Conduction is the process of transferring heat energy from a hot body to a cooler one (or from the hot part of a body to a cooler part). Radiation and convection are the other two ways heat flows. Conduction is the transfer of thermal energy by collisions between surrounding atoms or molecules.

The process of Conduction results from particle motion about their actual position. Fast or vigorously moving particles with large kinetic energy bump into less energetic particles and transfer their kinetic energy, making them move faster or vibrate more vigorously. Still, the particles do not leave their places.

What is Thermal Conduction?

Conduction is the heat transfer between two adjacent parts of a body because of their temperature difference. Heat transfer takes place from a region of higher temperature to an area of lower temperature, without the actual displacement of the atoms.

In this heat transfer method, the atoms or molecules closer to the hot end absorb the energy, and hence their amplitude of vibration about their mean position increases. Thus, they collide more frequently with their neighbouring atoms and transfer a part of their energy. In metals, the conduction occurs by the electrons present in them.

Types of Thermal Conduction

Conduction involves the transfer of heat from the hotter end of a conducting surface to its colder end. There are two types of conduction:
a. Transient Conduction
b. Steady-state Conduction

Transient or Non-steady State Conduction

Non-steady-state conduction usually occurs when a temperature change is introduced within the outer areas of an object or inside. The temperature change is brought about by the sudden entry of a new heat source within the object. To understand this, consider the case of starting an engine in a car. As the engine starts, it uses fuel and converts it into energy. Hence, a new heat source is added when the engine is turned on. This transient state stays only for a brief period before the steady-state is achieved.
Now take the example as the conduction of heat starts across a metal rod. It is the initial state when the transfer of heat begins across the conducting surface; in this state temperature of every part of the rod increases.

Heat received by each cross-section of the rod from the hotter end is used in three ways:
(i) A portion of this heat increases the temperature of the cross-section itself.
(ii) Another part is transferred to a neighbouring cross-section.
(iii) Remaining part of the heat gets radiated into the surrounding.
\(θ_1 > θ_2 > θ_3 > θ_4 > θ_5\)
Or, \(θ →\) Changing

Steady-State Conduction

After some time, a state is reached when the temperature at each cross-section becomes steady. This state is known as the steady-state. In this state, any heat received by any cross-section is partly conducted to the next section and is partially radiated, i.e., the cross-section absorbs no heat.

By steady state, it does not mean that the temperature of the whole bar has become the same; it simply means that in a steady-state, the temperature of the different parts of the bar is different, but the temperature of each part remains the same—the temperature decreases as one move away from the hot end of the metal bar.

1. In this state, the rate at which heat flows depends on the thermal conductivity of the bar’s material, so it remains constant.
2. If the metal bar is in contact with its surroundings and is heated at one end, the distance vs. temperature curve can be given as:
   
3. If the metal bar is insulated from its surroundings, there will be no loss of heat due to radiation, then the distance vs. temperature graph is a straight line as shown:
   

Isothermal Surface: Isothermal means that the temperature remains constant. If the heat in radiation or convection is nearly non-existent in a steady-state, the temperature of every transverse section of the rod is the same over the whole area of the cross-section. This is because every cross-section is perpendicular to the direction of the heat flow, and hence there will be no heat flow from one point to the other on the cross-section. Thus, every transverse section of the rod behaves as one isothermal surface.
Depending upon the direction of heat flow, we can get the following three types of isothermal surfaces:

1. Plane
2. Cylindrical
3. Spherical

Temperature Gradient: It is defined as the temperature change with the distance between two isothermal surfaces.

If the temperature of the isothermal surfaces be \(θ\) and \(θ\,-\,∆θ\) and the perpendicular distance between them is \(∆x\), then:
Temperature Gradient \( = \frac{{{\text{Change in temperature}}}}{{{\text{distance}}}} = \frac{{(\theta – {\text{∆ }}\theta ) – \theta }}{{{\text{∆ }}x}} = – \frac{{{\text{∆ }}\theta }}{{{\text{∆ }}x}}\)
The negative sign indicates that the temperature \(θ\) decreases as the distance \(x\) increases in the flow direction.
The SI unit of the temperature gradient is \(\rm{K}/\rm{m}\)

Thermal Conductivity

Let us visualize how the actual transfer takes place over a metal bar to understand the heat flow. Consider a metallic bar of length \(L\) and uniform cross-section \(A\) with its two ends maintained at different temperatures.

