自簡單系統的可測擴散參數估計複雜系統的擴散參數

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自簡單系統的可測擴散參數估計複雜系統的擴散參數

計畫類別： 個別型計畫 計畫編號： NSC91-2113-M-009-018- 執行期間： 91 年 08 月 01 日至 92 年 12 月 31 日 執行單位： 國立交通大學應用化學系 計畫主持人： 陳振興 計畫參與人員： 高大宇、張文益，吳承昌 報告類型： 精簡報告 處理方式： 本計畫可公開查詢

中 華 民 國 93 年 2 月 4 日

行政院國家科學委員會補助專題研究計畫

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自簡單系統的可測擴散參數估計複雜系統的擴散參數

計畫類別：■個別型計畫 □ 整合型計畫

計畫編號：NSC 91－2113－M－009－018

執行期間：91 年 08 月 01 日至 92 年 12 月 31 日

計畫主持人：陳振興

國立交通大學應用化學系

計畫參與人員：高大宇、張文益，吳承昌

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國立交通大學應用化學系

中 華 民 國 93 年 02 月 04 日

附件一

中文摘要

關鍵詞:傳輸矩陣、泰勒級數展開、平均首度通過時間、延滯時間

我們先行求出 Laplace domain 下的傳輸矩陣的形式是 Laplace variable s 的泰勒級 數展開式，展開係數可用成份薄板的可測擴散參數表示，如延滯時間、滲透係數正向及反 向前導時間，及其高階矩擴散參數。有了這個工具之後，複合體擴散諸參數就可以用成份 薄板的擴散參數表示。於是我們不必要直接測量複合體的擴散係數而由成份薄片的擴散參 數組合即可。以平均首度通過時間為例，闡明此新法之應用。

Abstract

Key word: transmission matrix, Taylor’s expansion, mean first passage time, time lag

The membrane transport in terms of transmission matrix, T(s), in the Laplace domain has been extended to a more useful formulation. This is achieved with the help of a combination of the uses of the matrix transport equations appropriate for initial conditions of void concentration or saturated equilibrium within the membrane and the theorem det[T(s)] = 1. This new formulation gives T(s) in power series of the Laplace variable, s, with the expansion coefficients as the algebraic functions of the experimentally measurable parameters: permeability, time lag, forward and backward time leads and their higher moments. The utility of this new formulation of the transmission matrix is illustrated in the estimation of the time moments for the first passage and residence times, which are not measurable directly from the experiments.

Background, Purpose and Significance of the Investigation

Permeation transport across membranes is of great importance in science and technology, and has played a crucial role in such diversed fields as chemical sensors,1,2 controlled release,3,4 separation processes,5 protection against pollution,6 electrodialysis,7 to name just a few. In practical applications, three modes of membrane transport are employed:8 (a) absorptive permeation, zero and the activities at the upstream and downstream faces are at a constant level

0

a (b) forward and/or backward desorptive permeation, (c) desorption, For heterogeneous membranes, where the diffusivity D(x) and partition coefficientK(x) depending on position, the permeation can be mathematically described by the Smoluchowski equation9-11

) x ( K ) t , x ( x ) x ( K ) x ( D x ) t , x ( t ρ ∂ ∂ ∂ ∂ = ρ ∂ ∂

. The complete knowledge about the membrane permeation entails the full time-dependent solution to the diffusion equation subject to appropriate boundary and initial conditions. Unfortunately such a solution is seldom obtained except for some simple, trivial cases. Thus one is usually satisfied with a few diffusion parameters characteristic of the permeation such as permeation (P), time lag (t ) for absorptive and time _L lead (t ) for desorptive permeation and their higher moments.₊ 12,13 All these permeation parameters can be obtained directly from the suitably designed experiments. Theoretically, they can also be formulated via the Taylor expansion of the transmission matrix, T(s), in power series of s with the expansion coefficients expressed in terms of the repeated integrals of

1 )] x ( D ) x ( K

[ − and K(x)14 or by the method of repeated integration of the diffusion equation.9,15 However, these formulations are useful only in the cases where the functions K(x) and D(x) are known beforehand.

Another concern about diffusion transport is the moments of the first passage time10,11,16 and residence time.17,18 Ordinarily, neither are obtainable directly from the experiments. Mathematically the former can be obtained by solving the adjoint (or backward) diffusionequation10,11 and the latter by the Green's function of the diffusion equation.17,18 Again these tasks are feasible only when K(x) and D(x) are known ahead of time.

