Rigorous implementation of continuous thermodynamics using orthonormal polynomials

In this work, a theoretical framework of using orthonormal polynomial distribution in continuous thermodynamics is derived. The functional equalities of phase equilibrium and material balance of continuous thermodynamics can be transformed into a system of non-linear algebraic equations. This corresponds exactly to the phase equilibrium problem using discrete Gaussian quadrature pseudocomponents. A Gibbs-Duhem equation in terms of coefficients of the orthonormal polynomial distribution is derived. Hence, the failure to satisfy conservation of mass and phase equilibrium criteria in a rigorous functional sense caused by use of fixed form distribution can be corrected. Examples of flash and distillation calculation are presented.