Modeling and Analysis
An extended Peng-Robinson
equation of state for carbon dioxide
solid-vapor equilibrium
Sergey Martynov, Solomon Brown and Haroun Mahgerefteh, University College London, UK
Abstract: The Peng-Robinson equation of state (PR EoS) for liquid-vapor equilibrium is extended to
model the solid-vapor (sublimation) and solid-liquid (melting) phase equilibria for carbon dioxide (CO2).
The sublimation behavior is modeled through the re-formulations of the empirically based analytical
expressions for the two temperature dependent parameters, a and b in the PR EoS. The melting phase
behavior on the other hand is modeled by the coupling of the original and the extended PR EoS and
equalization of solid and liquid phase fugacities. Analytical expressions derived based on the extended
PR EoS are used to determine thermodynamic and phase equilibrium derivative properties for solid/
vapor CO2. These include internal energy, enthalpy, heat capacity, thermal expansion, and isothermal
compressibility coefficients as well as the adiabatic speed of sound. In most cases good agreement
with the available experimental data is obtained covering the pressure and temperature ranges 0.1–
100 MPa and 100–300 K. A pressure/temperature phase equilibrium diagram for solid-liquid-vapor
CO2 is constructed to demonstrate the overall performance and the limitations of the two EoS as
compared to the experimental data spanning the triple point up to 100 MPa pressure. It is shown that
the application of the PR EoS along the CO2 sublimation line gives rise to significant errors.
© 2013 Society of Chemical Industry and John Wiley & Sons, Ltd
Keywords: equation of state; solid-fluid equilibria; carbon dioxide; CO2 transportation; pipeline safety
Introduction
s part of the carbon capture and sequestration
(CCS) chain, pressurized pipelines are considered as the most practical and efficient means
for transportation of the large amounts of CO2
captured from fossil fuel power plants for subsequent
sequestration.1 Given that CO2 gas is an asphyxiant at
concentrations higher than 7%,1 the safety of CO2
pipelines is widely considered to be of paramount
importance and indeed pivotal to the public acceptability of CCS as a viable means for reducing the
impact of global warming.2
A
Central to the hazard assessment of such pipelines
are the predictions of the transient discharge rate and
atmospheric dispersion of the escaping inventory in
the event of pipeline failure using reliable validated
mathematical models. This data governs all the
consequences associated with the pipeline failure,
including the minimum safe distances to populated
areas and emergency response planning.
A key feature governing the efficacy of the pipeline
outflow and dispersion models is the use of an appropriate equation of state (EoS) for the accurate determination of the thermo-physical and phase equilibrium properties of the escaping CO2. Given that a
Correspondence to: Haroun Mahgerefteh, Department of Chemical Engineering, University College London, London WC1E 7JE, UK.
E-mail: h.mahgerefteh@ucl.ac.uk
Received September 21, 2012; revised October 27, 2012; accepted October 29, 2012
Published online at Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/ghg.1322
136
© 2013 Society of Chemical Industry and John Wiley & Sons, Ltd | Greenhouse Gas Sci Technol. 3:136–147 (2013); DOI: 10.1002/ghg
Modeling and Analysis: An extended Peng-Robinson equation of state for CO2 solid-vapour equilibrium
robust pipeline outflow model invariably requires
time consuming (up to a few hours on a modern PC)
iterative numerical solution of the conservation
equations,3-6 the computational efficiency of the EoS
employed is an essential and critically important
additional requirement.
The majority of EoS thus far developed are capable
of handling two-phase liquid-vapor mixtures. Although this suits most hydrocarbon mixtures, in the
case of CO2, its unusually high Joule-Thomson
expansion coefficient may result in pipeline release
temperatures as low as –70 oC.7 Given that the triple
point for CO2 is –56.6 oC at 0.518 MPa,8 solid CO2,
also known as dry ice, is expected to form during
pipeline discharge.9
The failure to model solid CO2 formation poses a
number of technical and safety issues. Solid CO2
discharge and its subsequent sublimation can dramatically alter the cloud dispersion characteristics
and, as a result, the pipeline hazard profile. Also, the
instantaneous freezing of the vapor-liquid mixture
containing large volume fraction of the liquid phase
may cause clogging of the pipeline during controlled
blowdown.10-12 Finally, erosion of surrounding
structures by high-speed jets carrying solid CO2
particles is widely recognized as a serious potential
hazard, particularly during accidental rupture of CO2
pipes in close-spaced structures such as off-shore
platforms.13 Indeed high pressure CO2 solid particles
are used as a means for dislodging debris from
structures.14
Two main approaches exist to model solid, liquid,
and vapor phase equilibria. The first uses a unified
EoS for all three phases of interest. The other involves
the application of separate EoS to describe the different phases which are then jointed at the phase transition boundaries.
