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NeutronDiffusion.jl
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1072 lines (865 loc) · 37 KB
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### A Pluto.jl notebook ###
# v0.17.7
using Markdown
using InteractiveUtils
# ╔═╡ 1addcc10-5304-11eb-06d7-e1c69bd44c54
using Plots # plotting backend: takes a sec for first compile
# ╔═╡ 54f41f50-530c-11eb-1a8e-0b08dda08164
md"# Neutron Diffusion
*Modeling the diffusion of a U-235 sample*
Under what conditions will a fission chain reaction be sustained in a uranium 235 sample? Let's assume that the neutron density, $n(\vec{r}, t)$ obeys ordinary diffusion theory:
$\frac{\partial}{\partial t} n(\vec{r}, t)=D \nabla ^{2}n+Cn$
Where $D$ is the diffusion constant and $C$ is the creation rate for neutrons. For $U ^{235}$, $D \approx 10 ^{5} m ^{2}/s$ and $C \approx 10 ^{8} s ^{-1}$. To simplify the analysis, consider a one-dimensional system of length $L$, so the diffusion equation is
$\frac{\partial}{\partial t} n(x, t)=D \frac{\partial ^{2}n}{\partial x ^{2}}+Cn$
with $-L/2≤x ≤L/2$. If all neutrons that reach the boundary escape from the system, then
$n(x=-L/2,t)=0; \hspace{1em} n(x=L/2,t)=0$
are the (Dirchlet) boundary conditions.
---"
# ╔═╡ 9a0fece0-530c-11eb-15bd-29ebb4adb032
md"The struct `Args` contains our parameters:"
# ╔═╡ c3554ecf-b07b-4b21-9485-1a5e37745a66
Base.@kwdef struct Args
# Physical parameters
D = 1e5 # Diffusion constant for U-235 [m²/s]
C = 1e8 # Creation rate for U-235 [1/s]
L = 1.0 # Lenth of our 1D space [m]
# Parameters for our simulation
N = 100 # Number of grid points
τ = 1e-10 # Time step [s]
end
# ╔═╡ b539f410-5308-11eb-16eb-5354e7f4bae2
md"Check stability of FTCS algorithm:"
# ╔═╡ db60f84b-2d63-41fd-8b29-99d2fbd39e6c
let args = Args(), h = args.L / (args.N - 1) # h → grid spacing
args.τ ≤ h^2 / 2args.D
end
# ╔═╡ 115a22b0-52ff-11eb-288b-f1f22602c21b
md"The reaction will grow exponentially if the critical length, $L_c$, is less than the length of our 1-D geometry. This relation establishes the *critical mass* of our 1-D geometry: if $L<L_c$, then the flux of neutrons escaping at the boundary will damp out the neutron density."
# ╔═╡ fcf86f10-52ff-11eb-30b7-bb29ca63f98d
let args = Args()
(L_critical = π * √(args.D / args.C)) < args.L
end
# ╔═╡ 65f6b782-530d-11eb-1105-c1ede0ba670a
md"## FTCS algorithm
This function advances the state of the neutron density vector using a Forward in Time, Centered in Space algorithm (basically averaging neighboring points to solve the diffusion equation iteratively)"
# ╔═╡ 895801fb-e8ca-4290-9cb2-e0b0ee59a265
function step!(ρ::Vector, args::Args)
""" FTCS step - see Garcia's Numerical Methods for Physics """
h = args.L / (args.N - 1) # grid spacing
# Dirchlet Boundary Conditions are n[1] = n[end] = 0
ρ[2:end-1] = ρ[2:end-1] .+
( args.D * args.τ / h^2 ) * ( ρ[3:end] .+ ρ[1:end-2] .- 2ρ[2:end-1] ) .+
args.C * args.τ * ρ[2:end-1]
end
# ╔═╡ 6a57f8ac-cb6e-4e0d-bba0-e7c3e5e04e73
md"We'll make another convenience struct for position and neutron density arrays. This one will be mutable so we can update the density element."
