Newton Cooling Calculator
Predict cooling with the exponential law in metric or imperial units.
What it does
Hot objects cool fast at first and then creep toward room temperature. The gap between object and air shrinks by a fixed fraction per minute, which traces an exponential curve rather than a straight line. That single fact predicts coffee, castings, and server rooms with surprising accuracy.
The tool above applies the curve directly. Choose metric or imperial, enter start and air temperatures, the cooling constant k, and the elapsed minutes. It returns the final temperature, how far it fell, and what share of the original gap is left. Defaults cool 90 C toward 20 C with k = 0.05, so the page opens solved at 62.46 C.
k carries the physics. A fan, a heat sink, or a small mass raises k; still air, insulation, or a large thermal mass lowers it. Because k folds many effects into one number, best practice is to fit it: take one timed reading, solve k from the observed drop, then forecast the rest of the shift with that fitted value.
Both unit modes share the same k and time and mirror each other exactly. Metric 90 C toward 20 C lands at 62.46 C, while imperial 194 F toward 68 F lands at 144.42 F. Every row on screen follows the active system, so compare runs only within one mode.
Use the table for common fractions of the gap left at whole time constants. Then run your case. The page assumes still, fixed air and one k; gusty air, evaporation, freezing, or an internal heater all bend the curve away from this clean form.
How to use
- Pick metric (C) or imperial (F).
- Enter the start temperature of the object.
- Enter the steady air temperature.
- Enter k per minute and the elapsed minutes.
- Read the final temperature, the drop, and the gap left.
- Fit k from one timed reading before trusting a forecast.
Formula
- T(t)
- Temperature at time t, in C or F per the active mode.
- Start
- Starting temperature of the object.
- Air
- Surrounding air temperature, held fixed.
- k
- Cooling constant per minute, set by surface and airflow.
- t
- Elapsed time in minutes.
k uses per minute and t uses minutes, so the exponent stays unit free. The same k works in both unit modes because only temperatures convert. The model assumes fixed air temperature, one constant k, no phase change, and no extra heat source.
Worked example
Example: cooling from 90 C toward 20 C
A hot part at 90 C cools in 20 C air with k = 0.05 per minute. Find its temperature after 10 minutes.
- List inputs: start 90 C, air 20 C, k = 0.05 per minute, time 10 minutes.
- Exponent = -0.05 x 10 = -0.5. e to that power is 0.60653.
- Gap closes to 60.653% of the start gap: (90 - 20) x 0.60653 = 42.46 C.
- Add back air: 20 + 42.46 = 62.46 C.
- Drop so far = 90 - 62.46 = 27.54 C, with 60.65 % of the gap left.
- In Fahrenheit this is 194 F falling toward 68 F and landing at 144.42 F.
Reference table
| k x t | Gap left | Cooled share |
|---|---|---|
| 0.5 | 60.65 % | 39.35 % |
| 1 | 36.79 % | 63.21 % |
| 2 | 13.53 % | 86.47 % |
| 3 | 4.98 % | 95.02 % |
| 5 | 0.67 % | 99.33 % |
Limits
- Fixed air temperature only; drifting rooms break it.
- One constant k; fans that cycle need staged models.
- No phase change, evaporation, or internal heat.
- Food safety still needs a probe, not a forecast.
Common mistakes
- Mixing per second k with minute time.
- Entering air as the start and start as the air.
- Expecting a straight line drop per minute.
- Reusing a still air k for a fanned part.
- Reading the gap left as the temperature itself.
Try it yourself
Quick checks. Answers sit tucked under each question.
Start 80 C toward 20 C, k = 0.1, time 5 min?
Final is 56.39 C. Drop is 23.61 C. Gap left is 60.65 %.
Start 200 F toward 70 F, k = 0.05, time 20 min?
Final is 117.82 F. Drop is 82.18 F. Gap left is 36.79 %.
What share of the gap is left when k x t = 1?
e to the -1 is 0.3679, so 36.79 % of the gap is left after one time constant.
Questions
What is Newton law of cooling?
The exponential cooling rule: T(t) = air + (start - air) x e^(-kt). The gap to air shrinks by the same fraction each minute.
What is k?
k sets speed. It folds together surface area, airflow, and heat capacity. Double k and the gap halves in half the time.
What happens at time zero?
Set time to 0. The exponential equals 1, so the tool returns the start temperature with zero drop.
Does it ever reach air temperature?
It approaches air temperature but never quite lands. After five time constants (t = 5/k) only 0.67% of the gap is left.
Does it cover warming too?
No. The page fits one constant k with fixed air. Changing airflow, evaporation, or phase change breaks the simple curve.
How do I work in Fahrenheit?
Yes. Pick imperial and enter 194 F toward 68 F with the same k and time to land at 144.42 F.