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3. THE ARRHENIUS EQUATION AND MACROKINETIC
3.1 The Macrokinetic Approach
There is also so called macrokinetic approach allowing describing chemical reactions simultaneously with some physical processes. The term for new science was introduced by known soviet scientist Frank-Kamenetskyi [6] and firstly it was used for description of the various combustion and explosion effects. Later it was used for description of other similar processes, where the chemistry and physics
is combined by any way. The new science has allowed describing the lot of very complex system if the characteristic times of chemical and physical processes are of one order values, but this condition is not simple, because in the macrokinetic model frame many processes can be described by the different ways. You can simplify the chemical “part” of problem, changing the many chemical reaction by one or two reaction of the zero or first order? You can simply use another set of kinetic equations. For physical “part” you can forget some processes, but again to get a “satisfying” interpretation of the real experiments data due to mentioned properties of the Arrhenius exponential function and the manipulations with the corresponding coefficients.
The particular mathematical essence of problem is very trivial one, because the model basic equations (derived from the equations of energy, matter and components conservation) with an Arrhenius conjecture can be written in the following form:
Σ F(phys) = Σ F(chem) (exp(-E/2RT) (3)
On the right hand side Fchem are the functions responsible for the chemical processes, including the Arrhenius exponential term for the rate of the chemical reactions. The left side includes the functions Fphys responsible for the physical processes: thermal conductivity, diffusion, convection, radiation, etc. It follows, in practice, that the mathematical forms of the functions used make it possible to describe any experimental data in the field under consideration. Indeed, if the right and left parts of such expression are expressed as logarithms, this operation, without changing the essence of the relationship, "smoothes out" the contributions from those functions which did not initially contain an exponential term, so that they can often be neglected. In other words, the exponential function is dominant in representing the data so that, when using such equations, the possible accuracy of the contributory physical processes can be discounted. It is obvious, therefore, that together with such primitive approaches and results from inappropriate usage of rate expressions, there have been many reported attempts to interpret experimental data which, in consequence, remain only superficial. Specific examples are given below.
The author doesn't raise a problem to systemize or describe all the possible advantages or disadvantage of the macrokinetic approach as a hole (it can be a theme of a serious special investigation!). It is a very complicated scientific “tree” trunk or stem with lots of branches and leaves of «false science». This can be compared with the creepers wrapping up the stem. Author is specialist in the combustion theory and therefore below namely this science is considered.
Thus, as was said above the mathematical combustion theory is connected, from the one side, with a set of the chemical kinetic equations, and, with a physical formula for the heat conductivity, diffusion, phase transitions, etc., from other hand. The combustion processes models are based at well-known conservation laws for the mass, impulse and energy of the given multi-component and reacting mixture.
The required function (usually, temperature) is essentially non-linear: the dependence can be the Arrhenius exponent or have a power form. The main condition is that there was any "strong" function, but with the linear derivatives of temperature. Such equations are called in mathematics “quasi-linear” and they often have in combustion theory some analytical (in the middle of 20-th century) or numerically (present practice).
Еhe combustion theory equations are the “continuity equation” for a mixture as a hole and the individual components; Navies-Stokes equation for a compressible medium and the heat transfer equation with the chemical sources. In this paper we will not touch all nuances of the complex gas dynamics theory and will describe only some simplest examples.
3.2 The Thermal Explosion
This problem relates to the ignition in the media, that is usually considered stationary (immobile), and v=0 in equation (4). The theory of thermal explosion was proposed by N. N. Semenov (I will tell it about a little below, writing about some social aspects role in the history of science) and become a base for all further works in the combustion mathematical theory. It was a simplest theory where with account of the main fact (without which the thermal expansion notion has not a sense). N. N. Semenov takes the Arrhenius law for the heat release, using the reaction model of zero order (the simplest kinetics) and the Newton's law of heat removal, supposing that the temperature is the same over the all volume points (the simplest variant of heat release). In practice the temperature is not the same in all volume of the explosion vessel, and the ignition process is beginning only in a one point source. With account of this features were derived many other ignition process models (stationary, non-stationary, quasi- stationary, etc.). As a results there were solved a problem, described in “Galwey case” section. At professional slang it was called the “thermal explosion”, when for hundreds systems, ready to an exothermic reaction, were found the characteristics of a pre-explosion state. A nonlinear heat conductivity equation of type (4) with chemical heat sources has not an analytical solution in general form. The partial differential equations system is very complicated, and until now we have not the general schemes of the numerical calculations, that can be used at all combustion modes ant temperature ranges.
All approaches are connected with the various expressions and forms of the ignition criteria (thermal explosion). For some of forms we have even the analytical solutions of problem.