Following the first approach, Wenzel and Schmidt15
introduced an additional high-power attractive term
in the cubic Redlich-Kwong EoS.16 Four parameters of
this equation, two of which are temperature dependent, were fitted to the subliming solid density curve,
sublimation and melting pressures and latent heats of
melting and evaporation at the triple point. In the
case of CO2, solid-vapor equilibrium was characterized only for temperatures near the triple point (down
to 205 K), while the range of practically relevant
temperatures extends to 197.5 K (the sublimation
temperature of CO2 at atmospheric pressure).8 Also,
the authors did not apply the extended EoS to
S Martynov et al.
calculate the derivative thermodynamic properties
such as heat capacity and enthalpy all of which are
required in the modeling of the transient release and
dispersion.
Lang and Wenzel17 applied the concept of cluster
formation in the solid phase equilibrium calculations,
with the fluid phase described using a cubic EoS.
Geana and Wenzel18 used this method to compute the
solid-vapor and solid-liquid equilibria for CO2. Given
that in this approach, the chemical equilibrium
equation for the cluster formation is essentially
non-linear, its solution requires a time-consuming
iterative numerical procedure. This significantly
compromises the practical usefulness of the EoS if
implemented in pipeline rupture numerical CFD
codes.
Yokozeki19 developed an elegant extension of the
Van der Waals EoS, proposing a fourth order EoS
capable of correctly predicting the topology of the
solid, liquid, and vapor phase equilibria. The equation
containing four parameters, two of which were
temperature dependent, was applied to predict the
sublimation and melting phase behavior for pure CO2
and its mixtures with methane and water.19,20 Although the EoS was shown to describe well the solid,
liquid, and vapor phase equilibrium for pure CO2, its
performance beyond the triple point was considered
uncertain by the authors.
Other approaches for developing unified EoS for
three phases based on statistical mechanics have also
been reported.21,22 However, these have not been
validated for CO2 through comparison with experimental data. In addition, these EoS are highly computationally demanding. The same applies to the empirical unified multi-parameter EoS developed for
CO2,23,24 which have superior accuracy for prediction
vapor-liquid equilibria for pure CO2 (e.g. the data
tables25 are generated based on the Span and Wagner
EoS23), but apart from being exceptionally computationally demanding are not capable of handling the
solid phase.
Alternative approaches based on the application of
separate equations to describe the different phases
have been primarily developed using the framework
of cubic EoS.26-28 Most of these combine a classical
EoS with a general mathematical artifice for the
fugacity of the solid phase29 to predict the sublimation and melting phase equilibria. In the work
reported by Salim and Trebble27 for example, the
Trebble-Bishnoi-Salim (TBS) cubic EoS was
© 2013 Society of Chemical Industry and John Wiley & Sons, Ltd | Greenhouse Gas Sci Technol. 3:136–147 (2013); DOI: 10.1002/ghg
137
S Martynov et al.
Modeling and Analysis: An extended Peng-Robinson equation of state for CO2 solid-vapour equilibrium
employed to characterize separately the saturation
and the sublimation phase equilibria. The methodology for determining the cubic EoS parameters
describing the solid-vapor transition based on
applying four constraints at the triple point to defi ne
the triple point constants was presented. The above
was used to obtain six constants in the four-parameter TBS EoS for various pure substances, including
CO2. However, the authors did not apply their EoS
to determine CO2 derivative properties to assess its
efficacy.
Based on this review, it is reasonable to conclude
that so far most of the EoS developed in the past for
modeling of the solid phase equilibria have either
been limited to temperatures above those of interest
during rapid decompression of CO2 or not extended
for determining its pertinent derivative properties.
The majority involve iterative numerical solution
techniques which makes them unsuitable for application in CFD codes.
Cubic EoS admitting closed form analytical solutions
are most attractive for routine use in CFD codes.30
This particularly applies to the Peng-Robinson (PR)
EoS31 given its proven accuracy in modeling vaporliquid behavior of CO2,32,33 its relative simplicity and
computational efficiency. It is also one of the most
widely used and reliable EoS in the hydrocarbon
industry.34
This paper presents the application, extension and
validation of the PR EoS for predicting sublimation
and melting phase equilibria for CO2 and its derivative
properties. The sublimation phase equilibrium is
modeled by adjusting the EoS parameters using
experimental data along the solid-vapor equilibrium
line. The melting line on the other hand is predicted by
the merging of the original and the extended PR EoS
through equalization of solid and liquid fugacities.
attraction forces and the molecular volume respectively. For vapor-liquid equilibrium, the parameters, a
and b henceforth termed aLV and bLV, are defined in
the Appendix.
