An effective Lewis number has been defined for two-fluid mixtures under conditions of (1) supercritical temperature and pressure and (2) large gradients of temperature and composition. The Lewis number is a measure of the ratio between characteristic lengths for diffusion of heat and diffusion of mass. The traditional definition of the Lewis number for a fluid is straightforward under subcritical conditions, in which the molar flux depends only on mole-fraction gradients and the heat flux depends only on the temperature gradient. Under supercritical conditions, the traditional definition of the Lewis number does not account for additional heat- and mass-transfer effects and thus leads to inaccurate estimates of heat- and mass-transfer scales. Accurate estimates of these scales are needed for designing combustors that operate under supercritical conditions; for example, combustors in rocket, gas turbine, and Diesel engines.
The need for an effective Lewis number (as distinguished from the traditional Lewis number) becomes apparent in the context of the fluctuation-dissipation theory described in the preceding article. In that theory, the Soret and Dufour effects are described by terms that include the off-diagonal elements of the transport matrix. The differential operators and equations for mass fractions and temperature are coupled through the off-diagonal elements in that the diffusion terms in each equation include derivatives of both dependent variables. This coupling through the off-diagonal elements prohibits a simple definition of diffusion length scales for heat and mass transfer and represents additional contributions to heat and mass transfer that are not considered in the traditional Lewis number. These observations suggest the need for an effective Lewis number that is valid under general (including supercritical) conditions.
The traditional Lewis number relates the diffusion lengths of the mass fractions and temperature as given by the coefficients of the diffusive terms. In the classical situation in which the traditional Lewis number is defined, the differential operators for mass fractions and species are uncoupled, and so the diffusion terms in the differential operators can be expressed as the product of a diagonal matrix and a spatial derivative. In order to be able to define an effective Lewis number in the general case, one must find equivalent variables for which the matrix of the differential equations assumes a diagonal form.
Given the complexity of the equations, a simple, accurate combination of variables cannot be found a priori. Therefore, a solution was sought for the special case of a drop of liquid (e.g., liquid oxygen) surrounded by a gas (e.g., hydrogen) under simplifying assumptions of (1) a boundary-layer spatial variation under subcritical conditions that exist in a thin radial interval at the surface of the drop and (2) quasi-steady behavior. The adoption of this special case makes it possible to define equivalent variables in the forms of linear combinations of the temperature and the mass fraction of one species, that yield the desired decoupling between differential equations for temperature and mass fractions.
Numerical results from calculations for binary fluid systems, variously involving isolated or interacting fluid drops, show that the effective Lewis number can be as much as 40 times the traditional Lewis number, and that the spatial variations of the two numbers are different (see figure). Thus, the traditional Lewis number cannot be relied upon to give even a qualitatively correct approximation of heat- and mass-transfer scales under supercritical conditions.
This work was done by Josette Bellan and Kenneth Harstad of Caltech for NASA's Jet Propulsion Laboratory. NPO-20256
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The Lewis number under supercritical conditions
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