How Rayleigh’s number links conduction and convection

The concept

To understand the importance and significance of Rayleigh’s number, we must initially focus on the concept of natural convection and when this occurs.

The term convection comes from the Latin convectare, which means to take to a place. In heat transfer, the concept of convection refers to the process of moving thermal energy from or to a solid by means of an adjacent fluid in motion, in the presence of a temperature gradient.

The overall phenomenon involves two mechanisms:

• Phase 1: heat conduction, between the solid and the adjacent layer of fluid at rest, which we could call the initial phase.

• Phase 2: Movement of the particles in the fluid, away from the solid and letting others from the same fluid pass, which we could properly call convection.

Two main types can be considered depending on the source of the fluid motion:

• Forced convection, where the fluid movement is due to some external factor, such as a fan, extractor or pump.

• Natural convection, where the fluid movement is due exclusively to properties of the fluid itself between two determined points of the process.

Natural convection

Natural convection occurs in fluids in the presence of a gravitational field due to differences in density as a result of temperature differences within the fluid.

Normally, density variations are due to temperature gradients, although it is also common for them to occur due to concentration gradients, and natural convection phenomena can occur in electrically charged fluids when they are subjected to electric or magnetic fields.

The simplest and most common phenomenon is that of a fluid in a gravitational field in which there are density variations due to temperature gradients.

Movements within the fluid in these circumstances constitute the phenomenon known universally as Rayleigh-Bénard natural convection.

The movement of fluids due to natural convection occurs both in nature and in industrial processes. In nature, it occurs, for example, in the process of mixing ocean waters and storms.

It is constantly present in industry. In electronic equipment or industrial electric motors, the use of natural convection for heat dissipation is a determining factor.

Also important is its contribution in the ventilation of boiler or production rooms and in the insulation and heating of buildings.

Given its importance, a broad knowledge of its properties is necessary to optimize its application. Obviously, among these features is the determination of when the change occurs between the initial and second phases.

When heat or energy is transferred by forced convection, there are no doubts: the start-up of the external component that produces the movement, the fan or pump. But what about natural convection?

The theory

In his 1916 article “On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side”, Lord Rayleigh developed a theory from the previous experiments of Henri Bénard in 1906, where he explained all the physical mechanisms involved in the phenomenon of natural convection and also theoretically determined the existence of a critical temperature increase below which there can be no convection movement in the fluid due exclusively to variations in properties.

Until this critical value is exceeded, there is a “steady state” in the fluid.

A situation is stable if it is stationary; that is, if its evolution over time is not altered by a small modification in its initial conditions.

Or, formulating it inversely, a situation is unstable if a slight alteration of the initial conditions causes, in turn, a significant alteration in the evolution of the system.

When a fluid, whose density is not uniform, is in the presence of gravity, the force of buoyancy, from Archimedes’ principle, makes a portion of fluid denser than its environment descend, while one of lower density tends to ascend.

Therefore, heating a fluid and thus changing its density at different points should immediately cause convective movement.

However, in addition to buoyancy forces, mechanisms that tend to counteract their effect must be taken into account: friction, caused by viscosity forces, and heat conduction or diffusion, which tend to homogenize the temperature field within a fluid and, therefore, that of density.

Thus, if a disturbance is made in a fluid initially at rest with a density distribution increasing upwards, part of the fluid will return to its steady state or will keep on moving (unsteady state), depending on the relative importance of the three aforementioned effects.

Only an unstable situation gives rise to the natural convection phenomenon known as Rayleigh-Bénard convection.

This is because the density of the liquid decreases with increasing temperatures and therefore the upper layers are denser than the lower liquid layers.

The stratification leads to a configuration that could become potentially unstable under a vertical gravitational field.

Fig 1. Steady state, with stratification according to temperature/density

Since each layer is at a higher temperature and therefore a lower density than the upper layer, it undergoes a differential flotation force that pushes it towards the upper part.

Finding even colder and denser fluid in its path, the movement is reinforced until it becomes a destabilizing element in the evolution of the system.

However, as already mentioned, the two stabilizing processes are opposed.

First, the induced velocity naturally tends to decay due to the friction caused by the viscosity. Second, the thermal diffusion leads to a tendency for the temperature of the layers to equilibrate.

Therefore, the fluid layer remains at rest while the stabilising processes dominate until the perturbation – the difference in densities caused by the temperature – is sufficiently high for it to develop:

A ΔT greater than a critical value ΔTc can be termed the instability threshold

In the aforementioned model of the two plates, there is no stratification and the distribution of the fluid by temperature shows an ascending/descending circuit as shown in Fig 2, forming the so-called convection cells.

Fig 2. Convection cells. The flotation force is superior to the resistance from the viscosity and thermal diffusion.

To determine this instability threshold, Rayleigh based his study on the stability of small perturbations with respect to the situation at rest.

Thus he linearized the Navier-Stokes equations that explain the fluid movement between plates around the equilibrium solution, and discovered that, in the resulting linear system of partial differential equations, the dimensionless parameter that summarizes all the parameters that participate in this phenomenon appears naturally.

This means it can be determined when a certain fluid under the effects of a temperature gradient will go to an unsteady state and heat transfer will start by natural convection.

This dimensionless parameter, the Rayleigh number, was named in his honour and has the formula:

where:

• ß, coefficient of thermal expansion of the fluid
• ?, fluid kinematic viscosity
• α, thermal diffusivity of the fluid
• g, acceleration due to gravity
• h, thickness of the fluid layer – or characteristic length of the fluid domain –
• ΔT, temperature difference between plates

The Rayleigh number can be interpreted conceptually as the dimensionless parameter that measures the relative importance between the effects of the buoyancy forces and the effects of the viscosity forces and thermal conduction.

For Rayleigh number values lower than a certain critical value, the system is in steady state. For values higher than that value, the system becomes unstable and natural convection movements take place.

In 1926, Jeffreys first determined the critical Rayleigh number for the case of infinite parallel plates, Ra=1708.

Given that, as determinants in the Rayleigh number expression, a geometric component “h” is observed, significant differences may exist for this concept.

Subsequent to Jeffreys’ determination and using theoretical reasoning and experimental testing, Ra numbers were determined for different geometries and features of the temperature gradient – two-dimensional or three-dimensional, – or of the contour.

Although we have focused on the phenomenon of natural convection due to variations in associated densities with temperature gradients in a homogeneous fluid, as indicated above there are many circumstances in which natural convection is caused by density variations associated with concentration gradients in heterogeneous fluids.

An example of this situation can be seen in the landscapes of Salinas Grandes in Salta (Argentina) in Fig. 3.

Fig 3. Grandes Salinas, Salta (Argentina)

They are closed basins where water accumulates in the rainy season. In the dry season, the water is evaporated slowly by the sun, leaving thin layers of highly concentrated brackish water in which Bénard convection occurs.

The flow of liquid drains the salt and the convection cells are formed, where the fluid rises through the centre of the cell and descends by the perimeter and crystallizes as the water evaporates. When completely dry, you can see a geometric structure of salt in a hexagonal shape which characterises the phenomenon of natural convection in systems with a free surface.