Heat Conduction Basic Research Part 13 doc

13 Particle Transport Monte Carlo Method for Heat Conduction Problems Nam Zin Cho Korea Advanced Institute of Science and Technology KAIST, Daejeon, South Korea 1.. This chapter des

Self-Similar Hydrodynamics with Heat Conduction 289 quantities just for readers' comprehension The behavior of the velocity for → ∞ may seem physically unacceptable at least in a rigorous sense As a matter of fact, however, there are a number of examples for implosions and explosions in which the velocity profile is approximately linear with the radius (Sedov, 1959; Bernstein, 1978) In addition, the physical condition at enough large radii ( ≫ 1) will not affect the core dynamics for an intermediate time period Therefore, when we restrict our considerations to a finite closed volume containing the core, the present self-similar solution is expected to be an approximation of the core evolution at higher densities and temperatures

Fig 7 g - diagram showing the optimization process of the eigenvalue,

Fig 8 Eigenstructure of the self-similar solution

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Under the condition that the right integrated curve is to converge to = = 0, each curve has already optimized with respect to as was shown in Fig 6 Other fixed parameters are the same as in Fig 6

The normalized physical quantities are obtained as a result of the two-dimensional eigenvalue problem with fixed parameters, = 2, = 13/2, and = 5/3

5 Conclusions

The crucial role of dimensional analysis and self-similarity are discussed in the introduction and the three subsequent examples Self-similar solutions for individual cases have been demonstrated to be derivable by applying the Lie group analysis to the set of PDE for the hydrodynamic system, taking nonlinear heat conductivity into account as the decisive physical ingredient The scaling laws for thermally conductive fluids are conspicuously different from those for adiabatic fluids (not discussed in the present chapter; see references

by Murakami et al., 2002, 2005 for details) The former has one freedom less than the latter due to the additional constraint of thermal conductivity If a thermo-hydrodynamic system comprises multiple heat conduction mechanisms, self-similarity cannot be expected in a vigorous sense except for special cases However, self-similarity and scaling laws can always

be found at least in an approximate manner, by shedding light on the dominant conduction mechanism, which should give the basis of system design and diagnostics for scaled experiments for individual cases The necessity of dimensional analysis and finding self-similar solutions is encountered in many problems over wide ranges of research The simple general scheme and the examples mentioned in this chapter will help the reader who encounters a similar situation in his or her investigation find the underlying physics and prepare further theoretical and experimental setup

6 References

Antonova, R.N & Kazhdan, Y.M (2000) “A self-similar solution for spherically symmetric

gravitational collapse ” Astronomy Letters, Vol 26, pp 344 - 355

Barenblatt, G.I (1979) Similarity, Self-Similarity, and Intermediate Asymptotics (New York:

Consultants Bureau)

Basko, M.M & Murakami, M., (1998) “Self-similar implosions and explosions of radiatively

cooling gaseous masses” Phys Plasma, Vol 5, pp 518 – 528

Bernstein, I.B ＆ Book, D.L (1978) “Rayleigh-Taylor instability of a self-similar spherical

expansion” Astrophysical Journal, Vol 225, pp 633 – 640

Gitomer, S.J.; Morse, R.L & Newberger, B.S (1977) “Structure and scaling laws of

laser-driven ablative implosions”, Phys Fluids Vol 12, pp 234 - 238

Guderley, G (1942) “Starke kugelige und zylindrische Verdichtungsstoße in der Nahe des

Kugelmittelpunktes bzw der Zylinderachse” Luftfahrtforschung Vol 19, pp 302–

312

Gurevich, A.V.; Parrska, L.V & Pitaevsk, L.P (1966) “Self-similar motion of rarefied

plasma” Sov Phys JETP, Vol 22, pp 449 - &

Kull, H.J (1989) “Incompressible Description of Rayleigh-Taylor Instabilities in

Laser-Ablated Plasmas” Phys Fluids, Vol B1, pp.170 - 182

Kull, H.J (1991) “Theory of Rayleigh-Taylor Instability” Phys Reports, Vol.206, pp.197 - 325

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Landau, L.D & Lifshitz, E.M (1959) Fluid Mechanics (New York: Pergamon)

Larson, R.B (1969) ”Numerical calculations of the dynamics of collapsing proto-star” Mon

Not R Astr Soc., Vol 145, pp 271-&

Lie, S (1970) Theorie der Transformationsgruppen (New York: Chelsea)

London, R.A & Rosen, M.D (1986) “Hydrodynamics of Exploding Foil X-ray Lasers” Phys

