Thermodynamic Modeling of Hydrogen Sulfide Solubility in Weak

Electrolyte Solutions

Master Thesis

The Technical University of Denmark

Date of delivery: 31st of January 2008

Asger Lindholdt (s021870)

Supervisor: Kaj Thomsen

Problem formulation

I

Problem formulation The title of the projects is: Thermodynamic Modeling of H2S in Weak Electrolyte

Solutions.

The project consists of a theoretical and practical part where the systems H2S, H2S-H2O,

H2S-H2O-NH3 and H2S-H2O-CO2 are treated.

Theoretical part

The theoretical part consists of a literature study with focuses on vapor-liquid equilibrium

(VLE) and gas hydrate formation for the treated systems. General gas hydrate theory

and thermodynamic modeling of gas hydrate formation described by Munck (1988) with

emphasis on hydrogen sulfide is described. The Extended UNIQUAC model is

described. Phase diagrams are described with emphasis on the H2S and H2S-H2O

systems.

Practical part

The practical part consists of estimating parameters for hydrogen sulfide in the gas

hydrate model and estimating parameters for hydrogen sulfide species in the Extended

UNIQUAC for the systems H2S, H2S-H2O, H2S-H2O-NH3.

Time schedule and content of status reports

The project starts the 3rd of September 2007 and is to be handed in the 31st of January

2008. Through out the period of the project three status reports are to be handed in.

Status report 1

The date of delivery is the 12th of October. A literature survey is carried out with focuses

on the H2S and H2S-H2O systems. A general description of gas hydrates, modeling of

gas hydrate formation and VLE for the H2S and H2S-H2O systems are described. The

Extended UNIQUAC model and the article “Computations of the formation of gas

hydrates” by Munck (1988) are described.

Status report 2

The date of delivery is the 16th of November. The parameter estimation in the gas

hydrate model and the Extended UNIQUAC is estimated for H2S and HS- by using

experimental data from the systems H2S and H2S-H2O. The Langmuir constants in the

Problem formulation

II

gas hydrate model are estimated for H2S and for the Extended UNIQUAC model the

volume, surface and interaction parameters are estimated for the H2S species.

Status report 3

The date of delivery is the 15th of January. The parameter estimation in the Extended

UNIQUAC model is carried out for the H2S species in the systems H2S-H2O-NH3 and

H2S-H2O-CO2.

Preface

III

Preface

This thesis of 30 ECTS points is submitted as partial fulfillment of the requirements for

the Masters degree at the Technical University of Denmark. The work was carried out at

the Department of Chemical Engineering, from September 2007 to January 2008 under

the supervision of Associate Professor Kaj Thomsen.

This thesis deals with thermodynamic modeling and a standard graduate course in

thermodynamics should be sufficient to easily understand the major contents of the

report. The different chapters can be read independently, but in most cases the best

understanding is obtained when the previous chapters have been read.

In this work the references used are presented by their last name and year of

publication. The complete reference is then found in the list of references, which is at the

end of the thesis.

I thank my supervisor for guidance as well as all the time he used to discuss matters

concerning my project.

I would also like to thank Post Doc. Philip L. Fosbøl for helping me with the thesis and

my friend Kristian E. Nørgaard for assisting in the proof reading process.

Lyngby, January, 2008

Asger Lindholdt

Abstract

IV

Abstract

The Extended UNIQUAC model and the SRK EoS were successfully applied to the

systems H2S, H2S-H2O, H2S-H2O-gas hydrates, and H2S-H2O-NH3. Extended UNIQUAC

parameters for H2S and HS- were successfully estimated in the systems H2S, H2S-H2O,

H2S-H2O-NH3.

Langmuir constants in the gas hydrate model presented by Munck (1988) were

estimated for H2S. The gas hydrate model, the Extended UNIQUAC model and the SRK

EoS were applied and successfully correlate the experimental data points found in the

literature for the system H2S-H2O where gas hydrates are present.

A major review of the open literature for the systems H2S, H2S-H2O (with and without

gas hydrates), H2S-H2O-NH3, and H2S-H2O-CO2 were carried out. The review includes

collection of a large amount of experimental data and a presentation of the most

important VLE models. A presentation of the differences between the gas hydrate model

presented by Munck (1988) and other authors in the open literature are presented.

Nomenclature

V

Nomenclature

Notation

Symbol Explanation Unit A Avogadro’s number mol-1

A Debye-Hückel parameter kg1/2 mol1/2

A, B Langmuir parameters K b Constant kg mol-1 C Chemically independent

number of components -

C Langmuir adsorption constant

-

∆Cp Heat capacity difference CP Pure component critical

point -

d Density kg m-3 f Fugacity Pa

1f , 2f Functions in Pitzer’s equation

-

F Degrees of freedom - F Faradays constant C mol-1 H Hydrate - ∆H Enthalpy difference J I Ionic strength mol kg water-1 I Ice - K Three-phase critical end

point -

LA Aqueous liquid - LS H2S-rich liquid m Molality mol kg water-1 M Molecular weight Mol kg-1 n Number of moles mol P Pressure Pa P Number of phases - Q Quadruple point - R Universal gas constant -1 -1J mol K S Solid - T Temperature K TP Pure component triple

point

V Molar volume of water (ice or liquid)

m3 mol-1

∆V Volume difference m3

Y Probability of a filled cavity

-

Nomenclature

VI

Z Coordination number -

Greek letters

Symbol Explanation Unit ( ) ( )0 1

,β β Binary interaction parameters in Pitzer’s equation

-

rε Relative permittivity -

0ε Vacuum permittivity C2 J-1 m-1

π Mathematical constant - σ Standard deviation - τ Ternary interaction parameters µ Chemical potential J υ Number of cavities - γ Activity coefficient -

