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FLOW ANALYSIS FOR HYDROCARBON PIPELINE ENGINEERING

ALESSANDRO TERENZI Saipem S.p.A., Fano, Italy

Alessandro Terenzi

Gulf Professional Publishing is an imprint of Elsevier 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB, United Kingdom Copyright Ó 2022 Elsevier Inc. All rights reserved.

No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to se2ek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/ permissions.

This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein).

Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.

To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein.

ISBN: 978-0-12-822466-3 For information on all Gulf Professional Publishing publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Charlotte Cockle Senior Acquisitions Editor: Katie Hammon Editorial Project Manager: Jai Marie Jose Production Project Manager: Kamesh Ramajogi Cover Designer: Matthew Limbert Typeset by TNQ Technologies

Contents 1. Basics of hydrocarbons thermodynamics 1.1 Introduction 1.2 Equation of state 1.3 Other real fluid properties 1.4 The principle of corresponding states 1.5 Equilibrium and stability of one-component fluids 1.6 Vaporeliquid equilibrium for one-component fluids 1.7 Vaporeliquid equilibrium for multicomponent fluids 1.8 Equation of state for multicomponent mixtures 1.9 Phase diagrams of reservoir fluids References Further reading

2. Steady compressible flow 2.1 Introduction 2.2 1D flow equations in conservative form 2.3 1D flow equations for steady adiabatic pipe flow 2.4 The velocity of sound 2.5 Isentropic flow correlations 2.6 Stagnation and critical properties 2.7 Flow in area change regions 2.8 The convergent nozzle 2.9 The convergent-divergent nozzle 2.10 Performance of real flow restriction devices 2.11 Adiabatic flow with friction in a constant area duct 2.12 Real gas pipeline flow properties References Further reading

3. Transient compressible flow 3.1 3.2 3.3 3.4 3.5 3.6

Introduction Small amplitude waves in gases High amplitude waves in gases Discontinuities and the Riemann problem Types of transient operations in pipeline systems Examples of blowdown system simulations

1 1 2 4 5 10 14 21 29 32 38 39 41 41 43 47 48 50 51 52 54 57 58 60 65 82 83 85 85 86 90 95 100 109

v

j

vi

Contents

References Further reading

4. Multiphase gaseliquid flow 4.1 Introduction 4.2 Definitions 4.3 Flow regimes 4.4 Flow pattern map 4.5 Basic equations and models 4.6 Flow regime transitions 4.7 Slug flow 4.8 Pressure drop calculations for two-phase pipelines 4.9 Sound velocity in two-phase fluids 4.10 Design of multiphase pipelines References Further reading

5. Pipeline heat transfer 5.1 5.2 5.3 5.4

Introduction Steady state heat transfer in cylindrical geometry Steady state external heat transfer modelingdfully buried pipeline Steady state external heat transfer modelingdpartially buried or exposed pipeline 5.5 Transient external heat transfer modelingdfully buried pipeline 5.6 Insulation and heat tracing References

6. Flow assurance issues 6.1 Introduction 6.2 Hydrates 6.3 Wax deposition in pipelines 6.4 Asphaltene deposition in pipelines References Further reading

7. Flow properties affecting steel pipe material assessment 7.1 Introduction 7.2 Low temperature operation 7.3 Ductile fracture propagation analysis 7.4 Post-Weld Heat TreatmentdPWHT References

118 119 121 121 122 124 127 129 132 134 137 141 146 154 155 157 157 158 163 168 173 175 180 183 183 184 199 209 214 216 217 217 217 225 234 240

Contents

vii

8. Special pipelines

243

8.1 Introduction 8.2 CO2 pipelines 8.3 H2 pipelines References

243 244 253 267

9. Case Studies 9A. Precommissioning and commissioning simulations 9a.1 Introduction 9a.2 Precommissioning and commissioning scenarios 9a.3 Pipeline dewatering 9a.4 Pipeline air drying 9a.5 Pipeline first filling by inert slug purging 9a.6 Pig stuck and dislodging References 9B. Development of an overpressure protection system 9b.1 Introduction 9b.2 Pressure control systems and Australian standard requirements 9b.3 GLNG pipeline operating mode and pressure control system 9b.4 Determination of pressure control system setpoints by transient hydraulic simulations 9b.5 Conclusions References 9C. Solar radiationeinduced pressurization 9c.1 Introduction 9c.2 System description 9c.3 Thermal model and tuning 9c.4 Simulation results and discussion References 9D. Multiphase flow through restrictions 9d.1 Introduction 9d.2 Multiphase flow modeling in restrictions 9d.3 Model matching with lab and field data References 9E. Hydrate plug formation analysis in gas-condensate pipe 9e.1 Introduction 9e.2 Model for hydrate and ice plug formation 9e.3 System description 9e.4 Simulation results and discussion References

