Fusion engine module functions. View the published FusionEngine Demos. More...

Directories
directory	Cryo
	Cryocooler sizing.

directory	Economics
	Cost modeling.

directory	FRC
	FRC relations.

directory	Magnets
	Magnet fields and masses.

directory	PlasmaPhysics
	Plasma physics functions.

directory	Reactor
	Fusion reactor modeling.

directory	RFSystem
	Inductor and amplifier modeling.

directory	Shielding
	Shielding attenuation.

directory	Tokamak
	Tokamak beta and point modeling.

directory	Transport
	Transport loss relations.

Functions
function	Cryocooler (in qC, in tC)
	Cryocooler mass, efficiency, and performance coefficients. More...

function	CryocoolerBrake (in qC, in tC)
	Mass, efficiency, and performance coefficients of low-power cryocoolers. More...

function	HeatCapacityAluminum (in T)
	Heat capacity and enthalpy of aluminum at a given temperature. More...

function	ThermalConductivityG10 (in T)
	Thermal conductivity of G-10 Cryo at a given temperature. More...

function	ThermalConductivityInconel (in T)
	Give the thermal conductivity of Inconel at a given temperature. More...

function	CapacityFactor (in type)
	Capacity factor for a variety of power sources. More...

function	CRF (in dR, in n)
	Computes the capital recovery factor. More...

function	FuelCost (in type)
	Fuel cost for various energy sources. More...

function	LCOE (in oCC, in fOM, in vOM, in dR, in n, in cF, in hR, in fC, in t, in dPV)
	Computes the levelized cost of electricity (LCOE) More...

function	LCOF (in oCC, in fOM, in vOM, in dR, in n, in cF, in tC, in fP, in pS)
	Computes the levelized cost of fuel production. More...

function	MachineCost (in p, in pF, in cF, in fun)
	Infer the cost of a machine size given a few sizes and costs. More...

function	OvernightCapitalCost (in type)
	Returns Overnight Capital Cost of power plants and other parameters. More...

function	BetaBarnes (in rS, in rFC)
	Compute average beta for an FRC using Barnes relation. More...

function	BVacuum (in rMagnet, in rSeparatrix, in B0)
	Vacuum field calculated from magnet and separatrix radii. More...

function	FRCFieldLines (in r, in z, in B0, in rs, in kappa)
	FRC equilibrium flux and field strength. More...

function	FRCInertia (in rX, in rY, in rZ, in n, in mI)
	Inertia and mass of an FRC plasma. More...

function	MagneticFieldCurrentLoop (in i, in a, in p, in q, in x, in doPlot)
	Computes fields for multiple current loops. More...

function	MagnetMass (in rhoM, in sigmaM, in bM, in rM, in beta)
	Compute the magnetic structural mass using a shell method. More...

function	MagnetMassVirial (in rhoM, in sigmaM, in bM, in v)
	Compute the magnetic structural mass using a Virial method. More...

function	MagnetStoredEnergy (in b, in v)
	Energy stored in a magnet from field and volume. More...

function	NozzleRadius (in bAx, in bN, in rInner, in tSOL)
	Compute the nozzle coil inner radius from flux conservation. More...

function	AlfvenSpeed (in b, in rho)
	Computes the Alfven speed. More...

function	AtomicNumber (in r)
	Returns the atomic number for common fusion fuels. More...

function	BremsstrahlungFrequency (in nu, in tE, in nI, in zI)
	Computes the bremsstrahlung radiation as a function of wavelength. More...

function	BremsstrahlungWavelength (in lambda, in tE, in nI, in zI)
	Computes the bremsstrahlung radiation as a function of wavelength. More...

function	BurnupTime (in ni, in Ti, in r1, in r2)
	Reactant burnup time. More...

function	CollisionTimes (in Te, in ni, in Ti, in Z, in mu)
	Ion collisions in two-component plasma. More...

function	CoulombLog (in ne, in Te, in Z)
	Coulomb log for electron-ion collisions. More...

function	CoulombLogIons (in n, in T, in Z, in mu)
	Coulomb log for ion-ion collisions. More...

function	CrosssectionClassical (in e, in rB, in vB)
	Classical crosssection. More...

function	CrossSectionFromRest (in r1, in r2, in E)
	Cross section assuming the second particle is at rest. More...

function	CyclotronFrequency (in b)
	Computes the cyclotron frequency. More...

function	DebyeLength (in tE, in n0)
	The Debye Length. More...

function	DHe3Reaction (in t)
	Computes the fraction of D going to each reaction. More...

function	EffectiveCharge (in nI, in zI)
	Compute the effective charge in a plasma. More...

function	FusionReaction (in r1, in r2)
	Finds products of fusion reactions. More...

function	FusionReactionDD (in r)
	Finds products of D+D fusion reactions. More...

function	GasPressure (in n, in kT, in V)
	Computes Gas pressure. More...

function	GyroRadius (in t, in b, in mu, in z)
	Computes gyroradius. More...

function	IonCyclotronFrequency (in b, in z, in m)
	Computes the ion cyclotron frequency. More...

function	IonizationEnergy (in r)
	Returns the ionization energy per nuclei. More...

function	IonizationEnergyPlasma (in r1, in r2, in n1, in n2)
	Returns the ionization power per nuclei. More...

function	MaxwellianV (in r1, in r2, in tK)
	Computes Maxwellian thermonuclear reaction rates. More...

function	NumberOfElectrons (in nI, in zI)
	Computes the number of electrons. More...

function	PhotonEnergy (in frequency)
	Computes the photon energy. More...

function	PlasmaConstant (in q)
	Constants useful in plasma physics. More...

function	PlasmaFrequency (in n, in m)
	Returns the plasma frequency. More...

function	PlasmaHeating (in n, in kT)
	Computes plasma heating per unit volume. More...

function	PlasmaPressure (in nI, in zI, in tI, in tE)
	Computes fusion plasma pressure. More...

function	PlasmaRadius (in pNet, in pD, in aR)
	Computes the Plasma Radius from the desired power and net power density. More...

function	PlasmaVolume (in pNet, in pD)
	Computes the plasma vadius from the desired power and net power density. More...

function	PowerIonsToElectrons (in nE, in zI, in nI, in aI, in tI, in tE)
	Power lost by the ions to the electrons, Dawson model (relativistic). More...

function	ReactionString (in r1, in r2)
	Create a string with a reaction. More...

function	SigmaV (in r1, in r2, in tK)
	Computes thermonuclear reaction reaction rates. More...

function	SynchrotronFrequency (in b, in v)
	Computes the synchrotron frequency. More...

function	Beta (in b, in p)
	Compute the ratio of magnetic field and pressure. More...

function	BField (in p, in beta)
	Computes the magnetic field from plasma pressure and beta. More...

function	Bremsstrahlung (in nI, in zI, in tE, in method)
	Computes 0-D Bremsstrahlung radiation, NRL or McNally models. More...

function	CentralPressure (in t, in nI, in zI)
	Computes the central plasma pressure from temperature. More...

function	CentralTemperature (in p, in nI, in zI)
	Computes the central temperature from plasma pressure. More...

function	DensityFromFusionPower (in p, in r1, in r2, in tK)
	Calculate the ion density for volumetric fusion power. More...

function	DivertorCLR (in nUp, in qPar, in L, in unitIndex)
	Computes temperature drop over a conduction-limited divertor. More...

function	DMassFromPower (in energy, in time)
	Deuterium Reactant mass flow. More...

function	FirstWallThermalPower (in pLoss, in rWall)
	Computes the first wall thermal power. More...

function	FusionPower (in n1, in n2, in r1, in r2, in tK)
	Fusion power per reactants. More...

function	FusionPowerDHe3 (in ni, in Ti)
	Compute volumetric power for a D-He3 reactor ignoring tritium. More...

function	He3MassFromPower (in energy, in time)
	Helium-3 (He3) and Deuterium Reactant mass flow. More...

function	MagneticPressure (in beta, in b)
	Computes the magnetic pressure from beta and field. More...

function	MagnetRadius (in mM, in rhoM, in sigmaM, in bM)
	Computes magnet radius from mass, density, stess, and magnetic field. More...

function	NetFusionPower (in nI, in zI, in tI, in tE, in rW, in L, in b, in r1, in r2)
	Computes net fusion power after Synchrotron and Bremsstrahlung. More...

function	NumberDensityFromBeta (in beta, in b, in kT)
	Compute number density from beta, b and kT. More...

function	PlasmaInternalEnergy (in n, in t, in v)
	Computes the plasma internal energy. More...

function	PointFusionReaction (in r1, in r2, in nI, in v, in beta, in tI, in pType)
	Computes the net power from a fusion reaction. More...

function	PowerDD (in d)
	Compute the power for a He3-catalyzed D-D reactor. More...

function	PowerDHe3 (in d)
	Compute the power for a D-He3 reactor including Bremsstrahlung. More...

function	PowerDT (in d)
	Compute the power for a D-T fusion reactor. More...

function	ReactorField (in d, in beta)
	Calculate magnetic field from plasma pressure and beta. More...

function	Synchrotron (in nI, in zI, in b, in tE, in L, in rW)
	Computes synchrotron radiation. More...

function	VolumeAveragedPressure (in t, in nI, in zI)
	Computes the volume averaged pressure. More...

function	ClassERFAmplifier (in f, in p, in vCE, in vCC, in qL, in l1, in r)
	Class E Amplifier design tool. More...

function	Inductor (in d, in n)
	Model an inductor. More...

function	InductorDesign (in d)
	Design an inductor (fmincon) More...

function	MacroCrossSection (in rho, in formula, in M, in sigma)
	Compute a macroscopic cross-section for a material. More...

function	NumberDensity (in rho, in formula, in M)
	Number density of a molecule from its constituents. More...

function	ShieldingThickness (in limit, in years, in flux, in Sigma)
	Shielding thickness from neutron attenuation model. More...

function	XRayMassAttenCoeff (in el, in e, in file)
	Computes the X-ray mass attenuation coefficient. More...

function	EdgeLocalizedModes (in tau1, in tau2, in k, in t)
	Disturbance in a Tokamak plasma due to edge localized modes (ELMs). More...

function	FusionReactorDesign (in d, in mode, in pmin, in pmax, in n)
	FUSION REACTOR DESIGN FUNCTION (tokamaks) More...

function	SteadyStateTokamak (in d)
	Generates a statespace model of a tokamak. More...

function	TokamakBeta (in aR)
	Computes the tokamak beta from aspect ratio. More...

function	TPVHRHS (in x, in t, in d)
	Simulates a model of the vertical position of a plasma in a Tokamak. More...

function	ClassicalConfinementTime (in ni, in Ti, in Z, in mu, in B, in r_s)
	Classical ion confinement time. More...

function	CollisionsTrubnikov (in E1, in T2, in n2, in m1, in m2, in Z1, in Z2)
	Collision rates calculated using the Trubnikov potentials. More...

function	IonElectronEquilibration (in Te, in ni, in Ti, in Z, in mu)
	Ion-electron collision rate with different temperatures. More...

function	IonIonCollisionRate (in n, in T, in Z, in mu)
	Ion-ion collision rate. More...

function	LawsonProduct (in Te, in ni, in Ti, in Z, in mu, in B, in rS)
	Compute triple product using classical confinement. More...

function	PowerIE (in zI, in nI, in aI, in Ti, in Te)
	Power from ions to electrons, NRL model. More...

function	TransportPower (in n, in T, in tau, in v)
	Compute transport power loss given confinement time. More...

