5. lekce

Transkript

5. lekce
```Fyzikainerciálnífúze
5.lekce-Zisk
•
Začneme s předpokladem, že konfigurace paliva v čase zapálení se skládá z
centrálního sférického hot spotu obklopeného slupkou z velmi hustého
paliva. Obě oblasti paliva (hot spot i slupka) jsou v klidu a mají rovnoměrnou
hustotu, teplotu i tlak.
As
in
Section
4.2
V izobarickém případě je navíc v obou oblastech stejný tlak, p = ph = pc.
fuel configuration
central hot spot su
fuel regions are at
p
sure. For reference
Th
in Fig. 5.1. In the
pressure, p = ph =
Tc
tions, such as the i
throughout the fuel
!c
!h
Our presentation
(Meyer-ter-Vehn 19
r
Rh
Rf
model was later e
Fig. 5.1 The isobaric ignition
configuration with radial profiles of
configurations with
T, !, P
•
Energetickýzisk-hotspotmodel
Hot spot ignit
5.1
pressure, temperature, and density at the
time of implosion stagnation. The suffix h
Target
energy
gain
fus
DT f
" fraction.
#\$F
! " is the burn
on.
The
suffix
time
of implosion
stagnation. The suffix h
he
central
hothspot,
=
, burnt and
G=
3
3
3
hotquantities
spot,
E
E
refers
to
hot
spot,
d
d
R
=
M
+
M
=
(4π/3)
ρ
R
+
ρ
−
R
M
sentral
to those
of the of the central
5.1.1 Target gain, fuelfgain, coupling
ηhbe hwritten
h
c efficiency
c asf
mass
can
h
those
while of
thethe
index c refers5.1.1
to those Target
of the
gain,5.1.1
fuel gain,
coupling
efficiency
η
Target
gain, fuel
gain, coupling
efficiency η
!
11
surrounding cold fuel.
The target energywhere
gain isqDT
defined
as × 10 J/g is the fusion energy released pe
= 3.37
Mhburn
+ Mfraction
ρha
M
The
target
energy
gain
is fuel
defined
as f =the
The target energy gain is and,
defined
as
c = (4π/3)
provided
the
ignites,
can
be
Energetický zisk terče je
definovaný
and
" isjako the burn fraction. For the considered configurat
M
"
qburnt
Efus
DT fE
qDT Mf "
Target energy gain fus
M
" "
q
=
,
5.1
GE=
fus
DT mass
fG
can
be
written
as
and,
provided
the
fuel
ignites,
the
/(H
+
H
),
=
H
=
,
=
f
B
f
=
,
5.1
G
=
E
E
d Ed
"
#\$
!
Ed
Ed d Ed
3
3
3
R
=
M
+
M
=
(4π/3)
ρ
R
+
ρ
−
R
M
kde where
qDT = q337 =MJ/mg
je
fúzní
energie
uvolněná
na
jednotku
hmotnosti
f 11 J/g
hB is
c fusion
h released
c Bper
+f unit
Hinf ),
"=
H
hintroduced
h
f /(H
is
the
burn
parameter
Section
2.5
where
H
11
3.37
×
10
the
energy
mass
DT
11
=
3.37
×
10
J/g
is
the
fusion
energy
released
per
unit
where
q
DT
where q = 3.37 × 10 J/g is the fusion energy released per unit mass
•
•
•
Energetickýzisk-hotspotmodel
spáleného
DTpaliva a Φ je zlomek vyhoření paliva.
and
2.28)
and
further
discussed
in
Section
4.6,
burnt and " is the
burn
fraction.
For
the
considered
configuration
the
fuel
burnt
and
"
is
the
burn
fraction.
For
the
considered
configuration
the
is
the
burn
parameter
in
where
provided
theconsidered
fuel ignites,
theHburn
fraction
can
be approx
burnt and " is the burnand,
fraction.
For the
configuration
the
fuel
B
canwritten
be written
mass
canhmotnost
be writtenpaliva
as zapsatand
Pro mass
naší
můžeme
v podobě 2.28)#\$and
further
discussed in
mass
cankonfiguraci
be
as as
=
H
+
ρ
(R
−
R
),
H
"
#\$
!
"
!
f
h
c
f
h
Hf ),
" =!Hf /(HB +3 "
#\$3 3
3
3
3
R
=
M
+
M
=
(4π/3)
ρ
R
+
ρ
−
R
M
M = Mh +=M(4π/3)
3 ρch Rh + ρ
3 c Rfh3 −
f (4π/3)
h
c
c =
hR
f ρ (R
h − R5.2
hH
=
+
H
R
5.2
ρ
R
+
ρ
−
R
f
h
c
f
h ),
Mf = fMh + M
c
h
c
h
f
h
and
where HB is the burn parameter introduced in Section 2.5.2 (se
a v případě, že dojdeand,
k zapálení
paliva,
můžeme
zlomek vyhoření
provided
thethe
fuel
ignites,
the
burn
fraction
can be approximate
and,
provided
the
fuel
ignites,
burn
fraction
can
be
approximated
by
and
and
2.28)
and
further
discussed
in
Section
4.6,
and,
provided
the
fuel
ignites,
the
burn
fraction
can
be
approximated
by
paliva vztahem Hh = ρh Rh .
5.3
" = H /(H +"H=),Hf /(HB + Hf ),
B f ), Hf f = Hh + ρc (Rf − Rh ), Hh = ρh Rh .
" = Hf /(HfB + H
5.3
Within the model, all the processes connecting the incide
the burn
parameter
introduced
in Section
2.5.2
(see eqns
where
HB is
V
našem
modelu
jsou
všechny
procesy
přeměny
energie
na
terč
z
the burn
parameter
introduced
inWithin
Section
2.5.2
(see
eqns
2.26
where
B isburn
the
model,
all
the
processe
and
to
the
fuel
energy
at
ignition
E
are
absorbed
into a s
E
parameter
introduced
in
Section
2.5.2
(see
eqns
2.26
where
HB isHthe
d
f
and
2.28)
and
further
discussed
in
Section
4.6,
driveru E do energie paliva E v době zapálení obsaženy v jednom
d
and further
discussed
in Section
4.6,
Ed to the fuel energy at ignition E
and and
2.28)2.28)
and further
discussed
inf Section
4.6, efficiency
the
overall
coupling
parametru, který nazýváme
celková
přeměny (overall coupling
=
. účinnost
hR
ρhc (R
HfH=h H
h ρ+
f − Rh ), the overall coupling efficiency
Hρhc+
ρf c−
(RRf h−
R=h ),E /E .
5.4
HfH=
efficiency)
=
+
(R
),
5.4
H
f
h
η
f
d all the processes connecting the incident dr
Overall coupling efficiency
Within the model,
and
η = Ef /Ed .
Overall coupling efficiency
and
V případě přímé fúze
tento
parametr
obsahujeEfabsorpci
energie
the fuel
energyηat ignition
are absorbed
into a single
Ed to
• and
Hzáření
h = ρhaRpřeměnu
h.
této
energie do energie paliva.
overall coupling
efficiency
=
ρ
R
.
5.5
H
5.5
Hh = hρh Rh .h h
Within the model, all the processes connecting the incident driver en
= fuel
E
f /E
d.
Overall coupling efficiency
toηprocesses
the
energy
at ignition
Ethe
absorbed
into aenergy
single param
Within
the model,
the
processes
connecting
driver
d all
f areincident
Within
the model,
allEthe
connecting
the incident
driver energy
the
coupling
tofuel
the fuel
energy
at ignition
are absorbed
into a single
parameter,
Edthe
energy
at overall
ignition
Ef areEabsorbed
into a single
parameter,
Ed to
fefficiency
Fuel energy gain
For
indirect
this includes the intermediate steps of c
in the
range ofdrive,
3–10%.
In
direct
drive
targets,
η
driver
energy
and
of
latter
to
Beside
the to
target
defined
by coupling
eqn
5.1, output
wethe
introduce
asthermal
thegain,
ratioX-rays
of the fusion
energy
to
the fu
tion (see Section 11.2) and c
energy
gain
5.1.2
Hot9).
spot
Chapter
In real
one mayGain
hopecurves
for overall
) and G
justtargets,
before ignition.
G(Edcouplin
For mezistádia,
indirect drive,
tento parametr obsahuje i efektivitu
kteréthis include
• Pro nepřímou fúziThe
and
are
related
by
in
the
range
of
3–10%.
fuel
in
the
hot
spot
is
described
as RTG
an
ideal
spočívá v přeměně energie laserového záření do tepelného
záření
naof density
driver
energy
togas
thermal
X-rayρ
/E
G
f = Efus
ftarget
Beside
the
gain,
defined
by9).eqn
5.1,targets,
we introd
.
It
has a pressure
temperature
T
el energy gain
h
stěnách
hohlraumu
a
dále
absorpci
energie
RTG záření
zahrnující
Chapter
Inrovněž
real
one
spot fuel as an ideal gas
G(Ed ) = ηGin
ztráty způsobené transportem
v hohlraumu.
f (ηE
energy gain
thed ).
range of 3–10%.
Energetickýzisk-hotspotmodel
aspthe
ratio
ofTthe fusion energy output to the fuel energy content
=
ρ
#
h
h B h
Beside the target gain, de
just
before
ignition.
Gain
curves
G(E
Můžeme doufat, že celková účinnost přeměny energie bude
reálných
d ) andv G
f (Ef ) are both of
energy gain
=
E
/E
G
f
fus
f
terčích
v
rozmezí
3–10%.
and
are
related
by
nergy gain
and a specific internal energy
Zavedeme zde také energetický zisk samotného
paliva Efus
/E
Gf = to
5.1.2
Hot
spot output
f energy con
Fuel energy
)
=
ηG
(ηE
).
as G(E
thegain
ratio
of
the
fusion
energy
the
fuel
d
f
d
eh = 23 #B Th . The fuel in the hot spot is described as an id
Gf (Ef ) are bo
just
beforez ignition.
Gain vcurves
G(E
d ) andenergii
jako poměr výstupní
energie
fúzních reakcí
celkové
the ratio
of the fusion energy
.
It
has
a
pressure
temperature
T
h
and
are related
by energetického zisku G(Ed) a Gf(Ef)
palive
těsně
před
jeho
zapálením.
Křivky
Hot spot
fuel
as an
ideal
gas
For equimolar deuterium–tritium (DT)
fuel, theignition.
gas constant
just before
Gain is
curv
jsou obě zajímavé a jsou spojeny vztahem
and are related by
=
ρ
#
T
p
h ). h B h
5.1.2
Hot
spot
)
=
ηG
(ηE
G(E
d
f
#B = 4kB /(mD + mTd) = 0.766 × 1015 erg g−1 keV−1 ,
The fuel in the hot spot is described
as an
ideal
gas
of).density
)
=
ηG
(ηE
G(E
d
f
d
Hot spot - palivo uvnitř
hot
spotu
je
popsáno
stavovou
rovnicí
ideálního
and
a
specific
internal
energy
temperature Th . It has a pressure
t spot fuel as an ideal gas
tritium nucle
the masses mD vand
T of tlak
plynu s hustotou ρwith
němmtedy
specifickáand
vnitřní
h a teplotou Th. Jsou
the3 two electrons and two ions of each D
energie dané vztahyfactor four accounts for
ph = ρh #B Th eh = 2 #B Th .
Multiplying the pressure in eqn 5.9 by the radius Rh of the hot spot,
•
•
•
•
5.1.2 Hot
Vynásobením tlaku poloměrem
hotspot
spotu Rh dostáváme
pot fuel as an ideal gas
5.1.2 Hot spot (DT) fuel, th
and
a
specific
internal
energy
For
equimolar
deuterium–tritium
The
the
hot
spot
is
described
as
an
ideal
gas
of
de
= pin
R
=
H
#
T
.
FDTfuel
h h
h B h
The fuel in the hot spot is de
.
It
has
a
pressure
temperature
T
h
3
15
.
It
has
a
pressu
temperature
T
h
=
#
T
.
e
h
B
h
=
4k
/(m
+
m
)
=
0.766
×
10
erg
#
Hot spot
as2an is
ideal
gas B
B
D by nuclear
T
Thisfuel
quantity
completely
determined
fusion ignition ph
5.1 Hot spot ignition modelThis quantity is completely determined by nuclear fusion ignition physic
which has been discussed in Chapter 4. There, we have seen th
an ignition condition for both isobaric and isochoric ignition condition
in model
theapproximated
temperature
interval 5 ≤ Th ≤jaderné
15 keV.
Since
one typically h
Tato ignition
veličina
je
kompletně
určená
fúze.
Podmínka
5.1 Hot spot
by fyzikou zapálení
/ρhizobarické
≈ 10 in icentrally
igniting
isobaric
fuels,
we use in the followi
ρcpři
pro zapálení
izochorické
konfiguraci
paliva
je aproximována
5.1 Hot vztahem spot ignition
model approximate
! "interval
0.5ignition
the in
simpler
the temperature
5 ≤\$condition
Th ≤ 15 keV. Since one typically ha
#
ρ
h
2
on condition
g/cm
keV
5.1
T
≃
6
H
igniting
isobaric
fuels,
we
use
in
the
followin
ρc /ρh h≈h10 in centrally
ρc
in
the
temperature
interval
5condition
≤ Th ≤ 15 keV. Since one typic
"
the simpler! approximate
ignition
2T ≤ 15 keV relativně přesný.
který je v teplotním
intervalu
5
≤
HhρTh/ρ
≃ 2≈ g/cm
hkeV,
10 in centrally
igniting isobaric fuels, we use in the5.f
•
Energetickýzisk-hotspotmodel
c
h
•
! approximate
"
the
simpler
ignition
conditionje ρc/ρh ≈ 10,
2
Protože typický poměr hustot v případě izobarické
konfigurace
Hh Th ≃ 2 g/cm keV,
můžeme tuto
podmínku
from
which používat
we havev jednodušším přibližném tvaru !
"
2
g/cm keV,
h Th ≃
fromHwhich
we2 have
z čehož dostáváme FDT = Hh "B Th = ph Rh ≃ 15 Tbar µm.
•
FDT =
Hh "Bwe
Th have
= ph Rh ≃ 15 Tbar µm.
from
which
5.
5.1
Tato podmínka
platí především
hodnotyto the
odpovídající
našemu
This condition
applies, inpro
particular,
pair of values
referenčnímu bodu
který odpovídal
s minimální
energií This A,
condition
applies, zapálení
in particular,
to the pair
of values
•
5.1
FDT = Hh "B Th = ph Rh ≃ 15 Tbar µm.
Th = 8 keV, and Hh = 0.25 g/cm22,
5.
