Group interaction parameters of the PPR78 model:
1. Introduction
Carbon dioxide is an extremely important product of the chemical, pharmaceutical and petrochemical industries. Its main applications are production of coal liquids, petroleum processes such as enhanced oil recovery and separation and supercritical fluid extraction. CO_{2} is however a greenhouse gas that affects the Earth's temperature and many efforts are devoted to the reduction of CO_{2} emissions. The design and operation of many processes dealing with the CO_{2} capture and storage (CCS) greatly depend on knowledge about pressure-volume-temperature-composition (PVTxy) and mixing properties (
Carbon dioxide captured from an energy-conversion process always contains impurities that impact the design and operation of CCS systems. The type and amount of the impurities contained into the CO_{2} depend on the fuels used and the type of capture technology. A non-exhaustive list of these impurities is: CH_{4}, H_{2}S, N_{2}, O_{2}, CO, H_{2}, COS, Ar, SO_{x}, NO_{x}, NH_{3}, SO_{2}, amines and H_{2}O.
Because of the various natures of the impurities encountered in CCS processes and because these processes cover a large range of operating conditions (from atmospheric pressure to supercritical states), it is often necessary to fully guesstimate the phase equilibrium of mixtures CO_{2} + impurities in both the sub-critical and critical regions by using an appropriate thermodynamic model (this is detailed in section 2.1). In order to meet this objective, Jaubert and Privat developed a group-contribution method (GCM) allowing estimation of the temperature-dependent binary interaction parameters
In this chapter, the ability of the PPR78 model for prediction of CO_{2} + impurity binary mixtures is discussed and analyzed in terms of phase-equilibrium properties and enthalpies of mixing. The PPR78 model is also applied to the prediction of multicomponent CO_{2}-containing mixtures.
In the first part, the theoretical foundations of the PPR78 equation of state are carefully described. In the second part, the capabilities of the PPR78 model to predict both the global phase behavior (i.e., the phase behavior in both the sub and supercritical area) and the enthalpy data of various binary systems involved in CCS processes are graphically shown and quantitatively assessed.
It is often written in the literature that the greater the number of compounds in a mixture, the more a thermodynamic model may fail in representing its behavior. In that regard, the last part of the chapter deals with multicomponent systems (containing more than two pure species) encountered in CCS processes.
2. Presentation of the PPR78 thermodynamic model
2.1. On the need for an accurate thermodynamic model
Today, the synthesis design and optimization of CCS processes is carried out with the help of process simulators such as PRO/II, ProSim, HYSIS, ASPEN, UniSim, etc. It is however well known that the accuracy of the simulated results mainly depends on the quality of the chosen thermodynamic model. In most cases, the phase behavior of multicomponent systems, for which nearly no data are available, has to be known. The phase behavior can obviously be measured, but measurements are very time consuming (the VLE measurement of a 10-component system at atmospheric pressure in 10 mole% steps - a total of 92, 378 data points - would require 37 years!).
When dealing with fluids involved in CCS processes, many difficulties appear. Indeed, such mixtures contain a large number of various compounds, the proper representation of which involves accurately quantifying the interactions between each pair of molecules. This quantification obviously becomes increasingly difficult, if not impossible, as the number of molecules grows. To avoid such a time-consuming effort, an alternative solution lies in using a thermodynamic model able to estimate the interactions from mere knowledge of the structure of the molecules contained in the blend.
2.2. The Equation of State (EoS)
A few decades ago, Peng and Robinson [1] published their well-known equation of state, called in this chapter PR76. Some years later, the same authors published an improved version of this equation [2], which yields more accurate vapor pressure predictions for heavy hydrocarbons than those obtained by using PR76. This improved equation is called PR78 in this paper. For a pure component, the PR78 EoS is:
with:
where P is the pressure, R the gas constant, T the temperature, a and b are EoS parameters, v the molar volume, T_{c} the critical temperature, P_{c} the critical pressure and
A widely employed way to extend the cubic EoS to mixtures, the mole fractions of which are
where
Through this, two new parameters, the so-called
Note that the chosen mixing rules are used by most petroleum companies and above all are available in any computational package.
