Transport rate dependence on $\varPhi$

To test the dependence of the overall vapour and energy transport rates on the mass transfer rate factor, a series of Fastflo solutions were obtained for a square cavity ( $\mbox{$\mathcal A$}=1$) by the method described in chapter 5. The mass transfer rate factor was varied from 0.105 to 0.693. All other parameters were kept constant, except the buoyancy ratio, $N$, which, like $B$, is proportional to the mass fraction difference, $\Delta m_{*}$, for fixed reference mass fraction level, $m_{*r}$. The overall transport rates are listed in tables 6.1 - 6.2.

Table: Dependence of overall vapour transport rate on $\varPhi$. Mean Sherwood numbers from Fastflo runs for $\mbox{$\mathcal A$}=1, \mbox{\textit{Pr}}_r=0.71, \mbox{\textit{Pr}}_I=\mbox{\t... ...}=0.61, \mbox{\textit{Gr}}(1+N)\mbox{\textit{Pr}}_r=10^{3}, N=1-\exp(-\varPhi )$.

$\varPhi$	$\frac{\varPhi }{1-\exp(-\varPhi )}$	$\overline{\mbox{\textit{Sh}}}$
0.105	1.054	1.088
0.223	1.116	1.087
0.693	1.386	1.080

Table: Dependence of overall energy transport rate on $\varPhi$. Mean sensible Nusselt numbers from same the Fastflo runs as table 6.1.

$\varPhi$	$\varPhi _T$	$\frac{\varPhi _T}{1-\exp(-\varPhi _T)}$	$\overline{\mbox{\textit{Nu}}}_{sen}$
0.105	0.228	1.118	1.122
0.223	0.483	1.261	1.124
0.693	1.500	1.931	1.11

The mass transfer rate correction factors are also listed in the second last column of each table.

It will be noticed that the mean Sherwood and Nusselt numbers vary by less than 0.8% and 1.3%, respectively, in spite of the fact that the mass transfer rate correction factors vary by 31% and 73%. The variability in the predicted Sherwood and Nusselt numbers is close to the limits of accuracy of the numerical solutions and may also be affected by the changes in the buoyancy ratio, $N$, though this parameter generally has little effect on the transport rates at these Schmidt and Prandtl numbers if $\mbox{\textit{Gr}}(1+N)$ is fixed (McBain 1995, 1997b; §3.3.16).

The corresponding wide variation in the mass transfer rate correction factors is exactly reflected in the dimensional transport rates; thus, this series of runs strongly supports the contention (§6.1.2) that the present nondimensionalization of the vapour and energy fluxes accounts for the effect of the mass transfer rate factor, $\varPhi$.

The energy transfer results (table 6.2) may be compared with de Vahl Davis's (1983) bench-mark solution for the analogous single fluid heat transfer problem ( $\varPhi =N=0, \mbox{$\mathcal A$}=1, \mbox{\textit{Pr}}_r\equiv\mbox{\textit{Pr}}=0.71, \mbox{\textit{Gr}}(1+N)\mbox{\textit{Pr}}\equiv\mbox{\textit{Ra}}=10^{3}$). He estimated $\mbox{\textit{Nu}}=1.118\pm0.001$, which lies within 1% of each of the three present results for finite $\varPhi$.

Another example of the effect of $\varPhi$ is provided by the numerical solutions of Markham and Rosenberger (1980; §3.2.3) and Greenwell et al. (1981; §3.2.4) for isothermal vapour transport in cylinders. Both studies found that moderately large values of $\varPhi$ (1.0 and 0.944) led to strong radial variations in the vapour transport, but had no discernible effect on the mean Sherwood number.

The results of this section suggest that in order to calculate the overall vapour and (sensible) energy transport rates it may be possible to use a solution with a different value of $\varPhi$; in particular, one with $\varPhi =0$. The latent energy transport rate can then be immediately calculated from the Sherwood number and (2.68). This, of course, has great computational advantages: the transpiration boundary condition (2.59) becomes trivially homogeneous, the mixture specific heat capacity in the advective energy flux (2.49) becomes constant, and the interdiffusion flux (2.50) vanishes. The last two changes greatly simplify the energy equation (2.55), which then takes on the form (2.70).

This procedure is not exact. The mass transfer rate factor, $\varPhi$, does appear in the governing equations and cannot be completely dismissed as a parameter. Deviations from the mass transfer rate correction factors predicted by film theory have been noted for the forced convection laminar boundary layer by Bird et al. (1960, pp. 672-6), Spalding (1963, pp. 148-51) and Rosner (1966). These errors are small, however, unless the mass transfer rate is high, and vanish as $\varPhi \rightarrow 0$. The use of $\varPhi$ as the driving force is certainly expected to be superior to its alternatives, but the primary purpose of the present chapter is the formulation of a simplifying approximation for small values of $\varPhi$; any degree of accuracy at high values should be regarded as a bonus.

If, as is the case in the growth of crystals from the vapour, the local values of Sh are important, rather than just their mean, solution of the full equations is certainly indicated. Even in this case, though, the low mass transfer rate approximation may be useful for understanding qualitative features of the flow, particularly those due to buoyancy, and as a good initial guess for iterative solution procedures. Being rational, it could also be taken as the zeroth term in an asymptotic expansion.