Fourier number. 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. The time scale for diffusion characterizes the time needed for heat to diffuse over a distance, . For a medium with thermal diffusivity, , this time scale is , so that the Fourier number is . The Fourier number is often denoted as  or .

The Fourier number can also be used in the study of mass diffusion, in which the thermal diffusivity is replaced by the mass diffusivity.

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.

Definition 

The general definition of the Fourier number, , is:

For heat diffusion with a characteristic length scale  in a medium of thermal diffusivity , the diffusion time scale is , so that

where:
  is the thermal diffusivity (m2/s)
  is the time (s)
  is the characteristic length through which conduction occurs (m)

Interpretation of the Fourier number

Consider transient heat conduction in  a slab of thickness  that is initially at a uniform temperature, . One side of the slab is heated to higher temperature, , at time .  The other side is adiabatic. The time needed for the other side of the object to show significant temperature change is the diffusion time, .  

When , 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 .

When , significant temperature change occurs all the way through the thickness . None of the slab remains at temperature .

When , enough time has passed for the slab to approach steady state. The entire slab approaches temperature .

Derivation and usage 

The Fourier number can be derived by nondimensionalizing the time-dependent diffusion equation.  As an example, consider a rod of length  that is being heated from an initial temperature  by imposing a heat source of temperature  at time  and position  (with  along the axis of the rod). The heat equation in one spatial dimension, , can be applied 

where  is the temperature for  and . The differential equation can be scaled into a dimensionless form. A dimensionless temperature may be defined as , and the equation may be divided through by :

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. Together, the Fourier number and the Biot number determine the temperature response of a solid subjected to convective heating or cooling.

Application to mass transfer 

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:

where:
  is the Fourier number for mass transport
  is the mass diffusivity (m2/s)
  is the time (s)
  is the length scale of interest (m)

The mass-transfer Fourier number can be applied to the study of certain time-dependent mass diffusion problems.

See also 
 Biot number
 Convection
 Heat conduction
 Heat equation
 Molecular diffusion
 Dimensionless numbers in fluid mechanics

References 

Dimensionless numbers of thermodynamics