**1. Introduction**

204 Mass Transfer in Chemical Engineering Processes

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(2011). The effective recovery of praseodymium from mixed rare earths via a hollow fiber supported liquid membrane and its mass transfer related. *J. Alloy.*  Membranes become the key component of modern separation technologies and allow exploring new opportunities and creating new molecular selective processes for purification, concentration and separation of liquids and gases (Baker, 2002, 2004). Particularly the development of new highly effective processes of gas separation with application of existing materials and membranes takes specific place. In present time special attention devotes to purification of gas and liquid waste streams from ecologically harmful and toxic substances such as greenhouse gases, VOCs and others. From the fundamental point of view the development on new highly effective processes of gas separation demands the investigation of mass transfer in the unsteady (kinetic) area of gas diffusion through a membrane. This approach allows in some cases to obtain much higher selectivity of separation (using the same membrane materials) compared to traditional process where steady state conditions are applied. First studies of membrane separation processes under unsteady state conditions have demonstrated both opportunities and problems of such approach (Beckman, 1993; Hwang & Kammermeyer, 1975; Paul, 1971).

It was shown that effective separation in unsteady membrane processes is possible if residence times of mixture components significantly differ from each other that is the rare situation in traditional polymeric materials but well known for liquid membranes with chemical absorbents (Shalygin et al., 2006). Nevertheless similar behavior is possible in polymeric membranes as well when functional groups which lead to partial or complete immobilization of diffusing molecules are introduced in polymer matrix. Moreover the functioning of live organisms is related with controllable mass transfer through cell membranes which "operate" in particular rhythms. For example scientific validation of unsteady gas transfer processes through membranes introduces particular interest for understanding of live organisms' breathing mechanisms.

It can be noticed that development of highly effective unsteady membrane separation processes is far from systematic understanding and practical evaluation. Therefore the evolution of investigations in this area will allow to accumulate new knowledge about unsteady gas separation processes which can be prototypes of new pulse membrane separation technologies.

Theoretical description of unsteady mass transfer of gases in membranes is presented in this work. Examples of binary gas mixture separation are considered for three cases of gas

Particularities of Membrane Gas Separation Under Unsteady State Conditions 207

the output membrane surface the partial gas pressure is keeping close to zero during whole diffusion experiment. At the beginning the gas transfer is unsteady and then after definite

In the frames of "classical" diffusion mechanism (that is the diffusion obedient to Fick's law and the solubility – to Henry's law) the unsteady distribution of concentration of diffusing gas *C*(*x,t*) across the flat membrane with thickness *Н*, is determined by the 2nd Fick's law:

> *Cxt Cxt* ( ,) ( ,) *<sup>D</sup> t x*

Standard initial and boundary conditions are: *C*(0,*t*)=*Cu*; *C*(*H,t*)=0; *C*(*x,*0)=0, where *Сu* is the concentration of gas in membrane respected to partial pressure of gas at the upstream side

The unsteady gas flux through membrane follows from the solution of Eq. (5) and can be

 <sup>2</sup> <sup>2</sup> <sup>2</sup> 0 4 2 1 4 4 *ss m*

 *Dt Dt* 

The series of the Eq. (7) is converged at small values of time and the series of the Eq. (7') is

Traditionally, membrane gas transfer parameters *Р*, *D* and *S* can be found from two types of experimental time dependencies: (1) the dependence of gas volume *q*(*t*) or (2) the dependences of gas flow rate *J*(*t*), permeated through a membrane. The pulse function variation of gas concentration in upstream is applied enough rare in experimental studies and corresponding

*n Dt D S*

1 2 1 exp

1

 

1

*n Dt D S*

1 2 1 exp

2 2

 

*H m H*

2

*H* 

2

2

(5)

*C Sp u u* , (6)

(7)

, (7')

( ) ( ) ( ) *dJ t dqt j t dt dt* (8)

*A*

(9)

2 2 2

2 2 2

*H*

*H*

time the steady-state gas transfer is achieved.

where *S* is solubility coefficient of gas in polymer.

*Jt J*

*DC <sup>А</sup> PAp <sup>J</sup> H H* is steady-state gas flux.

response function *j*(*t*) in downstream relates with other functions as follows:

The unsteady selectivity for a gas pair can be expressed using Eq. (7) as follows:

*A A n n US <sup>B</sup> B B n n*

*ss*

1 1 2 1 exp *<sup>n</sup>*

*<sup>n</sup> Jt J Dt*

*n*

in accordance with Henry's law:

expressed in two forms:

where *<sup>u</sup> <sup>u</sup> SS*

converged at high values of time.

concentration variation on membrane: step function, pulse function and harmonic function. Unsteady gas flow rates and unsteady separation factors are calculated for all cases. Amplitude-frequency, phase-frequency and amplitude-phase characteristics as well as Lissajous figures are calculated for harmonic functions. The comparison of mixture separation efficiency under steady and unsteady mass transfer conditions is carried out. Calculations were performed for oxygen-nitrogen and oxygen-xenon gas mixtures separation by membranes based on polyvinyltrimethylsilane and for CO2 transfer in liquid membrane with chemical absorbent of CO2.