The extended PR EoS for solid-vapor
phase equilibrium
The application of Eqn (1) to solid (S) and vapor (V)
phases requires the expression of the parameters a
and b, henceforth termed, aSV and bSV, as a function of
temperature along the sublimation curve. To undertake this two constraints are applied along the CO2
sublimation pressure/temperature curve, psubl(T).
The first constraint simply requires that the mathematical expression for the specific volume of the
subliming solid phase, vs(psubl) correctly matches the
experimental values, vs,exp(psubl) for solid CO2:
v s ( psubl ) = v s ,exp ( psubl )
(2)
The second constraint is equal solid and vapor
fugacities at equilibrium:
f s psubl ) = f v psubl )
(3)
The fugacities, fs and fv are expressed as functions of
pressure, p and compressibility, Zi as follows:31
ln
fi
= Zi
p
l (
ln(
i
)−
⎛Z
. B⎞
ln ⎜ i
B ⎟⎠
2 2 B ⎝ Zi − 0.414B
A
(4)
where i is the phase index referring to either the solid
or the vapor phase. A and B on the other hand are
dimensionless parameters defined as:
A=
aSV p
R 2T 2
(5)
bSV p
(6)
RT
The constraints set by Eqns (2) and (3) lead to a set
of algebraic equations which can be solved for asv and
bsv at a given temperature. The procedure to obtain
these equations is described as follows.
First, it is convenient to rewrite Eqn (1) in terms of
compressibility:
B=
Theory
The PR EoS for liquid-vapor phase
equilibrium
In its most convenient form, the PR EoS31 may be
written as:
p=
RT
a
−
v b v(v + b ) b(v b )
where p, T and v are pressure, temperature and
specific volume respectively. a and b are empirical
parameters accounting for the intermolecular
138
(1)
Z 3 (1 B)Z 2 ( A 3B2 2B
B)Z ( AB B2 B3 ) 0 (7)
Expressing this equation for the solid phase gives:
Z s3 (1 B)Z s2 ( A 3B2 2B
B)Z s ( AB B2 B3 ) 0 (8)
© 2013 Society of Chemical Industry and John Wiley & Sons, Ltd | Greenhouse Gas Sci Technol. 3:136–147 (2013); DOI: 10.1002/ghg
Modeling and Analysis: An extended Peng-Robinson equation of state for CO2 solid-vapour equilibrium
Substitution of Eqn (4) for the solid and vapor
phases into Eqn (3) produces:
⎛ Z B⎞
⎛Z
. B Z v 0.414 B ⎞
A
Z s Z v − ln ⎜ s
ln ⎜ s
⋅
=0
⎟−
4 B Z v 2.414 B ⎟⎠
⎝ Z g B ⎠ 2 2 B ⎝ Z s − 0.414
(9)
In the framework of two-parameter EoS, A and B
uniquely define all three roots of Eqn (4), only two of
which have physical significance. Thus, knowing the
root Zs, analytical expressions for the other two roots
can be obtained by factorization of Eqn (7):
s
2
)((
q
where the subscript ig represents ideal gas. R, cV and cP
are the universal gas constant and the specific heat
capacities at constant volume and pressure respectively.
In Eqn (14), cP,ig is a function of temperature,
calculated for CO2 following Poling, Prausnitz, and
O’Connell:35
(
R 3.259 + 1.356 ⋅10−3 T
cP ,iig
− 2.374⋅
.374
374 10−8
3
⋅T 4
)
(15)
Also,
(10)
cP cV = −
where
p B − 1 + Zs
and
)Z s + Z s2
The largest root of the quadratic part of Eqn (10)
corresponds to the vapor compressibility given by:
q
q2
Zv = − +
−p
2
4
(11)
Equations (8), (9) and (11) can be numerically solved
for Zv, A and B at any given point along the sublimation phase curve. The parameters asv and bsv can then
be calculated using Eqns (5) and (6), respectively.
Depending on the availability of the relevant experimental data, the above procedure can be used to
determine the functions asv(T) and bsv(T) along the
sublimation curve for any pure substance.
Derivative thermo-physical properties
The equation of state (Eqn (1)) may be used to determine derivative thermodynamic properties such as
internal energy, U, enthalpy, H and isobaric heat
capacity cp. These are usually defined in terms of the
residual properties:35
⎡ ⎛ ∂pp ⎞
⎤
U U ig = ∫ ⎢T ⎜ ⎟ − p ⎥d
dv
⎝ ∂T ⎠ V
∞⎣
⎦
(12)
H H ig = U U iig + ( Z
(13)
v
) (
+ 1.056
056 ⋅10
1.502 10 5 T 2
2
p) = 0
q A − 3B 2 2 B + ( B
S Martynov et al.
