Процессы массопереноса с участием твердой фазы. Учебное пособие

concentration of component i in the liquid phase, called the concentration of saturation, or solubility:

In the particular case of two-component two-phase system, the number of degrees of freedom is equal to two. Taking into consideration a small dependence of properties of condensed media from the pressure, temperature can be considered as the only parameter, that defines solubility of solid substances in a mono-component solvent. Solubility of solid substances in various solvents in certain temperatures are given in the reference literature.

1.5. Various Modifications of Mass Delivery and Mass Transfer

Equations

The equation of mass transfer (1.59), obtained in the section 1.3, contains the moving (driving) force, which is determined as the difference of chemical potentials of components in the nuclei of different phases. Calculation of chemical potential is a rather complex task, as evidenced by the content of the previous section. That’s why in practice are usually used equations of mass delivery and mass transfer, containing the difference of concentrations of a component as the moving force of a process. Often a big problem of usage of the integral form of equations of mass delivery and mass transfer ((1.49), (1.62)) consists in the definition of the surface of phases (interfacial) contact in a real machine, which can be formed of the surfaces of sprays, drops, bubbles, foam. In this case are applied the modified equations, which do not contain the amount of interfacial surface. Before proceeding to the derivation of modified equations of mass delivery and mass transfer, we shall get some necessary for it ratios. In addition, further we will consider two-component mixtures.

Equations of material balance, workers and equilibrium lines of mass-transfer processes. Imagine, that two phases I and II with the flows G and L respectively, move counter-currently to each other in a typical cylindrical vertical machine for mass-transfer processes. Let us denote concentration of the distributable component in them as y and x. Let us assume, that concentration may be changed only with the height of apparatus and being permanent or averaged for each cross-section, i.e. we simplify the task to one-dimensional. Units of measurement of flows is

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better to choose in such a way, that G and L does not change with the height of apparatus (kg/s of inert component in case of absorption, kmol/s of a mixture in case of rectification, etc.).

Fig. 1.3. Scheme of mass exchange process in a vertical counter-current machine: L, G – flows of phases;

x, y - the concentration of distributable component in phases; indices «in» and «fin» - initial and final state; I, II – number of phases; A – A is an arbitrary cross-section of an apparatus

In stationary conditions, the law of conservation of mass (matter) for the whole apparatus (Fig. 1.3) can be recorded in accordance with (1.27) in the form of the equation of material balance: total input of mass (matter) should be equal to its total output (consumption):

You can write the material balance for distributable component in the absence of chemical reactions:

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Fig. 1.4. Working (1 and 2) and the equilibrium line on the x-y plot: 1 – counter-current and

2 – forward-current movement of phases

In Fig. 1.4 are shown the working line and the equilibrium line in case, when the working concentration in the phase I exceeds the equilibrium concentration. With the aspiration of the system for a state of equilibrium the working concentration in each of the phases is moving closer to equilibrium. If the concentration of distributable component in the phase is above equilibrium, then this component will go away from this phase to another, where its concentration is below the equilibrium. In this case, the distributed component will move from phase I to phase II, as y > y* , x < x* . In case, when the concentration of a component is

equal to equilibrium, the interfacial transfer of a substance is absent. Thus, by mutual orientation of the working and equilibrium lines we can make a conclusion about the existence or absence of mass transfer process, as well as about its direction. We can also assume that the value of the interfacial flow of a component will be proportional to the deviation of the system from the equilibrium state, i.e., the difference of the working and equilibrium concentrations. Let us confirm this assumption.

Mass transfer equation in the local form. Let us write down the equations of mass delivery for the two phases I and II and mark them by indices y and x, respectively. We shall use the difference in concentrations as the moving (driving) forces. In order to simplify the record of equations, we shall omit the upper index «d» and lower «y» in the designation of interfacial flow, the upper index «I» when referring to concentration, the

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lower index «i», which corresponds to the number of component. Let us suppose, that the distributed component moves from phase I to phase II:

where x, y – working concentrations of the distributable component in the phases.

