# Fourier number

In physics and engineering, the Fourier number (Fo) or Fourier modulus, named after Joseph Fourier, is a dimensionless number that characterizes heat conduction. Together with the Biot number, it characterizes transient conduction problems. Conceptually, it is the ratio of diffusive/conductive transport rate by the quantity storage rate and arises from non-dimensionalization of the heat equation. The transported quantity is usually either heat or matter (particles).

The general Fourier number is defined as:

$Fo = \dfrac{ \mbox{diffusive transport rate} }{ \mbox{storage rate} }$

The thermal Fourier number is defined by the conduction rate to the rate of thermal energy storage.

$\mathit{Fo}_h = \frac{\alpha t}{L^2}$

where:

• α is the thermal diffusivity [m2/s]
• t is the characteristic time [s]
• L is the length through which conduction occurs [m]

For transient mass transfer by diffusion, there is an analogous mass Fourier Number (also denoted Fo) defined as:

$\mathit{Fo}_m = \frac{D t}{L^2}$

where:

• "D" is the Diffusivity (m2/s)
• "t" is the characteristic timescale (s)
• "L" is the length scale of interest (m)

## Using Fourier number

Together with the Biot number, the Fourier number can be used to solve unsteady state conduction problems. The Fourier number is frequently used as a nondimensional time parameter. If the Biot number is less than 0.1, then the entire system can be treated at uniform temperature. The following equation derived from the Biot and Fourier numbers can be used to find the time, where T0 is the initial temperature and T is the temperature of the object at time t.

$t = \frac{\rho {{c}_{p}}V}{hA} \ln\frac{{{T}_{0}}-{{T}_{\infty }}}{T-{{T}_{\infty }}}$

## References

• Incropera, Frank P.; DeWitt, David P. Fundamentals of Heat and Mass Transfer (5th Edition ed.). Wiley.