Th = 8 keV, and Hh = 0.25 g/cm ,
5.1
This
condition
applies,
in
particular,
to
the
pair
of
values
Podmínka FDT = phRh = constant nám říká, že hot spot je plně určen buď
samotným corresponding
tlakem nebo samotným
poloměrem.
to reference
point A in Fig. 4.5 and to the simulati
corresponding to reference point A in Fig. 4.5
and to the simulatio
2
discussed
Fig.
alsoapplies
appliestoto
Th in=Chapter
keV, 3 (see,
= 0.25 g/cm
,
hparticular,
discussed
in8 Chapter
3and
(see,inH
inparticular,
Fig. 3.11).
3.11).
ItIt also
FDT = phRh = constant condition in eqn 5.15 implies that the hot
fully specified as soon as either the pressure or the spark radius is
Taking the pressure as independent variable, one obtains
•
Energetickýzisk-hotspotmodel
Budeme nyní pokládat tlak za nezávislou proměnnou Rh = FDT /ph ∝ ph−1 ,
ρh = Hh /Rh ∝ ph ,
Mh = (4π/3)Rh3 ρh ∝ ph−2 ,
Eh = eh Mh ∝ ph−2 ,
a Mh a Eh jsou hmotnost a vnitřní energie hot spotu.
•
•
•
where Mh and Eh are mass and internal energy of the hot
izentropický parametr α - aby bylo dosaženo komprese
paliva s minimální energií driveru, měl by být rezervoár paliva obklopující
hot spot udržován v průběhu imploze na nízké entropii.
je tlak
ve fuel:
studeném
stlačeném
palivu α
dán především
5.1.3
Cold
isentrope
parameter
degenerací elektronů.
It has been discussed in the previous chapters that high fuel compr
to a few
thousand
times
density
is a crucial
requirement fo
palivo
popisovat
jakoliquid
částečně
degenerovaný
Fermiho
tial fusion. The fuel reservoir surrounding the hot spot should the
plyn.
be kept at low entropy during implosion to achieve the compress
minimum cost of driver energy. Under such conditions, the press
the compressed hydrogen is predominantly due to degenerate elec
We therefore describe the cold fuel as a partially degenerate Ferm
a partially
gas
a
partially
•
as
rameter
rameter
•
•
•
•
of totally
degenerate
fuel at samearedensity.
In pressure
the following,
the
cold
fuel
quantities
needed 1a
5.1 Hot spot ignition model
In
theisentrope
following,
the
cold that
fuel
quantities
are need
(see Section
10.2.3).
The
pressure
is written
in the
form
pressure
and
parameter,
is,
and pressure
isentropeis parameter,
that
is,
(see Sectionpressure
10.2.3). The
written in the
form
Energetickýzisk-hotspotmodel
Tlak můžeme zapsat
3/5
T ) = vztahu αp (ρ−3/5
) = αA
p (ρ ,pomocí
5/3
ρc
5.
c cρc c= (αAdeg
c
deg
)
p
;
deg
c
−3/5 3/5
5/3
=
(αA
)
p
;
ρ
(ρ
,
T
)
=
αp
(ρ
)
=
αA
ρ
5.1
p
c
deg
c
c
c
c
deg
c
deg
c
2/5
3/5
3 internal energy
and the specific
in. the form
p
=
(αA
)
e
c
deg
c
a specifickou vnitřní energie2 pomocí
3 vztahu 3/5 2/5
p
.
=
(αA
)
e
and
the
specific
internal
energy
in
the
form
c
deg
c
2
2/3
3
ec (ρ
,
T
)
=
αe
(ρ
)
=
αA
ρ
, will be determined by5.E
The
energy
of2 the deg
coldc fuel
c internal
c
deg
c
2/3
3
The
internal
energy
of
the
will
be determined
ec (ρ
,
T
)
=
αe
(ρ
)
=
αA
ρ
, fuelE
asc the
fuel
energy
over after5.2
b
c surplus
deg ofc the 2totaldeg
c cold
f left
the as
subscript
‘deg’ of
labels
the quantities
fortlaku
a fully
degenera
Parametr α where
je tzv.
izentropický
parametr.
Je
to
poměr
současného
ku
the
surplus
the
total
fuel
energy
E
left
= Eaft
hot spot. The cold fuel mass then follows from
f Mcover
c/
tlaku odpovídajícímu
úplně
degenerovanému
materiálu
při stejnéby
hustotě.
electronthegas
(see
Section
10.2.3),
characterized
the
constant
where
subscript
‘deg’
labels
the
quantities
for
a
hot spot. The cold fuel mass then followsfully
fromdegenera
Mc =
electron gas (see Section
10.2.3),
characterized by the constant
!
"
2/3
Vnitřní energie chladného paliva
je určená
3π 2 z Ec = Ef − Eh, jako část celkové
2 h̄2
!
"
energie paliva A
Efdeg
,5.1.4
která
po vytvoření
hot spotu.
= zbyde
2/3
Isobaric
configuration:
pressure p
2
2
5/3
h̄ e [(mD 3π
25 2m
+ mT )/2]
5.1.4
Isobaric
configuration:
pressure
p
For=paliva
the
isobaric
configuration
we
have
Hmotnost studeného
pak
vyplývá
za
vztahu
M
=
E
/e
.
5/3
!
"
!
"
c
c c5/3
5 2me [(m12
+ mT )/2]3
3
D
erg/cm
/
g/cm
= 2.17
×
10
.have
5.
For the isobaric
configuration
we
!
" !
"
12
3 máme 3 5/3
tlak
p
pro
izobarický
10pc , erg/cm / g/cm
.
5.2
p 2.17
= ph×=
The quantity α is the isentrope parameter we have introduced
p = ph = pc ,
Chapter
3. It implies
isαtheisratio
the actualparameter
fuel pressure
the correspondi
The
quantity
the ofisentrope
weand
have
introduced
which
z čehož plyne pressure3.ofIttotally
degenerate
fuel atfuel
same
density.and the correspondin
Chapter
is
the
ratio
of
the
actual
pressure
which
implies
3 at
3 degenerate
In theEoffollowing,
the
cold
fuel
quantities
are
needed
as
functions
pressure
totally
same
density.
=
pV
=
2π
pR
.
f
f
f
2
pressure
isentrope
that
3 is,
3 parameter,
In the and
following,
the
cold
fuel
quantities
are needed as functions
Ef = 2 pVf = 2πpRf .
pressure
isentrope
parameter,
that is, the total fuel volume Vf
Thisand
relation
allows
to
determine
3/5
−3/5
= (αA
)
pc of; p and E . It is also worth observing5.t
ρc R
deg
as functions
Energetickýzisk-hotspotmodel
which
in Fig. 5.1. In the isobaric case, both
hot implies
and cold fuel have the same
pressure, p = ph = pc . But also the gain of3 other hot spot 3configuraEf = 2 pVf = 2π pRf .
tions, such as the isochoric ignition configuration with uniform density
throughout
fuel,
willdovoluje
be discussed.
Tento the
vztah
nám
určit
celkový
objem
paliva
Vf a jeho poloměr
f
This
relation
allows
to determine
the total Rfuel
volume
Our
presentation
follows
the original
derivation
ofof
thepisobaric
jako
funkci
tlaku
p a isobaric
energie
paliva
Efunctions
5.2
Gain
curves
of
the
model
f.
and Ef model
. It is also worth observin
Rf as
(Meyer-ter-Vehn 1982). A similar model
for isochoric
initial
conditions
3
uniform,
we have E
h = (3/2)pVh = 2π pRh , and hence
Je také
zajímavé
si všimnout,
že
protože
tlakBodner
pthe
rovnoměrný,
máme
developed
by
(1976b)jeand
(1981). determining
The 5.2
Gain
curves
of earlier
the
isobaric
model
LetKidder
us
summarize
here
equations
the main qu
3, aRosen
# (1984)
\$3
= (3/2)pV
tak modelEhwas
later extended
and
Lindl
to account
for
h = 2πpRhby
of the fuel assembly
and
the
gain.
Taking
as free parameters the
Eh
R
h
= η,
.c and
configurations with arbitrary
values
of the ratios
ρthe
ph /pc . determining
h /ρ
Let
us summarize
here
the
equations
the main pres
qua
coupling
efficiency
fuel
isentrope
α and the stagnation
Ef
Rf
ofwe
thecan
fuel
assembly
and
the
gain.
Taking as free parameters the o
write
the
gain
in
the
form
Předpokládejme dále volné parametry celkovou účinnost přeměny energie η,
Of course,
isobaric
condition
is only
a roughpress
app
coupling
efficiency
η, thethe
fuel
isentrope
αpro
and
the
stagnation
izentropu
paliva
α
a
tlak
při
stagnaci
paliva
p.
Pak
můžeme
energetický
5.1.1 Target gain, fuel gain, coupling
efficiency
ηin "
!
#\$at ignition.
actual
situation
a
real
target
Limitations
we
can
write
the
gain
in
the
form
q
H
+
H
4π
DT
h
c
zisk psát 3
3
3
ρ
R
R
+
ρ
−
R
,5.3.4.
G
=
The target energy gain is defined as possible improvements
h h
c
are
discussed
in
Section
f
h
3 Ed
HB + H h + H c
!
"
#\$
H h + Hc
4π qDT
qDT Mf "
Efus
3
3
3
=
, G=
5.1
G=
ρh Rh + ρc Rf − Rh
,
Ed
Ed where 3 Ed
HB + H h + H c
wherekde qDT = 3.37 × 1011 J/g is the fusion energy released per unit mass
where
ρh For
= Hthe
burnt and " is the burn fraction.
considered
configuration the fuel
h p/F
DT ,
mass can be written as
R!h = FDT /p,"
#\$
ρ
=
H
p/F
,
3
3
5.2
−
R
Mf = Mh + Mc = (4π/3)h ρh Rh3h+ ρc DTR
−3/5
3/5
f p h,
ρ
=
(αA
)
deg
Rh c= FDT /p,
and, provided
the fuel ignites,Rthe
burn fraction1/3
can be approximated by
f = (ηEd /2πp)
ρc = (αAdeg )−3/5 p3/5 ,
5.3
" = Hf /(HB + Hf ),
Hc = ρc (Rf − R1/3
h ).
Rf = (ηEd /2πp)
where HB is the burn parameter introduced in Section 2.5.2 (see eqns 2.26
•
6
Rf
r
of
y at the
e suffix h
hot spot,
of the
•
•
Energetickýzisk-hotspotmodel
•
•
5.2 Gain curves of the isobaric model
Energetickýzisk-hotspotmodel
Weurčené
are now
calculate
the energy
Konstanty
fyzikou
fúze
v DT palivu
jsougain
qDT, HB, Hh, a FDT.
We are now ready to calculate the energy gain
Jsme tedy již
připraveni
energetický
p, α; FDT
, Hh , HB , qzisk
G=
G(Ed ; η, počítat
DT ), v této podobě •
5.33
G = G(Ed ; η, p, α; FDT , Hh , HB , qDT ),
5.33
as a function of the driver beam energy Ed , of the free parameters η, p, α,
kde je funkcí
driveru F
EDT
tříh ,volných
η, p, physics.
α a závisí
d a, H
fixed by fusion
As na
it
HB , qDT ,parametrů
and ofenergie
the parameters
dalších čtyřech
parametrech
, Hhbeam
,toHother
qDT
, které
jsou
všakDT.
proNevernámi η, p, α,
the free
parameters
as athe
function
of the
driver
Ed , of
DT
B, energy
stands,
model can
beFapplied
fuels
than
equimolar
zvolenoutheless,
reakci
pevné.
fixed
fusion
physics.
Asisobari
it
HB , qDT , are
andthe
of examples
the parameters
FDTin, H
discussed
theh ,following
given
equimolar
5.2by for
Gain
curves
of the
DT fuel,
which
certainly
fuel of primary
for the
time being.
stands,
the ismodel
can the
be applied
to otherinterest
fuels than
equimolar
DT. NeverObecně budeme
normovat
všechny
parametry
jejich
referenční
hodnoty
a all param
As atheless,
set of
reference
valuesdiscussed
for
theand,
fixed
DTgeneral,
parameters,
wegiven
use
infor
this
in
weare
shall
normalize
the examples
in na
the
following
equimolar
psát
chapter
writinginterest for the time being.
DT fuel, which is certainly thevalues,
fuel ofbyprimary
As a set of reference values for the fixed DT parameters, we use in this
FDT = F̂DT fF ,
5.34
Fchapter
DT = F̂DT ≡ 15 Tbar µm,
Hh = Ĥh ≡ 0.25 g/cm2 ,
≡ 15 Tbar µm, H = Ĥ f ,
B
B HB
2
11
H
=
Ĥ
≡
0.25
g/cm
,
qDT = q̂DT fqDT ,
qDT =hq̂DT ≡h 3.374 × 10 J/g,
F
= F̂
Hh = Ĥh fHh ,
ĤB ≡ 7 DT
g/cm2 ,
HB = DT
HB = ĤB ≡ 7 g/cm2 ,
5.35
5.36
5.34
5.37
5.35
5.36featu
with obvious meaning of the f ’s. Basic
by this model are illustrated5.37
in the
qDT = q̂DT ≡ 3.374 × 1011generated
J/g,
Figures 5.2–5.4 are taken from the original public
(1982) and are based on the ignition parameters
along the dotted line. In the double-logarithmic plot, this follows from
generated by this model
are illustrated
the
following
subsections.
definition
(5.33) andinthe
scaling
relation
Gf (ηEd ) = G(Ed )/η. In Fig
Figures 5.2–5.4 are taken
from the
original
publication
of located
Meyer-ter-Vehn
ignition
points
for different
η are
on the dotted line, where b
(1982) and are based onenergy
the ignition
given
by eqnup5.17.
is justparameters
large enough
to build
the ignition spark at the g
Závislost na η, α, a pressure.
p - začneme
analýzou
závislosti
G nais celkové
Within
the model,
this line
given byúčinnosti
přeměny energie η a izentropickém parametru α pro daný tlak p = 0.15 Tbar,
ign
kterýDependence
odpovídá realistické
velikosti
hot
3 spotu
2 Rh = 100 μm.
5.2.1
on η,Eα,
and
p
=
2πF
/(ηp
),
DT
d
•
Energetickýzisk-hotspotmodel
ign
start by analysing
the G
dependence
on /(H
the overall
coupling
effiv závislosti
na =účinnosti
posunují
křivky
zisku
energie
(ηqDTof/eG
)H
+
H
).
•WeVýsledky
h
h
B
h
po
ciency
η anddané
the tečkovanou
isentrope parameter
α for a fixed pressure of p =
diagonále
čárou.
