Thermal-Hydraulic in nuclear reactor

THERMAL-HYDRAULIC IN NUCLEAR REACTOR GS. Trần Đại Phúc THERMAL-HYDRAULIC IN NUCLEAR REACTOR Summary Introduction Energy from fission Fission yield Decay heat Spatial distribution of heat sources Coolant flow & heat transfer in fuel rod assembly Enthalpy distribution in heated channel Temperature distribution in channel in single phase Heat conduction in fuel assembly Axial temperature distribution in fuel rod Void fraction in fuel rod channel Heat transfer to coolant THERMAL-HYDRAULIC IN NUCLEAR REACTOR I. Introduction An important aspect of nuclear reactor core analysis involves the determination of the optimal coolant flow distribution and pressure drop across the reactor core. On the one hand, higher coolant flow rates will lead to better heat transfer coefficients and higher Critical Heat Flux (CHF) limits. On the other hand, higher flows rates will also in large pressure drops across the reactor core, hence larger required pumping powers and larger dynamic loads on the core components. Thus, the role of the hydrodynamic and thermal-hydraulic analysis is to find proper operating conditions that assure both safe and economical operation of the nuclear power plant. THERMAL-HYDRAULIC IN NUCLEAR REACTOR This chapter presents methods to determine the distribution of heat sources and temperatures in various components of nuclear reactor. In safety analyses of nuclear power plants the amount of heat generated in the reactor core must be known in order to be able to calculate the temperature distributions and thus, to determine the safety margins. Such analyses have to be performed for all imaginable conditions, including operation conditions, reactor startup and shutdown, as well as for removal of the decay heat after reactor shutdown. The first section presents the methods to predict the heat sources in nuclear reactors at various conditions. The following sections discuss the prediction of such parameters as coolant enthalpy, fuel element temperature, void fraction, pressure drop and the occurrence of the Critical Heat Flux (CHF) in nuclear fuel assemblies THERMAL-HYDRAULIC IN NUCLEAR REACTOR I.1. Safety Functions & Requirements The safety functions guaranteed by the thermal-hydraulic design are following: Evacuation via coolant fluid the heat generated by the nuclear fuel; Containment of radioactive products (actinides and fission products) inside the containment barrier. Control of the reactivity of the reactor core: no effect on the thermal-hydraulic design. Evacuation of the heat generated by the nuclear fuel: The aim of thermal-hydraulic design is to guarantee the evacuation of the heat generated in the reactor core by the energy transfer between the fuel THERMAL-HYDRAULIC IN NUCLEAR REACTOR Rods to the coolant fluid in normal operation and incidental conditions. The thermal-hydraulic design is not under specific design requirements. However, the assured safety functions requires the application of a Quality Assurance programme on which the main aim is to document and to control all associated activities. Preliminary tests: The basic hypothesis on scenarios adopted in the safety analyses must be control during the first physic tests of the reactor core. Some of those tests, for example the measurements of the primary coolant rate or the drop time of the control clusters, are performed regularly. Other tests are performed in totality only on the head of the train serial. For the following units, only the necessary tests performed to guarantee that thermal-hydraulic characteristics of the reactor core are identical to the ones of the head train serial. THERMAL-HYDRAULIC IN NUCLEAR REACTOR The primary coolant rate and the drop time of the control rod clusters must be measured regularly. The main aim of the thermal-hydraulic design is principally to guarantee the heat transfer and the repartition of the heat production in the reactor core, such as the evacuation of the primary heat or of the safety injection system (belong to each case) assures the respect of safety criteria.   I.2. Basis of thermal-hydraulic core analysis The energy released in the reactor core by fission of enriched uranium U235 and Plutonium 238 appears as kinetic energy of fission reaction products and finally as heat generated in the nuclear fuel elements. This heat must be removed from the fuel and reactor and used via auxiliary systems to convert steam-energy to produce electrical power. THERMAL-HYDRAULIC IN NUCLEAR REACTOR I.3. Constraints of the thermal-hydraulic core design The main aims of the core design are subject to several important constraints. The first important constraint is that the core temperatures remain below the melting points of materials used in the reactor core. This is particular important for the nuclear fuel and the nuclear fuel rods cladding. There are also limits on heat transfer are between the fuel elements and coolant, since if this heat transfer rate becomes too