Abstract
Theoretical development of a new experimental method for investigation of mass transport in porous membranes, based on the principle of the modified Wicke-Kallenbach diffusion cell and the nonlinear frequency response analysis is presented. The method is developed to analyze the transport of a binary gas mixture in a porous membrane. The mixture is assumed to consist of one adsorbable and one inert component. Complex mass transfer mechanism in the membrane, where bulk or transition diffusion in the pore volume and surface diffusion take place in parallel, is assumed. Starting from the basic mathematical model equations and following a rather standardized procedure, the frequency response functions (FRFs) up to the second order are derived. Based on the derived FRFs, correlations between some characteristic features of these functions on one side, and the whole set of equilibrium and transport parameters of the system, on the other, are established. As the FRFs can be estimated directly from different harmonics of the measured outputs, these correlations give a complete theoretical basis for the proposed experimental method. The method is illustrated by quantifying the transport of helium (inert gas) and C3H8 and CO2 (adsorbable gases) through a porous Vycor glass membrane.
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Abbreviations
- A :
-
Input amplitude, general and of the dimensionless partial pressure of the adsorbable component in the feed stream
- a i,j :
-
Dimensionless first derivative of the pore diffusivity
- a s :
-
Dimensionless first derivative of the surface diffusivity
- a Φ :
-
Dimensionless first derivative of the adsorption isotherm
- B :
-
Output amplitude, general
- b Φ :
-
Dimensionless second derivative of the adsorption isotherm
- D p , m2/s:
-
Pore diffusivity
- D s , m2/s:
-
Surface diffusivity
- F n :
-
n-th order FRF corresponding to the dimensionless partial pressure of the adsorbable component in the closed chamber
- G n :
-
n-th order FRF corresponding to the dimensionless partial pressure of the inert component in the closed chamber
- H n :
-
n-th order FRF corresponding to the dimensionless total pressure in the closed chamber
- J, mol/m2/s:
-
molar flux
- K :
-
Auxiliary parameter (Table 1)
- L, m:
-
Membrane thickness
- P, Pa:
-
Pressure
- p :
-
Dimensionless pressure
- Q, mol/m3:
-
Concentration in the solid phase
- q :
-
Dimensionless concentration in the solid phase
- R, J/mol/K:
-
Gas constant
- T, K:
-
Temperature
- V, m3:
-
Volume
- x :
-
Input, general
- y :
-
Output, general
- z, m:
-
Spatial coordinate
- α :
-
Auxiliary parameter (Table 1)
- β :
-
Auxiliary parameter (Table 1)
- ε :
-
Porosity
- Φ:
-
Adsorption isotherm relation
- φ :
-
Auxiliary parameter (Table 1)
- γ :
-
Auxiliary parameter (Table 1)
- ϕ, rad:
-
Phase shift
- τ :
-
Tortuosity factor
- ω, rad/s:
-
Frequency
- 1:
-
Adsorbable component
- 2:
-
Inert component
- I :
-
Open chamber, first harmonic
- II :
-
Closed chamber, second harmonic
- III :
-
Third harmonic
- ads :
-
Adsorbed phase
- atm :
-
Atmospheric
- g :
-
Gas phase
- i :
-
Component i
- s :
-
Steady-state
- tot :
-
Total
- 0:
-
Feed
- ∗:
-
Auxiliary FRF
- FR:
-
Frequency response
- FRF:
-
Frequency response function
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Petkovska, M., Markovic, A., Lazar, M. et al. Investigation of gas transport through porous membranes based on nonlinear frequency response analysis. Adsorption 17, 75–91 (2011). https://doi.org/10.1007/s10450-010-9293-3
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DOI: https://doi.org/10.1007/s10450-010-9293-3