# Critical Point Analysis of Phase Envelope Diagram

```Critical Point Analysis of Phase Envelope Diagram
Darmadi Soetikno1 , Rudy Kusdiantara2∗ , Dila Puspita2 , Kuntjoro A. Sidarto2 ,
Ucok W. R. Siagian1 , Edy Soewono2 and Agus Y. Gunawan2
1 Department of Petroleum Engineering, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung 40132, Indonesia
2 Department of Mathematics, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung 40132, Indonesia
∗ [email protected]
Abstract. Phase diagram or phase envelope is a relation between temperature and pressure that shows the condition of
equilibria between the different phases of chemical compounds, mixture of compounds, and solutions. Phase diagram is an
important issue in chemical thermodynamics and hydrocarbon reservoir. It is very useful for process simulation, hydrocarbon
reactor design, and petroleum engineering studies. It is constructed from the bubble line, dew line, and critical point. Bubble
line and dew line are composed of bubble points and dew points, respectively. Bubble point is the first point at which the gas
is formed when a liquid is heated. Meanwhile, dew point is the first point where the liquid is formed when the gas is cooled.
Critical point is the point where all of the properties of gases and liquids are equal, such as temperature, pressure, amount of
substance, and others. Critical point is very useful in fuel processing and dissolution of certain chemicals. Here in this paper,
we will show the critical point analytically. Then, it will be compared with numerical calculations of Peng-Robinson equation
by using Newton-Raphson method. As case studies, several hydrocarbon mixtures are simulated using by Matlab.
Keywords: Critical Point, Equation of State, Fugacity Equation, Phase Envelope
PACS: 02.60.Cb, 64.60.fh, 64.60.Bd
INTRODUCTION
Ideal gas is composed of random particles, which is constructed by classical thermodynamics equation PVm =
nRT . In reality, real gases behave qualitatively like an
ideal gas only at certain normal condition of temperature
and pressure. Here, we simulate phase envelope using
real gas model. Real gas exhibit properties that cannot be
explained entirely using ideal gas law. Several aspects of
real gas are considered here such as: interactions of gas
molecules, compressibility effects, specific heat capacity,
Van der Waals force, and composition of the gas[3].
Peng-Robinson real gas equation is[1]:
P=
RT
aα
−
,
Vm − b Vm (Vm + b) + b(Vm − b)
(1)
where P is the pressure, T is the temperature, R the
ideal gas constant, Vm the molar volume, a and b are
parameters that determined empirically for each gas
and from their critical temperature (Tc ), and critical
pressure (Pc ), α = (1 + m(1 − Tr0.5 ))2 with m is the
parameter related to acentric factor ω, which is fulfill
m = 0.379642 + 1.48503ω − 0.1644ω 2 + 0.016667ω 3
and Tr = TTc is temperature reduction. The constant a and
2 2
REAL GAS EQUATION AND EQUATION
OF STATE
The Equation of State (EOS) here is given by thermodynamic equation. It is describing state of matter in a
certain physical conditions. This equation provides a
mathematical relationship between two or more state
variables, such as temperature, pressure, volume, or
energy of the system. EOS is useful to describe physical
properties of liquid, gas, or gas mixture. In general,
real gas equation is modified in such a way to simplify
the computational process. It is modified into a third
order polynomial equation by using compressibility
m
factor Z = PV
RT . Here, we use Peng-Robinson real gas
equation to simulate the Phase Envelope in this paper.
c
b in PR real gas equation are a = Ωa R PTc c and b = Ωb RT
Pc ,
where Ωa = 0.42747 and Ωb = 0.07780. By subtitute
Vm = ZRT
P to Equation(1), we obtain PR EOS cubic
equation[3]:
Z 3 − (B − 1)Z 2 + (A − 3B2 − 2B)Z
−(AB − B2 − B3 ) = 0,
(2)
bP
where A = (aα)P
.
and B = RT
(RT )2
Meanwhile, mixing rule for variables A and B satisfy
mP
m P)
A = (aα)
and B = (bRT
. Variables (aα)m and bm
(RT )2
apply[1]:
i
n n h
(aα)m = ∑ ∑ xi x j (ai a j αi α j )0.5 (ki j − 1)
i=1 j=1
(3)
Bubble Line
and
n
bm = ∑ [xi bi ]
(4)
i=1
R2 Tc2i
,
Pci
RTci
Pci
h
i2
T
αi = 1 + mi (1 − Tri0.5 ) , Tri =
Tci
mi = 0.379642 + 1.48503ωi − 0.1644ωi2
ai = Ωa
bi = Ωb
+0.016667ωi3
(5)
Bubble point which forms bubble line in FIGURE 1
is a point separating the liquid phase and the two phases
region, namely the liquid phase and the gaseous phase.
