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The Grind > FRP2 > Flashcards

FRP2 Flashcards

1
Q

Which are the variables that describe neutron distribution in a reactor?

A

position, time, direction of flight and energy(spectrum)
3 + 2 + 1 + 1 = 7 variables

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2
Q

Why we want to know the neutron distribution in the reactor?

A
  • assess the stability of the fission chain reaction
  • calculate multiplication factor
  • estimate neutron flux
  • compute fuel burn-up (long term behaviour)
  • simulate accident condition (short term behaviour)
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3
Q

Why the diffusion equation is not satisfactory to describe neutron distribution?

A

It assumes high collsion frequency between neutrons (not true) and it is not locally valid near neutron sources, sinks and boundaries.

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4
Q

What is the Boltzmann equation?

A

Also known as Transport equations it was developed for rarified gas but it’s appliable for neutrons. It is intrinsically non-linear but in the case of neutronsthe mutual interaction term can be neglected.
There are two main formulation: integral and integrodifferential. It can be solved both numerically and with MC methods

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5
Q

What are the main assumptions of the problem?

A
  • neutron mass at rest is 939 MeV
  • neutron’s energy is 10 MeV, we’ll say 15 MeV to be conservative
  • No relativistic effects since 15«939
  • neutrons are point particles described by the cross section
  • even if neutrons are fermions the density is too low for Pauli’s prinicple to have a relevant effect
  • Magnetic moment is neglected
  • neutrons are assumed stable since half life in void is minutes order while in therma reactors it’s 10^{-3;-5} and in fast reactor is 10^{-6}
  • No wave behaviour so they are described by position and velocity
  • Heisenberg principle is not a problem
  • Since burn-up is a long scale effect we consider timesteps in wich we consider burn-up constant
  • No temperature effects on cross sections
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6
Q

Why can we consider no wave behaviour for neutrons in the core?

A

pag.1

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7
Q

Does heisenberg principle count for neutrons in the reactor core?

A

pag.2

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8
Q

Why we do not consider scattering between neutrons?

A

pag.3

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9
Q

Why we assume burn-up to be constant when solving Boltzamnn transport equation?

A

Because it would introduce non-linearity since
burn-up -> Number of fissionable nuclei -> number of fissions -> increase burn-up
So we consider steps for increasing burn-up

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10
Q

Is the neutron density a good indicator for the actual neutron distribution?

A

It is a statistical quantity so it is good only if the variance is low. It’s a Markovian process so the neutron distribution is a poissonian. So the std = sqrt(n) = sqrt(10^{10}) = 10^{5}.
This is not true at start up when the neutron density is lower

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11
Q

What kind of variable is the flight direction?

A

A versor \omega with modulus = 1.
It is defined in polar coordinates.
Theta is the polar angle
Phi is the azimuthal angle
pag.4

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12
Q

How is the neutron velocity defined?

A

v- = v-(E,\omega) = v \omega-

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13
Q

What is the neutron angluar density?

A

N(r-,\omega-,E,t)
describes the neutron population in position r- with flight direction \omega-, with energy E at time t
It’s the unknown of Boltzmann equation

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14
Q

What is the neutron density?

A

n(r-,E,t)
describes the neutron population in position r- with energy E at time t.
It is obtained by integrating the neutron angular density in the angle.

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15
Q

What is the neutron flux?

A

phi(r-,E,t) = n(r-,E,t) v(E)
is the product between the neutron density and the neutron velocity v(E).

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16
Q

What is the Scalar angular flux?

A

Phi(r-,\omega-,E,t) = v N(r-,\omega-,E,t)
It is also called scalar flux or simply flux

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17
Q

What is the Density of neutron flux?

A

Phi-(r-,omega-, E,t) = v- N(r-,\omega-,E,t)
It is also called the vectorial flux.

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18
Q

During a differential time dt, how many neutrons with direction \omega- in d\omega- and with energy E in dE will cross the differential area dA?

A

pag.5

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19
Q

What it the total neutron flux?