Detailed Description

Fusion engine module functions. View the published FusionEngine Demos.

Function Documentation

◆ AlfvenSpeed()

function AlfvenSpeed	(	in	b,
		in	rho
	)

Computes the Alfven speed.

The Alfv�n speed is the speed of the waves resulting from the mass of the ions and the restoring force of the magnetic field: The Alfv�n speed is the speed at which hydromagnetic waves travel along a magnetic field line and is the characteristic speed at which perturbations of magnetic field lines travel.

Type AlfvenSpeed for a demo

Form

v = AlfvenSpeed( b, rho )

Inputs

b              (1,:) Magnetic field (T)
rho            (1,1) Density (kg/m^3)

Outputs

v              (1,:) Velocity (m/s)

References and().

◆ AtomicNumber()

function AtomicNumber ( in r )

Returns the atomic number for common fusion fuels.

Type AtomicNumber for a demo.

Form

a = AtomicNumber( r )

Inputs

r         (:}    Ion name (or array of names)

Outputs

a         (1,:)  Atomic numbers

◆ Beta()

function Beta	(	in	b,
		in	p
	)

Compute the ratio of magnetic field and pressure.

Type Beta for a demo.

Form

beta = Beta( b, p )

Inputs

b           (:)   Magnetic field (T)
pPlas       (:)   Pressure of plasma (N/m^2)

Outputs

beta        (:)   Beta

References: Glasstone, S., and R. H. Lovberg, Controlled Thermonuclear

Reactions, Krieger, 1975, pp. 53

References and().

◆ BetaBarnes()

function BetaBarnes	(	in	rS,
		in	rFC
	)

Compute average beta for an FRC using Barnes relation.

The Barnes relation does not include plasma pressure or radial field beyond the X point, hence gives a lower limit on beta.

Form

beta = BetaBarnes( rS, rFC )

Inputs

rS    (1,:)    Separatrix  radius
rFC   (1,:)    Wall (flux conserving) radius

Outputs

beta  (:,:)    Average beta

Reference:

W. T. Armstrong et al, "Field-reversed experiments (FRX) on compact toroids," The Physics of Fluids 24, 2068 (1981)

◆ BField()

function BField	(	in	p,
		in	beta
	)

Computes the magnetic field from plasma pressure and beta.

Type BField for a demo.

Form

bField = BField( p, beta )

Inputs

p            (1,:)     Plasma Pressure (N/m^2)
beta         (1,1)     Beta constant

Outputs

b             (1,:)    Magnetic Field (T)

References: Schaffer, M.J., "Considerations for Steady-State FRC-Based Fusion Space Propulsion", GA-A23579, December 2000. General Atomics, San Diego, CA

References and().

◆ Bremsstrahlung()

function Bremsstrahlung	(	in	nI,
		in	zI,
		in	tE,
		in	method
	)

Computes 0-D Bremsstrahlung radiation, NRL or McNally models.

Includes e-e bremsstrahlung and relativistic effects. This is radiation lost by the electrons. This model is for hydrogen-like plasmas.

Type Bremsstrahlung for a demo.

Form

p = Bremsstrahlung( nI, zI, tE, method )

- ----------------------------------------------------------------------—

Inputs

nI        (1,m)    Ion densities (#/m^3)
zI        (1,m)    Ion atomic numbers
tE        (1,n)    Electron temperature (keV)
method    (1,1)    1 = NRL, 2 = McNally (relativistic terms)

Outputs

p         (1,n)  Bremsstrahlung (MW/m^3)

References: 2016 NRL Plasma Formulary, Office of Naval Research;

McNally, Physics of Fusion Fuel Cycles, Nuclear Technology/Fusion, Vol. 2, Jan 1982

References and(), isempty(), and Update().

◆ BremsstrahlungFrequency()

function BremsstrahlungFrequency	(	in	nu,
		in	tE,
		in	nI,
		in	zI
	)

Computes the bremsstrahlung radiation as a function of wavelength.

Calls BremsstrahlungWavelength. Type BremsstrahlungWavelength for a demo.

Form

e = BremsstrahlungFrequency( nu, tE, nI, zI )

Inputs

nu        (1,:)  Frequency (Hz)
tE        (1,1)  Electron temperature (eV)
nI        (1,2)  Number density of ions (cm^-3)
zI        (1,2)  Ion charges

Outputs

e         (1,:)  Bremsstrahlung   (W/cm^3 nm)

◆ BremsstrahlungWavelength()

function BremsstrahlungWavelength	(	in	lambda,
		in	tE,
		in	nI,
		in	zI
	)

Computes the bremsstrahlung radiation as a function of wavelength.

Type BremsstrahlungWavelength for a demo.

Form

e = BremsstrahlungWavelength( lambda, tE, nI, zI )

Inputs

lambda    (1,:)  Wavelength (nm)
tE        (1,1)  Electron temperature (eV)
nI        (1,2)  Number density of ions (cm^-3)
zI        (1,2)  Ion charges

Outputs

e         (1,:)  Bremsstrahlung   (W/cm^3 nm)

Reference: Xianli Huang, Shigeru Morita, Tetsutarou Oishi, Motoshi

Goto and Hongming Zhang,"Evaluation of Zeff Profile in Low-Density and High-Ti Discharges with Carbon Pellet Injection based on Space-Resolved EUV Spectroscopy in LHD," Plasma and Fusion Research: Regular Articles, Volume 10, 3402036 (2015).

◆ BurnupTime()

function BurnupTime	(	in	ni,
		in	Ti,
		in	r1,
		in	r2
	)

Reactant burnup time.

This is a simple function of the fusion cross-section and ion density. Note that if the two species have different densities, the burnup rate for reactant 1 should use the reactant 2 density, and vice versa.

Form

tau = BurnupTime( ni, Ti, r1, r2 )

Inputs

ni  (1,n)  Particle density (/m3)
Ti  (1,m)  Ion temperature (keV)
r1   ''    Reactant 1, default 'd'
r2   ''    Reactant 2, default 'he3'

Outputs

tau (m,n)  Burnup time, s

◆ BVacuum()

function BVacuum	(	in	rMagnet,
		in	rSeparatrix,
		in	B0
	)

Vacuum field calculated from magnet and separatrix radii.

Form

[f,B] = BVacuum(rMagnet,rSeparatrix,B0)

Inputs

Outputs

References and().

◆ CapacityFactor()

function CapacityFactor ( in type )

Capacity factor for a variety of power sources.

Returns capacity factor. Coal and natural gas use the same factor as nuclear.

Types are not case sensitive. Numbers are from 2018.

Type CapacityFactor for a demo that will list all the types.

Form

[cF,t] = CapacityFactor(type)

Inputs

type    (1,:) Type

Outputs

cF      (1,1) Capacity factor
t       {:}   Type array if 'all' entered

Reference: EIA, "Electric Power Monthly," 2018. https://www.eia.gov/electricity/monthly/epm_table_grapher.php?t=epmt_6_07_b

References and().

◆ CentralPressure()

function CentralPressure	(	in	t,
		in	nI,
		in	zI
	)

Computes the central plasma pressure from temperature.

Type CentralPressure for a demo.

Form

pC = CentralPressure( t, nI, zI )

- ----------------------------------------------------------------------—

Inputs

t          (1,:)     Temperature (keV)
nI         (1,n)     Ion density per species (#/m^3)
zI         (1,n)     Atomic number per species

Outputs

pC         (1,:)     Central Plasma Pressure (Pa)

◆ CentralTemperature()

function CentralTemperature	(	in	p,
		in	nI,
		in	zI
	)

Computes the central temperature from plasma pressure.

Form

tC = CentralTemperature( p, nI, zI )

Inputs

p          (1,:)     Plasma Pressure (N/m^2)
nI         (1,:)     Ion density (#/m^3)
zI         (1,:)     Atomic number

Outputs

tC         (1,:)     Temperature (keV)

◆ ClassERFAmplifier()

function ClassERFAmplifier	(	in	f,
		in	p,
		in	vCE,
		in	vCC,
		in	qL,
		in	l1,
		in	r
	)

Class E Amplifier design tool.

Produces the inductor and capacitor values for the circuit. qL is the quality. infinite quality means that the circuit is tuned exactly to the frequency f.

Type ClassERFAmplifier for a demo.

Form

d = ClassERFAmplifier( f, p, vCE, vCC, qL, l1, r )

Inputs

f   (1,1) Frequency (kHz)
p   (1,1) Power (W)
vCE (1,1) Collector to emitter voltage (V)
vCC (1,1) Collector to ground voltage  (V)
qL  (1,1) Quality
l1  (1,1) Choke inductor (H)
r   (1,1) Use R if entered

Outputs

d   (.)   Data structure output
.c1 Capacitor 1 (F)
.c2 Capacitor 2 (F)
.l1 Inductor 1 (H)
.l2 Inductor 2 (H)
.r  Load (Ohms)
s   {:,2} Latex array for latex documents

Reference: Nathan O. Sokal,"Class-E High-Efficiency RF/Microwave Power Amplifiers: Principles of Operation, Design Procedures, and Experimental Verification."

References and().

◆ ClassicalConfinementTime()

function ClassicalConfinementTime	(	in	ni,
		in	Ti,
		in	Z,
		in	mu,
		in	B,
		in	r_s
	)

Classical ion confinement time.

The collision rates must be summed over each species collision, for example: D-D, D-He3, He3-D, He3-He3. The confinement time is ti*S^2, where ti is the ion-ion collision time and S is the ratio of the plasma length to gyroradius. For an FRC, S = 0.3*rS/rho. Classical confinement considers only the collision rate, so tau = 1/nu, and not other forms of transport. tau scales inversely with Z^2.

Form

[taui,ti,rho] = ClassicalConfinementTime( ni, Ti, Z, mu, B, r_s)

Inputs

ni   (1,2) Ion density   /m3
Ti   (1,2) Ion temperature(s) keV
Z    (1,2) Ion charges
mu   (1,2) Masses of ions in AMU
B    (1,1) Field strength (T)
r_s  (1,1) Separatrix radius (m)

Outputs

taui (1,2) Ion confinement time (s)
tii  (1,2) Ion collision time (s)
rho  (1,2) Gyroradii (m)

Reference: NRL Plasma Formulary, 2016

See also: IonIonCollisionRate, GyroRadius

References and().

◆ CollisionsTrubnikov()

function CollisionsTrubnikov	(	in	E1,
		in	T2,
		in	n2,
		in	m1,
		in	m2,
		in	Z1,
		in	Z2
	)

Collision rates calculated using the Trubnikov potentials.

Collision between a beam and a Maxwellian. Has a demo.

