## Reactivity Coefficients – Reactivity Feedbacks

Up to this point, we have discussed the response of the **neutron**

** population** in a **nuclear reactor** to an **external reactivity input**. There was applied an assumption that the level of the neutron population **does not affect** the properties of the system, especially that the neutron power (power generated by chain reaction) is sufficiently **low** that the reactor core does not change its **temperature** (i.e. **reactivity feedbacks may be neglected**). For this reason such treatments are frequently referred to as the **zero-power kinetics**.

However, in an operating **power reactor** the neutron population is always large enough to generated heat. In fact, it is the main purpose of power reactors **to generate large amount of heat**. This causes the temperature of the system changes and material densities change as well (due to the **thermal expansion**).

Demonstration of the** prompt negative temperature coefficient** at the **TRIGA reactor**. A major factor in the prompt negative temperature coefficient for the TRIGA cores is the core spectrum hardening that occurs as the fuel temperature increases. This factor allows TRIGA reactors to operate **safely** during either **steady-state** or **transient conditions**.

Source: Youtube

See also: General Atomics – TRIGA

Because macroscopic cross sections are proportional to densities and temperatures, **neutron flux spectrum** depends also on the density of moderator, these changes in turn will produce some changes in reactivity. These changes in reactivity are usually called the **reactivity feedbacks** and are characterized by **reactivity coefficients**. This is very important area of reactor design, because the reactivity feedbacks influence the** stability of the reactor**. For example, reactor design must assure that under all operating conditions the temperature feedback will be **negative**.

## How negative feedback acts against power excursion

### Example: Change in the moderator temperature.

**Negative feedback** as the moderator temperature effect influences the neutron population in the following way. If the temperature of the moderator is increased, negative reactivity is added to the core. This negative reactivity causes reactor power to decrease. As the thermal power decreases, the power coefficient acts against this decrease and the reactor returns to the critical condition. The reactor power stabilize itself. In terms of multiplication factor this effect is caused by significant changes in the resonance escape probability and in the total neutron leakage (or in the thermal utilisation factor when chemical shim is used).

**↑T**

_{M}⇒ ↓k_{eff}= η.ε. ↓p . ↑f . ↓P_{f }. ↓P_{t }(BOC)**↑T _{M} ⇒ ↓k_{eff} = η.ε. ↓p .f. ↓P_{f }. ↓P_{t } (EOC)**

**Resonance escape probability.** It is known, the resonance escape probability is dependent also on the **moderator-to-fuel ratio**. As the moderator temperature increases the ratio of the moderating atoms (molecules of water) decreases as a result of the **thermal expansion** of water. Its density decreases. This, in turn, causes a **hardening of neutron spectrum** in the reactor core resulting in higher resonance absorption (lower p). Decreasing the 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 two processes (or three if chemical shim is used), which determine the moderator temperature coefficient.

**↑T**

_{M}⇒ ↓k_{eff}= η.ε. ↓p . ↑f . ↓P_{f }. ↓P_{t }(BOC)**↑T _{M} ⇒ ↓k_{eff} = η.ε. ↓p .f. ↓P_{f }. ↓P_{t } (EOC)**

**Thermal utilization factor. **The impact on the thermal utilization factor depends strongly on the amount of boron which is diluted in the primary coolent (chemical shim).** **As the moderator temperature increases the density of water decreases due to the **thermal expansion** of water. But along with the moderator also **boric acid is expanded** out of the core. Since boric acid is a neutron poison, and it is expanding out of the core, positive reactivity is added. The positive reactivity addition due to the expansion of boron out of the core offsets the negative reactivity addition due to the expansion of the moderator out of the core. It is obvious this effect is significant **at the beginning of the cycle** (BOC) and gradually loses its significance as the boron concentration decreases.

**↑T**

_{M}⇒ ↓k_{eff}= η.ε. ↓p . ↑f . ↓P_{f }. ↓P_{t }(BOC)**↑T _{M} ⇒ ↓k_{eff} = η.ε. ↓p .f. ↓P_{f }. ↓P_{t } (EOC)**

**Change of the neutron leakage. **Since both (**P _{f} and P_{t}**) are affected by a change in

**moderator temperature**in a heterogeneous water-moderated reactor and the directions of the feedbacks for both negative, the resulting

**total non-leakage probability**is also sensitive on the change in the moderator temperature. In result, an

**increase in the moderator temperature**causes that the probability of

**leakage increases**. In case of

**the fast neutron leakage,**the moderator temperature influences macroscopic cross-sections for elastic scattering reaction (Σ

_{s}=σ

_{s}.N

_{H2O}) due to the thermal expansion of water, which results in an increase in the moderation length. This, in turn, causes an increase of the leakage of fast neutrons.

- For
**the thermal neutron leakage**there are two effects. Both processes have the same direction and together causes the increase in the thermal neutron leakage. This physical process is a part of the**moderator temperature coefficient (MTC).**- Macroscopic cross-sections for elastic scattering reaction
**Σ**which significantly changes due to the_{s}=σ_{s}.N_{H2O,}**thermal expansion**of water. As the temperature of the core increases,**the diffusion coefficient**(**D = 1/3.Σ**) increases._{tr} - Microscopic cross-section (
**σ**) for neutron absorption changes with core temperature. As the temperature of the core increases, the absorption cross-section decreases._{a}

- Macroscopic cross-sections for elastic scattering reaction

This figure shows the power excursion as a result of positive reactivity on a logarithmic scale. There is a curve without feedback, along with a curve for the same reactivity insertion but for which the effects of negative temperature feedback are included. It can be seen both curves initially follow the same, but as the power becomes larger the curve with feedback becomes concave downward and stabilizes at a constant power. At this point the negative feedback has completely compensated for the initial reactivity insertion.

### Examples: Change in the reactor power

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 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%**

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 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%**

In order to describe the influence of all these processes on the reactivity, one defines the **reactivity coefficient α**. A reactivity coefficient is defined as the change of reactivity per unit change in some operating parameter of the reactor. For example:

α = ^{dρ}⁄_{dT}

The amount of reactivity, which is inserted to a reactor core by a specific change in an operating parameter, is usually known as the **reactivity effect** and is defined as:

dρ = α . dT

**The reactivity coefficients** that are important in power reactors (PWRs) are:

**Moderator Temperature Coefficient – MTC****Fuel Temperature Coefficient or Doppler Coefficient****Pressure Coefficient****Void Coefficient**

As can be seen, there are not only **temperature coefficients** that are defined in reactor dynamics. In addition to these coefficients, there are two other coefficients:

The total power coefficient is the combination of various effects and is commonly used when reactors are at power conditions. It is due to the fact, at power conditions it is difficult to separate the moderator effect from the fuel effect and the void effect as well. All these coefficients will be described in following separate sections. The reactivity coefficients are of importance in safety of each nuclear power plant which is declared in the **Safety Analysis Report** (SAR).

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