UNIVERSITY OF CAPE TOWN EEE4101F / EEE4103F: Basic Nuclear Physics Problem Set 07 solutions (dga) Due 12:00 (!) Wednesday 29 April 2015 Some solutions are brief, and are pointers towards full solutions. To answer some of these questions you will have to do the course reading, since the necessary material may not have been fully covered in the lectures. 1) Why does k depend on whether the fuel and moderator in the core of a reactor is homogeneous or clumped (ie heterogeneous). SOLUTION: Lumping (or clumping) the fuel, ie keep the fuel separate from the mod- erator allows the neutrons a good chance to moderate all the way from fast to thermal without re-entering the fuel, and being captured by the narrow but high resonance peaks in the cross-=section for 238U(n; γ). This increases the factor p, and hence k.... 2) Calculate the power density of a typical PWR reactor such as Koeberg Unit 1 SOLUTION: Core height H = 3:7 m Height to diameter ratio is typically 1 : 1, thus core radius R = 1:7 m Volume of cylindrical core is V = πR2H = (3:14)(1:7)2(3:7) = 32 m3 Power of PWR is P = 922 MWe ≈ 2766 MWt The cooling system must cope with a power density of p = P=V = 2766=32 = 86 MW m−3 3) What is meant by a reactor poison? SOLUTION: From Wikipedia: A neutron poison (also called a neutron absorber or a nuclear poison) is a substance with a large neutron absorption cross-section, in applications such as nuclear reactors. In such applications, absorbing neutrons is normally an undesirable effect. However neutron- absorbing materials, also called poisons, are intentionally inserted into some types of reactors in order to lower the high reactivity of their initial fresh fuel load. Some of these poisons deplete as they absorb neutrons during reactor operation, while others remain relatively constant. Some of the fission products generated during nuclear reactions have a high neutron ab- 6 sorption cross-section, such as xenon-135 with cross-section σc = 2×10 b, and samarium- 3 149 with σc = 74:5 × 10 b. Because these two fission product poisons remove neutrons from the reactor, they will have an impact on the thermal utilization factor and thus the reactivity. The poisoning of a reactor core by these fission products may become so serious that the chain reaction comes to a standstill. From Wikipedia: Burnable poisons: To control large amounts of excess fuel reactivity without control rods, burnable poisons are loaded into the core. Burnable poisons are materials that have a high neutron absorption cross section that are converted into materials of relatively low absorption cross section as the result of neutron absorption. Due to the burn-up of the poison material, the negative reactivity of the burnable poison decreases over core life. Ideally, these poisons should decrease their negative reactivity at the same rate that the fuel's excess positive reactivity is depleted. Fixed burnable poisons are generally used in the form of compounds of boron or gadolinium that are shaped into separate lattice pins or plates, or introduced as additives to the fuel. Since they can usually be distributed more uniformly than control rods, these poisons are less disruptive to the core's power distribution. Fixed burnable poisons may also be discretely loaded in specific locations in the core in order to shape or control flux profiles to prevent excessive flux and power peaking near certain regions of the reactor. Current practice however is to use fixed non- burnable poisons in this service. Burnable poisons are loaded in fuel, such as gadolinium in the form of Gd2O3. 155 157 The isotopes Ga and Gd have higher σa than U-235 and thus are burned up faster than the U-235 fuel. This will cause f and hence keff to increase with time, and thus slightly over-compensate for the depletion of U-235 with time. 4) Explain how control rods can be used to control the power level of a reactor. SOLUTION: Control rods made of material with a large thermal neutron absorption cross section, eg boron (containing boron-10) or cadmium (containing cadmium-113) are inserted into the core, and so reduce f, the thermal utilization factor, and thus k. Boron may also be added to the coolant in the form of a dissolved boron salt. This is known as a chemical shim, or a boron shim. 5) What is meant by the term \the worth of a control rod." ? SOLUTION: ... The reactivity ρ is defined in terms of k as ρ = (keff − 1)=keff . Reactivity is a fraction, a pure number. The effect of introducing a control rod is to reduce k. If the reactor moves from k to k0, then (k0 − 1 k − 1 k0 − k ∆ρ = ρ0 − ρ = − = k0 k k0k If k = 1 and thus ρ = 0, then ∆ρ = ρ0. Thus the worth of a control rod is the ∆ρ0 that it causes. It is usually measured in units of percent mille (called pcm) which is (10−3)(10−2) = 10−5 In the US, the worth is sometimes measured in dollars. If ∆ρ = β where β = 0:0065 is the delayed neutron fraction, then the worth is said to be one dollar. If the reactivity is ρ = 0 and one dollar worth of control rod is removed, the reactor will go prompt critical, and will usually undergo what is euphemistically termed \disassembly". 