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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
Article

Abstract

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.

Keywords

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

D(xt)

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

F

Time gradient of Q (Eq. 8)

G

Mass flux (Eq. 9)

L

Length of the vessel

\(N^\mathrm{I}\)

Number of temporal increment

\(N_x \)

Number of grid points along the x-direction

\(N_\mathrm{T}\)

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

p

Dimensionless pressure

\(\bar{{p}}\)

Pressure

\(p_i \)

Dimensionless initial pressure

\(p_{\mathrm{max}}\)

Maximum pressure

Q

Dimensionless adsorption capacity

\(Q_o \)

Dimensionless initial adsorption capacity

\(r_i \)

Radius of the vessel

\(R_p \)

Adsorbent particle radius

t

Dimensionless time

\(t_\mathrm{f} \)

Final time of discharge

\(\bar{{t}}\)

Time

T

Temperature

\(\bar{{x}}\)

Axial coordinate variable

x

Dimensionless axial coordinate

Y

Desired pressure

Greek symbols

\(\alpha _i \)

Weighting coefficient

\(\beta \)

Search step size

\(\varepsilon \)

Convergence criteria

\(\varepsilon _b \)

Porosity

\(\gamma \)

Conjugate coefficient

\(\lambda \)

Lagrange multiplier

\(\mu \)

Viscosity

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

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

Notes

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.

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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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