Fluids, Vol 29, pp 3813 - 3822

Mora, P (2003) “Plasma Expansion into a Vacuum” Phys Rev Lett Vol.90, 185002 (pp 1 -

4)

Murakami, M.; Meyer-ter-Vehn, J & Ramis, R (1990) ”Thermal X-ray Emission from

Ion-Beam-Heated Matter” J X-ray Sci Technol., Vol 2, pp 127 - 148

Murakami, M & Meyer-ter-Vehn, J (1991) “Indirectly Driven Targets for Inertial

Confinement Fusion” Nucl Fusion, Vol 31, pp 1315 – 1331

Murakami, M., Shimoide, M., and Nishihara, K (1995) “Dynamics and stability of a

stagnating hot spot” Phys Plasmas, Vol.2, pp 3466 - 3472

Murakami, M & Iida, S., (2002) “Scaling laws for hydrodynamically similar implosions

with heat conduction”, Phys Plasmas, Vol.9, pp.2745 - 2753

Murakami, M.; Nishihara, K & Hanawa, T (2004) “Self-similar Gravitational Collapse of

Radiatively Cooling Spheres”, Astrophysical Journal, Vol 607, pp.879 - 889

Murakami, M.; Kang, Y.-G.; Nishihara, K.; Fujioka, S & Nishimura, H (2005) “Ion energy

spectrum of expanding laser-plasma with limited mass”, Phys Plasmas, Vol.12, pp

062706 (1-8)

Murakami, M & M M Basko (2006) “Self-similar expansion of finite-size non-quasi-neutral

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Murakami, M.; Fujioka, S.; Nishimura, H.; Ando, T.; Ueda, N.; Shimada, Y & Yamaura, M

(2006) “Conversion efficiency of extreme ultraviolet radiation in laser-produced

plasmas”, Phys Plasmas, Vol.13, pp 033107 (1-8)

Murakami, M.; Sakaiya, T & Sanz, J (2007) “Self-similar ablative flow of nonstationary

accelerating foil due to nonlinear heat conduction”, Phys Plasmas, Vol 14, pp

022707 (1-7)

Pakula, R & Sigel, R., (1985) “Self-similar expansion of dense matter due to heat-transfer by

nonlinear conduction ” Phys Fluids, Vol 28, pp 232 - 244

Penston, M.V (1969) “Dynamics of Self-Gravitating Gaseous Spheres – III Analytical

Results in the Free-Fall of Isothermal Cases” Mon Not R astr Soc., Vol 144, pp 425

- 448

Sedov, L.I (1959) Similarity and Dimentional Methods in Mechanichs (New York : Academic)

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the deflagration regime: Thermal smoothing”, Phys Fluids, Vol 31, pp 2320 – 2326

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Hydrodynamic Phenomena (New York: Academic Press)

Part 4

Numerical Methods

13

Particle Transport Monte Carlo Method

for Heat Conduction Problems

Nam Zin Cho

Korea Advanced Institute of Science and Technology (KAIST), Daejeon,

South Korea

1 Introduction

Heat conduction [1] is usually modeled as a diffusion process embodied in heat conduction equation The traditional numerical methods [2, 3] for heat conduction problems such as the finite difference or finite element are well developed However, these methods are based on discretized mesh systems, thus they are inherently limited in the geometry treatment This chapter describes the Monte Carlo method that is based on particle transport simulation to solve heat conduction problems The Monte Carlo method is “meshless” and thus can treat problems with very complicated geometries

The method is applied to a pebble fuel to be used in very high temperature gas-cooled reactors (VHTGRs) [4], which is a next-generation nuclear reactor being developed Typically, a single pebble contains ~10,000 particle fuels randomly dispersed in graphite–matrix Each particle fuel is in turn comprised of a fuel kernel and four layers of coatings Furthermore, a typical reactor would house several tens of thousands of pebbles in the core depending on the power rating of the reactor See Fig 1 Such a level of geometric complexity and material heterogeneity defies the conventional mesh–based computational methods for heat conduction analysis

Among transport methods, the Monte Carlo method, that is based on stochastic particle simulation, is widely used in neutron and radiation particle transport problems such as nuclear reactor design The Monte Carlo method described in this chapter is based on the observation that heat conduction is a diffusion process whose governing equation is analogous

to the neutron diffusion equation [5] under no absorption, no fission and one speed condition, which is a special form of the particle transport equation While neutron diffusion approximates the neutron transport phenomena, conversely it is applicable to solve diffusion problems by transport methods under certain conditions Based on this idea, a new Monte Carlo method has been recently developed [6-8] to solve heat conduction problems The method employs the MCNP code [9] as a major computational engine MCNP is a widely used Monte Carlo particle transport code with versatile geometrical capabilities

