Influence of Multi-Dimension Heat Conduction on Heat Flux

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Influence of Multi-Dimension Heat Conduction on Heat Flux Calculation for HFIR LEU Analysis

Nuclear Engineering Division

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ANL/RTR/TM-17/18

Influence of Multi-Dimension Heat Conduction on Heat Flux Calculation for HFIR LEU Analysis

prepared by Cezary Bojanowski and Aurelien Bergeron Nuclear Engineering Division, Argonne National Laboratory September 2017

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Influence of Multi-Dimension Heat Conduction on Heat Flux Calculation for HFIR LEU Analysis i

Abstract

Steady-state safety margins calculations are an important step in the evaluation of research reactor design. Margins for phenomena such as onset of nucleate boiling and critical heat flux are evaluated by comparing the heat flux leading to those phenomena to a local heat flux. Evaluating properly the magnitude of the heat flux is therefore important and can affect the design. Plate-type research reactors often exhibit significant power peaking on the edges of the plate, near the fuel/cladding interface. The magnitude of the heat flux calculated in these regions can be influenced by the size of the region in which they are evaluated but also by the ability to model heat transport in one or multiple dimensions. The present analysis studies the influence of these two parameters on the heat flux magnitude for the HFIR reactor. From this analysis, a fuel discretization grid is proposed that can generate one-dimension heat flux values relatively close to best estimate values obtained with a high-resolution grid and complex three-dimensional CFD modeling.

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Influence of Multi-Dimension Heat Conduction on Heat Flux Calculation for HFIR LEU Analysis ii

Acknowledgements This work was sponsored by the U.S. Department of Energy, Office of Material Management and Minimization in the U.S. National Nuclear Security Adminstration Office of Defense Nuclear Nonproliferation Office under Contract DE-AC02-06CH11357. The authors would like to thank the Oak Ridge National Laboratory (ORNL) High Flux Isotope Reactor (HFIR) conversion team lead by David Renfro for sharing their models.

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Influence of Multi-Dimension Heat Conduction on Heat Flux Calculation for HFIR LEU Analysis iii

Table of Contents Abstract ........................................................................................................................................... i Acknowledgements ......................................................................................................................... ii 1 Introduction .............................................................................................................................. 1 2 Power Distribution Generation ............................................................................................... 2 3 Three-Dimensional Computational Fluid Dynamics Model ................................................... 6

3.1 Geometry and Mesh ...................................................................................................................... 6 3.2 Material Properties ....................................................................................................................... 9 3.3 Base Case Results ....................................................................................................................... 12 3.4 Sensitivity Study ......................................................................................................................... 17

3.4.1 Mesh Sensitivity .............................................................................................................. 17 3.4.2 Thermal Boundary Conditions Sensitivity ...................................................................... 17 3.4.3 Turbulence Model Sensitivity ......................................................................................... 18 3.4.4 COMSOL Verification Simulations ................................................................................ 19 3.4.5 Summary of Sensitivity Analysis .................................................................................... 19

4 Suitable Fuel Discretization for 1D Analysis ..........................................................................20 4.1 3D versus 1D Heat Flux ............................................................................................................. 20 4.2 Coarse Fuel Discretization Suitable for 1D Analysis ................................................................. 24

5 Conclusions ..............................................................................................................................30 References ......................................................................................................................................31

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Influence of Multi-Dimension Heat Conduction on Heat Flux Calculation for HFIR LEU Analysis iv

