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THERMAL-HYDRAULIC IN NUCLEAR REACTOR GS. Trần Đại Phúc THERMAL-HYDRAULIC IN NUCLEAR REACTOR Summary Introduction 2.Energy from fission 3.Fission yield 4.Decay heat 5.Spatial distribution of heat sources 6.Coolant flow & heat transfer in fuel rod assembly 7.Enthalpy distribution in heated channel 8.Temperature distribution in channel in single phase 9.Heat conduction in fuel assembly 10.Axial temperature distribution in fuel rod 11.Void fraction in fuel rod channel 12.Heat transfer to coolant 1. 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 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 thermalhydraulic 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 uranium235. 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, plutonium239 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 shutdown, 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. 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 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 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-channelanalysis code, • Complex three-dimensional analysis using Computational Fluid Dynamics 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 channel cross section is defined in such a way that it can be obtained from the energy balance over a portion of the channel. For an arbitrary velocity, temperature and fluid property distribution across the channel cross-section, the bulk temperature is given by: Tlb = ƒACρlCpl V ldA / ƒAρlCplVldA (VIII.2) The temperature distribution along the channel is represented in figure VIII.1. THERMAL-HYDRAULIC IN NUCLEAR REACTOR Figure VIII.1: Bulk temperature distribution in a uniformly heated channel with constant heated perimeter. THERMAL-HYDRAULIC IN NUCLEAR REACTOR        In nuclear reactor cores the axial power distribution may have various shapes. The cosine-shaped power distribution is obtained in cylindrical homogeneous reactors, as previously derived using the diffusion approximation for the neutron distribution calculation. Using Eq. (V.2) and the coordinate system as indicated in figure VIII.2, the power distribution may be expressed as follows: Q”(z) = q”0cos(πz / H) (VIII.3) Equation (VII.2) becomes: Dil(z) / dz = q”0PH(z)/W.cos(πz / H) or dTlb(z) /dz = q”0PH(z) /W.Cp.cos(πz /H) . (VIII.4) THERMAL-HYDRAULIC IN NUCLEAR REACTOR  Figure VIII.2: Heated channel with cosines power distribution THERMAL-HYDRAULIC IN NUCLEAR REACTOR  After intergration, the coolant enthalpy and temperature distribution are as follows:  il(z) = q”0PH / W x H/π [sin (πz / H) + sin (πH /2H)] + ili  or     Tlb(z) = q”0PH / W.Cp x H/π [sin (πz / H) + sin (πH /2H)] + Tlbi (VIII.5) The channel exit temperature and enthalpy can be found by substituting z = H/2 into equation VIII.5: ilex(z) = il(H/2) = 2q”0.PH.H /πW.sin(πH /2H) + ili or Tlbex = Tlb(H/2) = 2q”0.PH.H / π.W.Cp.sin(πH /2H) + Tlbi (VIII.6) THERMAL-HYDRAULIC IN NUCLEAR REACTOR  Figure VIII.3 Represents the axial distribution of the coolant temperature with cosines heat flux distribution. THERMAL-HYDRAULIC IN NUCLEAR REACTOR  IX. Heat conduction in fuel assembly   Modern nuclear power reactors contain cylindrical fuel elements that are composed of ceramic fuel pellets located in metallic tubes (so-called cladding). A cross-section over a square lattice of fuel rods is shown in FIGURE 4-10. For thermal analyses it is convenient to subdivide the fuel rod assembly into sub-channels. The division can be performed in several ways; however, most obvious choices are socalled coolant centered sub-channels and rod-centered sub-channels. Both types of sub-channels are equivalent in terms of major parameters such as the flow cross-section area, the hydraulic diameter, the wetted perimeter and the heated perimeter. In continuation, the thermal analysis will be performed for a single sub-channel. THERMAL-HYDRAULIC IN NUCLEAR REACTOR    The stationary (time independent) heat conduction equation for an infinite cylindrical fuel pin, in which the axial heat conduction can be ignored is as follows: q” = (- 1/r).d/dr[λFr.dTF /dr] (IX.1) where F T is the fuel temperature, [K], F l is the thermal conductivity of the fuel material, [W m-1 K-1], q ¢ is the density of heat sources, [W m-3] and r is the radial distance. Here the angular dependence of the temperature is omitted due to the assumed axial symmetry of the temperature distribution. THERMAL-HYDRAULIC IN NUCLEAR REACTOR      Assuming that q”’ is constant a cross-section equation (IX.1) could be integrated to obtain: λFrdTF / dr = - r2 /2q”’ (IX.2) If the fuel conductivity was constant, Eq. (IX.2) could be integrated and the temperature distribution would be obtained. However in typical fuel materials the fuel thermal conductivity strongly depends on the temperature and this is the reason why the temperature distribution can not be found from Eq. (IX.2) in a general analytical form. Instead, Eq. (IX.2) is transformed and integrated as follows: λFrdTF = - r2 /2q”’dr → ƒTFcTF0λFdTF = - q” /2ƒ0TF0rdr = (r2F0 /4)q”’ (IX.3) where the integration on the left-hand-side is carried out from the temperature at the centerline, TFc, to the temperature on the fuel pellet surface TFo=TF (rFo). THERMAL-HYDRAULIC IN NUCLEAR REACTOR         λF = 1 / (TFc – Tfo)ƒTFoTFcλFdTF (IX.4) The temperature drop across the fuel pellet can be found as follows: ΔTF = TFc - TFo = q”’r2Fo /4λF (IX.5) In the thermal analysis of reactor cores, the power is often expressed in terms of the linear power density, that is, the power generated per unit length of the fuel element: Q’ = πr2Foq”’ (IX.6) By combining equations ( IX.5) &( IX.-6), we have: ΔTF = q’ /4π.λF (IX.7) Equation (IX.7) reveals that the fuel temperature drop is a function of the linear power density and the averaged fuel thermal conductivity. THERMAL-HYDRAULIC IN NUCLEAR REACTOR       In a similar manner the temperature drop across the gas gap can be obtained. In particular, Eq. (IX.1) can be used to describe the temperature distribution in the gas gap, however, unlike for the fuel pellet, the heat source term is equal to zero and the gas thermal conductivity can be assumed constant, thus: -1/2 (d/dr)λGr(dTG/dr) = 0 → TG(r) = C1 /λGlnr +C2 (IX.8) The integration constant C, can be found from condition of the heat flux continuity at r = rFo: -λG(dTG/dr) = -C1 / rFo = q’ / 2πrFo → C1 = -q’/2π (IX.9) And the temperature drop in the gap can be expressed as follows: ΔTG = TG(rGi) –TG(rGo) = q’/2πλGlnrGo/rGi (IX.10) THERMAL-HYDRAULIC IN NUCLEAR REACTOR         Equation (IX.10) is applicable to the clad material as well, since the assumptions on the heat generation and the thermal conductance are valid in this case as well. Substituting the proper dimensions and property data yields, ΔTc = Tc(rci) - Tc (rco) = q’ /2πλcln rco/rci (IX-11) where rCo is the outer clad radius and _C is the clad thermal conductivity. Heat transfer from the clad surface to the coolant is described by the following Equation: q” = h(Tco – Tlb) (IX.12) where h is the convective heat-transfer coefficient. Taking into account that ( ) Co q¢¢ = q¢ 2p r , the temperature drop in the coolant THERMAL-HYDRAULIC IN NUCLEAR REACTOR      the total temperature drop from the center of the fuel pellet to coolant is expressed as follows: ΔT = ΔTF + ΔTG + ΔTc + ΔTl = q’/2π [1/2λF + 1/λG lnrGo/rGi + 1/λClnrCo/rCi + 1/rCoh] (IX.14) The total temperature drop in a fuel rod cross-section is represented in following figure IX.2. THERMAL-HYDRAULIC IN NUCLEAR REACTOR Figure IX.1: Cross section of a square fuel lattice. THERMAL-HYDRAULIC IN NUCLEAR REACTOR  X. Axial temperature distribution in fuel rods       In the previous section expressions for the axial distribution of coolant temperature have been derived. It has been shown that the axial distribution of coolant temperature varies with the shape of the axial heat flux distribution. In particular, substituting Eqs. (VIII.3) and (VIII.4) into (IX.12) gives the following expression for the temperature of the clad outer surface: TC0(z) = (q”0.PH.H) /(π.W.Cp) x [sin(πz /H) + sin(πH /2H)] + q”0 /h.cos (πz / H) + Tlbi (X.1) THERMAL-HYDRAULIC IN NUCLEAR REACTOR Figure X.1: Repartition of the temperature across the fuel rod. THERMAL-HYDRAULIC IN NUCLEAR REACTOR Figure X.1: Represents the temperature of the cladding outer surface as function of axial distance. THERMAL-HYDRAULIC IN NUCLEAR REACTOR       It should be noted that the temperature of the clad outer surface gets i/ H)ts maximum value TCo,max at a certain location zCo,max. This location can be found from Eq. (X.1) using the following condition: dTC0(z) /dz = 0 (X.2) It is convenient to represent the cladding outer temperature as: TC0 (z) = A + Bsin (πz /H) + CC0.cos(πz /H) (X.3) Where, A = Bsin(πz / 2H) + Tlbi  B = (q”0.PH.H) / (π.W.Cp) (X.4)  CC0 = q”0 / h  By combining the equations (X.2) & (X.3), we have: THERMAL-HYDRAULIC IN NUCLEAR REACTOR Bcos(πzCO,Max / H) - CC0sin(πzCo,Max /H) = 0 (X.5)          Which is equivalent to the following equation: Tan(πzCo,Max / H) Co = B / C (X.6) Thus: zCo,Max = (H/π)arctan(B /CC0) It should be noted that a physically meaningful solution of the above equation should be positive and less than H. Noting that: sin(πzCo,Max /H) = ± tan(πzC0,Max /H) / (1 + tan2(πzCo,Max /H))1/2 = ± ((B/CC0) / (1 + (B /CC0)2)1/2    (X.7) and cos (πzC0,Max / H) = ± ( 1/ (1 + tan2(πzC0,Max / H) THERMAL-HYDRAULIC IN NUCLEAR REACTOR       The maximum temperature of the cladding outer surface becomes (taking only + sign ): TC0,Max = A + (B2 + CC02)1/2 (X.8) Using constants A, B and CCo given by Eq. (X.4), the maximum clad outer temperature is obtained as: TC0,Max = (q”0.PH.H) / π.W.Cp)sin(πH / 2H) + Tlbi + ((q”0.PH.H /π.W.Cp)2 + (q”0 /h)2)1/2 (X.9) or (π.W.Cp(TC0,Max - Tlbi)) / q”0.PH.H = sin(πH /2H) + ((1 + (π.W.Cp) / PH.H.h)2)1/2 THERMAL-HYDRAULIC IN NUCLEAR REACTOR     Since the clad maximum temperature is located on the inner surface, it is of interest to find it as well. The axial distribution of the clad inner temperature can be obtained from Eqs. (IX.11) and X.1) as: TCi = ΔTC + TC0(z) = (q’/2λC)lnrC0/rCi + (q”0.PH.H / π.W.Cp).