Fusion Reactor Technology I

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F usion Reactor Technology I

(459.760, 3 Credits)

Prof. Dr. Yong-Su Na

(32-206, Tel. 880-7204)

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Week 1. Magnetic Confinement

Week 2. Fusion Reactor Energetics (Harms 2, 7.1-7.5)

Week 3. How to Build a Tokamak (Dendy 17 by T. N. Todd) Week 4. Tokamak Operation (I): Startup

Week 5. Tokamak Operation (II):

Basic Tokamak Plasma Parameters (Wood 1.2, 1.3) Week 7-8. Tokamak Operation (III): Tokamak Operation Mode Week 9-10. Tokamak Operation Limits (I):

Plasma Instabilities (Kadomtsev 6, 7, Wood 6) Week 11-12. Tokamak Operation Limits (II):

Plasma Transport (Kadomtsev 8, 9, Wood 3, 4) Week 13. Heating and Current Drive (Kadomtsev 10)

Week 14. Divertor and Plasma-Wall Interaction

Tokamak Equilibrium

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•

- 1/R force

0 2 2 2

2 2

π a B μ F

_NET

=

II II I

I

II I

A B A

B

A A

B B

2 2

,

>

<

>

= φ φ

- Hoop force

0 2

2

) / 2

(

~

_R _I _I _II _II

μ

NET

e B A B A

F −

II II I

I

II I

A B A B

A A B

B

2 2

,

>

<

>

=φ

φ

Tokamak Equilibrium

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7

•

- External coils required to provide the force balance

v p

v

BIL R I B

F = = 2 π

₀

B

v

J G G

×

X

●

X

X X

X

Force PF coils

q B q

B B

B

v v

2

: : 1 :

: ε ε

θ φ

≈

>

Tokamak Equilibrium

(8)

•

The Shafranov Shift

- outward shift of the flux surfaces - consequences of toroidicity

Tokamak Equilibrium

β_p = 0.2 β_p = 1.2 β_p = 3.2

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⎭ ⎬

⎫

⎩ ⎨

⎧ ⎟⎟ ⎠ +

⎜⎜ ⎞

⎝

⎛ −

⎟ ⎠

⎜ ⎞

⎝

⎛ + −

Δ =

a b b

a l

R b b

i p

a 1 ln

2 1 2

2 ²

2

0

β

•

The Shafranov Shift

- outward shift of the flux surfaces - consequences of toroidicity

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ −

−

⎟⎟ +

⎠

⎜⎜ ⎞

⎝

⎛ − Δ ∝

Δ ∝

<<

Δ

2 2 2

2 )

2 (

) 1 (

0

2 1 ln 1

2 1

1

~

b a a

b b

a l

b b

R b b

i a

p a

a

β

ε

small shift outward shift due to

the tire tube force and the 1/R

internal field external field hoop force

outward shift due to the hoop

force

Tokamak Equilibrium

9

(10)

- the new boundary condition including the vertical field

•

Toroidal Force Balance by Means of a Vertical Field

) ln (

2 1 1 2

2 ²

2

0 B b

B a

b b

a l

R b b

v i

p a

θ

β −

⎭ ⎬

⎫

⎩ ⎨

⎧ ⎟⎟ +

⎠

⎜⎜ ⎞

⎝

⎛ −

⎟ ⎠

⎜ ⎞

⎝

⎛ + −

Δ =

- new shafranov shift

⎭ ⎬

⎫

⎩ ⎨

⎧ + − +

=

a

R l

R

B_v I _p ⁱ ⁰

0

0 8

2 ln 3 2

4

β

π μ

- How much vertical field do we need to keep the plasma centered?

Tokamak Equilibrium

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11

• Functions

- To produce the magnetising flux

- To produce the main equilibrium field, shaping fields including divertor configuration

- Fast position feedback

• Iron core Discussion Time (5 min) - Simpler magnetising system

- Difficult stray fields and equilibrium modelling

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Poloidal Field Coils

• Acting forces

- Principal forces: self- (hoop) and vertical and radial forces arising from other PF coils and the plasma current

- Most highly stressed coil: OH solenoid due to the requirement of the large volt-seconds swing

• Magnetising winding

- Providing the flux swing necessary to produce and sustain the plasma current for the desired pulse duration

- The volt-second consumption (empirical)

loop pulse

V t

IL +

≈

ΔΦ 1 . 5

for small machines

loop pulse

V t

IL + +

≈

ΔΦ 2

for large machines

- Total inductance of the plasma loop (L)

Internal inductance + External inductance to the plasma CX

( )

[ l R a ] R

R L

L

L =

_int

+

_ext

≈ μ

₀ _i

/ 2 + ln 1 . 3 / κ

¹^/²

≈ 2

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13

Poloidal Field Coils

• Flux swing with an iron-cored transformer - Simple transformer design:

very small net ampere-turns required for magnetising the iron good coupling between the primary and the plasma

(total primary ampere-turns ~ plasma current)

- The primary windings can be placed almost anywhere.

