Mass and energy fluxes

Define the reduced vapour mass flux by:

$$\mathbf{n} \equiv \frac{b}{\rho D\varPhi }\mathbf{n}_*,$$ (2.32)

where

$$\varPhi \equiv \ln\left( \frac{1-m_{*r}} {1-m_{*r}-\Delta m_{*}} \right)$$ (2.33)

is the mass transfer rate factor. This apparently complicated choice of a driving force arises naturally in one-dimensional problems (ch. 4) and will be much discussed, particularly in chapter 6.

The vapour flux is the sum of advective and diffusive fluxes:

$$\mathbf{n} = \mathbf{n}_{adv} + \mathbf{n}_{\mathit{diff}}$$ (2.34)

$$\mathbf{n}_{adv} = \left[ m_{*r}+m(1-m_{*r})\left(1-\mathrm{e}^{-\varPhi }\right)\right] \frac{\mbox{\textit{Gr}}(1+N)\mbox{\textit{Sc}}}{\varPhi }\;\mathbf{u}$$ (2.35)

$$\mathbf{n}_{\mathit{diff}} \equiv \mathbf{j} = -{}(1-m_{*r})\frac{1-\mathrm{e}^{-\varPhi }}{\varPhi }\mbox{\boldmath$\nabla$}m$$ (2.36)

where

$$\mbox{\textit{Gr}} \equiv \frac{g\beta\Delta T_*b^3}{\nu^2},$$ (2.37)

$$N \equiv \frac{\zeta\Delta m_*}{\beta\Delta T_*}, \quad\mbox{and}$$ (2.38)

$$\mbox{\textit{Sc}} \equiv \frac{\nu}{D}$$ (2.39)

are dimensionless parameters (see §1.3). Note that the reference vapour mass fraction $m_{*r}$ appearing here is not covered by the set of governing dimensionless parameters. This is unimportant, though, as it cancels out of the field equations, the boundary conditions and the expressions for the wall fluxes.

Define the reduced energy flux by

$$\mathbf{e} \equiv \frac{b[1-\exp(-\varPhi _T)]}{\lambda\Delta T_*\varPhi _T} \mathbf{e}_*,$$ (2.40)

where

$$\varPhi _T\equiv\frac{\varPhi (\mbox{\textit{Pr}}_r+\mbox{\textit{Pr}}_I)}{\mbox{\textit{Sc}}}$$ (2.41)

is the thermal mass transfer rate factor, and

$$\mbox{\textit{Pr}}_r \equiv \frac{\rho[c_{pB}+m_{*r}(c_{pA}-c_{pB})]\nu}{\lambda}, \quad\mbox{and}$$ (2.42)

$$\mbox{\textit{Pr}}_I \equiv \frac{\rho(c_{pA}-c_{pB})(1-m_{*r})\nu}{\lambda}$$ (2.43)

are the reference and interdiffusion Prandtl numbers, respectively.

The arbitrary reference enthalpy of the gas may be set to zero,

$$h_{Br} = 0,$$ (2.44)

for the reasons stated above (§2.1.3), and the reference enthalpy of the vapour equated to its heat of vaporization at the reference temperature,

$$h_{Ar}=h_{fg}.$$ (2.45)

The reduced heat of vaporization of the vapour, or the latent heat factor, is defined:

$$\varLambda \equiv \frac{h_{fg}}{c_{pA}\Delta T_*}.$$ (2.46)

The energy flux has components due to advection and conduction. The advective fluxes are split into a bulk advective flux, the interdiffusion flux and the latent heat flux:

$$\mathbf{e} = \mathbf{e}_{cond} + \mathbf{e}_{adv} + \mathbf{e}_{int} + \mathbf{e}_{lat}$$ (2.47)

$$\mathbf{e}_{cond} = -{} \frac{1-\exp(-\varPhi _T)}{\varPhi _T} \mbox{\boldmath$\nabla$}T$$ (2.48)

$$\mathbf{e}_{adv} = \frac{1-\exp({-\varPhi _T})}{\varPhi _T} \mbox{\textit{Gr}}(1+N) ... ..._r+\mbox{\textit{Pr}}_I\left(1-\mathrm{e}^{-\varPhi }\right)m\right]T\mathbf{u}$$ (2.49)

$$\mathbf{e}_{int} = -{} \frac{1-\exp(-\varPhi _T)}{\varPhi _T} \frac{\mbox{\textit{Pr... ...ox{\textit{Sc}}}\left(1-\mathrm{e}^{-\varPhi }\right)T\mbox{\boldmath$\nabla$}m$$ (2.50)

$$\mathbf{e}_{lat} = \frac{1-\exp(-\varPhi _T)}{\varPhi _T} \varPhi _T\varLambda \mathbf{n}$$ (2.51)

The conservation of species (2.5) and energy (2.11) then requires simply that $\mathbf{n}$ and $\mathbf{e}$ are solenoidal. These equations may be obtained thus or by nondimensionalizing equations (2.7) and (2.16); the result is given in §2.3.2. The latent heat flux, $\mathbf{e}_{lat}$, is solenoidal, being proportional to $\mathbf{n}$, and so makes no contribution to the energy equation. It must be retained, however, for calculation of the total flux at the boundaries.

The intrinsic link between the variability of the mixture specific heat and the interdiffusion term in the energy equation is made even more obvious by the above reduction: both depend on the same dimensionless parameter, the interdiffusion Prandtl number, $\mbox{\textit{Pr}}_I$.