INRODUCION O INERIAL CONFINEMEN FUSION R. Bett Lecture 1 Formula or hot pot temperature Reved dyamc model ad gto codto Etropy
he ormula below wa derved Lecture 9. It repreet the maxmum value o the cetral (r=) hot pot temperature MAX.81 κ /7 ( ρ ) /7 6/7 h Subttute cotat to d value o temperature Sptzer coducto or lλ=5 κ = κ = 3.74 1 = 5/ 69 5/ Joule = 3.74 1 (1.6 1 ) = 1.1 1 m MAX 69 5/ 16 5/ 3 5/ 1 1 kev kev Maxmum cetral temperature ( ke ) 6.4 6/7 / ( ) /7 km ρ g / cm 3
he eutro averaged temperature the temperature averaged over the uo rate. It the temperature erred by the uclear dagotc expermet dtd σv dtd σv 4 4 = Yeld dtd σ v 4 Neutro averaged temperature dtp() t dtp() t σ v d σ v d
Ue the depedece o reactvty (vald betwee 8-3ke) Deto σ v = () () dtp t ds dtp t d = S dtp() t ds dtp() t () t 1 /5 (1 x ) = = 1.15x d () t ()3 t x dx.7 () t () t dtp() t ds.7 dtp() t () t () t dtp() t ds dtp() t () t =
Ue the tme depedet varable ((t), P(t), R(t) or olume(t)) derved Lecture 8 ad 9 ˆ () = /7 t 7 ˆ ˆ t ˆ π + 4 + + 1 tˆ + (1 t ) Arc a[ t ] 5/ ˆ ˆ Pt ( ) (1 + t ) ˆ ˆ 3/ t ( ) (1 + t) MAX.7 dtp() t () t () t.47 =.6 MAX dtp() t () t Neutro averaged temperature or ( ke ) 3.8 ρ /7 km/ ( ) g / cm σ v = S 3 6/7.78
Ue the 3 depedece o reactvty vald betwee 3-8ke σ v = S 3 1 /5 (1 x ) = = 1.15x d () t ()3 t x dx.53 () t () t dtp() t dtp() t d d 1 /5 (1 x ) d = () t ()3 t x dx =.7 () t () t 1.15x.53 dtp( t) ( t) ( t).53 =.68 MAX.7 dtp() t () t () t 3 Neutro averaged temperature or σ v = S km/ /7 ( ke ) 4.35( ρ ) g / cm 3 6/7
Ue the 4 depedece o reactvty vald below 3ke σ v = d S 4 =.43 ( t) ( t) 3 3 d =.53 ( t) ( t) dtp() t 3 dtp() t 3.43 dtp( t) ( t) ( t).58 =.74 MAX.53 dtp( t) ( t) ( t) Neutro averaged temperature or ( ke ) 4.76 ρ /7 km/ ( ) g / cm σ v = S 3 6/7 4 d d
he mot relevat rage or curret ICF expermet 3-8ke. We wll ue the 3 depedece σ v = ρ =.19 g / cm = 37 / S Ue the 3 depedece ormula to etmate emperature or OMEGA mploo km km/ ( ) /7 ( ke ) 4.35 ρ 3.ke g / cm = 3 6/7 3 ρ =.16 g / cm = 45 / km km/ ( ) /7 ( ke ) 4.35 ρ 3.7ke g / cm = 3 6/7
Alo the areal dety ote meaured ug the eutro pectrum. Neutro Averaged Areal Dety. Same average a temperature ρ ( ρ ) dt d ρ ( ) 4 Yeld σv dtp() t d dtp() t () t () t ρ = h hell approx ( ρ ) ( ρ ) dtp() t d dtp() t () t () t ( ρ ) M h = 4 π Rt () Rt ˆ() From Lecture 8 σ v = ˆ ˆ Rt () = 1+ t S 3 ρ =.88( ρ ) Neutro Averaged Areal Dety
