A new theoretical investigation has been conducted for the prediction of the critical height at the onset of gas entrainment during single discharge from a stratified two-phase region through a branch installed on an inclined flat wall. Two models have been developed; a simplified point-sink model and a more-accurate finite-branch. The predicted critical height at the onset of gas entrainment was proven to be a function of Froude number (Fr) and density ratio of the interface fluids. The results of the predicted critical height at the onset of gas entrainment, at low values of Fr (>10), were found to be more accurate when using the finite-branch analysis compared to the results found using the pink-sink analysis. Whereas, with increasing Fr, the predicted values of both models converged to the same value. Furthermore, the point-sink analysis was demonstrated to be independent of wall inclination angle, while the finite-branch analysis showed a slight decrease in the value of the critical height with increasing wall inclination angle. Three different experimental data sets at wall inclination angles of zero, 45 and 90 degrees (i.e. side, inclined and bottom branches) were used in the following study for the comparisons between the experimental and theoretically predicted results. A good concurrence was illustrated between the experimental and theoretical values.