As for execution of practical calculations of mentioned processes it is necessary to dispose thermodynamics data on equilibriums, which are determined strictly only for steady equilibriums conditions, theory has postulated some hypothetical quasistatic processes, at which the system passes through continuous sequence of steady equilibriums conditions.
It is accepted simultaneously, that the process run infinitely slowly, allows by this opportunity to be established equilibrium after each elementary process.
The direction of these equilibrium's processes in phase space of
concentration is in the end determined by some variational principle,
which permits distinguish the true direction of evolution thermodynamic
system from other physically possible. In considered case on each
elementary step of process the choosing of direction should supply the
extreme change free enthalpy of system, which is directly connected with
extreme size and sign "useful work" doing by system
1.
1 In
thermodynamic under maximal "useful work'' understand summary value of
work without work, produced against constant pressure of environment,
A¢ = S A - p D V [1,2]
Footnotes:
At p, T = const the equilibrium's processes run spontaneously only in direction of reduction free enthalpy of system and its isobar-isothermal potential. For example, to number of processes with sufficient approach it is possible to add processes, convertible running in galvanic elements.
As opposed to spontaneous processes, in equilibrium process of opened evaporation to system need be delivered heat. The part of heat has been spent on completely convertible work against of external forces, connected with phase conversion of liquid in vapor that is removed in environment. Another part of heat is spent on heating of liquid phase, which temperature of boiling monotonic grows, as the phase is enriched by heavy volatile components. The remained part of thermal energy quantitatively transformed at p, T = const in "minimal useful work" that on physical sense is work of separation. In result of this work take place some increase of partial molecular pressure (compression) one and reduction of partial molecular pressure (expansion) other components of mixture. In the end, with account of above mentioned heating, it taking place an increasing of Gibbs' free enthalpy system of and isobar-isothermal's potential of liquid phase.
The outlined theory be consistent with fact, that in new conditions on the way of achievement of equilibrium at more high temperature and modified composition of liquid phase its free enthalpy again accepts the minimal significance. Thus between liquid and practically absented vapor is established in limit steady thermodynamics equilibrium, which in isolation conditions can be preserved no limit how long.
All said permits to consider the quasistatic processes of opened evaporations, in limit not connected with production entropy, as the stationary equilibrium processes, and to describe the sequence realizing composition of liquid with help of following autonomous system of ordinary differential equations. In right part of this equation it is absent a time and they have the general form [3]:
| (1) |
or in a vector form:
| (2) |
Although specific form of system of autonomous differential equations (1) and (2) remains unknown, in despite of this mathematical theory give proof that possible decisions of system of equations should be described in phase space of concentration by set of trajectories (including cyclic), which cannot intersect each other and themselves [3].
Thermodynamically steady or stable equilibrium should be corresponded absolute conditional maximum entropy and absolute conditional minimum of Gibbs' free enthalpy of system on which degrees of freedom have been imposed restrictions and connections. Both these criteria of equilibrium are equal in value and interchangeable, but use second is preferable, as at its use is not required during calculation to consider system as isolated [4].
At equilibrium in multi-phase and multi-component system these requirements are executed under condition of equality of temperature, pressure and chemical potentials of all components in coexist together phases [5]. This takes place independently on selection of calculation's method for determination of equilibrium.
Accordingly, in dependence on character of equilibrium in points, belonging to trajectories or their parts, the latter can be classified on steady, half-steady and unstable, and each degree of stability of different trajectories can be strong differ.
If there is data, necessary for analytical calculation of phase equilibrium in system vapor - liquid for multi-component mixes the arrangement of distillation lines on phase diagrams can be determined by numerical integration of systems of differential equations, describing ran of quasistatic processes of opened evaporation. In this process on system's degrees of freedom have not been imposed restrictions and connections.
The conclusion of these equations from conditions of material balance is described in some detail in literature [6].
| (3) |
Assumed, that lim\limitsD N ® 0(xi - xi*) ® - d xi® 0 the equations (3) are transformed in equation (4).
| (4) |
Here d t¢ = d ln N - so named dimensionless time; N - initial quantity of evaporating solution, mole; xi and yi* - contents i-ts component in liquid and vapor to be in equilibrium to liquid, mole/mole.
Further, for exception of quantity d t¢ divide n - 1 the equations(4) on one of belonging to this system of equations. On reasons of convenience, it is appropriate to choose the equation relating to light volatility or to heavy volatility components, contents of which in liquid phase monotonic changes as process is run. Having act in such way it is received the system n - 1 of differential equations [6]:
| (5) |
that allows the numerical decision if there is data on phase equilibrium or analytical methods for their calculation.
It will be noted, that equations (4) and (5) are formed the same typical autonomous system of ordinary differential equations.
This opens the opportunity of construction of multitude trajectories for processes of opened evaporation in phase space of concentration. In order for search one of trajectories among infinitely large number possible enough to know the location one point belonging to them.
Analyzing the current and subsequent steps of numerical integration of equation (5) in phase space of concentration, it is not difficult to see, that all steps are inseparable connected with analytical determination of phase equilibriums, each of which correspond the minimum Gibbs' free enthalpy of system.
In other words, taking into account equations (3) on each step of numerical integration it is found some vector connected two adjacent knot points of trajectory. Its module (length) has been defined on each step integration for n equations (3) by fixed quantity that equal to product unity vector on dimensionless factor. The last size is chosen from condition of achievement of required accuracy at numerical integration.
The direction of this vector in phase space of concentration by its fixed length is defined from conditions of achievement by system thermodynamics equilibrium or, that is the same, the minimum of Gibbs' free enthalpy of system. It is possible only in the event if on the short length of trajectory that is coincident with module of vector has been made the maximal useful work. In the most strict form the stated laws can be described with use of mathematical apparatus of theory of field [7].
If at p = const to each point of phase space of concentration of liquid phase heated up to temperature of boiling to set in conformity significances of temperature, enthalpy, entropy, and free enthalpy of Gibbs, the multitudes of these quantities are formed the corresponding scalar fields.
In each of these fields has determined the level lines of potential , bat at number of components n > 3 - equipotential surface. Simultaneously for each point of space can be found the gradients of change of named quantities, i.e., determined direction, in which change sizes of these quantities run with most speed.
As for each point of concentration's phase space a value of free enthalpy may be determine as if the value of potential energy, it is observed the total analogy, for example, between physical nature of distillation's lines and lines of force in any potential fields.
It follows from the facts that one and other lines are described mathematically by autonomous systems of ordinary differential equations, one and other lines are perpendicular to equipotential surfaces, along one and other lines on unit of their lengths are made the maximal work [7] and their directions in each point of potential field are coincident with direction of corresponding gradient.