Newton’s Law of Cooling

You will learn in detail about Newton’s Law of Cooling in this article. The transfer of heat can have various effects on the system that we learnt in thermodynamics, but how does the transfer of heat take place? Does it require a medium or not? On what factors does the rate of heat transfer depend?
Is there more than one way of transfer of heat?

The rate of heat loss from a body is directly proportional to the difference in temperature between the body and its surroundings, according to Newton’s law of cooling. In this article, we will explore the modes of transfer of heat, and we will discuss in detail Newton’s law of cooling and its limitations. Continue reading to know more.

Modes of Transfer of Heat

The heat from the sun reaches the earth even though there is no medium, but when a utensil is kept on the stove, the metal part of the utensil gets heated due to the transfer of heat through the metal medium. Now, suppose we have some water in the utensil. In that case, the water molecules rise from the base of the utensil to the top as they get heated, and new molecules get near the source, and again, they rise when they are heated, and the process continues, and the whole bulk of water gets heated up.
In the above three cases, the heat is transferred through three different modes,

1. Radiation
2. Conduction
3. Convection

Conduction

In the case of the metal part getting hotter when placed on the stove, the heat is transferred through the metal’s bulk, which means the heat travels through the medium. This is known as conduction.
Conduction: Conduction is a process of transfer of heat in which the heat is transferred through the bulk of matter or medium. This process is similar to that of the conduction of electric current through a conductor.

Convection

In the case of water, the heat gets transferred by the movement of water molecules. This method of transfer of heat is known as convection.
Convection: Convection is the process of transfer of heat in the case of fluids where the molecules are to move. The heated molecules rise, allowing the new molecules to get the heat, and this process is repeated to heat the whole bulk of the matter or fluid.
This mode of heat transfer only exists in fluids.

Radiation

Heat transfer from the sun to the earth without the help of any medium is known as Radiation.
Radiation: Radiation is the process through which the heat is transferred without the requirement of any medium is known as radiation.
In radiation, the heat is transferred as electromagnetic waves.

Terms Related to Radiation

Emissive Power: The thermal energy radiated in all directions by a body per unit time per unit area is known as the emissive power of that body.
Absorptive coefficient: When a body is radiated with thermal energy, then some part of the energy is reflected, some part of it is transmitted, while some part of it gets absorbed by the body to raise its temperature.
Absorptive coefficient is given by the ratio of the energy absorbed to the total incident energy.
Black body: Black body is a hypothetical body that absorbs all the radiation radiated to it. Its absorptive power or coefficient of absorption is unity.
Emissivity: It is the constant we introduce when we extend Stefan’s law for any general body. Its value is equal to the coefficient of absorptivity.

Kirchhoff’s Law

Kirchhoff’s law of radiation states that for any arbitrary body, the ratio of the emissive power to the absorptive power (keeping the temperature and wavelength of the radiation constant) is constant and is equal to the emissive power of the black body at that temperature for the given wavelength of the radiation.

Stefan’s Law

Stefan’s law for black body radiation states that the energy radiated per unit time by black body radiation is proportional to the temperature of the black body raised to the power four.

\(\mu \propto {T^4}\)
\(\mu = \sigma A{T^4}\)
Where,
\(\sigma \) is the Stefan-Boltzmann constant.
\(\sigma = 5.6 \times {10^{ – 8}}\;{\rm{W}}\;{{\rm{m}}^{ – 2}}\;{{\rm{K}}^{ – 4}}\)
\(A\) is the area of the black body radiating the energy.
When we extend the above law to any general body, a constant known as emissivity is multiplied to the above expression.

\(\mu = \sigma A{T^4}\)
Where,
\(e\) is the emissivity, and its value lies between zero to one.
Emissive power is given by,
\(E = e\sigma {T^4}\)
When we apply Kirchhoff’s law we get,
\(\frac{{{e_A}\sigma {T^4}}}{{{a_A}}} = \frac{{\sigma {T^4}}}{1}\)
\({e_A} = {a_A}\)
Emissivity and coefficient of absorption are equal for a body.

