Charging models for airborne suspended particles around HVDC lines

Abstract

Ground total electric field (GTEF) is a vital parameter in the field of electromagnetic environment of high voltage direct current (HVDC) lines. Some experimental works reported that airborne suspended particles can be charged in the corona ion flow of the lines and thus affect GTEF. This paper aims at studying the charging models for particles. Equivalent unipolar charging models (EUCMs) and bipolar charging models (BCMs) for describing field and diffusion charging mechanisms in bipolar ion environments are introduced and the Maxwell‐Wagner relaxation is proposed to be incorporated. Besides, the EUCMs and BCMs are analysed and validated through comparisons with experimental data. The results show that BCMs are applicable to describing the particle charging in a bipolar ion environment, while EUCMs are not. The predictions made by BCM that takes into account the Maxwell‐Wagner relaxation are in good agreement with experimental results. Moreover, the BCMs are proposed to analyse the experimental results about GTEF in the presence of particles, which are not understood before. It is demonstrated that the BCMs are able to qualitatively explain these results. Finally yet importantly, a new charging model combining field and diffusion charging mechanism is proposed for airborne suspended particles around HVDC lines and validated by experiments. The results show that the proposed hybrid field‐diffusion charging model is able to give predictions consistent with experiments.

INTRODUCTION

The ultra high voltage direct current (UHVDC) transmission technology in China has witnessed huge progress and massive achievements in the past decades, evolving from transmitting hydropower in the South at the very beginning to transmitting thermal power and clean energy in the North at present [1]. The environment and climate of the North and South are quite distinct, leading to some new issues confronted in developing high voltage direct current (HVDC) transmission in the North. It is found by using electromagnetic environment experiments that under the HVDC transmission lines in the North the ground total electric field (GTEF) at the side of negative pole shall be abnormally enhanced and the increase can be twice to three times as high as that in the normal case, when it is dry and air polluted [2, 3]. This phenomenon has rarely been observed in the South. It is believed to be associated with the airborne suspended particles which are charged and hardly moving in the ion flow field [2, 3]. Total electric field is the superposition of space-charge-free electric field stimulated by the stressed conductors and the space charge electric field generated by the ions. The GTEF is directly connected with the environment protection and sustainable development of UHVDC projects [4]. Hence, it is of key importance to control and mitigate the adverse effects of suspended particles on the total electric field of HVDC transmission lines. This paper is devoted to studying the particle charging mechanisms in the ion flow field of HVDC lines and their models, in order to lay foundations for constructing a reasonable mathematical model to predict GTEF. In addition, particle charging models also have a value in studying the deposition of charged particles onto the conductor surfaces, which changes the surface morphologies and thus affects the corona characteristics of conductors [5] and corona audible noise [6].

When a particle is exposed to ions in an electric field, these ions will collide with the particle and transfer their charges as a result of Brownian motion and electric-field driven motion. These two charging mechanisms can be found in lots of circumstances, such as electrostatic precipitation [7], electrical measurement of aerosols [8, 9] and charging of droplets and aerosols in cloud [10]. To comprehend and model these two mechanisms, some models have been brought forward in the literature. For field charging, Pauthenier and Moreau-Hanot derived the Pauthenier's model in 1932 [7] and it has been widely used since their work. However, some experimental results indicate that there exist appreciable discrepancies between this model and the experimental data [11‒13]. Besides, no one in aerosol science knew how to deal with a lossy particle which has permittivity and conductivity at the same time since Pauthenier's model is applicable to either purely dielectric or ideally conductive particle. These issues were addressed in Reference [14] by using Maxwell-Wagner relaxation theory and the so-called Maxwell-Wagner relaxation model was proposed. The validity of this new model has been demonstrated in unipolar case [14], but not in bipolar case yet.

For diffusion charging, the most widely used model is Fuchs' model and its improvements (hereafter all of them are called Fuchs' model). This type of model is first proposed by Fuchs [15] based on the work of Natanson [16] and later improved by Hoppel and Frick [17, 18], Stommel and Riebel [19]. The basic idea of Fuchs' model is that there is a limiting spherical shell dissecting the space around a particle into two non-overlapping zones, in one of which tightly close to the particle the ions behave as free molecules and in the other of which ions are treated as continuum, and on the interface of the two zones, namely the limiting spherical shell, the ionic fluxes in the two zones are matched. Fuchs' model has been validated by many experimental works [20‒23]. The other model is approximation solution of the Boltzmann equation based model [24‒26], which is too complex and rarely used relative to Fuchs' model.

