# Rayleigh Number

on . Posted in Dimensionless Numbers

Rayleigh number, abbreviated as Ra, a dimensionless number, is used in fluid dynamics and heat transfer to predict the onset of convection in a fluid or gas.  It characterizes the relative importance of buoyancy forces due to temperature differences and the dissipative effects of viscosity and diffusivity.  It is used particularly in the context of natural convection, which is the process where fluid motion is induced by temperature differences within the fluid itself, without the need for external mechanical forces.

The Rayleigh number indicates whether convection currents will form due to temperature differences.  When the Rayleigh number exceeds a certain critical value, it signifies the transition from a steady, laminar flow to an unstable, convective flow regime.

### Key Points about Rayleigh number

• For Rayleigh numbers below the critical value, heat transfer occurs mainly through conduction.
• For Rayleigh numbers above the critical value, fluid motion and heat transfer through convection become dominant.
• The Rayleigh number provides insights into the behavior of fluids in various natural convection phenomena, such as the cooling of a heated surface or the rising of hot air.
• In geophysics, the Rayleigh number is used to study processes like mantle convection and plate tectonics in the Earth's interior.

The Rayleigh number is a fundamental parameter in understanding the transition between different modes of heat transfer and fluid motion, making it valuable in various scientific and engineering fields.

### Rayleigh Number formula

$$Ra \;=\; \rho \; g \; \alpha_c \; \Delta T \; l^3 \;/\; \mu \; \alpha$$
Symbol English Metric
$$Ra$$ = Rayleigh number $$dimensionless$$
$$\rho$$ (Greek symbol rho) = density of fluid $$lbm \;/\;ft^3$$ $$kg \;/\; m^3$$
$$g$$ = gravitational acceleration  $$ft \;/\; sec^2$$ $$m \;/\; s^2$$
$$\alpha_c$$ (Greek symbol alpha) = thermal expansion coefficient $$in \;/\; in\;F$$ $$mm \;/\; mm\;C$$
$$\Delta T$$ = temperature differential $$F$$ $$K$$
$$l$$ = length $$ft$$ $$m$$
$$\mu$$ (Greek symbol mu) = absolute viscosity $$lbf - sec \;/\; ft^2$$ $$Pa - s$$
$$\alpha$$ (Greek symbol alpha) = thermal diffusivity $$ft^2 \;/\; sec$$ $$m^2 \;/\; s$$