A

phi(r-,E,t) = n v
Is the integral in 4\pi di (N v = Phi)

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20
Q

What is the neutron current?

A

J-(r-E,t)
is the integral of (\omega Phi(r-\omega-,E,t) ) in d\omega [4\pi]
Also known as net density current, represents the net flux of neutrons corssing a unit area in position r- at time t with energy E

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21
Q

What is the average velocity vector?

A

Is the average respect to all directions of flight
<v(r-,E,t)> =INT[4pi] v-(r-,\omega-,E,t) N(r-,\omega-,E,t)/n(r-,E,t) d\omega
where N(\omega-)/n = probability of neutrons to have \omega- flight direction

This allows us to have a consistent current defintion with the elctrical analogy:

<v-> = J-(r-,E,t) / n(r-,E,t)
</v->

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22
Q

Talk about the crossection.

A

It express the probability of interaction between two particles.
In ur case the interaction between neutron and nuclei.
We write with small sigma(r-,E) and is cm^-1. It’s a pdf per unit length, if multiplied by the velocity it is a pdf per unit time.
specific reaction (n,x) has cross section = sigma_x
We neglect dependency of cross section on Omega introducing the Isotropic Media hypotesis

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23
Q

What is the number of reactions (n,x) during time dt in dV with neutrons of energy dE and flight directions d\omega?

A

N(r,om,E,t) sigma_x v dt dV dE dOmega
If we want to consider multiple reaction we just perform a sum on sigma_i

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24
Q

What kind of reactions can neutron go through?

A
  • Elastic scattering (n,n)
  • Anelastic scattering (n,n)
  • Fission (n,f)
  • production of 2 neutrons (n, 2n)
  • radiative capture (n,gamma)
    We also have independent sources like(alfa,n), spontaneous fission or cosmic rays
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25
Q

What is the probability density function of appearance of a neutron for a source?

A

Q(r-,omega-,E,t)
Is the pdf of appearance of 1 neutron in position r-, with flight direction \omega-, energy E at time t.
the rate of appeareance is
Q dV dOmega dE

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26
Q

What is the transfer function?

A

The transfer function f is a conditional pdf that, knowing that a neutron with energy E’, direction \omega’- and position r’- has reacted somehow, gives the probability of n neutronsto exit with direction \omega- and energy E.
f(r’-, \omega’-, E’ –> omega-, E)
Can be infinitesimal if multiplied by dOmegadE.
It takes into account only absorption (0), scattering (1) and fission (2,3)

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27
Q

How to describe the probability of elastic scattering?

A

The probability that a neutron emerges within dE, dOmega, dt and dV from a scattering is:
pag.6

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28
Q

How does the transfer function modify for isotropic medium (our case)?

A

We do not care anymore about the incident angle but only on the difference between incident and exiting angles, defined as theta_0.
We use mu_0 = cos(theta_0):
c_n = Int( f_n(r-, E –> E, mu_o) dOmega)
For heterogeneous medium we can use a weighted average of the transfer function

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29
Q

What hypotesis did we introduce for fission transfer function and what is it?

A
  • Homogenous medium
  • Neutrons are independent from each other, so they are isotropically distributed
  • All neutrons are prompt, no delayed
    pag.7
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30
Q

What is the Total Transfer function?

A

A transfer function f(r,omega’,E’ –> omega, E) that takes into account all possible reactions.
pag.8

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31
Q

What is the total macroscopic cross section?

A

It’s the sum of fission, scattering and absorption cross sections

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32
Q

Botlzmann equation in Lagrangian form?

A

We start considering some neutrons in dV, flight direction dE, time t and look in t+dt.
Our volume will be moved by v(\omega-)dt. In this new volume in t+dt the neutrons will be due to 3 contributions:
- Not interacted neutrons from dV
- Neutrons colliding towards our volume, direction and energy
- Independent sources.
pag.9

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33
Q

Boltzmann equation in Eulerian form?

A

Starting from the neutron populartion we taylor expand the position
pag.10

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34
Q

What is the Boltzmann equation in Eulerian form express as functionof flux?

A

pag.11

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35
Q

What is the Transport operator?

