Contents Contents

(1)

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)

( , )

(2)

Contents Contents

Week 1 Magnetic Confinement

Contents Contents

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

2

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Contents Contents

Week 1 Magnetic Confinement

Contents Contents

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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Heating and Current Drive Heating and Current Drive Heating and Current Drive Heating and Current Drive

http://iter.rma.ac.be/en/img/Heating.jpg

4

(5)

Ohmic

Ohmic HeatingHeating Ohmic

Ohmic HeatingHeating

Electric blanket

5

(6)

Ohmic

Φ

I_CS I_p

I_p L_pI_p + I_pR_p =V_p = −φ

V

I_CS V_p

http://en.wikibooks.org/w/index.php?title=File:Transformer.svg&filetimestamp=20070330220406 6

(7)

Ohmic

φ _p + _p _p = _p = −

pI I R V

L

Total change in magnetic flux needed to induce a final current

f p f i

p p t

ind l I

k R R

I L

f dt

⎥⎦

⎢ ⎤

⎣

⎡ ⎟+ −

⎠

⎜ ⎞

⎝

≈ ⎛

=

Δφ_ind ∫0 φ _p _p μ⁰ ⁰ ^ln ⁸ ⁰ ₂ ² _p k

a ⎥

⎢ ⎦

⎣ ⎝ ⎠

∫0 ⁰ ⁰ ₂

[¹^.⁶⁵ ⁰^.⁸⁹⁽ ¹⁾]

ln + ₉₅ −

≈ q

l_i internal inductance

4 . 0

0 ,

0 ≈

=

Δφ_res C_Eμ R I_p^f C_E

Additional magnetic flux needed to overcome resistive losses during start up Ejima coefficient

0

0 p E

E

res μ

φ

dt'

R

t I

f

b ⁼ ∫

Δφ

Further change in magnetic flux needed to maintain I_p after start up

0 I_p R_pdt

burn ⁼ ∫

Δφ

ΔB

Δφ ²

Technological limit to the maximum value of B_OH

T k k i i h tl l d d i

7 OH

v B

r Δ

≈

Δφ π ² Tokamak is inherently a pulsed device.

(8)

Ohmic

j2

P_Ω =η

Ohmic

Ohmic heating density

Neoclassical resistivity

1

2 2 1

⎟⎟

⎞

⎜⎜

⎛

⎟⎞

⎜⎛

−

=

r

s n

η η

: Spitzer resistivity

) 3 /

2 / (

/ 10

8 1

2 3 8

⎟⎟⎠

⎜⎜⎝

⎟⎠

⎜⎝

− Z T R

R ^η^s: Spitzer resistivity

) 3 /

, 2 / (

/

10

8× ⁸ ² = =

≈ Z_eff T_e r a R a

η

) ) / ( 1 ( )

(r = j₀ − r a ²

j ^v

⎥⎤

⎢⎡

⎟⎟⎞

⎜⎜⎛

−

=

2 +1 0

2

0 1 1

) (

r v

j r a

B μ

Ampère’s law

) 1 2

2 /(

0

2 = j v+

j ⎥⎥

⎢ ⎦

⎢

⎣ ⎜⎜⎝ ⎟⎟⎠

= + 1 1 ₂

) 1 (

) 2

(r v r a

B_θ Ampère s law

0 0 0

0 1, 2 /

/ ,

/ _θ _φ μ

φa B R q q v j B Rq

B

q_a = _φ _θ , q_a q₀ = + , j₀ = _φ q₀μ₀

q_a _a

⎟⎞

⎜⎛

⎟⎟⎠

⎜⎜ ⎞

⎝

= ⎛

2

0 2

1 2 1

R j B

μ

φ

8

⎟⎠

⎜ ⎞

⎝⎛ −

⎠

⎝ 0 0

0

2 q q

R q

a

μ

(9)

Ohmic

L

E P

nT j

P_Ω =η ² = 3 /τ =

Ohmic

) 2 / (

/ ) / ( 10

0 .

1 ⁵ ³^/² ²

2

o a

o

effT B R q q q

Z

j = × ⁻ _φ −

η

5 4 5

2

0

108

7 .

