INTRODUCTION TO INERTIAL CONFINEMENT FUSION
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1 INTODUCTION TO INETIAL CONFINEMENT FUSION. Bei Lecure 7 Soluion of he imple dynamic igniion model
2 ecap from previou lecure: imple dynamic model
3 ecap: 1D model dynamic model ρ P() T enhalpy flux ino ho po 5 PV b dt hea flux ou of ho po κ dr Low deniy ho po Shell Hea-flux-ou Enhalpy-flux-in -κ T Ho po 5/PV b () P() Hea loe from ho po caue ablaion off he inner hell urface + Blow-off velociy V b Ablaed maerial blow off ino ho po wih velociy V b (blow-off velociy) V a Ablaion velociy Ablaion fron penerae ino he dene hell wih velociy V a (ablaion velociy) Becaue he hea loe are recycled back ino he ho po hrough ablaion, he ho po preure (energy per uni volume) i unaffeced. However hoe looe end o decreae he emperaure while increaing he deniy o ha P ρt i unchanged
4 ecap: 1D model dynamic model P P () From momenum equaion in ho po 3 1 d ( 5/3) /3 PV P Π T dv ε α S P V V d 16 f V () ( ) () () Energy equaion Π( T) ε α S 16 f Neglec radiaion loe and keep alpha heaing for T>6keV Aume <σv> quadraive behavior wih T for 8<T<3keV σ v ST f For pherically ymmeric imploion: V 4 π ε α 5 d ( P ) S P d 4 f 1D ho po energy equaion
5 ecap: 1D model dynamic model Two equaion for he wo unknown () and P() M h d 4 d π P 1D hell momenum (Newon law) d ( P ) ε α S P d 4 f 5 5 1D ho po energy equaion Iniial condiion a he ime 0 of peak imploion velociy (V i ). Saring from ha ime he hell op being acceleraed and ar coaing and deceleraing. The radiu a he ime of peak velociy i denoed a * (0) d d * V P(0) P* (0) i Iniial condiion
6 Soluion of he 1D dynamic model
7 ewrie he equaion in dimenionle form uing he iniial condiion * and V i * V i Dimenionle equaion and iniial condiion d ε M 0 P hvi ε 0 3 4π P * * d ( P ) ε P 5 5 * ε ε α d (0) 1 (0) 1 P (0) 1 f P 4V S P * * i P P *
8 Solve for large ε 0 >>1 and find criical value of ϒ ha yield a ingular oluion α d d 3 ϒ d ε 0 d ϒ εε 0 d d 1 (0) 1 (0) 1 (0) ε ϒ Soluion for 0 ε ( ) P ( ) almo ingular almo ingular
9 Soluion of he 1D dynamic model: Ue parameer o derive dimenionle equaion
10 Ue a differen approach o rewrie he equaion in dimenionle form Fir olve he equaion wihou alpha M h d 4 d π P d d ( P 5 ) 0 P P 5 5 * * P P 5 * * 5 Subiue oluion of energy equaion ino momenum equaion M h d 5 1 4π P * * 3 d
11 d d 4π P 1 d d d M d 5 * * 3 h Muliply by d/d 5 π P * * 1 d d d 1 d d M d h Bring under derivaive 5 P * * Vi h d 4π 1 1 d M 0 Inegrae in ime A agnaion d/d0 V 5 4π P * * 1 1 i Mh * 4π P 5 * * MhVi Small becaue agnaion radiu << iniial radiu
12 P P 5 5 * * Ue and rewrie 5 4π P 3 4π P MhVi MhVi ewrie 4π 3 1 P M V 3 3 h i Thi i imply energy conervaion: hell kineic energy convered ino ho po inernal energy 4π 3 3 ag 1 hell P EHS MhVi Ekin ( 0) 3