In our approach and in order to define a predictive model, the binary interaction parameters appearing in the mixing rules are calculated by the group-contribution method (GCM), which means that a
In Equation (5),
In this setup,
In order to deal with mixtures containing CO_{2}, light alkanes, N_{2}, H_{2}, H_{2}S and H_{2}O, a set of
the methane group (CH_{4})
the ethane group (C_{2}H_{6})
the carbon dioxide group (CO_{2})
the nitrogen group (N_{2})
the hydrogen sulfide group (H_{2}S)
the hydrogen group (H_{2})
the water group (H_{2}O)
Since the carbon dioxide and the impurities are all individually considered as single groups, Equation (5) can be simplified as follows:
where
For these 7 groups, a total of 42 parameters (expressed in MPa) were determined. They are summarized in Table 1 [3, 6-9, 13, 15].
CH_{4} | - | - | - | - | - | - | |
C_{2}H_{6} | - | - | - | - | - | ||
CO_{2} | - | - | - | - | |||
N_{2} | - | - | - | ||||
H_{2}S | - | - | |||||
H_{2}O | - | ||||||
H_{2} |
Note that this formulation is very useful for someone having commercial software working with the PR EoS. As shown in one of our previous studies [4], working with hydrocarbon binary mixtures, Equations 4 and 5 are able to predict the different
2.3. On the importance of enthalpy-of-mixing (h ^{M}) data
Engineers use principles drawn from thermodynamics to analyze and design industrial processes. The application of the first principle (also named energy rate balance) to an open multi-component system at steady state is written as:
where
where
From Equations (7) and (8), it thus clearly appears that good estimations of
3. Predicting the phase equilibrium behavior of systems containing CO_{2} + impurities
The results obtained by using the PPR78 EoS to reproduce the phase behavior of such systems are graphically shown in this section. In order to numerically evaluate the performances of the model, absolute and relative deviations between experimental data points and their prediction are systematically provided. We now define the different quantities we will refer to hereafter:
the absolute mean deviation on the liquid phase composition:
the relative mean deviation on the liquid phase composition:
with
the absolute mean deviation on the gas phase composition:
the relative mean deviation on the gas phase composition:
with
3.1. Phase diagrams of CO_{2} + light alkane systems
Mixtures of CO_{2} + methane have been measured extensively and a huge amount of reliable experimental phase equilibrium and critical data are available in the open literature. To our best knowledge, 424 bubble points, 418 dew points and 17 critical points have been published in the literature. An comprehensive list of references can be found in [6].
Figure 1a shows the isothermal phase diagrams for the system methane(1) + CO_{2}(2) at six different temperatures. For this system, from low to high temperature the binary interaction parameter varies from 0.093 to 0.112.
Figure 1b is a projection of the
Mixtures of CO_{2} + ethane have also been measured extensively. The open literature even offers a few more data than for the CO_{2} + methane system: 483 bubble points, 438 dew points and 22 critical points. For more details about the references, see reference [6].
Figure 2a, 2b and 2c show the isothermal phase diagrams for the system CO_{2}(1) + ethane(2) at twelve different temperatures. For this system, a homogeneous positive azeotrope always exists. The PPR78 model is able to perfectly predict the phase behavior of this system including the temperature minimum on the critical locus (see Figure 2d).
The deviations observed in both these systems (i.e., CO_{2} + methane and CO_{2} + ethane) are:
n_{bubble} | n_{dew} | |||||
CH_{4}(1) + CO_{2}(2) | 0.012 | 4.87 | 0.008 | 3.13 | 424 | 418 |
C_{2}H_{6}(1) + CO_{2}(2) | 0.020 | 6.58 | 0.015 | 4.16 | 483 | 438 |
3.2. Phase diagrams of CO_{2} + N_{2} system
Mixtures of nitrogen + carbon dioxide have been measured extensively and there is a vast amount of reliable experimental phase equilibrium and critical data (no fewer than 635 VLE data points were found for this system, including the critical points). An accurate list of references can be found in references [7, 8]. This system is so asymmetrical in terms of size and interactions that its critical locus is no longer continuous (see Figure 3e). Such behavior is called Type III according to the classification of Van Konynenburg and Scott [16]. It can be seen that the PPR78 model is able to predict the experimental data with relatively good accuracy. A few examples are shown Figure 3.
