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Nuclear Reactor Theory, JU, First Semester, (Saed Dababneh). 1 k = fp, Fast from thermal, as defined in HW 11. Fast from fast,. Thermal from fast, p. Thermal available for fuel Thinking QUIZ For each thermal neutron absorbed, how many fast neutrons are produced? Will need this when discuss two-group diffusion. Controlled Fission

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2 x 1.03 Fast fission factor Fast fission factor x 0.9 Resonance escape probability p x 0.9 Thermal utilization factor f x What is: Migration length? Critical size? How does the geometry affect the reproduction factor? Neutron reproduction factor k eff = Nuclear Reactor Theory, JU, First Semester, (Saed Dababneh). Neutron Life Cycle

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3Nuclear Reactor Theory, JU, First Semester, (Saed Dababneh). Neutron Life Cycle Why should we worry about these? How?

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Nuclear Reactor Theory, JU, First Semester, (Saed Dababneh). 4 k = fp (1- l fast )(1- l thermal ) Controlled Fission 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. Not fixed…!

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Nuclear Reactor Theory, JU, First Semester, (Saed Dababneh). 5 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 and over-moderation.

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6 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 Reactor Theory, JU, First Semester, (Saed Dababneh).

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7 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 Reactor Theory, JU, First Semester, (Saed Dababneh). Reactivity.

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8 Fission Reactors Essential elements: Fuel [fissile (or fissionable) 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 Reactor Theory, JU, First Semester, (Saed Dababneh). Chapter 4 in Lamarsh

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9 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 Reactor Theory, JU, First Semester, (Saed Dababneh). What are neutron generators?

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10 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 Reactor Theory, JU, First Semester, (Saed Dababneh).

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11 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 (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? Nuclear Reactor Theory, JU, First Semester, (Saed Dababneh).

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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). ….. 12 More on Moderators Nuclear Reactor Theory, JU, First Semester, (Saed Dababneh).

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13 More on Moderators Nuclear Reactor Theory, JU, First 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

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Nuclear Reactor Theory, JU, First Semester, (Saed Dababneh). 14 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.

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Nuclear Reactor Theory, JU, First Semester, (Saed Dababneh). 15 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

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Nuclear Reactor Theory, JU, First Semester, (Saed Dababneh). 16 More on Moderators

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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: Flat-top probability Obviously Nuclear Reactor Theory, JU, First Semester, (Saed Dababneh). 17 More on Moderators After one collision.

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Nuclear Reactor Theory, JU, First Semester, (Saed Dababneh). 18 More on Moderators (Re)-verify For doing so, you need to verify and use HW 13 (or 6 \ ) 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.

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Nuclear Reactor Theory, JU, First Semester, (Saed Dababneh). 19 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

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Nuclear Reactor Theory, JU, First Semester, (Saed Dababneh). 20 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.

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Nuclear Reactor Theory, JU, First Semester, (Saed Dababneh). 21 More on Moderators Moderator-to-fuel ratio Moderator-to-fuel ratio N m /N u. Ratio p 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. Self regulation.

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Nuclear Reactor Theory, JU, First Semester, (Saed Dababneh). 22 One-Speed Interactions Particular general. Recall: Neutrons dont have a chance to interact with each other (BAU 2007 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

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