The fully developed solution

Being linear and only weakly coupled, these equations are easily solved.

The equation of continuity of the mixture (4.13) can be immediately integrated to give $u=u_\infty$, a constant.

This may be substituted into the equation of continuity of vapour (4.14), which can then be integrated twice to give:

$$m=\frac{1-\exp[\mbox{\textit{Gr}}(1+N)\mbox{\textit{Sc}}\;u_... ...{1-\exp[\mbox{\textit{Gr}}(1+N)\mbox{\textit{Sc}}\;u_\infty]},$$ (4.22)

once the vapour mass fraction boundary conditions (4.18) have been satisfied. Differentiating this expression with respect to $X$ and substituting into either of the transpiration boundary conditions, (4.19) or (4.20), leads to:

$$u = \frac{-\varPhi }{\mbox{\textit{Gr}}(1+N)\mbox{\textit{Sc}}}$$ (4.23)

and

$$m = \frac{1-\mathrm{e}^{-\varPhi X}}{1-\mathrm{e}^{-\varPhi }}.$$ (4.24)

The other transpiration boundary condition is automatically satisfied, which is to be expected since the formulation of the equations in §2.1.1 guarantees that mass is conserved.

Now that $u$ and $m$ are known, the energy equation simplifies to:

$$-\varPhi _T \frac{\partial T}{\partial X}=\frac{\partial^2 T}{\partial X^2}.$$ (4.25)

Equation (4.25) is of the same form as the species equation (4.14), so that the solution satisfying the boundary conditions (4.18) is

$$T = \frac{1-\exp(-\varPhi _T X)}{1-\exp(-\varPhi _T)}.$$ (4.26)

The solutions for the pressure gradient and vertical component of velocity are;

$$v = \frac{\mbox{\textit{Sc}}}{1+N}\left\{ \frac{(\sigma-T)}{\varPhi \... ...textit{Pr}}_I-1)} + \frac{N(\sigma-m)}{\varPhi ^2(\mbox{\textit{Sc}}-1)}\right.$$

$$+ \left.\frac{\sigma-X}{\varPhi }\left[ \frac{1}{\varPhi _T} + \frac{N}{\varPhi } -c(1+N) \right]\right\}$$ (4.27)

and

$$\frac{\,\mathrm{d}p}{\,\mathrm{d}Y} = \frac{\overline{T}+N\overline{m}}{1+N}+c,$$ (4.28)

where $c$ is indeterminate, as discussed in §4.4.2,

$$\sigma = \frac{1-\exp[-\varPhi X/\mbox{\textit{Sc}}]}{1-\exp[-\varPhi /\mbox{\textit{Sc}}]},$$ (4.29)

and the quantities with overbars are averages across the cavity, given by

$$\overline{m} \equiv \int_{0}^{1} m \;\,\mathrm{d}X$$

$$= \frac{1}{1-\mathrm{e}^{-\varPhi }}-\frac{1}{\varPhi }$$ (4.30)

$$\overline{T} \equiv \int_{0}^{1} T \;\,\mathrm{d}X$$

$$= \frac{1}{1-\exp(-\varPhi _T)}-\frac{1}{\varPhi _T}$$ (4.31)

The vapour mass fraction and temperature profiles are illustrated in figure 4.1, and the velocity profile in figures 4.2 - 4.4.

Figure 4.1: Profiles of the vapour mass fraction (4.24), with parameter $\varPhi$, and temperature (4.26), with parameter $\varPhi _T$, in the narrow cavity limit.

Figure 4.2: Profiles of the vertical component of velocity (4.27), with parameter $\varPhi$, in the narrow cavity limit, for $N=1$, $\mbox{\textit{Sc}}=0.6$, $\mbox{\textit{Pr}}_r=0.7$, $\mbox{\textit{Pr}}_I=1$ and $c=0$.

Figure 4.3: Profiles of the vertical component of velocity (4.27), with parameter $N$, in the narrow cavity limit, for $\varPhi =2$, $\mbox{\textit{Sc}}=0.6$, $\mbox{\textit{Pr}}_r=0.7$, $\mbox{\textit{Pr}}_I=1$ and $c=0$.

Figure 4.4: Profiles of the vertical component of velocity (4.27), with parameter $c$, in the narrow cavity limit, for $N=1$, $\mbox{\textit{Sc}}=0.6$, $\mbox{\textit{Pr}}_r=0.7$, $\mbox{\textit{Pr}}_I=1$ and $\varPhi =2$.

As in the heat transfer only solution of Aung (1972), the temperature (and here also mass fraction) distribution is unaffected by the vertical velocity; this can be seen from the fact that the combined Grashof number, $\mbox{\textit{Gr}}(1+N)$, does not appear in the solutions (4.26) or (4.24).

With the substitution of mole fractions for mass fractions (because of the assumption of constant density rather than constant total molar concentration), the distributions of vapour mass fraction (4.24) and temperature (4.26) are identical to those given by Bird et al. (1960, pp. 572-4) for simultaneous heat and mass transfer across a stagnant film of noncondensable gas (see also Bird et al. 1960, pp. 522-6; Rohsenhow & Choi 1961, pp. 398-400; Greenwell et al. 1981 for derivations of the same mass fraction profile in other configurations).

