\varPsi }}\) :
-
Deviation from ideality of isotherms in the TD model (dimensionless)
- \(\omega \) :
-
Mole adsorbed per mass of adsorbent (mol/kg)
- a:
-
Adsorbed phase
- cal:
-
Calculated
- exp:
-
Experimental
- g:
-
Gas phase
- st:
-
Isosteric
- a:
-
Adsorbed phase
- \(\textit{i,j}\) :
-
Components \(\textit{i,j}\) in mixture
- mx:
-
Mixture
References
Bakhtyari, A.; Mofarahi, M.: Pure and binary adsorption equilibria of methane and nitrogen on zeolite 5A. J. Chem. Eng. Data 59(3), 626–639 (2014)
Mofarahi, M.; Bakhtyari, A.: Experimental investigation and thermodynamic modeling of CH\(_4\)/N\(_2\) adsorption on zeolite 13X. J. Chem. Eng. Data 60(3), 683–696 (2015)
Danish, M.; et al.: Response surface methodology approach for methyl orange dye removal using optimized Acacia mangium wood activated carbon. Wood Sci. Technol. 48(5), 1085–1105 (2014)
Danish, M.; et al.: Application of optimized large surface area date stone (Phoenix dactylifera) activated carbon for rhodamin B removal from aqueous solution: Box–Behnken design approach. Ecotoxicol. Environ. Saf. 139, 280–290 (2017)
Nasrullah, A.; et al.: High surface area mesoporous activated carbon-alginate beads for efficient removal of methylene blue. Int. J. Biol. Macromol. 107, 1792–1799 (2018)
Myers, A.; Prausnitz, J.M.: Thermodynamics of mixed-gas adsorption. AIChE J. 11(1), 121–127 (1965)
Myers, A.L.: Adsorption of gas mixtures—a thermodynamic approach. Ind. Eng. Chem. 60(5), 45–49 (1968)
Suwanayuen, S.; Danner, R.P.: A gas adsorption isotherm equation based on vacancy solution theory. AIChE J. 26(1), 68–76 (1980)
Suwanayuen, S.; Danner, R.P.: Vacancy solution theory of adsorption from gas mixtures. AIChE J. 26(1), 76–83 (1980)
Cochran, T.W.; Kabel, R.L.; Danner, R.P.: Vacancy solution theory of adsorption using Flory–Huggins activity coefficient equations. AIChE J. 31(2), 268–277 (1985)
Bering, B.; Dubinin, M.; Serpinsky, V.: Theory of volume filling for vapor adsorption. J. Colloid Interface Sci. 21(4), 378–393 (1966)
Moon, H.; Tien, C.: Further work on the prediction of gas-mixture adsorption equilibrium using the potential theory. Sep. Technol. 3(3), 161–167 (1993)
Shapiro, A.A.; Stenby, E.H.: Potential theory of multicomponent adsorption. J. Colloid Interface Sci. 201(2), 146–157 (1998)
Prausnitz, J.M.; Lichtenthaler, R.N.; de Azevedo, E.G.: Molecular Thermodynamics of Fluid-Phase Equilibria. Pearson Education, London (1998)
de Boer, J.H.: The Dynamical Character of Adsorption, vol. 76. Oxford University Press, Oxford (1953)
Do, D.D.: Adsorption Analysis: Equilibria and Kinetics, vol. 2. Imperial College Press, London (1998)
Ross, S.; Olivier, J.P.: On Physical Absorption. Interscience, New York (1964)
Hoory, S.; Prausnitz, J.: Monolayer adsorption of gas mixtures on homogeneous and heterogeneous solids. Chem. Eng. Sci. 22(7), 1025–1033 (1967)
Haydel, J.J.; Kobayashi, R.: Adsorption equilibria in methane–propane–silica gel system at high pressures. Ind. Eng. Chem. Fundam. 6(4), 546–554 (1967)
Payne, H.; Sturdevant, G.; Leland, T.: Improved two-dimensional equation of state to predict adsorption of pure and mixed hydrocarbons. Ind. Eng. Chem. Fundam. 7(3), 363–374 (1968)
Friederich, R.O.; Mullins, J.C.: Adsorption equilibria of binary hydrocarbon mixtures on homogeneous carbon black at \(25^{\circ } \text{ C }\). Ind. Eng. Chem. Fundam. 11(4), 439–445 (1972)
Sloan Jr., E.D.; Mullins, J.: Nonideality of binary adsorbed mixtures of benzene and Freon-11 on highly graphitized carbon at 298.15 K. Ind. Eng. Chem. Fundam. 14(4), 347–355 (1975)
DeGance, A.; Morgan, W.; Yee, D.: High pressure adsorption of methane, nitrogen and carbon dioxide on coal substrates. Fluid Phase Equilib. 82, 215–224 (1993)