List of Figures Figure 2-1 – Inner fuel element power density profile (left) and as a function of the distance along the arc length (right) taken at three axial position (top, midplane, bottom) ............................................................. 4 Figure 2-2 – Inner fuel element power density profile (middle) and as a function of the fuel length taken on the inner edge (left) and outer edge (right) ................................................................................................... 5 Figure 3-1 – Dimensions of the cross section of the model (top) symmetry plane in coolant (bottom) symmetry plane in the fuel. (Drawing not to scale) ...................................................................................... 7 Figure 3-2 – Cross-section through the Star-CCM+ model .......................................................................... 7 Figure 3-3 – Mesh distribution detail ............................................................................................................ 7 Figure 3-4 – Side view of the model ............................................................................................................. 8 Figure 3-5 – Dynamic viscosity as a function of temperature .................................................................... 10 Figure 3-6 – Density of water as a function of temperature ....................................................................... 10 Figure 3-7 – Thermal conductivity of water as a function of temperature ................................................. 11 Figure 3-8 – Specific heat of water as a function of temperature ............................................................... 11 Figure 3-9 – Convergence of residuals in the base simulation ................................................................... 12 Figure 3-10 – Map of wall Y+ on the interface between the clad and coolant ........................................... 13 Figure 3-11 – Velocity of the coolant .......................................................................................................... 14 Figure 3-12 – Temperature on the interface between the clad and the coolant ........................................... 14 Figure 3-13 – Boundary heat flux on the boundary between the clad and coolant ..................................... 15 Figure 3-14 – Boundary heat flux on the clad surface in the middle of the model ..................................... 15 Figure 3-15 – Boundary heat flux on the clad surface in the axial cross-section ....................................... 16 Figure 3-16 – Boundary heat flux in the base, coarse, and dense mesh models ......................................... 17 Figure 3-17 – Boundary heat flux in the base model and the model with adiabatic boundary conditions on the external walls ........................................................................................................................................ 18 Figure 3-18 – Boundary heat flux in the base mode with k-epsilon turbulence model as well in the model with k-omega turbulence model .................................................................................................................. 18 Figure 3-19 – Comparison of Results between Star-CCM+ and COMSOL ............................................... 19 Figure 4-1 – Comparison 1D heat flux (left) / 3D heat flux (middle) and ratio 1D/3D heat flux with ratio color scale legend (right) ............................................................................................................................ 20 Figure 4-2 – Comparison 3D / 1D heat flux along the fuel width at three axial location: top of the fuel (top), fuel midplane (middle) and bottom of the fuel (bottom) ............................................................................ 22 Figure 4-3 – Comparison 3D / 1D heat flux along the fuel height on the inner (left) edge ........................ 23 Figure 4-4 – Comparison 3D / 1D heat flux along the fuel height on the outer (right) edge ...................... 23 Figure 4-5 – Variation of the local peak 1D heat flux with node width in the inner edge midplane .......... 24 Figure 4-6 – Variation of the local peak 1D heat flux with node width in the outer edge midplane .......... 25 Figure 4-7 – Variation of the local peak 1D heat flux with node size (width and length) in the inner edge bottom ......................................................................................................................................................... 25

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Influence of Multi-Dimension Heat Conduction on Heat Flux Calculation for HFIR LEU Analysis v

Figure 4-8 – Variation of the local peak 1D heat flux with node size (width and length) in the outer edge bottom ......................................................................................................................................................... 26 Figure 4-9 – Comparison 3D / 1D / 1D coarse heat flux along the fuel width at three axial location: top of the fuel (top), fuel midplane (middle) and bottom of the fuel (bottom)...................................................... 28 Figure 4-10 – Comparison 3D / 1D / 1D “coarse” heat flux along the fuel height on the inner (left) edge 29 Figure 4-11 – Comparison 3D / 1D / 1D “coarse” heat flux along the fuel height on the outer (right) edge .................................................................................................................................................................... 29

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Influence of Multi-Dimension Heat Conduction on Heat Flux Calculation for HFIR LEU Analysis vi

List of Tables Table 2-1 – Inner Element Fuel Node Width ................................................................................................ 2 Table 2-2 - Inner Element Fuel Node Height ................................................................................................ 3 Table 3-1 – Material constants for the solid parts ......................................................................................... 9 Table 3-2 – Properties of coolant .................................................................................................................. 9 Table 4-1 – Coarse node width proposed for 1D calculation ...................................................................... 26 Table 4-2 – Coarse node height proposed for 1D calculation ..................................................................... 27

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Influence of Multi-Dimension Heat Conduction on Heat Flux Calculation for HFIR LEU Analysis 1

1 Introduction The Oak Ridge National Laboratory (ORNL) High Flux Isotope Reactor (HFIR) is a multi-purpose nuclear research reactor that has been in operation since 1965 [ORNL, 2017]. The core is made of two annular fuel elements containing fuel plates curved as circle involutes. HFIR currently uses a Highly Enriched Uranium (HEU) fuel (235U/U ≥ wt. 20%). Like many other facilities worldwide and in agreement with the international community goals to minimize or even eliminate if possible the use of HEU fuel in civilian activities, HFIR is actively engaged in an effort to convert the reactor to Low Enriched Uranium (LEU) fuel (235U/U < wt. 20%) [RERTR, 2017]. Steady-state thermal-hydraulic safety analyses for HFIR are currently performed with the code HSSHTC [McLain, 1967]. This code is not able to model heat conduction in multiple dimensions. Instead, all the heat is assumed to be deposited directly in the coolant channel. In reality, heat is transported in three dimensions (3D) and near the fuel edges, a substantial amount of heat is transported to the unfueled region of the plate. A direct consequence of it is that the heat flux at the cladding/coolant interface calculated in 3D near the edges can be substantially lower than when calculated with a one-dimension (1D) code (which by definition ignores this edge effect). These heat flux differences can lead to very different safety margins. Modeling heat transport in one or multiple dimensions can therefore have implications for the design. In addition, the profile of the fuel heat generation rate across the fuel width is typically not flat and tends to exhibit peaks near the edges. Using a 1D code, the magnitude of these power density peaks and the corresponding heat flux will depend on the size of the region (called node in the remaining of this report) in which they are calculated. Node size will therefore also have an impact on safety margins and design. Consequently, if one-dimensional codes have to be used for fuel element design and safety analysis, it is important to understand the effect of node size and the unfueled region on heat flux. HFIR being actively engaged in LEU design activities, the fuel discretization used to calculate the heat flux that is used as input for safety margin calculations must be chosen carefully. The goal of this work is to: - Study the differences between 1D and 3D (best estimate) heat transfer in HFIR plate geometry and; - Propose a reasonable fuel discretization for 1D heat transfer analysis to avoid unnecessary over-

