Engineering
Newton's Law of Cooling formula: transient temperature & time to cool
Calculate temperature and target time in a constant-ambient, constant-rate lumped cooling or warming model, with explicit reachability checks.
An exponential model with constant surroundings
The model assumes a spatially uniform object temperature, constant ambient temperature and a constant positive rate k. Under a lumped convection model k=h*A/(m*c_p); substantial internal temperature gradients or changing heat-transfer conditions require a different model.
Use k in inverse minutes when time is in minutes. Initial, ambient and target temperatures must use the same temperature scale; this example uses Celsius. The computed time is an ideal-model estimate, not a validated food-safety or safe-handling schedule.
Cooling and warming both approach ambient asymptotically. A displayed rounded temperature equal to ambient does not mean the mathematical trajectory has reached ambient at finite time.
| Thermal Decay Metric | Formula | T_0 = 92°C, T_amb = 22°C, k = 0.045/min | Target 20 C for this cooling example |
|---|---|---|---|
| Temp after 15 Minutes | B3 + (B2 - B3) * EXP(-B4 * B5) | 57.64 °C | Asymptotes to 22.00 °C |
| Time to Cool to 57 °C (Halfway) | -LN((B6 - B3)/(B2 - B3)) / B4 | 15.40 minutes (LN(2)/k) | No finite target time |
| Thermal Half-Life t_1/2 | LN(2) / B4 | 15.40 minutes per 50% gap | 15.40 minutes |
Transient Temperature T(t) = T_amb + (T_0 - T_amb) * EXP(-k * t)
Elapsed time must be nonnegative and k must be positive. Both warming and cooling use the same expression. An object initially at ambient stays at ambient.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(OR(B4<=0,B5<0),"Invalid inputs",ROUND(B3 + (B2 - B3) * EXP(-B4 * B5), 2)))Target time for cooling or warming
The initial temperature is reached at time zero. Otherwise the target must be strictly between initial and ambient temperatures. Targets at ambient, beyond ambient, or beyond the initial temperature in the other direction have no finite nonnegative target time under this model. If initial equals ambient, only that same target is reachable.
=IF(COUNT(B2:B4,B6)<>4,"Invalid inputs",IF(B4<=0,"Invalid inputs",IF(B6=B2,0,IF(OR(B2=B3,B6<=MIN(B2,B3),B6>=MAX(B2,B3)),"No finite target time",ROUND(-LN((B6-B3)/(B2-B3))/B4,2)))))Half-life of the temperature difference
Each half-life halves the remaining difference from ambient. This applies to warming as well as cooling.
=IF(NOT(ISNUMBER(B4)),"Invalid inputs",IF(B4<=0,"Invalid inputs",ROUND(LN(2)/B4,2)))Try the example
Use the original values or change them to test your understanding.