Reactivity Balance & Reactor Control System K.S. Rajan Professor, School of Chemical & Biotechnology SASTRA University Joint Initiative of IITs and IISc Funded by MHRD Page 1 of 6
Table of Contents 1 MULTIPLICATION FACTOR... 3 2 FIVE- FACTOR AND FOUR- FACTOR FORMULAE... 4 2.1 REACTIVITY... 4 2.1.1 Reactivity coefficients... 5 3 INTRODUCTION TO REACTOR CONTROL SYSTEM... 6 4 REFERENCES/ADDITIONAL READING... 6 Joint Initiative of IITs and IISc Funded by MHRD Page 2 of 6
In this lecture, we shall discuss four-factor formula for multiplication factor along with its utility in determining reactivity in a reactor. The later part of this lecture will include introduction to reactor control system. At the end of this lecture, the learners will be able to (i) understand the four-factor formula for multiplication factor (ii) list the factors influencing the multiplication factor (iii) relate multiplication factor with reactivity (iv) understand the basic functions of a control system 1 Multiplication factor In a nuclear reactor, self-sustaining chain reaction takes place. In other words, once the fission is initiated with neutrons in a critical mass of nuclear fuel, the neutrons obtained after every fission event is used to sustain the reaction, without the use of external source of neutrons. The neutrons produced per fission event must be large enough to sustain the chain reaction, despite the loss of neutrons due to leakage and by absorption in non-fissile materials. A criticality factor or effective multiplication factor (k) is defined to denote the relative number of neutrons produced in successive fission events. In effect, k is the ratio of number of neutrons produced by fission in one generation to the number of neutrons produced by fission in the preceding generation. If a system is critical, the multiplication factor is 1. Supercritical systems have multiplication factors greater than 1, while subcritical systems have multiplication factors lesser than 1. The important parameters to note in this context are: (i) average number of neutrons produced per fission event (ν) (ii) average number of fission neutrons produced per thermal neutron absorbed in the fuel (η) The above two factors are different, but are related. Let us take that during one fission event ν number of neutrons are produced. When these neutrons are absorbed in the fuel, some of them may induce fission while some of them may not. This depends on the ratio of fission to absorption cross sections. For a pure fissile material, the number of fissions produced per neutron absorbed is related to the average number of neutrons produced per fission event as follows:!! =!!!! (1) Joint Initiative of IITs and IISc Funded by MHRD Page 3 of 6
2 Five-factor and Four-factor formulae Apart from η, there are other factors that influence k. For instance, all the neutrons released during a fission event are fast neutrons. There are slowed down by elastic collision with moderator before they induce fission. A fraction of neutrons released that is thermalized can induce fission while the other fraction is captured. This fraction of neutrons released and thermalized is p. This fraction p is called resonance escape probability. The word escape is used as these neutrons escape being captured by fertile material like U-238 and undergo thermalization. Though the fission cross section of U-235 is low in the fast spectrum, the fission cross section is still finite. A part of neutrons that are not thermalized may induce fast fission. The contribution from fast fission is taken as a multiplication factor (ε). The value of ε =1 implies that there is no contribution due to fast fission. Typically the value of ε is taken as 1.02. Neutrons can be lost, if the reactor core is too small. P L is used to denote the neutron loss from core. The maximum value of P L is 1, when the core is large enough to prevent neutron leakage. The effective multiplication factor, k can be related to all the five factors (η, p, f, P L, ε) as k= pηfp L ε (2) Equation (2) is called five-factor formula. If diameter of the core is large enough to prevent any neutron leakage, P L =1 and hence Eq. (2) becomes k inf = pηfε (3) Equation (3) is called four-factor formula. 2.1 Reactivity Recall the discussion that we had at the beginning of the module with respect to effective multiplication factor. The state of a reactor with respect to its criticality can be judged from the value of effective multiplication factor. To maintain a constant power, k must be equal to 1. Value of k lesser than one results in reduction of power, while k>1 lead to power excursions. Hence the deviation of k from one is a significant factor. Reactivity (ρ) is defined as the ratio of deviation of k from 1 to the value of k. Joint Initiative of IITs and IISc Funded by MHRD Page 4 of 6
ρ =!!!! (4) The reactivity (ρ) is expressed in dollars ($). Mean neutron lifetime (l): It is the average time between the emission of a fission neutron and its absorption. The mean neutron lifetime (l) is the average of lifetime of prompt neutrons (l p ) and that of delayed neutrons (l d ). If β is the fraction of delayed neutrons then, l=(1-β)l p +βl d (5) Typically, β = 0.0065; l p is of the order of 10-5 s; l d is 12.7 s Reactor period (T): It is obtained from mean neutron lifetime (l) and multiplication factor (k) as follows: If (k-1)< β; T=βl d /(k-1) (6) If (k-1)>β; T=l/(k-1) (7) Both prompt and delayed neutrons contribute to criticality of a reactor. If a reactor is critical by prompt neutrons alone, it is called prompt critical. With the arrival of delayed neutrons, the reactor is likely to become supercritical. The following can be used distinguish different scenario of criticality: ρ = 0; critical or delayed critical 0< ρ < β: supercritical ρ < β; prompt critical ρ >= β; super prompt critical. Hence for the safe and steady-state operation, ρ must be equal to 0 2.1.1 Reactivity coefficients The reactivity in a system may be influenced by temperature, power, void fraction etc. depending on the reactor configuration and other parameters. The influence of these factors on reactivity is normally expressed in terms of coefficients as defined below: Temperature coefficient of reactivity = α = ρ/δt (8) Joint Initiative of IITs and IISc Funded by MHRD Page 5 of 6
If α is negative, it indicates that an increase in temperature will reduce reactivity (ρ) and multiplication factor (k). The reverse holds good if α is positive. Power coefficient of reactivity = a = ρ/δp (9) If a is negative, it indicates that an increase in power will reduce reactivity (ρ) and multiplication factor (k). The reverse holds good if a is positive. 3 Introduction to Reactor control system As discussed earlier in the lecture, the multiplication factor and hence the reactivity are functions of temperature, concentration of poisons, etc. For a reactor to operate at constant power, a system must be available to correct for changes in reactivity. Also, provisions must be available to modulate reactor power as required. The reactor control system with the following functions takes care of these requirements: (i) (ii) (iii) To maintain the multiplication factor at 1 (i.e. k =1) and dk = 0 during steady state operation. Any changes to dk during the normal course of operation must be adjusted by the system to maintain k at 1. When there is a requirement for increase in power, the control system must allow dk to attain positive value. Similarly, during the requirement of lower power, the system must allow dk to attain negative value. During shutdown, the reactor control system must decrease k by appreciable amount to allow a large negative value for dk to enable rapid shutdown. The above functions may be categorized as regulation (i & ii) and protection functions (iii). 4 References/Additional Reading 1. Nuclear Energy: An Introduction to the Concepts, Systems, and Applications of Nuclear Processes, 5/e, R.L. Murray, Butterworth Heinemann, 2000 (Chapter 19). 2. David Bodansky, Nuclear Energy: Principles, Practices, and Prospects, 2/e, Springer-Verlag, USA, 2004 (Chapter 7) 3. https://canteach.candu.org/content%20library/20050317.pdf Joint Initiative of IITs and IISc Funded by MHRD Page 6 of 6