The homogeneous equilibrium model (HEM) using the so-called omega method of Leung and Epstein has the advantage of providing a readily programmed analytical solution for predicting discharge rates from orifices and pipes. A further advantage ascribed to the method has been that it requires information on physical properties only for the source conditions. Recent papers have acknowledged that an alternative definition can improve upon the original defining equation for omega. This paper provides a comparison of the accuracy of the HEM using the alternative formulae for omega. It is shown that the original formula for omega tends to have a bias which underestimates the specific volume. This bias is reflected in predictions of discharge rates as a compensating error, since it tends to increase the predictions which tend to be low, at least for orifices and short pipes where the assumption of homogeneous equilibrium is not justified. The alternative formula for calculating omega has less bias in estimating specific volume (in absolute magnitude), but, since this bias tends to overestimate specific volume, it does not provide the compensating error which benefits discharge rate predictions.