To predict the GTEF in the presence of particles, some works incorporated the Pauthenier's model and Fuchs' model into the ion flow field of HVDC lines [27‒30]. There is a controversy on how to deal with the bipolar ions in the charging models. Some previous works intuitively thought a particle charging in the space full of bipolar ions is equivalent to one charging in a virtual space filled with net ion density, namely the algebraic sum of positive and negative ion densities. Models using this idea is termed equivalent unipolar charging models (EUCM). This idea may stem from the charging models in electrostatic precipitation where ions are unipolar. While some other works believe the net density flux, namely the algebraic sum of positive and negative ion density fluxes to the particle, should be used. Models using this idea is termed bipolar charging models (BCM). Since these two types of models have been applied in the charging of particles suspended around HVDC lines, it is of need to systematically compare them and justify their validities. While this point has not been addressed in previous works. It should be noted that in previous BCM models in References [27‒29], the Pauthenier's model, instead of the Maxwell-Wagner relaxation model, is employed.

In addition to those above-mentioned issues, experimental results show that GTEF in the presence of airborne suspended particles depends on relative humidity (RH). This coupling effect of airborne particles and humidity on GTEF has rarely been identified and reported in the literature and the physical mechanism underlying it needs to be explored.

To deal with the forgoing issues, firstly the EUCM and BCM for modelling field and diffusion charging mechanisms in bipolar ion environment are introduced in this paper, among which the Maxwell-Wagner relaxation mechanism is proposed to be inserted. Secondly, the EUCM and BCM are analysed and validated by comparisons with experimental data. Thirdly, the experimental result that GTEF in the presence of airborne suspended particles depends on RH is qualitatively analysed using the BCM. Lastly, a new charging model combining field and diffusion charging mechanism is proposed for airborne particles suspended around HVDC lines, which is validated by experimental results reported previously.

FUNDAMENTAL MODELS FOR PARTICLE CHARGING BY UNIPOLAR IONS

Diffusion charging – the Fuchs' model

When the electric field is ignored and a particle is suspended in a medium full of ions, some of these ions collide with the particle under the action of Brownian motion and thus charge the particle without the help of electric field. Fuchs' model has been proposed to describe the diffusion charging, in which the particle-ion attachment coefficient for diffusion charging R _D is [31, 32] 1 $R_{\text{D}}\left(a,q_{\text{D}}\right)=\frac{F_{\text{i}}\exp \left(-{\varphi }^{\left(q_{\text{D}}\right)}\left(\delta \right)/k_{\text{b}}T\right)}{1+\frac{F_{\text{i}}\exp \left(-{\varphi }^{\left(q_{\text{D}}\right)}\left(\delta \right)/k_{\text{b}}T\right)}{4\pi D_{\text{i}}}{\displaystyle {\int }_{\delta }^{\infty }\frac{1}{{\rho }^2}\exp \left({\varphi }^{\left(q_{\text{D}}\right)}\left(\delta \right)/k_{\text{b}}T\right)d\rho }}$

where R _D(a, q _D) is the attachment coefficient between ions and particles with charge q _D and radius a; D _i is the diffusion coefficient of ions and its value can be derived by use of Stokes-Einstein Relationship k _i = eD _i/k _b T in which k _i is the ion mobility; k _b is the Boltzmann constant; T is the gas temperature in unit K; δ is the radius of limiting sphere, which is given by [31] 2 $\delta =\frac{a^3}{{\lambda }_{\text{i}}^2}\left[\frac{1}{5}{\left(1+\frac{{\lambda }_{\text{i}}}{a}\right)}^5-\frac{1}{3}\left(1+\frac{{\lambda }_{\text{i}}^2}{a^2}\right){\left(1+\frac{{\lambda }_{\text{i}}}{a}\right)}^3+\frac{2}{15}{\left(1+\frac{{\lambda }_{\text{i}}^2}{a^2}\right)}^{2.5}\right]$