271 272 272 273 274 276 283 288 291 292 292 293 294 297 302 303 303 303 304 305 307 309 309 309 310 315 317 318 318 319 322 324 326

viii

Contents

9F. Depressurization of flashing liquids 9f.1 Introduction 9f.2 Pipeline depressurization of an HVP liquid 9f.3 Calculation model and sensitivities 9f.4 Simulation results References 9G. Postweld heat treatment experimental verification 9g.1 Introduction 9g.2 Field measurements 9g.3 Measurements and comparison with calculations 9g.4 Conclusions References

Index

327 327 327 329 332 335 335 335 336 337 342 342

343

CHAPTER ONE

Basics of hydrocarbons thermodynamics Contents 1.1 Introduction 1.2 Equation of state 1.3 Other real fluid properties 1.4 The principle of corresponding states 1.5 Equilibrium and stability of one-component fluids 1.6 Vaporeliquid equilibrium for one-component fluids 1.7 Vaporeliquid equilibrium for multicomponent fluids 1.8 Equation of state for multicomponent mixtures 1.9 Phase diagrams of reservoir fluids References Further reading

1 2 4 5 10 14 21 29 32 38 39

1.1 Introduction Hydrocarbons transported by pipeline systems are usually composed by a large number of components; at ambient temperatures, some of them are present in the gas phase, while others are liquids. Some heavier components may also appear as solid in some critical operating conditions (hydrates, waxes). Water is often present in the transported fluid mixture, and it is usually treated as a separate phase, even if a hydrocarbon liquid is present. The behavior of hydrocarbon mixtures depends on their compositions as well as the pressure and temperature conditions encountered.

The thermodynamics of mixtures is determined by the contributions of single components; for this reason, the properties of single components are covered first. The treatment begins with the description of the equations of state, defining the relationship between the main fluid properties; starting with the simplest approach based on ideal gas model, more complex and realistic models are introduced. Thermodynamic equilibrium conditions are discussed by considering the relevant trends of entropy

Flow Analysis for Hydrocarbon Pipeline Engineering ISBN: 978-0-12-822466-3 https://doi.org/10.1016/B978-0-12-822466-3.00001-1

© 2022 Elsevier Inc. All rights reserved.

1

j

2

Flow Analysis for Hydrocarbon Pipeline Engineering

and free energies, for specific systems. Stable equilibrium occurs when it is independent on time and its previous history, and it is able to resist to small thermodynamic parameter fluctuations. These concepts are applied to the analysis of vaporeliquid phase transformations, referred to both single components and binary mixtures. The discussion is further extended to multicomponent mixtures, as typical of hydrocarbon fluids. The description of their properties is developed by treating the interaction of different components, the phase behavior including a vaporeliquid coexistence condition occupying a finite region of the pressureetemperature diagram, retrograde condensation phenomena, the different behaviors of reservoir fluid as depending on their dominant components, and the interaction with the water phase.

1.2 Equation of state The thermodynamic properties of a fluid can be defined by using a volumetric equation of state, providing information about the relationship between pressure, temperature, and volume (or density).

The simpler EoS (Equation of Sate) is that relevant to the ideal gas model, which assumes no interaction among fluid particles and neglects their volume. It is expressed by the following: PV ¼ RT

(1.1)

where: P ¼ fluid pressure V ¼ fluid specific volume R ¼ universal gas constant T ¼ fluid absolute temperature In term of compressibility factor Z, defined in this way, Z¼

PV RT

(1.2)

the ideal gas model is represented by the condition Z ¼ 1.

