**Experiments carried out by RG Cunningham and published by ASME in July 1951 clearly demonstrated that the assumption of a fixed limit to critical flow through thin square edged orifice plates is not correct.**

**The flow continued to increase as P**_{2}** was reduced below the expected critical condition. Limiting flow was not evident even with P**_{2}** as low as 0.1 x P**_{1}**.**

**Cunninghamâ€™s work included tests with air and steam with the results and conclusions presented as tables, charts and formulas. Limited information is provided for the tests with steam.**

**The results demonstrated that with suitable corrections to the Expansion Factor Y, the formula for non-critical flow should be used in all cases for thin square edge orifice plates. Critical flow can, however, be expected for thick orifice plates with t ? 6 x the orifice diameter.**

**Cunninghamâ€™s paper also includes an equation for the Flow Coefficient, though this appears to provide only a rough approximation and other methods may be preferable (e.g. AGA 3):**

Eq. (7) |

**The ASME formula for Y was shown to be appropriate only down to P _{2} = 0.63 x P_{1;} (not the normally expected 0.528 from thermodynamic analysis) at which point there is a distinct discontinuity in the flow to lower discharge pressures. Continued use of the ASME formula for Y produces errors of up to 12% if used for lower discharge pressures. Alternative methods reviewed involved errors of up to 40%.**

**Analysis of the Cunningham data suggests the following formula may be used to determine Y for flange taps at discharge pressures below 0.63 x P _{1}:**

Eq. (8) |

**where Y _{0.63} is Y from the ASME formula at P_{2} = 0.63 x P_{1}.**

**The use of a formula similar to the form of the ASME equation is based on an expectation that there is a reasonable probability that the flow to lower pressures will be similarly sensitive to the same geometric and process parameters. The use of ? (beta) to the 4th power provides a reasonable fit to the experimental data. Since the relationship between Y and the pressure ratio is linear the (0.63-P _{2}/P_{1}) component is clearly appropriate. The inclusion of k as a direct divisor in the equation is less obvious and difficult to confirm from the limited data available.**

**The chart first chart below is extracted directly from the Cunningham report and clearly shows the discontinuity at a pressure ratio of 0.63 and the potential for error if the ASME formula is used beyond this point. The chart second chart below is generated using the proposed method.**

Figure 1: Results with Cunningham Method | Figure 2: Results with Proposed Method |