Inhour Equation — One of Seven Roots Sets the Stable Period

Principle and mechanism

In short: it is the reactivity–inverse-period relation obtained from the exponential-solution condition of the point kinetics equations, and the largest root determines the stable period.

It follows directly from point kinetics

For constant reactivity and no feedback, the point kinetics equations are:

• dn/dt = [(ρ−β)/Λ]n + Σλ_ic_i
• dc_i/dt = (β_i/Λ)n − λ_ic_i (i = 1–6)

Setting n = n̂e^ωt and c_i = ĉ_ie^ωt, the second equation gives ĉ_i = β_in̂/[Λ(ω+λ_i)]. Substituting into the first and using β − Σλ_iβ_i/(ω+λ_i) = Σβ_iω/(ω+λ_i) yields the inhour equation:

ρ = ωΛ + Σ_i β_iω/(ω+λ_i)

Within the point kinetics assumptions it is exact, not an approximation. Textbooks that use the prompt neutron lifetime ℓ_p (Lamarsh) write ρ = ωℓ_p/(1+ωℓ_p) + [1/(1+ωℓ_p)]Σβ_iω/(ω+λ_i); substituting Λ = ℓ_p/k and ρ = (k−1)/k gives the same equation. Written in terms of the period, ρ ≈ ℓ_p/T + Σβ_i/(1+λ_iT).

Physical meaning of each parameter

At small ω the delayed term Σβ_iω/λ_i is large and "absorbs" most of the reactivity, which is why the period becomes long.

Seven roots, one survivor

Multiplying out all denominators gives a seventh-degree polynomial in ω. The curve ρ(ω) consists of seven branches separated at ω = −λ_i, and because the derivative dρ/dω = Λ + Σβ_iλ_i/(ω+λ_i)² is always positive, each branch increases monotonically. Each branch runs from −∞ to +∞, so the horizontal line ρ = ρ₀ crosses every branch exactly once and all seven roots are real. They are ordered ω₇ < −λ₆ < ω₆ < … < ω₂ < −λ₁ < ω₁.

• ρ > 0: only ω₁ is positive; the other six are negative. In the solution n(t)/n₀ = ΣA_je^ω_jt the negative terms decay, so at long times n(t) ≈ A₁e^ω₁t, and T = 1/ω₁ is the stable period.
• ρ < 0: all seven roots are negative, and ω₁ lies between −λ₁ and 0. The negative stable period is therefore T₁ < −80.6 s, i.e. |T₁| > 1/λ₁ = 80.6 s (ρ = −0.01 → −82.6 s; −0.1 → −80.8 s). This is a property of the long-term dominant mode at constant reactivity without an external neutron source; it does not limit the speed of the prompt drop right after shutdown.
• Leftmost root ω₇: when |ω₇| ≫ λ₆, ω₇ ≈ −(β−ρ)/Λ. It is a prompt transient term that dies out within a few ms and produces the prompt jump and prompt drop.

Solving for ρ = +0.001 (+100 pcm) with the Keepin U-235 six-group data (β = 0.006502) and Λ = 2×10⁻⁵ s, and computing the amplitudes for a step insertion from critical equilibrium, gives:

A₇ ≈ −ρ/(β−ρ), and the sum A₁–A₆ = 1.18 equals the post-prompt-jump value β/(β−ρ). A₁ = 1.447 exceeds 1 because it is the intercept obtained by extrapolating the stable exponential line back to t = 0. For −650 pcm (about −1$) the roots are −0.0119, −0.0249, −0.0950, −0.242, −1.078, −2.950 and −650.3 s⁻¹, all negative, and T₁ = −83.9 s.

Approximations and where they apply

The one-group inhour equation ρ = ωΛ + βω/(ω+λ) is quadratic and has two roots (stable + prompt transient). One-group calculations are common, but the result is sensitive to the choice of λ.

Worked example 1 — period for ρ = 0.002

ρ = 0.002, β = 0.0065, λ = 0.08 s⁻¹, ℓ_p = 2×10⁻⁵ s.

• One-group equation ℓ_pω² + (ℓ_pλ + β − ρ)ω − ρλ = 0 → ω₁ = 0.03554 s⁻¹, T = 28.1 s (matches the Λ-neglected approximation (β−ρ)/(λρ) = 28.1 s); the other root, −225 s⁻¹, is a 4.4 ms prompt transient.
• Delayed neutrons ignored: T = ℓ_p/ρ = 0.010 s, a difference of about 2,800 times.
• For reference, the six-group Keepin constants give 17.3 s.

