Magnetized Target Fusion: Insights from Mathematical Modelling

Transcription

1 : Insights from Mathematical Modelling, UBC, PhD Candidate - Applied Math Brian Wetton, UBC, Supervisor Michael Ward, UBC, Committee Member Sandra Barsky, General Fusion, Committee Member November 28, 2014

2 Context A local Canadian research company is designing a reactor to produce fusion energy, which has never been achieved before. Success with this could hold immense promise for clean and sustainable energy sources in the future. In this talk, we will look into the mathematical analysis used to yield insights into the design work. Key components: Reactor model Numerical finite volume approach Formal asymptotic approach

3 Thanks to... Thanks to the following people for helpful discussions, collaboration, and support: Sandra Barsky and Aaron Froese, General Fusion George Bluman, Michael Ward, and Brian Wetton, UBC Math Department Randy LeVeque, University of Washington Applied Mathematics Department My colleagues and fellow graduate students

4 Fusion Fusing atomic nuclei yield new nuclei plus energy Occurs naturally in sun: pressure 300 G atm, temperature 10 7 K Potential clean energy source Tritium-deuterium plasma reaction: 3 1 H +2 1 H 4 2 He +n MeV of energy Depends on temperature, thermal distribution, etc.

5 Challenges Immense temperature and pressure must be sustained Thermal and radiative losses Lawson criterion 1 for energy yield: density temperature time cm 3 KeV s

6 General Fusion design General Fusion founded by Michel Laberge in 2002 Design magnetized target fusion reactor Isolate spheromak plasma with magnetic field Crush plasma in imploding metal cavity

7 General Fusion design How feasible is such a design?

8 Lead-Lithium Euler equations for mass and momentum conservation With density ρ, velocity v, and pressure P: ρ t + (ρv) = 0 (ρv) t + (ρv v) + P = 0 P = P(ρ) (mass) (momentum) (equation of state) Nonlinear coupled system of PDEs

9 Lead-Lithium Quadratic fit to high-pressure Lead experiments 2

10 Piston and plasma pressure Gaussian piston pressure: P(t) = P atmospheric + (P impact P atmospheric )e t2 /t 2 0 θ(t) Plasma has gas and magnetic pressure Ideal gas of volume V : Magnetic pressure 3 : P gas V γ, γ 1 P magnetic R 4 major

11 Assumptions and simplifications Spherical symmetry Isentropic (reversible) conditions so γ = 5/3 Initially P gas 0.1P magnetic Initial plasma temperature 100 ev System starts in equilibrium No mixing implies free boundary problem at plasma-metal and metal-piston interfaces: d dt r boundary(t) = v(r boundary (t), t) Neglect thermal and radiative energy losses

12 Overall model

13 Overall model In r L (t) < r < r R (t), t > 0, dimensionless system has form: ρ t + 1 r 2 (r 2 ρv) r = 0, (ρv) t + p r + 1 r 2 (r 2 ρv 2 ) r = 0 p = p(ρ), p(r L (t), t) = p L (r L (t)), v(r, ) = 0, dr L,R = v(r L,R (t), t) dt p(r R (t), t) = f (t) p(r, ) = p L (r L ( )) r L ( ) = χɛ 1/2, r R ( ) = 1

14 First-order finite volumes Given u t + (f (u)) x = 0 denote ū(x i, t) = 1 xi +h/2 h x i h/2 u(x, t)dx dū(x i,t) dt = 1 xi +h/2 h x i h/2 u t(x, t)dx = f (u(x i+1/2,t)) f (u(x i 1/2,t)) h Choose F L (u i 1, u i ) = f (u i 1 ) or f (u i ) depending on f Stable time-step k < h/ sup f, u j+1 i = u j i k h (F L (u j i, uj i+1 ) F L (u j i 1, uj i ))

15 Second-order flux-limited finite volume idea Smoothness: u t + f (u)u x = 0 Use u tt = f (u)u t u x f (u)u tx = (f (u) 2 ) x u x + f (u) 2 u xx Lax-Wendroff: u(x, t + k) = u(x, t) + ku t (x, t) + k 2 2 u tt(x, t) + O(k 3 ) Corresponding high-resolution flux F H Flux limited: take flux F = φf L + (1 φ)f H Limiter φ largest where derivative u (x) largest

16 Convergence Numerical solution u num (x, t) and exact solution u ex (x, t): E = u num (x, t) u ex (x, t) dx 0, as N Near shocks, E = O(h) Near smooth regions E = O(h n ) for n th -order method

17 Coordinate transformation Transform from moving to fixed computational domain Let τ = t and define (τ) = r R (τ) r L (τ), Γ(τ) = v R (τ) v L (τ) Set y = r r L(τ) (τ), y [0, 1] p(r R (t), t) = p(1, τ), etc. New conservation laws e.g. mass: ρ τ + { 1 [ (v L + Γy)ρ + ρv]} y = 2 r L + y Γ ρ

