Int. J. Nuclear Energy Science and Technology, Vol. 11, No. 2, 2017 Study on the temperature distributions in fuel assemblies of lead-cooled fast reactors Gilberto Espinosa-Paredes Área de Ingeniería en Recursos Energéticos, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, México DF 09340, México and Facultad de Ingeniería, Universidad Nacional Autónoma de México, Programa de Estancias Sabáticas del CONACyT, Morelos, México Email: [email protected] Juan-Luis François* Departamento de Sistemas Energéticos, Facultad de Ingeniería, Universidad Nacional Autónoma de México, Paseo Cuauhnáhuac 8532, 62550 Jiutepec, Morelos, México Email: [email protected] *Corresponding author Heriberto Sánchez-Mora Área de Ingeniería en Recursos Energéticos, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, México DF 09340, México and Departamento de Sistemas Energéticos, Facultad de Ingeniería, Universidad Nacional Autónoma de México, Paseo Cuauhnáhuac 8532, 62550 Jiutepec, Morelos, México Email: [email protected] Alejandría D. Pérez-Valseca Área de Ingeniería en Recursos Energéticos, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, México DF 09340, México Email: [email protected] Copyright © 2017 Inderscience Enterprises Ltd. 183 184 G. Espinosa-Paredes et al. Cecilia Martín-del-Campo Departamento de Sistemas Energéticos, Facultad de Ingeniería, Universidad Nacional Autónoma de México, Paseo Cuauhnáhuac 8532, 62550 Jiutepec, Morelos, México Email: [email protected] Abstract: The aim of this paper is to make a comparative study of two concepts of Lead-Cooled Fast Reactor (LFR) fuel assemblies, from a point of view of the thermofluids performance. The sub-channel analysis approach was applied to determine the temperature distribution in the fuel, in the cladding and in the lead-coolant. The mathematical model is fully transient and takes into account the heat transfer in an annular fuel pellet design. The thermofluid is modelled with a mass, energy and momentum balance with thermal expansion effects. The neutronic processes are modelled with point kinetic equations for power generation with feedback fuel temperature and expansion effects. The numerical experiments consider steady-state and transient behaviours. The numerical comparison shows that a hexagonal assembly is an option to compact the size of the LFR core design. This option leads to higher temperature in the fuel and the cladding than in the case of a rectangular assembly design. Results show the LFR with square array is more sensitive to power changes than the hexagonal array at the same nominal power and with the same transient conditions. Keywords: lead-cooled reactor; fast reactor; ELSY; annular fuel; fuel assembly; sub-channel analysis. Reference to this paper should be made as follows: Espinosa-Paredes, G., François, J-L., Sánchez-Mora, H., Pérez-Valseca, A.D. and Martín-del-Campo, C. (2017) ‘Study on the temperature distributions in fuel assemblies of lead-cooled fast reactors’, Int. J. Nuclear Energy Science and Technology, Vol. 11, No. 2, pp.183–203. Biographical notes: Gilberto Espinosa-Paredes received his bachelor in Energy Engineering in 1984, his master in Mechanical Engineering in 1992 and his PhD in Chemical Engineering (two-phase flow and heat transfer in nuclear reactors) in 1998. Currently he is Professor at the Division of Basic Sciences and Engineering of the Metropolitan Autonomous University, Campus Iztapalapa. Juan-Luis François received his bachelor in Energy Engineering in 1979, his master in Nuclear Engineering in 1984 and his PhD in Physical Sciences (Reactor Physics) in 1987. Currently he is Professor at the College of Engineering of the National Autonomous University of Mexico. Heriberto Sánchez-Mora received his bachelor in Energy Engineering in 2015. Currently he is doing his master studies in Energy Engineering at the National Autonomous University of Mexico. Alejandría D. Pérez-Valseca received her bachelor in Energy Engineering in 2010 and her Master of Science (Energy and Environment) in February of 2017. Currently she is a student of PhD in Sciences (Energy and Environment) at the Metropolitan Autonomous University. Study on the temperature distributions 185 Cecilia Martín-del-Campo is Professor at the Faculty of Engineering of the National Autonomous University of Mexico since 1999. She received her bachelor in Energy Engineering in 1979, her master in Nuclear Engineering in 1984 and her PhD in Physical Sciences (Reactor Physics) in 1987. 