Effect of pore size on effective conductivity of UO 2 : A - PDF document

Transactions of the Korean Nuclear Society Virtual Spring Meeting July 9-10, 2020 Effect of pore size on effective conductivity of UO 2 : A computational approach Bohyun Yoon, Kunok Chang  Department of Nuclear Engineering, Kyung Hee University, Yong-in city, Korea * Corresponding author: kunok.chang@khu.ac.kr 1. Introduction Miller et al. proposed effective conductivity model of polycrystalline UO 2 with pores as follows: [9] 𝑙 0 The thermal conducting properties of UO 2 pallet (1 − 𝑄) 𝛾 Ψ ⋯ (3) 𝑙 𝑓𝑔𝑔 = 1 + 𝑙 0 𝐻 𝑙 𝑒 ⁄ degrade over lifetime of a nuclear power plant and it can where G k is a Kapitza conductance, d is an average be a critical limiting factor of the safety and efficiency of grain diameter, β is a fitting parameter and Ψ =1-P is a the reactor [1-3]. A commercial UO 2 pallet contains correlation factor that relates 2-D to 3-D heat transport in microstructural inhomogeneities, such as grain boundaries, voids and He/Xeon bubbles [4-15]. Since porous media (for P<10.0%). those microstructural defects seriously affect thermal In their model, they incorporated phonon scattering at conductivity of the nuclear fuel, understanding the grain boundary, therefore, they assumed that the effective heat conductivity at the grain boundary is lower correlation between effective thermal conductivity and temporal distribution of the imperfections is a quite than the value in the matrix [9]. In Eq. 4, so called “Schulz equation” [18], β is important task. For decades, the prediction of effective conductivity of porous nuclear fuel largely depend on determined by geometry of pores. Maxwell-Eucken (hereafter ME) model [8] which is Nikolopoulos and Ondracek proposed the model for adopted in FRAPCON [16]. Including ME model, the effective conductivity of porous material (k eff ) at given porosity (P) with conductivity of nonporous material (k 0 ) effective thermal conductivity models of nuclear fuel do not reflect the effect of pore size [4-15]. There have been [14]. 𝑙 𝑓𝑔𝑔 = 𝑙 0 (1 − 𝑄) 𝛾 ⋯ (4) an widely accepted assumption that the pore size is much larger than the average phonon wavelength [4], therefore, Nikolopoulos and Ondracek predicted β =2.5 for spherical pores and β =1.667 for cylindrical pores which the effect of pore size on the effective thermal conductivity has not been of much concern for decades. statistically directed to the field direction in isotropic However, whether the assumption in the previous materials [14]. sentence is valid has not been thoroughly examined in an experimental study or continuum modeling. We 2.2 Steady-state thermal conduction analysis investigated a role of pore size by means of the continuum-level simulations. We performed the steady- We solved steady-state heat conduction equation as state heat conduction analysis in 2-D and 3-D systems belows: and the effect of the pore size on the effective ∇k(r)∇T(r) = 0 ⋯ (5) conductivity was evaluated systematically. In 2-D, we simulated 10.24 µm (L x ) × 10.24 µm (L y ) system. The simulation cell size is 1024×1024 grid 2. Methods and Results ∆x=∆y=10 nm. For boundary points, therefore conditions, Dirichlet boundary condition of T=800K is 2.1 Microstructure and temperature dependence on local applied on the line of x=0 and Neumann boundary conductivity condition of j = k(r) 𝜖𝑈(𝑠) = 50𝑁𝑋/𝑛 2 , constant heat 𝜖𝑦 flux condition is applied across the line x=10.24 µm. We We introduced the non-conserved structural order 𝜖𝑈(𝑠) parameter η i (r) which the value is 0 in the He bubble = 0 when y=0, applied the adiabatic condition, 𝜖𝑦 region and 1 at the matrix [10]. Within an interfacial 10.24 µm. The effective thermal conductivity is region, its parameter value diffuses smoothly. The evaluated by the relation [12]: thermal