Newton's Law of Cooling

What is Newton's Law of Cooling?

The rate, where a body that is exposed, changes the temperature via radiation is approximately proportional to the difference between the object’s temperature and its surroundings, and the provided difference is low. This is known as Newton's Law of cooling.

To define Newton's Law of cooling, the rate of heat loss from a body is directly proportional to the difference in body temperature, and its surroundings.

In a convective heat transfer, Newton's Law is followed for pumped fluid cooling or forced air, where the fluid properties do not strongly vary with the temperature. However, only approximately true for the buoyancy-driven convection, where the velocity of the flow increases with the difference in temperature. Finally, in the heat transfer case, by the thermal radiation, Newton's Law of cooling holds only for very slight temperature differences.

Newton's Law of cooling can be given by,

\[\frac{d_{T}}{d_{t}}\] = k(Tt – Ts)

Where, 

  • Ts = temperature of the surrounding

  • Tt = temperature at time 't'

  • k = Positive constant depends on the nature and area of the body surface under consideration

Newton's Law of Cooling Formula

The formula for Newton's Law of Cooling can be defined as the greater the temperature difference between the system and its surrounding; the heat is transferred more rapidly; it means the body temperature changes more rapidly. Newton's Law of the cooling formula is expressed by the formula given below.

T(t) = Ts + (To – Ts) e-kt

Where,

  • T(t) = body’s temperature at time ‘t’,

  • Ts = surrounding temperature,

  • To = initial temperature of the body,

  • t = time

  • k = constant

Newton's Law of Cooling Derivation

Let us look at the derivation of Newton's law of cooling. For small temperature differences between a body and its surrounding, the rate of body cooling is directly proportional to the difference in temperature and the surface area that is exposed.

 \[\frac{d_{Q}}{d_{t}}\] ∝ (q – qs)

Where q and qs are temperature corresponding to the object and its' surroundings.

From the above expression,

 \[\frac{d_{Q}}{d_{t}}\] = -k[q – qs)] ---- (1)

This expression represents Newton's Law of cooling. It can be derived directly from the Stefan's Law, that gives,

k =  \[\frac{4eq\timesθ_{0}^{3}}{mc}\]A ---- (2)

Now,  

 \[\frac{d_{θ}}{d_{t}}\] = -k[θ – θo]

\[\int_{θ1}^{θ1}\] \[\frac{d_{θ}}{θ-θ_{0}}\] = \[\int_{0}^{1}\]-kdt

[Image to be added Soon]

where,

  • qi = initial temperature of the object

  • qf = final temperature of the object

\[\frac{q_{f}-q_{0}}{q_{i}-q_{0}}\] = kt

(qf – q0) = (qi – q0) e-kt

qf = q0 + (qi – q0) e-kt ---- (3)

Methods to Apply Newton's Law of Cooling

If we need only approximate values sometimes from Newton's Law, we can assume a constant rate of cooling, equal to the rate of cooling corresponding to the average temperature of the body during the interval.

That is, \[\frac{d_{θ}}{d_{t}}\]  dθ\ = k(<q> – q0) ---- (4)

If qi and qf are the initial, final temperature of the body respectively, then,

<q> = \[\frac{q_{i}-q_{f}}{2}\]  ---- (5)

Note that equation (5) is only an approximation, and equation (1) must be used to exact values.

Limitations of Newton's Law of Cooling

Some of the limitations of Newton's Law of Cooling are listed below.

  • The loss of heat from the body should only be happened by radiation.

  • The temperature differences between the body and its surroundings must be small.

  • The main limitation of Newton's Law of Cooling is that the temperature of surroundings must remain constant during body cooling.

Newton's Law of Cooling Example

Consider Newton's Law of Cooling graph given below that states Newton's Law of Cooling.

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This process of cooling data can be measured and plotted, and the results are used to compute the unknown parameter 'k.' Sometimes, the parameter can also be derived mathematically.

Example of solving a problem using Newton's Law of Cooling

A body with a temperature of 40º C is kept in the surroundings of a constant temperature of 20º C. If its temperature falls to 35º C in 10 minutes, find how much excess time it will take for the body to attain the temperature of 30º C.

Solution

From Newton’s law of cooling, qf = qi e-kt

Now, for the interval where the temperature falls from 40º C to 35º C,

(35 – 20) = (40 – 20) e-k.10

e-10k = 3/4

Therefore, k = [ln 4/3]/10 ---- (a)

Now, for the next interval, 

(30 – 20) = (35 – 20)e-kt

So, e-kt = 2/3

kt = ln 3/2 ---- (b)

From equation (a) and (b), we get,

t = 10 × [ ln(3/2) / ln(4/3) ]= 14.096 min.

Applications of Newton's Law of Cooling

A few and important applications of Newton's Law of Cooling are listed below.

  • Using Newton's Law of Cooling, we can find the temperature of a soda that is placed in a refrigerator with a certain amount of time

  • Used to predict how long it will take for a hot object to cool down at a certain temperature

  • It also helps to indicate the death time given by the probable body temperature at the death time and the current body temperature

FAQ (Frequently Asked Questions)

1. Explain the Physical Meaning of Newton's Law of Cooling?

Ans: A body is hotter than its surroundings and cools according to how high its temperature is for the surroundings. Thereby, a hot body cools faster compared to a warm body. The same body quickly cools initially and then more and more slowly.

As an example, if there is a brick at 100 degrees, at room temperature, it is 20, and it cools to 60 degrees (halfway) in 5 minutes (for suppose), and it will take another 5 minutes to reach to 40 degrees and another 5 minutes to reach to 30 degrees, and again reaches to 25 degrees in another 5 minutes. So, every halfway step towards 20 degrees takes 5 minutes.

2. Are Newton's Law of Cooling and Stefan–Boltzmann Law Related to Each Other?

Ans: By performing many pieces of research, it is said that they both are related.

Stefan's Law: The total radiant energy per second per unit surface area of a perfectly black body is always directly proportional to the fourth power of its absolute temperature.

Newton's Law of Cooling: The rate of loss of heat by a body, due to the radiation is directly proportional to the differences in temperature between the body and its surroundings, and the temperature difference is less.

Newton's Law of cooling doesn't consider the fact that a body can cool by both radiation and convection; it only takes radiation (a limitation of this Law).

This Law can also be derived from the Stefan-Boltzmann law. If the temperature difference is very less, Stefan's Law can be altered to Newton's Law of cooling.