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The Fourier number compares elapsed time with the characteristic time for heat conduction over a length.
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Formula
Fo dimensionless; a in m2/s; t in s; Lc in m
| Symbol | Meaning | Unit |
|---|---|---|
| Fourier number | dimensionless | |
| Thermal diffusivity | m2/s | |
| Elapsed time | s | |
| Characteristic conduction length | m |
\mathrm{Fo}=\frac{a t}{L_c^2}Fo=at/Lc2 is a dimensionless time in transient conduction. It measures how far thermal diffusion has progressed relative to the selected body length. Characteristic length depends on geometry and boundary conditions. Fourier number alone does not give temperature distribution; it is used with Biot number, initial and boundary conditions in analytical or numerical solutions.
Fo=at/Lc2 is a dimensionless time in transient conduction. It measures how far thermal diffusion has progressed relative to the selected body length. Characteristic length depends on geometry and boundary conditions. Fourier number alone does not give temperature distribution; it is used with Biot number, initial and boundary conditions in analytical or numerical solutions.
The Fourier number compares elapsed time with the characteristic time for heat conduction over a length.
Base equation for the target quantity
Preserve signs, boundary conditions and units when rearranging.
Work through the numerical example for “Fourier number for transient heat conduction” and check its unit and magnitude.
For a=5×10−5 m2/s, t=20 s and Lc=0.01 m, Fo=10.
Substituting a related quantity that shares the same symbol.
Use variable definitions and check: Fo dimensionless; a in m2/s; t in s; Lc in m.
Formula cluster
The Fourier number compares elapsed time with the characteristic time for heat conduction over a length.
For a=5×10−5 m2/s, t=20 s and Lc=0.01 m, Fo=10.
Normalize transient heat-conduction time scales and compare thermal solutions.
Here is how to work through a typical Fourier number for transient heat conduction (Fo = at/Lc²) task step by step:
Task
Work through the numerical example for “Fourier number for transient heat conduction” and check its unit and magnitude.
Solution path
For a=5×10−5 m2/s, t=20 s and Lc=0.01 m, Fo=10.
A large Fo indicates that substantial time has passed for heat diffusion over the characteristic length. The temperature field has had more time to respond to the boundary. Fo alone does not give local temperature or internal uniformity; geometry and Biot number are also needed. Fo is a dimensionless diffusion time over the selected length. Interpret it with geometry, initial state and boundary condition; equal Fo does not ensure equal temperature fields if Biot number or shape differs. Use a in m2/s and Lc in metres. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
The heat equation contains second spatial derivatives, giving a characteristic diffusion time L2/a. Dividing elapsed time by that scale yields Fo=at/L2. Thus doubling length makes the characteristic time four times larger with other conditions fixed. Fo is a dimensionless diffusion time over the selected length. Interpret it with geometry, initial state and boundary condition; equal Fo does not ensure equal temperature fields if Biot number or shape differs. Use a in m2/s and Lc in metres. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
No. Thermal similarity requires matching relevant dimensionless groups and boundary conditions. For transient conduction, Biot number, geometry and dimensionless temperature profile also matter. Two bodies with the same Fo can have very different surface temperatures because their Biot numbers differ. Fo is a dimensionless diffusion time over the selected length. Interpret it with geometry, initial state and boundary condition; equal Fo does not ensure equal temperature fields if Biot number or shape differs. Use a in m2/s and Lc in metres. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
Write the quantity and unit from the variable list next to every value: Fo dimensionless; a in m2/s; t in s; Lc in m. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example For a=5×10−5 m2/s, t=20 s and Lc=0.01 m, Fo=10. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. Fo is a dimensionless diffusion time over the selected length. Interpret it with geometry, initial state and boundary condition; equal Fo does not ensure equal temperature fields if Biot number or shape differs. Use a in m2/s and Lc in metres. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
The background states this starting condition: Fo=at/Lc2 is a dimensionless time in transient conduction. It measures how far thermal diffusion has progressed relative to. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. Fo is a dimensionless diffusion time over the selected length. Interpret it with geometry, initial state and boundary condition; equal Fo does not ensure equal temperature fields if Biot number or shape differs. Use a in m2/s and Lc in metres. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
Quanta exam set
Curated exam set for "Fourier number for transient heat conduction":
Question (front)
Which formula describes Fourier number for transient heat conduction?
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How do you rearrange Fourier number for transient heat conduction for Base equation for the target quantity?
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Which common mistake happens with Fourier number for transient heat conduction?
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