Now, we can describe heat conduction quantitatively as the time rate of heat flow in a material for a given temperature difference. To achieve this, put the ends of the metal bar in thermal contact with large reservoirs at temperatures, say, \(T_H\) and \(T_C\) respectively, where \(T_H > T_C\). Under the ideal condition, the sides of the bar are fully insulated so that no heat is exchanged between the sides and the surroundings. When a steady state is reached after some time, the temperature of the bar decreases uniformly with distance from the end at \(T_H\) to \(T_C\).

The reservoir at \(H\) supplies heat at a constant rate, which gets transferred at the same constant rate throughout the metal bar and reaches the reservoir at \(C\). Experimentally in the steady-state, the rate of flow of heat (or heat current) \(H\) is found to be proportional to the temperature difference \((T_H – T_C)\) and the area of cross-section \(A\) and is inversely proportional to the length \(L\). Thus, the formula of conduction of heat through the bar,
\(H = \frac{{KA}}{L}\left({{T_H} – {T_C}} \right)\)
\(K\) is the constant of proportionality known as the material’s thermal conductivity; the greater the value of \(K\) for a material, the more rapidly it will conduct heat.

Coefficient of Thermal Conductivity \((K)\)

The coefficient of thermal conductivity, \(K\), of a material is defined as the amount of heat that flows through the given material having a unit length and a unit area of the cross-section in the steady-state when the difference between the temperature at the hotter and colder end is \(1℃\). Thus, the flow of heat is perpendicular to the ends of the material.
From above, \(K = \frac{{HL}}{{A\left({{T_H} – {T_C}} \right)}}\)
The SI unit of thermal conductivity is: \(\rm{Watt/m.K}\)
Its MKS unit: \(\rm{Kcal/m.s.K}\)
Its CGS unit is: \(\rm{cal/cm.s.℃}\)
Dimensions of thermal conductivity: \(\left[{{\text{ML}}{{\text{T}}^{ – 3}}{\theta ^{ – 1}}} \right]\)
For an ideal conductor, \(K\) is \(∞\)
For an ideal insulator, \(K = 0\)
In general, solids are better conductors than liquids, liquids are better conductors over gases, and metals are better conductors than non-metals. That’s why being a good conductor of heat, copper promotes heat distribution over the bottom of a pot for uniform cooking. On the other hand, plastic foams are good insulators, mainly because they contain pockets of air.

Fourier Law

The law of heat conduction is also known as Fourier’s law. According to this law of thermal conduction, the time rate of heat transfer through a material is proportional to the negative gradient in the temperature and the area.

Therefore, heat transfer processes can be quantified in terms of appropriate rate equations. The rate equation in this heat transfer mode is based on Fourier’s law of thermal conduction. This law states that the time rate of heat transfer through a material is proportional to the negative gradient in the temperature and the area, at right angles to that gradient, through which the heat flows. Its differential form is:
\(Q = – K∆T\)
Where \(Q\): is the local heat flux density, \((\rm{W/m}^2)\), \(K\): is the material’s conductivity, \((\rm{W/(m \cdot K)})\), and \(∆T\) : is the temperature gradient, \((\rm{K/m})\).

Thermal Resistance

Thermal resistance of a material is a measure of its opposition to the flow of heat through it. It is defined as the temperature difference ratio to the heat current (rate of heat flow). If a rod of length \(l\) and area of cross-section \(A\) has a temperature \(T_1\) at one end and \(T_2\), if \(T_1 < T_2\), then,
Temperature difference \(= T_1 – T_2\)
Heat Current, \(H = \frac{Q}{t}\)
Thermal resistance, \( {R_ {th}} = \frac{ { {T_1} – {T_2}}}{H} = \frac{ { {T_1} – {T_2}}}{ {Q/t}}\) ……….(i)
\( {R_ {th}} = \frac{ { {T_1} – {T_2}}}{ {KA\left({ {T_1} – {T_2}} \right)/l}} = \frac{l}{ {KA}}\)
\(\therefore \,{R_{th}} = \frac{{{\text{∆}}T}}{H} = \frac{l}{{KA}}\)
Let thermal current, \(i = \frac{Q}{t}\)
Then, by equation (i), \(i = \frac{T_1 – T_2}{R_{th}}\)
This is mathematically equivalent to Ohm’s law, with temperature donning the role of electric potential. Hence results derived from Ohm’s law are also valid for thermal conduction.
The SI unit of thermal resistance is \(\rm{K.s/kcal}\), and its dimension is \(\left[{{{\text{M}}^{ – 1}}{{\text{L}}^{ – 2}}{{\text{T}}^3}\theta}\right]\)

