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Subcritical Multiplication & Approach to Criticality. B. Rouben McMaster University EP 4D03/6D03 Nuclear Reactor Analysis 2008 Sept-Dec. Contents. We study subcritical multiplication and its application in estimating reactivity in the approach of a reactor to criticality. Reactor Startup.
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Subcritical Multiplication & Approach to Criticality B. Rouben McMaster University EP 4D03/6D03 Nuclear Reactor Analysis 2008 Sept-Dec
Contents • We study subcritical multiplication and its application in estimating reactivity in the approach of a reactor to criticality.
Reactor Startup • Starting up the reactor is a significant procedure in running a nuclear plant, and being able to have an estimate of how far the reactor is from criticality at any time is very important. • Subcritical multiplication provides a way to estimate how far the reactor is from criticality at any time during the startup.
The Reactor Equation • Let us write the diffusion equation in 1 energy group for a configuration with an external neutron source, which could be an independent source used to get a measurable flux during reactor startup, or could even simply be the spontaneous-fission source in the uranium fuel.
Subcritical Reactor • Now, during the approach to critical, the reactor is subcritical. So, without the external source S we would have • We can use this information to rewrite Eq. (2) as
Subcritical Multiplication • Eq. (10) says that the fission source is larger than the external source by a factor of (-1/). It is as if the external source has been multiplied by this factor in the subcritical reactor. • The flux is proportional to S and inversely proportional to : • The smaller the subcriticality, i.e., the smaller the (negative) value of , the larger is the subcritical multiplication.
Interactive Discussion/Exercise • Can the foregoing analysis be applied to a supercritical reactor? Explain why or why not. • What happens physically when an external source is placed in a supercritical reactor? • Can a time-independent situation occur for a supercritical reactor? • Do not turn the page until you have attempted/done this discussion/exercise.
Supercritical Configurations • The analysis of subcritical multiplication depends on subcriticality, i.e., on the reactivity having a negative value: • From a mathematical point of view, a positive reactivity would result in negative values for the fission source and for the flux (Eqs. 10, 11), which is not logical. • Physically, a supercritical reactor cannot give rise to a time-independent situation, even by itself, much less with an external source present. • A reactor can be in steady state only for a zero or negative core reactivity (i.e., only in criticality or subcriticality).
Applying Subcritical Multiplication • The approach to critical most often involves starting from a highly subcritical configuration (a state of “guaranteed shutdown”) and approaching criticality by removing absorption, e.g., withdrawing poison from the moderator. • The fact that the subcritical multiplication increases as the reactor gets closer to critical and that the flux is inversely proportional to can be applied by plotting the inverse flux at (a) detector position(s) against time. • This should give a straight line (assuming a constant rate of change in reactivity), and the time of criticality can be estimated by interpolating to a zero flux.
But How Far Are We? • The method in the previous slide is in traditional use to estimate the time of criticality. • But it does not give an estimate of reactivity at any given time. • The method explained in the following slides can provide a concrete estimate of the core reactivity in a subcritical situation, or at any given time in an approach to critical.
Estimate of Reactivity • We start with Eq. (11): • We may not know the exact value of S, or of f, but we’ll see that that’s O.K. We rewrite Eq. (11) as • K and S are not known, but we want to know .
Estimate of Reactivity (cont.) • Now suppose we add a known (or fairly well known) reactivity to the reactor. This can be the reactivity of a control rod, which we may know from a previous exercise, e.g., add (or subtract) a certain amount of known reactivity, rod. • For instance, in CANDU, we can add/remove water from the liquid zone controllers. It is known that 10% of water is worth about 0.7 mk, 0.7 mk can be added/subtracted by reducing (increasing) the zone-controller fill by 10%. With the new system reactivity, the new flux ’ would be:
Estimating the Reactivity • Dividing Eq. (12) by Eq. (13): • So, for instance, if the flux doubles upon a +0.7-mk reactivity insertion, we can calculate: • This method can be applied if we have a “known” reactivity that we can insert or remove.
Determining a Rod’s Reactivity Worth • Subcritical multiplication can also be used to determine the reactivity worth of a device or rod. This does require having a “calibrated” rod whose reactivity worth is known. • Steps: • Measure the flux 0in a subcritical assembly, without either rod inserted: • Now insert rod with known worth 1 into the core, and measure the flux 1: • Then insert the rod whose reactivity 2 is to be determined, and measure 2:
Interactive Discussion/Exercise • Manipulate Eqs. (15)-(17) to isolate the unknown reactivity 2 in terms of known quantities. • The method of determining the system reactivity or a rod’s reactivity worth by means of subcritical multiplication relies on a certain assumption regarding the effect of the various rods. Can you explain what assumption that is? • Do not turn the page until you have attempted/done this discussion/exercise.
Determining a Rod’s Reactivity Worth • Divide Eq. (15) by Eq. (16): • In this procedure, the measurement can be of the “total” flux, or of the flux at any point, as long as the devices do not (significantly) perturb the flux shape, i.e., the ratio of local flux at the measurement point to the total flux.