Form

[nu0,nuS, nuPerp, nuPar, nuE] = CollisionsTrubnikov( E1, T2, n2, m1, m2, Z1, Z2)

Inputs

E1    (1,1) Energy of test particle (eV)
T2    (1,1) Temperature of field particle (eV)
n2    (1,1) Number density of field particle (/cc)
m1    (1,1) Mass of test particle (electron masses)
m2    (1,1) Mass of field particle (electron masses)
Z1    (1,1) Charge state of test particle (elementary charges)
Z2    (1,1) Charge state of field particle (elementary charges)

Outputs

nu0   (1,1) Base Lorentz collision rate (/s)
nuS   (1,1) Slowing down (self-velocity) collision rate (/s)
nuPerp(1,1) Transverse diffusion collision rate (/s)
nuPar (1,1) Parallel diffusion collision rate (/s)
nuE   (1,1) Energy loss collision rate (/s)

References: NRL Plasma Formulary 2016 p. 31

B. A. Trubnikov, "Particle Interactions in a Fully Ionized Plasma," Reviews of Plasma Physics, Vol 1 (Consultants Bureau, New York, 1965), p 105

References and().

◆ CollisionTimes()

function CollisionTimes	(	in	Te,
		in	ni,
		in	Ti,
		in	Z,
		in	mu
	)

Ion collisions in two-component plasma.

This is for a simple two-component plasma (electrons and singly charged ions).

Cohen adds a factor of Z^2 to the denominator for multiply charged target ions.

Form

[tauE, tauI] = CollisionTimes( Te, ni, Ti, Z, mu )

Inputs

Te   (1,1) Electron temperature(s) keV
ni   (1,1) Ion density   /m3
Ti   (1,1) Ion temperature(s) keV
Z    (1,:) Ion charges
mu   (1,:) Masses of ions in AMU

Outputs

tauE  (1,1)   Electron-electron collision time, sec
tauI  (1,1)   Ion-ion collision time, sec

Reference: NRL Plasma Formulary 2016 p. 28, 33, 37

◆ CoulombLog()

function CoulombLog	(	in	ne,
		in	Te,
		in	Z
	)

Coulomb log for electron-ion collisions.

For D and He, this is good for Te>40 keV:

Ti*me/mi < 10Z^2 eV < Te

Type CoulombLog for a demo.

Form

lnL = CoulombLog( ne, Te, Z )

Inputs

ne (1,:) Electron density (/m3)
Te (1,:) Electron temperature (keV)
Z  (1,1) Ion charge

Outputs

lnL (1,:)  Coulomb log, ln(Lambda) == lambda

Reference: NRL plasma formulary, 2016, p. 34

References and().

◆ CoulombLogIons()

function CoulombLogIons	(	in	n,
		in	T,
		in	Z,
		in	mu
	)

Coulomb log for ion-ion collisions.

This is the multi-species formulation for mixed collisions. Type CoulombLogIons for a demo.

Form

lnL = CoulombLogIons(n, T, Z, mu)

Inputs

n  (1,1) or (1,2)  Ion density (/m3)
T  (1,1) or (1,2)  Ion temperature (keV)
Z  (1,2)           Ion charges
mu (1,2)           Masses of ions in AMU

Outputs

lnL (1,:)  Coulomb log, ln(Lambda) == lambda

Reference: NRL plasma formulary, 2016, p. 34

◆ CRF()

function CRF	(	in	dR,
		in	n
	)

Computes the capital recovery factor.

Type CRF for a demo.

Form

cRF = CRF(dR,n)

Inputs

dR      (1,1) Discount rate (fraction)
n       (1,1) Number of periods (years)

Outputs

cRF    (1,1) Capital recovery factor

Reference: NREL Energy Analysis https://www.nrel.gov/analysis/tech-lcoe-documentation.html

◆ CrosssectionClassical()

function CrosssectionClassical	(	in	e,
		in	rB,
		in	vB
	)

Classical crosssection.

Type CrosssectionClassical( for a demo.

Form

igma = CrosssectionClassical(e,rB,vB)

Inputs

e      (1.:) Energy (in units of vB)
rB     (1,1) Distance of closest approach
vB     (1,1) Barrier height

Outputs

sigma  (1,:) Crosssection (in units or rB)

Reference: https://www.physicsforums.com/threads/fusion-cross-sections.805866/

◆ CrossSectionFromRest()

function CrossSectionFromRest	(	in	r1,
		in	r2,
		in	E
	)

Cross section assuming the second particle is at rest.

Computes the total cross section as a function of the energy in keV of the incident particle with Duane coefficients.

Form

sT = CrossSectionFromRest( r1, r2, E )

Inputs

r1       (1,:) Reactant name ('D','T','He3')
r2       (1,:) Reactant name ('D','T','He3')
E         (:)  Kinetic temperature (keV)

Outputs

sT        (:)  Cross section in barns (1e-24 cm^2)

References: J.D. Huba, NRL Plasma Formulary, p. 44 (2011) G. H Miley, Fusion Cross Sections and Reactivities, U. Illinois, Report COO-2218-17 (1974)

◆ Cryocooler()

function Cryocooler	(	in	qC,
		in	tC
	)

Cryocooler mass, efficiency, and performance coefficients.

Returns cryocooler data from curve fits. This applies to single-stage Stirling cryocoolers in the range of 50K to 250K cold temperature. A heat sink temperature of 300 K is assumed for the Carnot efficiency,

COPcarnot = Tc/(Th - Tc)

Type Cryocooler for a demo

Form

[m,etaFC,cOP] = Cryocooler( qC, tC )

Inputs

qC      (1,:) Cooling power (capacity) (W)
tC      (1,:) Cold temperature   (K)

Outputs

m       (1,:) Mass (kg)
etaFC   (1,:) Efficiency (% of Carnot, COPcooler/COPcarnot)
cOP     (1,:) Coefficient of performance (Qc/Pin)

Reference: D. R. Ladner, "Performance and Mass vs. Operating Temperature for Pulse Tube and Stirling Cryocoolers," International Cryocoooler Conference (16), 2008.

References and().

◆ CryocoolerBrake()

function CryocoolerBrake	(	in	qC,
		in	tC
	)

Mass, efficiency, and performance coefficients of low-power cryocoolers.

Returns crycooler data from curve fits. This applies to GM, pulsed tube, and Stirling cryocoolers in the range of 4K to 80K cold temperature. A heat sink temperature of 300 K is assumed for the Carnot efficiency,

COPcarnot = Tc/(Th - Tc)

Note that while any power may be input, coolers at low temperature can only move a small amount of power. Typical ranges are:

4 K: max 2 W
10 K: max 10 W
20 K: max 20 W
40 K: max 500 W
80 K: max 50000 W

Type CryocoolerBrake for a demo

Form

[m,eta,cOP] = CryocoolerBrake( qC, tC )

Inputs

qC      (1,:) Cooling power (capacity) (W)
tC      (1,:) Cold temperature   (K)

Outputs

m       (1,:) Mass (kg)
eta     (1,:) Efficiency (% of Carnot, COPcooler/COPcarnot)
cOP     (1,:) Coefficient of performance (Qc/Pin)

Reference: Low-power cryocooler survey, H.J.M. ter Brake, G.F.M Wiegerinck,

Cryogenics 42 (2002) 705-718.

References and().

◆ CyclotronFrequency()

function CyclotronFrequency ( in b )

Computes the cyclotron frequency.

Type CyclotronFrequency for a demo.

Form

omega = CyclotronFrequency( b )

Inputs

b        (1,:) Magnetic field (T)

Outputs

omega    (1,:) Frequency (rad/s)

◆ DebyeLength()

function DebyeLength	(	in	tE,
		in	n0
	)

The Debye Length.

The Debye length is the radius of influence of the electrostatic charge of a particle.

Type DebyeLength for a demo.

Form

lD = DebyeLength( tE, n0 )

Inputs

tE    (1,n) Electron temperature (eV)
n0    (1,m) Number density (m^-3)

Outputs

lD    (m,n) Debye length (m)

◆ DensityFromFusionPower()

function DensityFromFusionPower	(	in	p,
		in	r1,
		in	r2,
		in	tK
	)

Calculate the ion density for volumetric fusion power.

Form

n = DensityFromFusionPower( p, r1, r2, tK )

Inputs

p         (1,1)    Fusion power (MW/m^3) 
r1        (1,:)    Reactant name ('D','T','He3','B11','Li6','He4')
r2        (1,:)    Reactant name ('D','T','He3','B11','Li6','He4')
tK        (:)      Kinetic temperature (keV)

Outputs

n         (1,1)    Ion density (#/m^3)

References: Dawson, J. M., "Advanced Fusion Reactors," Fusion,

Teller, E. Editor, Academic Press, 1981, pp. 457.

◆ DHe3Reaction()

function DHe3Reaction ( in t )

Computes the fraction of D going to each reaction.

Type DHe3Reaction for a demo.

Form

pR = DHe3Reaction( t )

Inputs

t        (1,:) Temperature

Outputs

pR       (3,:)  [He3, Dp, Dn]

References: http://home.earthlink.net/~jimlux/nuc/reactions.htm

◆ DivertorCLR()

function DivertorCLR	(	in	nUp,
		in	qPar,
		in	L,
		in	unitIndex
	)

Computes temperature drop over a conduction-limited divertor.

Implements the two-point divertor model in the Conduction Limited Regime (CLR) of Stangeby. The solver is original. Assumes deuterium SOL.

Type DivertorCLR for a demo.

Form

 [Tu, Tt] = DivertorCLR( nUp, qPar, L, unitIndex )

Inputs

nUp       (1,1) Density upstream in the SOL (1/m^3)
qPar      (1,1) Energy flux down the divertor (W/m^2 or MW/m^2)
L         (1,1) Connection length of the divertor (m)
unitIndex (1,1) (optional) Which set of units to use. 
1: (default) eV, 1/m^3, W/m^2.
2: keV, 1/m^3, MW/m^2.

Outputs

Tu        (1,1) Temperature upstream in the SOL (eV or keV)
Td        (1,1) Temperature downstream in the divertor, just before the
sheath (eV or keV) 
nt        (1,1) Density downstream in the divertor, just before the
sheath (1/m^3)
LRec      (1,1) Recommended connection length for large temperature
drop, given your nUp and qPar. Original work. (m)

References: Stangeby, P. C. "The Plasma Boundary of Magnetic Fusion Devices." Taylor & Francis, 2000.

◆ DMassFromPower()

function DMassFromPower	(	in	energy,
		in	time
	)

Deuterium Reactant mass flow.

Compute the fuel mass required to produce a specific amount of power. This is for a D-D reactor. Both branches are considered and the product helium-3 is assumed to fuse. The T is assumed to be captured and put on the shelf; the energy it eventually releases as a He3 is NOT considered.

Type DMassFromPower for a demo of 1 to 10 MW for 10 years.

Form mass = DMassFromPower( energy ) mass = DMassFromPower( power, time )

Inputs

energy   (1,:)  Total energy (Watt-s)
-OR-
power    (1,:)  Total power  (W)
time     (1,:)  Total energy (s)

Outputs

mass     (2,:)  Mass deuterium (kg)

◆ EdgeLocalizedModes()

function EdgeLocalizedModes	(	in	tau1,
		in	tau2,
		in	k,
		in	t
	)

Disturbance in a Tokamak plasma due to edge localized modes (ELMs).

These disturbances repeat. Thus, every tau3, set t = 0.

Type EdgeLocalizedModes for a demo.

Form

eLM = EdgeLocalizedModes( tau1, tau2, k, t )

Inputs

tau1        (1,1) Time constant 1 (s)
tau2        (1,1) Time constant 2 (s)
k           (1,1) Gain
t           (1,:) Time (s)

Outputs

d            (1,:1) State derivative

Reference: Scibile, L. and B. Kouvaritakis (2001.) "A Discrete Adaptive Near-Time Optimum Control for the Plasma Vertical Position in a Tokamak.

