Other low mass transfer rate limits

If, following Spalding (1960, 1963), $B$ had been used as the small parameter in the low mass transfer rate limit, the set of governing equations (6.16)-(6.19) would have been identical, since $-B\sim\varPhi +O(\varPhi ^2)$ as $\varPhi \rightarrow 0$, by (6.7). The equations of chapter 2 could also have been written more compactly, since $\varPhi$ more often appears there in the form (6.1). Then, however, $B$ would have been the driving force in the definition of the Sherwood number (Spalding 1963, p. 38).

In either low mass transfer rate limit, the narrow cavity mass fraction profile would be given by (4.34), and so the Sherwood number (6.21) would have the value unity. If the dimensional vapour transport rate were to be calculated from this, with $B$ as the driving force, the result would be

$$-\mbox{\boldmath$\hat{\imath}$}\cdot\mathbf{n}_{*,\infty} = -\frac{\rho D}{b} B.$$ (6.25)

Successive approximations would add further terms in the (absolutely convergent) asymptotic series

$$-\mbox{\boldmath$\hat{\imath}$}\cdot\mathbf{n}_{*,\infty} = \frac{\rho D}{b} \sum_{n=1}^{\infty}\frac{(-B)^n}{n},$$ (6.26)

but this is simply (6.5), since the series is just the Maclaurin expansion for $\varPhi$ in terms of $B$. Thus, using $\varPhi$ instead of $B$ leads to one of those happy instances of the telescoping of terms of an asymptotic series by transformation of the perturbation quantity (for other examples, see Van Dyke 1975, pp. 22-3, 244); in this case an infinity of terms becomes one. The practical benefit of the collapse of the series is that described in §6.1.2: the first approximation, i.e. that obtained from the solution of the low mass transfer limit equations (6.16)-(6.19), is either exact, in the narrow cavity limit, or very close to it, for the square cavity cases considered in §6.1.3. If $B$ had been used, the first term would have differed by the mass transfer rate correction factor (6.6).

Another advantage of $\varPhi$ over $B$ is its symmetry properties. Recall the transformation of §2.6.3. The condition for invariance on the mass transfer rate factor was

$$\tilde{\varPhi } = -\varPhi$$ (6.27)

or

$$\tilde{B} = \frac{-B}{B+1}.$$ (6.28)

Since this transformation is merely that of looking at the cavity from behind rather than from in front, the simple odd parity of the mass transfer rate factor seems more satisfying than the more complicated (6.28). Associated with this is the fact that the physically significant range of $B$ is $(-1,\infty)$, whereas $-\infty<\varPhi <\infty$. In general, for equal but opposite values of $\varPhi$, similar but reversed behaviour can be expected; whereas positive and negative values of $B$ must be compared with (6.28) in mind. Further, one would quite naturally expect the dimensional mass transfer rate to be an odd function of its driving force, and this is the case if $\varPhi$ is used; however, inspection of (6.26) shows that $B$ fails this test for the narrow cavity limit. The simplest function of $B$ with the required parity properties is $\ln(1+B)\equiv-\varPhi$.

The energy transfer rate is not an odd function of the driving force implicit in the definition of the Nusselt number (2.64), although it is an odd function of the temperature difference. The chosen form of the Nusselt number is preferred in spite of this because of its reduced dependence on $\varPhi _T$ (§6.1.3).

In a brief remark on page 156 of his book, Spalding (1963) came close to questioning the utility of his driving force, $B$, noting in particular the fact that the ratio of the dimensional mass transfer rate to $B$, i.e. the `conductance' or `mass transfer coefficient', depended on $B$. On page 159, the use of $\varPhi$ is mentioned as a simplifying alternative. This option has been wholly adopted in the present work.

If the mass fraction difference were employed instead of $\varPhi$ or $B$, it would be very difficult to calculate the higher approximations, since the transpiration boundary condition cannot be written in terms of it alone; the reference mass fraction, $m_{*r}$, is also required. Consider, for example, (2.59) for the simple case $m=0$,

$$\mbox{\boldmath$\hat{n}$}\cdot\mathbf{u} = \frac{-\left(1-\m... ...{*r})}\mbox{\boldmath$\hat{n}$}\cdot\mbox{\boldmath$\nabla$}m.$$ (6.29)

The mass transfer rate factor, $\varPhi$, can be developed in a double power series:

$$\varPhi = \Delta m_{*} + m_{*r}\Delta m_{*} + (\Delta m_{*})^2 + \mbox{h.o.t.};$$ (6.30)

thus, if $\Delta m_{*}\rightarrow 0$ and $m_{*r}\rightarrow 0$, then $\varPhi \sim\Delta m_{*}+O((\Delta m_{*})^2+m_{*r}\Delta m_{*})$. In other words, the mass fraction difference is an acceptable approximation for the driving force only if it and the reference mass fraction level are small; i.e. if the vapour is everywhere dilute. Using $\Delta m_{*}$ as driving force in the Sherwood number, the calculated dimensional vapour transport rate would vary by the factor,

$$\frac{\varPhi }{(1-m_{*r})\left(1-\mathrm{e}^{-\varPhi }\right)},$$ (6.31)

or the same mass transfer correction factor encountered for $B$ increased by the factor $(1-m_{*r})^{-1}$. Further disadvantages of the use of the mass fraction difference are discussed by Spalding (1963, p. 66 f.). The most serious problem here is that in order to obtain higher approximations, the reference mass fraction level must be introduced as a extra parameter.

These fairly obvious considerations notwithstanding, many authors have employed the mass fraction difference as the driving force for mass transfer (e.g. Rohsenhow & Choi 1961, p. 385; Bejan 1985; Prata & Sparrow 1985; Nunez & Sparrow 1988; McBain 1995, 1997b).