0.15 Tbar and a corresponding hot spot radius Rh = 100 µm (compare
zapálení
pro in
různé
eqnPodmínky
5.15). This
is shown
Fig. 5.2, for a parameter range relevant to
hodnoty
η leží právě na této reactor
targets.
p = 0.15 Tbar
"=1
1000
tečkované
kde je energie " = 2 ! = 15%
The
result čáře,
of varying
the overall coupling efficiency, while keeping
"=4
driveru
právě
dostatečná
k
tomu,
all other parameters fixed, is a simple diagonal shift of each gain "curve
=1
" = 2 ! = 5%
aby
vytvořila
hot
spot
along the dotted line. In the double-logarithmic plot, this follows from the
"=4
100
o
potřebném
tlaku.
definition (5.33) and the scaling relation Gf (ηEd ) = G(Ed )/η. In Fig. 5.2
ignition points for different η are located on the dotted line, where beam
V našem modelu je tato energy is just large enough to build up the ignition spark at the given
čára dána eliminováním 10
Isobaricpressure.
gain curves Within
for
model, this line is given by
parametru η z the
rovnic
•
Fig. 5.2
different overall coupling efficiencies η
and isentrope parameters α in the range
ign
3and
elevant to reactor targets.
= 2πF
/(ηp2 ),
Ed Spark
pressure p are kept fixed. Ignition points
or different η are located
the(ηq
dotted
Gignon =
DT /eh )Hh /(HB
ine (Meyer-ter-Vehn 1982).
Target gain G
•
5.42
1
+ Hh ).
0.1
1
10
Ed (MJ)
100
5.43
1000
•
•
Energetickýzisk-hotspotmodel
ign
3
2
=
2πF
/(ηp
), velmi strmě roste v důsledku
E
DT
d energetického
To vede k tomu, že křivka
zisku
rychlého růstu plošné hustoty
zvyšujícího
seh ).prohoření paliva.
Gign =paliva
(ηqDTρR
/eha)H
h /(HB + H
Pro vysoké energie naopak dochází k saturaci energetického zisku, protože dochází ke spálení 1000
většiny paliva.
•
Vyšší hodnoty izentropického parametru α způsobené předohřevem paliva v průběhu imploze degradují zisk především v oblasti vysokého Fig. 5.2 Isobaric gain curves for
zisku.
Závislost
je G ∝ α−3/5
different overall coupling
efficiencies
η
and isentrope parameters α in the range
elevant to reactor targets.
anddvojnásobek Zvýšením
α na
pressure p are kept fixed. Ignition points
vede ke snížení energetického
or different η are located on the dotted
zisku o zhruba 35%.
ine (Meyer-ter-Vehn 1982).
•
Target gain G
•
along the dotted line. In the double-logarithmic plot, this follows from
definition (5.33) and the scaling relation Gf (ηEd ) = G(Ed )/η. In Fig
ignition points for different η are located on the dotted line, where b
energy is just large enough to build up the ignition spark at the g
Pokud je k dispozicipressure.
více energie
vytvoření
palivaby
okolo hot
Withink the
model, stlačeného
this line is given
spotu, vzniká vlna hoření.
p = 0.15 Tbar
"=1
" = 2 ! = 15%
"=4
"=1
" = 2 ! = 5%
"=4
100
10
1
0.1
1
10
Ed (MJ)
100
1000
•
observed in Fig. 5.2. Increasing α by a factor of two leads to about 35%
analytically in Section 5.3.2. The figure also shows (dashed curves) g
reduction in gain.
curves
forgain
constant
mass
fuel, which
willofbethe
discussed
Next, Fig. 5.3
shows
curves
forof
different
values
pressurein Section 5.
5.3spot
spans
the R
parameter
space in which microfusion has
p, or equivalently Figure
of the hot
h = FDT /ph , and fixed values
Křivky zisku prodevelop.
různé hodnoty
p,end,
neboitpoloměru
hotregion
spotu Atshows
thetlaku
low
the
of α and η. The figure also
(thick
solidcovers
line) the
envelopebetween
of these 10 and 100
Rh = FDT/ph, a dané
hodnoty
αoptimism
a η jsou nalocated
obrázku
níže. for power production in relativ
where
early
options
∗
curves, which gives the limiting gain G (Ed ), that is the maximum energy
small
reactor
units.
Indeed,
the
modelThe
allows
for gain
50isat a beam ene
gain
that
can
be
achieved
for
a
given
driver
energy.
limiting
gain
Obrázek rovněž znázorňuje (silnou čarou) obálku těchto křivek zisku, která
ofby
about∗ 20 kJ, which amounts to 1 MJ of fusion energy per shot or 1 M
accurately
fitted
udává tzv. limitní zisk G (Ed). To je maximální zisk energie pro danou energii
Energetickýzisk-hotspotmodel
•
driveru. •
"0.3
ηEd
∗
,
G = 6000η
α
!
10,000
5.45
"=2
Obrázek pokrývá prostor units of!MJ.
This scaling will be derived
where the driver energy Ed is in1000
=
10%
parametrů, v němž se analytically in Section 5.3.2. The figure also shows (dashed curves) gain
mikrofúze zkoumá.
•
10
Target gain G
curves for constant mass of fuel, which will be discussed
ain in Section 5.3.1.
g
g
itin
m
i
L
Figureokraji
5.3 spans
the parameter
Na levém
pokrývá
100 space in which microfusion has to
develop.
lowkJ,end,
it covers the region between 10 and 100 kJ
oblast
meziAt
10the
a 100
.3 Isobaric gain curves
G(E
)early
for optimism
where
kde
se dnacházely
dříve located options for power production in relatively
nt values of the pressuresmall
of thereactor units. Indeed, the model allows for gain 50 at a beam energy
navržené
optimistické essed fuel (and therefore of the
10
of below
20 kJ,
which amounts
to 1 MJ of fusion energy per shot or 1 MW
předpovídaly
of the hot spot, see labels
the
mg
g
mg
1000
1m
0.1
g
1m
0.0
urves) at fixed values of
α, η, andenergie
of
produkci
i v takto ition parameters. The malých
envelope reaktorových
of
mily of curves defines the limiting
10,000
jednotkách.
The figure also shows (dashed
) gain curves at constant fuel mass
r-ter-Vehn 1982).
"=2
10 µm 25 µm 50 µm 100 µm 200 µm
1
0.01
0.1
1
Ed (MJ)
10
500 µm
100
•
•
•
•
•
•
Energetickýzisk-hotspotmodel
Skutečně nám náš model umožňuje zisk 50 s energií driveru pouze 20 kJ, což
vede na produkci 1 MJ fúzní energie na výstřel a 1 MW výkonu pro
frekvenci výstřelů jednou za sekundu.
V tomto případě by každý terč obsahoval pouze 10 μg paliva, ale většina
paliva by musela být stlačena 2 × 104 krát při izentropickém parametru α = 2,
aby bylo dosaženo tlaku při stagnaci 5 Tbar a velikosti hot spotu 3 μm.
To je dnes považováno za nemožné v důsledku omezení symetrií imploze.
Pro produkci energie při inerciální fúzi musí terč obsahovat nejméně 1 mg
fúzního paliva DT.
Fakt, že zvýšení tlaku také snižuje zisk může být na první pohled překvapivý.
Je to z toho důvodu, že v asymptotickém případě limitního zisku máme hot
spot obklopen tlustou slupkou chladného paliva a zvyšování tlaku v
izobarickém případě vede ke zbytečnému investování energie do vnitřní
energie tohoto chladného paliva.
the parameters η and α fit the LLNL results accurately. This is shown i
Fig. 5.4, where the curves of the isobaric model have been computed b
choosing p = 0.2 Tbar, and ignition parameters Hh = 0.4 g/cm2 , an
keV. The simulace
conservative
band
is then
spanned by overall cou
Th = 5 detailní
Modelové křivky zisku versus
- Vgain
roce
1979
výzkumnící
pling efficiencies ranging between 5% < η < 10% and taking α = 3. Th
z LLNL shrnuly své předpoklady o energetickém zisku v podobě tzv.
optimistic gain curve is fitted with η = 15% and α = 1. One also find
konzervativního pásma zisku
(conservative
band) a optimistické
křivky
that the
upper limit ofgain
the conservative
gain curve can
be fitted by taking
zisku (optimistic gain curve).
Překvapivě
tytoclose
křivky
z výpočtů
veachievable
shodě in the nea
smaller
value of η jsou
= 3.5%,
to what
is believed
LLNL gain curves and gain band
•
Energetickýzisk-hotspotmodel
s naším jednoduchým izobarickým modelem (zafixováním počátečního tlaku
a parametrů η and α ).
•
stic
imi
h
Target gain G
h
100
ive
last ttwo
"=3
! = 5%
dec-
the LLNL group on the basis of extensive
50 <
Gain < 100,
numerical simulations, and
curves
generated by the isobaric
model,
for<
fixed
1
MJ
<
E
10 MJ,
d
values of the pressure of the compressed
et window fuel (Meyer-ter-Vehn
50 µm
1982).< Rh < 200 µm,
erv
a
Although much has been learned about targets over the
Optimistická křivka zisku je pro ades, the gain expectations concerning reactor targets have not changed
η = 15% a α = 1.
very much. One still expects that driver energy in the range 1–10 MJ
10
is
required
to
achieve
target
gain
G
=
50–200.
According to Figs 5.3
Okno pro parametry terčů použitelných 5.4, thejewindow
for reactor
targets is then expected in the region
vand
reaktoru
nyní očekáváno
v oblasti
Con
s
•
"=1
! = 15%
kde předpokládáme celkovou účinnost přeměny
v rozmezí
future in energie
hohlraum
targets, and α = 1, p = 0.25
Tbar
"=3
LLNL
gain (Rh = 49 µm),
! = 10%
5%
< η0.2
< 10%
= 3.T = 8 keV.
predictions
g/cma2 ,αand
H =
1000
Opt
•
Konzervativní pásmo zisku je oblasti, Limiting gain curves
Rh = 75 µm
p = 0.2 Tbar
1
0.1
1
Ed (MJ)
10
5.46
1 mg < Mf < 10 mg.
The upper limit is given by the requirement of containing the microexplosion in a reactor cavity and is not well defined. At E = 10 MJ and
mass and derive analytical expressions for the limiting gain and for the
minimum energy required to assemble an igniting configuration.
r fixed fuel mass
nd α. Pressure p
along the gain
puted from the
p and Rh are
e. Maximum gain
mum drive energy
mparison of gain
odel and
ynamics
tions.
of a given mass of fuel as the pressure is increased (see Fig. 5.5).
Při
nízkém
udržení
došlo kfrom
zapálení.
The
cold tlaku
fuel není
isentrope
is dostatečné,
kept fixed.aby
Starting
low pressure, at
first confinement is not sufficient to achieve ignition. At some point
Se zvyšujícím se tlakem pak v nějakém bodě dojde k zapálení a to nejprve
the ignition condition 5.15 is marginally met by a homogeneous fuel
pro konfiguraci, kdy je homogenní palivo. Zisk je v tomto případě malý,
protože chybí rezervoár studeného paliva pro pozdější hoření.
(a)
(b)
500
500
GfM
†
Gf
50
100
20 10
1 2
0.1
Rh (µm)
5
100
Model
(HB = 9 g/cm2)
Gf
•
•
Další vhled do křivek energetického zisku můžeme získat studiem zisku
5.3.1 o Gain
curve for av given
fuel na
mass
paliva
dané hmotnosti
závislosti
zvětšujícím se tlaku. Izentropický
parametr
přitom
zafixujeme.
insight
into the gain curves is obtained by studying the gain
Fuel gain Gf
•
Limitnízisk
p (Tbar)
200
0.05
10
10
†
Ef
EfM
100
Ef (kJ)
10
10
100
Ef (kJ)
mass and derive analytical expressions for the limiting gain and for the
minimum energy required to assemble an igniting configuration.
r fixed fuel mass
nd α. Pressure p
along the gain
puted from the
p and Rh are
e. Maximum gain
mum drive energy
mparison of gain
odel and
ynamics
tions.
of a given mass of fuel as the pressure is increased (see Fig. 5.5).
M (a energie paliva EfM) dosáhne zisk maxima GfM.
VThe
nějakém
cold bodě
fuel pisentrope
is kept fixed. Starting from low pressure, at
first confinement is not sufficient to achieve ignition. At some point
Při ještě silnější kompresi se již zisk snižuje, protože používáme ke kompresi
the ignition condition 5.15 is marginally met by a homogeneous fuel
paliva zbytečně příliš mnoho energie.
(a)
(b)
500
500
GfM
†
Gf
50
100
20 10
1 2
0.1
Rh (µm)
5
100
Model
(HB = 9 g/cm2)
Gf
•
•
Při dalším zvětšováním tlaku jsou již generovány konfigurace paliva s hot
5.3.1 a Gain
curve
for a given
fuelTymass
spotem
okolním
studeným
palivem.
vyžadují méně energie na zapálení a
zároveň
dochází
ke spálení
více
paliva,
takže
je větší zisk.
insight
into the
gain
curves
is obtained
by studying the gain
Fuel gain Gf
•
Limitnízisk
p (Tbar)
200
0.05
10
10
†
Ef
EfM
100
Ef (kJ)
10
10
100
Ef (kJ)
•
•
Limitnízisk
Další důležitý aspekt se týká investované energie k vytvoření dané
konfigurace zapálení paliva. Tato energie nejprve klesá ke své minimální
hodnotě E† a potom dále roste s hustotou paliva.
Všimněme si, že E† ≃ 0.8EM a G†f = Gf (E†) je pouze o 20% menší, než G∗f = Gf (EM).
•
Tento model je v souladu se simulacemi právě v té nejdůležitější oblasti
okolo optimálního výkonu terče, zatímco velké rozdíly jsou pozorovatelné v
oblasti malého tlaku a malého zisku.
•
Samozřejmě, v tomto případě předpoklad konfigurace terče jako hot spotu
obklopeného studeným palivem nefunguje, a proto jsou vztahy pro zapálení a
prohoření paliva použité v modelu nesprávné.
5.3.2 Analytic derivation of the limiting gain
Limitnízisk-analytickýmodel
•
In this section, the limiting gain curve appearing in Fig. 5.3 and corresAnalytické odvození limitního zisku - hledáme maximální zisk energie pro
ponding scaling relations are investigated analytically, following Rosen
energii paliva Ef = ηEd a izentropický parametr α tím, že měníme tlak p.
and Lindl (1984). To this purpose, we search for maximum gain at fixed
α, byjevarying
p.
f = ηEd andzisku
V okolí E
maximálního
konfigurace
paliva charakterizovaná tím, že
maximum gain, fuel configurations are characterized by
Mh≪Mf a HNear
h≪Hf ≈HB. Většina paliva je tedy ve formě studeného paliva a
≪M
Hf ověříme
≈ HB . Most
of the
fuel odvození.
mass appears as cold
Mh část
h ≪ To
jenom malá
jef vand
hot H
spotu.
na konci
našeho
fuel, and only a tiny fraction is used for the hot spot; this will be checked
a posteriori
in this
subsection.