13 Previou agnaion condiion and P were derived wihou alpha, call hem Ue hem o rewrie he equaion in dimenionle form V i P no α P ( ) 3 P 4π P M V h i New dimenionle equaion d P ( ) d P ξ P 5 5 ξ εα S f P ( / Vi) P 4 V 4 / ( ε S ) i α f
14 New model wih iniial condiion d Combine P ( ) * 1 (0) >> no 1 α d (0) 1 P * 5 (0) << no 1 P α P d P ξ P d d 3 d ξ 5 5 Iniial condiion where i an arbirary mall number P * 5 (0) << no 1 P α P << 1 d 3 (0) << 1
15 ewrie energy equaion dp 1 5 d + P ξ P Define new variable and find linear equaion Φ 1 P dφ 1 d 5 Φ ξ no α Soluion aring from he agnaion ime d Φ ξno α
16 Soluion aring from agnaion ime. Aume alpha dominan afer agnaion. A agnaion, he condiion are approximaely righ (aumpion) d Φ ξno α Condiion for ingular preure or Φ0 5 5 d d ξ 0 1 ξ 5 5 ewrie ingulariy condiion 1 ξ d 5 ( d / d )
17 Expand abou agnaion (auming alpha are dominan afer agnaion) ( ) 1+ d d ( ) ( 1) ( ) d d 1 ξ Eimae inegral 1 ξ ( ) d 5 ( d / d ) d ( )( 1) Since inegrand decay rapidly afer agnaion hen exend inegral o infiniy d ξno d α ( )( 1) ( ) 1 ( 1)
18 Approximae ingulariy condiion 1 ξno α ( ) d 5 1 ( 1) Ue approximaion ha alpha are relevan afer agnaion (no a agnaion) ( ) d 35π ( 1) 18 Igniion condiion ξ ξ 18 35π 1.7 εα S f P ( / Vi) P 4 V 4 / ( ε S ) i α Noe: hi i a Pτ erm a in Lawon crierion f Noe: hi i ame erm a in previou Lawon crierion
19 Igniion condiion χ > 1 χ ξno α P ( / Vi ) > / ( ε S ) α f I /V i he confinemen ime? The hell confine he ho po preure. Sar from he hell Newon law M h d 4 d π P Scale i o find he confinemen ime a agnaion M ~4 π P τ h M h τ ~ 4 π P
20 ewrie he confinemen ime τ ~ M M V h h i i π i 4πP V 4 P V 1 MhVi 4π 3 3 P 3 Ye indeed, he confinemen ime cale a he agnaion radiu over he imploion velociy τ ~ Vi aio of iniial hell kineic energy wih ho po inernal energy a agnaion ~ 1
21 Igniion condiion
22 ewrie he igniion condiion χ P ( / Vi) > / ( ε S ) α f Manipulae erm ( ) 3 ( ) 3 P MhVi P h i Vi V M i MhVi hvi P M V ( ) ( ) Ue energy conervaion: iniial hell kineic energy agnaion ho po energy M V 4π P h i ( ) 3 P 1 M V V 4π h i i ( )
23 M ewrie hell ma a agnaion for a hin hell of hickne and radiu 4πρ ( ) no no no h P 1 MhVi ρ V 4π i Subiue ino Obain form of Pτ wrien in erm of velociy and areal deniy ( ) P 1 M V V 4π h i i V i ewrie igniion condiion in erm of velociy and areal deniy ( ) χ P ( / Vi) / ( ε S ) α f ρ Vi / ( ε α Sf ) Ue ICF uni: g/cm and km/ 10ρ / ( ε α f ) 3 V ( g / cm ) i( km/ ) S
24 Subiue value of S f S f 4 10 / / 1keV m kev 16 J S f σ v T ε α 3.5MeV T ( kev ) Igniion condiion for: (a) hin hell, (b) T>8keV where <σv>~t, (c) all alpha depoi heir energy inide he ho po ρ V ( / ) ( / ) 00 g cm i km Shell areal deniy Imploion velociy Igniion require areal deniie of order 1g/cm and velociy of hundred km/
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