)RT
)(
cP cP ,igig = ( cP cV + cV − cV ,ig − cP ,iig cV ,ig
)
(14)
⎛ ∂pp ⎞
T⎜ ⎟
⎝ ∂T ⎠ V
(16)
⎛ ∂pp ⎞
⎜⎝ ∂v ⎟⎠
T
⎛ ∂(U − U ig ) ⎞
cV cV ,ig = ⎜
⎟
∂T
⎝
⎠V
(17)
cP ,iig cV ,ig = R
(18)
Given that the parameter, b is a function of temperature in the solid and vapor phase regions, respective
revised expressions for the residual properties U – Uig,
H – Hig, cP – cP,ig and cV – cV,ig are required. These are
derived as follows. ∂p
First, to calculate
in Eq. (12), Eq. (1) for p is
∂T V
differentiated with respect to temperature at constant
volume:
∂p
R
T b′
=
1+
v −b
∂T V v − b
(19)
a
a′
2 (v b )b ′
−
+
v(v + b ) + b(v − b ) a v(v b ) b(v b )
where a′ = da/dT and b′ = db/dT.
Substitution of Eqn (19) into Eqn (12) and integrating gives the residual internal energy:
U U ig =
(a ′ T − a ) ⎛ v (
ln ⎜
2 2b
⎝v (
)b ⎞
⎟
)b ⎠
ab′T
RT 2 b ′
+
−
v(v b ) b(v b ) v b
(20)
Differentiation of Eqn (1) with respect to specific
volume, v at a constant temperature, produces:
© 2013 Society of Chemical Industry and John Wiley & Sons, Ltd | Greenhouse Gas Sci Technol. 3:136–147 (2013); DOI: 10.1002/ghg
139
S Martynov et al.
Modeling and Analysis: An extended Peng-Robinson equation of state for CO2 solid-vapour equilibrium
⎛ ∂p ⎞
2a(b + v )
RT
−
⎜⎝ ∂v ⎟⎠ =
2
⎡⎣v(v + b ) + b(v − b )⎤⎦ ( v − b
T
)
2
(21)
Substituting the above equation into Eqn (16), gives:
2
⎛ ∂pp ⎞
−T ⎜ ⎟
⎝ ∂T ⎠ V
2a(b + v )
cP cV =
⎡⎣v(v + b ) b(v b )⎦⎤
2
−
(22)
RT
(v − b )
2
∂p
is given by Eqn (19).
∂T V
Taking the temperature derivative of U – Uig given
by Eqn (20) and substituting in Eqn (17) produces the
following expression for the residual specific heat
capacity at constant volume:
where
cV cV ,ig =
⎡
b′ ⎤ ⎛ v + ( +
⎢T a ′′ + (a T a ′ ) b ⎥ ln ⎜
2 2b ⎣
⎦ ⎝v (
1
)b ⎞
⎟+
)b ⎠
⎞
(a ′ T + a)b ′+ Tab ′′ RT ⎛ T b ′ 2
−
+ 2b ′ T
Tb ′′⎟ +
⎜
v(v + b ) b(v − b ) v b ⎝ v b
⎠
(a ′ T a ) v
b′
2T ab ′ 2 (v − b )
⋅ −
v(v + b ) + b(v − b ) b ⎡v(v + b ) + b(v − b )⎤ 2
⎣
⎦
(23)
where a″ = d2a/dT2 and b″ = d2b/dT2.
The residual heat capacity at constant pressure,
cp – cp,ig may now be determined by substituting
Eqn (23) into Eqn (14).
These expressions for the partial derivatives,
(∂p/∂T)V and (∂p/∂v)T respectively given by Eqns (19)
and (21) can also be used to calculate other derivative
thermo-physical properties such the thermal expansion coefficient:
1 ⎛ ∂v ⎞
αV = ⎜ ⎟
v ⎝ ∂T ⎠ P
Figure 1. The variation of the parameter a with temperature.
Curve A: aLV defined by Eqn (A1) of the PR EoS for liquid
and vapor phases. Curve B: aSV defined by Eqn (27) of the
extended PR EoS for solid and vapor phases.
Results and discussion
Parameters aSV and bSV for the extended
PR EoS
Figures 1 and 2, respectively, show the variations of
the parameters a and b with temperature in the range
from 100 K to 300 K for CO2. The triple point temperature, Ttr (216.58 K) for CO2 is also indicated in
the figures to define the various fluid phases.
Returning to Fig. 1, Curve A and Curve B are
respectively aLV and aSV values calculated based on
the original PR EoS (Eqn (A1)) and the extended PR
EoS (Eqn (5)). Curves A and B in Fig. 2 on the other
hand respectively show the bLV and bSV values calculated based on the original PR EoS (Eqn (A2)) and the
extended PR EoS (Eqn (6)).