At bubble point conditions apply[1]:
xi = zi , 1 ≤ i ≤ n
nL = 1, nv = 0
(6)
n
The parameter ki j is a correction factor called the binary
interaction coefficient, which can be determined empirically, characterizing the binary formed by i-component
and j-component in the hydrocarbon mixture. The other
parameter which is important to understand two-phase
behavior is fugacity. The fugacity (f) is a represent the
molar Gibbs energy of a real gas. It used to calculate bubble and dew point. In a mathematical form, the fugacity
of a component is defined by
Z P Z −1
dP
(8)
f = P · exp
P
0
The ratio of the fugacity to the pressure is called the
fugacity coefficient Φ is calculated from equation(8)
Z P f
Z −1
Φ = = exp
dP
(9)
P
P
0
PHASE DIAGRAM
According to thermodynamic definition phase diagram
(phase envelope) is a graph showing the pressure at
which transition of different phases from a compound,
respect to temperature[1]. Here an example of phase envelope of a compound and its region phases.
n
∑ yi = ∑ [zi Ki ] = 1,
(7)
i=1
(10)
(11)
(12)
i=1
where xi is liquid mole fraction of i-component, yi is
gaseous mole fraction of i-component, zi is mixing mole
fraction of i-component, Ki is gas-liquid equibrium ratio
for i-component, nL is total number of mole in the liquid
phase, and nv is total number of mole in the gaseous
phase. While, Ki satisfy[4]:
Ki =
ΦL
fL /(xi Pb ) fLi /(zi Pb )
yi
= iv = iv
= v
xi
Φi
fi /(yi Pb )
fi /(yi Pb )
Equation (12) can be rewritten as:
n n L n zi fLi
f
zi ΦLi
=
=
∑ zi Pb Φv ∑ Pb Φi v = 1
∑ Φv
i
i
i
i=1
i=1
i=1
(13)
(14)
or
n
Pb = ∑
i=1
fLi
Φvi
So, equation(15) can be simplify as[1]:
n L f
f (Pb , T ) = ∑ i v − Pb = 0,
Φ
i
i=1
(15)
(16)
where Pb is bubble point pressure, T is temperature, fLi is
fugacity component i in liquid phase, and Φvi is fugacity
coefficient component i in gaseous phase. We need to
calculate Bubble Pressure(Pb ) in Equation(16). Equation
(16) can be solved by using Newton-Raphson method[5]:
Pbr+1 = Pbr −
f (Pb , T )
∂ f (Pb , T )/∂ Pb
(17)
To this end, first derivative of f (Pb , T ) respect to Pb . We
carry out as follow:
n v
Φi (∂ fLi /∂ Pb ) − fLi (∂ Φvi /∂ Pb )
∂f
=∑
− 1 (18)
∂ Pb i=1
(Φvi )2
FIGURE 1. Simple Phase Envelope
Using center difference, discretize (18) as:
L
∂ fLi
fi (Pb + ∆Pb ) − fLi (Pb − ∆Pb )
≈
∂ Pb
2∆Pb
v
v
∂ Φi
Φi (Pb + ∆Pb ) − Φvi (Pb − ∆Pb )
≈
∂ Pb
2∆Pb
(19)
(20)
Dew Line
where
Dew point which forms dew line in FIGURE 1 is a
point separating the gaseous phase and the two phases
region, namely the liquid phase and the gaseous phase.
At dew point, the follow conditions must be satisfied[1]:
yi = zi , 1 ≤ i ≤ n
nL = 0, nv = 1
n
n zi
∑ xi = ∑ Ki = 1
i=1
i=1
(21)
(22)
(23)
Using gas-liquid equilibrium ratio, we simplify Equation
(23) to become[2]:
n n n zi Φvi
zi fvi
fvi
∑ L = ∑ zi Pd ΦL = ∑ Pd ΦL = 1 (24)
i
i
i=1
i=1
i=1 Φi
or
n
Pd = ∑
i=1
fvi
ΦLi
(25)
Since Pd is independent of n. For the need of computation, we rewrite (25) as[1]:
n v f
(26)
f (Pd , T ) = ∑ iL − Pd = 0
i=1 Φi
where Pd is dew point pressure, T is temperature, fvi is
fugacity component i in gaseous phase, and ΦLi is fugacity coefficient component i in liquid phase. Equation
(26) can be solved for determine Pd by using NewtonRaphson method as (17) to (20)[5]. We replace the variables Pb with Pd , fLi with fvi , and Φvi with ΦLi .