The compressibility factor is a parameter giving the indication of the deviation of a real gas thermodynamic behavior from the ideal gas approximation; actually, the real substances can be described by Eq. (1.1) only in a limited range of conditions, i.e., at low pressures and moderate temperatures. Hence, other equations of state have been developed on the basis

Basics of hydrocarbons thermodynamics

3

of experimental data of real fluids; the most important from a historical point of view is the classical Van der Waals equation, given by the following:

a P þ 2 ðV bÞ ¼ RT (1.3) V In this expression, two parameters have been introduced to represent real fluid effects, namely: a ¼ it is a parameter relevant to the Van der Waals intermolecular forces, which are included as an equivalent pressure term proportional to the inverse square of the volume b ¼ it is a parameter representing the finite volume of molecules Eq. (1.3) is a so-called cubic EoS, since a cubic algebraic equation must be solved in order to obtain the volume.

The Van der Waals equation is not very accurate, but it was the first equation capable of predicting the transition between vapor and liquid; furthermore, it has also been the prototype for modern, more accurate equations of state as the PengeRobinson one (Peng and Robinson, 1976), given by: P¼

RT aðT Þ

V b V 2 þ 2bV b2

(1.4)

or the SoaveeRedlicheKwong (Soave, 1972): RT aðT Þ

(1.5) V b V ðV þ bÞ The parameters of the above equations can be obtained or by fitting the equations to PVT data for the fluid of interest or by using general relations between them and critical point properties (see Section 1.4).

All the above expressions can be written in the following form: P¼

Z 3 þ aZ 2 þ bZ þ g ¼ 0 (1.6) This is again a cubic algebraic equation that must be solved to get the compressibility factor Z, as must be for the fluid volume.

All cubic equations of state are approximate; in general, they provide a reasonable description of thermodynamic properties of vapor and liquid phases of hydrocarbons, and of vapor region only for many other pure fluids.

The PengeRobinson and SoaveeRedlicheKwong are, at present, among the more popular cubic equations of state in oil and gas applications.

4

Flow Analysis for Hydrocarbon Pipeline Engineering

A different type of equation of state is the virial equation (Sandler, 1999): PV BðT Þ CðT Þ þ ::: ¼1 þ þ RT V V2

(1.7)

where BðT Þ and CðT Þ are the temperature-dependent second and third virial coefficients. This expression is of theoretical interest since it can be derived from statistical mechanics with explicit formulations of the virial coefficients in terms of particle interactions. It is a power series expansion in specific volume about the ideal gas result. With a sufficient number of coefficients, the virial equation can predict the vapor phase properties with good precision, but it is not applicable to the liquid phase. However, even in the case of the gases, it is not recommended for pressure above 10 bar.

1.3 Other real fluid properties One of the quantities appearing in the fundamental thermodynamic relationships is the heat capacity. It is possible to show that, given a volumetric EoS, and heat capacity data as a function of temperature at a single value of pressure P1 or volume V1, the value of the heat capacity in any other state (represented by pressure P2 or volume V2) can be computed by the following (Sandler, 1999): ZP2 2 v V dP CP ðP2 ; T Þ ¼ CP ðP1 ; T Þ T vT 2 P

(1.8)

ZV2 2 v P CV ðV2 ; T Þ ¼ CV ðV1 ; T Þ T dV vT 2 V

(1.9)

P1

V1

where CP and CV are the constant pressure and constant volume heat capacities, respectively.

In practice, heat capacity data are usually collected for low pressure conditions or large specific volumes where all fluids are ideal gases. Hence, if P1 or V1 are taken as 0 and N, respectively in Eqs. (1.8) and (1.9) above, it is possible to write: CP ðP; T Þ ¼ CP* ðT Þ T

ZP 2 v V dP vT 2 P 0

(1.10)

5

Basics of hydrocarbons thermodynamics

CV ðV ; T Þ ¼ CV* ðT Þ T

ZV 2 v P dV vT 2 V

(1.11)

N

where the asterisk denotes the ideal gas heat capacity. Usually, the temperature dependence of the ideal gas heat capacity is given in polynomial form as follows: CP* ðT Þ ¼ A þ BT þ CT 2 þ DT 3

(1.12)

Other thermodynamic variables that allow to calculate the deviations of the real fluid behavior from the ideal gas status are the “Departure Functions.”

In particular, it is possible to show that, for what concerns the enthalpy H and the entropy S, their changes for a real fluid are equal to that of an ideal gas undergoing the same transformation plus the departure of the fluid from the ideal gas behavior at the end state less the departure from the ideal gas behavior at the initial state. These “Departure Functions” can be computed from the equation of state, based on the following: ZP H H

IG

0

S S

IG

V T

¼

vV vT

dP

(1.13)

P

ZP vV R ¼ dP vT P P

(1.14)

0

1.4 The principle of corresponding states The analysis presented in Sections 1.2 and 1.3 has shown that it is possible to calculate the thermodynamic characteristics of a real substance given only the ideal gas heat capacity and the volumetric EoS. However, the necessary information on the EoS is not always available for all fluids.