- Soft iron saturates at around 2 T

→ required cross-sectional area of the core ≈ ΔФ/4 (m²), assuming a bidirectional flux swing

- Introducing toroidal asymmetries including RMPs

- Loss of equilibrium when saturated and sharp increase of the stray fields

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Poloidal Field Coils

• Flux swing with an air-cored transformer

- Overcoming disadvantages of iron-cored transformer

- Extremely poor coupling generally → large primary ampere-turns, and strong constraints on the primary winding distribution to avoid the generation of stray fields in the plasma

- The volt-seconds produced by a simple long solenoid (bidirectional swing)

sol sol sol

sol

Jr r f

r

B

_max

π

²

8 π

² ²

δ

2 =

=

ΔΦ

^r_δr^sol_sol^{: radius}: thickness

f_sol: packing fraction

- The average hoop stress

sol sol

sol

r f

r

J δ

π σ = 20

²

σ_max ≈ 30 MPa for OFHC Cu, 200 MPa for special alloys due to the fatigue-failure limit for the envisaged life

(

³

)

¹^/²

10 )

( σ ≈ r δ r f σ ΔΦ

- The stress-limited flux swing

( / 20 r r f )

¹^/²

J = σ π δ

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15

Poloidal Field Coils

Heating rate for an epoxy-resin insulated OFHC copper solenoid

2 / 1

2

/

1600 )

( θ ≈ r

_sol

δ r

_sol

f

_sol

t

_pulse

ΔΦ

s dt

J

²

≤ 133 kA

²

cm

⁻⁴

∫

2 / 1 max

( ) 20 / t

_pulse

J θ ≤

3

2

/

max 2

pulse

t J dt

J =

∫

For a triangular current waveform

- Juggling with the r_sol and δr_sol to obtain the desired volt-seconds swing without breaking or overheating the magnetising solenoid

sol sol

sol

r f

r

J δ

π σ = 20

²

(

max

)

¹^/²

10

3

)

( σ ≈ r

_sol

δ r

_sol

f

_sol

σ

ΔΦ

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• Vertical field requirement

( )

[ ] ( R a )

R l I

a R R

B

_v

I

_p _i

ln 6 /

2 10 / 3 2 / /

8

10

₀

ln

⁰

+ + − ≈

₀ ⁰

≈ β

coil coil

v

I R

B ≈ / 1 . 1

Vertical field produced by a Helmholtz coil pair What is a Helmholtz coil pair?

a device for producing a region of nearly uniform magnetic field.

It is named in honor of the German physicist Hermann von Helmholtz.

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17

Poloidal Field Coils

• Vertical field requirement

- Minimum cross-sectional area for this coil if constructed with epoxy-resin insulated copper

( )

[ ] ( R a )

R l I

a R R

B

_v

I

_p _i

ln 6 /

2 10 / 3 2 / /

8

10

₀

ln

⁰

+ + − ≈

₀ ⁰

≈ β

coil coil

v

I R

B ≈ / 1 . 1

( R a )

R I

I

_coil

0 . 11 R

^coil

ln 6

₀

/

0

≈

( )

_coil

coil pulse

coil

t R I R a R f

A ≥ 10

¹^/²

ln 6

₀

/ /

₀

Vertical field produced by a Helmholtz coil pair

2 / 1 max

( ) 20 / t

_pulse

J θ ≤

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• Flux swing with an iron-cored transformer - The primary voltage

pri pri loop

t

pri

N V I R

V = +

t

pri

I N

I = /

( ) I L ( 1 ~ 2 ) 2 I R

₀

V

_loop

=

_resistive

+ ≈ +

N_t: number of primary turns

• Power supply

sol sol sol

tot

N r l

L ≈ 1 . 5 × 3 . 9

² ²

2 )

(

5 r f r l J

P

_resistive _sol

≈ π

_sol _sol

δ

_sol _sol

( )

_v ² _coil _coil _coil

/ 400

resistive

B J f A R

P ≈

( ) ⎟⎟

⎠

⎜⎜ ⎞

⎝

≈ ⎛

coil coil coil

coil coil v

resistive

a R R

I I B

P 1 . 1

ln

3

for flattop

: including the ramp-up phase

- Feedback systems: relatively lower power but fast, typically based on thyrister choppers or linear amplifiers of 10-500 kW

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19

Poloidal Field Coils

• Alignment

- The PFCs have to be circular and well aligned to the TFC to avoid producing RMPs and possible islands.

- The PFCs also have to be positioned in radius and height so as to minimise stray perpendicular fields.

- The required positional tolerance is ~ 10^-3 of the major radius of the machine for each type of error.

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Support Structure

- Need to accommodate the toppling forces on the TFCs and the vertical forces on the PFCs

- Responsible for maintaining the alignment of all the TFCs and PFCs - Stainless steel commonly used to obtain high strength with low

magnetic permeability

Cf. Permeability of stainless steel increases where worked, cut or welded, and so one (sometimes even after heat treatment).

→ generating nonaxisymmetric and potentially RMPs - Any volume of unsaturated magnetic material creates a

disturbance in the field breaking symmetry and likely to generate islands.

- Critical volume to generate 10^-4 of the PF at the plasma edge

- Critical volume to generate 10^-4 of the PF at the machine centre

( 1 )

/

300

⁶ ₀²

−

≤

_R

c

R R a

V μ

( 1 )

/

250

²

−

≤ IRR aB

V μ

R: range of the offending volume from the machine centre

Fusion Reactor Technology I