introduction to inertial confinement fusion

INTRODUCTION TO INERTIAL CONFINEMENT FUSION

R. Betti

Lecture 6

Ignition of a simple DT plasma

Recap from previous lecture • Quasi-static approximation of a stagnating plasma where the alpha heating starts at stagnation leads to an ignition condition known as the Lawson ignition criterion • Minimum Pτ for ignition is

• This is called Lawson triple product

• A better ignition model uses a dynamic evolution of a thin and dense cold shell/piston compressing a hot “hot spot” plasma

MHS THS VHS

Hot spot

Cold fuel (piston)

Vf

124 /

nono

PSα

αα

τχε

≡ >2 at about 8keVv

STσ

≡

[ ] 24no ignition

PSα

α

τε

≡

[ ] 61.1 10 11 11no ignitionP Pa s atm s Gbar nsατ ≡ × ≈ ≈

2P nTτ τ=

Hot spot ignition: dynamic model

The isobaric approximation

for the hot spot

Hot spot

Cold fuel (piston)

α ν

Vi

• The shell has been accelerated (by the laser) to a peak implosion velocity.

• The shell is compressing the hot spot and raising its pressure by converting its kinetic energy into hot spot internal energy. • The hot spot is gaining energy from the shell and from alpha heating while loosing energy through radiation emission and loosing heat through heat conduction

Mshell

• The simplest shell model is the one of a thin dense shell (high aspect ratio) whose motion (after being accelerated by the laser) is governed by Newton’s law.

Hot spot

R(t)

24shM R PRπ=

*(0)R R=(0) iR V= −

• t=0 is now the time of peak implosion velocity or beginning of the coasting/deceleration phase

• P(t) is the hot spot pressure pushing on the shell surface 4πR2 P

From previous lecture

• Need to couple Newton’s law for the shell to the energy equation and momentum equation of the hot spot because there are two unknowns: P(r,t) and R(t). Note that in general the pressure inside the hot spot is a function

of radius and time

• Start with the hot spot momentum equation (Euler equation)

U U PUt r r

ρ ∂ ∂ ∂ + = − ∂ ∂ ∂

• Scale each term. Here ↔ means “compare”

U U PUt r r

ρ + ⇔ −

• Time scale: t ~ R/Vi; velocity: U ~ Vi; Spatial scale r ~ R

2 2i iV V PR R R

ρ

+ ⇔


2 2 22

2~ ~/ /i i i

i th

V V VMachP T mρ υ

=

• Define the Mach number of the hot spot

• Therefore the LHS is of order Mach2 with respect to the RHS

22~iV PMach

R Rρ

• Hot spot is multi keV and the thermal velocity of its plasma is >> implosion velocity of the shell, therefore Mach<1: SUBSONIC HOT SPOT

22

2~ 1i

th

VMachυ

<<

• Therefore neglect LHS of momentum equation leads to P=P(t): ISOBARIC HOT SPOT

0Pr

∂≈

∂( )P P t≈


The dynamic energy balance for the hot spot

( )HSHS rad

e U e P T q qt ακ∂ = ∇• − + + ∇ + − ∂

• Start with the exact specific energy equation (energy per unit volume) for an ideal plasma

23 12 2HSe P Uρ= + Total specific energy: internal + kinetic

Rate of change Flux of enthalpy Heat flux Alpha heating

Radiation losses

• Note the pdV work is not fully into this equation. Since the pdV work acts on the surface of the plasma, its contribution comes when the above equation is integrated over the hot spot volume

25 12 2HS HSh e P P Uρ≡ + = + Enthalpy for unit volume

• Using the results of previous section: isobaric approximation and subsonic hot spot flow leads to

22 23 1 3 3( ) 1 ( ) 1 ( )