Hence, it is possible to resort to the principle of corresponding states, which allows to predict thermodynamic properties of fluids from generalized property correlations based on experimental data for similar fluids.

We concentrate our attention on a volumetric EoS, which is determined by the intermolecular interactions. From the study of molecular behavior, it has been found that molecules can be grouped into classes,

6

Flow Analysis for Hydrocarbon Pipeline Engineering

such as spherical molecules, nonspherical molecules, molecules having a permanent dipole, and so on, and that within any one class molecular interactions are similar.

It has been also found that all the members of a class obey the same volumetric EoS, i.e., the volumetric data of each member are fitted by simply changing the parameters of the EoS. The fact that several different molecular species may be described by a volumetric EoS of the same form suggests that it might be possible to construct generalized correlations for both the EoS and density-dependent contribution to enthalpy, entropy, and other thermodynamic variables. The first historical generalized correlation arose from the study of the Van der Waals EoS. Fig. 1.1 shows the isotherms of this equation in a P versus V diagram, for five values of temperature including the critical temperature TC, which is defined as the maximum temperature at which a liquid phase can exist. To be noted that this definition holds for a single component fluid only.

It can be observed in this figure that the isotherms with temperature below the critical one exhibit a non-monotone behavior, with a local minimum followed by a local maximum over part of the specific volumed pressure range. Such trend is associated with a liquidevapor transition.

While this behavior is absent for T > TC, in such a way that the liquid phase cannot exist in this range, in the critical temperature curve, the two extremes coincide, and this point (called critical point) is an inflection point,

T1 T2 TC T3

P

T4

V

Figure 1.1 Isotherms of the Van der Waals EOS in a plane (P,V).

Basics of hydrocarbons thermodynamics

7

where the first and second derivative of P with respect to V vanishes simultaneously. This is expressed analytically by the following requirements:

2 vP v P ¼ 0 and ¼ 0 at PC and VC (1.15) vV TC vV 2 TC By replacing Eq. (1.3) in the relations (1.15), the following expressions of the Van der Waals parameters a and b are obtained: VC 9VC RTC a¼ (1.16) 3 8 Using (1.16) in the Van der Waals Eq. (1.3), it is possible to get the pressure PC and the compressibility factor ZC at the critical point: b¼

a P C VC 3 ZC ¼ ¼ (1.17) 2 RTC 27b 8 By using the above expressions and by defining the dimensionless variables called reduced temperature Tr, reduced pressure Pr and reduced volume Vr PC ¼

Pr ¼

P T V ; Tr ¼ ; Vr ¼ PC TC VC

(1.18)

the following form of the Van der Waals equation of state is obtained:

3 Pr þ 2 ð3Vr 1Þ ¼ 8Tr (1.19) Vr From (1.19) it is concluded that all the fluids obeying the Van der Waals equation of state have the same numerical value of reduced volume for given values of Pr and Tr. Two fluids having the same values of reduced pressure and temperature are said to be in corresponding states.

The principle of the corresponding states has been historically represented by drawing the compressibility factor Z as function of reduced pressure and temperature, as shown in Fig. 1.2, where the compressibility data for several fluids have been reported.

Fig. 1.2 demonstrates that the idea of the corresponding states is reasonable, since the general trend of the data confirms the similarity represented by (1.19). However, it can be observed that compressibility factors for inorganic fluids are almost always below those for hydrocarbons. Furthermore, if (1.19) were universally valid, all fluids would have the same value of the critical compressibility ZC ¼ 0.375, as shown by (1.17), while for most fluids, the critical compressibility is in the range 0.23e0.31.

8

Flow Analysis for Hydrocarbon Pipeline Engineering

Figure 1.2 Compressibility factor for several fluids as a function of reduced pressure and temperature. From Su and Chang (1946).

These deviations have led to the development of more complicated corresponding states principles. The simplest generalization is based on the adoption of a family of different relations Z ¼ Z(Pr,Tr), with different values of ZC, i.e., the critical compressibility factor is used as an additional parameter of the compressibility law: Z ¼ ZðTr ; Pr ; ZC Þ (1.20) In fact, other quantities have been considered as additional corresponding states parameters, since the critical compressibility factor ZC is not known with enough accuracy for many substances.

