# Eulerian Modeling of Monodisperse Gas-Particle Flow with Electrostatic Forces

- Conference: Fluidization
- Year: 2019
- Proceeding: Fluidization XVI
- Group: General Paper Pool
- Session:
- Time: Wednesday, May 29, 2019 - 2:31pm-2:43pm

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

cases.

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

relevant.

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="https://www.aiche.org/sites/default/files/aiche-proceedings/conferences/..." class="documentimage"> |
(1) |

where F_{e} is the electrostatic force, q_{p} is the particle charge, and

α_{p} the solid volume fraction. The electrical potential ϕ is

governed by

âˆ‡ â‹…(Ïµm âˆ‡Ï•) = -p-p Ïµ0 " src="https://www.aiche.org/sites/default/files/aiche-proceedings/conferences/..." class="documentimage"> |
(2) |

where ϵ_{m} is the mixture permittivity and ϵ_{0} 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 q_{p} is known a

priori.

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:

âˆ‚- Î±pQp + âˆ‚-- Î±pUiQp = âˆ‚t Vp âˆ‚xi Vp ( ) -âˆ‚-(Ïƒ E )+ âˆ‚-- (Îº +Îº *) âˆ‚Qp- âˆ‚xi p i âˆ‚xi q q âˆ‚xi " src="https://www.aiche.org/sites/default/files/aiche-proceedings/conferences/..." class="documentimage"> |
(3) |

where V _{p} is the particle volume, U the fluid velocity, σ_{p} the

triboelectric conductivity, κ_{p} the diffusion coefficient and κ_{p}^{*} 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 (κ_{p}).

However, this assumption significantly underestimates the

magnitude of diffusion, and therefore the self-diffusion

coefficient (κ_{p}^{*}) was added a posteriori in analogy to heat

conduction.

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 (κ_{q}∇Q_{p}) 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 (κ_{q}^{*}∇Q_{p}) from the transport equation governing the electrical charge due to the fluctuating particle velocity.

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).

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