large, critical heat flux may be approached leading to boiling transition. This, in turn, will result in a rapid increase of the clad temperature of the fuel rod. THERMAL-HYDRAULIC IN NUCLEAR REACTOR The coolant pressure drop across the core must be kept low to minimize pumping requirements as well as hydraulic loads (vibrations) to core components. Above mentioned constraints must be analyzed over the core live, for all the reactor core components, since as the power distribution in the reactor changes due to fuel burn-up or core management, the temperature distribution will similarly change. Furthermore, since the cross sections governing the neutron physics of the reactor core are strongly temperature and density dependent, there will be a strong coupling between thermal-hydraulic and neutron behaviour of the reactor core. II. Energy from nuclear fission  Consider a mono-energetic neutron beam in which n is the neutron density (number of neutrons per m3). If v is neutron speed then Snv is the number of neutron falling on 1 m2 of target material per second. THERMAL-HYDRAULIC IN NUCLEAR REACTOR Since s is the effective area per single nucleus, for a given reaction and neutron energy, then S is the effective area of all the nuclei per m3 of target. Hence the product Snv gives the number of interactions of nuclei and neutrons per m3 of target material per second. In particular, the fission rate is found as: Σf nv = ΣfФ , where Σf =nv is the neutron flux (to be discussed later) and Σf= Nσf , N being the number of fissile nuclei/m3 and σf m2/nucleus the fission cross section. In a reactor the neutrons are not mono-energetic and cover a wide range of energies, with different flux and corresponding cross section. In thermal reactor with volume V there will occur V Σf Ф fissions, where Σf and Ф are the average values of the macroscopic fissions cross section and the neutron flux, respectively. THERMAL-HYDRAULIC IN NUCLEAR REACTOR To evaluate the reactor power it is necessary to know the average amount of energy which is released in a single fission. The table below shows typical values for uranium-235. Table II.1: Distribution of energy per fission of U-235. 10-12 J = 1 MeV Kinetic energy of fission products 26.9 168 Instantaneous gamma-ray energy 1.1 7 Kinetic energy of fission neutrons 0.8 5 Beta particles from fission products 1.1 7 Gamma rays from fission products 1.0 6 Neutrinos 1.6 10 Total fission energy 32 200 THERMAL-HYDRAULIC IN NUCLEAR REACTOR As can be seen, the total fission energy is equal to 32 pJ. It means that it is required ~3.1 1010 fissions per second to generate 1 W of the thermal power. Thus, the thermal power of a reactor can be evaluated as: P (W) = VΣfФ / 3.1x1010 (W) Thus, the thermal power of a nuclear reactor is proportional to the number of fissile nuclei, N, and the neutron flux f . Both these parameters vary in a nuclear reactor and their correct computation is necessary to be able to accurately calculate the reactor power. Power density (which is the total power divided by the volume) in nuclear reactors is much higher than in conventional power plants. Its typical value for PWRs is 75 MW/m3, whereas for a fast breeder reactor cooled with sodium it can be as high as 530 MW/m3.   THERMAL-HYDRAULIC IN NUCLEAR REACTOR III. Fission yield   Fissions of uranium-235 nucleus can end up with 80 different primary fission products. The range of mass numbers of products is from 72 (isotope of zinc) to 161 (possibly an isotope of terbium). The yields of the products of thermal fission of uranium-233, uranium-235, plutonium-239 and a mixture of uranium and plutonium are shown in following figure III.1. THERMAL-HYDRAULIC IN NUCLEAR REACTOR Figure III.1: Fission yield as a function of mass number of the fission product. As can be seen in all cases there are two groups of fission products: a “light” group with mass number between 80 and 110 and a “heavy” group with mass numbers between 125 and 155. THERMAL-HYDRAULIC IN NUCLEAR REACTOR Figure III.2: Illustration of the 6 formula: THERMAL-HYDRAULIC IN NUCLEAR REACTOR IV. Decay heat   A large portion of the radioactive fission products emit gamma rays, in addition to beta particles. The amount and activity of individual fission products and the total fission product inventory in the reactor fuel during operation and after shut-down are important for several reasons: namely to evaluate the radiation hazard, and to determine the decrease of the fission product radioactivity in the spent fuel elements after removal from the reactor. This information is required to evaluate the length of the cooling period before the fuel can be reprocessed. Right after the insertion of a large negative reactivity to the reactor core (for example, due to an injection of control rods), the neutron flux rapidly decreases according to the following equation, THERMAL-HYDRAULIC IN NUCLEAR REACTOR Φ(t) = Ф0{(β / β – ρ) e (λρ / β – ρ)t - (ρ / β – ρ))e (β – ρ / l)t } (IV.1) Here f (t ) is the neutron flux at time t after reactor shut-down, 0 f is the neutron flux during reactor operation at full power, r is the step change of reactivity, β is the fraction of delayed neutrons, l is the prompt neutron lifetime and l is the mean decay constant of precursors of delayed neutrons. For LWR with uranium-235 as the fissile material, typical values are as follows: l = 0.08 s-1, β = 0.0065 and l = 10-3s. Assuming the negative step-change of reactivity r = -0.09, the relative neutron flux change is given as: Ф(t) / Ф0 = 0.067 e -0.075t + 0.933 e -96.5t (IV-2) THERMAL-HYDRAULIC IN NUCLEAR REACTOR The second term in Eq. (4-3) is negligible already after t = 0.01s and only the first term has to be taken into account in calculations. As can be seen, the neutron flux (and thus the generated power) immediately jumps to ~6.7% of its initial value and then it is reduced e-fold during period of time T = 1/0.075 = 13.3 s. After a reactor is shut down and the neutron flux falls to such a small value that it may be neglected, substantial amounts of heat continue to be generated due to the beta particles and the gamma rays emitted by the fission products. FIGURE 4-2 shows the fission product decay heat versus the time after shut down. The curve, which covers a time range from 1 to 106 years after shut down, refers to a hypothetical pressurized water cooled reactor that has operated at a constant power for a period of time during which the fuel (with initial enrichment 4.5%) has reached 50 GWd/tU burn-up and is then shut down instantaneously. Contributions from various species which are present in the spent fuel are indicated. THERMAL-HYDRAULIC IN NUCLEAR REACTOR Figure IV.1: Fission product decay heat power (W/metric ton of HM) versus time after shutdown. THERMAL-HYDRAULIC IN NUCLEAR REACTOR Figure IV.2: Relative decay power versus relative time after reactor shutdown for various operation periods from 1 month to 12 months. THERMAL-HYDRAULIC IN NUCLEAR REACTOR The power density change due to beta and gamma radiation can be calculated from the fllowing approximate equation [IV-1], q” / q”0 = 0.065 { t - top) -0.2 - t -0.2} (IV.3) Here q”0 is the power density in the reactor at steady state operation before shut down, q” is the decay power density, t is the time after reactor shut down [s] and top is the time of reactor operation before shut down [s]. Equation (IV-3) is applicable regardless of whether the fuel containing the fission products remains in the reactor core or it is removed from it. However, the equation accuracy and applicability is limited and can be used for cooling periods from approximately 10 s to less than 100 days. THERMAL-HYDRAULIC IN NUCLEAR REACTOR Equation (IV.3) can be transformed to: q”’ / q”0 = 0.065 / top 0.2{ 1 / (t – top / top) 0.2 - 1 / (t / top) 0.2} (IV.4) Here ʘ = (t – top) / top is the relative time after reactor shut down. Equation (IV.4) is shown in FIGURE IV.2 for the reactor operation time top from 1 month to 1 year.   V. Spatial distribution of the heat sources The energy released in nuclear fission reaction is distributed among a variety of reaction products characterized by different range and time delays. Once performing the thermal design of a reactor core, the energy deposition distributed over the coolant and structural materials is frequently reassigned to the fuel in order to simplify the thermal analysis of the core. The volumetric fission heat source in the core can be found in general case as: THERMAL-HYDRAULIC IN NUCLEAR REACTOR q”’ (r) = Σi wf (i) Ni (r) ƒ0∞ dEσf(i) (E)Ф (r,E) (V.1) Here (i ) f w is the recoverable energy released per fission event of i-th fissile material, (r) i N is the number density of i-th fissile material at location r and (E) i f s is its microscopic fission cross section for neutrons with energy E. Since the neutron flux and the number density of the fuel vary across the reactor core, there will be a corresponding variation in the fission heat source.   The simplest model of fission heat distribution would correspond to a bare, homogeneous core. One should recall here the one-group flux distribution for such geometry given as: THERMAL-HYDRAULIC IN NUCLEAR REACTOR Ф(r, z) = Ф0J0 {2.405r / R}cos{πz / H} (V.2) Here 0 is the flux at the center of the core and R and H are effective (extrapolated) core dimensions that include extrapolation lengths as well as an adjustment to account for a reflected core. Having a fuel rod located at r = rf distance from the centerline of the core, the volumetric fission heat source becomes a function of the axial coordinate, z, only: q”’(z) = wfΣfФ0J0{2.405rf / R}cos{πz / H} (V.3) There are numerous factors that perturb the power distribution of the reactor core, and the above equation will not be valid. For example fuel is usually not loaded with uniform enrichment. At the beginning of core life, higher enrichment fuel is loaded toward the edge of the core in order to flatten the power distribution. Other factors