Subscripts

Symbol Explanation A Aqueous Liquid C Combinatorial i Type of cavity K Component L Large cavity m molar 0 Reference temperature

273.15 K R Residual S Small cavity S Sulfide rich liquid w Water

Superscripts

Symbol Explanation α Non-hydrate phase β Hypothetical empty lattice H Hydrate , ,i j k Component , ,i j k

0 Pure ice or liquid water 0 Reference temperature

298.15 K ∞ Infinite solution * Rational, unsymmetrical

Nomenclature

VII

Abbreviation

EoS Equation of state LLE Liquid-liquid-equilibrium LM Levenberg-Marquardt NM Nelder-Mead NP Number of experimental data points PR Peng-Robinson SLE Solid-liquid-equilibrium SLVE Solid-liquid-vapor-equilibrium SRK Soave-Redlich-Kvong VLE Vapor-liquid-equilibrium

Summary

VIII

Summary This thesis deals with thermodynamic modeling of hydrogen sulfide systems containing

the weak electrolyte solutions CO2 and NH3. The Extended UNIQUAC model is used to

describe the excess Gibbs energy in these systems. The Extended UNIQUAC

parameters are estimated for H2S and HS- in the systems H2S, H2S-H2O, H2S-CO2-H2O,

H2S-NH3-H2O.

Langmuir gas hydrate parameters for the model presented by Munck (1988) were

estimated for H2S in the system H2S-H2O.

Chapter 1: Introduction to aqueous electrolytes and the importance of thermodynamic

models containing H2S and weak electrolytes are presented.

Chapter 2: Thermodynamic concepts pertinent to thesis including the chemical potential,

activity coefficients and the Extended UNIQUAC model are described.

Chapter 3: General gas hydrate theory including structures, characteristics of guest

molecules, H2S gas hydrates and thermodynamic models for gas hydrates are

presented.

Chapter 4: Phase diagrams for the systems H2S and H2S-H2O and Gibbs phase rule are

presented.

Chapter 5: Different thermodynamic VLE models from the literature used to describe

weak electrolytes systems containing H2S is described.

Chapter 6: Calculations of approximate concentrations of H2S species in the H2S-H2O

system are presented.

Chapter 7: The principal method used to estimate the Extended UNIQUAC parameters

including description of the objective functions, the Levenberg-Marquardt and the

Nelder-Mead algorithms are presented

Chapter 8: Estimated Extended UNIQUAC parameters for H2S species for the systems

H2S and H2S-H2O are presented

Chapter 9: Estimated Langmuir gas constants for H2S for the H2S-H2O system are

presented.

Chapter 10: The Extended UNIQUAC parameters estimated for the H2S-NH3-H2O

system are presented. It is argued that the very scarce experimental data points for the

system H2S-CO2-H2O are wrong.

Summary

IX

Chapter 11: Is the conclusion, summarizing the results of the project.

Chapter 12: Future work related to the project, which is relevant to investigate is

presented.

Table of Contents

X

Table of Contents Problem formulation ............................................................................................................ I

Theoretical part ............................................................................................................ I

Practical part ................................................................................................................ I

Time schedule and content of status reports ................................................................ I

Status report 1 .............................................................................................................. I

Status report 2 .............................................................................................................. I

Status report 3 .............................................................................................................II

Preface............................................................................................................................... III

Abstract ............................................................................................................................. IV

Nomenclature..................................................................................................................... V

Notation.......................................................................................................................... V

Greek letters .................................................................................................................. VI

Subscripts...................................................................................................................... VI

Superscripts................................................................................................................... VI

Abbreviation ................................................................................................................VII

Summary.........................................................................................................................VIII

Table of Contents............................................................................................................... X

1 Introduction................................................................................................................. 1

2 Thermodynamic model ............................................................................................... 2

2.1 Chemical potential and activity coefficients....................................................... 2

2.1.1 Chemical potential ...................................................................................... 2

2.1.2 Excess chemical potentials and activity coefficients .................................. 3

2.2 The Extended UNIQUAC model........................................................................ 4

3 Gas hydrates................................................................................................................ 9

3.1 General gas hydrate theory ................................................................................. 9

3.1.1 Structure.................................................................................................... 10

3.1.2 Characteristics of Guest Molecules .......................................................... 12

3.2 Hydrogen Sulfide .............................................................................................. 13

3.3 Thermodynamic model for gas hydrates........................................................... 13

4 Phase diagrams.......................................................................................................... 22

4.1 Phase rule .......................................................................................................... 22

