Процессы массопереноса с участием твердой фазы. Учебное пособие

→

mechanisms. We can name it diffusion flow of mass jdi . In contrast to the

total flow, this value does not take into account convective mechanism of mass transfer, in spite of the fact that the latter can be present. Projection of the flux of mass by means of diffusion can be recorded on the Y-axis (when considering the change of this magnitude only in the direction of the ) in the form of

Let us introduce the value ji *(y), which characterizes the ratio

between the magnitudes of the flow at a distance y from the interfacial boundary and on the phase boundary:

Equation (1.41) can be rewritten, using the value ji *(y), then it can be solved in respect to d i and integrated, using the model of diffusion

boundary layer (area near the phase boundary, in which occur 99% of changes in concentration), from the phase boundary up to the outer boundary of the layer (δd). Respectively, concentration will change from the magnitude on the interfacial (boundary (b)) surface ( i b) up to the

magnitude in the nucleus (n) of the phase ( i n). It should be borne in mind,

that coefficient of turbulent diffusion is a function of the distance from the phase boundary D (y) and may not be taken out from under the integral

We can obtain the equation of mass delivery by the way of solving the equation (1.43) in respect to jdbiy . The expression in its right side, which

stands before the difference of concentrations, is called coefficient of mass delivery:

21

The difference of the numbers of chemical potentials or concentrations at a phase boundary and in the nucleus of a phase is called moving (driving) force of mass delivery. Difference of the moving force value from the zero is the necessary condition of the process. The meaning of coefficients in equation (1.45) can be easy understood, if we will solve the equation for obtaining this coefficients:

Coefficient of mass delivery – amount of substance of component i, transferred from the boundary of the phase into the nucleus of the phase or in the opposite direction per unit time, through the unit of the interfacial

surface per unit of the moving force. Coefficients β i and β I are defined by

equations (1.47) and (1.48), depending on the use as a moving force of a process, respectively, the difference of chemical potentials or concentrations. In future we will use mainly coefficients β i as the most frequently used in practice. Equations (1.41) - (1.48) are valid for twocomponent media. In case of multicomponent systems should be used matrixes of coefficients of mass delivery.

22

Integral form of mass delivery equation. Equations of interfacial transfer of substances in the integral form are more often used in practice in case, when they are obtained by means of averaging of local equations for a part or for overall interfacial surface F:

In general, such a record is conditional in case of a simultaneous variation of coefficient of mass delivery and of moving force over the interfacial surface, because it is impossible to divide the procedure of averaging of coefficient and of moving force (integral of a product is not equal to a product of integrals). In extreme cases, we can conduct an independent averaging of one value, but then the averaged value of the second variable will depend on the nature of variation and on the method of averaging of the first value. As a rule, moving force undergoes a much more change, than coefficient of mass delivery. Therefore, kinetic coefficient can be considered constant and then the moving force can be averaged as one variable.

Determination of coefficients of mass delivery, similarity of the corresponding processes. Mass delivery coefficients cannot be found theoretically from (1.46 ) for most practically important cases, therefore, they are usually determined by using method of physical modeling. It is based on a generalization of experimental data with the use of the theory of similarity in the form of criteria equations. In case of mass delivery, the determinate will be diffusion Nusselt criterion (number) Nud , also called

Sherwood criterion (number) Sh, and the determinative criteria - Reynolds number Re, diffusion Prandtl criterion Prd , also called Schmidt number

Sc, and diffusion criterion (number) of Fourier Fod :

Equations of mass transfer. Local form of equations. In this section transfer of mass component i from the phase I through the phase

23

barrier surface into the phase II by means of molecular and turbulent mechanisms will be discussed. Let us suppose, that interfacial surface resistance towards the transfer of substances can be neglected. This is equivalent to the assumption about the establishment of equilibrium at a phase boundary. Then you can write the equality of chemical potentials of component i from both sides of interfacial surface

Let us consider the derivation of mass transfer equation. If we denote by index I the first phase, out of which the transfer of substance occurs, then µiI > µiII . Then we shall direct Y-axis from phase I to II. In view of the above, we can write equations of mass delivery of each of the phases (1.45), dividing them on the corresponding coefficients:

βiII

We can add up these two equations, bearing in mind the assumption (1.54), and solve them in respect to the substance flow of component i, which passes through the interfacial surface:

before the difference between chemical potentials in its right-hand part is called coefficient of mass transfer. Thus,

24

00

In general, like for equation (1.49), such a record is conditional in case of a simultaneous change of kinetic coefficient and moving force on the interfacial surface, because it is impossible to divide the procedure of the averaging of kinetic coefficient and driving force. Similarly to mass delivery, it is possible to average one of the values independently, but then the magnitude of the second value will depend on the nature of variation of the first value.

