What You Need to Know - Reactor Physics
Terms in this set (30)
What are some common fission products, and what are their atomic numbers?
Caesium -133 -->134
Iodine -135, decays in 6 hrs to Xenon -135 (easily goes to 136, most important poison)
Zirconium - 93
molybdenum - 99
Caesium - 137
Technetium - 99
Strontium - 90
Iodine - 131
Promethium - 147
Samarium - 149/151 - poison
How does the buildup of fission products affect performance?
Poisons - absorb neutrons, reduce power, eventually so many poisons in fuel that it is uneconomical, and reactor needs to be refueled.
Products in general - alter thermal behaviour of fuel
Derive an equations to give the concentration of primary fission product buildup. When does saturation occur? What is inventory proportional to at saturation?
Similar to equations for decay, but with absorption terms.
DN/dt = rate of production from fission - decay rate - rate of losses from transmutation/absorption.
Dn/dt = Y(fp)Σ𝑓φ - λN(t) - σaN(t)φ, where Y(fp) is fraction of fission that produces the product, σaN(t)φ is rate of losses from transmutation/further fission.
Simplify λ' = λ +σaφ
Dn/dt = Y(fp)Σ𝑓φ - λ'N(t)
Using Integrating Factor (practice this), becomes
N(t) = (Y(fp)Σ𝑓φ / λ' )* (1 - e^(-λ't))
λ't<<1, increases linearly, λ't>>1, e^(-λ't) = 0, static value (saturation)
Inventory proportional to reactor power at saturation
What are the equations for the iodine -> xenon fission product system? How would you find max/min, or time when it was at a max? When reactor is run at steady state, what are equilibrium values?
For Iodine σa is very small, can assume to be 0.
DIodine(t)/dt = Y(iodine)Σ𝑓φ - λ(iodine)N(t)
DXenon(t)/dt = Y(xenon)Σ𝑓φ + λ(iodine)N(t) - λ(xenon)N(t) - σaN(t)φ
(production from fission and from iodine decay = xenon decay and absorption)
max when one or the other DN/dt is = 0.
Find Equilibrium when dI/dt = 0, dX/dt = 0 and solve to find time.
Derive an equation to show what happens to the population after shutdown?
Assume shutdown instantaneous, at shutdown I = I(0) and X = X(0), (these can be equilibrium amounts if reactor had been running for several days)
di(t)/dt = λi
I(t), I(t) = I(0)
dx(t)/dt = λi
t) - e^λx
use IF to solve, AGAIN DO THIS
X(t) = X0
t) + (λi/(λi - λx))
0) * (e(-λx*t)-e(-λi(t))
first term from decay of Xenon, second from production of xenon from iodine and subsequent decay
How do you calculate cross-sections for composites of materials? e.g. H2O
Σ = N * σ, Where N is number density (atoms/m^3) and σ is microscopic x-section in barns (m^2).
N = (density * Avagadros constant)/Atomic Number
For composites, find macroscopic value for every part, and add as formula. For example ΣH2O = 2* ΣH + ΣO
What is the equation that describes absorption/scattering of a neutron beam in a material? How would you find probability of colliding, probability of leaving (without colliding), and what is the relationship to the mean free path?
Beam intensity at distance x from initial intensity I(0) is I(x) = I(0)e^(-Nσx)
Prob of travelling without collision is I(x)/I(0) = e^(-Nσx), not colliding is just 1- this.
To derive mean free path λ, collision in dx after travelling distance x. Integrate p(x)dx = Σe^(-Σx)dx, to find λ = 1/Σs.
Draw graphs to show the distribution of neutrons as a function of energy for a thermal and fast reactor. What are key values (e.g. average energies)
Fast, spiky (because of resonance) and around MeV range with small trailing towards intermediate. (most neutrons absorbed before getting scattered down)
Thermal - peak at high (2MeV), bigger peak at low (0.025eV), very low at intermediate
range of energies 0.001eV --> 10MeV
What is the equation for collision parameter alpha?
Alpha = ((A+1)/(A-1))^2
What is the equation for the slowing down decrement? Slowing down power? Slowing down ratio?
