## Total Power Coefficient

**The total power coefficient** – TPC is defined as the change in reactivity per percent change in the reactor power.

α_{P} = Δρ/Δ% power

It is expressed in units of pcm/% power. It is defined for all states, in which the reactor power changes. **The power coefficient** combines all the **Doppler**, **moderator temperature**, and** void coefficients **and is commonly used when reactors are at **power operation (mode 1). **It is due to the fact, at power conditions it is difficult to separate (when the reactor power changes) the **moderator effect** from the** fuel effect** and the **void effect** as well.

In PWRs, the total power coefficient can range, for example, from **-20 pcm/% to -30 pcm/%**. The value of the power coefficient is always negative in core life, but it is more negative at the end of the cycle primarily due to the **decrease in the moderator temperature coefficient**.

It was written the total power coefficients combines all the **Doppler**, **moderator temperature**, and** void coefficients. **It is obvious, it is a significant difference in power coefficients of PWRs and BWRs. The main difference is in the void effect. In following points, the effects and its contribution to the power coefficient of PWRs will be discussed:

**Doppler effect.**About**78%**of power coefficient. In PWRs, the Doppler coefficient can range, for example, from**-5 pcm/°C to -2 pcm/°C**. It seems to be a small value, but it is fact the power changes cause significant changes in the fuel temperature. The changes in the fuel temperature may be of the order of hundreds of**°C**. It must be added the doppler coefficient is also called the**prompt temperature coefficient**because it causes an**immediate response**on changes in fuel temperature. It is of the highest importance in the reactor stability.**Moderator temperature effect.**About**20%**of power coefficient. The value of moderator temperature coefficient usually ranges from 0 pcm/°C to -80 pcm/°C (depending on the boron concentration).**Void effect.**About 2**%**of power coefficient. In pressurized water reactors, the void content of the core may be about one-half of one percent. Therefore this effect is marginal for the power coefficient in**PWRs**.

## Power Defect

**For power reactors** at power conditions the reactor can behave **differently** as a result of the **presence of reactivity feedbacks**. Power reactors are initially started up from **hot standby mode **(subcritical state at 0% of rated power) to **power operation mode** (100% of rated power) by withdrawing control rods and by boron dilution from the primary coolant. During the reactor startup and up to about 1% of rated power, the reactor kinetics is **exponential** as in zero power reactor. This is due to the fact all temperature reactivity effects are minimal.

On the other hand, during further power increase from about 1% up to 100% of rated power, **the temperature reactivity effects play very important role**. As the neutron population increases, the fuel and the moderator increase its **temperature**, which results in **decrease in reactivity** of the reactor (almost all reactors are designed to have the **temperature coefficients negative**).

**The negative reactivity coefficient** acts against the initial positive reactivity insertion and this positive reactivity is **offset** by negative reactivity from **temperature feedbacks**. 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 on the power level 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 is known as **the power defect**. **The power defects** for PWRs, graphite-moderated reactors, and sodium-cooled fast reactors are:

- about
**2500pcm**for PWRs, - about
**800pcm**for graphite-moderated reactors - about
**500pcm**for sodium-cooled fast reactors

**The power defects** slightly depend on the fuel burnup, because they are determined by the **power coefficient** which depends on the fuel burnup. **The power coefficient** combines the **Doppler**, **moderator temperature**, and** void coefficients**. 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**.

It is logical, as the power coefficient acts against power increase, it acts also **against power decrease**. When reactor power is decreased **quickly**, as in the case of **reactor trip**, power coefficient causes a positive reactivity insertion (as a result of the fuel temperature and the moderator temperature decrease) and the initial rod insertion must be sufficient to make the reactor **safe subcritical**.

It is obvious, if the power defect for **PWRs** is about **2500pcm** (about 6 βeff), the control rods must weigh **more than 2500pcm** to achieve the **subcritical condition**. To ensure the **safe subcritical condition**, the control rods must weigh more than 2500pcm plus value of **SDM **(SHUTDOWN MARGIN). The total weigh of control rods is design specific, but, for example, it may reach about 6000pcm. To ensure that the control rods can **safe shut down the reactor**, they must be maintained above a minimum rod height (rods insertion limits) specified in the technical specifications.