Periodic Reporting for period 1 - NI2PhORC (Numerical Investigation of Two-Phase flows for ORC applications)
Período documentado: 2023-09-01 hasta 2025-10-31
Compressible two-phase flows play a crucial role in many technical applications, where they can manifest as condensation and evaporation in unsteady flows, liquid accumulation in pipelines, boiling and cavitating flows. A relevant example is found in Organic Rankine cycle (ORC) systems, one of the most promising technologies for waste heat recovery. Unlike conventional Rankine cycles, ORCs use organic fluids, which makes them particularly suited for power generation from thermal sources below 300°C. These fluids may exhibit the so-called wet-to-dry expansion: starting within the two-phase region at sufficiently high pressure, an isentropic expansion can end in the dry-vapor region. This distinctive thermodynamic behavior exhorts the investigation of ORCs operating with two-phase expansion.
The overarching goal of the NI2PhORC project is to deliver a computational fluid dynamics (CFD) tool tailored to compressible unsteady non-equilibrium two-phase flows. In CFD, diffuse-interface methods are effective approaches for compressible multiphase flows, able to handle the complexities of evolving material interfaces that separate distinct fluids. At the core lies the Baer-Nunziato (BN) model, which describes compressible two-phase flows in full non-equilibrium, with each phase evolving according to its own pressure, velocity, temperature, and Gibbs free energy.
Various BN-type models have been proposed, incorporating different transfer terms for mass, momentum, and heat exchange between phases. Relaxation terms may also be added to model how the mechanical equilibrium is reached at the interfaces and to include heat and mass transfer. With suitable closures, BN-type models have proven effective in simulating a wide range of multiphase regimes, from dispersed to resolved-interface flows. Moreover, they are particularly suited for fluids described by different equations of state (EOSs), since each phase is treated as a separate continuum with its own thermodynamic model. This capability is essential when working near the saturation curve, where organic fluids often deviate strongly from ideal-gas behavior.
Nevertheless, open questions remain regarding the proper definition of finite relaxation parameters in BN-type models. While instantaneous mass transfer has been shown to be inadequate to match experimental data, an unambiguous formulation for finite relaxation times is still lacking. The NI2PhORC project will aim to extend modeling capabilities for compressible two-phase flows by developing a CFD tool with relaxation parameters that can be tuned based on physical insight or application-specific needs.
The overarching goal of the NI2PhORC project is to deliver a computational fluid dynamics (CFD) tool tailored to compressible unsteady non-equilibrium two-phase flows. In CFD, diffuse-interface methods are effective approaches for compressible multiphase flows, able to handle the complexities of evolving material interfaces that separate distinct fluids. At the core lies the Baer-Nunziato (BN) model, which describes compressible two-phase flows in full non-equilibrium, with each phase evolving according to its own pressure, velocity, temperature, and Gibbs free energy.
Various BN-type models have been proposed, incorporating different transfer terms for mass, momentum, and heat exchange between phases. Relaxation terms may also be added to model how the mechanical equilibrium is reached at the interfaces and to include heat and mass transfer. With suitable closures, BN-type models have proven effective in simulating a wide range of multiphase regimes, from dispersed to resolved-interface flows. Moreover, they are particularly suited for fluids described by different equations of state (EOSs), since each phase is treated as a separate continuum with its own thermodynamic model. This capability is essential when working near the saturation curve, where organic fluids often deviate strongly from ideal-gas behavior.
Nevertheless, open questions remain regarding the proper definition of finite relaxation parameters in BN-type models. While instantaneous mass transfer has been shown to be inadequate to match experimental data, an unambiguous formulation for finite relaxation times is still lacking. The NI2PhORC project will aim to extend modeling capabilities for compressible two-phase flows by developing a CFD tool with relaxation parameters that can be tuned based on physical insight or application-specific needs.
A new solver for the BN model under generic EOS has been implemented, verified, and validated against reference Riemann problems. The solver adopts a splitting strategy based on two operators:
i) a hyperbolic operator, built on a second-order finite-volume scheme with explicit time integration;
ii) a relaxation operator, based on an efficient and adaptive time integrator for the ordinary differential equations (ODEs).
The solver is coupled with the thermodynamic library CoolProp, which provides state-of-the-art EOSs for a wide variety of fluids. The hyperbolic operator ensures the necessary efficiency while working with complex EOSs thanks to a primitive-variable update scheme. Rather than evolving the partial total energies, as in standard BN models, the phasic temperatures are advanced, reducing the computational costs of evaluating the thermodynamic state of each component. A proper choice of the linearization coefficients—related to the thermodynamic derivatives—guarantees conservation of total energy.
