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Applied Mathematical Sciences, Vol. 8, 2014, no. 124, 6191 - 6203
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2014.47508
The Mathematical Models of Gas Transmission
at Hyper-Pressure
G. I. Kurbatova
Saint-Petersburg State University, Russia
N. N. Ermolaeva
Saint-Petersburg State University, Russia
Copyright © 2014 G. I. Kurbatova and N. N. Ermolaeva. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction
in any medium, provided the original work is properly cited.
Abstract
Different approaches to the modeling of thermodynamic processes in a gas flow at
hyper-pressure are discussed and their equivalence is proved for any form of equation of
state. The article presents arguments in favor of the approach in which temperature and
density are chosen as independent thermodynamic variables and equation of sate is
written in one of its analytical forms, for example, in the form of Redlich-Kwong
equation of state. Explicit analytical expressions are obtained for the dependences of
internal energy and specific heat at constant volume on gas temperature and density. It
is proved that simplifications of the model of thermodynamic processes, which are
based on the assumption of incompressibility of gas flow, are inadmissible, even in case
of negligibly small inertial forces.
Keywords: mathematical models, gas-pipelines, thermodynamic processes, equation of
state, Joule-Thompson effect
1. Introduction
Modern gas-pipelines operate at hyper-pressure conditions (~25 MPa) and this
complexifies modeling of thermodynamic processes, which is responsible for adequacy
of the mathematical model used.
6192 G. I. Kurbatova and N. N. Ermolaeva
In the 90’s of last century in St.-Petersburg State University (Russia) a quasi-one-
dimensional model of steady flow of a real multicomponent gas mixture in a sea hyper-
pressure gas-pipeline was developed, with the model taking into consideration radial
distribution of gas velocity, relief of the ground, construction of pipeline coating, and
possibility of pipeline glaciation. This model was described in details in the book
“Models of sea gas-pipelines” [1]. The model was successfully used in simulating gas
flow in pipelines running from Shtokman gas-field in the Barents Sea and in the North
European Gas Pipeline (Nord Stream) in the Baltic Sea. In [2], the validity of reduction
of a quasi-one-dimensional model to one-dimensional one was proved for the case of a
high value of the Reynolds number (Re>106). In [3], the model was generalized for the
case of unsteady processes.
The goal of the present study is to analyze approaches to the modeling of
thermodynamic processes in different mathematical models of gas transmission1, as
well as to demonstrate how simplifications of mathematical models can result in
principally wrong inferences about temperature distribution in a gas flow.
2. General model
One of rather general mathematical models of the above-mentioned processes is the
VBVK (Vasil’ev-Boldyrev-Voevodin-Kanibolotsky) model published in 1978 in [5],
and up to now this model is often in use [6,7,8]. Our mathematical model developed in
[1,3] mainly differs from the VBVK model in the description of thermodynamic
processes, while in both models the presentations of continuity equation (mass
conservation), equation of motion (or momentum equation), and energy equation (law
of conservation of energy) coincide with an accuracy of up to symbols:
continuity equation
( ) 0,ut z
(1)
equation of motion
2( ) | |( ) cos ( ),
4
u u up u g z
t z R
(2)
energy equation
1 For example, a brief review of such models is presented in [4].
Mathematical models of gas transmission 6193
*
2
( ) 2( ) cos ( ),
e pu e T T ug z
t z R W
(3)
and relations between energy, internal energy, and enthalpy are
2 / 2, / .e u i p (4)
The set of Eqs. (1)-(4) should be supplemented by equation of state
( , )p p T (5)
and caloric equation of state either in terms of enthalpy
( , ),i i p T (6)
or in terms of internal energy
( , ).T (7)
We use the following designations: u, ρ, p, and T are the flow velocity, the density, the
pressure, and the temperature of a gas mixture, correspondingly, which are functions of
time t and coordinate z coinciding with the gas-pipeline axis; e, ε, and i are the mass
densities of energy, internal energy, and enthalpy which are also functions of t and z; R
is the inner radius of the gas-pipeline; W is the explicit function of the coefficients of
inner and external heat exchange, the thicknesses of the layers comprising the coat2 of
the gas-pipeline, and the heat conductivities of these layers (examples of function W are
presented in [1]); g is the gravity acceleration; α(z) is the angle between the gravity
vector and Z-axis; λ is the coefficient of hydraulic resistance, which can be expressed in
terms of the Reynolds number Re and the coefficient of relative roughness of inner wall
of the pipeline k (λ= λ(Re,k), for example, see the empirical equation of Colebrook-
White [1]); T* is the ambient temperature.