Combination of Conductors

Depending on the arrangement of conductors with respect to each other, the following two combinations are possible:

Series Combination

Suppose two conductors of areas of cross-sections \(A_1\), \(A_2\) and lengths \(l_1\), \(l_2\), coefficients of thermal conductivities \(K_1\), \(K_2\) are connected in series. Assume no heat loss to the surrounding through curved surface.

Let heat is allowed to flow through this combination after the steady-state reached. Let the temperatures of the two outer faces be \(θ_1\) and \(θ_2\) and the temperature of the junction is \(θ\). The heat current in them will be,
\({H_1} = \frac{{{\theta _1} – \theta }}{{{R_1}}}\) and \({H_2} = \frac{{\theta – {\theta _2}}}{{{R_2}}}\)
But here, i.e., in series \(H_1 = H_2 = H\)
So, \(θ_1 – θ = HR_1\) and \(θ – θ_2 = HR_2\)
Or \(θ_1 – θ_2 = H(R_1 + R_2)\)
\(H = (θ_1 – θ_2)/(R_1 + R_2)\)
\(H = (θ_1 – θ_2)/R\) where \(R = R_1 + R_2\)
i.e., in series, the total internal resistance is equal to the sum of individual resistances.
This case is similar to two resistances in series. The temperature difference replaces the potential difference. The electrical resistance is replaced by thermal resistance, and the electrical current (rate of flow of charge) is replaced by heat current (the rate of heat flow).
\({R_1} = \frac{{{l_1}}}{{{K_1}{A_1}}}\) and \({R_2} = \frac{{{l_2}}}{{{K_2}{A_2}}}\)
\(\frac{{{l_{eq}}}}{{{K_{eq}}A}} = \frac{{{l_1}}}{{{K_1}{A_1}}} + \frac{{{l_2}}}{{{K_2}{A_2}}}\)
\(\frac{{{l_1} + {l_2}}}{{{K_{eq}}A}} = \frac{{{l_1}}}{{{K_1}{A_1}}} + \frac{{{l_2}}}{{{K_2}{A_2}}}\)
\({K_{eq}} = \frac{{\left({{l_1} + {l_2}} \right)/A}}{{\left({\frac{{{l_1}}}{{{K_1}{A_1}}} + \frac{{{l_2}}}{{{K_2}{A_2}}}} \right)}}\)
If, \(l_1 = l_2 = l\), and \(A_1 = A_2 = A\), Then, \({K_{eq}} = \frac{{2{K_1}{K_2}}}{{{K_1} + {K_2}}}\)
If we have more than two slabs, then the total resistance of the slabs connected in series,
\(R_{eq} = R_1 + R_2 + R_3 +……\)

Parallel Combination

Similar to the series case, if the two slabs are connected in parallel (i.e., one placed on top of the other) as shown in the figure:
The thermal resistance \(R\) of the parallel combination is given by:
\(\frac{1}{R} = \frac{1}{{{R_1}}} + \frac{1}{{{R_2}}}\)
\({R_1} = \frac{{{l_1}}}{{{K_1}{A_1}}}\) and \({R_2} = \frac{{{l_2}}}{{{K_2}{A_2}}}\)
\( \Rightarrow \frac{{{K_{eq}}{A_{eq}}}}{l} = \frac{{{K_1}{A_1}}}{{{l_1}}} + \frac{{{K_2}{A_2}}}{{{l_2}}}\)
\( \Rightarrow {K_{eq}} = \frac{{\left({\frac{{{K_1}{A_1}}}{{{l_1}}} + \frac{{{K_2}{A_2}}}{{{l_2}}}} \right)}}{{\frac{{{A_1} + {A_2}}}{l}}}\)
If, \(l_1 = l_2 = l\), and \(A_1 = A_2 = A\), Then, \(K_{eq} = K_1 + K_2\)
For more than two slabs, the total resistance of the slabs will be:
\(\frac{1}{{{R_{eq}}}} = \frac{1}{{{R_1}}} + \frac{1}{{{R_2}}} + \frac{1}{{{R_3}}} + ……\)

