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Newton's Law of Cooling

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Last updated date: 13th Jul 2024
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What is Newton's Cooling Law?

The rate at which an exposed body changes temperature through radiation is approximately proportional to the difference between the object's temperature and its surroundings, provided the difference is small, according to Newton's law of cooling.


The rate of heat loss from a body is directly proportional to the temperature differential between the body and its surroundings, according to 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,


Q = h. A. (Tt – Tenv)


Where, 

  • Q = rate of heat transfer out of the body

  • H = heat transfer coefficient

  • A = heat transfer surface area

  • T = temperature of the object's surface

  • Tenv  = temperature of the environment

  • Tt = time-dependent temperature

Formula 2:

 \[\frac {dT}{dt}\] = k (Tt – Ts


Where,

Tt = temperature of the body at time t and

Ts  = temperature of the surrounding,

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

The Formula for Newton's Law of Cooling

The greater the temperature difference between the system and the surrounding environment, the faster heat is transmitted and the body temperature changes. The formula for Newton's law of cooling is,


Ts + (To – Ts) e-kt = T(t)


Where,

t stands for time, and

T(t) is the temperature of a particular body at a given time t.

Ts denotes the temperature of the environment.

to = body's starting temperature,

k stands for constant.


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

Methods of Newton's Law of Cooling

We can assume a constant rate of cooling, which is equal to the rate of cooling related to the average temperature of the body during the interval, when we just need approximate values from Newton's law.


i.e. \[\frac {d \theta}{dt}\] = k (q – qs) . . . . . . . (4)   


If q and qs are the body's starting and ultimate temperatures, then


<q> = \[\frac {(q_i + q_f)}{2}\] . . . . . (5)  


Remember that equation (5) is simply an approximation, and that for exact numbers, equation (1) must be used.


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,  

\[\frac {d \theta}{dt}\] = 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.

Newton's Law of Cooling's Limitations

The temperature difference between the body and the environment must be minimal.


Only radiation should be used to lose heat from the body.


The temperature of the surroundings must remain constant while the body cools, which is a critical constraint 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 caused 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.

Derivation of Newton's Law of Cooling

The rate of cooling of a body is directly related to the temperature difference and the surface area exposed when the temperature difference between the body and its surroundings is modest.


\[\frac {dQ}{dt}\] (q – qs), where q and qs represent the temperature of the item and its surroundings, respectively.


According to the preceding expression, \[\frac {dQ}{dt}\] = -k[q – qs] . . . . . . . . (1)


Newton's law of cooling is represented by this statement. Stefan's law, which gives, can be immediately deduced. 


k = \[\frac {4e \sigma \times \theta_{0}^{3}}{mc}\] A . . . . . (2)


Now, \[\frac {d \theta}{dt}\] = - k[θ – θo]    


\[\Rightarrow \int_{\theta_1}^{\theta_2} \frac {d \theta}{(\theta - \theta_0)} = \int_0^1 -kdt\]


(Image will be Uploaded Soon)


where,

qi = the object's starting temperature,

qf is the object's final temperature.

ln \[\frac {(q_f -q_0)}{(q_{f}i - q_0)}\] = kt

(qf – qo) = (qf – qo) e-kt

qo + (qi – qo) e-kt = qf . . . . . . (3).

Check: Conduction heat transfer

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


\[\frac {dQ}{dt}\] ∝ (q – qs)


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

From the above expression, 


\[\frac {dQ}{dt}\]= -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 =  \[\frac {4eq \times \theta_0^3}{mc}\] A ---- (2)


Now,  

\[\frac {d \theta}{dt}\]= -k[θ – θo

⇒ \[\Rightarrow \int_{\theta_1}^{\theta_1} \frac {d \theta}{(\theta - \theta_0)} = \int_0^1 -kdt\]

(Image will be Uploaded Soon)

where,

  • qi = initial temperature of the object

  • qf = final temperature of the object

\[\frac {(q_f -q_0)}{(q_{f}i - q_0)}\] = kt


(qf – qo) = (qf – qo) e-kt


qo + (qi – qo) e-kt = qf . . . . . . (3).

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 = \[\frac {3}{4}\]


Therefore, k = ln \[\frac {4}{3}\] ln \[\frac{\frac{4}{3}}{10}\] ---- (a) 


Now, for the next interval, 


(30 – 20) = (35 – 20)e-kt


So, e-kt = \[\frac {2}{3}\]


kt = ln\[\frac {3}{2}\] ---- (b) 


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


t = \[ 10 \times In (\frac {3}{2})\]= 14.096 min.

Applications of Newton's Law of Cooling

A few 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

FAQs on Newton's Law of Cooling

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

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?

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.

3. What does Newton's Law of Cooling mean in terms of Physics?

A body is hotter than its surroundings and cools in proportion to how hot the surroundings are. As a result, a hot body cools faster than a warm body. The same body cools down swiftly at first, then gradually.


For example, if a brick is at 100 degrees and is 20 degrees at room temperature, it cools to 60 degrees (halfway) in 5 minutes (for example), then another 5 minutes to reach 40 degrees, another 5 minutes to reach 30 degrees, and another 5 minutes to reach 25 degrees. As a result, every half-step closer to 20 degrees takes 5 minutes.

4. Is there a link between Newton's Law of Cooling and the Stefan–Boltzmann Law?

It has been determined that they are both related through extensive investigation. Stefan's Statute: A totally black body's total radiant energy per second per unit surface area is always proportional to the fourth power of its absolute temperature. Newton's Law of Cooling states that the rate of heat loss by a body owing to radiation is directly proportional to temperature differences between the body and its surroundings, and that the smaller the temperature difference, the faster the body loses heat. The notion that a body can cool through both radiation and convection is ignored by Newton's Law of Cooling; it only takes radiation (a limitation of this Law).

5. Is the free PDF of Newton’s Law of Cooling – Explanation, Methods to Apply, Limitations and Solved Examples helpful?

Yes, the free PDF of Newton’s Law of Cooling – Explanation, Methods to Apply, Limitations and Solved Examples is very helpful. It’s been explained with the help of examples, methods to apply, and limitations. The concepts given in the free PDF are easy to read and provide learning throughout the session. You can solve the problem questions related to them and will be able to score examinations. To know more you can download the PDF from Vedantu.

6. What is the equation of Newton’s law of cooling?

The rate of cooling of a body is directly related to the temperature difference and the surface area exposed when the temperature difference between the body and its surroundings is modest. \[\frac {dQ}{dt}\] (q – qs)], where q and qs represent the temperature of the item and its surroundings, respectively. Students are required to practise the problem sums based on this equation with the help of the free PDF. 

7. What are the factors on which cooling depends?

The temperature of a body varies at a rate proportional to the difference in temperature between the body and its surroundings, according to Newton's law of cooling (or heating). In some cases, it is a reasonably realistic estimate. Then, with the same temperature differential, the rate of cooling is determined by: I The area of the body's surface. (ii) The nature of the body's surface. (iii) The material of the body's surface (material effects conductivity through walls of the body)