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Abstract:

A mathematical model is developed for stationary nonisothermal unsaturated vapour transport. It incorporates the Boussinesq approximation, but includes the finite mass transfer rate effects of nonzero interfacial velocity and the enthalpy carried by diffusive fluxes. It is proved that (under conditions more general than those assumed in the model) strong local extrema of temperature or vapour mass fraction are impossible. The model is used to investigate the effect of finite mass transfer rates and bounding solid surfaces on vapour transport.

A solution is presented for the space bounded only by two parallel plane vertical walls held at different uniform temperatures and vapour mass fractions. This is used to demonstrate many of the basic features of vapour transport. In particular, it shows that the total energy transfer across a cavity may be greatly affected by the enthalpy carried by vapour.

Numerical solutions are obtained using the Fastflo finite element package. These demonstrate the existence of a fully developed region in plane vertical rectangular cavities with vertical aspect ratios as low as 5. The criterion for this is shown to depend on both the vertical aspect ratio and the combined Grashof number, $\mbox{\textit{Gr}}(1+N)$.

In many applications, such as those involving humid air at ordinary temperatures, the mass transfer rates are low. A simplified but rational approximation to the model is derived that is valid under these conditions. The contribution of vapour gradients to buoyancy forces and particularly the effect of evaporative cooling must be retained in this limit.

The infinite vertical plane cavity solution is extended to horizontally bounded cavities with rectangular or elliptic sections for low mass transfer rates. It is found that the flow in the vertical midplane of a tall narrow cuboid is only effectively two-dimensional if the span of the cavity is at least 1.7 times its breadth.

An asymptotic solution, to first order in $\mbox{\textit{Gr}}(1+N)$ and zeroth order in $\varPhi$, the mass transfer rate factor, is presented for spherical cavities. This follows from a new general solution of the inhomogeneous Stokes problem in the sphere. In addition to the primary buoyancy-driven circulation, the solution shows two secondary flows. These correspond to the `convective' and `inertial' mechanisms for three-dimensional flow in a side-heated cuboid identified by Mallinson and de Vahl Davis (1973, 1977). The analytical form of the solution clarifies the nature of these mechanisms. In particular, it is shown that the inertial mechanism vanishes quadratically as the primary flow tends to zero and also vanishes if the stream-lines are straight.