Transport in Porous Media

, Volume 110, Issue 3, pp 613–626 | Cite as

Application of Inverse Method to Estimation of Gas Adsorption Isotherms

  • H. Rahideh
  • M. Mofarahi
  • P. Malekzadeh
  • M. R. Golbahar Haghighi


As a first step to analyze the inverse kinetic of adsorption, an inverse algorithm is developed to estimate equilibrium adsorption isotherm in a gas storage vessel by using the dynamic transient internal pressure. In the present study, no prior information is need for the functional form of the unknown isotherm equation to solve continuity equation. The conjugate gradient method is employed for optimization procedure. The incremental differential quadrature method as a computationally efficient and accurate numerical tool is applied to solve the corresponding direct, sensitivity and adjoint problems. The accuracy of the presented approach is examined by simulating the exact data of known model. Good accuracy of the obtained results validates the presented approach.


Gas adsorption isotherm Inverse problem Differential quadrature Conjugate gradient method 

List of symbols

\(A_{ij}^{(\alpha )}\)

The first-order weighting coefficient \(({\alpha =x,t})\)

\(B_{ij}^{(x)} \)

The second-order weighting coefficient along the x-axis


Direction of descent (i.e., search direction)


Time gradient of Q (Eq. 8)


Mass flux (Eq. 9)


Length of the vessel


Number of temporal increment

\(N_x \)

Number of grid points along the x-direction


Total number of grid points in the t-direction

\(N_t^I \)

Number of grid points in the t-direction for the ith time interval


Dimensionless pressure



\(p_i \)

Dimensionless initial pressure


Maximum pressure


Dimensionless adsorption capacity

\(Q_o \)

Dimensionless initial adsorption capacity

\(r_i \)

Radius of the vessel

\(R_p \)

Adsorbent particle radius


Dimensionless time

\(t_\mathrm{f} \)

Final time of discharge






Axial coordinate variable


Dimensionless axial coordinate


Desired pressure

Greek symbols

\(\alpha _i \)

Weighting coefficient

\(\beta \)

Search step size

\(\varepsilon \)

Convergence criteria

\(\varepsilon _b \)


\(\gamma \)

Conjugate coefficient

\(\lambda \)

Lagrange multiplier

\(\mu \)


\(\bar{{\rho }}_\mathrm{b}\)

Bulk density

\(\bar{{\rho }}_\mathrm{g} \)

Gas density

\(\bar{{\rho }}_l \)

Solid sorbent density

\(\sigma \)

Coefficient in Darcy equation

\(\sigma _s \)

Standard deviation of the measurements

\(\upsilon \)

