› Newton's Law of Cooling — Coffee Simulator Back
Heat Transfer

Newton's Law of Cooling — Coffee Simulator

Quick answer
Newton's law of cooling dT/dt = −k·(T − T_env) has the solution T(t) = T_env + (T₀ − T_env)·e^(−k·t) (k = cooling constant, T_env = room temperature). The temperature-difference half-life is t₁/₂ = ln2/k, and the drinkable time is t* = −(1/k)·ln[(T* − T_env)/(T₀ − T_env)].

Vary initial temperature, room temperature, and cooling constant k to calculate how long until your coffee reaches drinking temperature. Experience exponential cooling described by Newton's law in real time.

Parameter Settings

Container preset
Initial temperature T₀
°C
Ambient temperature T_env
°C
Cooling constant k
/min
Drinking temperature T_drink
°C
Playback speed (× real time)
×
☕
Time until drinkable
—
Minutes
Cooling
—
Current temperature (°C)
—
Elapsed time (min)
—
Ambient temperature T_env (°C)
—
Cooling constant k (/min)
—
Time constant τ (min)
—
Half-life t½ (min)
—
Initial cooling rate (°C/min)
—
Time until ready to drink (min)
Compare
Log
Theory & Key Formulas
Compare
Log scale
Theory & Key Formulas
Differential equation: \(\dfrac{dT}{dt} = -k(T - T_{env})\)

Analytical solution: \(T(t) = T_{env} + (T_0 - T_{env})e^{-kt}\)

Time constant: \(\tau = 1/k\), half-life: \(t_{1/2}=\ln 2/k\)
Time to drinking temperature: \(t^* = -\dfrac{1}{k}\ln\dfrac{T^* - T_{env}}{T_0 - T_{env}}\)

🎓 Learn Newton's Law of Cooling Through Conversation

🙋
When you brew coffee, it cools quickly at first, then seems to slow down. Can physics explain that?
🎓
That is exactly Newton's law of cooling. The cooling rate is proportional to the temperature difference: $dT/dt = -k(T - T_{env})$. If coffee is 90°C and the room is 20°C, the difference is 70°C, so it cools quickly. At 60°C the difference is 40°C, so the cooling rate falls to about 57% of the initial value.
🙋
What is the time constant τ? If the simulator says τ = 50 minutes, how should I read that?
🎓
τ = 1/k is the time required for the temperature difference to fall to about 37% (1/e) of its initial value. If the initial difference is 70°C, after one τ it becomes about 26°C. A smaller τ means faster cooling. Paper cups may be around 15 minutes, ceramic cups 30-50 minutes, and insulated tumblers can exceed 100 minutes.
🙋
So what about adding milk now versus later? Mathematically, which keeps the coffee warmer when I drink it?
🎓
If you drink immediately, adding milk later keeps it hotter. But if you drink in five minutes, adding milk early can leave it warmer then. Adding milk early lowers the initial temperature difference, reducing the absolute heat loss during the waiting period.
🙋
Interesting. Is this law useful beyond coffee? Does it appear in CAE too?
🎓
Very often. It appears in lumped thermal analysis for small parts, electronic cooling models, building heat storage, forensic time-of-death estimation, annealing control, and duct-flow temperature response. In CAE terms, it is the basic first-order transient thermal model.