A

It’s the sum of an algebraic operator (L_sigma = -sigma) a differential operator (L_Omega = Omega- * grad) and an integral operator (L_t=int(sigma’ f dOmega’,dE’))
L = L_sigma + L_Omega + L_t
Pay attention that L_t acts on Phi’, not Phi.
pag.11

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36
Q

What is needed to close the boltzmann equation problem?

A
  • The independent source term Q
  • The geometry
  • The cross sections
  • The transfer function
  • The initial distribution of neutrons N(t=0) = N0
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37
Q

What kind of boundaries can u have in a reactor?

A
  • Boundaries between different materials, so different crossections. neutron angular density and flux are continuous or we have sinks/sources. the derivative may not be continuous though
  • External surfaces, must be convex, to avoid re-entering of neutrons, and isolated. If a source is outside the domain I change the domain
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38
Q

Proof of consistency for Boltzmann equation

A

We perform a balance on the entire reactor integrating for Volume, Energies and flight directions
pag.12

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39
Q

Since L, transport operator, is linear, property will the solution have?

A

Solution can be superposed. If I have a set of Q_i source terms I can obtain a set of phi_i solutions. I fnow I consider Q = sum_i c_i Q_i the soultion is phi = sum_i c_i phi_i where c_i are a dimensionless coefficents.
proof at pag.13

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40
Q

What is a Green function?

A

Given a differential linear operator L acting on a suitable functional space and given the differential equation:
Lu(x-) = f(x-)
then the response of the system to an impulse (Dirac delta) is called Green function G(x-;x0-)
LG(x-;x0-) = \delta(x- - x0-)

41
Q

What is the green function of Boltzmann equation?

A

I get it by considering a point source in all variables, like a dirac delta.
pag.14
… non chiaro

42
Q

How to obtain the integral form opf the Boltzmann equation?

A

Two ways, one more mathematical, one more physical. We choose the physical.
We define the generalized source q:
pag.15
The advantage is that this one can be solved with montecarlo methods.

43
Q

What is the probability of non interaction in the path s’ in the integral form of the Boltzmann equation?

A

For homogeneous medium it’s
p = e^(-s’ \sigma(E))
for non-homogeneous medium it can be calculated in slices so that
p = e^(-int( sigma(r- -s’Omega, E) ds’))

44
Q

What is the Optical path?

A

The optical path or free flight kernel or attenuation term is defined as:
tau(r- -s’\omega- –> r-,E) = int[0,s’]( \sigma(r- - s’‘sigma, E)ds’’

45
Q

What is the Von Neumann Expansion method?

A

Starting from the integral form of B.E. we explicit the generalized source term.
It requires a stationary case and a subcritical system where the steady state is obtained thanks to the independent source Q.
pag.16

46
Q

What are the two components ^Kphi and Q* in the Von Neumann expansion method?

A

Q* = phi0 is the neutron flux that did not experience any collision
^Kphi = phi1 is the neutron flux which has experienced 1 collision.
If we apply it recursively:
^Kphi_i = phi_(i+1).
pag.17

47
Q

What is the alpha eigenvalue model?

A

It studies the criticality of the system through the asyntotic behaviour of the neutron axngular density. It starts from the integro differential formulation and treat it as an initial value problem.
pag.18

48
Q

What Hps does the alpha eigenvlaues method use?

A
  • Q is negligible at criticality

After math passagges
- Eigenvalues are discrete (guarantes convergence)
- Eigenvalues are set to be at the left of b
- Eigenvalues have finite value

49
Q

Does the transport equation always have solutions?

A

It is demonstrated only to have solution for physical cross section values

50
Q

What can we say about eigenvalues in the alpha eigenvalues method?

A

They can be ranked from lower to higher according to the real part and the highest one only has the real part.
This has a physical explanation. The higher term is the dominating one (fundamental eigenvalue) and since the system is not oscillating but exponential it must be real.
Also from his sign we can determine if the system is supercritical (alpha0>0), critical (alpha0=0) or subcritical (alpha0<0)

51
Q

What considerations can be made on the asymptotic case for the alpha eigenvalues method solution?

