Operator-Based Linearization approach for flow and transport with equilibrium and kinetic reactions

In my talk, I will present an efficient element-based reduction technique which can significantly decrease the number of conservation equations and thereby reduce the computational time. The proposed formulation is based on the consistent element balance reduction of the molar (overall composition) formulation. To predict the complex phase behaviour in such systems, we include the chemical equilibrium constraints in the multiphase multi-component flash calculations and solve the thermodynamic and chemical phase equilibrium simultaneously. In this solution, the phase equilibrium is represented by the partition coefficients whereas the chemical equilibrium reaction is represented by the activity coefficients model. Using the Equilibrium Rate Annihilation matrix allows us to reduce the governing unknowns to the element conservation equations only while the coupling between chemical and thermodynamic equilibrium is captured by a simultaneous solution of modified multiphase flash equations. The element composition of the mixture serves as an input for these computations whereas the output is fractions of components in each phase, including solids. 

Next, a finite-volume unstructured discretization in space is applied together with a backward Euler approximation in time. The resulting complex nonlinear system is parameterized using the Operator-Based Linearization (OBL) approach. The OBL framework transfers the governing nonlinear Partial Differential Equations into a linearized operator form where the Jacobian is constructed as a product of a matrix of derivatives with respect to state variables and discretization operators. The state-dependent operators are only evaluated adaptively at vertices of the mesh introduced in the parameter space. The continuous representation of state-dependent operators as well as their derivatives is achieved by using a multi-linear interpolation in parameter space. This means that the usually time-consuming phase and chemical equilibrium computations, performed on each nonlinear iteration and in every control volume, are only executed when evaluating the operators in the new supporting points, thereby significantly reducing both the linearization time and the number of nonlinear iterations. The simulation of multidimensional problems of practical interest has been performed using the proposed technique.