## Example – Infinite Multiplying System Without Source and Delayed Neutrons

An equation governing the neutron kinetics of the system without source and with the absence of delayed neutrons is **the point kinetics equation** (in a certain form). This equation states that the time change of the neutron population is equal to the **excess of neutron production** (by fission) **minus neutron loss** by absorption** in one prompt neutron lifetime**. The role of prompt neutron lifetime is evident. Shorter lifetimes give simply faster responses to multiplying systems.

If there are neutrons in the system at t=0, that is, if n(0) > 0, the solution of this equation gives the simplest form of point kinetics equation (without source and delayed neutrons):Let us consider that **the prompt neutron lifetime is ~2 x 10 ^{-5}** and k (k

_{∞}– neutron multiplication factor) will be step increased

**by only 0.01%**(

**i.e., 10pcm or ~1.5 cents**), that is k

_{∞}=1.0000 will increase to k

_{∞}=1.0001.

It must be noted such reactivity insertion (10pcm)** is very small** in case of LWRs. The reactivity insertions **of the order of one pcm** are for LWRs **practically unrealizable**. In this case the reactor period will be:

**T = l / (k _{∞ }– 1) = 2 x 10^{-5 }/ (1.0001 – 1) = 0.2s**

**This is a very short period.** In one second, the reactor’s neutron flux (and power) would increase by a factor of e^{5} = 2.718^{5}. In 10 seconds, the reactor would pass through 50 periods, and the power would increase by e^{50} = ……

Furthermore, in the case of fast reactors in which prompt neutron lifetimes are **of the order of 10 ^{-7} seconds**, the response of such a small reactivity insertion will be even more unimaginable. In the case of 10

^{-7}, the period will be:

**T = l / (k _{∞ }– 1) = 10-7 / (1.0001 – 1) = 0.001s**

**Reactors with such kinetics would be very difficult to control.** **Fortunately, this behavior is not observed** in any multiplying system. Actual reactor periods are observed to be considerably longer than computed above, and therefore the nuclear chain reaction can be **controlled more easily**. The longer periods are observed due to the presence of **the delayed neutrons.**

## Interactive chart – Infinite Multiplying System Without Source and Delayed Neutrons

Press the “**clear and run**” button and try to stabilize the power at 90%.

Look at the reactivity insertion you need to insert to stabilize the system (of the order to the tenth of pcm).

Do you think that such a system is controlable?