1 Introduction The Generation IV reactor concepts must meet the following criteria: (1) economic competitiveness, (2) inherent safety, (3) minimisation of waste, (4) non-proliferation enhancement, and (5) social acceptance. The Lead-Cooled Fast Reactor (LFR) has been proposed as a possible answer to the compelling need for safer and sustainable nuclear energy production (Generation IV International Forum, 2002; Generation IV International Forum, 2015). The LFR has been included in GIF-IV because of its fastneutron spectrum, which allows for a closed fuel cycle for efficient conversion of fertile uranium and the management of actinides. The LFR design increased safety margins, even under the most severe accidental conditions and has the propitious characteristics of an inert coolant such as lead and bismuth. Liquid metal cooled reactors are expected to play an important role in the future of nuclear energy production because of their possible efficient use of uranium, and have the ability to reduce the radioactive lifetime and volume of nuclear waste. Thermofluids is recognised as a key scientific topic in the development of such reactors. Two important challenges for the design of liquid metal fast reactors are the fuel assembly and the analysis of thermofluids (Roelofs et al., 2013). The high boiling point and relatively low melting point make lead attractive as a coolant in fast reactor designs. Lead is chemically inert in water, which may be regarded as an advantage. However, the main drawbacks are the corrosion and erosion issues on the reactor internal components (Wolniewicz et al., 2015). The LFR has been the subject of several studies, and numerous investigations have been carried out to assess their effectiveness in the light of the requirements of the GIF-IV (Loewen and Tokuhiro, 2003; Chandra and Roelofs, 2010; Colombo et al., 2010; Alemberti et al., 2011; Bandini et al., 2011; Aufiero et al., 2013; Bortot et al., 2013; Wang et al., 2013; Chen et al., 2015; Guo et al., 2015; Sartori et al., 2016; Yang et al., 2016). LFRs offer great potential for plant simplifications and higher operating efficiencies compared to other coolants. However, the safety characteristics and design-basis represent challenges, which need to be analysed and studied, in depth, using numerical analysis and computational tools. The thermofluids and point kinetics model for the ELSY lead fast reactor, with an open square core option, was used to verify the core temperatures at beginning- and end-of-cycle conditions for a nominal power of 1500 MWth (Bandini et al., 2011). In this work we are comparing the thermofluids processes in the core, the fuel heat transfer, and the thermal power, in two fuel assemblies in an LFR: rectangular and hexagonal. In LFR core design, the prediction of the maximum fuel temperature and coolant are crucial to security requirements. In previous work, an ELSY reactor model, which considers thermofluids and reactor point kinetics, was developed with RELAP5 code and has been used to verify the core temperatures at a nominal power of 1500 MWth (Bandini et al., 2011). 186 G. Espinosa-Paredes et al. In this study the sub-channel analysis approach was applied to determine the temperature distribution in the fuel, the cladding and lead. The sub-channel analysis is the most suitable approach in the design of fuel assemblies (Wang et al., 2013; Chen et al., 2015). The sub-channel model developed in this work is fully transient. The mathematical models govern the different physical phenomena occurring in a subchannel of the LFR. The fuel pin is annular with a distribution of temperatures in a onedimensional radial direction and is transient. The thermofluids in the sub-channel is onedimensional in the axial direction and is also transient, taking into account the thermal expansion effects of lead. The neutronics power is modelled with point kinetics equations for power generation with fuel temperature feedback, and the power distribution in the axial direction is non-uniform. The coupling of the neutron processes, heat transfer in the fuel, and thermofluids, is discussed. 