Fusion Reactor Technology I

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FF usion Reactor Technology I usion Reactor Technology I

(459.760, 3 Credits) (459.760, 3 Credits)

P f D Y S N Prof. Dr. Yong-Su Na (32-206, Tel. 880-7204)

( , )

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Contents Contents

Week 1 Magnetic Confinement

Week 1. Magnetic Confinement

Week 2-3. Fusion Reactor Energetics

Week 4 Basic Tokamak Plasma Parameters Week 4. Basic Tokamak Plasma Parameters Week 5. Plasma Heating and Confinement Week 6 Plasma Equilibrium

Week 6. Plasma Equilibrium

Week 8. Particle Trapping in Magnetic Fields Week 9 10 Plasma Transport

Week 9-10. Plasma Transport

Week 11. Energy Losses from Tokamaks Week 12 The L and H modes

Week 12. The L- and H-modes Week 13-14. Tokamak Operation

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Week 1 Magnetic Confinement

Week 1. Magnetic Confinement

Week 2-3. Fusion Reactor Energetics

Week 4 Basic Tokamak Plasma Parameters Week 4. Basic Tokamak Plasma Parameters Week 5. Plasma Heating and Confinement Week 6 Plasma Equilibrium

Week 6. Plasma Equilibrium

Week 8. Particle Trapping in Magnetic Fields Week 9 10 Plasma Transport

Week 9-10. Plasma Transport

Week 11. Energy Losses from Tokamaks Week 12 The L and H modes

Week 12. The L- and H-modes Week 13-14. Tokamak Operation

3

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Objectives of the Tokamak

Objectives of the Tokamak OperationOperation Objectives of the Tokamak

Objectives of the Tokamak OperationOperation

Hi h /

• High <n_e>/n_GW

• High β_N

• High H₉₈(y,2)

• Pulse length

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I_p (MA) D_α

P_NBI (MW)

α

High <n >/n

4xl_i

• High <n_e>/n_GW

• High β_N

Hi h H ( 2)

β_N

H₉₈(y 2)

• High H₉₈(y,2)

• Pulse length

1 0

H₉₈(y,2)

<n_e>/n_GW 0.0

1.0 H₉₈(y,2)

<n_e>/n_GW

0.0 1.0

-1.0

Even n

Odd n

0 2 4 6 8

Time (s)

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Basic Tokamak Variables Basic Tokamak Variables Basic Tokamak Variables Basic Tokamak Variables

• Cylindrical and local coordinates for a tokamak

• Aspect ratio: R₀/a ~ 3-5 ex) KSTAR: 3.6, ITER: 3.1

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• Separation of plasma from wall by a limiter and a divertor

Strike point

• Advantage of the divertor configuration

- First contact with material surface at a distance from plasma boundary R d i th i fl f i i d i iti i t th i t i f th l

7

- Reducing the influx of ionized impurities into the interior of the plasma by diverting them into an outer „SOL“

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• Plasma equilibrium parameters

• Elongation: k

• Triangularity: d

• Squareness: ζ

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Elongation

• Elongation

• Triangularity

= b

κ δ = ^c ⁺ ^d

g y

= a

κ δ = 2a

9

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• Outer and inner squareness: ζζ_o,i_o,i What is the squreness?

) i i

(θ ¹δ θ

R

R ⁻

) 2 sin sin(

) sin sin

cos(

,

1 0

θ ζ

θ κ

θ δ

θ

i

a o

Z

a R

R

+

=

+ +

=

(11)

Parameters KSTAR ITER • Plasma shape

Major Radius, R₀ Minor Radius, a Plasma Current I

1.8 m 0.5 m 2 0 MA

6.2 m 2.0 m 15 MA Plasma Current, I_P

Elongation, κ_x Triangularity, δ_x T id l Fi ld B

2.0 MA 2.0 0.8 3 5 T

15 MA 1.85 0.5 5 3 T Toroidal Field, B₀

Pulse Length Fuel

3.5 T 300 s H, D

5.3 T 500 s D, T

11

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• Beta

plasma pressure magnetic

p

magnetic pressure

2 0 / 2μ p B β =

z The ratio of the plasma pressure to the magnetic field pressure

z A measure of the degree to which the magnetic field is holding f

a non-uniform plasma in equilibrium.

