4.3. Counter-current limitation

By eliminating the pressure gradients from Eq. (4.17), we obtain

(4.21)\[ \tau_{i} \frac{Pe_{i}}{A \alpha_{L} \alpha_{G}} + \tau_{W} \frac{Pe_{W}}{A \alpha_{L}} = \Delta \rho g\]

We assume \(u_{L}/u_{G} \ll 1\); hence,

(4.22)\[ \tau_{i} \sim f_{i} \frac{\rho_{G}}{2} u_{G}^{2} = f_{i} \frac{\rho_{G}}{2} \frac{j_{G}^{2}}{\alpha_{G}^{2}}\]

The wall shear stress is

(4.23)\[ \tau_{W} = f_{W} \frac{\rho_{L}}{2} u_{L}^{2} = f_{W} \frac{\rho_{L}}{2} \frac{j_{L}^{2}}{\alpha_{L}^{2}}\]

Substituting the shear stress expressions into the momentum equation gives

(4.24)\[ \frac{f_{i}}{2} \frac{Pe_{i}}{A \alpha_{L} \alpha_{G}} \frac{u_{G}^{2}}{\Delta \rho g / \rho_{G}} + \frac{f_{W}}{2} \frac{Pe_{W}}{A \alpha_{L}} \frac{u_{L}^{2}}{\Delta \rho g / \rho_{L}} = 1\]

(4.25)\[ \frac{2 f_{i} \sqrt{\alpha_{G}}}{\alpha_{L} \alpha_{G}^{3}} \frac{j_{G}^{2}}{\Delta \rho g D / \rho_{G}} + \frac{2 f_{W} }{\alpha_{L}^{3}} \frac{j_{L}^{2}}{\Delta \rho g D / \rho_{L}} = 1\]

When \(\delta / R \ll 1\), \(\sqrt{\alpha_{G}} \sim 1\); therefore,

(4.26)\[ \frac{2 f_{i}}{\alpha_{L} \alpha_{G}^{3}} j_{G}^{*2} + \frac{2 f_{W} }{\alpha_{L}^{3}} j_{L}^{*2} = 1\]

where \(j_{k}^{*}\) is the so-called Wallis parameter:

(4.27)\[\begin{split}\begin{split} &j_{G}^{*} = \frac{u_{G}}{\sqrt{\Delta \rho g D / \rho_{G}}} \\ &j_{L}^{*} = \frac{u_{L}}{\sqrt{\Delta \rho g D / \rho_{L}}} \\ \end{split}\end{split}\]

We assume the proportionality relation, \(f_{i} \propto \alpha_{L}\), and \(f_{W} = \text{const.}\) for turbulent film condition, and therefore,

(4.28)\[ \frac{F}{\alpha_{G}^{3}} j_{G}^{*2} + \frac{G}{\alpha_{L}^{3}} j_{L}^{*2} = 1\]

where \(F\) and \(G\) are constants. Differentiating this with respect to \(\alpha_{L}\) gives

(4.29)\[ \frac{3 F}{(1-\alpha_{L})^{4}} j_{G}^{*2} - \frac{3 G}{\alpha_{L}^{4}} j_{L}^{*2} = 0\]

and

(4.30)\[ \frac{\alpha_{L}}{1 - \alpha_{L}} = \left( \frac{G j_{L}^{*2}}{F j_{G}^{*2}} \right)^{1/4}\]

(4.31)\[ F^{1/4} j_{G}^{*1/2} + G^{1/4} j_{L}^{*1/2} = 1\]

One can rewrite this result as

(4.32)\[ j_{G}^{*1/2} + m j_{L}^{*1/2} = C\]

The \(m\) and \(C\) are determined by experimental data. This equation was derived by Wallis [Wal69] and has been used to correlate flooding (flow reversal) condition and counter-current flow limitation. A more detailed analysis can be found in Goda et al. [GHVK+18] and a series of works by Murase (e.g., [MUK+17]).