### A Pluto.jl notebook ###
# v0.20.4

#> [frontmatter]
#> order = "27"
#> title = "5. Optimisation wastewater treatment"
#> date = "2025-08-06"
#> tags = ["exercises"]
#> description = "Optimisation wasterwater treatment"
#> layout = "layout.jlhtml"
#> 
#>     [[frontmatter.author]]
#>     name = "Gauthier Vanhaelewyn"


using Markdown
using InteractiveUtils

# ╔═╡ 2b26c3b2-df08-4b24-a08d-23717248c10d
using Pkg; Pkg.activate("../../pluto-deployment-environment")

# ╔═╡ f08fa69c-a744-11ef-0e79-3daf5bf297ea
using Markdown

# ╔═╡ e6c97e37-d062-4b65-96a4-bac0dab220d8
using InteractiveUtils

# ╔═╡ 0407d891-a46d-4deb-a21a-23833acbcb87
using Catalyst, OrdinaryDiffEq, StatsPlots

# ╔═╡ e48dc930-be03-47b2-b9e3-16e854782aec
using Turing, StatsBase, Optim

# ╔═╡ 6458329f-73dd-4cb0-8da4-90678875a1f0
using PlutoUI; TableOfContents()

# ╔═╡ c1bc698d-41ee-45e6-b17d-29f0d53557a1
md"""
# Exercise: Wastewater treatment - Optimisation
"""

# ╔═╡ dfe77a8c-a8db-46d7-9a4d-3b00413d383b
md"""
Consider a wastewater treatment plant where wastewater circulates in cylindrical tanks so that microorganisms can break down the organic material present. At the top of such a tank with volume $V\;[m^3]$, wastewater enters at a flow rate $q\;[m^3/h]$. The concentration of organic material in the inflow is known and equal to $C_{in}\;[kg/m^3]$. At the bottom of the tank, wastewater and microorganisms leave the tank at the same flow rate $q\;[m^3/h]$ so that the volume of wastewater in the tank remains constant.

The concentration of organic material in the tank is denoted as $C\;[kg/m^3]$ and the concentration of microorganisms is denoted as $X\;[kg/m^3]$. The microorganisms in the tank break down the organic material at a rate proportional to $r\cfrac{K_s}{K_s+C}\;[m^3\,h^{-1}\,kg^{-1}]$ with yield coefficient $Y$. The factor $K_s\;[kg/m^3]$ is the concentration of $C$ where the rate is half its maximum rate and $r\;[m^3\,h^{-1}\,kg^{-1}]$ is the maximum growth rate coefficient. Furthermore, the microorganisms degrade with rate coefficient $k_d$. In the middle of the tank there is a mixing system that ensures that the wastewater and microorganisms are well mixed. This means that the concentration in the outflow is equal to the concentration in the tank: $C_{out} = C$ and $X_{out} = X$. The system of differential equations describing the change in the concentrations $C(t)$ and $X(t)$ is given by:

$$\cfrac{dC}{dt} = \cfrac{q}{V}\left(C_{in} - C\right) - r\cfrac{K_s}{K_s+C}\,C\,X$$
$$\cfrac{dX}{dt} = -\cfrac{q}{V}X -k_d\,X + Y\,r\cfrac{K_s}{K_s+C}\,C\,X$$

The initial concentrations and the parameter values are summarised in the following tables:

|   $C_0$   |   $X_0$   |
|:---------:|:---------:|
|   $4.0$  |   $0.01$  |

|   $q$    |   $V$    |   $r$    | $C_{in}$  |   $K_s$   |   $k_d$   |   $Y$    |
|:--------:|:--------:|:--------:|:---------:|:---------:|:---------:|:--------:|
|   $5.0$  |   $50$   |  $0.4$   |   $3.0$   |   $5.2$   |  $0.10$   |   $1.2$  |

The amount of organic waste that is being broken down by the microorganisms depends on the flow rate $q$. First (Part 1), we will simulate the system with the parameters given above. Secondly (Part 2), we will optimize the value of the flow rate $q$ so that the concentration of organic waste in the tank is at most $0.28\; kg\,m^{-3}$. Take a simulation time of $72\;hours$ in both cases.
"""

# ╔═╡ c0c83df7-a9cc-4bde-b6ec-038a423b0d90
md"""
## Part 1 - Simulation

In this part, we will simulate the system with the parameters given above.
"""

# ╔═╡ bbd50cc5-032a-4219-bcee-91145935a7c4
md"""
### Implementation of the system
"""

# ╔═╡ a5f79c20-f62e-4df2-be79-b4f2141ced5e
md"""
Create a *reaction network object* model for the aforementioned problem. Name it `wastewater_treatment`.

