Rayleigh number

The Rayleigh number (Ra) is a dimensionless parameter that expresses the relationship between all the actions that positively contribute to motion in a fluid (floating force) and everything that is negatively opposed (viscosity and thermal diffusivity):

\[Ra=\dfrac{F_g}{\mu\alpha}\]

It is, therefore, possible to understand whether there is motion, or not, in conditions of natural convection; or if, due to the effect of the floating force resulting from the density field, which in turn is a consequence of the temperature field, motion in the fluid is triggered or not. The Rayleigh number, for Newtonian and Fourier fluids, can be calculated through the Oberbeck-Boussines approximation. It can be expressed as the product of Grashof number and Prandtl number:

\[Ra=Gr\cdot Pr=\dfrac{g\beta_0 (T_s-T_i)L^3}{\nu\alpha_0}\]

Where: \(\beta_0\) is the thermal expansion at temperature \(T_0\); \(T_s-T_i\) is the temperature difference between the upper and lower surfaces; \(g\) is the modulus of gravity acceleration; \(L\) is a linear dimension characteristic of the phenomenon under consideration; \(\nu\) is the kinematic viscosity of the material; \(\alpha_0\) is the thermal diffusivity at temperature \(T_0\).

In the case where the Rayleigh number is lower than the critical value (equal to 1708), the buoyancy thrusts due to density gradients within the fluid do not succeed in overcoming the opposition of the kinematic viscosity: motion does not manifest itself, and the heat exchange takes place by simple conduction.

Vice versa, if Ra > 1708, the thrusts are such as to allow the overcoming of the resistance imposed by the product \(\mu\alpha\) and, therefore, the convection motion manifests itself: we are in a regime of natural convection.

In the industrial field there are many situations in which a fluid moves inside a duct, in a compressible regime, exchanging energy in the form of heat and, therefore, varying its total enthalpy. Typical examples are heat exchangers and combustion chambers of open systems. The Rayleigh model of motion is based on the following assumptions:

  • the motion is quasi-one-dimensional and quasi-stationary;
  • the cross-sectional area of the fluid passing through the duct is assumed to be constant;
  • the fluid does not exchange work with the environment and both viscous effects and gravitational forces are negligible;
  • the production of entropy is negligible, in other words the thermofluidodynamic transformation is considered reversible;
  • the only pushing force is the heat exchange along the duct, which gives rise to a variation of the total enthalpy H.

For high values of Froude number, it is possible to neglect gravitational forces, while it is less likely to disregard entropy production because, in addition to viscous stresses not considered here, it is associated with both heat transfer and chemical reactions.

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