References set().

◆ EffectiveCharge()

function EffectiveCharge	(	in	nI,
		in	zI
	)

Compute the effective charge in a plasma.

Zeff = sum( ni*Zi^2 )/ne

Type EffectiveCharge for a demo.

Form

zEff = EffectiveCharge( nI, zI )

- ----------------------------------------------------------------------—

Inputs

nI        (:)    Ion densities (#/m^3)
zI        (:)    Ion atomic numbers

Outputs

zEff      (1,1)  Effective charge

References: Santarius, J.F. and B.G. Logan, "Generic Magnetic Fusion Rocket", UWFDM-914, University of Wisconsin, February 1998.

◆ FirstWallThermalPower()

function FirstWallThermalPower	(	in	pLoss,
		in	rWall
	)

Computes the first wall thermal power.

Assumes a spherical shell for the wall.

Type FirstWallThermalPower for a demo.

Form

 pW = FirstWallThermalPower( pLoss, rWall )

Inputs

pFus    (1,:)     Loss (MW)
rW      (1,1)     Radius of the wall (m)

Outputs

pW      (1,:)   Wall Power (MW/m^2)

References: Schaffer, M.J., "Considerations for Steady-State FRC-Based Fusion Space Propulsion", GA-A23579, December 2000. General Atomics, San Diego, CA.

◆ FRCFieldLines()

function FRCFieldLines	(	in	r,
		in	z,
		in	B0,
		in	rs,
		in	kappa
	)

FRC equilibrium flux and field strength.

A Solov'ev model is used inside the elliptical separatrix. The O point is at r0 = rs/sqrt(2). The X-points are at +/- zs where zs = kappa*rs.

Form

[psiIn,psiOut,B] = FRCFieldLines( r, z, B0, rs, kappa )

Inputs :

r       (:,:) Radial position(s)
z       (:,:) Axial position (along elongation)
B0      (1,1) Nominal field
rs      (1,1) Separatrix radius
kappa   (1,1) Elongation factor

Outputs :

psiIn   (:,:) Magnetic flux inside the separatrix
psiOut  (:,:) Magnetic flux outside the separatrix
B       (:,:) Magnitude of the magnetic field inside the separatrix

References and().

◆ FRCInertia()

function FRCInertia	(	in	rX,
		in	rY,
		in	rZ,
		in	n,
		in	mI
	)

Inertia and mass of an FRC plasma.

z is the long axis of the cylinder.

Type FRCInertia for a demo.

Form

[inr, mass] = FRCInertia( rX, rY, rZ, n, mI )

Inputs

aX      (1,1)  X-axis (m)
aY      (1,1)  Y-axis (m)
aZ      (1,1)  Z-axis (m)
n       (1,:)  Number density /m^3
mI      (1,:)  Mass ion (kg)

Outputs

inr     (3,3)  Inertia matrix  (kg-m^2)
mass    (1,1)  Mass (kg)

References: Yamada, M, H. Ji, R. Kulsrud, N. Pomphrey and H. Himura, "Studies of Global Stability of Field-Reversed Configuration Plasmas using a Rigid Body Model," April 22, 1998. Weisstein, E. W. "Moment of Inertia-Ellipsoid", http://scienceworld.wolfram.com/physics/ MomentofInertiaEllipsoid.html

References and().

◆ FuelCost()

function FuelCost ( in type )

Fuel cost for various energy sources.

Returns fuel cost. type is not case sensitive. Numbers are from 2018. Type FuelCost for a demo that will list all the types.

Form

[fC,t] = FuelCost(type)

Inputs

type    (1,:) Type

Outputs

fC      (1,1) $/BTU
t       {:}   Type array if 'all' entered

Reference: http://www.amsenergy.com/fuel-cost-calculator/

◆ FusionPower()

function FusionPower	(	in	n1,
		in	n2,
		in	r1,
		in	r2,
		in	tK
	)

Fusion power per reactants.

Calculate the fusion power for the given reactants assuming ion density and temperature. Returns ion and neutron power separately.

Type FusionPower for a demo of the D-T reaction.

Form

[pI, pN] = FusionPower( n1, n2, r1, r2, tI )

Inputs

n1        (1,1)    Ion density of reactant 1 (#/m^3)
n2        (1,1)    Ion density of reactant 2 (#/m^3)
r1        (1,:)    Reactant 1 name ('D','T','He3','B11','Li6','He4')
r2        (1,:)    Reactant 2 name ('D','T','He3','B11','Li6','He4')
tK        (:)      Kinetic temperature (keV)

Outputs

pI        (:)      Ion power (MW/m^3)
pN        (:)      Neutron power (MW/m^3)

References: Dawson, J. M., "Advanced Fusion Reactors," Fusion, Teller, E. Editor, Academic Press, 1981, pp. 457.

References and().

◆ FusionPowerDHe3()

function FusionPowerDHe3	(	in	ni,
		in	Ti
	)

Compute volumetric power for a D-He3 reactor ignoring tritium.

Computes the D-D and D-He3 reactions for the input density and temperature. Ignores tritium.

Type FusionPowerDHe3 for a demo.

Form

[pF, pN] = FusionPowerDHe3( ni, Ti  )

Inputs

ni      (1,2)  [nD nHe3]  Number densities (#/m3)
Ti      (2,:)  [TD;THe3]  Temperature      (keV)

Outputs

pF       (1,1)  Total fusion power (MW/m^3)
pN       (1,1)  Neutron power (MW/m^3)

See also MaxwellianV, FusionReaction, AtomicNumber, PlasmaConstant

References and().

◆ FusionReaction()

function FusionReaction	(	in	r1,
		in	r2
	)

Finds products of fusion reactions.

D-D, D-T, p-B11 and D-He3 are included. For D-D, you must specify which reaction path you want with either 'D','Dp' for 2D->p+T or 'D','Dn' for 2D->n+He3

Type FusionReaction for a demo.

Form

x = FusionReaction( r1, r2 )

Inputs

r1       (1,:) Reactant name ('D','T','He3','B11','Li6','He4')
r2       (1,:) Reactant name ('D','T','He3','B11','Li6','He4')

Outputs

x             (:)  Data structure
.product
.particle    (:,:) names of particles
.energy      (:,:) energies of particles (MeV)
.mass        (:,:) mass of each particle
.totalEnergy (1,1) total energy released (MeV)
.totalMass   (1,1) total mass of the products
.massNeutrons(1,1) total mass of the neutrons
.charge      (1,:) charge of each product
.ions        (1,:) index to the ions
.neutrons    (1,:) index to the neutrons
.netEnergy   (1,1) Energy - neutron energy
.reactant
.particle    (:,:) names of particles
.mass        (:,:) mass of each particle
.charge      (1,:) charge of each product
.totalMass   (1,1) total mass of the reactants

References: 2016 NRL Plasma Formulary, Office of Naval Research;

McNally, Physics of Fusion Fuel Cycles, Nuclear Technology/Fusion, Vol. 2, Jan 1982

References and().

◆ FusionReactionDD()

function FusionReactionDD ( in r )

Finds products of D+D fusion reactions.

Reactions are specified below:

'D-DTHe3': 6D -> 2n+2p+2He4 'D-DT': 5D -> 2n+p+He3+He4 'D-DHe3': 5D -> n+2p+He4+T 'D-Dnp': 4D -> n+p+He3+T 'D-Dn': 2D -> n+He3 'D-Dp': 2D -> p+T

The product 'thermal' indicates that primary products / secondary reactants have some energy even before they fuse.

Type FusionReactionDD for a demo.

Form

x = FusionReactionDD( r1, r2 )

Inputs

rS       (1,:) Reaction name (See above, 'D-Dp','D-Dn','D-DT',etc)

Outputs

x             (:)  Data structure
.product
.particle    (:,:) names of particles
.energy      (:,:) energies of particles (MeV)
.mass        (:,:) mass of each particle
.totalEnergy (1,1) total energy released (MeV)
.totalMass   (1,1) total mass of the products
.massNeutrons(1,1) total mass of the neutrons
.charge      (1,:) charge of each product
.ions        (1,:) index to the ions
.neutrons    (1,:) index to the neutrons
.netEnergy   (1,1) Energy - neutron energy
.reactant
.particle    (:,:) names of particles
.mass        (:,:) mass of each particle
.charge      (1,:) charge of each product
.totalMass   (1,1) total mass of the reactants

References: 2016 NRL Plasma Formulary, Office of Naval Research;

McNally, Physics of Fusion Fuel Cycles, Nuclear Technology/Fusion, Vol. 2, Jan 1982

◆ FusionReactorDesign()

function FusionReactorDesign	(	in	d,
		in	mode,
		in	pmin,
		in	pmax,
		in	n
	)

FUSION REACTOR DESIGN FUNCTION (tokamaks)

The function checks that the reactor satisfies key operational constraints for tokamaks. Selecting a mode ('a', 'H', B_max', 'P_E', or 'P_W') sets the corresponding variable as the dependent variable over which to perform a parameter sweep, also referred to as the parameter vector. Default mode is 'a'. Default vector is a = linspace(0.5,1.7,100).

Type FusionReactorDesign for a demo.

Form

FusionReactorDesign;           % run demo
d    = FusionReactorDesign;    % default data
d    = FusionReactorDesign( d )
d    = FusionReactorDesign( d , mode )
d    = FusionReactorDesign( d , mode , pmin , pmax , n )

Inputs

d (.) Input data structure
.B_max (1,1) Maximum magnetic field at the coil [T]
.H     (1,1) H-mode enhancement factor
.P_E   (1,1) Electric power output [MW]
.P_W   (1,1) Maximum neutron wall loading [MW/m^2]
.a     (1,1) Minor radius of the plasma [m]
mode ''  'a'*, 'H', 'B_max', 'P_E', 'P_W'   * is default
pmin (1,1) Minimum of parameter vector
pmax (1,1) Maximum of parameter vector
N    (1,1) Number of elements in parameter vector

Outputs

d (.) Data structure additions
.xvec       (1,n) Vector for selected dependent variable mode
(x-axis)
.n_nG       (1,n) Ratio of plasma density to the Greenwald
density limit
.beta_betaT (1,n) Ratio of normalized plasma pressure to the
Troyon beta limit
.qK_qstar   (1,n) Ratio of normalized inverse current to kink
safety limit
.fB_fNC     (1,n) Ratio of required to achievable bootstrap 
current
.parameters (22,2) Table listing the tokamak parameter outputs

Reference: Freidberg, Mangiarotti, and Minervini, "Designing a tokamak fusion reactor--How does plasma physics fit in?" Physics of Plasmas 22, 070901 (2015).

◆ GasPressure()

function GasPressure	(	in	n,
		in	kT,
		in	V
	)

Computes Gas pressure.

Type GasPressure for a demo.

Form

p = GasPressure( n, kT, V )

Inputs

n         (1,1)   Number of molecules or #/volume^3
kT        (1,:)   Temperature (eV)

Outputs

p         (1,:)    Pressure (N/m^2)

◆ GyroRadius()

function GyroRadius	(	in	t,
		in	b,
		in	mu,
		in	z
	)

Computes gyroradius.

If two arguments are entered it will compute the electron gyro radius.

Type GyroRadius for a demo.