!!! tip
	You can use the repressive Michaelis-Menten function `mmr(C, r, Ks)` for $r\cfrac{K_s}{K_s+C}$.
"""

# ╔═╡ 8708de16-3532-4352-b211-c092f95c82d3
# Uncomment and complete the instruction
# wastewater_treatment = @reaction_network begin
#     @parameters missing
#     @species missing
#     missing
#     missing
#     missing
#     missing
# end

# ╔═╡ 10c73294-a32b-4aa3-80a8-10785e5eab8f
md"""
Convert the system to a symbolic differential equation model and verify your system of differential equations.
"""

# ╔═╡ fee917dd-7ab5-4fda-b1b7-87ee61e21f19
# osys = missing            # Uncomment and complete the instruction

# ╔═╡ 08ebcb95-8603-4579-879e-810b1494b013
md"""
##### Setting up initial conditions, timespan and parameter values
"""

# ╔═╡ d248f64e-ebba-4443-9c49-ff0290aa7810
md"""
Initialize a vector `u0` with the initial conditions:
"""

# ╔═╡ fe02a755-5b00-4d80-a511-fec115b42964
# u0 = missing              # Uncomment and complete the instruction

# ╔═╡ 67481927-0d03-4da9-af6c-9afa409fc006
md"""
Set the timespan:
"""

# ╔═╡ fadd372a-a665-4b16-9b6d-e32cb7f25d7f
# tspan = missing           # Uncomment and complete the instruction

# ╔═╡ 734e4d51-95a7-464e-9a23-5ad6c8715d65
md"""
Initialize a vector `params` with the parameter values:
"""

# ╔═╡ 15ce9889-a437-46c8-9062-74b8d234a8bd
# params = missing          # Uncomment and complete the instruction

# ╔═╡ b8a48461-3882-45f6-980c-38d650ac52c7
md"""
### Creating an ODE problem, solve the problem and plot results
"""

# ╔═╡ a4c57b64-6a7d-4bd4-8bb2-578923e184d2
md"""
Create the ODE problem and store it in `oprob`:
"""

# ╔═╡ b1e18139-5277-4f06-b1f8-b0f5f11c41d8
# oprob = missing           # Uncomment and complete the instruction

# ╔═╡ ad6d8fe6-e62f-4c67-8d63-4ee13b928ad0
md"""
Solve the ODE problem. Use `Tsit5()` and `saveat=0.1`. Store the solution in `osol`:
"""

# ╔═╡ e2ffba9e-aaf2-4540-84cf-8b7297ae9285
# osol = missing             # Uncomment and complete the instruction

# ╔═╡ 70871ee8-b0a4-4a9a-af39-5a63459b55f7
md"""
Plot the results. Use `ylim=(0, 3)` and `lw=2` (or `linewidth=2`) as options.
"""

# ╔═╡ 34309734-3751-47e0-a602-d113ffaae510
# Uncomment and complete the instruction
# begin
#     missing
#     hline!([0.28], ls=:dash, lw=2, lc=:green, lab="C=0.28")
# end

# ╔═╡ a0e735ad-09c2-4aa8-bc41-b294a9d56ea8
md"""
Check out the end value of the organic waste.
"""

# ╔═╡ f62898d5-1b8d-4350-8655-78aa3decb2a2
# missing               # Uncomment and complete the instruction

# ╔═╡ 7eb5df9c-a475-4812-81c3-e43484c82242
md"""
## Part 2 - Optimization

In this part, we will optimize the value of the flow rate $q$ so that the concentration of organic waste in the tank is at most $0.28\; kg\,m^{-3}$.
"""

