Thermal theories

The first quantum mechanical treatment for the radiationless electron transfer was developed by Gurney [68] for hydrogen evolution on mercury. In his model, Gurney assumed that the intermediate H is not 

Replacing the Fermi—Dirac distribution by the Boltzman distribution (for U below the Fermi level) and integrating eqn. (151), a current maximum close to the Fermi level of the electrode is predicted with the major contribution within kB T around UF. 

Butler [33] extended the treatment for the case where the intermediate H is adsorbed on the electrode surface. In these theories, the electron transfer is considered to take place at a unique point in time and space. 

Gerischer developed a complete theory for redox systems at metal and semiconductor electrodes on the basis of Gurney s treatment [69], The difference between the metal and a semiconductor is the integration over the electronic states in the electrode [17]. 

The transition state theory was applied to the proton transfer reaction at electrodes by Horiuti and Polanyi [58] and Eyring et al. [31]. The stretching of the H+—OH2 bond gives rise to the activated complex by a gradual transition in time and space. Details of this model were discussed in Sect. 3.1. 

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Thermal Theory. The thermal approach to flame retardancy can function in two ways. Eirst, the heat input from a source may be dissipated by an endothermic change in the retardant such as by fusion or sublimation. Alternatively, the heat suppUed from the source maybe conducted away from the fibers so rapidly that the fabric never reaches combustion temperature. 

Law C.K. and Egolfopoulos F.N., A unified chain-thermal theory of fundamental flammability limits, Proc. Combust. Inst., 24 137-144,1992. 

Sivashinsky, G.L, Diffusional-thermal theory of cellular flames. Combust. Sci. Technol., 15,137,1977. 

Formally the thermal theory can be established, via TFD, within c algebra (I. Ojima, 1981 A.E. Santana et.al., 1999) and symmetry groups (A.E. Santana et.al., 1999), opening a broad spectrum of possibilities for the study of thermal effects. For instance, the kinec-tic theory has been formulated for the first time as a representation theory of Lie symmetries (A.E. Santana et.al., 2000) and elements of... 

In this section we review the construction of thermal theories starting with TFD to show the connection among different methods. A mechanism for space and time compactification of a quantum field theory is then discussed in this context. 

Since the non-tilde operators describe physical variables, G(k (3)11 is the physical propagator to be used to treat the properties of the thermal bosonic system. It is interesting to observe that, except for the non-diagonal elements, this TFD-propagator is equal to the one introduced in the Schwinger-Keldysh approach, which is claimed to be (in this equivalence with TFD) a thermal theory describing linear-response processes only (H. Chu et.al., 1994). 

It is worth emphasizing that for the fermionic field, we have antiperi-odic KMS conditions, which when considered in terms of space com-pactification, coincide withthe physical bag-model conditions (A.P.C. Mal-bouisson et.al., 2004 A. Chodos et.ah, 1974 C.A. Lutken et.al., 1988). Such a result will be used in Section 5. Now we will be concerned with the algebraic elements of the thermal theories. 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

The initial theoretical analyses for the determination of the laminar flame speed fell into three categories thermal theories, diffusion theories, and comprehensive theories. The historical development followed approximately the same order. 

Later, there were improvements in the thermal theories. Probably the most significant of these is the theory proposed by Zeldovich and Frank-Kamenetskii. Because their derivation was presented in detail by Semenov [4], it is commonly called the Semenov theory. These authors included the diffusion of molecules as well as heat, but did not include the diffusion of free radicals or atoms. As a result, their approach emphasized a thermal mechanism and was widely used in correlations of experimental flame velocities. As in the... 

The theory of particle diffusion was first advanced in 1934 by Lewis and von Elbe [5] in dealing with the ozone reaction. Tanford and Pease [6] carried this concept further by postulating that it is the diffusion of radicals that is all important, not the temperature gradient as required by the thermal theories. They proposed a diffusion theory that was quite different in physical concept from the thermal theory. However, one should recall that the equations that govern mass diffusion are the same as those that govern thermal diffusion. 

