In this article, we will delve into the basics of thermal conductivity, including its definition, units of measurement, factors affecting it, and applications in different fields, derivation of the thermal conductivity formula in a detailed manner.
What is Thermal Conductivity?
Thermal conductivity is defined as the rate of heat flow across a unit area of a conductor per unit temperature gradient.
It is defined as the amount of heat that flows through a unit area of a material in unit time, per unit temperature difference across the material.
Thus, \boxed{\mathbf{K = - \frac{Q'}{A\left( \frac{dT}{dx} \right)}}}
Where,
- Q’ is that rat of heat flow given by \left( \frac{dQ}{dt} \right)
- A is the cross-sectional area of the conductor
- \left( \frac{dT}{dx} \right) is the temperature gradient.
The -ve sign is optional, which indicates that the health flows from a higher to a lower temperature side.
The unit of Thermal conductivity is W m-1 K-1.
In solids, heat transfer takes place by conduction. In the process of heat transfer, both electrons and phonons take part. hence in general, thermal conductivity can be written as
K_{Total} = K_{electron} + K_{phonon}
Since thermal conductivity due to electrons is greater than the thermal conductivity due to phonons in the case of metals, the total thermal conductivity is given by
K_{Total} = K_{electron}
Derivation of Thermal Conductivity Formula
Consider a uniform metallic rod AB. Let surface A be at a higher temperature T, and surface B be at a lower temperature T-dT, as shown in the figure below.
let the distance of separation between the surface be \lambda(mean free path). the electrons conduct heat from A to B. During a collision, the electrons near A lose kinetic energy while the electrons near B gain energy.
Let,
The density of electrons = n
The average thermal velocity = v.
Based on kinetic theory,
The average kinetic energy of an electron at A = \frac{3}{2}k_BT
Similarly, the average kinetic energy of an electron at B = \frac{3}{2}k_B(T-dT)
Therefore, the excess kinetic energy carried by electrons from A to B
= \frac{3}{2}k_BT - \frac{3}{2}k_B(T-dT)
= \frac{3}{2}k_BdT
There is an equal probability for the electrons to move in all six directions(x,y,z) and (-x,-y,-z).
The number of electrons crossing unit area in unit time from A to B =\frac{1}{6}nv
Therefore, the excess energy transferred from A to B per unit area in unit time
=\frac{1}{6}nv \times \frac{3}{2}k_bdT=\frac{1}{4}nvk_BdT
Similarly, the deficiency of energy carried from B to A per unit area in unit time
=-\frac{1}{4}nvk_BdT
Since the net energy transferred from A to B per unit area per unit time is the rate of heat flow Q. Then,
Q=\frac{1}{4}nvk_BdT- \left(- \frac{1}{4}nvk_BdT \right )
Q=\frac{1}{2}nvk_BdT
But from the definition of thermal conductivity
Thus, K = \frac{Q}{\left( \frac{dT}{dx} \right)}
In this case, dx=\lambda
\therefore K = \frac{Q}{\left( \frac{dT}{\lambda} \right)}
hence, substituting the value of Q from above, we get
K = \frac{\frac{1}{2}nvk_BdT}{\left( \frac{dT}{\lambda} \right)}
\boxed{\mathbf{K=\frac{nvk_B\lambda}{2}}}
The value of K obtained with the above expression is verified experimentally and the free electron theory is found to be successful in explaining the thermal conductivity.
Effect of Temperature on Thermal Conductivity
The effect of temperature on thermal conductivity for different types of materials:
| Material | Effect of Temperature on Thermal Conductivity |
|---|---|
| Metals | Generally increases with temperature over a certain range, following the Wiedemann-Franz law |
| Insulators | May decrease with increasing temperature due to increased thermal vibrations leading to more scattering of heat-carrying phonons |
| Semiconductors | May increase or decrease depending on the specific material and temperature range |
| Polymers | Generally decreases with increasing temperature due to increased molecular motion leading to more scattering of heat-carrying phonons |
| Liquids | Generally increases with temperature due to increased molecular motion leading to more efficient heat transfer |
| Gases | Generally increases with temperature due to increased molecular motion and collisions leading to more efficient heat transfer |
It’s important to note that these are general trends and the actual behavior of a specific material may depend on a variety of factors, such as its crystal structure, impurities, and defects.
Factors Affecting Thermal Conductivity
Temperature is not the only factor that causes a variance in the thermal conductivity of a material. Some other important factors that influence the heat conductivity of substances are tabulated below.
| Factor | Effect on Thermal Conductivity |
|---|---|
| Material composition | Different materials have different thermal conductivities due to differences in their molecular structures |
| Temperature | Generally, thermal conductivity increases with increasing temperature for most materials |
| Pressure | For gases, thermal conductivity increases with increasing pressure due to increased molecular collisions |
| Density | Generally, higher density materials have higher thermal conductivities |
| Porosity | Higher porosity generally leads to lower thermal conductivity due to the presence of air pockets |
| Moisture content | Generally, higher moisture content leads to lower thermal conductivity due to the presence of water molecules |
| Crystal structure | Materials with a more ordered crystal structure tend to have higher thermal conductivities |
| Impurities | The presence of impurities can lead to increased scattering of phonons and lower thermal conductivity |
| Grain boundaries | The presence of grain boundaries can lead to increased scattering of phonons and lower thermal conductivity |
| Phonon scattering | Processes that cause phonon scattering, such as impurities, grain boundaries, and defects, can lead to lower thermal conductivity |
Again, it’s important to note that the actual behavior of a specific material may depend on a variety of factors, and these are just general trends.
Applications of Thermal Conductivity
Thermal conductivity plays a vital role in various fields, including:
1. Building and Construction: Thermal conductivity is an essential property in building and construction materials. Insulating materials with low thermal conductivity are used to reduce heat loss in buildings and homes. Examples include fiberglass, mineral wool, and foam insulation.
2. Engineering: Thermal conductivity is important in the design and operation of heat exchangers, refrigeration systems, and air conditioning units. The thermal conductivity of the materials used in these systems affects their efficiency and performance.
3. Manufacturing: Thermal conductivity is a critical property in many manufacturing processes. For example, in the production of electronic devices, the thermal conductivity of the materials used in heat sinks and cooling systems affects the performance and reliability of the devices.
4. Materials Science: Thermal conductivity is an essential property in materials science. It is used to characterize and study the thermal properties of materials, including metals, ceramics, polymers, and composites.
FAQs on Thermal Conductivity
-
What is thermal conductivity?
Thermal conductivity refers to the ability of a given material to conduct/transfer heat. It is generally denoted by the symbol ‘k’ but can also be denoted by ‘λ’ and ‘κ’.
-
What is Fourier’s law of thermal conduction?
Fourier’s law of thermal conduction states that the rate at which heat is transferred through a material is proportional to the negative of the temperature gradient and is also proportional to the area through which the heat flows.
-
What is the SI unit of thermal conductivity?
The SI unit of thermal conductivity is watts per meter per Kelvin or Wm-1K-1.
-
What materials have the highest thermal conductivity?
Materials with the highest thermal conductivity include diamond, copper, silver, and aluminum.