2 2 2 3 / 2HSUe P U P t P t O MachP

ρρ

= + ≈ + ≈ +

Neglect for Mach<<1

2 25 1 5 ( ) 1 ( )2 2 2HSh P U P t O Machρ = + ≈ +

• Use the bremmstrahlung formula for DT (Z=1) derived in Lecture 3

2rad bq C n T=

• Combine with alpha heating

22

2 3/24 4 4b

rad b

v CPq q n v C TT T

α αα

σε εσ − = − = −

2 ( )radq q P Tα − = Π 2 3/2

1( )4 4

bv CTT T

α σε Π ≡ −

• Simplified energy equation for subsonic/isobaric hot spots. Valid in 3D

23 ( ) 5 ( ) ( ) ( )2 2

dP t U P t T P t Tdt

κ ≈ ∇• − + ∇ + Π

• Integrate over volume V enclosed by a surface S and apply Gauss theorem

23 ( ) 5 ( ) ( ) ( )2 2S V

dP tV U n P t n T dS P t T dVdt

κ ≈ − + ∇ + Π ∫ ∫

• Consider following simple model of hot spot hydrodynamic profiles

P(t)

ρ

T

High density shell

Low density hot spot

• Note while the flat pressure profile was derived from the subsonic flow approximation, the flat profiles of T and ρ are not derived. For now, they are

simply assumed. IMPORTANT: the product of the profiles T×ρ must be flat to satisfy the ideal gas EOS inside the hot spot

r R(t)

2( ) 2i

P t nT Tm

ρ= =

• The continuity of the pressure across the hot spot-shell interface implies that the shell temperature must be a lot less than the hot spot temperature (cold shell)

HS HS Sh ShT Tρ ρ≈ HSSh HS HS

Sh

T T Tρρ

≈ <<

• Since the thermal conductivity of a plasma is proportional to T5/2 ,the shell behaves like a thermal insulator when compared to the hot spot

5/2 5/2

( ) ( ) ( ) ( )Sh HSSpitzer Sh Spitzer HS Spitzer HS Spitzer HS

HS sh

TT T T TT

ρκ κ κ κρ

≈ ≈ <<

• This implies we can neglect the heat flux inside the shell

• Consider mass conservation equation

( ) 0Utρ ρ∂+∇ =

∂

Hot spot R+

ρ

R-

• Integrate the equation through the volume ∆V across the hot spot-shell interface. Note ∆V can be any volume element across that interface

• Bring volume integral inside time derivative ∆V

∆S+

∆S-

0S

SV

dV U n dSdtρ ρ

+

−

∆

∆∆

∂ + = ∫ ∫

0S S

VS SV

dV U n dS U n dSdt

ρ ρ ρ+ +

− −

∆ ∆

∆ ∆∆

∂ − + = ∫ ∫ ∫

( ) 0S

VSV

dV U U n dSdt

ρ ρ+

−

∆

∆∆

∂ + − = ∫ ∫

• Combine integrals

n

• Take limit of ∆V0

HS b Sh aV Vρ ρ=

• Valid for arbitrary ∆S, therefore the equality below applies to every point on the surface and implies that the mass flow across the interface is continuous (NOT SURPRISING!)

• Since S- is inside the hot spot and S+ is inside the shell then and n is radial in 1D

( ) 0S

VSU U n dSρ

+

−

∆

∆ − = ∫

( ) ( )V VS SU U n U U nρ ρ

− + − = −

b V HSV U U≡ −

a V ShV U U≡ −

• Note that Va and Vb are flow velocities relative to the interface moving with UV.

Hotspot-shell interface Frame of Reference.

Hot spot

Shell aV

bVFlow across hotspot-shell Interface

HSa b b

Sh

V V Vρρ

= << HS Shρ ρ<<

• Note that Va and Vb are flow velocities relative to the interface moving with UV.


Hot spot

Shell aV

bVFlow across hotspot-shell Interface

HSa b b

Sh

V V Vρρ

= <<

HS Shρ ρ<<

• Continuity of the mass flow across the interface and the fact that the shell density is much greater than the hot spot density leads to a relative flow much faster in the hot spot than in the shell (If the flow exists) Here relative means relative to the interface.

