In the present study, we develop a theoretical model for predicting hydrogen jet ignition in a shock tube with a partially opened diaphragm for hydrogen as a driver gas and air as a driven gas. Effects of pressure losses associated with the gradual diaphragm opening process are taken into account by developing a new discharge coefficient model. The discharge coefficient for incompressible flows is first calculated and then converted into the discharge coefficient for compressible flows. Three regimes are identified for the discharge coefficient, which depend on whether the ruptured portion of the diaphragm, which remains attached to the orifice, is negligible or not. We extensively validate the new model against experimental results from the literature. We show that the critical initial pressure ratio across the diaphragm and the shock-wave strength required for ignition can be calculated in a conservative manner with a maximum relative error equal to 9% and 10%, respectively, for the ratio of diaphragm opening area to driven-section cross-sectional area from 0.125 to 1.0. Finally, we explore, via our new model, the influence of different parameters on the shock-wave Mach number and ignition limits. We show that all our model predictions can be generalized into two simple dimensionless correlations.