Fourier number

# Fourier number

Discussion

Encyclopedia
In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

and engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

, the Fourier number (Fo) or Fourier modulus, named after Joseph Fourier
Joseph Fourier
Jean Baptiste Joseph Fourier was a French mathematician and physicist best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations. The Fourier transform and Fourier's Law are also named in his honour...

, is a dimensionless number that characterizes heat conduction. Conceptually, it is the ratio of the heat conduction rate to the rate of thermal energy storage. Together with the Biot number
Biot number
The Biot number is a dimensionless number used in non-steady-state heat transfer calculations. It is named after the French physicist Jean-Baptiste Biot , and gives a simple index of the ratio of the heat transfer resistances inside of and at the surface of a body...

, it characterizes transient conduction problems. It is defined as: $\mbox\left\{Fo\right\} = \frac\left\{\alpha t\right\}\left\{R^2\right\}$ where: NEWLINE
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• α is the thermal diffusivity
Thermal diffusivity
In heat transfer analysis, thermal diffusivity is the thermal conductivity divided by density and specific heat capacity at constant pressure. It has the SI unit of m²/s...

[m2/s]
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• t is the characteristic time [s]
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• R is the length through which conduction occurs [m]
NEWLINE For transient mass transfer by diffusion, there is an analogous mass Fourier Number (also denoted Fo) defined as: $\mbox\left\{Fo\right\} = \frac\left\{D t\right\}\left\{L^2\right\}$ where: NEWLINE
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• "D" is the Diffusivity
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• "t" is the characteristic timescale
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• "L" is the length scale of interest
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## Using Fourier number

Together with the Biot number
Biot number
The Biot number is a dimensionless number used in non-steady-state heat transfer calculations. It is named after the French physicist Jean-Baptiste Biot , and gives a simple index of the ratio of the heat transfer resistances inside of and at the surface of a body...

, 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, the following equation derived from the Biot and Fourier numbers can be used to find the time. Where $T_0$ is the initial temperature and T is the temperature at the center. $t=\frac\left\{\rho \left\{\left\{c\right\}_\left\{p\right\}\right\}V\right\}\left\{hA\right\}ln\frac Inphysics$
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

and engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

, the Fourier number (Fo) or Fourier modulus, named after Joseph Fourier
Joseph Fourier
Jean Baptiste Joseph Fourier was a French mathematician and physicist best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations. The Fourier transform and Fourier's Law are also named in his honour...

, is a dimensionless number that characterizes heat conduction. Conceptually, it is the ratio of the heat conduction rate to the rate of thermal energy storage. Together with the Biot number
Biot number
The Biot number is a dimensionless number used in non-steady-state heat transfer calculations. It is named after the French physicist Jean-Baptiste Biot , and gives a simple index of the ratio of the heat transfer resistances inside of and at the surface of a body...

, it characterizes transient conduction problems. It is defined as: $\mbox\left\{Fo\right\} = \frac\left\{\alpha t\right\}\left\{R^2\right\}$ where: NEWLINE
NEWLINE
• α is the thermal diffusivity
Thermal diffusivity
In heat transfer analysis, thermal diffusivity is the thermal conductivity divided by density and specific heat capacity at constant pressure. It has the SI unit of m²/s...

[m2/s]
• NEWLINE
• t is the characteristic time [s]
• NEWLINE
• R is the length through which conduction occurs [m]
NEWLINE For transient mass transfer by diffusion, there is an analogous mass Fourier Number (also denoted Fo) defined as: $\mbox\left\{Fo\right\} = \frac\left\{D t\right\}\left\{L^2\right\}$ where: NEWLINE
NEWLINE
• "D" is the Diffusivity
• NEWLINE
• "t" is the characteristic timescale
• NEWLINE
• "L" is the length scale of interest
NEWLINE

## Using Fourier number

Together with the Biot number
Biot number
The Biot number is a dimensionless number used in non-steady-state heat transfer calculations. It is named after the French physicist Jean-Baptiste Biot , and gives a simple index of the ratio of the heat transfer resistances inside of and at the surface of a body...

, 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, the following equation derived from the Biot and Fourier numbers can be used to find the time. Where $T_0$ is the initial temperature and T is the temperature at the center. $t=\frac\left\{\rho \left\{\left\{c\right\}_\left\{p\right\}\right\}V\right\}\left\{hA\right\}ln\frac Inphysics$
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

and engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

, the Fourier number (Fo) or Fourier modulus, named after Joseph Fourier
Joseph Fourier
Jean Baptiste Joseph Fourier was a French mathematician and physicist best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations. The Fourier transform and Fourier's Law are also named in his honour...

, is a dimensionless number that characterizes heat conduction. Conceptually, it is the ratio of the heat conduction rate to the rate of thermal energy storage. Together with the Biot number
Biot number
The Biot number is a dimensionless number used in non-steady-state heat transfer calculations. It is named after the French physicist Jean-Baptiste Biot , and gives a simple index of the ratio of the heat transfer resistances inside of and at the surface of a body...

, it characterizes transient conduction problems. It is defined as: $\mbox\left\{Fo\right\} = \frac\left\{\alpha t\right\}\left\{R^2\right\}$ where: NEWLINE
NEWLINE
• α is the thermal diffusivity
Thermal diffusivity
In heat transfer analysis, thermal diffusivity is the thermal conductivity divided by density and specific heat capacity at constant pressure. It has the SI unit of m²/s...

[m2/s]
• NEWLINE
• t is the characteristic time [s]
• NEWLINE
• R is the length through which conduction occurs [m]
NEWLINE For transient mass transfer by diffusion, there is an analogous mass Fourier Number (also denoted Fo) defined as: $\mbox\left\{Fo\right\} = \frac\left\{D t\right\}\left\{L^2\right\}$ where: NEWLINE
NEWLINE
• "D" is the Diffusivity
• NEWLINE
• "t" is the characteristic timescale
• NEWLINE
• "L" is the length scale of interest
NEWLINE

## Using Fourier number

Together with the Biot number
Biot number
The Biot number is a dimensionless number used in non-steady-state heat transfer calculations. It is named after the French physicist Jean-Baptiste Biot , and gives a simple index of the ratio of the heat transfer resistances inside of and at the surface of a body...

, 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, the following equation derived from the Biot and Fourier numbers can be used to find the time. Where $T_0$ is the initial temperature and T is the temperature at the center. $t=\frac\left\{\rho \left\{\left\{c\right\}_\left\{p\right\}\right\}V\right\}\left\{hA\right\}ln\frac\left\{\left\{\left\{T\right\}_\left\{0\right\}\right\}-\left\{\left\{T\right\}_\left\{\infty \right\}\right\}\right\}\left\{T-\left\{\left\{T\right\}_\left\{\infty \right\}\right\}\right\}$

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• Convection
Convection
Convection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....

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• Heat conduction
Heat conduction
In heat transfer, conduction is a mode of transfer of energy within and between bodies of matter, due to a temperature gradient. Conduction means collisional and diffusive transfer of kinetic energy of particles of ponderable matter . Conduction takes place in all forms of ponderable matter, viz....

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• Heat equation
Heat equation
The heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...

• NEWLINE
• Biot number
Biot number
The Biot number is a dimensionless number used in non-steady-state heat transfer calculations. It is named after the French physicist Jean-Baptiste Biot , and gives a simple index of the ratio of the heat transfer resistances inside of and at the surface of a body...

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