# Fourier number

Physical key figure
Surname Fourier number
Formula symbol ${\ displaystyle {\ mathit {Fo}}}$ dimension dimensionless
definition ${\ displaystyle {\ mathit {Fo}} = {\ frac {at} {L ^ {2}}}}$ ${\ displaystyle a}$ Thermal diffusivity ${\ displaystyle t}$ characteristic time ${\ displaystyle L}$ characteristic length
Named after Jean Baptiste Joseph Fourier
scope of application transient heat conduction,
mass transfer processes

The Fourier number  (after Jean Baptiste Joseph Fourier ) is a dimensionless number used to describe problems of transient heat conduction or general mass transfer processes. It can be interpreted as the ratio of the transport rate to the storage rate. In the case of transient heat conduction, it is the ratio of the rate at which sensible heat is transported to the rate at which it is absorbed. ${\ displaystyle {\ mathit {Fo}}}$ ## definition

### Thermal Fourier number

The Fourier number for heat conduction results from the de-dimensionalization of the heat conduction equation . The thermal diffusivity is used as the transport coefficient : ${\ displaystyle a}$ ${\ displaystyle {\ mathit {Fo}} = a \ cdot {\ frac {t} {L ^ {2}}} = {\ frac {\ lambda} {c _ {\ mathrm {p}} \ cdot \ rho} } \ cdot {\ frac {t} {L ^ {2}}}}$ in which

• the thermal diffusivity the quantities ${\ displaystyle a}$ • ${\ displaystyle \ lambda}$ ( Thermal conductivity )
• ${\ displaystyle c _ {\ mathrm {p}}}$ ( specific heat capacity at constant pressure ) and
• ${\ displaystyle \ rho}$ ( Density ) summarizes while
• ${\ displaystyle t}$ for the time and
• ${\ displaystyle L}$ represent a characteristic length of the problem.

It describes the duration of a thermal process in relation to the duration of the heat transport and is therefore used as a dimensionless time parameter.

### Mass Fourier number

In mass transfer processes in mechanical process engineering such as B. When mixing , the Fourier number is used together with the Bodenstein number . Instead of the thermal diffusion coefficient (also known as the "thermal diffusion coefficient"), the (mass) diffusion coefficient or the dissipation coefficient is used as the transport coefficient . ${\ displaystyle D}$ ${\ displaystyle M}$ ## Applications

• Problems of unsteady heat conduction of different sizes, but geometrically similar, show an identical development of the temperature field if the Fourier number is used as the time coordinate.
• In the case of a periodic, one-dimensional thermal wave , the Fourier number has the value π if the reciprocal value of the excitation frequency and the penetration depth into the homogeneous material are used.${\ displaystyle t}$ ${\ displaystyle L}$ • During the exponential cooling of a body with an insulating layer, the Fourier number, together with the Biot number, determines the size of temperature differences within the body to the temperature difference to the outside.
• If the unsteady heat conduction comes about through a stationary pipe flow (unsteady in the reference system moving with the flow ), the reciprocal of the Fourier number, called Graetz number , is the ratio of convective heat transport to heat conduction in the stationary reference system.

## Individual evidence

1. ^ Josef Kunes: Dimensionless Physical Quantities in Science and Engineering . Elsevier, 2012, ISBN 0-12-391458-2 , pp. 175 ( limited preview in Google Book search).
2. Matthias Bohnet (Ed.): Mechanical process engineering. Wiley-VCH, Weinheim 2004, ISBN 3-527-31099-1 , pp. 213-229.