IMPUESTO A LA PROPIEDAD RURAL

Prompt Neutrons

• The great majority (over 99%) of the neutrons produced in fission are released within about 10^-14 seconds of the actual fission event. These are called prompt neutrons.

• ₀n¹ + ₉₂U²³⁵ → (₉₂U²³⁶)* → ₃₈Sr⁹⁰ + ₅₄Xe¹⁴⁴ + 2n

• For thermal fissions the average number of

prompt neutrons emitted per fission is 2.43 for U²³⁵

• For Pu²³⁹ n = 2.89

• For Pu²⁴¹ n = 2.93

Delayed neutrons

E.g.

⁸⁷

Br

Delayed neutron precursor: ⁸⁷Br

Delayed Neutrons (Cont.)

Photoneutrons

• Neutrons are also produced as a result of (γ, n) reactions called photoneutrons

γ + ₁H² → ₁H¹ + ₀n¹

• Energy of γ > binding energy of ₁H²

Neutron Generation Time

• The neutron generation time is the time required for neutrons from one generation to cause the fissions that produce the next

generation of neutrons.

• The generation time for prompt neutrons involves three time intervals

Parameters of delayed neutrons

• Delay time from fission to release of delayed neutrons:

• average delay time  ~ 13 s

• Delayed neutron fraction :

• fraction of delayed neutrons in the total number of neutrons produced per fission

•  < 1%, but crucial for successful reactor control

•  depends on nuclide and neutron energy

• thermal fission of ²³⁵U:  = 0.0065

• thermal fission of ²³⁹Pu:  = 0.0021

• fast fission of ²³⁸U:  = 0.0164

• due to mixture of nuclides in the fuel, the average delayed neutron fraction changes (decreases) with fuel burn-up

Distribution of energy released by

²³⁵

U fission

Prompt energy release range

fission fragments 168 MeV ~ 10 μm

prompt neutrons 5 MeV 0.1 – 1 m

prompt gamma rays 7 MeV 0.1 – 1 m

gamma rays from (n, ) reactions 6 MeV 0.1 – 1 m

total prompt energy release 186 MeV

Delayed energy release

β from fission fragment decay 7 MeV ~ mm γ from fission fragment decay 6 MeV 0.1 – 1 m β and  from nuclei produced by (n, ) 1 MeV 0.001 – 1 m

Consumption of fissile material

• reactor operates 1 day at power at 1 MW power

energy produced = 1 MWd

• number of fissions = number of fissioned ²³⁵U nuclei

• the uranium mass that corresponds:

m = 2.7·10²¹ · 235 · 1.66·10^-27 kg = 1.05·10^-3 kg ≈ 1 g

• example:

• 30 days of operation at power of 3000 MW: 90 kg of ²³⁵U is consumed

Decay heat

• Energy released in the core after reactor shutdown

• a consequence of ^- and  decay of fission products

• Decay heat proportional to reactor power before shutdown

• with higher power, more fission products are produced

• Immediately after shutdown, the decay heat diminishes rapidly

• consequence of quick decay of short-lived fission products

Decay heat after a long operation at 3000 MW

_th

Time after shutdown Full power fraction Decay heat

1 s 6.2 % 185 MW

1 min 3.6 % 107 MW

1 hour 1.3 % 38 MW

8 hours 0.6 % 19 MW

1 day 0.4 % 13 MW

1 week 0.2 % 7 MW

1 month 0.1 % 4 MW

Division of neutrons in terms of their energy

• Basic division:

• fast: E > 0.1 MeV

• epithermal: 1 eV < E < 0.1 MeV

• thermal E < 1 eV

• fast neutrons are produced by fission

• epithermal neutrons are in the slowing-down process

• slowing-down takes place in the moderator

Slowing-down of neutrons in the reactor core

• fast neutron loses its energy by scattering on nuclei in the matter

• most effective is elastic scattering on light nuclei

• moderator: material that slows down neutrons in the reactor core

• in light-water reactors the moderator is water (hydrogen)

• other possible moderators:

• heavy water (deuterium)

• graphite (carbon)

Characteristics of moderator

• A good moderator has:

• large probability of scattering, i.e. large scattering x-section _s,

• large average energy loss per collision

• small probability for neutron absorption, i.e. small absorption x-section _a,

• non-nuclear properties: stable, good thermo-hydraulic properties, low price.

• Ordinary (light) water is relatively good moderator:

• however, it is corrosive at high temperatures

• activated in neutron flux

• good thermal/hydraulic properties

• cheap

Lifetime of a generation of neutrons

• The majority of fissions in light water reactors is induced by thermal neutrons.

• In fission, fast neutrons are produced.

• What happens to a generation of fast neutrons, born in thermal fission?