include the influence of the control rods and variation of the coolant density. THERMAL-HYDRAULIC IN NUCLEAR REACTOR All these power perturbations will cause a corresponding variation of temperature distribution in the core. A usual technique to take care of these variations is to estimate the local working conditions (power level, coolant flow, etc) which are the closest to the thermal limitations. Such part of the core is called hot channel and the working conditions are related with so-called hot channel factors. One common approach to define hot channel is to choose the channel where the core heat flux and the coolant enthalpy rise is a maximum. Working conditions in the hot channel are defined by several ratios of local conditions to core-averaged conditions. These ratios, termed the hot channel factors or power peaking factors will be considered in more detail in coming Chapters. However, it can be mentioned already here that the basic initial plant thermal design relay on these factors. THERMAL-HYDRAULIC IN NUCLEAR REACTOR In thermal reactors it is assumed that 90% of the fission total energy is liberated in fuel elements, whereas the remaining 10% is equally distributed between moderator and reflector/shields.  VI. Coolant flow and heat transfer in fuel rod assembly  Rod bundles in nuclear reactors have usually very complex geometry. Due to that a thorough thermal-hydraulic analysis in rod bundles requires quite sophisticated computational tools. In general, several levels of approximations can be employed to perform the analysis: • Simple one-dimensional analysis of a single sub-channel or bundle, • Analysis of a whole rod bundle applying the sub-channel-analysis code, • Complex three-dimensional analysis using Computational Fluid Dynamics (CFD) codes. THERMAL-HYDRAULIC IN NUCLEAR REACTOR In this chapter only the simples approach is considered. In this approach, the single sub-channel or rod bundle is treated as a one-dimensional pipe with a diameter equal to the hydraulic (equivalent) diameter of the sub-channel or bundle. The hydraulic diameter of a channel of arbitrary shape is defined as: Dh = 4A / Pw (VI.1) where A is the channel cross-section area and Pw is the channel wetted perimeter.figure VI.1shows typical coolant sub-channels in infinite rod lattices. Figure VI.1: Typical coolant sub-channels in fuel rods assembly. THERMAL-HYDRAULIC IN NUCLEAR REACTOR Figure VI.1: Typical coolant sub-channels in fuel rods assembly. The subchannel flow area is expressed as following: A = p2 - πd2 / 4 for square lattice A = (31/2 / 4)p2 - πd2 / 4 (VI.2) THERMAL-HYDRAULIC IN NUCLEAR REACTOR And the wetted perimeter (part of the perimeter filled with heated walls) is given by: Pw = πd for square lattice Pw = 1/2 πd for triangular lattice (VI.3)  Where p is the lattice pitch and d is the diameter of fuel rods. The hydraulic diameter is expressed as: Dh = d{4 / π(p / d)2 – 1} for square lattice Dh = d{2x31/2 / π (p / d)2 – 1 } for triangular lattice (VI.4) In case of fuel assemblies in Boiling Water Reactors (BWR), the hydraulic diameter should be based on the total wetted perimeter and the total cross-section area of the fuel assembly. Assuming fuel assembly as shown in FIGURE 4-5, the hydraulic diameter is as follows: THERMAL-HYDRAULIC IN NUCLEAR REACTOR Dh = 4A / Pw = (4w2 – Nπd2) / (4w + Nπd) (VI.5) Where N is the number of rods in the fuel assembly, w is the width of the box (m) and d is the diameter of fuel rods(m). Figure VI.2: Cross-section of a BWR fuel assembly.   THERMAL-HYDRAULIC IN NUCLEAR REACTOR VII. Enthalpy distribution in heated channel   Assume a heated channel with an arbitrary axial distribution of the heat flux, q’’(z), and an arbitrary, axially-dependent geometry, as shown in figure VII.1. The coolant flowing in the channel has a constant mass flow rate W. As follow THERMAL-HYDRAULIC IN NUCLEAR REACTOR The energy balance for a differential channel length between z and z + dz is given as follows: ΔH = W . il (z) + q”(z).PH(z).dz = W[il (z) + dil] (VII.1) Which to the following differential equation for the coolant enthalpy: ΔH = dil(z) / dz = q”(z) PH(z) / W (VII.2) Where PH(z) is the heated perimeter of the channel. Integration of Eq. (4-13) from the channel inlet to a certain location z yields: i l(z) = ili + 1/W ƒ-H/2 z q”(z).PH(z)dz (VII.3)   THERMAL-HYDRAULIC IN NUCLEAR REACTOR where il(z) is the coolant enthalpy at location z and ili is the coolant enthalpy at the inlet to the channel (z = -H/2).   VIII. Temperature distribution in channel in single phase   For low temperature and pressure changes the enthalpy of a single-phase (non-boiling) coolant can be expressed as a linear function of the temperature. Assuming a uniform axial distribution of heat sources and a constant heated perimeter, Eq. (VII.3)) yields, Tlb (z) = Tlbi + q”PH (z + H/2) / CpW (VIII.1) THERMAL-HYDRAULIC IN NUCLEAR REACTOR Here Tlb(z) is the coolant bulk temperature at location z. The bulk temperature in a cha