conductivity of He gas is fixed at 0.152 W/(K·m), 𝑙 𝑓𝑔𝑔 = 𝑘 × 𝑀 𝑦 ∆𝑈 ⋯ (6) and the conductivity of UO 2 crystal is taken from the model suggested by Harding and Martin which has the where j is the heat flux. following temperature dependence in the unit of W/(K·m) To calculate ∆ T, we evaluate the average temperature [17]. of the line, x=0 and x=L x and find the difference of them. 4.715×10 9 1 −16361 𝑙 𝑑𝑠𝑧𝑡𝑢𝑏𝑚 = 0.0375+2.165×10 −4 𝑈 + 𝑓𝑦𝑞 ( ) 𝑈 2 𝑈 2.3 Crank-Nicolson method ··· (1) The effect of structural order parameter η i (r) on the We solve steady-state heat conduction equation using local heat conductivity is a finite-difference approximation based on the Crank- 6 (𝑠) × 𝑙 𝑑𝑠𝑧𝑡𝑢𝑏𝑚 k(r) = ∑ 𝜃 𝑗 ⋯ (2) Nicolson scheme (CN) and alternating-direction-implicit (ADI) method. The CN scheme is an implicit method 𝑗

Transactions of the Korean Nuclear Society Virtual Spring Meeting July 9-10, 2020 which discretizes time and space. And we apply Douglas- Gunn’s ADI splitting method to decompose the discretization matrix of the CN scheme. This ADI method splits the matrix into three simple matrices introducing intermediate time steps between n and n+1. With the notations δ 2 and δ defined as: 2 𝑣(𝑠) = 𝑣(𝑠) 𝑗−1,𝑘,𝑙 − 2𝑣(𝑠) 𝑗,𝑘,𝑙 + 𝑣(𝑠) 𝑗+1,𝑘,𝑙 ⋯ (7) 𝜀 𝑦 𝜀 𝑦 𝑣(𝑠) = 𝑣(𝑠) 𝑗+1,𝑘,𝑙 − 𝑣(𝑠) 𝑗−1,𝑘,𝑙 ⋯ (8) And letting C as: 𝐷 𝑦 = 𝐸(𝑣)∆𝑢 2 + (𝜀 𝑦 𝐸(𝑣)∆𝑢 (∆𝑦) 2 𝜀 𝑦 4(∆𝑦) 2 ) 𝜀 𝑦 ⋯ (9) The CN method applied steady-state heat conduction equation can be arranged to: [1 − 𝐷 𝑦 2 − 𝐷 𝑧 2 − 𝐷 𝑨 2 ] 𝑣(𝑠) 𝑜+1 = [1 + 𝐷 𝑦 2 + 𝐷 𝑧 2 + 𝐷 𝑨 2 ] 𝑣(𝑠) 𝑜 ⋯ (10) Fig. 1. Effective conductivity (k eff with function of porosity at different pore sizes in 2D. This equation can be factorized and using a Douglas- Gunn method: (1 − 𝐷 𝑦 3 = (1 + 𝐷 𝑦 2 ) 𝑣(𝑠) 𝑜+1 2 + 𝐷 𝑧 + 𝐷 𝑨 ) 𝑣(𝑠) 𝑜 (1 − 𝐷 𝑧 2 ) 𝑣(𝑠) 𝑜+2 3 = (1 + 𝐷 𝑦 2 + 𝐷 𝑧 2 + 𝐷 𝑨 ) 𝑣(𝑠) 𝑜 + 𝐷 𝑦 2 𝑣(𝑠) 𝑜+1 3 (1 − 𝐷 𝑨 2 ) 𝑣(𝑠) 𝑜+1 = (1 + 𝐷 𝑦 2 + 𝐷 𝑧 2 + 𝐷 𝑨 2 ) 𝑣(𝑠) 𝑜 + 𝐷 𝑦 3 + 𝐷 𝑧 2 𝑣(𝑠) 𝑜+1 2 𝑣(𝑠) 𝑜+2 3 ⋯ (11) 2.4 Porous material with different pore sizes Fig. 2. Effective conductivity (k eff with function of porosity at According to our best knowledge, the effect of different pore sizes in 3D. porosity and temperature on the effective conductivity of UO 2 has been studied in depth, while the effect of pore 3. Conclusions size has not been studied. [5-15] To examine the pore size effect on effective In this work, we have investigated a role of pore size conductivity in 2D and 3D, we measured effective on effective conductivity of UO 2 using the computational conductivity with different pore sizes. In 2D, we method in 2D and 3D systems at continuum-level. The examined 8 different pore sizes, which is homogeneous calculations reveal that small pores are more effective in in the system. The system size is 2048 ∆x × 2048∆y reducing effective conductivity rather than large pores at (10.24 µm × 10.24 µm). given porosity in both 2D and 3D systems. We found the exponent value β of Schulz equation and we found that We introduced the structural order parameter phi there is a dependence of pore size on exponent β value. which value is 1.0 in non-porous region and 0.0 in porous region. The parameter varies smoothly at the interface We assumed all pore size is same in the system and we found that β exponent decreases as the pore size region to enhance the numerical stability of the effective conductivity calculation. We found that the exponent β increases in 2D and 3D. in Eq. 4 increases as the pore size decreases, which REFERENCES means small pore-sized structure has stronger dependency of porosity in effective conductivity rather than large pore-sized structure. [1] P. Lucuta, H. Matzke, I. Hastings, A pragmatic approach to modeling thermal conductivity of irradiated uo2 fuel: review and recommendations, Journal of nuclear materials 232 (2-3) 166-180 (1996).