pore size determination by NKP method

modified Gas Translation (GT)-model

\( P_i = \frac{k_{0,i}}{\sqrt{M_iRT}} \exp (-\frac{E_{P,i}}{RT}) \) (1)

where
\( k_{0,i} = k_{0}(d_p - d_{i})^3 \), \( k_0 = \frac{\sqrt{2 \pi} N_p}{6 \tau L A_m} \)

\(P_i\):permeance of i-th component \(\rm{[mol/(m^2 \ s \ Pa)]}\) \(E_{p,i}\):activation energy of permeation for the i-th component \(\rm{[J/ \ mol]}\)

\(R\):gas constant \(\rm{[J \ mol^{-1} \ K^{-1}]}\) \(T\):absolute temperature \(\rm{[K]}\) \(k_{0,i}\):permeation constant for the i-th component [-]

\(d_p\):pore diameter \(\rm{[m]}\) \(d_i\):diameter of the i-th component \(\rm{[m]}\) \(k_0\):structural constant \(\rm{[m^3]}\) \(N_p\):number of pores [-]

\(\tau\):tortuosity [-] \(L\):membrane thickness \(\rm{[m]}\) \(A_m\):membrane area \(\rm{[m^2]}\) \(M_i\):Molecular weight of i-th component \(\rm{[kg \ mol^{-1}]}\)

\( P_{He} = \frac{k_{0,He}}{\sqrt{M_{He}RT}} \exp (-\frac{E_{P,He}}{RT}) \) (2)

Therefore, gas Permeance normalized by He is obtained by Eq.(1) and Eq.(2).

\( \frac{P_i}{P_{He}} = \frac{(d_0 - d_i)^3}{(d_0 - d_{He})^3} \frac{\sqrt{M_{He}}}{\sqrt{M_i}}\exp (-\frac{E_{P,i}-E_{P,He}}{RT}) \) (3)

When the activation energy can be neglected for the sake of simplicity, Eq.(3) can be expressed as follows.

\( \frac{P_i}{P_{He}} = \frac{(d_0 - d_i)^3}{(d_0 - d_{He})^3} \frac{\sqrt{M_{He}}}{\sqrt{M_i}} \) (4)

\( f = \frac{P_i}{P_{He}\sqrt{\frac{M_{He}}{M_i}}} =\frac{(1 - \frac{d_i}{d_0})^3}{(1 - \frac{d_{He}}{d_0})^3} \) (5)

\(f\): Knudsen-based gas permeance [-]

The Knudsen-based gas permeances are applied to Eq.(5) to estimate the pore size.
Equation(5) shows that the value of \(d_i\) at \(f\)(Knudsen-based gas permeance) = 0 corresponds to the pore size.

Reference

1) Yoshioka T, Kanezashi M, Tsuru T. Micropore size estimation on gas separation membranes: a study in experimental and molecular dynamics. AIChE J. 2013;59:2179–2194.
2) H.R. Lee, T. Shibata, M. Kanezashi, T. Mizumo, J. Ohsita, and T. Tsuru, Pore-size-controlled silica membranes with disiloxane alkoxides for gas separation. J. Membr. Sci., 383 (2011) 152-158

How to use this web

1. Please select "Gas type" and "Molecular size"

2. Please input permeance \(P_i\) \(\rm{[mol/(m^2 \ s \ Pa)]}\)

3. Please click the button

pore size determination by NKP method

Gas type
Molecular weight,\(M_i\) [g/mol]
Molecular size,\(d_i\) [nm]
Type of Molecular size
Permeance [mol/(m2 s Pa)]
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pore size

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