Monte Carlo techniques for heat conduction have been reported [10-13] in the past But most of the earlier Monte Carlo methods for heat conduction are based on discretized mesh systems, thus they are limited in the capabilities of geometry treatment Fraley et al[13] uses a

“meshless” system like the method in this chapter but does not give proper treatment to the boundary conditions, nor considers the “diffusion-transport theory correspondence” to be described in Section 2.2 in this chapter Thus, the method in this chapter is a transport theory treatment of the heat conduction equation with a methodical boundary correction The

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transport theory treatment can easily incorporate anisotropic conduction, if necessary, in a

future study

(c) A pebble-bed reactor core

(a) A pebble fuel element

(b) A coated fuel particle

Fig 1 Cross-sectional view of a pebble fuel (a) consisting several thousands of coated fuel

particles (b) in a reactor core (c)

2 Description of method

2.1 Neutron transport and diffusion equations

The transport equation that governs the neutron behavior in a medium with total cross

section ( , )t r E and differential scattering cross sections( ,r E E,   )is given as [5]:

Particle Transport Monte Carlo Method for Heat Conduction Problems 297

where

r neutron position,

E neutron energy, neutron direction,

S neutron source, (r ,E, ) neutron angular flux.

Fig 2 Angular flux and boundary condition

Fig 2 depicts the meaning of angular flux (r ,E, )   and boundary condition In the

special case of no absorption, isotropic scattering, and mono-energy of neutrons, Eq (1)

s (r , ) for n ,

An important result of the asymptotic theory provides correspondence between the

transport equation and the diffusion equation, i.e., the asymptotic (  solution of Eq (3) )

satisfies the following diffusion equation:

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s

(r ) S(r ), (r ) 

It is known that, between the two solutions from transport theory and from diffusion

theory, a discrepancy appears near the boundary Thus, the problem domain is extended

using an extrapolated thickness (typicallyd one mean free path  1/t) for boundary layer

correction, as shown in Fig 3

Fig 3 Boundary correction with an extrapolated layer

2.2 Monte Carlo method for heat conduction equation

wherek(r ) is the thermal conductivity and q (r ) is the internal heat source

If we compare Eq (5) with Eq (4), it is easily ascertained that Eq (4) becomes Eq (5) by

setting

Particle Transport Monte Carlo Method for Heat Conduction Problems 299

s (r ) , k(r )

with a largeand the problem domain extended byd

The Monte Carlo method is extremely versatile in solving Eqs (1), (2) and (3) with very

complicated geometry and strong heterogeneity of the medium Thus, Eq (3) is solved by

the Monte Carlo method (with a large) to obtain(r ) The result of(r ) is then translated

to provide T(r ) (r ) (r )s as the solution of Eq (5) [See Fig 3.]

Here,  1 is a scaling factor rendering the transport phenomena diffusion-like A large

scaling factor plays an additional role of reducing the extrapolation distance to the order of a

mean free path To choose a proper value for, we introduce an adjoint problem to perform

sensitivity studies, specific results for a pebble problem provided later in this section

Proof of principles of the method

In order to confirm or provide proof of principles of the Monte Carlo method described in

Section 2.2, first we consider a simple heat conduction problem which allows analytic

solution The problem consists of one-dimensional slab geometry, isotropic solid, and

uniformly distributed internal heat source under steady state The left side has reflective

boundary condition and the right side has zero temperature boundary condition Fig 4(a)

shows the original problem and Fig 4(b) shows the extended problem to be solved by the

Monte Carlo method, incorporating the boundary layer correction Table 1 provides the

Table 1 Calculation Conditions for Simple Problem

Figs 5 and 6 show the Monte Carlo method results with and without the extension by

extrapolation thickness in comparison with the analytic solution Note that the result of the

Monte Carlo method with boundary layer correction is in excellent agreement with the

analytic solution

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Fig 5 Monte Carlo heat conduction solution with extrapolated layer

Fig 6 Monte Carlo heat conduction solution without extrapolated layer

To test the method on a realistic problem, the FLS (Fine Lattice Stochastic) model and CLCS (Coarse Lattice with Centered Sphere) model [14] for the random distribution of fuel particles in a pebble are used to obtain the heat conduction solution by the Monte Carlo method Details of this process are described in Table 2 and Fig 7 The power distribution generated in a pebble is assumed uniform within a kernel and across the particle fuels The pebble is surrounded by helium at 1173K with the convective heat

transfer coefficient h=0.1006( W / cm2 K) A Monte Carlo program HEATON [15] was written to solve heat conduction problems using the MCNP5 code as the major computational engine