Heat and mass transfer boundary conditions

For simplicity, either Dirichlet (fixed value), Neumann (fixed normal derivative) or Robin (a linear combination of the previous two) boundary conditions will be specified for the vapour mass fraction and temperature, depending on the particular problem. Ideas for how these idealizations might be implemented have been presented elsewhere (McBain, Close, Suehrcke, Harris & Brandemuehl 1998).

Appropriate thermal boundary conditions for the analogous single fluid heat transfer problem (§2.4) have been discussed by many authors, including Batchelor (1954), Mallinson (1987) and Leong, Hollands and Brunger (1998), but comparatively little attention has been paid to the mass transfer boundary conditions. The only investigation I am aware of is the simple parametric study of Costa (1997). This is probably because the nonisothermal transport of moisture (or other condensed vapour) through porous media is in itself an extremely difficult problem, indeed, the fundamentals seem to be less well understood than in the present problem (see, for example, Dahl et al. 1996; Hens 1996; Künzel & Kiessl 1997). At some stage, once the transport of vapour through gaseous and porous phases is better understood, it will be possible to properly consider the conjugate heat and mass transfer problem.

No vapour passes an impermeable wall so that, on eliminating the normal component of velocity by (2.20) from (2.4),

$$\mbox{\boldmath$\hat{n}$}\cdot\mathbf{n}_* = 0$$

$$\mbox{\boldmath$\hat{n}$}\cdot\mbox{\boldmath$\nabla$}_* m_* = 0.$$ (2.24)

Possibly the simplest thermal boundary condition for these walls is the adiabatic, which results from setting the normal component of the absolute energy flux (2.15) to zero, and eliminating the mass fluxes by (2.20) and (2.24):

$$\mbox{\boldmath$\hat{n}$}\cdot\mathbf{e}_* = 0$$

$$\mbox{\boldmath$\hat{n}$}\cdot\mbox{\boldmath$\nabla$}_* T_* = 0.$$ (2.25)

The adiabatic condition is not meaningful at a mass transfer interface, since the enthalpy carried by the mass fluxes is arbitrary. Its relevance for any wall bounding a gas-vapour phase is questionable (see studies cited above) since the thermal conductivity of any solid is higher than that of most gas-vapour mixtures. A better idealization, first suggested by Batchelor (1954), is to treat the solid wall as a perfect conductor. This uncouples the heat conduction in the solid, which sees the boundary with the gas-vapour phase as adiabatic, and leads to a Dirichlet condition for the gas-vapour phase temperature. An example is described in §7.5.2.