Analytický
přístup jelater
možný,
když
zanedbámeAnManalytic
porovnáníbecomes
s těmito posh a Hh v approach
sible
neglecting
Mh and
Hh in comparison
parametry
pro when
celé palivo
a budeme
tak používat
Mf ≃ Mc a to
Hf the
≃ Hfull
c. fuel values,
and thus taking Mf ≃ Mc and Hf ≃ Hc , and approximating the burn-up
Dále aproximujeme
parametr
mocninnou funkcí, což je přesné s
fraction by the
power prohoření
law expression
chybou maximálně 20% v rozmezí 0.3 HB ≤ Hf ≤ 3.5 HB •
•
•
1
Hf
≃
#=
HB + H f
2
!
Hf
HB
"1/2
1
≃
2
!
Hc
HB
"1/2
,
which is 20% accurate for 0.30HB ≤ Hf ≤ 3.5HB .
5.47
!
"1/2
withwith
Mc inHthe
1 qDT
#
DT M
f ! is then
c approximate
Theqfuel
gain
written
form
#
"
!3 \$ "
!
"p3/5 3/5 2E 5.48
! "
≃
,!
Gf =
Rh
f
R
2E
p
!
"
EThe
2
E
H
f
h
f
f
B
1/2
fuel
gain
isρ
then
written
in
the
approximate form1 − 1 −
=
ρ
(V
−
V
)
=
M
c
c
f
h
=
(V
−
V
)
=
M
H
q
M
!
M
q
1
c
c
f
h
Zisk paliva můžeme
DT f aproximovat
DT jako c
c
αAdeg 3p 3p 5.48Rf Rf
αA
=
≃
,
G
deg
f
!
"
with
Ef qDT M2f !Ef 1 qDT
HBMc Hc 1/2
#
5.48
Gf = and and
! , "3 \$
! ≃ "3/5
Ef
Rh
p 2 Ef 2Ef HB
with
M
"3/5 !"1/35.49
"1/3 ! "
c = ρc (Vf − Vh ) = !
#
\$!
!
!1 − !! ""
3/5
"
3/5 3p
Rfp 3 Ef Ef
Rh
p
with
R
2E
p
f
h
(RRf h−) =
Rh )1=− #
Hρ
1− 1−
(Rρf c−
c c=
) c==
5.49
Mc = ρc (Vf − VhH
\$
! 2πp"32πp
!
αA
R
deg
f
αA
3p
R
deg
f
and
Rh
2Ef
p
1− "
− Vh ) = " !
5.49
Mc = ρc (Vf !
!
"
3/5
1/3
Here,Here,
eqnspeqns
5.22 5.22
and
5.25
have
been been
used
tof express
the cold
and
5.25
used
to express
thef
3phave
R
and
E
R
f
h
H = ρ (R − Rthe
. functions
5.50
fuel
volume,
and!the
as functions
of theofpres
c
c f
h) =
the!fuel
volume,
andfuel
the
fuel
th
"3/5
" as
"
1/3
αA
2πp
R
EE
Rhf
p
deg
and
f = ηE . Varying
fixed
Ef , η
the
fuel
internal
energy
G at fixed
f Ef =
1d −ηEd . Varying
. G at5.50
= fuel internal energy
Hc = ρc (Rf − Rh ) the
"3/5 !
!
"
"1/3
!
αA
2πp
R
deg
f
to Eintroduce
Here,
5.22G and
5.25
have been
tojeexpress
the
cold
fuel density,
convenient
Rh
pa α,
Pokud eqns
hledáme
proconvenient
dané
hodnoty
,used
η,introduce
dobréE
zavést
proměnnou f to
f
1 − and .of 5.50
ρc (R
= as functions of the pressure
Hc =and
f −
the
fuel volume,
the
fuel
Here,
eqns 5.22
and
5.25
have
beenαA
used
to
express
the
cold
fuel
density,
2πp
Rf
deg
x
=
R
/R
h ηE
fh /R
=
f as functions
=
G at fixed
α, itofis
thethe
fuelfuel
internal
energy
Efxfuel
volume,
and the
of theEpressure
d . Varying
f , η, andand
eqns
5.22 E
and=5.25
have
been G
used
to express
the
cold
fuel
density,
convenient
to introduce
ηE
.
Varying
at
fixed
E
,
η,
and
α,
it
is
the fuel Here,
internal
energy
f substitute
dvariable
f the condition
as a as
substitute
for p.forUsing
of isoba
a
variable
p.
Using
the
condition
of
S použití podmínky
izobaricity
a
the
fuel
volume,
and
the
fuel
as
functions
of
the
pressure
and
of
convenient to introduce
eqn
5.25,5.25,
and the
ignition
condition
5.15, 5.15,
one has
podmínky
zapálení
dostáváme eqn
and
the
ignition
condition
hasα, it is
x = Rhpro
/R
η,5.51
and
thef fuel internal energy Ef = ηEd . Varying G at fixed Ef ,one
\$1/2 \$
# #
x = Rconvenient
5.51
h /Rf
to introduce
1/2
3
3 condition of isobaricity 5.24,
FDT F the
as a substitute variable for p. Using
DT
p =p
2π
. condition
as
a
substitute
variable
for
p.
Using
the
of isobaricity 5.24,
=
2π
.
3
eqn 5.25, andx the
condition
5.15,3 one has
= Rignition
5.51
Ef x E
h /Rf
f x one has
eqn 5.25,
and
the
ignition
condition
5.15,
\$1/2
#
a 3substitute
variable
forfor
p.eqn
Using
the
condition
of isobaricity
5.24,
\$
#as F
Also
notice
that
5.26
the
ratio
of
the
hot
spot
energy
1/2
DT3
notice condition
that for eqn
5.26
the
ratio of the5.52
hot spot e
p = 2π
eqn 5.25,
and.Also
the
ignition
5.15,
one
has
3
F3DT fuel
gain can then be5.52
written as
energy
is Eh /Ef = x . The
3
Ef x
p = 2π
.
fuel energy is E /E = x . The gain can then be writte
5.3 Limiting gain curves
•
•
•
Limitnízisk-analytickýmodel
in
3
FDT
p = 2π
Ef x 3
•
•
.
where
where
Limitnízisk-analytickýmodel
3(2π
)
H
AG =
AG =
5.5
qDT
3/10
1/2
B
9/10
deg
2/5
DT
1/2 9/10 32/5
3/10
notice
that energie
for eqnhot
5.26
thekuratio
of the
hot
spot
energy
3(2π)
Hje
VšimněteAlso
si také,
že poměr
spotu
energii
paliva
EA
x DT
. to the tota
h/E
f=F
B
deg
contains numerical constants and the fusion physic
3
fuel energy is Eh /Ef = x . The gain can then be written as
Zisk pak můžeme zapsat jako 2/5
contains
constants
f (x) = xnumerical
(1 − x 3 )(1
− x)1/2.and the fusion p
"3/10
Ef
2/5
3
1/2 to zero or
The
function
f (1
(x)−vanishes
as x)
x goes
f
(x),
5.5
Gf = AG
f
(x)
=
x
x
)(1
−
.
3
α
for x = x ∗ ≃ 0.3485, with f (x ∗ ) = 0.507. For a rat
however,
the
function
f isvanishes
close to its
The
function
f (x)
as maximum
x goes to zv
vymizí
s
x
jdoucím
k
0
nebo
k
1
a
je
maximální
pro • Funkce∗ f(x)
where
∗ leads to the
∗
∗
for
0.12
<
x
<
0.6.
Choosing
x
=
x
∗
0.3485,
(x ) = 0.507. Fo
=0.507.
x ≃ Pro
x = x ≃ 0.3485, s maximální hodnotou for
f(x x) =
dost with
velkýfinterval
qDT limiting gain
however,
the function
hodnot x, je A
funkce
hodnoty,
např.
f(x) > 0.4 fprois close to its maxim
5.5
G = f poblíž maximální
1/2
9/10
2/5
3/10 H
! "3/10
3(2π)
A
F
0.12
<
x
<
0.6.
for
0.12
<
x
<
0.6.
Choosing
x
=
x
B
deg ∗ DT
Ef
Limiting fuel gain
−2/5
−1/2
Gf ≃ 6610
f
f
f
5.3 Limiting
gain
curves
113
qDT HB ,
F
3
limiting
gain
α
∗ vede na
Volba
x
=
x
vztah
contains
numerical
constants
and
the
fusion
physics parameters, and
•
10,000
3/10 and fF , fqDT , and fHB
where Ef is taken!inEMJ"units
pro limitní zisk
1/2
Limiting fuel gain
f
−2/5
−1/2
∗ .parameters around
!
=
1 values
f (x) = x 2/5 (1 − x 3 )(1 the
−
x)
5.5(
Isochoric
fixed
their
reference
Gf ≃ 6610
f
f
f
,
q
DT
F
H
B
! = 1.3
α3
19,200 [Ef (MJ)/!3]0.4
!=2
The function f (x) vanishes as x goes to zero or to 1 and !is= maximum
4
is taken
inaMJ
unitslarge
and interval
fF , fqDTof
, ax
Ef 0.507.
(x ∗ ) =
For
rather
for x = x ∗ ≃ 0.3485, with fwhere
fixed
parameters
around
their
reference
v
Fig. 5.6 Scaling of the limiting
fuel gain the function f is the
however,
close
to
its
maximum
value,
e.g.
f
(x)
>
0.
versus the parameter (Ef /α 3 ), tested by
Isobaric
to the
for
0.12 < x < 0.6. 1000
Choosing x = x ∗ leads
1D hydrodynamics simulations
performed
3 0.3scaling relation fo
5600
[E
(MJ)/
!
]
f
by the IMPLO-upgraded code (Atzeni
limiting
1995). The figure refers to both
isobaric gain
(see Sections 5.2 and 5.3) and isochoric
! "3/10
fuel configurations (to be discussed in
Ef
−2/5
−1/2
∗
Section 5.5.1). The functional dependences
1
100
G
≃
6610
f
f
f
5.5
qDT HB10 ,
F
predicted by the model are recoveredf with
3
Ef (kJ)
α
high accuracy.
Gf
!
Limitnízisk-analytickýmodel
Limitnízisk-analytickýmodel
(Atzeni
a smaller1995).
front factor, corresponding to burn parameter HB ≃ 9.5 g/cm2
Ignition configuration at maximum
gain
Let us1995).
now characterize
at ∗maximum
gain in
∗
∗
∗ 3
∗
∗ the ignition configuration
(Atzeni
Rh /Rf ≃ 0.35 and Eh /Ef = (Rh /Rf ) ≃ 0.042.
more
Wecharacterize
thatconfiguration
the ratio of spark
Letdetail.
us now
thefound
ignition
at maximum
gain inis
fixed
and,
forWe
eqn
5.26,
this hold
forthat
thethe
energies
too.
One
has therefore
more
detail.
have
found
ratio
of
spark
to
fuel
Podívejme se na konfiguraci
při maximálním
Hence,paliva
at maximum
gain, onlyzisku 4% of the fuel energy
and,
for eqn
5.26,
this ∗hold
for the∗energies
too.
One
has M
therefore
3
guration at maximum gain fixed
∗ 5.3contained
∗ 3 ingain
∗
∗
The
fraction
of
mass
the
spark,
/M
=
ρ
R
114
Limiting
curves
h
f
h
h /ρf R
Rh /Rf ≃ 0.35 and Eh /Ef = (Rh /Rf ) ≃ 0.042.
5.57
even smaller and typically below 1%, depending on the density
figuration at maximum gain
∗ energie
∗
∗
∗ 3 pro zapálení.
∗
∗
Při maximálním
je
tedy
jen
4%
použito
Rh /Rfzisku
≃ 0.35
and
E
/E
=
(R
/R
0.042.
5.57
h ρf fis the average
h
f) ≃
/ρ
,
where
fuel
density.
Useful
scaling
ρ
h
f
pressure
p, andrelation
implo
Hence, at maximum gain, only 4% of the fuel energy is used for ignition.
other
quantities
are
listed
as follows:
3
3
3
3
The
fraction
of
mass
contained
in
the
spark,
M
/M
=
ρ
R
/ρ
R
, ais 2
h
f
h
f
Množství
paliva
v
hot
spotu
je
M
/M
=
ρ
R
/ρ
R
,
a
je
tedy
ještě
menší
h
f
h h
f f
h
f
Hence, at maximum gain, only 4% of the fuel energy is used forEignition.
1/p .
f ∝ratio
even
smaller
and
typically
below
1%,
depending
on
the
density
typicky The
pod fraction
1% v závislosti
na poměru
hustot
ρhM
/ρf,/M
kde=ρρf je
3 průměrná
3
3/2
of mass
in
the
spark,
R
/ρ
R
, is
∗ contained−1/2
h
f
h
f
h
f
p
≃
0.22
E
f
Tbar,
/ρ
,
where
ρ
is
the
average
fuel
density.
Useful
scaling
relations
for
ρ
h f
f
hustota paliva.
F
f
Concerning
even smaller5.3
and typically gain
belowcurves
1%, depending on the density
ratio the implo
114other
quantities areLimiting
listed
as
follows:
1/2 −1/2
∗ average
simulated
fuel
density.
Useful scaling relations
forin Chapter 3
ρh /ρf , where ρf isRthe
≃
67
E
f
µm,
h
F
f
Můžeme také získat vztahy mezi energií paliva při zapálení a tlakem paliva
the implosion
ofu a bar
other∗ quantities are
listed
−1/2
3/2 as follows:
−1/2
1/2
pressure
p,
and
implosion
velocity
∗
3
p, nebo implozní
rychlostí
u
imp . F
imp.
p ≃ 0.22 Ef ρh f
5.58
≃
Ef
fF fHh g/cm ,
F 37Tbar,
imploding fuel is conv
•
Limitnízisk-analytickýmodel
•
•
•
•
•
2
3/2
1/2−1/2
−1/2
2
∗∗
9/10
≈
M
u
tentatively
E
∗
2
−3/10
3
f i
≃ 67
0.22
Tbar,
5.58
EfEf fρFc f≃
µm,
5.59 f
. ,
Ze vztahu Rph ≃
(α máme
Ef ) EffF∝ 1/p
g/cm
F 1011
1/2 −1/2
−1/2
∗∗
6/5 3 ,
∗1/2 fµm,
≃
37
E
f
g/cm
5.60
ρ
R
≃
67
f
2
H
α 3 /u10
implosion
velocity,
)−3/5 mg, cožthe
Dále Ef ≈ Mfhhu imp /2 fa MFf F≃ 29h Ef (αfFConcerning
vede
na Ef ∝5.59
imp .we re
9/10 3 na1/5
Z toho plyne
silná
energie
implozní
rychlosti.