where λ_i is the mean free path. F _i is a ratio constant given by 3 $F_{\text{i}}=\pi {\delta }^2\overline{c_{\text{i}}}\alpha$ where ${\overline{c}}_{\text{i}}$ is the mean thermal velocity and can be calculated by 4 $\overline{c_{\text{i}}}=\sqrt{\frac{8k_{\text{b}}T}{\pi m_{\text{i}}}}$ where m _i is the mass of ions; α is the collision probability and defined as the ratio of the ions collided with and deposited onto the particle surface to those entering into the limiting sphere, namely 5 $\alpha ={\left(\frac{b_{\min }}{\delta }\right)}^2$ where b _min is the impact parameter corresponding to the minimum apsoidal distance r _min, and these two can be calculated by finding the minimum of 6 $b^2=r^2\left\{1+\frac{2}{3k_{\text{b}}T}\left[\varphi \left(\delta \right)-\varphi \left(r\right)\right]\right\}$

In Equations (1) and (6), ϕ(r) is the electrostatic potential of the ions in the field of the particle at the distance r 7 $\varphi \left(r\right)=\frac{1}{4\pi {ϵ}_1}\left[\frac{q_{\text{D}}e}{r}-\frac{{ϵ}_2-{ϵ}_1}{{ϵ}_2+{ϵ}_1}\frac{e^2{a}^3}{2r^2\left(r^2-a^2\right)}\right]$ where q _D is the particle charge and considered positive if the charges of the ions and the particles are of the same sign; ε₁ and ε₂ are the permittivity of the medium and the particles, respectively. The first term of Equation (7) represents the Coulomb potential, and the second term represents the image potential. Using the particle-ion attachment coefficient R _D, the particle diffusion charging equation can be given as 8 $\frac{d{q}_{\text{D}}}{dt}=e{n}_0{R}_{\text{D}}$ where n ₀ and q _D are the number density and the charge of the particle, respectively. Normally, numerical solutions are sought for Equation (1), and an analytical solution to it exists only if the image potential is neglectable. This condition is fulfilled when the particle radius is sufficiently large as the impact of the image potential goes down with the increase of particle radius [8, 23]. In this case R _D can be simplified as 9 $R_{\text{D}}\left(a,q_{\text{D}}\right)=\frac{D_{\text{i}}e{q}_{\text{D}}}{{ϵ}_1{k}_{\text{B}}T\left[\exp \left(\frac{e{q}_{\text{D}}}{4\pi {ϵ}_{1}a{k}_{\text{B}}T}\right)-1\right]}$

In Fuchs' model, the values of mass, mobility, diffusion coefficient, mean free path and mean thermal velocity need to be known to calculate the particle charge q _D. These parameters are called ion properties. They are dependent on each other and given any two of them, the other three can be derived. Normally, mass and mobility are set to be known, and diffusion coefficient can be derived by the Stokes-Einstein Relationship, mean thermal velocity by Equation (4), and mean free path by 10 ${\lambda }_{\text{i}}=1.329\frac{k_{\text{i}}}{e}\sqrt{\frac{m_{\text{i}}{m}_{\text{a}}}{m_{\text{i}}+m_{\text{a}}}{k}_{\text{b}}T}$

For particle charging in air, m _a = 28.8 amu. amu is short for atomic mass unit and 1 amu = 1.66 × 10⁻²⁷ kg.

Field charging – the Maxwell-Wagner relaxation model

When the Brownian motion of ions is ignored and a particle is suspended in a medium full of ions and stressed with an external electric field, these ions are driven by the field to deposit on the particle and thus it gets charged. The charging equation in Maxwell-Wagner relaxation model is formulated as 11 $\frac{d{q}_{\text{E}}}{dt}=R_{\text{E}}\left(a,q_{\text{E}}\right)e{n}_0$ where q _E is the particle charge; the particle-ion attachment coefficient for field charging R _E is given by [14] 12 $R_{\text{E}}\left(a,q_{\text{E}}\right)=\pi a^2{k}_{\text{i}}{E}_{0}f\left(a,t\right){\left(1-\frac{q_{\text{E}}}{4\pi a^2{ϵ}_1{E}_{0}f\left(a,t\right)}\right)}^2$

