Fusion Engineering and Design

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1 Fusion Engineering and Design Contents lists available at ScienceDirect Fusion Engineering and Design journal homepage: Multi-dimensional computational model of the movement of the solid gas interface during the layering process in inertial confinement fusion targets in a non-uniform thermal environment K.-J. Boehm, A.R. Raffray Center for Energy Research, M/C D EBU II, University of California, San Diego, CA , United States article info abstract Article history: Received 11 February 2010 Received in revised form 16 June 2010 Accepted 5 August 2010 Keywords: Target layering Numerical model Heat- and mass-transfer The redistribution of deuterium DD or of a deuterium tritium mixture DT to form a layer on the inside of spherical inertial confinement fusion ICF capsules is a challenging problem because of the symmetry requirements of the fuel layer thickness, the smoothness requirement of the inside target surface, and the time restriction on the production process. Heat- and mass-transfer processes have been identified to interact with one another to influence the outcome of the layering process. For example, the mass redistribution speed of the fuel inside the shell towards a uniform layer and the final layer thickness uniformity depend on the variation in local heat transfer coefficient along the outer target surface. The focus of this work was to develop a numerical tool to help understand the physics involved in the layering process to be able to assess the influence of key parameters on the transient layer formation. The coupled mass and heat transfer processes governing target layering have been studied numerically, implementing unique boundary conditions to track the movement of the gas solid boundary on the inside of the shell. The model was validated through comparison with theoretical results and laboratory-scale experiments. With this model, a window of parameters can by identified, under which layering experiments are likely to be successful Elsevier B.V. All rights reserved. 1. Introduction The time-dependent formation of frozen deuterium or deuterium tritium layers inside small spherical capsules 2 4 mm for inertial confinement fusion experiments has been investigated in this study for a non-uniform thermal environment. In previous work by Martin et al. [1], the effectiveness of beta heating on driving deuterium tritium DT ice layers towards a uniform thickness was studied analytically for a simplified one-dimensional case. While describing the underlying physics of mass redistribution towards a spherical layer inside a hollow sphere, the one dimensionality of the solution does not allow studying the effect of a non-uniform heat transfer coefficient and/or temperature field on the outer surface of the pellet, does not account for the spherical shape of the target, and is unable to model inner surface roughness features. A two-dimensional numerical study was published by Martin et al. [2], in which the authors estimated the final layer formation of non-constant heat transfer coefficient imposed on the outside surface of the shell by a membrane holding the target in place dur- Corresponding author. Tel.: address: kuboehm@gmail.com K.-J. Boehm. ing layering. In this work, the expected instantaneous mass transfer into the void space based on the computed temperature profile was calculated in two dimensions. Based on this information, the instantaneous rate of redistribution could be predicted; however, details of the 2D movement of the interface over time based on sublimation and re-condensation was not included. Instead, scenarios including a non-uniform convective cooling on the target surface and a nonuniform outer temperature on the pellet are mentioned as possible shortfalls of this model. Later, Harding et al. [3] published results of a 3D model that computed the temperature profile inside a DT-filled shell exposed to a certain thermal environment, but the authors did not publish any results on the transient mass-transfer processes governing the layer formation. In the production process, these targets are generally gas-filled at room temperature e.g. by a fill tube or a permeation process, and then cooled past the triple point DT K and DD K leading to a non-uniform accumulation of mass inside the pellet. Through beta layering, a mass-transfer phenomenon that is driven by the energy deposited in the solid by the beta decay of tritium or by another volumetric heat source infra-red IR light, the formation of a layer of uniform thickness is observed, provided that the outer surface of the shell is held at a constant temperature. Spherical symmetry of the fuel layer thickness inside the target has /$ see front matter 2010 Elsevier B.V. All rights reserved. doi: /j.fusengdes

2 52 K.-J. Boehm, A.R. Raffray / Fusion Engineering and Design Nomenclature Symbols A surface area over which the phase change occurs A s surface area of the particle C p,s specific heat capacity of the solid CG void center of gravity of the void d equilibrium layer thickness ı difference between equilibrium layer thickness and actual layer thickness ı speed of the interface h heat transfer coefficient h 1, h 2 thickness of the interface H s heat of sublimation I xx, I yy, I zz values of the mass moment of inertia tensor around principle axes k ice, k gas, k thermal conductivity of ice/gas/general M A molecular mass of species A ṁ gas mass flux into the void void f fluid viscosity n, n coordinate normal to the surface, increment N A molecular flux of species A Nu, Pr, Re Nusselt, Prandtl and Reynolds Number P g pressure in the gas phase P v temperature-dependent vapor pressure over solid q surface heat flux q volumetric heating rate R radius of the sphere r, z, r, z radial and axial coordinates, increments in axial and radial direction r i,cg, z j,cg axial, and radial coordinate of the center of gravity of the gaseous fraction at grid point r i, z j R gas universal gas constant s density of the solid S coordinate along the interface T temperature t, t time, time step size T g, T gas temperature of the gas T h1, T h2 temperature of the inner interface T 0 outer surface temperature T temperature of the cooling gas total layering time constant U gas flow speed V n velocity of the interface normal to the interface V solid fuel total volume of the solid fuel V Total void, V Outer sphere total volume of the void and of the outer sphere V change in volume of the void x, x* space variable X distance between center of gravity and center of the sphere Z cg axial coordinate of the overall center of gravity Superscrpit n, n +1,n 1 nth, n 1th and n + 1th time step Subscript i, i +1,i 1, j +1,j 1 location in finite difference formula been established as a requirement by target implosion physicists to minimize Rayleigh Taylor instabilities [4]. This uniformity in the layer thickness is achieved by the layering process after the fuel has been frozen at the bottom of the shell. Experimentally, this has first been demonstrated by Hoffer and Foreman [5] on a cylindrical geometry and has been used since by a number of researchers for producing inertial confinement fusion pellets. For example, Harding et al. [3] demonstrated experimentally that the production of spherical deuterium ice layers for direct drive targets is possible. In these experiments, the targets were placed in the center of a copper layering-sphere to provide spherical isotherms. In the absence of tritium, whose radioactivity makes it difficult to handle, an infrared heating source was used to produce volumetric heating in the deuterium. The wavelength of the IR light is matched with the absorption spectrum of deuterium to provide uniform volumetric heating. The target support in this experimental setup, which was made out of 4- m diameter spider silk, and asymmetries in the retractable copper cooling sphere were shown to prevent the required layer uniformity indicating that small changes in the thermal environment have a significant impact on the final result. It seems technically challenging to provide a highly isothermal environment [3,6] such that the layering process results in layer thickness uniformity as required for a high gain inertial confinement fusion event. Instead, this layering process is likely to impose a transient non-uniform heat transfer coefficient on the outer target surface, the impact of which needs to be assessed numerically. The interest on the effects of a non-uniform, time-dependent heat transfer coefficient on layer formation have been mentioned by Harding for single shell layering in a stationary copper sphere and in conjunction of using a fluidized bed as a mass production layering device for inertial confinement energy power plants [6,7]. In the present study, a numerical method was developed to track the location of the solid-to-gas interface for different non-uniform thermal environment scenarios. Upon validation, the model can be used to study the feasibility of different target production concepts provided the variations of the local heat transfer coefficient and outside temperature are known. This paper describes the development and application of this numerical model. The relevant equations and physical background for the layering process are given in Section 2; details of the transient two-dimensional layering model are presented in Section 3; validation of the model by comparison to simple test cases is discussed in Section 4; benchmarking the modeling results against an experiment performed in a controlled temperature environment is described in Section 5; finally, results from an example layering scenario are presented and discussed in Section Physical background and equations The underlying physics of mass redistribution towards a spherical layer inside a hollow sphere, the layering process, has been described theoretically by Martin et al. [1] and later by Bernat et al. [8]. The analyses show that if the surface of spherical targets filled with frozen DT, is kept in a highly isothermal environment, the volumetric heating from the beta decay 0.05 W/cm 3 for solid DT at triple point of the tritium can drive non-uniform DT ice layers towards uniformity. The bulk heating will induce sublimation of fuel into the gas phase at the inner surface of thicker ice layer regions and condensation of gaseous fuel on the inner surface of thinner regions. The speed of this process depends on the magnitude of the bulk heating. The main difference between the two authors [1] and [8] lies in the modeling of the movement of the gas inside the gaseous void. If the time between filling the capsule and layering is long enough 7 days as reported by Hoffer and Foreman [5], the buildup of 3 He from the tritium decay in the capsules impedes the movement of the DT-gas through the 3 He in the void, making it a two-species diffusion problem. Boehm [7] has analyzed the theories behind the two models, and has compared the results for one specific DT layering case.