Note that in the vicinity of the critical temperature of pure carbon dioxide, the critical locus is perfectly predicted. Similarly to what was observed in the CO_{2} + methane system, the overestimation of the mixture’s critical pressure by the PPR78 model increases when the temperature decreases. In the present case, the binary interaction parameter
The deviations observed in the N_{2} + CO_{2} system are:
(absolute) |
(relative) |
(absolute) |
(relative) |
n_{bubble} | n_{dew} | |
N_{2}(1) + CO_{2}(2) | 0.008 | 6.03 | 0.008 | 2.32 | 320 | 301 |
3.3. Phase diagrams of CO_{2} + H_{2}S system
For this system, 177 bubble points, 176 dew points and 10 critical points were found in the open literature. These references are reported in [9]. Eight predicted isobaric dew and bubble curves are shown in Figure 4a and 4b. As can be seen, the PPR78 model is able to predict the behavior of this system with relatively good accuracy. However, as shown by the critical curve displayed in Figure 4c and in accordance with what we previously observed, the PPR78 model tends to overestimate the critical pressures although the absolute deviations remain quite small (< 5 bar).
The deviations observed in this system are:
n_{bubble} | n_{dew} | |||||
CO_{2}(1) + H_{2}S(2) | 0.009 | 5.13 | 0.008 | 3.06 | 177 | 176 |
3.4. Phase diagrams of CO_{2} + H_{2} system
Mixtures of H_{2}(1) + CO_{2}(2) present many reliable experimental data (302 bubble points, 300 dew points and 11 critical points). For further details, see reference [13]. As shown in Figure 5a, a type III phase behavior is observed for this system. In the vicinity of the critical point of the least volatile component (CO_{2}), the critical locus is perfectly predicted (for T > 250 K), as well as the corresponding isothermal P–xy phase diagrams (see Figures 5b and 5c). Once again, critical pressures are systematically overestimated, as highlighted by the isotherms in Figure 5c. Generally, liquid-liquid equilibrium (LLE) and vapor-liquid equilibrium (VLE) data associated with these two binary systems are fairly well predicted over wide ranges of temperature and pressure.
The deviations observed in this system are:
n_{bubble} | n_{dew} | |||||
H_{2}(1) + CO_{2}(2) | 0.018 | 12.8 | 0.018 | 7.11 | 302 | 300 |
3.5. Phase diagrams of CO_{2} + H_{2}O system
3.5.1. On the difficulty to model aqueous systems
Binary aqueous systems containing hydrocarbons or permanent gases all exhibit vapor–liquid, liquid–liquid and liquid–liquid–vapor equilibria. Here we want to graphically illustrate the well–known dilemma that at a given temperature, the
At this temperature, both pure components are subcritical and a three–phase line (locus of liquid-liquid-vapor equilibrium) is found since the temperature of the three–phase critical endpoint, noted CEP (which is the terminating point of the three-phase line) is about 308 K. A negative
3.5.2. What the PPR78 model can do for such complex systems
The behavior of the CO_{2} + H_{2}O system has been studied experimentally by different authors (as reported by Qian et al. [15]) in the high–temperature/high–pressure region with value of 539 K for the minimum temperature of the critical locus beginning at the water critical point. As shown in Figure 7 f, reliable predictions of the critical coordinates by the PPR78 EoS are obtained until 4000 bar. As previously discussed, at T = 298.15 K (see Figure 7a), a
The deviations observed in this system are:
n_{bubble} | n_{dew} | |||||
CO_{2}(1) + H_{2}O(2) | 0.006 | 12.9 | 0.027 | 15.8 | 1068 | 543 |
4. Predicting enthalpies of mixing of CO_{2} + impurities systems with the PPR78 model
When molecules are few-polar and few-associating (e.g., in mixtures of alkanes), pure-component terms provide an excellent estimation of the molar enthalpy of the mixture. Therefore, the enthalpy-of-mixing terms can be seen as a correction, just aimed at improving the first estimation given by the pure-component ground terms. In other words, with few-polar and few-associating mixtures,
When the parameters involved in the
However, since
In order to assess how errors in
where
For all the data, it was possible to find for the considered systems (references are reported in reference [17]), that the deviations are:
Number of data n_{hM data} |
||
CO_{2} + methane | 0.63 | 636 |
CO_{2} + ethane | 1.35 | 408 |
CO_{2} + N_{2} | 1.01 | 693 |
CO_{2} + H_{2}O | 6.88 | 539 |
Satisfactory results are thus obtained for the first three systems. This leads us to claim that accurate energy balances can be performed with the PPR78 model in processes involving mixtures of carbon dioxide, nitrogen and light alkanes as well.