where 13 $\begin{array}{l}f\left(r,t\right)=1+\frac{2a^3}{r^3}\frac{{ϵ}_2-{ϵ}_1}{{ϵ}_2+2{ϵ}_1}\exp \left(-\frac{t}{{\tau }_{\text{MW}}}\right)\\ +\frac{2a^3}{r^3}\frac{{\sigma }_2-{\sigma }_1}{{\sigma }_2+2{\sigma }_1}1-\exp \left(-\frac{t}{{\tau }_{\text{MW}}}\right)\end{array}$

and 14 ${\tau }_{\text{MW}}=\frac{{ϵ}_2+2{ϵ}_1}{{\sigma }_2+2{\sigma }_1}$

where σ₁ and σ₂ are the conductivities of the medium and particle, respectively. This τ _MW is called Maxwell-Wagner relaxation time constant. More discussions on the Maxwell-Wagner relaxation model can be found in Reference [14].

FUNDAMENTAL MODELS FOR PARTICLE CHARGING BY BIPOLAR IONS

In this section, we will first introduce the two types of models. Especially the bipolar charging model that includes Maxwell-Wagner relaxation for bipolar ion environment is proposed.

Equivalent unipolar charging models

The EUCM for diffusion charging is formulated as 15 $\frac{\text{d}{q}_{\text{D}}}{\text{d}t}=e\left(n_{\text{i}+}-n_{\text{i}-}\right)R_{\text{D}}\left(a,q_{\text{D}}\right)$ where n _i+ and n _i‒ denote the positive and negative ion density, respectively; R _D is given by Equation (1). The EUCM for field charging is proposed based on the Maxwell-Wagner relaxation model and it is formulated as 16 $\frac{d{q}_{\text{E}}}{dt}=e\left(n_{\text{i}+}-n_{\text{i}-}\right)R_{\text{E}}\left(a,q_{\text{E}}\right)$ where R _E is given by Equation (12). The physical meaning of these two equations are that particles charging in dielectrics full of bipolar ions is equivalent to those charging by unipolar ions of density n ₀ = n _i+ ‒ n _i‒.

Bipolar charging models

Diffusion charging

The BCM for diffusion charging is formulated as 17 $\frac{\text{d}{q}_{\text{D}}}{\text{d}t}=e{n}_{\text{i}+}{R}_{\text{D}+}\left(a,q_{\text{D}}\right)-e{n}_{\text{i}-}{R}_{\text{D}-}\left(a,-q_{\text{D}}\right)$ where R _D is given by Equation (1). The two terms on the right hand side of the equation represent positive and negative charging currents, respectively. The physical meaning of this equation is that in dielectrics full of bipolar ions particles are charged by positive and negative ions simultaneously and the total charging current fluxing into the particles is equal to the algebraic sum of positive and negative charging currents driven by diffusion charging.

Let the total charging current entering into the particle be zero, namely dq _D/dt = 0, we have 18 $\frac{R_{\text{D}+}\left(a,q_{\text{D}}\right)}{R_{\text{D}-}\left(a,-q_{\text{D}}\right)}=\frac{n_{\text{i}-}}{n_{\text{i}+}}=\frac{1}{\gamma }$ where γ is the ratio of positive ion density to negative ion density. Define the state where dq _D/dt = 0 as an equilibrium one. The charge q _Deq at the equilibrium state can be computed by Equation (18). Normally, this equation has to be solved numerically. Only when the particle radius if sufficiently large to neglect the influence of image potential, an analytical solution can be found [33, 34]: 19 $q_{\text{Deq}}=\frac{4\pi {\epsilon }_{1}a{k}_{\text{B}}T}{e}\text{ln}\mathrm{\Gamma }$ where Γ is the ratio of positive ionic conductivity σ_i+ to negative ionic conductivity σ _i‒. The ionic conductivity σ_i is defined as σ_i = en _i k _i. It can be inferred from Equation (19) that the q _Deq at equilibrium state is proportional to the natural logarithm of the ionic conductivity ratio Γ. When Γ = 1, q _Deq = 0 even if the particle is surrounding by ions.