Pitzer (1995) has proposed the so-called acentric factor u as third correlative parameter, which is defined in this way: u ¼ 1:0 Log10 ðP vap ðTr ¼ 0:7Þ = PC Þ Here P vap ðTr ¼ 0:7Þ is the vapor pressure of the fluid at Tr

(1.21)

¼ 0:7.

Even these extensions of the corresponding state concept have been found not suitable to represent the thermodynamic behavior of some kinds of molecules, in particular that having dipoles or quadrupoles. Hence, this application remains limited to some classes of molecules only, as in case of most hydrocarbons.

9

Basics of hydrocarbons thermodynamics

The modern version of the corresponding states idea is the use of generalized EoS. The Van der Waals equation is the first example of application of this approach. In fact, the expressions (1.16), rewritten in terms of the critical pressure and temperature in this way, a¼

27R2 TC2 RTC b¼ 64PC 8PC

(1.22)

demonstrate that the thermodynamic properties of a class of fluids with parameters a and b that have not been derived from a set of experimental data can be described by knowing the fluid critical properties only. As it was said before, the Van der Waals equation is not accurate enough, and other more sophisticated equations are used.

In particular, we consider here the PengeRobinson equation already presented in Section 1.1, by showing that its generalized form is given by the following: P¼

RT aðT Þ 2 V b V þ 2bV b2

(1.23)

R2 TC2 aðT Þ PC

(1.24)

with: aðT Þ ¼ 0:45724

RTC PC rffiffiffiffiffiffiffi 2

T aðT Þ ¼ 1 þ c 1 TC b ¼ 0:0778

(1.25) (1.26)

c ¼ 0:37464 þ 1:54226u 0:27u2 (1.27) The above relationships were obtained by Peng and Robinson in order to improve the predictions of the boiling point pressure versus temperature, and the function aðT Þ was chosen by fitting the vapor pressure values for many fluids. With respect to the Van der Waals scheme, the generalized PengeRobinson approach uses one parameter more other than PC and TC, namely the acentric factor u.

10

Flow Analysis for Hydrocarbon Pipeline Engineering

This equation can be used to calculate not only the compressibility factor, but also the departure functions for the other thermodynamic properties.

Another very popular cubic equation of state used in the oil and gas industry is the SRK (SoaveeRedlicheKwong): P¼

RT aðT Þ

V b V ðV þ bÞ

(1.28)

with: aðT Þ ¼ 0:42748

R2 TC2 aðT Þ PC

RTC PC rffiffiffiffiffiffiffi 2

T aðT Þ ¼ 1 þ c 1 TC b ¼ 0:08664

(1.29) (1.30) (1.31)

c ¼ 0:37464 þ 1:54226u 0:27u2 (1.32) The current industrial practice is based on the application of the corresponding states concept, by assuming that different fluids are described by the same generalized equations of state, and that their thermodynamic behavior in an extended pressure and temperature range is described by knowing the generalized parameters only. Such kind of equations can give a very accurate thermodynamic picture of many substances, in particular hydrocarbons.

1.5 Equilibrium and stability of one-component fluids The equilibrium state of a closed thermodynamic system can be derived by using the energy and entropy balances derived from the first and second principles of thermodynamics: dU dV ¼ Q_ P dt dt

(1.33)

dS Q_ ¼ þ S_ irr dt T

(1.34)

11

Basics of hydrocarbons thermodynamics

where: U ¼ specific internal energy Q_ ¼ heat transfer rate S_ irr ¼ rate of internal generation of entropy by irreversible processes 0 In (1.33) only work due to deformation of the system boundaries is considered.

For an adiabatic (Q_ ¼ 0) and constant volume system, the system of energy and entropy balances equations becomes dU ¼0 dt

(1.35)

dS _ (1.36) ¼ Sirr 0 dt Since the entropy function can only increase during the approach to equilibrium due to (1.36), the entropy must be a maximum at equilibrium.

Hence, the equilibrium criterion for a closed and isolated system is as follows: S ¼ maximum (1.37) This principle can be illustrated by referring to the simple system shown in Fig. 1.3.

