Reactivity control

Teodora gives Hana a sheet with three columns: what changes reactivity, how fast and by how much, and what the control systems use to counter it. "Every reactor has the same problem," she says. "Fuel burns, xenon comes and goes, temperatures change with power, and every so often we need to shut down fast. A CANDU and a PWR solve it with completely different hardware, and you need to understand both, because our safety analysis people will ask you about both."

What has to be controlled

Reactivity control serves four purposes. It must compensate for slow changes such as burnup and fission product buildup over weeks and months. It must follow intermediate changes such as xenon transients over hours and power manoeuvres over minutes. It must shape the power distribution so that local limits are met. And it must shut the reactor down quickly and keep it shut down under all conditions. Different devices are suited to different purposes, and a well-designed reactor uses each for what it does best.

Control rods and their worth

A control rod or blade contains a strong absorber, such as silver-indium-cadmium alloy, boron carbide or hafnium. Its worth, the reactivity change when it is fully inserted, depends on the flux where it sits. To first order, perturbation theory says that the worth of a small absorber is proportional to the square of the flux at its location, weighted by the importance of neutrons there. Rods near the centre of the core, where flux is highest, are worth much more than rods near the edge. Rods also shadow each other: inserting one rod depresses the flux near its neighbours and lowers their worth.

The worth of a rod partly inserted from the top follows an S-shaped curve. When the tip is near the top of the core, where the axial flux is low, moving it further adds little worth. When the tip is near the core midplane, where the flux peaks, each centimetre adds the most worth. Near the bottom, the differential worth falls again. For a cosine axial flux shape, the differential worth varies as sin² of the relative position, and the integral worth as:

where W is the total worth, x the insertion and H the core height. Measured curves differ from this ideal shape, because the axial flux is not a pure cosine and the inserted rod itself distorts it, but the S shape is always present.

Soluble boron in PWRs

In a pressurized water reactor, boric acid dissolved in the primary coolant provides the main compensation for burnup. Its great advantage is that it is uniform: it absorbs neutrons everywhere in the core without distorting the power distribution. The chemical and volume control system adds borated water or pure water to change the concentration slowly, at rates of a few ppm per minute. At the start of a cycle, the critical boron concentration might be over a thousand ppm; as the fuel burns, the operators dilute the coolant, and the boron falls toward a few tens of ppm at the end of the cycle. The curve of critical boron against burnup is called the boron letdown curve, and predicting it accurately is one of the core designer's main tasks.

The program below calculates an integral rod worth curve, a simplified boron letdown curve and a shutdown margin check.

# Reactivity control: rod worth curves, soluble boron letdown and shutdown margin.
# All values are illustrative teaching numbers.
import math
H = 366.0                         # active core height, cm
W = 1200.0                        # total worth of a control bank, pcm
def integral_worth(x):
    """Worth inserted when the bank tip is x cm into the core (sine-squared differential worth)."""
    return W * (x / H - math.sin(2 * math.pi * x / H) / (2 * math.pi))
print("insertion (cm) | integral worth (pcm) | differential worth (pcm/cm)")
for x in (0, 60, 120, 183, 240, 300, 366):
    dw = W / H * (1 - math.cos(2 * math.pi * x / H))
    print(f"{x:14.0f} | {integral_worth(x):20.0f} | {dw:12.2f}")
# Soluble boron letdown: excess reactivity falls with burnup; boron holds it down.
boron_worth = -9.0                # pcm per ppm (illustrative; becomes less negative with burnup)
excess0, loss_per_efpd, xe_sm = 16500.0, 26.0, 2800.0
print("\n   EFPD | excess (pcm) | critical boron (ppm)")
for d in range(0, 541, 90):
    excess = excess0 - (xe_sm if d > 0 else 0.0) - loss_per_efpd * d
    print(f"{d:7d} | {excess:12.0f} | {max(0.0, excess / -boron_worth):10.0f}")
# Shutdown margin with the most reactive rod stuck out
total_rods, stuck_rod_worth = 6800.0, 1150.0
required_power_defect, margin_required = 1600.0, 1300.0
sdm = total_rods - stuck_rod_worth - required_power_defect
print(f"\nshutdown margin with the strongest rod stuck out: {sdm:.0f} pcm (target at least {margin_required:.0f} pcm)")

insertion (cm) | integral worth (pcm) | differential worth (pcm/cm)
             0 |                    0 |         0.00
            60 |                   33 |         1.59
           120 |                  225 |         4.82
           183 |                  600 |         6.56
           240 |                  945 |         5.11
           300 |                 1157 |         1.89
           366 |                 1200 |         0.00