HS b Sh aV Vρ ρ=

• But does mass exchange really occur between shell and hot spot or both Va and Vb are zero? More later

Hot spot

R+

P

• Consider the integration of the energy equation up to a surface S+ and radius R+ just inside the shell so we can neglect the heat flux

23 ( ) 5 ( ) ( ) ( )2 2 VS

dP tV U n P t n T dS P t T dVdt

κ+

≈ − + ∇ + Π ∫ ∫

neglect

23 ( ) 5 ( ) ( ) ( )2 2 S V

dP tV P t U n dS P t T dVdt + +

+ ≈ − + Π ∫ ∫

• Valid in 3D

VS

dV U n dSdt +

+

= ∫

• Use the general volume evolution equation. UV is the velocity at which the volume is changing

( )VS S

dVU n dS U U n dSdt+ +

+

= − + ∫ ∫

• Rewrite:

( )23 ( ) 5 5( ) ( ) ( ) ( )2 2 2 V

V S

dP t dVV P t P t T dV P t U U n dSdt dt + +

++ ≈ − + Π + − ∫ ∫

• Rewrite energy equation:

• Combine terms (and drop superscript + from V but not from S):

( ) ( )5/3 22/3

3 1 5( ) ( ) ( )2 2 V

V S

d PV P t T dV P t U U n dSV dt +

≈ Π + − ∫ ∫

This term represents the flow Va across the surface S+ inside the shell just outside the hot spot

( ) [ ]5/3 22/3

3 1 5( ) ( ) ( )2 2 a

V S

d PV P t T dV P t V dSV dt +

≈ Π + ∫ ∫

HSa b

Sh

V Vρρ

=• From two previous slides:

( )5/3 22/3

3 1 5( ) ( ) ( )2 2

HSb

ShV S

d PV P t T dV P t V dSV dt

ρρ+

≈ Π + ∫ ∫

• Substitute into energy equation and take limit for HS Shρ ρ<<

neglect

( )5/3 22/3

3 1 ( ) ( )2 V

d PV P t T dVV dt

≈ Π∫

• Energy equation for isobaric hot spot surrounded by a shell with “infinite” density

Ablation off the inner shell surface

• To estimate the flow across S+, let’s take the integral of the energy equation over ∆V across interface i.e. from just inside to just outside the hot spot

3 ( ) 5 ( ) ( ) ( )2 2

S

S

dP t V U n P t n T dS P t T Vdt

κ∆ +

∆ −

∆ ≈ − + ∇ + Π ∆ ∫

• In the limit of ∆V0 we find (this limit is taken by setting ∆V= ∆Sdr and taking dr0)

5 5( ) ( )2 2S S

U n P t n T U n P t n Tκ κ∆ + ∆ −

− + ∇ = − + ∇

Neglect this because inside the shell where κ0

( ) [ ]5 ( )2 S

U U n P t n Tκ− +−

− = ∇

Heat flux leaving the hot spot <0 because ∇T negative at R-.

Therefore flow is inward

Enthalpy flux through hot spot boundary

• For arbitrary ∆S, heat losses from hot spot compensated by enthalpy flux

Hot spot R+

ρ

R-

∆V ∆S+

∆S-

ndr

• Replace from before (relative flow is inward, i.e. negative er)

( ) ( )b aU U n V V− +− = − −

[ ]5( ) ( )2b a S

V V P t n Tκ−

− − = ∇

• Rewrite previous equation. Heat losses are compensated by enthalpy flux across the interface

HSa b b

Sh

V V Vρρ

= <<• Since

[ ]25 ( )

HS Sa

Sh

n TV

P tκρ

ρ−

∇≈ −

[ ]25 ( )

