Reactor BehaviorPart 3 of 12

Navy Recruiting District DenverCDR Mike Wenke – CO

ET1 (SS) Matt Byron – Nuke CoordinatorENS Titus Reed

OC Kellan Downing

Criticality• Critical

– core can sustain reaction with no net change in number of free protons in the core

– keff = 1• Supercritical

– more neutrons are created than needed

– power increases with time– keff > 1

• Subcritical– power decreases with time– fewer neutrons are created than

needed– power decreases with time– keff < 1

• keff is the ratio of the number of neutrons in one generation to the number of neutrons in the preceding generation

• For a critical system

igenration in neutrons

1)(i generationin neutronskeff

absorptionleakage

productionneutron 1keff

Reactivity• Reactivity (δk “delta kay”)

– Is useful way of specifying the extent that a reactor departs form criticality

– Typically in units of 10-4

• ex. δk = +10 x 10-4 δk• ex. δk = -20 x 10-4 δk

– Can be reported in percent• δk = +10 x 10-4 = 0.1%

1)(i Generation of Population

i Generation

of Population

1)(i Generation

of Population

δk

k

1kδk

eff

eff

Definitions Utilizing Reactivity• Excessive Reactivity

– At any time a core has excess reactivity that varies with conditions– Once a core has lost its positive reactivity it is no longer operable– Must be compensated for with control rod– δkex= core (positive)reactivity with all rods out

• Reactivity Margin– Is the excess reactivity (for the core) under hot, maximum xenon plus

samarium conditions• Shut Down Margin

– Is the amount of negative reactivity when control rods are fully inserted

– Usually 1 to 2%

Core Reactivity

• A core’s reactivity (δk) is made up of the reactivity of its components

Equation 3.1

Core Reactivity Terms• Each reactivity term in Eqn. 3.1 is based on the

normal operating temperature (NOT) of the primary coolant• However, the actual average coolant temperature

varies from the NOT, so a temperature reactivity term is placed into equation 3.1 to account for this difference

Equation 3.2: General Equation for Determining Core Reactivity

Temperature Reactivity Term

Change in Core Reactivity• If a reactivity term changes based on varying

circumstances, an overall change of core reactivity occurs

• The amount of temperature reactivity added by ΔδkTave is:

• αT is called the temperature coefficient of reactivity, and designates the response of a reactor to a change in temperature

Equation 3.3: Reactivity Balance Equation

Positive Value of αT

• If αT is positive, and there is an increase in T, there is in increase in overall core reactivity, δk

• An increase in core reactivity causes an increase in power level of the reactor, which further increases the temperature, and thus increases reactivity once again (reactor needs to be shutdown or accident will occur

• If αT is positive and T decreases, core reactivity decreases (reactor eventually shuts down)

Negative Value of αT

• A negative value for αT results in very different behavior

• If αT is negative, and there is an increase in T, there is a decrease in overall core reactivity, δk, which decreases the power and thus the temperature: the reactor returns to its original state

• If αT is negative, decrease in T causes increase in δk, then increasing T, reactor returns to original state

δk

Equations for αT

Equation 3.4 αT based on temperature change and reactivity

Equation 3.5 αT based on Keff and reactivity change with respect to temperature

Equation 3.6 αT based on Keff and six factor formula terms, where dk/dt is sum of changes each

of the six factors

Summary of αT • A positive value for αT results in unstable

reactor behavior• If αT is negative, reactor is inherently stable.