2 Lead-cooled fast reactor (LFR) description The ELSY reactor with a thermal power of 1500 MWth, which is a pool type reactor cooled with pure lead, was applied in this study, in order to perform the comparison between the two fuel assemblies: rectangular and hexagonal. The European Lead Fast Reactor has been developed in the framework of the European lead system project funded by the Sixth Framework Programme of EURATOM. The project points out the possibility of designing a competitive and safe fast critical reactor using simple engineered technical features, while fully complying with the Generation IV goals (Alemberti et al., 2011). Table 1 shows the specifications of the ELSY reactor. Table 1 Main design data of ELSY Feature Thermal power Value 1500 MWth Efficiency 40% Coolant Lead Length 1.2 m Coolant inlet temperature 673.15 K Coolant outlet temperature 750.15 K Speed of lead 2 m/s In this work we analyse and compare the temperature distribution between the hexagonal and square assemblies in the ELSY core reactor. Figure 1 shows a schematic illustration of the core and fuel assembly for both arrangements, as well as each zone of the core, control rods and reflectors. The characteristics of the fuel assemblies studied in this work are presented in Table 2. In this table the hexagonal array has more fuel assemblies than the square array; however, the square assembly has more fuel rods per assembly. The difference between the number of rod fuels of the square and the hexagonal arrays implies that the masses of fuel and coolant are different for each configuration; the comparison is based on the same thermal power for both reactors. Study on the temperature distributions Figure 1 187 Scheme of LFR: (a) core in hexagonal array, (b) core in square array, (c) hexagonal fuel assembly, (d) square fuel assembly Inner zone Middle zone Control rods Outer zone Reflectors (a) (b) (c) (d) In each of these arrangements, the core distribution assemblies depend on the Pu enrichment. In the case of the square array, enrichments are: 14.45% Pu in the inner zone, 17.53% Pu in the middle zone and 20.50% Pu in the outer zone (Artioli et al., 2009). While those for the hexagonal array are: 14.9% Pu in the inner zone%, 15.5% Pu in the middle zone and 17.4% Pu in the outer zone (Sobolev at al., 2009). Table 2 Features of the core in the hexagonal and square arrays Feature Hexagonal array Square array Total assemblies in the core 325 170 Rods in the assembly 165 428 27.31 20.62 Thermal power/rod (kW/rod) 188 3 G. Espinosa-Paredes et al. Sub-channel analysis The LFR uses two liquid metal coolant materials. The first is Pb which has a melting temperature of 327.45°C, and an atmospheric boiling temperature of 1743°C. The second is lead-bismuth eutectic (LBE) which is composed of 45.0 at% Pb and 55.0 at% Bi with a melting temperature of 124.5°C and an atmospheric boiling temperature of 1670°C. The Pb and LBE densities, at 480ºC, are 10,470 kg/m3 and 10,100 kg/m3, respectively. Pb has a high boiling temperature that is well above the temperatures at which the steel cladding and structures lose their strength and melt. 3.1 Neutronic power The thermal power in the sub-channel is given by: P t , z P0 n t z (1) where P0 is the nominal thermal power per fuel rod, n(t) is the neutron density, and (z) is the axial power distribution. According to Table 2 P0 is 27.31 kWth/sub-channel for the hexagonal array and 20.62 kWth/sub-channel for the square array. The axial power distribution (z) used in this work is shown in Figure 2. The axial distribution of power in the hottest and coldest pins of the warmest assembly in the inner zone is similar (Sobolev at al., 2009). The axial neutron flux profiles in LFR within the active height were obtained with the SERPENT code and compared with the COMSOL code results (Aufiero et al., 2013), which are very close to trend shown in Figure 2. The axial distribution is the same for all the fuel rods and is estimated for each node in de rod. Figure 2 Axial power distribution in the sub-channel (see online version for colours) The neutron density is calculated with neutron point kinetics equations with six precursors of delayed neutrons: dn t dt t 6 n t i Ci t i 1 (2) Study on the temperature distributions dCi t dt i 189 n t i Ci t , for 1, 2,3, , 6 T fuel (3) f T fuel c Tclad l Tlead T fuel in Clad expansion Lead expansion Fuel expansion 0 1.1K D ln out (4) Doppler where is the reactivity, is the total fraction of delayed neutrons, is the mean neutron generation time, is the decay constant of delayed neutron precursor, Ci is the concentration of the i-th delayed neutron precursor. The first and second terms on the right side of equation (3) represents the rate of formation of the precursors and radioactive decay of the i-th