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Newton's Law of Cooling Calculator

Use the given online tool to determine the rate at which an object cools in a surrounding environment according to Newton’s Law of Cooling.

Initial Temperature:

°C ▾

Ambient Temperature:

°C ▾

Area:

m² ▾

Cooling Constant:

J/K ▾

Heat Transfer Coefficient:

W/(m²·K) ▾

Cooling Constant:

sec ▾

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Using the Newton's Law of Cooling calculator, you can quickly determine how long it takes for an object to cool from one temperature to another.

What Is Newton's Law of Cooling?

“The rate of heat loss of a body is directly proportional to the difference between its temperature and the surrounding (ambient) temperature.”

In simple terms, it describes how the temperature of an object changes when exposed to surroundings with a different temperature. Heat transfer occurs mainly through conduction and convection for this law to be applicable.

Newton's Law of Cooling Formula

The differential form of Newton's Law of Cooling is:

\(\dfrac{dT}{dt} = -k (T - T_s)\)

T = temperature of the object at time t
Tₛ = surrounding (ambient) temperature
k = cooling constant

Solving this differential equation gives the temperature as a function of time:

\(\displaystyle T(t) = T_s + (T_0 - T_s)e^{-k t}\)

T₀ = initial temperature of the object
t = time elapsed
k = cooling constant

This allows you to calculate the temperature of a body at any given time. For more details, see Wikipedia.

How to Calculate Newton’s Law of Cooling

Follow these steps:

Determine the Ambient Temperature: Measure the temperature of the surrounding air or medium.
Determine the Initial Temperature: Measure the object's initial temperature (°C or K).
Determine the Cooling Coefficient (k): Estimate based on material properties and exposed surface area.
Measure the Final Temperature: Use the formula \(T(t) = T_s + (T_0 - T_s)e^{-k t}\) to find the temperature at time t.

Example:

Find the final temperature of a body after 3 seconds using Newton’s Law of Cooling. Given:

Ambient temperature (Tₛ) = 20 °C
Initial temperature (T₀) = 3 °C
Surface area (A) = 0.003 m²
Heat capacity (C) = 4 J/K
Heat transfer coefficient (h) = 1 W/(m²·K)

Solution:

Calculate the cooling constant: \(k = \dfrac{hA}{C} = \dfrac{1 \times 0.003}{4} = 0.00075\)
Use the cooling formula: \(\displaystyle T(t) = T_s + (T_0 - T_s)e^{-k t}\)
Substitute values: \(\displaystyle T(3) = 20 + (3 - 20)e^{-0.00075 \times 3}\)
Calculate: \(\displaystyle T(3) = 20 - 17 \times e^{-0.00225} \approx 20 - 16.96 = 3.04 °C\)

Instead of manual calculations, the Newton’s Law of Cooling calculator can provide accurate results instantly.

Limitations of Newton’s Law of Cooling

The temperature difference between the object and surroundings should be small.
Heat loss should primarily occur through convection and conduction.
The surrounding temperature must remain constant during the process.

Frequently Asked Questions (FAQs)

Why Is Newton's Law of Cooling Important?

Predicts how the temperature of an object changes over time
Analyzes heat transfer between an object and surroundings
Useful in practical engineering applications

What Factors Affect Newton's Law of Cooling?

Surface area of the object
Temperature difference between object and surroundings
Heat transfer coefficient (k)

How Do You Find k in Newton's Law of Cooling?

The cooling constant k can be determined in two ways:

Using temperature difference over time: \(\displaystyle k = \frac{T_1 - T_2}{t}\), where t = t₂ - t₁
Using heat transfer coefficient and area: \(\displaystyle k = \frac{hA}{C}\), where h = heat transfer coefficient, A = area, C = heat capacity

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