Weaver and Viskanta (1991a)

Weaver and Viskanta (1991a) reported numerical solutions for plane vertical square cavities. Their principal intention was to demonstrate the effects of spatial variations in the thermophysical properties due to temperature and vapour mass fraction gradients.

There is a serious inconsistency in the energy equation (their equation 5), which was formulated with the mixture specific enthalpy as the dependent variable rather than temperature. Fourier's Law was employed for the conduction flux, but in writing this in terms of the gradient of the mixture enthalpy, $h$ was taken as a function only of temperature. When they calculated the temperature field from the mixture specific enthalpy field, however (their equation 18), $h$ was taken as a function of both temperature and vapour mass fraction. Consistency would have required the temperature gradient in the conduction term to have been replaced by the weighted sum of mixture specific enthalpy and vapour mass fraction gradients. While possible, this would certainly have been inconvenient.

A consistent formulation of the energy equation in terms of temperature was derived in §2.1.3 (see also Bird et al. 1960, pp. 560-6). The general difficulty encountered both here and by Weaver and Viskanta is that the energy equation contains advection terms, which are naturally expressed in terms of enthalpy, and a conduction term, which must be expressed in terms of temperature. Since temperature is a measurable quantity whereas enthalpy is not, and since expressing the conduction in terms of enthalpy would require two sets of second order derivatives (as noted above), temperature seems the natural choice for the dependent variable of the energy equation. This was also the choice in the detailed and rigorous numerical model of Rosenberger et al. (1997; §3.3.18).

Note that the formulation for the mixture enthalpy used in §2.1.3 is identical with Weaver and Viskanta's (their equation 18), except that in §2.1.3, the partial specific heats are taken to be independent of temperature. This difference in no way affects the consistency or otherwise of the equations.

The inconsistency of their treatment of the mixture enthalpy led Weaver and Viskanta to a gross error. In one of their numerical solutions, a strong interior relative minimum appeared in temperature field. This is impossible: it contradicts Theorem 1 (p. ).