Moderator Temperature Coefficient (MTC) — Sign Set by the Water-Density Effect, and the Positive MTC at Beginning of Cycle

Principle and mechanism

Density, not temperature, does the work

When the moderator temperature rises in a light-water reactor, the water expands and the hydrogen number density falls. The MTC is set mostly by this density effect; the pure temperature (spectrum-hardening) effect is small. The MTC is therefore linked to the moderator density coefficient α_D = ∂ρ/∂ρ_w by α_M ≈ α_D·(∂ρ_w/∂T), with ∂ρ_w/∂T < 0. A drop in density does three things at once:

1. Less moderating power → the resonance escape probability p falls (negative).
2. Longer slowing-down and diffusion lengths → the non-leakage probability falls (negative).
3. Less absorption in the water and the dissolved boron → the thermal utilization factor f rises (positive). This term grows in proportion to the boron concentration.

The sum of the three sets the sign of the MTC. A PWR places its design point in the under-moderated region on a boron-free basis (moderator-to-fuel ratio smaller than the ratio at the k∞ peak). Terms 1 and 2 then outweigh term 3 and the MTC is negative.

More boron moves the peak to the left

Dissolved boron increases absorption on the moderator side, so the k∞ peak moves toward lower moderator-to-fuel ratios. If the peak moves to the left of the design point, the design point becomes over-moderated and a temperature rise (density drop) actually increases k∞. This is the mechanism behind the positive MTC at beginning of cycle (BOC).

A simple two-factor model (f = 1/(1 + b·x), p = exp(−0.25/x), design point x = 1) puts the boron-free peak at x ≈ 1.96. At 1,500 ppm the peak is at x ≈ 1.06, and at 2,000 ppm it is at x ≈ 0.97, past the design point. This is a conceptual model without leakage or spectrum changes; the actual boron concentration at which the MTC turns positive comes from core design calculations.

What it means in a plant

How the MTC changes over the cycle

Typical ranges for commercial PWRs (illustrative): about 0 to +a few pcm/°F at BOC hot zero power (HZP), about 0 to −10 pcm/°F at BOC full power, and about −30 to −45 pcm/°F at EOC full power.

Both extremes are used in safety analysis

• Least negative (BOC, low power): conservative for heat-up events such as loss of load, loss of feedwater and rod withdrawal at low power, and for the ATWS pressure peak. APR1400 DCD Sections 15.4.1 and 15.4.2 assume the most positive BOC MTC.
• Most negative (EOC): conservative for cooldown events such as a main steam line break, where it drives return to criticality and return to power.
• In normal operation a negative MTC supports power self-regulation and load following. The prompt feedback on the fuel side is the FTC; the combined power feedback of both is the power coefficient.

Minimum temperature for criticality

Reasons for keeping the cold-leg temperature at or above the technical-specification minimum before criticality:

1. At lower temperature the MTC is less negative (possibly positive), so the limit keeps the MTC within the range assumed in the safety analysis.
2. Protection-system instruments operate within their normal measuring range.
3. The pressurizer steam bubble (operable condition) is maintained.
4. The reactor vessel goes critical above the minimum RT_NDT (P-T limits).

Worked examples

• ITC and power coefficient: isothermal temperature coefficient ITC = MTC + FTC. With ITC = −12 and MTC = −10 pcm/K, FTC = −12 − (−10) = −2 pcm/K, and power coefficient = MTC·dT_M/dP + FTC·dT_F/dP = (−10)(2) + (−2)(10) = −40 pcm/%power. With ITC = −6 and MTC = −5 pcm/K, FTC = −1 pcm/K and power coefficient = (−5)(1) + (−1)(5) = −10 pcm/%power.
• MTC and stable period: −0.2 %Δk/k for a 100 °C rise gives α_M = −2 pcm/°C. Cooling from 350 to 320 °C gives Δρ = +60 pcm (+6×10⁻⁴). With the one-group approximation (β = 0.0065, λ = 0.08 s⁻¹) the stable period is T = (β − ρ)/(λρ) = 0.0059/(0.08 × 0.0006) ≈ 123 s, and keff = 1/(1 − ρ) = 1.0006.

Regulatory and design criteria

The criteria below are examples from the United States (US NRC) and the IAEA.

• 10 CFR 50 Appendix A, GDC 11 (reactor inherent protection): in the power-operating range, the prompt inherent nuclear feedback must tend to compensate for a rapid increase in reactivity. Negative MTC and FTC are what this requirement means in practice.
• NUREG-0800 SRP 4.3 (nuclear design): reviews the calculation of reactivity coefficients and their uncertainties.
• STS LCO 3.1.3 (Westinghouse type, NUREG-1431, current Rev. 5): the MTC is kept within the limits in the core operating limits report (COLR). The upper limit applies in MODE 1 and MODE 2 (keff ≥ 1), the lower limit in MODES 1–3. If the upper limit is not met, administrative rod-withdrawal limits are set within 24 hours, otherwise the plant goes to MODE 2 (keff < 1) within 6 hours. If the lower limit is not met, the plant goes to MODE 4 within 12 hours (example from a plant's technical specifications).
• Example limit values: the Ginna COLR (Westinghouse type, Cycle 39) gives a BOL HZP upper limit of +5 pcm/°F (below 70 % RTP) and 0 pcm/°F (70 % and above), and an EOL full-power lower limit of −44.6 pcm/°F. The CE-type STS (NUREG-1432, LCO 3.1.4) sets the positive limit at [+0.5×10⁻⁴ Δk/k/°F] (70 % RTP or below) and [0] (above 70 % RTP). Actual values follow each unit's COLR for each cycle.
• Measurement: in the startup physics tests (HZP), Tavg is changed slightly to measure the ITC, and MTC = ITC − FTC (calculated). Near the end of cycle a surveillance measurement confirms that the lower limit is met.
• LCO 3.4.2 (minimum temperature for criticality): based on Tavg for Westinghouse plants and on Tcold for CE plants and the APR1400, with the value in the technical specifications themselves (APR1400 DC example: Tcold ≥ 548 °F, with increased surveillance below 553 °F).
• IAEA: SSR-2/1 (Rev. 1, 2016) core design requirements; SSG-52 (2019) evaluation of reactivity coefficients.
• Revisions: the STS moved from Rev. 2 (2001) to the current Rev. 5 (2021), but the MTC concept and the LCO structure are the same.

Easily confused pairs

The moderator density coefficient and the void coefficient describe the same density effect through a different variable.

Key points at a glance

1. Definition: α_M = ∂ρ/∂T_M, the change in reactivity per degree of moderator temperature (pcm/°C).
2. Main mechanism: temperature↑ → water density↓ (the pure temperature effect is small).
3. Negative effects: moderating power↓ → p↓; diffusion length↑ → leakage↑.
4. Positive effect: absorption by water and dissolved boron↓ → f↑, proportional to the boron concentration.
5. Design: under-moderated on a boron-free basis, so the MTC is negative in the power-operating range.
6. Why BOC can be positive: large excess reactivity means maximum boron → the k∞ peak shifts left → the design point falls on the over-moderated side.
7. Low temperature and low power: smaller thermal expansion of water makes the MTC less negative → minimum temperature for criticality limit.
8. Countermeasures: burnable absorbers to limit boron, COLR upper limit with HZP measurement, rod-withdrawal limits if the upper limit is exceeded.
9. EOC: boron dilution makes the MTC most negative → conservative for cooldown events (MSLB).
10. Units and signs: convert pcm/°F to pcm/°C (×1.8) before combining values; a cooldown inserts positive reactivity when the MTC is negative.