[sin(πz /H) + sin (πH 2/H] + (q"0 /h).cos(πz /h) + Tlbi = (q”0.PH.H / π.W.Cp).[sin(πz /H) + sin (πH /2H] + q”0(rC0/λC)ln(rC0/rCi) +1/h)cos(πz/H) + Tlbi (X.10)    Equation (X.10) ca,n be expressed: TCi(z) = A + Bsin(πz / H) + CCicos(πz /H) (X.11) THERMAL-HYDRAULIC IN NUCLEAR REACTOR          Where A & B are given by equation (X.3) and: CCi = q”0(rC0/λC)ln(rC0/rCi +1/h) (X.12) Using the same approach as in the case of the clad outer temperature, the location of the maximum temperature on the clad inner surface is found as: zCi,Max = (H/π)arctyan(B/CCi) (X.13) and the maximum corresponding temperature is: TCi,Max = (q”0.PH.H) / (π.W.Cp)sin(πH/2H) + Tlbi + ((q”0.PH.H)/(π.W.Cp)2 + [q”0(rC0/λC)ln(rC0/rCi) + 1/h]2)1/2 (X.14) In a similar manner the fuel maximum temperature at the center of the fuel pellet is given by: TFc(z) = A + Bsin (πz/H) + CFccos(πz/H) (X.15) THERMAL-HYDRAULIC IN NUCLEAR REACTOR        where CFC = q” [(rC00/λc)ln(rC0/rCi) + (rC0/λG)ln(rG0/rGi) + rCo/2λF + 1/h (X.16) The maximum fuel temperature is located at: zFc,Max = (H/π)arctan(B/CFc) (X.17) and its value is: TFC,Max = (q”0.PH.H/π.W.Cp).sin (πH/2H) + Tlbi + ((q”0.PH.H/π.W.Cp)2 + [q”0{(rC0/λC)ln(rC0/rCi) + (rC0/ λG)lnrG0/rGi +rC0/2λF + 1/h}]2)1/2. (X.18)   XI. Void fraction in boiling channels.   The characteristic feature of boiling channels is the presence of two phases: the liquid and the vapor phase. THERMAL-HYDRAULIC IN NUCLEAR REACTOR   Clearly, the presence of two phases changes the fluid flow and heat transfer processes as compared to the nonboiling channels. In addition, the density changes of coolant are more significant in boiling channels due to the dramatic change of density once liquid transforms into vapor. Thus, to be able to predict the local value of the coolant density it is required to determine the local volume fraction of both phases. Typically, the void fraction (that is the volume fraction of the vapor phase) is determined using various models, as described below. The various two-phase flow and heat transfer regimes in a boiling channel, such as BWR fuel assembly, is shown in figure( XI.1). THERMAL-HYDRAULIC IN NUCLEAR REACTOR Figure XI.1: Two-phase flow & heat transfer regime in a boiling channel ONB (Onset of Boiling), OSV (Onset of Significant Void, OAF (Onset of Annular Flow). THERMAL-HYDRAULIC IN NUCLEAR REACTOR The heat exchange coefficient depends on the local properties of coolant flow, which evolve all along the hot channel. This coefficient is characterized by its FΔh and the coolant by:  - Ti: the inlet temperature  - g: the mass flow rate  - Tsat: the saturation temperature  - X: the quality = steam mass/ mixture mass  - α: the void fraction = steam volume/mixture volume * A first assumption is made that the channel is isolated & exchanges neither mass nor energy with neighboring channels. This hypothesis, in fact highly penalizing, is not verified in a real PWR.  THERMAL-HYDRAULIC IN NUCLEAR REACTOR    As the coolant rises along the channel (see following figure), its physical properties are modified because its temperature increases, along with the temperature of the channel wall. The height of the channel can be divided into a certain number of zones with different properties: - 1) A lower zone, in which the wall temperature and the coolant temperature are below the saturation temperature. In this zone, the flow is single phase and the heat exchange regime is one of forced convection. The heat exchange between the cladding & the coolant is good & the temperature difference ΔT remains small, not exceeding tens of degrees. THERMAL-HYDRAULIC IN NUCLEAR REACTOR   2) Starting from a certain length of the tube, the wall temperature exceeds the coolants saturate temperature, Tsat, whereas the coolant remains at a temperature less than Tsat. Bubbles then begin to appear along the cladding wall, while the coolant remains strongly under saturation. These bubbles improve the thermal exchange, because they do not stuck to the wall but are carried along by coolant flow. Consequently they transmit calories from the wall to the coolant. - 3) Since the coolant continues to heat up, the density & the size of the bubbles increase. Suddently, there is a coalescence of the bubbles and the creation of a stable vapor film along the cladding wall. From this moment on, the heat exchange degenerates (h decreases, Tcladding increases). THERMAL-HYDRAULIC IN NUCLEAR REACTOR    This degraded heat exchange is explained, among other reasons, by the fact that steam has lower thermal conductivity than water. It occurs when a certain value of thermal flux has been reached, and leads to « Departure from Nucleate Boiling » & burn-out. Burn-out - Zone 0 to 1: corresponds to the forced convection regime. The fluid is under the unsaturated liquid form (quality equal zero), its exchange coefficient is relatively constant at given mass flow. The temperature difference between the surface and the center of the fluid flow is proportional to calorific flux to be transferred. THERMAL-HYDRAULIC IN NUCLEAR REACTOR    Zone 2: corresponds to nucleate boiling regime, which is interacting because the sleep slope (large heat-exchange coefficient) shows that a considerable amount of heat is extracted by means of a small increase in the wall temperature. The overheating of the liquid at the wall is sufficient to permit locally the creation of bubbles; the center of the fluid flow is being under the sub-saturated liquid. Theoretically, due to the forces of the superficial stresses, the over-pressure and thus the over-heating inside the bubbles of radius R is proportional to 1/R. When the R is nil or in order of value of inter-molecular, the over-heating is very great at which the prohibition of the bubble formation. Practically, on the wall there are germs permitting the bubbles to be created for the overheating very limited to some degrees. THERMAL-HYDRAULIC IN NUCLEAR REACTOR     Since the apparition of the bubbles, there is stabilization of the temperature at the wall which became less dependent of the heating flux. There is nucleate boiling. However the mean temperature of the se fluid remains inferior to the saturation, these bubbles after being separated from the wall, are re-condensate in the center of the fluid flow. Cladding temperature = Tsat + Δtsat With Δtsat = k Φ 0.25 e- p/p. The difference between the cladding temperature and the fluid temperature does not vary linearily with the heat flux Φ; thus allows the passing to higher flux Φ without excessive increasing of the wall temperature wich depends essentially to the pressure. Per example, at fixed pressure, the increase of 100 % of the flux Φ induces an increase of 20 % of the Δtsat. THERMAL-HYDRAULIC IN NUCLEAR REACTOR   In the zone 3, 4, 5; the mean temperature of the fluid has reach the saturation (α ˃ 0), we are in boiling phase. The precedent correlation still applicable because the wall temperature is always less dependent of the heat flux Φ. - Zone 3: This boiling is done always by the creation of the bubbles at the wall, with a flow in liquid phase containing the bubbles of vapor more or less important. Starting at point 3 in the curve, which is the critical temperature, the exchange of heat becomes less efficient. Although the heat continues to be removed, the temperature rises quickly, which risks damaging the cladding material (as well as the oxide pellets). Furthermore, this zone is unstable, with oscillations of the flow rate and the temperature. The regime in this zone is called « transition boiling » (region 3) THERMAL-HYDRAULIC IN NUCLEAR REACTOR     - Zone 4: There is some separation of liquid and vapor phase, the vapor phase is concentrated at the center mixed to the droplets of liquid, the liquid phase forming a film wetting the wall. As the thickness of the vapor film decreases the exchange at the wall is done by the direct vaporization of the film and the number of the bubbles created at the wall is decreasing. In that zone, there is essentially vaporization by nucleate boiling by stabilization of temperature at the wall slightly above the saturation temperature. - Zone 5: The liquid at the wall is disappeared, inducing the brutal degradation of the exchange coefficient. One passes by nucleate boiling by liquid film (the difference temperature at the wall has tendency to decrease). THERMAL-HYDRAULIC IN NUCLEAR REACTOR   - Zone 6: The liquid at the wall is disappeared, inducing the brutal degradation of the exchange coefficient. One passes by liquid film to the vaporization of the vapor film. There is dry-out of the wall. However, it remains liquid in gaseous core. The vapor is still very Beyond the point 4, stable film boiling is achieved, with forced convection in a single phase vapor flow. The value q’’c corresponding to the point 3 is thus a threshold value, beyond which there is a risk of dangerous high wall temperatures. This is therefore called « the Critical flux ». Theoretically this is a limiting value never to be exceeded under normal operating conditions. As we will see later, however, in PWR a rule of always remaining well below this critical value is imposed. THERMAL-HYDRAULIC IN NUCLEAR REACTOR        The dry-out and the burn-out could induce the degradation of the cladding, or the partial fusion of the nuclear fuel pellets. In the simplest two-phase flow model it is assumed that both phases are in the thermodynamic equilibrium and that they move with the same velocity. These assumptions are the basis of the Homogeneous Equilibrium Model (HEM), in which the local, channel-average void fraction is determined from the corresponding local value of the equilibrium thermodynamic quality. XI.1. Homogeneous Equilibrium Model (HEM) The HEM