3 K.-J. Boehm, A.R. Raffray / Fusion Engineering and Design by Eq. 6. Fig. 1. 1D schematic illustrating the temperature distribution during layering. The present work was focused on the heat transfer aspect of the redistribution process for cases without any or with very low 3 He concentration in the capsule, such as for experimental layering of pure deuterium or for DT layering with short fill and cooling time of the shells. A fill time shorter than 5 days keeps the effects of the 3 He buildup low enough less than 50% to justify the simplification of omitting the multi-species diffusion in the capsule for the current model. Starting with the general heat diffusion equation with a heat generation term, the main points of the layering model derivation are highlighted here for a simplified 1D case and will be the starting point of the development of a 2D layering model. In the 1D representation, the layers on opposite sides of the hollow shell are represented by two slabs of finite thickness, but infinite length and width as shown in Fig. 1. Eq. 1 shows the heat diffusion equation in Cartesian coordinates in one dimension using the x-coordinate in the radial direction, Eq. 2 describes the boundary conditions applied at the inner and the outer surface. q d 2 T + k ice dx 2 = 0 1 dt dx = N A H s Evaporation flux at the interface x = h1, and 2a k ice T = T 0 at x = R 2b Integrating Eq. 1 twice and applying the boundary condition from Eqs. 2 lead to a temperature distribution in the target layer on the left in Fig. 1 according to: T h1 T 0 = N A H s h 1 + q h 2 k ice 2k 1 3a ice Applying the same equations to the layer on the right hand side leads to: T h2 T 0 = N A H s h 2 + q h 2 k ice 2k 2 3b ice The movement of the interface ı can be related to the molar flux and the molar density, as shown in Eq. 4, which, in combination with Eqs. 3, leads to Eq. 5. ı = N A s 4 T h1 T h2 = 2d k ı s H s + q ı 5 ice Assuming no temperature difference between the two walls idealized scenario, the movement of the interface can be determined ı = q ı 6 s H s As shown by Martin et al. [1], this equation is a 1st order ODE with the solution: ıt = ı0 exp q t 7 s H s In this special case, a very small temperature difference between the two interfaces assumed to be exactly zero for the calculations above is the driving force for the molecules to move through the vapor space. This is a good approximation if there is pure fuel vapor present in the gas phase, as the gas molecules are assumed to flow through the vapor space instantaneously. This follows from the pressure of the gas close to the interface in steady state being very close to the vapor pressure at the surface temperature. In a small void, large pressure differences resulting from large temperature differences would equalize in a time frame of the order of the diameter of the shell divided by the speed of sound which is much smaller than the layering time. This results in sublimation at the surface until the equilibrium temperature is reached T 0 = TP v. Both the characteristic time of the sublimation flux at the surface and characteristic time of the pressure equilibration in the void are very small compared to the layering times, making this assumption acceptable. However, if a non-participating gas species such as 3 He is present, the flow of vapor molecules will be slowed down. The total pressure in the gas will be constant in the entire void, but the values of the two partial pressures will be different at the surface depending on the surface temperature. The speed of redistribution for this case will depend on the conditions of the non-participating gas species as well. The problem becomes both a heat transfer and mass-transfer problem. The mass diffusion of species A fuel through species B non-participating gas needs to be accounted for. This difference in partial pressure can be computed by solving both mass and heat equation simultaneously. Two different models have been applied for this case [1,8]. Boehm [7] compared results from both models, and described a simple experiment which was set up to extend their validity from hydrogen diffusion through a void filled with helium-3 to different gas species. In the following analysis, the influence of a non-participating gas has not been taken into account as the model was focused on cases of layering pure deuterium under IR irradiation and of layering DT within 5 days of filling the target. There is a substantial advantage in studying the effects of the second dimension on the layering process. Since small spherical shells are being layered, the approximation of looking at two infinite thin plates in Cartesian coordinates, as assumed in previous theoretical analyses [1,8], is expected to break down as two-dimensional effects become increasingly important. In addition, some of the inner surface-roughening phenomena observed by Hoffer et al. [9] likely resulted from 2D effects. These surface features have been studied in solid liquid systems [10] but have yet to be analyzed in the context of target layering. While numerical descriptions of solid liquid phase changes can easily be found in the literature [10 12], a two-dimensional description of a solid to gas phase transformation could not be found. The main reason for this lies in the limited application of a sublimation and re-condensation system. Furthermore, since the total volume of the system is limited to the volume of the shell, the large density changes between the gas and the solid phase make the boundary condition at the inside surface difficult to apply. Additional reasons for studying this problem in two dimensions come from the time-dependent local heat transfer coefficient at the outer surface of the sphere imposed on the shells during

4 54 K.-J. Boehm, A.R. Raffray / Fusion Engineering and Design the layering process. This temporal and local change in heat transfer coefficient can only be studied by looking at a twodimensional case. Since there is no easy theoretical solution for the two-dimensional layering problem as there is in 1D, the heat conduction equation Eq. 1 is solved numerically to study the influence of different outer boundary conditions on the resulting mass redistribution on the inside of the shell. 3. Development of a 2D layering model The extension of the system of equations presented above Eqs. 1 5 to the second dimension is in principle a straightforward task; since the temperature and the molar density of the gas phase throughout the void are taken to be uniform as a function of the inner surface temperature, solving the heat conduction equation in the solidified portion of the fuel will suffice to describe the physics of layering. Of course the proper boundary conditions need to be applied: conservation of mass in the gaseous void, combined with the appropriate boundary conditions at the inner and outer surface of the fuel. A Gauss-Red-Black algorithm had been developed and tested successfully in previous work on target survival studies [13]. Similar to the target survival studies, the response of the temperature field in the fuel layer to different heat flux scenarios will be studied. However, the treatment of the inner and outer boundaries will be significantly more involved, as the heat flux applied on the outer surface will cause the inner surface to change its shape over the time frame studied in this case. The entire fuel layer needs to be studied as the targets are not spherically symmetric as has been assumed in previous work. The main obstacle in developing the two-dimensional layering model is the treatment of the solid gas boundary. Several concepts modeling a moving interface are presented in Minkowycz et al. [11]. A mixed Eulerian Lagrangian model, the sharp interface description, first presented by Udaykumar et al. [12], was chosen to describe the layering process as it is capable to compute the exact position of the interface over long periods of time, well suited to account for the large movements of the interface expected in the layering case. In this description, the heat equation is solved on a fixed grid, while the interface is treated as a sharp discontinuity that is moving through the grid and is tracked by recording the coordinates of a number of marker points along the interface. These marker points are treated in a Lagrangian frame, while the