lesson 4: nuclear reactors, neutron propagation

Reactors, Neutron Propagation.. 1

Laboratory for Reactor Physics and Systems Behaviour

Neutronics

Lesson 4: Nuclear Reactors, Neutron Propagation

Classification of Reactors (according to power)

Common Characteristics of Nuclear Power Plants

Neutronics Modelling of a Reactor, Desired Results

Visit to CROCUS, Typical Experiments

Neutron Propagation

Multiple Scattering, Angular and Scalar Flux

Transport and Diffusion Theory



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U238 / U235 Cycle

(to direct storage)

Current-day situation in most countries (without recycle; “once-through”)



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U238 / U235 Cycle (with recycling)



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Classification of Reactors (according to power) … Zero Power Reactors

Critical assemblies, low neutron flux (106 to 108 n/cm2-s) • Power only of several watts, natural convection sufficient for cooling

Allow detailed reactor physics (neutronics) studies to be carried out for advanced systems, e.g. PROTEUS reactor at PSI • Used for different experimental programmes (GCFR, HTR, LWR,…)

Main purpose is to generate experimental (integral) data for verification and validation of reactor physics caculational codes • Critical size, detailed neutron balance, etc.

Another important application: teaching and training reactor operators, e.g. CROCUS at EPFL



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PROTEUS Zero-Power Research Reactor at PSI



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“Cold” Power Reactors (Medium, High Flux Research Reactors)

Neutron flux levels typically ~ 1013 to 1014 n/cm2-s (similar to power plants) • Cooling by forced convection • Modest coolant temperature (<100°C, i.e. near to ambient) • No energy conversion (“cold” calories) • Technology, relatively simple • Wide range of applications possible

Common type: “swimming pool” reactors, e.g. SAPHIR at PSI • First reactor in Switzerland (1960) • Upgraded to 10 MWth (flux ~ 1014 n/cm2-s) • Fuel : MTR (materials test reactor) plate-type, 20% enr. uranium (LEU) • Operated until 1994



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Research Reactor Applications: Example of SAPHIR

Basic physics (condensed matter) research • Strong neutron source, as needed for neutron

scattering experiments

Radioisotope production • For medicine, industry

Radiation damage studies

Radiochemistry applications

Si transmutation doping (industrial use)

Neutron radiography

View into the pool of the blue Cerenkov radiation from the SAPHIR core



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Other “Cold” Power Reactor Applications

District heating

Desalination of sea water

Military applications

R: Reactor, P: Pump, H.E.: Heat Exchanger



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“Hot” Power Reactors: Nuclear Power Plants (NPPs)

Almost all commercial production of nuclear energy today, in the form of electricity • High-quality utilisation

Demanding technology needed for energy conversion • Coolant temperature of at least ~ 300°C • Thermal efficiency ~ 30 to 35% for LWRs, 40 to 45% for HTRs 0°C

– Typical thermodynamic cycle shown below – Other, more complicated schemes possible, e.g. co-generation of district heat

R: Reactor P: Pump SG: Steam generator HP: High pressure turbine stage LP: Low pressure turbine stage C. Condenser A: Alternator



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Nuclear Power Plants Current nuclear energy scene dominated by

LWRs (85% of all NPPs) • 65% PWRs, 20% BWRs • Others: GCRs: 9%, CANDUs: 5%

Principal “vendors” • Framatome ANP, Westinghouse, GE

NPP specificities, cf. fossil plants • “Boiler”, a nuclear reactor (keff=1,..) • Strong neutronics-thermalhydraulics coupling

(static, dynamic behaviour) • Large radioactive inventory of core, importance of

nuclear safety analysis (reactivity effects, decay heat removal,..) • Material behaviour under strong irradiation

conditions, crucial aspect

PWR

BWR



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Typical PWR Components Containment, inside view Reactor Pressuree Vessel, RPV

(155 bars, 13m ht, 20cm steel) Fuel Assembly :

1 of 157, 17x17 rods incl. guide tubes, 3-4% enr. UO2 pellets in zircaloy clad



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Characteristic NPP Components