expression for the void fraction takes the following form: α = 0 for xe ˂ 0 α = 1/ 1 + ρg/ρf x (1 – xe)/xe) for 0 ˂ xe ˂ 1 (XI.1) α =1 for xe ˃ 1 THERMAL-HYDRAULIC IN NUCLEAR REACTOR       Where xe is the equilibrium thermodynamic quality, which is determined from the energy balance of the coolant in the heated channel. Equation (XI.1) strongly over-predicts the coolant density (that is it gives a higher value than the actual one) in the region of sub-cooled boiling, since it assumes only liquid, whereas in reality both the liquid and the vapor co-exist in that region. XI.2. Drift-flux model Once applying the Drift-flux model, the void fraction is found as: α = Jv / C0J + Uvj (XI.2) Equation (XI.3) expresses the cross-section mean void fraction α in terms of channel mean superficial velocity of gas, Jv, total superficial velocity, J, and two parameters, C0 and Uvj. THERMAL-HYDRAULIC IN NUCLEAR REACTOR   The first parameter is the so-called drift-flux distribution parameter and is simply a covariance coefficient for crosssection distributions of void fraction and total superficial velocity. The second coefficient is the so-called drift velocity and can be interpreted as cross-section-averaged difference between the gas velocity and the superficial velocity, using local the void fraction as a weighting function. The drift-flux parameters are not constant and depend on flow conditions. Table XI.1 gives expressions for drift-flux parameters, which are valid in a wide range of flow conditions. THERMAL-HYDRAULIC IN NUCLEAR REACTOR Table XI.1: Distribution parameter & drift velocity used drift flux model. THERMAL-HYDRAULIC IN NUCLEAR REACTOR XI.3. Sub-cooled boiling region  It is commonly accepted that a significant void fraction in a boiling channel appears at locations where bubbles depart from heated walls. The void fraction between that point, referred often as the Onset of Significant Void fraction (OSV) point, and the  Onset of Nucleate Boiling (ONB) point is very small and can be neglected. To establish the location of the OSV point it is recommended to use a correlation proposed by Saha and Zuber (1974), which states that OSV point is located at such position in a channel, where the local equilibrium quality is as follows: xe,OSV = -0.0022 q”.Dh.Cpf / 1fg.λf for Pe ˂ 70000 (XI.3)   xe,OSV = -154q”/G.ifg for Pe ˃ 70000 THERMAL-HYDRAULIC IN NUCLEAR REACTOR       where Pe is the Peclet number defined as follows: Pe = G.Dh.Cpf / λf (XI.4) For an uniform heat flux distribution, the location of the OSV point is given from the energy balance as: zOSV = (xe,OSV - xe)[W.ifg / q”.PH] (XI.5) Several models have been proposed to predict the flow quality downstream of the OSV point. Levy (1966) proposed a fitting relationship, which satisfy a condition at z=zNVG, where x = 0 and also which will predict the flow quality to approach the equilibrium quality when z is increasing downstream of the OSV point. The Levy’s relationship is as follows: Xα(z) = xe(zOSV).e[xe(z)/xe(zOSV) – 1] (XI.6) THERMAL-HYDRAULIC IN NUCLEAR REACTOR    Having the flow quality given by Eq. (4-57), one can apply the general drift flux model to calculate the void fraction distribution. The recommended expression for thedistribution parameter for sub-cooled boiling is as follows: C0 = β[1 + (1/β)b] (XI.7) Where  Β = 1/(1 + (ρg/ρf.(1 – xα(z)/xα(z) (XI.8)  B = (ρg/ρf)0.1    (XI.9) The recommended by Lahey and Moody (1977) drift velocity is as follows: Uvj = 2.9{σg(ρf - ρg) /ρf2)}0.25 (XI.10) THERMAL-HYDRAULIC IN NUCLEAR REACTOR          XII. Heat transfer to coolant XII.1. Single phase The heat transfer coefficient h for coolant flow in a rod bundle is calculated from the Nusselt number Nu as follows,: H = Nu.λ / Dh (XII.1) where < is the fluid thermal conductivity and Dh is the bundle hydraulic diameter. For laminar flow far from the inlet to a channel, the Nusselt number is as follows: Nu = 4.364 (XII.2) In the inlet region of the channel the following expression is valid: Nu = 1.31 (1 + 2ζ) /.ζ3/2 (0 ˂ ζ ˂ 0.4) (XII.3) THERMAL-HYDRAULIC IN NUCLEAR REACTOR  Where  ζ = (z / Dh)Pe (XII.4) Pe = UDh/a (XII.5) For turbulent flow in a pipe the Nusselt number can be calculated from the Dittus- Boelter correlation: Nu = 0.023Re0.8 Prn (XII.6)      Where Pr is the Prandtl number; Pr = /a, and is the kinematic viscosity of liquid. The formula is valid for Re > 104 and 0.7 < Pr < 100, n = 0.4 for fluid heating and n = 0.3 for fluid cooling. Petukhov [4-6] proposed the following semi-empirical expression for the Nusselt number for turbulent flow in pipes: THERMAL-HYDRAULIC IN NUCLEAR REACTOR        Nu = ((Cf,p/2)RePr) /(1 +13.6Cf,p + (11.7 +1.8Pr -1/3)(Cf,p/2)1/2 (Pr2/3 -1)) (XII.7) Where : Cf,p = 0.25(1.82lg10Re – 1.64)-2 (XII.8) For rod bundles with triangular lattice and with 1.1 < p/d < 1.8 Ushakov (presented in [4-7]) proposed the following correlation: Nu = {0.0165 + 0.02[1 – 0.91/(p/d)2](p/d)0.15}Re0.8 Pr0.4 (XII.9) where the correlation is valid for 5c103 < Re < 5c105 and 0.7 < Pr < 20. Similar correlation was derived by Weissman[4-14], Nu = CRe0.8Pr3/2 (XII.10) THERMAL-HYDRAULIC IN NUCLEAR REACTOR          Where C = 0.026(p/d) – 0.024 for triangular lattices with 1.1 < p/d < 1.5, C = 0.042(p/d) – 0.024 for square lattices with 1.1 < p/d < 1.3. Subbotin et al. recommended for heat transfer to liquids flowing in a bundle with triangular lattice the following correlation: Nu = A Re0.5 Pr0.4 (XII.11) Where: A = 0.0165 + 0.02[1 - (0.91/(p/d)2](p/d)0.15 (XII.12) The correlation is valid for 1.1 < p/d < 1.8, 1.0 < Pr < 20 and 5.103 < Re < 5.105. For gas flow in tight rod bundles Ajn and Putjkov give: THERMAL-HYDRAULIC IN NUCLEAR REACTOR         Nu /NuDB = 1.184 + 0.35.lg(p/d - 1) (XII.13) bundle The correlation is valid for 1.03 < p/d < 2.4 and DB Nu is found from the Dittus-Boelter correlation given by Eq. (XII.6). Markoczy performed a study of experimental data obtained in 63 rod bundles with different geometry details and proposed the following relationship: Nubundle/NuDB = 1 + 0.91Re -0.1 Pr0.4(1 -2e-B) (XII.14) Where B = (2x3 1/2/π)(p/d)2 for triangular lattice (XII.15) B = 4/π(p/d)2 - 1 for square lattice Here again DB Nu is found from the Dittus-Boelter correlation given by Eq. (4-67). THERMAL-HYDRAULIC IN NUCLEAR REACTOR       The correlation is applicable in the following range of parameters: 3c103 < Re < 106 0.66 < Pr < 5 1.02 < p/d < 2.5. Another approach was proposed by Osmachkin [4-4], who recommended to calculate the Nusselt number from correlations which are valid for pipes, replacing however the hydraulic diameter with the “effective”: Deff = 2ε/(1 – ε)2{ε/2 – 3/2 –lnε/1 –ε}Dh (XII.16) THERMAL-HYDRAULIC IN NUCLEAR REACTOR   where ε is the fraction of the cross-section of the bundle which is occupied by rods. The formula is applicable for rod bundles with triangular lattice and for p/d > 1.3.       XIII. Two-Phase flow Heat transfer coefficient for two-phase boiling flow can be predicted from various correlations, for example from the Jens Lottes (subcooled boiling) and the Chen (saturated boiling) correlations, described in [4-1]. A simple estimation of the boiling heat transfer coefficient can be obtained from a correlation proposed by Rasohin [48], h = 5.5p0.25(q”)2x(3)1/2 for 0.1 ˂ p ˂ 8 H = 0.577p1.33(q”)2x(3)1/2 for 8 ˂ p ˂ 20 (XIII.1) THERMAL-HYDRAULIC IN NUCLEAR REACTOR  where h is heat transfer coefficient [W/m2K], p is pressure [MPa] and q’’ is heat flux [W/m2].   XIV. Pressure drops     Calculation of pressure drops in a reactor core is important since they influence the flow distribution in sub-channels and thus affect the local thermal margins. In addition, the total pressure drop over the coolant circulation loop has to be known in order to determine the needed pumping power. XIV.1. Single-phase flows One can identify several mechanisms that will cause a pressure drop along the fuel assembly: THERMAL-HYDRAULIC IN NUCLEAR REACTOR         * Friction losses from the fuel rod bundle * Local loses from the spacer grids *. Local loses at the core inlet and exit (contraction and expansion) *. Elevation pressure drop The total pressure drop in a channel with a constant crosssection area can be calculated from the following equation: Δptotal = Δpfric + Δploc + Δpelev = {(4CfL / Dh) + Σiζi}G2/2ρ + Lρgsinφ (XIV.1) Where Cf is the Fanning friction coefficient, L is the length of the channel, G is the massflux, Dh is the channel hydraulic diameter and is the coolant density. THERMAL-HYDRAULIC IN NUCLEAR REACTOR       The friction coefficient for laminar flow can be written in a general form as, Cf = a.Re-b (XIV.2) where a and b are constants, which for the laminar flow in a pipe are equal to 16 and - 1, respectively. For laminar flow in rod bundles, Osmachkin proposed to use equations. (XIV.1) and (XIV.2)), where hydraulic diameter Dh is replaced with an “effective” diameter given as: Deff = 2ε/(1 – ε)2{ε/2 – 3/2 –lnε/1 –ε}Dh (XIV.3) where ε is the fraction of the cross-section of the bundle which is occupied by rods. The formula is applicable for rod bundles with triangular lattice and for p/d > 1.3.For turbulent flow the coefficients in equation. (XIV.2) are obtained experimentally. For flow in a rod bundle with triangular lattice (see figure VI.1) and 1.0 < p/d < 1.5, the Fanning friction coefficient can be THERMAL-HYDRAULIC IN NUCLEAR REACTOR        Cf,b = {0.96(p/d) + 0.63)Cf,p (XIV.4) where Cf,p is the friction factor proposed by Filonenko [47], which is valid for tubes and annuli for Re > 4000: Cf,p = 0.25(1.82lg10Re – 1.64)2 (XIV.5) For rod assemblies Aljoshin et al. [4-5] proposed a general correlation as follows: Cf = Ax Pw,ch /PW,r (Ach /Ar)mRe-0.25 (XIV.6) where Pw,ch and Pw,r – are the wetted perimeters of the channel and rods, respectively; Ach and Ar – are the crosssection areas of the channel and rods, respectively. The formula is valid for rod bundles with triangular lattice, for which A = 0.47, m = 0.35 and 4·103 < Re < 105, and for rectangular lattice, for which A = 0.38, m = 0.45 and 103 < Re < 5·105. THERMAL-HYDRAULIC IN NUCLEAR REACTOR         Additional pressure losses are associated with spacer grids, the coolant inlet and exit of the bundle as well as the sudden area changes of the bundle cross-section area. Such losses are classified as the local pressure losses and are calculated according to the following general expression: Δploc = ζloc (G2 /2ρ) (XIV.7) where loc x is the local pressure loss coefficient. The local loss coefficient for grid spacers is in general dependent on the spacer geometry and is usually determined in an experimental way. Typical spacer loss coefficient is expressed as: ζspacer = a + b.Re-c (XIV.8) where a, b and c a are constants determined experimentally. For sudden enlargement and contraction of the channel, the local pressure losses can be calculated according procedures described in [4-1]. THERMAL-HYDRAULIC IN NUCLEAR REACTOR       XIV.2. Two-phase flows Pressure drop in fuel assemblies with two-phase flow can be calculated according to the procedures described in [VI-1], using the hydraulic diameter as described by equations. (VI.1) and (VI.4) with some modifications appropriate to the fuel assembly design. As shown in [4-1], the total two-phase flow pressure drop in a channel with a constant cross-section area can be calculated as: -Δp = r3Cf,loc ((4L/D)(G2/2ρl) + r4Lρlgsinφ + r2(G2/ρl) + {Σi=1 N Ф2loc,i ζi} (XIV.9) where r2, r3 and r4 are two-phase pressure drop multipliers (acceleration, friction and gravitation, respectively) and 2 lo,i f , ix are local loss multiplier and local pressure loss coefficient, respectively, at location i in the channel. THERMAL-HYDRAULIC IN NUCLEAR REACTOR  XV. Critical Heat Flux (CHF)     The conditions at which the wall temperature rises and the heat transfer decreases sharply due to a change in the heat transfer mechanism are termed as the Flux (CHF) conditions. The nature of CHF, and thus the change of heat transfer mechanism, varies with the enthalpy of the flow. At sub-cooled conditions and low qualities this transition corresponds to a change in boiling mechanism from nucleate to film boiling. For this reason the CHF condition for these circumstances is usually referred to as the Departure from Nucleate Boiling (DNB). THERMAL-HYDRAULIC IN NUCLEAR REACTOR  At saturated conditions, with moderate and high qualities, the flow pattern is almost invariably in an annular configuration. In these conditions the change of the heat transfer mechanism is associated with the evaporation and disappearance of the liquid film and the transition mechanism is termed as dry-out. Once dry-out occurs, the flow pattern changes to the liquid-deficient region, with a mixture of vapor and entrained droplets. It is worth noting that due to high vapor velocity the heat transport from heated wall to vapor and droplets is quite efficient, and the associated increase of wall temperature is not as dramatic as in the case of DNB. THERMAL-HYDRAULIC IN NUCLEAR REACTOR   The mechanisms responsible for the occurrence of CHF (DNB- and dry-outtype) are not fully understood, even though a lot of effort has been devoted to this topic. Since no consistent theory of CHF is available, the predictions of CHF occurrence relay on correlations obtained from specific experimental data. LWR fuel vendors perform their own measurements of CHF in full-scale mock-ups of fuel assemblies. Based on the measured data, proprietary CHF correlations are developed. As a rule, such correlations are limited to the same geometry and the same working conditions as used in experiments. Most research on CHF published in the open literature has been performed for upward flow boiling of water in uniformly heated tubes. The overall experimental effort in obtaining CHF data is enormous. It is estimated that several hundred thousand CHF data points have been obtained in different labs around the world. THERMAL-HYDRAULIC IN NUCLEAR REACTOR        More than 200 correlations have been developed in order to correlate the data. Discussion of all such correlations is not possible; however, some examples will be described in this section. XV.1. Departure from Nucleate Boiling (DNB) The usual form of a DNB correlation is as follows: q”critical = q”critical (G,p,Dh, L, …) (XV.1) Which means that the main parameters that influence the occurrence of DNB are mass flux, G, pressure, p, as well as the hydraulic diameter, Dh and length L of the heated channel. For upflow boiling of water in vertical 8-mm tubes with constant heat flux, Levitan and Lantsman recommended the following correlation for DNB: THERMAL-HYDRAULIC IN NUCLEAR REACTOR       q”critical = [10.3 – 7.8(p/98) + 1.6(p/98)2](G/1000)1.2{[0.25(p – 98] – xe}e-1.5xe (XV.2) where cr q¢¢ is the critical heat flux [MW m-2], p is the pressure in [bar], G is the mass flux in [kg m-2 s-1]. The correlation is valid in ranges 29.4 < p < 196 [bar] and 750 < G < 5000 [kg m-2 s-1] and is accurate to ±15%. The correlation can be applied to channels with arbitrary diameters if the following correction factor is applied: Q”critical = q”critical (8mm) x (8 / D)0.5 (XV.3) where D is the tube diameter in [mm] and cr mm q¢¢ is the critical heat flux obtained from Eq. (XV.2). There are several semi-empirical correlation used by reactor core designers such as: Westingouse (W3 & WRB1); GE; Babcock & Wilcock; CE,ect.. THERMAL-HYDRAULIC IN NUCLEAR REACTOR    XV.2. Dry-out The usual form of a “dryout" correlation is as given: xcritical = xcritical (G,p,Dh, L, …) (XV.4)      which means that the main parameters that influence the occurrence of dryout are mass flux, G, pressure, p, hydraulic diameter Dh, boiling length (that it the distance from the beginning of saturated flow to the dryout point), LB, and possibly other. For dryout predictios in 8-mm pipes Levitan and Lantsman recommended the following expression: xcritical (8mm) = [0.39 + 1.57(p/98) – 2.04(p/98)2 + 0.68(p/98)3](G/1000)-0.5 (XV.5) where xcr is the critical quality, p is the pressure in [bar] and G is the mass flux in [kg m-2 s-1]. The application region of the correlation is 9.8 < p < 166.6 [bar] and 750 < G < 3000 [kg m-2 s-1] and the accuracy of xcr is ±0.05. THERMAL-HYDRAULIC IN NUCLEAR REACTOR          The critical quality given by Eq. (4-114) can be used for other tube diameters with the following correction factor: xcritical = xcritical(8mm) x (8/D)0.15 (XV.6) Here cr mm x 8 is the critical quality obtained from Eq. (4-114) and D is the tube diameter in [mm]. For fuel rod bundles the following correlation was proposed by General Electric: xcritical = ((A x L*B)/B + L*B)(1.24/Rf) (XV.7) Where L*B :LB/0.0254 (LB is boiling length I meter) Rf: radial peaking factor A = 1.055 - 0.013{pR -600/400)2 - 1.233GR + 0.907GR2- 0.285GR3  B = 17.98 +78.873GR – 35.464GR2  GR = G/1356.23 (G mass flux in (kgm-2s-1)  p = p)/6894.757 (p: pressure in (Pa) THERMAL-HYDRAULIC IN NUCLEAR REACTOR     The correlation is valid for 7x7 bundles. It can also be applied to 8x8 bundles once replacing B with B/1.12. XV.3. Protection against boiling crisis The thermal-hydraulic design is such that the probability of nonappearance of a boiling crisis during the normal operation, during the normal transients and during all transient conditions resulting anomalies with moderated frequency (events classes 1 & 2), is at least equal to 95% with a confidence level of 95%. By preventing the boiling crisis, one assures sufficient transfer of heat between the fuel rod cladding to the primary coolant fluid, thus assures the integrity of the fuel cladding. The maximal temperature of the cladding could not be constituted as a criterion because it is situated at some degrees under the coolant fluid temperature under nucleate boiling. The limits assured by the control systems, nuclear limitations and protections are such that criterion will be respect for the transients associated to events of class 2. THERMAL-HYDRAULIC IN NUCLEAR REACTOR    There exists a supplementary margin to the DNBR in case of operating at nominal power and during the normal transients. The utilization of a simplified calculation algorithm of the DNBR in the protection system and in the surveillance system allow the respect of the design criteria by defining an automatic shutdown of the reactor core at low level DNBR (DNBR(shutdown)) and a limit condition of operation (DNBR(lco)) associated to the DNBR. The on-line calculation resulted on elaborated by systems which utilizes measurements to reconstitute the local conditions by mean of an algorithm and to apply a select experimental correlation of CHF to determine the DNBR. THERMAL-HYDRAULIC IN NUCLEAR REACTOR  A. Fuel temperature  The thermal-hydraulic design is such that, in operation conditions associated to events classes 1 and 2, there is a minimal probability of 95% with confidence level of 95%, fuel rods having maximal power density (W/cm) does not exceed the melting point of the nuclear fuel. The melting point admitted for UO2 is 2800°C for a non-irradiated fuel element, the melting point admitted for MOX fuel is 2737°C for nonirradiated fuel element. These values decrease with the burn-up (-32°C/ 10000 MWd/tHM). By preventing the fusion of the fuel, one preserves the geometry of the later and eventual unfavourable effects of the fuel melting to the cladding are eliminated.   