Form

r = GyroRadius( t, b, mu, z )

- ----------------------------------------------------------------------—

Inputs

t         (1,:)  Temperature (kEV)
b         (1,1)  Magnetic Field (T)
mu        (1,1)  Ratio of mass to proton mass
z         (1,1)  Charge

Outputs

r         (1,:)  Gyro radius (m)

References: http://www.ipp.mpg.de/~dpc/nrl/28.html

NRL Plasma Formulary, 2011

◆ He3MassFromPower()

function He3MassFromPower	(	in	energy,
		in	time
	)

Helium-3 (He3) and Deuterium Reactant mass flow.

Compute the fuel mass required to produce a specific amount of power. Also calculates the volume of 3He and D - as a liquid, for sizing tanks. The volume at STP is useful for cost estimations.

Type He3MassFromPower for a demo of 1 to 10 MW for 10 years.

Form [mass,volume,volSTP] = He3MassFromPower( energy ) [mass,volume,volSTP] = He3MassFromPower( power, time )

Inputs

energy   (1,:)  Total energy (Watt-s)
-OR-
power    (1,:)  Total power  (W)
time     (1,:)  Total energy (s)

Outputs

mass     (2,:)  Mass [3He;D] (kg)
volume   (2,:)  Liquid volume [3He;D] (m^3)
volSTP   (2,:)  Liters at STP [3He;D] (L)

◆ HeatCapacityAluminum()

function HeatCapacityAluminum ( in T )

Heat capacity and enthalpy of aluminum at a given temperature.

Form

[cp,H] = HeatCapacityAluminum( T )

Inputs

T           (1,:) Temperature (K)

Outputs

cp          (1,:) Specific heat capacity (J/gK)
H           (1,:) Enthalpy (J/g). Integrates cp, so answer is only good
to 1%

Reference: Buyco, Edgar H., and Fred E. Davis. "Specific Heat of Aluminum from Zero to Its Melting Temperature and beyond. Equation for Representation of the Specific Heat of Solids." Journal of Chemical & Engineering Data 15, no. 4 (October 1, 1970): 518?23. https://doi.org/10.1021/je60047a035.

References and().

◆ Inductor()

function Inductor	(	in	d,
		in	n
	)

Model an inductor.

Has a built-in demo.

Form

[p, m, l, r, lWire] = Inductor( d, n )

◆ InductorDesign()

function InductorDesign ( in d )

Design an inductor (fmincon)

Has a built-in demo.

Form

[m, p, l, a] = InductorDesign( d )

Inputs

Outputs

References and().

◆ IonCyclotronFrequency()

function IonCyclotronFrequency	(	in	b,
		in	z,
		in	m
	)

Computes the ion cyclotron frequency.

Type IonCyclotronFrequency for a demo of Helium-3.

Form

omega = IonCyclotronFrequency( b, z, m )

Inputs

b        (1,:) Magnetic field (T)
z        (1,1) Number of electrons
m        (1,1) Ion mass (kg)

Outputs

omega    (1,:) Frequency (rad/s)

Reference: http://en.wikipedia.org/wiki/Ion_cyclotron_resonance

◆ IonElectronEquilibration()

function IonElectronEquilibration	(	in	Te,
		in	ni,
		in	Ti,
		in	Z,
		in	mu
	)

Ion-electron collision rate with different temperatures.

Compute the collision rate. Include all species in the plasma for the correct computation of the electron density. The default ion is deuterium.

Type IonElectronEquilibration for a demo.

Form

[nu,nuT] = IonElectronEquilibration( Te, ni, Ti, Z, mu )

Inputs

Te (1,1)  Electron temperature (keV)
ni (1,:)  Ion density (/m3)
Ti (1,:)  Ion temperature (keV)
Z  (1,:)  Ion charge
mu (1,:)  Ion mass (amu)

Outputs

nu  (1,1) Equilibration coeffient e|i, /s
nuT (1,1) Simplfied form, Te~Ti, /s

Reference: NRL Plasma Formulary, 2016 p. 34

◆ IonIonCollisionRate()

function IonIonCollisionRate	(	in	n,
		in	T,
		in	Z,
		in	mu
	)

Ion-ion collision rate.

Compute the collision rate for a test particle with a target particle. This is for near-Maxwellian distributions. Any input may be scalar if it is the same for both species.

Form

nu = IonIonCollisionRate( n, T, Z, mu )

Inputs

n  (1,2)  [Test Target] particle density (/m3)
T  (1,2)  [Test Target] particle temperature (keV)
Z  (1,2)  [Test Target] particle charge
mu (1,2)  [Test Target] particle AMU

Outputs

nu  (1,1) Collision rate, /s

Reference: NRL Plasma Formulary, 2016 p. 28, 32/33

References scalar().

◆ IonizationEnergy()

function IonizationEnergy ( in r )

Returns the ionization energy per nuclei.

Type IonizationEnergy for a demo.

Form

e = IonizationEnergy( r )

Inputs

r        (1,1) Reactant name ('H', 'D','T','He3','B11','Li6')

Outputs

e        (1,:) Energy (eV)

◆ IonizationEnergyPlasma()

function IonizationEnergyPlasma	(	in	r1,
		in	r2,
		in	n1,
		in	n2
	)

Returns the ionization power per nuclei.

Type IonizationEnergyPlasma for a demo.

Form

e = IonizationEnergyPlasma( r1, r2, n1, n2 )

Inputs

r1       (1,1) Reactant name ('H', 'D','T','He3','B11','Li6')
r2       (1,1) Reactant name ('H', 'D','T','He3','B11','Li6')
n1       (1,1) Number density /m^3
n2       (1,1) Number density /m^3

Outputs

e        (1,1) Energy MJ/m^3

◆ LawsonProduct()

function LawsonProduct	(	in	Te,
		in	ni,
		in	Ti,
		in	Z,
		in	mu,
		in	B,
		in	rS
	)

Compute triple product using classical confinement.

Form

nTt = LawsonProduct( Te, ni, Ti, Z, mu, B, rS )

Inputs

Te      Electron temperature, keV
ni      Ion density, /m3
Ti      Ion temperature, keV
Z       Ion charge(s)
mu      Ion atomic weight(s)
B       Magnetic field (T)
rS      Separatrix radius (m)

Outputs

nTt     Triple product

See also IonIonCollisionRate, PlasmaInternalEnergy, ClassicalConfinementTime

◆ LCOE()

function LCOE	(	in	oCC,
		in	fOM,
		in	vOM,
		in	dR,
		in	n,
		in	cF,
		in	hR,
		in	fC,
		in	t,
		in	dPV
	)

Computes the levelized cost of electricity (LCOE)

Type LCOE for a demo from the second reference. It matches the online calculator. The heating rate can have any units as long as fuel cost and heating rate units are consistent. This has additional inputs that that are optional.

O&M is Operations and Maintenance. n is number of years and dR is the interest per year.

See https://openei.org/apps/TCDB/levelized_cost_calculations.html for values of dPV.

Form

[lCOE, lCF, lCI, lCVOM] = LCOE(oCC,fOM,vOM,dR,n,cF,hR,fC,t,dPV)

Inputs

oCC     (1,1) Overnight capital cost ($/kW, see OvernightCapitalCost.m)
fOM     (1,1) Fixed O&M ($/kW-yr, see OvernightCapitalCost.m)
vOM     (1,1) Variable O&M ($/kWh, see OvernightCapitalCost.m)
dR      (1,1) Discount rate (fraction/year)
n       (1,1) Number of periods (years)
cF      (1,1) Capacity factor (0 to 1, see CapacityFactor.m)
hR      (1,1) Heating rate (Btu/kWh, OvernightCapitalCost.m)
fC      (1,1) Fuel cost ($/Btu)
t       (1,1) Tax rate (0.397, optional)
dPV     (1,1) Present value of depreciation (optional)

Outputs

lCOE    (1,1) Levelized cost of electricity ($/kWh)
lCF     (1,1) Cost of fuel ($/kWh)
lCI     (1,1) Cost of the investment, including fixed O&M ($/kWh)
lCVOM   (1,1) Cost of the variable O&M, ($/kWh)

See also OvernightCapitalCost, CapacityFactor, CRF

Reference: NREL Energy Analysis https://www.nrel.gov/analysis/tech-lcoe-documentation.html https://www.nrel.gov/analysis/tech-lcoe.html https://openei.org/apps/TCDB/levelized_cost_calculations.html

References and().

◆ LCOF()

function LCOF	(	in	oCC,
		in	fOM,
		in	vOM,
		in	dR,
		in	n,
		in	cF,
		in	tC,
		in	fP,
		in	pS
	)

Computes the levelized cost of fuel production.

Type LCOF for a demo

O&M is Operations and Maintenance. The number of periods needs to be in years and the discount rate in fraction/year.

Form

lCOF = LCOF(oCC,fOM,vOM,dR,n,cF,hR,fC,fP,pS)

Inputs

oCC     (1,1) Overnight capital cost of fuel production plant ($)
fOM     (1,1) Fixed O&M ($/kg)
vOM     (1,1) Variable O&M ($/kg)
dR      (1,1) Discount rate (fraction/year)
n       (1,1) Number of periods (years)
cF      (1,1) Capacity factor
tC      (1,1) Transportation cost ($/kg)
fP      (1,1) Feedstock price ($/kg)
pS      (1,1) Production plant scale (kg/yr)

Outputs

lCOE    (1,1) Levelized cost of fuel ($/kg)

Reference: NREL Energy Analysis https://openei.org/apps/TCDB/levelized_cost_calculations.html

References and().

◆ MachineCost()

function MachineCost	(	in	p,
		in	pF,
		in	cF,
		in	fun
	)

Infer the cost of a machine size given a few sizes and costs.

Fits curve

The machine is modeled by the function

c = a1*fun(a2*p) + a3

where c is the cost and p is the size. The default fun is asinh. pF and cF are row vectors, at least 3 in length. The function uses fminsearch and a quadratic cost functional to find the a1, a2 and a3.

pF, cF and fun are optional. Type MachineCost for a demo.

The plot assumes that the size parameter is power in MW and cost is in units of $M.

Form

mC = MachineCost(p,pF,cF,fun)

Inputs

p   (1,:) Size (ex: Power in MW)
pF  (1,:) Example size vector (ex: Power in MW)
cF  (1,:) Cost for whatever pFs are specified (ex: $M)
fun (1,1) Pointer to your nonlinear function (optional)

Outputs

mC  (1,:) Machine cost (ex: $M)

References and().

◆ MacroCrossSection()

function MacroCrossSection	(	in	rho,
		in	formula,
		in	M,
		in	sigma
	)

Compute a macroscopic cross-section for a material.

The macroscopic cross-section is the sum of the component microsections, scaled by their number density. The microscopic cross-sections must be for the same energy, for example fast or thermal neutrons.

The mean free path is 1/Sigma.

Form

Sigma = MacroCrossSection( rho, formula, M, sigma )

Inputs

rho      (1,1)   Material density, in g/cm3
formula  (1,:)   Chemical formula, i.e. [1 1] for LiH
M        (1,:)   Atomic weight of constituents
sigma    (1,:)   Microscopic cross-sections at energy of interest (barn)

Outputs

Sigma    (1,:)   Macroscopic cross-section (1/cm)

◆ MagneticFieldCurrentLoop()

function MagneticFieldCurrentLoop	(	in	i,
		in	a,
		in	p,
		in	q,
		in	x,
		in	doPlot
	)

Computes fields for multiple current loops.

