General formulation structure for steady-state process models: Guaranteeing solvability

Current steady-state process modeling software suffers from two long-standing problems: lack of a method for diagnosing specification feasibility, and use of inadequate numerical methods. Both problems arise from the complex nonlinearity of and lack of common structure between process unit models, making investigation of numerical properties of connected systems difficult. To alleviate both problems we introduce a variational modeling framework for dissipative processes and a numerical method for solving models in this form.;The model of any real process can be put in the form of an unconstrained positive definite quadratic variational principle in the thermodynamic potentials. For this model to correspond with real physical behavior the variations must be taken assuming the coefficients in the quadratic form are not functions of the potentials. Often, however, the coefficients are strong nonlinear functions of the potentials. Therefore, a two part model separating the variational principle from the thermodynamic model must be used. The variational principle maps boundary conditions, transport coefficients, and conserved quantity densities to thermodynamic potentials; the thermodynamic submodel maps thermodynamic potentials to densities and transport coefficients. A standard fixed point theorem in conjunction with the properties of variational principles is used to prove the existence of a simultaneous solution under the proper boundary conditions. Using the model structure we can then deduce the feasible set of boundary conditions.;To facilitate numerical solution, the variational models are simplified by approximating the potential fields with basis functions. The variational objective is then integrated, reducing the variable space to weights on the basis functions. For well studied geometries, basis functions can be used for which the coefficients in the resulting nonlinear program are already tabulated (friction factors, heat transfer coefficients, etc).;The numerical method we propose for these problems is a homotopy method which manipulates the link between the thermodynamic and variational subproblems. The starting point is the variational model with trivial constant densities. As the homotopy parameter is increased the dependence of the densities and transport coefficients on the potentials is returned. The same properties that ensure existence of a solution also ensure existence of a bounded homotopy path.