conservatism To perform the work, a LEU model has been created using a HFIR neutronic model in order to generate a power distribution (see section 2). This power distribution is “converted” into a heat flux profile assuming only mono-directional heat transfer (later simply referred as 1D heat flux). The same power distribution is then used in a CFD model (see section 3). A new heat flux profile is obtained (later simply referred as 3D heat flux) and compared to the previous one. From there, a new “coarse” fuel discretization is proposed to generate a 1D heat flux profile of magnitude reasonably close the 3D one (section 4).

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2 Power Distribution Generation In order to study the influence of the unfueled region of the plate on heat flux, the HFIR neutronic MCNP model described in [Chandler, 2016] has been used to calculate a highly detailed power distribution. In both fuel elements, the fuel has been replaced by LEU UMo monolithic fuel. Unlike the current HEU plates, the LEU fuel has not been contoured so that power peaking on the plate edges was maximized. In addition, the inner element plates did not contain a neutronic absorber, again to artificially increase power peaking at the edges. Such a configuration is not realistic but considered useful for this academic exercise. The heat flux analysis presented here focused only on the inner element. More details on the fuel and plate geometry are provided in section 3.1. The inner element fuel has been discretized in a set of 40 lateral x 188 axial = 7,520 regions. Node width for all nodes is provided in Table 2-1 and node height is provided in Table 2-2. The power density profile obtained with this design is presented in Figure 2-1 and Figure 2-2. As expected, along the fuel width, the power density exhibits a “U-shape” with significant power peaking on both the inner and outer edges. In the axial direction, the power density has a cosine shape but also exhibits significant power peaking on the very top and very bottom.

Table 2-1 – Inner Element Fuel Node Width

lateral region

# node width

(cm) lateral

region # node width

(cm) 1 0.052 21 0.433 2 0.052 22 0.633 3 0.053 23 0.669 4 0.053 24 0.705 5 0.053 25 0.741 6 0.054 26 0.778 7 0.054 27 0.814 8 0.054 28 0.850 9 0.055 29 0.087

10 0.055 30 0.087 11 0.055 31 0.088 12 0.056 32 0.088 13 0.056 33 0.088 14 0.057 34 0.089 15 0.057 35 0.089 16 0.057 36 0.090 17 0.058 37 0.090 18 0.058 38 0.090 19 0.058 39 0.091 20 0.059 40 0.095

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Table 2-2 - Inner Element Fuel Node Height

axial region

#

node height (cm)

axial region

#

node height (cm)

axial region

#

node height (cm)

axial region

#

node height (cm)

axial region

#

node height (cm)

1 0.05 41 0.05 81 0.5 121 0.5 161 0.05 2 0.05 42 0.05 82 0.5 122 0.5 162 0.05 3 0.05 43 0.05 83 0.5 123 0.5 163 0.05 4 0.05 44 0.05 84 0.5 124 0.5 164 0.05 5 0.05 45 0.05 85 0.5 125 0.5 165 0.05 6 0.05 46 0.05 86 0.5 126 0.5 166 0.05 7 0.05 47 0.05 87 0.5 127 0.5 167 0.05 8 0.05 48 0.05 88 0.5 128 0.5 168 0.05 9 0.05 49 0.5 89 0.5 129 0.5 169 0.05

10 0.05 50 0.5 90 0.5 130 0.5 170 0.05 11 0.05 51 0.5 91 0.5 131 0.5 171 0.05 12 0.05 52 0.5 92 0.5 132 0.5 172 0.05 13 0.05 53 0.5 93 0.5 133 0.5 173 0.05 14 0.05 54 0.5 94 0.5 134 0.5 174 0.05 15 0.05 55 0.5 95 0.5 135 0.5 175 0.05 16 0.05 56 0.5 96 0.5 136 0.5 176 0.05 17 0.05 57 0.5 97 0.5 137 0.5 177 0.05 18 0.05 58 0.5 98 0.5 138 0.5 178 0.05 19 0.05 59 0.5 99 0.5 139 0.5 179 0.05 20 0.05 60 0.5 100 0.5 140 0.5 180 0.05 21 0.05 61 0.5 101 0.5 141 0.05 181 0.05 22 0.05 62 0.5 102 0.5 142 0.05 182 0.05 23 0.05 63 0.5 103 0.5 143 0.05 183 0.05 24 0.05 64 0.5 104 0.5 144 0.05 184 0.05 25 0.05 65 0.5 105 0.5 145 0.05 185 0.05 26 0.05 66 0.5 106 0.5 146 0.05 186 0.05 27 0.05 67 0.5 107 0.5 147 0.05 187 0.05 28 0.05 68 0.5 108 0.5 148 0.05 188 0.05 29 0.05 69 0.5 109 0.5 149 0.05 30 0.05 70 0.5 110 0.5 150 0.05 31 0.05 71 0.5 111 0.5 151 0.05 32 0.05 72 0.5 112 0.5 152 0.05 33 0.05 73 0.5 113 0.5 153 0.05 34 0.05 74 0.5 114 0.5 154 0.05 35 0.05 75 0.5 115 0.5 155 0.05 36 0.05 76 0.5 116 0.5 156 0.05 37 0.05 77 0.5 117 0.5 157 0.05 38 0.05 78 0.5 118 0.5 158 0.05 39 0.05 79 0.5 119 0.5 159 0.05 40 0.05 80 0.5 120 0.5 160 0.05

