### A Pluto.jl notebook ###
# v0.17.1

using Markdown
using InteractiveUtils

# ╔═╡ c7b4078a-5508-4a7c-9c4f-faf1e49d34ff
begin
	using Plots
	using PlutoUI
	using DifferentialEquations
end

# ╔═╡ ddaf7fe6-0b1a-4983-a0a6-b82dea16a160
md"""
### Steady-state versus Equilibrium: Can we have equilibrium in an open system?
We started this unit by introducing two seemingly similar, but in actuality significantly different, concepts: steady-state versus equilibrium. Let's further explore these concepts by considering an example. 

Suppose we wanted to produce an _unstable_ product $B$ from precursor compound $A$ in a continuous stirred tank reactor (CSTR)  according to the reaction scheme:

$$A~{\rightleftharpoons}~B~{\rightleftharpoons}~{C}$$

where the compound $C$ is an _undesired_ degradation product. Further, suppose that no reaction can occur outside of the reactor (e.g., there is a catalyst resident in the reactor that is required for reaction).

__Assumptions__
* Ideal liquid phase behavior
* Constant T,P in the reactor
* Reactor is well-mixed
* The CSTR has a single input ($s=1$), a single output ($s=2$) and constant V
* All reactions follow mass-action kinetics and are _first_ order

__Objectives__: By the end of this lecture you should be able to:
* Reason about the time constants of transport versus reaction in a continuous stirred tank reactor (CSTR)
* Understand the difference between equilibrium and steady-state in an open system
* Have a _basic_ understanding of how to numerically solve systems of Ordinary Differential Equations (ODEs)

"""

# ╔═╡ b9f8c24b-642c-4940-a8a4-82d4c57d25ad
md"""
#### Objective 1: Time constants in a CSTR

The concentration balance for species $i=1,2,\cdots,\mathcal{M}$ in a CSTR with up to $j=1,2,\cdots,\mathcal{R}$ reactions is given by:

$$V\frac{dC_{i}}{dt} = \dot{F}C_{i,1} - \dot{F}C_{i,2} +\sum_{j=1}^{\mathcal{R}}\sigma_{ij}\hat{r}_{j}V$$

Dividing both sides by $\dot{F}$ and introducing the Dilution rate $D=\dot{F}/V$ gives the expression:

$$\tau\frac{dC_{i}}{dt} = \left(C_{i,1}-C_{i,2}\right) + \tau\sum_{j=1}^{\mathcal{R}}\sigma_{ij}\hat{r}_{j}$$

where $\tau\equiv{D}^{-1}$ is the _transport_ time constant (also called the residence time) (units: time). 
The value of $\tau$ indicates how long the average particle of fluid spends in the reactor. Let's consider some limiting cases for $\tau$:

###### Case 1: $\tau\gg{1}$ (large residence time)
In the limit of $\tau\gg{1}$ the concentration balance behaves like a _batch_ reactor (no inputs or outputs):

$$\frac{dC_{i}}{dt}\sim\sum_{j=1}^{\mathcal{R}}\sigma_{ij}\hat{r}_{j}$$

When $\tau\gg{1}$, the volume $V$ is large compared to the volumetric flow rate $\dot{F}$, thus particles of fluid that enter the reactor stay nearly forever.

###### Case 2: $\tau\sim{1}$ (just right)
When $\tau\sim{1}$ we consider both the reaction terms and the transport terms in the balance equations:

$$\frac{dC_{i}}{dt} \sim \left(C_{i,1}-C_{i,2}\right) + \sum_{j=1}^{\mathcal{R}}\sigma_{ij}\hat{r}_{j}$$

In case 2, both the transport and reaction terms are important.

###### Case 3: $\tau\ll{1}$ (small residence time)
In the limit of $\tau\ll{1}$ the system behaves like:

$$C_{i,1} \sim C_{i,2}$$

In other words, the volumetric flow rate $\dot{F}$ is large compared to the volume, so particles of fluid carrying the chemical components enter the reactor and quickly leave (giving them little time to react).