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Reactor Core: Regular Lattice of “Unit Cells”

Multiplying medium, in practice, is heterogeneous • Associated to each “unit cell” volume of fuel, certain volumes (in given geometry) of

structural material (cladding) and moderator • Simple unit cell for LWR lattice • For CANDU, with cluster of fuel rods, effectively a “supercell”

coolant, moderator (H2O)

cladding

Unit Cell (repeated pattern)

pressure, calandria tubes

moderator (D2O)

coolant (D2O)

cladding

UO2

LWR (PWR, BWR)

CANDU (larger scale)



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Simplifying Steps for a Thermal Reactor

Spatial “homogenisation” of the unit cell • Explicit treatment of the heterogeneity of fuel/cladding/moderator, etc. ⇒ Σa(E), Σf(E), Σs(E), … (effective macroscopic cross-sections for the “quasi-homogeneous” medium)

Consideration of neutron slowing down in the medium • From fission energies to thermal ⇒ definition of thermal “sources”

Monoenergetic treatment of the thermal neutrons (propagation, balance) ⇒ Critical state, etc.

In the course, we start the other way around… - First: propagation (diffusion, transport), balance of monoenergetic neutrons - Next: slowing down of fast neutrons



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Desired Results

Criticality condition • Critical size, critical mass

Spatial distribution of the neutron flux • Power density • Fuel burnup

Kinetic (time-dependent) behaviour • Control of the chain reaction • Safety studies

⇒ First, a visit to a reactor some of you already know: CROCUS...



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Teaching Reactor CROCUS at EPFL



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Simple Experiments on CROCUS… Approach-to-Critical

Neutron flux in a sub-critical reactor with an external source

• For criticality, keff = (Productions) / (Absorptions + Leakage) = 1

• For a shutdown reactor (absorptions too high in control rods), keff<1 → Φ ~ 0

• With an external, e.g. isotopic, source (S n/cm3-s), Φ ~ S/(1- keff )

• The response of a neutron detector (C counts/s) is a measure of the subcriticality 1/C ∝ 1/Φ ∝ (1- keff )

• If one increases keff (by withdrawing control rod, i.e. reducing absorptions), C ↑

• At critical, 1/C → 0 … One may conduct an experiment to predict the criticality…



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Approach-to-Critical (contd.)

One determines C for different insertion positions of the control rod, e.g. • Using a Pu-Be neutron source and a fission chamber as detector

Extrapolation of the curve 1/C-vs.-position

yields the critical control-rod insertion

Confirmation of the critical state: • Flux remains constant, independent of the presence of the external source • Possibility to establish any desired flux level (reactor power)

– Regulatory limit for CROCUS ~ 2.5.109 n/cm2-s (~ 100 W)

critical position

Control rod position →

↑ 1/C



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Kinetic Behaviour

For increasing the flux level, the reactor has to be made supercritical (keff>1)

→ Φ(t) = Φ0 . eωt = Φ0 . et/τ

The reactor “period”, τ = 1/ ω (time for a flux increase by factor e) depends on

• Reactivity, ρ = (keff - 1) / keff

• Kinetic parameters of CROCUS

– Λ, prompt neutron generation time (~ 10-4s for LWRs)

– Delayed neutron parameters (crucial for reactor control) βeff , Teff … values for CROCUS: ~ 0.7%, 7s , respectively



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Reactor Period in function of Reactivity

For ρ < βeff , τ much greater than 1s

For ρ = βeff (~0.6%) , reactivity of 1$ , τ becomes very short (“prompt critical”)

• Extremely important safety restriction

Possibility of a “reactivity accident” (cf. Chernobyl) can be avoided through appropriate reactor design

• Adequate control system, negative reactivity coeficients (feedbacks), etc.