# ╔═╡ 8d6b2ab3-94a0-42d6-9137-6ca786237a26
mutable struct NeutronDiffuser
x::Vector # X positions
ρ::Vector # Neutron Density
end
# ╔═╡ 37f699c3-9af7-4cef-9cdd-0f1ef241dc63
md"We'll also make another helper function `neutrondiffuser` to make sure we initialize our `NeutronDiffuser` struct correctly with respect to our `Args` and some initial condition."
# ╔═╡ 5b4c4cb3-fa31-43c0-ad47-d09dd32f7a29
function neutrondiffuser(args::Args; ic=:delta)
x = range(-args.L/2, args.L/2; length=args.N)
ρ = zeros(args.N)
# Our Initial Condition is a delta spike in the middle
ic == :delta && (ρ[args.N÷2] = args.N / args.L)
NeutronDiffuser(x, ρ)
end
# ╔═╡ 41758b00-e8a6-4ad4-80d0-9a57460bb739
md"We can iterate over a given number of steps, producing a given number of data and writing that data to a $N×M$ matrix for neutron density, and a vector of length $M$ for time, where $M$ is the total number of samples we'll record (which is less than the total number of steps we'll take; perhaps a rate of one recording to every 5-10 FTCS steps) and $N$ is the number of grid points from our `Args` struct, i.e. `args.N`."
# ╔═╡ b3a20600-08dc-4bd3-95bf-e2fe7cc38b02
function iterate(nd::NeutronDiffuser, args::Args; steps=300, samples=50)
logstep = steps ÷ samples # Iterations between logs
ρ = zeros(args.N, samples) # Neutron density result matrix
t = zeros(samples) # Time result vector
logᵢ = 1
for i ∈ 1:steps
step!(nd.ρ, args)
if i % logstep == 1 # logging occasionally
ρ[:, logᵢ] = copy(nd.ρ)
t[logᵢ] = i*args.τ
logᵢ += 1
end
end
(ρ, t)
end
# ╔═╡ 5c911c31-629b-402b-8ed0-ad010f281542
md"## Plotting
We're all set up; let's go ahead and create our `Args` and `NeutronDiffuser` structs, then iterate using our `iterate` function."
# ╔═╡ de8e6fc3-029c-4d01-a003-ca9948f7b835
begin
args = Args()
diffuser = neutrondiffuser(args)
(ρ, t) = iterate(diffuser, args)
end;
# ╔═╡ 3b84522d-4878-422d-a298-3811dd9df281
contourf(t * 1e9, diffuser.x, ρ, title="Neutron Diffusion",
xlabel="Time (ns)", ylabel="Position (m)", c=:thermal)
# ╔═╡ fc5f0250-530c-11eb-15d9-bb6c5204bde9
surface(t*1e9, diffuser.x, ρ, c=:thermal, title="Neutron Diffusion",
xlabel="Time (ns)", ylabel="Position (m)", zlabel="Neutron Density")
# ╔═╡ f5f2acf3-a066-4cf0-8264-a6a0bc14bf7d
md"Let's make an animation just for funzies."
# ╔═╡ de6ac937-5d2d-45d7-b4c3-081b65bd5252
let x = range(-args.L/2, args.L/2; length=args.N), ρ = zeros(args.N)
ρ[args.N÷2] = args.N / args.L
steps = 300
totaltime = round(args.τ*steps * 10^9, digits=2)
@gif for i ∈ 1:steps
step!(ρ, args)
plot(x, ρ, lw=3, lab="$(round(args.τ*i * 10^9, digits=2)) ns")
plot!(title="Neutron Diffusion Over $(totaltime) Nanoseconds")
plot!(xlabel="Position (m)", ylabel="Neutron Density", ylims=(0, 100))
end every 6
end
# ╔═╡ ca6f707b-0743-4899-a86e-7f25834d82e3
md"## Source
Alejandro L. Garcia. [Numerical Methods for Physics (Python), Second Edition (Revised)](http://algarcia.org/nummeth/nummeth.html). Chapter 6.3: *Critical Mass* (156-159)."
# ╔═╡ 00000000-0000-0000-0000-000000000001
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