THERMAL-HYDRAULIC IN NUCLEAR REACTOR    B. Reactor core coolant mass flow rate A minimal value of 94.5% of thermal-hydraulic design of the primary circuit flows across different fuel elements and constituted effectively the cooling of the fuel rods. The coolant fluid pass through the guide tubes and the coolant mass rate leakage pass through reactor core baffles, are not be considered as efficiency for the heat evacuation. The thermal-hydraulic studies utilize the thermal-hydraulic rate (minimal rate) in the inlet of the reactor pressure vessel. In the hot condition of the upper-plenum, a maximal of 5.5% for that value is allocated to the by-pass flow. This included the cooling mass flow rate of the control clusters system, the cooling mass flow rate of the upperplenum, the leakages between baffles and leakages to the outlet of the pressure-vessel. THERMAL-HYDRAULIC IN NUCLEAR REACTOR        C. Hydro-dynamic stability of the reactor core The operating modes associated to events of classes 1 and 2 do not induce hydro-dynamic instability of the reactor core. D.Technology of the DNBR or Critical Heat Flux ratio and Mixing effect The minimal DNBR in the hot channel is located at the down-stream at point of maximal thermal flux (hot point) due to the elevation of the enthalpy at down-stream of that point. D.1.Technology of the DNBR a) Critical Heat Flux (CHF) – Correlation The earlier CHF tests have been performed with a fluid pass through simple heated tubes and in annular configurations with one or two heated walls. The results obtained from such tests have been analyzed and correlated to different physical models to describe the CHF phenomenon. The correlations obtained are consequently, by nature, strongly empirical. THERMAL-HYDRAULIC IN NUCLEAR REACTOR    The development of testing methods has conduct to the utilization of rod bundle instead of isolated channels, it is known that the mean rates of the rods bundle could not be utilised in the CHF correlations. The results obtained from the tests have shown that the CHF correlations could not be based on the mean conditions. Thus, it is necessary to known the local conditions in the sub-channels. To determine the local conditions in the sub-channels, many thermalhydraulic computer codes have been developed. In these computer codes, a rods bundle is considered as a sub-channels mesh, each of them has have the passing surface the cross-section delimited by four adjacent fuel rods. The sub-channels are divided in axial meshes which defined the reference volumes. The local conditions of the fluid in each reference volume are calculated by resolving simultaneously the mass equation and energy equations and the quantity of movement. The predicted CHF is elaborated by utilizing the local conditions of the fluid in the sub-channels calculated by the computer code and the correlation. THERMAL-HYDRAULIC IN NUCLEAR REACTOR    Utilization of data obtained from CHF tests The experimental tests are based on results obtained from fuel assembly bundle. The tests have been performed under following conditions:  Axial distribution of uniform flux;  Axial distribution of non-uniform flux;  On typical cells;  On guide tubes cells; The tests performed under following parameters:  Pressure (20.7< p < 170.6 bar);  Mass rate (980 < G < 4790 kg/m2/s);  Quality (-0.22 < X < 0.44 ) THERMAL-HYDRAULIC IN NUCLEAR REACTOR       Those limits are representatives of the operating conditions of the PWR. C) The form of the correlation for axial uniform thermal flux The CHF correlation is given under analytic form and function of:  Thermal-hydraulic variables: pressure (p), mass flow rate (G) and quality (X);  Fuel assembly geometry (grid spacers distances and form);  Cell types (typical or guide tube) The principal term of correlation of uniform flux does not depend on the fuel assembly geometry. It depends only on the thermal-hydraulic variables. It is supposed to depend to the linear variable X and expressed as following: Ф (CHF) = Ạ(p,G) – B(p,G)*X The other terms associated to the fuel assembly depend on geometrical effects as following: THERMAL-HYDRAULIC IN NUCLEAR REACTOR   Distance between grid spacers; Distance between the location of the CHF and the location of the grid spacer at up-stream.      The CHF correlation is represented by: CHF(p, G, X, d(g), g(sp), r(g)), with: CHF = ạ(p,G,X,dg) + c(p,G,X, g(sp)) + d(p,X,g(sp),r(g) d) Form of the correlation for axial non-uniform thermal flux The values of the CHF measured in the rods bundle with an axial distribution non-uniform of the thermal flux are lowers than the ones obtained with uniform distributions. The application of the CHF correlation has shown that the predicted flux is higher than the measured flux. The predicted value must be reset-up. To this effect, one applied the correction factor of non-uniform flux of L.S.Tong. The reset flux is expressed by following: THERMAL-HYDRAULIC IN NUCLEAR REACTOR             Ф = Фu/ F(nu) Where : - Ф is the reset value of the flux: - Фu is the predicted value of the flux for the axial distribution of the uniform flux. - F(nu) is the non-uniform factor. e) Definition of the DNBR The DNBR, in typical cell and in tube guide cell with cold wall is defined as: DNBR = q”(CHF,N)/ q”(local) Where : q”(local) is the real thermal flux q”(CHF,N) = q”(CHF(u)/ F q”(CHF,u) is the thermal critical uniform flux which is predicted by the correlation CHF. F is Tong’s form factor for the non-uniform flux distribution. THERMAL-HYDRAULIC IN NUCLEAR REACTOR     f) Mixing effect between sub-channels In a rods bundle, the channels formed by four adjacent fuel rods are interconnected together by intermediary space between two neighbouring fuel rods. There is cross-flow between the channels due to their difference in pressure between them. The effects of the cross decrease the enthalpy elevation in the hot channel. In the energy balance equation of the computer code, a term permitting the creation the turbulence enthalpy exchange model between the neighbouring channels is taking into consideration. It is proportional to the enthalpy difference between the channels. In the proportional factor which is defined as the turbulence exchange coefficient. The value of this coefficient is determined after series of specific tests performed with the corresponded spacer grids. THERMAL-HYDRAULIC IN NUCLEAR REACTOR      g) Uncertanties relative to the manufacturing parameters These uncertainties are taking into account the manufacturing variations in term of materials, fuel rods and assembly geometries. There are two types of manufacturing uncertainties:  Effect of the eccentricity of the fuel pellets and the ovality of the fuel rod cladding on the CHF;  Effect on the manufacturing tolerances of the spacer grids on the CHF. h) Effect of the eccentricity of the fuel pellets and the ovality of the cladding on the CHF. Some fuel pellets could be eccentred with regard to the cladding in the beginning of life. The cladding could be ovalized with the time. In these cases, there is a variation of the axial flux at a small axial distance. THERMAL-HYDRAULIC IN NUCLEAR REACTOR     In the case of eccentric pellets, the local flux peak, under a given angle, will axially extend to a distance corresponding to maximal length of some pellets due to the random contact between the pellets and the cladding at the beginning of life. This random character of the contact point is induced by the variations of the format pellet extremities and by their variations of diameters. In the case of the ovality of the cladding, the local peak of the thermal flux, under a given angle, will axially extend to a distance corresponding to maximal length of some pellets, due to the random axial distribution of chips of the fissured fuel pellets. The uncertainties relatives to the hot channel are taking into account of the non-perfect of geometries and materials of fuel rod and fuel assembly. One distinguishes the following uncertainties concerning the hot channel: THERMAL-HYDRAULIC IN NUCLEAR REACTOR   Technological factor of hot point (F(q,E): this incertitude is utilized to evaluate the maximal local power peak (the hot point) and it is determined by the statistic combination of tolerances relative to the diameter, the density and the enrichment level of the fuel pellet. However, the CHF tests with the local peaks thermal flux have shown it is not necessary to take into account a particular incertitude concerning the local flux. Nuclear factor of enthalpy elevation of the hot channel (FΔH, E): It is determined by statistic combination of the effects on the enthalpy elevation of manufacturing tolerances relative to the nuclear fuel density and enrichment and to the position of the fuel rods in the reactor core. THERMAL-HYDRAULIC IN NUCLEAR REACTOR      i) Effect of the manufacturing tolerances of the spacer grids on CHF This uncertainty represents directly the effect of manufacturing tolerances of the spacer grids on the CHF and, more precisely, effect of tolerances relative to singular pressure-drops of the spacer grids on the redistribution the coolant mass flow rate in the reactor core. For the fuel assembly with the same design, the effect on the dispersion of the values of CHF is negligible. j) Effect of the fuel rod bowing in the reactor on the CHF The CHF could be influenced by the bowing phenomenon of the fuel rods which has been detected on irradiated examinations. This phenomenon consists to a displacement of the fuel rod with regard to it nominal position in the channel. It depends strongly to the nuclear fuel rods. THERMAL-HYDRAULIC IN NUCLEAR REACTOR     The modification of the flow rate due to the fuel rods bowing induces a decrease of the CHF. The penalty which is resulted is quantified by two unfolding models: An envelope law defining the bowing order of the fuel rods, based on the measurements of the bowing of the fuel rods in irradiated fuel assembly. A law defining the penalty to the CHF versus the closing of the channel cross-section. The penalty law utilised is the one approved by the NRC in 1979. It consists to differentiate the operating with nominal mass flow rate to the operating with reduce flowing rate. The resulting model gives the penalty versus the burn-up of the fuel assembly. The experimental results obtained have shown that the bowing penalty is nil under burn-up of 16000 MWday/tHM. Upper that burn-up limit, the penalty increase linearly, but could be limited if the burn-up