Slowing down decrement is the mean value of the logarithm of the energy loss ratios ξ = 𝑙𝑛(𝐸/𝐸′) ->
ξ = 1 +(𝛼/(1−𝛼))*𝑙𝑛𝛼, approx as 2/(A + 2/3)
Slowing down power ξΣs
Slowing down ratio ξΣs/Σa
What is k? how do you calculate k from k(infinity)?
k = neutrons in ith +1 generation/ neutrons in ith generation
k = k (inf) * Pnl
What is the eta term (η) for neutron economy?
η = fission neutrons produced/neutrons absorbed = νΣ𝑓(𝐸)/Σ𝑎(𝐸)
What is Vm/Vf (m can be mod or coolant) for a PWR, a AGR, and a Na cooled fast reactor? What are the rough volumes of these reactors and why?
Thermal reactors require thermal neutrons, so need a moderator with a large slowing down ratio ξΣs/Σa, with hence high slowing down power and low absorption.
Fast reactors want no absorption and low as possible scattering to not lose energy, but need cooling to carry away heat/produce electricity.
AGRs have a much bigger volume as graphite is really bad at slowing down neutrons (much higher atomic weight)
PWR - Vm/Vf = 1.95 Volume = 30m^3
AGR - Vm/Vf = 135 Volume = 450m^3
NaF - Vm/Vf = 1.25 Volume = 14m^3
Find the closest way you pack square and hexagonal fuel rods.
Find area of fuel, for square pi
r^2 for hex 3
r^2. Then find area of moderator = area of shape (l^2 for square, 6
root(3)*r^2 for hex) - area fuel. Then you have terms for vm, and vf. Compare these to given vm/vf ratios to solve.
What are the components of the 4 factor formula? What do the components of the individual equations mean? (include dankov correction)
Fast fission factor ε = total number of fission neutrons/total number of fission neutrons from thermal fission
Resonance escape probability p = number of neutrons that reach thermal E, number of neutrons that start to slow down
Thermal or fuel utilisation f = number of thermal neutrons absorbed in fuel/total number of thermal neutrons absorbed
Reproduction factor η(t) = number of fast neutrons produced by thermal fission/number of neutrons absorbed
What is the one delayed group approximation?
Delayed neutrons are produced from only one fission product, spatially averaged over lattice cell
what is the prompt jump approximation?
Initially jump from prompt neutrons as they have short lifetime, but change in delayed slow due to lifetime of precursor.
what is the reactivity change approximation?
first jump from prompts, as they have short lifetime, but delayed respond slowly due to much longer lifetime
what is the rod drop approximation?
rod immediately dropped in. Can find dollar value of rod by using eqn p/B = n0/n1-1
What is buckling? What is the equation for buckling in a cylindrical reactor?
Buckling is a measure of neutron leakage. Can have material buckling (which is from production -absorption in material) or geometric buckling, which is to do with the shape of the reactor. High buckling = lots of leakage, low bulking = lots of absorption etc.
B^2 = (2.405/R)^2 + (pi/H)^2, where R is distance from centreline radially, and H is difference from centreline vertically (max H/2, min -H/2)
How do you use an integrating factor to solve a differential equation? (in general)
1) put in form dy/dx + Py = Q
2) Find I.f. such that I.f. = e^int(Pdx)
3) From general case, Iy = int(I*Q)dx
4) solve this to find y in terms of x
What is neutron lifetime? and equation
𝑙∞ = 1/λ = 1/(Σ𝑎*v(avg velocity))
What is the formula for the area of a hexagon? What is the area for the fuel in a hexagonal lattice?
r^2. A = 3
What is the equation for number of collisions to slow a particle from energy e0 to eN?
n = 1/slowing down decrement * ln (E0/En)
What is l'hopitals rule?
As lim tends to infinity, 1/0 tends to 0. This is useful for the slowing down decrement eqn when A = 1 (hydrogen).
What does tent represent?
Prompt generation lifetime = l/k
What is capital gamma?
Fraction of neutrons that would have been absorbed, that instead fission
What is reactivity?
What is the dancov correction?
Self-Shielding from other fuel pins in close pakced lattice affects neutron population, affects p
What is the average energy and velocity of a thermal neutron?
E = 0.025eV
v = 2200m/s