4. The mathematically strict proof of scientific groundlessness a theoretical plate conception.

I.R.Prigogine.

Introduction to Thermodynamics of Irreversible Processes. Charles C.Thomas. Springfield. Illinois. USA. 1955, (back translation from Russian).

"The stationary irreversible states ... may be described by the extreme principle, according to which the entropy increment per every second has a minimum value, being the state stationary. Let such stationary irreversible states not be muddled with the equilibrium states, where the entropy enlargement per every second is zero."

In monograph of Fr.Bosnjakovic foregoing statement was proofed applied to processes of division of mixtures by means of rectification.

Dr.-Ing., Dr.h.c., Dr.-Ing. E. h. Fr.Bosnjakovic.

Technische thermodynamic, II Teil, 5., durchgesehene Auflage. Verlag Theodor Steinkopff, Dresden, 1971,
s.148ё149. The subsequent text, allocated with blue colour, in German.

"Eine Stofftrennung in einer noch so hohen Kolonne ist nur innerhalb desjenigen Zusammen-setzungsbereiches möglich, innerhalb dessen jede Querschnittsgerade steiler verläuft als die von ihr geschnittenen Naß="19ßdampfisothermen. Die Läuterung verläuft immer in dem Sinne, der durch die küzeste Drehung der Querschnittsgerade in die Isothermenrichtung angegeben wird (s.die Pfeile in Abb. IV/30). Diese Regel stellt eigentlich den Ausdruck des zweiten Hauptsatzes für den Läuterungsvorgahg dar, sie hat also den Rang einer Gesetzes (Die Bemercung: Die analytische Begründung hierzu wird im Abschnitt ``Nichtumkehrbarkeit des Zerlegungsvorganges"gegeben.) Aus diser Regel können wir nur eine gance Reihe Folgerungen von grundlegender Bedeutung ziehen.". For example, as follows from Fig. 1, at concurrence secant lines pF0, pF1, pFm... with isotherms 1, 2, 3.... in indefinitely small volume affiliated to this cross-section of rectifications column the total size increase of entropy describing irreversibility of process becomes equal zero. Accordingly possible size \f work of division in this volume also becomes equal to zero. Conditions, at which in a column there is a section, which temperature sated pair is equal to temperature heated up to boiling temperature of liquid, determine minimal size of reflux ratio, maximal thermodynamic efficiency of dividings process and establishes borders of his practicability. Going from purely geometrical consideration laws of thermodynamics, determined evolution of systems in stationary processes of rectification, to their analytical methods of mathematical description it is possible to formulate the law Bosnjakovic in the form of following disparity:

Fig. 1.Losses owing to irreversibility of process of division in the dual column

TG > TL (1)

The very strict observance of this disparity determine ranges of composition, in which it is possible a separation of binary and multicomponent mixes by their rectification in a column apparatuses with a countercurrent movement of phases.

The observance of this condition is directly connected with necessity of existence in process rectification a heat flow from vapor phase to liquid and maintenance of liquid phase at temperature of boiling, which continuously grows along trajectory of process.

At infringement of this rule in one point all trajectory of process become thermodynamics unstable, and the overall process has been made thermodynamics impossible and unrealizable.

The theoretical plate conception a priori assumes the existence of steady thermodynamics equilibrium between the liquid and the vapor leaving a plate. The mathematical deduction of conditions, determining existence of steady thermodynamics equilibrium, proposed by Gibbs, is described below.

I.R.Prigogine, R.Defay.

Chemical thermodynamics. In Russian translation: И.Р.Пригожин, Р.Дефей Химическая термодинамика. Новосибирск: Изд. Наука, Сибирское отделение, 1966, с. 93.

For the determination of the state of a thermodynamics equilibrium between vapor and liquid Gibbs formulated a equation, based on the finding of minimum of free enthalpy of multicomponent two-phases system. This equation have a following view.