Theoretically, the model based on Eqs. (1)-(7) is equal to the one which, instead of the
energy equation Eq.(3), includes balance equation for internal energy
2*
2
' 2 | |( )
4
d u u up T T
dt z R W R
, (8)
where d
udt t z
is the operator of material derivative. Eq.(8) can be derived from
Eqs. (2)-(4) using the well-known procedure described in [9]. From the view-point of
the simplicity of numerical simulation the model including energy equation, rather than
2 One of these layers could be ice that is typical for sea gas-pipelines.
6194 G. I. Kurbatova and N. N. Ermolaeva
balance equation for internal energy, appears to be preferable because it tolerates the
usage of conservative difference schemes.
3. Modeling of thermodynamic processes
There are two approaches to the modeling of thermodynamic processes in gas flow.
In the first one [5], p and T are chosen as independent thermodynamic variables; the
equation of state Eq.(5) and the caloric equation in terms of enthalpy Eq.(6) take the
forms
( , ) gpV Z p T R T , (9)
0 0
( , )
pT
p
pT p
Vi p T c dT V T dp
T
, (10)
where Z(p,T) is the compressibility factor, V is the specific volume (V=1/ρ), Rg is the
constant equal to Ro/M (Ro is the gas constant and M is the molecular weight of the gas
mixture). cp(T) is the temperature dependence of specific heat at constant pressure,
which is considered unknown. Numerous works based on this approach differ just in the
dependence Z(p,T) used [5,10,11].
In the first approach the balance equation for internal energy Eq.(8) is rewritten as
2*
2
2 | |( ) .
4p
p
d T V d p u uc T T T
dt T dt R W R
(11)
The derivation of Eq.(11) from Eq.(8) is based on the following relations:
p T
d d p i d T i d p d pi
dt dt T dt p dt dt
, (12)
, p
p pT
i V iV T c
p T T
, (13)
1 1.
d u
dt z
(14)
Eqs. (13) are the well-known thermodynamic equalities [12], Eq. (14) ensues from the
continuity equation Eq. (1) and definition of material derivative. Eqs. (1), (2), (11), and
(9) form a closed system of equations for unknown functions ρ, u, T, and p, which,
being supplemented by initial and boundary conditions, enables one to calculate all
characteristics of gas flow.
Mathematical models of gas transmission 6195
In the second approach [1], the independent thermodynamic variables are ρ and T, while
one of the analytical forms of the generalized Van der Waals equation is used as
equation of state. By now, a vast number of such analytical forms has been proposed,
but the most known ones are the Redlich-Kwong, Soave, Peng-Robinson, Benedict-
Webb-Rubin equations of state [7, 11, 13]. In our works [1,3] as well as in [4,7] the
two-parameter Redlich-Kwong equation of state is used, for this equation is considered
one of the best which works in a wide range of variations of p, ρ and T, including hyper-
pressure range. For a gas mixture it can be written as [13]:
2/1
2
0
11 T
c
M
TRp
, (15)
where cca pMTRc 22/52
0 and ccb MpTR0 , Ωa and Ωb are the known constants,
pc and Tc are the critical pressure and temperature of a gas mixture of given chemical
composition. The values pc and Tc can be determined using the tables presented in [13].
In the second approach, caloric equation is written in terms of internal energy ε(T,V),
and the balance equation of internal energy Eq. (8) is transformed into
2*
2
2 | |( ) .
4V
V
d T p u u uc T T T
dt T z R W R
(16)
The derivation of Eq.(16) from Eq.(8) is based on the following relations:
V T
d d T d V
dt T dt V dt
,
, V
T V V
pp T c
V T T
, (17)
1d V u
dt z
,
where cV is the specific heat at constant volume, which for a real (unideal) gas mixture
is a function of ρ and T. The derivation of the first relation of the set of equations
Eq.(17) can be found in [1]. In accordance with the Redlich-Kwong equation of state the
derivative (∂p/∂T)V is determined as
2
3/ 2
1.
1 2 (1 )V
p h c
T T
(18)
Eqs. (1), (2), (16), and (15) form a closed set of equations for unknown functions ρ, u,
T, and p.
6196 G. I. Kurbatova and N. N. Ermolaeva
It is arguable that Eq. (11) and Eq. (16) are equivalent for any equation of sate. To make
sure that it is true let us show that
.p V
p V
d T V d p d T p uc T c T
dt T dt dt T z
(19)
Using the known thermodynamic relations [12]
,
,
V p
V p
p V T
p Vc c T
T T
V p V
T T p
(20)
Eq. (20) and Eq. (14), and introducing the following designations
, V p
p Va b
T T
, (21)
let us transform the right side of Eq. (19) as follows:
( )
.
