Eulerian Modeling of Monodisperse Gas-Particle Flow with Electrostatic Forces

Montilla, C., Université de Toulouse, CNRS-Toulouse
Nasro-Allah, Y., Université de Toulouse, CNRS-Toulouse
Ansart, R., Université de Toulouse, CNRS-Toulouse
Simonin, O., Université de Toulouse, CNRS-Toulouse
Fox, R. O., Iowa State University
Eulerian modeling of monodisperse gas–particle flow with electrostatic forces
Carlos Montilla1 , Youssef Nasro-Allah1 , Renaud Ansart1 , R. O. Fox2 and O. Simonin3
1Laboratoire de Génie Chimique, Université de Toulouse, CNRS–Toulouse. France.

2Department of Chemical and Biological Engineering, Iowa State University, Ames, IA, USA.

3Institut de Mécanique des Fluides de Toulouse, Université de Toulouse, CNRS–Toulouse. France.

Keywords: gas–particle flows, kinetic theory, electrostatic force, CFD simulation


Due to triboelectric charging, the solid phase in gas–particle flows can become electrically charged, which induces an electrical
interaction between all the particles in the system. Because this force decays very rapidly, many current models neglect the contribution
of the electrostatic interaction between the particles. Nevertheless, the impact that this force can have in many industrial configurations is
well documented. In this work an Eulerian–Eulerian model for gas–particle flow is proposed in order to take into consideration the
electrostatic interaction between the particles. We use the kinetic theory of granular flows to derive the transport equation for the
electrical charge for dense gas–particle systems. In order to close the collision integrals in the Boltzmann–Enskog equation, we employ
the Grad’s 13-moments theory to approximate the non-equilibrium probability distribution. The model is validated using two different
problems from the literature: a fully periodic box, for which an analytical solution can be found for some special cases;
and a periodic channel. The Eulerian–Eulerian simulation results are compared against DEM simulations for the same


Nowadays, gas–particle flows play an extremely important role in
many industrial technologies. Fluidized beds, cyclonic separators
and the transport of air pollutants are just a few examples of this
type of flow. In some configurations the particles collide with
other solid materials (either another particle or a solid boundary).
During these particle–particle or particle–wall interactions, an
electron can move from one surface to the other inducing
an electrical charge on the particles. This effect is called
triboelectrification (Matsusaka, S. & Masuda, H.  2003). The
electrically charged particles can now interact with other charged
particles via the Lorentz force. Because the particle velocity is
very small compared to the speed of light, the magnetic
contribution can be dropped, and only the electrostatic term is

Currently, many Eulerian-Eulerian gas–particle models
neglect the effect of this force, because the electrostatic
interaction decays like the inverse square law of the distance between two particles. Nevertheless, it has been known for many decades that this type of interaction can modify the dynamics of the solid phase (Miller C. & Logwinuk, A.  1951). In many cases, the presence of electrical charges is undesirable. For example, they can reduce the efficiency of many industrial reactors, or even increase safety hazards due to the rise in the electric potential (Hendrickson, G.  2006).

Eulerian modeling of the electrostatic force

Recently some attempts have been made to include the electrostatic force into an Eulerian–Eulerian model. One of the first attempts was made by Rokkam, R. G. et al.  (2013). These authors added an extra force term to the solid momentum equation:

" src="" class="documentimage">

where Fe is the electrostatic force, qp is the particle charge, and
αp the solid volume fraction. The electrical potential ϕ is
governed by

∇ ⋅(ϵm ∇ϕ) = -p-p ϵ0
" src="" class="documentimage">

where ϵm is the mixture permittivity and ϵ0 the vacuum
permittivity. This model was applied to a fluidized-bed reactor,
successfully capturing the bed height and solid volume fraction
distribution, as well as the segregation of the solid phases
and the wall-adhesion phenomenon. The model, however,
assumes that the solid phases have a constant electrical
charge, limiting its application to cases where qp is known a

More recently, a more complete approach was proposed by
Kolehmainen, J. et al.  (2018). These authors derived the charge
transport equation using the kinetic theory:

∂- αpQp + ∂-- αpUiQp =
∂t Vp ∂xi Vp ( ) -∂-(σ E )+ ∂-- (κ +κ *) ∂Qp- ∂xi p i ∂xi q q ∂xi
" src="" class="documentimage">


where V p is the particle volume, U the fluid velocity, σp the
triboelectric conductivity, κp the diffusion coefficient and κp* the
self-diffusion coefficient. Assuming an uncorrelated Maxwellian
probability distribution for the velocity and the electrical charge,
they found an expression for the diffusion coefficient (κp).
However, this assumption significantly underestimates the
magnitude of diffusion, and therefore the self-diffusion
coefficient (κp*) was added a posteriori in analogy to heat

In our work, we propose a closure for the collisional and
kinetic dispersion terms in the mean charge balance (3) derived in
the framework of the kinetic theory of granular medium (Jenkins,
J. T. & Richman, M. W.
 1985). In particular, we show that the
closure assumption for the collisional contribution (κq∇Qp) can
be derived without presuming an uncorrelated Maxwellian
probability distribution for the particle electrical charge. In
addition, we derive a closure for the dispersion term due to transport by the random velocity (κq*∇Qp) from the transport equation governing the electrical charge due to the fluctuating particle velocity.

Numerical simulations

In order to test the proposed model, we conduct the two numerical experiments described by Kolehmainen, J. et al.  (2018). First, we solve a fully periodic box with a known initial charge distribution. Making some simplifying assumptions, an analytical solution can be derived for this configuration. The second problem we address is a periodic dense channel with conducting walls. Due to the complexity of this problem, the exact solution cannot be found, and therefore we solve it using CFD simulations. Our results for both problems are compared against the DEM simulations from Kolehmainen, J. et al.  (2018).

The CFD simulations are conducted using NETPUNE_CFD, a state-of-the-art code developed by EDF (Électricité de France). This software is capable of doing transient multiphase flow simulation using an Eulerian n-fluid approach for each phase. The governing equations are solved using a cell-centered finite-volume method. Furthermore, NEPTUNE_CFD can handle structured and unstructured meshes, and can also take advantage of parallel architectures using MPI protocol (Hamidouche et al.  2018).


   Hamidouche, Z., Masi, E., Fede, P., Ansart, R., Neau, H., Hemati, M. & Simonin, O. Numerical simulation of multiphase reactive flows. Advances in Chemical Engineering. Vol. 52, 51–124 (2018)

   Hendrickson, G. Electrostatics and gas phase fluidized bed polymerization reactor wall sheeting. Chemical Engineering Science. Vol 61, 1041–1064 (2006)

   Jenkins, J. T. & Richman, M. W. Grad’s 13-moment system for a dense gas of inelastic spheres. Archive for Rational Mechanics and Analysis. Vol 87, 355–377 (1985)

   Kolehmainen, J., Ozel, A. & Sundaresan, S. Eulerian modelling of gas–solid flows with triboelectric charging. Journal of Fluid Mechanics. Vol 848, 340–369 (2018)

   Matsusaka, S. & Masuda, H. Electrostatics of particles.
Advanced Powder Technology. Vol 14, 143–166 (2003)

   Miller C. & Logwinuk, A. Fluidization studies of
solid particles. Industrial & Engineering Chemistry. Vol 43,
1220–1226 (1951)

   Rokkam, R. G., Sowinski, A., Fox, R. O., Mehrani,
P. & Muhle, M. E. Computational and experimental study
of electrostatics in gas-solid polymerization fluidized beds.
Chemical Engineering Science. Vol 92, 146–156 (2013)