Let us use the assumption (1.54) about the absence of resistance of interfacial surface towards the transfer of a substance, or about equilibrium at the phase boundary, expressing it in the form of

(1.102)

Further derivation of the mass transfer equation is similar to (1.55) - (1.58). We can express xb from (1.102) and substitute it in (1.101). We can solve equations (1.100) and (1.101) relative to the difference of concentrations, summarize both equations and then find from the received equation the expression for the flow:

It is easy to see that, if the distribution coefficient does not depend on the composition of the phase, m(xb) = m(x) = m, then the equation (1.103) can be simplified:

K y βy βx

In general, equation (1.103) can be transformed to a traditional record of the equation of mass transfer with the help of (1.100) - (1.102). However, coefficient of mass transfer Ky, in case of dependence of m on the composition, will be determined as follows:

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xb, which is determined from the solution of the system of equations (1.100) - (1.102). If the equilibrium line for the cross-section of an apparatus - in the area from xA xA* can be approximated by a straight line, then for finding m1y there is no need to solve the system of equations and determine xAb, because in this case

If the value yb of (1.102) substitute into (1.100) and repeat the above transformations, then the equation of mass transfer will take the form of

moving forces are used the differences of the working and equilibrium concentrations of a component in one of the phases. The usage of one of two mass-transfer coefficients Ky or Kx depends on the choice of the phase,

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with the help of whose concentrations the moving (driving) force has been recorded. During the calculation and usage of coefficients of mass delivery and mass transfer we must monitor their accordance with the dimensionalities of flows, moving forces, coefficients of distribution, mass delivery and mass transfer. If the moving force is expressed in mole fractions, and the flow of a matter - in kmol/(m2s), coefficients of mass delivery and mass transfer will have the dimension of kmol /(m2s · mole fraction). In this case distribution coefficient must also connect the equilibrium concentrations of a component, expressed in mole fractions. From the equations (1.104) and (1.109) it is easy to establish the link between these coefficients:

K y

In case of approximation of the line of equilibrium by a straight line in the area from x to x*, we can get from (1.107) the ratio

K y

= m1

Integral form of mass transfer equation. We can obtain equations of mass transfer in the integral form by the way of integrating the equations (1.104), (1.109) over the whole interfacial surface of the apparatus or of its part:

These equations acquire the practical meaning, only if the values of coefficients of mass transfer can be considered constant (Ky, Kx= const) at the considered area of integration. Then they can be withdrawn out from under integral and rewritten as

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Equations (1.117) and (1.119) bear the name of basic equations of mass transfer. We can define the average moving forces of mass transfer process for the model of ideal displacement and constancy of the flows along the height of an apparatus (G, L = const). The quantity of distributable component, which moves out of the phase I (y) into the phase II (x) per unit of time d & b across an elementary part of the interfacial surface dF, can be expressed either from the equation of material balance (1.90), or from the equation of mass transfer (1.104):

Let us divide the variables and then integrate them over the surface of interfacial contact in the analyzed apparatus (part of apparatus):

Using the integral form of equations of material balance, you can

write:

We can define G from (1.123) and substitute it in (1.122), then change the limits of integration in the left part of this equation to get rid of the sign «minus», and solve the equation for & b :

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Comparing (1.124) and (1.117), we find:

Similarly, you can obtain the expression:

The calculation of the average moving (driving) forces of mass transfer provides for a finding of a definite integral. In particular case, when the distribution coefficient m = const within the limits of integration, or the line of equilibrium can be approximated by a straight line, the average moving (driving) force is determined by the average logarithmic value. This can be seen, substituting the corresponding dependence in (1.125) or (1.126):

where yup and ylw - the moving force of mass transfer in the upper (up) and lower (lw) sections of an apparatus (part of an apparatus). A similar ratio is true for xm.

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