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Figure 2-1 – Inner fuel element power density profile (left) and as a function of the distance along the arc length (right) taken at three axial position (top, midplane, bottom)

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Figure 2-2 – Inner fuel element power density profile (middle) and as a function of the fuel length taken on the inner edge (left) and outer edge (right)

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3 Three-Dimensional Computational Fluid Dynamics Model

In this section, the fuel heat generation rate distribution generated with the neutronic model and described in the previous section is implemented into a Computational Fluid Dynamics (CFD) model. This CFD model is able to model heat transport in three dimensions (3D). Unless stated otherwise, the CFD code used through this analysis is the code Star-CCM+ [Star-CCM, 2017].

3.1 Geometry and Mesh The HFIR core consists of two annular fuel elements containing plates curved as circle-involute. Curvilinear shapes usually contribute to the complexity of the meshing process. The involute shape assures constant cross section of the fuel as well as the coolant channels between all the fuel plates. For this initial analysis, abovementioned geometric features allow for assuming a flattened-out shape of a repeatable fuel plate and adjacent coolant channels. This leads to the final assumption that only one fuel plate with one coolant channel can be represented in the model. The repeatability of the geometry can be implemented through the symmetry planes either in the middle of the coolant channel or in the middle of the fuel element. For turbulent flows, the numerical solution is more stable if the symmetry plane is enforced in the middle of the solid plate instead of the coolant. Figure 3-1 shows cross sections through the geometries of the two possible uses of symmetry. Figure 3-2 shows a cross section through the Star-CCM+ model, which is true to the scale. The Star-CCM+ model has been extended beyond the geometry of the fuel element adding an inlet and an outlet plenum. The inlet plenum assures proper flow development before the fuel plate is reached. The outlet plenum prevents recirculation to interfere with the pressure outlet boundary. In these plenums, only the coolant is present. The external boundaries of the solid parts of the model were modeled as walls with constant temperature boundary conditions to simulate pool like conditions. The side view of the model is show in Figure 3-4. Majority of the CFD work described here was performed in Star-CCM+. However, verification runs were performed additionally in COMSOL [COMSOL, 2017]. The mesh built in Star-CCM+ CAD module with Directed Mesher cannot be exported in any format recognizable by COMSOL. Thus, the geometry of the model was created separately within CAD modules of Star-CCM+ and COMSOL. Directed mesher in Star-CCM+ (Swept mesh in COMSOL) takes as input a cross section of a model that is then divided into multiple rectangles. The edges of each rectangle allows for specification of number of seed points for the mesh. One way as well as two-way zoning of the mesh is possible and was extensively used in the model to keep small cell sizes near the material interfaces and larger cells away from them. Altogether 31 mesh regions were defined in the Star-CCM+ model that were tied by conformal in-place interfaces for internal connections within one material or contact interfaces between different materials. Overall, the base model consisted of nearly 2.9 M hexahedral elements. The number has shown to be optimal and further increase in mesh density did not provide any difference in the results. The elements of the mesh sensitivity study will be presented later in this report. Figure 3-3 shows a close up view to the edge of the fuel meat where the mesh is the densest. Directed mesher (as opposed to the automatic mesher) allows for considerable savings of the computational resources by placing densest mesh only in the areas where it is needed i.e. in the transitions from the fuel meat to the cladding. 10 elements per thickness of the fuel meat were used (5 on each side). 22 elements were used through the thickness of the cladding (11 on each side). 25 elements were used through the thickness of the coolant channel. In the longitudinal (z)

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direction the model consisted of 260 elements in the plate section and additional 80 in the extensions (360 total).