group, respectively. The nuclear parameters used in this study are for the dominant fissile nuclide Pu239 and are shown in Table 3. Nuclear parameters for Pu239 (Waltar et al., 2012) Table 3 Group i s 1 i 1 0.0129 0.0000817 2 0.0311 0.000602 3 0.134 0.0004644 4 0.331 0.0007052 5 1.26 0.00022145 6 3.21 0.00007525 6 The total fraction of delayed neutrons is given by i 0.00215 , and the mean i 1 neutron generation time is 6.116 107 s (Ponciroli et al., 2014). Equation (4) computes the total reactivity that assumes various contributions: initial reactivity margin (0), Doppler effect, fuel expansion, cladding expansion, and the effect of lead density, whose parameters are: Doppler constant KD = –555 pcm, reactivity coefficients f = –0.232 pcm/K, c = 0.045 pcm/K, and l = –0.247 pcm/K (Ponciroli et al., 2014; Waltar et al., 2012). It is important to note that the c includes the effects of the axial and radial expansion of the clad. The changes in the average temperature in equation (4) are defined as: Tfuel Tfuel Tfuel , Tclad Tclad Tclad 0 , and Tlead Tlead Tlead 0 , where 0 the subscript 0 represents the reference temperature. The average temperatures to compute equation (4) are defined as follows: T fuel in T fuel out 1 zL T fuel L z 0 1 zL T fuel L z 0 r ra dz r rf dz (5) (6) 190 G. Espinosa-Paredes et al. T fuel Tclad Tlead 1 V fuel 1 Vclad T fuel dV (7) Tclad dV (8) V fuel Vclad 1 zL Tead dz L z 0 (9) where ra and rf are radiuses of the inner annular region and the fuel region of the pin, respectively, as is illustrated in Figure 3. In these equations L represents the active length of the core, and V is the volume. The numerical solution of the neutronics power considers two methods. The first is the Runge–Kutta 4th order method, and is applied for the numerical solution of neutrons density, given by equation (2), and the second is the Euler method, and is applied for the numerical solution of concentration of the i-th neutron delayed precursor, given by equations (2 and 3). The step length used in this work was of 0.0001 s. 3.2 Fuel heat transfer The fuel mathematical model calculates the heat transfer in annular fuel pellets of hexagonal and square arrays, as is illustrated in Figure 3. The fuel pellets have differences in dimensions for each fuel-assembly design, as presented in Table 4. Figure 3 Annular fuel pellets (see online version for colours) In Figure 3, there is a gap void between the fuel and the cladding. Although the thickness of the gap is quite thin, the low thermal conductivity of gases causes a large temperature drop across the gap. In the real case the gap spacing is not uniform and the heat conduction transfer process is very complex, but in this work, in order to simplify the Study on the temperature distributions 191 calculations, it is considered uniform. The cladding temperature is determined in terms of coolant temperature, and, eventually, in terms of the coolant inlet conditions and the reactor power level. Table 4 Fuel pellets dimensions (Figure 3) Hexagonal fuel assembly (Sobolev et al., 2009) Radius (mm) Square fuel assembly (Bandini et al., 2011) ra 1 1 rf 4.5 4.44 rg 4.65 4.55 rcl 5.25 5.25 The fuel heat transfer formulation is based on the following fundamental assumptions: (1) axis-symmetric radial heat transfer, (2) the heat conduction in the axial direction is negligible with respect to the heat conduction in the radial direction, (3) the volumetric heat rate generation in the fuel is uniform in the radial direction, (4) the gap spacing is uniform, and (5) the annular region is in thermodynamic equilibrium with the temperature of the inner surface of the fuel. Under these assumptions, the transient temperature distribution in the annular fuel pin, initial and boundary conditions is given by: T fuel Cp fuel Cp gap T Tgap Cp clad T k fuel T fuel r r r r k gap Tgap r , Gap rf r rg r r r Tclad kclad Tclad r T r r r q t , z , Fuel ra r rf , Clad rg r rcl (10) (11) (12) The initial condition is given by T(r, 0) = f(r), and the boundary conditions are: dT fuel dr k gap 0, at r ra dTgap dr hgap T fuel Tgap , at r rf (13) (14) kclad dTclad hgap Tgap Tclad , at r rg dr (15) kclad dTclad hlead Tclad Tlead , at r rcl dr (16) In these equations is the density, Cp is the specific heat, k is the thermal conductivity, hlead is the lead heat transfer coefficient, hgap is the gap conductance, and q t , z is the heat source given by: q t , z P t, z Vf (17) 192 G. Espinosa-Paredes et al. where