A

We call Nas the asymptotic solution.
Nas = N0 e^(alpha0 t)
If we substitute Nas in the B.E. I.D. form we get a stationary BE (with one unique solution if Nas != 0) with a modification at the sink term due to an addition of a “temporal cross section”. If alpha0=0 then there is no change, if >0 then cross section increase and means we need to reduce cross section to reach criticality and viceversa for <0

alpha0 / v + sigma

52
Q

What are alpha eigenvalues method issues?

A

It looks at an isolated system and cannot take into account multiple cells close to each other for example.

53
Q

What is K-eigenvalue method?

A

Also callled Iterated Fission Source method, it uses a generational scheme approach. Starting from B.E. in I.D. form. We substitute the fission term with 1/4pi * X * nu
pag.19

54
Q

Chosen b_n with a solution, is it a unique solution?

A

If b_n is the highest eigenvalue.
Let’s consider b_(n-1)<b_n.
Then nu(n-1) = nu/b(n-1) > nu_n.
Then we would have more neutrons that are not getting subtracted anywhere which is unphysical.

Let’s consider b_(n+1)>b_n.
Then nu(n+1) = nu/b(n+1) < nu_n.
We would have less neutrons which does not satisfy stationary B.E.

55
Q

How does the generation scheme works in the K Eigenvalue method?

A

The first generation of neutrons are the ones inside the reactor from the start or produced from an external source. They will undergo fission, scatter or absorption. We can write the transport for this generation.
pag.20

56
Q

Why b and K are equivalent in the K Eigenvalue method?

A

For i > i* (and t>t) we can define a normalized time function f_i(t) that expresses the shape of N_i in time so that N_i is not dependent on time. We can then raccoglierlo and we are left with an infinite sum:
sum_i[i inf] (k^(i-i*) f_i(t)).
If you sub it you get b = k

57
Q

What is the Fission Iterated Source Method?

A

Based on the K eigenvalue method, It is a recursive method
- Start from an initial guess N0(r,omega,E) and K0
- Defin the source for the next generation as:
Q1 = 1/(4pi k0)int[omega’E’] N0 sigma’ v’ f_f domega’ dE’
- Solve B.E. for N1
- Assume that the number of entering neutrons is the same and find k1. We are basically asking the system to be stationary.
We iterate this until k_i-k_(i-1) is below a threshold.
It works well near criticality

58
Q

What kind of numerical methods can be used for solving B.E.?

A
  • Derivative discretization like finite differences or finite elements
  • Series expansions of orthonormal functions
59
Q

How does a series expansion work?

A

A generic function
u(xi, x-) = sum_l [0 inf] U_l(x-) P_l(xi) = sum_l [0 L] U_l(x-) P_l(xi)
(troncation of infinite at L)
U are the coefficents, P the functions

60
Q

How can we use Legendre polynomial expansion to describe a 2d infinite piled slabs reactor?

A

Due to the symmetry we only need two coordinates (x,theta) to describe the system. We can replace mu = cos(theta) so that:
phi(r-, omega-, e) = phi(x, mu, E)
We call s the position along a fixed flight direction. The flux variation along s is only due to the streaming term:
omega- grad(phi) = … = mu dphi/dx
(for sphere or cylinder would be more complicated)
pag.21

61
Q

What are Legendre Polynomials?

A

Starting from indepepndent variable mu (|mu|<1) we write the polynomials P_n:
P_n(mu) = sum_i[0 n] c_(n,i)mu^i
This polynomial are such that
- P_n(1) = 1
- P_n and P_m are orthogonal for n!=m, so they are a basis

62
Q

What are the properties of Legendre Polynomials?

A
  • Recurrence law
    (2n+1) mu P_n = (n+1) * P_(n+1) + n * P_(n-1), for n>=0
  • Addition Theorem
    If theta0 is the angle between two vectors and mu0 = cos(theta0)
    P_n(mu0) = P_n(mu)P_n(mu’) + 2* sum_m[1 n] (n-m)!/(n+m)! P_n^m(mu)P_n^m(mu’)
63
Q

What is P_n aproximation?

