Under-moderated vs Over-moderated Reactor

From the moderator-to-fuel ratio point of view, any multiplying system can be designed as:

Under-moderated. Under-moderation means less than the optimum amount of moderator between fuel plates or fuel rods. An increase in moderator temperature and voids decreases the k_eff of the system and inserts negative reactivity. An under-moderated core would create a negative temperature and void feedback required for a stable system.

Over-moderated. Over-moderation means a higher than optimum amount of moderator between fuel plates or fuel rods. An increase in moderator temperature and voids increases the k_eff of the system and inserts positive reactivity. An over-moderated core would create a positive temperature and void feedback. It will result in an unstable system unless another negative feedback mechanism (e.g., the Doppler broadening) overrides the positive effect.

Reactor engineers must balance the composite effects of moderator density, fuel temperature, and other phenomena to ensure system stability under all operating conditions. Most light water reactors are therefore designed as so-called under-moderated, and the neutron flux spectrum is slightly harder (the moderation is slightly insufficient) than in an optimum case. But this design provides an important safety feature. An increase in the moderator temperature results in negative reactivity, which tends to make the reactor self-regulating. It must be added the overall feedback must be negative. Still, local positive coefficients exist in areas with large water gaps that are over-moderated such as near control rods guide tubes.

Another phenomenon associated with an under-moderated core is called the neutron flux trap effect. This effect causes an increase in local power generation due to better thermalization of neutrons in areas with large water gaps (between fuel assemblies or when the fuel assembly bow phenomenon is present). Note that “flux traps” are a standard feature of most modern test reactors because of the desire to obtain high thermal neutron fluxes for the irradiation of materials. Still, it can also occur in PWRs.

On the other hand, also under-moderation has its limits. In general, it causes a decrease in overall k_eff. Therefore more fissile material is needed to ensure criticality of the core. Moreover, there is also a limit on the minimal value of MTC (most negative). It is because the negative temperature feedback also acts against a decrease in the moderator temperature. Consider what happens when moderator temperature is decreased quickly, as in the case of the main steamline break (MSLB – standard initiating event for PWRs). The steamline break causes the steam pressure the saturation temperature in the steam generators to fall rapidly. As a result of the falling saturation temperature in the steam generators, the moderator temperature will rapidly decrease. The rapid moderator temperature drop causes a positive reactivity insertion. The amount of reactivity inserted also depends on the magnitude of the MTC, and therefore it must be limited. The typical values for lower limit are MTC = -80 pcm/°C, but it is a plant-specific value limited in technical specifications.

Moderator-to-fuel Ratio

As was written, the moderator temperature coefficient is primarily a function of the moderator-to-fuel ratio (N_H2O/N_Fuel ratio). The moderator-to-fuel ratio is the ratio of the number of moderator nuclei within the reactor core volume to the number of fuel nuclei. As the core temperature increases, fuel volume and number density remain essentially constant. The volume of moderator also remains constant, but the number density of moderator decreases with thermal expansion. As the moderator temperature increases, the ratio of the moderating atoms (molecules of water) decreases due to the thermal expansion of water (especially at 300°C; see: Density of Water). Its density simply and significantly decreases. This, in turn, causes hardening of neutron spectrum in the reactor core resulting in higher resonance absorption (lower p). The decreasing density of the moderator causes neutrons to stay at a higher energy for a longer period, which increases the probability of non-fission capture of these neutrons. This process is one of three processes that determine the moderator temperature coefficient (MTC). The second process is associated with the leakage probability of the neutrons and the third with the thermal utilization factor.

The moderator-to-fuel ratio strongly influences especially:

Resonance escape probability. An increase in the moderator-to-fuel ratio causes an increase in resonance escape probability. As more moderator molecules are added relative to the number of fuel molecules, then it becomes easy for neutrons to slow down to thermal energies without encountering a resonance absorption at the resonance energies.
Thermal utilization factor. An increase in the moderator-to-fuel ratio causes a decrease in the thermal utilization factor. The value of the thermal utilization factor is given by the ratio of the number of thermal neutrons absorbed in the fuel (all nuclides) to the number of thermal neutrons absorbed in all the material that makes up the core.
Thermal and fast non-leakage probability. An increase in moderator-to-fuel ratio causes a decrease in migration length, which in turn causes an increase in non-leakage probability.

As can be seen from the figure, at low moderator-to-fuel ratios, the product of all the six factors (k_eff) is small because the resonance escape probability is small. At the optimal value of the moderator-to-fuel ratio, k_eff reaches its maximum value. This is the case of so-called “optimal moderation”. At large ratios, k_eff is again small because the thermal utilization factor is small.