Superficial velocity


Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Alifanov, O.M., Artyukhin, E., Rumyantsev, A.: Extreme methods for solving ill-posed problems with applications to inverse heat transfer problems. Begell House, New York (1995)Google Scholar
  2. Alifanov, O.: Solution of an inverse problem of heat conduction by iteration methods. J. Eng. Phys. Thermophys. 26, 471–476 (1974)CrossRefGoogle Scholar
  3. Bastos-Neto, M., Torres, A.E.B., Azevedo, D.C., Cavalcante Jr, C.L.: A theoretical and experimental study of charge and discharge cycles in a storage vessel for adsorbed natural gas. Adsorption 11, 147–157 (2005)CrossRefGoogle Scholar
  4. Beck, J.V., Blackwell, B., Clair, C.R.S.: Inverse heat conduction: Ill-posed problems. Wiley-Interscience, New York (1985)Google Scholar
  5. Folly, F., Neto, A.S., Santana, C.: An inverse mass transfer problem for the characterization of simulated moving beds adsorption columns. In: 5th international conference on inverse problems in engineering: theory and practice (2005)Google Scholar
  6. Gao, W., Engell, S.: Estimation of general nonlinear adsorption isotherms from chromatograms. Comput. Chem. Eng. 29, 2242–2255 (2005)CrossRefGoogle Scholar
  7. Golbahar Haghighi, M., Malekzadeh, P., Rahideh, H.: Three-dimensional transient optimal boundary heating of functionally graded plates. Numer. Heat Transf. Part B Fundam. 59, 76–95 (2011)CrossRefGoogle Scholar
  8. Haghighi, M.G., Eghtesad, M., Malekzadeh, P., Necsulescu, D.: Two-dimensional inverse heat transfer analysis of functionally graded materials in estimating time-dependent surface heat flux. Numer. Heat Transf. Part A Appl. 54, 744–762 (2008)CrossRefGoogle Scholar
  9. Hahn, T., Sommer, A., Osberghaus, A., Heuveline, V., Hubbuch, J.: Adjoint-based estimation and optimization for column liquid chromatography models. Comput. Chem. Eng. 64, 41–54 (2014)CrossRefGoogle Scholar
  10. Hashemi, M., Abedini, M., Malekzadeh, P.: Numerical modeling of long waves in shallow water using incremental differential quadrature method. Ocean Eng. 33, 1749–1764 (2006)CrossRefGoogle Scholar
  11. Khajehpour, S., Hematiyan, M., Marin, L.: A domain decomposition method for the stable analysis of inverse nonlinear transient heat conduction problems. Int. J. Heat Mass Transf. 58, 125–134 (2013)CrossRefGoogle Scholar
  12. Lugon Jr, J., Silva Neto, A.J., Santana, C.C.: A hybrid approach with artificial neural networks, Levenberg-Marquardt and simulated annealing methods for the solution of gas-liquid adsorption inverse problems. Inverse Probl. Sci. Eng. 17, 85–96 (2009)CrossRefGoogle Scholar
  13. Malekzadeh, P., Rahideh, H.: IDQ two-dimensional nonlinear transient heat transfer analysis of variable section annular fins. Energy Convers. Manag. 48, 269–276 (2007)CrossRefGoogle Scholar
  14. Marin, L., Hào, D.N., Lesnic, D.: Conjugate gradient-boundary element method for the Cauchy problem in elasticity. Q. J. Mech. Appl. Math. 55, 227–247 (2002)CrossRefGoogle Scholar
  15. Marin, L.: Numerical solution of the Cauchy problem for steady-state heat transfer in two-dimensional functionally graded materials. Int. J. Solids Struct. 42, 4338–4351 (2005)CrossRefGoogle Scholar
  16. Medi, B., Kazi, M.-K., Amanullah, M.: Nonlinear direct inverse method: a shortcut method for simultaneous calibration and isotherm determination. Adsorption 19, 1007–1018 (2013)CrossRefGoogle Scholar
  17. Mofarahi, M., Gholipour, F.: Gas adsorption separation of \(CO_2\)/CH 4 system using zeolite 5A. Microporous Mesoporous Mater. 200, 1–10 (2014)CrossRefGoogle Scholar
  18. Mofarahi, M., Seyyedi, M.: Pure and binary adsorption isotherms of nitrogen and oxygen on zeolite 5A. J. Chem. Eng. Data 54, 916–921 (2009)CrossRefGoogle Scholar
  19. Ozisik, M.N.: Inverse heat transfer: fundamentals and applications. CRC Press, Boca Raton (2000)Google Scholar
  20. Sacsa Diaz, R., Sphaier, L.: Development of dimensionless groups for heat and mass transfer in adsorbed gas storage. Int. J. Therm. Sci. 50, 599–607 (2011)CrossRefGoogle Scholar
  21. Vasconcellos, J.F.V., Silva Neto, A.J., Santana, C.C.: An inverse mass transfer problem in solid-liquid adsorption systems. Inverse Probl. Sci. Eng. 11, 391–408 (2003)CrossRefGoogle Scholar
  22. Woodbury, K.A.: Inverse engineering handbook. CRC press, Boca Raton (2002)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of Chemical Engineering, School of EngineeringPersian Gulf UniversityBushehrIran
  2. 2.Department of Mechanical Engineering, School of EngineeringPersian Gulf UniversityBushehrIran

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