A  =  -  е k = G,L    n е i = 1   nki ж з и ¶ G ¶ nki ц ч ш T,p,j  №  i    =  -  ж и   n е i = 1   nGi mGi +  n е i = 1   nLi mLi ц ш   =  0

(2)

The equation (2) has homogeneous character, as in processes of a phase conversion the molecules of different components lead itself independently each with other. Therefore all sizes of stoichiometric coefficients refered to different phases, n_Gi and n_Li, correspondingly are equal  +1  and  -1 . This allow by p,T  =  const for each components to formulate a following condition of phases equilibrium :

Ai  =  (nGi mGi-nLi mLi)p,T  =  0     (3i)

or

mGi  =  mLi     (4i)

Thus the condition of establish a thermodynamics equilibrium in coexistent phases are equality of its pressures and its temperatures, as well as equality chemicals potentials of all components that are able to change from one phase to another.

Comparing of requirements to states parameters of the systems, formulated Gibbs, which that determine conditions of initiation and maintenance of thermodynamics equilibrium, including the requirement of equality of temperatures of equilibrium phases, T_G = T_L, it is no difficult to see, that they are not agreed with requirements, which has formulated Fr.Bosnjakovic in form of the disparity (1), which is the thermodynamics condition of practicability of process rectification.

From it directly follows, that all knot points which at calculation of rectifications processes on the basis of concept about theoretical plates (stages) are determined from condition of equilibrium the vapor and liquids. At numbers of components more two this point can not belong to real trajectory of rectifications processes, i.e. they can not determine the sequence of composition of liquid phase on height of column and location of trajectory of process in phase space of concentration. As a result it is appear mistakes in determination of number of theoretical steps of division, which, on my accounts, reach a hundred and more percents in dependencies on size reflux number.

Z.T.Fidkowski, M.F.Malone, M.F.Doherty.

Nonideal Multicomponent Distillation - Use of Bifurcation-Theory for Design. AICHE Journal 1991, Vol. 37, No. 12, p. 1761-1779.

In one of a number latest articles Doherty were made ``the following assumptions:

1. All stages are theoretical stages: the liquid and vapor leaving a stage are in equilibrium.
2. The process is isobaric at a known pressure.
3. There is constant molar overflow in each section of the column.
4. Both of products are saturated liquids.

The last two assumptions are made only for simplicity of presentation and are not necessary in the method."

From said follows, that in work M.F.Doherty and coworkers a priori are accepted such initial assumptions, which have the thermodynamics faulty character, at which they can not lead to correct results.

       29.10.1996.

Confirmation from an independent source.

Multicomponent Rectification. Theory and Calculation. M.:"Chemistry", 1983, p. 166, p. 185, p. 247.

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The occurrence of bulges on trajectories of process rectification naturally follows from the fact of mutual crossing of trajectories of rectification processes. These laws are analysed earlier in the chapter "Consistent application methods of the irreversible thermodynamics effectively enhances the accuracy and the reliability of engineering calculation of rectification" and are shown on the Fig. 1. , placed in it. As it is visible from Fig. 2. the set of points was placed on a bulge in a zone of constant concentration. By definition one theoretical plate should be put in conformity to each such point. The quantity of these points allows to estimate size of mistakes arising at definition of number of theoretical plates. It is self-evident, that mistakes at a finding of a form and position of a trajectory of process in space of concentration conduct to mistakes at definition of compositions of products of separation.

Fig. 2.Calculating trajectories rectification of mix a methanol - isopropanol - water in a mode, close to a mode minimal reflux. [The named book, p. 168.] (Illegible inscriptions on figure: "Zone of constant concentration", "These bulges of trajectories are evident mistakes", above "R > Rmin" and below "R < Rmin").

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Fig. 3. Crossing areas border of rectification by trajectories designed with help theoretical plates at finite reflux. [The named book, p. 169.] (b - boundary line; c - vector, connecting equilibrium composition of a liquid and pair; d - Line of construction). At calculating separating of multicomponent azeotropic mixes the mentioned above mistakes get especially serious character. At casual confluence circumstances can result in crossing borders, dividing areas continuous rectification. Last is connected with infringement of monotonous

character of change temperatures of liquid along a trajectory process and non-observance of second beginning of thermo- dynamics. It occurs because these borders are formed depressions or spinal lines on the diagrams, describing at P = const dependence temperatures of the liquid, heated up to boiling, from its composition. It is nothing surprising in that in the extensive literature on a question separations multicomponent azeotropic mixes by a way rectification, describing technical feature of practical use of this process during several centuries, is not present one indication that such separating is possible to reach at constant pressure only owing to change reflux or steam numbers. All told above can result to designing disabled technological installations for separating of azeotropic mixes.

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