A

After Legendre expansion I have a system of infinite equations which is not closed.
The system is obtained by analitical procedure so it has no apporximation but we introduce it now.
We say:
dphi_n/dx = 0 (the higher n equation) This closes the system.
We then choose n, tipically 3 or 5 is enoguh since the flux does not fluctuate that much.

64
Q

In the P_n approx what is Phi0?

A

Since P0 = 1
Phi0 = int4pi = phi(x, E)
This is the scalar flux

65
Q

In the P_n approx what is Phi1?

A

Since P1 = mu
Phi1 = int4pi = J(x, E)
This is the neutron current
The higher order Phi do not have a physical meaning

66
Q

In the P_n approx what is P_1?
What other Hp we need?

A

It can be reconduced to the diffusion equation
pag.22
Hp:
- isotropic source term
- thermal neutrons
- BIlancio dettagliato
- Homogeneous slab
- Group constants

67
Q

How can the diffusion coefficent be written in the framework of the P_1 legendre approx?

A

D(x,E) = 1/(3sigma(1-mu-0*c))
where mu-0 is an average and c is the average number of neutron produced per collision

68
Q

What is Sn approximation?

A

It consists in a discretization of the angle in N slices so that mu is discrete. Higher N means higher accuracy

69
Q

What is multigroup approximation?

A

We can not treat the energy dependence same as the direction dependence mainly due to resonances.
We then split the energy domain in many groups (20/100 groups), ranging from thermal (0.025 eV) up to fast spectrum (10MeV).
We reduce the complexity of the equation deleting the integral but increasing the number of equation that need to be solved, G*(N+1)

70
Q

What group constant we define to get rid of energy integrals in the Lagrange expansion approach?

A
  • Group flux, integral of flux Phi_n
  • Group cross section, int(sigma*Phi)/int(Phi)
  • Group transfer cross section, int(Phi(int(sigma))/int(Phi)
  • Group source term, Int(Q)
    The equation becomes:
    pag.23
    The inconvenient is that we define the groups through the flux which is the unknown of the problem
71
Q

In the Lagrange expansion approach after multigroup approach what Hp can be used to simplify?

A

If we add Hp of homogeneous medium then sigma does not depend anymore on the position and becomes a constant for each energy group. To estimate it we can use an estimation of the flux profile and eventually use a trial and error iterative procedure

72
Q

What is the fine discretization approach?

A

In the Lagrange expansion with multiple groups you do not discretize the cross section but the flux and consider a flat flux in each interval.
Let’s consider 10^3 groups so that phi_n depends only on x, not on E. Then we can directly compute the group constants since phi is a constant and can be simplified.

73
Q

Estimate the computational cost?

A
  • Space: 100 cell per dimension
  • Angle: 2/4 components with Lagrange polynomial
  • Energy: 10^3 with fine discretization
    So the unknows are around 10^8, 10^9.
    Nowadays it can be solved also because the matrix is sparse but it’s not what is done
74
Q

How does the Lgrange expansion multigroup apporach change in three dimension?

A

J is a vector so n=1 results in three equations instead of 1.
Also the group constants values are placed in diagonal matrices since we can’t divide by a vector. We divide them by components resulting in multiple values.

75
Q

How can we solve the French PWR case?

A

We perform different steps with different discretization. small geometry allows for high number of energy groups, large geometry less groups.
- First step is calculating a single cell in an assembly with BE. Taking into acccount control rods absorption (increasing water absorption) we impose symmetric boundary condition and discretization will be
(radial 10, angular 100, vertical 50, energy 300)
- Second step is assembly calculation with BE. Only 26 energy group. The key point is homogenization from previous step to the new step.
- Third step is calculation of all reactor with Diffusion equation with 2 or 3 energy groups. Once again homogenization is key.

76
Q

What is the adjoint problem?

A

Is a way of describing the neutrons in terms of their importance for something that will happen in the future.
If B.E. is the direct problem, the adjoint problem is the inverse one and the I.C. is placed in the future.