18 Finite volume overview Pseudocode 4 : - eigen analysis for system - constant extrapolation to ghost-points - use projections for limiters - homogeneous Riemann problem then use source - update time and boundary data Verify L 1 -convergence Table: Convergence of Numerical Scheme at fixed time N Velocity Error Density Error Mass Error Rate

19 Pulse profiles

20 Abridged sensitivity analysis Table: Min radius R min, Lawson triple product Π L, impact pressure P impact, initial plasma radius R plasma,0, initial sphere radius R lead,0, time scale t 0. System R min (cm) Π L (10 15 kev s cm 3 ) Baseline R plasma, P impact R lead, t P impact R lead, t

21 Big picture Modifications: - Gaussian impulse pressure without heaviside - linearized P(ρ) equation of state - only magnetic pressure Relevant scales: - asymptotic parameter ɛ 1 - big sound speed: bɛ 1/2 - impulse pressure: O(1) - small initial plasma radius: χɛ 1/2 - very small impulse time: O(ɛ) - very, very small initial pressure: µɛ 3/2 Matched asymptotics 5

22 Asymptotic regimes

23 Pulse formation and focusing Formation: - ρ(y, τ) 1 + ɛρ 1 + ɛ 3/2 ρ 2, v(y, τ) ɛ 1/2 v 0 + ɛv 1 - Riemann Invariants 6 - plane wave solutions in (, 0] [0, ): ρ 1 (y, τ) = 1 b 2 e (τ+y/b)2, - need divergent ρ 2, v 1 for matching Focusing: - amplitude growth: v 0 (y, τ) = 1 2 b e (τ+y/b) ρ 1 + ɛρ 1 + ɛ 3/2 ρ 2, v ɛ 1/2 v 0 + ɛv 1 (outer) ρ 1 + ɛ 1/2 ρ 1 + ɛρ 2, v v 0 + ɛ 1/2 v 1 (inner) - linear acoustic limit 7 : ρ 1,ˆt + v 0,σ + 2 σ v 0 = 0, v 0,ˆt + b2 ρ 1,σ = 0

24 Pulse reflection Complete reflection of ρ 1 and v 0, negligible compression Next order: v 1,ˆt + b2 ρ 2,σ = (ρ 1 v 0 )ˆt (v 2 0 ) σ 2 σ v 2 0 Careful integration and boundary condition trickery gives v 1 (σ, ) = 2π b 2 χσ 2 Residual velocity field v ɛ 1/2 v 1 compresses plasma! Rayleigh energy argument 8 or more dynamics

25 Plasma compression Integration of PDEs 9 with reformulation gives free boundary ODE Initially - ρ 1 + ɛ 2 ˆρ, v ɛ 1/2ˆv - O(ɛ 1/2 ) space scale, O(1) time scale - find ˆv(σ L ) = ˆv L = 2π ; inner wall at σ L b 2 χ 3/2 σ 3/2 L Finally - ρ 1 + ɛ 1/2 ˆρ, v ɛ 1/4ˆv - O(ɛ) space scale, O(ɛ 5/4 ) time scale - find ˆv(z L ) = ˆv L = 2Az L 2µ ; inner wall at z zl 2 L Matching gives A Turnaround point when ˆv L = 0

26 Minimum radius Minimum radius: r min b4 χ 3 µ ɛ π Agreement with numerics as ɛ 0 with b = 1.05, χ = 0.937, µ = π: ɛ r min,num r min,asy

27 Key insights Dimensional minimum radius R min C4 s P plasma,0 R 7 plasma,0 ϱ3 0 πp 4 impact R4 lead,0 t2 0 = 1.6 cm Symbol Meaning Symbol Meaning C s lead sound speed P plasma,0 initial pressure R plasma,0 initial plasma radius ϱ 0 lead density P impact piston pressure R lead,0 initial lead radius t 0 impulse time scale Qualitative consistency with numerics

28 Key insights consistent compression profile wave-like behaviour: sound speed dominates almost all input energy reflected: E input 8π 3 b ɛ3/2, E compression 4π2 b 4 χ 3 ɛ5/2

29 Results and future work Results: - numerics and asymptotics consistent - sensitivity to parameters, energy yield may be within reach - larger outer sphere radius and impact pressure Future directions: - incorporate more plasma physics - consider angular instabilities - more precise assessment of design 1 Lawson, Some Criteria for a Power..., Proc. Phys Society B, Rothman et al., Measurement of principal isentropes..., J. Phys D, Woodruff et al., Adiabatic compression of a doublet field..., JOFE, LeVeque, Finite Volume Methods for Hyperbolic Problems, Hinch, Perturbation Methods, Whitham, Linear and Nonlinear Waves, Landau and Lifschitz, Fluid Mechanics, Lord Rayleigh, On the Pressure Developed..., Phil. Mag. Series 6, Brennan, Cavitation and Bubble Dynamics, 1995.