Tritium burnup fraction

C. Kessel, PPPL

ARIES Project meeting, July 27-28, 2011Gaithersburg, MD

Tritium burnup fractionFor MFE we must inject more tritium into the plasma chamber than is consumed in fusion reactions, because all the fuel is not consumed before it is pumped out of the chamber

The rate at which all different particles are pumped out of the chamber is determined so that we do not get a build up of He (product of D + T) and impurities, and the fusion power is what we desire

ST = nDnT<σv> + nT/τT* simple balance of tritium density inside the plasma

The burnup fraction is the ratio of the amount of tritium consumed in fusion reactions every second to the total tritium we lose every second, this is always < 1.0

fb = nDnT<σv> / (nDnT<σv> + nT/τT*)

Any value of burnup fraction is not accessible, since we must keep the density of He and other impurities sufficiently low

nHe/τHe* = nDnT<σv> balance of He density inside the plasma

We do not know what to assume for this

Tritium burnup, cont’d

€

fb =1

1+1

nDτ T * σvDT

€

σvDT

(m3 /s) = 3.68 ×10−18 1

Ti2 / 3

exp −19.94

Ti1/ 3

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Volume average density<nD> ~ 0.8-1.25x1020 /m3

Volume average temperature<Ti> ~ 20 keV

Global energy confinement timeτE ~ 2 s

Ti is in keV

From experiments, the once through particle confinement time is about the same as the global energy confinement timeτT ~ τE

To include tritium re-entering the plasma, its total particle confinement time is larger than the once through particle confinement timeτT* > τT , often expressed as τT /(1-R), where R is a recycling coefficient, which is a severe simplification of the physics

For a power plant, using only τT* = τT we can get a lower bound to fb, which is about 5.6%

If we assume τT* = 5 xτT, we get 23%If we assume τT* = 10 xτT, we get 37%

Experiments indicate that the residence time of He in the plasma before being pumped out of the plasma chamber (τHe*), and hydrogen is expected to be similar (τHe* ~ τT* ~ τD*), is about 3-10 x τE, with the lower values well established

Measurements of the He density in the plasma are made, so the time constant for the decay of this helium density are made after the source (He neutral beam) is turned offThe τHe*/τE, decreases with increasing plasma density, since the recycling of the He back into the plasma, from the walls, is decreased

It is possible to get preferential removal of DT fuel relative to He, or visa versa, based on atomic physics in the divertor, but this is observed to be quite variable

In experiments this is described by (nHeo/2nD2

o) / (nHe/ne) , that is a ratio of helium gas relative to hydrogen in the divertor, divided by the helium ion density relative to electron density in the plasma

We want this total residence time to be short enough that the helium concentration in the plasma is not too large, BUT we also want to take any credit for fuel (D and T) to re-enter the plasma and undergo fusion, so we like it to be longer if possible and increase our T burnup fraction

Tritium burnup fractionWe also introduce impurities to help radiate power, while other impurities will come from eroded materials inside the plasma chamber

The He and impurities reduce the fusion power by diluting the D and T fuel

There are densities of these “pollutants” that we can tolerate, and this will determine a self-consistent balance of all particles in the plasma, and the injection of fuel and impurities, and the pumping of fuel, He, and impurities

In the plasmas we are always quasi-neutral ne = nD + nT + 2nHe +ΣiZinZ,i

The physics of the particles inside the plasma chamber is a bit more complicated that this….

The plasma particles diffuse out of the plasma and enter a region between the plasma and the solid material wall, we call the scrape off layer

The particles can diffuse to the first wall or flow into the divertor, they impact the solid walls and re-emerge as neutral atoms

Some of these particle can also be absorbed or implanted into the FW or divertor, or pumped out of the chamber through the divertor

The neutral atoms can find their way back to the plasma and re-enter it, quickly becoming ionized

This re-entering population can bring unburned fuel (D and T) back where it could undergo fusion…..so when we try to figure out how much time a tritium ion spends in the plasma we must include this effect

How efficiently particles from the scrape off layer can fuel the plasma is a critical research area, we characterize this by the neutral penetration depth, which means how far into the plasma does a neutral atom get before it is ionized

We know that fueling from the edge is always going on, but we also know that this fueling can be reduced by operating the plasma with high densities, typical of the values we want in a fusion power plant (even ITER will get to this level)…..this is called screening

So low density plasmas typically have high edge fueling efficiencies (or high neutral penetration), and high density plasmas have low edge fueling efficiencies (or short neutral penetration)

Tritium burnup fraction, cont’d

If a particle is ionized near the plasma edge, it has a good chance of just diffusing back out of the plasma, and not getting into the hot center of the plasma where fusion is going on

Recall some of the particles end up in the divertor where they are pumped out of the plasma chamber, and there may be preferential pumping of some types of particles relative to others depending on atomic physics in the divertor

wall

Pellet injection, deep penetration

Neutral tritium atoms re-entering the plasma, weak penetration

Tritium ions becoming atoms after hitting a material wall

Tritium ions go to the divertor and hit the divertor surface becoming atoms, and some re-enter the plasma

Some basic definitionsST = source term for tritium (fueling source)

nDnT<σv> = fusion reaction rate, consumption of tritium

τT = mean time a triton that is injected into the plasma (pellet) takes to leave the plasma (how long does a triton that is deposited deep in the plasma take to get out)

