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. The general definition of the Fourier number, Fo, 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) 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 . The Fourier number can be derived by nondimensionalizing the time-dependent diffusion equation.