Project 2: Cooling of Coffee in a Mug

Lumped thermal modeling, radiation, evaporation, and parameter fitting

1 Problem Statement

A mug filled with hot coffee cools in a room. The goal is to predict the coffee temperature as a function of time and estimate the influence of mug properties, convection, radiation, and evaporation.

2 Suggested Parameters

Quantity	Symbol	Value
Coffee mass	(m_c)	(0.25 )
Coffee heat capacity	(c_{p,c})	(4180 )
Initial coffee temperature	(T_0)	(90^)
Room temperature	(T_)	(22^)
Mug inner radius	(r_i)	(0.04 )
Coffee height	(H)	(0.08 )

3 Basic Model

[ m c_p = -hA(T-T_) ]

with solution:

[ T(t)=T_+(T_0-T_) (-t) ]

4 Julia Example

using DifferentialEquations
using Plots

function cooling!(du,u,p,t)
    T = u[1]
    Tinf, hA, mcp = p
    du[1] = -hA/mcp*(T-Tinf)
end

u0 = [90.0]
p = (22.0, 0.25, 0.25*4180.0)
prob = ODEProblem(cooling!, u0, (0.0, 3600.0), p)
sol = solve(prob)

plot(sol.t ./ 60, sol[1,:],
     xlabel="Time [min]",
     ylabel="Temperature [°C]",
     label="Coffee")

5 ML/PINN Extension

Estimate unknown cooling parameters from measured data:

[ __i ^2 ]

Possible unknowns:

(h)
(UA)
evaporation coefficient
emissivity
mug heat capacity

6 Student Tasks

Implement one-node and two-node models.
Add radiation.
Add evaporation.
Fit parameters from data.
Compare simulation with experiment.

---
title: "Project 2: Cooling of Coffee in a Mug"
subtitle: "Lumped thermal modeling, radiation, evaporation, and parameter fitting"
---

## Problem Statement

A mug filled with hot coffee cools in a room. The goal is to predict the
coffee temperature as a function of time and estimate the influence of mug
properties, convection, radiation, and evaporation.

## Suggested Parameters

| Quantity | Symbol | Value |
|---|---:|---:|
| Coffee mass | \(m_c\) | \(0.25\ \text{kg}\) |
| Coffee heat capacity | \(c_{p,c}\) | \(4180\ \text{J/(kg K)}\) |
| Initial coffee temperature | \(T_0\) | \(90^\circ\text{C}\) |
| Room temperature | \(T_\infty\) | \(22^\circ\text{C}\) |
| Mug inner radius | \(r_i\) | \(0.04\ \text{m}\) |
| Coffee height | \(H\) | \(0.08\ \text{m}\) |

## Basic Model

\[
m c_p\frac{dT}{dt}
=
-hA(T-T_\infty)
\]

with solution:

\[
T(t)=T_\infty+(T_0-T_\infty)
\exp\left(-\frac{hA}{mc_p}t\right)
\]

## Julia Example

```julia
using DifferentialEquations
using Plots

function cooling!(du,u,p,t)
    T = u[1]
    Tinf, hA, mcp = p
    du[1] = -hA/mcp*(T-Tinf)
end

u0 = [90.0]
p = (22.0, 0.25, 0.25*4180.0)
prob = ODEProblem(cooling!, u0, (0.0, 3600.0), p)
sol = solve(prob)

plot(sol.t ./ 60, sol[1,:],
     xlabel="Time [min]",
     ylabel="Temperature [°C]",
     label="Coffee")
```

## ML/PINN Extension

Estimate unknown cooling parameters from measured data:

\[
\min_\theta \sum_i
\left[
T_\theta(t_i)-T_i^{meas}
\right]^2
\]

Possible unknowns:

- \(h\)
- \(UA\)
- evaporation coefficient
- emissivity
- mug heat capacity

## Student Tasks

1. Implement one-node and two-node models.
2. Add radiation.
3. Add evaporation.
4. Fit parameters from data.
5. Compare simulation with experiment.