Source multiplication is a topic that is usually developed early-on in the study of reactors. Unfortunately, it is developed a bit too early. Conventional source multiplication is based on a crude, over simplified, chain reaction model, but then is not revisited later when dealing with a more appropriate model. The use of two quite different models is seldom, if ever, noted ... which surely is not helpful to the student. The first crude model we label as the "generation time model", or the GTM model for short. The second realistic model we label as the "balance equation model", or BEM model for short. Key features of these models are as follows:
1. The GTM model - is based on an average fission neutron. The lifetime of the the average neutron is the population weighted average of the prompt and delayed neutron lifetimes, called the "generation time". And although the generation time varies with the mix of prompt and delayed neutrons in the population, it is of the order of 0.1 second. But because of the skewed mix of prompt and delayed neutrons and the extremely long lifetime of the delayed neutron when including time of precursor decay, the generation time differs greatly from the actual lifetime of either the prompt or delayed neutron. The generation time is taken as the average time between fissions in the chain reaction. The GTM model is typically used to demonstrate that, for exponential power change, it is the relatively long generation time that makes reactors controllable, as compared to the extremely short lifetime of an all prompt neutron model. This is a false conclusion ... generation time is NOT what make the reactor controllable ... as will be discussed in a future essay, generation time is a fiction. As such, the GTM model cannot reproduce operational type reactor behavior. Unfortunately, it is also the GTM model from which the definition of keffective is derived.
2. The BEM model - maintains the identity of prompt neutrons, delayed neutrons, and non-fission source neutrons ... by treating them separately.There is no fictitious average neutron involved. The unit time interval of the model, of the balance equations, is the prompt neutron lifetime. Solving the equations of the BEM model yields the reactor rate equation, the single most important relationship ever discovered for defining, and understanding, transient reactor behavior. Depending on form, the reactor rate is commonly expressed either as the startup rate or as the reactor period. The BEM model also provides the source multiplied form of the power equation, which applies from shutdown to full power. The balance equations are widely used in computer programs that simulate reactor behavior ... because they can reproduce all operational reactor behavior, from shutdown to full power. Of necessity, the BEM model uses a modified multiplication factor called kprompt, which will be discussed in a future essay.
Construction of a set of chain reactions using the GTM model appears as follows:
By convention, this equation is said to define equilibrium subcritical multiplication. Because of certain mathematical substitution involved in its derivation, it is valid only for values of keff that are less than one ... i.e., it is only valid for the subcritical reactor.
The neutron balance equation yields a definition of source
multiplication, which is:
Again, because of a mathematical substitution in derivation, the BEM expression for multiplication contains a restriction, but one that does not effect application to operational conditions. The BEM expression is valid for reactivity conditions less than prompt criticality ... i.e., for reactivity less than +beta, the numerator (beta - rho) is positive and the expression is valid. Thus, for the BEM model, source multiplication is a process that extends from shutdown to full power ... it is not restricted to subcriticality (see the Power Diagram below). This is not surprising since source multiplication is a manifestation of many ongoing chain reactions. The reactor does not operate by one physical process while subcritical and another when critical and beyond. The chain reaction is always the underlying physical process and it applies up to prompt criticality.
Notice especially, as presented in NUKEFACT #2, that the numerator of the BEM power equation is the total neutron source strength (actually the power produced by source neutrons that cause fission). The denominator is the fraction of the total neutron population that consists of source neutrons. This makes sense. The reactor is strictly a machine that multiplies source neutrons. If we know the number of fissions produced by source neutrons, and the fraction of fissions that are due to source neutrons, then the total fissions, and power, is determinable. The inverse of the total neutron source fraction, (beta - rho), is the source multiplication factor, represented as:
Whoever, ever, told you that?
Is this important? Absolutely! Recognizing that source multiplication is always the underlying process provides a unifying element to the topic of reactor behavior. Source multiplication is always the physical process that determines the magnitude of reactor power. The relationship between the total neutron source fraction and the source multiplication factor is absolutely essential to understanding reactor power production.