Development of an Electrostatic Precipitator with Porous Carbon Electrodes to Collect Carbon Particles
2. Calculation of the Corona Discharge Model 1. Calculation Model
The space charge and electrical field distributions were calculated by the finite-element method (FEM) using COMSOL Multiphysics FEM software [14,15]. The calculation model is shown in Figure1.
The air gap between the high voltage wire electrode and the grounded electrode was 9 mm the voltage supplied to the wire electrode was+8 kV, and the grounded electrode was assumed to be 0.1 mm thick in the case of stainless steel and 10 mm thick in the case of woodceramics. The electrical potential at the bottom of each electrode was 0 V. The overall mesh size was 0.4 mm or less, and the mesh above the grounded electrode was further divided of 0.08 mm in width or less. In this model, it was assumed that the woodceramics electrode was a solid material. The stainless-steel material was not defined in this model but, as a substitute a boundary condition between d and c in Figure1was set as 0 V. In the woodceramics, the gas-flow velocity was set to zero, other parameters of the woodceramics are shown
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in Table1[16–19]. The diffusion constant of the woodceramics was calculated using the Einstein’s equation for Brownian motion as follows:
D μ =k·T
e , (1)
whereDis the diffusion constant,μis the mobility,kis the Boltzmann constant (1.380662×10−23J/K),e is the elementary charge (1.602×10−19C), andTis the temperature (278 K).
(a) Stainless-steel electrode model (b) Woodceramics electrode model x
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Figure 1.Calculation models for (a) the grounded electrode with a stainless-steel electrode and (b) the grounded electrode with a woodceramics electrode (which is defined as a solid substance of 10 mm thickness).
Table 1.Calculation parameters in the air and woodceramics domains.
In Air Woodceramics Mobility (m2/V·s) 2.34×10−4 1.00 Diffusion constant (m2/s) 2.89×10−6 0.256
Relative permittivity 1.00 5.68
Gas flow (m/s) - 0
The space charge was analyzed by the FEM with reference to [14,15]. The spatial electric field was calculated by the following Poisson equation:
− ∇ ·ε0εr∇V=ρ, (2)
whereε0is the dielectric constant,Vis the electric potential, andεris the relative permittivity of air (1.0). The space charge density,r, is expressed by the following equation:
ρ=−eNp, (3)
whereNpis the number density of positive ions (in Num/m3) andeis the elementary charge (in C). The applied voltage and number density of charges were multiplied by tanh (105t) (wheretis the time in seconds) to obtain 90% of the applied voltage at 15μs and 99.5% of the number density at 30μs. The charge densityNpis given around the wire. However, the calculations revealed that there was a gap
Energies2019,12, 2805
between high density and zero density in the unmodified model. Hence, given the central position of the wire wasx=0,Npwas multiplied by the following parameter:
α=1−x2
l2, (4)
wherexis the position in the horizontal direction, andlis half of the wire radius. It should be noted that this calculation method does not result in attenuation compared with the real charge density. The number density of positive ions,Np, was calculated as follows:
Np= Id
Lw× 1
μp×Lg×LAir
Vap×1 e×1
2, (5)
whereIdis the corona discharge current,Vapis the applied voltage,Lwis the length of each wire electrode (88 mm×2), andμpis the mobility of the positive ions (2.34×10−4m2/V·s).
The transport equation for positive ions, including the ion wind and the gas flow, is given in Equation (6).
∂Np
∂t +∇ ·
−Dp∇Np−μpENp
+∇ · NpUg
=0, (6)
whereDpis the diffusion constant of the positive ions in the air (2.89×10−6m2/s),Ugis the gas flow (in m/s), andEis the electric field strength (in V/m):
E =−∇V. (7)
Note that this equation does not account for thermal diffusion.
The Navier–Stokes equation is used to model the gas flow and ion wind as follows:
ρg∂Ug
∂t +ρg
Ug· ∇
Ug=−∇P+μg∇2Ug+F, (8) wherePis the pressure (in Pa),rgis the air density (1.205 kg/m3),μgis the dynamic coefficient of air viscosity (1.822×10−5Pa·s), andFis the external Coulomb force (in N), which is the product of the space charge density,ρand the electric fieldE.
2.2. Numerical Results
The potential distributions are shown in Figure2. The contour lines represent the equipotential line from 500 to 4000 V. With the woodceramics electrode, the potential is high near the grounded electrode. In addition, the ion distributions over 2.0×1014Num/m3are shown in Figure3. The two graphs are similar in shape demonstrating that the ion distribution area in the corona discharge has not changed. Due to the influence of the electrode material, the electric field and ion number distribution were evaluated directly below the wire electrode. The electric fields between the wire and the grounded electrode are shown in Figure4a. The electric field at the surface of the woodceramics electrode was approximately 10% less than that at the surface of the stainless-steel electrode. The ion distribution between the wire and the grounded electrode is shown in Figure4b. This curve was smoothed with a moving average algorithm over an averaging distance of 0.48 mm for approximately 20 sampling points. With the stainless-steel electrode, the ion density slightly fluctuated. The ion density increased close to the electrodes but remained comparable for the two types of electrodes.
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(a) Stainless-steel electrode (b) Woodceramics electrode
Figure 2.Electrical potential of the corona electrode structure with the (a) stainless-steel grounded electrode and (b) woodceramics grounded electrode. The bottom side of the woodceramics was grounded, so, the potential near the woodceramics surface was approximately 480 V because of the current flow in the woodceramics.
(a) Stainless-steel electrode (b) Woodceramics electrode Figure 3. Illustration of the charge number densities of charges at the (a) stainless-steel grounded electrode and (b) the woodceramics electrode. These distributions are similar in shape to the charge distribution. It can be seen that the resistivity of the grounded electrode does not have strong influence on the overall charge distribution.
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Figure 4.(a) Electric field in the y direction and (b) charge density between the wire electrode and grounded electrode. The horizontal axes represent the distance from the wire electrode. With the woodceramics electrode, the electric field decreased by 10% and the charge density was slightly lower than that with the stainless-steel electrode.
Energies2019,12, 2805
These results shed light on the particle-collection process; the particle charge and immigration velocity decrease when using a woodceramics electrode, unlike with a stainless-steel electrode, because of the decrease in the electric field.
3. Experimental Methods