The system is composed by two subsystems containing different amounts of the same molecular species of particles of a single-component fluid (N represents the number of moles). They are connected by a channel of infinitesimal volume equipped by an ideal valve opening instantaneously. The system is assumed adiabatic and constant volume, exchanging neither mass neither energy with the external environment. However, the two subsystems can exchange heat and mass across the communication channel when the valve is open. For the global system, the total number of moles

N1, S1, U1, V1

N2, S2, U2, V2

Figure 1.3 Isolated nonequilibrium system.

12

Flow Analysis for Hydrocarbon Pipeline Engineering

N, the total internal energy U, the total volume, and the total entropy are given by the addition of the two subsystems contributions: N ¼ N1 þ N2

(1.38)

U ¼ U1 þ U 2

(1.39)

V ¼ V 1 þ V2

(1.40)

S ¼ S1 þ S2 (1.41) By using the fundamental thermodynamic relationships, the entropy change of each i-th subsystem after communication and mixing of particles can be expressed as a function of changes in the number of moles, volumes, and internal energy by the following:

vSi vSi vSi 1 Pi Gi dSi ¼ dUi þ dVi þ dNi ¼ dUi þ dVi dNi vUi vVi vNi Ti Ti Ti (1.42) where Gi is the i-th subsystem molar Gibbs energy.

The entropy change dS of the overall system can be easily obtained by summing the two subsystems contributions, by considering that the total quantities given by (1.38), (1.39), and (1.40) are constant:

1 1 P1 P2 G1 G2 dS ¼ dU1 þ dV1 dN1

(1.43) T1 T2 T1 T2 T1 T2 Since the entropy must be a maximum at equilibrium, the (1.25) expression must vanish for system variations, thus obtaining the following: 1 1 ¼ T1 T2

0

T1 ¼ T2

(1.44)

P1 P2 ¼ T1 T2

0

P1 ¼ P2

(1.45)

G1 G2 ¼ 0 G1 ¼ G 2 (1.46) T1 T2 Hence, the equilibrium condition for an adiabatic isolated system states that all its subsystems must have the same temperature, the same pressure, and the same molar Gibbs energy.

The above discussion is based on the condition dS ¼ 0, which is necessary but not sufficient for S to reach a maximum value. The further

13

Basics of hydrocarbons thermodynamics

condition d 2 S 0 assures that a maximum entropy value, giving a true equilibrium state, has been identified. On the contrary, a minimum of the entropy corresponds to an unstable state. Therefore, the sign of d2 S determines the stability of the thermodynamic state found from dS ¼ 0.

It is possible to determine the equilibrium and stability for other kinds of systems also.

For instance, for a closed system kept at constant temperature and volume, the energy and entropy balance equations become dU ¼ Q_ dT

(1.47)

dS Q_ (1.48) ¼ þ S_ irr dt T By combining the above equations, the following relationship, written in term of the molar Helmholtz free energy F, is obtained: dðU TSÞ dF (1.49) ¼ ¼ T S_ irr 0 dt dt Since the molar Helmholtz free energy F can only decrease during the approach to equilibrium due to (1.49), it must be a minimum at equilibrium.

Hence, the equilibrium criterion for a closed system kept at constant temperature and volume is the following: F ¼ minimum (1.50) For a closed system kept at constant temperature and pressure, the energy and entropy balance equations become the following: dU dðPV Þ ¼ Q_ dt dt

(1.51)

dS Q_ (1.52) ¼ þ S_ irr dt T By combining the above equations, the following relationship, written in term of the molar Gibbs free energy G, is obtained: dU dðPV Þ dðTSÞ dG þ

¼ ¼ T S_ irr 0 dt dt dt dt

(1.53)

which leads to the following equilibrium criterion: G ¼ minimum

(1.54)

14

Flow Analysis for Hydrocarbon Pipeline Engineering

Table 1.1 Equilibrium and stability conditions for selected systems.

Equilibrium Stability Kind of system Constraints condition condition

Isolated, adiabatic system with constant volume Isothermal, closed system with constant volume Isothermal, isobaric closed system

U ¼ constant, V ¼ constant

S ¼ maximum, dS ¼ 0

d2 S 0

T ¼ constant, V ¼ constant

F ¼ minimum, dF ¼ 0

d2 F 0

T ¼ constant, P ¼ constant

G ¼ minimum, dG ¼ 0

d2 G 0

Actually for all the systems considered here, the application of the equilibrium criteria determines that they are at uniform temperature, pressure, and molar Gibbs free energy.