Figure 3-1 – Dimensions of the cross section of the model (top) symmetry plane in coolant (bottom) symmetry plane in the fuel. (Drawing not to scale)

Figure 3-2 – Cross-section through the Star-CCM+ model

Figure 3-3 – Mesh distribution detail

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Figure 3-4 – Side view of the model

Extended inlet section

Extended outlet section

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3.2 Material Properties Two solid materials (for fuel and aluminum clad) and one fluid material were defined in the model. The material constants in the base model for the solid parts are shown in Table 3-1.

Table 3-1 – Material constants for the solid parts

Material constant Fuel Aluminum Clad Density (kg/m^3) 17,200.0 2,700.0

Specific heat (J/kg-K) 903.0 903.0 Thermal conductivity

(W/m-K) 15.0 150.0

For the fluid (water), K-Epsilon turbulence model was selected in the base model. The density, dynamic viscosity, specific heat as well as thermal conductivity of water were defined as polynomial functions of temperature of different orders. These polynomials have been generated from the water properties found in [NIST, 2017]. Below, in Table 3-2 these quantities are defined in a following manner, the second column denotes the number of components, the third one denotes the temperature range for which the dependency is defined, the fourth one contains all the coefficients for the polynomial, and the fifth one contains the exponents of temperature associated with the coefficients. The entire function may be built of several polynomials defined for different ranges of temperature according to Equation 1. Figure 3-5 to Figure 3-8 show these properties of coolant in a form of plots.

(1)

Table 3-2 – Properties of coolant

# of polynomial

coefficients Interval ranges Coefficients Exponents

dynamic viscosity [Pa-s] 7 273, 450 6.118072E-16, -1.399347E-12, 1.330595E-9, -6.735074E-7,

1.914979E-4, -0.02902184, 1.833873 6, 5, 4, 3, 2, 1, 0

Density [kg/m^3] 5 273, 450 -4.72297E-8, 7.403817E-5, -0.04562428, 12.21753, -178.6494 4, 3, 2, 1, 0

specific heat [J/kg-K] 6 273, 450 -5.537245E-9, 1.062099E-5, -0.008063263, 3.040666, -570.526,

46809.37 5, 4, 3, 2, 1, 0

thermal conductivity

[W/m-K] 4 273, 450 1.875566E-8, -2.772204E-5, 0.01319425, -1.358133 3, 2, 1, 0

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Figure 3-5 – Dynamic viscosity as a function of temperature

Figure 3-6 – Density of water as a function of temperature

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0.0018

0.002

273 293 313 333 353 373 393 413 433

Dyna

mic

visco

sity (

K Pa

s)

Temperature (K)

880

900

920

940

960

980

1000

1020

273 293 313 333 353 373 393 413 433

Dens

ity (k

g/m

^3)

Temperature (K)

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Figure 3-7 – Thermal conductivity of water as a function of temperature

Figure 3-8 – Specific heat of water as a function of temperature Inlet coolant mass flux at the rate of 6700 kg/m2/s was used as an input to the model. The water entering the domain was set at 322 K. The temperature on the other external boundaries of the model were assumed to be 343 K which is the temperature of the surrounding pool. The power density of fuel was defined in the model as the Volumetric Heat Source.

0.56

0.58

0.6

0.62

0.64

0.66

0.68

0.7

273 293 313 333 353 373 393 413 433

Ther

mal

cond

uctiv

ity (W

/m-K

)

Temperature (K)

4150

4200

4250

4300

4350

4400

4450

273 293 313 333 353 373 393 413 433

Spec

ific h

eat (

J/kg-

K)

Temperature (K)

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3.3 Base Case Results Steady-state type of analysis was performed for all the cases. The convergence progression in the base analysis is shown in Figure 3-9. Significant drop in all the residuals is observed. Although approximate results are obtained within couple of thousands of iterations, at least 20,000 iterations are needed for full convergence of the results. As a measure of convergence change in the peak value of boundary heat flux on the interface between the fuel and the coolant was assumed.

Figure 3-9 – Convergence of residuals in the base simulation The standard K-epsilon turbulence model was used in the base case and most parametric runs. The runs with K-omega model did not show noticeable differences in the results. In addition, the convergence rate was similar for both models. The mesh in the model went through several updates to satisfy the requirement of wall Y+ parameter being between 30 and 100, which assures correctness of the approximations of near-wall flows. Figure 3-10 shows the map of wall Y+ in the base model on the interface between the clad and the coolant. For majority of the area, Y+ is between 50 and 100.

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Figure 3-10 – Map of wall Y+ on the interface between the clad and coolant Figure 3-11 shows the velocity profile of the coolant in the middle of the coolant channel (i.e. on the symmetry plane). This cross section shows the main part of the model as well as the extensions (plenum) added to the model at the inlet and outlet. These extensions are entirely filled with the coolant. This addition was necessary to obtain stable results. In reality, the entrance to the water channels is curved and smooth. For simplicity, these edges were modeled straight (no curvature). These edges add a significant local disturbance to the flow but keep the meshing process simple and allow using only the directed mesher with conformal interfaces and regular hexahedral mesh. These simplifications are not expected to impact the heat flux calculations.