Summary

Conduction is the heat transfer between two adjacent parts of a body because of their temperature difference. Heat transfer takes place from a region of higher temperature to an area of lower temperature, without the actual displacement of the atoms. There are two types of conduction:
a. Transient Conduction: Non-steady-state or Transient conduction usually occurs when a temperature change is introduced within the outer areas of an object or inside.
b. Steady-state Conduction: After some time during heat transfer, a state is reached when the temperature at each cross-section becomes steady. This state is known as the steady-state.
1. Isothermal Surface: In a steady-state, if heat due to radiation or convection is almost non-existent, then the temperature of every transverse section of the rod is the same on the cross-section’s whole area, making it an isothermal surface.
2. Temperature gradient is defined as the temperature change with the distance between two isothermal surfaces.
The formula of conduction of heat through the bar, \(H = \frac{{KA}}{L}\left({{T_H} –  {T_C}}  \right)\) where \(K\) is the constant of proportionality known as the material’s thermal conductivity; the greater the value of \(K\) for a material, the more rapidly it will conduct heat.
1. For an ideal conductor, \(K\) is \(∞\)
2. For an ideal insulator, \(K = 0\)

Fourier Law: According to this law of thermal conduction, the time rate of heat transfer through a material is proportional to the negative gradient in the temperature and the area.
Thermal Resistance: If a rod of length \(l\) and area of cross-section \(A\) has a temperature \(T_1\) at one end and \(T_2\), if \(T_1 < T_2\), then, \({R_{th}} = \frac{{\Delta T}}{H} = \frac{l}{{KA}}.\)

1. For two slabs connected in series, \({K_{eq}} = \frac{{\left({{l_1} + {l_2}} \right)/A}}{{\left({\frac{{{l_1}}}{{{K_1}{A_1}}} + \frac{{{l_2}}}{{{K_2}{A_2}}}} \right)}}\) and if \(l_1 = l_2 = l\), and \(A_1 = A_2 = A\), Then, \({K_{eq}} = \frac{{2{K_1}{K_2}}}{{{K_1} + {K_2}}}.\)
2. For two slabs connected in parallel, \({K_{eq}} = \frac{{\left({\frac{{{K_1}{A_1}}}{{{l_1}}} + \frac{{{K_2}{A_2}}}{{{l_2}}}} \right)}}{{\frac{{{A_1} + {A_2}}}{l}}}\) and if, \(l_1 = l_2 = l\), and \(A_1 = A_2 = A\), Then, \(K_{eq} = K_1 + K_2\)

FAQs

Q.1. What are three examples of conduction?
Ans: 1. Heat transfer through conduction from the iron to the clothes.
2. Heat is transferred from hands to ice cubes resulting in the melting of an ice cube when held in hands.
3. Heat conduction through the sand at the beaches. This can be experienced during summers. Sand is a good conductor of heat.

Q.2. What are conduction and convection?
Ans: 1. Conduction is a heat transfer method that describes how heat flows from objects with higher temperatures to objects with lower temperatures.
2. Convection is defined as the movement of fluid molecules from higher temperature regions to lower temperature regions.

Q.3. What is conduction explain with an example?
Ans: Conduction is the transfer of heat from the hotter part of the material to its colder part without the actual movement of the particles. The example of conduction that we see in our day-to-day lives is that most cooking utensils come with wooden handles. This ensures that despite the uniform heating of the utensil due to conduction, we can use it safely, with the wood handle being an insulator. Interestingly, the bottom of cooking utensils is made from copper because copper is an excellent conductor, ensuring maximum heat transfer via conduction.

Q.4. What is conduction in the body?
Ans: Conduction is the transfer of heat from the hotter part of the material to its colder part without the actual movement of the particles.

Q.5. What is thermal conductivity?
Ans: Thermal conductivity is the ability of a material to conduct heat. It represents the quantity of thermal energy that flows per unit time through a unit area with a temperature gradient of \(1°\) per unit distance.

Q.6. What are the two types of conduction?
Ans: Transient conduction: During transient conduction, the temperatures can change or vary at any part within an object at a given time.
Steady-State conduction: For a conductor in a steady-state, there is no absorption or emission of heat at any cross-section (as the temperature at each point remains constant with time).