# Fourier Number

on . Posted in Dimensionless Numbers

Fourier number, abbreviated as Fo, also called Fourier modulus, a dimensionless number, is the ratio of heat conduction rate to the rate of thermal energy storage in a solid.  This is mainly used in unsteady state heat transfer.

In heat transfer analysis, the Fourier number is used to determine the relative importance of conduction within a material compared to the time and length scales involved.  It helps in understanding the transient behavior of heat conduction and predicting how quickly temperature changes will propagate through a solid or fluid.  A smaller Fourier number indicates that heat conduction is dominant compared to the time and length scales involved, meaning that temperature changes will propagate rapidly.  A larger Fourier number suggests that the effects of conduction are less significant, and temperature changes will propagate more slowly.

The Fourier number is particularly useful in the analysis of unsteady state or transient heat conduction problems, where temperature changes occur over time.  By considering the Fourier number, engineers and researchers can make predictions about the thermal behavior of materials and optimize the design of heat transfer systems.

## Fourier number formula

$$\large{ Fo = \frac{\alpha \; t_c }{l_c^2 } }$$

### Fourier Number - Solve for Fo

$$\large{ Fo = \frac{\alpha \; t_c }{l_c^2 } }$$

 thermal diffusivity, α characteristic time, tc characteristic length, lc

### Fourier Number - Solve for α

$$\large{ \alpha = \frac{Fo \; l_c^2 }{ t_c } }$$

 Fourier number, Fo characteristic length, lc characteristic time, tc

### Fourier Number - Solve for tc

$$\large{ t_c = \frac{Fo \; l_c^2 }{ \alpha } }$$

 Fourier number, Fo characteristic length, lc thermal diffusivity, α

### Fourier Number - Solve for lc

$$\large{ l_c = \sqrt{ \frac{\alpha \; t_c }{ Fo } } }$$

 thermal diffusivity, α characteristic time, tc Fourier number, Fo

Symbol English Metric
$$\large{ Fo }$$ = Fourier number $$\large{ dimensionless }$$
$$\large{ \alpha }$$  (Greel symbol alpha) = thermal diffusivity $$\large{\frac{ft^2}{sec}}$$ $$\large{\frac{m^2}{s}}$$
$$\large{ t_c }$$ = characteristic time $$\large{ sec }$$ $$\large{ s }$$
$$\large{ l_c }$$ = characteristic length $$\large{ ft }$$ $$\large{ m }$$