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 an
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NUCLEAR REACTOR
GS Trần Đại Phúc
Trang 25. 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
Trang 3in 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.
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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
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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
Trang 6 However, the assured safety functions requires the
application of a Quality Assurance programme on which the main aim is to document and to control all associated
activities.
Preliminary tests: The basic hypothesis on scenarios
adopted in the safety analyses must be control during the first physic tests of the reactor core Some of those tests, for example the measurements of the primary coolant rate
or the drop time of the control clusters, are performed
regularly Other tests are performed in totality only on the head of the train serial.
For the following units, only the necessary tests performed
to guarantee that thermal-hydraulic characteristics of the reactor core are identical to the ones of the head train
serial.
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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.
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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.
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The coolant pressure drop across the core must be kept low to
minimize pumping requirements as well as hydraulic loads
(vibrations) to core components.
Above mentioned constraints must be analyzed over the core live, for all the reactor core components, since as the power
distribution in the reactor changes due to fuel burn-up or core
management, the temperature distribution will similarly change.
Furthermore, since the cross sections governing the neutron
physics of the reactor core are strongly temperature and density dependent, there will be a strong coupling between thermal-
hydraulic and neutron behaviour of the reactor core.
II Energy from nuclear fission
Consider a mono-energetic neutron beam in which n is the
neutron density (number of neutrons per m3) If v is neutron
speed then Snv is the number of neutron falling on 1 m2 of target material per second.
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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: Σ f nv = Σ f Ф ,
where Σ f =nv is the neutron flux (to be discussed later) and
Σ f = Nσσ f , N being the number of fissile nuclei/m3 and σ f 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 Σ f Ф
fissions, where Σf and Ф are the average values of the
macroscopic fissions cross section and the neutron flux,
respectively.
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To evaluate the reactor power it is necessary to know the average amount of energy which is released in a single fission The table below shows typical values for uranium- 235.
Table II.1: Distribution of energy per fission of U-235.
10 -12 J = 1 MeV
Kinetic energy of fission products 26.9 168
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As can be seen, the total fission energy is equal to 32 pJ It means that it is required ~3.1 10 10 fissions per second to generate 1 W of the thermal power Thus, the thermal
power of a reactor can be evaluated as:
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.
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.
Trang 13of thermal fission of uranium-233, uranium-235,
plutonium-239 and a mixture of uranium and plutonium are shown in following figure III.1.
Trang 14125 and 155
Trang 15THERMAL-HYDRAULIC IN
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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,
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Φ(t) = Ф 0 {(β / β – ρ) e ) e (λρ) e / β – ρ) e )t - (ρ) e / β – ρ) e ))e (β – ρ) e / l)t } (IV.1)
Here f (t ) is the neutron flux at time t after reactor
shut-down, 0 f is the neutron flux during reactor operation at full power, r is the step change of reactivity, β is the fraction of delayed neutrons, l is the prompt neutron lifetime and l is the mean decay constant of precursors of delayed
neutrons For LWR with uranium-235 as the fissile material, typical values are as follows: l = 0.08 s-1, β = 0.0065 and l
= 10-3s.
Assuming the negative step-change of reactivity r = -0.09, the relative neutron flux change is given as:
Ф(t) / Ф0 = 0.067 e -0.075t + 0.933 e -96.5t (IV-2)
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The second term in Eq (4-3) is negligible already after t = 0.01s and only the first term has to be taken into account in calculations As can be seen, the neutron flux (and thus the generated power) immediately jumps to ~6.7% of its initial value and then it is reduced e-fold during period of time T = 1/0.075 = 13.3 s.
After a reactor is shut down and the neutron flux falls to
such a small value that it may be neglected, substantial
amounts of heat continue to be generated due to the beta particles and the gamma rays emitted by the fission
products FIGURE 4-2 shows the fission product decay heat versus the time after shut down The curve, which covers a time range from 1 to 106 years after shut down, refers to a hypothetical pressurized water cooled reactor that has
operated at a constant power for a period of time during
which the fuel (with initial enrichment 4.5%) has reached
50 GWd/tU burn-up and is then shut down instantaneously Contributions from various species which are present in the spent fuel are indicated.
Trang 19THERMAL-HYDRAULIC IN
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of HM) versus time after shutdown.
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radiation can be calculated from the fllowing approximate equation [IV-1],
q” / q” 0 = 0.065 { t - t op ) -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.
Trang 22 V Spatial distribution of the heat sources
The energy released in nuclear fission reaction is
distributed among a variety of reaction products
characterized by different range and time delays Once
performing the thermal design of a reactor core, the energy deposition distributed over the coolant and structural
materials is frequently reassigned to the fuel in order to
simplify the thermal analysis of the core The volumetric
fission heat source in the core can be found in general case as:
Trang 23 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,
flux distribution for such geometry given as:
Trang 24 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) = wfΣfФ 0 J 0 {2.405rf / R}cos{πz / H} (V.2)z / H} (V.3)
There are numerous factors that perturb the power
distribution of the reactor core, and the above equation will not be valid For example fuel is usually not loaded with
uniform enrichment At the beginning of core life, higher
enrichment fuel is loaded toward the edge of the core in
order to flatten the power distribution Other factors
include the influence of the control rods and variation of
the coolant density.
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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.
Trang 26 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
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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.
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Figure VI.1: Typical coolant sub-channels in fuel rods assembly.
A = p 2 - πz / H} (V.2)d 2 / 4 for square lattice
A = (3 1/2 / 4)p 2 - πz / H} (V.2)d 2 / 4 (VI.2)
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And the wetted perimeter (part of the perimeter filled with heated walls) is given by:
Pw = πz / H} (V.2)d for square lattice
Pw = 1/2 πz / H} (V.2)d 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 / πz / H} (V.2)(p / d) 2 – 1} for square lattice
Dh = d{2x3 1/2 / πz / H} (V.2) (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:
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Dh = 4A / Pw = (4w 2 – Nπz / H} (V.2)d 2 ) / (4w + Nπz / H} (V.2)d) (VI.5)
Where Nσ 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.
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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
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The energy balance for a differential channel length
between z and z + dz is given as follows:
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) = il i + 1/W ƒ -H/2 z q”(z).P H (z)dz (VII.3)
Trang 33 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,
T lb (z) = T lbi + q”P H (z + H/2) / CpW (VIII.1)
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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:
T lb = ƒ AC ρ) e lC pl V l dA / ƒ A ρ) e l C pl V l dA (VIII.2)
The temperature distribution along the channel is
represented in figure VIII.1.
Trang 35THERMAL-HYDRAULIC IN
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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:
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Figure VIII.2: Heated channel with cosines power distribution
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Figure VIII.3 Represents the axial distribution of the coolant temperature with cosines heat flux distribution.
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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
so-called coolant centered channels and rod-centered channels Both types of sub-channels are equivalent in
sub-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.