Sb

n TV

P tκ

−∇

≈ −


Hot spot

bV

Flow across hotspot-shell interface in the frame of reference of interface

aV

Shell

In the shell Frame of Reference the hot spot boundary penetrates with velocity Va

aVShell

Hot spot

• Va is the penetration velocity of the hot spot boundary into the shell, also called ABLATION VELOCITY

Flow in the frame of reference of the inside of the shell


Hot spot

bVaV

Shell

• Vb is the velocity at which the ablated shell plasma blows off into the hot spot, also called the BLOW-OFF VELOCITY

• Ablation (and blow off) of the shell into the hot spot is driven by the heat flux from the hot spot

Heat flux from hot spot

• Heat flows out of the hot spot following the temperature gradient (high T inside hot spot and low T inside the shell)

P(t)

ρ

T High density shell


r

heat flux out of hot spot= dTdr

κ−

5enthalpy flux into hot spot2 bPV= No heat goes through shell

(cold shell = thermal insulator)

P(t)

ρ

T


heat flux out of hot spot= dTdr

κ−

5enthalpy flux into hot spot2 bPV=

Locating hotspot-shell interface at R+ inside cold shell where there is not heat flux makes the hot spot ADIABATIC (in the absence of alpha heating and radiation losses)

R+

( )5/3 22/3

3 1 ( ) ( )2 V

d PV P t T dVV dt

≈ Π∫• If RHS = 0 (no alpha heating and no radiation losses) then

( )5/3 0d PVdt

≈

ADIABATIC here means there is no heat flux across the boundary

5/3PV const≈

• Important difference: an Adiabatic hot spot does NOT mean that the plasma inside the hot spot has constant Entropy

Adiabatic isentropic≠• Entropy of a plasma (or gas) is a local quantity related to a fluid element. Below

is the entropy per unit mass. The constant cp is the specific heat.

5/3logpPs cρ

=

• In a closed system, mass M is conserved and 5/3

5/3 5/3

P PVMρ

=

Therefore if M is constant and PV5/3 is constant then Entropy is constant. This however does NOT apply to the hot spot which is an open system receiving mass from the shell through ablation while losing heat. In the hot spot M is not constant

MV

ρ =

High temperature limit and the simplest dynamic model

b

qqα

( )T keV 4.4keV

Alpha Heating dominates for T>4.4keV

Radiation Losses dominates for T<4.4keV

• It is safe to state that if the hot spot is heated by compression to temperatures of about 6keV or higher then the alpha heating greatly exceeds the radiation losses and the rad losses can be neglected

2 3/2

1( )4 4

bv CTT T

α σε Π ≡ −

2 ( )radq q P Tα − = Π

• From before

Neglect rad losses

2

vTσ

( )T keV

• If we consider temperatures in the 8 to 23 keV range, then <σv>≈Sf T2

• In this high temperature case the energy equation is only dependent on pressure and volume without temperature dependence

( )5/3 22/3

3 1 ( ) ( )2 V

d PV P t T dVV dt

≈ Π∫ ( )16 fT SαεΠ ≈

( )5/3 22/3

1 ( ) ( )24 f

d PV S P t V tV dt

αε≈ 343

V Rπ=In 1D

22

2 4shd RM R Pdt

π=

• In 1D, couple energy equation to Newton’s law for thin shell (shell momentum equation)

( )5 2 5

24 fd PR S P Rdt

αε≈

22

2 4shd RM R Pdt

π=

*(0)R R= (0) idR Vdt

= −

• Simplest DYNAMIC model of a thin shell compressing the hot spot in the high temperature limit where alpha heating exceeds rad losses and <σv>∼T2

• Two equations for the two unknowns R(t) and P(t)

• Initial conditions at the time t=0 of peak implosion velocity (Vi). Starting from that time the shell stops being accelerated and starts coasting and decelerating. The radius at the time of peak velocity is denoted as R*

introduction to inertial confinement fusion

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