Neutron reproduction. factor ε. k eff = Neutron Life Cycle. x η

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1 Neutron reproduction factor k eff = What is: Migration length? Critical size? How does the geometry affect the reproduction factor? x 0.9 Thermal utilization factor f x 0.9 Resonance escape probability ii p x 1.03 Fast fission factor ε Neutron Life Cycle x η 1

2 Neutron Life Cycle Why should we worry about these? f How?

3 Controlled Fission k = fpεη(1-l fast )(1-l l thermal ) Not fixed! Thermal utilization factor f can be changed, as an example, by adding absorber to coolant (PWR) (chemical shim, boric acid), or by inserting movable control rods in & out. Poison. Reactors can also be controlled by altering neutron leakages using movable neutron reflectors. f and p factors change as fuel is burned. f, p, η change as fertile material is converted to fissile material. 3

4 Controlled Fission Attention should be paid also to the fact that reactor power changes occur due to changes in resonance escape probability p. If Fuel T, p due to Doppler broadening of resonance peaks. Under-moderation d and over-moderation moderation. 4

5 Controlled Fission Time scale for neutron multiplication Time constant τ includes moderation time (~10-6 s) and diffusion time of thermal neutrons (~10-3 s). Time Average number of thermal neutrons t n t+τ τ kn t + τ k n For a short time dt Show that n ( t ) dn dt = = kn n τ n 0 e ( k 1) t τ 5

6 Controlled Fission n ( t ) = n e 0 ( k 1) t k = 1 n is constant (Desired). Reactivity. k < 1 n decays exponentially. k > 1 n grows exponentially with time constant τ / (k-1). k = 1.01 (slightly( supercritical..!) ) e (0.01/0.001)t = e 10 = 06 in 1s. Design the reactor to be slightly subcritical for prompt neutrons. The few delayed neutrons will be used to achieve criticality, allowing enough time to manipulate the control rods (or use shim or ). Cd control rods τ 6

7 Fission Reactors ssential elements: Fuel [fissile (or fissionable) material]. Core Moderator (not in reactors using fast neutrons). Reflector (to reduce leakage and critical size). Containment vessel (to prevent leakage of waste). Core catcher. Shielding (for neutrons and γ s). Coolant. Control system. mergency systems (to prevent runaway during failure). Chapter 4 in Lamarsh 7

8 Fission Reactors Types of reactors: Used for what? Power reactors: extract kinetic energy of fragments as heat boil water steam drives turbine electricity. Research reactors: low power (1-10 MW) to generate neutrons (~10 13 n.cm -.s -1 or higher) for research. Converters and breeders: Convert non-thermallyfissionable material (non-fissile) to a thermallyfissionable material (fissile). ADS. Fusion. What are neutron generators? 8

9 Fission Reactors What neutron energy? Thermal, fast reactors. Large, smaller but more fuel. What fuel? Natural uranium, enriched uranium, 33 U,, 39 Pu, Mixtures. How??? From converter or breeder reactor. 9

10 Fission Reactors What assembly? Heterogeneous: moderator and fuel are lumped. Homogeneous: moderator and fuel are mixed together. In homogeneous systems, it is easier to calculate l p and f for example, but a homogeneous natural uranium- graphite mixture (for example) can not go critical. Why? What coolant? Coolant prevents meltdown of the core. It transfers heat in power reactors. Why pressurized-water reactors. Why liquid sodium? 10

11 More on Moderators What moderator? 1. Cheap and abundant.. Chemically stable. 3. Low mass (high ζ logarithmic energy decrement). 4. High density. 5. High Σ s and very low Σ a. Graphite (1,,4,5) increase amount to compensate 3. Water (1,,3,4) but n + p d + γ enriched uranium. D O (heavy water) (1!) but has low capture cross section natural uranium, but if capture occurs, produces tritium (more than a LWR)... 11

12 More on Moderators ζ s Moderating ratio HW * 7 a B + n B Li + α Calculate both moderating power and B-10 B ratio for water, heavy 1/v region water, graphite, polyethylene y and boron. Tabulate your results and comment. 1

13 More on Moderators HW 1 (continued) Calculate the moderating power and ratio for pure D O as well as for D O contaminated t with a) 0.5% and b) 1% H O. Comment on the results. In CANDU systems there is a need for heavy water upgradors. 13

14 More on Moderators \ ln( / ) ln \ nζ n n = ln n = ζ Recall n = ln( f ζ / th ) u After n collisions After one collision = ζ = ln \ av = 1+ ( A 1) A ln A 1 A + 1 Total mean free path = n λ s Is it random walk or there is a preferred direction??? f th 14

15 More on Moderators 15

16 More on Moderators A 1 A + 1 After one collision. \ Recall min = α (head-on). Then the maximum energy loss is (1-α), or α \. For an s-wave collision: 1 \ \ \ P ( ) d = 1 P ( ) = α (1 α) Assumptions: Flat-top probability \ 1 = (1 +α) Obviously σ s ( ) dσ s σ s ( ) = (1 α) d 0 \ α 1. lastic scattering.. Target nucleus at rest. 3. Spherical symmetry in CM. otherwise 16

17 HW 13 (or 6 \ ) (Re)-verify \ = A More on Moderators Scattering Kernel. Slowing down density. Migration length. Fermi age and continuous fermi model. CM A cos θ 1 = (1 + α) + (1 α) cosθ ( A + 1) [ ] cosθ + A sin θ [ ] CM = ( A + 1) For doing so, you need to verify and use cosθ = A Acos θ CM Acosθ CM 17

18 More on Moderators HW 13 (or 6 \ ) continued Forward scattering is preferred for practical moderators (small A). If isotropic neutron scattering (spherically symmetric) in the laboratory frame average cosine of the scattering angle is zero. Show that µ = cos ( θ ) = 3A 18

19 More on Moderators HW 13 (or 6 \ ) continued dσ s dω CM 1 = σ s ( θ ) = σ s ( 4π Spherically symmetric in CM ) CM Show that σ σ ( ) 4π ( A + A 1+ A 1 s s ( θ ) = 1 cosθ cosθ CM 3 CM + 1) Neutron scattering is isotropic in the laboratory system?! valid for neutron scattering with heavy nuclei, which is not true for usual thermal reactor moderators (corrections are applied). Distinguish from Angular neutron distribution. ib ti 19

20 More on Moderators Moderator-to-fuel ratio N m /N u. Self regulation. Ratio p Σ a of the moderator f (leakage ). Ratio p f (leakage ). T ratio (why). Other factors also change. Temperature coefficient of reactivity. Moderator temperature coefficient of reactivity. 0

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