Процессы массопереноса с участием твердой фазы. Учебное пособие

dynamic characteristics of molecules of this components and, therefore, of

Then the diffusion flux of the first component will be greater than the

Molecular mechanism will initiate the resulting transfer of a substance from the first part of the system to the second, which in the closed device (system) will lead to a rise of gradient of the density of the number of

w

Fig. 1.2. Diffusion flow (g) and convective speed, caused by diffusion in the closed device

Thus, it is difficult under the non-equilibrium conditions observe and study in pure form the molecular transfer of mass, because it requires artificial maintenance of the constancy of a pressure in a system. The difficulty exists in experimental determination of the values of Di and

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convective velocity ^w. Even if you have measured the flows of all

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components ^ ji and the field of concentrations ci in laboratory system of

reference, you can't solve a system of n equations (1.6), since it contains

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n+1 of unknown values (Di, w ). So, usually the diffusion flows are determined in the reference system, whose velocity relatively to laboratory system can be set quite easily, and, as a rule, in this case are used the frames of reference of medium-mass or medium-volume type.

Reference system is given by the condition of equality to zero of the total flow of corresponding feature (let us denote it zi) in this frame of reference (z):

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i=1

In the medium-mass reference system zi = mi (molar mass of a component), and in the medium-volume reference system zi = Vi (partial molar volume of a component Vi, m3/kmol):

Then the mole flow of component i in laboratory reference system can be presented in the form of

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It is more convenient to use in practice the diffusion coefficients, which are linking the flows not with gradients of chemical potentials, but with gradients of concentration. So, we can express chemical potentials through mole concentrations and use the ratio of

which allows to reduce the number of independent variables on one. Then it is possible to write:

system z depends on gradients of concentrations of all components. Coefficients of proportionality in (1.12) is called the matrix of multicomponent diffusion coefficients, and they are defined by the properties

closed device. In these conditions, total flow of the volume is equal to zero, i.e. the laboratory system of reference coincides with the medium volume. Therefore, the experimental data on coefficients of diffusion is usually cited for the medium-volume (v) frame of reference. In the special case of a two-

This ratio is called the first Fick`s law. Using concept of the independent diffusion, it's possible to express coefficients of mutual diffusion through the Einsteinian coefficients:

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remembered about the dependence of coefficients of diffusion (in addition to Einsteinian) on the choice of reference frame. So, for example, coefficients of binary diffusion in medium-mass system of reference (m) do not have the property of symmetry (mD*ij ≠ mD*ji) and can be expressed through vDij:

The flows of substance, in case of interfacial transfer, are often written on the phase boundary. Thus, as a rule, one or several components are not migrated through the phase interface (inert components). In a binary mixture, whose first component crosses the phase boundary (b) and another component does not, flows in one phase can be written in the frame of reference, where the flow of the second component is equal to zero :

obtain relations for coefficients of binary diffusion in the frame of reference, corresponding to the zero flow of the second component,

When c2 → 0 then 2 D12 → ∞ , that clearly demonstrates the

infinitely large range of changes of coefficients of binary diffusion depending on the choice of reference frame. They can take both positive and negative values according to the choice of reference frame and elements of z Dij . Only the Einsteinian diffusion coefficients of Di (1.5) do

not depend on the choice of the frame of reference, always positive, and have a clear physical meaning.

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Turbulent mechanism. Turbulent transfer of mass can be considered by the analogy with molecular and can be respectively represented as a result of the chaotic movement of vortices. Then can be introduced coefficient of turbulent diffusion D , which depends on the

properties of the medium, of heterogeneity of velocity and distance from interfacial surface. Total turbulent substance flux, concerning laboratory system of reference, can be written as

where δij is Kronecker symbol.

You can enter coefficients of turbulent diffusion in two-component mixture in the corresponding reference system:

medium-mass system: m D 1 = m 2 D (V2ρ)≠m D 2 = m1D (V1ρ). Because volumes of medium, which participate in the turbulent

pulsations, significantly exceed molecular dimensions, the intensity of the turbulent mass transfer may be significantly higher than of molecular mass

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transfer. The ratio of coefficients of turbulent and molecular diffusion in

1.2. The Law of Mass Conservation

In the analysis of technological processes and calculation of devices the laws of conservation of mass, momentum and energy are used. It should be recalled, that fundamental laws are formulated on the basis of a vast experimental material and do not involve any theoretical justification. Relativistic effects of the relationship of mass and energy in the chemical technology, as a rule, are negligible. Conservation laws can be recorded with regard to the entire system or its parts (integral form), as well as to individual points of space (local form), in addition, can be used for the environment in general (medium) or individual components.

