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. Heat Conduction – Basic Research 290 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 Verdichtungsstoße in der Nahe 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. Self-Similar Hydrodynamics with Heat Conduction 291 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 plasmas into vacuum: Relation to the problem of ion acceleration”, Phys. Plasmas, Vol. 13, pp. 012105 (1-7). 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). Sanz, J.; Nicolás, J.A.; Sanmartín, J.R. & Hilario, J. (1988). “Nonuniform target illumination in the deflagration regime: Thermal smoothing”, Phys. Fluids, Vol. 31, pp. 2320 – 2326. Takabe, H; Montierth, L. & Morse, R.L. (1983). ”Self-consistent Eigenvalue Analysis of Raileigh-Taylor Instability in an Ablating Plasma”, Phys. Fluids, Vol. 26, pp. 2299 - 2307. True, M.A.; Albritton, J.R. & Williams, E.A. (1981). “Fast Ion Production by Suprathermal Electrons in Laser Fusion Plasmas”, Phys. Fluids Vol. 24, pp. 1885 - 1893. Heat Conduction – Basic Research 292 Zel'dovich, Ya.B. & Raizer, Yu.P. (1966). Physics of Shock Waves and High Temperature 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 Heat Conduction – Basic Research 296 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 rE and differential scattering cross section (, , )       s rE E is given as [5]: ts (r,E, ) (r,E) (r,E, ) dE d (r,E E, ) (r,E , ) S(r ,E, )                             (1a) with boundary condition, for n,    0   s s (E, ), given, (r ,E, ) , if vacuum,         0    (1b) Particle Transport Monte Carlo Method for Heat Conduction Problems 297 where r neutron position, E neutron energy, neutron direction, Sneutron 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) becomes           1 44 ss S(r ) (r, ) (r) (r, ) (r) (r) ,       (2a) with vacuum boundary condition, s (r , ) f or n ,    00     (2b) where scalar flux is defined as (r) d (r, ).          (2c) Let us now consider a “scaled” equation of (2a),           11 44 ss S(r ) (r, ) (r) (r, ) (r) (r) .       (3) 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: Heat Conduction – Basic Research 298 s (r) S(r), (r)    1 3    (4a) with vacuum boundary condition s (r d) , d extrapolation distance.   0  (4b) 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 (typically t d one mean free path /   1  ) 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 Correspondence The steady state heat conduction equation for a stationary and isotropic solid is given by [1]: k(r) T(r) q (r) ,      0   (5a) with boundary condition s T(r ) , 0  (5b) where k(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 [...]... 300 Heat Conduction – Basic Research 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. ..  3.1 , x, y and z in centimeters The results are shown in Figs 13, 14 and 15 (15c) 308 Heat Conduction – Basic Research Fig 13 Temperature distribution along x -direction with y  z  0 in Test Problem 1 Fig 14 Temperature distribution along z -direction with x  y  0 in Test Problem 1 Particle Transport Monte Carlo Method for Heat Conduction Problems 309 Fig 15 Comparison of Test Problem 1 and a... 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 301 Particle Transport Monte Carlo Method for Heat Conduction Problems Material Thermal Conductivity... equivalent conduction medium preserving heat flux Particle Transport Monte Carlo Method for Heat Conduction Problems 307 Examples The method is applied to a pebble fuel with Coarse Lattice with Centered Sphere (CLCS) distribution of fuel particle [14] The description of a pebble fuel is shown in Fig 12 and Table 2 The pebble fuel is surrounded by helium at given bulk temperature with convective heat transfer... there is no heat source in the fluid The transformed problem can then be solved by the Monte Carlo method in Section 2.1 with replacement of r0 by rs and rs by rb , and Tb as the boundary condition Eq (13) with the right-hand side replaced by Eq (14) is no more than a continuity expression of heat flux on the interface Fig 11 shows the concept in this transformation 306 Heat Conduction – Basic Research. .. x, y and z in centimeters The results are shown in Figs 16, 17, and 18 310 Heat Conduction – Basic Research Fig 16 Temperature distribution along x -direction with y  z  0 in Test Problem 2 Fig 17 Temperature distribution along z -direction with x  y  0 in Test Problem 2 Particle Transport Monte Carlo Method for Heat Conduction Problems 311 Fig 18 Temperature distribution along y -direction with... graphite-moderator temperature Essentially, if the lattice has a kernel (heat source), the tally is done over the kernel volume and over the moderator (graphite and layers) volume separately Otherwise, if the lattice consists of only graphite, the tally is done over the cubical volume Particle Transport Monte Carlo Method for Heat Conduction Problems 313 Fig 21 Tally regions depending on the geometries In this... Radius ( cm ) Number of Triso Particles Power/pebble (W ) Kernel Buffer Inner PyC SiC 0.0346 0.0100 0.0400 0.1830 0.02510 0.03425 0.03824 0.04177 Outer PyC Graphite-matrix Graphite-shell 0.0400 0.2500 0.2500 0.04576 2.5000 3.0000 9394 1893.95 Table 2 Problem Description for a Pebble Fig 7 A planar view of a particle random distribution for a pebble problem with the FLS model Heat conduction solutions for... (diffusion solution) However, the computational time increases rapidly as the scaling factor increases In Table 3 and Fig 8, it is shown that a scaling factor of 10 or 20 is not large enough 302 Heat Conduction – Basic Research Scaling Factor 1 10 20 50 a Maximum Temp ( K ) 1674.21 1556.96 1558.54 1553.22 Relative Errora (%) 1.59 1.14 1.12 1.11 Graphite Temp Near Center( K ) 158.33 1533.53 1531.67 1527.07... reduced as the calculation transports particles backward from the detector region (at the center of the pebble) to the source region Additionally, it is possible that the changed tally regions used in the adjoint calculation allow effective particle tallies Scaling Factor 1 20 50 80 100 120 Maximum Temp ( K ) 1685 .131 1558.817 1553.931 1553.586 1552.995 1552. 713 Standard Deviation( K ) 0.409 0.308 . analytic solution. Heat Conduction – Basic Research 300 Fig. 5. Monte Carlo heat conduction solution with extrapolated layer Fig. 6. Monte Carlo heat conduction solution without. treatment of the heat conduction equation with a methodical boundary correction. The Heat Conduction – Basic Research 296 transport theory treatment can easily incorporate anisotropic conduction, . , , x y and z in centimeters. The results are shown in Figs. 13, 14 and 15. Heat Conduction – Basic Research 308 Fig. 13. Temperature distribution along x -direction with 0  yz in