4.2 The H2S system................................................................................................. 23

4.3 The H2S-H2O system ........................................................................................ 24

4.3.2 Summary of the three-phase loci .............................................................. 33

5 H2S-H2O-weak electrolyte systems .......................................................................... 35

5.1 Solid-liquid-vapor equilibrium ......................................................................... 35

5.1.1 The H2S-NH3-H2O system........................................................................ 37

5.1.2 The H2S-CO2-H2O System ....................................................................... 38

5.1.3 Vapor-liquid equilibrium models.............................................................. 39

5.1.4 Vapor-liquid equilibrium model by Edwards ........................................... 40

6 Concentration calculations for the H2S-H2O system ................................................ 43

7 Parameter estimation and data description ............................................................... 46

7.1 Minimization..................................................................................................... 46

7.1.1 Levenberg-Marquardt Algorithm.............................................................. 47

Table of Contents

XI

7.1.2 Nelder-Mead Algorithm............................................................................ 47

7.2 Confidence limit for estimated parameters ....................................................... 47

7.3 Collection and review of data ........................................................................... 48

8 H2S and HS- parameter estimation............................................................................ 49

8.1.1 Objective function..................................................................................... 49

8.1.2 Vapor-liquid equilibrium data................................................................... 50

8.1.3 Estimated UNIQUAC parameters............................................................. 51

9 Gas hydrate parameter estimation............................................................................. 60

10 Parameter estimation for ternary systems ............................................................. 65

10.1 Description of the H2S-CO2-H2O system ......................................................... 65

10.2 Parameter estimation of the H2S-NH3-H2O system .......................................... 67

11 Conclusion ............................................................................................................ 79

12 Future work........................................................................................................... 81

References.......................................................................................................................... A

13 Appendices............................................................................................................... i

13.1 Appendix A.......................................................................................................... i

13.1.1 Concentration calculations for the H2O-H2S system ................................... i

Introduction

1

1 Introduction

Accurate knowledge of the phase behavior in aqueous systems containing weak

electrolytes like hydrogen sulfide, ammonia and carbon dioxide is crucial, since these

systems are encountered in a variety of fields. Typical examples are the cleaning of raw

gases in power stations, the production of fertilizers, the oil industry, and in the field of

environmental protection. The volatile weak electrolytes of greatest industrial importance

are ammonia, carbon dioxide and hydrogen sulfide and good models are therefore of

industrial importance for these components. A typical example is the process of

gasification of coal where nitrogen, sulfur and carbon are liberated as ammonia,

hydrogen sulfide and carbon dioxide. The investigated phase equilibrium of these weak

electrolyte gases in aqueous solutions is of interest to the industry when designing the

separation process of these pollutant components.

Many sour reservoir fluids contain hydrogen sulfide and it is in general desirable to avoid

the formation of condensed water to reduce the risk of gas hydrate formation and ice

formation. It is therefore important to have a reliable model to predict the gas hydrate

formation for the binary system hydrogen sulfide and water.

Furthermore, hydrogen sulfide and water are encountered in many natural hydrocarbon

reservoirs. The phase equilibrium of mixtures of these two components is therefore

important in petroleum systems in general. The H2S-H2O system is also encountered in

the important field of geochemistry. Hot, aqueous (hydrothermal) reservoirs that contain

hydrogen sulfide are important in the formation of some sulfide and sulfate minerals. The

hydrogen sulfide-water system is also used to produce heavy water (D2O).

Thermodynamic models describing the phase equilibrium for systems containing weak

electrolytes like hydrogen sulfide, ammonia and carbon dioxide are therefore important

in a wide variety of areas in the chemical industry.

Thermodynamic model

2

2 Thermodynamic model

The thermodynamic models used in this work are the Extended UNIQUAC (Universal

Quasi Chemical), the Soave-Redlich-Kvong (SRK) equation of state (EoS) and the gas

hydrate model presented by Munck (1988).

The Extended UNIQUAC model is used to calculate the activity coefficient in the liquid

phase, the SRK EoS is used to calculate the fugacity in the gas phase and the gas

hydrate model presented by Munck (1988) is used to calculate the gas hydrate formation

(solid). The gas hydrate model is presented in details in section 3.3 (Thermodynamic

model for gas hydrates) even though it is part of the thermodynamic model. The SRK

EoS is not described further in this work, because it is well described in the literature and

an already well-know EoS.

A review of the Extended UNIQUAC model and a minor review of the chemical potential

and activity coefficients are presented in this section.

2.1 Chemical potential and activity coefficients

2.1.1 Chemical potential

For a substance i in a mixture its chemical potential i

µ is defined as the partial molar

derivative of the total Gibbs energy G where the temperature T, pressure P, and amount

of j are held constant.

, , j

i

i T P n

G

nµ

∂≡

∂

(2-1)

The chemical potential is used to determine if the system is at a state of equilibrium

since the chemical potential of each substance is the same at equilibrium. For ideal

solutions the chemical potential is

( )id

,0 lni i iRT xµ µ= + (2-2)

Thermodynamic model

3

where ,0iµ is the chemical potential of component i at standard state, R is the gas

constant, T is the absolute temperature, and i

x is the mole fraction of component i .

2.1.2 Excess chemical potentials and activity coefficients

Mixtures deviate from ideality, and in order to describe this deviation the excess Gibbs

energy is used. Gibbs excess energy is the difference between the chemical potential of

a real solution and that of an ideal solution. The excess chemical potential for

component i is

( )E lni iRTµ γ= (2-3)

where i

γ is the symmetrical activity coefficient of i .

The chemical potential of component i , in a real solution (not ideal), is the summation of

the excess and ideal Gibbs energy

( )id E

,0 lni ii i i iRT xµ µ µ µ γ= + = + (2-4)

where E

iµ is the excess chemical potential. For pure solutions, the mole fraction and the

symmetrical activity coefficient is unity, and the excess term therefore vanishes for pure

solutions. In general the purer the solution is, the more ideal the behavior is (e.g. the

symmetrical activity coefficient equal to unity).