2 ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

× ⎛

= R

B q

nq T Z

a E

effτ φ

0 ⎠ ⎝ ⎠

⎝ q_aq

/ ) /

( ²⁰ ²

5 .

=1

Zeff q_aq_o =1.5 2 / ) 10 /

(n ²⁰ a²

E =

τ

4

Alcator scaling

5 4

87 .

0 B_φ T =

Average temperatures above ~ 7keV are necessary before alpha heating is

9

Average temperatures above 7keV are necessary before alpha heating is large enough to achieve a significant fusion rate

(10)

Ohmic

Plasma stability requires:

R j B a R

j a B I R

a B R

B a q B

p a

0 2

0 0

2 2 μ π

μ ^φ μ ^φ ^φ

θ

φ = = = >

=

j B

R j a

p 0 0

0

2 2

μ π

μ

φ θ

< R j

μ0

<

10

(11)

Neutral Beam Injection Neutral Beam Injection

Injection of a beam of neutral

Neutral Beam Injection Neutral Beam Injection

B fi ld fuel atoms (H, D, T)

at high energies (E_b > 50 keV)*

B-field

Ionisation in the plasma⇓

H,D,T

Beam particles confined⇓ Beam particles confined Collisional slowing down⇓

11

* E_b = 120 keV and 1 MeV for KSTAR and ITER, respectively

(12)

Neutral Beam Injection Neutral Beam Injection Neutral Beam Injection Neutral Beam Injection

Neutralizer

Extraction Vacuum valve

• Generation of a neutral fuel beam

Neutralizer

Ion source H

Extraction acceleration grids

Deflection t

Vacuum valve

Beam

H H₂

Low H⁺

magnet _duct

o H

temperature H plasma,

5 eV

H⁺ H⁺

B

H⁺, e^- H₂⁺ H

H⁺

100 kV -3 kV O kV Beam dump

Ex) W7-AS: V=50 kV, I=25 A, power deposited in plasma: 0.4 MW 12

(13)

• Ion source

13

(14)

• Ion acceleration

14

(15)

• Ion neutralization

15

(16)

• Ion deflection

16

(17)

• Neutral injection

17

(18)

• JET NBI System

18

(19)

• JET NBI System

19

(20)

• JET NBI System

JET with machine and Octant 4 Neutral Injector Box 20

(21)

• JET NBI System

Octant 4 Neutal Injector Box 21

(22)

• Ion sources

RF Source Duopigatron

0.4 - 0.8 m

Cathodes difficult to replace, finite life time 22

(23)

3 lens system

• Extraction and steering

• 3-lens system

potential lines 0.2m

H⁺ 85%

H 85%

H₂⁺12%

H₃⁺ 3% 0.5m

100 kV -3kV 0kV

Typically 800 extraction

Grid system at ASDEX Upgrade

23

holes per source

(24)

• Neutralizer

N N N N

fast slow fast gas

D D

D

D⁺ + ₂ → + ₂⁺

Charge exchange:

Re-ionization: ND + ND → N ND⁺ + D +e⁻

fast gas fast gas

2 2

Efficiency:

24

(25)

• Negative ion beam development in JT-60U

25

(26)

Charge exchange: H + H ⁺ → H ⁺ + H

• Ionisation in the plasma

Charge exchange: H _fast + H → H _fast + H

Ion collision: H _fast + H ⁺ → H ⁺_fast + H ⁺ + e⁻

Electron collision: H + e⁻ → H ⁺ + 2e⁻

Electron collision: H _fast + e → H _fast + 2e

Attenuation of a beam of neutral particles in a plasma

NBI

Andy Warhol

n: density

ti

beam energy

http://www.nasa.gov/mission_pages/galex/20070815/f.html 26

σ: cross section

(27)

Charge exchange: H + H ⁺ → H ⁺ + H

Charge exchange: H _fast + H → H _fast + H

Ion collision: H _fast + H ⁺ → H ⁺_fast + H ⁺ + e⁻

Electron collision: H + e⁻ → H ⁺ + 2e⁻

Electron collision: H _fast + e → H _fast + 2e

b x N

dN ( ) ( ) ( )

Ex) beam intensity: Î⁽^x⁾ ⁼ Î₀ ^⋅êxp(⁻ ^x ^/^λ)

tot b

b N x n x

dx x

dN ( ) ( ) ( )σ

−

=

) y ⁽ ⁾ ₀ ^p( )

E_b0 = 70 keV σ_tot = 5⋅10⁻²⁰m²

3

1020

5 ₁₀²⁰ ⁻³ 1 5⋅

= m

n m

n 1_tot ≈ 0.4

= σ λ

In large reactor plasmas

27

In large reactor plasmas, beam cannot reach core!

