PTQ Q4 2023 Issue

1.00

1200

Eq. 11 with B&BZ, Fixed Cv

0.90

Simulator SRK Simulator PR Vdp method Eq.12

1000

0.80

800

0.70

600

0.60

400

0.50

200

0.40

200

400

600

100

200

300

400

500

Time, seconds

PR-Z

B&G-Z PR-Cp

SRK-Cp PR-Cv

SRK-Cv

Figure 7 Comparison of non-adiabatic pressure profiles

SRK-Z

Figure 6 Transient non-adiabatic Z and C p , C v from simulator

value generated from the simulator. Based on the C p value from the simulator, C p in Equation 12 can be approximated close to ZC p = C v +(R/Z) and also assumed constant in the spreadsheet. Good estimates of Z s as a function of pressure and tem- perature, which also characterises the deviation of real gas from ideal gas, are essential for the spreadsheet model for predicting the depressurisation pressure profiles. Beggs and Brill (B&B) method 3 is used for calculating Z s based on reduced pressures and temperatures. Figure 6 displays the Z s from the Beggs and Brill method and shows the results in fairly good agreement with those from PR or SRK. Figures 7 and 8 show the non-adiabatic transient P and T profiles calculated from Equations 11 and 12 using spreadsheet and those from the simulator using PR and SRK. The same depressuring system with a fixed heat transfer rate is used for all the profiles. In addition to assuming a constant C v for calculating the T profile (Figure 8 red solid line) from Equation 11, m D v 2 / 2g c term is also assumed negligible and not accounted for. For systems with high diameter ratios of the depressurised segment to the outlet flow nozzle, m D V²/2g c may no longer be negligible. Figure 7 shows fairly good agreement among all of the non-adiabatic P profiles. As shown in Figure 8, T profiles from Equation 11 or 12 noticeably deviate from those gen- erated by the PR or SRK package of the simulator. In addi - tion to differences in Z s, the deviation is expectedly due to the constant C v assumed in Equation 11 and constant C p in Equation 12. Realistically, as shown in Figure 6, C v and C p values generated by the PR and SRK packages of the sim- ulator vary with P and T and increase significantly, start - ing at 400 seconds of the depressuring time. This appears consistent with the T profile from Equation 12, showing sharper or faster temperature increases starting at 400 seconds relative to the profiles from PR or SRK. As such, for the LNG feed gas examples, Equations 11 and 12, with the simplifying assumptions, result in non-adiabatic P and T profiles consistent with those estimated by the simulator and can be expanded to use P and T dependent C v and C p values which likely improve agreement.

estimated by transient energy balance (see Equation 11 ) or alternately Equation 12 derived from the transient enthalpy balance of the segment:

(Eq 11)

(Eq 12)

Where: u – internal energy, Btu/lb q – net heat input, Btu/sec h – enthalpy, Btu/lb P o – back pressure (psia) at outlet v – gas exit velocity, ft/sec g c – dimensional constant, lb ft/sec 2 lb/F C p – heat capacity at constant pressure BTU/lb/F C v – heat capacity at constant volume, BTU/lb/F

As dM/dt=m, h =u+P u, and d u/ d t = C v (T i+1 – T i ), T i+1 can be solved from Equation 11. Transient data for C v (or C p ) are mainly required for solving the transient temperature profile and can be directly generated from the simulator by selecting an appropriate package. For cases where a simu- lator may not be available or needs to be verified, alternative methods for estimating C v or C p can be useful. For real gases, however, C p and C v vary with compositions, T and P, and the available equations/correlation for estimating C p or C v from T s and P s are generally complex or only applicable to cer- tain compositions. Correlations for C p or C v of a specific gas composition at a range of temperature and pressure can be developed time-effectively using an equation of state. Primarily for simplification purposes, an estimate of C v value in Equation 11 is used in the spreadsheet and assumed unchanged during the depressurisation. C v , which depends on the type of gas molecules and the degrees of freedom in the molecular motion, is roughly close to 7R/2 for the LNG feed gas example, and the C v value estimated from 7R/2 is reasonably close to initial C v (at time = 0 second)

106

PTQ Q4 2023