field equations are solved on an Eulerian grid. The stencils of grid points which will be affected by the interface are adjusted to impose the influence of the interface to the temperature of the adjacent grid points. Udaykumar et al. [10,12] illustrate the advantages of this sharp interface description and compare the results for a liquid solidification process to other sources. This model has been adapted to compute the movement of a gas solid interface by following the outline presented in [12], while certain modifications to the model had to be implemented in order to describe the specifics of the layering process. These modifications include describing a solid gas boundary instead of a solid liquid boundary, the main difference of which lies in the density change in the gas solid case, which is much larger than in the solid liquid case. Other differences include the heat flux boundary condition applied on the outer shell surface, implementing temperature-dependent coefficients, and writing the model in cylindrical coordinates to better model the target geometry. In the following section, the main features of the moving phase front on a fixed grid are highlighted, and the modifications to the boundary conditions are described. Fig. 2. Schematic of the two-dimensional layering problem. The blue markers schematically represent the inner layer initial condition, which will move through the grid. The red markers represent the outer shell boundary; despite being immobile, the local heat transfer coefficient is applied at these points. For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article. As has previously been exemplified in the 1D case, the layering process can be modeled by solving the energy transport equation, Eq. 8, in the solid part of the fuel while applying the boundary conditions from Eq. 9 on the inner boundary and from Eq. 10 on the outer boundary. It is worth noticing that Eq. 8 accounts for temperature-dependent thermal conductivity since this value changes significantly for DD or DT close to the triple point, as shown in [14]. In this sublimation and re-sublimation problem, the interface velocity is computed directly from the Stefan condition and the gradient of the temperature normal to the interface, as shown in Eq. 9, while a known heat flux is imposed on the outer boundary Eq. 10. In contrast to Udaykumar et al. [12], these boundary conditions must be applied while simultaneously conserving the total mass of the system. This means that the sum of sublimation and re-condensation at the interface, and the accumulation of mass in the gaseous state in the void must be balanced. A small change in solid volume will cause a significant increase in pressure in the void. This change in pressure, in turn, affects the sublimation flux at the surface Eq. 19. These conditions originate from the fact that we are modeling an enclosed void containing a pure species. T t = 1 s C p,s 1 r 1 V n = s H s k ice r T r r k ice T n solid + z k T ice z + T k gas n gas q s C p,s T k ice n = ht 0 T 10 The starting point of modeling the layering process in a sphere is the development of a rectangular grid representing the volume containing the sphere see Fig. 2. Two interfaces enclose the solid fuel domain in this volume. The outer surface describes the location of the thin plastic shell containing the fuel. This interface is fixed 8 9

5 K.-J. Boehm, A.R. Raffray / Fusion Engineering and Design on the grid, only the heat flux or surface temperature applied at the surface will vary. The inner surface represents the solid gas boundary, and it is free to move through the grid as the mass redistribution is modeled. The location of both interfaces is described by two sets of markers along the interfaces, which record the coordinates of the interfaces in certain intervals. By applying cylindrical coordinates, the entire sphere can be modeled; the axis of rotation runs along the centerline of the sphere. In applying this equation, some assumptions are being made about the symmetry of the initial layer shape, but they seem reasonable considering the complexity of modeling a 3D sphere and the small additional benefit one would get out of such a model. In applying cylindrical coordinates, zero gradient boundary conditions have to be imposed in the radial direction at the origin. The interface is also assumed to cross the z-axis perpendicularly. The temperature field of the medium around the shell is not computed. Instead, a certain heat flux is applied on the outer surface, which is assumed to be due to the convective or conductive cooling on the outer surface, which is ultimately determined by the technology used for the layering process. The temperature of the vapor inside the pellet is considered homogenous throughout the void, but it does depend on the temperature of the inner surface. Thus, Eq. 8 is only solved for grid points in the solid domain; these can be divided between those whose stencils are affected by the interfaces and those whose stencils are not affected. As a general form for grid points not affected by the interface, Eq. 8 can be discretized according to the rules of numerical differentiation using finite difference formulation for pure implicit solution of the problem, resulting in T n+1 T n i,j i.j t = 1 sc p,s [k n i.j T n+1 T n+1 i.j+1 i.j z T n+1 T n+1 i+1.j i 1.j + 2 r T n+1 + T n+1 n+1 2T T n+1 + T n+1 n+1 2T i.j+1 i.j 1 i.j i+1.j i 1.j i.j + z 2 r 2 k n+1 i.j+1 kn+1 i.j 1 2 z ] k n+1 i+1.j kn+1 k n+1 i 1.j i.j + + q 2 r R i For all the grid points, which are in close proximity to the interface, the stencils are modified to account for the presence of the phase boundary. As described in Udaykumar et al. [12], the location of the interface is described by recording the coordinates of a number of marker points along the interface. A mathematical description of the form rs and zs can be found. The spacing between two markers rs k, zs k and rs k+1, zs k+1, has to be of the order of the grid spacing. It is worth noticing that the markers of the interface do not necessarily coincide with the grid points. If the position of the interface between two markers is needed, it is computed by a cubic spline interpolation. After identifying all grid points in immediate proximity of the interface, we need to determine whether they lie in the solid or the gaseous domain see Fig. 3. For each grid point we thus need to find the line that passes through the grid point r i, z j and is perpendicular to the surface. Once the intersection of the surface normal and the interface s SN is found, the scalar product of the vector Eq. 12 and the surface normal n Eq. 13 will determine which side of the interface the grid point lies on based on the sign convention implemented here. The radial and axial components of are given by: r = r SN r i z SN z z = i r SN r 2 + z SN z j 2 r SN r i 2 + z SN z j Fig. 3. Schematic illustrating how the surface normal through each grid point is found to determine weather a grid point lies in the solid or the gaseous domain. The radial and axial component of the surface normal, n, can be computed by n r = z/ s r/ s n z = 13 z/ s 2 + r/ s 2 z/ s 2 + r/ s 2 Once all the grid points affected by the interface and lying on the solid side of the interface are identified, their stencils need to be adjusted. For these grid points, the discretization of Eq. 8 will be different from Eq. 11 to accommodate the presence of the interface. While the temperature field of all grid points unaffected by the interface is determined by the iterative Gauss-Red-Black GRB algorithm, the computation of the temperatures of the grid points affected by the interface using the modified stencils is performed after every Gauss-Red-Black GRB iteration. The stencils marked by a 2 in Fig. 4 will need to be adjusted due to the presence of the interface. Grid point r i, z j is chosen as an example to describe the modifications to the stencil in the r- and in the z-direction, since two of its neighbors r i 1, z j and r i, z j+1 lie in the gaseous domain. In the positive z-direction, grid point r i, z j+1 needs to be replaced in Eq. 11 by the point r i, z SZ, the intersection of the grid line with the interface, and the temperature T i,j+1 needs to be replaced by T SZ, the temperature of the interface at the intersection with the grid line. This leads to the following discretization: k 2 T z 2 + T z k z = [ 2 kti,j z SZ z j 1 + TSZ T i,j 1 z SZ z j 1 TSZ T i,j z SZ z j ] ktsz kt i,j 1 z SZ z j 1 T i,j T i,j 1 z j z j 1 14 In the r-direction, the discretization looks similar but with r SR, z j substituting for r i 1, z j and T SR for T i 1,j : Ti+1,j T i,j k 2 T r 2 + T r k r = [ 2 kti,j r i+1 r SR + Ti+1,j T SR r i+1 r SR r i+1 r i kti+1,j kt SR r i+1 r SR T i,j T SR r i r SR + kt i,j r i ] 15