V p
p
d T u d T d Vc Ta c Tab Ta
dt z dt dt
d T d T a d Vc Tb a
dt dt b dt
(22)
For any equation of state V=V(p,T) the equality
,pT
d V V d p V d T
dt p dt T dt
is valid. Given Eq. (20) and Eq. (21), this equality can be rewritten as
d V b d p d Tb
dt a dt dt
that allows writing the right side of Eq. (22) as follows
p p
d T d T a d V d T d pc Tb a c Tb
dt dt b dt dt dt
.
Thus, Eq. (19) is valid and, therefore, the equivalence of Eq.(11) and Eq.(16) is proved. This evidences the theoretical parity of the two approaches to the modeling of thermodynamic processes in gas flow. The choice of either one approach or another is related to the choice of equation of state. If the equation of state in the form of Eq. (9) is used, specific volume V can be presented as an explicit function of p and T and, therefore, the derivative (∂V/∂T)p can be easily calculated, that makes the first approach expedient. If an analytical form of equation of state is used, pressure p is an explicit
Mathematical models of gas transmission 6197
function of V and T and, therefore, the derivative (∂p/∂T)V can be calculated. In this case the second approach is more expedient. If both equations of state are capable of giving similar and adequate interrelations
between values p, ρ, and T, both approaches can be used. However, the problem to find
an expression which could adequately describe the compressibility factor Z(p,T) in a
wide range of p and T appears to be rather difficult. That is why the use of suitable
analytical forms of equation of state and, therefore, the second approach is more
preferable, especially in the case of hyper-pressure.
4. Dependences ε(ρ,T) and cV(ρ,T)
Within the frames of the second approach let us find the dependences of internal energy
and specific heat on density and temperature. Internal energy ε being a function of V and
T, that is ε=ε(T,V), its total differential dε can be written as
.V
T
d c dT dVV
Above we have written Eq. (17) from which one can obtain the well-known
Helmholtz’s equation
2 .T V
pT
V T T
(23)
After integration of Eq. (23) we have
0
2
0( , ) ( , ) .
V
VV
pT V T V T dV
T T
(24)
Let V0 →∞, then any gas mixture can be considered an ideal gas and, therefore, ε(T,V0)→ ε0(T), where ε0(T) is the internal energy of an ideal gas [14].
Now let us calculate the integral in the right side of Eq. (24). For the Redlich-Kwong
equation of state the following expression
2
1/ 2
3,
2 ( )V
p cT
T T V V T
is valid, and the integral can be calculated as
2
1/ 2 1/ 2
3 3ln(1 ),
2 ( ) 2
V V
V
p c dV cT dV
T T T V V T
6198 G. I. Kurbatova and N. N. Ermolaeva
so that the internal energy ε(ρ,T) of a gas mixture obeying the Redlich-Kwong equation
of state becomes equal to
0 1/ 2
3( , ) ( ) ln(1 ).
2
cT T
T
This expression for ε(ρ,T) allows finding cV as an explicit function of ρ and T:
).1ln(4
3),(
2/3
0 pT
c
dT
d
TTcV
Internal energy of an ideal gas ε0 is known to be a function of temperature only: ε0(T)=
Vc T, where Vc is the specific heat of an ideal gas (including ideal gas mixtures). Thus,
the specific heat and the internal energy of a real gas mixture obeying the Redlich-
Kwong equation of state take the forms
),1ln(4
3ˆ),(
2/3p
T
ccTc VV
(25)
).1ln(2
3ˆ),(
2/1p
T
cTcT V
(26)
Numerical simulations of the model based on Eqs. (1), (2), (16), and (15) show that
allowance for the dependence cV(ρ,T) (Eq. (25)) in solving Eq. (16) mostly influences
the resultant temperature field and it proves to be rather important in the computer
modeling and engineering of sea gas-pipelines in the North.
Above it is mentioned that for numerical simulations the model based on the energy
equation Eq. (3) is more preferable. In this case the functions to be found are ρ, u, ε, and
p. Eq. (26) enables one to express T in the right side of Eq. (3) via functions ρ and ε. Eq.
(26) is a cubic equation with respect to T1/2, with only one root having physical
meaning. This root can be easily found by means of standard techniques.
In the first approach to the modeling of thermodynamic processes the functions to be
found are ρ, u, i, and p. The dependence i(p,T) can be found from Eq. (10) and Eq. (9)
and this makes possible to present T (in the right side of Eq. (3)) as a function of
enthalpy i and temperature T.
5. Simplification of mathematical models
In solving certain problems, simplified variants of the model Eqs. (1)-(7) are often used.
