The Reynolds Analogy is popularly known to relate turbulent momentum and heat transfer.
[Geankoplis, C.J. ''Transport processes and separation process principles'' (2003), Fourth Edition, p. 475.] That is because in a turbulent flow (in a pipe or in a
boundary layer
In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a Boundary (thermodynamic), bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces ...
) the transport of momentum and the transport of heat largely depends on the same turbulent
eddies: the velocity and the temperature profiles have the same shape.
The
main assumption is that
heat flux
In physics and engineering, heat flux or thermal flux, sometimes also referred to as heat flux density, heat-flow density or heat-flow rate intensity, is a flow of energy per unit area per unit time (physics), time. Its SI units are watts per sq ...
q/A in a turbulent system is analogous to momentum flux τ, which suggests that the ratio τ/(q/A) must be constant for all radial positions.
The complete Reynolds analogy* is:
Experimental data for gas streams agree approximately with above equation if the
Schmidt and
Prandtl
Ludwig Prandtl (4 February 1875 – 15 August 1953) was a German Fluid mechanics, fluid dynamicist, physicist and aerospace scientist. He was a pioneer in the development of rigorous systematic mathematical analyses which he used for underlyin ...
numbers are near 1.0 and only
skin friction is present in flow past a flat plate or inside a pipe. When liquids are present and/or
form drag is present, the analogy is conventionally known to be invalid.
[
In 2008, the qualitative form of validity of Reynolds' analogy was re-visited for laminar flow of incompressible fluid with variable dynamic viscosity (μ).][Mahulikar, S.P., & Herwig, H., 'Fluid friction in incompressible laminar convection: Reynolds' analogy revisited for variable fluid properties,' ''European Physical Journal B: Condensed Matter & Complex Systems'', 62(1), (2008), pp. 77-86.] It was shown that the inverse dependence of Reynolds number (''Re'') and skin friction coefficient(''c''f) is the basis for validity of the Reynolds’ analogy, in laminar convective flows with constant & variable μ. For μ = const. it reduces to the popular form of Stanton number (''St'') increasing with increasing ''Re'', whereas for variable μ it reduces to ''St'' increasing with decreasing ''Re''. Consequently, the Chilton-Colburn analogy of ''St''•''Pr''2/3 increasing with increasing ''c''f is qualitatively valid whenever the
Reynolds’ analogy is valid. Further, the validity of the Reynolds’ analogy is linked to the applicability of Prigogine's Theorem of Minimum Entropy Production.[Prigogine, I. ''Introduction to Thermodynamics of Irreversible Processes'' (1961), Interscience Publishers, New York.] Thus, Reynolds' analogy is valid for flows that are close to developed, for whom, changes in the gradients of field variables (velocity & temperature) along the flow are small.[
]
See also
* Reynolds number
In fluid dynamics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between Inertia, inertial and viscous forces. At low Reynolds numbers, flows tend to ...
* Chilton and Colburn J-factor analogy
References
Transport phenomena
Analogy