The Mathematical Models of Gas Transmission at Hyper · PDF fileThe Mathematical Models of Gas...

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.


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.


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.


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


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


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.


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


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


,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


this work and support, as well as Professor D. K. Tagantsev for fruitful discussions and

assistance in writing this article.

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http://www.mathnet.ru/php/person.phtml?option_lang=eng&personid=63741








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Received: June 15, 2014

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