Energy and entropy equations in coupled nonequilibrium thermal mechanical diffusive chemical heterogeneous system

A general theory for coupled chemo-electro-thermo-mechanical heterogeneous system

“…The mass conservation equation of the species k is the same for CETM and CTM systems because the macroscopic mass is independent to the electromagnetic field. For easy to read the mass conservation equation given in literatures [1][2][3][4] is repeated here. Let a CETM system be consisted of N = N 1 + N 2 species with N 1 reactants and N 2 products, the total mass M , total volume V , total density and chemical reaction number L .…”

“…Let a CETM system be consisted of N = N 1 + N 2 species with N 1 reactants and N 2 products, the total mass M , total volume V , total density and chemical reaction number L . The j th chemical reaction equation can be written as [1,2,[17][18][19][20] where B (k) is the chemical formula of species k , (jk) is the stoichiometric constant of a species k in the chemical reaction j , which is positive if species k is a product and negative if k is a reactant. If B (m) is not appear in the j th chemical reaction, then (jm) = 0 .…”

“…Let (k) , c (k) , v (k) , (jk) , (jk)̇(j) and Θ (k) be the partial density or the apparent density, the mass fraction, the velocity, the reaction rate, the mass production rate per volume in j th chemical reaction and in all chemical reactions of the species k respectively; v =u and u are the barycentric velocity and the mechanical displacement vector of a representative element respectively, J (k) is the mass diffusion flow of the species k . The mass conservation equation of the species k is [1][2][3][4]:…”

“…The Gibbs function and the Gibbs equation of the entire system for gas are respectively [1][2][3][4][5][6][7][8][9][10][11][12] where (k) is called the electrochemical potential of spe-…”

“…All these systems are complex thermodynamic systems. A general theory for CTM system has been discussed in papers [1,2], where a complete energy conservation equation, an appropriate entropy equation and the governing equation system have been given and modified the previous theories [3][4][5][6][7][8]. Most of these previous theories are for electrically neutral system.…”