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Neutronics (Reactor Physics)… Problems to be Treated

Due to neutron slowing down, a given reactor has a characteristic energy spectrum, Φ(r, E)

If appropriate, energy-averaged σ’s are available, the neutrons may be considered monoenergetic, e.g. average thermal σ’s for a Maxwellian spectrum

⇒ One may accordingly commence on a very simple basis:

Propagation (diffusion) of monoenergetic neutrons

Thereafter, neutron slowing down … provides the thermal-neutron “source”

Finally multiplying media • Steady state… criticality condition, flux (power) distribution • Kinetic behaviour • Reactivity changes in a reactor, feedback effects, control



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Nuclear Data

Cross-sections characterise the different, possible neutron interactions • Obtained on the basis of both experiments and nuclear theory

Neutronics… “bridge” between microscopic interactions & macroscopic effects

For “passive” media… σs , σc

For mutiplying media… σs , σc , σf ( σa = σc + σf )

For mixtures of nuclei… , , etc.

€

Σs = (Ni .σ ii∑ )

€

ν .Σ f = (Ni .ν ii∑ .σ fi)



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Neutron propagation: Beam

Semi-infinite, homogeneous, passive medium

Upon integration, ⇒ Exponential attenuation law

(analogous to photons)

In terms of probabilities,

⇒ Probability of a collision between x and x+dx

⇒ Probability of arriving at x without a collision



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Mean Free Path, λt

One may use p1, p2 for defining…

→ Probability to have a collision between x and x+dx

Thus,

Consider the average distance traversed for one collision (mean free path) €

P(x) = e−Σt x .Σt€

P(x) = p2(x).p1(x)dx



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Point Source

For a point source of neutrons in a vacuum,

With interposition of a material (Δρ, Σt),

With the above, notion of Φ is same as before, i.e • Flux is unidirectional, collisions correspond to “disappearance”

→ Expression correct if Σt = Σa , but not for the case of scattering…

€

Φ =S

4πρ2

€

Φ(ρ) =S

4πρ2.e−ΣtΔρ



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Successive Scattering Events

For a neutron beam incident on a slab of scattering material (moderator) • Neutrons arrive at B also after one or more

scattering collisions

The exponentisal attenuation law applies only to neutrons arriving with their initial direction (“virgin” neutrons) • Actual “flux” is much greater

What remains valid: • The neutrons traverse, without collision,

a distance x = MM’ with probability • Σt : probability of a collision per cm •

€

e−Σt x



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Angular Flux

In practice, the n’s move in all directions • Similar to molecules of a gas

For mono-directional neutrons ⇒ Angular flux ϕ

N’s are incident on dS with a specific direction ϕ dΩ = n v dΩ

Corresponding reaction rate dR = Σ ϕ dΩ , where Σ is independent of

⇒ with

Comment 1… A mono-directional beam ⇒ an angular flux

Comment 2… For obtaining a reaction rate, one needs the scalar flux

€

Ω ( )

€

Ω ( )

€

Ω ( )

€

Ω ( )

€

Ω

Scalar flux n, total density



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Example: Power Density

Macroscopic effects have to be described in the reactor core, e.g. • Heat generation rate at a particular location (power density, or specific power)

One needs the scalar flux in the fuel, to obtain the thermal power density as:

with Rf = ΣfΦ = NcσfΦ

In general, one needs to have the angular flux, in order to obtain the scalar flux…



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Transport Theory, Diffusion Theory

For an explicit treatment of the directional dependence of the neutron flux: Neutron Transport Equation • Boltzmann Equation* … applied also in several other domains of physics

– Kinetic theory of gases, astrophysics, plasma physics, etc. • Analytical solution difficult, possible only in a few special cases

In many situations, one can take an approximative approach: Neutron Diffusion Equation • Direct consideration of the scalar flux • Simplified treatment of the “neutron current” (utilisation of Fick’s Law)

______ *



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Summary, Lesson 4

Classification of reactors according to power

Common characteristic features

Simplifying steps for a thermal reactor, desired results

Zero-power reactor CROCUS, typical experiments

Neutronics: treatment steps

Neutron propagation, mean free path

Multiple scattering

Angular, scalar flux

Neutron transport theory, diffusion theory

lesson 4: nuclear reactors, neutron propagation

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