Short Answer:

Rayleigh number is a dimensionless number used in heat transfer and fluid mechanics to determine the nature of natural convection. It represents the combined effect of buoyancy and thermal diffusion in a fluid. The Rayleigh number helps to predict whether the fluid flow will remain stable, laminar, or become turbulent due to temperature differences.

In simple words, Rayleigh number shows how temperature difference, fluid properties, and size of the system affect the motion of fluid. When the Rayleigh number exceeds a certain critical value, natural convection starts, and the fluid begins to move, enhancing heat transfer.

Detailed Explanation :

Rayleigh Number

The Rayleigh number (Ra) is an important dimensionless parameter in the study of natural convection heat transfer. It gives a measure of how strongly buoyancy forces (caused by temperature differences) compete against viscous and thermal diffusion effects in a fluid. The Rayleigh number is used to predict the onset of convection, the type of flow (laminar or turbulent), and the rate of heat transfer in systems where natural convection occurs.

The Rayleigh number is mathematically defined as:

Where,

= Rayleigh number
= Grashof number (represents the ratio of buoyancy to viscous force)
= Prandtl number (represents the ratio of momentum diffusivity to thermal diffusivity)

By substituting the expression of Grashof number, we get:

Here,

= acceleration due to gravity (m/s²)
= coefficient of thermal expansion (1/K)
= temperature difference between surface and surrounding fluid (K)
= characteristic length (m)
= kinematic viscosity (m²/s)
= thermal diffusivity (m²/s)

Physical Meaning of Rayleigh Number

The Rayleigh number combines the effects of buoyancy (which promotes fluid motion) and diffusion (which resists it). In other words, it measures how much the temperature difference can drive the fluid motion against the resistance caused by viscosity and thermal conductivity.

If the Rayleigh number is small, it means that heat transfer mainly occurs by conduction, and there is little or no fluid motion.
When the Rayleigh number is large, buoyancy becomes strong enough to create circulation, and convection begins.
When it becomes very large, the flow becomes turbulent, leading to a high rate of heat transfer.

The Rayleigh number thus helps engineers and scientists decide whether a system will remain stable or develop convective motion.

Critical Rayleigh Number

Natural convection starts only when the Rayleigh number exceeds a certain critical value.

For a fluid layer heated from below, convection begins when Ra > 1700.
Below this value, heat transfer happens mainly through conduction because buoyancy forces are not strong enough to move the fluid.
As the Rayleigh number increases beyond this limit, convection cells form and eventually become turbulent at very high Rayleigh numbers.

Flow Regimes Based on Rayleigh Number

Ra < 10³:
Heat transfer is purely by conduction, no motion occurs.
10³ < Ra < 10⁷:
Fluid flow is laminar, and steady convection currents form.
10⁷ < Ra < 10⁹:
Transition region between laminar and turbulent convection.
Ra > 10⁹:
Flow becomes turbulent, and heat transfer is greatly enhanced.

These ranges may vary depending on the geometry and boundary conditions of the surface.

Importance of Rayleigh Number

The Rayleigh number is one of the most important dimensionless parameters in thermal engineering because it helps:

Predict the onset of convection in fluids heated from below or cooled from above.
Determine whether the heat transfer mode is conduction or convection.
Classify the flow regime as laminar or turbulent.
Estimate heat transfer coefficients in natural convection systems.
Analyze thermal stability in engineering and environmental systems.

Relation with Other Dimensionless Numbers

The Rayleigh number is a combination of the Grashof and Prandtl numbers:

The Grashof number (Gr) represents the effect of buoyancy compared to viscous forces.
The Prandtl number (Pr) shows the relationship between momentum and thermal diffusivities of the fluid.

Thus, the Rayleigh number provides a complete picture of how a fluid behaves under temperature differences and gravitational forces.

Examples of Rayleigh Number in Practice

Boiling Water:
When water in a pot is heated from below, convection currents form after the Rayleigh number exceeds its critical value.
Atmospheric Circulation:
Air movement in the atmosphere is influenced by temperature gradients and buoyancy, represented by large Rayleigh numbers.
Cooling of Electronic Devices:
Heat generated in components is transferred to the surrounding air by natural convection, analyzed using the Rayleigh number.
Solar Collectors and Radiators:
The performance of natural convection in heating and cooling systems depends on the Rayleigh number.
Ocean Currents:
Temperature and salinity differences in ocean water generate motion related to high Rayleigh numbers.

Factors Affecting Rayleigh Number

Temperature Difference (Tₛ – T∞):
A higher temperature difference increases buoyancy, leading to a higher Rayleigh number.
Fluid Properties:
Viscosity, thermal expansion, and thermal diffusivity have direct effects on the Rayleigh number value.
Characteristic Length (L):
Larger systems or surfaces increase the Rayleigh number because of greater volume of fluid involved.
Gravity (g):
The Rayleigh number is directly proportional to gravitational acceleration; without gravity, convection cannot occur.

Conclusion

The Rayleigh number is a vital concept in mechanical and thermal engineering that combines the effects of buoyancy and thermal diffusion to describe natural convection. It helps determine when a fluid starts to move due to temperature differences and predicts the type of flow that will occur. A higher Rayleigh number means stronger convection and more efficient heat transfer. Therefore, it plays a central role in analyzing natural convection processes in engineering systems, environmental studies, and everyday thermal applications.