Fourier number

 ( Dimensionless Numbers Of Thermodynamics ) 

In the study of heat conduction, the Fourier number, is the ratio of time, $t$, to a characteristic time scale for heat diffusion, $t\_d$. This dimensionless group is named in honor of Joseph 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, $L$. For a medium with thermal diffusivity, $\backslash alpha$, this time scale is $t\_d\; =\; L^2/\backslash alpha$, so that the Fourier number is $t/t\_d\; =\; \backslash alpha\; t/L^2$. The Fourier number is often denoted as $\backslash mathrm\{Fo\}$ or $\backslash mathrm\{Fo\}\_L$.

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.

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Definition The general definition of the Fourier number, , is:

(2025). 9781107012172, Cambridge University Press. ISBN 9781107012172

$$\backslash mathrm\{Fo\}\; =\; \backslash frac\{\; \backslash text\{time\}\; \}\{\; \backslash text\{time\; scale\; for\; diffusion\}\; \}\; =\; \backslash frac\{\; t\; \}\{\; t\_d\}$$

For heat diffusion with a characteristic length scale $L$ in a medium of thermal diffusivity $\backslash alpha$, the diffusion time scale is $t\_d\; =\; L^2/\backslash alpha$, so that

$$\backslash mathrm\{Fo\}\_L\; =\; \backslash frac\{\backslash alpha\; t\}\{L^2\}$$

where:

• $\backslash alpha$ is the thermal diffusivity (meters²/second)
• $t$ is the time (s)
• $L$ is the characteristic length through which conduction occurs (m)

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Interpretation of the Fourier number Consider transient heat conduction in a slab of thickness $L$ that is initially at a uniform temperature, $T\_0$. One side of the slab is heated to higher temperature, $T\_h\; >\; T\_0$, at time $t=0$. The other side is adiabatic. The time needed for the other side of the object to show significant temperature change is the diffusion time, $t\_d$.

When $\backslash mathrm\{Fo\}\; \backslash ll\; 1$, 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 $T\_0$.

When $\backslash mathrm\{Fo\}\; \backslash cong\; 1$, significant temperature change occurs all the way through the thickness $L$. None of the slab remains at temperature $T\_0$.

When $\backslash mathrm\{Fo\}\; \backslash gg\; 1$, enough time has passed for the slab to approach steady state. The entire slab approaches temperature $T\_h$.

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Derivation and usage The Fourier number can be derived by nondimensionalizing the time-dependent diffusion equation. As an example, consider a rod of length $L$ that is being heated from an initial temperature $T\_0$ by imposing a heat source of temperature $T\_L>T\_0$ at time $t=0$ and position $x=L$ (with $x$ along the axis of the rod). The heat equation in one spatial dimension, $x$, can be applied

$$\backslash frac\{\backslash partial\; T\}\{\backslash partial\; t\}\; =\; \backslash alpha\; \backslash frac\{\backslash partial^2\; T\}\{\backslash partial\; x^2\}$$

where $T$ is the temperature for

$$\backslash frac\{\backslash partial\; \backslash Theta\}\{\backslash partial\; (\backslash alpha\; t/L^2)\}\; =\; \backslash frac\{\backslash partial^2\; \backslash Theta\}\{\backslash partial\; (x/L)^2\}$$

The resulting dimensionless time variable is the Fourier number, $\backslash mathrm\{Fo\}\_L\; =\; \backslash alpha\; t\; /\; L^2$. The characteristic time scale for diffusion, $t\_d\; =\; L^2/\backslash alpha$, 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.

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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, $\backslash mathrm\{Fo\}\_m$ defined as:

(2025). 9780444633033, Elsevier. . ISBN 9780444633033

$$\backslash mathrm\{Fo\}\_m\; =\; \backslash frac\{D\; t\}\{L^2\}$$

where:

• $\backslash mathrm\{Fo\}\_m$ is the Fourier number for mass transport
• $D$ is the mass diffusivity (m²/s)
• $t$ is the time (s)
• $L$ 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.

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See also

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

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