Calculate cooling time to a target temperature with Newton’s law of cooling, or find the final temperature or cooling constant k, in °C, °F or K.

k = hA/(mc), not the convection coefficient h. Unknown? Measure the temperature twice and use the Cooling coefficient mode.

Optional settings

Cooling Time Formula

The following formula is used to calculate the cooling time for Newton’s law of cooling (lumped-capacitance model), assuming a constant ambient temperature and heat transfer coefficient (and that the object’s internal temperature is approximately uniform).

Tc = (m c) / (h A)ln((Tᵢ - T_∞) / (Tf - T_∞))

Formula source:  A Heat Transfer Textbook by John H. Lienhard IV and John H. Lienhard V (5th ed., 2019)

Variables:

  • Tc is the cooling time (s)
  • m is the mass of the object or substance (kg)
  • c is the specific heat capacity of the substance (J/kg·K)
  • Ti is the initial temperature (°C or K)
  • Tf is the target final temperature (°C or K)
  • T∞ is the ambient (surrounding fluid) temperature (°C or K)
  • h is the heat transfer coefficient (W/m²·K)
  • A is the surface area of the object or substance (m²)

For the calculator, enter k = hA/(mc) = 1/τ in a compatible reciprocal time unit. Then Tc = ln((Ti − T∞)/(Tf − T∞))/k. A different target must lie strictly before ambient. The initial temperature itself has earliest time zero. The same model describes warming toward a hotter ambient.

What is a Cooling Time?

Cooling time refers to the period required for an object or substance to decrease in temperature to a desired level. This term is often used in various fields such as physics, engineering, and cooking. The duration of the cooling time can depend on several factors including the initial temperature, the desired final temperature, the properties of the substance, and the cooling method used.

How to Calculate Cooling Time?

The following steps relate the physical model to the calculator’s coefficient input:

  1. First, gather initial, ambient and target temperatures, plus a measured or independently estimated cooling coefficient k. If deriving k physically, also obtain mass, specific heat, surface area and heat transfer coefficient.
  2. If using physical properties, compute τ = (m · c)/(h · A) and k = 1/τ. With SI inputs, k is in s⁻¹.
  3. Select Time to target and enter the temperatures and k. Time is reported in minutes; coefficient units convert the physical rate.
  4. Check that the target is reachable before ambient and that the constant-coefficient, uniform-temperature assumptions apply. Ambient is approached as a limit.

Example Problem:

Use the following values as an example problem to test your knowledge:

mass (m) = 2 kg

specific heat capacity (c) = 500 J/kg°C

initial temperature (Ti) = 50°C

ambient temperature (T∞) = 10°C; target final temperature (Tf) = 20°C

heat transfer coefficient (h) = 100 W/m²°C

surface area (A) = 5 m²

These values give τ = 2 s and k = 0.5 s⁻¹. Enter initial 50°C, ambient 10°C and target 20°C: t = 2 ln(4) ≈ 2.773 s = 0.04621 min. The calculator’s decimal-place setting controls displayed precision.