of the fuel rods increases. THERMAL-HYDRAULIC IN NUCLEAR REACTOR    The anterior analyses have shown that beyond the burn-up of 35000 MWd/tHM, the fuel rods do not present maximal value of FΔH. k) Correlation between the heat transfer and the void coefficient to the radial distribution of the nuclear power The flowing model is based on the double-phase flowing by taking into account the thermal unbalance of the liquid phase and the rate differences of the liquid and vapour phases. This model is obtained from the mass equation, the quantity of movement and the energy balance for the double-phase flowing. The equation of the enthalpy balance permits to the calculations of the local boiling. These equations necessitated a physical model to describe the phase interactions, the turbulence mixing and interactions between fluid and wall. Models using in these equations are: THERMAL-HYDRAULIC IN NUCLEAR REACTOR      Friction model at the walls: Heat transfer model; Slip model to take into account the mass flow rate different rates for the liquid and vapour phases: Turbulent viscosity and diffusivity coefficients which are calculated with an algebraic model permitting the description of the mixing effect. To effectuate the calculations of the thermal-hydraulic design of the reactor core of the PWR or precisely to calculate the necessary local properties of the fluid for to predict the CHF margins, the computer codes (THINC-IV, FLICA, etc.) with their proper models of heat transfer are utilized by the designer THERMAL-HYDRAULIC IN NUCLEAR REACTOR     E. Hydro-dynamic instability The boiling flows could be subjected to the thermal-hydrodynamic instabilities. Those instabilities are not acceptable in the reactor core because they could induce a modification of thermalhydraulic conditions which are leading to a reduction of the CHF with regard to the one observed in the permanent flow conditions or to the undesirable vibrations of reactor components. Consequently, a thermal-hydraulic criterion has been established, to guarantee the operating modes in case of events classes 1 and 2 do not induce thermal-hydro-dynamic instabilities. Two types of specific instabilities have been taking into account for the operation of the PWR. It is the permanent instability Ledinegg (rate oscillation) and dynamic instability with density of wave. Ledinegg’s instability implies a sudden variation of the rate flow of a permanent flow to another. THERMAL-HYDRAULIC IN NUCLEAR REACTOR   This instability appears when the slope of the curve loss “pressuremass flow rate” (δP/δG internal) of the primary circuit become lower than of the curve loss “pressure- flow rate” (δP/δG external) of feedwater loop (load high). The stability criterion is: δP/δG internal) > (δP/δG external). The mechanism of mass flow rate oscillation in a heated channel could be described as following: Briefly, an inlet mass flow rate fluctuation produces a perturbation to the enthalpy. Thus perturbed along the length and the loss-pressure of the single-phase zone and induces perturbations of the quality or void coefficient in the doublephase zone which rise up the channel with the mass flow. The perturbations of the quality and the double-phase create the perturbations of the loss-pressure of the double-phase. THERMAL-HYDRAULIC IN NUCLEAR REACTOR    However, as the total loss-pressure of the reactor core is maintained by the characteristics of the external fluid circuit of the reactor core, thus results the perturbation of the loss-pressure of the double-phase rise up to the single-phase zone. The induced perturbations could be attenuated or self-generated. The cooling mass flow rate passes through the adjustable inlet of the pressure-vessel. It goes down to the down-comer via the annular space formed by the pressure-vessel and an envelope of the reactor core and then goes up to upper- plenum of the reactor core. It gets out the pressure-vessel via adjustable pressure-vessel outlet. There are many by-pass ways: • Cooling mass flow rate passes through the upper-plate, it constitutes of water in annular down stream; the mass flow rate is then directed from the pressure-vessel dome to the upper-plate. In configuration “hot dome”, which is the retained design option, THERMAL-HYDRAULIC IN NUCLEAR REACTOR      * This mass flow rate is directed down to some guide tubes, in normal operation. In other guide tubes there exist an up-stream circulation. * The mass flow rate gets in the guide tubes of control clusters to cool down the control rods, the burnable poison (if utilized) or instrumentation sources. * Leakage mass flow rate directly from the adjustable inlet to adjustable outlet of the pressure-vessel via annular space between the pressure-vessel and core envelope. * Mass flow rate passing between the hard reflector and core envelope and inside the hard reflector to cool down these components and is not considered as available to refrigerate the reactor core. * Mass flow rate flows between the peripheral fuel assemblies and the adjacent hard reflector. THERMAL-HYDRAULIC IN NUCLEAR REACTOR    The maximal value of the by-pass fluid is 5.5% of the total rate. On the total, 2.2% is associated to the reactor core, the rest is associated to internal components. The calculations have been effectuated by utilizing of tolerances in the direction the most penalizing and by taking into account uncertainties on pressure drops. F. Defect of distribution of the rate at pressure-vessel inlet Generally, the distribution of the inlet mass flow rate is non-uniform. Studies with thermal-hydraulic computer codes by decreasing the mass flow rate of the fluid in a limited zone inlet of the reactor core have shown that a rapid readjustment of one third of reactor core high has been observed, thus the defect on inlet mass flow rate distribution, in practice, has a negligible effect on the CHF of the hot channel. That redistribution of the mass flow rate is due to the readjustment of the fluid rate. Consequently, the defect of the distribution of inlet mass flow rate induces no penalties on the CHF and its location; none of penalty has been taking into account in the design. THERMAL-HYDRAULIC IN NUCLEAR REACTOR      G. Pressure drops in the pressure-vessel Pressure drops are due to the friction on walls and to changes of geometry of walls guiding the fluid. One supposes that the mass flow rate is in single-phase, turbulence and the fluid is incompressible. These hypotheses apply to calculations of pressure drops in the reactor core and the pressure-vessel performed to evaluate the losspressure in the pressure-vessel because the mean void coefficient of the reactor core is negligible. The character double-phase of the mass flow rate has been taking into account in the thermal-hydraulic analyses of sub-channels. Due to the complexity of the geometry of the reactor core, one could not get the precise analytic values of the form and friction coefficients. The experimental values have been obtained on similar geometrical model The reactor core pressure drops have been determined during the the thermal-hydraulic tests on the fuel assembly. THERMAL-HYDRAULIC IN NUCLEAR REACTOR       These tests have been performed in a loop test within large range of Reynolds numbers, including the ones observed in the PWRs reactor core. The pressure-vessel pressure drops are obtained by combination of reactor core loss pressure observed on hydraulic moke-up tests on reduced scale of pressure-vessel and correlated to the pressure drops models. Measurements of primary circuit rate flow have been performed at the start-up tests of the NPP to verify the design flow rate. H. Hydraulic forces The maximal hydraulic forces exercised on the internal components of the pressure-vessel are reached for the nominal mass flow rate. In nominal operating condition, the hydraulic forces are determined with the mechanical design mass flow rate, by the taking into account the minimal value of the by-pass mass flow in the reactor core. THERMAL-HYDRAULIC IN NUCLEAR REACTOR      At cold-shutdown, the hydraulic forces are determined with the same mass flow rates (pressure-vessel and core by-pass) but by taking into account the difference of density of the cooling water. Thus is corresponded to the enveloped value for normal operating condition. The transient conditions such as over-speed of primary pumps, capable to product a mass flow rate of 20% superior to the mechanical design mass flow rate are utilized to determine the envelope of hydraulic forces in the transient conditions. The hydraulic tests have been performed to verify the hydraulic loads during the over-speed of the primary pumps to the calculated mechanical mass flow rate at hot and cold conditions. I. Hydraulic Dimensioning of the internal components The dimensioning of the internal components is related to the specific design characteristics of the reactor core and the pressure-vessel (structural components, guide tubes, baffles, reflectors, etc.). THERMAL-HYDRAULIC IN NUCLEAR REACTOR     Tests have been performed to determine the inlet and outlet mass flow rate in the pressure-vessel. J. Thermal effects during the normal transients. The safety limits of the reactor with regard to DNBR are defined in function of the cooling water temperature, of the pressure, of the reactor core power and of the axial and radial distribution of the power. An operation conditions superior to these limits must guarantee that the DNBR criteria are respected. Preventions have been taken at operation by adopting the protection chain at “low DNBR threshold” and by setting the automatic shutdown of the reactor at “low DNBR threshold”. Thus assure sufficient in the same time the protection for the steady state and incidental transients which are sufficiently low with regard to the delay of fluid transport in the primary circuit. For the rapid transients and transients at hot condition with