def flow_balance(ne, TeV, ng, args):
'''
Function: flow_balance
Difference of "charge flow rate" between inlet and outlets of the orifice.
Should be zero (what comes in = what comes out)
Inputs:
- ne: plasma density (1/m3)
- TeV: electron temperature (eV)
- ng: neutral density (1/m3)
- args: list of arguments necessary for the subfunctions called
Outputs:
- "Charge" flow rate balance (C/s = A)
'''
TgV, r, M, mdot = args
mdot_A = mdot * cc.sccm2eqA # Flow rate (eq-A)
Ao = np.pi * r**2 # Orifice area (m2)
# Neutral particle flux (A)
vg = np.sqrt(cc.me / M) * cp.mean_velocity(TgV, 'e')
gam_g = 1 / 4 * cc.e * ng * vg
gam_g *= Ao
# Ion flux (A)
gam_i = J_i(ne, TeV, M)
gam_i *= Ao
# Balance
ret = gam_g + gam_i
ret -= mdot_A
return ret
def ambipolar_diffusion_model(rc, ng, TiV, species='Xe'):
"""
Implementation of Goebel's ambipolar diffusion model used for determination
of the orifice- or insert-region plasma electron temperature based on the
cathode geometry, neutral density and ion temperature (usually taken to be
some constant value 2-4x the insert temperature).
Inputs:
- rc: cathode radius (m)
- ng: neutral density (#/m^3)
- TiV: ion temperature (eV)
Optional Input:
- species, defaults to 'Xe'
-NOTE: currently only works for xenon
Output:
Electron temperature (eV)
"""
#NOTE: THIS SECTION WILL CURRENTLY ONLY WORK FOR XENON!!
lhs = lambda TeV: (rc / cc.BesselJ01)**2 * (ng * xsec.ionization_xe_mk(TeV)
* cp.mean_velocity(TeV, 'e'))
rhs = lambda TeV: ((cc.e / cc.M.species(species)) * (TiV + TeV) /
(ng * xsec.charge_exchange(TiV, species) * cp.
thermal_velocity(TiV, species)))
goal = lambda x: lhs(x) - rhs(x)
TeV = fsolve(goal, x0=5.0)
return TeV
def orifice_plasma_density_model(Id, TeV, TeV_insert, L, d, ne, ng, eps_i):
Rp = plasma_resistance(L, d, ne, ng, TeV)
Vori = cc.pi * d**2 * L / 4
ve = cp.mean_velocity(TeV, 'e')
num = Rp * Id**2 - 5 / 2 * Id * (TeV - TeV_insert)
den = cc.e * ng * xsec.ionization_xe_mk(TeV) * ve * eps_i * Vori
ne_bar = num / den
return ne_bar
def nu_en_xe_mk(ng, TeV, xsec_type='variable'):
"""
Mandell and Katz' expression for the electron-neutral collision frequency
of xenon. Can either use the simple collision cross section (constant) or
the variable one.
Inputs:
- ng: neutral density (1/m3)
- TeV: electron temperature (eV)
- xsec_type: The type of cross section model to use. Can either be
"constant" or "variable". If "variable" uses the model from year 1999 and
above. Otherwise uses a set value of 5 10^{-19} m2.
Ouputs:
- Electron-neutral collision frequency (s)
References:
- Mandell, M. J. and Katz, I., "Theory of Hollow Cathode Operation in Spot
and Plume Modes," 30th AIAA/ASME/SAE/ASEE Joint Propulsion Conference &
Exhibit, 1994.
- Katz, I., et al, "Sensitivity of Hollow Cathode Performance to Design and
Operating Parameters," 35th AIAA/ASME/SAE/ASEE Joint Propulsion Conference
& Exhibit, 1999. http://arc.aiaa.org/doi/pdf/10.2514/6.1999-2576
- Goebel, D. M. and Katz, I., "Fundamentals of Electric Propulsion,"
Appendix D p.475, John Wiley and Sons, 2008.
"""
# Depending on the type of cross-section the velocity is different
# Mandell and Katz with a constant cross section use the thermal velocity
# of electrons. Later models use the Maxwellian-averaged cross section and
# therefore the mean Maxwellian velocity.
if xsec_type == 'constant':
# Thermal velocity of electrons
ve = cp.thermal_velocity(TeV)
elif xsec_type == 'variable':
ve = cp.mean_velocity(TeV, species='e')
else:
raise ValueError
# Cross section
en_xsec = xsec.electron_neutral_xe_mk(TeV, xsec_type)
# Collision freq.
nu_en = ng * en_xsec * ve
return nu_en
def ion_production(ne, TeV, ng, L, r, sigma_iz):
'''
Function: ion_production
Calculate the total amount of ions produced in the volume by direct-impact
ionization.
Inputs:
- ne: plasma density (1/m3)
- TeV: electron temperature (eV)
- ng: neutral density (1/m3)
- L,r: length and radius of plasma column (m)
- sigma_iz: ionization cross-section function (m2)
Outputs:
- Ion production (1/s)
'''
vol = np.pi * r**2 * L # Volume
sig_iz = sigma_iz(TeV) # Cross-section term
ve = cp.mean_velocity(TeV, 'e')
return vol * sig_iz * cc.e * ne * ve * ng
def excitation_loss(ne, TeV, ng, L, r, eps_x, sigma_ex):
'''
Function: excitation_loss
Calculates the total amount of power spent in excitation.
Inputs:
- ne: plasma density (1/m3)
- TeV: electron temperature (eV)
- ng: neutral density (1/m3)
- L,r: length and radius of plasma column (m)
- eps_x: excitation energy (eV)
- sigma_iz: ionization cross-section function (m2)
Outputs:
- Excitation power loss (W)
'''
vol = np.pi * r**2 * L # Volume
sig_ex = sigma_ex(TeV) # Cross-section term
ve = cp.mean_velocity(TeV, 'e')
el = eps_x * vol * sig_ex * cc.e * ne * ve * ng
return el
def average_plasma_density_model(Id,
TeV,
phi_wf,
L,
d,
ne,
ng,
phi_p,
eps_i,
h_loss=heat_loss()):
phi_s = sheath_voltage(Id, TeV, phi_wf, L, d, ne, ng, h_loss)
Rp = plasma_resistance(L, d, ne, ng, TeV)
f_n = np.exp(-(phi_p - phi_s) /
TeV) #edge-to-average ratio as defined by Goebel
Aemit = L * cc.pi * d
Vemit = L * cc.pi * d**2 / 4
### Calculate average plasma density
# Numerator
num = (Rp * Id**2 - 5 / 2 * TeV * Id + phi_s * Id)
# Denominator
ve = cp.mean_velocity(TeV, species='e')
t1 = 1 / 4 * cc.e * f_n * TeV * ve
t1 *= Aemit * np.exp(-phi_s / TeV)
t2 = cc.e * ng * Vemit * (eps_i + phi_s) * xsec.ionization_xe_mk(TeV) * ve
den = t1 + t2
# Results
ne_bar = num / den
return ne_bar, phi_s
def random_electron_current_density(ne, TeV, phi_s):
return cc.e * ne * cp.mean_velocity(TeV, 'e') * np.exp(-phi_s / TeV) / 4
def J_i(ne, TeV, M):
ve = cp.mean_velocity(TeV, 'e')
return 1 / 4 * cc.e * np.sqrt(cc.me / M) * ve * ne