6 56 K.-J. Boehm, A.R. Raffray / Fusion Engineering and Design Fig. 4. Schematic illustrating a segment of the interface, the markers of which are represented by the blue squares. The distinction between the red and black points for the GRB algorithm is depicted along with the classification of the true neighbor points 2 and the regular solid domain 1. For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article. Thus, instead of using the temperature of the neighboring grid point to solve Eq. 11 at the present point, the temperature of the interface at the intersection with the grid line is used. In the case where the interface separates the point r i, z j inthe solid domain from r i, z j 1, in the gaseous domain in the negative direction, Eq. 14 needs small adjustments, but the underlying discretization would be the same. A similar argument holds true for Eq. 15 and the interface passing between point r i, z j solid and r i 1, z j gas. The position SR and SZ in coordinates along the interface at which the interface intersects the grid lines in the radial and axial directions, respectively, can be found by using a cubic spline interpolation of the interface. In the case where the phase boundary crosses a grid line more than once in the neighborhood of one grid point, the intersection that lies closest in the corresponding direction along the grid line has to be found. In order to complete the description of the layering problem, the appropriate boundary conditions need to be implemented at both the inner and the outer interfaces. In the present study, the effects of a certain distribution in heat transfer coefficient along the outer surface of the pellet will be investigated. Applying a non-constant heat transfer coefficient on the outer surface leads to a variation in outer temperature; a cubic spline interpolation was implemented to determine the temperature of the interface between marker points. When using Eqs. 14 and 15 to compute the temperatures of the true neighbor grid points, the temperature on the interface is required, which depends on the temperature field of the solid domain AND on the heat flux boundary condition on the outer surface. As a result both the temperature field and the heat flux boundary condition have to be computed simultaneously. By iterating between first computing the temperature field in the solid domain based on previously computed temperatures at the interface markers, and then updating the temperature at the interface markers based on the temperature field, the system temperature can be taken to desired levels of convergence. In this work, the temperature field at the interface and in the solid domain is obtained Fig. 5. Schematic illustrating the temperature distribution computed along the surface normal in order to relate the heat flux from the cooling gas outside the shell to the surface temperature of the shell. iteratively by coupling the computation of the interface temperature into the GRB algorithm. When computing the temperature gradient normal to the interface at each interface marker, a forward differencing formula is applied using two points along the surface normal. The temperature at these points is determined through bilinear interpolation of the four neighboring grid points. The distance between the two points along the surface normal is chosen to be of the order of the grid spacing, as illustrated in Fig. 5. The temperature of the interface marker is then found by applying q = k ıt 4 ın = k TN1 T N2 3 T n q = ht 0 T 17 where T is the gas temperature of the cooling gas around the sphere, and h is the local heat transfer coefficient. The interface temperature can be computed directly by applying: T 0 = T + 1 h/k + 3/2 n 4 TN1 T N2 2 n 18 The inner boundary poses some complex difficulties, which originate from the geometry of the problem and the nature of the solid gas phase change that we are trying to model. In contrast to the solidification/melting problem as modeled by Udaykumar et al. [10,12], the sublimation condensation problem at hand is accompanied with a large change in density between the two phases. This, in combination with the fact that the gaseous phase is enclosed in a cavity, poses a challenging problem: the mass flux at the interface, coupled to the temperature field defined by the Stefan Condition Eq. 9 on the solid side, has to be matched to the condensation and sublimation flux defined by Eq. 19, as described by Collier [15], on the gas side, and to the physical law of mass conservation. It is the last part, the mass conservation equation, which makes this problem difficult, as it causes a pressure increase in the gaseous void in the case of a net sublimation flux along the entire interface and a net decrease in pressure in the case of a net condensation flux. H s P g q = 2MA Tg P vt h 19 R gas T h

7 K.-J. Boehm, A.R. Raffray / Fusion Engineering and Design Due to the high density difference and the small size of the void, an excess sublimation or condensation of fuel leads to large changes in pressure, which, from Eq. 19, in turn leads to a large change in the sublimation/condensation flux. This means that Eq. 19 and Eq. 16 have to be matched for the heat flux across the interface, and this heat flux has to satisfy Eq. 9 Stefan Condition for the movement of the interface; this movement of the interface has to result in a pressure change, such that Eq. 19 is still satisfied. Similarly to the outer boundary case, this problem is solved through the GRB-iterations of the main transport equation. Although the pressure and temperature in the void are subject to changes, Eq. 19 indicates that the pressure in the void and the vapor pressure at the interface temperature have to be very close to each other. The following sequence has been implemented in order to satisfy all of the above conditions: After each GRB-iteration, the temperature at the boundary is found from the temperature field of the solid domain by using Eqs. 14 and 15. Based on this temperature field, the heat flux at the surface is found from Eq. 16. The same forward difference formula in combination with bilinear interpolation as illustrated in Fig. 5 has been implemented for this purpose. The velocity of the interface is then computed from Eq. 20, under the assumption of a very small temperature gradient in the gas. Next, the net mass flux based on this heat flux is found by applying Eq. 21. V n = 1 T k s H ice s n solid as k gas T n gas 0 20 ṁ gas void = V n s A 21 where A is the area of the surface over which the phase change occurs. This is found for each marker point k by computing the length of the interface between k 0.5 and k This length is then multiplied with the arc length 2r k to get the area. Once the total volume of the void is known, the change in pressure resulting from the change in mass in the void can be computed. Ideally, one would use this value in the next iteration specifically as P g in Eq. 19; however, this results in a very stiff set of nonlinear differential equations making this approach highly unstable. Instead, the pressure after each GRB-iteration is raised by a small value if the computed change in pressure is positive, or is decreased by a small value if the change is negative. Once the change in pressure switches signs, the interval by which the pressure is changed is decreased, converging in the real value of the pressure. This method, though crude, worked reliably in the simulations. It depends on a good initial guess of the temperature and pressure at the interface within a few degrees K or on a very small time step for the first few seconds of the simulation. This boundary condition imposes a very strong time step restriction on the model. If the time step is too large, the resulting movement of the interface is too large, causing strongly oscillating values of the pressure as the interface temperature is adjusted. In applying this