Newton's Law of Cooling Calculator: Temperature Against Time, Time to Cool and Cooling Constant

Find how long an object takes to cool down or warm up to the temperature around it.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Enter the starting temperature, the temperature of the surroundings and the cooling constant. Or give one measured temperature and its time and let the calculator find the constant.

It gives the temperature after the time you choose, T = Ta + (T0 - Ta) e to the minus kt, and the time to reach a temperature you want.

It also gives the time constant and the half-life of the temperature difference.

Formulas used

  • T(t) = T_a + (T_0 - T_a)e^{-kt}
  • t = \frac{1}{k}\ln\frac{T_0 - T_a}{T - T_a}
  • k = \frac{1}{t_1}\ln\frac{T_0 - T_a}{T_1 - T_a}
  • \tau = 1/k

Worked example

With the default numbers in the calculator above:

  1. Time constant and half-life of the temperature difference: 20.00 min and 13.86 min
  2. Temperature after the time you gave: 62.46 C
  3. Time to reach the temperature you gave: 25.06 min (0.42 hours)

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

What does the law say?

The rate of cooling is proportional to the difference between the temperature of the object and its surroundings.

Does the object reach the room temperature?

Only in the limit. The difference falls by the same fraction in each equal time, and never quite reaches zero.

How do I find k?

Measure the temperature at the start and after some time, or use k = h A over m c from the heat transfer coefficient.

When is it accurate?

For small temperature differences, with natural convection, and for objects that are all at the same temperature.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Incropera and DeWitt, Fundamentals of Heat and Mass Transfer (lumped capacitance method).

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