"""

# ╔═╡ 5efd5f95-ba77-44e1-b5a0-d29a77f0740f


# ╔═╡ 8d23ab99-6b7f-4212-991b-0d51c93b961d
md"""
#### Objective 2: Steady-state versus equilibrium and the role of $\tau$
"""

# ╔═╡ 45903db9-2b98-4cb8-8bcb-cdceef733383
begin
	# Setup the problem parameters -
	volumetric_flow_rate = 0.0 		# units: L/hr
	V = 14.0 						# units: L
	τ = V/volumetric_flow_rate 		# units: hr

	with_terminal() do
		println("Time constant τ = $(τ) hr")
	end
end

# ╔═╡ 74e2338b-619d-465e-8da8-14d9e089eb7d
md"""
##### Ok: so we may be at steady-state, but are we at equilibrium?
Steady-state requires that all the time derivatives for all the chemical species, volume, temperature etc are equal to zero. In other words, nothing is changing in time. Equilibrium on the other hand requires this __and__ one additional condition: the net reaction rate for all chemical reactions is equal to zero, ie., there is no net chemical reaction. In our example system, is this true?
"""

# ╔═╡ 0264f8da-d3c5-4702-8520-9ad64b687532


# ╔═╡ c5b78908-00dc-4ab8-98ef-6ee9cbe5aa14
md"""
#### Objective 3: Numerical solution of Ordinary Differential Equations (ODEs)
Suppose we had a _scalar_ ordinary differential equation of the form (initial value problem):

$$\frac{dx}{dt} = f\left(x,k\right)$$

where $x$ denotes what we are trying to estimate e.g., a concentration, 
$k$ denotes parameters (e.g., rate constants), $t$ denotes time and 
$f\left(\cdot\right)$ denotes the right-hand side of the differential equation e.g., a mole or mass balance. 
The differential equation above is subject to the initial condition $x\left(t_{0}\right)=x_{o}$.

To estimate the time behavior of the variable $x$, we multiply through by $dt$, and integrate both sides from some time $t_{1}$ $\rightarrow$ $t_{2} = t_{1}+\Delta{t}$:

$$\int_{t_{1}}^{t_{1}+\Delta{t}}dx = \int_{t_{1}}^{t_{1}+\Delta{t}}f\left(x,k\right)dt$$

The left-hand side becomes:

$$\int_{t_{i}}^{t_{i}+\Delta{t}}dx = x_{t_{i}+\Delta{t}} - x_{t_{i}}$$

which gives:

$$x_{t_{i}+\Delta{t}} = x_{t_{i}} + \int_{t_{i}}^{t_{i}+\Delta{t}}f\left(x,k\right)dt$$

This is the basis for _most_ numerical methods for the solution of ordinary differential equations; these methods differ in how they approximate the integral of $f\left(\cdot\right)$ on the right-hand side. One simple approach (which is not accurate unless the step-size $\Delta{t}$ is small) is to assume $f\left(\cdot\right)$ is constant across the time interval $\Delta{t}$ which gives:

$$x_{t_{i}+\Delta{t}} \simeq x_{t_{i}} + f\left(x_{t_i},k\right)\Delta{t}$$

This is known as the [forward Euler method](https://en.wikipedia.org/wiki/Euler_method) and it has error (at each step) of order $\mathcal{O}\left(\Delta{t}^{2}\right)$. Other more sophisticated methods do a better job at approximating the integral, thus they have better error performance.