The loops are nominally in the yz-plane. You rotate them with the quaternion q(:,k) and translate the rotated coils with p(:,k). Rotation is done first and then translation of the coil center.

If using this function to plot field lines you should arrange the coils so that the point [0;0;0] is in the center.

When plotting field lines the accuracy of the computation will cause the field lines not to close. If the computation is continued, the field lines will spiral.

Type MagneticFieldCurrentLoop for a demo. It will plot the field vectors for two coaxial coils of different radii and different x positions and also the field lines.

Form

[b,l] = MagneticFieldCurrentLoop( i, a, p, q, x, doPlot )

Inputs

i           (1,:)  Currents in each loop
a           (1,:)  Loop radii
p           (3,:)  Loop positions
q           (4,:)  Loop quaternions from fixed to the loop frame
x           (3,:)  Points at which to compute b
or
d             (.)  Data structure
.xL     (1,1) Initial points
.nP     (1,1) Number of points per line
doPlot      (1)   Boolean to create plot

Outputs

b           (3,:)  Field (T)
l           (1,:)  Line handles

Reference: http://www.netdenizen.com/emagnettest/offaxis/?offaxisloop

References and(), close(), and quaternion().

◆ MagneticPressure()

function MagneticPressure	(	in	beta,
		in	b
	)

Computes the magnetic pressure from beta and field.

Type MagneticPressure for a demo.

Form

p = MagneticPressure( beta, b )

Inputs

beta             (1,1)  Beta
b                (1,:)  Magnetic Field (T)

Outputs

p               (1,:)   Pressure (Pa)

References and().

◆ MagnetMass()

function MagnetMass	(	in	rhoM,
		in	sigmaM,
		in	bM,
		in	rM,
		in	beta
	)

Compute the magnetic structural mass using a shell method.

Computes the magnetic mass from density, stress, magnetic field, and radius of the magnet. Uses a spherical shell model. The mass scales with radius cubed and field strength squared.

Note that the reference does not give a complete equation.

Type MagnetMass for a demo.

The demo uses Titanium. The default beta is 1.

Form

mM = MagnetMass( rhoM, sigmaM, bM, rM, beta )

Inputs

rhoM             (1,1)          Density (kg/m^3)
sigmaM           (1,1)          Stress (N/m^2)
bM               (1,1)          Magnetic field (T)
rM               (1,:)          Radius (m)
beta             (1,1)          Plasma beta

Outputs

mM                (1,:)         Magnet Mass (kg)

References: Schaffer, M.J., "Considerations for Steady-State FRC-Based Fusion Space Propulsion", GA-A23579, December 2000. General Atomics, San Diego, CA

References and().

◆ MagnetMassVirial()

function MagnetMassVirial	(	in	rhoM,
		in	sigmaM,
		in	bM,
		in	v
	)

Compute the magnetic structural mass using a Virial method.

Computes mass from density, stress, magnetic field, and volume enclosed by the magnet. This is an energy method. Note that the reference does not give a complete equation.

Type MagnetMassVirial for a demo.

The demo uses carbon fiber.

Form

mM = MagnetMassVirial( rhoM, sigmaM, bM, v )

Inputs

rhoM             (1,1)          Density (kg/m^3)
sigmaM           (1,1)          Stress (N/m^2)
bM               (1,1)          Magnetic field (T)
v                (1,:)          Volume (m^3)

Outputs

mM                (1,:)         Magnet Mass (kg)
eM                (1,:)         Stored energy (J)

Reference: Santarius, J.F. and B.G. Logan, "Generic Magnetic Fusion Rocket Model", UWFDM-914, University of Wisconsin, February 1998.

References and().

◆ MagnetRadius()

function MagnetRadius	(	in	mM,
		in	rhoM,
		in	sigmaM,
		in	bM
	)

Computes magnet radius from mass, density, stess, and magnetic field.

Type MagnetRadius for a demo.

Form

rM = MagnetRadius( mM, rhoM, sigmaM, bM )

Inputs

mM            (1,:)        Mass (kg)
rhoM          (1,1)        Density (m^-3)
sigmaM        (1,1)        Stress (N/m)
bM            (1,1)        Magnetic field (T)

Outputs

rM            (1,:)        Radius (m)

Reference: Schaffer, M.J., "Considerations for Steady-State FRC-Based Fusion Space Propulsion", GA-A23579, December 2000. General Atomics, San Diego, CA

References and().

◆ MagnetStoredEnergy()

function MagnetStoredEnergy	(	in	b,
		in	v
	)

Energy stored in a magnet from field and volume.

Type MagnetStoredEnergy for a demo.

Form

e = MagnetStoredEnergy( b, v )

Inputs

b           (1,1) Magnetic field (T)
v           (1,:) Volume (m^3)

Outputs

e           (1,1) Stored energy (J)

Reference: Santarius, J.F. and B.G. Logan, "Generic Magnetic Fusion Rocket", UWFDM-914, University of Wisconsin, February 1998.

References and().

◆ MaxwellianV()

function MaxwellianV	(	in	r1,
		in	r2,
		in	tK
	)

Computes Maxwellian thermonuclear reaction rates.

Reaction rates averaged over Maxwellian energy distributions. Add n or p to one of the d's to get the neutron or proton variant. d-d gives you the proton variant. These are accurate to within about 2% for 1<T<100 keV and 10% for 2<T<150 keV.

Type MaxwellianV for a demo.

Form

sV = MaxwellianV( r1, r2, tK )

Inputs

r1       (1,:) Reactant name ('D','T','He3')
r2       (1,:) Reactant name ('D','T','He3')
tK       (:)   Kinetic temperature (keV)

Outputs

sV       (:)   Reaction rate <ov> in m^3/sec

Reference: "A Simple Computational Form for Maxwellian Reactivities", L. Hively, Plasma Engineering, Technical Note, 1982.

References Add().

◆ NetFusionPower()

function NetFusionPower	(	in	nI,
		in	zI,
		in	tI,
		in	tE,
		in	rW,
		in	L,
		in	b,
		in	r1,
		in	r2
	)

Computes net fusion power after Synchrotron and Bremsstrahlung.

Point source model. Assumes electron and ion temperatures are the same.

Type NetFusionPower for a demo.

Form

[pN, pF, pB, pS] = NetFusionPower( nI, zI, tI, tE, rW, L, b, r1, r2, )

- ----------------------------------------------------------------------—

Inputs

nI        (1,2)  Ion densities (#/m^3)
zI        (1,2)  Ion atomic numbers
tI        (1,:)  Ion temperature (keV)
tE        (1,:)  Electron temperature (keV)
rW        (1,1)  Wall reflection
L         (1,1)  Average Plasma Radius (m)
b         (1,1)  Magnetic Field (T)
r1        (1,1)  Reactant 1
r2        (1,1)  Reactant 2

Outputs

pN         (1,:)  Net power (MW/m^3) 
pF         (1,:)  Fusion power (MW/m^3) 
pB         (1,:)  Bremsstrahlung power (MW/m^3) 
pS         (1,:)  Synchrotron power (MW/m^3)

◆ NozzleRadius()

function NozzleRadius	(	in	bAx,
		in	bN,
		in	rInner,
		in	tSOL
	)

Compute the nozzle coil inner radius from flux conservation.

The radius and thickness must be the same units, which will be the units of rN.

Form

rN = NozzleRadius( bAx, bN, rInner, tSOL )

Inputs

bAx      (1,1)   Axial magnetic field (T)
bN       (1,1)   Nozzle magnetic field (T)
rInner   (1,1)   SOL inner radius
tSOL     (1,1)   SOL thickness

Outputs

rN      (1,1)    Nozzle inner radius

References and().

◆ NumberDensity()

function NumberDensity	(	in	rho,
		in	formula,
		in	M
	)

Number density of a molecule from its constituents.

Form

N = NumberDensity( rho, formula, M )

Inputs

rho      (1,1)   Material density, in g/cm3
formula  (1,:)   Chemical formula, i.e. [1 1] for LiH
M        (1,:)   Atomic weight of constituents

Outputs

N        (1,:)   Number density of constituents, atoms/cm3

◆ NumberDensityFromBeta()

function NumberDensityFromBeta	(	in	beta,
		in	b,
		in	kT
	)

Compute number density from beta, b and kT.

Type NumberDensityFromBeta for a demo.

Form

n = NumberDensityFromBeta( beta, b, kT )

Inputs

beta          (1,1)  Plasma beta (0 to 1)
b             (1,1)  Magnetic field (Tesla)
kT            (1,:)  Energy (kEV)

Outputs

n             (1,:)  Number density (/m^3)

Reference: Szirmay, L. V., "Nuclear Fusion Reactor Design Fundamentals",

Colas Pub, 1993, p. 137.

References and().

◆ NumberOfElectrons()

function NumberOfElectrons	(	in	nI,
		in	zI
	)

Computes the number of electrons.

Typing NumberOfElectrons gives the electrons in a D-T reaction.

Type NumberOfElectrons for a demo.

Form

nE = NumberOfElectrons( nI, zI )

Inputs

nI        (1,2)  Number densities of the constituents
zI        (1,2)  Atomic numbers

Outputs

nE         (1,1) Number of electrons

◆ OvernightCapitalCost()

function OvernightCapitalCost ( in type )

Returns Overnight Capital Cost of power plants and other parameters.

Returns overnight capital cost, heating rate, variable O&M and fixed O&M. If type = 'all' oCC will be a latex table and hR will be a cell array for all of the power sources.

type is not case sensitive. Numbers are from 2016.

Type OvernightCapitalCost for a demo that will list all the types.

Form

[oCC,hR,fOM,vOM] = OvernightCapitalCost(type)

Inputs

type    (1,:) Type

Outputs

oCC     (1,1) Overnight capital cost ($/kW)
hR      (1,1) Heating rate (Btu/kWh)
fOM     (1,1) Fixed O&M ($/kW-yr)
vOM     (1,1) Variable O&M ($/kWh)

Reference: EIA, "Capital Cost Estimates for Utility Scale Electricity Generating Plants," November 2016.

References and().

◆ PhotonEnergy()

function PhotonEnergy ( in frequency )

Computes the photon energy.

Type PhotonEnergy for a demo from 1 Hz to 1 EHz.

Form

energy = PhotonEnergy( frequency )

Inputs

frequency  (1,:)  Frequency (Hz)

Outputs

energy    (1,:)   Energy (eV)

Since Version 2018.1

◆ PlasmaConstant()

function PlasmaConstant ( in q )

Constants useful in plasma physics.

If no inputs are entered it prints the list in the catalog.

Type PlasmaConstant to get a catalog of constants.

Form

[x, u] = PlasmaConstant( q )

Inputs

q          (1,:)    String describing the constant

Outputs

x          (1,1)    The value
u          (1,:)    Units (if applicable)

References get().

◆ PlasmaFrequency()

function PlasmaFrequency	(	in	n,
		in	m
	)

Returns the plasma frequency.

The default mass is the electron mass.

Type PlasmaFrequency for a demo.

Form

omega = PlasmaFrequency( n, m )

Inputs

n        (1,1) Number density
m        (1,1) Mass

Outputs

e        (1,:) Energy (eV)

Reference: Stacey, W. "Fusion", John Wiley & Sons, 1984, p. 25.