Using the ion density ratio γ, the diffusion charging Equation (17) can be reformulated as 20 $\frac{d{q}_{\text{D}}}{dt}=e{n}_{\text{i}+}\left(R_{\text{D}+}\left(a,q_{\text{D}}\right)-\frac{1}{\gamma }{R}_{\text{D}-}\left(a,-q_{\text{D}}\right)\right)$

It can be inferred from this equation that given gas temperature, ion mobilities, particle radius, and positive or negative ion density, the ion density ratio γ is the decisive factor during the diffusion charging process. When γ→0 or +∞, Equation (20) reduces to Equation (8).

Field charging

The BCM for field charging is proposed based on the Maxwell-Wagner relaxation model and it is formulated as 21 $\frac{\text{d}{q}_{\text{E}}}{\text{d}t}=e{n}_{\text{i}+}{R}_{\text{E}+}\left(a,q_{\text{E}}\right)-e{n}_{\text{i}-}{R}_{\text{E}-}\left(a,-q_{\text{E}}\right)$ where R _E is given by Equation (12). The physical meaning of this equation is similar to that of Equation (17).

Let the total charging current entering into the particle be zero, namely dq _E/dt = 0, we have 22 $\frac{R_{\text{E}+}\left(a,q_{\text{E}}\right)}{R_{\text{E}-}\left(a,-q_{\text{E}}\right)}=\frac{n_{\text{i}-}}{n_{\text{i}+}}=\frac{1}{\gamma }$

Define the state where dq _E/dt = 0 as an equilibrium one. The charge q _Eeq at the equilibrium state can be computed by Equation (22), which is given by 23 $q_{\text{Eeq}}=4\pi a^2{ϵ}_1{E}_{0}f\left(a,t\right)\frac{\sqrt{\mathrm{\Gamma }}-1}{\sqrt{\mathrm{\Gamma }}+1}$

Since dq _Eeq/dt = 0, f(a,t) must be independent of time t. When the relaxation time τ _MW is finite and the charging time t _pc→∞, the Maxwell-Relaxation process is over and f(a,t) is constant in this case and given by 24 $f\left(a,t\right)=\frac{3{\sigma }_2}{{\sigma }_2+2{\sigma }_1}$

In practical cases, the charging time t _pc is finite. If τ_MW << t _pc, the Maxwell-Relaxation process is finished very quickly comparing to the charging process and f(a,t) is constant in the end and given by equation. If τ_MW >> t _pc, the Maxwell-Relaxation process is just started by the end of charging, f(a,t) is considered to be constant and given by 25 $f\left(a,t\right)=\frac{3{ϵ}_2}{{ϵ}_2+2{ϵ}_1}$

If τ _MW is comparable to t _pc, namely τ _MW ∼ t _pc, f(a,t) can be even considered as constant, and thus Equation (23) is not valid in this case. Based on forgoing analysis, our bipolar field charging model predicts that only when the relaxation time constant τ _MW is incomparable to the charging time t _pc, an equilibrium can be reached. In Pauthenier's model, Maxwell-Wagner relaxation is not existent, which can be considered as a limiting case where τ_MW >> t _pc. Hence, in this model, the equilibrium state can always be reached. This is one of the differences between Pauthenier's Model and Maxwell-Wagner relaxation model, which is not covered in Reference [14].

Under the condition that f(a,t) can be considered constant, Equation (23) can be reformulated as 26 $q_{\text{Eeq}}=q_{\text{s}}\frac{\sqrt{\mathrm{\Gamma }}-1}{\sqrt{\mathrm{\Gamma }}+1}$

where q _s is the saturation charge given as 27 $q_{\text{s}}=4\pi a^2{ϵ}_1{E}_{0}f\left(a,t\right)$ where f(a,t) is given by Equation (24) or (25). From Equation (26), one can find the charge q _Eeq at the equilibrium state is decided by the square root of the ionic conductivity ratio Γ, further the square root of the ion density ratio γ. When Γ = 1, q _Eeq = 0 even if the particle is surrounding by ions. Using the ion density ratio γ, the field charging Equation (21) can be reformulated as 28 $\frac{\text{d}{q}_{\text{E}}}{\text{d}t}=e{n}_{\text{i}+}\left(R_{\text{E}+}\left(a,q_{\text{E}}\right)-\frac{1}{\gamma }{R}_{\text{E}-}\left(a,-q_{\text{E}}\right)\right)$

It can be inferred from this equation that given gas temperature, ion mobilities, particle radius, and positive or negative ion density, the ion density ratio γ is the decisive factor during the field charging process. When γ→0 or +∞, Equation (28) reduces to Equation (11).

