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cocktails.jl
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cocktails.jl
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### A Pluto.jl notebook ###
# v0.19.0
using Markdown
using InteractiveUtils
# ╔═╡ 268f2426-e2e4-45b5-9d34-279dc5de5182
using ModelingToolkit
# ╔═╡ 8d26386d-33f6-4a57-91b9-221329fc0e91
using DifferentialEquations, Plots
# ╔═╡ 785a59ba-c716-11ec-3225-e5ec94a651a2
md"""
# Modelling cocktail shaking
Plenty of cool stuff going on in the Julia community. One of the language's strengths is that it allows one to write tools that can work on the language itself (e.g. its metaprogramming capacities). Recently, the community has invested in building [the foundation for a computer algebra system](https://symbolicutils.juliasymbolics.org/), allowing tools that reason about equations. One of the flagship packages is [ModelingToolKit](https://github.com/SciML/ModelingToolkit.jl), a library that can help to construct, modify and analyse complex models. By way of exploration, let us use this software to generate a model for cocktail shaking!"""
# ╔═╡ e85060ec-e334-4b2d-844f-8e748b532b34
md"""
## Fundamentals of cocktail making
As I discussed in [an earlier post](https://michielstock.github.io/posts/2020/2020-05-21-compuational-mixology/), cocktail making is quite instructive from a thermodynamics point of view. After mixing, the bartender typically shakes or stirs the cocktail with ice (which is usually discarded afterwards). The reaction with ice has two interconnected effects. Firstly, the ice melts, diluting the cocktail. Secondly, ice melting is a strongly endothermic reaction, absorbing heat from the liquid and cooling your cocktail. Cooling is diluting, and diluting is cooling. This is called the *First Law of Cocktail Making* by Dave Arnolds.
The dilution and cooling will also impact the perceived flavour of the cocktail. Most of its balance is determined by the sweetness and the acidity. Sweetness, in particular, depends in a complex way on the sucrose concentration and the temperature of the solution. This is why shaken cocktails, which are drunk more chilled, typically contain more sugar.
"""
# ╔═╡ b298c1d2-8ecf-459b-b238-64714923086d
md"""
## Using ModelingToolKit to model how a cocktail changes
The change of the cocktail while shaking can be described by elementary chemical and physical laws. Typically, one would use the mass and energy balances of the quantities of interest to derive a suitable system of differential equations. Let us try to automate this process using ModelingToolKit. We only provide the conservation laws and some behaviours and let the CAS figure it out for itself.
To keep it simple, let us treat a cocktail as a mere sugar solution (a virgin cocktail, if you will) with sufficient ice; it cannot completely melt. First, we define the variables.
"""
# ╔═╡ c07f3dd4-8983-4bc9-b864-8df9c664e740
@variables t I(t) V(t) S(t) Z(t) T(t)
# ╔═╡ 579ced03-c8eb-4fb1-80ac-66c3fba92b7f
md"""
These correspond to:
- the time (in seconds): $t$;
- the amount of ice in kg $I(t)$;
- the volume of the cocktail, in litres: $V(t)$;
- the concencentration of sucrose in grammes per litre: $S(t)$;
- the temperature of the cocktail, in degrees Celcius: $T(t)$.
"""
# ╔═╡ 37d0eb59-1fb7-4832-8aba-dbf4d95aef4e
md"We also define the derivative w.r.t. time operator:"
# ╔═╡ 7e627856-f165-4f29-b500-ea08c8c1330e
D = Differential(t)
# ╔═╡ ce939f44-8fe0-4080-b4c2-8208fc8cd7b9
md"Firstly, we define the melting of ice. This process depends on shaking intensity and the temperature of the liquid. Based on *Liquid Intelligence*, the following rate works well."
# ╔═╡ 954505b3-77b0-482a-8ad0-71af7288f114
İ = (-5e-4(T + 7)) * (I > 0)
# ╔═╡ a84b89dc-18a5-4afe-aae1-1bbe7973dba2
md"We set this rate equal to the change in ice mass."
# ╔═╡ d54b27c0-2292-4e13-a427-ace53856c119
melting = D(I) ~ İ
# ╔═╡ 08985029-5100-4571-96d4-2d8fba4c7a68
md"Next, we have our *conservation of water*. Any change in volume $\frac{\text{d}V}{\text{d}t}$ can only take place because a corresponding quantity of ice has melted. So we have:"
# ╔═╡ e6b01aa7-0aa0-42b8-aac4-e95cf167a5e9
water_balance = İ + D(V) ~ 0
# ╔═╡ 6f57980d-2cc1-4115-9b9b-eb7c49a8590e
md"Next, we consider the heat balance. During the short time period, the liquid does not absorb heat from the surroundings, the liquid only cools because the melting of ice is endothermic. Our heat balance is given by"
# ╔═╡ 3d183855-d846-4e5b-bfd4-810364418f34
md"For simplicity's sake, we use the thermodynamic properties of water. Here, `Cmelt` is the [enthalpy of fusion for water](https://en.wikipedia.org/wiki/Enthalpy_of_fusion) while `Cₚ` is the [specific heat capacity](https://en.wikipedia.org/wiki/Specific_heat_capacity) of water. Considerting the heat of water-ethanol mixtures correctly is rather [messy](https://en.wikipedia.org/wiki/Ethanol_(data_page))."