The CO_{2} + H_{2}O system shows a much more important deviation of around 7 K, which may certainly introduce inaccuracy in an energy balance. However, in mixtures mainly containing CO_{2} and small proportions of water, errors in the estimation of
5. Predicting the phase behavior of CO_{2}–containing multicomponent systems
Many phase-equilibrium data have been published on multicomponent CO_{2}-containing mixtures. These mixtures are generally synthetic natural gases containing not only the few molecules studied in this chapter but also light alkanes (generally from methane to hexane). The capability of the PPR78 model is now illustrated in such mixtures. All the references of the experimental datapoints can be found in [12].
Yarborough et al.'s fluids.
In 1970, working with 45 different natural gases containing between 2 and 7 components (N_{2}, CO_{2}, methane, ethane, propane, n-butane, n-pentane), Yarborough et al. measured 52 bubble and dew-point pressures (the compositions of Yarborough et al.'s fluids are given in [12]). The PPR78 model is able to predict these 52 pressures with an absolute average deviation of 2.0 bar (i.e., 3.2%) which is certainly close to the experimental uncertainty.
Jarne et al.'s fluids.
In 2004, Jarne et al. measured 110 upper and lower dew-point pressures for two natural gases containing nitrogen, carbon dioxide and alkanes up to n-C_{6}. The composition of the fluids and the accuracy of the PPR78 model can be seen in Figure 9. The average deviation of these 110 pressures is only 2.0 bar. This is an extremely good result because many data points are located in the vicinity of the cricondentherm, where the slope of the dew curve is very steep.
Zhou et al.'s fluid.
In 2006, Zhou et al. measured 6 dew point-pressures for a natural gas containing N_{2}, CO_{2} and 7 alkanes. Figure 10 shows that with an average deviation lower than 1.2 bar (i.e., 1.5%), the PPR78 model is able to accurately predict these data.
6. Conclusion
This chapter has demonstrated that the PPR78 model is capable of predicting the phase equilibrium behavior of mixtures containing carbon dioxide, light alkanes, nitrogen, hydrogen sulfide, hydrogen and water with good accuracy. This model can also be used to perform energy balance calculations and can be successfully extended to multicomponent mixtures.
Although this chapter shows phase-equilibrium predictions obtained with the PPR78 EoS over the whole composition range (i.e., from CO_{2}-rich to CO_{2}-poor mixtures), CCS processes essentially involve mixtures containing high proportion of CO_{2} and small fractions of impurities.
Because PPR78's authors did not want to limit the range of applicability of their model, the A and B parameters involved in the
We believe that the PPR78 model can be safely used to model CCS processes. Whereas interactions between CO_{2} and impurities were thoroughly described in this chapter, interactions between impurities themselves were not (and are still present in multicomponent mixtures). A proof that a similar quality of predictions can actually be obtained is given in some of the papers mentioned in the bibliographic section [3, 6, 7-9, 13, 15, 17].
Finally, as a limitation on the use of the PPR78 model, some molecules such as NH_{3}, SO_{2} or NOx cannot be presently described by the model. The PPR78 model is currently still under development by its authors and the missing groups should be added in the near future.
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