−1/2
1/2
simulated
in Chapter 3,2 in which
the ignit
2závislost
3 2/5
−3/5
ρρc∗h∗velmi
≃
E
f
g/cm
,
5.61
≃ 1011
37 Ef(αH
f∗Ff )≃−3/10
f12.8
g/cm
,
5.60
Hh F
α
Ef thefFimplosion
+ 0.25fof
This,fuel
indicates
very
Hh ag/cm
f
bare
shell.aIn
thiss
6/5
3
will come
back
to this
Ef (α 2(αf
)−3/5fF9/10
mg, g/cmimploding
5.62
Mρf∗c∗ ≃
≃ 29
1011
Ef )F−3/10
,
5.61
fuel is converted
into
internal
e
2 /2 and using eqn
with Ef 1/5
= ηE
of MJ.
d in unitstentatively
≈
M
u
E
2/5
f
f
6/5
∗∗
−3/5
2
−3/5
imp 5.62
E
f
+
0.25f
g/cm
,
5.63 front c
H
≃ 12.8
29 Eα
(αf
)
mg,
Mff ≃
H
Notice
that
the
scalings
5.58–5.62
apply,
with
different
h
F
F
f
f
5.3.3
cients,
configurationsEscaled
of x.Minimum
We can alsoene
us
3at fixed
10 value
1/5all 2/5
∗
−3/5 to
2
∝
α
/u
.
f
αin units
Ef offFMJ.+ 0.25fHh g/cm
,
5.63
HfE ≃=12.8
imp
with
ηE
f
dabove relations to obtain relations between fuel energy at ignition
Another interesting
Notice that the scalings 5.58–5.62 This
apply,
with different
†
† = coeffiindicates
a veryEfront
strong
dependence
oft
E
/η
required
f the
with Eto
f =
d in units of MJ.
cients,
allηE
configurations
scaled at fixed
value
of
x.
We
can
also
use
will come back to this
belowfor
in Sect
Anissue
expression
E † (M
†
† =
fuel
energy.
Solving
eqnmass
5.49ofwi
and use eqn 5.49
toErelate
Mc to the
/η
required
to
ignite
and
burn
a
given
f
E
fto burn a given fuel mass
5.3.3
Minimum
energy
to theenergy
energytoweburn
obtain
5.3.3 respect
Minimum
a given
† fuel mass
Anenergy
expression
for
) form
is minimum
easily
As in
the prev
f fuel
5.3.3fuelinteresting
Minimum
to the
burn
aEgiven
mass derived.
f (Mthe
Another
question
concerns
driver
energy
The
gain
is
then
written
in
approximate
Another †interestingwe
question
concerns
the 5/6
minimum
driver
energy
1/2
5/6
1/6
1/2
in
comparison
to
M
,
thus
writing
M
=
M
neglect
M
h
c
f
†
=
(2π)
(αA
)
M
F
g(x),
5.6
E
(3/2)
!
"
Another
interesting
question
concerns
the
minimum
driver
energy
†
Minimální
pro zapálení
dané
hmotnosti
- zajímavá
je
také
otázka
f/ηrequired
deg a given
=energie
Eqf/η
required
toignite
ignite
and
burn
a given
mass
of
fuel;
see
Fig.
5.5.
E† =
1/2
DT
f mass
E
to
and
burn
of
fuel;
see
Fig.
5.5.
E
H
q
M
!
M
1
DT† f
DT
ceqn†5.49
c
to of
thefuel;
fuel
energy.
Solvin
and
use
relate
Mmass
†==f E
†to
c zapálení
†
týkající
se
minimální
energie
driveru
E
=
E
/η
potřebné
k
a
hoření
≃
,
5.48
G
/η
required
to
ignite
and
burn
a
given
see
Fig.
5.5.
E
f
f
†
fE for
Anexpression
expression
forEE2f(M
(M
easily
derived.
As
in
the
previous
subsection,
ff )isiseasily
−1/2
3
−5/6
E
H
An
)
derived.
As
in
the
previous
subsection,
f
B
f
†
respect
to
the
energy
we
obtain
fxEzanedbáme
with
g(x) Opět
=
(1
−
x M) h vderived.
. TheAsfunction
g(x) subsection,
is minimum f
dané hmotnosti
paliva.
porovnání
sM
c. previous
An
expression
for
(M
)
is
easily
in
the
f
comparison
toMM
, thuswriting
writing
M
M
M≃c ≃
M
we neglectM
Mhhinincomparison
f
cthus
f =
h+
c,
to
,
M
=
M
+
M
M
,
we
†
−1/3
†
c
f
h
c
c
withneglect
6 in comparison
= 0.55, with
g(x
)writing
≃ 1.57.
The
corresponding
valu
xneglect
=x =
to
M
,
thus
M
=
M
+
M
≃
M
,
we
M
h
c
f
h
c
c
5/6
1/2
thefuel
fuel
energy.
Solving
eqn
5.49
with
and use
use
eqn5.49
5.49
torelate
relate
M
5/6
1/6
1/2 eqn
ctotothe
# energy.
\$
Solving
5.49
with
and
eqn
to
M
Vyjádřením
energie
z
rovnice c
"
"
!
!
=
(2π)
(αA
)
M
F
g(x),
E
(3/2)
f and
deg
3/5
3Solving eqn
ofuse
the eqn
ratios
energies
are energy.
DT with
f
to
the
fuel
5.49
and
5.49
relate
M
c
R
2E
p
f
h
respect
to
the
energy
we
obtain
respect
toρcthe
energy
we obtain
1−
=
(V
−
V
)
=
5.49
M
c
f
h
respect†to the
energy
we
obtain
R
†
†3p †
−1/2
3 )f −5/6
3
with
g(x)
=
x
(1
−
x
The function g(x)5.6
i
=
0.55,
and
E
/E
=
0.55
≃ .0.17
Rh /R5/6
5/6
1/2
5/6
1/6
1/2
5/6
1/2
1/6
1/2
f
h
f
(2π) 1/6
(αA
) 1/2
M
FDT
g(x),†
5.66
Eff ==(3/2)
(3/2) 5/6
dostaneme 5/6
1/2
(2π)
(αA
)
M
F
g(x),
5.66
E
deg
†
−1/3
deg
f
DT
f
=
6
=
0.55,
with
g(x
)
≃
1.57.
The
x
=
x
5.66corre
and Ef = (3/2) (2π) (αAdeg ) Mf FDT g(x),
and the minimum
is
"3/5
!
!
"are
"1/3 energies
! energy
of
the
ratios
of
and
−1/2
3
−5/6
−1/2
3
−5/6
Ef function
Rg(x)
with g(x)
g(x)==xx −1/2
The
functiong(x)
minimum
s
with
(1(1−−xxp) )3 −5/6
. .The
minimum
forfor
h is is
5.3 Limiting gain curves
•
•
•
•
Limitnízisk-analytickýmodel
= Rxh ) = (1 − x )
. †The function
for
1 − g(x). is minimum
5.50
ρg(x)
Hwith
c =
c (R
f −
†
−1/3
†
−1/3
†
5/6g(x
1/2 ) 2πp
†66 −1/3
1/2
αA
Rcorresponding
=
=
0.55,
with
g(x
≃
1.57.
The
values 5.6
=xxx= =
†
†)
=
0.55,
with
≃
1.57.
The
corresponding
values
xx =
deg
fcorresponding
†
†
†
†
3
E
≃
0.054
α
M
f
MJ,
=
6
=
0.55,
with
g(x
)
≃
1.57.
The
values
x
−1/3 = 0.55,
†)/E
F
f
f† =
/R
0.55,
and
E
≃ 0.17
R
Funkce
g(x)
je
minimální
pro
x
=
x
=
6
s
g(x
Příslušné
h
f
h ≃ 1.57.
f = 0.55
of
the
ratios
of
and
energies
are
of
the
ratios
of
and
energies
are
of
the
ratios
of
and
energies
are
Here,
eqns
5.22
and
5.25
have
been
used
to express the cold fuel density,
hodnoty poměrů poloměrů a energií jsou where
the and
mass
M
mg.
All
quantities
referring
to the
minimum a
f is
the
fuel
volume,
the
fuel
as
functions
of
the
pressure
and
of
†
†
†
†
and
the
minimum
energy
is
†
†
†
†
3
3
† =
† 0.55,
†/E =
†=
3≃
/R
and
E
/E
0.55
≃
0.17
R
/R
=
0.55,
and
E
0.55
0.17
5.67
R
/R
=
0.55,
and
E
/E
=
0.55
0.17
5.67
R
hhlabelled
f f f by
h dh.h Varying
f ff
a
dagger.
G at fixed
Ef , η, and α, it is 5.67
the fuel
internal
energy
Ef = ηE
h
† = E † /η is found by evaluatin
5/6 E1/2
†
convenient
to introduce
1/2
The
gain
for
minimum
beam
energy
Ef ≃
0.054 α Mf fF MJ,f
a minimální
energie
je energy
and
the
minimum
isisis
the
minimum
energy
andand
the
minimum
energy
† , which gives
5.53
for
x
=
x
x =eqn
Rh /R
5.51
f
5/6
1/2
†† †
5/6
1/2
1/2
5/6
1/2
Zisk proE
minimální
energii
svazku
je1/2
1/2
where
the
mass
Mf is mg. All quantities referring
to th
≃
0.054
α
M
f
MJ,
5.68
E
≃
0.054
α
M
f
MJ,
5.68
E
≃
0.054
α
M
f
MJ,
5.68
F
F
f
f
f
f
!
"
f
f † p.F3/10
as a substitute
variable for
Using the condition of isobaricity 5.24,
labelled
by a−2/5
dagger.
Econdition
−1/2has
† the ignition
eqn
5.25,
and
5.15,
one
≃mass
5740
fquantities
fHB referring
fqreferring
,
5.6
Gthe
†are
where
is
mg.
All
quantities
referring
to
the
minimum
are
where
the
MMfM
is
mg.
All
to
the
minimum
DT
†
f
F
fmass
3
where
the
mass
is
mg.
All
quantities
to
the
minimum
are
f The
α gain for minimum beam energy E = Ef /η is fou
\$
#
1/2
labelled
a dagger.
labelled
by
dagger.
3dagger.
† , which gives
labelled
byaby
a
FDT
eqn
5.53
for
x
=
x
† ††/η is found by
† =E
†E†=
pThe
=where
2πgain
.
5.52
The
for
minimum
beam
energy
evaluating
gain
for
minimum
beam
energy
E
E
/η
is
found
by
evaluating
is
in
units
of
MJ,
or
using
eqn
5.68,
the
energy
E
3
f
f
f
The gainEfor
minimum
beam
energy
E
=
E
/η
is
found
by
evaluating
x
f
f
†
, which
gives! † "3/10
eqn
5.53
for
x x=† ,†!
xwhich
gives
eqn
5.53
for
x
=
"
eqn 5.53 for x = x , which1/4gives E
Limitnízisk-analytickýmodel
•
•
•
•
•
Závislost veličin s křížkem na energii je stejná jako pro maximální zisk co se
týká mocnin jednotlivých členů.
Tyto vztahy ukazují, že zisk v tomto bodě je blízký maximálnímu zisku,
který jsme schopni dostat s použitou energií, ale je toho dosaženo pro
mnohem menší tlak.
Tím výrazně snižujeme nároky na tlak vytvořený ozařováním terče a zároveň
na symetrii tohoto tlaku.
Můžeme proto přemýšlet o tomto bodu s minimální energií jako o ideálním
pracovním bodu.
Dvě důležité veličiny charakterizující implozi slupky, která vytvoří
konfiguraci stlačeného paliva, jsou implozní rychlost a konvergenční poměr
hot spotu.
which
generates
the compressed
configuration
are the
implosion
Two important
quantities
characterizing
the implosion
of the
shell ve
which
the spot
compressed
configuration
are the
implosioncan
velocitygenerates
and the hot
convergence
ratio. Both
quantities
cityobtained.
and the hot
convergence
ratio.
quantities
Thespot
implosion
velocity
is Both
computed
by equating
the maxim
obtained.
The implosion
velocity
isucomputed
by equating
the maximum
2
fuel
kinetic
energy
(1/2)M
to
the
fuel
energy
at ignition.
f improvnítka mezi maximální
2
Implozní rychlost
můžeme
vypočítat
položení
fuel kinetic energy (1/2)Mf uimp to the fuel energy at ignition. We
then paliva
have (1/2)Mf u2imp a energii paliva při zapálení. Pak
kinetickou energii
Limitnízisk
•
then have
máme •
5.78
5
†
† †
†
=
R
/R
of
a
thin
fuel
shellfuel sh
Concerning
the
hot
spot
convergence
ratio
C
a thin
Concerning the hot spot†convergence
ratio
C0h =hR0 /Rh of
h
†
Co se týká konvergenčního poměru Ch = R0/Rh tenké slupky s3 počátečním
3 find:
with
initial
aspect
ratioratio
Ar0 , such
that
M
=
4πρ
R
/A
,
we
f
DT
r0
with
initial
aspect
A
,
such
that
M
=
4πρ
R
/Ar0 , we find:
0
r0
aspect ratio Ar0, pro hmotnost paliva máme
Mf = 4πρ
r0, a0 z ní
můžeme vyjádřit †
1/3
−1/12 1/4
f−1/12
5.79
Ch =† 22.6 Ar0 α −1/4
1/3 M
−1/4
F . 1/4
f
•
•
−1/12
1/4
7 1/4
−1/12
1/4
† u†
7
1/4
= 3.28
α MffF cm/s.
fF cm/s.
= 3.28
× 10×α10 M
uimp imp
f
Ch = 22.6 Ar0 α
Mf
fF .
5
From eqns 5.68 to 5.78 we see that the main parameters of an optimal
From eqns
to 5.78
wefuel
see mass
that the
parameters
of the
an optim
configuration
only5.68
depend
on the
andmain
the isentrope,
once
Vidíme, že hlavní
parametry only
optimální
konfigurace
závisí
jenand
na the
hmotnosti
configuration
depend
on
the
fuel
mass
isentrope,
factors f ’s parametrizing fusion physics are fixed. Only the
hot spotonce
paliva a izetropě jakmile jsou známy faktory f , které jsou spojené s fúzní
convergence
also depends on
the initial
aspect
ratio
of theOnly
fuel shell.
factors f ratio
’s parametrizing
fusion
physics
are
fixed.
the hot s
fyzikou.