◆ PlasmaHeating()

function PlasmaHeating	(	in	n,
		in	kT
	)

Computes plasma heating per unit volume.

Type PlasmaHeating for a demo.

Form

q = PlasmaHeating( n, kT )

Inputs

n        (1,1) Number density (1/m^3)
kT       (1,1) Temperature (keV)

Outputs

q        (1,:) Energy (MJ/m^3)

◆ PlasmaInternalEnergy()

function PlasmaInternalEnergy	(	in	n,
		in	t,
		in	v
	)

Computes the plasma internal energy.

May enter multiple species.

Type PlasmaInternalEnergy for a demo.

Form

iE = PlasmaInternalEnergy( n, t, v )

Inputs

n             (1,:)      Density (#/m^3)
t             (1,:)      Temperature (keV)
v             (1,:)      Volume (m^3)

Outputs

iE            (1,:)      Plasma Internal Energy (J)

◆ PlasmaPressure()

function PlasmaPressure	(	in	nI,
		in	zI,
		in	tI,
		in	tE
	)

Computes fusion plasma pressure.

The ion constituents are assumed to be at the same temperature. Type PlasmaPressure for a demo.

Form

p = PlasmaPressure( nI, zI, tI, tE )

Inputs

nI        (1,2)  Number densities of the constituents (1/m^3)
zI        (1,2)  Atomic numbers
tI        (1,:)  Temperature of the ions (keV)
tE        (1,:)  Temperature of the electrons (keV)

Outputs

p         (:)    Pressure (N/m^2)

◆ PlasmaRadius()

function PlasmaRadius	(	in	pNet,
		in	pD,
		in	aR
	)

Computes the Plasma Radius from the desired power and net power density.

aR is entered for a cylindrical plasma.

Type PlasmaRadius for a demo.

Form

r = PlasmaRadius( pNet, pD, aR )

Inputs

pNet           (1,:)    Net power (MW/m^3)       
pD             (1,1)    Desired power (MW)
aR             (1,1)    Aspect ratio length/r

Outputs

r              (1,:)    Plasma Radius (m)

References and().

◆ PlasmaVolume()

function PlasmaVolume	(	in	pNet,
		in	pD
	)

Computes the plasma vadius from the desired power and net power density.

Type PlasmaVolume for a demo.

Form

v = PlasmaVolume( pNet, pD )

Inputs

pNet           (1,:)    Net power (MW/m^3)       
pD             (1,1)    Desired power (MW)

Outputs

v              (1,:)    Plasma Volume (m^3)

References and().

◆ PointFusionReaction()

function PointFusionReaction	(	in	r1,
		in	r2,
		in	nI,
		in	v,
		in	beta,
		in	tI,
		in	pType
	)

Computes the net power from a fusion reaction.

Calculates the power output of a fusion reactor. Assumes that profiles inside the core volume (v) are flat.

Also calculates the electron temperature self-consistently by balancing the power collisionally transferred from ions to electrons with power lost from electrons via Bremsstrahlung. Synchrotron loss is not included. Electron particle loss is not included.

Type PointFusionReaction for a demo.

Form

[p, tE, b, pB, pF] = PointFusionReaction( r1, r2, nI, v, beta, tI, pType )

Inputs

r1       (1,:)   Reactant name ('p' 'D','T','He3','B11','Li6','He4')
r2       (1,:)   Reactant name ('p' 'D','T','He3','B11','Li6','He4')
nI       (1,:)   Ion density (#/m^3) for each species
v        (1,1)   Volume (m^3)
beta     (1,1)   Plasma beta
tI       (1,:)   Ion temperature
pType    (1,:)  'lin' or 'log'

Outputs

p        (1,:)  Power output         (MW)
tE       (1,:)  Electron temperature (kEV)
b        (1,:)  Magnetic field       (T)
pB       (1,:)  Bremsstrahlung power (MW)
pF       (1,:)  Fusion power         (MW)

◆ PowerDD()

function PowerDD ( in d )

Compute the power for a He3-catalyzed D-D reactor.

Computes the D-D, D-3He and D-T side reactions. fT reduces the D-T side reactions for non-homogeneous plasmas. fHe sets the helium-3 density as a fraction of the D density. Synchrotron is neglected. The rate at which 3He is produced is the same as the neutron flux from D-D reactions.

Type PowerDD for a demo for temperatures from 20 keV to 200 keV.

Form

d = PowerDD
[pF, pN, pB, rHe, nNd, nNt] = PowerDD( d )

Inputs

d        (.)    Data structure
.nD       (1,1)  Number density deuterium (#/m3)
.tD       (1,:)  Temperature deuterium    (keV)
.tE       (1,:)  Electron temperature     (keV)
.fT       (1,1)  Factor reducing D-T reactions (0-1)
.fHe      (1,1)  Factor recycled 3He      (0-0.5)

Outputs

pF       (1,1)  Total fusion power (MW/m^3)
pN       (1,1)  Neutron power (MW/m^3)
pB       (1,1)  Bremsstrahlung power (MW/m^3)
rHe      (1,1)  Rate of Helium-3 consumption (#/s)
nNd      (1,1)  Neutron flux from D-D reactions (#/s)
nNt      (1,1)  Neutron flux from D-T reactions (#/s)


See also MaxwellianV, FusionReaction

Reference: Santarius, J.F. and B.G. Logan, "Generic Magnetic Fusion Rocket", UWFDM-914, University of Wisconsin, February 1998.

References and().

◆ PowerDHe3()

function PowerDHe3 ( in d )

Compute the power for a D-He3 reactor including Bremsstrahlung.

Computes the D-D, and D-T side reactions. fD and fT reduce the D-D and D-T side reactions for non-homogeneous plasmas. Synchrotron is neglected.

Type PowerDHe3 for a demo for temperatures from 20 kEV to 200 kEV.

Form

d = PowerDHe3
[pF, pN, pB, nNd, nNt] = PowerDHe3( d )

Inputs

d        (.)    Data structure
.nD       (1,1)  Number density deuterium (#/m3)
.nHe3     (1,1)  Number density helium-3  (#/m3)
.tD       (1,:)  Temperature deuterium    (keV)
.tHe3     (1,:)  Temperature helium-3     (keV)
.tE       (1,:)  Electron temperature     (keV)
.fD       (1,1)  Factor reducing D-D reactions (0-1)
.fT       (1,1)  Factor reducing D-T reactions (0-1)

Outputs

pF       (1,1)  Total fusion power (MW/m^3)
pN       (1,1)  Neutron power (MW/m^3)
pB       (1,1)  Bremsstrahlung power (MW/m^3)
nNd      (1,1)  Neutron flux from D-D reactions (#/m^3-s)
nNt      (1,1)  Neutron flux from D-T reactions (#/m^3-s)


See also MaxwellianV, FusionReaction

Reference: Santarius, J.F. and B.G. Logan, "Generic Magnetic Fusion Rocket", UWFDM-914, University of Wisconsin, February 1998.

References and(), and Update().

◆ PowerDT()

function PowerDT ( in d )

Compute the power for a D-T fusion reactor.

Computes the D-T and D-D side reactions. fT reduces the D-T side reactions for non-homogeneous plasmas. fHe sets the helium-3 density as a fraction of the D density. Synchrotron is neglected. The rate at which 3He is produced is the same as the neutron flux from D-D reactions.

Type PowerDT for a demo for temperatures from 20 keV to 200 keV.

Form

d = PowerDD
[pF, pN, pB, rHe, nNd, nNt] = PowerDT( d )

Inputs

d        (.)    Data structure
.nD       (1,1)  Number density deuterium (#/m3)
.nT       (1,1)  Number density tritium (#/m3)
.tD       (1,:)  Temperature deuterium    (keV)
.tE       (1,:)  Electron temperature     (keV)

Outputs

pF       (1,1)  Total fusion power (MW/m^3)
pN       (1,1)  Neutron power (MW/m^3)
pB       (1,1)  Bremsstrahlung power (MW/m^3)
nNd      (1,1)  Neutron flux from D-D reactions (#/s)
nNt      (1,1)  Neutron flux from D-T reactions (#/s)


See also MaxwellianV, FusionReaction

Reference: Santarius, J.F. and B.G. Logan, "Generic Magnetic Fusion Rocket", UWFDM-914, University of Wisconsin, February 1998.

References and().

◆ PowerIE()

function PowerIE	(	in	zI,
		in	nI,
		in	aI,
		in	Ti,
		in	Te
	)

Power from ions to electrons, NRL model.

This is a simplified form suitable when Te ~= Ti. Nonrelativistic.

Form

[p,nu] = PowerIE( zI, nI, aI, Ti, Te )

- ----------------------------------------------------------------------—

Inputs

zI        (m)    Ion atomic numbers
nI        (m)    Ion densities (#/m^3)
aI        (m)    Ion atomic weight (amu)
Ti        (1,1)  Ion temperature (kev)
Te        (n)    Electron temperature (kev)

Outputs

p         (m,n)  Ion to Electron Loss (MW/m^3)    
nu        (m,n)  nuE/I, equilibration

References: NRL Plasma Formulary, 2016.

◆ PowerIonsToElectrons()

function PowerIonsToElectrons	(	in	nE,
		in	zI,
		in	nI,
		in	aI,
		in	tI,
		in	tE
	)

Power lost by the ions to the electrons, Dawson model (relativistic).

This is the power lost to electrions via collisions, when ions are hotter than electrons.

Type PowerIonsToElectrons for a demo.

Form

p = PowerIonsToElectrons( nE, zI, nI, aI, tI, tE )

- ----------------------------------------------------------------------—

Inputs

nE        (1,1)  Electron density (#/m^3)
zI        (:)    Ion atomic numbers
nI        (:)    Ion densities (#/m^3)
aI        (:)    Ion atomic weight
tI        (1,1)  Ion temperature (kev)
tE        (n)    Electron temperature (kev)

Outputs

p         (m,n)  Ion to Electron Loss (MW/m^3)

References: Dawson, J. M., "Advanced Fusion Reactors," Fusion, Teller,

E. Editor, Academic Press, 1981, pp. 488.

◆ ReactionString()

function ReactionString	(	in	r1,
		in	r2
	)

Create a string with a reaction.

Reactants : 'P' 'H', 'D','T','He3','B11','Li6','He4' H and P mean the same.

Type ReactionString for a demo.

Form

s = ReactionString( r1, r2 )

Inputs

r1        (1,:)    Reactant name
r2        (1,:)    Reactant name

Outputs

s         (1,:)    String

References and(), and Create().

◆ ReactorField()

function ReactorField	(	in	d,
		in	beta
	)

Calculate magnetic field from plasma pressure and beta.

Form

b = ReactorField( d, beta )

- ----------------------------------------------------------------------—

Inputs

d        (.)    Input data structure
.nD    (1,1) Number density D (m^-3)
.nHe3  (1,1) Number density He3 (m^-3)
.tHe3  (1,1) Temperature He3 (kEV)
.tD    (1,1) Temperature D (kEV)
.tE    (1,1) Temperature e- (kEV)
beta    (1,1)   Ratio of pressure to magnetic pressure

Outputs

b       (1,1)   Field strength (T)

See also PowerDHe3, PlasmaConstant, NumberOfElectrons, BField

References and().

