A rational approximation
for low mass transfer rates

In the field equations, $\varPhi$ enters only in the energy equation in the factor $[1-\exp(-\varPhi )]$, which has the Maclaurin series

$$1-\mathrm{e}^{-\varPhi } \equiv -B = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}\varPhi ^n}{n!} \qquad(\vert\varPhi \vert<\infty)$$ (6.7)

and vanishes like $O(\varPhi )$.

From its definition (2.33), it is clear that for $m_{*r}<1$, $\varPhi$ only vanishes when the vapour mass fraction difference, $\Delta m_{*}$, does. If this difference were zero, the buoyancy ratio, $N$, would be too. Note that $N$ can be expressed

$$N \equiv \frac{\zeta\Delta m_*}{\beta\Delta T_*}$$ (6.8)

$$= \frac{1-\mathrm{e}^{-\varPhi }}{[\beta/\zeta(1-m_{*r})]\Delta T_*}$$ (6.9)

$$\sim \frac{\varPhi }{[\beta/\zeta(1-m_{*r})]\Delta T_*} \left[1+O(\varPhi )\right]\qquad(\varPhi \rightarrow 0).$$ (6.10)

It would be inappropriate to require $N$ to vanish in the low mass transfer rate limit, however, since it is inversely proportional to temperature difference, $\Delta T_{*}$. In applying the Boussinesq approximation in chapter 2, $\beta\Delta T_{*}$ was assumed to be a small quantity. Moreover, in applications, there will often be a direct link between the temperature difference and the mass fraction difference, and so the mass transfer rate factor. In physical vapour transport, for example, the vapour mass fraction difference is created by differentially heating two opposing surfaces of pure solidified vapour (Jhaveri & Rosenberger 1982). Indeed, quite generally, the partial pressure of the vapour over the condensed phase will increase with temperature, as may be quantified by Clapeyron's relation (Guggenheim 1959, p. 148). Consider, for example, an ideal saturated mixture: the limiting value of the buoyancy ratio is (Close & Sheridan 1989)

$$\lim_{\Delta T_*\rightarrow 0} N = \frac{m_*(M_B-M_A)h_{fg}}{\Re T_*},$$ (6.11)

where $\Re$ is the universal gas constant and $M_A$ and $M_B$ are the molar masses of the vapour and gas, respectively. This limit is finite; substituting in the properties of air and water vapour at 30$\circ$C, for example, gives a limiting value of $N$ of 0.28, which is hardly negligible compared to unity.

In the unsaturated case there is no such general relation between $\varPhi$ and $\Delta T_*$. Nevertheless, the buoyancy ratio, and any other terms containing a similar ratio of $\varPhi$ and $\Delta T_{*}$ should not be eliminated in the low mass transfer rate limit.

Another example is furnished by the product $\varLambda \varPhi _T$, appearing in the expression for the latent Nusselt number (2.68). It can be expressed (cf. equation 4.32) as

$$\varLambda \varPhi _T = \frac{\varPhi }{(\lambda/\rho Dh_{fg})\Delta T_*},$$ (6.12)

or

$$\varLambda \varPhi _T = \frac{\mbox{\textit{Pr}}_r}{\mbox{\textit{Sc}}}\frac{\varPhi }{c_{pr}\Delta T_*/h_{fg}},$$ (6.13)

which shows that, by the rule established above, $\mbox{\textit{Nu}}_{lat}$ should be retained in the low mass transfer rate limit. To take a numerical example, for humid air at 30$\circ$C, $h_{fg}/c_{pr}\approx 2400$K, which will make the denominator of (6.13) small indeed.

In summary, the rational approximation for low mass transfer rates is obtained by taking the limit $\varPhi \rightarrow 0$ with

$$\frac{\Delta T_*}{\zeta(1-m_{*r})/\beta} = O(\varPhi )$$ (6.14)

$$\frac{\Delta T_*}{\mathit{Pr}_r\;h_{fg}/(\mathit{Sc}\;c_{pr})} = O(\varPhi ).$$ (6.15)