model, a value of the pressure in the capsule is found that satisfies both the mass flux boundary condition from Eq. 19 and the rise in pressure due to mass accumulation/loss. Simultaneously, the temperature field corresponding to this mass flux is determined. After each time step, the interface makers are moved according to Eq. 9. The change in total volume is computed by Eq. 22 as it will influence the absolute pressure for the next time step. V = V n t A 22 As a final step to close this system of equations, the total volume of the gaseous void needs to be initialized. The change in volume can be determined from the movement of the interface, but the total volume at the beginning of the simulation needs to be computed, Fig. 6. Schematic illustrating the computation of the mass moments of inertia for an un-layered sphere. The x-moment is computed around the x-axis passing through the origin, while the y and z moments are computed around the axes parallel to y and z, but passing through the center of gravity. as it quantifies the rise in pressure due to a change in the number of moles in the void. In order to keep the initial conditions arbitrary, the area of the gaseous fraction and the coordinates of the center of gravity for each cell are computed before the first time step. The volume, V i,j, corresponding to the gaseous fraction of the grid can be computed by applying the symmetry condition along the centerline of the cylinder. V i,j = 2 A i,j r i,cg 23 where r i,cg is the radial coordinate of the center of gravity of the gaseous portion at grid point r i, z j, and A i,j is the respective area. During the simulation, some mechanical properties, such as the overall center of mass and the mass moments of inertia around the main axes are computed. The overall center of gravity of the fuel layer in the axial direction, Z cg, is computed under the assumption that the gas does not contribute to the overall weight which is a good approximation considering the large difference in density: V i,j z j,cg i,j Z cg = 24 V Outer sphere V Total void where z j,cg is the axial coordinate of the center of gravity of the gaseous fraction at grid point r i, z j, V Outer sphere is the total volume of the sphere that is being layered, and V Total void is the total volume of the void determined by summing Eq. 23 over all grid points in the gaseous domain. Two different values for the three different mass moments of inertia of an unbalanced sphere can be found. Assuming that the center of gravity is shifted in the negative axial direction as indicated in Fig. 6, we can compute I xx from Eq. 25 by subtracting the moment of inertia of the void from the moment of inertia of the solid sphere. The other two moments I zz and I yy are computed by first determining the moments for the void around the x = 0 axis Eq. 26a and applying the equation corresponding to the moment of inertia of a ring to each individual area volume segment and parallel axis theorem. The moment of the void needs then to be subtracted from the moment of the solid sphere and simultaneously the parallel axis theorem needs to be applied again to account for the shift in center of gravity away from the center of the sphere from Eq. 26b see Fig. 6 for the nomenclature. Here, we assume

8 58 K.-J. Boehm, A.R. Raffray / Fusion Engineering and Design Fig. 7. Two different initial conditions implemented in the model for preliminary testing. Both assume the presence of a thin film, covering the inside of the plastic shell completely. This approximation can be justified by the zero degree wetting angle reported by Harding 60 in his single sphere layering studies. that the geometrical center of the sphere is located at z = 0 in the coordinate system of the layering model. I xx = 8 15 sr 5 V i,j i,j r 2 i,cg 25 i,j I yy,void = I zz,void = V i,j i,j z 2 j,cg r i,cg i,j I yy = sr 3 5 R2 + X 2 I yy,void V i,j 2 X CGvoid + X 2 i,j 26a 26b In general any interface shape can be imposed as an initial condition to the layering problem. As an input, the model requires the coordinates of a certain number of marker points. Two different initial conditions have been implemented for preliminary testing, as depicted in Fig. 7. The spherical shape is chosen to reproduce redistribution speeds similar to the 1D case, while the frozen puddle represents more closely the initial conditions after freezing the fuel to the bottom of the shells. Results are discussed in the following section. It was reported by Harding et al. [16] that the liquid deuterium or deuterium tritium mixture will wet the entire inside surface of the shell due to its zero degree wetting angle. After freezing, the initial layer is expected to look similar to the ones represented in Fig. 7; however, the initial conditions presented here are just suggestions, and can certainly be changed. The model described above is suitable to simulate the layering process under the given assumptions. However, before showing results from test cases, a couple of important points must be made: As the movement of the interface through the grid is modeled, we need to have an interface present before the layering begins. This is done by assuming that a layer of finite thickness is present before the layering is initiated. Numerically, the minimum thickness of this initial layer depends on the grid spacing. Otherwise the freezing and crystal growth would have to be modeled first, which is outside the scope of this study. The zero degree wetting angle of deuterium and deuterium tritium reported in the literature [16] allows the assumption of a thin initial layer. In phase change problems the solid gas interface rarely maintains a planar state as material is deposited or evaporated from the interface. Small perturbations of the surface smoothness tend to grow into bigger disturbances [9,10,12]. These disturbances seem to grow at first, but smoothen out again, when the layer is close to equilibrium due to the bulk heating. These disturbances can be explained by the temperature gradient along a surface normal that is only locally normal to the surface, but not globally due to the initial slight deviation from the planar state. This is not an error or instability of the model, but the result of a physical instability. Ref. [10] shows that local disturbances to the shape of the interface push the isotherms closer together, leading to an increase of the local heat flux. These instabilities lead to the development of inner surface roughness features during the initial layering phase. These features will disappear as the layering process continues. Similar roughening has been observed and reported in freezing and layering experiments at the Los Alamos National Laboratories [9]. In order to avoid numerical difficulties resulting in the development of long fingerlike features in the layering process, the maximum allowable curvature of the interface was limited, following the arguments presented in Udaykumar and Shyy [10]. 4. Validation and test cases The first tests after developing the model consisted of convergence for decreasing time steps and grid spacing, which were successfully performed. A number of simple test were then performed to ensure that the basic principles of mass and energy conservation have been followed. Next, we compared the model results to the 1D case that can be solved analytically. As a final step in the model validation, a mass redistribution experiment was performed providing a test case under controlled conditions, which will be described in the following section. The inner boundary condition, treating both the phase change and the change in pressure originated from a net flux of gas to or from the void Eqs , imposed a strong time step restriction on the model. Since time steps larger than 3.0 s resulted in numerical instabilities, the maximum time step allowed by the solid gas phase change boundary condition is significantly smaller than the stability criterion of the thermal model by almost one order of magnitude. Thus, it is sufficient to choose a time step size sufficiently small for the inner boundary to be stable to compute a converging solution. Because of the strong coupling of net mass flux into the void from a net evaporative and condensing flux and the change in pressure, an inaccurate account of the total mass would substantially affect the outcome of