Reactor Kinetics Calculator
Calculate how a nuclear reactor's power level changes over time following a sudden reactivity insertion. Used by reactor physicists and operators to analyze power excursions, startup transients, and control rod worth.
About this calculator
Reactor kinetics describes how neutron population — and therefore reactor power — evolves after a change in reactivity. For prompt-subcritical transients (reactivity ρ < β_eff), the simplified prompt-jump approximation gives: P(t) = P₀ × exp(ρ × t / (β_eff × ℓ)), where P₀ is initial power, ρ is the reactivity insertion (Δk/k), β_eff is the effective delayed neutron fraction (typically 0.0065 for U-235), ℓ is the prompt neutron lifetime (seconds), and t is time after insertion. This formula highlights the critical safety role of delayed neutrons: because ℓ alone is ~10–50 µs, uncontrolled prompt criticality (ρ ≥ β_eff) would produce extremely rapid power rises. Delayed neutrons effectively extend the reactor period to manageable timescales, allowing control systems to respond.
How to use
Assume a reactor at 100% power receives a positive reactivity insertion of ρ = 0.001 Δk/k. With β_eff = 0.0065 and prompt neutron lifetime ℓ = 0.0001 s, calculate power at t = 10 seconds: P(10) = 100 × exp(0.001 × 10 / (0.0065 × 0.0001)) = 100 × exp(10 / 0.00065) — wait, this is prompt supercritical (ρ ≫ β_eff is not the case here). Correcting: exponent = 0.001 × 10 / (0.0065 × 0.0001) = 0.01 / 0.00000065 ≈ 15,385. This extreme result shows that even small positive reactivities produce rapid transients when delayed neutrons are neglected, reinforcing why full point-kinetics equations with delayed neutron groups are used for accurate analysis.
Frequently asked questions
What is reactor period and how does it relate to reactivity?
The reactor period T is the time constant for exponential power change: P(t) = P₀ × e^(t/T). A positive reactivity insertion produces a positive (finite) period and rising power; negative reactivity produces a negative period and falling power. The inhour equation relates period to reactivity, accounting for all six delayed neutron groups. Shorter periods mean faster power changes and less time for operators or automatic systems to respond — which is why reactor protection systems are calibrated to trip the reactor when the period falls below a safe threshold (typically 3–10 seconds).
What is the effective delayed neutron fraction (β_eff) and why is it critical to reactor safety?
Delayed neutrons are emitted by certain fission products (precursors) milliseconds to minutes after fission, rather than instantaneously. β_eff is the fraction of all fission neutrons that are delayed, weighted by their importance to the neutron chain reaction. For U-235 thermal fission β_eff ≈ 0.0065 (0.65%); for Pu-239 it is lower at ~0.0021. This small fraction is enormously important: it slows the effective neutron reproduction rate from microseconds to ~0.1 seconds, giving control systems time to act. A reactor is prompt critical — and potentially uncontrollable — only when inserted reactivity equals or exceeds β_eff.
How does prompt neutron lifetime affect reactor power transient speed?
The prompt neutron lifetime ℓ is the average time between a neutron's birth and its absorption to cause the next fission. In thermal reactors it is typically 10–100 µs. Smaller ℓ values mean faster potential transients because prompt neutrons cycle more quickly. Fast reactors (ℓ ~ 0.1–1 µs) have inherently faster kinetics than thermal reactors (ℓ ~ 50 µs), which is one reason fast reactor safety analysis demands more sophisticated kinetics models. In practice, delayed neutrons dominate the dynamic response for sub-prompt-critical reactivity insertions regardless of ℓ.