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  • Neutron reproduction

    Neutron Life Cycle

    x x 00..99

    factor k eff = 1.000

    Thermal Thermal utilization utilization factor “f”factor “f”

    x η

    x x 00..99 Resonance Resonance

    escape escape i i ” ”i i ” ”

    x x 11..0303 Fast fissionFast fission

    probability ”p”probability ”p” What is: • Migration length? Fast fission Fast fission

    factor “factor “εε””

    g g • Critical size? How does the

    t ff t thgeometry affect the reproduction factor?

    1Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

  • Neutron Life Cycley

    Why should we b t th ?worry about these?

    How? f

    How?

    2Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

  • Controlled Fission

    k = fpεη(1-lf )(1-l h l) Not fixed !k fpεη(1 lfast)(1 lthermal) • Thermal utilization factor f can be changed, as an

    Not fixed…!

    g , 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 y g leakages using movable neutron reflectors. • f and p factors change as fuel is burned.p g • f, p, η change as fertile material is converted to fissile material.

    Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

    3

  • 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.

    U d d tiUnder-moderation and

    over-moderationover moderation.

    Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

    4

  • Controlled Fission Time scale for neutron multiplicationTime 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 neutronsTime Average number of thermal neutrons t n

    t + τ knt + τ kn t + 2τ k2n

    nkndn• For a short time dt τ

    nkn dt dn −

    =

    •• Show thatShow that τtkentn )1(0)( −=

    5

    0)( Nuclear Reactor Theory, BAU, Second Semester, 2011-2012

    (Saed Dababneh).

  • Controlled Fission τtkentn )1()( −=

    • k = 1 n is constant (Desired). • k < 1 n decays exponentially

    entn 0)( = 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 = e10 = 22026 in in 11s. s. ( g y p ) • Design the reactor to be slightly subcritical for prompt neutrons. • The “few” “delayed” neutrons will be used to achieve criticality, allowing enough time toallowing enough time to manipulate the control rods (or use shim or …).

    6

    ( ) Cd control rodsCd control rods

    Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

  • Fission Reactors Essential elements:Essential elements: • Fuel [fissile (or fissionable) material]. • Moderator (not in reactors using fast neutrons). Core

    • 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. • Emergency systems (to prevent runaway during failure).

    Chapter 4 in Lamarsh

    7Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

    p

  • Fission Reactors Types of reactors:Types of reactors: Used for what?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 (~1013 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. What are neutron generators?What are neutron generators?

    8Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

    What are neutron generators?What are neutron generators?

  • Fission Reactors What neutron energy?What neutron energy? • Thermal, fast reactors. • Large, smaller but more fuel. What fuel?What fuel? • Natural uranium, enriched uranium, 233U, 239Pu,, , , , Mixtures.

    From converter or breeder reactorHow??? breeder reactor.How???

    9Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

  • Fission Reactors What assembly?What assembly?

    H t d t d f l l d• Heterogeneous: moderator and fuel are lumped. • Homogeneous: moderator and fuel are mixed together.

    I h t it i i t l l t d f• 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?graphite mixture (for example) can not go critical. Why?

    What coolant?What coolant? • Coolant prevents meltdown of the core. • It transfers heat in power reactors. • Why pressurized-water reactors. • Why liquid sodium?

    10Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

  • More on Moderators What moderator?What moderator? 1. Cheap and abundant. 2. Chemically stable. 3. Low mass (high ζ logarithmic energy decrement). 4. High density. 5. High Σs and very low Σa. • Graphite (1,2,4,5) increase amount to compensate 3. • Water (1,2,3,4) but n + p → d + γ enriched uranium. • D2O (heavy water) (1!) but has low capture cross

    section natural uranium, but if capture occurs, produces tritium (more than a LWR).

    • ….. 11Nuclear Reactor Theory, BAU, Second Semester, 2011-2012

    (Saed Dababneh).

  • More on Moderators

    Moderating ratio ≡ s ∑ζ

    g

    Calculate both a ∑HW HW 1212 α+→→+ LiBnB 7*1110

    moderating power and ratio for water, heavy

    B-101010BBy water, graphite, polyethylene and boronboron.

    1/v region

    p y y Tabulate your results and comment.

    12Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

  • More on Moderators

    HW HW 12 12 (continued)(continued)

    Calculate the moderating power and ratio for pure D O ll f D O t i t d ith ) 0 25%D2O as well as for D2O contaminated with a) 0.25% and b) 1% H2O. C t th ltComment on the results. In CANDU systems there is a need for heavy water

    dupgradors.

    Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

    13

  • More on Moderators

    ζnEE −= lnln \ )/ln( \ nEEn =Recall ζnEEn −= lnln ζ

    n =Recall

    After n collisionsAfter n collisions

    )/ln( thf EE After one collision 1 1ln

    2 )1(1ln

    2

    \ + −−

    +=⎥⎦ ⎤

    ⎢⎣ ⎡==∆

    A A

    A A

    E Eu

    av

    ζζ )( thfn =

    ζ

    Total mean free path = n λ f

    Total mean free path = n λs Is it random walk or there is a

    f d di ti ??? Nuclear Reactor Theory, BAU, Second Semester, 2011-2012

    (Saed Dababneh). 14

    preferred direction??? th

  • More on Moderators

    Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

    15

  • More on Moderators

    A ⎞⎛ 21 After one collision.

    Recall (head-on). Then the maximum energy loss is (1-α)E, or αE ≤ E\ ≤ E.

    EE A AE α≡⎟

    ⎠ ⎞

    ⎜ ⎝ ⎛

    + −

    =min\ 1 1

    For an ss--wavewave collision: EEPdEEEP

    E 1)(1)( \\\ =→∴=→∫ Assumptions:Assumptions: 1.1. Elastic scattering.Elastic scattering. E↓Flat-top probability

    E EEPdEEEP

    E )1( )(1)(

    αα − →∴→∫

    1.1. Elastic scattering. Elastic scattering. E↓ 2.2. Target nucleus at rest. Target nucleus at rest.

    E↑ 3.3. Spherical symmetry in Spherical symmetry in

    CMCM EE )1(21

    \ α+= CM.CM.

    Obviously

    2

    ⎪ ⎨ ⎧ ≤≤

    −=≡→ EEE

    E E

    dE dEE

    s s

    s )1( )(

    )( |

    | \ αα

    σσσ

    Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

    16

    ⎪⎩ otherwise dE 0

  • More on Moderators HW HW 13 13 (or (or 66\\)) •• Scattering Kernel.Scattering Kernel.•• Slowing down density.Slowing down density.

    [ ] 2\ 1cos21++ AAE CMθ

    (Re)-verify g yg y

    •• Migration length.Migration length. •• Fermi age and continuous fermi model.Fermi age and continuous fermi model.

    [ ]

    [ ]2 2 cos)1()1(2

    1 )1( cos21

    −++= +

    ++ =

    A AA

    E E CMθααθ

    [ ] 2

    2 22

    )1( sincos

    + −+

    = A A θθ

    )( For doing so, you need to verify and use

    CMA θ1 CM

    CM

    AA A

    θ θθ cos21

    cos1cos 2 ++

    + =

    Nuclear Reactor Theory, BAU, Second Semester, 2011-2012 (Saed Dababneh).

    17

  • More on Moderators HW HW 13 13 (or (or 66\\) ) continued…continued… • Forward scattering is pre

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