## Isoentropic outflow through a nozzle

The formula [1] that is used in the collection “E”, as well as in other Italian [2] and foreign [3] standards, for the calculation of safety valves that must discharge gases or vapours, is that of the isoentropic outflow through a nozzle under critical jump conditions, which for an ideal gas is:

where the expansion coefficient C is given by:

being k the exponent of the isoentropic expansion equation:

#### Table 1

Fluid |
P1 (bar) |
T1 (°C) |
q’ (kg/h) |
q (kg/h) |
(q’/q) x 100 |

Methane | 12 | 50 | 1472 | 1466 | 100,4 |

Methane | 23 | 200 | 2314 | 2267 | 102,1 |

Propane | 12 | 100 | 2261 | 2181 | 103,7 |

Hexane | 12 | 178 | 3099 | 2740 | 113,1 |

Hexane | 23 | 220 | 6519 | 5111 | 127,5 |

Heptane | 12 | 215 | 3232 | 2821 | 114,4 |

q’= flow rate calculated with *k = Cp/Cv* (20 °C, 1 atm)

q = flow rate calculated with *k = (Cp/Cv) • (Z/Zp)*

By introducing the experimental coefficient k of safety valve outflow, which globally considers the real outflow performance of the valve, a safety coefficient of 0.9 and the compressibility factor Z_{1} for the real fluid, we arrive at the formulation of the collection “E”:

*The isoentropic exponent k can be expressed as:*

For an **ideal gas**, for which** P x V / R x T =1** , it is demonstrated that k is equal to the ratio Cp/Cv between the specific heats at constant pressure and volume.

For a **real gas**, k can be expressed (see Appendix B) by:

where Z is the compressibility factor defined by Z=**P x V / R x T** and Zp is the “derived compressibility factor”. When applying formula [3], according to collection “E”, the values of Cp/Cv, Z and Zp must be evaluated at discharge conditions P_{1} and T_{1}.

The derived compressibility factor Zp is defined in formula [4] as:

The compressibility factor Z can be expressed as:

and similarly, can be expressed as:

where the values of Z^0, Z^1, Zp^0, Zp^1 are tabulated in Appendix A as a function of Pr and Tr.

In [4] and [5], Ω is Pitzer’s acentric factor defined by:

Where is the reduced vapour pressure corresponding to a reduced temperature value Tr=T/Tc=0,7. Appendix A shows the Ω values of some fluids. Z e Zp can also be derived directly from an analytical equation of state.