1.4. Phase Equilibrium

Main purpose of this section is to establish the nature of relationship between concentrations of components in phases in equilibrium. Mass-exchange processes in chemical technology are taking place in systems, consisting of two or more components, so here will be considered binary and multi-component mixtures. The material is presented taking into account a detailed study of this issue in the courses of physics, thermodynamics and physical chemistry.

Conditions of equilibrium in a hetero phase system, which is not under the influence of external forces, are the equality of pressure, temperature and chemical potentials of components in all phases. The chemical potential of component i in each of phases is defined as the partial derivative of Gibbs energy with respect to the number of moles of the i-th component at the conditions of fixed pressure, temperature and quantity of matter of other components:

(1.63), then equation (1.64) establish a connection between these values, reducing the number of independent variables on the number of equations.

26

With this in mind, a number of degrees of freedom C (a number of independent variables) in the equilibrium hetero phase (F) system will be equal to

Thus we have received the ratio, known as Gibbs phase rule.

Now we proceed to establish the connection between concentrations of components in the phases in equilibrium. Formal relations, which replace (1.64), can be written very simply, they are called equations of equilibrium:

ci I = mIi,,cIIcIIi , cIi = mIi,,cIII cIIIi ,... (1.66)

The proportionality coefficients between concentrations in different phases bear the name of coefficients of distribution. The problem is to find them. Using Gibbs phase rule (1.65), you can determine the number of independent variables, from which will be dependent coefficients of distribution. Specific values can be found from (1.66), using experimental data about equilibrium concentrations, or from (1.64), if the nature of the concentration dependence of chemical potential is already set.

It should be noted, that the number of distribution coefficient depends on the way of expressing of concentrations in (1.66). So, if instead

systems, the upper indices of distribution coefficients can be omitted. If in each of the phases is not more than two components, then usually the corresponding them values are not marked by the lower index i, which shows the number of components. The implication is that this values refer to the distributed component. If interfacial boundary comes through the both components, then this values refer to the easy-flowing component. Concentration of the component in phases can be indicated by different letters, what allows not to mark a number of the phase. So, for the designation of mole fraction in a gaseous phase is usually used letter y, and in a liquid phase - x. In this case, equilibrium equations can be written in the form

27

For liquid mixtures are usually used two different standard statuses. If temperature and pressure of a mixture correspond to the liquid state of a pure component i, then just it must be selected as the standard status. If a pure component exists in the gaseous or solid state at the same temperature and pressure of a mixture, then the status of infinitely diluted system must be chosen for him as the standard:

Here and hereinafter, the values, characterizing pure component i (xi → 1), are indicated by one stroke, in the state of infinite dilution (xi → 0) - by two strokes, except the activities i and coefficients of activity γi , for which a stroke indicates only the choice between the first or the second standard status.

With the help of the notion of volatility, the condition of phase equilibrium can be formulated in another form, than in (1.64). Taking into account the equality of temperatures of phases, and, consequently, µi+, we obtain from (1.64) and (1.69):

Now we shall find the coefficients of distribution mi for different cases of equilibrium in two-phase systems.

Balance in a liquid - liquid system. Some liquids have limited solubility, which leads to the disintegration of a system into phases, for example, oil - water. Only strongly non-ideal mixtures, in which the energy of attraction of heterogeneous molecules is much less than

29

homogeneous, can exfoliate. Exfoliation can be sometimes observed only within a certain range of temperature and concentration values, and outside of this field the system will be a single phase. Pressure has little effect on the equilibrium in a liquid - liquid system. Condition for equilibrium in two-phase liquid system has the form

Thus, finding of distribution coefficient is reduced to the calculation of activity coefficients of component i in different phases.

Determination of coefficients of activity is a complex problem. In practice the model equations, describing the deviation of a liquid mixture from the ideal, are often used. They are based on various models of determination of Gibbs energy, excessive in relation to the ideal mixture, what allows you to calculate coefficients of activity, using several parameters for each pair of components, which compose the mixture. These parameters are extracted from experimental data on phase equilibrium of binary systems and are listed in the reference literature. The most universal can be considered the uniform chemical equation UNIQUAC. In addition, for this purpose are also widely used more simple equations of Wilson and NRTL. For mixtures of substances, for which there are no experimental equilibrium data, it is possible to forecast the calculation of coefficients of activity with the help of equations UNIFAC. It is based on the equation of UNIQUAC and the method of group components, the essence of which consists in the determination of the substance properties by the way of analysis of its structural groups (methyl, carbonyl, etc.).

Equilibrium in a solid - liquid system. Let us consider the simplest case, when a pure component i is a solid (s). The condition of its equilibrium with a mixture of liquid components will have the form

The use of distribution coefficient in this case does not have any sense, as the mole fraction of component i in the solid phase is known (is equal to unity). The value, which have to be determined, is the equilibrium

30