Essence of the law of conservation of mass is that mass (weight) cannot disappear or emerge, i.e. the total number of mass in a closed system is constant (closed system does not exchange by mass with the environment), therefore, M = 0 or dM/dt = 0. Let us consider mass conservation law for open systems.

Integral form of the law of mass conservation (material balance).

Change in mass in a fixed volume V is caused by the difference between input and output of mass inside a selected volume:

use the notion of mass consumption (flow) G, which corresponds to the

If the density of a substance does not change (the environment is incompressible) or the process takes place in stationary conditions (steadystate), then material balance can be simplified:

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This equation is not universal and is valid only in the absence of chemical reactions in a system, as in the latter case some components can transform into the other. In general case, equation of material balance for each component will have the form

time (source of mass). Summing up equation (1.29) for all components, we must obtain equation (1.25) for the whole mass in general. Hence arise the natural condition for the sources of mass of individual components (negative sources of mass sometimes are called sinks):

Local form of the law of mass conservation (continuity equation). Law of mass conservation in the local form may be formulated similarly to material balance. The difference consists only in the fact, that in this case we analyze not the finite volume V, but an infinitely small volume dV:

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This formula can be represented, using differential operator , as well as the expression for mass flux, in the form of

This equation, that expresses the law of mass conservation in the local form, bears the name of continuity equation. Left part of this equation describes the change in time of the density of a moving medium in a fixed point of space (elementary volume dV), which is motionless in respect to laboratory system of reference.

The movement of a medium as a whole has been analyzed earlier. It is easy to get the law of conservation of mass for multi-component systems in the local form for each component:

In general, the law of mass conservation in relation to a single volume can be formulated as follows:

Equation of the type (1.35), when considering multiphase systems, can be written for each phase separately. If there occurs a transfer of mass component from one phase to another, then it may be taken into account in the source of mass rm,i .

It was said earlier, that for multi-component medium are usually used flows not of a mass, but substance, and respectively, instead of densities of the components are used their mole concentrations. Dividing equation (1.35) on mole mass of the component mi, we get

Let us consider the simplest case of mass transfer in twocomponent medium in the absence of chemical reactions. Using expression

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for the flow of a component in medium-mass frame of reference, you can write the equation of unsteady-state convective diffusion:

change in concentration over time to an observer, moving together with a

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medium, with the velocity m − W .

In case, when the medium-mass velocity in laboratory frame of

If we assume the stationarity (steady state) of the process, then it can be more simplified:

Thus, here on the basis of the law of conservation and equations of mass transfer are received differential equations, by solving which we can define the fields of concentrations and mass flow components in any machine. Integration of differential equations gives a common solution for a whole class of processes. For obtaining a specific private decision the equation should be supplemented with the single-valuedness conditions.

1.3. Interfacial Mass Transfer

Conducting of basic processes of chemical technology is accompanied by transfer of substances from the nucleus of one phase through the phase boundary to another phase. We can identify the transfer

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of mass, heat and momentum according to the type of transferred parameter of a substance.

Process of interfacial mass transfer can be divided into three stages: transfer of mass from nucleus of the first phase towards the phase boundary, transfer directly through the phase boundary, and transfer from the phase boundary towards nucleus of the second phase.

Transfer of mass from the boundary to the nucleus of a phase or from the nucleus to the boundary is called mass delivery (internal mass transfer). Transfer of mass directly through the phase boundary is called mass transfer (external mass transfer).

Decision of the engineering tasks often does not require knowledge of magnitudes of the fields of velocity, pressure, temperature, concentrations, as well as of the flows of substances in the whole volume of a device. For such purposes would be enough to obtain the magnitudes of these variables, averaged over cross-section of each of the phases in the output from machine, especially when the input magnitudes of these variables are known. This can be done with the help of laws of conservation of mass, momentum and energy in the integral form, knowing the quantity of substance and (or) the values of its parameters, transferred from one phase to another. For determination of the latter the equations of interfacial transfer of a substance are used.

Thus, in engineering practice, solution of a system of differential equations with partial derivatives is replaced by solution of a system of linear algebraic equations of the balance of substances and interfacial transfer, which is the significant simplification. However, theoretical definition of coefficients in equations of interfacial transfer of substances, derivation of equations and their proper application, can be made only on the basis of transfer equations.

Mass delivery equation. Local form of equation. Imagine an elementary, infinitely small section of an interfacial surface dF, and the rectangular reference system, which has been oriented so, that the plane X - Z coincides with the surface of the phase, and the X-axis - with the direction of the movement of the marked phase (because the interfacial surface area is selected infinitely small, then the interfacial surface can be considered flat).

Flow of mass through the boundary of a phase will occur along the normal to the boundary, i.e. along the Y axis. Let us consider a flow of mass, which exists at the expense of molecular and turbulent transfer

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