τT* = mean time a triton that was originally injected into the plasma (pellet) spends in the plasma before being pumped out of the plasma vacuum chamber

€

dnTdt

= ST − nDnT <σv > −nTτ T *

Balance in the plasma

The term nT/τT* is the loss of tritons from the plasma before they undergo fusion

The time constant τT* looks simple but actually represents a mean time spent inside the plasma, which is very hard to measure, while having hidden in it

the particle transport time inside the plasma (τT) the pumping of tritons out of the plasma chamberthe recycling of tritons from the wall the neutral penetration distance from the plasma edge into the plasmathe depth that a tritium ion penetrates toward the hot plasma center after re-entering the plasma from the edgethe implantation and subsequent loss of tritons thru the FW or divertor materials

It is normally written as τT* = τT / (1-R), where R is the recycling coefficient (although it actually has all the other physics in it too from above)

Reiter, et al, examined the allowed helium concentration in a plasma for DT fusion in 1990, and also derived a particle confinement time that accounted for the physics that we expect to be governing this situation, albeit for He….but we expect the same physics to govern the hydrogen isotopes as well

With this basic physics picture in mind, we can derive an expression for the time constant, τHe* or τT*

τHe* = τHe,1 + (Reff/(1-Reff)) τHe,2 Reff/(1-Reff) is the mean # of recycling events back into the plasma experienced by the particle before being pumped

The τHe,1 is the mean time a He ion spends in the plasma after being produced from a fusion reaction (its first introduction to the plasma), this is typically ~ τE

The τHe,2 is the mean time a He ion spends in the plasma after re-entering from the plasma edge region, which is expected to be < τE

Some fairly old simulations for ITER indicated that τHe,1 ~ 8 s, while τHe,2 ~ 0.5 s, using a transport model in BALDUR

Some other efficiencies can be defined to help relate our expression to real systems, and uncover the physics in the effective recycle coefficient, Reff

Exhaust efficiency = 1-Reff, probability a particle will be pumped whenever it crosses the plasma boundary in outward direction

εcoll = collection efficiency, εcollφout fraction of particles leaving plasma that are collected by pumping system, and (1-εcoll)φout hit the wall

εrem = pump removal efficiency, εremεcollφout fraction of particles leaving plasma that are collected by pumping system and pumped away

εpump = exhaust efficiency = εremεcoll, so εremεcollφout are removed from the plasma chamber

R(1-εcoll) φout particles will recycle from the wall (R is the true recycling coefficient)

(1-R) (1-εcoll) φout will be absorbed in the wall (whether it is implantation or other mechanism)

γ = refueling efficiency of recycled particles

τHe* = τHe,1 + (Reff/(1-Reff)) τHe,2

τHe* = τHe,1 + γ[R(1-εcoll)+εremεcoll]/{1- [R(1-εcoll)+εremεcoll]} τHe,2

γ<< 1, low re-fueling efficiency

R ~ 1, stationary wall conditions

εcoll = 0.5

εrem = 0.02

γ~ 1, high re-fueling efficiency

Others the same

τHe* = τHe,1

τHe* = τHe,1 + τHe,2

Further efforts on tritium burnup fraction

• Relate these quantities back to fueling and pumping rates, and neutral gas enrichment in divertor

• Examine more experimental results, gather results together

• Simulation results? B2-Eirene?

• Correlate the re-entering tritium with a probability of fusion depending on penetration

• Model expansion to include more stuff

Use 2 population model for tritium

€

dnTdt

= ST − nDnT <σv > −nTτ T *

€

dnT1

dt= ST − nDnT1

<σv > −nT1

τ T1

€

dnT2

dt= R

nT1

τ T1

− nDnT2<σv > −

nT2

τ T2

+ RnT2

τ T2

nT1 = density of original tritons injected via pellets into the plasmanT2 = density of tritons which subsequently left the plasma and re-entered via recyclingnT = nT1 + nT2

τT1 = mean residence time for tritons originally injected via pelletsτT2 = mean residence time for tritons that re-enter the plasma via recyclingR = an effective recycling coefficient, it contains other physics besides just recycling

Assume nD is fixed, although it would follow similar equations, and introduce a nonlinearity

2 population model, cont’d

€

nT1=

STτ T1

1+ nDτ T1σv

= STτ T1(1− fb1

)

€

nT2=nT1Rτ T2

τ T1(1− R)

1

1+nDτ T2

σv

(1− R)

=nT1Rτ T2

(1− fb2)

τ T1(1− R)

€

nT = ST (1− fb1) τ T1

+τ T2R(1− fb2

)

(1− R)

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⎦ ⎥

€

nT = ST η f (1− fb1)τ T1

+η f (1− fb1

)(1− fb2)τ T2R

(1− R)+

(1−η f )(1− fb2)τ T2

(1− R)

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⎣ ⎢

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⎦ ⎥

Including an original pellet fueling efficiency ηf, then

Just the same as we have for helium, τT,1 ~ τE

and τT,2 < τE

Unfortunately, this global particle model does not tell us the probability that the re-entered triton has of getting into the hot fusion zone in the plasma center

Or

The chance of experiencing a fusion event regardless of where it is in the plasma

We don’t know where it gets to, this is a particle transport question

We might characterize this with the neutral penetration depth and the local fusion reactivity associated with that depth