Tcm
Tc j
Vc j
Θj
τi j
:
:
:
:
:
Mixture critical temperature
Critical temperature of component j
Critical volume of component j
Surface fraction of component j
Binary interaction coefficient
In the next section, we are simulating phase envelope
of hydrocarbon mixtures, by solving Equation(16)
and Equation(26) which satisfy Equation(2) and
Equation(28).
NUMERICAL SIMULATION
In the following simulation, we use composition:
TABLE 1.
Composistion data
No
Component
Mole Fraction
Case 1 Case 2
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
CO2
N2
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7
C8
C9
C10
C11
C12
C13
C14
C15
1.857
2.201
84.796
6.437
2.73
0.598
0.643
0.278
0.164
0.198
0.098
0
0
0
0
0
0
0
0
1.087
0.535
84.148
6.681
3.727
0.913
0.996
0.471
0.303
0.336
0.391
0.148
0.189
0.039
0.008
0.007
0.009
0.005
0.006
Critical Point
State of pressure and temperature at which all properties of the gas and liquid phases are equal at a certain
point is known as critical point. It is satisfing[1]:
2 ∂ P
∂P
=
=0
(27)
∂V T
∂V 2 T
From condition (27), Chueh and Prausnitz simplified
Critical Temperature as follow:
n
Tcm =
n
n
∑ (Θ j Tc j ) + ∑ ∑
j=1
(Θi Θ j τi j )
(28)
i=1 j=i+1
with
2
Θj =
x j (Vc j ) 3
2
∑ni=1 xi (Vci ) 3
(29)
Hydrocarbon composition of the first case is a mixture
of hydrocarbons with light component to moderate component, while the second case of hydrocarbon composition is a mixture of light component to heavy component. Here, we will see the difference of the phase envelope properties between the first and second case such
as Critical Pressure, Critical Temperature, Cricondenbar
and Cricondenterm.
CONCLUSION
The simulation result are given:
Case 1
We calculated bubble point and dew point, which build
bubble line and dew line. Critical temperature was calculated using Chueh and Prausnitz correlation. While, critical pressure was calculated using dew function. The results of numerical simulation for Peng-Robinson real gas
equation obtained:
The greater number of C components are involved
in a composition of hydrocarbons, critical temperature, Cricondenbar and Cricondentherm are greater.
• The greater number of C components are involved
in a composition of hydrocarbons, more extensive
two-phase region is formed.
•
ACKNOWLEDGMENTS
We acknowledge all of RC-OPPINET ITB’s for providing financial support and feedbacks. Our gratitude also
goes to Center for Mathematical Modeling and Simulation (PPMS) which has been facilitating the OPPINET
activities.
FIGURE 2. Case 1
Case 2
REFERENCES
1. T. Ahmed, Hydocarbon Phase Behavior, Gulf Publishing
Company, Houston, 1989.
2. Li and Nghiem, The Development of a General Phase
Envelope Construction Algorithm for Reservoir Fluid
Studies, Society of Petroleum Engineers of AIME, New
Orleans, 1982.
3. W.D. McCain, The Properties of Petroleum Fluids 2nd
Edition, PennWell Publishing Company, Tulsa, 1990.
4. D.V. Nichita, Phase Envelope Construction with Many
Component, Energy & Fuel 22, pp. 488-495, 2008.
5. R.G. Ziervogel and B.E. Polling, A Simple Method for
Construction Phase Envelopes for Multicomponent
Mixture, Fluid Phase Equilibria 11, pp. 127-135,
Amsterdam. 1983.
FIGURE 3. Case 2
Properties
Case 1
Case 2
Critical Pressure
Critical Temperature
Cricondenbar
Cricondentherm
1149.17 psia
408.66 R
1428.23 psia
515.97 R
1694.61 psia
430.14 R
2298.9 psia
677.08 R
Simulation results show that the composition involved
will affect the shape and properties of phase envelope is
formed. From the simulation above, we can determine in
which areas, liquid, gas, or a mixture of liquid and gas
are formed in both cases.
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