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 had been develope 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 odhadnout 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 dopadající 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. dopadajícího laserovéhothe 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 terčiasku 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 the adeuterium 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 Chladné palivo: respectively. 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. V takovém případě 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 Proto budeme chladné 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 Adeg 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 Izobarický případ:= tlak p pro izobarický případ 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 had been 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 ready to 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. radiusDT = 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. Spark radius 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 radius 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 aboutkteré 20 kJ, which amounts to 1 MJ of fusion energy per shot or 1 MW odhady, 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- Fig. 5.4 Gain predictions published by 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. Additional 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. Additional 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 αAdeg 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 $ "3/5αAdeg ! 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 αAdeg 3phave R and E R f h H = ρ (R − Rthe . functions 5.50 fuel volume, and!the radius as functions of theofpres c c f h) = the!fuel volume, andfuel the fuel th "3/5 !1 −radius " 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 Rh )radius = 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 = Rradius 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 radius 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 qADT F 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 ∗ leads 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 have already thatconfiguration the ratio of spark to fuel radius 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 already found ratio of spark to fuel radiusisisused for ign 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 oftoradii 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 αAdeg 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 radii 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 radii and energies are of the ratios of radii and energies are of the ratios of radii 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 radius 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 be read cityobtained. and the hot convergence ratio. quantities can be readily 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 f DTR03/ADT 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; Případ (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 případech 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 Hot spot radius 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 Omezenékřivkyziskuadesignterčů • • • 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 předpokladům (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 leads to aponechat 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. Omezenékřivkyziskuadesignterčů 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 Omezenékřivkyziskuadesignterčů • • • • • 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 first task is200 toablation relate the ab For Be of ablators approximately Mbar are temperatures 300 eV.pressures For directoflaser irradiation, 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 impirradiation, 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 = admitted by0.35 laserµm. plasma instabilities. 2/3 ∝ ablators (IL /λL ) pressures . ForpaBe of závislost • Při přímé fúzi máme by laserparameter plasma instabilities. Theadmitted 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 admitted 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 admitted by 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 Omezenékřivkyziskuadesignterčů Omezenékřivkyziskuadesignterčů • • • • • • 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 výhradně 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 . Omezenékřivkyziskuadesignterčů • 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 Omezenékřivkyziskuadesignterčů 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 vyjadřuje 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 flight-askect-ratio, 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 radius Rcap as cap implode capsule of radius R laserový výkon potřebný k implozi kapsle sof poloměrem Rcapas Omezenékřivkyziskuadesignterčů • 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 Theradius radiusinineqn 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 radττimp ,, Ecap ≈ 4πRcap FFrad imp • • 4 is the thermal radiation flux, with σ the where F = σ T rad 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 theadmitted 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 Omezenékřivkyziskuadesignterčů • • • omezeno plazmovými nestabilitami a Aif by zase hydrodynamic Here, the in-flight-aspect-ratio Aif is restricted nestabilitami, existuje8),omezení na radiation 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, že admitted 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 radiation 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 Omezenékřivkyziskuadesignterčů 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 of the radiation d , different can be solvedSince with respect to temperature Tignition byradiation 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 . to radiation 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 Omezenékřivkyziskuadesignterčů • 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, instead 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 a chladného 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 ρc instead 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 IMPLO-upgraded simulations. 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