◆ ShieldingThickness()

function ShieldingThickness	(	in	limit,
		in	years,
		in	flux,
		in	Sigma
	)

Shielding thickness from neutron attenuation model.

This is a simple model for the thickness of neutron shielding given a flux and the shielding material properties. The attenuation equation is:

I/I0 = exp(-Sigma*t)

The cross-section must be calculated for the energy of the incoming flux. The macroscopic cross-section for LiH is about 0.27.

Form

thick = ShieldingThickness( limit, years, flux, Sigma )

Inputs

limit         (1,1)  Lifetime fluence limit, n/m2
years         (1,:)  Lifetime in years
flux          (1,1)  Flux at the wall, % n/m2/s
Sigma         (1,1)  Macroscopic cross section, 1/cm (default: 0.27)

Outputs

thick         (1,:)   Shielding thickness

References and().

◆ SigmaV()

function SigmaV	(	in	r1,
		in	r2,
		in	tK
	)

Computes thermonuclear reaction reaction rates.

Energy-averaged fusion reaction rates. Add n or p to one of the d's to get the neutron or proton variant. d-d gives you the proton variant.

Type SigmaV for a demo.

Form

sV = SigmaV( r1, r2, tK )

Inputs

r1       (1,:) Reactant name ('D','T','He3','B11','Li6','He4', 'p')
r2       (1,:) Reactant name ('D','T','He3','B11','Li6','He4', 'p')
tK        (:)  Kinetic temperature (keV)

Outputs

sV       (:)  Reaction rate in m^3/sec

References: http://home.earthlink.net/~jimlux/nuc/reactions.htm

References Add().

◆ SteadyStateTokamak()

function SteadyStateTokamak ( in d )

Generates a statespace model of a tokamak.

The output is the vertical current moment of the plasma current centroid. This system is unstable.

Type SteadyStateTokamak for a demo.

Form

g = SteadyStateTokamak( d )

Inputs

d           (1,1) Data structure
.rAA (1,1) Active coil resistance (Ohm)
.rVV (1,1) Passive coil resistance(Ohm)
.lAV (1,1) Active-passive coil mutual inductance (H)
.lAA (1,1) Active coil self-inductance (H)
.lVV (1,1) Passive coil self-inductance  (H)
.lAP (1,1) Mutual change inductance between the active coils and plasma displacement (H/m)
.lVP (1,1) Mutual change inductance between the passive coils and plasma displacement(H/m) 
.aPP (1,1) Normalized destabilizing force 0.4 to 0.6 e-6 H/m^2

References statespace().

◆ Synchrotron()

function Synchrotron	(	in	nI,
		in	zI,
		in	b,
		in	tE,
		in	L,
		in	rW
	)

Computes synchrotron radiation.

Synchrotron is radiation lost by the electrons. The NRL formulary provides a global loss model that assumes a constant field and temperature. Bingren provides a model that accounts for wall reflection for a toroidal field.

Type Synchrotron for a demo comparing the two models.

Form

p = Synchrotron( nI, zI, b, tE )         % global model (NRL)
p = Synchrotron( nI, zI, b, tE, L, rW )  % wall reflection (Bingren)

- ----------------------------------------------------------------------—

Inputs

nI        (1,m)  Ion densities (#/m^3)
zI        (1,m)  Ion atomic numbers
b         (1,1)  Magnetic Field (T)
tE        (1,n)  Electron temperature (keV)
L         (1,1)  Average plasma radius (m)
rW        (1,1)  Wall reflection (0-1)

Outputs

p         (1,n)  Synchrotron radiation (MW/m^3)

References: NRL Plasma Formulary, 2016. Bingren, S., "Core Plasma Characteristics of a Spherical Tokamak D-3He Fusion Reactor", Plasma Science & Technology, Vol. 7 No. 2 April 2005. Bing-Ren, S. "Possibility of Establishing D-3He Fusion Reactor Using Spherical Tokamaks", Chinese Physics Letters, 2004

References and().

◆ SynchrotronFrequency()

function SynchrotronFrequency	(	in	b,
		in	v
	)

Computes the synchrotron frequency.

Enter one argument for the cyclotron frequency (non-relativistic). MKS units.

Type SynchrotronFrequency for a demo.

Form

omega = SynchrotronFrequency( b, v )

Inputs

b        (1,:) Magnetic field (T)
v        (1,1) Electron speed (m/s)

Outputs

omega    (1,:) Frequency (rad/s)

◆ ThermalConductivityG10()

function ThermalConductivityG10 ( in T )

Thermal conductivity of G-10 Cryo at a given temperature.

It's different in the warp direction than in the normal directon, so give both.

Form

[KWarp,KNormal] = ThermalConductivityG10( T )

Inputs

T           (1,:) Temperature (K)

Outputs

KWarp       (1,:) Thermal conductivity in the warp direction (W/K*m)
KNormal     (1,:) Thermal conductivity in the normal direction (W/K*m)

Reference: Bradley, Peter, Ray Radebaugh, and Gary Hardin. "NIST Thermodynamics Research Center Cryogenic Technology Resources: Cryogenic Material Properties." NIST Thermodynamics Research Center Cryogenic Technology Resources: Cryogenic Material Properties. Accessed November 16, 2020. https://trc.nist.gov/cryogenics/materials/materialproperties.htm.

◆ ThermalConductivityInconel()

function ThermalConductivityInconel ( in T )

Give the thermal conductivity of Inconel at a given temperature.

From 4 K to 300 K

Form

K = ThermalConductivityG10( T )

Inputs

T           (1,:) Temperature (K)

Outputs

KWarp       (1,:) Thermal conductivity (W/K*m)

Reference: Bradley, Peter, Ray Radebaugh, and Gary Hardin. "NIST Thermodynamics Research Center Cryogenic Technology Resources: Cryogenic Material Properties." NIST Thermodynamics Research Center Cryogenic Technology Resources: Cryogenic Material Properties. Accessed November 16, 2020. https://trc.nist.gov/cryogenics/materials/materialproperties.htm.

◆ TokamakBeta()

function TokamakBeta ( in aR )

Computes the tokamak beta from aspect ratio.

The input is the minor radius/major radius. The minor radius is the radius of the cylinder that makes up the toroid. The major radius is from the center of the toroid to the centerline of the cylinder that makes up the torus.

Type TokamakBeta for a demo.

Form

beta = TokamakBeta( aR )

Inputs

aR        (1,:)   Aspect ratio, minor radius/major radius

Outputs

beta      (1,:)   Toroidal beta

References: Szirmay, L. V., "Nuclear Fusion Reactor Design Fundamentals", Colas Publishing, 1992, p. 146.

◆ TPVHRHS()

function TPVHRHS	(	in	x,
		in	t,
		in	d
	)

Simulates a model of the vertical position of a plasma in a Tokamak.

Form

xDot = TPVHRHS( x, t, d )

Inputs

x           (3,1) State [wA4;wB4;h]
t           (1,1) Time
d           (1,1) Structure
.a         (1,1) State space model
.b         (1,1)
.c         (1,1)
.d         (1,1)
.v         (1,1)
.eLM       (1,1) Edge limited mode disturbance

Outputs

xDot        (3,1) State derivative

Reference: Scibile, L. and B. Kouvaritakis (2001.) "A Discrete Adaptive Near-Time Optimum Control for the Plasma Vertical Position in a Tokamak." IEEE Transactions on Control System Technology. Vol. 9, No. 1, January 2001.

◆ TransportPower()

function TransportPower	(	in	n,
		in	T,
		in	tau,
		in	v
	)

Compute transport power loss given confinement time.

This is the ratio of internal plasma energy to the confinement time.

Form

[Pt,iE] = TransportPower( n, T, tau, v )

Inputs

n   (1,:)   Ion densities     (/m3)
T   (1,:)   Ion temperatures  (keV)
tau (1,:)   Confinement time  (s)
v   (1,1)   Volume, default 1 (m3)

Outputs

Pt   (1,:)  Power (MW)
iE   (1,:)  Plasma internal energy (J)

◆ VolumeAveragedPressure()

function VolumeAveragedPressure	(	in	t,
		in	nI,
		in	zI
	)

Computes the volume averaged pressure.

Compute the electron density from the ion density to compute the plasma pressure.

Type VolumeAveragedPressure for a demo.

Form

rMS = VolumeAveragedPressure( t, nI, zI )

- ----------------------------------------------------------------------—

Inputs

t (1,:) Temperature (keV) nI (1,:) Ion density (#/m^3) zI (1,:) Atomic number

Outputs

p          (1,:)     Volume Averaged Pressure (N/m^2)

References: M.J. Schaffer, "CONSIDERATIONS FOR STEADY-STATE FRC-BASED FUSION SPACE PROPULSION", General Atomic GA?A23579,2000.

◆ XRayMassAttenCoeff()

function XRayMassAttenCoeff	(	in	el,
		in	e,
		in	file
	)

Computes the X-ray mass attenuation coefficient.

Uses NIST data. d has one cell row per element in the database.

file needs to be an ASCII text file of the form:

%% element
e (MeV), mu, mu Energy
e (MeV), mu, mu Energy
e (MeV), mu, mu Energy
...

You can have multiple elements in one file. The default file has helium, xenon, hydrogen, titanium and tungsten.

Some of the NIST data has discontinuities, two values of mu at the same energy. If the file reader finds an element like that it adjusts the energy slightly.

This function will extrapolate past the limits of the input data.

Type XRayMassAttenCoeff for a demo for a mixture of helium and xenon.

Form

[m, mEn, d] = XRayMassAttenCoeff( el, e, file )

Inputs

el   {}               Elements {name, fraction ... }
e    (1,:)            Energy (MeV)
file (1,:)            File Name (Default is MassAttenCoeff.txt)

Outputs

m    (1,:)            Mass attenuation coefficients (cm^2/g)
mEn  (1,:)            Mass attenuation energy coefficients (cm^2/g)
d    {name,e,m,mEn}   Data

Reference: https://www.nist.gov/pml/x-ray-mass-attenuation-coefficients

References and().

Directories

Functions

Detailed Description

Function Documentation

◆ AlfvenSpeed()

Form

Inputs

Outputs

◆ AtomicNumber()

Form

Inputs

Outputs

◆ Beta()

Form

Inputs

Outputs

◆ BetaBarnes()

Form

Inputs

Outputs

◆ BField()

Form

Inputs

Outputs

◆ Bremsstrahlung()

Form

- ----------------------------------------------------------------------—

Inputs

Outputs

◆ BremsstrahlungFrequency()

Form

Inputs

Outputs

◆ BremsstrahlungWavelength()

Form

Inputs

Outputs

◆ BurnupTime()

Form

Inputs

Outputs

◆ BVacuum()

Form

Inputs

Outputs

◆ CapacityFactor()

Form

Inputs

Outputs

◆ CentralPressure()

Form

- ----------------------------------------------------------------------—

Inputs

Outputs

◆ CentralTemperature()

Form

Inputs

Outputs

◆ ClassERFAmplifier()

Form

Inputs

Outputs

◆ ClassicalConfinementTime()

Form

Inputs

Outputs

◆ CollisionsTrubnikov()

Form

Inputs

Outputs

◆ CollisionTimes()

Form

Inputs

Outputs

◆ CoulombLog()

Form

Inputs

Outputs

◆ CoulombLogIons()

Form