4/14/2015 Reactor Theory 1. One Group Reactor Equation a) Assumptions {bare reactor} (1) Rx is a homogeneous fuel & moderator (2) Rx consists of one region and has neither a blanket or reflector. b) Diffusion Equation Nuclear Reactor Theory Chapter 6 ∅ − Σ ∅ + = 1 Eq. 1 D – Diffusion coefficient Σ - x section for absorption v- n yield S- source density (n/cm3 –s) = νΣ ϕ Reactor Theory 2 Reactor Theory So we have 1 ∅ − Σ ∅ + Σ ∅ = 0 If the fission source neutrons do not balance the leakage and absorption terms, then the RHS of Eq. 1 is non-zero. Let = { Σ - Σ } then ∅ = − ∅ substituting − ∅ for ∅ In this case we could balance the equation by multiplying the source term by a constant say “1/k ”. We get − ∅ − Σ ∅ + Σ ∅ = 0 Eq.2 If the source is too small then k <1. If the source is too large then k >1. ∅ Solving for k : = ∅ ! ∅ = ! 3 Reactor Theory k = the multiplication factor 4 Reactor Theory Notice that = νΣ ϕ = ηΣ# ϕ = η !$ Σ ϕ ! ∅ = ∅ ! ∅ = #n/fiss ! Numerator = neutrons born via fission Denominator = neutrons lost to leakage and absorption. f= !$ ! Fuel f n per abs = fuel utilization: the fraction of all neutrons absorbed in the reactor that are absorbed in the fuel. = η%Σ ϕ 5 Eq. 3. 6 1 4/14/2015 Reactor Theory C. Multiplication factor in the infinite reactor. Assumptions: 1. All neutrons are absorbed. No neutrons leak. 2. The neutron flux φ is constant everywhere. Is independent of position. Concept: 1. Since all neutrons born are eventually absorbed, then Σ ϕ is the total number of neutrons. 2. Of these %Σ ϕ are absorb in the fuel, and release η%Σ ϕ in the next generation Reactor Theory Multiplication factor in the infinite reactor. Reactor Theory D. Buckling B2 in the Critical bare reactor. Reactor Theory D. Buckling B2 in the Critical bare reactor. Concept: 3. Dividing the number of neutrons in one generation by those in the next gives effective yield/absorp & = '! ( ! ( = η% Fuel utilization 4. & is k for and infinite reactor. 5. Since η and % are constants that depend on the material properties of the reactor, & is the same for a bare reactor as for an infinite reactor of the same composition. 6. Thus & refers to a Rx in which no neutrons leak. 7 1. Since & = '! ( ! ( = η% 8 4. If Rx is critical, k = 1, and = 0, − ∅ + (& −1)Σ ∅ =0 5. Let / = 0! = diffusion area, then (& − 1)∅ − ∅ + = 0 / 2 6. Solving for B (& − 1) = / 2. And the source term = η%Σ ϕ, then = & Σ ϕ 3. Using this in the one-group Rx eqn. {use Eqn. 2 & 3} 1 *+ − ∅ − Σ ∅ + = *, 1 *+ − ∅ − Σ ∅ + (& Σ ∅) = *, 4. If Rx is critical, k = 1, and = 0, − ∅ + (& −1)Σ ∅ =0 9 10 Reactor Theory Reactor Theory Ex 6.1 … Find f and k∞for a mixture of U-235 and sodium (Na) in which the U is present to 1 w/0. Solution: %= = !$ ! = !$ !2 !$ = 3!2 03 !$ $ !$ 67 &6# =atomic concentrations of Na and U So A2 A$ = B2 C$ C2 B$ where But : B$ B$ B2 B C$ V!2 $ C2 V!$ = 1 + B2 in 9:;<=/?9@ H DE=F<+=E,G{IJ@} So: B = 0.01 ⇒ B 2 = 99 Thus: % U = 1 + 99 4 5 4 25!2 Now: % U $ @Z @ [.[[[\ .]Z = 1.48 % = 0.671 Next: & = +% = 2.2 0.671 = 1.48 and LE=MNO9O,:9E?PM,{H⁄JQRS} 11 Thus an infinite reactor of this composition would be super critical. 12 2 4/14/2015 Reactor Theory Reactor Theory c ∅ 2. Rx Eqn. is cd + ∅ = 0 With the general solution ∅ e = f cos e + f sin e Where C1 and C2 are to be determined using Ficks law. II. Solutions to the one group reactor equation. The Slab Reactor 3. l = − Assume: 1. Infinite bare reactor of thickness ‘a’. 2. Is critical (flux in steady-state) a c∅ = cd 0 at x = 0 due to symmetry of slab, so 0 a/2 c∅ Since: ∅ x Finally: and: = f cos ∅ e = f cos 13 Reactor Theory ∅ e = f f:=(e) = 0 and = n e n + = 1,3,5, … + = 1,3,5, … 14 Reactor Theory The Slab Reactor ∅ e = f cos = 0. Thus: cd = −f sin 0 + f cos 0 = 0O,e = 0 So: f = 0! -a/2 c∅ cd The Slab Reactor The value of C1 depends on the reactor power level (P). From the text page 274, +r e + = 1,3,5, … O For a critical reactor only the n = 1 eigenvalue will exist. n f = s t ! +r ∅ e = f cos e O u P = power per unit area {v7⁄IJw } xy =constant conversion factor of 3.2 x 10U n=1 n ∅ e = s n=3 t ! -a/2 0 a/2 n u?:=( ) x 15 16 Reactor Theory Reactor Theory C. Buckling Recall the solution for the slab reactor ∅ e = f cos( e) The square of the first eigenvalue is called the buckling of the reactor. B. Other Reactor Shapes 1. Sphere of radius R ∅(N) = sin(rN0z ) u 4xy Σ z z 2. Infinite Cylinder of radius R ∅ N = Since [.{@\| .