Moded Dyamc Model cludg emperature ad µ depedece o reactvty M h d R = 4 dt π RP Shell mometum σ v = S µ 5 ε η 5 d ( PR ) S P R Hot Spot Eergy η = µ dt 4 d P 3 5/ R.86κ R dt 1 /5 (1 ) () ()3 x η η η η ds = t t x dxs = () t () t S 1.15x 1 /5 η (1 x ) S = 3S x dx 1.15x Hot Spot emperature
We have already olved th model wthout alpha ad oud the o-alpha tagato value. Ue thoe to rewrte dmeole parameter Rˆ R ˆ t = t = Pˆ R o R P = ˆ P o New dmeole model ˆ d R dtˆ = ˆ ˆ RP d ( PR ˆˆ5) ˆ ˆ5 ξ ˆ P R dtˆ η ξ New parameter ξ P R ε S ( ) 4 η ˆ ˆ 3 ˆ 5/ ˆ d P R R dtˆ ˆ
Same oluto a beore: d crtcal value o ξ that lead to a gular oluto χ ( ) ξo o R ε S P 1 crt crt ξ = 4ξ η Igto codto P From Lecture 8 From Lecture 9 χ R ( ρ ) = h o 1 κ ( ) /7 6/7 ρ h /7 /7.86 New Igto codto term o areal dety ad mploo velocty ξ ε S 1 = 1 6 1 1+ η + 7 ( ) 7 o η ρ crt h η/7 crt η/7 ξo 4κ ξo.86
Rewrte gto codto ug temperature tead o velocty =.86 1/3 7/6 1/3 κ ( ) ( ρ ) 1/3 h ξ ε S /3.95 ( ) ( ) 7/6 + η χo = κ ρ > 1 ξ 1/3 crt crt h 4ξo Replace velocty wth t ucto o temperature From Phyc Plama 17, 581 (1) η = 1.1 Rewrte gto codto S 7.5 1 m ke 6 3 1 3.1 New Igto codto term o areal dety ad hot pot temperature ξ ε S /3.95 ( ) ( χ ).18 κ ρ 1 ξ = 1/3 crt crt h 4ξo
χ Rewrte ug eutro averaged quatte.53 /3.18 ξ ε S o 1/3 ρ κ crt crt ξo 4ξ.88.53 χ =.95 1 ξ 1. ρ crt ξo g / cm ρ =.88( ρ ) /3 o ke 4.18 New gto codto wth eutro averaged areal dety ad temperature ρ g / cm Igto Rego the curret ICF program amg at ke 4ke ρ 1 g / cm ke
Areal dety v emperature: IGNIION PLOS Smulato (LILAC) From Phyc Plama 17, 581 (1) Our model ρ g / cm Igto Ft o mulato ke
1D Dyamc o a mplodg hell: Etropy Shock Wave
What the etropy o a deal ga/plama? he etropy S a property o a ga jut lke P, ad ρ p S cv l cot = c / 3 ρ l [ ] p ρ = = cot 5 v 5/ 3 We call the adabat. It ha othg to do wth alpha partcle or alpha heatg. cv the pecc heat at cotat volume For a deal ga c v = 3(1 + Z) m
Startg rom ma, mometum ad eergy coervato or deal ga ρ + ρu = t u ρ + u u = P+ µ u t vcoty ε + u + P = + w w t ( ε ) κ rad Heat coducto Source ad k 3 1 ε = P+ ρu Oe ca combe the above equato to how that (try to do th): S DS u κ ρ u S ρ µ + = = + + ource/k t Dt
he etropy/adabat S/ chage through dpato or heat ource or k ρ S DS u κ u S µ + = = + + ource/k t Dt I a deal ga (o dpato o vcoty ad o heat coducto) ad wthout ource ad k, the etropy/adabat a cotat o moto o each lud elemet DS Dt = S, = cot p ~ ρ 5/3