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• Beta

r B NI

π μ

= 2

2 0 / 2μ p B β =

z The power output for a given magnetic field and plasma assembly is proportional to the square of beta.

13

z In a reactor it should exceed 0.1 – economic constraint

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• Beta

2 a

Assuming that the magnetic surfaces have concentric, circular CXs and that conditions are independent of ϕ.

∫ ∫ ⁼ ∫

= ^a p r rdr

dS a pdS

p

0

2 ( )

/ 2

π π G

G

0 0

0

' ' ) ' (

, )

1 ( μ μ

μ

ϕ θ

r

dr r r j B

j rB

j B

∫

=

∂ =

∂

=

×

∇G G G

Ampère’s law

0 0

0

/ 2

2π _ϕ π _θ μ

ϕ ϕ

a a

p j rdr aB

I

r r

r

∫

=

∂

0

2 2 2 2

0 2

0 2 8

2 ,

p

t I

p a B

p B

p

μ π β μ

β = μ = =

0 p

a I

B

B_ϕ _θ μ

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• Normalized beta – stability limit

p t t

N I

β aB β =

• Fundamental elements for the β limit

for the β_N-limit 1. Current profile 2 Pressure profile 2. Pressure profile 3. Plasma shape 4. Stabilising wall

5. Resistive instability

15

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• Normalized beta – stability limit

• High β_N in KSTAR

1. Strong plasma shaping 2. Passive stabilizers

3. PF/CS system capability 4 High heating power

4. High heating power 5. RWM control coils

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• Safety factor q = number of toroidal orbits per poloidal orbit

ϕ

θ B

BK G +

∫

= ^r j r r dr B_θ μr⁰ _ϕ( ') ' '

r ₀

Magnetic field lines Magnetic flux surfacesg

number of toroidal windings b f l id l i di

=

q number of poloidal windings

q

17

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q-profile

Baseline scenario

q₉₅

5

Advanced scenario Baseline scenario

4

Hybrid scenario

3

1 2

0 0 5 1

1

0 0.5 1

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ϕ ε

) rB (

Large aspect ratio tokamak with a circular CX

• General definition

θ θ ϕ

ε B μ j r r dr

r s

s B

r R q

r 0 0

' ' ) ' ( )

(

=

≡

=

∫

B ds R q ⁼ ∫ B

θ ϕ 0

ϕ ϕ

ϕ

π ε

j I B

a q aB

dr r r rB j

s B R

p 2

0 0

2

) (

,

=

≡

≡ ∫ Integral is along a closed path

enclosing the minor axis and lying on a specific magnetic

θ 0

ϕ ϕ

ϕ θ

μ

π μ

q B j B

j a R

I B

q R

p a

a

0 0

2 0

0 0

, 2 2

,

=

Derive this!

y g p g

surface; thus q is a surface quantity.

ϕ

ϕ μ

μ

q j

R q j

q j R

a

a 0

0 0 0 0

0

0 ,

=

Homework

Current profile peakedness

jϕ

q₀ ^p ^p

19

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• Z-effective: a convenient measure of the extent to which the plasma contaminated to which the plasma contaminated

2 , ∑

∑ ⁼

= _s _s _e _s _s

eff

eZ n Z n n Z

n ∑ ∑ ^Zs : charge number for the s-type ion

s s

ff

• Z_eff = 1 in a pure hydrogen plasma

• Method to determine Z_eff

-Impurity concentration determined by analyzing resonance line

intensities in the vacuum UV supplemented by measurements of soft intensities in the vacuum UV, supplemented by measurements of soft X-ray spectra; this data, coupled with a theory for ionization rates - visible Bremsstrahlung radiation

- Spitzer’s formula for the parallel resistivity

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Basic Tokamak Variables Basic Tokamak Variables

Temperature

• Energy confinement time

Temperature

Time

21

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Temperature

1/e

τ_E

Time

• τ_E is a measure of how fast the plasma looses its energy.

• The loss rate is smallest, τ_E largest

if the fusion plasma is big and well insulated if the fusion plasma is big and well insulated.