"""

# ╔═╡ ba11e8aa-3d4a-41d8-ac8d-ed53d37a94e7


# ╔═╡ 9bc6e553-32af-4086-8b32-2674a21661ff
md"""
ODE solution code below here (if you are interested)
"""

# ╔═╡ 668d00a7-3f7f-43a8-8aa3-8c33d04d6e37
begin

	# initialize -
	parameter_dictionary = Dict{String,Any}()

	# What is the composition of the feed?
	A₀ = 10.0 	# 1 units: mmol/L
	B₀ = 0.0 	# 2 units: mmol/L
	C₀ = 0.0 	# 2 units: mmol/L

	# what is the initial condition in the reactor?
	initial_condition_array = [
		10.0 	# 1 units: mmol/L
		0.0 	# 2 units: mmol/L
		0.0 	# 3 units: mmol/L
	];
	
	# rate constants -
	rate_constant_array = [
		2.0 	; 	# 1: k1+ units: hr^-1
		1.1 	; 	# 2: k1- units: hr^-1
		10.0 	; 	# 3: k2+ units: hr^-1
		6.0 	; 	# 4: k2- units: hr-1
	];

	# let's create the stoichiometric matrix -
	# Rxn 1: A = B
	# Rxn 2: B -> C (optional)
	S = [

		# r₁ r₂ 
		-1.0 0.0 ; # A
		1.0 -1.0 ; # B
		0.0  1.0 ; # C
	]

	# put data in the parameter_dictionary -
	parameter_dictionary["F_dot"] = volumetric_flow_rate
	parameter_dictionary["V"] = V
	parameter_dictionary["A₀"] = A₀
	parameter_dictionary["B₀"] = B₀
	parameter_dictionary["C₀"] = C₀
	parameter_dictionary["initial_condition_array"] = initial_condition_array
	parameter_dictionary["rate_constant_array"] = rate_constant_array
	parameter_dictionary["stoichiometric_matrix"] = S
	parameter_dictionary["number_of_species"] = length(initial_condition_array)
	
	# show -
	nothing
end

# ╔═╡ d94d3b0c-b551-41bd-ba78-03c11a373472
function compute_kinetics(x,parameter_dictionary)

	# which phase r we?
	does_reaction_occur_flag = parameter_dictionary["does_reaction_occur"]
	
	# alias -
	A = x[1]
	B = x[2]
	C = x[3]
	
	# get some data from the problem_dictionary -
	rate_constant_array = parameter_dictionary["rate_constant_array"];
	k1f = rate_constant_array[1]
	k1r = rate_constant_array[2]
	k2f = rate_constant_array[3]
	k2r = rate_constant_array[4]

	# compute the kinetics -
	r1_net = 0.0
	r2_net = 0.0
	if (does_reaction_occur_flag == true)
		r1_net = k1f*A - k1r*B
		r2_net = k2f*B - k2r*C
	end
	rV = [r1_net ; r2_net]

	# return -
	return rV
end

# ╔═╡ 6b0f4437-dfba-4c60-85fc-1b4d314adbff
function concentration_balances(dx, x, problem_dictionary, t)

	# alias -
	A = x[1]
	B = x[2]
	C = x[3]

	# feed et al -
	F = parameter_dictionary["F_dot"]
	V = parameter_dictionary["V"]
	A₀ = parameter_dictionary["A₀"]
	B₀ = parameter_dictionary["B₀"] 
	C₀ = parameter_dictionary["C₀"]
	S = parameter_dictionary["stoichiometric_matrix"]
	number_of_species = parameter_dictionary["number_of_species"]

	# what is D? units: 1/hr
	D = F/V

	# compute the kinetics - 
	rV = compute_kinetics(x, parameter_dictionary)

	# transport terms -
	transport_terms = [
		D*(A₀ - A) ;
		D*(B₀ - B) ;
		D*(C₀ - C) ;
	]

	# reaction terms -
	reaction_terms = S*rV

	# compute the balances -
	dxdt = transport_terms .+ reaction_terms
	for species_index = 1:number_of_species
		dx[species_index] = dxdt[species_index]
	end
end

# ╔═╡ fd731a95-9cf6-4e27-b035-10ceb8251dd5
begin

	# set the time span -
	T_end_phase_1 = 10.0
	tspan = (0.0, T_end_phase_1)

	# during phase 1 - no reaction is occurring -
	parameter_dictionary["does_reaction_occur"] = false
	
	# get the initial conditions -
	xinitial = parameter_dictionary["initial_condition_array"]
	ode_prob = ODEProblem(concentration_balances,xinitial,tspan)

