Wall fluxes

In this project, the primary interest is in vapour transport between parallel vertical walls. Denoting their common unit normal by $\hat{\imath}$ (see §2.5), the reduced vapour flux at these walls, the (local) Sherwood number, is

$$\mbox{\textit{Sh}}\equiv -\mbox{\boldmath$\hat{\imath}$}\cdot\mathbf{n}.$$ (2.60)

Since the walls are impermeable to the gas, (2.21) holds. In dimensionless form this is

$$\frac{\mbox{\textit{Gr}}(1+N)\mbox{\textit{Sc}}}{\varPhi }\m... ...{n}$}\cdot\mathbf{u}=\mbox{\boldmath$\hat{n}$}\cdot\mathbf{n},$$ (2.61)

so that the Sherwood number can be calculated from the transverse component of velocity:

$$\mbox{\textit{Sh}}= -\frac{\mbox{\textit{Gr}}(1+N)\mbox{\tex... ...c}}}{\varPhi }\;\mbox{\boldmath$\hat{\imath}$}\cdot\mathbf{u};$$ (2.62)

or, because of the transpiration boundary condition (2.59), from the mass fraction field:

$$\mbox{\textit{Sh}}= \frac{1-\mathrm{e}^{-\varPhi }} {\varPh... ... \mbox{\boldmath$\hat{\imath}$}\cdot\mbox{\boldmath$\nabla$}m.$$ (2.63)

This definition of the Sherwood number (2.60, see also equation 2.32), using the mass transfer rate factor, $\varPhi$, instead of the mass fraction difference, $\Delta m_{*}$, for the driving force, is unconventional, though it was used by Jhaveri and Rosenberger (1982). Spalding (1960, 1963) has explained in detail the disadvantages of using the mass fraction difference as a driving force, but instead proposed the use of a quantity equivalent to $[\exp(-\varPhi )-1]$. The superiority of the present choice will become apparent once solutions of the system of equations are found and examined (ch. 4), especially when a rational approximation for the system at low mass transfer rates is considered (ch. 6).

The (local) Nusselt number, Nu, is defined at the parallel vertical walls by:

$$\mbox{\textit{Nu}} \equiv -\mbox{\boldmath$\hat{\imath}$}\cdot\mathbf{e}$$ (2.64)

$$= \frac{1-\exp(-\varPhi _T)}{\varPhi _T}\bigg\{ \mbox{\boldmath$\hat{\imath}$}\cdot\mbox{\boldmath$\nabla$}T$$

$$- {}\mbox{\textit{Gr}}(1+N) \left[\mbox{\textit{Pr}}_r+\mbox{\tex... ...rm{e}^{-\varPhi }\right)m\right] T\mbox{\boldmath$\hat{\imath}$}\cdot\mathbf{u}$$

$$+ \frac{\mbox{\textit{Pr}}_I}{\mbox{\textit{Sc}}}(1-\mathrm{e}^{-... ...\cdot\mbox{\boldmath$\nabla$}m +\varPhi _T\varLambda \mbox{\textit{Sh}}\bigg\}.$$ (2.65)

The interfacial velocity and the mass fraction gradient can be eliminated from this expression by the equations for the Sherwood number (2.62) and (2.63):

$$\mbox{\textit{Nu}}= \frac{1-\exp(-\varPhi _T)}{\varPhi _T}\l... ...abla$}T +(T + \varLambda )\varPhi _T\mbox{\textit{Sh}}\right]$$ (2.66)

Since the latent heat factor, $\varLambda$, does not appear in the field equations (2.52)-(2.55), it will occasionally be convenient to consider that part of the energy transfer rate independent of it--the sensible Nusselt number:

$$\mbox{\textit{Nu}}_{sen}= \frac{1-\exp(-\varPhi _T)}{\varPhi... ...\boldmath$\nabla$}T + T \varPhi _T \mbox{\textit{Sh}}\right).$$ (2.67)

The remainder,

$$\mbox{\textit{Nu}}_{lat}\equiv\mbox{\textit{Nu}}-\mbox{\textit{Nu}}_{sen}=[1-\exp(-\varPhi _T)]\varLambda \mbox{\textit{Sh}},$$ (2.68)

is the latent Nusselt number.

The above comments on the unconventionality of the definition of the Sherwood number apply equally to the Nusselt number. As will be seen in §2.4, the present Nusselt number reverts to the conventional definition (e.g. Incropera & DeWitt 1990, p. 347) in the absence of mass transfer ($\varPhi =0$).