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• Energy confinement time r rd T n T

n k pdS

W

a

i i e e

∫ B

∫ ⁼ ⁺

=

0

) 2 (

3 2

1

π ~ total thermal energy in the torus

p h

p u

L E

j Q

hv r r

r t

u

r

D + = − = =

∂ + ∂

∂

∂ )

2 5

2 , ( 3

)}

( 1 {

)

(ρ ρ ρ ρ

ϕ ϕ

total heat flux radiation energy loss rate

t W

r r t

R E E

E

−

=

∂ +

∂

*

1 1

) 1 (ln

2 2

τ τ

τ

internal

energy enthalpy density

r Lrd

W r

rd E j

W Q

pv r

W

a R a E

E r

D

E ∫ ∫

≡

⎥⎦

⎢ ⎤

⎣

⎡ ⎟

⎠

⎜ ⎞

⎝

⎛ +

≡ , ^* ,

2

5 τ τ

τ

ϕ

jϕ a

r ∫ ∫

⎥⎦

⎢⎣ ⎝ 2 ⎠ ₌ ₀ ^ϕ ^ϕ ₀

Energy

confinement Energy

replacement Radiation loss

time p

time time

23

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d d

power heating

applied

energy stored

*

=

≈

∂

− ∂

=

∂

− ∂

=

in in E

E P

W t

P W W t

W W

W τ

τ In steady conditions,

neglecting radiation loss, Ohmic heating replaced by

total input power

∂

E ∂t t

τ ^p ^p

• To predict the performance of future devices, the energy confinement time is one of the most important parameter.

• Since tokamak transport is anomalous, empirical scaling laws for energy confinement are necessary

for energy confinement are necessary.

• Empirical scaling laws: regression analysis from available experimental database. p

ακ αε

α α

α

α ε κ

τ_th^fit_,_E = CI ^I B ^BP ^Pn ⁿM ^M R ^R

in engineering variables

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78 0 58 0 97 1 19 0 41 0 69 0 15 0 93 0 )

2 (

98 ^y 0 0562 I B P M R

IPB −

78 . 0 58 . 0 97 . 1 19 . 0 41 . 0 69 . 0 15 . 0 93 . 0 )

2 (

98 ^y, 0.0562 I B P n M R a

IPB

th,E ε κ

τ = ⁻

i KSTAR d ITER?

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τ_E in KSTAR and ITER?

Why should ITER be large?

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• Particle confinement time )

( )

1 (

r S v

r rn r

t n

e D

e

e =

∂ + ∂

∂

∂ electron number density source

*

a a

p p =τ

τ In steady state

[ ⁰ ] ^, ^* ⁰

d S

r rd n v

rn

r rd n

a e p

D e

e

p ∫

∫

∫ ≡

≡ τ

τ [ ]

0

r rd S_e

a D r

e ⁼ ∫

particle

confinement particle replacement confinement

time replacement

time

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• Momentum confinement time

⋅

=

⋅ Π

⋅

∇

∂ + + ∂

∂

b

rv F

v r r

v r

t 1 ( )

)

(ρ _ϕ ρ _ϕ ϕ ϕ Momentum equation having

the toroidal component

=

H a

R H

H ² ² ²

*

2

ϕ

ϕ τ

τ In steady state

[ ] _∫ ^∫

∫

≡

=

⋅

≡ +

⋅ Π

⋅

∇

≡

=

a a

a b r r

a R a H H v rdr

r rd F

H v

v r r rd

H

0 2

2 0 2

* ,

torque Beam

2

, _ϕ ^ϕ ^ϕ _ϕ _ϕ

ϕ ϕ

ϕ π ρ

ϕ τ

ρ ϕ

τ

0 0

Toroidal momentum confinement

Toroidal momentum replacement

time p

time

27

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• Momentum confinement time

• Toroidal angular momentum confinement time of thermal particles during NBI versus simultaneously measured energy confinement time for steady state L-mode and ELMy H-mode discharges (crosses), and for for steady state L mode and ELMy H mode discharges (crosses), and for transient ELM free phase of hot ion H-mode discharges (squares) in JET

(29)

• Intrinsic rotation

J E Ri t l N l F i 47 1618 (2007)

29

J. E. Rice et al, Nucl. Fusion 47 1618 (2007)