P(t, z) is the sub-channel power given by equation (1), and Vf is the fuel volume. Table 4 presents the fuel pellet dimensions for each core array. The physical properties of fuel consisting of 0.15 molar fraction of plutonium oxide, and 0.85 of U-238 as a function of temperature are presented in Table 5. Table 6 shows the physical properties of the gap (helium) and cladding (T91). Table 5 Fuel properties (Carbajo et al., 2001) Density kg m3 ; Specific heat Cp J kgK ; Thermal conductivity k W mK fuel 9.9672 10 1 11043.5 1.179 105 T 2.429 10 9 T 2 1.219 1012 T 3 90.998 106 A 1.620 1012 Cp fuel 0.85 1.6926 102 T 2 T2 A 1 T 111.275 106 B 0.15 2.9358 102 T 2 B 1 T 1 6400 e 16.35 k fuel 1.158 ; 4 2 5 2 0.1205 2.6455 10 T A e548.68 T ; B e18541.7 T ; T 1000 hgap 6000 W m 2 K (Glasstone and Sesonske, 1981). The heat transfer coefficient of equation (16) is given by hlead klead Nu Dh , where klead is the thermal conductivity (given in Table 7), Gh is the equivalent diameter (see Section 3.3), and the Nusselt number for each fuel-assembly arrangement, is given by Todreas and Kazimi (1990): 4 0.025 Pr Re 0.8 , for square array Nu 0.8 7 0.025 Pr Re , for hexagonal array (18) where Pr and Re are the Prandtl and Reynolds numbers, respectively. Table 6 Gap and clad properties (Glasstone and Sesonske, 1981) Property Density kg m 3 Specific heat Cp J kgK Thermal conductivity k W mK Gap Cladding 2.425 7700.0 5191.0 622.0 15.8 10–4T0.7 26.0 The annular fuel pellet temperature distribution is obtained considering 19 radial nodes at each of the 24 axial nodes in the core. Ten nodes were considered in the fuel, four nodes in the gap, and five in the cladding. The differential equations described previously are transformed into discrete equations using the control volume formulation technique in an Study on the temperature distributions 193 implicit form. Application of the control volume formulation enables the equations for each region (fuel, gap and clad) to be written as a single set of algebraic equations for the sweep in the radial direction: t a j T jt t b j T jt1t c j T jt 1 d j for j 1, 2,3, , M (19) where aj, bj, cj and dj are coefficients that are computed at the time t. The procedure solution applied in this work is the Thomas algorithm (Patankar, 1980). 3.3 Thermofluids analysis The thermofluids are modelled with mass, energy and momentum balance that takes into account thermal expansion effects. th lead dTlead G 0, Mass balance z dt (20) Tlead Pm hlead Tclad Tlead G dTlead , Energy balance t Af lead Cplead lead dz (21) fr G 2 G 2 G lead g , Momentum balance t 2 lead L z lead (22) In these equations lead lead T , th is the thermal expansion coefficient, G is the mass flux, Pm is the wetted perimeter (given by d rod 4 4l p ; see Figure 4), and Af is the flow area (cross-sectional area). In the momentum balance given by equation (16), the friction coefficient is calculated with the following relation: fr 0.32 0.210 L l p 1 1 Re0.25 Dh d rod (23) where the rod pitch for the hexagonal array is lp = 15.5 mm (Sobolev et al., 2009), and for the square array is lp = 13.9 mm (Bandini et al., 2011), and drod is the rod diameter (see Table 4) as is illustrated in Figure 4, as well as the flow area (Af) indicated by the red line. The hydraulic diameter for each array is given by: 4 d rod Dh 4 d rod 2 3 2 d rod lp 4 2 2 2 d rod lp , 4 , for hexagonal array (24) for square array The physical properties of the lead used in this work are given in Table 7. The numerical solution applied to balance equations of mass, energy and momentum was the Euler method. 194 4 G. Espinosa-Paredes et al. Multiphysics coupling Numerical simulation of the nuclear reactors is a technological tool that allows for the safety analysis and reliability of future reactor designs. Nuclear reactors are multiphysics and multiscale systems that involve the nuclear process, fuel heat transfer, and thermofluids analysis. The multiphysics coupling between different mathematical models (presented in the previous section) allows the simulation of the behaviour of the reactor core of an LFR. The coupling of the physical processes involves a complex dynamic interaction of variables among nuclear processes, of fuel heat transfer, and thermofluids. The simulation of nuclear processes with the neutron point kinetics approach is coupled with fuel heat transfer, through average temperatures of the fuel and cladding, and at the same time is coupled with thermofluids through