The sublimation pressure psubl(T) and the specific
volume of subliming solid as a function of temperature data required to determine aSV and bSV were
respectively obtained from data published by Angus
et al.8 and Din.36
(24)
isothermal compressibility coefficient:
1 ⎛ ∂v ⎞
βT = − ⎜ ⎟
v ⎝ ∂pp ⎠ T
(25)
and adiabatic speed of sound:
cS =
140
cP 1
cV ρβT
(26)
Figure 2. The variation of parameter b with temperature.
Curve A: bLV defined by Eqn (A2) of the PR EoS for liquid
and vapor phases. Curve B: bSV defined by Eqn (28) of the
extended PR EoS for solid and vapor phases.
© 2013 Society of Chemical Industry and John Wiley & Sons, Ltd | Greenhouse Gas Sci Technol. 3:136–147 (2013); DOI: 10.1002/ghg
Modeling and Analysis: An extended Peng-Robinson equation of state for CO2 solid-vapour equilibrium
As can be seen from Figs 1 and 2, both aSV and bSV
show the expected sudden change in their values at
the triple point as compared to the aLV and bLV.
Also, aSV and aLV (Fig. 1) vary almost linearly with
temperature. The observed increase in their value
albeit at different degrees as temperature decreases
may be interpreted by the increase in the molecular
attraction forces. Figure 2 shows that the molecular
exclusion diameter bSV decreases with temperature as
opposed to remaining constant in the case of bLV.
The data points representing aSV and bSV values
given in Figs 1 and 2 may be respectively approximated by the following linear and exponential
functions:
aSV (T ) = a0 (
T / Ta )
bSV (T ) = b0 ⎡⎣
cb exxp(T Tb ⎤⎦
(27)
)
(28)
where a0, b0, cb, Ta and Tb are the approximation
constants, in turn determined using a least-square
method to produce the following values:
a0 = 1143 (GPa cm6/mol2), Ta = 343.55 (K)
b0 = 24.45 (cm3/mol), cb = 0.00035, Tb = 39 (K)
In this study, the approximating functions in Eqns
(27) and (28) were chosen such that they are simple in
form, involving the minimum number of constants.
They also produce monotonous variations of aSV and
bSV with temperature beyond the ranges shown in
Figs 1 and 2.
As discussed by Trebble and Bishnoi,37 and may also
be seen directly from Eqns (14), (22), (19) and (23), the
non-linear variation of parameter bSV as a function of
temperature has a major impact on the residual heat
capacity cP – cP,ig. Accordingly, the determination of
the approximation constants in Eqns (27) and (28)
using the least-square method involved minimisation
of the fitting errors in the calculation of cP – cP,ig to
produce accurate heat capacity data for solid CO2.
S Martynov et al.
Figure 3. The variation of density of subliming solid CO2
with pressure. Curve A: Calculated from the PR EoS.
Curve B: Calculated from the extended PR EoS. Curve C:
Experimental data.38
using the PR EoS. Curve B on the other hand shows the
solid density predictions using the extended PR EoS
incorporating the derived parameters aSV (Eqn (27))
and bSV (Eqn (28)). Figure 4 shows the corresponding
data as in Fig. 3 but for subliming vapor CO2 density.
Returning to Fig. 3, it is clear that the extended PR
EoS (Curve B) produces excellent agreement with the
experimental data (Curve C) for subliming solid CO2.
The maximum density difference is 0.4%. This finite
difference is primarily due to fitting errors in the
approximating Eqns (27) and (28) for aSL and bSL
respectively. In contrast, the PR EoS produces very poor
performance, under-predicting the solid density by as
much as 25% in the pressure range under consideration.
Density predictions
Figure 3 shows the variation of the predicted subliming
solid CO2 density with pressure as compared to the
experimental data from Anwar and Carroll38 up to the
triple point pressure, ptr (0.518 MPa). For the sake of
comparison, Curve A shows the solid density data
Figure 4. The variation of density of subliming vapor CO2
with pressure. Curve A: Calculated from the PR EoS.
Curve B: Calculated from the extended PR EoS. Curve C:
Experimental data.38
© 2013 Society of Chemical Industry and John Wiley & Sons, Ltd | Greenhouse Gas Sci Technol. 3:136–147 (2013); DOI: 10.1002/ghg
141
S Martynov et al.
Modeling and Analysis: An extended Peng-Robinson equation of state for CO2 solid-vapour equilibrium
In the case of saturated vapor density however, both
EoS provide accurate predictions as compared to the
experimental data (ca. 2% for PR EoS and 3.3% for
extended PR EoS at the triple point).
