(380bb) A Penalization Method for the Direct Numerical Simulation of Weakly Compressible Reacting Gas-Solid Flows

1. Hardy¹, J. De Wilde², G. Winckelmans³

¹Institute of Mechanics, Materials and Civil Engineering (iMMC), Université Catholique de Louvain, baptiste.hardy@uclouvain.be

²Institute of Mechanics, Materials and Civil Engineering (iMMC), Université Catholique de Louvain
³Institute of Mechanics, Materials and Civil Engineering (iMMC), Université Catholique de Louvain

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) describes both the gas and the solid phases as continua while the Discrete Element Model (CFD-DEM) tracks each particle (or group of particles) in a Lagrangian manner. However, these large-scale models rely on closure relations to characterize the impact of the unresolved scales of the flow onto the large resolved scales. 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 solving external flows past bluff bodies, 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. The proposed method allows the treatment of general boundary conditions for the transported scalars: Dirichlet-type boundary conditions for infinitely fast surface reaction and very rapid solid phase temperature homogenization, Neumann-type boundary conditions for finite rate surface reaction and coupled heat release and finally an intra-particle description with coupled heat and mass transfer between the solid and the fluid phase. Different reaction scenarios are investigated: heat consumption or release 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.

References

[1] C. S. Peskin, “Flow patterns around heart valves: A numerical method,” J. Comput. Phys., vol. 10, no. 2, pp. 252–271, Oct. 1972.

[2] J. Mohd-Yusof, “Combined immersed-boundary/B-spline methods for simulations of fow in complex geometries,” Cent. Turbul. Res. Annu. Res. Briefs, 1997.

[3] M. Uhlmann, “An immersed boundary method with direct forcing for the simulation of particulate flows,” J. Comput. Phys., 2005.

[4] N. G. Deen and J. A. M. Kuipers, “Direct Numerical Simulation of Fluid Flow and Mass Transfer in Dense Fluid–Particle Systems,” Ind. Eng. Chem. Res., 2013.

[5] Y. Tang, S. H. L. Kriebitzsch, E. A. J. F. Peters, M. A. van der Hoef, and J. A. M. Kuipers, “A methodology for highly accurate results of direct numerical simulations: Drag force in dense gas-solid flows at intermediate Reynolds number,” Int. J. Multiph. Flow, vol. 62, pp. 73–86, 2014.

[6] N. G. Deen, S. H. L. Kriebitzsch, M. A. van der Hoef, and J. A. M. Kuipers, “Direct numerical simulation of flow and heat transfer in dense fluid-particle systems,” Chem. Eng. Sci., vol. 81, pp. 329–344, 2012.

[7] F. Municchi and S. Radl, “Consistent closures for Euler-Lagrange models of bi-disperse gas-particle suspensions derived from particle-resolved direct numerical simulations,” Int. J. Heat Mass Transf., vol. 111, pp. 171–190, 2017.

[8] B. Sun, S. Tenneti, and S. Subramaniam, “Modeling average gas-solid heat transfer using particle-resolved direct numerical simulation,” Int. J. Heat Mass Transf., vol. 86, pp. 898–913, 2015.

[9] M. Mehrabadi, E. Murphy, and S. Subramaniam, “Development of a gas-solid drag law for clustered particles using particle-resolved direct numerical simulation,” Chem. Eng. Sci., vol. 152, pp. 199–212, 2016.

[10] E. Arquis and J. P. Caltagirone, “Sur les conditions hydrodynamiques au voisinage d’une interface milieu fluide-milieu poreux: application à la convection naturelle,” CR Acad. Sci. Paris II, vol. 299, pp. 1–4, 1984.

[11] B. Lessani and M. V Papalexandris, “Time-accurate calculation of variable density flows with strong temperature gradients and combustion,” 2005.

Acknowledgements

Baptiste Hardy is a research fellow of the F.N.R.S-Fonds de la Recherche Scientifique, under grant n° 1.A.700.18F. This work was completed with the help of the computational resources provided by the Consortium des Equipements de Calcul Intensif (CÉCI).