The comparisons of the models and their validities

The biggest difference between EUCM and BCM is treating the charging current flowing into the particle in a distinct way. EUCM states that the net charge forms the charging current, while BCM argues that both positive and negative ion charge the charge at the same time and the net charging current equals to the total charging current. To deepen our understanding of this difference, a numerical study is conducted in the following to give a quantitative impression. The parameters used in the numerical study are listed in Table 1.

TABLE 1 The parameters used in this section

Parameter	Symbol	Unit	Value
Vacuum permittivity	ε 0	F/m	8.85 × 10−12
Air permittivity	ε 1	F/m	1ε 0
Particle permittivity	ε 2	F/m	2.46ε 0
Air conductivity	σ 1	S/m	2 × 10−14
Particle conductivity	σ 2	S/m	4 × 10−16
Particle radius	a	μm	0.075, 0.15, 5.5
Electric field	E 0	kV/cm	0.5
Particle charging time	t cp	s	0.69
Air molecule mass	m a	amu	29
Positive ion mass	m i+	amu	109
Negative ion mass	m i‒	amu	95
Positive ion mobility	k i+	m2/(V s)	1.33 × 10−4
Negative ion mobility	k i‒	m2/(V s)	1.85 × 10−4
Ionic density ratio	γ	/	1∼+∞

Define dimensionless time τ and dimensionless charge ν as follows 29 $\tau =\frac{e}{{ϵ}_1}{k}_{\text{i}+}{n}_{\text{i}+}t$ 30 $\nu =\frac{eq}{4\pi {ϵ}_{1}a{k}_{\text{b}}T}$ where q is a function of time t, and consequently ν is a function of τ. This q could be the diffusion charge q _D, the field charge q _E or the particle's total charge. Correspondingly, dimensionless diffusion charge ν_D and dimensionless field charge ν_E can be defined. Let t→∞, we have q _D(∞) = q _Deq and thus ν_D(∞) = eq _Deq/(4πε ₁ ak _b T), if the diffusion charging is considered. Similarly, we have q _E(∞) = q _Eeq and ν_E(∞) = eq _Eeq/(4πε ₁ ak _b T), if the field charging is dealt with.

Let γ varies from 1 to 10⁶ and calculate the particle charges. The results are shown in Figure 1. It can be seen from Figure 1 that there are huge differences between the results predicted by EUCM and BCM. Firstly, when γ = 1, EUCM predicts ν_D = 0 and ν_E = 0 for the positive and negative ion densities cancelled each other and the net charge density n ₀ = 0. However, BCM predicts ν_D = ‒0.24 and ν_E = ‒0.20 by τ = 100. Secondly, comparing to the case where γ = 1, the charging dynamics predicted by EUCM changes by a greater extent than that by BCM when γ = 3. Thirdly, equilibrium states can be clearly seen in BCM under the conditions γ = 3 and 10, while they do not appear in EUCM. And the time duration for reaching equilibrium state depends on γ. The larger the γ is, the longer time it takes for reaching the equilibrium state. Lastly, when γ ≥ 10, EUCM predicts that γ impacts the charging dynamics insignificantly, while BCM predicts that γ affects it appreciably.

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In the following part, both EUCM and BCM are compared with experimental data to check their validities. References [33, 35] reported the bipolar charging characteristics of particles. Figure 2 plots the experimental results for particles of radius a = 0.15 μm and γ = 4.17, 13.91 and +∞. In this case, particles are mainly charged by diffusion. The predicted results given by EUCM and BCM are presented in Figure 2. Note field charging is omitted. It can be seen from Figure 2 that although there are deviations from the experimental data when γ = 13.91 and +∞, the predicted numerical results given by BCM agree well with them qualitatively for all γs. Especially, experimental results show that γ impacts the charging dynamics and equilibrium states exist, which are in good consistency with the predictions by BCM. On the contrary, it appears that γ has an insignificant effect on the charging and no equilibrium states occur.

