Effect of Compressibility on the Discharge Coefficient of Orifices and Convergent Nozzles

Abstract

It has been known for some time that when a gas flows through an orifice the coefficient of discharge increases as the ratio of the downstream pressure to the upstream pressure is reduced (Stanton (1)).† This increase continues even after the critical (sonic) pressure ratio is passed, since, unless the discharge coefficient of the orifice is already unity, the mean velocity in the plane of the orifice is still subsonic and can be influenced by the downstream pressure. In a paper to this Institution, Jobson (2) suggested a method by which the variation of discharge coefficient with pressure ratio could be calculated. The method was based on the assumption that the velocity pattern at the walls, upstream of the orifice, was independent of flow rate: it yielded results that were in good agreement with experiments on sharp-edge orifices. The assumption breaks down, however, and leads to demonstrably false results, when the discharge coefficient is greater than about 0·65 and the pressure ratio well away from unity, since the upstream velocity pattern is then affected by compressibility. In the present paper, a simple assumption about the flow pattern at the walls enables allowance to be made for this additional compressibility effect. The resulting curves of discharge coefficient against pressure ratio are then correct for a perfect nozzle, with a discharge coefficient of unity, as well as for a Borda mouth-piece, and so might be expected to be reasonable approximations in between these extremes. Experimental results from a variety of sources have been compared with the theoretical predictions. The agreement is good at pressure ratios up to the critical but very few results at really low pressure ratios are available. It should perhaps be emphasized that neither Jobson's nor the present method of analysis enables the discharge coefficient of a particular orifice to be calculated ab initio. Their object is to predict how the discharge coefficient of a particular nozzle, known under one set of flow conditions, will vary under others. It is therefore to be hoped that the curves presented will prove particularly useful for design purposes, and for performance correlations.