Fast fission factor ε

• Some fast neutrons induce fission, mainly on ²³⁸U in low-enriched fuel.

• neutrons, born in fast fission, are fast neutrons, as well

• fast fission increases the number of fast neutrons for a factor of ε:

• The initial generation of n fast neutrons, born in thermal fissions, is

Fast fission: factor ε

• Fast fission factor is always greater than one.

• LWRs: ε ~ 1.1

n   n

fast fission

238U thermal

fission

235U

239Pu

Fast non-leakage factor P

_f

• Probability that a fast neutron will not escape from the core:

• From (ε ∙ n) fast neutrons, (P_f ∙ ε ∙ n) remain in the core.

91

Fast neutron leakage: factor P

_f

• Fast non-leakage factor is always smaller than one

• typical PWR: P_f  0.98

fast neutrons which escape from the core

  n P_f    n

Resonance escape probability p

• As fast neutrons slow down, same of them are absorbed.

• The highest probability for neutron absorption is in the epithermal energy range (resonance absorption)

• most resonance absorption occurs in ²³⁸U and ²⁴⁰Pu

• Probability that fast neutrons that slow down will not be absorbed in resonances:

Resonance escape: factor p

• resonance escape probability is always smaller than 1

• depends on the fuel enrichment P_f    n

p . P_f    n

resonance absorption

E

_a

Thermal non-leakage factor P

_t

• Probability that a thermal neutron will not escape from the core:

• From (p ∙ P_f ∙ ε ∙ n) neutrons that slowed down to thermal energies, (P_t

∙ p ∙ P_f ∙ ε ∙ n) remain in the core.

Thermal neutron leakage: factor P

_t

• Thermal non-leakage factor is always smaller than one

• typical PWR: P_t  0.99

p P _f    n P_t  p P_f    n

thermal neutrons which escape from the core

Thermal utilisation factor f

• Fraction of thermal neutrons absorbed in fuel relative to absorption elsewhere the core:

• From (P_t ∙ p ∙ P_f ∙ ε ∙ n) thermal neutrons that remain in the core, (f ∙ P_t ∙ p ∙ P_f ∙ ε ∙ n) thermal neutrons are absorbed in the fuel

Utilisation of thermal neutrons: factor f

• The value of factor f can change significantly

• it is changed intentionally, in order to control the operation of reactor

P_t  p P_f    n

f P _t  p P_f    n

absorption in const. mater.

absorption in boron

absorption in control rods

Neutron yield per absorption 

• Number of fast neutrons which are released in thermal fission per thermal neutron absorbed in fuel:

• From (f ∙ P_t ∙ p ∙ P_f ∙ ε ∙ n) thermal neutrons absorbed in the fuel, (

Neutron reproduction: factor 

• Neutron yield per absorption is always greater than one.

• depends significantly on fuel enrichment:

• natural uranium:  = 1.34,

• 3% enrichment:  = 1.84,

• pure ²³⁵U:  = 2.08.

  f P_t  p P_f    n f P _t  p P_f    n

fuel

235U

238U

239Pu

Neutron cycle

n

generation of fast neutrons by thermal fission

new generation of fast neutrons by thermal

fission

fast fission thermal

utilization

 •f P p P• _t• • _f ••n

f P p P• _t• • _f ••n

P p Pt• • f ••n Pf ••n

•n

Multiplication factor k

• Multiplication factor, k, is the defined as the radio of the number of neutrons in one generation to the number of neutrons in the previous generation:

• In the neutron cycle, as described previously, k can be expressed as a product of six factors: _{k =  ∙ f ∙ P}

t ∙ p ∙ P_f ∙ ε

Chain reaction

• each generation of neutrons leads to the birth of next generation of neutrons

• chain reaction in the case k = 2:

Criticality

• k = 1: critical chain reaction

• the number of neutrons in the core does not change with time,

• the number of fissions per unit time is constant,

• reactor power is constant.

• k > 1: supercritical chain reaction

• the number of neutrons in the core increases with time,

• the number of fissions per unit time increases with time,

• reactor power increases.

• k < 1: Sub-critical chain reaction

• the number of neutrons in the core does decreases with time,

• the number of fissions per unit time decreases with time,

• reactor power decreases.

Reactivity

• We are usually interested in reactor conditions where k  1

• we want to avoid using a large number of decimal places

• Let us introduce reactivity, which represents departure from criticality:

k < 1  ρ < 0  reactor is subcritical,

k k

k 1

1 1



 

 

Notations for reactivity

• reactivity is dimensionless quantity

• several notations are used:

• k/k: reactivity is calculated and the number is followed by k/k

• % k/k: 1% k/k = 0.01 k/k

• pcm: 1 pcm = 0.00001 k/k

• $: 1 $ = k/k

Positive and negative reactivity

• Any change of the core properties modifies k and ρ:

• burnup of fuel, production of fission products,

• operator actions (boron concentration, control rods).

