Процессы массопереноса с участием твердой фазы. Учебное пособие

Vector – a value, characterized by module (length) and direction. In the Cartesian system of coordinates the vector is given by three numbers - projections on the appropriate axis :

→→ →

where i , j , k are the unit vectors, directed along X, Y, Z axes

respectively.

Vector can be represented in form of a row or a column:

1.2. Some Operations with the Values

Multiplication of a scalar value on a vector - all projections are multiplied by a scalar (has the property of commutativity):

Scalar product of vectors - the corresponding projections of two vectors are multiplied and the results are added (possesses the properties of commutativity):

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→

Differential Hamilton operator (inverted delta) is a vector operator. It is not a value itself and can be only used to indicate the operation with the value, recorded on the right of it, as well as any other derivative. In the Cartesian coordinate system it can be represented in the form of

The usage of 2 towards scalar values and vectors formally means the multiplication of operator on corresponding value:

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2.1. Expression for the Matrix of Multicomponent Diffusion Coefficients in Medium Volume System

According to the approach of independent diffusion, the matrix of coefficients of multicomponent diffusion in liquid mixture can be expressed through the Einsteinian diffusion coefficients [12 - 14]:

n−1

Dij = ∑ ((Diδik − ci (Vk Dk − Vn Dn )) • k =1

Dij = Di δij − xi (Dj − Dn ).

2.2.Calculation of Einsteinian Coefficients of Diffusion

in Gas Mixtures

In the gases of a moderate density the Einsteinian coefficients of diffusion can be determined on the basis of molecular characteristics. So, for a spherically symmetric potential of the intermolecular interaction of Lennard-Jones they can be found from the expression [12]

component i; σi , εi - parameters of the Lennard-Jones potential, the values

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of which for a number of substances are presented in [3], and can also be found on the critical parameters [3] ( σi - a distance between the molecules,

in which ϕ(σi ) = 0 - see Fig. 1.1), and εi the - depth of a potential hole);

T, p is pressure and temperature in the system; xi - mole fraction of component i; Ωij (Tij* ) is a function, taking into account the difference of

Lennard-Jones potential from the model of solid spheres, which convenient approximation is shown in [3]; n - the number of mixture components.

In the highly rarefied gases, where the length of a free path significantly exceeds the size of a vessel (Knudsen`s diffusion in pores) diffusion coefficients depend on the pores radius r:

Di = 2 r(8kT / πmmi )1/ 2 3

2.3. Calculation of Einsteinian Diffusion Coefficients

in Liquid Mixtures

A large set of experimental data and semi-empirical ratios exist for the values of Dij0 , for example, obtained by Wilkie-Cheng [3, 15]:

where m j , µ j - molar mass [kg/kmol] and coefficient of dynamic viscosity of the solvent [MPa·s]; βj - coefficient, which takes into account the association of the solvent molecules; υi - molar volume at the normal boiling point of a pure component i [cm3/mol].

BIBLIOGRAPHY: REFERENCIES AND FURTHER READING

94

Liquids. - Mc Graw Hill, New York, 1977 ( ., .,

. # ' & : #. / .

>. . &. - /.: % ", 1982, - 592 .)

4. Y.I. Dytnerskiy. Processes and Apparatuses of Chemical Technology. – Vol. 2. Mass Exchange Processes and Apparatuses. – M: Chemistry, 2002

5. Adsorption and Diffusion. - Series: Molecular Sieves. Science and Technology. Vol. 7. / Editors: H.G. Karge, J. Weitkamp. – Springer, 2008.

7.Ion Exchange Membranes: Fundamentals and Applications. - Membrane Science and Technology Series. Vol. 12 / Edited by Yoshinobu Tanaka. – Elsevier, 2007.

8.Fr. Rouquerol, L. Rouquerol, K. Sing. Adsorption by Powders and Porous Solids. Principles, Methodology and Applications. – Academic Press, 1999.

9.Ramm V.M. Absorption of Gases. – M: Chemistry, 1976. – 655 p.

( . . " #. - M: % ", 1976. – 655 .)

10.V.G. Levich. Physico-chemical hydrodynamics. 1: Fizmatlit, 1959 –

670p. ( . . ( &- & " # &. –

1: (, 1959. – 670 .)

11.Aleksandrov I.A. Mass Transfer in Distillation and Absorption of Multicomponent Mixtures. L.: Chemistry, 1975. - 320 p.

( . . 1 & & # & . /.: % ", 1975. 320 .)

12.Diakonov S.G., Razinov A.I. Kinetic Description of Multicomponent diffusion in Gases and Liquids. // J. Theor. Phys. - 1980. - V.50. - @9. - P.1948-1954. ("& .)., . . & # & # ' & ". // A!(. - 1980. -

!.50. - @ 9. - C.1948-1954).

13.Diakonov S.G., Razinov A.I. Description of the Isobaric-Isothermal Diffusion in an Ideal Liquid Mixtures. - Theor. Fund. Chem. Techn. - 1982.

95

- V.16. - @1. - P.105-109. ("& .)., . . *- & ' & ". //

!*%!.- 1982. - !.16. - @1. - .105-109).

14. Diakonov S.G.,, Kazantsev S.A., Manevich G.SH., Razinov A.I.

Forecasting of Diffusion Coefficients in Liquid Mixtures on the Basis of Nitrogen Dioxide // Bulletin of the Academy of Sciences of BSSR. Ser. Phys.- Energ. - 1989. - @1. - P. 58-63 (# & .' & " & / "& .).,. ., 1 ).;., . . // 4 & >. .

.-. #. - 1989. - @1. - . 58-63).

15. Pavlov C.P., Romankov P.G., Noskov A.A. Examples and Tasks for the Course of Processes and Devices of Chemical Technology. - L.: Chemistry, 1987. - 574 p. ( .(., & .)., & . .& & #. - /.:

% ", 1987. - 574 c.)

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