6) What is meant by the term prompt critical? SOLUTION: The reactor can achieve criticality, k = 1, with prompt neutrons alone. The results in a very short reactor period, of the order of a fraction of a second, which is too fast for reactor control by control rod insertion. From Wikipedia article on \prompt critical" Criticality: An assembly is critical if each fission event causes, on average, exactly one additional such event in a continual chain. Such a chain is a self-sustaining fission chain reaction. When a uranium-235 (U-235) atom undergoes nuclear fission, it typically releases between 1 and 7 neutrons (with an average of 2.4). In this situation, an assembly is critical if every released neutron has a 1/2.4 = 0.42 = 42% probability of causing another fission event as opposed to either being absorbed by a non-fission capture event or escaping from the fissile core. The average number of neutrons that cause new fission events is called the effective neu- tron multiplication factor, usually denoted by the symbols k-effective, k-eff or k. When k-effective i is equal to 1, the assembly is called critical, if k-effective is less than 1 the assembly is said to be subcritical, and if k-effective is greater than 1 the assembly is called supercritical. Critical versus prompt-critical: In a supercritical assembly the number of fissions per unit time, N, along with the power production, increases exponentially with time. How fast it grows depends on the average time it takes, T , for the neutrons released in a fission event to cause another fission. The growth rate of the reaction is given by: kt=T N(t) = N0e Most of the neutrons released by a fission event are the ones released in the fission itself. These are called prompt neutrons, and strike other nuclei and cause additional fissions within microseconds. [. ] a supercritical assembly is said to be prompt-critical if it is critical without any contribution from delayed neutrons and super-prompt-critical if it is supercritical without any contribution from delayed neutrons[clarification needed]. In this case the time between successive generations of the reaction, T , is only limited by the lifetime of the prompt neutrons, and the increase in the reaction will be extremely rapid, causing a rapid release of energy within a few milliseconds. Prompt-critical assemblies are created by design in nuclear weapons and some specially designed research experiments. 7) Describe the role of delayed neutrons in enabling the control of a nuclear reactor. SOLUTION: See Lilley Ch 8. Or, from Wikipedia: However a small additional source of neutrons is the fission products. Some of the nuclei resulting from the fission are radioactive isotopes with short half-lives, and nuclear reac- tions among them release additional neutrons after a long delay of up to several minutes after the initial fission event. These neutrons, which on average account for less than one percent of the total neutrons released by fission, are called delayed neutrons. The relatively slow timescale on which delayed neutrons appear is an important aspect for the design of nuclear reactors, as it allows the reactor power level to be controlled via the gradual, mechanical movement of control rods. Typically, control rods contain neutron poisons (substances, for example boron or hafnium, that easily capture neutrons without producing any additional ones) as a means of altering k-effective. With the exception of experimental pulsed reactors, nuclear reactors are designed to operate in a delayed-critical mode and are provided with safety systems to prevent them from ever achieving prompt criticality. In a delayed-critical assembly, the delayed neutrons are needed to make k-effective greater than one. Thus the time between successive generations of the reaction, T , is dominated by the time it takes for the delayed neutrons to be released, on the order of seconds or minutes. Therefore the reaction will increase slowly, with a long time constant. This is slow enough to allow the reaction to be controlled with electromechanical control sys- tems such as control rods, and as such all nuclear reactors are designed to operate in the delayed-criticality regime.