(28)

Charge exchange:

H H

H

H + ⁺ → ⁺ +

Charge exchange:

Ion collision:

Electron collision:

H H

H

H _fast + → _fast +

− +

+

+ → + +

+ H H H e H _fast _fast

− +

− → +

+ e H e

H 2

Electron collision:

b x N

dN ( ) ( ) ( )

+

→

+ e H e H _fast _fast 2

tot b

b N x n x

dx x

dN ( ) ( ) ( )σ

−

=

28

(29)

• Slowing down

⎞

⎛

81 4

/ 3

, 6

ln

2 ¹₂ ₂ ₂³ ¹₂

2 1 2

3 2 2

3

2 1 4 2 2

1

≈

⎟ =

⎟⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛ Λ +

−

= ^e ^b ^e ^b _b _e _i

b C Z M m m

W C T

W M

m e Z n dt

dW π

ε π

6π ²ε₀M^b ⎜⎝T^e² W^b² ⎟⎠

3 19

2

, C ≈

W_b _crit

Critical energy W : ^W ¹⁴ ⁸ ^A^b ^T [ ]^keV

3 ≈19

= C T_e

Critical energy W_b,crit: ^T [ ]^keV

W A _e

i b crit

b_, =14.8⋅ ₂ ₃ ⋅

The electron and ion heating rates are equal.g q

29

(30)

• Slowing down

2 1

/ 3 ⎟⎞

⎜⎛ ⎛ξ ⎞

ξ ₂

2 1

b b

b = m v

ξ

] [keVs

ˆ 1 10

71 .

1 ¹

2 / 3

18 −

− ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎠

⎜⎜ ⎞

⎝ +⎛

×

=

b c e

b b e

T A P n

ξ ξ ξ

Ion h

3 / 2

8 ˆ .

14 A_bT_e

ξ = ₂_/₃ ^heating f

) ( _i _i

c Z A

ξ

raction

e 30

b /Tˆ ξ

(31)

v

v⊥

v_b

• Slowing down

1. ξ_b > ξ_c: Slowing down on electrons no scatter

v_||

v_b

v_||

v⊥

v_b^b v_||

2. ξ_b < ξ_c: Slowing down on ions scattering of beams

1

Fraction of initial beam energy going to ions

31

0 ξ_b/ξ_c ¹⁰

(32)

• Injection angle

Radial (perpendicular, normal) B injection

Tangential injection injection

t d d t Radial injection: B

• standard ports

• shine-through

• particle loss

∇B

.

32

∇B

(33)

• Injection angle

Radial (perpendicular, normal) B injection

Tangential injection injection

KSTAR NB hi th h

33

KSTAR NB shine-through armor

(34)

• Injection angle

I

. B I_p

P j t d

Better heating efficiency Worse

heating efficiency

Projected particle

drifts

efficiency efficiency

34

At low magnetic fields heating efficiency depends on NBI direction.

(35)

ASDEX Upgrade

JET

35

(36)

Electromagnetic Waves Electromagnetic Waves Electromagnetic Waves Electromagnetic Waves

Tuning forkg

Resonance

36

(37)

Tacoma Narrows Bridge (1940 11 4)

37

(1940. 11. 4)

(38)

Microwave oven

http://blog.naver.com/rlhyuny27?Redirect=Log&logNo=30029307561 38

http://cafe.naver.com/nadobaker.cafe?iframe_url=/ArticleRead.nhn%3Farticleid=82

(39)

• Electron Cyclotron Resonance Heating (ECRH)

Linear / Circular polarization

39

polarization

http://ko.wikipedia.org/wiki/%ED%8E%B8%EA%B4%91

(40)