Figure 3-12 shows the map of the temperatures in the middle of the fuel plate. The temperature at the entrance to the domain is at 322 K. In the simulation, the temperature at the outlet rose to about 354 K.

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Figure 3-11 – Velocity of the coolant

Figure 3-12 – Temperature on the interface between the clad and the coolant

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Out of all results, of highest interest was the heat flux on the interface between the plate and the coolant. Figure 3-13 presents the boundary heat flux on that surface obtained in the base simulation. The maximum heat flux was obtained near the midplane in vicinity of the side plate where the power density was the highest.

Figure 3-13 – Boundary heat flux on the boundary between the clad and coolant Figure 3-14 shows the distribution of the boundary heat flux along the plate width and taken at the mid height of the fuel element. The maximum of 584.6 W/cm2 occurred near the inner (left) edge of the fuel where the power input was the highest. Unlike in 1D analysis, the heat flux has a local maximum not at the interface but approximately 2.5mm away from the interface but then drops significantly when approaching the fuel edge. The maximum is not significantly dependent on the mesh density.

Figure 3-14 – Boundary heat flux on the clad surface in the middle of the model

1.00E+06

1.50E+06

2.00E+06

2.50E+06

3.00E+06

3.50E+06

4.00E+06

4.50E+06

5.00E+06

5.50E+06

6.00E+06

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Boun

dary

Hea

t Flu

x (W

/m^2

)

Position X (m)

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Figure 3-15 shows the boundary heat flux in the axial direction. Zero on the horizontal axis denotes the bottom of the fuel. The maximum is recorded near the central point with two local peak values near the inlet and the outlet to the domain.

Figure 3-15 – Boundary heat flux on the clad surface in the axial cross-section

1.00E+06

2.00E+06

3.00E+06

4.00E+06

5.00E+06

6.00E+06

7.00E+06

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55

Boun

dary

hea

t flu

x (W

/m^2

)

Position Z (m)

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3.4 Sensitivity Study Sensitivity of results was tested in respect to several model aspects including mesh density, type of boundary conditions and turbulence model.

3.4.1 Mesh Sensitivity A coarser and a denser mesh models were built starting from the base model. The number of cells was reduced nearly by a factor of two to 1.5M cells in the coarse model. In the dense mesh model, the number of cells was increased by a factor over two to 7M cells. These changes in the model have proven to have minimal impact on the peak value of the boundary heat flux as illustrated in Figure 3-16. The only difference was noticed in the smoothness of the curve near the peak. It means that the computational mesh used in the base model is dense enough and further increase in the density does not bring significant improvement of the results. There is lack of smoothness of the curve near the base of the peaks. This is because Star-CCM+ does not use an interpolation for coarse data provided as heat source (power) in that region. For the dense mesh model that data was smoothly interpolated outside of the Star-CCM+.

Figure 3-16 – Boundary heat flux in the base, coarse, and dense mesh models

3.4.2 Thermal Boundary Conditions Sensitivity As a next sensitivity analysis parameter, the temperature boundary conditions on the external walls have been replaced with adiabatic conditions. Figure 3-17 presents the boundary heat flux in the middle of the model for this case and the base model with constant temperature boundary conditions. The curves nearly overlap indicating that minimal heat exchange occurs trough these boundaries and most of it is extracted out of the domain by the flowing coolant.

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Figure 3-17 – Boundary heat flux in the base model and the model with adiabatic boundary conditions on the external walls

3.4.3 Turbulence Model Sensitivity As indicated earlier, another model was tested with the use of k-omega turbulence model for the coolant instead of the standard k-epsilon model. Figure 3-18 presents the heat flux on the interface of clad and the coolant. The peak value for the k-omega model grew slightly to 586 W/cm2. Overall, very similar results have been obtained. Thus, the model appears insensitive to the choice of the turbulence model.

Figure 3-18 – Boundary heat flux in the base mode with k-epsilon turbulence model as well in the model with k-omega turbulence model

1.00E+061.50E+062.00E+062.50E+063.00E+063.50E+064.00E+064.50E+065.00E+065.50E+066.00E+06

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3.4.4 COMSOL Verification Simulations A limited number of simulations using the COMSOL software was performed for the same model. A model with 5M of degrees of freedom was built. Identical initial and boundary conditions were modeled in COMSOL as in Star-CCM+. The results from the COMSOL model were in almost perfect agreement with the results obtained with the dense mesh model in Star-CCM+ (see Figure 3-19).

Figure 3-19 – Comparison of Results between Star-CCM+ and COMSOL

3.4.5 Summary of Sensitivity Analysis Based on the results described in sections 3.4.1 to 3.4.4, it can be concluded that the profile and magnitude of the 3D heat flux calculated with the CFD base model are not sensitive to:

- The density of the mesh (above reasonable number of elements) - The turbulence model - The thermal boundary conditions - The CFD code used

All the above tends to demonstrate that the heat flux calculated in the reference model is reliable.