Method of Investigation and Results

Starting with the matrix transport equation and the appropriate initial and boundary conditions and a further use of the definitions of the experimentally accessible diffusion parameters for P(permeability), tL(time lag), t+(forward time lead), t-(backward time lead) and their higher

series of Laplace variable, s, with the expansion coefficients as functions of the above mentioned parameters . Namely             =       =       ) s ( Jˆ ) s ( aˆ ) s ( T ) s ( T ) s ( T ) s ( T ) s ( Jˆ ) s ( aˆ ) s ( T ) s ( Jˆ ) s ( aˆ u u 22 21 12 11 u u d d (1) With Λ − + − − − − − − = 3 ) 3 ( L ) 2 ( L L 3 L 2 ) 2 ( L 2 L L 12 s P 6 t t t 6 t 6 s P 2 t t 2 s P t P 1 ) s ( T

Λ

−

β

−

β

−

β

−

β

−

=

3 3 2 2 1 0

s

s

s

(2) Λ + − − + + − + + = ₊ + + + + + + 3 ) 2 ( L ) 2 ( L 2 L ) 3 ( 2 ) 2 ( L 22 s 6 t t 3 t t 3 t t 6 t s 2 t t t 2 s t 1 ) s ( T Λ + δ + δ + δ + δ = 3 3 2 2 1 0 s s s (3) Λ + − − + + − + + = ₋ − − − − − − 3 ) 2 ( L ) 2 ( L 2 L ) 3 ( 2 ) 2 ( L 11 s 6 t t 3 t t 3 t t 6 t s 2 t t t 2 s t 1 ) s ( T =α₀ +α₁s+α₂s2 +α₃s3 +Λ (4) 2 ) 2 ( ) 2 ( 21 [2t t t t ]s 2 P s ] t t [ P ) s ( T =− ₊ + ₋ − _{+ −} − ₊ − ₋ Λ + + − − + − _{+ − + − + − + −} 3 L ) 2 ( ) 2 ( ) 3 ( ) 3 ( s ] t t t 6 t t 3 t t 3 t t [ 6 P (5) where t₊ =t_L −t₊ , t₊(2) =t(_L2) −t(₊2) , t(₊3) = t(_L3) −t(₊3) , t₋ =t_L −t₋ , t₋(2) =t(_L2) −t(₋2) and ) 3 ( ) 3 ( L ) 3 ( t t

t₋ = − ₋ . The super index number in the parenthesis denotes the respective rank of moment, and the definition of αi, βi, γi and δi are self-explanatory.

Application

As an example, we consider the first passage time for a particle initially located at x ₀ between a reflecting face x and an absorbing face _u x , _d x_u <x₀ <x_d.11,16 Solving this problem requires the matrix transport equation for an initial condition of δ-function typel4,19

      +       =       1 0 ) s ( T ) s ( Jˆ ) s ( aˆ ) s ( T ) s ( Jˆ ) s ( aˆ _* u u d d (6)

where T*(s) is the transmission matrix for the subdomain from x to ₀ x . After _d

substitution of Jˆ_u(s)=0 and aˆ_d(s)=0 into eqn. (6), the flux escaping the face x is _d calculated to be ) s ( T ) s ( T ) s ( T ) s ( T ) s ( T ) s ( Jˆ 11 * 12 21 * 22 11 d − = (7)

The mean first passage time can be represented by

) s ( Jˆ ds d 0 s lim dt ) t ( J dt ) t ( J t d 0 d 0 d 1 = =− _→ µ

∫

∫

∞ ∞ (8)

where the denominator represents the initial total amount which is equal to unity. Putting eqn. (7) into eqn. (8) followed by expanding the transition matrix elements in terms of eqns. (2), (3), (4) and (5), we obtain * * 1 t P P ] t t [ ₊ + ₋ − ₊ = µ (9)

Here the quantity associated with the superscript * denotes the fact this quantity is to be specified to the subdomain, x₀ <x<x_d. The second moment of first passage time is also easily calculated by

) s ( Jˆ ds d 0 s lim dt ) t ( J dt ) t ( J t d 2 2 0 d 0 d 2 2 = = _→ µ

∫

∫

∞ ∞ )] t t ( t 2 t 2 t t [ * P P t t t 2 *_L *₊ − ₊(2)*+ (₊2) + (₋2) + 2₋ − *_{L +} + ₋ = (10)

where t*₊ =t*_L −t*₊ and t₊(2)* =t(_L2)* −t(₊2)*. Thus with each quantity on the right-hand sides of eqs 9 and 10 being experimentally accessible, µ₁ and µ₂ can be estimated.

Conclusion

With such a Taylor expansion of transmission matrix as a tool and in cooperation with appropriate matrix transport equations, the problems of time moments for first passage time, and residence time can be solved algebraically. As a result, various time moments can be expressed by the algebraic function of the above-mentioned measurable parameters. Thus, some experimentally inaccessible time moments can be estimated from permeation parameters measurable in appropriate permeation experiments. This methodology for the estimation of the time moments would provide a useful vehicle for the treatment of the membrane permeation and other problems in diffusive transports.

Acknowledgment

We are grateful to the National Science Council of Taiwan for the financial support of this work under Grant Nos.NSC 91－2113－M－009－018.

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