DeGance, A.E.: Multicomponent high-pressure adsorption equilibria on carbon substrates: theory and data. Fluid Phase Equilib. 78, 99–137 (1992)
Garbacz, J.; et al.: Equation of physical adsorption on a homogeneous surface based on a two-dimensional analog of the Beattie–Bridgeman equation of a real gas state. Colloids Surf. A Physicochem. Eng. Aspects 119(2–3), 215–220 (1996)
Zheng, Y.; Gu, T.: Modified van der Waals Equation for the prediction of multicomponent isotherms. J. Colloid Interface Sci. 206(2), 457–463 (1998)
Zhou, C.: Modeling and Prediction of Pure and Multicomponent Gas Adsorption. Oklahoma State University, Stillwater (1994)
Zhou, C.; et al.: Predicting gas adsorption using two-dimensional equations of state. Ind. Eng. Chem. Res. 33(5), 1280–1289 (1994)
Sudibandriyo, M.; et al.: Adsorption of methane, nitrogen, carbon dioxide, and their binary mixtures on dry activated carbon at 318.2 K and pressures up to 13.6 MPa. Langmuir 19(13), 5323–5331 (2003)
Fitzgerald, J.; et al.: Adsorption of methane, nitrogen, carbon dioxide and their mixtures on wet Tiffany coal. Fuel 84(18), 2351–2363 (2005)
Peng, X.; et al.: Adsorption separation of CH\(_4\)/CO\(_2\) on mesocarbon microbeads: experiment and modeling. AIChE J. 52(3), 994–1003 (2006)
Mofarahi, M.; Hashemifard, S.: New mixing rule for predicting multi-component gas adsorption. Adsorption 17(2), 311–323 (2011)
Fotoohi, F.; et al.: Predicting pure and binary gas adsorption on activated carbon with two-dimensional cubic equations of state (2-D EOSs) and artificial neural network (ANN) method. Phys. Chem. Liq. 54(3), 281–302 (2016)
Van Ness, H.: Adsorption of gases on solids. Review of role of thermodynamics. Ind. Eng. Chem. Fundam. 8(3), 464–473 (1969)
Talu, O.; Kabel, R.: Isosteric heat of adsorption and the vacancy solution model. AIChE J. 33(3), 510–514 (1987)
Nieszporek, K.: Theoretical description of the calorimetric effects accompanying the mixed-gas adsorption equilibria by using the ideal adsorbed solution theory. Langmuir 18(24), 9334–9341 (2002)
Nieszporek, K.: The application of NIAS approach to describe enthalpic effects accompanying mixed-gas adsorption. Appl. Surf. Sci. 207(1–4), 208–218 (2003)
Nieszporek, K.: Potential Theory approach as a powerful tool in theoretical prediction of mixed-gas adsorption equilibria and their accompanying calorimetric effects. Chem. Eng. Sci. 60(10), 2763–2769 (2005)
Nieszporek, K.: Application of the vacancy solution theory to describe the enthalpic effects accompanying mixed-gas adsorption. Langmuir 22(23), 9623–9631 (2006)
Bhadra, S.J.; Ebner, A.D.; Ritter, J.A.: On the use of the dual process Langmuir model for predicting unary and binary isosteric heats of adsorption. Langmuir 28(17), 6935–6941 (2012)
Calleja, G.; Pau, J.; Calles, J.: Pure and multicomponent adsorption equilibrium of carbon dioxide, ethylene, and propane on ZSM-5 zeolites with different Si/Al ratios. J. Chem. Eng. Data 43(6), 994–1003 (1998)
Talu, O.; Zwiebel, I.: Multicomponent adsorption equilibria of nonideal mixtures. AIChE J. 32(8), 1263–1276 (1986)
Chen, Y.; Ritter, J.; Yang, R.: Nonideal adsorption from multicomponent gas mixtures at elevated pressures on a 5A molecular sieve. Chem. Eng. Sci. 45(9), 2877–2894 (1990)
Hyun, S.H.; Danner, R.P.: Equilibrium adsorption of ethane, ethylene, isobutane, carbon dioxide, and their binary mixtures on 13X molecular sieves. J. Chem. Eng. Data 27(2), 196–200 (1982)
Ritter, J.; Yang, R.: Equilibrium adsorption of multicomponent gas mixtures at elevated pressures. Ind. Eng. Chem. Res. 26(8), 1679–1686 (1987)