However if αT is too negative or not negative enough, undesirable consequences result– If αT is not sufficiently negative, a change in

reactor power in response to a change in steam demand may be too slow

– If αT is too negative, reactor power can increase too quickly due to a sudden change in demand (i.e. a steam line rupturing)

Parameter Trends as Temperature Increases

• Factors from the six-factor formula that decrease outweigh the factors that increase, thus αT decreases or becomes more negative as temperature increases

Behavior of Factors with Lifetime• , reproduction factor, remains same• , fast fission factor, decreases due to resonance fission

cross section decreasing• f , thermal utilization factor, increases because lumped

poison burns out• Nf , Fast Non-Leakage Factor, increases due to decrease

in buckling and increase in slowing down length• Nth , Thermal Non-Leakage Factor, decreases because

there is an increased chance in diffusion of neutrons from core to reactor

• P , resonance escape probability, increases due to less absorption of fast neutrons in the fuel and lumped poisons

Xenon-135

• Produced from:– Fission of Uranium (yield of 0.3%)– Decay of Iodine-135 (yield of 6.1%)

• Thermal Absorption Cross Section 2.6x106 barns

• 9.2 hr half-life• Decays to 135Cs

Changes in 135Xe Concentration

• As a reactor runs the concentration of Xenon will increase until equilibrium is reached

• The equilibrium concentration is dependant on the neutron flux

• Build up to equilibrium takes 40 to 50 hr

Peak Xenon• On shutdown

xenon concentration increases

• Magnitude of the peak is dependant on NI

Eq/NXeEq

• Time until peak is a function of neutron flux

Samarmium-149

• Produce from decay of Promethium-149 (1.07% yield 53.1 hr half-life)

• Is stable and is only removed by neutron absorption

• Thermal absorption cross section is 4.08x104 barns

Equilibrium Concentration

• The equilibrium concentration of Promethium increase with increasing neutron flux

• The equilibrium concentration of Samarium– Independent of neutron flux– Proportional to the fuel concentration

149Sm Concentration Upon Shutdown

• When the reactor is shut down the concentration for 149Sm increase to its maximum

• The 149Sm will remain until reactor operation continues

Other Fission Products

• Almost 1000 fission products are know– ≈180 are considered in reactor core design

• The effect of all fission poisons (excluding 135Xe & 149Sm) can lumped together– σa

fpp is the fission product poison microscopic neutron absorption cross section• Units of barns/fission

– Σafpp= σa

fpp NfU235

• Σafpp is the fission product poison macroscopic cross sections

• NfU235 cumulative number of fission per cm3

General Reactor Kinetics Equations

• Reactor kinetics deal with changes in reactor power

• Reactor kinetics must account for rate of change of reactor power, but also for delayed neutron precursor concentrations, which affect the next generation of reactor power

First Kinetics Equation• Expresses how the change in total fission rate

per second is related to core reactivity

• Which can be interpreted as:

Equation 3.7: First Kinetics Equation

First Kinetics Equation Terms

Second Kinetics Equation• Expresses how the rate of change of the fission-equivalent

delayed neutron precursor concentration is related to the fission rate and the rate of precursor delay

• The production term can vary rapidly with time, while the loss term generally varies more slowly

Expressing How QuicklyReactor Power Changes

• Reactor power is directly related to neutron population

• 2 ways to define rate of reactor power change:– Reactor Period (T):

• If reactor period (T) is constant:

– Startup Rate (SUR):• Defined as the powers of ten by which the reactor

power changes in 1 minute. units are expressed in decades per min (DPM)• SUR = 26.06/T

Prompt Criticality

• Defined as the reactivity condition wherein the reactor would be just critical in the absence of both source neutrons and delayed neutrons

• Fission rate is such that the resulting prompt neutrons alone are sufficient to offset neutron losses due to absorption and leakage, and maintain power

• In the first kinetics equation, this means the sum of the effect of delayed neutrons and effect of neutron sources equals zero

• Mathematically defined as:

Power Turning• Occurs when the rate at which the total fission rate

changes with time is zero– i.e: = 0

• This means that reactor power is constant

Power turning point for decreasing reactor power

Power turning point for increasing reactor power

Time to Power Turning

• The time delay between the time corrective reactivity insertion occurs and the time power actually turns

• Time delay is defined mathematically as:

• Where:– is the amount of reactivity added per second– is the initial value of core reactivity