Electron Thermal Conductivity

HallThruster has a few electron thermal conductivity models built in and allows users to define their own. This page describes these models and the process by which algebraic transport models can be added by the user.

Built-in Models

Braginskii

This form of the thermal conductivity follows the result of S. I. Braginskii, in Reviews of Plasma Physics, edited by M. A. Leontovich (Consultants Bureau, New York, 1965), Vol. 1, p. 205.:

\[\begin{aligned} \kappa_{e\perp} & \approx C \frac{neTe (eV) \nu}{m_e \omega_{ce}^2} \\ \nu = \nu_{ei} + \nu_{en} + \alpha \nu_{AN} \\ \end{aligned} `C` is a charge-dependent coefficient given by is ``C = 13/4 + \sqrt{2}/Z_{\mathrm{eff}}``, where ``Z_{\mathrm{eff}} = \sum_s n_s Z_s^2 / n_e``. This analytic form is given by [Hunana et al. (2022), equation (60)](https://arxiv.org/abs/2201.11561), equations (88)–(94) The coefficient $\alpha$ ranges from zero to one and scales the amount that anomalous collisions contribute to heat flux. This defaults to 1.0. A `Braginskii` model can be initialized in the user-defined configuration as: \]

julia conductivitymodel = Braginskii(anomscale = 1.0)

### `Mitchner()`

This model employs the form described in Mitchner & Kruger "Partially Ionized Gases" (1973), given below

math \begin{aligned} \kappa{e\perp} & \approx \frac{1}{1+\OmegaH^2} \frac{2.4}{1+ \frac{\nu{ie}}{\sqrt{2}\nu}} \frac{neTe (eV)}{\nu me} \ \nu = \nu{ei} + \nu{en} + \alpha \nu_{AN} \ \end{aligned}

The model can be initialized in the configuration as

julia conductivitymodel = Mitchner(anomscale = 1.0)

In the user-defined configuration


## Custom thermal conductivity models

The procedure for adding additional thermal conductivity models generally follows that of adding anomalous transport models, as described on the [Anomalous transport](@ref) page.
The two main differences is that the model type is of 'ThermalConductivityModel' rather than 'AnomalousTransportModel' and that the function needs to return ``\kappa`` rather than ``\nu an``.
For demonstration pursposes however, let's say we wanted to implement the Braginskii model but without the anomalous collision frequency and a coefficient of 4.66 (singly charged ions only).
We first define the type:

julia using HallThruster

struct Braginskii_Classical <: ThermalConductivityModel end

With the type defined, we then need to define the function that describes the model. This function should take two arguements, the vector of thermal conductivity values and the solver params.

julia function (model::Braginskii_Classical)(κ, params) (;B, ne, Tev, νc) = params.cache me = HallThruster.me e = HallThruster.e

for i in eachindex(κ)
    ωce = e * B[i] / me
    κ[i] = 4.66 * ne * e * Te * νc[i] / (me * ωce^2)
end

return κ

end ```

We could now employ this model by setting `conductivitymodel = BraginskiiClassical()' in the user-defined configuration.