Rayleigh number

In fluid mechanics, the Rayleigh number (Ra) for a fluid is a dimensionless number associated with buoyancy-driven flow, also known as free convection or natural convection. When the Rayleigh number is below a critical value for that fluid, heat transfer is primarily in the form of conduction; when it exceeds the critical value, heat transfer is primarily in the form of convection.

The Rayleigh number is named after Lord Rayleigh and is defined as the product of the Grashof number, which describes the relationship between buoyancy and viscosity within a fluid, and the Prandtl number, which describes the relationship between momentum diffusivity and thermal diffusivity. Hence the Rayleigh number itself may also be viewed as the ratio of buoyancy and viscosity forces multiplied by the ratio of momentum and thermal diffusivities.

Classical Definition

For free convection near a vertical wall, the Rayleigh number is defined as:

\mathrm {Ra} _{x}={\frac {g\beta }{\nu \alpha }}(T_{s}-T_{\infty })x^{3}=\mathrm {Gr} _{x}\mathrm {Pr}

where:

x is the characteristic length

Ra_x is the Rayleigh number for characteristic length x

g is acceleration due to gravity

β is the thermal expansion coefficient (equals to 1/T, for ideal gases, where T is absolute temperature).

$\nu$ is the kinematic viscosity

α is the thermal diffusivity

T_s is the surface temperature

T_∞ is the quiescent temperature (fluid temperature far from the surface of the object)

Gr_x is the Grashof number for characteristic length x

Pr is the Prandtl number

In the above, the fluid properties Pr, ν, α and β are evaluated at the film temperature, which is defined as:

T_f = \frac{T_s + T_\infin}{2}.

For a uniform wall heating flux, the modified Rayleigh number is defined as:

{\mathrm {Ra}}_{{x}}^{{*}}={\frac {g\beta q''_{o}}{\nu \alpha k}}x^{4}

where:

q"_o is the uniform surface heat flux

k is the thermal conductivity.^[1]

For most engineering purposes, the Rayleigh number is large, somewhere around 10⁶ to 10⁸.

Other Definitions

The Rayleigh number can be also used as a criterion to predict convectional instabilities, such as A-segregates, in the mushy zone of a solidifying alloy. The mushy zone Rayleigh number is defined as:

\mathrm{Ra} = \frac{\frac{\Delta \rho}{\rho_0}g \bar{K} L}{\alpha \nu} = \frac{\frac{\Delta \rho}{\rho_0}g \bar{K} }{R \nu}

where:

K is the mean permeability (of the initial portion of the mush)

L is the characteristic length scale

α is the thermal diffusivity

ν is the kinematic viscosity

R is the solidification or isotherm speed.^[2]

A-segregates are predicted to form when the Rayleigh number exceeds a certain critical value. This critical value is independent of the composition of the alloy, and this is the main advantage of the Rayleigh number criterion over other criteria for prediction of convectional instabilities, such as Suzuki criterion.

Torabi Rad et. Al. showed that for steel alloys the critical Rayleigh number is 17.^[3] Pickering et. al explored Torabi Rad's criterion, and further verified its effectiveness. Critical Rayleigh numbers for lead–tin and nickel-based super-alloys were also developed.^[4]

Geophysical applications

In geophysics, the Rayleigh number is of fundamental importance: it indicates the presence and strength of convection within a fluid body such as the Earth's mantle. The mantle is a solid that behaves as a fluid over geological time scales. The Rayleigh number for the Earth's mantle due to internal heating alone, Ra_H, is given by:

{\mathrm {Ra}}_{H}={\frac {g\rho _{{0}}^{{2}}\beta HD^{5}}{\eta \alpha k}}

where:

H is the rate of radiogenic heat production per unit mass

η is the dynamic viscosity

k is the thermal conductivity

D is the depth of the mantle.^[5]

A Rayleigh number for bottom heating of the mantle from the core, Ra_T, can also be defined as:

\mathrm {Ra} _{T}={\frac {\rho _{0}^{2}g\beta \Delta T_{sa}D^{3}C_{P}}{\eta k}}

where:

ΔT_sa is the superadiabatic temperature difference between the reference mantle temperature and the core–mantle boundary

C_P is the specific heat capacity at constant pressure.^[5]

High values for the Earth's mantle indicates that convection within the Earth is vigorous and time-varying, and that convection is responsible for almost all the heat transported from the deep interior to the surface.

Notes

↑ M. Favre-Marinet and S. Tardu, Convective Heat Transfer, ISTE, Ltd, London, 2009
↑ Torabi Rad, M; Kotas, P; Beckermann, C (2013). "Rayleigh number criterion for formation of A-Segregates in steel castings and ingots". Metall. Mater. Trans. A. 44A: 4266–4281.
↑ Torabi Rad, M; Kotas, P; Beckermann, C (2013). "Rayleigh number criterion for formation of A-Segregates in steel castings and ingots". Metall. Mater. Trans. A. 44A: 4266–4281.
↑ Pickering, EJ; Al-Bermani, S; Talamantes-Silva, J (2014). "Application of criterion for A-segregation in steel ingots". Materials Science and Technology.
1 2 Bunge, Hans-Peter; Richards, Mark A.; Baumgardner, John R. (1997). "A sensitivity study of three-dimensional spherical mantle convection at 10⁸ Rayleigh number: Effects of depth-dependent viscosity, heating mode, and endothermic phase change". Journal of Geophysical Research. 102 (B6): 11991–12007. Bibcode:1997JGR...10211991B. doi:10.1029/96JB03806.

References

Turcotte, D.; Schubert, G. (2002). Geodynamics (2nd ed.). New York: Cambridge University Press. ISBN 0-521-66186-2.

Dimensionless numbers in fluid mechanics

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Rayleigh number

Classical Definition

Other Definitions

Geophysical applications

See also

Notes

References