average temperature of the lead. The fuel heat transfer requires a nuclear heat source and the thermal properties of lead. And the thermofluids in the core requires the cladding wall temperature. Figure 4 Schematic sub-channel: (a) hexagonal fuel-assembly arrangement, (b) hexagonal fuel-assembly arrangement (see online version for colours) (a) Table 7 (b) Lead physical properties (Sobolev et al., 2008) Property Relation Density kg m3 lead 11441 1.2795T Heat capacity J kgK Cplead 175.1 4.961 102 T 1.985 105 T 2 2.099 109 T 3 Thermal conductivity W mK klead 9.2 0.011T 1.524 106 T 2 Thermal expansion coefficient (K–1) th Viscosity (Pa s) 1 8942 T 1069 lead 4.55 104 exp T Study on the temperature distributions 5 195 Results and discussion The numerical analysis shows the comparison of hexagonal fuel assemblies versus square fuel assemblies in steady-state analysis and transient behaviour. 5.1 Steady-state analysis Figure 5 shows the temperature field (axial and radial) in the fuel rod at different powers: 100%, 75%, 50%, 25% of rated power, in which can be clearly seen the zones of high, medium and low temperatures. Table 8 shows the cladding and fuel pin peak temperatures at 25%, 50%, 75% and 100% of rated power that are representative results of Figure 5. According with this table the cladding temperatures for both arrays are close with a maximum difference of 1.52 K. However, the difference maxima of the peak temperatures between square and hexagonal arrays are 304 K, which represents a crucial difference that must be considered in the LFR core design. Results of the temperature distributions in the axial direction are presented in Figure 6, where the average definitions applied are given in equations (6)–(9). It is important to note that there are significant differences in temperature in the fuel and in the gap, but in the cladding they were very close in both arrangements. This is important from the point of view of safety. When comparing the steady-state results between arrays, the square array design is attractive from the point of view of temperatures, but it would be difficult to make a competitive compact design in comparison to the hexagonal array owing to the triangular features of the hexagonal array design (see Table 2 and Figure 3). Table 8 Temperature comparison in the fuel rod Rated power *Clad temperature *Clad temperature (%) square (K) hexagonal (K) Peak temperature square (K) Peak temperature hexagonal (K) 25 675.95 676.09 821.80 868.18 50 679.06 679.39 993.16 1102.31 75 682.35 682.94 1191.87 1385.28 100 685.83 686.75 1423.99 1727.51 Note: *Obtained with equation (8). 5.2 Transient analysis Transient responses of the two LFR fuel assemblies (square and hexagonal arrangements) were investigated and compared. Three transients were simulated: (1) transient due to velocity changes in the inlet lead-coolant, (2) transient due to temperature changes in the inlet lead-coolant, and (3) transient due to reactivity changes. These transients were performed instantaneously based on the steady-state condition. The first transient considers effects in the reduction and in the increase of the mass flux in the LFR core due to the main coolant pump action; the second transient considers effects due to possible faults in the steam generator; and the third transient corresponds to the extraction and insertion of a control rod. All the transients started at nominal power steady-state conditions (Table 1). 196 Figure 5 G. Espinosa-Paredes et al. Temperature field at 100%, 75%, 50%, and 25% of rated power Study on the temperature distributions Axial temperatures distribution Figure 6 Fuel temperature 1.2 1 0.8 z [m] z [m] 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 700 Cladding temperature 1.2 Hex 100% Sq 100% Hex 25% Sq 25% 1 800 900 1000 1100 T [K] 1200 1300 1400 680 700 720 T [K] 740 760 780 Lead temperature 1.2 Hex 100% Sq 100% Hex 25% Sq 25% 1 Hex 100% Sq 100% Hex 25% Sq 25% 0 660 1500 Gap temperature 1.2 1 0.8 z [m] 0.8 z [m] 197 0.6 0.6 0.4 0.4 0.2 0.2 0 650 700 750 800 850 900 950 0 670 Hex 100% Sq 100% Hex 25% Sq 25% 680 690 700 710 T [K] 720 730 740 750 760 T [K] The numerical results of these transients are shown in Figure 7, where Sq and Hex were used to indicate square and hexagonal, respectively. The steady-state condition (at time initial, which was used as a reference) was indicated using a continuous black line, and was the reference case at 100% of rated power. In this figure the behaviour of normalised power, and total reactivity, for both fuel-assembly arrangements is presented. 