power equal zero, specific protection functions have been provided. THERMAL-HYDRAULIC IN NUCLEAR REACTOR       K. Uncertainties K.1/ Critical Heat Flux Ratio (DNBR) Uncertainties treatment in the calculations One utilizes a statistical approach to combine the uncertainties affecting the DNBR. The uncertainties representing a random character and a well probability law are treated with statistic methods, the others are treated with the deterministic methods. This approach is utilizes to guarantee the respect of the DNBR criteria for all transients excepted the transient of uncontrolled-withdrawal of control rod cluster in the case of reactor core un-critical or at low power and for transient of steamline break on which the uncertainties are combined under deterministic manner. THERMAL-HYDRAULIC IN NUCLEAR REACTOR        b) Statistical approach To establish the relation between the uncertainties affecting the DNBR and the variation of DNBR, one utilizes a variable defined by the following equation: Y = DNBR(r) /DNBR(c) Where DNBR(r) is the real DNBR and DNBR(c) is the calculated value, determined by taking into account all the parameters associated to the calculation of the DNBR to their most probable value. DNBR(c) is the calculated DNBR in operating by an algorithm set in place on the I&C. Prob (DNBR(r) > T) = 95% with a confidence level of 95% and is equivalent to: Prob(DNBR(c) x Y > T) = 95% with a confidence level of 95%. If m and σ are the mean value and standard deviation of the distribution of probability for the random variable Y, the Prob(DNBR(c) x Y > T) = 95% with a confidence level of 95% is guaranteed if DNBR(c) THERMAL-HYDRAULIC IN NUCLEAR REACTOR            DNBR(r) is a random variable could be decomposed in random variable products as following: DNBR = Ф(rc)/Ф(cp) x Ф(cp)/Ф(LDC) x Ф(LDC)/Ф(rl) x P Where: Ф(rc) is the real CHF Ф(cp) is the predicted CHF determined by the CHF correlation Ф(LDC) is the local CHF calculated by the computer code Ф(rl) is the local real CHF in the same thermal-hydraulic conditions P is the penalty factor Ф(cp)/Ф(LDC) is DNBR(DC) is the DNBR caculated by the computer code DNBR(DC) is a random variable which is function of variavles of the system (temperature, local power, etc.). DNBR(DC) could be decomposed as following: DNBR(DC) = DNBR(DC)/ DNBR(DCO) x DNBR(DCO)/ DNBR(AO) x THERMAL-HYDRAULIC IN NUCLEAR REACTOR        Where: DNBR(DCO) is the calculated DNBR by the computer code to the most probable values. DNBR(AO) is the one-line calculated DNBR by the algoritm in the I&C to the same most probable values. Consequently: Y = DNBR(r)/DNBR(AO) = Ф(rc)/ Ф(cp) x DNBR(DC)/DNBR(DCO) xDNBR(DCO)/DNBR(AO)/ Ф(LDC)/ Ф(rl) x P Y is the product of P factor with the following variables: Ф(cp)/ Ф(LDC) : Distribution of probabilities of that variable, is provided by the correlation of CHF. It is a normal distribution characterized by a mean value m(c) and a standard deviation σ(c). DNBR(DC)/DNBR(DCO): This random variable is function of independent random variables: THERMAL-HYDRAULIC IN NUCLEAR REACTOR   • Operating parameters of NPP and measured on site (temperature, reactor pressure, local power, relative measured primary mass flow rate); • Parameters which are not detected but influencing the DNBR (uncertainties related to pellets enrichment, to diameter and to the dishing) The distribution of probabilities of that variable representative of the global uncertainties of the system is characterized by a value m(s) and a standard deviation (σ(s)). DNBR(DCO)/DNBR(CAO) : This random variable is taking into account the uncertainty of the computer code. The distribution of probabilities is characterized by two parameters: m(DC) and σ(DC). * A supplementary uncertainty must be taking into account: transient uncertainty with regard to the steady state. THERMAL-HYDRAULIC IN NUCLEAR REACTOR    This uncertainty gives all the discordance introduced by the utilization of the local properties of the fluid resulting analyses of accidental transients for the determination of the DNBR in steady state. It is independent of uncertainties mentioned above. The parameters characterizing the distribution of probabilities are: m(tss) and σ(tss). The P factor corresponds to the all uncertainties which are treated under deterministic way: • Absolute total mass flow rate; • Core by-pass mass flow rate; • Fuel rods bowing effect ; • Neutron data; • Set-point limit of automatic shutdown of the reactor, etc. THERMAL-HYDRAULIC IN NUCLEAR REACTOR      As all variables mentioned above (different measurements, I&C algorithm, DNBR correlation, computer code, manufacturing uncertainties) are independent and as the perturbations with regard to the mean values are small, the coefficient taken into account the distribution of uncertainty associated to the DNBR could be calculated as follow: V(r)expo2 = (σ(Y)/m(Y))expo2 = (σ(c)/m(c))expo2 + (σ(s)/m(s))expo2 + (σ(a)/m(a))expo2 (σ(tss)/m(tss))expo2 +(σ(DC)/m(DC))expo2 All the terms of the above equation are determined separately excepted σ(s)/m(s) is determined by Monte Carlo method. On the other hand, the probability distribution function of Y is close to the normal distribution with: Mean value: m(Y) = m(c) x m(s) x m(a) x m(tss) x m(DC) x P THERMAL-HYDRAULIC IN NUCLEAR REACTOR      In consequence, the probability that the DNBR is superior to the threshold T is 95% with the confidence of 95% if the DNBR is superior to the threshold of theoretical DNBR(th) defined as following: DNBR(th) = T/m(Y)(1-1,645V(Y) c) Deterministic approach All the uncertainties mentioned above are treated by deterministic way. As the simplified calculation of the DNBR on site is utilized to protect the reactor against the low DNBR for the concerned transients by deterministic approach, even for the action of the protection system or by utilizing the surveillance of limit operating conditions (LCO) of the DNBR, each parameter has effect on the DNBR must be controlled by a specific limit operating condition (LCO) and, per consequence, its uncertainty must be taking into consideration. Those parameters are: THERMAL-HYDRAULIC IN NUCLEAR REACTOR • • •     Mean temperature of the primary circuit; The reactor pressure; The local power. About the the power distribution, the analyses of the transients will be effectuated by utilizing the one most unfavourable. d) Uncertainties relative to computer code and mixing coefficient The results obtained from a sensibility study with the computer code has shown that the minimum DNBR in the hot channel is relatively less sensible to the variations of axial power distribution for the hole reactor core. (for the same value of FΔH(N). Studies have been performed to determine the sensibility of minimum DNBR in the hot channel at axial and radial meshes, to the inlet mass flow rate, to the loss-pressure, to the power distribution, to the mixing coefficient and to the void model. THERMAL-HYDRAULIC IN NUCLEAR REACTOR     Results obtained have shown the minimum DNBR in the hot channel is sensible with the three of them: mixing coefficients, double-phase model and the radial distribution of press-drop coefficients of the grid spacers. e) Justification of the statistic combination of the uncertainties. As explained above, one utilizes a statistic approach to combine the following uncertainties which have effect of the DNBR: • Uncertainty related to the CHF correlation (m(c), σ(c)); • Uncertainty related to the complete system (m(s), σ(s)); • Uncertainty related to the algorithm (m(a), σ(a)); • Uncertainty related to the computer code (m(DC), σ(DC)); • Uncertainty of transient regime in function of the steady state (m(tss), σ(tss). The independent parameters on which the uncertainty presents a random character and a well know probability law are treated by the deterministic method. THERMAL-HYDRAULIC IN NUCLEAR REACTOR     f) Uncertainties relative to the CHF correlation The evaluation of the characteristics of the CHF correlation on the basis of the comparison of the results obtained from the tests led to the definition of the distribution of probabilities of the measured CHF to the predicted CHF. The latter presents a normal distribution. g) Uncertainty relative to the whole system Two principal uncertainties are defined that each of them could be divided in several uncertainties:  • Uncertainties related to the physical measured parameters in operation: The following operating parameters of the reactor core are used to calculate the DNBR: the inlet temperature, the pressure of the pressurizer, the relative measured primary mass flow and the local power. THERMAL-HYDRAULIC IN NUCLEAR REACTOR   The inlet temperature is obtained by the pyrometric gage at the cold leg, the pressure of the pressurizer is obtained by the primary pressure gage, the primary mass flow obtained by the mass flow gage at the primary pumps and the power distribution of the hot channel is obtained directly from the in-core measurement of self-powered detectors. Each measurement is independent to each others. An uncertainty, for example, on the pyrometric due to the scaling does not have any relation with the pressure gauge of the pressurizer neither to the mass flow rate gage of the primary pumps. Between the gage and the utilized signal in the protection system, certain dispositive are intercalated (for example for the temperature: convector ohms-amps, convector amps-volts, isolation module if necessary and convector analogue/numerical), each of these dispositive has independent uncertainty and random, it is treated statistically. THERMAL-HYDRAULIC IN NUCLEAR REACTOR     The power distribution of the hot channel induces uncertainties on the accuracy of the aeroball measurement system (taking into account the accuracy of the activation rate, the reconstruction of the relative density of power, the discretization of the burn-up and the number of instrumented fuel assembly) and on the accuracy of signals obtained from self-powered detectors (derivation, provision relative to the burnable poisons). The total uncertainty could be divided in several distributions of probabilities (uncertainties relative to the gage, to the scaling of gagetransmitters, etc.). The resulting distribution of probabilities of such a great number of random variables is a normal distribution, as generally observed at the measurement uncertainties. * Uncertainties relative to the manufacturing tolerances The FΔH(E) taking into account the manufacturing variables which affect the thermal power along the channel.