Examples: Reactor Stability

Self-regulating properties of nuclear reactor

Operating with a negative MTC is favorable operational characteristics also during power changes. There is an exact energy balance between the primary and secondary circuits at normal operation. Therefore when the operator decreases the load on the turbine (e.g., due to a grid requirement), the steam demand decreases (see the initial electrical output decrease in the picture). At this moment, the reactor will produce more heat than the steam turbine can consume. This disbalance causes the steam pressure the saturation temperature in the steam generators to increase (see II. pressure at the picture). As a result of increasing saturation temperature in the steam generators, the moderator temperature will simply increase (see inlet temperature). Increasing the temperature of the moderator adds negative reactivity, which reduces reactor power (without any operator intervention). As can be seen, to a certain extent, the reactor is self-regulating, and the reactor power may be controlled via the steam turbine and via grid requirements. This feature is limited because the range of allowable inlet temperatures is limited. It is power plant-specific, but in general, power changes of the order of units of % are common.

Power increase: from 75% to 100%

During any power increase, the temperature, pressure, or void fraction change and the reactivity of the core changes accordingly. It is difficult to change any operating parameter and not affect every other property of the core. Since it is difficult to separate all these effects (moderator, fuel, void, etc.), the power coefficient is defined. The power coefficient combines the Doppler, moderator temperature, and void coefficients. It is expressed as a change in reactivity per change in percent power, Δρ/Δ% power. The value of the power coefficient is always negative in core life. Still, it is more negative at the end of the cycle primarily due to the decrease in the moderator temperature coefficient.

Let’s assume that the reactor is critical at 75% of rated power and that the plant operator wants to increase power to 100% of rated power. The reactor operator must first bring the reactor supercritical by inserting a positive reactivity (e.g., by control rod withdrawal or boron dilution). As the thermal power increases, moderator temperature and fuel temperature increase, causing a negative reactivity effect (from the power coefficient), and the reactor returns to the critical condition. Positive reactivity must be continuously inserted (via control rods or chemical shim) to keep the power to be increasing. After each reactivity insertion, the reactor power stabilizes itself proportionately to the reactivity inserted. The total amount of feedback reactivity that must be offset by control rod withdrawal or boron dilution during the power increase (from ~1% – 100%) is known as the power defect.

Let assume:

the power coefficient: Δρ/Δ% = -20pcm/% of rated power
differential worth of control rods: Δρ/Δstep = 10pcm/step
worth of boric acid: -11pcm/ppm
desired trend of power decrease: 1% per minute

75% → ↑ 20 steps or ↓ 18 ppm of boric acid within 10 minutes → 85% → next ↑ 20 steps or ↓ 18 ppm within 10 minutes → 95% → final ↑ 10 steps or ↓ 9 ppm within 5 minutes → 100%

reactor power - 75 to 100 of rated power — Power increase. Let’s assume that the reactor is critical at 75% of rated power and that the plant operator wants to increase power to 100% of rated power.

Power decrease: from 100% to 75%

During any power decrease, the temperature, pressure, or void fraction change and the reactivity of the core changes accordingly. It is difficult to change any operating parameter and not affect every other property of the core. Since it is difficult to separate all these effects (moderator, fuel, void, etc.), the power coefficient is defined. The power coefficient combines the Doppler, moderator temperature, and void coefficients. It is expressed as a change in reactivity per change in percent power, Δρ/Δ% power. The value of the power coefficient is always negative in core life. Still, it is more negative at the end of the cycle primarily due to the decrease in the moderator temperature coefficient.

Let’s assume that the reactor is critical at 100% of rated power and that the plant operator wants to decrease power to 75% of rated power. The reactor operator must first bring the reactor subcritical by inserting a negative reactivity (e.g., by control rod insertion or boric acid addition). As the thermal power decreases, moderator temperature and fuel temperature decrease, causing a positive reactivity effect (from the power coefficient), and the reactor returns to the critical condition. Negative reactivity must be continuously inserted (via control rods or chemical shim) to keep the power to be decreasing. After each reactivity insertion, the reactor power stabilizes itself proportionately to the reactivity inserted.

Let assume:

the power coefficient: Δρ/Δ% = -20pcm/% of rated power
differential worth of control rods: Δρ/Δstep = 10pcm/step
worth of boric acid: -11pcm/ppm
desired trend of power decrease: 1% per minute

100% → ↓ 20 steps or ↑ 18 ppm of boric acid within 10 minutes → 90%→ next ↓ 20 steps or ↑ 18 ppm within 10 minutes → 80% → final ↓ 10 steps or ↑ 9 ppm within 5 minutes→ 75%

References:

Nuclear and Reactor Physics:

J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.

Advanced Reactor Physics:

K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.
K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.
D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2.
E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.

See above:

Reactor Dynamics