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Figure 3 plots the experimental results for particles of radius a = 5.5 μm and γ = 4.17, 13.91 and +∞. In this case, particles are mainly charged by electric field. The predicted results given by EUCM and BCM are presented in Figure 3. Note diffusion charging is omitted. It can be seen from Figure 3 that although there are appreciable deviations from the experimental data, the predicted numerical results given by BCM agree well with them qualitatively for all γs. Especially, experimental results show that γ impacts the charging dynamics and equilibrium states exist, which are in good consistency with the predictions by BCM. On the contrary, it appears that γ has an insignificant effect on the charging and no equilibrium states take place.

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From the experimental results in Figures 2 and 3, it can be seen that there are two prominent features with regard to the charging behaviours of particles in bipolar ion environment. When γ ≥ 1,

• the ion density ratio γ is a key parameter controlling the acquired charge of particles. Under given conditions, the particle charge goes up with increasing γ starting from 1;

• as long as the charging time t _pc is large enough, the particle charge can reach steady values, and the time duration for reaching the steady values are dependent on γ. The larger the γ is, the more time it costs.

When γ < 1, after replacing the γ with 1/γ in the two points above, they remain correct.

BCM is able to explain these features. It claims that the physical mechanism behind them is that positive and negative ion charging currents flow into the particle simultaneously. In the following, this point is illustrated in detail. Due to the fact that the positive ion mobility k _i+ is a bit smaller than negative ion mobility k _i‒, the ratio k _i+/k _i‒ is a bit less than 1. If the ratio γ is sufficiently large to ensure the conductivity ratio Γ > 1, at the very beginning of the charging process, the positive ionic charging current is larger than the negative ionic charging current, and consequently the particle is positively charged. As time goes by, the more positive charge the particle has, the greater repulsion on positive ions and larger attraction on negative ions it enforces, and thus the positive ion charging current goes down and negative ion charging current goes up. At some time constant, the two charging currents cancel each other, leading to the total charging current being zero, and thus an equilibrium state is reached and a steady charge is obtained. The bigger the γ is, the larger the ratio of the positive ion charging current to the negative one. As a consequence, the more time it takes in reaching an equilibrium state.

Based on the forgoing comparisons and analysis, it can be concluded that EUCM is not applicable to model the particle charging in mediums full of bipolar ions, such as particles suspended around HVDC lines, while BCM has the ability to do so.

Applications of the bipolar charging models

China Electric Power Research Institute has conducted a large number of experiments on the effects of airborne suspended particles on the GTEF under UHVDC lines. In hazy days, the GTEF could increase quite a lot and this increase depends heavily on the RH. As RH goes up, the increased GTEF goes down. Figures 4 and 5 display two typical results. They are obtained under the full-scale UHVDC test line at Beijing by using a GTEF measurement system. The detailed descriptions of this line and the GTEF measurement system can be found in References [36, 37]. In Figure 4a,b the GTEF distributions under RH = 80% in a heavily hazy day and RH = 40% in a lightly hazy day are presented, respectively. In Figure 5 the time series data of GTEF varying with RH in a hazy day are shown.

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It can be seen from Figure 4a that in a heavily hazy day, the GTEF distribution is slightly symmetric under RH = 80%, but as can be seen from Figure 4b that it becomes extremely asymmetric under RH = 40% in the other slightly hazy day and both the maximum absolute negative GTEF and the one at ‒60 m increase appreciably. Besides, the electric field at ‒60 m is greater than that at ‒50 m, which rarely occurs in clear days. In a word, comparing with that in highly humid and heavily hazy days, particles affect the GTEF in a greater extent in dry and lightly hazy days. It can be seen from Figure 5 that the maximum absolute GTEF at the negative side reduces from 30 to 15 kV/m as RH increases from 35% to 65% in a hazy day. All these experimental results indicate that airborne suspended particles have an impact on GTEF and humidity can control this influence. To the authors' knowledge, this phenomenon has never been identified and reported in the literature. In the following part, a qualitative explanation is given by using BCM.

