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The mathematically strict proof of scientific groundlessness a theoretical plate conception
"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. It is impossible to mix up such stationary irreversible states with the equilibrium states, where the entropy enlargement per every second is zero."
Prigogin I. Introduction to Thermodynamics of Irreverible Processes. Charles C.Thomas. Springfield. Illinois. USA. 1955, (back translation from Russian)
The theoretical propositions, formulated in the classical works of the Nobel Prize winner I.Prigogin, were used for further researches. It was comprehensively studied the physical laws determining the beginning and maintenance of the stationary irreversible states taking place in the rectification column at continuous separation of mixtures.
The performed analytical work allowed to remove the thermodynamics errors of traditional methods of engineering calculation of the rectification processes, caused by ungrounded using of the stable thermodynamics equilibrium state as final state of the thermodynamics system's evolution. The final state of the evolution of a system enclosed in the rectification column may be only a stationary irreversible state, as they continuous exchanging mass and heat with environment at constant conditions at their boundaries.
The thermodynamics condition of practicability of process rectification that has be formulated by Fr.Boshnjakovich (See Boshnjakovich Fr. Technische Thermodynamic. II. Teil. 6., verbesserte Auflage Verlag Theodor Steinkopff, Dresden, 1972. s. 148) in form of the disparity T_{G} > T_{L}. Comparing it with states parameters that determine conditions of beginning and maintenance of thermodynamics equilibrium, including the requirement of equality of temperatures in equilibrium phases, T_{G} = T_{L}, it is no difficult to see, that they have mutually incompatible character.
From this directly follows, that all knot points determined at calculation of rectifications processes on the basis of concept of theoretical plates or stage (vapor and liquid leaving a stage are in equilibrium) at numbers of components more two can not belong to real trajectory of rectifications processes. In other word 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.
There are known that stationary quasistatic reversible and irreversible processes mathematically describe by autonomous systems of the ordinary differential equations. The solution of these equations may be kinematically interpreted in the concentrations phase space in such a way, that the kinetic of stationary process is put into the correspondence the movement of a point on the trajectory, which describes a liquid compositions sequence through the rectification column's height. The mathematical theory rigorously proves that the bunches of trajectories cannot intersect themselves or one another (See Pontryagin,L.S. The Ordinary Differential Equations. Addison-Wesley, 1962 (Adiwes international series in mathematics). Quoted on the edition 2. Moscow. The Science Publishing House, the Main Editorial Office of the Phusics and Mathematics literature, 1965, p. 103.
As result in each point direction of trajectory (without sign) coincide with direction of gradient of free enthalpy of Gibbs field, that has been formed by liquid heated to boiling in any point of concentrations phase space. This determine in the end extreme character of "useful" work of division and minimal rate of entropy production.
By this means on using mathematical apparatus of field theory it was found the only possible rigorous solutions of the problem.
The following Fig. 1. shows that the traditional methods of multicomponent rectification calculations, based on the theoretical plates conceptions lead to the intercrossing of the calculated trajectories at changes of the reflux ratio or vapor ratio. It directly contradicts the above mentioned conclusions of the mathematical theory.
Fig. 1. Reciprocal intersections of calculated trajectories in the concentration phase space with the alteration of reflux ratio or the vapor ratio. The calculations were based on the theoretical plates conception.
The arising errors are quite considerable both in regard determination of the rectification column size and the correctness of selection of a technological flow's scheme for separation of azeotropic mixtures.
When using the purely analytical methods of calculation of a separation of multicomponent azeotropic mixtures, these errors may cause under certain circumstances a creation of the technologically inoperative schemes.
Taking in account this circumstance it made possible to determine with considerably higher accuracy the location trajectory, assumed coincidence trajectories processes of open evaporation and rectification that mathematically describe sequences of the liquid phase composition in concentrations phase's space.
Eliminating above-mentioned errors enlarge accuracy and reliability of determination of trajectories locations and engineering calculation number of theoretical stages as well as enhancing efficiency definite composition of divided products and selection of a technological flow's scheme for separation of azeotropic mixtures.
On the other hand, the refined theory makes it possible to use the multicomponent rectification stationary processes modeling to eliminate errors caused by a lack of accuracy of existing methods of predicting of the phase equilibriums data.
The refined theory considerably extends the possibility to generalize experimental data on the kinetics due to taking into account of the dependent between components and heat flows. This allows to obtain the mathematically single kinetic equation that relates the value of one independent flow to the value of the conjugated generalized thermodynamics motive force by means of a single independent phenomenologics coefficient.
