The narrow cavity and `film theory'

In the narrow cavity limit, the wall fluxes of vapour and sensible and latent energy are:

$$\mbox{\textit{Sh}}_{\infty} = 1;$$ (6.2)

$$\mbox{\textit{Nu}}_{sen,\infty} = 1;$$ (6.3)

$$\mbox{\textit{Nu}}_{lat,\infty} = \left(1-\mathrm{e}^{-\varPhi _T}\right)\varLambda ,$$ (6.4)

or, in dimensional terms:

$$\left. \begin{array}{rccccc} -\mbox{\boldmath$\hat{\imath}$}... ...frac{\varPhi }{1-\mathrm{e}^{-\varPhi }} \end{array}\right\}_.$$ (6.5)

There is a striking similarity between these three fluxes: each being the product of a specific capacity, a conductance, a driving force and a mass transfer correction factor. The factor $[1-\exp(-\varPhi )]$ is Spalding's driving force for mass transfer, $-B$ (see equation 6.1). These expressions are formally identical to those arising from the `film theory' of simultaneous heat and mass transfer between a bulk fluid and a surface developed by Colburn and Drew (1937) and refined by Bird et al. (1960, ch. 21), except that there the conductances are unknown functions of the flow field, considered to be the ratio of a conductivity ($D$ or $\lambda$) and the `film thickness'. Here, the fictitious film thickness is simply the width of the air-space, $b$. It is interesting to note that the associated mass fraction (4.24) and temperature (4.26) profiles are also generally fictitious, but that here they are exactly realized.

This formal similarity suggests a means of extending the range of validity of the analysis to cavities that are not narrow. The film theory approximately separates the effects of the bulk flow field and interfacial mass transfer. The former is lumped into the conductances, or film thicknesses, while the latter is given by corrections of the form

$$\frac{\varPhi }{1-\exp(-\varPhi )} \equiv \frac{\varPhi }{-B}.$$ (6.6)

The film theory has been used successfully in the study of both laminar boundary layers and fully turbulent systems with simultaneous heat and mass transfer (Sherwood & Pigford 1952, ch. 4). Thus, even if the conditions for the narrow cavity solution are not met, so that the film thickness differs from the cavity width, the relative effects of interfacial velocity and interdiffusion may still be assessable from the same mass transfer rate factors.

Rather than being continually encumbered with the mass transfer rate correction factors, however, it would seem to be much simpler to combine these into the driving forces; i.e. for the vapour transport, to use $\varPhi$ instead of Spalding's $B$, and for energy transport, to use $\Delta T_*\varPhi _T/[1-\exp(-\varPhi _T)]$, instead of the temperature difference, $\Delta T_*$. This is precisely what was done in nondimensionalizing the vapour and energy fluxes in §2.3.1.

The intended result is that the Sherwood and (sensible) Nusselt numbers should be much less dependent on $\varPhi$. This was perfectly achieved in the narrow cavity limit (§4.4.1).