For Semenov such criterion was connected with the excess of heat, released at chemical reaction, over the heat release. At other criteria the ignition begins when the heat coming from the external source and the chemical reaction heat release become compared by value. For example, for the so called energetic condensed systems (ECS) (rocket fuels, powders, pyrotechnics compounds, etc.), the ignition begins as a results of the surface heating by different mechanisms (heat radiance, convective heating, heating by a hot block, etc.) is exceeding at 4 times the sample inert heating (without chemical reaction). It turns out that practically all criteria give the near results. The proposed models and critical ignition conditions allow predicate the real situations with the rocket fuels, powders, pyrotechnics compounds, etc. ignition and combustion). Apparently the “thermal explosion” is one of the best illustrations of the AE usage for an approximation of the complex chemical processes, having a practical meaning. It is worthy to mark that the experimenters in this studies are watching for the minimal number of parameters (usually the registered only the temperature dependence on time) and in this situation the above described striking and strange properties of AE are appearing in a full scale.
3.3 The Problem of the Propagation Velocity of a Burning Wave
If the medium is capable of chemical reaction (burning) in it, the propagation of the corresponding waves is possible. For stationary propagation of the flame front with velocity U, it is convenient to choose a coordinate system associated with the combustion front. In this system, the front is stationary, and the initial mixture is blown through it with velocity U. In the stationary state, the partial derivatives with respect to time in equation (4) are zero and in the one-dimensional case, neglecting the temperature dependence of thermal conductivity, we get the expression.
Further, with the help of various assumptions for gases, taking into account the similarity of the fields of concentration and temperature, when the difference between the diffusion coefficients and the thermal diffusivity of the components of the mixture can be neglected or even these coefficients are equal for close molecular weights. As well as neglecting the chemical reaction at low temperatures, using the methods of decomposition of the exponent and others, we obtain an analytical expression for the velocity U. For a zero-order reaction:
Thus, the burning propagation velocity turns out to be in the "Arrhenius" dependence on the combustion temperature.
U ~ exp (-E/RT)
The results obtained for the simplest reactions in the gaseous medium have been used to burn many systems, condensed, homogeneous, heterogeneous, explosive substances, etc.
It is necessary to note the importance of many results obtained with the help of these approaches: a correct qualitative understanding of the various moments of the propagation of combustion waves, the theory of limits, and the stability of combustion regimes.
By selecting the coefficients for the exponents and other "constants" in the equations, it is possible to obtain satisfactory agreement with the experiments in terms of the agreement between the calculated and experimental speed and temperature. However, other parameters - real kinetics, activation energies and other parameters of the kinetic triad, the structure of the front, the size of the zones (warm-up, reaction) are very different from reality. And this is a serious problem.
3.4 The Global Kinetic Mechanism and Software Packages with Various Kinetic Schemes
Realistically, reactions between valence-saturated molecules have very much larger activation energies (by tens of kJ/mole, and more) compared with reactions involving free atoms and radicals. In real systems, the mechanism of interaction is determined by the fast reactions of the chain carriers and by how rapidly these multiply and perish. However, the desire was to simplify analyses of the systems of equations for physical and chemical combustion processes, sometimes using analytical methods (including simplified methods of integrating exponentials, narrow-band methods). These approaches led to the use of expressions for the overall reaction rate, applicable to only the chemical part of the process, the Arrhenius exponents for the zero or first order reactions and ignoring the real dependencies on reagent concentrations. These yield meaningless activation energies and preexponential factors chosen only to satisfy the experimental data but provide no insights into factors controlling rates and/or mechanisms. This approach has since become very widespread under the name: "global kinetic mechanism". For example, in [7], an empirical kinetic scheme is used for the single-stage combustion of methane with oxygen. The kinetic parameters are chosen from the experimentally measured flame velocity, and this allowed calculations of two- and three-dimensional flows, including turbulent systems.
Of course, with the development of supercomputers, scientists are trying to move away from the use of global kinetic schemes, replacing calculations with abbreviated kinetic schemes and detailed
schemes (new kinetic scheme packages appear), but problems remain. If we “simplify” the physical part of the problem by ignoring the real physical processes, then the chemical part of the problem with Arrhenius exponents will result in a relatively acceptable fit in some parameters, but this may differ significantly from what happens in other parameters. If we took into account the basic physical processes and took the kinetics with a hundred or more reactions, firstly, it is not obvious that these are the right reactions, much also depends on how well the parameters of the kinetic triad are selected for each of the reactions. As for the kinetic reaction packages, it is not important how many reactions are in a package: thirty or one hundred, but how processes with each of the important radicals are reflected in them. If there are no reactions with an active particle that is rare for this system, then it is not as important as, for example, if for the important radical in the process there was no death or birth reaction, as a result, it would start to accumulate or, on the contrary, disappear. But the properties of the exhibitors, even in this case, can give the appearance of a successful simulation. Among the comic laws of Murphy, there is the so-called "law of reliability": "It is human nature to be mistaken, but only a computer can finally confuse everything."
