252200 Development of Realistic Liquid Equations of State for Multiphase CFD Modelling
Trickle-bed reactors are broadly defined as those in which gas and liquid flow co-currently through a packed bed. The understanding of the characteristic hydrodynamics of trickle-bed reactors have been a subject of long-standing interest in both industry and academia due to the complex coupling of reaction kinetics, mass transfer, global- and local-scale hydrodynamics and multiphase interactions (Satterfield, 1975; Blok et al., 1983; Ng, 1986; Iliuta et al., 1999).
In previous work we validated a single-phase code using high-resolution magnetic resonance imaging (MRI) data (Robbins et al., 2012), and it is now desired to extend the code to simulate a full multi- phase flow inside a trickle-bed reactor. As a step towards achieving this, a one-dimensional multi-phase algorithm based on the AUSM+-upwind discretisation scheme has been tested in order to evaluate the impact of the selection of liquid equation of state and to explore multi-phase numerical phenomena such as interfacial pressure and presence of discontinuities.
For a density-based solver, selection of an appropriate equation of state (EOS) is of paramount importance as it couples the primitive variables (pressure, volume fraction, velocity and temperature) to the thermodynamic variables (density, enthalpy, speed of sound) that define and discretise the conservative variables. Whilst the gas phase is adequately described with an ideal gas assumption, the liquid phase is more difficult to model. One option is to assume that the liquid is incompressible, reducing the mass balance equation to one of a criterion of non-divergence for the velocity in the flow. For a density-based solver, the pressure then has to be solved by rearranging the governing equations in terms of the time-derivatives of the primitive variables. This then requires an artificial compressibility-style method in order to alleviate the mathematical stiffness of the system, particularly at the low-Mach number flows that are of interest (Weiss and Smith, 1995). This stiffness is also exhibited by modelling the essentially-incompressible liquid with a compressible equation of state. The assumption of incompressibility, however, is not valid in trickle-bed reactors where thermal gradients, especially as a cause of endo- or exothermic reactions, drive density changes across the flow. As such, several equations of state have been developed to couple the thermodynamic variables for a compressible liquid phase.
Cubic equations of state such as the Peng-Robinson or the Soave modification of the Redlich-Kwong equation of state (Peng and Robinson, 1976; Soave, 1972) are common candidates for representation of liquid properties. However full evaluation of properties such as enthalpy and speed of sound require calculation of complex departure functions involving second derivatives of the p − v − T state equation, which is computationally expensive and generally inaccurate for liquids. We have therefore coupled the equations of state to group contribution theory (Ru ̆i ̆ka and Domalski, 1993) for the determination of specific heat capacity and subsequent fluid enthalpy along with an assumed constant adiabatic index to simplify the speed of sound calculation. A second option that has been investigated is to use the theory of corresponding-states to derive equations to calculate the density, speed of sound and enthalpy as functions of critical fluid properties in order to couple density to temperature and pressure for a generic liquid.
In this work we have used these newly-developed equations of state to evaluate the flow profiles in three test cases for the system of n-hexane and air. The water-air system has also been considered, as the accurate IAPWS-IF97 (Wagner et al., 2000) equation of state is available to thoroughly compare and mathematically validate the new coupled EOSs. By using a novel method for discretisation of the convective fluxes and source terms over all Mach numbers we have developed a simple and robust method that ensures sharp resolution of discontinuities, eliminates unphysical solutions associated with the non-hyperbolic multi-phase equations and is mathematically amenable to low-Mach number flow of realistic fluids. Agreement of the new EOSs with both analytical solutions and a highly spatially-resolved IAPWS-IF97 EOS reference is excellent, demonstrating that the coupling of group-contribution theory serves to both simplify and accelerate the solution of the EOS.
Ransom’s faucet problem (Ransom, 1987) was used as one of the test cases for the code. This test case is indicative of a common problem encountered in reaction columns - resolution of a sharp discontinuity of volume fraction due to, for example, a gas bubble passing through a computational cell which previously contained liquid. The solutions for a water-air system are compared to a reference EOS, comparing the new algorithms to one using the stiffened-gas equation of state, commonly used as a simple liquid equation of state for evaluation of newly-developed algorithms (Paill`re et al., 2003; Liou et al., 2008).
In the test case, major deviation between the newly-developed liquid EOSs and the stiffened-gas EOS lies in the liquid mass flux, with the over-prediction of liquid density of the stiffened-gas EOS contributing to an overly predicted liquid mass flux. Gas volume fraction and velocity profiles are nearly identical due to the ideal-gas equation of state used for the air in each case, with a slight deviation in simulated pressure at the inlet caused by the differing gas-liquid interaction terms, correlated to liquid density. The newly-developed algorithm allows sharp resolution of flow variable discontinuities, is robust and maintains accuracy even at lower-resolution computational grids.
Thorough testing of the developed equation of states are ongoing with other test cases and fluids to test the robustness of the algorithm and its ability to maintain discontinuity sharpness when confronted with more challenging flow conditions. Once the algorithm has been explored for hexane (exploratory water test cases have been completed), it shall be extended to higher dimensions and implemented into a 3D unstructured parallel code. Other models for source terms such as those for surface tension and reaction are also being investigated for inclusion, with the overall aim of the project being to successfully simulate flow in a trickle-bed reactor using a novel high-fidelity,
parallelised density-based CFD solver.
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