The necessary software was designed to provide practical use in industry the new methods of engineering calculation and physical modeling of the rectification processes. The programs make it possible to do direct design calculations of rectification processes to realize the uncontradicted formulated technology requirements, in place of use of trial -- and -- error method.
In addition, the program's algorithm reduces the dimension of the task, providing for the general calculation convergence and makes it by using different numerical methods. The described advantages of the algorithm are quite considerable in terms of optimization of the design decisions on separation of the complicated mixtures in column systems.
Table 1. T H E C O M P A R I S O N O F T H E R E S U L T S of the direct engineering calculations of the theoretical and thermal stages numbers needed to reach the prescribed separation. Set: 1 mole of the initial mixture: acetone : methanol : ethanol The mole percentage composition: 45.629 : 24.630 : 23.741 mole % The distillate output: 0.400 mole The acetone concentration in the distillate: 80.000 mole %
No | Reflux or vapour ratio |
Theoretical plates | Thermal plates | Distillate composition | Distillate composition | ||||||
Recti- fying section of column |
Strip- ping section of column |
Recti- fying section of column |
Strip- ping section of column |
Mole % calculated by theoretical plates |
Mole % calculated by thermal plates |
||||||
Ace- tone |
Metha- nole |
Etha- nole |
Ace- tone |
Metha- nole |
Etha- nole |
||||||
1 | 2.0 1.33 |
16 |
3 |
9 |
3 |
80.000 |
19.998 |
0.002 |
80.000 |
19.650 |
0.350 |
2 | 1.90 1.27 |
23 |
3 |
10 |
3 |
80.000^{ } |
19.999^{ } |
3·10^{-5} |
80.000^{ } |
19.665^{ } |
0.335^{ } |
3 | 1.8 1.20 |
R < R_{min} |
- |
10 |
3 |
- |
- |
- |
80.000 |
19.680 |
0.320 |
4 | 1.60 1.20 |
R < R_{min} |
- |
12 | 4 |
- |
- |
- |
80.000 |
19.714 |
0.286 |
The data presented in Table 1 are directly shown the merits of the refined theory and the followed from it calculation methods.
As the calculated number of the theoretical stages and the number of the thermical stages in the case of binary mixtures separation are identical, the failure of their coincidence in the case of multicomponent mixtures separation should be completely attributed to errors of the traditional methods of calculations. Taking this into account makes it possible to cut the investment and maintenance costs through the reduction of the constructive dimensions of the rectifying column, as a result of decrease their hydraulic resistance and energy consumption, as well as a result of certain raising of the output in case on the thermally unstable substances separation.
All taken together has objectivity particularly importance for science, education, industry and economics many countries.
The same Articel is available in LaTex 2.09 format in saplement.
1. Vigdorov A.S., Malyusov V.A. and Zhavoronkov N.M. On Analytical Description of the Processes Running along Phase Transition Trajectory, Teoret. Osnovy Khim. Teknol., 1975, v.9, No. 2, p.178. (in English translation: Theoretical Foundations of Chemical Engineering p. 163)
2. Vigdorov A.S., Malyusov V.A. and Zhavoronkov N.M. Analysis of Continuous Process of Multi-Component Rectification, ibid., 1976, v.10, No. 3, p. 340. (in English translation: Theoretical Foundations of Chemical Engineering p. 312) Analysis of Continuous Process of Multi-Component Rectification, ibid., 1976, v.10, No. 3, p. 340. (in English translation: Theoretical Foundations of Chemical Engineering p. 312)
3. Vigdorov A.S. Mathematical Description of Kinetics of Binary Mixture Continuous Rectification Basing on the Laws of Nonequlibrum Thermodynamics, Khim. Prom., 1981, No. 8, p. 487 (in English translation: The Soviet Chemical Industry, p. 1057).
4. Vigdorov A.S. Phase Transitions and Technical Stability of the Trajectories of Stationary Processes in Multi-Component Rectification, Zhurnal Phisicheskoj Khimia, 1986, v. 60, No. 6, p. 1541 (in English translation: Russ. J. of Phys. Chem., p. 922)
5. Vigdorov A.S. Thermodynamic (Mathematical) Errors in Film Model for Describing Heat and Mass Transfer in Fraction Processes. Mendeleev Chemistry Journal, Vol. 36, No. 1, pp. 59 - 66. (Zhurnal Vses. Khim. Ob-va im. D.I. Mendeleeva, Vol. 36, No 1, pp 117 - 121, 1991)
6. Vigdorov A.S. Revision of theory and engineering methods of calculating rectification processes to achieve the consistency with the most important concepts of thermodynamics, irreversible thermodynamics and mathematics. Dep. VINITI No 4279--B91.
