The structure of divergence(s) in stationary state of irreversible heat conduction processes and their partial differential equations of elliptic type

Abstract

Irreversible processes mean entropy production or simply energy dissipation. This is true for stationary states too. The Laplace’s equation for heat conduction as an elliptic linear second order partial differential equation does not express any energy dissipation in the conservative potential field according to the minimum principles. A new quasilinear elliptic type second order partial differential equation to stationary state heat conduction process was analyzed with the aid of minimum principles (and also on the base of the divergence term). Investigations made for Onsager [1,2] and Prigogine [3,4] principles showed the deciding role of local dissipation potentials. The existence of these potentials is basically a crucial point for real processes. The new quasilinear elliptic type partial differential equation of second order is in total agreement with Gyarmati’s [5] integral principle for stationary state too. Treating the above questions the proper Lagrange densities and the Euler-Lagrange differential equations must be applied [16] in the different representational pictures (to treat the variational problems). On the base of the new equation(s) the different non-equilibrium temperatures can be determined for steady state irreversible processes but it cannot be done for the Laplace’s equation. The structure of the divergence shows all these features. And what is more important one can find a connecting equation between the internal energy and the entropy (entropy production!) considering the steady state irreversible process. The new equation(s) interprets in a special way the results of the so-called dimensional analysis for nonlinear heat conduction in stationary state too. Boundary conditions were also taken into consideration. Discussion with heat reservoirs helps to expose the questions on the classical thermodynamic level too.