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Commercial Program Model
Commercial Program 93 %PV WT Model
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Time, seconds
Time, seconds
Figure 2 Pressure vs time
Figure 3 Flow rates vs time
– R), the ratio of heat capacity at constant pressure to heat capacity at constant volume. To evaluate the value of n for real gases, the actual P and u data along the isentropic path are needed. They are available from several sources, includ - ing published data or estimated from the selected applicable equation of state or other property packages. For the option to generate isentropic P and u data from a known or commercially proven simulator, a dry LNG feed gas at initial conditions of 1,000 psig and 80°F is used as an example for illustrating the calculation models for depres - surisation. This feed gas is primarily methane, and other components in small percentages are hydrocarbons from ethane to iso-pentane (C₂ to i-C₅), nitrogen, and traces of CO₂. Figure 1 shows the isentropic P and u data generated from a commercial simulator using Peng-Robinson (PR) property package. As shown, the P and u data along the isentropic path can be well correlated with Equation 4 , and the resulting R-squared (R2) or coefficient of determination essentially equals to one, indicating a good fit, at least for this LNG feed gas example.
100 psig, and the flare backpressure during the depressur - isation is typically much lower than 100 psig. Higher flare backpressures generally could occur during major or multi - ple relief scenarios. However, the associated backpressures typically stay below 100 psig, and activating the depres - surising system in these relief events may be considered less likely. For cases where the depressurising orifice back pressure is well below the choked or critical flow pressure, m can be estimated by assuming choked flow across a resistance coefficient/factor. The depressurisation feature in the com - mercial simulator provides several options for flow rate calculation, and using a flow coefficient CV for a valve is one of these options. For consistent comparison, the same CV equation ( Equation 2 ) in the simulator is used in the spreadsheet models for calculating the depressurised rates in the choked flow region.
(Eq 2)
Equation 2 provided by a control valve vendor contains the control valve coefficient (C v ), P pressure (PSIA), G – specific gravity relative to air, T – temperature (°R), and Z – compressibility factor. K is the product of a unit conversion constant and a valve critical flow factor from the vendor. The same type of equa - tion is also used in the commercial simulator to get the results for comparison purposes. For adiabatic depressurisation, the gas in the segment can be modelled as essentially reversible, isentropic or polytropic expansion. For real gases, pressure and specific volume ( u, ft³/lb) are often related by Equation 3 ¹, especially for some limited pressure ranges (1,000-100 psig in this example), and the gas pressures ( P ) and the specific vol - umes ( u, ft3/lb) can be generally expressed as Equation 3.
(Eq 4)
P u 1.32498 = 202.18
With Equation 4 showing correlation of P (t) with u (t) and assuming equilibrium or average bulk u in the isolated volume ( V ) being depressurised, M | t+ D t can be calculated from Equation 1 with m from Equation 2. M | t+ D t corresponds to u| t+ D t value, which equals to V / M | t+ D t , and P| t+ D t can be calculated from Equation 4. As such, the time-dependent pressure of the depressurised segment can be calculated using Equations 1 to 4 in a spreadsheet. Note that Z and T data are both required to calculate m (t). With the calculated specific volume u i and P i at time t i (i = time increment number) calculated from V / M | i+1 , the values of Z i+1 and T i+1 at t i+1 should result in the same value of u i+1 . Correlations to estimate Z are algebraically less straightfor - ward and require estimates of other parameters, such as critical pressure, critical temperature, and eccentric factor. For isentropic paths where P u n is constant, T i+1 can be solved by iteration to reach a consistent Zi+1 value from Equations 5 and 6 , where R is gas constant. With an assumed T i+1 , Z i+1
(Eq 3)
n is the isentropic expansion coefficient or reversible polytropic exponent (RPE) derived from actual P and u data of real gases. For ideal gas, n equals to C p / C v or C p / (C p
104
PTQ Q4 2023
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