Derivative thermo-physical properties
The derivative thermo-physical properties of solid
phase CO2, namely residual enthalpy (H – Hig),
residual heat capacity (cP – cP,ig), heat capacity difference (cP – cV), thermal expansion coefficient (αV) and
speed of sound (cS) are respectively defined by Eqns
(13), (14), (16), (24) and (26). The calculation of these
properties involves the solution of Eqns (17) to (23)
incorporating the first and second temperature
derivatives of aSV (Eqn (27)) and bSV (Eqn (28)) given
by the following equations:
aSV
′ = − a0 / Ta
(29)
aSV
′′ = 0
(30)
bSV
′ =−
b0cb
exp
x (T / Tb
Tb
)
(31)
bSV
′′ = −
b0cb
)
(32)
Tb2
exp
x (T / Tb
Figure 5 shows the variation of cP/R with temperature for CO2 in the temperature range 100 to 300 K.
The triple point temperature, Ttr is also shown for
reference. Curves A and B show the predicted data for
Figure 5. The variation of heat capacity of saturated liquid
and subliming solid CO2 with temperature. Curve A:
Calculated based on the PR EoS. Curve B: Calculated
based on the extended PR EoS. Curve C: Experimental
data.38 Curve D: Calculated using Eqn (33).
142
the saturated liquid and the subliming solid using the
PR and the extended PR EoS, respectively. For the
sake of comparison, the data predicted using the PR
EoS (Curve A) is also presented along the sublimation
line in order to demonstrate the equation’s applicability in this region. Curve C shows the corresponding
saturated liquid experimental data from Anwar and
Carroll.38 Curve D on the other hand for the subliming solid is generated using the empirical formula
recommended by the Design Institute for Physical
Properties:39
cP
3
T − 12
12.152
152 ⋅T 2
+ 0.05158
5 ⋅T 3 − 7.77 ⋅10 ⋅T 4 ( J/kmol/K
)
(33)
As can be seen from Fig. 5, the predicted cP/R data
for the subliming solid phase (Curve B) monotonously
increases with temperature producing increasing good
agreement with the DIPPR Eqn (33) (Curve D) as the
triple point temperature is reached (maximum
discrepancy 15%). Also the cross over at Ttr results in
the expected step change in cP/R values.
Figure 5 also shows that the cP/R predictions using
PR EoS for the saturated liquid phase (Curve A) are in
a very good agreement with the experimental data
(Curve C) throughout, with the maximum discrepancy corresponding to ca. 4%. However, the PR EoS
fails to capture the expected step change in heat
capacities at Ttr, producing poor predictions in the
sublimation region .
Figure 6 shows the corresponding data as in Fig. 5
but for the variations of enthalpies of evaporation,
Figure 6. The variation of the enthalpy of sublimation and
evaporation of CO2 with temperature. Curve A: Calculated
based on the PR EoS. Curve B: Calculated based on the
extended PR EoS. Curve C: Experimental data.38
© 2013 Society of Chemical Industry and John Wiley & Sons, Ltd | Greenhouse Gas Sci Technol. 3:136–147 (2013); DOI: 10.1002/ghg
Modeling and Analysis: An extended Peng-Robinson equation of state for CO2 solid-vapour equilibrium
S Martynov et al.
Figure 7. The variation of heat capacity difference
(cP – cV )/R of subliming solid and saturated liquid CO2 with
temperature. Curve A: Calculated based on the PR EoS.
Curve B: Calculated based on the extended PR EoS.
Figure 8. The variation of adiabatic speed of sound cs of
subliming solid and saturated liquid CO2 with temperature.
Curve A: Calculated based on the original PR EoS. Curve
B: Calculated based on the extended PR EoS.
Hv – Hl and sublimation, Hv – Hs with temperature in
comparison with the experimental data.38 Very
similar trends as those for the specific heat capacity
data may be observed with the exception of sublimation phase change enthalpies, Hv – Hs decreasing
marginally with temperature. Also, based on comparison with the limited range of the experimental
data available (Curve C), the performance of the
extended PR EoS in predicting Hv – Hs (Curve B) is
about the same as that in predicting the cP/R values
for the solid phase (cf. 13% with 14% maximum
errors). Furthermore, the application of the PR EoS in
the sublimation region (Curve A) results in significant
errors (ca. 32% maximum) when compared with the
experimental data (Curve C).