Newton’s Law of Cooling

The phenomenon of absorption and emission happens simultaneously. 
Energy radiated by the body per second is given by,
\({\mu _1} = e\sigma A{T^4}\)
Energy absorbed is given by,
\({\mu _2} = e\sigma AT_0^4\)
Where,
\(T\) is the temperature of the body.
\({T_0}\) is the temperature of the surrounding
If the temperature of the body is more than the surrounding, then the heat will flow from the body towards the surrounding,
\(T > {T_0}\)
The change in temperature of the body and the amount of heat energy per unit time is related by,
\(\Delta U = {\mu _2} – {\mu _1} = mS\left( { – \frac{{dT}}{{dt}}} \right)\)
\( \Rightarrow – \frac{{dT}}{{dt}} = \frac{{e\sigma A}}{{mS}}\left( {{T^4} – T_0^4} \right)\)
Let, \(\Delta T = T – {T_0}\)
\( \Rightarrow – \frac{{dT}}{{dt}} = \frac{{{e\sigma A}}}{{mS}}\left( {{{\left( {{T_0} + \Delta T} \right)}^4} – T_0^4} \right)\)
\( \Rightarrow – \frac{{dT}}{{dt}} = \frac{{e\sigma AT_0^4}}{{mS}}\left( {{{\left( {1 + \frac{{\Delta T}}{{{T_0}}}} \right)}^4} – {1^4}} \right)\)
We consider the difference temperature is small when compared to the temperature of the surrounding,
\(\Delta T \ll {T_0}\)
Using binomial expansion we get,
\( – \frac{{dT}}{{dt}} = \frac{{e\sigma AT_0^4}}{{mS}}\left( {\left( {1 + \frac{{4\Delta T}}{{{T_0}}} – 1} \right)} \right)\)
\( \Rightarrow – \frac{{dT}}{{dt}} = \frac{{4e\sigma AT_0^3}}{{mS}}\Delta T\)
\( \Rightarrow – \frac{{dT}}{{dt}} = K\left( {T – {T_0}} \right)\)
The above equation represents Newton’s law of cooling.
The rate of cooling of a body is directly proportional to the temperature difference between the body and the surrounding. It is important to note that the temperature difference is small compared to the temperature of the surrounding.
\(\Delta T \ll {T_0}\)
The heat should only be dissipated through radiation.

Sample Problems

Q.1. The temperature of the body is \({\theta _i}\) and the temperature of the surrounding is \({\theta _0}\) Find the temperature of the body as a function of time.\({\theta _0} < {\theta _i}\)
Ans: Let the temperature of the body at any time \(‘t’\) be, \(\theta .\)
From Newton’s law of cooling, we have,
\( – \frac{{d\theta }}{{dt}} = K\left( {{\theta _i} – {\theta _0}} \right)\)
\(\int_{{\theta _i}}^\theta {\frac{{d\theta }}{{\theta – {\theta _0}}}} = \int_0^t K dt\)
\( \Rightarrow \theta = \left( {{\theta _i} – {\theta _0}} \right){e^{ – Kt}} + {\theta _0}\)

Q.2. What is the difference between emissivity and emissive power?
Ans: Emissivity is a dimensionless quantity that is introduced when we extend the Stefan’s to a general body, whereas the emissive power is the thermal energy radiated in all directions by a body per unit time per unit area.
\(E = \frac{{\Delta U}}{{\Delta A\Delta t}}\)

Summary

In this article, we learnt about different modes of transfer of heat. We discussed radiation in detail. We learnt about Kirchhoff’s law, Stefan’s law for radiation. We also discussed important terms related to the radiation then we also derived Newton’s law of cooling. We discussed the limitations of Newton’s law of cooling. Newton’s law of cooling is valid when the difference in temperature is very small compared to the surrounding temperature. The temperature of the surround should not change.

Frequestly Asked Questions

Q.1. Does heat require a medium to travel?
Ans: The requirement of a medium for the transfer of heat depends on the mode of transfer of heat. For convection and conduction, the medium is essential, furthermore, for convection, the medium should be fluid. For heat transfer through radiation, a medium is not required as heat is transferred through electromagnetic waves.

Q.2. When is heat transfer through convection possible?
Ans: In convection, the medium particles itself move to transfer heat. So, to transfer heat through convection, the medium particle should be able to movely, which only exists for fluids; thus, convection mode of transfer of heat can only exist in fluids.

Q.3. What is the relation between emissivity and coefficient of absorption?
Ans: For a body, the emissivity and the coefficient of absorption is equal. Or a black body, the value of emissivity or coefficient of absorption is equal to one.

Q.4. What is a black body?
Ans: A Black body is a hypothetical matter which absorbs all the radiation incident on its surface, and since it doesn’t reflect any radiation, it will appear black in colour hence the name black body.

Q.5. What are the limitations of Newton’s law of cooling?
Ans: Limitation of Newton’s law of cooling are as follows:
a. The temperature difference should be small as compared to the temperature of the surrounding
b. The mode of transfer of heat should only be through radiation.
c. The temperature of the surrounding should be constant.

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