Connected equations of heat and mass transfer in a chemically reacting solid mixture with allowance for deformation and damage

Abstract

536.46+539.3+531Changes in the properties of a substance due to a chemical reaction (change in the concentration of components), thermal expansion upon heating, and inhomogeneities in the structure of the substance (both in the initial state and during heating and reaction) determine the volume changes dV/Vo = (V -Vo)/Vo of the substance. For many solids, these changes are small and have no effect on heat and mass transfer processes. For high heating rates or significant differences in the properties of the reactants and products, a volume change dV can lead to the occurrence of stresses in the reaction zone and even to damage (accumulation of defects and inhomogeneities in the structure compared with the initial state). We write these changes in the formVo k where aT = (1/3) (OV/OT)Vo 1 is the coefficient of linear thermal expansion; a} = (1/3)(OV/ON•)Vo-' is the coefficient of concentration expansion for each component; ctr = (1/3)(OV/c3vp)Vo 1 is the coefficient of structural expansion; Nk is the concentration of the kth component; vp is the volume of damage (discontinuities, pores, cracks, etc.). We can speak in general about a change in structure inhomogeneity of one or another type. For simplicity of analysis, we restrict the consideration to overall volume changes. In expansion (1), only first-order terms are taken into account, by virtue of the smallness of the coefficients aT, ak, and ctc in accordance with their meaning for a solid medium. Let us determine how small changes (1) manifest themselves in heat and diffusion equations. For this recall that with temperature and volume changes in the system, the free energy F is the main thermodynamic potet~tial [1]. Its total differential is written asHere dF ~ is the change in the free energy due to heating and chemical reaction:k where #k is the chemical potential of the kth component; S is the entropy of a unit volume; and dF' is the change in the free energy during deformation and damage:dF' = ~_, aodr j + ~_, X,,ndw,m,ij lmTomsk State University, Tomsk 634050.

Generalization of the Clausius-Clapeyron equation in a coupled thermomechanical model

“…aT is the coefficient of linear thermal expansion, ak are the coefficients of concentration expansion relative to each component (similar concepts were introduced in [4] for vacancies and dislocations), and as is the coefficient of structural expansion, is the generalization of (1.4). Using (1.2) and (1.4), we find…”

“…In terms of the model of [4], the basic equation of thermodynamics (Gibbs equation) for a local volume has the form n+2 du = T ds + p-1 aij dgij + ~ gk d_Yk + XP-1 d~h (1.1) k=l where u is the specific internal energy, s is the entropy, T is the temperature, p is the density of the medium, a~j and Eij are the components of the stress and strain tensors, gk [J/kg] are the chemical potentials of the components (or their specific partial Gibbs energies) for k = 1, 2,..., n, vacancies for k = n + 1, and dislocations for k = n+2, Ark are the corresponding mass concentrations, X [J/m3] is the energy potential of the macrodamages (or the structural potential), ~ = vp/v, vp [m3/g] is the specific volume of the macrodamages (cracks and pores), and v = p-1 is the specific volume of the medium. For small strains, the constant-density approximation p ~ const is valid and it is convenient to determine the thermodynamic potentials for a unit volume.…”

“…In particular, the model of the medium used in [4,8] admits a simple generalization to anisotropic media. In the present work, to describe phase transformations that are connected to the transition of one crystalline modification to another, we introduce the tensors of phase concentrations _ (k) The (or components) .~' (k).lm and the corresponding tensors of the coefficients of concentration expansion ~lm-structure of these tensors and the number of their independent components are determined by the type of symmetry of the crystal lattices of real crystals and their modifications similarly to the thermal properties of crystals (e.g., ST) in the models of anisotropic media.…”

“…However, in some situations, when the role of various energy effects or the dependence of the temperature of the phase transition on the parameters of the medium should be evaluated, the connectedness of various processes can be of primary importance. In the present study, the conditions of phase equilibrium and the possibility of determining the parameters that enter the model of [4] and characterize the medium and the first-order phase transition are analyzed. In the particular situations considered, the external load is absent, and the stress and strains are a consequence of the phase transformation.…”

Model of transition zone evolution between coating and substrate under intense short thermal impulse

“…(2) The mathematical model takes into account cross-diffusion fluxes, thermal diffusion (soret effect), diffusive thermal conductivity (Dyufor effect), relaxation time of heat and mass fluxes [18][19]. Taking into account the symmetry of the part in the cylindrical coordinate system, the model includes onedimensional nonlinear equations:…”