* THERMAL-HYDRAULIC IN NUCLEAR REACTOR      The FΔH(E) taking into account the manufacturing variables which affect the thermal power along the channel. Those variables are the diameter, the density and the enrichment rate of U235 of the pellets. The uncertainties relatives to those variables are determined by sampling measurements on the fabrication. The resulted uncertainty is independent of uncertainties notified, and it is a normal distribution. * Uncertainty relative to the algorithm This uncertainty taking into account the difference between the calculations obtained from the design computer code and the calculations obtained from the algorithm of the DNBR implemented on the site in the same thermal-hydraulic conditions. The algorithm is adjusted to the calculations of the design computer code. A statistical analyse allows to determine the distribution of probabilities differences between the algorithm and the computer code. It is a normal distribution. THERMAL-HYDRAULIC IN NUCLEAR REACTOR     * Uncertainty relative to the design computer code Uncertainty relative to the design computer code includes all the effects of the complete core analysis by means a numerical computer code. As the analyse of thermal flux tests are conducted to define the correlation characteristics is performed by the design computer code, the whole predicted thermal flux for each serial of experimental data includes the uncertainty relative to the calculation code and, consequently, also the parameters * Uncertainty of the transient conditions versus the steady state conditions This uncertainty taking into account all of discordance introduced by the calculation of the DNBR in steady state condition by utilizing local properties of the fluid obtained from the transient conditions; It is independent of the upper notified uncertainties. It is a normal distribution. THERMAL-HYDRAULIC IN NUCLEAR REACTOR    L. Temperatures of the fuel pellet and the cladding The temperature of the fuel pellet depends on the thickness of the corrosion layer of zirconium dioxide (ZrO2) on the cladding, on the pellet-cladding gap and the pellets conductance. The uncertainties relative to the calculations of the fuel temperature are essentially two types: • Manufacturing uncertainties (dimensional variations of pellets and cladding); • Uncertainties relative to the density and to pellet models (conductivity and gap conductance variations). These uncertainties have quantified by comparison of thermal model to measurements performed in the reactor core, and by results obtained from nuclear THERMAL-HYDRAULIC IN NUCLEAR REACTOR    fuel and cladding during the fabrication. The obtained uncertainties are then utilized in all the evaluations where intervened the fuel temperature. Other than the uncertainty relative to the temperature mentioned above, uncertainty of measurement during the determination of local power and the effect of the density variation and the enrichment rate on the local power are taking into account to establish the thermal flux factor of the hot channel. Uncertainty affecting the determination of temperature of the cladding results the uncertainty relative to thickness of the zirconium dioxide layer. Due to the excellent heat transfer between the surface of cladding and the cooling water, the temperature decrease on the cladding does not give an appreciable contribution to that uncertainty. THERMAL-HYDRAULIC IN NUCLEAR REACTOR      M. Hydraulic a) Uncertainties related to the pressure-drops in the reactor core and in the pressure-vessel b) The pressure-drops are calculated on the basis of the most probable mass flow rate. The attached uncertainties concern in the same time the test results and the extrapolation of these values to the reactor. The pressure-drops in the reactor and in the pressure-vessel have essential utilization in the determination of the primary mass flow rate. More than that, the tests will be performed on the primary pumps before the first initial start-up to verify the conservative value of the primary mass flow utilized during the design phase and the reactor analysis. c) Uncertainties due to the defect of the repartition of the inlet mass flow rate THERMAL-HYDRAULIC IN NUCLEAR REACTOR       The effects of the uncertainties relative to the distribution of the mass flow rate have been discussed above. d) Uncertainties relative to the mass flow rate The thermal-hydraulic mass flow rate is defined to be utilized in the evaluations of the thermal-hydraulic characteristics of the reactor core which are taking account on the uncertainties of the prediction and the measurement. More however, one admits a maximal 5.5% of thermal-hydraulic mass flow rate is not efficient in term of heat evacuation capacity of the reactor due to that part by-pass the reactor core in different ways in the pressure-vessel mentioned earlier. e) Uncertainties relative to the hydraulic forces As explained earlier, the envelop of the hydraulic forces on the fuel assembly is evaluated in normal operation in cold-shutdown conditions and in the transient conditions for a transient of over-speed of the primary pumps of 20% superior to the mechanical design mass THERMAL-HYDRAULIC IN NUCLEAR REACTOR       f) Uncertainties relative to the hydraulic dimensioning of the internal components The uncertainties are taking into account either by condering pessimistic hypotheses for the limit conditions of calculations, or with inherent concervatism to the computer code or to numerical schemas utilized to effectuate the calculations. N. Methods of analysis and study data N.1. Methods utilized to analyse the transients Set point of automatic shutdown of the reactor by the low level of DNBR and the limit conditions of operation (LCO). In the cases of sollicitation of the low level DNBR protection system, it must define the low level DNBR protection threshold. In the second case, the main goal is to define a value of the criteria of DNBR which must be respected for each transient of that category. THERMAL-HYDRAULIC IN NUCLEAR REACTOR       The transients are classified in three categories: Transient of type I: Transients of power of which the protection of low level DNBR is effective; Transients of type II: Transients of power of which the protection of low level DNBR is in-effective: Transients of type III: Transients studied and based hot-shutdown (withdrawal of a control rod cluster at nil power and stream line rupture). N.2. Transients of power (Set point of automatic shutdown threshold of the reactor by low level of DNBR and limit conditions of operation) As the the protection of the low level of DNBR and the surveillance of the LCO of the DNBR are based on the value of DNBR elaborated by the simplified algorithm, the threshold of automatic shutdown of the reactor at low level DNBR and the threshold of the LCO with regard to the DNBR(LCO) are defined by THERMAL-HYDRAULIC IN NUCLEAR REACTOR         taking into account the uncertainties related to the elaboration of the DNBR and the accuracy of the measurements. The boiling crisis (DNBR criteria been reached) is avoided by maintaining the calculated values of DNBR in the operation under the thresholds. The uncertainties could be different for the two systems because they are associated to the accuracy of the system to the relative operation conditions respectively to the latter. To define the two thresholds, it is necessary to repartite the transients in two classes: *The transients where the automatic shut-down of the reactor is effective (transients of type I): They are characterized by the following conditions: The parameters affecting the DNBR during the transient are utilized in the low level DNBR protection chain; The evolution of parameters is sufficiently low to be correctly THERMAL-HYDRAULIC IN NUCLEAR REACTOR       The automatic shut-down of the reactor at low level DNBR is initiated each time that the DNBR has reach the safety DNBR criterion. The variable Y = DNBR/ DNBR(PS) is characterised by a mean value (m(Y,PS) and a deviation standard (σ(Y,PS). DNBR(PS) is the value of the predicted DNBR by the algorithm of the protection system. The automatic shut-down of the reactor at low level DNBR is then initiated at the calculated value of the DNBR by the protection system reaches the threshold of the calculated DNBR(PS) as follow: DNBR(PS) = SC/ m(Y,PS)(1 – 1.645 V(Y,PS) This means that the probability to avoid the boiling crisis is then 95% with a confidence of 95%. THERMAL-HYDRAULIC IN NUCLEAR REACTOR     The transients on which the protection at low level DNBR is not effective (transients type II): In this case, the automatic shut-down is obtained unically by utilizing the specific parameters (in several cases, only one parameter, per example low mass flow rate of the pump). Consequently, it is necessary to survey other parameters which are not taking into account in the elaboration of the automatic shut-down and to determine the threshold of the survey system of the limit of operation conditions. The most critical point is associated to the power distribution and the best method is the surveillance of the initial DNBR. In normal operation, the value of the calculated DNBR by the algorithm must be maintained above the surveillance threshold of the DNBR(LCO) in order to avoid the boiling crisis d...