Frequently Asked Questions

What is Newton's law of cooling?
It is an empirical rule stating that the cooling rate of an object is proportional to the temperature difference between the object and its surroundings. It is expressed by the differential equation $dT/dt = -k(T - T_{env})$, with the solution being the exponential function $T(t) = T_{env} + (T_0 - T_{env})e^{-kt}$. This empirical law was observed by Newton in the 17th century, and in modern heat transfer theory, it corresponds to convective heat transfer $Q = hA(T - T_{env})$.
How is the cooling constant k determined?
It is expressed as $k = hA/(mc_p)$. Here, h is the heat transfer coefficient (W/m²K), A is the surface area (m²), m is the mass (kg), and $c_p$ is the specific heat (J/kgK). A container with insulation has a small h and thus a small k. A thin metal cup has a large h and also a small mass m, resulting in a large k. To determine k experimentally, plot temperature vs. time data on a logarithmic scale and find the slope.
Are there conditions where Newton's law of cooling does not hold?
When the Biot number (Bi = hL/λ) exceeds 0.1, the temperature gradient inside the object cannot be ignored, and the lumped capacitance assumption breaks down. Also, if the temperature difference is very large, radiative heat transfer becomes dominant (proportional to $T^4$), making the linear approximation invalid. In such cases, it is necessary to use transient heat conduction analysis with partial differential equations or combine the Stefan-Boltzmann radiation law.
How is 'time of death estimation' done in forensic medicine?
The Henssge and Marshall formula is used. By measuring the rectal temperature T of a corpse, the elapsed time is back-calculated from $T_{body} = T_{env} + (37 - T_{env})e^{-kt}$. The cooling constant k of the human body varies with body weight, clothing, room temperature, and ventilation conditions, but for a standard weight, k is approximately 0.05–0.1/h. However, this is only a guideline, and actual forensic medicine combines multiple indicators.
Why can an insulated tumbler keep drinks hot for hours?
A vacuum-insulated tumbler reduces heat loss due to convection and conduction to nearly zero by creating a vacuum between the inner and outer walls. The mirror finish on the inner wall also reflects radiative heat. As a result, k becomes extremely small (less than 1/10 that of a regular cup), and τ becomes several hours or more. The principle of vacuum insulation is the same as that of the thermos (invented by Dewar in 1892), and in CAE, it is also applied to the thermal design of spacecraft.

Related Tools

Cooling Coil Load Simulator — Sensible, Latent & Total Heat Cooling coil load simulator: compute sensible, latent and total heat loads, dehumidificatio... Wien Displacement Law Simulator — Peak Wavelength of Black-Body Radiation Wien displacement simulator: real-time peak wavelength from lambda_max*T=b and Stefan-Boltz... Bimetal Thermostat Simulator — Thermal Bending of a Two-Layer Strip The bimetal thermostat simulator uses the Timoshenko bimetal formula to compute how a two-l... Brayton Cycle Simulator — Thermal Efficiency of a Gas Turbine Brayton cycle simulator: computes thermal efficiency, T2, T4 and net work in real time from... Heat Of Combustion Calculate fuel heating values (LHV/HHV), thermal efficiency, SFC, and CO₂ emissions for eng... Jet Impingement Cooling Simulator — Heat Transfer on the Plate The jet impingement cooling simulator computes the Nusselt number and heat transfer coeffic...

How to Use

  1. Set the initial coffee temperature (t0) using the slider or numeric input—typical values range 85–95°C for freshly brewed espresso or filter coffee.
  2. Define the ambient room temperature (tenv) between 15–25°C depending on your environment; this is the baseline the liquid cools toward.
  3. Adjust the cooling constant (k), which depends on cup material and geometry; ceramic cups typically use k=0.08–0.12 min⁻¹, while thin paper cups exhibit k=0.15–0.20 min⁻¹.
  4. Enter the desired target time (td) in minutes to see the predicted temperature drop following the equation T(t)=tenv + (t0−tenv)·e^(−k·t).

Worked Example

A ceramic cup of 95°C coffee placed in a 20°C office (k=0.10 min⁻¹) will cool to approximately 67°C after 10 minutes: T(10)=20+(95−20)·e^(−0.10·10)=20+75·0.368≈47.6°C. After 30 minutes, the same coffee reaches roughly 32°C. Industrial beverage packaging—including thermal carafe design and insulated shipping containers—relies on precise k-values to maintain temperature tolerances during transport.

Practical Notes

  1. Lid presence significantly reduces k by 30–40%; sealed containers slow convection loss, extending drinkable temperature windows from 5 to 12 minutes.
  2. Stirring and surface exposure increase k by 15–25%; stagnant liquid in narrow vessels cools slower than agitated liquid in wide bowls.
  3. Material conductivity matters: stainless steel (k≈0.18 min⁻¹) sheds heat faster than thick ceramic (k≈0.08 min⁻¹) at identical volumes.