Charge Accumulation
The Charge Accumulation feature is used to compute the charge accumulated at a surface of a particle in a background of ions as a function of time. Charging models can be classified according to charging polarity (unipolar or bipolar), aerosol regime (continuum, free molecular, or transition regime), and charging mechanism (diffusion charging and/or field charging). The charging models used in this feature are unipolar, meaning that the ions in the background are either positively or negatively charged. The different models categories are summarized in Table 5-5 as a function of the particle diameter for STP conditions. The Knudsen number is defined as Kn = λ/rp, where λ (SI unit: m) is the ion mean free path and rp (SI unit: m) is the particle radius. For reference, the mean free path at atmospheric pressure of small ions such as oxygen and nitrogen is about 100 nm, and of larger ions such as water and methane agglomerates is around 10 nm.
0.5 μm
Ionic charging of particles is usually described in terms of diffusion charging or field charging. Diffusion charging occurs when the particle acquires a charge by virtue of the random thermal motion of ions and their collision and adherence to the particles. Field charging occurs when the electric flux lines deflect toward the particle resulting in the capture of ions. As the particle becomes charged, ions begin to be repelled by the particle, reducing the rate of charging. Eventually, the particle will reach a saturation charge and charging will cease (Ref. 43).
Diffusion charging is the dominant mechanism for small particles or low fields, while field charging is dominant for large particles and high fields. At intermediate sizes, diffusion and field charging can coexist; a simple way to estimate the total charging rate is to simply add the contributions from the two charging mechanisms. Lawless (Ref. 44) developed a more elaborate model to combine diffusion and field charging that reduces the diffusional component as the field becomes stronger, giving a distinctive diffusion charging rate above the saturation charge.
The charging models available in this feature are for the continuum and free molecular regime. Currently, this feature does not include models for the transition regime. The models are described in the below. To define the models the following dimensionless quantities are used:
.
The saturation charge of a dielectric particle is defined as
and the saturation charge of a conductive particle is defined as
.
Classical Diffusion charging model
Classical Field charging model
Classical diffusion and Field charging model
Where Rd and Rf use the same definitions as the Classical diffusion and Classical field charging models, respectively.
lawless charging model
White charging model
White’s charging model is one of the simplest models for the free molecular regime. Using kinetic arguments and assuming a Boltzmann distribution to describe the ion density around a particle it is possible to deduce the rate of charge accumulation to be
where the ion thermal velocity is given by
.
The following definitions are used in this section:
Z (dimensionless) is the total charge of the particle,
εr,p (dimensionless) is the particle permittivity,
rp (SI unit: m) is the particle radius,
Ti (SI unit: K) is the ion temperature,
mi (SI unit: K) is the ion mass,
E (SI unit: V/m) is the electric field,
e (SI unit: C) is the elementary charge, and
ρq (SI unit: C) is the ion charge density.