# Comparison of Uniform and Non-uniform Pressure Approaches Used to Analyze an Adsorption Process in a Closed Type Adsorbent Bed

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## Abstract

Heat and mass transfer in an annular adsorbent bed filled with silica gel particles is numerically analyzed by uniform and non-uniform pressure approaches. The study is performed for silica gel–water pair, particle radius from 0.025 to 1 mm and two bed radii of 10 and 40 mm. For uniform pressure approach, the energy equation for the bed and the mass transfer equation for the particle are solved. For non-uniform pressure approach, the continuity and Darcy equations due to the motion of water vapor in the bed are added, and four coupled partial differential equations are solved. The changes of the adsorbate concentration, pressure, and temperature in the bed throughout the adsorption process for both approaches are obtained and compared. The obtained results showed that the particle size plays an important role on the validity of uniform pressure approach. Due to the interparticle mass transfer resistance, there is a considerable difference between the results of the uniform pressure and non-uniform pressure approaches for the beds with small size of particles such as \(r_\mathrm{{p}} =\) 0.025 mm.

## Keywords

Heat and mass transfer Inter and intraparticle mass transfer resistance Adsorbent bed## List of Symbols

## Variables

- \(C_\mathrm{p}\)
Specific heat of adsorbent (Jkg\(^{-1}\) K\(^{-1}\))

- \(D_\mathrm{{eff} }\)
Effective mass transfer diffusivity (m\(^{2}\) s\(^{-1}\))

- \(D_\mathrm{K}\)
Knudsen diffusivity (m\(^{2}\) s\(^{-1}\))

- \(D_\mathrm{m}\)
Molecular diffusivity (m\(^{2}\) s\(^{-1}\))

- \(D_{\mathrm{{bed}}}\)
Effective diffusivity of adsorptive in adsorbent bed (m\(^{2}\) s\(^{-1}\))

- \(D_{\mathrm{{o}}}\)
Reference diffusivity (m\(^{2}\) s\(^{-1}\))

- \(E\)
Diffusional activation energy (J mol\(^{-1}\))

- \(K_\mathrm{{inh}}\)
Inherent permeability of adsorbent bed (m\(^{2}\))

- \(K_\mathrm{{app}}\)
Apparent permeability of adsorbent bed (m\(^{2}\))

- \(M\)
Molecular weight of adsorptive (kg mol\(^{-1}\))

- \(P\)
Pressure (Pa)

- r\(_\mathrm{{p}}\)
Radius of adsorbent granule (m)

- \(R\)
Radius of bed, m; ideal gas constant (J mol\(^{-1}\) K\(^{-1}\))

- \(T\)
Temperature (K)

- \(t\)
Time (s)

- \(V_{\mathrm{{r}}}\)
Adsorptive velocity (m s\(^{-1}\))

- \(\overline{{W}}\)
Average adsorbate concentration (kg\(_\mathrm{{l }}\)/kg\(_\mathrm{{s}}\))

- \(W_{\infty }\)
Local adsorbate concentration (kg\(_\mathrm{{l}}\)/kg\(_\mathrm{{s}}\))

## Greek Symbols

- \(\rho \)
Density (kg m\(^{-3}\))

- \(\Delta H_\mathrm{{ads}}\)
Heat of adsorption (J kg\(^{-1}\))

- \(\phi \)
Porosity

- \(\phi \)
A dependent variable

- \(\lambda _\mathrm{{eff}}\)
Effective thermal conductivity (W m\(^{-1}\) K\(^{-1}\))

- \(\mu \)
Adsorptive viscosity (Ns m\(^{-2}\))

- \(\sigma \)
Collision diameter for Lennard–Jones potential (A\(^{0}\))

- \(\varOmega \)
Collision integral

- \(\tau \)
Tortuosity

## Subscripts

- a, d
Final and initial conditions of adsorption

- i
Inner

- l
Adsorptive

- o
Outer

- s
Adsorbent

- sat
Saturation

- v
Adsorbate

- \(\infty \)
Equilibrium

## Notes

### Acknowledgments

The authors express their very sincere thanks to the reviewers for their valuable comments and suggestions.

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