A review of the dense Z-pinch

More documents

Recommendations

Info

Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review Figure 34. Trajectories calculated for the slug model with constant current [261]. trajectories of both the piston and the shock front. In many ways this aspect is similar to Kuwabara [262] who resorted to a numerical integration. Potter proceeds further to show that there is a final equilibrium, in which the ratio of the pinch radius to its initial radius becomes purely a function of Ɣ, the ratio of the principal specific heats. The model assumes again a sheet current piston, and a thin shock thickness with a uniform pressure between the piston and the shock. The thinness of the shock means that planar Rankine–Hugoniot conditions can be applied to the shock transition. For a strong shock these conditions directly relate the fluid velocity u s the density ρ s and pressure p s immediately behind the shock to the filling density ρ 0 and the speed of shock v s (= dr s /dt) given by u s = 2 Ɣ +1 v s, (4.4) ρ s = Ɣ +1 Ɣ − 1 ρ 0, (4.5) p s = 2 Ɣ +1 ρ 0vs 2 . (4.6) The condition of uniform pressure between the shock and piston relies on the sound transit time being short compared with the time taken for the shock to reach the axis. This condition becomes ( Ɣ +1 1 ≫ 1 − r ) [ s Ɣ [2Ɣ(Ɣ − 1)] 1/2 r p Ɣ +1− rs 2/r2 p ] Ɣ Ɣ−1 , (4.7) where the sound velocity immediately behind the shock is taken. As the shock converges to the axis (r s /r p > 0.1) the right-hand side is as high as 0.5 for Ɣ = 5/3 and 0.3 for Ɣ = 1.4. In addition, the sound speed decreases to about half its value towards the piston due to the temperature profile obtained by this model with no thermal conduction. Thus the assumptions are perhaps questionable. An interesting feature of the slug model is that because of the assumption of spatially uniform pressure between the piston and shock the piston comes uniformly to rest as the shock reaches the axis as shown in figure 34, and there is no reflected shock or bouncing. The ratio 59
Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review of the final piston radius r p to initial radius r 0 is found as a function of Ɣ and r s /r p to be [ ] Ɣ/Ɣ−1 r p Ɣ = (4.8) r 0 Ɣ +1− rs 2/r2 p and at r s = 0givesr p /a equal to 0.0263, 0.1516 and 0.3088 for Ɣ = 1.2, 1.4 and 5/3, respectively. To obtain greater compression Potter proposes adiabatic compression of a pinch by a carefully programmed current waveform, analogous to compression in inertial confinement fusion. The gas between the shock and piston behaves adiabatically, i.e. for a fixed mass, pV Ɣ is a constant in time where V is the volume per unit length while the uniform pressure is p s and exactly equal to the magnetic pressure, i.e. p s = B2 = µ 0I 2 . (4.9) 2µ 0 8π 2 rp 2 There is a subtlety in the calculation of dV and Potter shows that for an instantaneous fixed mass dV = 2π(r p dr p − r s dR s ), (4.10) where dR s /dt is equal to u s evaluated at the shock and is given by equation (4.4). The rate of change in pressure is related to the rate of change in the shock velocity by differentiating equation (4.6) to give, with equation (4.10), dp s = 4p s dv s =−Ɣ P s dV dt (Ɣ +1)v s dt V dt . (4.11) Figure 34 is calculated for constant current [261] to consider also the case of linearly rising current so that direct comparison can be made with the snowplough and shell models. Indeed from equations (4.6), (4.9) and (4.11) a coupled pair of nonlinear equations for r p (t) and r s (t) can be derived for any given I(t), namely ( ) dr s (Ɣ 1/2 +1)1/2 µ0 I = (4.12) dt 4π ρ 0 r p and [ dr p Ɣ r p − 2 ] ( ) dt Ɣ +1 r dr s s = (rp 2 dt − r2 s )r p d I . (4.13) I dt r p For the particular case of I = At the numerical solution of equations (4.12) and (4.13) is shown also in figure 33. 4.4. Accelerating hollow shell A zero-dimensional analytic model is a good guide for designing experiments using wire arrays, liners and hollow gas puffs. Here the mass involved in acceleration is fixed and the equation of motion is simply written as d 2 a M l dt =−µ 0I 2 2 4πa , (4.14) where M l is the mass per unit length and I(t) is either given or calculated from a circuit equation which includes the varying impedance of the load [263]. For direct comparison with the snowplough model figure 33 includes a plot of a(t) for the case of a linearly rising current I = At where now the dimensionless equation is d 2 x dy 2 =−y2 (4.15) x 60
Page 1 and 2:
Home Search Collections Journals Ab
Page 3 and 4:
Plasma Phys. Control. Fusion 53 (20
Page 5 and 6:
Page 7 and 8:
Page 9 and 10: Plasma Phys. Control. Fusion 53 (20
Page 59: Plasma Phys. Control. Fusion 53 (20
Page 111 and 112:
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
Page 125 and 126:
Page 127 and 128:
Page 129 and 130:
Page 131 and 132:
Page 133 and 134:
Page 135 and 136:
Page 137 and 138:
Page 139 and 140:
Page 141 and 142:
Page 143 and 144:
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
Page 151 and 152:
Page 153 and 154:
Page 155 and 156:
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
Page 167 and 168:
Page 169:
show all

A review of the dense Z-pinch

Create successful ePaper yourself

Delete template?

Save as template?