A temperature distribution with the same form as equation (4.26) was suggested by Ranganathan and Viskanta (1988; reviewed in §3.3.8) for a square cavity ( $\mbox{$\mathcal A$}=1,\mbox{$\mathcal S$}\rightarrow\infty$) with $\mbox{\textit{Pr}}_I=0$ and $N=-1$ in the special cases $\mbox{\textit{Sc}}=\mbox{\textit{Pr}}_r$, for which net buoyancy effects vanish everywhere, and $\varPhi \rightarrow -\infty$, for which the mass transfer-induced horizontal velocities are assumed to overwhelm the buoyancy-induced vertical velocities. Neither of these conclusions is quite correct.

In the first case, $\mbox{\textit{Sc}}=\mbox{\textit{Pr}}_r$, it results that the temperature distribution depends on the specific heat capacity of the mixture, whereas it is evident on expressing equation (4.26) in terms of primitive quantities that it only depends on the partial specific heat of the vapour, since

$$\varPhi _T \equiv \frac{(\nu\rho c_{pr}/\lambda) +[\nu\rho(c_{pA}-c_{pB})(1-m_{*r})/\lambda]}{\nu/D}\varPhi$$

$$= \frac{\rho Dc_{pA}}{\lambda}\varPhi ;$$ (4.32)

the reciprocal of the quantity multiplying $\varPhi$ being called the vapour Lewis number by McBain (1998), and McBain and Harris (1998). Thus, in the narrow cavity limit, the temperature distribution is independent of the partial specific heat capacity of the gas. The error of Ranganathan and Viskanta (1988) arose from their neglect of the interdiffusion term (and variation of mixture specific heat capacity) in the energy equation and so vanishes only if the gas and vapour partial specific heat capacities are equal, i.e. $\mbox{\textit{Pr}}_I=0$, an unlikely situation due to the fact that partial specific heat capacity of vapours is in general much higher than that of gases, as pointed out in §2.3.2. From this one can deduce that the mass transfer effects of transpiration and enthalpy interdiffusion (including variable mixture specific heat capacity) should be included together: one should only be included in a model without the other in the rare circumstance that the interdiffusion Prandtl number, $\mbox{\textit{Pr}}_I$, is numerically either very small or large.

In the second case, $\varPhi \rightarrow -\infty$, their solution is inconsistent with the no-slip boundary conditions at the floor and ceiling. These will affect a large proportion of the cavity, since this is not narrow ( $\mbox{$\mathcal A$}=1$) in their study. As shown by Meyer and Kostin (1975; reviewed in §3.2.2), the nonslip walls give rise to a circulation of the gas.

A concentration profile similar to equation (4.24) also occurs as the exact solution of the general one-dimensional advection-diffusion equation discussed by Patankar (1980, p. 86) in connection with upwinding in finite volume numerical schemes. There, the parameter $\varPhi$ of equation (4.24) plays the role of a cell Peclet number. Recall (§3.2.3) that Markham and Rosenberger (1980) called $\varPhi$ the Peclet number of a Stefan diffusion tube.

The velocity profile (4.27), illustrated in figures 4.2 - 4.4 is a new result, differing from the single fluid heat transfer cubic profile because of the horizontal advection of vertical momentum due to the vapour migration and the curved buoyancy distribution; the temperature and mass fraction varying differently and nonlinearly across the cavity.

The vapour mass fraction (4.24), temperature (4.26) and velocity (4.27) solutions all have singularities at $\varPhi =0$, but these are removable. This can be seen from the Maclaurin series in $\varPhi$ for $\sigma$, for example:

$$\sigma= X-\frac{X(X-1)}{2\mbox{\textit{Sc}}}\varPhi +\frac... ...c{X^2(X-1)^2}{24\mbox{\textit{Sc}}^3}\varPhi ^3 +O(\varPhi ^4)$$ (4.33)

Note the $\sigma$ has the same functional form as $m$ and $T$--with 1 and $\mbox{\textit{Sc}}/(\mbox{\textit{Pr}}_r+\mbox{\textit{Pr}}_I)$ replacing Sc, respectively. Thus, the solutions are made analytic for $-\infty<\varPhi <\infty$ by setting

$$m = X$$ (4.34)

$$u = 0$$ (4.35)

$$v = \frac{X(X-1)}{2} \left( \frac{1-2X}{6} + c \right)$$ (4.36)

$$\frac{dp}{dY} = \frac{1}{2} + c$$ (4.37)

$$T = X$$ (4.38)

for $\varPhi =0$, which is equivalent to the solution of Nelson and Wood (1989).

The dashed line in figure 4.2, drawn from equation (4.36), also represents the solution for the analogous single fluid heat transfer problem; i.e. Aung's (1972) solution, since then the buoyancy ratio, $N$, enters only through the speed scale in the definition of $v$ (2.27).

There are also removable singularities at $\mbox{\textit{Sc}}=1$ and $\mbox{\textit{Pr}}_r+\mbox{\textit{Pr}}_I=1$ which are less important. The nondimensionalization breaks down in the vicinity of $N=-1$, as discussed in §2.6.2. Note the dramatic difference between the velocity profiles in figure 4.3 for $N=-0.9$ and $-1.1$.