Fourier number

The general definition of the Fourier number, {{math|Fo}}, is:{{cite book |last1=Glicksman |first1=Leon R. |last2=Lienhard |first2=John H. |title=Modelling and Approximation in Heat Transfer |date=2016 |publisher=Cambridge University Press |location=New York, NY |isbn=978-1-107-01217-2 |page=67 |chapter=Section 3.2.4}}

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

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 (m²/s)
• $t$ is the time (s)
• $L$ is the characteristic length through which conduction occurs (m)

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 $T\_h$.

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.

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:{{cite book |last1=Ostrogorsky |first1=Aleks G. |last2=Glicksman |first2=Martin E. |editor1-last=Rudolph |editor1-first=Peter |title=Handbook of Crystal Growth |date=2015 |publisher=Elsevier |isbn=9780444633033 |page=999 |edition=Second |url=https://doi.org/10.1016/B978-0-444-63303-3.00025-0 |chapter=Chapter 25: Segregation and Component Distribution|doi=10.1016/B978-0-444-63303-3.00025-0 }}

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

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.

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

{{Reflist}}

Category:Dimensionless numbers of thermodynamics