Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh). 1 Controlled Fission Note that  is greater than 2 at thermal energies and almost 3 at high energies. These “extra” neutrons are Used to convert fertile into fissile fuel. Plutonium economy. India and thorium. Efficiency of this process is determined by neutron energy spectrum. Variations in 

Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh). 2 Controlled Fission Conversion ratio Conversion ratio CR is defined as the average rate of fissile atom production to the average rate of fissile atom consumption. For LWR's CR  0.6. CR is called BR for values > 1. Fast breeder reactors have BR > 1. They are called “fast” because primary fissions inducing neutrons are fast not thermal, thus η > 2.5 but σ f is only a few barns. Moderator??

3 Controlled Fission Time scale for neutron multiplication Time constant  includes moderation time (~10 -6 s) and diffusion time of thermal neutrons (~10 -3 s). TimeAverage number of thermal neutrons t n t +  kn t + 2  k 2 n For a short time dt Show that Show that Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

4 k = 1  n is constant (Desired). k < 1  n decays exponentially. k > 1  n grows exponentially with time constant  / ( k -1). in 1s. k = 1.01 ( slightly supercritical..! )  e (0.01/0.001) t = e 10 = 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 …). Controlled FissionDangerous Cd control rods Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh). Reactivity.

5 Fission Reactors Essential elements: Fuel (fissile material). Moderator (not in reactors using fast neutrons). Reflector (to reduce leakage and critical size). Containment vessel (to prevent leakage of waste). Shielding (for neutrons and  ’s). Coolant. Control system. Emergency systems (to prevent runaway during failure). Core Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh). Chapter 4 in Lamarsh

6 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 -2.s -1 or higher) for research. Converters and breeders: Convert non-thermally- fissionable material (non-fissile) to a thermally- fissionable material (fissile). ADS. Fusion. Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh). What are neutron generators?

7 Fission Reactors What neutron energy? Thermal, fast reactors. Large, smaller but more fuel. What fuel? Natural uranium, enriched uranium, 233 U, 239 Pu, Mixtures. From converter or breeder reactor. How??? Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

8 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 p and f for example, but a homogeneous natural uranium- graphite mixture 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? Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

What moderator? 1. Cheap and abundant. 2. Chemically stable. 3. Low mass (high  logarithmic energy decrement ). 4. High density. 5. High  s and very low  a. Graphite (1,2,4,5) increase amount to compensate 3. Water (1,2,3,4) but n + p  d +   enriched uranium. D 2 O (heavy water) (1!) but has low capture cross section  natural uranium, but if capture occurs, produces tritium (more than a LWR). ….. 9 More on Moderators Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh).

10 More on Moderators Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh). Moderating ratio  boron Calculate both moderating power and ratio for water, heavy water, graphite, polyethylene and boron. Tabulate your results and comment. Moderating power HW 12 Good absorber, bad moderator. Never consider this only! For a compound? 10 B

Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh). 11 HW 12 (continued) More on Moderators Calculate the moderating power and ratio for pure D 2 O as well as for D 2 O contaminated with a) 0.25% and b) 1% H 2 O. Comment on the results. In CANDU systems there is a need for heavy water upgradors.

Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh). 12 More on Moderators Recall  After n collisions Total mean free path = n s Is it random walk or there is a preferred direction??? After one collision f th

Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh). 13 More on Moderators

Assumptions: 1.Elastic scattering. 1.Elastic scattering. E  2.Target nucleus at rest. 2.Target nucleus at rest. E  3.Spherical symmetry in CM. Recall (head-on). Then the maximum energy loss is (1-  )E, or  E  E \  E. s -wave For an s -wave collision: show that Obviously Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh). 14 More on Moderators HW 13 (or 6 \ ) After one collision.

Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh). 15 More on Moderators (Re)-verify For doing so, you need to verify and use HW 13 (or 6 \ ) continued… Scattering Kernel. Scattering Kernel. Slowing down density. Slowing down density. Migration length. Migration length. Fermi age and continuous fermi model. Fermi age and continuous fermi model.

Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh). 16 More on Moderators HW 13 (or 6 \ ) continued… Forward scattering is preferred for “practical” moderators (small A). laboratory If isotropic neutron scattering (spherically symmetric) in the laboratory frame  average cosine of the scattering angle is zero. Show that

Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh). 17 More on Moderators Spherically symmetric in CM Show that Try to sketch. HW 13 (or 6 \ ) continued… scattering 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 distribution. Angular neutron distribution.

Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh). 18 More on Moderators Moderator-to-fuel ratio  Moderator-to-fuel ratio  N m /N u. Ratio  p   a of the moderator  f  (leakage  ). Ratio  slowing down time  p  f  (leakage  ). T  ratio  (why). Other factors also change. Temperature coefficient of reactivity. Moderator temperature coefficient of reactivity. Self regulation.

Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh). 19 One-Speed Interactions Particular  general. Recall: Neutrons don’t have a chance to interact with each other (review test!)  Simultaneous beams, different intensities, same energy: F t =  t (I A + I B + I C + …) =  t (n A + n B + n C + …)v In a reactor, if neutrons are moving in all directions n = n A + n B + n C + …  R t =  t nv =  t 

Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh). 20  Neutrons per cm 3 at r whose velocity vector lies within d  about . Same argument as before  One-Speed Interactions where

Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh). 21 Multiple Energy Interactions  Neutrons per cm 3 at r with energy interval ( E, E+dE ) whose velocity vector lies within d  about . Generalize to include energy Thus knowing the material properties  t and the neutron flux  as a function of space and energy, we can calculate the interaction rate throughout the reactor. Scalar

Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh). 22 Neutron Current Similarly and so on … Redefine as Scalar Neutron current density From larger flux to smaller flux! Neutrons are not pushed! More scattering in one direction than in the other.

Nuclear Reactors, BAU, 1st Semester, (Saed Dababneh). 23 Net Net flow of neutrons per second per unit area normal to the x direction: In general: Equation of Continuity Rate of change in neutron density Production rate Absorption rate “Leakage in/out” rate Volume Source distribution function Surface area bounding  Normal to A Equation of Continuity