77
Q

Which space do we work in for solving the adjoint problem?

A

We work in a Hilbert space, complete and with a norm and inner product. It is generically a complex space but in reactor physics all quantities are real.

78
Q

What is an adjoint operator?

A

Mathematically is an operator such that:
<A phi|psi> = <phi|A^+ psi>

79
Q

When is an operator self adjoint?

A
  • A = A^+
  • They share the domain
    D(A) = D(A^+)
80
Q

Is the L operator adjoint?

A

Since it’s linear we can prove it by verifying if each operator term is adjoint.

81
Q

Is the streaming term operator adjoint?

A

L_Omega = -Omega- grad(
pag.24
no

82
Q

What is the adjoint operator of L_Omega?

A

pag.25

83
Q

Is the cross section term self adjoint?

A

L_sigma = -sigma
Yes since it is just a multiplication by a scalar
pag.26

84
Q

Is the transfer operator adjoint?

A

No. We do not prove it.
His adjoint is:
pag.27

85
Q

What is the transport operator adjoint?

A

L^+ = -L_Omega + L_sigma + L^+_t

86
Q

What is the physical meaning of the adjoint flux?

A

It’s also called importance function and it’s the number of counts collected by the detector from time t to final time T_f due to the original neutron and its progeny.

87
Q

How can the importance be caluclated, which events must be summed?

A

assuming [t,t+dt] small enough that I have only one interaction
- Probability of non interaction in [t, t+dt] * importance in t+dt
- Probability of interaction with medium in [t,t+dt] * importance of neutron exiting integrated in E and Omega
- probability of interaction with detector * importance of detection event
pag.28

88
Q

What boundary condition can I use for the adjoint problem?

A

symmetric to B.E. is
phi^+ = 0
if Omega-*n- >= 0
Which means that an exiting neutorn has no importance

89
Q

What Initial Condition can I use for the adjoint problem?

A

Phi^+(r-, Omega-, E, T_f) = 0 since neutrons produced at final time give no counts

90
Q

How can we understand the meaning of the adjoint flux mathematically?

A

If we consider B.E. and the adjoint problem, multiply the first by Phi^+ and the second by Phi and integrate them we get that all operator aside from Q simplify. We then subtract them term by term:
<1/v dPhi/dt | Phi^+> +
<Phi |1/v dPhi^+/dt> = <Q|Phi^+> - <Phi | Q^+>
If we calculate the first term we get.
d/dt<1/v Phi | Phi^+> = …
pag.29

91
Q

In which case we can use perturbation theory for the adjoint problem? What Hp do we need?

A

If the parameters are not perfectly defined but have some small degree of variations.
Hp:
- Critical system (k-eigenvalue method)
- Stationary case

92
Q

Passagges of perturbation theory for the adjoint problem.

A

pag.30

93
Q

Why use adjoint problem with perturbation theory?

A

Because once the direct and inverse problems are solved calculating a perturbed state is just matter of performing some numerical integration(veryfast). We can then perform easily multiple perturbed configurations.
It is also very accurate

94
Q

What is point kinetic?

A

Using adjoint formalism we want to investigate the time behaviour of the system

95
Q

What Hp is fundamental for point kinetic?

A

The distinction between prompt and delayed neutrons.
nu(E) = nu_p(E) + sum_j nu_j(E)
The neutron fraction beta will be
Beta_j (E) = nu_j(E) / nu(E)

96
Q

How do we change the fission transfer function in the point kinetic derivation?

A

We divide prompt and delayed contribution
pag.32

97
Q

PK derivation

A

pag.33

98
Q

In the prompt jump derivation are CHI and beta function of E or E’?

A

Techniclly yes but Chi is very loosly depepndent on E’ so we neglect this dependence. hence beta is only a function of E as well

99
Q

How our PK formulation can be unified with the fission1 PK?

A

We define phi_0^+ = nu sigma_f E_f.
We also approximate rho = (k*-1) / k assuming k=1 == rho=0

The Grind (8 decks)

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