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4 Suitable Fuel Discretization for 1D Analysis In this section, the CFD-based 3D heat flux is compared with the 1D heat flux (recall, it was obtained assuming that all the heat is transported in the coolant channel). Then, node size used to calculate how the 1D heat flux is changed (coarsened) to better understand how node size can be used to obtain 1D heat flux values closer to what is predicted with 3D heat conduction models.

4.1 3D versus 1D Heat Flux Figure 4-1 illustrates the differences between 1D and 3D heat flux distribution at the cladding/coolant interface. It can be seen that the two profiles are very similar except for the very edges where the differences are substantial. The ability of modeling heat conduction in more than one dimension is therefore relevant only in regions near the interface of the fuel/unfueled region of the plate. The effect is strong because the cladding that surrounds the fuel is made of an aluminum alloy of high thermal conductivity.

Figure 4-1 – Comparison 1D heat flux (left) / 3D heat flux (middle) and ratio 1D/3D heat flux with ratio color scale legend (right)

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Looking at more details, Figure 4-2 presents the 3D and 1D heat flux profile along the fuel width calculated at three different axial locations: top, middle and bottom of the fuel. On these three plots, the 1D heat flux is represented by the solid blue line while the 3D heat flux is represented by a solid red line. At the midplane, 1D and 3D heat flues are compared at the exact same axial location. At the top and bottom, the provided 3D heat flux is taken approximately 2.5 mm from the very top or bottom edges. This is because at these locations, the axial conduction plays a significant role and the local peaks of the lateral heat fluxes are actually found 2.5 mm away from the edges. It can be seen on the three plots of Figure 4-2 that the 1D heat flux always presents a “U-shape”, having two local maximums at the very edges of the fuel region and exhibiting a minimum in the middle of the plate. The 3D heat flux always present an “M-shape” sharing many characteristic with the U-shape profile except for the fact that the local peak heat fluxes occur approximately 2.5 mm away from the edges and is decreasing sharply in the vicinity of the edges. These three plots show that a substantial amount of heat is actually transported to the lateral unfueled region of the plates (made of aluminum alloy), region of high thermal conductivity. At the midplane, it can be seen that the magnitude of the 1D and 3D heat flux is very similar everywhere except near the fuel edges for the reason mentioned above. At the top and bottom however, the 1D heat flux is substantially larger than the 3D heat flux everywhere. This is explained by the substantial amount of heat that is also transported in the axial direction where the top and bottom unfueled region of the plate are also made of aluminum alloy of high thermal conductivity. Similarly, Figure 4-3 and Figure 4-4 show the 3D and 1D axial heat flux profile along the fuel length calculated at the inner (left) and outer (right) edge. As previously, the presented 3D heat flux is taken 2.5 mm away from the edges, where it is maximum. As observed previously, even if the 1D and 3D heat flux profiles share some similarities, the magnitude is substantially different. Itis explained this time by the substantial heat conduction occurring in the lateral direction not present in 1D calculations.

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Figure 4-2 – Comparison 3D / 1D heat flux along the fuel width at three axial location: top of the fuel (top), fuel midplane (middle) and bottom of the fuel (bottom)

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Figure 4-3 – Comparison 3D / 1D heat flux along the fuel height on the inner (left) edge

Figure 4-4 – Comparison 3D / 1D heat flux along the fuel height on the outer (right) edge

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4.2 Coarse Fuel Discretization Suitable for 1D Analysis As explained previously, the 1D and 3D heat flux are very similar except on the edges where substantial heat flux differences exist between 1D and 3D heat transport calculations. In this section, the influence of node size on the magnitude of the 1D heat flux is analyzed. Where strong gradients exist, the magnitude of the heat flux will decrease as the size of the node in which it is calculated increases. The size of the node can therefore be used to decrease the magnitude of the 1D heat flux and obtain values closer to the ones obtained with 3D heat conduction code. The size of the node has been varied in four locations (midplane inner and outer edge, fuel bottom inner and outer edge) and the 1D heat flux recalculated and compared to the 3D heat flux obtained previously. The results are shown in Figure 4-5 to Figure 4-8. These figures show that the magnitude of the 1D heat flux decreases significantly with node size. The rate of the decrease is, however, not identical in all locations and seems to be correlated with the power peaking gradients. In this small set of data, it is found that node size as large as 0.72 to 1.25 cm would be required to match the magnitude of the local 3D heat fluxes. Smaller node size would lead to 1D heat flux values exceeding the 3D heat flux.

