Agriculture 4. Technologies of storage and processing of agricultural products

Assoc. Prof. V.Yu. Ovsyannikov, graduate Ya.I. Kondrateva,

graduate student Yu.S. Kraminova. student T.S. Kirichenko

Voronezh State University of Engineering Technologies, Russia

The partial crystallization kinetics of moisture in the apparatus

of periodic action when the concentration of liquid media

There are currently a sufficient number of papers devoted to mathematical simulation of processes of crystallization of ice in the apparatus of periodic and continuous action with a refrigerant concentration of liquid media, however, there is no single stochastic kinetic theory of the formation and growth of ice. In this paper an attempt is made to solve this problem [1, 2].

When building a mathematical description, it is assumed that the growth of ice crystals is carried out in the source liquid environment while maintaining the required cooling conditions occur the diffusion processes of mass transfer to the surface of an ice crystal.

We introduce the following notation, let – the density distribution function of the number of ice crystals weight m per unit volume of the apparatus at time ; - the concentration of dry soluble substances in the liquid phase at time ; - the growth rate of ice crystals; - the rate of formation of new centers of crystallization mass , formed by melting of the particles with mass ; - the initial mass of the ice crystal. Then, given the stochastic of the process of formation and growth of ice crystals, from the balance by the number of micro particles subject to the law of conservation of mass we write the following system of equations:

, (1)

, (2)

где - stochastic parameter (diffusion coefficient); - the number of ice crystals per unit volume of the apparatus at time ; - the Delta Dirac function; - the economic rate. For solving systems of equations we introduce the initial and boundary conditions

, (3)

для и , (4)

that is, the function exists only in the field , but outside this area identically equal to zero.

Multiplying the expression (1) on left and right and integrating by from zero to infinity; subject to the conditions (3) and (4) we get the equation for finding moment

(5)

From equation (5), when get

(6)

, (7)

(8)

Equation (2), (6) – (8) allow to determine such quantities as and variance of the distribution .

Theoretical description of the process of freeze – crystallization from a solution based on two main approaches. The first one considers the one-dimensional solidification front and continuous crystallization within the volume of the liquid phase is ignored, and the temperature at the interface is taken equal to the equilibrium temperature of the crystallization solution of a given concentration. The second approach is limited to consideration of the crystallization kinetics, neglecting the thermal processes inside the solid phase. The complexity of the problem due to the insufficient development of the theory only allows in some cases to reach a decision [3, 5].

Consider particular cases, which are of scientific and practical interest.

Let the process of crystallization is limited by the thermal processes inside the crystal. In this case there will be a fair equation

(9)

where - the maximum specific growth rate of an ice crystal; - constant. Considering equation (9) of the dependencies (2) and (7) follow the equations

, (10)

the solution of which can be represented in the form

(11)

. (12)

Let the crystallization process is limited by diffusion, then

(13)

where - the mass transfer coefficient, which depends on the temperature (through the diffusion coefficient) and the mixing conditions in the apparatus; - the surface area of the crystal.

Taking into account dependences (13), equations (2) and (7) is transformed to

(14)

Equation (14) contains an unknown ratio . However, calculations show that the value of

(15)

almost constant and close to unity. Given the above, equation (14) is solved in the form:

(16)

Thus, the proposed stochastic model of the formation and growth of ice crystals in cold concentration liquid media allows to solve wide range of problems of applied character [4, 5].

Literature

1. Pap L. Freeze concentration. Translation. with Hungarian, edited by O.G. Komyakova. - M.: Light and Food Industry, 1982. - 97 p.

2. Antipov S.T., Ovsyannikov V.Yu., Ryazanov A.N., Yashhenko S.M. Development of the model of analysis and forecast of the fundamental characteristics of the process of cryoconcentration. Storage and processing agricultural. 2001. № 4. pp. 36-38.

3. Ovsyannikov V.Yu. Study of the process of freezing moisture from the extracts of the endocrine and special raw material. Diss. cand. tech. the sciences. Voronezh. State. tech. Acad., 2003. 184 p.

4. Antipov S.T., Ovsyannikov V.Yu., Kondratyev Ya.I. Kinetics of the process of concentration by freezing the cherry juice. Herald of the Voronezh state university of the engineering technologies. 2014, № 4 pp. 44-48.

5. Antipov S.T., Dobromirov V.E., Ovsyannikov V.Yu. [Heat- and mass exchange with the concentration of liquid media by freezing. Voronezh. State. tech. Acad. Voronezh, 2004. 208 p.