The chemical potential of water in an aqueous solution, expressed by equation (2-4), is

given below.

( ),0 lnw w w wRT xµ µ γ= + (2-5)

The chemical potential for the ion i , expressed by equation (2-4), is as follows.

Thermodynamic model

4

( ),0 lni i i iRT xµ µ γ= + (2-6)

The activity coefficients of a solute, for example ions, are normalized so that the activity

coefficient is unity at infinite dilution. This is achieved by defining a rational,

unsymmetrical activity coefficienti

γ ∗

ii

i

γγ

γ∗

∞=

(2-7)

where i

γ ∞ is the symmetrical activity coefficient at infinite solution and i

γ is the

previously introduced symmetrical activity coefficient. The adjective “unsymmetrical”

refers to the fact that this activity coefficient is unity at infinite dilution where as the

symmetrical activity coefficient is unity for the pure component state.

The excess chemical potential of ion i expressed so that the activity coefficient is unity

at infinite dilution is

( )

( ) ( )

( ),0

,0

,0

ln

ln ln

lni

i i i i

i i i i

i i

RT x

RT RT x

RT x

µ µ γ

µ γ γ

µ γ

∞ ∗

∗ ∗

= +

= + +

= +

(2-8)

where ,0i

µ∗ is rational, unsymmetrical chemical potential of component i .

2.2 The Extended UNIQUAC model

The Extended UNIQUAC model is a local composition model. It is identical to the

original UNIQUAC model (Abrams and Prausnitz 1975, Maurer and Prausnitz 1978) with

the difference that a Debye-Hückel term has been added (Sander 1984, 1986a, 1986b).

For local composition models it is assumed that the composition on a molecular level is

different from the bulk composition. It is assumed that the molecules sizes, shapes, and

interaction energies determine the microscopic (local) composition. The Extended

UNIQUAC model consists of a combinatorial or entropic term, residual or enthalpic and

Debye-Hückel or electrostatic term, where the molecules sizes, shapes and interaction

Thermodynamic model

5

energies are included. The excess Gibbs energy is calculated by addition of the three

terms.

E E E E

Combinatorial Residual Debye-HückelG G G G= + + (2-9)

The combinatorial and the residual terms are identical to the terms used in the traditional

UNIQUAC equation. The combinatorial (entropic) term is

E

CombinatorialGln ln

2

i ii i i

i ii i

zx q x

RT x

φ φ

θ

= −

∑ ∑

(2-10)

where z=10 is the co-ordination number (assumption), i

x is the mole fraction, i

φ is the

volume fraction, and i

θ is the surface area fraction of component i .

The volume and surface fraction of component i is given by the following expressions

i ii

l l

l

x r

x rθ =

∑;

i i

i

l l

l

x q

x qφ =

∑

(2-11)

where

ir and

iq respectively are volume and surface parameters for component i .

The residual (enthalpic) term is

E

ResidualGln

i i k ki

k

x qRT

θ ψ

= − ∑

(2-12)

The shape parameter

kiψ is given by

u u

exp ki iiki

Tψ

− = −

(2-13)

where uki

and uii

are interaction energy parameters. The interaction energy parameters

temperature dependence is

Thermodynamic model

6

( )0u u u 298.15ki ki

t

ki T= + − (2-14)

where 0uki

is the temperature independent interaction parameter at 298.15 K and uki

t is

the temperature dependent parameter. Equation (2-14) is also used for the case i k= .

The electrostatic Debye-Hückel term is

( )E 2Debye-Hückel 1 2 1 2

3

G 4A b Iln 1 bI bI

b 2w wx M

RT

= − + − +

(2-15)

where w

x is the mole fraction of water, w

M is molar mass of water, I is the ionic strength

(def. in eq.(2-18)), b is considered a constant of 1.50 (kg mol-1)1/2 ,and A is the Debye-

Hückel parameter.

The Debye-Hückel parameter A is calculated from the following equation

( )

1 23

3

04 2A r

F dA

N RTπ ε ε

=

(2-16)

where F (C mol-1) is Faradays constant, NA (mol-1) Avogadro’s number, 0ε the vacuum

permittivity (C2 J-1 m-1), R the gas constant ( -1 -1J mol K ), T is the temperature (K), d is

the density (kg m-3) and r

ε the relative permittivity (dielectric constant, dimensionless) of

the solution. The temperature dependent parameters are the density and the relative

permittivity.

Based on the density and the relative permittivity of pure water, the Debye-Hückel

parameter A has been approximated in the temperature range 273.15 K - 383.15 K as

follows.

( ) ( )23 5 1 2 1 21.131 1.335 10 273.15 1.164 10 273.15A T T kg mol

− − − = + ⋅ ⋅ − + ⋅ ⋅ −

(2-17)

The ionic strength is

Thermodynamic model

7

21

2i i

i

I m z= ∑

(2-18)

where

im is the molality of ion i .