In Table 5.1 we
thedepends
values of
of of
optimal
convergence
ratiolistalso
onthe
themain
initialparameters
aspect ratio
the fuel sh
configurations
for
cases
with
values
of
mass
andmain
isentrope
parameter
In Table
5.1
wenalist
the
values
of the
parameters
of optim
Konvergence
hot spotu
závisí
také
počátečním
aspect
ratio
paliva.
0.2 mg and
representative
of widely
studied
targets.
Caseof(a)mass
with and
Mf =isentrope
configurations
for cases
with
values
parame
α = 1.5 refers to the point design for the ignition experiment on the NIF
mg a
representative of widely studied targets. Case (a) with Mf = 0.2
to
laser (Lindl 1995); case (b), with Mf = 1.7 mg and α = 3 corresponds
α = 1.5 refers to the point design for the ignition experiment on the N
laser (Lindl 1995); case (b), with Mf = 1.7 mg and α = 3 correspond
•
116
•
•
•
Limitnízisk
Případ (a) s Mf = 0.2 mg a α = 1.5 odpovídá bodovému designu, pro který se
připravoval experiment na laseru NIF;
Případ (b), s Mf =1.7 mg a α = 3 odpovídá parametrům přímé fúze;
(c) s M
5 mg a α = 1.5 může být považován za reprezentanta pro
f =curves
5.3 Limiting
gain
velký terč relevantní pro reaktor.
Table 5.1 Parameters of optimal igniting configurations for targets with
fuel mass and
Ve given
všechDT
jsmeisentrope
zvoliliparameter.
fiʹ = 1 a Ar0 = 10.
(a)
(b)
(c)
DT fuel mass
Isentrope parameter
Mf
α
mg
0.2
1.5
1.7
3
5
1.5
Fuel energy
Implosion velocity
Hot spot density
Reservoir density
Fuel pressure
Confinement parameter
H.S. convergence ratio
Fusion output
Target gain
Ef†
u†imp
ρh†
Rh†
ρc†
p†
Hc†
Ch†
†
Efus
G†
MJ
cm/s
g/cm3
µm
g/cm3
Tbar
g/cm2
0.0173
4.1 × 107
143
17.5
1780
0.85
2.8
50
20.4
1179η
0.146
4.1 × 107
49
51
620
0.29
2.8
35
174
1197η
0.253
3.1 × 107
37
67
796
0.22
4.6
38
666
2637η
MJ
eff
c
•
•
•
5.4
Constrained
gain curves
and target design
Nejpřesnější křivky zisku
pochází z numerických
simulací.
Tento obrázek odpovídá nepřímé fúzi a jsou na něm vyznačeny dvě šedé
oblasti s pásy zisku (gain bands), každý odpovídající různým implozním
most accurate gain curves are provided by numerical simulations
rychlostem a teplotám The
záření.
using highly detailed physical models. An example is shown in Fig. 5.7.
It presents the gain curves which were obtained by the Livermore group
Křivky omezující každou
oblast
různým
(Lindl
1995)odpovídají
and helped to
define the
parameters oftýkajícím
the US National
se nerovností na povrchu
znázorněn
Ignitionterče.
Facility.Na
The obrázku
figure refersjeto rovněž
indirect drive
targets andbod,
shows two
odpovídající referenčnímu návrhu terče pro laser NIF se ziskem G ≈ 10.
•
Target gain G
•
Fig. 5.7 GainNová
curves for
indirect-drivez těchto informace
targets developed at the LLNL (Lindl
simulací
zisku
1995). The figure
shows two je
gainzávislost
bands
(grey areas), referring
to different values of
na energii.
the implosion velocity. The limiting gain
(broken line) scales according to
2/3
simulace
G ∝ Ed . TheDetailní
black dot corresponds
to the NIF target point design. The
že G ∝ E2/3 dot-dashed linenaznačují,
refers to a direct-drive
target designed což
at the není
Laboratory
for Laser s v souladu
Energetics. These results are based on a
modelem,
fixed hohlraum izobarickým
efficiency of ηhohl =
15%.
For smaller efficiency,
they
have
0.3to .be
kde
G
∝
E
shifted along the constant yield lines, for
example to the ηhohl = 10% line. The
hohlraum efficiency is the product of the
absorption efficiency, and of the conversion
efficiency and transfer efficiency defined in
Chapter 9 (Lindl 1995).
100
10
M
10
100
MJ
yie
ld
Jy
ield
Direct-drive
15%
=
l
! hoh
1G
J
2/3
G
∝E
NIF target
point design
10%
=
l
! hoh
uimp = 4 × 107 cm/s
Tr = 300 eV
1
0.1
1
Laser energy Ed (MJ)
uimp = 3 × 107 cm/s
Tr = 225 eV
10
the laser intensity exists to secure sufficient laser absorption (compare
Section 11.2).
The second constraint relates to Rayleigh-Taylor instability (RTI). It
sets an upper bound on the in-flight-aspect-ratio of the imploding shell,
this vhodné
lower bound
of the laser power
required
for target
Z toho nám vyplývá, žeand
není
izentropický
parametr
jako
In Fig. 5.8, the admissible aspect ratio is expressed in terms of
nezávislou konstantu. ignition.
Pro reálné
terče závisí hodnota izentropického
ablator surface finish in shell fabrication, which sets the the size of the
parametru na energii driveru.
seeds from which RTI grows.
600
V
400
=
e
0
0
3
=
Tr
200 Å
r
T
=4
První omezení je spojeno s nestabilitami laserového plazmatu a absorpcí laserového záření při vysokých intenzitách.
00
eV
35
0
eV
Ta jsou znázorněna na obrázku.
r
•
•
Při návrhu terčů musíme vzít v úvahu dvě důležitá omezení, jež jsou
důsledkem nestabilit.
T
•
Laser power Pd (TW)
•
V
0
200
=
Tr
25
eV
Tr
e
25
2
=
Surface finish
•
Fig. 5.8 The grey area in the figure
Toregion
nastavuje
horní hranici indicates the
in the
laser-energy–laser-power
plane,
where
pro
výkon
laserového
záření.
indirect drive inertial confinement fusion
(ICF) targets can be ignited. The different
curves in the figure are discussed in
Section 5.4.3 (Lindl et al. 2004).
0
0
1
2
Laser energy Ed (MJ)
3
•
•
•
•
•
V hohlraum terčích je příslušnou nestabilitou především stimulovaný
Ramanův rozptyl a s ním spojený vznik horkých elektronů. Ke
stimulovanému Ramanovu rozptylu dochází během průletu svazku záření
řídkým plynem vyplňujícím hohlraum. Tato nestabilita omezuje dosažitelnou
teplotu záření.
Pro přímou fúzi existuje podobné omezení na maximální výkon nutné pro
zajištění dostatečné absorpce laserového záření.
Druhé omezení je spojeno s Rayleigh-Taylorou nestabilitou (RTI). To
určuje horní hranici na in-flight-aspect-ratio implodující slupky a vede na
omezení zdola pro laserový výkon potřebný k zapálení terče.
Přípustný aspect ratio závisí na míře nerovností na povrchu ablátoru terče při
jeho výrobě. Tyto nerovnosti určují míru amplitud poruchy, ze které následně
roste RTI.
Vystínovaná oblast obrázku ukazuje hodnoty energie a výkonu dostatečné
pro zapálení terče včetně uvažovaných omezení z hlediska nestabilit
laserového plazmatu a hydrodynamických nestabilit.
light for direct drive. The ablation
pressure
driven derived
by therm
of
scaling
relations
in
is derived in Section 7.7.2 and scales as
The first
is derived in Section 7.7.2 and scales
as task is to relate the ab
city uimp to the drive paramet
pa ∝ Tr3.5 . 3.5
∝
T
.
p
for
indirect
drive
and
the
inte
a
r
omezující tuto oblast odpovídá teplotě záření
300 eVpressure
a
5.4.1okolo
Ablation
an
• Horní křivkaFor
Benaablators
of approximately
Mbar
are The
obtaine
for200
direct
drive.
abl
spodní nerovnostem
povrchu pressures
terče o velikosti
20 nm.light
The
is200
toablation
relate
the
ab
For Be of
ablators
approximately
Mbar
are
temperatures
300 eV.pressures
For directoflaser
the
pres
is
derived
in Section
7.7.2
and
topochopili
the drivethe
paramet
city
ulépe
temperatures
of 300
eV.
For-the
direct
laser
ablati
rychlost
slupky
abychom
inimplodující
Section
7.8.2
with
scaling
• Ablační tlakisaobtained
3.5
∝
T
.
p
for
indirect
drive
and
a
is obtained
in zabývat
Sectionvztahem
7.8.2 with
scaling
r tlakem
zmíněná omezení, budeme
se nyní
mezithe
ablačním
pa the inten
2/3
∝
(I
/λ
)
p
lightteplota
for direct
drive.
a
L
L
a rychlostí implodující slupky uimp . a parametry
driveru jako
záření
Tr The abl
2/3
For
Be
ablators
pressures
of
∝
(I
/λ
)
.
p
a
L
L
derivedfúzi.
in Section 7.7.2 and
pro nepřímou fúzi nebo intenzita IL a vlnová délka λL proispřímou
For d2
Values of approximately 100 Mbar aretemperatures
found for ILof=300
1015eV.W/cm
3.5
15w
is
obtained
in
Section
7.8.2
∝
T
.
p
Values
of
approximately
100
Mbar
are
found
for
I
=
10
Ablační
tlak
závisí
na
teplotě
záření
při
nepřímé
fúzi
jako
to Lthe maxim
λL = 0.35 µm. These drive parameters aare close
•
r
These
drive parameters are close
to the
λL =
by0.35
laserµm.
plasma
instabilities.
2/3
∝ ablators
(IL /λL ) pressures
.
ForpaBe
of
závislost
• Při přímé fúzi máme
by laserparameter
plasma instabilities.
other important
characterizing shell implosion and
temperatures of 300 eV. For d
pressure
is the
implosion
velocity.
It isofpři
obtained
from
the
ro
The other
important
parameter
characterizing
shell
implosi
Values
approximately
100
velmi důležitý
parametr
spojený
s implozí
a
tlakem
stagnaci
• Jak již víme, nation
is obtained in Section 7.8.2 w
model
presented
in Section
7.10.3. Inλvelocity.
ofIt
the
in-flight
isent
nation
pressure
is the implosion
is obtained
from
paliva je implozní
rychlost.
= 0.35
µm.
These drive
Lterms
2/3
parameter
αifpresented
and the in-flight-aspect-ratio
we
find
7
∝if ,(I
/λ
)
.Section
paAIn
model
in Section 7.10.3.
terms
of
the
in-fligh
by
laser
plasma
inst
L L in
z modelu
urychlované rakety, který budeme řešit později.
• Tu můžeme získat
for indirect
drive
parameter
αif and the in-flight-aspect-ratio
in Sec
The otherAimportant
if , we findparame
V závislosti na izentropickém parametru v průběhu letu Values
αif a in-flight-aspectof approximately
100
nation
pressure
is
the
implosi
for
indirect
drive
3/5
9/10
ratio Aif je pro nepřímou
uimp ∝ αfúzi µm. These
drive
λL = 0.35
if Aif Tr
model
presented
in
Section
3/5
9/10
laser
plasma
insta
∝
α
A
T
u
imp
if
r
parameter
α
and
the
in-flight
if
if
and for direct drive
The other important parame
a pro přímou fúzi and for
! direct drive
"1/2 for indirect drive
3/5
nation pressure is the implosi
4/15
∝
α
A
(I
/λ
)
.
u
imp
3/5
9/10
if ! if L L
to implosion velocity
"1/2
∝ αif Aif in
Tr Section 7
uimp presented
model
3/5
4/15
.
u ∝ αif Aif (IL /λL )
parameter
and the in-flight
limit to implosion velocity Here, the imp
in
in-flight-aspect-ratio Aif is restricted byαifhydrodynamic
•
•
•
•
•
•
Hodnota in-flight-aspect-ratio Aif je omezena kvůli hydrodynamickým
nestabilitám, zatímco teplota záření a laserová intenzita jsou omezeny
nestabilitami v plazmatu.
Z toho vyplývá pro implozní rychlost omezení shora a obtížnost dosažení její
vysoké hodnoty.
Závislost energie potřebné pro zapálení na implozní rychlosti - v této
souvislosti je velmi důležitá otázka, jak se mění entropie paliva daná v
průběhu komprese parametrem αif, v průběhu stagnace.
Zatímco je entropie téměř konstantní během imploze, během stagnace začíná
určitě růst v důsledku průchodu silných rázových vln palivem. To dost
výrazně ovlivňuje energii potřebnou pro zapálení a energetický zisk.
Analytický model popisující stav paliva při stagnaci v závislosti na
parametrech implodující slupky je popsán dále v knize a budeme o něm
mluvit v dalších hodinách.
Tento model je použitelný na implodující slupkové terče s rovnoměrným
Machovým číslem M0 v době, kdy se slupka uzavírá a vzniká uvnitř stlačený
plyn s téměř rovnoměrnou hustotou p.
ity solutions.
It applies
to imploding
hollow6.7.13
shells on
having
the incoming
shell
is developed
in Section
the uni
bas
at the time
of void closure
andshells
creating
a comp
number MIt0 applies
ity solutions.
to imploding
hollow
having
u
in theM
centre
withtime
almost
uniform
pressure
p. The
gas dynam
at
the
of
void
closure
and
creating
a com
number
0
trated
Fig. 6.18.
It isv found
thattéměř
the stagnation
pressure
depe
Z modelu vyplývá, in
že the
tlak
při in
stagnaci
podstatě
na gas dyna
centre
with závisí
almost
uniform
pressure
p. The
on thevzorce implosion Mach number and scales accord
implozním Machovutrated
čísluexclusively
ainmění
se
podle
Fig. 6.18. It is found that the stagnation pressure de
p on the3implosion Mach number and scales acco
exclusively
≃ 3.6M .
•
p0
•
•
0
p
3
≃
3.6M
Machovo číslo je definováno
vztahem
M0 = uimp
/c
rychlost
if , kde zvuková
0 .relation
This
scaling
is
demonstrated
in
Fig.
6.19. The Ma
p
0
během imploze (in-flight) cif je spojena s vnitřní energií, tlakem a
is defined by M0 = uimp /cif . Here the in-flight sound vel
izentropickým parametrem
implodující
slupky.
related
to
internal
energy,
pressure, andinisentrope
parameter
This scaling relation
is
demonstrated
Fig.
6.19.