Electromagnetic Waves

Electromagnetic Waves _Resonance Electromagnetic Waves

Excitation of plasma wave

(f ) l d

Resonance zone (frequency ω) near plasma edge

Antenna

B m qB

c = ω ωc

ω =

R 40

ωc

ω

(41)

Electromagnetic Waves _Resonance Electromagnetic Waves

Excitation of plasma wave

(f ) l d

Resonance zone

(frequency ω) near plasma edge wave transports power ⇓

into the plasma center absorption near resonance, ⇓

e g ω ≈ ω

Antenna

e.g. ω ≈ ω_c ,

i.e. conversion of wave energy into kinetic energy of

resonant particles B

resonant particles

Resonant particles thermalise⇓

m qB

c = ω ωc

ω =

R 41

Resonant particles thermalise ω ω^c

(42)

Plasma Waves Plasma Waves Plasma Waves Plasma Waves

• Considering externally driven perturbations in the magnetic and electric fields and in the current, relative to an

0 2

1 t E j c

B ∂

+ ∂

=

×

∇

G G G μ

equilibrium condition for a cold plasma w/o external magnetic fields

t E B

t c

∂

− ∂

=

×

∇

G ∂

G Maxwell equation

2 2 0 2

2 1

) (

)

( t

E c

t E j

E

E ∂

− ∂

∂

− ∂

=

∇

−

⋅

∇

=

×

∇

×

∇

G G G

G

G μ

∑

≡

i

i ieu n j

G ⎞

⎛ ∂

e u

p B

u E e n u

t u n u

m

i

i i

G G

G G G G

G

∂

∇

−

× +

⎟ =

⎠

⎜ ⎞

⎝

⎛ + ⋅∇

∂

∂ ( ) ( ) Equations of motion:

isotropic pressure assumed

42

m E e t

u

i i =

∂

⇒ ∂

(43)

Plasma Waves Plasma Waves Plasma Waves Plasma Waves

Plane waves with space

d ti d d

E E e

E n

E ⁱ

G G G

G ⎟⎟⎞ − ∂

⎜⎜⎛

−

=

∇

−

⋅

∇

∇⁽ ⁾ ² ^μ⁰ ∑ ² ¹ ² and time dependences

e E E n

E k E

k k

t E c

E m E

i

i i

G G

G G G

G

⎟⎟⎞

⎜⎜⎛

−

= +

⋅

−

⎟⎟ ∂

⎜⎜ ⎠

∇ ⎝

∇

∑

2 2 0

2 2

2 0 2

) (

ω μ μ

[

⁽ ⁾

]

exp i kG⋅r −ωt m E

c E E

k E

k k

i i ⎟⎟⎠

⎜⎜⎝

+ ² ⁰ ∑

)

( μ

frequency :

vector wave

: ω

kG

B₀

k k_⊥

0

External magnetic field

k_||

43

(44)

Plasma Waves Plasma Waves

Plane waves with space

d ti d d

E E e

E n

E ⁱ

G G G

G ⎟⎟⎞ − ∂

⎜⎜⎛

−

=

∇

−

⋅

∇

∇⁽ ⁾ ² ^μ⁰ ∑ ² ¹ ²

and time dependences

e E E n

E k E

k k

t E c

E m E

i

i i

G G

G G G

G

⎟⎟⎞

⎜⎜⎛

−

= +

⋅

−

⎟⎟ ∂

⎜⎜ ⎠

∇ ⎝

∇

∑

2 2 0

2 2

2 0 2

) (

ω μ μ

[

⁽ ⁾

]

exp i kG⋅r −ωt m E

c E E

k E

k k

i i ⎟⎟⎠

⎜⎜⎝

+ ² ⁰ ∑

)

( μ

E kG G

|| :frequency

vector wave

: ω

kG E

k ||

2 2

0

2 1

p

i i

i

m e

n ω

ω = ε ∑ ≡ Plasma frequency

•

E kG G

⊥

2 2

•

2 2

k p

c ω

ω = +

44

(45)

• Waves applied to the edge of a plasma tend to be “shielded”

t f th hi hl d ti l

(^η ^μ_o) ^B

E t

B /∂ = −∇× = / ∇²

∂ G G G

out of the highly conducting plasma.

