A Rational Approximation
for Low Mass Transfer Rates

IT MUST be stressed that the value of the mass transfer rate factor ($\varPhi =0.69$) employed in chapter 5 was much larger than those values likely to be encountered in air-water vapour systems at ordinary temperatures. It was chosen large so as to show more clearly the effects of mass transfer (enthalpy interdiffusion and transpiration); to show that the numerical method was capable of handling these; and to show that the narrow cavity limiting solution (ch. 4) could apply--and, therefore, the conduction-diffusion regime exist--at a high mass transfer rates.

Indeed, in the experiments of McBain, Suehrcke & Harris (unpub.) on the evaporation of free water into air at 20-40$\circ$C and 75% relative humidity, $\varPhi$ did not exceed 0.012. Given the not inconsiderable difficulty added to the solution of vapour transport problems by the interdiffusion energy flux and the transpiration boundary condition (§5.1), and noting that both of these effects vanish with $\varPhi$, there is much motivation for constructing an approximation for low mass transfer rates. This is the task of the present chapter.

A rational approximation is one that is exact in some limit (Van Dyke 1964, pp 2-3). The limit considered here is, of course, small $\varPhi$. The superiority of this approach over the dropping, however intuitive or judicious, of `negligible' terms should be obvious; in particular, the accuracy of a rational approximation can always be investigated by examining more terms in the asymptotic expansion.

Several irrational approximations for vapour transport have already been uncovered in the literature:

• allowing the mixture specific heat capacity to vary with composition in the bulk advective energy flux, $\mathbf{e}_{adv}$, but neglecting the interdiffusion flux, $\mathbf{e}_{int}$, (§2.1.3);
• neglecting inertial terms in the momentum equation while retaining the advective flux of vapour, $\mathbf{n}_{adv}$, (§3.2.2);
• including transpiration but not interdiffusion (§3.3.8);

and it was shown what inconsistencies they can lead to. Actually even these approximations could be considered as rational, as each would become exact in the large Schmidt number limit. Since, however, this is not a physically appropriate approximation for gas-vapour mixtures, and since this was unlikely to have been the intention of the authors, the above distinction seems apt.

There are other possibilities for the small parameter with which to characterize low mass transfer rates, such as the maximum vapour mass fraction or Spalding's (1960) driving force for mass transfer,

$$B \equiv \mathrm{e}^{-\varPhi }-1 \equiv \frac{-\Delta m_*}{1-m_{*r}}.$$ (6.1)

Although in many ways equivalent, the present choice possesses several advantages over these, as will be demonstrated.