## Under-moderated vs Over-moderated

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

**Under-moderated**. Under-moderation means that there is**less than optimum**amount of**moderator**between fuel plates or fuel rods. An increase in moderator temperature and voids decreases 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 that there is**higher than optimum**amount of**moderator**between fuel plates or fuel rods. An increase in moderator temperature and voids increases 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 of 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 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, but** local positive coefficients** exists in areas with **large water gaps** that are over-moderated such as near control rods guide tubes.

Another phenomenon associated with under-moderated core is called the **neutron flux trap effect**. This effect causes an increase in local power generation due better thermalisation of neutrons in areas with large water gaps (between fuel assemblies or when 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, but basically it can occur also 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 due to the fact the negative temperature feedback acts also against 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 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 depends also on a magnitude of the MTC and therefore it must be limited. Typical values for lower limit is 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 volume of a reactor core 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 as a result of the

**thermal expansion of water**(especially at 300°C; see: Density of Water). Its density simply and significantly decreases. This, in turn, causes a

**hardening of neutron spectrum**in the reactor core resulting in

**higher resonance absorption**(lower p). Decreasing density of the moderator causes that neutrons 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, which 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**moderator-to-fuel ratio causes**an increase in resonance escape probability. As more moderator molecules are added relative to the amount of fuel molecules, than 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**moderator-to-fuel ratio causes**a decrease in 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 optimal value of **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

**negative MTC**is favorable operational characteristics also during

**power changes**. At normal operation there is an exact

**energy balance**between the primary circuit and secondary circuit. 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 at 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 also 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.

**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 but is more negative at the end of the cycle primarily due to the decrease in the moderator temperature coefficient.

Let 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 insertion of 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. In order to keep the power to be increasing, **positive reactivity must be continuously inserted** (via control rods or chemical shim). After each reactivity insertion, the reactor power **stabilize 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%**

**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 but is more negative at the end of the cycle primarily due to the decrease in the moderator temperature coefficient.

Let 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 insertion of a negative reactivity (e.g. by control rod insertion or boric acid addition). As the thermal power decreases, moderator temperature and fuel temperature decrease as well, causing a positive reactivity effect (from the power coefficient) and the reactor returns to the critical condition. In order to keep the power to be decreasing, **negative reactivity must be continuously inserted** (via control rods or chemical shim). After each reactivity insertion, the reactor power stabilize 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%**

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