In February 2019, I participated in one of the most important Russian conferences on combustion and explosion: the 12th Annual Scientific Conference of the Institute of Chemical Physics with a number of reports, one of which coincided with the theme of this text (about the features of the Arrhenius equation). In the competition for the best works, one of the first places was taken by the work: “Kinetics of pyrolysis and auto-ignition of acetone behind reflected shock waves: experiment and numerical simulation” A. M. Teresa, G. L. Agafonov, S. P. Medvedev, N. V. Nazarova, V.N. Smirnov (not yet printed). A feature of this theoretical work was that 4 different programs for the kinetics of acetone were sequentially taken, and the result (not trivial, there are features) was the same independently selected package of kinetic equations. The organizers of the conference, who distributed the prizes, liked this. This work will be printed in Russian, with an annotation in English in the journal of this conference until the end of 2019 and will be available at: http://combex.org/iournal/.
Unfortunately, most of the works to which I refer are written in Russian and it may be difficult to find them. But I am sure that such problems arise quite widely and I would be grateful to the readers if they would send me similar links and share my concern, as did the English scientist Galway A K.
The following is another example from the works of Russian combustion theorists. If in the previous examples and reasoning, strictly speaking, instead of the exponent of the Arrhenius equation, we could use other types of functions (power, polynomial), so long as there was a “strong” dependence on temperature, then the following example shows the property of exponentials to describe oscillatory phenomena. In problems of combustion, various oscillatory phenomena constitute one of the sections of the theory, and the Arrhenius equation turns out to be particularly suitable.
3.5 The Example of "Mathematical Modeling" for an Illustrative Experiment
The above mentioned degradation of science is demonstrated by examples where AE modeling opens the possibility of applying numerous alternative variants of methods used, thereby "deforming" the usual work and practice of scientific investigations in this field. This may result in time-expensive numerical analyses, requiring strict inspections of results, additional information, etc.
As an example, let us consider the mathematical modeling of the combustion experiment [8] illustrated in Fig. 1. During burning up and combustion the admixture gas, namely hydrogen (0.1- 0.3% of the mass), accumulated within the titanium due to the peculiarities of the metal oxidation within the titanium sponge is interfering with the oxygen supply. In this case the combustion can even be constrained or hindered. As a result the hydrogen release is decreasing, the combustion is again activating and so on. Process becomes similar to periodic one. An unusual type of "vibrating" combustion is occurs. What do the "theorists" do? They ignore everything that has been said above about the impurity gases evolution. A standard system of equations of the type (4) is written and the observed oscillations arise due to the properties of the exponentials. Thus, certain processes are observed, analyzed by formulas based on a very different process, but the result found is the same. This example demonstrates how the Arrhenius equation can "model" the oscillating processes only due to the properties of the exponential equations.
![Fig. 1. This figure illustrates a particular combustion experiment [7]. Titanium is poured into the boat and covered with quartz glass plate. Titanium is ignited in air, which passes through a narrow slit. An unusual type of "vibrating" combustion is observed. 1 - Ignition spiral, 2 - Quartz glass, 3 - Initial powder layer, U - Front velocity](https://avatars.dzeninfra.ru/get-zen_doc/1709225/pub_5e91d98c7ea04b16b5540111_5e91e14b6dfdce0b1380277e/scale_1200)
4. MACROKINETIC PLUS SOCIAL AND POLITICAL CIRCUMSTANCES
4.1 Some Curious Stories
Of course, such a cunning, slippery object as the Arrhenius equation is not the only one in the history of science. We have already mentioned the theory of phlogiston; we can recall the topic with the "perpetual motion", etc. The Schrodinger equation, which contains exponents, is also able to “approximate” and create objects that only imitate reality. Much depends on the specific scientists and specific circumstances. In a more detailed brochure, which I am preparing with Galwey AK and Khachoyan A V, there will be a historical section on the history of science and various scientists.
For example, if we recall the stories of Russian Tsar Peter the Great, who loved England very much and said that if he were not a king, he would have dreamed of becoming an “English admiral” and decided to establish the Academy of Sciences as in England. For this, he found the best scientist. At that time it was German Leibniz and began to pay him a lot of money. One of Leibniz's first assignments about Tsar Peter was to make a copy and assemble in Russia a “perpetual motion machine” that “functioned” then in Germany. Leibniz took up the solution of this question, but all the time he wrote to the tsar about various temporary problems, asked and received money from Russia. Soon, both Tsar Peter and Leibniz died ... and Russia was left without a "perpetual motion machine."
Or a real episode from the history of the 20th century, when academicians Kapitsa and Zel'dovich were summoned to the Central Committee (the highest governing body of the USSR) and asked what they thought about intelligence reports that the Americans invented the anti-gravity engine. Academics said they do not believe that this is true. Six months later, they were again summoned to the Kremlin and then science director Malenkov, reported that they were right, the American project manager, stole 10 million and fled with his secretary to Brazil (there from then there was no extradition of criminals in the USA). To which Zel'dovich replied: "Speak, 10 million ..., with the secretary ..., to Brazil ..., the anti-gravity engine ... Perhaps there is something in this idea".
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