the simulation, underlining the importance of conservation of mass. It is crucial for the accuracy of the prediction that the sum of all moles in the solid and the gas phase is as close to constant as possible and depends only on the initial condition. As a first test, a simulation was started using the parameters given in Tables 1 and 2 and arbitrary initial conditions. The total number of moles in the solid and the gas phase were added after each time step and compared to the initial number of moles. The error in mass was found to be less than four orders of magnitude lower than the total number of moles in the system. These small changes in mass can be attributed to the approximation that the gas temperature is equal to the temperature of the inner surface. The magnitude of this change in temperature depends on the difference in temperature chosen as initial condition and computed as a final inner surface temperature. In special instances when the curvature of the interface becomes too large, or in order to accommodate the boundary conditions at the axis of symmetry, the marker points are artificially moved leading to small disturbances in the total number of moles in the system. In these cases the change in total number of moles in the system is of the order of

9 K.-J. Boehm, A.R. Raffray / Fusion Engineering and Design Table 1 Input parameters and properties for DT, D 2 and water layering. DT-layering D 2-layering H 2O-layering T Temperature of the cooling gas H s Latent heat of fusion J/m J/m J/m 3 M Molecular mass kg/mol kg/mol kg/mol ice Density of the ice 251 Kg/m kg/m kg/m 3 P vt Temperature-dependent vapor pressure see Eqs. 27 and 28 A = A = A = B = B = B = C = C = C = q Volumetric heat W/cm W/cm 3 k ice, Thermal conductivity 0.31 W/m K 0.31 W/m K 2.25 W/m K k vapor Thermal conductivity of vapor 0.01 W/m K 0.01 W/m K W/m K C p,s Specific heat 4280 J/kg K 2675 J/kg K 2050 J/kg K 0.01% of the total mass in the system. However, the model recovers quickly from these disturbances and stays below the acceptable disturbance limit. P v,dt D2 = exp A B + C lnt T Kelvin 27 Kelvin 6150 P v,h2 OT = [A + B T Celsius + C T 2 Celsius ] exp T Kelvin in mbar 28 As a second test, the shape of the final layer in thermal equilibrium is compared between two cases with different initial conditions. These two cases were chosen such that the total mass in the shells was equal. The final outcome of the layer formation does not depend on its initial condition but only on the time it takes to develop the equilibrium position, as can be seen in Fig. 8. Since a constant heat flux of a given value is imposed on the outer surface in both cases, the equilibrium temperature field and the final location of the inner boundary are uniform and identical for the two cases. In addition, the mass moment of inertia was computed and is shown in Fig. 9 as a function of time. As the simulation continues and the fuel layer equilibrates to uniform thickness, the mass moments of inertia around the three main axes approach the same value in both cases, indicating that the model computes the same final layer distribution independent of the initial condition. As a third test, the global conservation of energy was verified. While applying different values for the heat transfer coefficient, h, and gas temperature, T, on the outer surface, the equilibrium temperature of the system changed, leading to slight changes in the location of the inner surface. After equilibrating, the temperature of the outer surface was used in Eqs. 29 to verify that the amount of cooling induced by the gas is equal to the total amount of heat produced inside the shell by the volumetric heat see Tables 1 and 2 for operating parameters. q h = T 0 T T 0 = T + q V solid fuel A s h 29a 29b In Eq. 29b, the product of the volumetric heat and the solid volume is used to compute the total amount of heat produced in the shell, while the heat flux coefficient and the outer surface area are used to compute the cooling provided by the gas. If the computed final temperature of the shell corresponds to the one computed in Eq. 29b, the total amount of heat is balanced, satisfying the conservation of energy. As an example studied within this series of tests, for a volumetric heating of W/m 3 in a 43% filled 4-mm shell, the difference in temperature between the cooling gas and the outer surface of the shell in steady state was computed to be K from Eqs. 29, and K from the time-step model, demonstrating that the total energy in the system is balanced. As a next test, the model s capabilities to compute the volume of the void, the mass moment of inertia and the distance between the center of mass and the geometrical center were tested in cases for which exact solutions exist. By using a sphere as the initial shape of the void the radius of the inner sphere had to be sufficiently small for it to fit within the boundaries of the outer sphere, but the two spheres were not concentric the quantities in question could be computed exactly by applying global equations of a twobody problem. We could then compare these results to the ones computed by the discretized model for validation. In the case of two non-concentric spheres, the volume of the void can be found by computing the volume of the inner sphere and the location of the center of gravity can be computed from Eq. 30. The mass moment of inertia can be calculated from Eq. 31 by applying the parallel axis theorem. For two spheres, whose centers are separated by S, the distance between the center of the outer sphere and the center of gravity Table 2 Input parameters for conservation of mass and energy tests. In order to simplify the theoretical computations, a constant heat flux has been applied on the outer surface of the shell. Parameter Value Unit Heat transfer coefficient 77.8 W/m 2 K Outer radius m Temperature cooling gas 18.9 K Initial target temperature K Volumetric heat W/m 3 Latent heat of sublimation J/m 3 Equilibration time 3000 s Volume of the void m 3 Initial offset m Theoretical temperature difference between the cooling gas and the target surface assuming spherical geometry R 3 R 3 inner q 3Rh K

10 60 K.-J. Boehm, A.R. Raffray / Fusion Engineering and Design Fig. 8. Layering process for two different initial conditions. On the left hand side, the initial inner boundary is defined by a non-concentric inner circle of radius 1.6 mm. On the right hand side, the initial layer is chosen to simulate the frozen fuel gathered at the bottom of the shell. Both initial condition lead to the same final fuel distribution since the shells contain the same amount of fuel, and the same outer boundary conditions constant heat flux are applied. X is given by: V innersphere S X = 30 V outersphere V innersphere Based on this result, we can compute the mass moments of inertia using the following equations: I xx = 8 15 R5 R 5 i I yy,zz = 4 3 R3 2 5 R2 + X R3 i 2 5 R2 i + X + S 2 31 The results from this test are summarized in Table 3. The decreasing difference between the analytical value and the value computed by the model using spatial discretization with decreasing grid spacing indicates spatial convergence. Fig. 9. Mass moment of inertia of the system during the layering process for two layering cases with the same void volume while a uniform temperature and heat transfer coefficient is applied. One-dimensional analytical results presented by Martin et al. [1] for constant local heat transfer coefficients or outer surface temperature and confirmed by Hoffer and Foreman [5] experimentally for cylindrical shapes were simulated with the new 2D spherical layering model. Due to the difference in geometry between the model sphere and the theoretical analysis planar geometry, the initial condition had to be picked carefully. The results from the model cannot be expected to perfectly match the theoretical results due to the difference in geometry. A spherical shape, the center of which is shifted in the positive axial direction, was chosen as the initial condition of the inner boundary. Thus, a gradual change in layer thickness is imposed to the model initially, while the maximum and minimum thicknesses correspond to the extreme angles 0 and 180 degrees. This leads to a distribution of mass between two non-concentric spheres as initial conditions, as shown in Fig. 8left hand side. In order to compare the results, the difference between the center of the sphere and the center of gravity is plotted in Fig. 10. The results from the simulated layer redistribution can be compared to the theoretical analysis by finding an exponential fit through the modeling results and comparing the results to the 1D theoretical predictions from Eq. 7. The heating rate chosen for this simulation is 200 mw/cm 3 in DT solid fuel, four times higher than the heating from the natural beta decay in DT, which would lead to a theoretical 1/e layering time constant of 384 s. In the simulation, the difference between the center of gravity and the center of the sphere show that in the spherical geometry a faster layering time can be observed until the unbalance is 10 m; the computed 1/e layering time was 312 s. The deviation of the simulated results from the exponential curve for unbalanced smaller than 10 m is found to be due to the surface roughness features that develop during the initial stage of the mass redistribution process. These features resolve at a much slower rate

