Electrostatic Precipitator

In this tutorial are modeled several aspects of an electrostatic precipitator. First, a simplified model for corona discharges coupled with the Laminar Flow interface is used to compute the fluid velocity, electric field, and space charge density, which are necessary to compute the particle charging and relevant forces acting on particles. After, the Particle Tracing for Fluid Flow interface is used to compute the particle collection efficiency as a function of the particle radius.

Model Definition

Figure 1 shows the simulation domain, which consists of a cross section of a rectangular electrostatic precipitator in a wire-to-plane configuration. The DC high voltage source is applied to the inner electrodes and the walls are grounded. The particles are released from the inlet at the left and are transported with the fluid. Particles accumulate charge along their trajectory and become susceptible to electric forces that deflect their trajectories in the direction of the collecting plates. The operation conditions of the electrostatic precipitator are presented in Table 1.

Table 1: Operation conditions of the electrostatic precipitator.
Inner electrode radius	0.5 mm
Inner electrode separation	15 cm
Inner electrode distance to wall	5 cm
Applied voltage V0	20 kV
Inlet fluid velocity	1m/s
Temperature	293.15 K
Pressure	1 atm

Figure 1: Simulation domain of the electrostatic precipitator (the inner electrodes are not to scale).

Corona Model

The simplified corona model is based on the conservation of current transported by the charged carriers. It should be emphasized that the model is not self-consistent in the sense that the both potential and the electric field need to be given at the corona electrode. In other words, the electric field necessary to sustain the discharge is not obtained from first principles: electron and ion transport, electrons gaining energy from the electric field, and electrons losing energy in collisions with the background gas.

Domain Equations

The simplified model for the corona solves the transport of a charge carrier using the charge conservation equation coupled with Poisson’s equation. The transport of the charge carriers includes drift in the electric field and convection. Without source terms the domain equations are

(1)

(2)

(3)

where J (SI unit: A/m2)is the current density, zq is the charge number, μ (SI unit: m2/V·s) is the mobility, ρq (SI unit: C/m3) is the space charge number density, E is the electric field, u is the fluid velocity (SI unit: m/s), V is electric potential, and ε0 is the vacuum permittivity. This set of equations can be manipulated to obtain the following transport equation

(4)

where it is assumed that the mobility is constant. It is interesting to note that the domain equations do not contain any information related to plasma creation and maintenance. All plasma physics is condensed in the boundary conditions for the inner electrodes.

Boundary conditions

The normal component of the electric field at the corona electrode is used as a boundary condition for Poisson’s equation

(5)

The other boundary conditions for Poisson’s equation are V = 0 at the collection plates and zero charge at the inlet and outlet. The boundary condition for Equation 4 involves in finding the space charge density ρq at the corona electrode, using a Lagrange multiplier, so that the imposed potential V0 is verified

(6)

In this model both potential and electric field are imposed at the corona electrode. To obtain predictive physical results the value of the electric field at the wire needs to be close enough to the real one. Here, it is used Peek’s law

(7)

where E0 (SI unit: V/m) is the breakdown electric field, δ is the gas number density normalized to the gas density at 760 torr and 293.15 K, ri is the radius of the corona electrode.

Laminar Flow Model

The Laminar Flow interface is used to solve for the fluid velocity and pressure

(8)

where μ is the dynamic viscosity (SI unit: kg/(m·s)), ρ is the fluid density (SI unit: kg/m3), p if the pressure (SI unit: Pa), and FEHD is the electrohydrodynamic force define as

(9)

Particle Tracing Model

The particle positions are computed by solving second-order equations of motion for the particle position vector components, following Newton’s second law,

(10)

where q is the particle position (SI unit: m), v is the particle velocity (SI unit: m/s), mp is the particle mass (SI unit: kg), and Ft is the total force (SI unit: N) acting on the particle. In this example the forces acting on the particles are the drag force and the electric force. Rarefaction effects need to be included in the drag force because the particle radius become very small. Here, the drag force FD (SI unit: N) is described with the Cunningham-Millikan-Davis model

(11)

where τp is the particle velocity response time (SI unit: s) define as

(12)

where ρp is the density of the particles (SI unit: kg/m3), dp is the particle diameter (SI unit: m), CD is the drag coefficient, and Rer is the relative Reynolds number given by the expression

(13)

and S is the drag correction coefficient defined as

(14)

where are dimensionless coefficients.

The electric force Fe (SI unit: N) acting on the particles is defined as

(15)

where e (SI unit: C) is the elementary charge, and Z is the accumulated charge number on each particle.

The charge accumulated on the particles is computed with the Lawless model

(16)

where τc is the characteristic charging time

(17)

where kB is the Boltzmann constant, and Ti is the ion temperature. Rf and Rd are the dimensionless charging rates due to field and diffusion transport, respectively, defined as

(18)

(19)

where

(20)

(21)

(22)

where εr,p is the relative permittivity of the particles. fa is a function used to join the diffusion and field charging rates defines as

(23)

Results and Discussion

Figure 2, Figure 3, and Figure 4 show the fluid velocity, the electrostatic potential, and the space charge density obtained with the corona model coupled with the Laminar Flow interface. It is with this information that the particles trajectories are computed.