( )

ω μ η δ

μ η

δ

o x

o

e B

BG ≈ K ⁻ ^/ , ≈ /

• Externally applied RF waves can penetrate the plasma only by coupling to natural waves in the plasma determined by the plasma dielect ic tenso

E K c

EG I G

⋅

=

×

∇

×

∇ ( ) (ω / )²

the plasma dielectric tensor.

) (

Dielectric tensor

45

Determinant = 0 Disperison relation D(ω, k) = 0

(46)

Dispersion Relation Dispersion Relation Dispersion Relation Dispersion Relation

Plasma waves are solutions of disperison relation D(ω, k)=0.

ll generally:

ω given by generator k_||given by antenna where k >k

solution: k_⊥ = k_⊥ (ω, k_||) where k_||>k_||,Vacuum

Special cases:p

1 k → 0

Perpendicular to magnetic field

cutoff“ ^reflection

1. k_⊥→ 0 „cutoff _tunnelling

http://www.srh.noaa.gov/srh/jetstream/atmos/ionosphere_max.htm

2. k_⊥→ ∞ „resonance“ mode converison

absorption

46

(47)

• Ion Cyclotron Resonance Heating (ICRH):

30 MHz 120 MHz ( 10 m) ω ~ ω_ci, 30 MHz – 120 MHz (~ 10 m)

eB z m

n m +n

ω

ω (1 / )

2

Ion-ion resonance frequency

Lower Hybrid (LH) Resonance Heating:

i i ci

c c

ii m

eB z Z

n Z n Z

m Z

m

m n m

n =

+

= ω ω + ω

ω ,

) /

/ (

) /

1 (

1 1 2 2 2

1 1 2

1 1 2 2 2

2 1

• Lower Hybrid (LH) Resonance Heating:

ω_ci ^<ω ^<ω_ce, 1 GHz – 8 GHz (~ 10 cm)

• Electron Cyclotron Resonance Heating (ECRH):

2 2

2

2 _pi /(1 _pi / _ce), _pi _ci

LH ω ω ω ω ω

ω ≈ + >>

Electron Cyclotron Resonance Heating (ECRH):

ω ^~ω_ce, 100 GHz – 200 GHz (~ mm)

2 2

2

47 2

2 2

ce pe

UH ω ω

ω ≈ +

(48)

ICRH

ICRH –– Wave PropagationWave Propagation ICRH

ICRH Wave PropagationWave Propagation

Loci of cut off and resonances in the poloidal cross section of a tokamak. The fast wave is

evanescent in the hatched region.

48

(49)

ICRH

ASDEX Upgrade Alcator C-Mod

Re(E_y)

Multiple current straps

49

(50)

ICRH

50

(51)

ICRH

ICRH –– Minority HeatingMinority Heating

Plasma ith mi t e of H and D

ICRH

ICRH Minority HeatingMinority Heating

• Plasma with mixture of H and D with n_H << n_D

n_D_D →→ polarizationpo a at o propagation n_H → absorption

B e⋅B

• Production of tail in H velocity distribution function

• Good single pass absorption

H

H m

B

= e

= ω ω

B e⋅

=

=ω ω

g p p

51 H

D = ⋅m

=ω 2 ω

(52)

Ion Cyclotron Resonance Heating Ion Cyclotron Resonance Heating

• JET ICRH System

Ion Cyclotron Resonance Heating Ion Cyclotron Resonance Heating

52

(53)

Ion Cyclotron Resonance Heating Ion Cyclotron Resonance Heating Ion Cyclotron Resonance Heating Ion Cyclotron Resonance Heating

• JET ICRH System

- 8 x 4 MW RF generators, each one has two 2 MW outputs giving a possible 32 MW total.

- Frequency range 23 MHz -57 MHz q y g (excluding 39-41MHz)

- 8 HVDC Power supplies.

- 4 x 4 strap antenna system with - 4 x 4 strap antenna system with

vacuum interspace, Getter pumping and Penning gauge protection.

53

(54)

Lower Hybrid Heating Lower Hybrid Heating Lower Hybrid Heating Lower Hybrid Heating

• JET LH System

54