Utility of the narrow cavity limit

The narrow cavity limit is useful here for two reasons.

Firstly, while some numerical studies (e.g. Weaver & Viskanta 1991b; Rosenberger et al. 1997) have included the transpiration boundary condition and the interdiffusion energy flux, there has been no attempt to quantify their influence on the overall transfer rates. This is not surprising, since even for plane laminar steady constant-property flow, the problem depends on eight dimensionless parameters. However, for the narrow cavity limit, the effects are precisely quantified.

Secondly, the concept of a narrow vertical air space is a very practical one. It has long been known (Batchelor 1954) that including an air cavity within a building wall is an economical means of reducing its thermal conductance, and the thickness of the air space is the parameter over which the designer has the most control. The importance of the narrowness of the space is that although the rate of radiative transfer is almost independent of it, if `the product of the temperature difference (in degrees Fahrenheit) and the cube of the space thickness (in inches) is less than 3 [3$\circ$F.in $^{3} \approx 3\times 10^{-5}$K.m$^{3}$]... convection is practically suppressed' (ASHRAE 1993, p. 20.7). This equates to about 1cm wall separation for a 30K temperature difference, or with the properties of dry air at 20$\circ$C, a Grashof number, Gr, of about 4000--about half the critical Grashof number usually (§5.4) associated with the transition to multicellular convection. Given the strong dependency of Gr on the space thickness, $b$, though, the ASHRAE rule of thumb represents a reasonable margin of safety. The suppression of convection also depends on the height of the cavity (§5.5), and a proper assessment of the strength of convection should also take into account the buoyancy force due to gradients in humidity (Wee et al. 1989). The narrow cavity limit is useful as it provides an upper bound on the thermal resistivity.