Figure 4-5 – Variation of the local peak 1D heat flux with node width in the inner edge midplane

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Figure 4-6 – Variation of the local peak 1D heat flux with node width in the outer edge midplane

Figure 4-7 – Variation of the local peak 1D heat flux with node size (width and length) in the inner edge bottom

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Figure 4-8 – Variation of the local peak 1D heat flux with node size (width and length) in the outer edge bottom After several iterations, a “coarse” fuel discretization has been selected so that the 1D heat flux is relatively close to the 3D heat flux but is still bounding. This new fuel discretization uses 12 lateral and 25 axial regions for a total of 300 fuel regions. The proposed discretization is described in Table 4-1 and Table 4-2. Figure 4-9, Figure 4-10, and Figure 4-11 are similar to Figure 4-2, Figure 4-3, and Figure 4-4, respectively, but in addition, they show the 1D heat flux obtained with the new proposed fuel discretization. It can be seen that this new “coarse” 1D heat flux is now quite similar to the 3D heat flux everywhere, except at the very edges where the 1D coarse heat flux remains above the 3D heat flux but is much closer to what was obtained with the previous high-resolution grid.

Table 4-1 – Coarse node width proposed for 1D calculation

lateral region # node width (cm) 1 0.5 2 0.25 3 0.35 4 0.35 5 1.05 6 1.05 7 1.05 8 1.05 9 0.7

10 0.7 11 0.25 12 0.5

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Table 4-2 – Coarse node height proposed for 1D calculation

axial region # node height (cm) 1 0.6 2 1.0 3 1.3 4 2.5 5 2.5 6 2.5 7 2.5 8 2.5 9 2.5 10 2.5 11 2.5 12 2.0 13 1.0 14 2.0 15 2.5 16 2.5 17 2.5 18 2.5 19 2.5 20 2.5 21 2.5 22 2.5 23 1.3 24 1.0 25 0.6

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Figure 4-9 – Comparison 3D / 1D / 1D coarse heat flux along the fuel width at three axial location: top of the fuel (top), fuel midplane (middle) and bottom of the fuel (bottom)

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Figure 4-10 – Comparison 3D / 1D / 1D “coarse” heat flux along the fuel height on the inner (left) edge

Figure 4-11 – Comparison 3D / 1D / 1D “coarse” heat flux along the fuel height on the outer (right) edge

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5 Conclusions The present report qualitatively describes the differences between heat fluxes evaluated with and without multi-dimensional heat conduction. It appears clear that the significant differences between 1D and 3D heat flux occur only on the fuel edges, at the interface between the fuel and the surrounding aluminum cladding of high thermal conductivity. In addition, the model used to evaluate the 3D heat flux appears quite reliable since the 3D heat flux profiles and magnitude are found to be quite insensitive to mesh density, turbulence model, boundary conditions or even the code used. The node size used to calculate the local 1D heat flux has been varied to understand how node size variation can be used to lower the heat flux magnitude and produce results closer to the values obtained with code able to model heat conduction in three dimension. From there, a “coarse” fuel discretization has been proposed that can generate 1D heat fluxes relatively close to the 3D ones while remaining reasonably conservative. Despite the fact that the power distribution used to perform these analyses is not realistic, the fact that the proposed grid is able to accommodate very different types of power peaking is an indication that it can likely accommodate most design and power distribution profiles. The proposed grid is recommended for HFIR exploratory design analysis performed at Argonne National Laboratory.

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References [Chandler, 2016] D. Chandler et al. “Modeling and Depletion Simulations for a High Flux Isotope

Reactor Cycle with a Representative Experiment Loading”, ORNL/TM-2016/23, September 2016

[COMSOL, 2017] COMSOL website, https://www.comsol.com/, last accessed August 2017 [McLain, 1967] H. A. McLain “HFIR Fuel Element Steady State Heat Transfer Analysis Revised

Version”, ORNL-TM-1903, December 1967 [NIST, 2017] National Institute of Standards and technology (NIST), thermophysical properties

of fluid systems website: http://webbook.nist.gov/chemistry/fluid/ [ORNL, 2017] ORNL website, http://neutrons.ornl.gov/facilities/HFIR/, consulted August 2017 [RERTR, 2017] Reduced Enrichment for Research and Test Reactor website,

http://www.rertr.anl.gov/, last accessed August 2017 [Star-CCM, 2017] StarCCM+ website, https://mdx.plm.automation.siemens.com/star-ccm-plus, last

accessed August 2017

https://www.comsol.com/

http://webbook.nist.gov/chemistry/fluid/

http://neutrons.ornl.gov/facilities/HFIR/

http://www.rertr.anl.gov/

https://mdx.plm.automation.siemens.com/star-ccm-plus

Nuclear Engineering Division Argonne National Laboratory 9700 South Cass Avenue, Bldg. 208 Argonne, IL 60439 www.anl.gov

Argonne National Laboratory is a U.S. Department of Energy laboratory managed by UChicago Argonne, LLC

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Influence of Multi-Dimension Heat Conduction on Heat Flux