Worked example 2 — reactivity from a power history

Data: ℓ_p = 1.0×10⁻⁴ s, β = 0.006 and λ = 0.0125 s⁻¹. The power history is: A (0–5 min) constant → B (5–10 min) rises by a factor of 1,000 → C (10–15 min) constant → D (15–25 min) falls to 1/1,000.

• A and C: T = ∞, ρ = 0, k_eff = 1
• B: T = 300/ln1000 = 43.4 s → ρ = ℓ_p/T + β/(1+λT) = 0.00389, k_eff = 1/(1−ρ) = 1.0039
• D: T = −600/ln1000 = −86.9 s → 1 + λT = −0.086 → ρ = −0.070, k_eff = 0.935

The reactivity for D comes out large (12 times β) because the one-group equation diverges at T = −1/λ = −80 s. Solving the same periods with six groups gives about 0.0012 for B and about −0.0039 for D: the one-group result is sensitive to the choice of λ, especially for negative periods near −1/λ.

Where the name "inhour" comes from

"Inhour" is short for "inverse hour". In early reactor work ω was written in h⁻¹, and the reactivity that gives a stable period of one hour (3,600 s) was defined as 1 inhour. With the constants of this note, ρ(ω = 1/3600) = 2.33×10⁻⁵, i.e. about 2.3 pcm (0.36 ¢). Present-day PWR documents mainly use pcm, $ and %Δk/k and hardly ever the inhour unit, but the name survives in the equation. For a sense of scale, 1 dpm (T = 26 s) corresponds to about 160 pcm, and T = 60 s to about 94 pcm.

What it means in a plant

• Principle of reactivity measurement: the positive period method withdraws a control rod slightly, measures the stable period (or doubling time) and converts it to reactivity with the inhour curve to calibrate rod worth. A reactivity computer uses inverse kinetics, solving point kinetics backwards, and in the steady exponential-rise region it gives the same value as the inhour equation.
• Zero-power physics tests: after refueling, the reactivity computer is used to measure the critical boron concentration, the isothermal temperature coefficient and control rod worth, which are compared with design values. The β_eff, Λ and λ_i fed to the reactivity computer are supplied cycle by cycle by the core design.
• Safety analysis: control rod withdrawal and ejection analyses start from point kinetics. Above 1$ the prompt term ωΛ dominates and Λ becomes the key parameter.

Regulatory and design criteria

Reactivity measurement and physics testing are anchored in design and test requirements. For example, in the United States:

• 10 CFR 50 Appendix A: GDC 26 and 27 (reactivity control systems), GDC 28 (reactivity limits).
• NUREG-0800: SRP 4.3 (nuclear design, kinetics parameters), SRP 14.2.1 (initial test program), SRP 15.4.1–15.4.8 (reactivity and power distribution anomalies).
• RG 1.68 (initial test programs for water-cooled reactors): low-power physics test items. Reload physics tests follow ANSI/ANS-19.6.1.
• Technical specifications: reactivity anomaly surveillance requirements rest on comparing measured reactivity with design values.
• Acceptance criteria: the physics of the inhour equation is unchanged; the US reactivity-accident acceptance criteria moved from RG 1.77 (1974) to RG 1.236 (2020).

Easily confused pairs

What the inhour equation gives is the stable period after the transients have died out; the instantaneous period n/(dn/dt) during a transient comes from integrating point kinetics.

Key points at a glance

1. Definition: reactivity–inverse-period relation obtained by substituting n ∝ e^ωt into the point kinetics equations.
2. Equation: ρ = ωΛ + Σβ_iω/(ω+λ_i), ω = 1/T (ℓ_p form: ρ ≈ ℓ_p/T + Σβ_i/(1+λ_iT)).
3. Parameters: Λ prompt neutron generation time, β_i group delayed fractions, λ_i precursor decay constants, ρ inserted reactivity.
4. Roots: six groups → seventh-degree equation, seven roots, asymptotes at ω = −λ_i.
5. ρ > 0: one positive root → stable period T = 1/ω₁; the other six are negative (decaying).
6. ρ < 0: all roots negative; |T| > 1/λ₁ ≈ 80 s (long-term dominant mode).
7. Prompt transient term: ω₇ ≈ −(β−ρ)/Λ, giving the prompt jump and drop.
8. Approximations: one group T ≈ (β−ρ)/(λρ); prompt supercritical T ≈ Λ/(ρ−β).
9. Example: ρ = 0.001 → six-group T = 55 s; prompt only 0.02 s.
10. Inhour: reactivity for a one-hour stable period = 1 inhour (about 2.3 pcm); the basis of the positive period method and the reactivity computer.