By partial molar differentiation of the combinatorial and the residual UNIQUAC terms, the

combinatorial and residual parts of the rational, symmetric activity coefficients are

obtained:

( )ln ln 1 ln 12

C i i i ii i

i i i i

zq

x x

φ φ φ φγ

θ θ

= + − − + −

(2-19)

( )ln 1 lnR k iki i k ki

k K l lk

l

qθ ψ

γ θ ψθψ

= − −

∑ ∑∑

(2-20)

The infinite dilution terms are obtained by setting 1w

x =

( )ln ln 1 ln 12

C i i i w i wi i

w w w i w i

r r rq rqzq

r r r q r qγ ∞

= + − − + −

(2-21)

( ) ( )ln 1 lni i wi iw

R qγ ψ ψ∞ = − − (2-22)

The electrostatic contributions to the symmetric water activity coefficients and the

unsymmetrical ionic activity coefficients are obtained by partial molar differentiation of

the extended Debye-Hückel term, giving the following expressions.

( ) ( )

( ) ( )

DH 3 2 1 2

w w

3

2ln

3

3 11 2ln 1

1

M AI bI

x x xx x

γ σ

σ

=

= + − − +

+

(2-23)

Thermodynamic model

8

( )1 2

DH 2

1 2ln

1i i

AIz

bIγ ∗ = −

+

(2-24)

The activity coefficient for water in the Extended UNIQUAC model is calculated by

summation of the combinatorial, residual and Debye-Hückel term.

( ) ( ) ( ) ( )w w w

C R DH

wln ln ln lnγ γ γ γ= + + (2-25)

The activity coefficient for ion i , obtained as the rational unsymmetrical activity

coefficient, is found by summation of the different contributions.

( ) ( ) ( ) ( ) ( ) ( )w

C C R R DHln ln ln ln ln lni i i i i

γ γ γ γ γ γ∗ ∞ ∞= − + − + (2-26)

To summarize, the activity coefficients for water and ions in the liquid phase are

calculated by use of the Extended UNIQUAC model, that is a local composition model

based on size, volume and interaction parameters (Thomsen 2005).

Gas hydrates

9

3 Gas hydrates

Systems containing hydrogen sulfide and water form gas hydrates under specific

conditions and this give rise to engineering problems in the chemical industry. Gas

hydrates can for example cause problems when plugging transmission lines and process

equipment. The system H2S-H2O is often encountered as a part of the natural gas

systems (Guo et al. (2003) and exhaustion gas of coal-fired power plants. A reliable

model able to predict the formation of gas hydrates is therefore important when, for

example, designing processing equipment and preventing plugging of pipelines.

In this section a general introduction to gas hydrates, with emphasis on hydrogen sulfide

gas hydrates, and the model used to calculate the gas hydrate formation by Munck

(1988) are presented.

3.1 General gas hydrate theory

Gas hydrates are ice-like crystalline compounds of water and gases of a rather modest

size such as, for example, light natural gas components. Many gases of small molecular

size form hydrates by becoming trapped in cavities in solid water. The crystal is held

together by hydrogen bonding between the water lattice and the trapped molecules. The

gas hydrates belong to a group called Clathrate hydrates. The name comes from

clathratus meaning enclosed or protected by cross bars of grating (Cady 1983).

The formation of gas hydrates is favored at high pressures and low temperatures. The

gas hydrates may form at temperatures below approximately 310 K. There must be a

sufficient amount of water of water present, but also not too much water present, in order

for gas hydrate formation to occur. Some of the most common gas hydrates formers, or

guest molecules as they are often named, are methane, ethane, propane, iso-butane,

butane, nitrogen, carbon dioxide, and hydrogen sulfide. Highly soluble gases such as for

example ammonia are not known to form gas hydrates. As gas hydrates may exist far

above the freezing point of water they can cause plugging in transmission lines and

production processing equipment. Adding an inhibitor may lower the hydrate formation

temperature. Inhibition is quite analogous to freezing point depression and the

compounds causing the largest freezing-point depressions for water are also the most

powerful inhibitors. Alcohols, glycols and salts are examples of good inhibitors (Munck

1988; Sloan 1990).

Gas hydrates

10

3.1.1 Structure

The hydrates are known to form the more common cubic structures I and II and the more

rare hexagonal structure H. The hydrate structure is stabilized by the van der Waals

forces arising from the guest molecule that occupies the cavity in the ice-like structure of

the hydrate. The pure natural gas components nitrogen, propane, and isobutene are

known to form structure II while methane, ethane, carbon dioxide, and hydrogen sulfide

all form structure I.

The unit cell of structure I contains 46 water molecules and has form as a cube with side

length of 12 Å. The total number of cavities is eight per unit cell. It is comprised of two

“small” 12-hedral and six “large” 14-hedral cavities per unit cell. The 12-hedral cavity

consists of all regular pentagons while the 14-hedral cavity consists of twelve sides that

are pentagonal and two (oppositely positioned) hexagonal sides. The 12-hedral and 14-

hedral are therefore respectively denoted 512 and 51262. If all eight cavities of the unit cell

were filled with the guest molecule M, the ratio of water and guest molecules would be

46 8 5.75= . The empirical formula for the structure I is therefore 2M 5.75H O⋅ .

The unit cell of structure II contains 136 water molecules and has form as a cube with

side length 17.3 Å. The total number of cavities is 24 per unit cell. It is comprised of

sixteen “small” 12-hedral cavities and eight 16-hedral cavities. The former are like those

in structure I. The latter consists of four hexagonal sides and twelve pentagonal sides

and denoted 51264. If all of the cavities were filled with the guest molecules, the ratio of

water and guest molecules would be 224M 136H O⋅ . The empirical formula is therefore

224M 136H O⋅ or 22M 5 H O

3⋅ (Sloan 1990).