1 Transient due to the velocity variation in the inlet lead-coolant The speed of the inlet lead-coolant is the boundary condition to obtain the flux mass in the LFR core, which is 2 m/s at steady-state condition (Figure 7 (a)). The velocity variation transient consists of the reduction and increases around 2 m/s. This figure shows that the hexagonal array has lower power changes (increase or decrease) relative to the square array. This is because the Doppler feedback contribution to reactivity is higher in the hexagonal array, mainly due to the greater fuel rods pitch of the hexagonal versus the square lattice, and the slightly higher volume of fuel in the rods of the hexagonal assembly. With respect to the transient due to temperature behaviour, a reduction of lead speed (mass flux) from 2 m/s to 1 m/s is due to a partial loss of cooling, and therefore higher temperatures in fuel and lead, where the power decreases by fuel temperature rise. An increase of lead speed from 2 m/s to 3 m/s increases cooling capacity, decreases the temperature and increases the power. 198 2 G. Espinosa-Paredes et al. Transient due to the temperature changes in the inlet lead-coolant This transient consists of changes in the inlet temperature of lead-coolant, maintaining the speed inlet (mass flux) invariable. This transient compares both fuel-assembly arrangements when the inlet temperature of lead-coolant suffers changes. The coolant inlet temperature in steady-state condition is 673.15 K, which corresponds to nominal power, and the transient applies 10 K at around this temperature in one time step. The transient behaviour of this scenario is shown in Figure 7 (b). A reduction of coolant temperature of 673.15 K to 663.15 K produces an insertion of positive reactivity due to the Doppler effect (being the main positive contribution in the reactivity followed by the lead temperature decrease), and therefore the power increases. When the coolant temperature increases from 673.15 K to 683.15 K the behaviour of power and reactivity is exactly opposite in regards to the reduction of coolant temperature, i.e. the power decreases due to negative reactivity. Regarding the comparison between the hexagonal and square arrangements, the numerical results show that the hexagonal array presents lower power changes than the square array, although these changes are marginal. This transient is important for the LFR, because it is an inherent alternative power control linked to the coolant temperature (lead) in the inlet of the core. 3 Transient due to reactivity changes This transient corresponds to the extraction and the insertion of a control rod equivalent to 20 pcm step reactivity (Figure 7 (c)). After the instantaneous step insertion of positive reactivity (+20 pcm) the power suddenly increases, after that, there is a small decrease to finally achieve the steady-state condition, as can be seen in Figure 7 (c). After the insertion of negative reactivity (–20 pcm) the behaviour of the power and reactivity is exactly the opposite. As in the previous cases (Figure 7 (a) and (b)) the power changes are greater for the square arrays. The main feedback is due to fuel temperature and lead temperature. The transient of insertion of positive reactivity produces a power peak of about 109.5% for the square array, and about 108% for the hexagonal array. The final power of steady state (107% and 105% for the square and hexagonal arrays, respectively) is greater than those obtained in transients of increase of lead speed (Figure 7 (a)), and a reduction of lead-coolant temperature. However, the lowest final power of steady state is obtained with a reduction of lead speeds from 2 m/s to 1 m/s (Figure 7 (a)), about 92% for the square array and about 94% for the hexagonal array. In general, according to results, the LFR with a square array is more sensitive to power changes than the hexagonal array at the same nominal power and transient conditions. Figure 8 presents the transient field temperature in the LFR (fuel, gap, clad and lead) for both fuel arrangements during insertion of a positive reactivity (+20 pcm), for elapsed times of simulation of 1 s, 5 s and 200 s. In this figure there is a temperature increase (the intensity of the colours is higher) as the time elapses. The clad and peak temperatures for 1 s, 2 s, and 200 s of elapsed time of simulation are