The
M
2 ∝ e ∝ α 3/5 p 2/5 , making use of eqn 5.23. Identifying t
by
c
is defined
if byif M0 if= uifimp /cif . Here the in-flight sound v
Konkrétně pressure
with the
ablationpressure,
pressure, and
pif ≈
pa , we find
from e
related
to internal
energy,
isentrope
paramete
3/5 2/5
2 ∝ e ∝ α −9/10
2/5
by
c
p
,
making
use
of
eqn
5.23.
Identifyin
3
if
if
if
if
p
∝
u
α
p
.
Stagnation •
pressure
vs implosion
S použitím
ablačního tlaku jako imp
aktuálního
tlaku
během imploze (in-flight
a
if
parameters
tlaku) dostáváme pressure with the ablation pressure, pif ≈ pa , we find from
ation pressure vs implosion
eters
This relation expresses the stagnation pressure in terms of
−9/10 2/5
p meters,
∝ u3impcharacterizing
αif
pa . the implosion and all three subject to the
discussed above:
This relation expresses the stagnation pressure in terms o
ablation pressure
pa , subjectand
to the
limitssubject
on lasertoint
meters,
the implosion
all three
• thecharacterizing
hohlraum
discussed
above:temperature due to plasma instabilities,
• the implosion velocity uimp , constrained in addition by the l
aspect ratio due to hydrodynamic instabilities,
pressure
pa ,parameter
subject toαifthe
limits toonpreheat
laser
• the• ablation
the in-flight
isentrope
, subject
hohlraum temperature due to plasma instabilities,
pressure vs implosion
rgy scaling
imp if
a
This relation expresses the stagnation pressure in terms of three parameters, characterizing the implosion and all three subject to the limitations
discussed
above:
Tento vztah
tlak během stagnace v závislosti na třech parametrech
charakterizujících implozi a všechny tři parametry jsou nějak omezeny:
the ablation pressure pa , subject to the limits on laser intensity and
ablační tlak
pa je omezen
v důsledku
laserové intenzity a teploty
hohlraum
temperature
due to omezení
plasma instabilities,
hohlraumu
důsledkuvelocity
plazmových
thevimplosion
uimp ,nestabilit,
constrained in addition by the limits on the
aspect ratio due to hydrodynamic instabilities,
implozníthe
rychlost
uimpisentrope
je ještě navíc
omezena
důsledkutozávislosti
na inin-flight
parameter
αif ,v subject
preheat and
pulse
které determines
je omezenothe
hydrodynamickými
shaping, which
shock and entropynestabilitami.
evolution.
•
•
•
•
•
•
•
•
in-flight izentropický parametr αif závisí na předohřevu terče a tvarování
Withinimpulsu,
the isobaric
the stagnation
is of central importlaserového
které model,
určují vývoj
rázových pressure
vln a entropie.
ance because it is directly connected to the drive energy. Making use
of eqnsmodelu
5.64 andje5.86,
obtainstagnace
the important
lawparametr,
(Atzeni and
V izobarickém
tlak we
během
velmi scaling
důležitý
et závislostí
al. 2001). (viz. dříve) protože jeMeyer-ter-Vehn
přímo spojen s 2001;
energiíKemp
driveru
Ef
1
1.8 −6 −0.8 −1
∝
α
η .
∝
Ed =
if uimp pa
2
η
ηp
5.87
It determines how the drive energy required for fuel ignition scales with
the implosion parameters discussed above.
At this point we want to emphasize that this scaling law was first
obtained by Herrmann et al. (2001), based on a broad numerical study
and completely independent of the model derivation presented here.
illustrated in Fig. 5.8. Here we follow the derivation originally p
by Lindl (1995) for indirect-drive targets. With simple mod
by Lindl (1995) for indirect-drive targets. With simple modific
can
targets.For
Forsimplicity,
simplicity,
canalso
alsobe
beapplied
applied to
to direct-drive
direct-drive targets.
wewe
ag
front
factors
ininthe
derivation.
Westart
startby
bywriting
writingthe
thelaser
laser
pow
front
factors
the
derivation.
We
power
Možné parametry laserové energie a výkonu - začneme s tím, že napíšeme
totoimplode
aacapsule
Rcap
as cap
implode
capsule
of
laserový výkon potřebný
k implozi
kapsle sof
poloměrem
Rcapas
•
E
EEdd
Edduuimp
imp
PP
≈
≈
,
≈
dd ≈
ττimp
R
Rcap
imp
cap
kde předpokládáme,
že délka
je přibližně
rovná
době isisofofthethe
where
weimpulsu
have assumed
assumed
that
the
pulse
ordo
where
we
have
that
theimplozní
pulseduration
duration
τimp ≈ Rcap / uimp. implosion
≈ Rcap /u
is is
elim
implosiontime
timeττimp ≈
/uimp ..The
eqn5.89
5.89
el
imp
cap
imp
•
follows.We
Wewrite
write the
the absorbed
absorbed capsule
energy
asas
follows.
capsule
energy
Zbavíme se nyní závislosti na poloměru. Zapíšeme energii absorbovanou v
2
2cap
≈
4π
R
E
kapsli jako cap
Ecap ≈ 4πRcap FFrad
imp
•
•
4 is the thermal radiation flux, with σ the
where
F
=
σ
T
B
4
where Frad = σB Trr is the thermal radiation flux, with BσB t
Boltzmann
kde Frad = σBTr4 je tok
tepelného constant.
záření. Here the absorbed capsule energy Ecap is r
i
Boltzmann
constant.
Here
the
absorbed
capsule
energy
E
cap
the laser energy by Ecap = ηhohl Ed , and ηhohl is the hohlraum
the
by
= ηhohl
ηhohl
is the
caps energií
d , and
Energie absorbovaná
vlaser
kapslienergy
Ecap
je spojena
Ecapηabs
=
ηηhohl
a ,hohlraum
=
where η
efficiency.
It can
be E
expressed
aslaseru
ηEhohl
conEηd,trans
= ηabs
ηconand
ηtrans
, wher
efficiency.
Itefficiency
can be expressed
as in
ηhohl
ηhohl je koeficient efektivity
hohlraumu.
and
absorption
(discussed
Section
11.2),
ηcon
an
absorption
in Section
11.2),
and ηconresp
the X-ray efficiency
conversion (discussed
efficiency and
the transfer
efficiency,
Tento koeficient můžeme
vyjádřit
jako ηhohl
= ηabs ηand
con ηtrans, kde ηabs je
the
X-ray
conversion
efficiency
the transfer
and efficiency,
approximatir
(see Chapter 9). Taking ηhohl as a parameter
koeficient absorpce a ηcon a ηtrans jsou konverzní účinnost do RTG záření a
as
a
parameter
and
approxim
(see
Chapter
9).
Taking
η
≈
R
/u
,
we
can
write
eqn
5.90
as
τ
hohl
imp
cap
imp
účinnost přenosu energie.
\$1/3 eqn 5.90 as
τimp ≈ Rcap#/uimp , we can write
Rcap ∝# ηhohl Ed uimp Tr−4 \$ .
Rcap ∝
Inserting
1/3
−4
ηhohl Ed uimp Tr
.
eqn 5.91 for R and eqn
cap
5.83 for the implosion velo
and ηto
are ma
absorption
efficiency
11.2), and
µm. (discussed
These
parameters
areηconclose
λL =
transthe
=in ηSection
efficiency. It
can
be0.35
expressed
as ηhohldrive
abs ηcon ηtrans , where ηabs is the
X-ray conversion
efficiency
and the transfer efficiency, respectively
by
laser
plasma
instabilities.
are
absorption (see
efficiency
(discussed
in Section
11.2), and
ηapproximating
con and ηtransagain
as
a
parameter
and
Chapter
9).
Taking
η
hohl
The other important
parameter characterizing shell implosion an
the X-ray conversion
efficiency
and
the
transfer
efficiency, respectively
≈
R
/u
,
we
can
write
eqn
5.90
as
τimp
cap
imp
nation pressure is the implosion velocity. It is obtained from the
tedyChapter
psát
and approximating again
9). Taking
ηhohl as a\$parameter
#
• Můžeme(see
1/3
model
presented
inT −4
Section
7.10.3. In terms of the in-flight
is
∝
η
E
u
.
5.91
R
cap
hohl
d
imp
5.90 as
τimp ≈ Rcap /uimp , we can write eqn
r
parameter
α
and
the
in-flight-aspect-ratio
Aif , we find in Section
if
\$1/3
#Inserting eqn 5.91 for
Rcap and eqn 5.83 for the implosion velocity into
for
indirect
drive
−4
a s použitím ηhohl
Ed u
Tr the expression
.
5.91
Rcap ∝ eqn
imp
5.89,
we
obtain
for the driver power
3/5
9/10
uimp ∝ αif Aif Tr −1/3 2/3 2/5 2/3 29/15
αif ,R
Acap
)∝η
Ed αfor
Tr
.
5.92
eqn
the
velocity into
Inserting eqnP5.91
d (Ed ,for
if , Trand
hohl 5.83
if A
if implosion
dostaneme
ve
tvaru for the driver power
and
fordriveru
direct
drive
eqnvztah
5.89,pro
we výkon
obtain
the
expression
! −1/3 2/3 2/5 2/3"1/229/15
Pd (Ed , αif , Auifimp
, Tr )∝∝ αη3/5
E(I
αifL )4/15
Aif Tr . .
5.92
A
/λ
hohl
d
if
L
if
Upper limit to implosion velocity
•
•
•
omezeno
plazmovými nestabilitami
a Aif by zase
hydrodynamic
Here, the in-flight-aspect-ratio
Aif is restricted
nestabilitami,
existuje8),omezení
výkon laserového
ilities (see Chapter
while the
temperature and th
intensity are limited by plasma instabilities. This sets an upper l
the implosion velocity.
Křivky závislosti Pd(Ed) jsou zobrazeny v obrázku na dalším slidu pro různé
teploty Tr a dané hodnoty parametrů ηhohl, αif, a Aif.
Protože Tr je
hydrodynamickými
záření Pd < Pcrit.
5.4.2 Scaling of ignition energy with implosion velocity
Šedá oblast představuje možné hodnoty v rovině laserové energie a výkonu s
The
model developed
in Sections 5.1–5.3 expresses the energy
maximem daným právě
maximální
možnou teplotou.
function of parameters of the stagnating fuel. The question sti
m allowed Tr , the laser
Fig. 5.9(a) we0show curves Pd (Ed ) for different values 0of the temperature
ave to lie in1 the grey 2
1
2
3 fixed values
0
1 parameters
2 η
3α , and A0 . The grey
T
and
of
the
,
area
r
hohl
if
if
ng curve. (b) Curves
Ed (MJ)
Ed (MJ)
E
(MJ)
d
r different values of
indicates the allowed temperature region in the laser energy–power plane
atio Aif . The arrow
with an
upperje bound
to the
maximum
of the
Na 5.92)
druhou
stranu
spodnídue
hranice
daná
podmínkou,
musí dojíttemperature.
k zapálení
Pd (E
n(a)ofCurves
increasing
Aifd .) (eqn
122
5.4 On
Constrained
gain
curves
and
target
design
the other
hand,
a lower bound is obtained
by the
to ignite
Aifof
, the
ntlowed
values
thelaser
pomocí imploze
s danou
in-flight-aspect-ratio
Aif < Aifmax
. condition
Since (b)Thaving
by plasma (c)instabilities
and Aif by hydrody
r is limited
ave
to lie
in the
greyratio. The
with implosions
in-flight-aspect-ratios
Aif < Amax
e and
fixed
aspect
(a)
if . In order to
ng curve.
(c) The 600of increasing
cates
the direction
<
P
instabilities,
eqn
5.92
sets
a
limit
to
the
laser
power,
P
600 minimum
600
show this,
we
start
from
eqn
5.87,
expressing
the
driver
energy
d
c
r
r
T
T
Ed , Pd plane
is
aemaximum
allowed
Tr , therequired
laserwed for
ed
(E
)
for
different
values
of the tempe
Fig.
5.9(a)
we
show
curves
P
w
ignition
by
o
d d
l
lo
ersection
of
the
grey
l
l
a
a
Ed , Pd have to lie
400
400in the grey
400
um
u,m α , and
m
nd (b).
T
and
fixed
values
of
the
parameters
η
m
m
i
i
r
hohl
if Maximu Aif . The gre
Pd (TW)
Pd (TW)
Pd (TW)
•
A if
this limiting curve. (b) Curves
−1 1.8 −6 −0.8 A
ax
ax
lowed 5.93
l
M
M
a
E
≥
E
∼
η
α
u
p
.
d
ign
if
if allowed
hohl
imp a temperature
qn 5.96) for different
values
of
indicates
the
region
in the laser energy–power
200
200
200
Maximum
t-aspect-ratio Aif . The arrow Tr
with
an5.83
upper
bound
due
the maximum
of the admitted tempera
allowed
Aeqn
if to5.81
Substituting
eqn
for
u
and
for
p
,
we
find
imp
a
he direction of increasing Aif .
0other hand, a lower bound0is obtained by the condition to
On
the
aximum allowed A0if0, the laser
0
1
2
3
1
2
3
0
1
2
3
−1 −1.8 −6 −8.2
max
Ed (MJ)
Ed .(MJ)
Ed , Pd have to lie in the greyEE
<
A
having
in-flight-aspect-ratios
A
ηhohl
αimplosions
A
T
5.94
d (MJ)
ign ∝ with
if
if
if r
if . In or
the limiting curve.
(c) The
show this, we start from eqn 5.87, expressing the minimum driver e
Fig. 5.9 (a) Curves Pd (Ed ) (eqn 5.92)
rtion of the Efor
Pd plane
isThis
values
d , different
can
be
solvedSince
with
respect
to
temperature
Tignition
plasmaenergii
instabilities
and Aif to
bygive
hydrodynamic
r is limited
required
for
by
Abychom
vyjádříme
minimální
driveru
potřebnou
k
temperature
and fixed aspect to
ratio.ukázali,
The
d as the intersection
of
the
grey
arrow indicates the direction of increasing
instabilities, eqn 5.92 sets a limit to the laser power, Pd < Pcrit . In
zapálení
jako ames (a) and (b).
