# Selecting the Proper Gas Compressibility Z for Relief Valve Sizing

- Type: Conference Presentation
- Conference Type:
AIChE Spring Meeting and Global Congress on Process Safety
- Presentation Date:
April 23, 2018
- Duration:
30 minutes
- PDHs:
0.50

**Selecting the Proper Gas
Compressibility Z for Relief Valve Sizing**

Freeman Self (feself@bechtel.com)

Satyajit Verma (sverma4@bechtel.com)

Bechtel Oil, Gas and Chemicals, Inc.

Prepared for

presentation at the

American

Institute of Chemical Engineers Spring Meeting

14^{th}

Global Congress on Process Safety

Orlando, Florida

April 2018

*Process Safety Spotlights(T1F)** - Chair and Contact: Wayne Chastain*

*- T1F00 Pressure
Relief Design - Oral Session -*

**Abstract**

**Introduction**

Gas flow sizing

of relief valves is commonly performed using the analytical gas formula for

choked flow. The gas compressibility is

an important parameter. The proper

selection of the value of Z will avoid possible over-sizing or under-sizing a

relief valve.

This paper

will explain when one should use the ideal gas compressibility Z (equal to 1.0)

or when to use the actual Z (at temperature and pressure) when sizing a relief

valve using the analytical gas choked flow formula.

**Overview **

Sizing of

relief valves in gas flow is commonly performed using the analytical gas

formula for choked flow. The common form

used in vendor catalogues and API Standard 520-I “Sizing, Selection and

Installation of Pressure-Relieving Devices” is:

W = A Kd C P Ö

( M / T Z ) (1)

Rearranging

using the formula for density results in:

W = A Kd C’ Ö

( P r )

(2)

W

is mass flow [lb/hr], A is orifice area [in2], Kd is discharge coefficient, P

is inlet pressure [psia], M is molecular weight [lb/mole], T is Rankin

temperature [°R], C is coefficient

containing only the parameter K ideal gas heat capacity ratio (K = Cp / (Cp –

R)), C’ is the same coefficient with revised conversion factors, r

is density [lb/cf] and Z is the gas compressibility

Although deceptively simple, the

assumptions inherent in the derivation may result in drastically oversized or

undersized relief valves. Yet, there

has been little elucidation of the issues.

The results may be surprising since there are no clear resolution for

certain situations.

To explain the problem, it is necessary

to understand two of the major assumptions utilized in the derivation of the

choked flow equation. (The derivation is

quite complex but may be found in advanced transport books.)

· Gas is ideal, which

means the compressibility Z is equal to 1.0 and the ratio of heat capacities is

constant

· The density (as a

function of pressure) is determined using a specific correlation called the

“isentropic expansion expression”. This expression states that the density at

the throat pressure, as the backpressure on the relief valve is reduced, may be

calculated from the previous pressure and density, ideal gas heat capacity ratio

K and utilizing a constant entropy process.

(The subscripts 1 and 2 are used for any two pressures along an

isentropic path.)

P / r^{K} = constant (3) or stated equivalently:

P_{1} / r_{1}^{K} = P_{2} / r_{2}^{K} (4) where K is the ideal gas heat capacity ratio

K = Cp / (Cp – R)

**The
Conundrum of Density**

The analytical gas formula for choked

flow - equation (2) - shows that the flow rate at choked condition through the

relief valve is a function of density. When

sizing the relief valve using the typical equation (1), the density at the

initial pressure is established by the compressibility. If the gas is close to

ideal, the ideal and actual densities will be comparable in value.

The densities at other pressures are

implicitly represented by the “isentropic expansion expression” equation (3 or

4) which is used in the derivation of equation (1 or 2). The density pressure relationship can be

demonstrated by calculating the densities utilizing the “isentropic expansion

expression” equation (3 or 4). In general, they are almost a linear function of

pressure since K is close to the value of one.

In contrast, the actual densities frequently do not usually exhibit a

linear relationship with pressure. Consequently

the densities from the “isentropic expansion expression” may not track the

actual densities.

Therefore accuracy of the choked gas flow

rate calculation depends both on:

· the initial value of

the compressibility,

· how well the actual densities

track the densities represented by the “isentropic expansion expression”.

Two examples will illustrate the

differences in results, and how they are affected by the densities. The graph plots the fluid’s density and pressure

at the relief valve orifice as the backpressure on the relief valve is reduced

from the initial pressure to a lower pressure. Three densities are plotted: 1. the actual

density (from a process simulator), 2. the density calculated from the

“isentropic expansion expression” equation (3) with the initial ideal gas

density (Z=1.0), and 3. the density calculated from the “isentropic expansion

expression” with the initial actual density.

· Example 1 provides an

illustration of the situation where the ideal gas compressibility provides an

acceptable sized relief valve. Although the initial ideal gas density is higher

than that calculated with the actual Z, the ideal gas density curve coincides

better with the actual densities.

· Example 2 illustrates

the opposite situation where the actual gas compressibility provides an

acceptable sized relief valve. Using the

ideal gas compressibility produces higher flowrates and an oversized relief

valve.

Additional examples will be provided in

the presentation that will provide guidance on proper selection of the gas

compressibility factor.

**Conclusions**

The homogenous direct numerical

integration method (HDI) will provide the best answer in all cases. (The HDI method is provided in API Standard

520-I “Sizing, Selection and Installation of Pressure-relieving Devices”.) However analytical gas equation is commonly

employed to size relief valves in gas service.

The proper selection of the value of Z when using the analytical

equation will avoid possible over-sizing or under-sizing a relief valve in gas

service.

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