Fourier's Law: Formula, Derivation & Equation

Fourier's law governs the process of transferring (conduction) heat energy from a higher to a lower temperature area. Conduction of heat occurs when molecules of matter vibrate and transmit energy to adjacent molecules. 

• As these molecules collide with each other, heat energy is transferred from a higher temperature area to a lower one. 
• Fourier's law is also known as the law of thermal conduction equations or the law of thermal conductivity.
• Its discrete and electrical analogues include Newton's law of cooling and Ohm's law.
• The chemical analogue of Fourier's law is Fick's law of diffusion.
• Mathematically, it can be represented as:

q = - k▽T

• where, ∇T is temperature gradient 
• k is the conductivity of the materials 
• q is the heat flux density vector  

Key Terms: Heat conduction, Thermal conductivity, Heat flux, Temperature Gradient, Fourier’s law, Heat transfer, Thermal conduction, Radiation, Fourier’s Law Derivation, Differential Law of Fourier’s Law

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What is Fourier’s Law?

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According to Fourier’s law or the law of thermal conduction, the rate of heat transfer through a material is proportional to the negative gradient in the temperature and the area (perpendicular to the gradient) of the surface through which the heat flows. 

• The law is defined for all states of matter, be it solid, liquid, or gas.
• The rate equation for heat transfer is deduced using Fourier’s law of thermal conduction. 
• In order to determine the solution for the law, the relationship between temperature difference, geometry, and thermal conductivity of the material must first determined. 
• The law was proposed by Joseph Fourier in 1822.
• He stated that the heat flux due to thermal conduction is proportional to the magnitude of the temperature gradient.
• Therefore, the rate equation derived using Fourier’s law is represented as:

q = - k\(\bigtriangledown\)T

where,

• ∇T is temperature gradient (K. m^-1)
• k is the conductivity of the materials (W. m^-1. K^-1)
• q is the heat flux density vector (W. m^-1)

Heat Flux 

Heat flux refers to the amount of heat transferred per unit area per unit time from or to a surface. It is a vector quantity i.e., it has both magnitude and direction. For a pure solid substance, the conductive heat flux (JHc) is:

JH_c=λ (dT/dZ)

Where,

• JH_c = conductive heat flux
• T = temperature
• λ = thermal conductivity constant

Thermal Conductivity

Thermal conductivity indicates the ability of a material to conduct/transfer heat. It is denoted by the symbol ‘k’, ‘K” or ‘λ’. The SI unit of conductivity is J S^-1 m^-1 K^-1 or W m^-1 K^-1

• Materials with high thermal conductivity are used as heat sinks while those with low thermal conductivity are used as thermal insulators.

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Fourier’s Law Derivation

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Fourier’s law derivation was explained through an experiment which stated that: 

Rate of heat conduction ∝ (area) (temperature difference)/thickness

• Suppose T1 and T2 are different temperatures through a short distance of an area, where the distance is Δx, the area is A and k is the thermal conductivity of the matter.
• Therefore, in one dimension, the equation can be represented as:

Q_cond= kA (T1 − T2 / Δx) = −kA (ΔT / Δx)

• If Δx 0, the equation in differential form can be written as:
• Q_cond = −kA (ΔT / Δx)
• Thus, the 3-dimensional form of Fourier’s law is:

q = - k\(\bigtriangledown\)T

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Differential Form of Fourier’s Law

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Some important forms of Fourier’s Law are as follows:

Differential Form

Fourier’s law differential form is given as:

q=−k\(\bigtriangledown\)T

where,

• ∇T is temperature gradient (K. m^-1)
• k is the conductivity of the materials (W. m^-1. K^-1)
• q is the heat flux density vector (W. m^-2)

One dimensional form

The one-dimensional form of the equation for Fourier’s Law is given as:

q_x=−k dT/dx

Integral Form

The integral form of Fourier's Law is s given as:

Where,

• ∂Q/∂t is the amount of heat transferred per unit time
• dS is the surface area element

Fourier’s law in terms of conductance

The fourier’s law for conductance is given as:

△Q / △t = UA(−△T)

where 

• U: Conductance

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Things to Remember

• Fourier’s law can easily be represented in terms of integral and differential forms.
• In this law, the conductive heat flux is linearly proportional to the temperature gradient .
• Thermal conductivity indicates the ability of a material to conduct/transfer heat. 
• Diamond is the hardest material and has the highest thermal conductivity.
• For a bar of length L and uniform cross-section A with its ends maintained at temperatures TC and TD, the rate of heat flow (H) is H = K A ((T_C - T_D)/L), where K is the thermal conductivity of the material of bar.

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