Analytic Expressions for Optimal ICF Hohlraum Wall Density and Wall Loss
Solutions to the radiation diffusion equation predict the absorbed energy (''wall loss'') within an inertial confinement fusion (ICF) hohlraum. Comparing supersonic vs. subsonic solutions suggests that a high Z metallic foam as hohlraum wall material will reduce hydrodynamic losses, and hence, net absorbed energy by {approx}20%. We derive an analytic expression for the optimal density (for any given drive temperature and pulse-length) that will achieve this reduction factor and which agrees well with numerical simulations. This approach can reduce the cost of a reactor driver. Radiation heat waves, or Marshak waves, play an important role in energy transport and in the energy balance of laser, z-pinch and heavy ion beam hohlraums for ICF and high energy density physics experiments. In these experiments, a power source, e.g. a laser, delivers energy to the interior of a high Z cavity that is converted to x-rays. Typically, most of the energy is absorbed in a thin, diffusively heated layer on the hohlraum interior surface, and re-emission from the heated layer sets the radiation temperature T achieved in the hohlraum. In our recent paper, (henceforward referred to as HR) we developed an analytic theory of Marshak waves via a perturbation theory using a small parameter {var_epsilon} = {beta}/(4 + {alpha}) where the internal energy varies as T{sup {beta}} and the opacity varies as T{sup -{alpha}}. A consistent theory was built up order-by-order in {var_epsilon}, with the benefits of good accuracy and order-by-order energy conservation. We first derived analytic solutions for supersonic Marshak waves, which remarkably allowed for arbitrary time variation of the surface temperature. We then solved the full set of subsonic equations, though specialized to the case that the surface temperature varies as t{sup k}, where self-similar solutions can be found. Our solutions compared very well with exact analytic solutions (for the specialized cases for which they exist) and with radiation-hydrodynamic simulations. In this paper we apply those results to the following question: Can we save on driver energy by making hohlraum walls out of low density high Z foams, which have less hydrodynamic motion and hence, reduced net absorbed energy by the walls? We answer this question using our HR analytic theory, as well as by numerical simulations. To the degree that the ''pure'' HR theory diverges from the simulations we derive non-ideal non-self-similar corrections to the theory that bring it into agreement with the simulations. We show that low-density high Z foams can indeed bring a savings of {approx}20% in the required driver energy. Remarkably, this reduction is universal- independent of drive T and its pulse-duration {tau}. We derive an analytic expression for the optimal density (for any given T and {tau}) that will achieve this reduction factor and which agrees very well with numerical simulations. For a nominal 5B$ ICF reactor driver of 5 MJ, this 20% savings could save 0.5-1B$. Reduced hydrodynamic motion of the wall material may also reduce symmetry swings, as found for heavy ion beam targets.
- Research Organization:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
- Sponsoring Organization:
- USDOE
- DOE Contract Number:
- W-7405-ENG-48
- OSTI ID:
- 878200
- Report Number(s):
- UCRL-JRNL-204370; TRN: US0602285
- Journal Information:
- Physical Review E, Vol. 72, Issue 5-2
- Country of Publication:
- United States
- Language:
- English
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71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
74 ATOMIC AND MOLECULAR PHYSICS
ACCURACY
DIFFUSION EQUATIONS
ENERGY BALANCE
ENERGY CONSERVATION
ENERGY DENSITY
HEAVY IONS
HYDRODYNAMICS
INERTIAL CONFINEMENT
OPACITY
PERTURBATION THEORY
PHYSICS
RADIATIONS
SYMMETRY
TARGETS