The equilibrium and stability conditions for the system under consideration, written for a single-component fluid, are summarized in Table 1.1. It is possible to show that they are also valid for multicomponent fluids.

From the stability criteria listed in Table 1.1, three fundamental thermodynamic inequalities can be derived for systems considered in a stable equilibrium state:

vP vU CV > 0; < 0; > 0 (1.55) vV T vT

1.6 Vaporeliquid equilibrium for one-component fluids Now we are going to analyze the equilibrium of two phases (vapor and liquid) on the basis of the discussion exposed in Section 1.5. In Fig. 1.4, some isothermal curves in a plane (P,V) are shown, as given by a typical cubic EoS. These curves are represented for increasing temperature (T1 < T2 < T3 < T4 < T5). We can observe that the isotherms labeled by T1 and T2 do not have a monotone trend, but they present a minimum followed by a maximum in a certain region of the (P,V) diagram. Therefore, vP Þ < while for the high-temperature isotherms the stability condition ðvV T

0 is fulfilled everywhere, this is not true for the low-temperature isotherms.

Actually, the regions of the diagram where the stability violation occurs are not physically realizable and cannot be observed in practice. Thus, in this

15

Basics of hydrocarbons thermodynamics

T1 T2 T3 T4

P

T5

V

Figure 1.4 Isotherms on a plane (P,V) predicted by a cubic equation of state.

particular thermodynamic region, the cubic equations of state fail to predict the real behavior of the fluid, and their trend must be replaced by another physical description. The isotherm at temperature T3 has a single point for which the first and second derivatives of pressure which respect to volume vanish simultaneously: this is called critical point and T3 is the critical temperature (usually identified as TC).

A low-temperature isotherm is represented alone in Fig. 1.5; it is shown that the intersection of a horizontal line at constant pressure with the curve gives three possible values for the volume of the fluid, V 0 , V 00 , and V 000 . One of these, V 00 , is on the part of the isotherm that is unphysical for the above mentioned stability reasons. The other two values have a physical meaning instead, and they represent the volumes of the liquid and vapor phase, respectively, at the specified pressure and temperature.

The true thermodynamic behavior in the region between V 0 and V 000 is shown in Fig. 1.6, where a phase change occurs along the considered isotherm isobar represented by the horizontal section which boundary volumes are the single-phase liquid and vapor volumes labeled as V L and V V .

This is the vaporeliquid coexistence region. The two-phase mixture specific volume V M varies in this region since, although the specific volumes of the single phases are fixed, the vapor fraction a in the mixture varies from 0 to 1, thus giving the following: V M ¼ aV V þ ð1 aÞV L

(1.56)

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Flow Analysis for Hydrocarbon Pipeline Engineering

P

Isotherm T1 < T3 Isobaric Line

B A

V'

V''

V'''

V

Figure 1.5 Low-temperature isotherm in a plane (P,V) predicted by a cubic EoS.

P

Isotherm T1 < T3

V

V

L

V

V

Figure 1.6 Real low-temperature isotherm in a plane (P,V) including the vaporeliquid phase transition.

Thus, the isotherm section predicted by a cubic EoS in the two-phase region of a single-component fluid must be replaced by the isobaric horizontal line shown in Fig. 1.6.

The question that remains is the value of pressure corresponding to the vaporeliquid coexistence region. The criterion to be applied in order to find this pressure is the equality of the two phases Gibbs free energies, as

17

Basics of hydrocarbons thermodynamics

derived from Section 1.5. From fundamental thermodynamic relationships, the free Gibbs energy variation DG along an isotherm is given by the following integral: ZP 00 DG ¼

VdP

(1.57)

P0

Thus, by using the equation of state which provides the relationship among V and P, the value of the phase change pressure P f is obtained by the following: ZP 00 GV GL ¼ 0 ¼

ZP 00 VdP þ

Pa

ZP a VdP þ

P0

VdP

(1.58)

P 00

where P 0 and P 00 are the minimum and maximum pressure values in the phase transition region. Graphically, this corresponds to the equality between the regions A and B of Fig. 1.5, comprised between the horizontal transition line and the isotherm curve given by the cubic EoS. Fig. 1.4 must be replaced by Fig. 1.7, showing the true thermodynamic behavior associated to the phase transition, in which the two-phase mixture region is included inside a domeshaped envelope curve crossing the low-temperature isotherms. For

T1 T2 TC T4 T5

P

Liquid-Vapour Coexistence Region

V

Figure 1.7 Real isotherms in a plane (P,V) including the vaporeliquid coexistence region.