Gas hydrates

11

Table 3-1. Description of gas hydrates with structure I and II (Sloan 1990).

Property Structure I Structure II

Cavity Small Large Small Large

Description 512 62 512 64

Number of

Cavities per unit

cell

2

6

16

8

Average Cavity

Radius / [Å]

3.91 4.33 3.902 4.683

Number of H2O

molecules per

unit cell

46

136

Crystal System Cubic Cubic

Lattice

Description

Body Centered Diamond

The unit cell of structure H contains 34 water molecules and the total number of cavities

is 6 per unit cell. It is comprised of three smaller 12-hedral cavities, two intermediate

cavities formed of three squared sides, six pentagonal sides and three hexagonal sides

and a large cavity comprised of twelve pentagonal sides and eight hexagonal sides. The

smaller cavity is identical to the one in structure I. The intermediate structure and the

larger structure are respectively denoted 435

66

3 and 5

126

8. The empirical formula

is 22M 5 H O

3⋅ .

The structure of the different hydrate types and the number of times each cavity occurs

in the complete structure is seen in Figure 3-1. Structure I is for example comprised of

the cavity 512 two times and six times the cavity 51262 (Sloan 1998).

Gas hydrates

12

Figure 3-1. The structure of the different hydrate types and the number of times each cavity occurs in the complete structure is seen. For example is structure I comprised of two times 5

12 and six times 5

126

2. A guest molecule in the form of methane is seen

occupied in a 12-hedral cavity (Sloan 1998).

3.1.2 Characteristics of Guest Molecules

The guest molecule is classified by its chemical nature, its size and to a lesser extent its

shape. The size of the guest molecule is directly related to the hydrate number and, in

most cases, to its non-stoichiometric value. Gas hydrates are non-stoichiometric and the

degree of saturation depends (number of guest molecules per cavity) on the temperature

and pressure (Carroll 1991). The fractional occupancy for the smaller cavities (512) from

structure I and II usually varies between 0.7 and 0.9. The gas hydrates stoichiometric

variation causes them to be called non-stoichiometric in order to distinguish them from

the stoichiometric salt hydrates.

The guest molecule must not contain either a single strong hydrogen-bonding group or a

number of moderately strong hydrogen bonding groups if gas hydrates are to be formed.

The natural gas components do not make hydrogen bonds (Sloan 1990).

Gas hydrates

13

3.2 Hydrogen Sulfide

Hydrogen sulfide forms structure I. Hydrogen sulfide is of such a size that it can occupy

both cavities (512 and 62) in structure I. If there is a mixture of gases forming structure II

then hydrogen sulfide can also be part of structure II. Of the components commonly

found in natural gas the hydrate of hydrogen sulfide forms at the lowest pressure and

persist to the highest temperature.

Hydrogen sulfide can be classified as a water-soluble acid gas and not a hydrophobic

compound like most other guest molecules. The hydrogen sulfide gas hydrate properties

are summarized in Table 3-2 (Sloan 1990).

Table 3-2. Hydrogen sulfide gas hydrate properties (Sloan 1990).

Component Hydrogen sulfide gas hydrate

Hydrate structure Structure I

Crystal System Cubic

Lattice Structure Diamond

Compound classification Water-soluble acid gas

Theoretical structure 2 2H S 5.75H O⋅

H2S diameter [Å] 4.58

3.3 Thermodynamic model for gas hydrates

The thermodynamic model presented by Munck (1988) is used in this work to compute

the gas hydrate formation. A review of the relevant parts of the article is therefore

presented in this section.

The condition for phase equilibrium in a closed system is that each component must

have the same chemical potential. The equilibrium condition of the model is based on

the chemical potentials of the hydrate phase H and a non-hydrate α-phase, which is

either ice (I) or an aqueous solution (LA).

Gas hydrates

14

H αµ µ= (3-1)

The chemical potential of water in the hydrate phase is calculated from the van der

Waals and Platteeuw (1959) adsorption model

H β

i

i

RT ln 1 Y

1,2,..., Number of cavities

1,2,..., Number of components

i K

K

i

k

µ µ υ

= + −

=

=

∑ ∑

(3-2)

where β refers to a hypothetical empty lattice state, i

υ is the number of cavities of type

i , and KYidenotes the probability of a cavity of type i being occupied by a hydrate-

forming molecule of type k.

The probability is calculated according to the Langmuir adsorption theory

Kk

j j

j

CY

1 C

j=1,2,...,Number of components

i ki

i

f

f=

+∑

(3-3)

where k

f is the fugacity of hydrate-forming component k , KCi is the adsorption

constant at the specified temperature, and j is the number of hydrate forming

components.

The Langmuir constants are considered to be temperature-dependent and are

calculated from a two-parameter approach

Ki KiKi

A BC exp

T T

=

(3-4)

where KiA and KiB are constants for cavity of type i and molecule of type k . The

parameters KiA and KiB are estimated from experimental data points.

Gas hydrates

15

There are other existing methods to estimate the adsorption coefficient KiC such as, for

example, the Kihara potential model used by John et al. (1985). The Kihara potential

model has a more theoretical approach, but it is somewhat more complicated than the

convenient two-parameter approach presented here and used by Munck (1988).

The chemical potential of water in the α-phase (ice or an aqueous solution) may in

general be written as

αα 0 w

0

w

RT lnf

fµ µ

= +

(3-5)

where 0µ is the chemical potential of pure water as ice, liquid water or gaseous water at

temperature T and pressure P, w

fα is the fugacity of water in the α-phase and 0

wf is the

fugacity of pure ice, liquid water or gaseous water at the reference temperature.