	# solve the ODEs numerically -
	soln_phase_1 = solve(ode_prob, Tsit5(), reltol=1e-8, abstol=1e-8)

	# show -
	nothing
end

# ╔═╡ 653936dd-8eba-47c7-a515-b1431ed35087
begin

	# set the time span -
	ΔT = 20.0
	tspan_phase_2 = (T_end_phase_1, T_end_phase_1 + ΔT)

	# during phase 1 - no reaction is occurring -
	parameter_dictionary["does_reaction_occur"] = true
	
	# get the initial conditions -
	A_phase_1 = [u[1] for (u,t) in tuples(soln_phase_1)]
	xinitial_phase_2 = [
		last(A_phase_1) ;
		0.0 			;
		0.0 			;
	]
	
	ode_prob_phase_2 = ODEProblem(concentration_balances,xinitial_phase_2,tspan_phase_2)

	# solve the ODEs numerically -
	soln_phase_2 = solve(ode_prob_phase_2, Tsit5(), reltol=1e-8, abstol=1e-8)

	# show -
	nothing
end

# ╔═╡ 3d2901ff-34bc-45c3-adae-71d730ed0e32
begin

	# smash the solns together -
	n_t_steps_phase_1 = length(soln_phase_1.t)
	n_t_steps_phase_2 = length(soln_phase_2.t)
	total_number_t_steps = n_t_steps_phase_1 + n_t_steps_phase_2

	# buld the T and X structures -
	T = vcat(soln_phase_1.t, soln_phase_2.t)
	X = vcat(soln_phase_1.u, soln_phase_2.u)

	# build the state array -
	x_soln_array = zeros(total_number_t_steps,4)
	for step_index = 1:total_number_t_steps

		x_soln_array[step_index,1] = T[step_index]
		x_soln_array[step_index,2] = X[step_index][1]
		x_soln_array[step_index,3] = X[step_index][2]
		x_soln_array[step_index,4] = X[step_index][3]
	end
end

# ╔═╡ 875e2cb4-3a91-4f7e-b1b4-2356f8a46b98
begin
	plot(x_soln_array[:,1], x_soln_array[:, 2], lw=2, label="A", legend=:topright)
	plot!(x_soln_array[:,1], x_soln_array[:, 3], lw=2, label="B")
	plot!(x_soln_array[:,1], x_soln_array[:, 4], lw=2, label="C")
	xlabel!("Time (hr)",fontsize=18)
	ylabel!("Concentration (mmol/L)",fontsize=18)
end

# ╔═╡ d1952491-6c26-4403-900b-cd67a969b72b
with_terminal() do

	# get the steady-state values from the soln array -
	A_ss = x_soln_array[end,2]
	B_ss = x_soln_array[end,3]
	C_ss = x_soln_array[end,4]

	# display -
	println("A_ss = $(A_ss) mmol/L B_ss = $(B_ss) mmol/L C_ss=$(C_ss) mmol/L")
end

# ╔═╡ 77778548-adf6-408e-917c-00611ace230f
rV = compute_kinetics(x_soln_array[end,2:end], parameter_dictionary)

# ╔═╡ 5f65d268-40db-11ec-318e-4bb375e1410b
html"""
<style>
main {
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}

a {
    color: blue;
    text-decoration: none;
}

.H1 {
    padding: 0px 30px;
}
</style>"""

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version = "0.1.0+3"

[[Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "cc4bf3fdde8b7e3e9fa0351bdeedba1cf3b7f6e6"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.0+0"

[[ZygoteRules]]
deps = ["MacroTools"]
git-tree-sha1 = "8c1a8e4dfacb1fd631745552c8db35d0deb09ea0"
uuid = "700de1a5-db45-46bc-99cf-38207098b444"
version = "0.2.2"

[[libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "c45f4e40e7aafe9d086379e5578947ec8b95a8fb"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+0"

[[nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

[[x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

# ╔═╡ Cell order:
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# ╠═6b0f4437-dfba-4c60-85fc-1b4d314adbff
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