A novel relaxation operator has also been developed, specifically tailored to non-equilibrium two-phase flows. It encompasses finite-rate mechanical and thermo-chemical relaxation to model how the mechanical equilibrium is reached at the interfaces and to include heat and mass transfer. Designed to operate with generic EOSs, the operator uses an adaptive sub-time step ODE integrator to ensure both robustness and efficiency. The formulation considers space- and time-dependent relaxation parameters. As a demonstration, an evaporating supersonic nozzle flow of siloxane MDM (Octamethyltrisiloxane) was studied, considering a 5% volumetric fraction of liquid droplets with diameters ranging from 1 to 200 µm. In this test case, all relaxation parameters were expressed as functions of the local fluid state and particle diameter.
Both operators, even when considered independently, introduce novel aspects that advance the simulation of two-phase flows. Taken together, they represent a significant extension of the capabilities of current CFD models and numerical methods.
i) a hyperbolic operator, built on a second-order finite-volume scheme with explicit time integration;
ii) a relaxation operator, based on an efficient and adaptive time integrator for the ordinary differential equations (ODEs).
The solver is coupled with the thermodynamic library CoolProp, which provides state-of-the-art EOSs for a wide variety of fluids. The hyperbolic operator ensures the necessary efficiency while working with complex EOSs thanks to a primitive-variable update scheme. Rather than evolving the partial total energies, as in standard BN models, the phasic temperatures are advanced, reducing the computational costs of evaluating the thermodynamic state of each component. A proper choice of the linearization coefficients—related to the thermodynamic derivatives—guarantees conservation of total energy.
A novel relaxation operator has also been developed, specifically tailored to non-equilibrium two-phase flows. It encompasses finite-rate mechanical and thermo-chemical relaxation to model how the mechanical equilibrium is reached at the interfaces and to include heat and mass transfer. Designed to operate with generic EOSs, the operator uses an adaptive sub-time step ODE integrator to ensure both robustness and efficiency. The formulation considers space- and time-dependent relaxation parameters. As a demonstration, an evaporating supersonic nozzle flow of siloxane MDM (Octamethyltrisiloxane) was studied, considering a 5% volumetric fraction of liquid droplets with diameters ranging from 1 to 200 µm. In this test case, all relaxation parameters were expressed as functions of the local fluid state and particle diameter.
Both operators, even when considered independently, introduce novel aspects that advance the simulation of two-phase flows. Taken together, they represent a significant extension of the capabilities of current CFD models and numerical methods.
The project has developed a numerical framework that advances current capabilities for the simulation of compressible two-phase flows, with particular relevance to non-ideal compressible fluid dynamics (NICFD) regimes. These flow conditions are challenging for existing approaches and are of interest for energy related applications, such as ORC systems. The proposed simulation tool has been shown to operate reliably across a wide range of physical regimes, including two-phase flows with varying droplet sizes as well as limiting cases corresponding to quasi-pure vapors. This versatility is essential for applications like ORC systems, where different flow topologies arise under standard operating conditions.
A key result beyond the state of the art is the formulation of a BN-solver that can handle mechanical and thermo-chemical relaxation processes simultaneously and at arbitrary rates, while remaining compatible with arbitrary equations of state. This combination has not been previously available in a single numerical framework and extends the applicability of full-disequilibrium two-phase flow models.
The solver is designed to operate with user-defined relaxation parameters that may vary in space and time or depend on the solution. While the detailed physical calibration of these parameters requires further investigation, the framework developed in this project provides an effective and reliable basis for such future studies.
Overall, the results contribute a flexible and robust numerical framework that supports further research on complex two-phase flow phenomena and facilitates the exploration of physical effects that were previously difficult to address within a unified modeling approach.
A key result beyond the state of the art is the formulation of a BN-solver that can handle mechanical and thermo-chemical relaxation processes simultaneously and at arbitrary rates, while remaining compatible with arbitrary equations of state. This combination has not been previously available in a single numerical framework and extends the applicability of full-disequilibrium two-phase flow models.
The solver is designed to operate with user-defined relaxation parameters that may vary in space and time or depend on the solution. While the detailed physical calibration of these parameters requires further investigation, the framework developed in this project provides an effective and reliable basis for such future studies.
Overall, the results contribute a flexible and robust numerical framework that supports further research on complex two-phase flow phenomena and facilitates the exploration of physical effects that were previously difficult to address within a unified modeling approach.