Reich, R.; Ziegler, W.T.; Rogers, K.A.: Adsorption of methane, ethane, and ethylene gases and their binary and ternary mixtures and carbon dioxide on activated carbon at 212–301 K and pressures to 35 atmospheres. Ind. Eng. Chem. Process Design Dev. 19(3), 336–344 (1980)
Gumma, S.: On Measurement, Analysis and Modeling of Mixed Gas Adsorption Equilibria. Cleveland State University, Cleveland (2003)
Nolan, J.T.; McKeehan, T.W.; Danner, R.P.: Equilibrium adsorption of oxygen, nitrogen, carbon monoxide, and their binary mixtures on molecular sieve type 10X. J. Chem. Eng. Data 26(2), 112–115 (1981)
Qiao, S.; Wang, K.; Hu, X.: Study of binary adsorption equilibrium of hydrocarbons in activated carbon using micropore size distribution. Langmuir 16(11), 5130–5136 (2000)
Loughlin, K.; Hasanain, M.; Abdul-Rehman, H.: Quaternary, ternary, binary, and pure component sorption on zeolites. 2. Light alkanes on Linde 5A and 13X zeolites at moderate to high pressures. Ind. Eng. Chem. Res. 29(7), 1535–1546 (1990)
Costa, E.; et al.: Adsorption equilibrium of ethylene, propane, propylene, carbon dioxide, and their mixtures on 13X zeolite. J. Chem. Eng. Data 36(2), 218–224 (1991)
Danner, R.P.; Choi, E.C.: Mixture adsorption equilibria of ethane and ethylene on 13X molecular sieves. Ind. Eng. Chem. Fundam. 17(4), 248–253 (1978)
He, Y.; Yun, J.-H.; Seaton, N.A.: Adsorption equilibrium of binary methane/ethane mixtures in BPL activated carbon: isotherms and calorimetric heats of adsorption. Langmuir 20(16), 6668–6678 (2004)
Al-Baghli, N.A.; Loughlin, K.F.: Adsorption of methane, ethane, and ethylene on titanosilicate ETS-10 zeolite. J. Chem. Eng. Data 50(3), 843–848 (2005)
Al-Baghli, N.A.; Loughlin, K.F.: Binary and ternary adsorption of methane, ethane, and ethylene on titanosilicate ETS-10 zeolite. J. Chem. Eng. Data 51(1), 248–254 (2006)
Abdul-Rehman, H.; Hasanain, M.; Loughlin, K.: Quaternary, ternary, binary, and pure component sorption on zeolites. 1. Light alkanes on Linde S-115 silicalite at moderate to high pressures. Ind. Eng. Chem. Res. 29(7), 1525–1535 (1990)
Nakahara, T.; Hirata, M.; Omori, T.: Adsorption of hydrocarbons on a carbon molecular sieve. J. Chem. Eng. Data 19(4), 310–313 (1974)
Nakahara, T.; Hirata, M.; Mori, H.: Adsorption of a gaseous mixture of ethylene and propylene on a carbon molecular sieve. J. Chem. Eng. Data 27(3), 317–320 (1982)
Nakahara, T.; et al.: Adsorption of binary gaseous mixtures of ethylene–ethane and ethylene–propylene on a carbon molecular sieve. J. Chem. Eng. Data 29(2), 202–204 (1984)
Nakahara, T.; Wakai, T.: Adsorption of ethylene/ethane mixtures on a carbon molecular sieve. J. Chem. Eng. Data 32(1), 114–117 (1987)
Wu, Q.; et al.: Adsorption equilibrium of the mixture \(\text{ CH }_4+ \text{ N }_2+ \text{ H }_2\) on activated carbon. J. Chem. Eng. Data 50(2), 635–642 (2005)
Dunne, J.; et al.: Calorimetric heats of adsorption and adsorption isotherms. 1. \(\text{ O }_2, \text{ N }_2, \text{ Ar }, \text{ CO }_2, \text{ CH }_4, \text{ C }_2\text{ H }_6\), and \(\text{ SF }_6\) on silicalite. Langmuir 12(24), 5888–5895 (1996)
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We thank the Persian Gulf University, for the financial support, and for granting the required approval for this study.
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Bakhtyari, A., Mofarahi, M. A New Approach in Predicting Gas Adsorption Isotherms and Isosteric Heats Based on Two-Dimensional Equations of State. Arab J Sci Eng 44, 5513–5526 (2019). https://doi.org/10.1007/s13369-019-03838-2
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DOI: https://doi.org/10.1007/s13369-019-03838-2