# ╔═╡ 5a695734-677f-4bf6-a703-8e22382b7529
md"""
First, declare the Turing model function. Sample the flow rate $q$ prior from an uniform distribution in the range $[0, 5]\;kg\,m^{-3}$. Suppose therein that the desired final value of the organic waste (i.e. $0.28\; kg\,m^{-3}$) is normally distributed with  mean the end value obtained from the solution and standard deviation $10^{-3}\; kg\,m^{-3}$.
"""

# ╔═╡ b6bac48a-4a3d-47e4-90ea-788ca20dadff
# Uncomment and complete the instruction
# @model function wastewater_treatment_inference()
#     q ~ missing
#     u0 = missing
#     tspan = missing
#     params = missing
#     oprob = missing
#     osol = missing
#     C_d ~ missing
# end

# ╔═╡ b3a40556-0c00-4f6d-8cd9-c5fca79d8bbf
md"""
Define the desired value for the organic waste with the variable name `C_val`.
"""

# ╔═╡ 2df409ef-bd95-4ac3-a2b8-c5e17c490eba
# missing                  # Uncomment and complete the instruction

# ╔═╡ 70cafd87-63f7-4674-ae49-43d422fdeae7
md"""
Now condition the model with the desired value:
"""

# ╔═╡ ef20f8b8-4527-4f02-b449-fa67b68bbf65
# wastewater_treatment_cond_mod = missing

# ╔═╡ ee1ffc12-55a1-47ef-ac5b-33148706a09b
md"""
Optimize the prior for $q$. Do this with `MLE` method and Nelder-Mead. Store the optimization results in `results_mle`.
"""

# ╔═╡ afc035be-075b-464b-8ba2-20235082f005
# results_mle = missing    # Uncomment and complete the instruction

# ╔═╡ 97a00511-93d4-45d6-b320-9bad1b102397
md"""
Check out the coefficient table.
"""

# ╔═╡ 64844b47-1578-4e47-9cc7-7f242a583067
# missing

# ╔═╡ 3ee8121e-3e78-4901-a32d-f04d0c6a0996
md"""
Get the optimized value for $q$ and assign it to `q_opt`.
"""

# ╔═╡ 98a157a1-8c20-474d-acb8-00373ee6d224
# q_opt = missing          # Uncomment and complete the instruction

# ╔═╡ ceb146c9-a09a-458b-b7d8-3bb7d3de38e0
md"""
Set up parameter values with the optimized parameter value.
"""

# ╔═╡ e275df05-5c77-4c17-ad2e-503574596c31
# params_opt = missing      # Uncomment and complete the instruction

# ╔═╡ cd515dad-44fb-4af2-b933-805ef76be9b3
md"""
Create an ODEProblem and solve it. Use `Tsit5()` and `saveat=0.1`.
"""

# ╔═╡ a4388f06-1223-4815-a557-9b9c3ec232bb
# oprob_opt = missing        # Uncomment and complete the instruction

# ╔═╡ ec7bf654-b275-4cfd-a819-d82bdc1be93b
# osol_opt = missing          # Uncomment and complete the instruction

# ╔═╡ 82809c26-4cab-405e-8107-a8a43e81f699
md"""
Plot $C$ and $X$ simulated with both the initial and the optimized parameter values. Use `ylim=(0, 3)` and `lw=2` (or `linewidth=2`) as options. The dashed line indicates $C = 0.28\; kg\,m^{-3}$.
"""

# ╔═╡ 81429279-4190-41d1-a72a-20da0ce90528
# Uncomment and complete the instruction
# begin
#     missing
# 	  plot!(osol, ls=:dash, lw=1, lab=:none)
#     hline!([0.28], ls=:dash, lw=2, lc=:green, lab="C=0.28")
# end

# ╔═╡ 6589acfd-1d81-4c10-adea-34ca7fa1ab5d
md"""
!!! question 
	Does the value of $C$ respect now the limit in the concentration? Draw your conclusion.
"""

# ╔═╡ 7c7d99b9-77b9-4c08-a74e-54eaa7d187ec
md"""
- Conclusion: missing
"""

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