Figures 7, 8, and 9, respectively, show the heat
capacity difference, (cP – cV)/R, the adiabatic speed of
sound, cs and the thermal expansion coefficient, αV of
saturated liquid and subliming solid CO2 as functions
of temperature. Unfortunately no experimental data is
available in the temperature range of interest (100 K
to Ttr) to enable the evaluation of the performance of
the two EoS. Nevertheless, it is interesting to note that
in the case of (cP – cV)/R (Curve B, Fig. 7) and αV
(curve B, Fig. 9) for the subliming solid, both equations of state produce similar predictions. In the case
of the adiabatic speed of sound for the solid however
(cf. Curves A and B, Fig. 8) the differences between
the two prediction is more marked (ca. 40–65 %). As
would be expected, the solid phase adiabatic speed of
sound predicted using the extended PR EoS is larger
than that predicted using the PR EoS (cf. Curves B
and A in Fig. 8). Similarly, the thermal expansion
coefficient of solid phase predicted using the extended
PR EoS is smaller than that obtained using the
original PR EoS (cf. Curves B and A in Fig. 9). Notably, unlike the other derivative properties examined
above, in the case of αV (Fig. 9) no discontinuity in its
value may be observed at Ttr.
Solid-vapor and solid-liquid phase
equilibria
Figure 10 shows the pressure-temperature phase
diagram for CO2 generated using the PR and the
extended PR EoS in comparison with the available
experimental data from Angus et al.8 in the respective
Figure 9. The variation of thermal expansion coefficient αv
of subliming solid and saturated liquid CO2 with temperature. Curve A: Calculated based on the PR EoS. Curve B:
Calculated based on the extended PR EoS.
© 2013 Society of Chemical Industry and John Wiley & Sons, Ltd | Greenhouse Gas Sci Technol. 3:136–147 (2013); DOI: 10.1002/ghg
143
S Martynov et al.
Modeling and Analysis: An extended Peng-Robinson equation of state for CO2 solid-vapour equilibrium
Figure 10. Pressure-temperature phase diagram for CO2.
Curve A: LVE predicted using the PR EoS. Curve B: SVE
predicted using the extended PR EoS. Curve C: SLE
predicted with the help of Eqn (34). Curve D: SVE experimental data.8 Curve E: SLE experimental data.8 Curve F:
LVE experimental data.38
pressure and temperature ranges of 0.001 to 100 MPa
and 150 to 300 K. The predicted solid-liquid equilibrium (SLE) or the melting curve was generated using
the iso-fugacity condition:
f s p ,Tm ( p ))
f l p ,Tm ( p ))
(34)
where f l(p,Tm(p)) and fs(p,Tm(p)) are fugacities of the
liquid and solid phases, respectively, each in turn
calculated using Eqn (4) incorporating the original
(vapor – liquid) and extended (solid–liquid) sets of
parameters for a and b. Equation (34) was solved
numerically to obtain the melting temperature, Tm at
a given pressure, p.
As it may be observed, the sublimation and melting
lines (Curves B and C) are in very good agreement
with the corresponding experimental data (Curves D
and E). However, although the PR EoS produces
excellent performance along the saturation line (cf.
Curves A and F) significant errors are encountered
when the equation is applied to describe the sublimation line (cf. Curves A and D) with the degree of
disagreement increasing as the temperature is
reduced.
Conclusions
In the CCS chain, the accidental rupture of a pressurized CO2 pipeline and the resulting expansion induced cooling of the escaping cloud may result in
solid CO2 release. The subsequent delayed sublimation
144
of any accumulated solid, particularly if present in
large quantities, will significantly modify the CO2
cloud dispersion behavior thus impacting the minimum safe distances to populated areas and emergency
response planning.
In this paper, given its popularity, relative mathematical simplicity and robustness, the PR EoS was
extended to model the sublimation phase transition
behavior of CO2. The above involved the modification
of the two parameters, a and b which were in turn
used to determine the pertinent thermo-physical and
thermodynamic properties for the subliming CO2
including density, specific heat capacities, speed of
sound as well as enthalpy and internal energy. In all
cases where the relevant experimental were available,
reasonably good agreement with the predicted properties were obtained.
In addition to modeling the sublimation phase
behavior, the liquid/vapor or the melting phase
behavior for CO2 was modeled by the merging of the
original and the extended PR EoS through equalization of solid and liquid fugacities.
The range of applicability of the original and the
extended PR EoS was demonstrated by constructing
the predicted solid/liquid/vapor pressure/temperature
phase diagram for CO2 in comparison with the
experimental data. It was found that although the PR
EoS provided very accurate prediction of the vapor/
liquid saturation line, its application along the sublimation line resulted in significant errors.
It is appreciated that depending on the capture
technology, the transported CO2, although forming
the major constituent, will contain a range of different
impurities such as N2, CH4, O2, CO, NOx, SOx.40
Some of these have already been shown41 to profoundly impact the CO2 saturation phase behavior.
The same may apply to its sublimation behavior.
Additionally, the direct application of an EoS for
predicting the fluid phase behavior is appropriate
provided the constituent phases are in thermal and
mechanical equilibrium. Clearly non-equilibrium
effects such as delayed nucleation, phase slip and
thermal stratification may become important during
pipeline decompression. As such the inclusion of an
EoS into CFD outflow and near-field dispersion
models should be coupled with the appropriate
closure models to account for such phenomena.