18

Flow Analysis for Hydrocarbon Pipeline Engineering

Liquid-Vapour Transition Curve

P

Critical Point (TC , PC )

T

Figure 1.8 Liquidevapor transition curve in a plane (P,T).

temperatures above the critical temperature, no phase transition can occur, and the fluid is said to be in the supercritical condition or dense phase gas.

A typical phase transition curve in a (P,T) diagram has the form shown in Fig. 1.8, ending with the critical point identified by the couple (TC, PC).

Again, for temperatures above the critical temperature, no phase transition can occur (supercritical fluid). For temperatures T < TC and pressures P > PC, the fluid is in the liquid phase, but its thermodynamic properties, close to the critical conditions, are very similar to that of the supercritical gas in such a way they are practically indistinguishable.

The equation P ¼ PðT Þ of the phase transition curve can be derived by applying the equality of the Gibbs free energies relevant to the two phases: G V ðT ; PÞ ¼ G L ðT ; PÞ (1.59) From (1.59), it descends that for small changes of pressure and temperature the changes of the Gibbs free energies are small and equal: dGV ¼ dGL (1.60) From fundamental thermodynamic relationships, the differentials of (1.60) can be expressed in terms of the other thermodynamic quantities in this way: V V dP S V dT ¼ V L dP SL dT

(1.61)

19

Basics of hydrocarbons thermodynamics

And, by indicating with DV ; DS the changes in specific volume and entropy when passing from one phase to the other, from (1.61), we obtain the following: vP DS DH ¼ ¼ vT DV T DV

(1.62)

where DH is the phase transition enthalpy variation, also called latent heat of vaporization (or condensation). The (1.62) is known as Clapeyron equation.

The pressure at which the liquid and vapor phases are in equilibrium is termed vapor or saturation pressure, here indicated as P vap . From (1.62), its trend as function of temperature P vap ¼ f ðT Þ can be obtained. At temperatures for which the vapor pressure is not very high, it results that V V >> V L and DV zV V . Furthermore, if the vapor can be treated as ideal, we have DV ¼ V V ¼ RT P , thus giving the following: dP vap P vap DH vap ¼ dT RT 2

(1.63)

that can be written also as follows: dðln P vap Þ DH vap ¼ RT 2 dT

(1.64)

where DH vap ¼ H V H L is just the latent heat of vaporization.

The integration of (1.64) gives the wanted vapor pressure temperature function; if it is assumed that the temperature dependence of DH vap is negligible, we get the following: ln P vap ¼

DH vap þC RT

(1.65)

where C is an integration constant.

A common correlation used to reproduce the vapor pressure as temperature function is the Antoine equation, which is very similar to (1.65) and it is expressed in this way: ln P vap ¼

D þC T þE

(1.66)

where D and E are correlating parameters.

The discussion reported so far has been concerned with the analysis of stable equilibrium states of a single-component fluid. However, metastable states as the superheated liquid and the subcooled vapor can occur in

20

Flow Analysis for Hydrocarbon Pipeline Engineering

Isotherm T1 < T3 Superheated Liquid

P

Subcooled Vapour

VV

VL

V

Figure 1.9 Metastable states along an isotherm in a plane (P,V).

experiments. In fact, by suitable experimental procedures in which the thermodynamic transformation along an isotherm is realized very slowly, by avoiding as much as possible the onset of bubble nucleation in case of liquid and of droplet nucleation in case of vapor, the one-phase isotherm sections can be prolonged outside the stability region as shown in Fig. 1.9. The liquid belonging to the extended left isotherm portion is said superheated liquid and the vapor belonging to the extended right isotherm portion is said subcooled vapor. Both these states are metastable since they exist as one-phase fluids in a region where the two-phase transition should have had occurred, and if slightly perturbed, the fluid experiences a sudden transition toward the stable condition.

Actually, in the study of liquidevapor equilibria, it is common practice to utilize the so-called fugacity in the equilibrium equation, instead of the Gibbs free energy.

The fugacity f is defined in this way: 0 1

ZP GðT ; PÞ G IG ðT ; PÞ 1 RT f ¼ P exp V ¼ P exp@ dP A RT RT P 0

(1.67) Another quantity correlated to the fugacity is the fugacity coefficient f ¼ Pf .