A number of experimental results obtained under the full-scale UHVDC test line in Beijing by China Electric Power Research Institute show that the GTEF distribution is nearly symmetric when RH = 60∼70% in clear summer days and the lower the RH is, the higher the asymmetry and the absolute GTEF at the negative side will be. It can be inferred from these observations that on the whole the ion density ratio γ in symmetric case is closer to one comparing to the asymmetric case, and the higher the asymmetry is, the more γ deviates from 1. One can deduce based on this inference and the forgoing analysis on BCM that as a whole, particles surrounding HVDC lines are charged at a lower level in symmetric case comparing to the asymmetric case, and the higher the asymmetry is, the higher level of charge the particles will have. Hence, at low RH, for example RH = 40%, particles tend to have more charges than that at high RH, for example RH = 80%. In addition, particle charge can be lowered by reducing the ion density γ on the whole as RH increases from 35% to 65%. As stated before, the physical mechanism behind this charge reduction is the adjustment of positive and negative ion charging current to the particles.

HYBRID CHARGING MODELS FOR PARTICLES SUSPENDED AROUND HVDC LINES

Normally, HVDC lines are in bipolar operation, and thus airborne suspended particles are charged by bipolar ions. Both diffusion charging and field charging mechanisms are existent for these particles around HVDC lines. In this section, an empirical field-diffusion charging model combining these two mechanisms is proposed, in which the Maxwell-Wagner relaxation model is adopted for field charging, Fuchs' model is used for diffusion charging, and charges obtained by these two models are summed: 31 $q=q_{\text{D}}+q_{\text{E}}$

This model is to be verified in the following part. Predictions of charging dynamics for particle radii of 0.15  and 5.5 μm given by our proposed field-diffusion charging model is presented and compared with experimental data in Figure 6.

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In Figure 6a the data are extracted from Reference [33] and parameters for calculation are the same as those for Figure 2; in Figure 6b the data are extracted from Reference [35] and parameters for calculation are the same as those for Figure 3. Note that dimensionless electric field defined in Fjeld et al. 1990 differs between Figure 6a,b since electric field strength is held constant for different particle diameters. As can be seen from Figure 6 that the theoretical values of particle charging dynamics given by this model are in good agreement with the experimental data. Therefore, the proposed field-diffusion charging model is preliminarily validated. Its effectiveness in a wide range of parameters will be systematically investigated in a work in the near future.

CONCLUSION

The aim of this paper is to develop charging models for airborne suspended particles around HVDC lines.

Firstly, Equivalent unipolar charging models (EUCMs) and bipolar charging models (BCMs) for describing field and diffusion charging mechanisms in bipolar ion environment are introduced and the Maxwell-Wagner relaxation is proposed to be incorporated into the EUCM and BCM for field charging.

Secondly, the EUCMs and BCMs are analysed and validated by comparing them with experimental data reported in the literature. It is found that in BCMs, the ion density ratio γ is a key parameter in controlling the particle charge and as long as the charging time t _pc is incomparable to the Maxwell-Wagner relaxation time τ _MW, the particle charge can reach steady values, and the time duration for reaching the steady values is dependent on γ. The larger the γ is, the more time it costs to reach a steady value. While EUCM does not have these features. The results also show that BCMs are applicable to modelling the particle charging in bipolar ion environment, while EUCMs are not. The predictions made by BCM that takes into account the Maxwell-Wagner relaxation are in good agreement with experimental results.

Thirdly, the BCM is proposed to analyse the experimental results that the GTEF could increase quite a lot and this increase depends heavily on the RH. As RH goes up, the increased GTEF goes down. These are not understood before. It is demonstrated that the BCMs is able to qualitatively explain these experimental results with the help of the fact that the ion density ratio γ decides the particle charge in bipolar ion environment. The ion density ratio γ at medium RH is much closer to 1 than that at low RH on the whole. As RH goes up, γ reduces and so does the particle charge. Hence, GTEF declines and compared with that in highly humid and heavily hazy days, particles affect the GTEF by a greater extent in dry and slightly hazy days.

Finally yet importantly, a new charging model combining field and diffusion charging mechanism is proposed for airborne particles suspended around HVDC lines and preliminarily validated by experimental results.

Further researches can be conducted in lots of aspects, for instance the systematic investigation on the effectiveness of the new field-diffusion charging model in a wide range of parameters, the coupling of the new model with the corona ion flow field model of HVDC lines and predicting the GTEF under the lines.

ACKNOWLEDGEMENTS

This work was supported by the State Grid Corporation of China (52,010,118,001P).

The authors would like to express their gratitude to the referees for their precious advices.

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