In conclusion, it is noteworthy that in principle,
depending on the availability of the relevant experimental data for deriving the modified expressions for
© 2013 Society of Chemical Industry and John Wiley & Sons, Ltd | Greenhouse Gas Sci Technol. 3:136–147 (2013); DOI: 10.1002/ghg
Modeling and Analysis: An extended Peng-Robinson equation of state for CO2 solid-vapour equilibrium
the parameters, a and b, the extended PR EoS described in this study may be used to model the
sublimation and the melting behavior for other fluids
of interest.
Notation
a = parameter in Eq. (1), (Pa m6/mol2)
a0 = parameter in Eq. (27), (Pa m6/mol2);
b0 = parameter in Eq. (28), (m3/mol)
ac = parameter in Eq. (2), (Pa m6/mol2)
b = parameter in Eq. (1), (m3/mol)
bc = parameter in Eq. (3), (m3/mol)
cb = parameter in Eq. (28)
cP = isobaric heat capacity (J/mol/K)
cS = adiabatic speed of sound (m/s)
cV = isochoric heat capacity (J/mol/K)
f = fugacity (Pa)
p = pressure (Pa)
R = universal gas constant = 8.3144621 (J/mol/K)
T = temperature (K)
Ta = parameter in Eq. (27), (K)
Tb = parameters in Eq. (28), (K)
U = internal energy (J/kg)
v = molar volume (m3/mol)
Z = compressibility
H = enthalpy (J/mol)
Greek letters
αV = thermal expansion coefficient (1/K)
βT = isothermal compressibility coefficient (1/Pa)
ω = acentric factor
Subscripts
c = critical point
igg = ideal gas
l = liquid
m = melting (SLE) line
P = pressure
s = solid
subl = sublimation (SVE) line
tr = triple point
T = temperature
v = vapour
V = volume
LV = refers to the Liquid and Vapour phases
SV = refers to the Solid and Vapour phases
Abbreviations
LVE = vapor-liquid equilibrium
SLE = solid-liquid equilibrium
S Martynov et al.
SVE = solid-vapor equilibrium
CCS = carbon capture and sequestration
CFD = computational fluid dynamics
PR = Peng-Robinson
EoS = equation of state
Acknowledgement
The research leading to this work has received funding from the European Union Seventh Framework
Programme FP7-ENERGY-2009-1 under grant
agreement number 241346.
Appendix
In the original PR EoS,31 parameters aLVV and bLVV are
defined as functions of temperature T
T, the critical
pressure pc and temperature Tc of a fluid and the
acentric factor ω:
aLV
.
bLV
.
Tc2 / pc
(T ),
(A1)
Tc / pc ,
(A2)
where
α( ) ⎡
⎣
(
κ (ω )
)
2
T / Tc ⎤ ,
⎦
37464 + 11.54226
54226ω − 0 26992ω 2 .
κ (ω ) = 00.37464
For pure CO2, pc = 7.382 (MPa) and Tc = 304.2 (K),
and ω = 0.228.
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Sergey Martynov
Sergey Martynov is a Research
Associate at UCL and received his
MSc in Technical Physics from
Moscow Power Engineering Institute in
1998. He then studied processes in
fuel injectors at the University of
Brighton to obtain his PhD in 2005.
Research interests are the mathematical modelling of
processes in multi-phase flows, breakup and atomization of liquid sprays and bubble dynamics.
© 2013 Society of Chemical Industry and John Wiley & Sons, Ltd | Greenhouse Gas Sci Technol. 3:136–147 (2013); DOI: 10.1002/ghg
Modeling and Analysis: An extended Peng-Robinson equation of state for CO2 solid-vapour equilibrium
Solomon Brown
Solomon Brown is a Research Associate at UCL and received a Master’s
degree in Mathematics from King’s
College London 2007. He went on to
work on modeling the consequences
of pipeline failure for which he obtained his PhD in 2011. His main
research interest is in the area of computational fluid
dynamics and uncertainty quantification applied to
safety and loss prevention.
S Martynov et al.
Haroun Mahgerefteh
Haroun Mahgerefteh is Professor of
Chemical Engineering at UCL. Research interests are hazard assessment and material selection for next
generation CO2 pipelines. He coordinates the CO2PipeHaz EC project in
collaboration with China and several
European countries, partner in EPSRC/E.On MATTRAN
and National Grid COOLTRANS projects on CO2
pipelines.
© 2013 Society of Chemical Industry and John Wiley & Sons, Ltd | Greenhouse Gas Sci Technol. 3:136–147 (2013); DOI: 10.1002/ghg
147