In the present, the corona and the fluid model are fully coupled. However, model results show that the fluid velocity is practically not influence by the electrohydrodynamic force, and the drift velocity is always much larger than the fluid velocity in the regions of interest. In future works in similar operation conditions it could be of interest to uncouple the two models since the computation times become considerable shorter.

The space charge density is more intense near the inner electrodes, as expected, where a corona discharge is luminous. It is in the regions near the inner electrodes that particles accumulate charge at a faster rate due to the combined effect of large space charge densities and intense electric fields.

Figure 5, Figure 6, Figure 7, and Figure 8 show particle trajectories and charge accumulated in the particle expressed in color for particles of several radius. Particles are released at the left and are transported in the fluid flow toward the right outlet. The particles become progressively charged along their trajectory resulting in electric forces that deflects their trajectory in the wall direction. The particle radius influences the balance of the drag and electric force felt by particles and consequently influences the particle trajectory and the electrostatic precipitator collection efficiency.

Figure 9 and Figure 10 present the particle collection efficiency and the average particle charge at the last time step as a function of the particle radius. The collection efficiency is larger at the extremes of the particle dimensions. Larger particles are collected more efficiently because they attain greater electric charge, while smaller particles are collected more efficiently because are subjected to less drag force. Between this two extremes, the drag force influences the most the particle trajectories resulting in almost straight lines parallel to the collection plates (see Figure 6) that result in low collection efficiency.

At the small particle branch partial charging becomes important. A model that correctly describes partial charging should capture the random nature in which a particle can have a charge of 1 or 0. In the present model, particles can be charged with a number between 0 and 1, which might result in inaccurate collection efficiency results for small particles.

Figure 2: Velocity magnitude of the flow in the electrostatic precipitator.

Figure 3: Electric potential.

Figure 4: Space charge density.

Figure 5: Particle trajectories with the charge number along the trajectory expressed in color for particles with a radius of 0.01 μm.

Figure 6: Same as in Figure 5 for a particle radius of 0.2 μm.

Figure 7: same as in Figure 5 for a particle radius of 2 μm.

Figure 8: same as in Figure 5 for a particle radius of 5 μm.

Figure 9: Particle collection efficiency as a function of the particle radius.

Figure 10: Average charge accumulated per particle at the last time step as a function of the particle radius.

Plasma_Module/Corona_Discharges/electrostatic_precipitator

References

1. M.A. Lieberman and A.J. Lichtenberg, Principles of Plasma Discharges and Materials Processing, John Wiley & Sons, 2005.

2. A.A. Kulikovsky, “Positive streamer between parallel plate electrode in atmospheric pressure air,” J. Phys. D: Appl. Phys., vol. 30, pp. 441–450, 1997.

3. LXCAT, see http://fr.lxcat.net for Phelps database (2016).

Modeling Instructions

The following instructions show how to create a 2D model of an electrostatic precipitator and how to obtain the particle collection efficiency as a function of the particle radius. Two studies are needed:

•

A study that couples the , and interfaces.

•

A study that solves for the particle trajectories using the interface to obtain the particle collection efficiency.

From the menu, choose .

New

In the window, click

Model Wizard

In the window, Select the interface and to compute the fluid velocity, the electric field, and the space charge density that are necessary for the interface to compute the particle charging and trajectories.

click

In the tree, select .

Click .

In the tree, select .

Click .

Click

In the tree, select .

Click

Add some parameters for the precipitator dimensions, the corona electrode configuration, and the ion reduced mobility.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
H	0.1[m]	0.1 m	Height
rin	0.5[mm]	5E-4 m	Electrode radius
W	0.7[m]	0.7 m	Width
sp	15[cm]	0.15 m	Electrode separation
muN	3e21[1/(Vms)]	3E21 1/(V·m·s)	Reduced ion mobility

Geometry 1

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type W.

In the text field, type H.

Locate the section. In the text field, type -H/2.

Circle 1 (c1)

In the toolbar, click

In the window for , locate the section.

In the text field, type rin.

Locate the section. In the text field, type W/2-sp.

Array 1 (arr1)

In the toolbar, click

and choose .

Select the object only.

In the window for , locate the section.

In the text field, type 3.

Locate the section. In the text field, type sp.

Difference 1 (dif1)

In the toolbar, click

and choose .

Select the object only.

In the window for , locate the section.

Find the subsection. Click to select the

toggle button.

Select the objects , , and only.

Click

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Add , and features. The feature is used to introduce the electrohydrodynamic force computed in the interface into the interface.

Laminar Flow (spf)

Inlet 1

In the window, under right-click and choose .

Select Boundary 1 only.

In the window for , locate the section.

From the list, choose .

Locate the section. In the Uav text field, type 1[m/s].

Outlet 1

In the toolbar, click

and choose .

Select Boundary 4 only.

Volume Force 1

In the toolbar, click

and choose .

Select Domain 1 only.

In the window for , locate the section.

Specify the F vector as


ct.Fehdx	x
ct.Fehdy	y

In the interface only the ground needs to be defined. The applied voltage is defined in the feature in the node.

Electrostatics (es)

In the window, under click .

Ground 1

In the toolbar, click

and choose .

Select Boundaries 2 and 3 only.

In the feature the electric potential for the charged species migration comes automatically from the interface. The ion mobility and the charge number of the ion need to be set.

Add a second coupling mechanism between the and the interfaces by adding convection to the transport mechanisms.

Charge Transport (ct)