In the study of heat conduction, the Fourier number, is the ratio of time,
, to a characteristic time scale for heat diffusion,
This dimensionless group is named in honor of J.B.J.
Fourier, who formulated the modern understanding of heat conduction.
[1] The time scale for diffusion characterizes the time needed for heat to diffuse over a distance,
The Fourier number is used in analysis of time-dependent transport phenomena, generally in conjunction with the Biot number if convection is present.
The Fourier number arises naturally in nondimensionalization of the heat equation.
The general definition of the Fourier number, Fo, is:[3] For heat diffusion with a characteristic length scale
, the diffusion time scale is
, so that where: Consider transient heat conduction in a slab of thickness
One side of the slab is heated to higher temperature,
The time needed for the other side of the object to show significant temperature change is the diffusion time,
, not enough time has passed for the other side to change temperature.
In this case, significant temperature change only occurs close to the heated side, and most of the slab remains at temperature
, significant temperature change occurs all the way through the thickness
, enough time has passed for the slab to approach steady state.
The entire slab approaches temperature
The Fourier number can be derived by nondimensionalizing the time-dependent diffusion equation.
by imposing a heat source of temperature
The heat equation in one spatial dimension,
The differential equation can be scaled into a dimensionless form.
: The resulting dimensionless time variable is the Fourier number,
The characteristic time scale for diffusion,
, comes directly from this scaling of the heat equation.
The Fourier number is frequently used as the nondimensional time in studying transient heat conduction in solids.
A second parameter, the Biot number arises in nondimensionalization when convective boundary conditions are applied to the heat equation.
[2] Together, the Fourier number and the Biot number determine the temperature response of a solid subjected to convective heating or cooling.
An analogous Fourier number can be derived by nondimensionalization of Fick's second law of diffusion.
The result is a Fourier number for mass transport,
defined as:[4] where: The mass-transfer Fourier number can be applied to the study of certain time-dependent mass diffusion problems.