The difference between the chemical potentials of pure water, as an empty lattice, and

as ice or liquid water at pressure P and temperature T is obtained by combining equation

(3-1), (3-2) and (3-5).

αβ 0 w

Ki0i Kw

ln ln 1 Yi

fRT RT

fµ µ υ

− = − −

∑ ∑

(3-6)

The temperature and pressure dependence for the chemical potential of pure water, as

an empty lattice, and liquid water (equation (3-6)) can be derived from the well-known

Gibbs-Helmholtz equation. The Gibbs-Helmholtz equation is as follows (Atkins 2002).

2

P

G H

T T T

∂ ∆ ∆ = − ∂

(3-7)

The temperature dependence of H∆ between the empty hydrate lattice and the non-

hydrate phase as a function of the difference of the heat capacity ∆Cp is as follows.

Gas hydrates

16

p

p

HC

T

∂∆ = ∆ ∂

(3-8)

By integration of equation (3-8) from the reference temperature T0 to temperature T, the

temperature dependence of the enthalpy difference of the system is obtained. The

enthalpy difference is assumed to be of linear dependence and is

( )0 0pH H C T T∆ = ∆ + ∆ − (3-9)

where T0 represents the reference temperature 273.15 K.

By substitution of equation (3-9) in to the Gibbs-Helmholtz equation (equation (3-7)) and

integration with respect to temperature, the following temperature dependence of Gibbs

energy is obtained.

( )0

0 0

2

Tp

T

H C T TGdT

T T

∆ + ∆ −∆= −∫

(3-10)

By setting 0dT = the pressure dependence for a closed system in the absence of non-

expansion work and at constant composition can be derived from the following well-

known thermodynamic function

dG Vdp SdT= − (3-11)

where S is the entropy and V is the volume of the system (Atkins 2002).

By integration with respect to pressure the following pressure dependence for the

system with respect to Gibbs energy is obtained

0

p

p

G Vdp∆ = ∆∫

(3-12)

Gas hydrates

17

where ∆V is the volume difference and 0p is the vapor pressure at the reference

temperature.

Equation (3-12) is expanded with the gas constant and a temperature parameter.

0

p

p

G Vdp

RT RT

∆ ∆= ∫

(3-13)

An average temperature T is introduced where 0T is the reference temperature.

The difference in volume ( V∆ ) between the hydrate phase and the non-hydrate phase

at the average temperature T is used to calculate the difference in Gibbs energy

between the hydrate and non-hydrate phase.

0

p

p

G Vdp

RT RT

∆ ∆= ∫

(3-15)

The temperature and pressure dependence for Gibbs energy is expressed in

respectively equation (3-10) and (3-15). By applying the definition m

Gµ = to these

equations a general expression for the difference between the chemical potentials of

pure water, as an empty lattice, and as ice or liquid water at pressure P and temperature

T is obtained.

( )

0 0

Pβ 00 p 00

2

0 P

VdT+ d

R

T

T

H C T TP

RT RT RT T

µµ µ ∆ + ∆ −∆− ∆= − ∫ ∫

(3-16)

0µ∆ denotes the difference between the chemical potential of water in the empty hydrate

lattice and ice or liquid water at reference temperature 0T (273.15 K). 0H∆ is the

corresponding enthalpy difference, pC∆ the heat capacity difference and V∆ the volume

( )0 2T T T= + (3-14)

Gas hydrates

18

difference. P0 is the vapor pressure at reference temperature T0 and since it is very small

compared to the pressure P it can be set to zero ( 0 0P = ).

V∆ and 0H∆ are both considered to be pressure-independent because the pressure

effects on condensed phases are small. V∆ is also considered to be temperature-

independent, while the temperature dependence of the enthalpy term is taken into

account by means of a constant molar heat capacity difference ∆Cp.

The gas hydrate model is obtained by combining equations (3-6) and (3-16).

( )

0 0

P α0 p 00 w

Ki2 0i K0 wP

VdT+ d ln ln 1 Y

R

T

i

T

H C T T fP

RT RT fT

µυ

∆ + ∆ − ∆ ∆ − = − −

∑ ∑∫ ∫

(3-17)

The mole fraction and the activity coefficient, for the solid or liquid phase, replace the

respective fugacity and equation (3-17) is given by the following equation.

( )

0 0

P α α0 p 00 w w

Ki2 0 0i K0 w wP

VdT+ d ln ln 1 Y

R

T

i

T

H C T T xP

RT RT xT

µ γυ

γ

∆ + ∆ − ∆ ∆ − = − −

∑ ∑∫ ∫

(3-18)

The fugacity of the α-phase consisting of ice is unity by definition of a solid and the

fugacity of a liquid phase without inhibitor is pure and therefore also unity. The first term

on the right side of equation (3-18) in these cases therefore cancels out. By including the

term describing the probability of cave YKi being occupied the complete gas hydrate

model is derived when no inhibitor is added.

