A penalization method for the Direct Numerical Simulation of low-Mach reacting gas-solid flows

Gas-solid flows are encountered in many natural and industrial phenomena. Fluidized beds are the most well-known application of gas-solid reactors in the chemical industry where they are valued for their drastic improvement of mass and heat transfer rates. Yet, the simulation of such equipment at industrial scale is still a challenge due to the tracking of billions of solid particles. In the past decades, different approaches have been addressed to model complex multi-scale and multi-phase reacting flows. The Two-Fluid Model (TFM) which describes both the gas and the solid phases as continua is currently very popular due to its lower computational cost with respect to the Discrete Element Model (CFD-DEM) in which each particle (or group of particles) is tracked in a Lagrangian manner. Direct Numerical Simulation (DNS) turns out to be a powerful tool in order to extract closure laws from fundamental principles for mass, momentum and heat transfer rates which can be subsequently inserted into the TFM or CFD-DEM models. In the recent years, different DNS methodologies for particle-resolved simulation have been investigated, mostly based on the Immersed Boundary Method (IBM) originally developed by Peskin [1] to study elastic boundaries. The direct forcing method introduced by Mohd-Yusof [2] and later improved by Uhlmann [3] allows a better numerical treatment of rigid body problems and is therefore very popular in the fields of particulate flows. DNS-IBM was then applied by several research groups to predict fluid-particle mass, momentum and heat transfer rates [4]–[9]. Among immersed boundary methods, the penalization method developed by Arquis and Caltagirone [10] models solid obstacles as porous media with close to zero porosity. Originally used for fluid-structure interaction problems, this method has been scarcely investigated for the study of heat and mass transfer problems in reactive gas-solid flows. The present study aims at combining the penalization method to account for the presence of the solid phase with the low-Mach number assumption for the gas phase. Indeed, strong thermal effects induced by chemical reactions can induce non-negligible density gradients at the surface of solid particles and affect interfacial transfer laws. The low-Mach number assumption is of high interest for gas-solid reactive flows in that it allows density fluctuations while removing the constraint on the time step imposed by the speed of sound in fully compressible flows. Here, we extend the methodology of Lessani et al. [11] for low-Mach number flows in order to incorporate the penalization of the solid phase for momentum, heat and species transport. Different reaction scenarios are investigated: heat consumption/production and gas expansion or compression. Finally, a comparison is established with the incompressible version of the penalization method to assess the impact of density fluctuations in view of building new closure laws for dense gas-solid flows