Global Journal of Science Frontier Research, A: Physics and Space Science, Volume 23 Issue 1

where j u (W·m -2 ) is the internal energy flux density through the vector element d f of the closed surface f of thesystem of constant volume V in the direction of the external normal n (Figure 1). Figure 1: Energy flux through the boundary of the system Unlike the later equation of J. Poynting (1884), this form of the energy conservation law takes into account the kinetics of real processes, without making any assumptions about the mechanism of energy transfer in a solid media or about the internal structure of the system. According to this equation about short-range effects, energy U does not simply disappear at some points of space and arise in others, but rather carries the i th energy carrier Θ i (with the number of moles k of the substance N k , their corresponding charges Q k , their entropies S k , the impulses P k , etc.) through the fixed boundaries of the system. Let us now find the expanded form of this law, which is valid for any i th material carrier of energy. For this, we will take into account that the energy flux j u is the sum of the fluxes j ui carried by each of them. These fluxes, in turn, are expressed by the product of the flux density of the i th energy carrier j i = ρ i υ i by its potential ψ i ≡ dU i / d Θ i , where ρ k = d Θ i / dV and υ i =dr i /dt are the density of the i th energy carrier and the rate of its transfer across fixed boundaries systems, resp., i,e. j ui = ψ i j i , so that j u = Σ i j ui = Σ i ψ i j i , (4) Using the Gauss-Ostrogradsky theorem, we transform the integral ∮ · f into a volume integral ∫ ∇ ⋅ j u dV. Then, after decomposing ∇ j u = ∇ ( ψ i j i ) into independent components Σ i ψ i ∇ · j i + Σ i j i · ∇ ψ ik , the energy conservation law (3) takes the form: dU / dt + Σ i ∫ ψ i ∇ ⋅ j i dV+ Σ i ∫ j i · ∇ ψ i dV = 0 (5) If we take the average value Ψ i of the potential ψ i and the average value X i of the potential gradient ∇ ψ i both from under the integral sign, then equation (5) can be expressed in terms of the parameters of the system as a whole, as is customary in classical thermo- dynamics: dU / dt+ Σ i Ψ i J i + Σ i X i ·J i = 0. (6) Here J i = ∫ ∇ ⋅ j i dV = ∫ j i d f is thescalar flux of the i th energy carrier through the boundaries of the system; J i = ∫ ρ i υ i dV= Θ i i υ is its vector flux (impulse). Unlike the Gibbs relation, Equation (6) contains 2 i terms ( i= 1,2, ... n ) and describes not only the processes of introducing the k th substance N k into the system as well as the series Q k , the entropy S k , the momentum P k , etc. in the homogenous system being investigated, but also the processes of redistribution of the system volume of overcoming the forces of the X i and the performance of work "against equilibrium" in it. Therefore, it is applicable to a wide class of open ( N k = var ), non-closed ( X i = var ) and non-isolated systems ( U = var ), which are the object of study in other fundamental disciplines. At the same time, it allows the irreversibility of the above processes. Indeed, considering (6) together with the integral equation of the energy carrier balance Θ i d Θ i / dt+ ∫ ∇ ⋅ j i dV= ∫ σ i dV . (7) In this case, the densities of local and substantial fluxes j u coincide. We find that, besides the energy carrier Θ i appearing in the Gibbs ratio, it takes into account the presence of these internal sources of density σ i . It is easy to see that under the conditions of local equilibrium ( X i = 0), Eq. (6) takes the form dU / dt = Σ i Ψ i d Θ k / dt- Σ i Ψ i ∫ σ i dV , (8) i.e., it transforms into a generalized Gibbs ratio for complex multivariable systems dU = Σ i ψ i d Θ i only when the internal sources of entropy d u S / dt = ∫ σ s dV and other energy carriers d u Θ i / dt = ∫ σ i dV (including products of chemical reactions d u N k / dt ) disappear. This testifies to the inconsistency of the hypothesis of local equilibrium, according to which the state of an element of the inhomogeneous continuum of the system is characterized by the same set of variables as in equilibrium. This follows from the fact that this assumption also means the absence of "production of entropy" ( d u S / dt >0). The latter makes it necessary to introduce the parameters of in homogeneity X i and the fluxes J i associated with them into the equations of nonequilibrium thermodynamics. It is remarkable that Equation (6) does not become an inequality despite the obvious inclusion ofthe dynamic (irreversible) processes under consideration. This solves a major "problem of thermodynamic inequalities" which until now has prevented any application of the mathematical apparatus of nonequilibrium thermodynamics to real processes (i.e., those with fluxes at finite speeds). It is also important that our derivation of an expanded form of the law of conservation of energy (6) © 2023 Global Journals 1 Year 2023 10 Global Journal of Science Frontier Research Volume XXIII Issue ersion I VI ( A ) New Applications of Non-Equilibrium Thermodynamics