rizer.plasma.constant_volume_reactor#

Attributes#

`logger`
`u`

Classes#

ConstantVolumePlasmaReactorOde

Parameters of the ODE system and auxiliary data are stored in the ReactorOde object.

Module Contents#

rizer.plasma.constant_volume_reactor.logger#

rizer.plasma.constant_volume_reactor.u#

class rizer.plasma.constant_volume_reactor.ConstantVolumePlasmaReactorOde(plasma: cantera.Solution, plasma_extension: rizer.plasma.reactor.PlasmaExtension, rho: float, generator_voltage: Callable[[float], float], generator_resistance: float = 1.0, cable_impedance: float = 75.0, cable_length: float = 0.3, cable_wave_speed: float = 200000000.0, gap: float = 0.003, radius: float = 0.0001, nb_reflexions: int = 1)#

Parameters of the ODE system and auxiliary data are stored in the ReactorOde object.

Parameters:

plasma (ct.Solution) – Cantera plasma object.
plasma_extension (PlasmaExtension) – Object that extends the plasma object to include properties not currently implemented in Cantera, such as cp_mole.
rho (float) – Density of the plasma in kg/m^3.
generator_voltage (Callable[[float], float]) – Function that returns the generator voltage at time t in V.
generator_resistance (float, optional) – Resistance of the generator in Ohm, by default 1.0.
cable_impedance (float, optional) – Impedance of the cable in Ohm, by default 75.0.
cable_length (float, optional) – Length of the cable in m, by default 0.3.
cable_wave_speed (float, optional) – Wave speed in the cable in m/s, by default 2.0e8.
gap (float, optional) – Gap between the electrodes in m, by default 3.0e-3.
radius (float, optional) – Radius of the plasma in m, by default 1.0e-4.
nb_reflexions (int, optional) –
Number of reflexions to take into account in the computation of the plasma voltage, by default 1.
- 0: No reflexion, i.e. the plasma voltage is equal to the generator voltage.
- 1: One reflexion, i.e. the plasma voltage is equal to the generator voltage, multiplied by
  the attenuation coefficient, and the transmission coefficient.
- 2: Two reflexions, i.e. the plasma voltage is the sum of two waves, one of generation 1 and
  one of generation 2. The first wave is the generator voltage multiplied by the attenuation coefficient, and the transmission coefficient, which after reflecting on the plasma, will reflect on the generator, and come back later to the plasma. The second wave is the generator voltage multiplied by the attenuation coefficient, and the transmission coefficient.

Notes

Equations solved implies no mass flow, and homogeneous quantities only. The equations solved are based on [Aurora].

Since there is no mass flow and the volume is constant, it implies a constant mass density.

The equation of state chosen is the multi-temperature ideal gas law, like in [Chemkin]:

\[P = \sum_k [X_k] R T_k\]

where:

\(P\) is the pressure, in Pa,
\(X_k\) is the mole fraction of species \(k\),
\(R\) is the ideal gas constant, in J/mol/K,
\(T_k\) is the temperature of species \(k\), in K.

START OF BUG/WARNING ———————————————————————— The Cantera plasma object may not allow this equation of state for now, so mistakes are possible. For instance:

enthalpy seems to be correct: https://cantera.org/stable/cxx/d5/dd7/classCantera_1_1PlasmaPhase.html#a0c618ae5a6d01ececdcc4da21cf2e1b5
molar enthalpy is not: https://cantera.org/stable/cxx/d5/dd7/classCantera_1_1PlasmaPhase.html#aba0353a9480e109384cabf2cf7754b47

END OF BUG/WARNING ————————————————————————

The conservation of species:

\[\frac{dY_k}{dt} = \frac{W_k \dot{\omega_k}}{\rho}\]

with:

\(Y_k\) the mass fraction of species \(k\),
\(W_k\) the molecular weight of species \(k\), in kg/kmol,
\(\omega_k\) the production rate of species \(k\), in kmol/m^3/s,
\(\rho\) the density, in kg/m^3.

The conservation of energy for the electrons:

\[\rho c_{v, e} Y_e \frac{dT_e}{dt} = \dot{\omega_e} M_e c_{v, e} (T_g - T_e) + \vec{j} \cdot \vec{E} - \dot{Q}_{elas} - \dot{Q}_{inel}\]

with:

\(Y_e\) the mass fraction of electrons,
\(c_{v, e}\) the specific heat at constant volume of the electrons, in J/kg/K,
\(T_e\) the electron temperature, in K,
\(\omega_k\) the production rate of electron \(k\), in kmol/m^3/s,
\(M_e\) the molecular weight of the electron, in kg/kmol,
\(\vec{j}\) the current density, in A/m^2,
\(\vec{E}\) the electric field, in V/m,
\(\dot{Q}_{elas}\) the heat production rate due to elastic collisions, in W/m^3,
\(\dot{Q}_{inel}\) the heat production rate due to inelastic collisions, in W/m^3.

Note that it is assumed that newly created electrons have the same temperature as the gas.

The conservation of energy for the gas:

\[\rho \bar{c_v} \frac{dT_g}{dt} = - \sum_{k \neq e} W_k u_k \dot{\omega_k} - W_e c_{v, e} T_g \dot{\omega_e} + \dot{Q}_{elas} + \dot{Q}_{inel} - \dot{Q}_{loss}\]

with:

\(\bar{c_v}=\sum_{k \neq e} c_{v, k} Y_k\) the mean specific heat at constant volume of heavies species in the gas, in J/kg/K,
\(u_k\) the internal energy of species \(k\), in J/kg,
\(\dot{Q}_{loss}\) the heat loss rate, in W/m^3.