   EFPD | excess (pcm) | critical boron (ppm)
      0 |        16500 |       1833
     90 |        11360 |       1262
    180 |         9020 |       1002
    270 |         6680 |        742
    360 |         4340 |        482
    450 |         2000 |        222
    540 |         -340 |          0

shutdown margin with the strongest rod stuck out: 4050 pcm (target at least 1300 pcm)

The letdown curve shows the early step from xenon and samarium buildup, then a steady fall as the fuel burns. The real curve is usually curved early in the cycle, rising or flattening as the gadolinia burns out and releases reactivity, which is one reason burnable absorber design and boron letdown are designed together. The cycle ends when the critical boron reaches a small value; after that, the unit can coast down as lesson 8 described.

Shutdown margin

Shutdown margin is the amount by which the reactor would be subcritical if all control rods were inserted, except the single rod of highest worth, which is assumed to stick out. It is calculated at the most reactive condition, usually at hot zero power, with no xenon and with conservative allowances. Technical specifications require a minimum margin, often about 1 300 pcm or more for a PWR, at all times. Because rods are worth less in a core with a harder spectrum and because the most reactive state shifts through the cycle, shutdown margin must be checked at several points in every cycle. The calculation in the program above shows a margin well above a typical requirement; in reload design, a pattern that meets every other goal but fails shutdown margin must be changed.

Reactivity devices in a CANDU reactor

A CANDU reactor has no soluble boron in the coolant during operation and uses a different set of devices, all inserted vertically into the low-pressure moderator between fuel channels rather than into the high-pressure coolant:

The two shutdown systems in a CANDU reactor are fully independent, diverse in design and each able to shut the reactor down on its own for any design-basis event. This design reflects the CANDU's positive coolant void reactivity: in a large loss-of-coolant accident, voiding the coolant adds reactivity, and the shutdown systems must act within a second or two to terminate the power pulse. In a light-water reactor, voiding the coolant removes moderator and reduces reactivity, which is one of the inherent safety features of that design. Both designs meet the same regulatory principle, that the reactor be safely shut down in all events, by different means.

Boiling water reactors use cruciform control blades inserted from below between groups of four assemblies, and no soluble boron during normal operation. Burnup is compensated by burnable gadolinia and by moving blades in a planned sequence through the cycle. Power changes can also be made by changing core flow, because more flow sweeps out steam voids, adding moderator and reactivity. The BWRX-300, as a natural-circulation design without recirculation pumps, cannot use flow control in the same way and relies on control blades and gadolinia for reactivity control.

Power coefficients and the power defect

Raising the power of a reactor heats its fuel and moderator. In a light-water reactor, both the Doppler coefficient and the moderator temperature coefficient are normally negative, so raising power removes reactivity. The total reactivity change from zero to full power is the power defect, which in a PWR is roughly 1 000 to 2 000 pcm. Rods or boron must supply that reactivity to raise power, and when power falls, the same reactivity is released. The power defect is a large share of the reactivity that a PWR must hold in reserve at hot zero power, and it is included in the shutdown margin calculation.

In a CANDU reactor, the power coefficient is small and can be slightly negative or slightly positive depending on burnup and conditions, because the negative fuel temperature coefficient is partly offset by a positive coolant temperature effect. CANDU control systems are designed for this: the zone controllers respond to power changes quickly and the shutdown systems are designed to act for any event in which reactivity rises. Core designers in both countries calculate these coefficients throughout every cycle and show that they remain within the limits assumed in the safety analysis.

Measuring rod worth and reactivity

Calculated rod worths and coefficients are checked by measurement. At the start of each PWR cycle, the operators measure the worth of control banks during the start-up physics tests, often by the boron dilution method, in which boron is diluted at a steady rate while a bank is inserted to hold the core critical, or by the rod swap method, in which a reference bank's measured worth is used to infer the others. A reactivity computer, which solves the inverse point kinetics equations from the measured flux, converts flux changes into reactivity. The measured worths must agree with the predictions within the acceptance criteria of the methodology, typically about ten percent for individual banks; larger differences are investigated before the reactor goes to power.

In a CANDU station, the worths of adjusters, zone controllers and shutoff rods are confirmed by measurements during commissioning and after major outages, and the zone controller response is checked continuously in operation. The reactivity balance, comparing zone levels and device positions with the core simulator's predictions, serves as a daily check on the calculated reactivity of the core.