Tr . Given a maximum allowed Tr , the laser
5.9(a)
we show
curves
T ≥ T ∝ (η Fig. E
)−0.12
α −0.22
AP−0.73
5.95
d (Ed ). for different values of the temperature
•
hohl d
−1if 1.8 if−6 −0.8
parameters Ed , Pd have tor lie in theign
grey
ignfixed values
area below this limiting curve. (b) Curves d Tr and
hohl ofifthe parameters
imp a ηhohl , αif , and Aif . The grey area
Pd (Ed ) (eqn 5.96) for different values of
indicates the allowed temperature region in the laser energy–power plane
the in-flight-aspect-ratio Aif . The arrow
a Aif . imp with an upper bound
r due to the maximum of the admitted temperature.
indicates the direction of increasing
a to ignite
On the other hand, a lowerimp
bound is obtained by the condition
Given a maximum allowed Aif , the laser
parameters Ed , Pd have to lie in the grey
with implosions having in-flight-aspect-ratios Aif < Amax
if . In order to
area above the limiting curve. (c) The
−1 we−1.8
−6eqn −8.2
−0.56
0.43expressing
0.02 −0.74
show this,
start from
5.87,
the minimum driver energy
allowed portion of the EdLaser
, Pd plane is ign
d if
ifif
ign required
if
hohl
if
hohl
if r d
for ignition
by
determined as the intersection of the grey
areas of frames (a) and (b).
•
E ≥E
∼η
α u
p
.
Substituting this last expression in eqn 5.92, we finally find that the laser
Substitucí za p Substituting
a u , řešením
pro5.83
T a for
substitucí
do předchozí
nerovnice
profind
eqn
u
and
eqn
5.81
for
p
,
we
power has to satisfy
výkon dostaneme
P
≥ PE (E∝, α
η
η , Aα ) ∝ A
TE .α
A
.
5.96
−1 1.8 −6 −0.8
E
≥
E
∼
η
αif respect
uimp pa laser
This equation
implies
that
the
required
power decreases
as5.93
the
d
ign
hohl
This can be solved with
temperature
to give
in-flight-aspect-ratio
Aif iseqn
increased
the5.81
laser
is reduced.
Substituting
5.83 for uimpand
and eqn
for penergy
a , we find
−0.22
−0.73 values of A
−0.12
) are
shown
in
Fig.
5.9(b)
for
different
Curves Pign (E
Tr d≥
Tign ∝ (η
A
.
if
d ) −8.2 α
−1 hohl
−1.8E−6
•
Tato rovnice nám říká, že potřebný laserový výkon klesá, když se zvyšuje inflight-aspect-ratio Aif a snižuje se laserová energie.
•
Protože Aif je omezeno hydrodynamickými nestabilitami na určitou
maximální hodnotu Amax, vyplývá z toho, že laserová energie a výkon musí
být v oblasti nad křivkou Pign(Ed) odpovídající Aif = Aifmax.
•
Povolená oblast v rovině laserové energie a výkonu je znazorněna v obrázku
zelenou barvou.
Gf
which
gives pressure,
assumption of uniform density ρ = ρh = ρmodel,
of
uniform
c
5.5.1 Isochoric assemblies with hot spot
3
1000
! "7/18
and the use of a different ignition condition.
Ef
qDTthat is, 1a mg
As
we
shall
discuss
in
Chapter
12,
an
isochoric
assembly,
con∗
Gain curves for different masses can be generated
by proceeding as in
Gf = 0.0828
3
7/6
1/2
2/9 density,
4/9
figuration
in
which
hot
spot
and
cold
fuel
have
the
same
uniform
α
hotisspotem
- vbyizochorickém
uspořádání
the Izochorické
isobaric case.uspořádání
The limitingsgain
computed
the hot A
spot
ignition
B FDT Hh je
deg H
be used
to modelpaliva
the fast-ignitor
Here an analytical
model
hustota
hotcan
spotu
stejná a vscheme.
obou oblastech
rovnoměrná.
model,
which
gives
for the gain použít
is developed
by rychlého
aAs
slight
of
the isobaric
Tento
model
k popisu
zapálení
fast-ignition.
we modification
have
seen- in
Section
4.2.2,model
a suitable ig
Fig.
5.10 můžeme
Gain curves at constant
mass
! "Rosen
100the obvious
2 )keV. Us
7/18 ignition
(Kidder
1976b;
Bodner
1981;
and
Lindl
1983),
with
and limiting gain for qinitially
isochoric
R
T
=
6
(g/cm
choric
is
ρ
E
h
h
h
50
100
DT
f
∗
GEnergetický
0.0828
5.97změnou
assumption
uniform
density
ρ Section
=jako
ρ. h =dříve
ofmenší
uniform
pressure,
zisk
získat
obdobně
jen
scan
assembly
with
α můžeme
= 1/2
2.ofThe
circles
f =
in
5.1.2,
we
write
3
7/6
2/9
4/9indicate
Ef
α
A
H
F
H
simulation
results
(Atzeni
1995).
and
the
use
of
a
different
ignition
condition.
oproti izobarickému modelu, která spočívá v předpokladu rovnoměrné
Křivkyziskuproneizobarickoukonf.
•
•
deg
B
DT
h
for different
masses
by
hustoty ρ = Gain
ρh =curves
ρc místo
rovnoměrného
tlaku
jinéproceeding
podmínkyas in
FDTcan
=be
phagenerated
Rpoužití
h ≃ 46 Tbar µm = 3.8F̂DT .
As zapálení.
we have seen
in Section
a suitable
ignition
condition
forhot
iso-spot ignition
the isobaric
case.4.2.2,
The limiting
gain
is computed
by the
Thethecorresponding
density
is
6 (g/cm2 )keV.
Using
same
notation
as
choric ignition
is ρh R
h Th =
model,
which
gives
This condition applies, for example, to the
Limitní5.1.2,
zisk vypočtený
s podmínkou zapálení
z hot spotu
je in Section
we can write
!
"!
Fig. 4.5terče
with
! "7/18
Hh
FDT
∗
E
q
DT
f
∗
ρ ≃ 67.2
G
=
0.0828
. 0.55.98
5.97
2
f
2
FDT = ph Rh ≃ 46 Tbar µm 7/6
= 3.81/2
F̂DT2/9
. T 4/9
3
g/cm
46 Tbar
µ
.
Adeg HB FDT Hhh = 12α keV, and Hh = 0.5 g/cm
•
izochorické
zapáleníforje použitelná
zapálení
ve tvaru Gain
curves
have
also been obtained b
applies,
example, podmínka
to
the reference
point
B of
•ThisProcondition
Equation
5.97
then
becomes
As we have seen in Section 4.2.2, a suitable ignition condition for iso-
2)keV. Tato podmínka odpovídá referenčnímu bodu B ρ4.5
= 6 (g/cm
hRhT
h gain
An example is shown in Fig. 5.10, prese
Fig.
with
Limiting
of
initially
the same
notation as
choric ignition is ρh Rh Th = 6 (g/cm2 )keV. Using !
"
isochoric
DT assemblies
7/18
of
the
fuel
mass
M
and fixed isentrope
f
E
in Section 5.1.2, we can write
f
4
2.18
× is
10fitted
T h = 12 keV, and Hh = 0.5 g/cm2 . G∗f =
5.99 .
gain
line
by
3
α
FDT = ph Rh ≃ 46 Tbar µm = 3.8F̂DT .
5.98
"
!
0.4
Equation 5.97 then becomes
Ef
∗
Limitní zisk ze simulací můžeme aproximovat Gf = 19,200
, B of
3
This condition
applies,
for
example,
to
the
reference
point
! "7/18
α
Ef
∗
4 with
Fig.
4.5
.
5.100
Gf = 2.18 × 10
3
α
in excellent agreement with eqn 5.97, o
2
Th = 12 keV, and Hh = 0.5 g/cm
5.99of eqn 5
HB =. 8.5 g/cm2 . The accuracy
•
Křivkyziskuproneizobarickoukonf.
124
jako v izobarickém případě závisí zisk na energii driveru E
• 5.5Stejně
Gain curves for non-isobaric configurations
d
a izentropickém parametru α v kombinaci Ed/α3.
Gf
•
Důležitá informace je, že v oblasti energií zajímavých z hlediska IFE je zisk v izochorickém případě vyšší než v izobarickém.
•
Maximální zisk pro danou energii Ed dostáváme v izochorickém případě pro nižší hustotu než v izobarickém, protože tlak v chladném palivu nemusí Fig. 5.10 Gain curves at constant mass
být takisochoric
vysoký, jako tlak v hot spotu. and limiting gain for initially
assembly with α = 2. The circles indicate
simulation results (Atzeni 1995).
•
10,000
! =2
19,200 [Ef (MJ)/!3]0.4
10 mg
3 mg
1000
1 mg
100
50
100
500
1000
Ef (kJ)
Důsledkem toho je, že pro kompresi paliva potřebujeme méně energie. Tento
pozitivní efekt překonává jiné negativní efekty v důsledku menšího udržení
corresponding
density is
paliva a náročnějších The
podmínek
pro zapálení.
!
"!
"1/2
Hh
FDT
−1/2
∗
ρ ≃ 67.2
E
.
f
2
0.5 g/cm
46 Tbar µm
Gain curves have also been obtained by 1D simulations (Atzeni 1
Křivkyziskuproneizobarickoukonf.
•
•
•
•
•
•
Objemové zapálení opticky tlustého DT paliva - při objemovém zapálení
máme homogenní hmotu paliva Mf, která se zapálí spíše v celém svém
objemu, než od centrálního hot spotu.
Velké a husté terče mohou být tak opticky tlusté, že k zapálení DT dochází i
při teplotách 1–1.5 keV. Zapálení při takto nízkých teplotách částečně
vynahrazuje nevýhodu, kterou představuje nutnost ohřevu celé hmoty paliva.
Zlomek vyhoření paliva nezávisí jenom na ρR ale podmínky zapálení a
hoření závisí také na střední volné dráze záření, která je úměrná ∝ ρ2R. Proto neexistuje pro objemové zapálení žádný jednoduchý analytický model.
Energetický zisk terče není funkcí jenom hmotnosti terče Mf a parametru
udržení Hf = ρf Rf , ale také počáteční teploty.
Když je překročena minimální hodnota teploty pro zapálení, zisk rychle roste
a dosahuje maxima pro optimální teplotu Topt = Topt(Mf,Hf), která bývá jenom
mírně nad hraniční minimální teplotou pro zapálení.
Při dalším zvyšování Tf se zisk již nezvyšuje, nejprve je konstantní, protože
větší spálení paliva vynahradí vyšší investovanou energii, a později zisk již
znatelně klesá.
ature Topt = Topt (Mf , Hf ) only slightly above such a threshold. Increasing
limiting gain (Atzeni 1995).
Tf further, we first observe a plateau, where higher burn-up counterbalances the higher internal energy, and then a significant gain decrease.
From Fig. 5.11, we see that substantial gain is only obtained for optically thick systems and
very large
of the
confinement
parameter
Vidíme,
že values
vysoký
zisk
dostáváme
jenom pro opticky tlusté terče a velmi
in turn
very high
densities. Similar
gain
Hf > 10 g/cm2 , implying
vysoké
hodnoty
parametru
udržení
Hfcurves
>10g/cm2, cožthe
značígain
vysoké
at hustoty.
the optimal tempe
have been published in the early years of ICF research by Fraley et al.
spark-isobaric and spark-isocho
(1974) for microgram-sized
fuels.
Results
for
milligram-sized
fuels
were
Jako v případě izobarického a izochorického systému zapáleného od hot
subsequently presented by Basko (1990).
providess the
limiting
gain, whic
spotu,
i
zde
existuje
limitní
zisk,
který
je
možné
dosáhnout
danou
energií
In analogy with the previous cases of central ignition, also for volIt is found that the limiting fue
ume ignition we can driveru.
draw gain curves for a constant mass of fuel. In
Fig. 5.12 we show a set of such gain curves, computed by a set of full 1D
∗
0.16
Tento limitní
zisk
se
dá
dobře
aproximovat
vztahem
Each
point
in
these
curves
represents
G
=
1000
E
,
f
f
Limiting gain of volume-ignited fuel
Křivkyziskuproneizobarickoukonf.
•
•
•
126
5.5 Gain curves for non-isobaric configurations
6
4
30 50
10
100
200
400
1000
Gf
600
2
=
l
P
800
f
R
Tf (keV)
Gf < 1
1
Gf < 1
Fig. 5.12 Volume ignition. Gain curves
for different values of the fuel mass, for
0.5
volume-ignited assemblies. The circles
1 hydrodynamics
4
10
represent results of 1D
Rf (g/cm2)
simulations. The thick line represents!fthe
limiting gain (Atzeni 1995).
1000
40
100
100
with Ef in units of MJ. This
the0.16 range of 1 ≤ Mf ≤ 10
1000 [Ein
(MJ)]
f
0.2 ≤ Ef ≤ 4 MJ. It even appli
fuel energies up240tomg30 MJ (Atze
As a variation of the homog
consider assemblies with more
for example, as obtained from s
1 mg ics
3 mg
10 mg
equations
(compare Sectio
Johzaki et al. (1998) refer to thi
Detailed
numerical
simulations
1000
10,000
has Ethe
same functional depend
(kJ)
f
coefficient replaced by 1700.
Křivkyziskuproneizobarickoukonf.
•
•
•
Porovnání různých konfigurací a paliv - pro jednoduchost budeme
porovnávat limitní zisky. Ty jsou definovány jako maximální zisk, který
můžeme dostat při dodání daného množství energie palivu se zanedbáním
problémů se stabilitou a symetrií.
IFE vyžaduje Gf ≥ 10/(ηdη). Pro ηd = η = 0.1, to odpovídá Gf ≥ 1000.
Pro terče zapálené od hot spotu, kde limitní zisky závisí na izentropickém
curves
for non-isobaric
configurations
parametru budeme kreslit dvě křivky5.5proGain
každou
konfiguraci
odpovídající
α = 1 a α = 2.
Křivka pro objemové zapálení odpovídá optimální zápalné teplotě.
Obrázek
ukazuje, že v oblasti energií •Fig. 5.13
Comparison of limiting gain
1
!=1
Isochoric
10,000
!=2
α=1
Gf
•
curves 50
for different
of
≤ Ef ≤initial
500configurations
kJ relevantních
pro IFE the compressed DT fuel assembly (Atzeni
pro
stejné
hodnoty
α, izochorické 1995). aThe
figure
also shows
the limiting
gain foruspořádání
a T-poor target paliva
(tritium content
má 2–3 krát vyšší zisk FT = 0.5%) which does not require
než
izobarické.
external
tritium
breeding; see Tato výhoda se může zdát
Sectionnepodstatná.
12.3.3 and AtzeniNa
and Ciampi
druhou stranu, pokud (1997). The value of the isentrope
budeme
uvažovat
energii, kterou parameter
α is indicated
for eachocurve.
Isobaric
!=2
1000
Volume (optimal)
T-poor (! = 1.3)
100
10
100
Ef (kJ)
1000
potřebujeme na dosažení daného zisku, pak zjistíme, že je 7–10 krát nižší!
curves for each configuration, corresponding to α = 1 and α = 2. T
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