• If changes in the core decrease the multiplication factor k:

• negative reactivity has been added in the core.

Reactivity and change of reactor power

• During operation at constant power:

• the reactor is critical,

• multiplication factor k = 1,

• core reactivity ρ = 0.

• If we want to increase the power, positive reactivity has to be added in the core.

• If we want to decrease the power, negative reactivity has to be added in the core.

Reactor operation at low power range

• Thermal power negligible  the temperature of fuel and coolant does not change perceptibly with change of power

• no temperature feedback effects on reactivity

• Any change of reactivity is cause exclusively with external actions, e.g.

withdrawal or lowering the control rods

Time dependence of reactor power

Reactor period T

Neutron lifetime

• Time from the birth to the disappearance of a neutron (escape, absorption).

• It is divided into:

• birth time: the time from fission to neutron release ~ 10^-14 s,

• slowing-down time: the average time from neutron’s release to its thermalization ~ 10^-5 s,

• diffusion time: the average time from a neutron’s thermalization to its disappearance ~ 10^-5 s.

Phases of neutron lifetime

lifetime birth time

birth release thermalization absorption

slowing-down time diffusion time

Chain reaction with prompt neutrons

Average lifetime of prompt and delayed neutrons

• Slowing-down and diffusion times of delayed neutrons are comparable to those of prompt neutrons

• Birth time of delayed neutron is the life time  of its precursor (for ²³⁵U fission ~ 13 s) and is much longer that diffusion or slowing/down time.

• Lifetime of delayed neutrons is essentially the lifetime  of their precursors.

• Average lifetime of all neutrons:

Chain reaction with prompt and

delayed neutrons

Prompt criticality

Reactor power response to a positive step

change in reactivity

Reactor response to a negative step change in

reactivity

Classification of reactivity changes

• During reactor operation, the core properties and consequently its reactivity change.

• Based on how fast the core properties or its reactivity change during reactor operation, reactivity changes are classified as:

• short-term changes (~ seconds, minutes)

• medium-term changes (~ hours)

• long-term changes (~ months)

Short-term reactivity changes

• when reactor is in operating power range (0% - 100%), fuel and coolant temperatures change

• change of temperature influences:

• density of the moderator

• resonance absorption in fuel

• void fraction in the coolant

Impact of coolant (moderator) temperature on reactivity

• increase in coolant (moderator) temperature decreases its density

• mass of water in the core decreases, mass of fuel remains constant:

1.increase in neutron absorption in fuel relative to absorption in water

increase of the thermal utilization factor f

2.slowing down distance of neutrons increases

resonance absorption increases: resonance escape factor p decreases

• Depending on which of the two effects prevails: α_m is negative or positive.

• LWRs usually constructed for a negative α_m

The moderator temperature coefficient, α_m, is defined as the change in reactivity per degree change in the average moderator temperature.

Impact of fuel temperature on reactivity

The fuel temperature coefficient, α_f, is defined as the fractional change in reactivity per unit change in the fuel temperature.

Impact of void fraction on reactivity

• Increase in fraction of voids (steam bubbles) in the core reduces the moderator density.

• Impact on reactivity is similar to the impact of the moderator temperature.

• Due to small total void fraction formed in the core of a PWR, the associated total reactivity change is small.

The void coefficient, α_v, is the reactivity change per percent change in the void fraction.

Impact of power change on reactivity

• Change of power changes the coolant temperature, the fuel temperature, and the void fraction.

• The combined effects of moderator, fuel and axial power

redistribution are accounted for in the total power coefficient.

• In PWRs, power coefficient is always negative.

The power coefficient, α_p, is defined as the change in reactivity per percent power change.

The power defect tells us how much core reactivity changes if reactor power is changed by a given value of P.

Response to a step change in reactivity at operating power

• At operating power range, temperature feedback effects are present (neglected in chapter

“Reactor kinetics”).

• Operator withdraws the control rods for a few steps

Mid-term changes of reactivity

• Fissions in the core result in over 200 different fission products.