The simplifications are based on the following assumptions; all the processes are steady,
isothermal, and adiabatic, as well as the inertial and gravity forces are ignored.
Mathematical models of gas transmission 6199
Let us dwell upon the assumption pertaining to the ignoring of inertial forces, which is used in many similar studies. Main conditions of the exploitation of different modern gas-pipelines are about the same; gas density in the hyper-pressure range is equal to about 160 kg/m3 and, in case of the gas rate Q in the pipe and the pipe diameter D being equal to about 450 kg/s and 1 m, correspondingly, such density conditions rather low gas flow velocity u (u≈3.5 m/s). Under these conditions Mach’s number is ≪1 that enables the gas compressibility not to be taken into consideration. However, complete ignoration of gas compressibility in the modeling of gas transmission processes is inadmissible, because, as it is shown below, it leads to absurd temperature distributions in gas-pipelines.
Let us consider a steady gas flow in a horizontal gas-pipeline thermally isolated from
outside ambient. Now we will use the first approach to the modeling of thermodynamic
processes. The above-mentioned assumptions allow reducing the model Eqs. (1), (2),
(11), and (9) to the following set of equations:
const,QuS (27)
,2
||)( 2
D
uuup
dz
d (28)
,2
|| 2
D
uu
dz
dp
T
VuT
dz
dTuc
p
p
(29)
( , )gp R Z p T T , (30)
where Eq. (27) is the integral of the continuity equation Eq. (1), S is the area of the
pipeline cross-section. Let us ignore the inertial forces in the equation of motion Eq.
(28) and write
pup 2 . (31)
Then the reduced equation of motion takes the form
.2
||
D
uu
dz
dp (32)
Using Eq. (32) we rewrite Eq. (29) as
,1
dz
dp
dz
dpV
T
VT
cdz
dT
pp
(33)
where μ is the Joule-Thompson coefficient. For the equation of state Eq. (30) this
coefficient is known [12] to be defined as
,
2
pp
g
T
Z
pc
TR
6200 G. I. Kurbatova and N. N. Ermolaeva
that allows rewriting Eq. (33) as follows:
.
2
dz
dp
T
Z
pc
TR
dz
dT
pp
g
(34)
The set of equations Eqs. (27), (32), (34), and (30) presents a simplified model of the
relevant processes, which results from the assumption Eq. (31).
In the second approach to the modeling of thermodynamic processes we have to use Eq.
(16), instead of Eq. (11) used in the first approach. Taking into account the reduced
equation of motion Eq. (32), Eq. (16) can be transformed into the form
.dz
dpu
dz
du
T
pT
dz
dTuc
V
V
(35)
It is above mentioned that in the second approach one of known analytical forms of
equation of state should be chosen as actual equation of state to be used in the modeling,
for example, the Redlich-Kwong equation of state Eq. (15). Note, Eq. (33) and Eq. (35)
are equivalent for any equation of state. The same way as in general case, this fact
ensues from the thermodynamic identities Eq. (20), the equality
dz
dpu
dz
du
,
and obvious relations
1
2 dz
d
dz
dV
and
dz
dp
p
V
dz
dT
T
V
dz
dV
Tp
.
It is essential that in both approaches, that is, in Eq. (34) and Eq. (35), the gas
compressibility is taken into consideration. The model based on Eqs. (27), (32), (34),
and (30) is used in many similar studies, for example [16]. Simulations based on the
general model as well as the simplified models are presented in our previous work [15]
where the following inference has been reasoned. If the equations of state Eq. (15) and
Eq. (30) give the same dependence p=(ρ,T), the results of simulations based on the
model Eqs. (27), (32), (34), (30) and the model Eqs. (27), (32), (35), and (15) coincide
and they provide qualitatively correct evolution of temperature field, that is, temperature
decreases with a drop in pressure.
We show what would happen if inertial forces were completely ignored. Simplification
of the equation of motion Eq. (28) resulting in the equation of motion Eq. (32) formally
validates the equality
Mathematical models of gas transmission 6201
,0dz
duu
from which for steady problems it follows that
constu and const . (36)
If these conditions are accepted as valid ones, we come to a principally incorrect result,
that is, principally inadmissible behavior of temperature. To make sure of this
incorrectness it is easy to use the second approach to the modeling of thermodynamic
processes. Indeed, if du/dz=0, from Eq. (35) it follows that
1,
V
dT dp
dz c dz (37)
that is, the derivatives of functions p and T with respect to z have different signs. It means that a decrease in pressure should lead to an increase in temperature. It is to this conclusion that the authors of several works come, for example [10]. However, this result contradicts well-known experimental data and, in addition, is not in agreement with neither the known equations of state nor the results of simulations based on rather general models in which the assumption p+ρu2≈p is not used.
