1. Introduction

2. Model Description

2.1. Electrostatic Precipitation Model

The physical fields in the AHPC include the fluid flow, the corona discharge, the particle charging, the particle dynamics and the filtration. The coupled effects between the fields should be considered too.

For the corona discharge, the equations can be expressed in dimensionless forms as follow [13]:

(1)

(2)

(3)

(4)

where $V$ is the electric potential ($V$), $V_{\mathrm{w}}$ the electric potential at the corona wire, $V^{\text{'}}=V/V_{\mathrm{w}}$, ${\rho }_{\mathrm{w}}$ the space charge density at the corona wire ($C/m^3$), ${\rho }_{\mathrm{s}}$ the space charge density and ${\rho }_{\mathrm{s}}^{\text{'}}={\rho }_{\mathrm{s}}/{\rho }_{\mathrm{w}}$, ${\epsilon }_0$ the air permittivity ($8.85\times {10}^{-12}{C}^2/N·m^2$), $\mathrm{J}$ the current density ($A/m^2$), $b_{\mathrm{i}}$ the ion mobility ($m^2/V·s$), $\mathsf{{\rm E}}$ the electric field ($V/m$).

As the pressure drop across the bag filter increase along with the filtration, the fluid flow is unsteady. Besides, the flow field is influenced by the electric field, namely the electro-hydrodynamic (EHD). This paper solves the full Navier-Stokes equations with the RNG turbulent equations [14]:

(5)

(6)

where $U_i(i=1,2,3)$ is the fluid velocity ($m/s$), ${\rho }_{\mathrm{f}}$ is the fluid density ($kg/m^3$), t is the time (s), $\mu$ is the laminar viscosity ($kg/m·s$), ${\mu }_{\mathrm{t}}$ is the turbulent viscosity ($kg/m·s$), $P$is the fluid pressure (Pa), $F_{ci}$ is the electric body force ($N/m^3$), $g_i$ is the acceleration of gravity ($m/s^2$).

The RNG turbulent model is derived from the application of a rigorous statistical technique to the instantaneous Navier-Stokes equations. It is suitable for the problems with high streamline curvatures [15]. An unstructured finite volume method has been developed to solve this equation system [16]. The fluid filed equations are solved by using the CFD package Fluent [17]. The inlet boundary condition is the velocity-inlet and the outlet one is the outflow. The boundary condition of the walls is the wall function. The program for solving the corona discharge equations is linked to the Fluent by using the user defined functions (UDF).

The particle dynamics is modeled by using Lagrangian method [18]. According to the Newton’s second law, the equation that describes the particle motion is written as:

(7)

where ${\mathrm{U}}_{\mathrm{p}}$ is the local particle velocity, $\mathrm{U}$ is the local fluid velocity, ${\rho }_{\mathrm{p}}$ is the particle density, $d_{\mathrm{p}}$ is the particle diameter, $q_{\mathrm{p}}$ is the particle charge ($C$), $C_{\mathrm{D}}$ is the drag force coefficient, $g$ is the gravity gradient.

The drag force coefficient $C_{\mathrm{D}}$ can be expressed as a function of the particle Reynolds number ${\mathrm{Re}}_{\mathrm{p}}$ as follows:

$$C_{\mathrm{D}}=a_1+a_2/{\mathrm{Re}}_{\mathrm{p}}+a_3/{\mathrm{Re}}_{\mathrm{p}}^2$$

(8)

$${\mathrm{Re}}_{\mathrm{p}}=\frac{{\rho }_f\left|\mathrm{U}-{\mathrm{U}}_{\mathrm{P}}\right|d_{\mathrm{P}}}{\mu }$$

(9)

The turbulence has strong effect on the particle motion, namely the turbulence dispersion. In the present work, the discrete random walk (DRW) model [19] is used.

The particle charge $q_{\mathrm{P}}$ in the Eq. (7) is calculated by using a field modified diffusion (FMD) model [20]. The charging rate is expressed in the following dimensionless form:

(10)

where $f(w)$ is the fraction of the surface covered by the diffusive band:

(11)

where $v$($=\frac{q_{\mathrm{p}}e}{2\pi {\epsilon }_0{d}_{\mathrm{p}}kT}$) is the dimensionless particle charge, $w$($=\frac{K_{\mathrm{p}}}{K_{\mathrm{p}}+2}\frac{E{d}_{\mathrm{p}}e}{2kT}$) is the dimensionless electric field, and $t_q(=\frac{{\rho }_s{b}_{\mathrm{i}}t}{{\epsilon }_0})$ is the dimensionless particle charging time, $k$ is Boltzmann’s constant ($1.38\times {10}^{-23}J/K$), $T$ is the temperature ($K$), $e$ is the charge on an electron ($1.6\times {10}^{-19}C$),$K_{\mathrm{p}}$ is the particle dielectric constant, $E$ is the local electric field ($V/m$), ${\rho }_s$ is the space charge density ($C/m^3$).

The particle motion equation (7) is solved by using the Fluent program and the program for solving the particle charge equations (10)-(11) is linked to the Fluent by using the UDF.

2.2. Filtration Model

In the simulation, the particles will deposit on the bag surface to form a cake. There are two pressure drops: the one across the filter body and the one across the cake (Figure 3):

(12)

where $\Delta P$ is the total pressure drop, $\Delta P_c$ the pressure drop across the cake and $\Delta P_f$ the pressure drop across the filter.

The pressure drop across the filter body dependents only on the filtration velocity. And the pressure drop across the cake can be related to the filtration velocity according to the Kozeny-Carman equation [21]:

where ${\epsilon }_c$ is the cake porosity, ${\rho }_{\mathrm{p}}$ the particle density, d_p the particle diameter, $V_f$ the filtration velocity, $\mu$ the air viscosity and $M_c$ the cake mass density.

The cake porosity ${\epsilon }_c$ can be related to the particle diameter as [22]:

(14)

Thus, the pressure drop can be written as:

(15)

In the simulation, the bag filter is modeled as a porous media. And the cake mass density is calculated by using the total mass of the particles that reach the bag surface piece.

3. Case Setup and Parameters

4. Results and Discussion

5. Conclusion

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