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

#> [frontmatter]
#> order = "6"
#> title = "EXTRA. ODE fermentor first order"
#> date = "2025-02-07"
#> tags = ["exercises"]
#> description = "ODE model of a first order fermentor"
#> layout = "layout.jlhtml"
#> 
#>     [[frontmatter.author]]
#>     name = "Gauthier Vanhaelewyn"

using Markdown
using InteractiveUtils

# ╔═╡ e99680dc-73af-40aa-bf57-a06d3a7372be
begin
	# add this cell if you want the notebook to use the environment from where the Pluto server is launched
	using Pkg
	Pkg.activate("../../pluto-deployment-environment")
end

# ╔═╡ 2e58f4ae-f711-11ee-2598-7f3a6f2e2013
using Markdown

# ╔═╡ 539c1823-16d0-4355-97b6-fe9f0b106864
using InteractiveUtils

# ╔═╡ e66518ee-b6f6-4cca-a224-30e01cffddbe
using Catalyst

# ╔═╡ bd648109-f042-42de-9e0e-017b502fab95
using OrdinaryDiffEq, StatsPlots

# ╔═╡ 7856d878-8586-4cfd-9cf6-d61234450e41
md"""
# Exercise: Fermenter - First order kinetics
"""

# ╔═╡ 8500c35e-6bc3-4900-81bf-7705ddd61532
md"""
In a fermenter reactor biomass $X$ grows on substrate $S$. The reactor is fed with a inlet flow rate $Q_{in}$ [$L/h$], which consist of a (manipulable) input concentration of substrate $S_{in}$ [$g/L$]. Inside the reactor, biomass, with a concentration of $X$ [$g/L$], is produced through first-order kinetics (first-order in $S$):

$$\begin{eqnarray*}
% S \xrightarrow[\quad\quad]{\beta} Y \, X
S \xrightarrow[\quad\quad]{\text{r}} Y \, X \quad\quad\quad\quad r = \beta \, S
\end{eqnarray*}$$

with $\beta$ [$h^{-1}$] the reaction rate constant, and $Y$ [$gX/gS$] the yield coefficient which is defined here by the amount of produced biomass by consumption of one unit of substrate. Futhermore, the reactor is drained with an outlet flow $Q$ [$L/h$], which consist of the current concentrations of substrate $S$ [$g/L$] and biomass $X$ [$g/L$] inside the reactor. The volume $V$ [$L$] of the reactor content is kept constant by setting $Q_{in} = Q$.
"""

# ╔═╡ f1350528-07a5-4860-ad2d-627588186abc
md"""
Create a *reaction network object* model for the aforementioned problem in order to simulate the evolution of substrate $S$ and biomass $X$ with time. Name it `fermenter_firstorder`.
"""


# ╔═╡ 331a34f4-89d4-4193-896c-c14ab0bf04e7
# Uncomment and complete the instruction
# fermenter_firstorder = @reaction_network begin
#     missing        # Y*X is created from one S at a rate β
#     missing        # S is created at a rate Q/V*Sin
#     missing        # S and X are degraded at a rate Q/V*S
# end

# ╔═╡ 55746566-2d46-4475-851a-02b7fad87a1a
md"""
Convert the system to a symbolic differential equation model and verify, by analyzing the differential equation, that your model is correctly implemented.
"""

# ╔═╡ ec9cb3bd-f5ed-4ab0-9b3d-b875692227ac
# osys = missing           # Uncomment and complete the instruction

# ╔═╡ 67117a27-dcea-4b43-b962-9ad9fd07f4f4
md"""
The parameter values are $\beta = 0.98$, $Y = 0.80$, $Q = 2.0$, $V = 40.0$ and $S_{in} = 2.2\;g/L$. With the latter values, the fermenter reactor is in steady state operation with concentrations for substrate $S = 0.1068\;g/L$ and biomass $X = 1.6746\;g/L$. Suppose that at timepoint $t = 20\;h$, the concentration of substrate in the inlet flow (cf. $S_{in}$) is suddently increased to $3.4\;g/L$. Simulate the evolution of $S$ and $X$ during 120 hours.
"""


# ╔═╡ d13e6e38-037e-4812-85e9-2c18bed360f6
md"""
Initialize a vector `u₀` with the initial conditions:
"""

# ╔═╡ 4b556cf0-8fad-434d-be56-dc1848d898ae
# u0 = missing            # Uncomment and complete the instruction

# ╔═╡ ea55d648-7575-43c3-a385-5f4979996ef2
md"""
Set the timespan for the simulation:
"""

# ╔═╡ 1365c12e-e662-4858-983b-02ba94cd9f0d
# tspan = missing         # Uncomment and complete the instruction

# ╔═╡ 3941bd60-a83c-4f72-84b3-28e28cb845d0
md"""
Initialize a vector `params` with the parameter values:
"""

# ╔═╡ d6c1316a-cf96-43d1-854a-f25925cf4a55
# params = missing         # Uncomment and complete the instruction

# ╔═╡ eeb8ec6e-154e-4fe9-8b5b-edbe71914985
md"""
Create the *condition* that contains the timepoint for the sudden change in $S_{in}$. Store it in `condition`:
"""

# ╔═╡ 7ca8efaa-97b6-46f2-b4d3-6ca8aa97dda7
# condition = missing

# ╔═╡ d864bfc3-05b2-483b-9a55-da026931703f
md"""
Make a new *reaction system* where the discrete event is included. Name it `fermenter_firstorder2`.
"""

# ╔═╡ d758b918-d184-473f-9b9e-ed6da7b0f088
# @named fermenter_firstorder_c = missing

# ╔═╡ ef596d8c-efdc-4b5c-9584-335d799acfe8
md"""
Complete the new *reaction system*. Name it `fermenter_firstorder_c_com`.
"""

# ╔═╡ c07b3121-4c27-454c-b9e4-7dfa27371ebd
# fermenter_firstorder_c_com = missing

# ╔═╡ c6e81f41-a244-48c9-9d18-b3b9e0984fbb
md"""
Create the ODE problem and store it in `oprob`:
"""

# ╔═╡ ed56f8d6-2260-4829-9190-69b60b7d7599
# oprob = missing

# ╔═╡ 8b73b16b-7f7d-4d2e-a1c2-7e1adf2336e9
md"""
Solve the ODE problem. Make a deepcopy and use `Tsit5()` and `saveat=0.5`. Store the solution in `osol`:
"""

# ╔═╡ 433db8d1-f038-4e7b-9133-90bfeccabd07
# osol = missing

# ╔═╡ fc824241-f718-46a8-b50b-a680470c062b
md"""
Plot the results:
"""

# ╔═╡ e2609226-5f2f-4802-89e3-0efeac740081
# missing

# ╔═╡ 5fa47281-4c1f-4b8e-ab99-c92f4dc7ec65
md"""
Interpret the results. Ask yourself the following questions:

1. Can you clearly see the effect of the increase in $S_{in}$?
"""

# ╔═╡ ae8e5a4a-59d8-4746-accb-c9b09099bc2b
md"- Answer: missing"

# ╔═╡ 822ac3d0-9430-4763-9e94-d77b9e77c79c
md"""
2. Can you argue, by means of reasoning or by determining and analyzing the operating point, why the increase of $X$ is larger than the increase of $S$?
"""

# ╔═╡ a5d1f9e9-cd84-4221-9295-5c739cb289b2
md"- Answer: missing"

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