presented in Table 9, for square and hexagonal arrangements. Study on the temperature distributions 199 Transients due to changes in: (a) inlet velocity of lead-coolant, (b) coolant inlet temperature, (c) reactivity Figure 7 (a) Variation of lead inlet velocity 1.04 Variation of lead inlet velocity 2 Power [normalized] 1.02 0 Reactivity [pcm] 1 Sq 3 m/s Sq 1 m/s Hex 3 m/s Hex 1 m/s 0.98 0.96 0.94 -2 -4 -6 Sq 3 m/s Sq 1 m/s Hex 3 m/s Hex 1 m/s 0.92 -8 0.9 0 20 40 60 80 100 120 140 160 180 0 200 20 40 60 80 100 120 140 160 180 200 t [s] t [s] (b) Variation of lead inlet temperature 1.02 Variation of lead inlet temperature 1.5 1.015 1 Reactivity [pcm] Power [normalized] 1.01 1.005 1 Sq 663K Sq 683K Hex 663K Hex 683K 0.995 0.99 0.5 0 -0.5 Sq 663K Sq 683K Hex 663K Hex 683K -1 0.985 0.98 0 20 40 60 80 100 120 140 160 180 200 -1.5 0 20 40 60 80 t [s] 100 120 140 160 180 200 t [s] (c) Variation of reactivity 1.1 20 15 1.04 10 Reactivity [pcm] Power [normalized] 1.06 1.02 1 Sq 20pcm+ Sq 20pcmHex 20pcm+ Hex 20pcm- 0.98 0.96 5 0 -5 -10 0.94 Sq 20pcm+ Sq 20pcmHex 20pcm+ Hex 20pcm- -15 0.92 0.9 Variation of reactivity 25 1.08 -20 0 20 40 60 80 100 t [s] 120 140 160 180 200 -25 0 20 40 60 80 100 t [s] 120 140 160 180 200 200 Figure 8 G. Espinosa-Paredes et al. Transient temperature fields during insertion of a positive reactivity As shown in this table the clad temperatures of both arrays are very close with a difference of less than 1 K at any simulation time. However, the maximum differences of the peak temperatures between the square and hexagonal arrays are 310.08 K at 1 s, 315.15 K at 5 s, and 310.73 K at 200 s. Table 9 Temperature comparison: transient insertion of positive reactivity Time (s) *Clad temperature square (K) *Clad temperature hexagonal (K) Peak temperature square (K) Peak temperature hexagonal (K) 1 686.08 687.03 1440.92 1751.50 5 686.61 687.46 1476.92 1792.07 200 686.81 687.56 1494.41 1805.14 Note: *Obtained with equation (8). Results of the clad and lead temperature distributions in the axial direction at different elapsed times of simulation are presented in Figure 9. In this figure, the temperature difference (both clad and lead) between 5 s and 200 s are very similar for the hexagonal array, but for the square array it is significantly higher. The physical explanation of these temperature differences, between clad and lead, at 5 s and 200 s, is that the thermal resistance of the LFR with the square array is greater than the hexagonal array, which is related to the thermal conductivity of the fuel, clad, lead and heat transfer coefficient (given by equation 18). Study on the temperature distributions Figure 9 Axial temperatures distribution: transient insertion of positive reactivity Square array 1 1 0.8 0.8 0.6 T lead,1s T clad,1s T lead,5s T clad,5s T lead,200s T clad,200s 0.4 0.2 0 660 680 700 720 T [K] Hexagonal array 1.2 z [m] z [m] 1.2 6 201 740 760 780 0.6 T lead,1s T clad,1s T lead,5s T clad,5s T lead,200s T clad,200s 0.4 0.2 0 660 680 700 720 740 760 780 T [K] Conclusions With the development of relatively simple mathematical models based on the subchannel analysis approach, interesting results were obtained for two concepts of LFR fuel assemblies, from a point of view of the thermofluids performance. Regarding the steady-state condition, comparisons show that the hexagonal assembly leads to higher temperature in the fuel than in the case of the rectangular assembly design, with a peak temperature difference of 304 K between both arrays at 100% rated power. This is important from the point of view of safety. Concerning the kinetic behaviour, for the transient due to the velocity variation in the inlet coolant, it was found that the hexagonal array has lower power changes relative to the square array, due to the higher Doppler contribution to reactivity in the hexagonal array, as it was explained in Section 5.2 (1). Regarding the other two analysed transients: the temperature changes in the inlet lead-coolant and the reactivity variation, similar behaviours were observed as far as the power changes are concerned. In conclusion: results show that the LFR with square array is more sensitive to power changes than the hexagonal array at the same nominal power and with the same transient conditions. Also, from the transient analysis it was found that the thermal resistance of the LFR with the square array is greater than the hexagonal array. 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