6. Conclusions
It is proved that even if gas flow velocity is very small, the assumption that ρ and u are
constants is inadmissible.
In our opinion there are no reasons to use simplifications leading to a decrease in the
accuracy of the calculations of main characteristics of gas flow, for the calculations
based on general models which take into consideration inertial forces is not critically
complicated.
Two approaches to the modeling of thermodynamic processes in a gas flow have been
analyzed, an equivalence of two balance equations for internal energy used in the
approaches has been proved, and it has been shown that the second approach is more
preferable in the case of modeling of gas flow taking place under hyper-pressure
conditions. Explicit dependences of internal energy and specific heat at constant volume
on temperature and gas density have been found for the two-parameter Redlich-Kwong
equation of state.
Acknowledgments
The author acknowledges Professor V. A. Pavlovsky for long-term collaboration,
Academician N. F. Morozov and Professor N. V. Egorov for their continuous interest in
6202 G. I. Kurbatova and N. N. Ermolaeva
this work and support, as well as Professor D. K. Tagantsev for fruitful discussions and
assistance in writing this article.
References
[1] G. I. Kurbatova, E. A. Popova, B . V. Filippov, V. B. Fillipov, K. B. Fillipov,
Models of sea gas-pipelines, St.-Petersburg State University, St.-Petersburg, 2005.
[2] G. I. Kurbatova, E. A. Popova, Problem of allowance for velocity profiles in
simulation of a turbulent flow in pipes, Gazette of St.-Petersburg State University,
series 10, (2) (2007) 18-28.
[3] E. V. Grunicheva, G. I. Kurbatova, E. A. Popova, Non-stationary non-isothermal
flow of gas mix in the sea gas pipelines, Matem. Mod., 23 (4) (2011) 141–153.
[4] V. E. Seleznev, G. S. Klishiv, V. V. Aleshin, S. N. Pryalov, V. V. Kiselev, A. L.
Boichenko, V. V. Motlokhov, Numerical analysis and optimization of gasdynamic
conditions of natural gas transmission, Moscow, URSS, 2003.
[5] O. F. Vasil’ev, E. A. Bondarev, A. F. Voevodin, M. A. Kanibolotsky, Non-
isothermal gas flow in pipes, Nauka, Novosibirsk S.O., 1978.
[6] V. I. Zubov, V. N. Koterov, V. M. Krivtsov, A. V. Shipilin, Unsteady gas
dynamic phenomena in pipeline through the Black sea, Matem. Mod., 13 (4)
(2001) 58–70.
[7] V. E. Seleznev, V. V. Aleshin, S. N. Pryalov, Basics of numerical modeling of
long-distance pipelines, KomKniga, Moscow, 2005.
[8] M. Chaczykowski, Transient flow in natural gas pipeline – The effect of pipeline
thermal mode l, Appl. Math. Model., 34 (2010) 1051-1067.
[9] C. Truesdell, A first course in rational continuum mechanics, Maryland, The
Johns Hopkins University, Baltimore, 1972.
[10] M. V. Lur’e, O. V. Pyatakova, Particularities of thermal calculations of long-
distance gas-pipelines with allowance for the inversion of the Joule-Thompson
effect, Gas Industry, (2) (2010) 16-19.
[11] J. L. Modisette, Equations of State Tutorial, Proceedings of the PSIG – The 32-
nd Annual Meeting, Savannach, (8) (2000) 1-21.
[12] Ch. Kittel, Thermal Physics, John Wiley and Sons, Inc., New York, 1969.
Mathematical models of gas transmission 6203
[13] R. C. Reid, J. M. Prausnitz, Th. K. Sherwood, The properties of gases and
liquids, MeGraw – Hill Book Company, New York. St. Louis, San Francisco,
1977.
[14] D. Kondepudi, I. Prigogine, Modern thermodynamics. From heat engines to
dissipative structures, John Wiley & Sons, Chichester, New York, Weinheim,
Brisbade, Singapore, Toronto, 1999.
[15] G. I. Kurbatova, E. A. Popova, On difference in mathematical models of gas
transmission in pipelines, Gazette of St.-Petersburg State University, series 10, (3)
(2011) 47-55.
[16] A.D.Tevyashev, V.S. Smirnova, The method of the approached decision of the
Cauchy problem for system of the equations of stationary flow of gas in the
pipeline, Radioelektronika i informatika, (1) (2009) 81-87.
Received: June 15, 2014