A theoretical investigation has been conducted for predicting the critical liquid height and the location of liquid entrainment during dual discharge from a stratified two-phase region through branches mounted on a vertical wall. The two branches are horizontal and their centrelines fall in a common inclined plane. Two models have been developed; a simplified point-sink model and a more-accurate finite-branch model. Predictions from the two models are shown to be in good agreement for the condition of high flow rates from the branches, while the finite-branch model is necessary for good predictions at low branch flow rates. Influences of the various independent parameters on the predicted onset are presented and discussed. Comparisons with experimental results are shown in Part 2 of this paper.