• Two fission products important for reactor operation:

• ¹³⁵Xe – absorption cross-section for thermal neutrons _{a = 2∙10}⁶ _b

• ¹⁴⁹Sm – absorption cross-section for thermal neutrons _a = 4∙10⁴ b

Production and removal of Xenon-135

135

Xe during reactor start-up

• After reactor start-up, ¹³⁵Xe concentration starts to grow

• Equilibrium concentration of ¹³⁵Xe approx. 40 - 50 h after start-up

• equilibrium conc. depends on reactor power, but dependence not linear

25%

00 -500 -1000 -1500 -2000 -2500 -3000

reactivity due to Xe [pcm]

10 20 30 40 50 60 70 80 90 100

50%

75%

100%

135

Xe during reactor shutdown

• after shut-down, ¹³⁵Xe in produced by the decay of ¹³⁵I (t_1/2

= 6.6 h), and is removed by its own decay (t_1/2 = 9.1 h)

• maximum ¹³⁵Xe concentration reached ~ 9 h after shutdown (depending on reactor power)

• after ~24 h ¹³⁵Xe concentration roughly equal to its level at shutdown

• after ~ 80 – 90 h there is practically no ¹³⁵Xe in the core

Reactivity due to

¹³⁵

Xe after shutdown

25%

0 0

500

1000

1500

2000

2500

3000 -500 -1000 -1500 -2000 -2500 -3000

reactivity due to Xe [pcm]

10 20 30 40 50 60 70 80 90 100

50%

75%

100%

Production and removal of Samarium-149

149

Sm during reactor start-up

• After reactor start-up and with a fresh core, ¹⁴⁹Sm concentration starts to grow.

• Equilibrium concentration of ¹³⁵Xe approx. 400 h after start-up and is independent of reactor power.

149

Sm during reactor shutdown

• after shut-down,

149Sm burn-up by neutron absorption ceases, but ¹⁴⁹Sm is still produced by the decay of ¹⁴⁹Pm

• maximum ¹⁴⁹Sm concentration reached ~ 400 h

149

Sm during reactor restart

• As the reactor is restarted on power, the equilibrium concentration of

149Sm equal to its value prior to shut-down is re-established.

Long-term changes of reactivity

• With operation of a LWR through a fuel cycle:

• burn-up of ²³⁵U

• burn-up of ²³⁸U (negligible importance)

• production of ²³⁹Pu and other Plutonium isotopes

• production of fission products, some of which are important neutron absorbers

• With core burn-up, the reactivity of core decreases.

Excess reactivity

• Excess reactivity is compensated by adding boron to the coolant and by burnable poisons.

• Since core reactivity decreases with burn-up, the concentration of boron C_B in the moderator is gradually reduced over the fuel cycle.

Excess reactivity is defined as the amount of surplus reactivity over that needed to enable reactor operation at zero power.

Reactor

Thermal-hydraulics

Definition of basic quantities

Heat is energy which passes from a point of higher temperature to a point of lower temperature.

If there is no difference in temperature, there can be no heat transfer.

Modes of heat transfer

1. heat conduction through matter (diffusion)

2. heat transfer by means of fluid flows (convection) 3. radiation

convection

Heat conduction

• process which takes place at the atomic level

Heat conduction equation

Temperature profile in a wall made of various

materials

Temperature profile in fuel rod

coolant fuel pellet gap cladding

T

T_W

T

Convection

Example of natural convection in PWR: core cooling after primary pumps have been shut down

Convection involves heat transferred by the flow of fluid.

Convection can be natural or forced.

heat source heat sink

Δh

Natural convection

• Convection is heat transfer by the flow of fluid.

• Natural convection: the fluid moves due to a difference in fluid density (temperature difference or phase change)

• Example of natural convection due to difference in liquid density is PWR core cooling after shut down of primary pumps.

Forced convection

• Fluid is forced to move by means of a pump or fan.

• Example in PWR: the flow of the primary coolant through the core driven by the operation of reactor coolant pumps.

Equation of heat transfer

Radiative heat transfer

Boiling heat transfer

• When heat is transferred from a solid body to a liquid and the temperature of solid body is high enough, the liquid may boil

phase change

• heat flux is significantly higher than in the case of natural or forced convection

• similar applies for condensation, as well

• saturated boiling: water is at the boiling point temperature throughout its

Bubble formation

steam liquid

Bubbles break away from the heating surface, “mixing” the liquid and

“breaking” the laminar layer of liquid which would otherwise cause much lower heat transfer.

Boiling curve

4 regions of the boiling curve

1. Natural convection: small T

2. Nucleate boiling: beginning of boiling

• subcooled nucleate boiling

• saturated nucleate boiling

3. Partial film boiling: bubbles combine into layers next to the wall

• this layer of vapour insulates the surface and reduces heat transfer

• in this region, the vapour layers are not stable

4. Film boiling and radiation: stable layers of vapour form next to the heating body

• poor heat transfer despite a relatively large temperature difference

Departure from Nucleate Boiling

• Point c:

boiling crisis or

Departure from Nucleate Boiling – DNB

• nucleate boiling switches over to film boiling

• heat flux at point c is called the critical or DNB heat flux.

• Departure from Nucleate Boiling Ratio – DNBR:

DNBR = critical heat flux

In document Análisis del desarrollo normativo del municipio de Pucará, encuadrado en el Código Orgánico de Organización Territorial, Autonomía y Descentralización - COOTAD (página 118-121)