# ╔═╡ b2f0dff9-0cb8-4ef3-9642-8ad1645dc61b
const Cmelt = 3.34e5 # J/kg
# ╔═╡ de6c1273-682e-461f-b182-103ad1878580
const Cₚ = 4.187e3 # J/kg * K
# ╔═╡ 2edeadc7-b5ba-4737-8ab9-95036ad5cd1a
heat_balance = Cmelt * İ ~ Cₚ * V * D(T)
# ╔═╡ 09195d0d-687c-4738-bb20-518b3d4aa8f4
md"The conservation of sugar is rather simple. Though the concentration might change, the total quanity of sugar is fixed."
# ╔═╡ f29fc313-355b-43ac-abca-90445844bf4b
md"Finally, the perceived sweetness depends on the sugar concentration and the temperature. Setting up such a relation is done in the field of psychophysics (e.g. [Steven's law](https://en.wikipedia.org/wiki/Stevens%27s_power_law)). Based on [this article](https://pubmed.ncbi.nlm.nih.gov/7100291/), we obtain a fairly simple emperical relation."
# ╔═╡ dfc5132b-e082-47c5-9a36-fbfaaae396c8
sweetness = Z ~ 14.9 * (S / 342.30)^(1.422 - 0.0146T)
# ╔═╡ afd38f19-41ee-4c52-aaef-c8239250c92f
md"Let us model a [Bee's Knees](https://www.liquor.com/recipes/bees-knees/) cocktail. Liquid Intelligence kindly provides the intial mixing volume and concentration:"
# ╔═╡ 0066669f-4541-4d32-b13b-fcc98c637de8
V₀ = 0.105 # L
# ╔═╡ e78790ab-31f3-46bf-8d62-96fb5b9d0ac6
S₀ = 10.1 # g / L
# ╔═╡ 2b7fe775-db50-4e83-aa79-334a771d4f0f
sugar_balance = S * V ~ V₀ * S₀
# ╔═╡ 9453473b-4b1a-493b-93b8-f71cf33a78d5
md"And let us use 150 g of ice."
# ╔═╡ 1c9e5668-8a96-487e-a001-7b041b044205
I₀ = 0.2 # kg
# ╔═╡ bf4c7a1c-ac8a-4bab-a2ae-5e4ccb05967a
md"Now for the cool part! Let us piece everything together and use `structural_simplify` to derive a simpler system a solver can handle."
# ╔═╡ cdf1ee0c-b250-45ef-a910-c3aa1aac9d4b
@named sys = ODESystem([melting,
water_balance,
heat_balance,
sugar_balance,
sweetness])
# ╔═╡ 1afbf3a1-56b3-46bd-b192-d3bbcfb11c6c
equations(sys)
# ╔═╡ 647a62bf-acb6-4008-84c3-0c220b1ef3e1
states(sys)
# ╔═╡ e13257e4-741b-4b44-be42-dfb1a618db5b
simpsys = structural_simplify(sys, simplify=true)
# ╔═╡ f167eb5a-8eb7-4da2-8cca-044c3318539e
equations(simpsys)
# ╔═╡ d8170974-0e42-49d4-b79e-e0647b9617c0
states(simpsys)
# ╔═╡ 88de08b6-ac27-4d39-9266-9769eb2d1002
md"""
We see that by simplification, the five laws are turned into three ODEs and one simple law from which sugar concentration can be derived. Perceived sweetness depends directly on everything above, so it does not need to be taken into account for the solver.
Now we add the initial conditions and turn the system in an ODE problem we can solve!
"""
# ╔═╡ d5276187-42c4-4e9e-a274-c298964ca53b
prob = ODEProblem(simpsys, [V=>V₀, S=>S₀, I=>0.20, T=>20], (0.0, 20.0))
# ╔═╡ d3c0b55d-9d6a-4c1b-84f5-cfc5d1f51d25
solution = solve(prob)
# ╔═╡ 90fe88dc-b8c7-4e36-9fc3-3e5e8d21a511
md"""
Finally, we plot our states (note that `z(t)` can still be extracted). All processes are simulated as we please, we can see how the composition, temperature and sweetness change with time. Most drinks are in their prime between -5 and -1 degrees of Celcius, so five seconds of shaking would do the trick! 👌
"""
# ╔═╡ 7c9b299f-2b94-4cc9-b05d-611678c311a7
plot(solution, vars=[I, V, Z])
# ╔═╡ f2ca523f-5153-4e56-97a5-e09881d6800b
plot(solution, vars=[T, S])
# ╔═╡ dca7ab9c-0ed8-4352-bab1-64ac891d55ff
let
temps = [44, 36, 28, 20, 12, 4]
slopes = [0.8, 0.89, 1.01, 1.02, 1.4, 1.31]
X = [ones(6) temps]
β̂ = X \ slopes
end;
# ╔═╡ 53eddad6-e5c3-41fe-8cc7-ab2e7fa33c59
perceived_sweetness(S, T) = 14.9 * (S / 342.30)^(1.42224 - 0.0146071T)
# ╔═╡ 36ee98f4-2476-470d-bb69-0f9d1d333a30
heatmap(1:200, 4:40, perceived_sweetness, xlabel="Sugar concentration (g/L)", ylabel="temperature (degree Celcius)", title="perceived sweetness")
# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
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[compat]
DifferentialEquations = "~7.1.0"
ModelingToolkit = "~8.8.0"
Plots = "~1.27.6"
"""
# ╔═╡ 00000000-0000-0000-0000-000000000002
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