The low mass transfer rate equations

Under the approximation defined above, the system (2.52)-(2.55) becomes to order O(1)

$$\mbox{\boldmath$\nabla$}\cdot\mathbf{u} = 0$$ (6.16)

$$\mbox{\textit{Gr}}(1+N)\mbox{\textit{Sc}}\;\mathbf{u}\cdot\mbox{\boldmath$\nabla$}m = \nabla^2m$$ (6.17)

$$\mbox{\textit{Gr}}(1+N) \mathbf{u}\cdot\mbox{\boldmath$\nabla$}\m... ...al A$}\mbox{\boldmath$\nabla$}p -\frac{T+Nm}{1+N}\mbox{\boldmath$\hat{\jmath}$} = \mbox{\boldmath$\nabla$}^2 \mathbf{u}$$ (6.18)

$$\mbox{\textit{Gr}}(1+N)\mbox{\textit{Pr}}_r\mathbf{u}\cdot\mbox{\boldmath$\nabla$}T = \nabla^2T,$$ (6.19)

the transpiration boundary condition (2.59) is replaced by

$$\mbox{\boldmath$\hat{n}$}\cdot\mathbf{u}=0$$ (6.20)

at the interfaces, and the equations for the Sherwood and Nusselt numbers become:

$$\mbox{\textit{Sh}} = \mbox{\boldmath$\hat{\imath}$}\cdot\mbox{\boldmath$\nabla$}m;$$ (6.21)

$$\mbox{\textit{Nu}}_{sen} = \mbox{\boldmath$\hat{\imath}$}\cdot\mbox{\boldmath$\nabla$}T;\quad\mbox{and}$$ (6.22)

$$\mbox{\textit{Nu}}_{lat} = [1-\exp(-\varPhi _T)]\varLambda \mbox{\textit{Sh}}.$$ (6.23)

To the order of the approximation, the factor in brackets in (6.23) can be replaced by $\varPhi _T$; here the stated form is retained, however, as it is exact.

Equation (6.21) is not quite the same as the one recommended in elementary mass transfer textbooks (e.g. Incropera & DeWitt 1990, p. 348). The right hand side is the same, and is also formally identical to the usual expression for the Nusselt number in single fluid heat transfer problems (Incropera & DeWitt 1990, p. 347; §2.4), but the left hand side uses for the driving force not the mass fraction difference, $\Delta m_{*}$, but the mass transfer rate factor, $\varPhi$.

Similarly, the sensible Nusselt number retains its earlier definition (2.64), which incorporates the mass transfer rate correction factor. Note that

$$\lim_{\varPhi \rightarrow 0} \frac{\varPhi _T\Delta T_*}{1-\exp(-\varPhi _T)} = \Delta T_*;$$ (6.24)

so that the low mass transfer rate Nusselt number (6.22) is correct for zero mass transfer rates; i.e. in homogeneous fluids.

The key to the success of the results of §6.1.3 is that the definitions from chapter 2 of the Sherwood and Nusselt numbers are used for both general mass transfer rates (e.g. chapters 4 and 5) and low mass transfer rates.