Fourier's law
The Fourier's Law, also known as the Fourier law of heat conduction, describes how heat is transferred by conduction within a body. It is one of the fundamental principles of thermodynamics and heat transfer.
Statement of Fourier's Law
It can be written considering the cross-sectional area \(S\), and using either derivatives or finite differences depending on the context.
Differential form (for small/local variations):
$$\Phi_{q} = -kS \frac{dT}{dx}$$
Finite difference form (for larger intervals):
$$\Phi_{q} = -kS \frac{\Delta T}{\Delta x}$$
where:
- \(\Phi_{q}\): amount of heat transferred per unit time (watts, W)
- \(k\): thermal conductivity of the material \((W/(m \cdot K))\)
- \(S\): cross-sectional area through which heat flows \((m²)\)
- \(\frac{dT}{dx}\) or \(\frac{\Delta T}{\Delta x}\): temperature gradient, either differential or finite
- The negative sign indicates that heat flows from hot regions to cold ones
Notes:
- Use the differential form \(\frac{dT}{dx}\) for mathematical or local analysis with continuous functions.
- Use the finite difference form \(\frac{\Delta T}{\Delta x}\) for practical estimates over defined distances (e.g., between two metal plates).
Interpretation
- The law states that the amount of heat that flows through a surface is proportional to the temperature gradient and the thermal conductivity of the material.
- A greater temperature difference or a better thermal conductor results in a higher heat flow.
Relation to thermal diffusivity
Thermal diffusivity \(\alpha\) describes how quickly a material responds to temperature changes. It is defined as:
$$\alpha = \frac{k}{\rho c}$$
where:
- \(\alpha\): thermal diffusivity \((m^{2}/s)\)
- \(\rho\): material density \((kg/m^{3})\)
- \(c\): specific heat capacity \((J/(kg \cdot K))\)
- \(k\): thermal conductivity \((W/(m \cdot K))\)
Thermal diffusivity combines material properties to indicate how fast heat spreads within a body. It plays a key role in the heat equation:
$$\frac{\partial T}{\partial t} = \alpha \frac{\partial^{2} T}{\partial x^{2}}$$
This equation connects Fourier’s Law with time-dependent temperature changes, modeling how heat diffuses through an object.
Applications
- Design of heating and cooling systems.
- Thermal analysis in buildings, engines, electronics, etc.
- Materials engineering (e.g., thermal insulation).
Example
If you have a metal rod with one end hot and the other cold, Fourier’s Law allows you to calculate how much heat is transferred per second along the rod, depending on the material, size, and temperature distribution.
Simulation
This interactive simulation shows how heat propagates through a bar using the one-dimensional Fourier's Law. You can adjust the position of the heat source with the "Heat source position" slider and set its intensity using the "Initial temperature \( \left( ^{\circ} C \right) \)" control. The "Thermal diffusivity \(\left( \alpha \right)\)" parameter determines how quickly the heat spreads along the bar. You can also modify the simulation’s resolution by changing the number of segments with the "Number of points" slider. Pressing the "Reset simulation" button will apply all the selected values and start a new thermal propagation. In the visualization, color represents temperature: red indicates hotter areas, and blue represents cooler areas.