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Dimensionless field equations

The dimensionless equations of continuity (2.6, 2.7), motion (2.10), and energy (2.16) are:

$\displaystyle \mbox{\boldmath$\nabla$}\cdot\mathbf{u}$ $\textstyle =$ $\displaystyle 0$ (2.52)
$\displaystyle \mbox{\textit{Gr}}(1+N)\mbox{\textit{Sc}}\;\mathbf{u}\cdot\mbox{\boldmath$\nabla$}m$ $\textstyle =$ $\displaystyle \nabla^2m$ (2.53)
$\displaystyle \mbox{\textit{Gr}}(1+N) \mathbf{u}\cdot\mbox{\boldmath$\nabla$}\m... A$}\mbox{\boldmath$\nabla$}p
-\frac{T+Nm}{1+N}\mbox{\boldmath$\hat{\jmath}$}$ $\textstyle =$ $\displaystyle \mbox{\boldmath$\nabla$}^2 \mathbf{u}$ (2.54)
$\displaystyle \mbox{\textit{Gr}}(1+N)\left[\mbox{\textit{Pr}}_r+\mbox{\textit{P...
...-\mathrm{e}^{-\varPhi }\right)m
$\displaystyle - \frac{\mbox{\textit{Pr}}_I}{\mbox{\textit{Sc}}}\left(1-\mathrm{e}^{-\varPhi }
\right)(\mbox{\boldmath$\nabla$}m)\cdot(\mbox{\boldmath$\nabla$}T)$ $\textstyle =$ $\displaystyle \nabla^2T.$ (2.55)

The dimensionless parameters appearing here are not all independent, since the interdiffusion Prandtl number, $\mbox{\textit{Pr}}_I$, and the mass transfer rate factor, $\varPhi $, only appear in the group $\mbox{\textit{Pr}}_I[1-\exp(-\varPhi )]$. The mass transfer rate factor, however, enters separately in the transpiration boundary condition (2.59), below.

The general roles of the various dimensionless parameters are as follows.

  1. Gr, the thermal Grashof number, indicates the strength of buoyancy forces due to temperature variations. It is a measure of the nonlinearity of the system. It has the same sign as the temperature difference (for fluids expanding on heating), which is usually taken as positive.
  2. $N$, the buoyancy ratio, indicates the strength of buoyancy forces due to vapour concentration variations relative to the thermal effects. Since buoyancy is usually the main driving force for the flow (rather than the interfacial velocity), the combined Grashof number, $\mbox{\textit{Gr}}(1+N)$, is a measure of the relative importance of advective and diffusive (viscous) processes. Thus, fluid motion instabilities are more likely at higher combined Grashof numbers. $N$ can range over $(-\infty,\infty)$.
  3. Sc, the Schmidt number, is the ratio of the diffusivities of momentum and species, a necessarily positive quantity. Its product with the combined Grashof number indicates the relative importance of advection and diffusion for species transport. Its numerical value for air-water vapour under normal conditions is about 0.61 (Gebhart et al. 1988, p. 943).
  4. $\varPhi $, the mass transfer rate factor, is central to all aspects of mass transfer. It is the driving force for vapour transport, and enters into the boundary conditions on velocity at the interfaces, as well as the interdiffusion term in the energy equation.
  5. $\mbox{\textit{Pr}}_r$, the reference Prandtl number, is the ratio of the diffusivities of momentum and heat, based on the (isobaric) specific heat capacity of the mixture at the reference composition. When mass transfer rate effects are small, this approaches the usual single fluid Prandtl number, and as such is proportional to the bulk advection term in the energy equation. When the diffusion flux is comparable to the mean mass flux, this interpretation must be modified. This quantity is necessarily positive. Its numerical value for dry air at 275-310K is 0.71 (Gebhart et al. 1988, p. 935).
  6. $\mbox{\textit{Pr}}_I$, the interdiffusion Prandtl number, has the form of a Prandtl number, except that in the usual place of a specific heat it has the difference of the specific heats of the vapour and gas. While this quantity is not necessarily positive (although $\mbox{\textit{Pr}}_r+\mbox{\textit{Pr}}_I>0$), it will often be so in practice since vapours are typically distinguishable from gases by a greater molecular complexity--this being related to the causes of the intermolecular attractions responsible for their relatively high boiling point. More complex molecules have more kinds of energy (e.g. rotational and vibrational) and so higher specific heats (Jeans 1940, pp. 33-6, 275 f.). This trend is somewhat offset by the fact that specific heat capacities are, all else being equal, inversely proportional to molar mass (Jeans 1940, pp. 35, 276). The alkanes form a counterexample: the boiling points (at 1atm) and specific heat capacities (at 298K and low pressure) of methane, ethane and propane are 112, 185 and 231K and 2219, 1746 and 1669J.K$^{-1}$.kg$^{-1}$, respectively (Dean, J. A. 1992, pp. 1.199 - 1.286, 6.19 - 6.33). In our principal example, however, the rule is true: the specific heat capacities of air and water vapour at 300K being 1003.5 and$^{-1}$.K$^{-1}$. At infinite dilution, 300K and 1atm, the value of the interdiffusion Prandtl number for air-water vapour is:
    $\displaystyle \mbox{\textit{Pr}}_I$ $\textstyle \equiv$ $\displaystyle \frac{\rho D(c_{pA}-c_{pB})(1-m_{*r})}{\lambda}$ (2.56)
      $\textstyle =$ $\displaystyle \frac{(1.1768)
(2.58\times 10^{-5})
{0.0263}$ (2.57)
      $\textstyle =$ $\displaystyle 1.004.$ (2.58)

    The interdiffusion Prandtl number occurs twice in the energy equation: in the interdiffusion term; and in the bulk advection term, where it is proportional to the variation in the mixture specific heat. As explained in §2.1.3, the variation of the mixture specific heat and the interdiffusion effect are two aspects of the same phenomenon. It is inconsistent to include one without the other. Including interdiffusion without variable mixture specific heat in the bulk advection term introduces a term into the energy flux divergence (the steady energy equation) which is dependent on the temperature datum for enthalpy, which must be arbitrary. If the mixture specific heat is treated as a constant in the flux divergence, a spurious source term occurs. This has not always been recognized in the literature, and is one of the important features of the present model.

next up previous contents
Next: Dimensionless transpiration condition Up: Nondimensionalization Previous: Mass and energy fluxes   Contents
Geordie McBain 2001-01-27