11 K.-J. Boehm, A.R. Raffray / Fusion Engineering and Design Table 3 Comparison of geometric parameters calculated analytically and from the model computations for 2 cases with inner and outer radii of 1.6 and 2.0 mm, respectively: 1 two concentric circles; and 2 distance between the two centers of the sphere, s = 0.2 mm. For increasing grid points, the model returns values closer and closer to the computed value indicating convergence Mark Mark Mark Mark 321 Exact calc. X m Volume of void 10 8 m I xx kg m I yy kg m Mark Mark Mark Mark 321 Exact calc. X m Volume of void 10 8 m I xx kg m I yy kg m than the overall equilibration, as also observed experimentally by Hoffer et al. [9]. Applying lower values for the volumetric heating leads to similar results; in each case, the redistribution speed from the model is slightly faster than the one predicted by the theory for the planar case. In each instance, the equilibration speed slows down significantly once the non-uniformities are of comparable size to the surface roughness features 10 m. 5. Verification of the model by comparison to experimental results In order to build higher confidence in the modeling results, a meaningful test case needed to be found, which provided a known, but non-uniform heat transfer coefficient along the outer surface in a thermally controlled environment. Thus, a redistribution experiment was set up in a laboratory environment. Due to the complexity of the apparatus required to fill spherical shells with a pure species generally, hollow plastic shells are filled by permeating gaseous fuel at high pressures into the shells which are then cooled past the triple point to reach a solid gas mixture, the possibility of using water as a surrogate was investigated. The redistribution of water in a partially water filled and otherwise evacuated volume was chosen to provide a test case against which the model could be benchmarked. This volume was to be cooled in a gas stream in order to provide a non-uniform local heat transfer coefficient. Fig. 11 shows a schematic of the test stand for the water redistribution experiment. A cuvette, a small tube of square cross-section 1 cm by 1 cm and 5 cm in length, is filled to a level of 0.5 cm with pure water and held in place inside a glass tube 2.4 cm in diameter by a thin steel tube. This steel tube feeds through a small hole in a thin 1 cm by 1 cm plate, which is glued to the top of the cuvette sealing the volume of the cuvette and the steel tube from the cooling gas stream. Through this tube, excess gas air in the cuvette can be evacuated. The cuvette is inserted upright into a glass tube and cooled by a temperature controlled nitrogen gas stream. In this experiment, the mass redistribution of water inside the cuvette under IR irradiation can be studied. While a known but non-uniform heat flux is imposed on the outer surface, the movement of the interface can be studied experimentally by analyzing pictures of the water level before and after layer redistribution and numerically by applying the corresponding initial and boundary conditions along with the appropriate geometric constraints as model input. The heating system providing bulk heating to the water in the cuvette consists of an IR halogen light bulb, which is mounted on the outside of the vacuum vessel. Honed copper tubes act as waveguides to deliver most of the IR radiation directly into the water. A narrow band pass filter is used to limit the incoming radiation to wavelengths where the absorption of water lies between 1 and 3cm 1. In order to quantify the total heat in the filtered spectrum provided by a 21 Watts halogen IR bulb, the heat flux through the entire setup of waveguides, filters and window glass was measured in front of and behind the water-filled cuvette. Measurements yielded mw in front of the cuvette and mw behind the cuvette, Fig. 10. Histories of the computed system unbalance distance between the center of mass of the entire system and the center of the outer sphere for a simulated case with q = 200 mw/cm 3. Fig. 11. Experimental setup used to perform a water surrogate layering experiment.

12 62 K.-J. Boehm, A.R. Raffray / Fusion Engineering and Design Fig. 12. Pictures from the water layering experiment at time = 0 and time = 64 h. thus the absorbed heat was estimated as 9.58 mw over the path length of 1 cm and an area of cm 2 area of the detector plate. This corresponds to an absorption coefficient of 2.7 cm 1, which is in good agreement with the reported absorption coefficient of water [17] at the wavelength = 1382 nm and the specifications of the optical filter used in this setup. The volumetric heating rate was W/cm 3 in the narrow band around 1380 nm. The properties that describe this experimental setup are presented in Table 1. The temperature-dependent vapor pressures over the solid are computed from Eq. 28, the coefficients A, B, and C are listed in Table 1. Due to the significant influence of air as a non-participating gas in the void space of the cuvette the experiment was performed under evacuated conditions. The partial pressure of air in the vapor phase needs to be lower than the vapor pressure of water at the operating temperature, which reduces the influence of the air as a non-participating gas to negligible levels. After evacuating the air from the vapor space, the water in the cuvette was cooled to just under the freezing point. A temperature sensor was installed in this setup, measuring the temperature of the cooling gas 20 cm below the cuvette. The absolute temperature reading of the gas stream at the freezing point of water was K. The freezing point was determined by observing the water inside the cuvette to turn opaque fast freezing. This absolute temperature reading might be distorted due to the distance of the location of measurement and the point of interest cuvette, and due to the heat connection between the temperature sensor and the surrounding tube. However, the temperature difference between the freezing point of water and the operating point is estimated to be accurate to within 0.1 K accuracy of the platinum resistance temperature detector RTD is 0.05 K. The operating point for the measured gas temperature for this experiment was chosen to be 0.3 K below the measured temperature of the freezing point. This temperature was maintained by a proportional-integral-derivative PID feedback control system which measures the temperature and adjusts the power delivered to a small trim heater 20 W to keep the temperature at a pre-set value K. The temperature control worked to within a ±20 mk of the set point. The gas stream velocity of the cooling nitrogen was measured beforehand in a separate experiment; it was determined that the gas flow speed was 6 ± 0.5 m/s at atmospheric pressure. In order to estimate the approximate time after which a significant redistribution can be expected, we applied the results from the heat transfer layering equations shown in the previous section for the mass redistribution between two parallel plates under completely evacuated conditions: ıt = ı 0 exp t 32a = H s s q = 39.4 h 32b Based on these results, a significant mass redistribution will occur within a couple of days, and we conservatively picked 64 h as the experimental layering time.

