Cooling coffee

A hot cup of coffee cools fast at first and then ever more slowly: its excess temperature over the room decays exponentially, Newton's law of cooling, and it approaches the room temperature without ever crossing it.

The coffee starts at 90 degrees in a room at 20 degrees, with the cooling rate k = 0.1 per minute. The default blueprint shows the temperature over one hour.

import matplotlib.pyplot as plt
import numpy as np

from skcomplex.engine.runtime import Simulation
from skcomplex.engine.viz import RerunObserver
from skcomplex.models import create

MODEL = 'cooling_coffee'

system = create(MODEL, T=90.0, room=20.0, k=0.1)
observer = RerunObserver(MODEL, save_path=f'{MODEL}.rrd', blueprint='default', interval=1)
simulation = Simulation(system, dt=0.5, observers=[observer]).check().prepare().run(until=60)

plot cooling coffee

Open the recording in the Rerun viewer

Exponential relaxation

The solution is T(t) = T_room + (T0 - T_room) exp(-k t), so the excess temperature T - T_room should fall on a straight line of slope -k on a logarithmic axis, for every starting temperature.

times = np.asarray(simulation.trajectory_.times)
fig, ax = plt.subplots(figsize=(6, 4))
for start in [90.0, 60.0, 40.0]:
    run = Simulation(create(MODEL, T=start, room=20.0, k=0.1), dt=0.5).check().prepare().run(until=60)
    line = ax.semilogy(times, np.ravel(run.trajectory_.values('coffee.T')) - 20.0, label=f'T0 = {start:.0f}')[0]
    ax.semilogy(times, (start - 20.0) * np.exp(-0.1 * times), '--', color=line.get_color())
ax.set_xlabel('time (minutes)')
ax.set_ylabel('T - T_room')
ax.set_title('Excess temperature decays as exp(-k t), dashed')
ax.legend()
plt.show()

Excess temperature decays as exp(-k t), dashed

Download Python source code: plot_cooling_coffee.py

Download Jupyter notebook: plot_cooling_coffee.ipynb

Gallery generated by mkdocs-gallery

About the model

A cup of coffee whose temperature relaxes to that of the room, T' = -k (T - T_room).

References

Newton 1701
Meadows 2008