}[Z~ l ( ) st y Q y Then 3. Solutions for other shapes listed in table 6.2. c ∅ cd + ∅ = 0 = U c ∅ ∅ cd The RHS is an expression that is proportional to the curvature of the flux, which in turn is a measure of how much the flux curves or “buckles”. 17 18 3 4/14/2015 Reactor Theory Reactor Theory C. Buckling D. Max. to Avg. Flux Ratio Ω (table 6.2) n Notice that = ( ) for a slab, and that decreases as ‘a’ increases. In the limit as ⇒ ∞, =0, φ = constant and thus has no buckle. For the reasons listed below, it is important that the flux distribution in an actual reactor be as flat as possible.. Ω= is a measure of the variation of the flux w/ in a reactor, and the extent to which the max. power density at the center exceeds the avg power density. e.g. For a bare infinite slab: ∅ e = f cos so ∅Jd = f – Reduced power spikes – Uniform fuel burn-up – Prevent ‘hot channel’ effects And / ∅H = U/ f cos( Thus Ω = 0 = 19 Reactor Theory E. Ex 6.2. A bare spherical rx radius R = 50 cm, operates at 100Mw=10\ J/s and Σ = 0.0047?9U . What are the max and avg. flux values? ∅Jd = lim }s ~→[ | t y [ n ( t ) ~ = =} @.d[ = [.[[}{(Z[)w ∅H = }.\d[ @. = 1.27e10Z IJ n nd )Fe = n e , n = 1.57 20 Reactor Theory III. k for a non- infinite reactor Recall a critical reactor has: = | n }st y y 4.18e10Z ∅Jd ∅H IJ U7 U rearranging : which was found for & =1 = 1 Eqn. 1 Since n’s either leak or are absorbed, the relative probability that a n will be absorbed is called the non-leakage probability u . U7 Table 6.2 21 Reactor Theory Reactor Theory III. k for a non- infinite reactor For other then critical conditions we can write: III. k for a non- infinite reactor u = = u = #:%+ =O=:N<F Σ ∅ = #:%+ =O=:N<F:N;<O<F Σ ∅ + ∅ 0 ! 3! ! 03! = 22 = & u Eqn. 2 Comparing Eqn. 1 and 2 & u = 1%:NO?NE,E?O;N<O?,:N. 23 24 4 4/14/2015 Reactor Theory III. k for a non- infinite reactor Ex…Given a reactor bare sphere of radius R = 48.5 cm, with / = 384?9 . What is the probability that a fission neutron will be absorbed? 1 1 u = = = 0.38 1 + / 1 + 384(r0 48.5) table 6.2 38% chance of absorption, 62% chance of leakage. 25 Reactor Theory IV. Thermal Reactors A. Recall that thermal rx’s contain a moderator to slow down fission neutrons to thermal energies. For convenience all materials in the reactor other than fuel are considered moderator. B. Thermal Utilization (f) {formerly the fuel utilization} is the fraction of all neutrons absorbed that are absorbed in the fuel in a thermal reactor. Σ %= Σ 26 Reactor Theory IV. Thermal Reactors D. The resonance escape probability (p) is the probability that a neutron is not absorbed while slowing down by nuclides having absorption resonances above thermal energies. E. Based on previous discussion, the absorption of Σ ∅ thermal neutrons leads to the production of η %Σ ∅ new neutrons in an infinite thermal reactor. The infinite rx multiplication factor is; η %Σ ∅ & = = η %ε Σ ∅ Reactor Theory IV. Thermal Reactors C. Fast fission factor (ε) is defined as the ratio of the total number of fission neutrons produced by both fast and thermal fission to the number produced by thermal fission alone. The value of ε (fast fission factor) ~ 1.02- 1.08. Thus about 2 – 8 % of fission are from fast neutron reactions. 27 Reactor Theory IV. Thermal Reactors F. Although in a thermal reactor most fissions occur with neutrons at thermal energies, all fission neutrons are born at fast energies. Therefore it is customary to describe a thermal reactor by two groups of neutrons (fast and thermal). Thus there are to groups of neutron flux terms: Let ∅ = ∅,<%O=,+<,N:+%;ex+1 And ∅ = ∅,<,<N9O;+<,N:+%;ex+2 28 Reactor Theory IV. Thermal Reactors Substituting and rearranging the diffusion equation leads to (see page 289 of text) The Two Group Critical Equation ( )( ) P<N</ = O+F = 29 =1 = ,<N9O;FE%%=E:+ON<O Σ = +<,N:+OM< Σ 30 5 4/14/2015 Reactor Theory Reactor Theory IV. Thermal Reactors Neutron life cycle = η %εu u# IV. Thermal Reactors /<,,E+Mu = ( probability O+Fu# = = ) ( ) thermal non-leakage = fast non-leakage probability Σ# ∅ → η → ¡ → u# → → u → % → n’s absorbed in fuel Which leads to & u u# = 1for a critical reactor. Or = & u u# = η %εu u# Which is the six factor formula for a thermal reactor! n’s via thermal fission Fast n’s that don’t leak n’s from thermal + fast fission Slow n’s that don’t leak Slow n’s Slow n’s absorb in fuel 31 32 33 34 35 36 6 4/14/2015 Reactor Theory IV. Thermal Reactors Reactor Theory IV. Thermal Reactors 37 Reactor Theory IV. Thermal Reactors 38 Reactor Theory IV. 39 40 Reactor Theory Reactor Theory IV. IV. 41 42 7
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