13 K.-J. Boehm, A.R. Raffray / Fusion Engineering and Design Fig. 13. Numerical results of the mass redistribution of ice in a gas cooled cylindrical cuvette. Each line on the left represents the location of the interface at layering time intervals of 8 h. The right hand side shows an overlay of the experimental and the numerical results. The location of the solid vapor interface inside the cuvette was recorded using a camera and a backlight. A significant amount of backlight was required to record an image with the camera. However, the backlight would also be absorbed in the water, and the results would be distorted if the backlight was used too long or too frequently. Fig. 12 shows two images taken at 0 h and 64 h of redistribution time. We established that the redistribution was indeed due to the IR light, as when the ice was cooled without the heat source in a separate set of experiments, no layer movement was observed. We developed an experimental case for which the parameters and boundary conditions are known with a non-uniform heat flux at the outer surface of the cuvette. Based on the flow information in combination with the gas temperature, the local heat transfer coefficient along the outer surface of the cuvette can be estimated. All other parameters of interest have been determined, and the layering model can be used to simulate the experiment. The model presented in Section 3 is suitable to model the experiment described above. However, some modifications needed to be implemented in order to accommodate the altered geometry. Furthermore, the properties of the redistributed mass needed to be changed from deuterium or deuterium tritium mixture to the water used in the experiment see Table 1. In order to limit the changes to the model presented and tested previously, a cylindrical cuvette with round cross-section was modeled instead of the square cross-section used in the experiment. This approximation will distort the final results slightly, but for the benchmarking purpose of this test, the results are expected to be sufficiently accurate. This simulation serves as a verification rather than as a complete validation. However, the modeling results can be used to show that the correct physical laws and processes have been modeled. The redistribution of the water from its initial configuration is shown in intervals of 8 h in Fig. 13 until a simulated run time of 64 h is reached. The heat flux boundary condition along the outside surface of the cuvette is approximated by the heat transfer found in the case of cooling of a flat plate with laminar flow, based on the local Nusselt number. The Nusselt number as a function of the axial coordinate, z, is given by: Nuz = Pr 1/3 Re 1/2 33a Fig. 14. Temperature distribution from the simulation of water layering experiment. The sub-cooled region close to the cooled surface is shown in blue, while areas of elevated temperatures are shown in red. For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article. The local heat transfer coefficient is then estimated as: 1/2 U hz = Pr 1/3 g k gas 33b f z Comparison of the simulated results in Fig. 13 to the experimental results in Fig. 12 shows a reasonably good agreement. The dark area underneath the surface in the experimental results see Figs. 12 and 13 on the right hand side can be explained from studying the thermal contour picture shown in Fig. 14. The region of elevated temperature shown as a red contour in Fig. 14 has the same shape as the dark spot seen experimentally after 64 h see Fig. 13, indicating that the elevated temperature in this region is causing the ice to melt. The experiment was conducted at a temperature very close to the freezing point, thus small increases above this temperature the model predictions are 10 mk will certainly lie in the liquid domain. The volume containing the melt forms a rough interface with the ice, causing it to become opaque. In summary, the attempt to provide a controlled experimental test case for the model under known conditions was successful. The model reproduced the mass redistribution reasonably well, considering the uncertainties of the experimental setup and the geometry approximation. The predicted surface roughness fea- Fig. 15. Temperature field of the fuel layer shown as a color coded 3D plot. Areas of elevated temperatures can be seen in thicker parts of the layer, while colder areas can be identified close to the cooled surface. For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.

14 64 K.-J. Boehm, A.R. Raffray / Fusion Engineering and Design Fig. 16. Influence of non-uniform local heat transfer coefficient on layering thickness for a single stationary sphere in a cooling gas stream. tures were very coarse 1.5 mm as compared to 0.5 mm in the experimental case, and the peak-to-valley-distances amplitudes of the surface instabilities were magnified 0.5 mm as compared to 0.1 mm. This is a result of the coarse grid that was chosen in this test in order to keep the computation time within reasonable margin <1 day. 6. Example input and layering simulation results As an input, the model requires the coordinates of a number of marker points along the inner and outer surfaces, the magnitude of the volumetric heating and thermal and mechanical properties of the fuel, including the vapor pressure curve over the solid, the density, latent heat and molecular mass. In addition the layering time needs to be specified along with the local heat transfer coefficient along the outer target surface and the temperature of the cooling gas. The grid spacing and the size of the overall domain also needs to be given. As output, the model returns the position of the markers of the inner interface after a certain number of time steps, along with the coordinates of the center of gravity and the mass moments of inertia. The temperature at the inner and outer surface and the entire temperature field in the fuel layer are part of the output see Fig. 15 for an example output showing the temperature field in the solid fuel layer in a 3D plot. The history of the vapor pressure in the void is also part of the model output. The model can be used to determine the equilibrium layer distribution for a specific case with non-uniform heat flux. Fig. 16 shows an example for a non-uniform heat flux around a stationary sphere in a cooling gas stream. The local heat flux around the sphere as a function of the angle from the leading edge was based on Ref. [18], and implemented to the model in form of a polynomial fit, as depicted on the left in Fig. 16. The results show that in thermal equilibrium, a non-uniform layer thickness results from the nonuniform heat flux as indicated at the two extreme angles = 0 and = infig. 16 on the right. 7. Summary and conclusions The mass redistribution process has been modeled numerically in two dimensions; the computational algorithm was verified by comparison to 1D results presented in the literature and to a water layering experiment performed as part of this study. This model delivers information about the layer formation in a sphere, which is exposed to a certain local heat flux distribution on the outer surface and which is partially filled with a volumetrically heated solid. The key findings from this study are summarized below: - The two-dimensional numerical description of a solid gas interface poses a challenge due to the significant change in density across the interface. Through careful application of the basic principles of physics in particular the conservation of mass and energy, a model derived in the literature describing a solid liquid phase change could be adapted to describe the sublimation and re-sublimation problem encountered in this application. - The model was developed to simulate deuterium layering in experiments and D T layering within a few days of having the target filled. Thus, the influence of a non-participating gas i.e. 3 He in the gaseous void was not included in the model. - By applying cylindrical coordinates in two dimensions, the movement of the interface in a spherical target could be modeled. The layering times computed in this analysis could be compared to the 1D theoretical results, which assumed planar geometry. - The expansion of the layering model to the second dimension enabled the analysis of the influence of a non-uniform heat flux on the outer shell of the target. The equilibrium layer thickness non-uniformity for a given non-uniform heat flux applied on the outer surface could be found. - The development of inner surface roughness features, previously reported in single sphere layering experiments, was observed and explained from the simulations, arising from the unstable growth of small surface perturbations. However, these roughness features subside resulting ultimately in a smooth inner surface,