These theories fostered a great deal of experimental research to determine the effect of temperature and pressure on the flame velocity and thus to verify which of the theories were correct. In the thermal theory, the higher the ambient temperature, the higher is the final temperature and therefore the faster is the reaction rate and flame velocity. Similarly, in the diffusion theory, the higher the temperature, the greater is the dissociation, the greater is the concentration of radicals to diffuse back, and therefore the faster is the velocity. Consequently, data obtained from temperature and pressure effects did not give conclusive results. 

Hirschfelder et al. [7] reasoned that no dissociation occurs in the cyanogen-oxygen flame. In this reaction the products are solely CO and N2, no intermediate species form, and the C=0 and N=N bonds are difficult to break. It is apparent that the concentration of radicals is not important for flame propagation in this system, so one must conclude that thermal effects predominate. Hirschfelder et al. [7] essentially concluded that one should follow the thermal theory concept while including the diffusion of all particles, both into and out of the flame zone. 

The ignition criteria of the thermal theory of ignition are then represented by... 

Thermal Theory proposed (but not actually developed) by J.H. Van t Hoff (1852-1911) was mathematically elaborated by N.N. Semenov. Accdg to this theory, the heat of reaction becomes, under certain coalitions (temperature, pressure, etc), highehthan losses of... 

G. Weingarten, A Thermal Theory for Rates of Propagative Burning , PicArsn FREL Tech Rept 2596 (June 1959). Ordnance Project TS5-5407 Dept of the Army Proj 504-01-027 6) Anon, Military Pyrotechnic... 

In my APL Report BBW/CGD/TR-11, on my trip to the Eleventh symposium on Combustion, there is a discussion of chain reactions in explosion on pages 17-19, and a comparison of the chain-reaction theory with the thermal theory on pages 19-23... 

Such reactions have been used to explain the three limits found in some oxidation reactions, such as those of hydrogen or of carbon monoxide with oxygen, with an "explosion peninsula between the lower and the second limit. However, the phenomenon of the explosion limit itself is not a criterion for a choice between the critical reaction rate of the thermal theory and the critical chain-branching coefficient of the isothermal-chain-reaction theory (See Ref). For exothermic reactions, the temperature rise of the reacting system due to the heat evolved accelerates the reaction rate. In view of the subsequent modification of the Arrhenius factor during the development of the reaction, the evolution of the system is quite similar to that of the branched-chain reactions, even if the system obeys a simple kinetic law. It is necessary in each individual case to determine the reaction mechanism from the whole... 

The smallness of the temperature coefficient in initiation and detonation has led to one objection to the thermal theory. 

Andreev Belyaev (Ref 25, pp 276-86), in the discussion of thermal theory of initiation of explns by means of "hot spots , described briefly, besides Bowden s work, the investigations done in Russia by N.A. Kholevo K.K. Snitko... 

Mathematical Theory of Thermal Explosions of Frank-Kamenetskii. See under Detonation (Explosion, Deflagration and Decomposition), Thermal Theories, and Thermochemistry of... 

Reynolds number, p 46), etc 61-72 (Shock relationships and formulas) 73-98 (Shock wave interactions formulas) 99-102 (The Rayleigh and Fanno lines) Ibid (1958) 159-6l(Thermal theory of initiation) 168-69 (One-dimensional steady-state process) 169-72 (The Chapman-Jouguet condition) 172-76 (The von Neumann spike) 181-84 (Equations of state and covolume) 184-87 (Polytropic law) 188, 210 212 (Curved front theory of Eyring) 191-94 (The Rayleigh transformation in deton) 210-12 (Nozzle thepry of H. Jones) 285-88 (The deton head model) ... 

In view of the assumption made, however, this agreement should not be regarded as providing positive evidence for the thermal theory of initiation... 

Gray fit Yang (Ref 1), a mathematical model was proposed to unity the chain and thermal mechanisms of explosion. It was shown that the trajectories in the phase plane of the coupled energy and radical concentration equations of an explosive system will oive the time-dependent behavior of the system when the initial temperature and radical concentration are given. In the 2nd paper of the same investigators (Ref 2), a general equation for explosion limits (P—T relation) is derived from a unified thermal and chain theory and from chis equation, the criteria of explosion limits for either the pure chain or pure thermal theory can be deduced. For detailed discussion see Refs... 

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