The author has a version of manuscript translated into English.
The additions to article of A.S.Vigdorov:
"REVISION OF THEORY AND ENGINEERING METHODS OF CALCULATING
RECTIFICATION PROCESSES TO ACHIEVE THE CONSISTENCY WITH THE MOST
IMPORTANT CONCEPTS OF THERMODYNAMICS, IRREVERSIBLE THERMODYNAMICS AND
MATHEMATICS"
Quasistatic equilibrium processes of free evaporation and force lines
of potential field formed
free enthalpy of Gibbs.
Physical mechanism of stationary processes multi-component rectification
and force lines
of potential field.
The mathematically strict proof of scientific groundlessness a theoretical plate conception.
Role of re mrictions and connections at Mathematical description of
stationnary multicomponent
rectification proccesses
Maxwell-Stefan multicomponent diffusion theory cannot be used
for mathematical description heat-
and mass transfer in continuous stationary rectification processes.
(Reply-To: Chemical Engineering List - CHEME-L@ULKYVM.LOUISVILLE.EDU.)
Download:
http://www.aha.ru/~vigdorov/addtex.zip
Sponsors of 6th World
Congress of Chem.Eng.,
The Meeting Planners
91 - 97 Islington Street
COLLINGWOOD
VICTORIA, AUSTRALIA 3066
Email: chemeng@meetingplanners.com.au
I very much regret, that the condition of my health does not allow me to visit so far from Moscow and so interesting country, what Australia is for me. Nevertheless largely being guided by interests of chemical engineers community I consider necessary, to address to you and to yours the person to all participants 6th World Congress of Chemical Engineering with following.
Our works under the common description of content "Revision of theory and engineering methods of calculating rectification processes to achieve the consistency with the most important concepts of thermodynamics, irreversible thermodynamics and mathematics has not only the paramount scientific significance for subsequent development of theoretical bases of chemical technology but also provides the reception of significant economic benefit" were published in Internet on ours Home Page - http://www.aha.ru/~vigdorov
The works have caused the certain interest of a scientific public (~2700 visitors of Home Page), have deserved a positive estimation (were recognised by best on two search categories; "Best Of The Web : Science : Technology : Academia" and "Best Of The Web :: Science : Technology : Chemical Engineering :") and were widely advertised by many search machines.
Our tasks are not limited to demonstration one only our modest theoretical achievement.
We filled by want to render the practical help to community of the engineers - chemists working as in a higher school, and in an industry in use of the new theory and engineering methods of calculating of rectification processes based on consecutive use of representations of nonequilibrum thermodynamics.
The Kurnakov Institute of General and Inorganic Chemistry of Russian Academy of Sciences is not commercial organisation, but in conditions very much limited state financing has not opportunity to transfer their elaboration's, such as algorithms and programs, capable to ensure large economic benefit, even without partial compensation of the expenses, made for 30 years carrying out works in this scientific direction.
By analysing various variants of the decision worth before us and community of engineers - chemists of tasks, connected with use of our scientific elaboration's, we came to a conclusion, that the optimum decision consist in a refusal from single sales of the programs to separate organisations and firms, bat in a finding the sponsors, which could pay partial expenses on all our work and to open free access to last part of our work, connected with her practical use. Such approach will ensure a simultaneous beginning of works above different tasks in many organisations also will promote effective information interchange. It is especially important in conditions, when the engineering calculation a priori assume necessity of use of the equations with phenomenological coefficients.
Therefore we actively search for state and private organisations, universities, scientific and design institutes, also large firm, which could partially pay the expenses, made by us, bat but also using the results, achieved by us, to organise in international scale works, connected to use of representations by nonequilibrium thermodynamics for development of theoretical bases of chemical technology in interests of higher school and industry.
We shall be very grateful to all who will state the remarks, advises, wishes and offers on the mentioned questions. All your reasons will be benevolently considered and are taken into account in the further work.
Write us to E-mail: vigdorov@aha.ru
With many thanks for your interest and your good cooperation.
Yours sincerely,
---------------------------------
Dr.Ph. A.S.Vigdorov
Kurnakov Institute of General
and Inorganic Chemistry RAS
Leninskiy prospekt 31,
117009, GSP-1, Moscow, Russia
Tel/Fax:7 095-954-1279
vigdorov@aha.ru
I consider my own debt to thank cordially my friend Ph.Dr. M.A.Golub
for large disinterested aid in assimilation different type of computing
equipments and its software as well as in advertising of my works.
In turn I counsel to all to be acquainted with his Home Page:
http://www.aha.ru/~golub