Fourier’s Law of Heat Conduction

  • Fourier’s law is the relationship between the negative vector gradient of temperature and the heat flux vector
  • Thermal conductivity is the flow of heat through any material
  • Heat flow is affected by the temperature difference, thickness of material and surface area

Fourier’s Law states that the time it takes for heat to travel through a material is proportional to the temperature’s negative gradient, and to the cross-sectional area perpendicular to the gradient through which the heat is flowing. Fourier’s law is simply an alternative name for the law of heat conduction.

Fourier’s law formula:

qi = -k dT/dxi

Where:

qi = the parts of the heat flux vector

k = the heat conductivity coefficient

T = the temperature

Heat Flux

The thermal conductivity of the material (k) is also known as the proportionality constant that is obtained in the formula. A high value of k denotes that the material is a good thermal conductor, and easily transfers energy through it.

This law was first used by Joseph Fourier in 1822 who stated that “the heat flux obtained from the thermal conduction is directly proportional to the size of the temperature gradient and opposite to it in sign.”

The heat flux through a slab is calculated with Fourier’s law, and it can also be used to obtain the temperature difference when q is known.

Thermal Conductivity

The thermal conductivity of a material is represented by k, and this stands for the heat transfer characteristics of any solid material. It is measured in W/m.K, and is the material’s ability to pass heat through itself by conduction. Fourier’s Law applies for all states, whether it is solid, liquid or gas. Therefore the thermal conductivity of most solids and liquids varies with temperature, for gases it is dependent on the pressure.

The majority of materials are very nearly homogeneous, so we can write k = k(T). Other definitions are synonymous with thermal conductivity in the y and z directions, however for an isotropic material the direction of transfer affects the thermal conductivity; kx = ky = kz = k.

Thermographic image of pipes
Thermographic image of pipes

Carrying on from this, increasing thermal conductivity results in the increase of the conduction heat flux, and both of these also increase with an increasing temperature difference. The thermal conductivity of a solid is typically larger than that of a liquid, which in turn is larger than that of a gas. Thefourierse relationships are mostly due to their differences in intermolecular spacing. For example, diamond has the highest thermal conductivity and hardness of any material.

In a composite material, the effective coefficients of thermal conductivity are very important and can be estimated from the phase properties of the material.

Temperature Difference

To explain these concepts in layman’s terms, for it to be possible for heat to flow at all, a temperature difference must be present. For example, the walls of every building you see has heat flowing through them, this is because of the temperature difference between the outside air and the inside of the building.

Tests have shown that the higher the temperature difference, the more heat that flows through the material. This is the reason that when it gets colder outside like in winter, that more heat is needed to heat a room rather than in summer or spring. If the difference in temperature is zero between both sides of the wall, then there is no need to heat the room, and therefore no heat passes through.

Thickness of the Material

The thickness of any material also affects heat’s ability to pass through. Think about it, would you rather put on a thick jacket in winter time or a thin one? You always put on the thicker jacket as it allows less heat to pass through it and it keeps you warmer. This also applies to houses and buildings, typically older buildings that were built with thicker blocks or bricks were better at insulating. This is of course because they relied on heat transfer technology for heat retention more than the modern world and all of the technology that is available today.

Surface Area

The cross-sectional area through which the heat flows is also very important in thermal conductivity. In the case of a building, it is the area of any exterior wall. The larger the surface area of the wall, the more heat that can flow through the material of said wall. This is also relevant when discussing windows, and can lead to much more heat flow as windows are better conductors of heat than walls.

So there you have it, Fourier’s law of heat conduction and the factors that contribute to it. Have you had any interesting experiences with heat flow in your career, or in general? We would love to hear about them in the comments below!

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