( )

0 0

Ki KiP

0 p 00 0

2i K ji ji0 P

j

j

A Bexp

V T TdT+ d ln 1

A BR1 exp

T T

T k

i

T

fH C T T

PRT RT T

f

µυ

∆ + ∆ −∆ ∆ − = − −

+

∑ ∑∫ ∫∑

(3-19)

The physical properties used in the gas hydrate model are seen in Table 3-3 and are

taken from Munck (1988). Only the physical properties for the liquid phase in this project

were used to estimate the Langmuir parameters, but the properties of ice are presented

Gas hydrates

19

for completeness. The physical properties used by Munck (1988) are not treated in the

article and the correctness of these parameters is therefore uncertain. The properties

used by Munck (1988) seem to be taken from Parrish and Prausnitz (1972), although

they are not referred to directly in the article, but listed in the reference list.

Table 3-3. Physical properties used in the gas hydrate model (equation (3-19)). ∆µ0 denotes the difference in the chemical potential of water in the empty hydrate lattice and in the liquid state (liq) or solid state (ice) at 273.15 K. ∆H0 and ∆V0 is the corresponding enthalpy difference. ∆Cp is the molar heat capacity difference (Munck 1988).

Property Structure I Unit

0 (liq)µ∆ 1264 J mol

0 (liq)H∆ - 4858 J mol

0 (liq)V∆ 4.6 3cm mol

p (liq)C∆ 39.16 J (mol K)⋅

0 (ice)µ∆ 0 J mol

0 (ice)H∆ 1151 J mol

0 (ice)V∆ 3.0 3cm mol

p (ice)C∆ 0 J (mol K)⋅

In Table 3-4 the values of the physical properties used in the gas hydrate model are

seen from several different authors. It is seen that there is a great deal of uncertainty of

the correct values. It is therefore important to investigate the physical properties, not only

for systems containing water and hydrogen sulfide, but also for gas hydrate systems with

structure I. However, investigating these physical properties further is beyond the scope

of this project.

Gas hydrates

20

Table 3-4. Physical properties used in the gas hydrate model from different sources in the literature. ∆µ0 denotes the difference in the chemical potential of water in the empty hydrate lattice and in the liquid state (liq) or solid state (ice) at reference temperature. ∆H0 and ∆V0 is the corresponding enthalpy difference. ∆Cp is the molar heat capacity difference. T0 I is the reference temperature 273.15 K. (Parrish et al. 1980; Parrish 1972; Holder 1980).

Property Value Reference Unit

0 (liq)µ∆ 1235±10 Holder (1980) J mol 1297 Parrish (1980) 1155 Holder (1976) 1264 Parrish (1972) 1255 Child (1964) a1264 Munck (1988)

0 (ice)H∆ 1684 Holder (1980) J mol 1389 Parrish (1980) 381 Holder (1976) 1151 Parrish (1972) a1151 Munck (1988)

0 (liq)H∆ -4327 Holder (1980) J mol - 4860 Parrish and Prausnitz

(1972)

- 4858 Munck (1988)

0 (ice)V∆ 3.0 Stackelber and Mller

a3.0 Munck (1988)

p (liq)C∆ -37.32+0.179(T-To) Holder (1980) J (mol K)⋅

0.565+0.002(T-To) Holder (1980) 9.11-0.0336(T-To) Weast (1968) a39.16 Munck (1988) aUsed by Munck (1988) and in this work. A short summary of the complete gas hydrate model (equation (3-19)) is given here. The

gas hydrate model is used to describe the SLVE between the gas hydrates (solid), the α-

phase that is a non-hydrate phase that is either ice (I) or a liquid aqueous phase (LA),

and the gas phase.

υ is the number of cavities, A and B are Langmuir constant that are estimated from

experimental data points, f is the fugacity, T is the average temperature and T0 is the

reference temperature.

Gas hydrates

21

For the α-phase consisting of ice, 0µ∆ , 0H∆ , pC∆ and V∆ denotes the differences

between the empty lattice and ice at reference temperature.

For the α-phase consisting of a liquid aqueous phase (LA), 0µ∆ , 0H∆ , pC∆ and

V∆ denotes the differences between the empty lattice and liquid water at reference

temperature (Munck 1988).

Phase diagrams

22

4 Phase diagrams

A phase of a substance is a form of matter that is uniform throughout in chemical

composition and physical state (Atkins 2002). In order to describe a given substance

phase diagrams are used to visualize the physical state and composition of a given

system. In this section an introduction to phase diagrams is presented with emphasis on

the H2S and H2S-H2O phase diagrams. No information concerning phase diagrams for

the ternary systems H2S-H2O-NH3 and H2S-H2O-CO2 was found in the literature.

4.1 Phase rule

The phase rule deduced by J.W. Gibbs is a general relation between the degrees of

freedom (F), the number of independent components (C), and the number of phases at

equilibrium (P) for a system of any composition.

F=C-P+2 (4-1)

The degrees of freedom F for a system are the number of variables that can be changed

independently without disturbing the number of phases in equilibrium. For a single-

component, single-phase system (C=1, P=1) such as pure water, the pressure and

temperature may be changed independently without changing the number of phases, so

the degrees of freedom is 2 (F=2). For pure water at conditions where water is in

equilibrium with vapor there is only one degree of freedom. Therefore, if the pressure is

fixed the temperature will automatically be determined and vice-versa. At the triple point

of pure water, ice, water, and steam are in equilibrium. According to the phase rule, a

one component system has zero degrees of freedom when three phases are in

equilibrium (F=0) and the system is by definition at an invariant point.

The number of components C in a system is the minimum number of independent

species necessary to define the composition of all the phases present in the system. An

aqueous solution of a pure salt contains three species which are water, cations and

anions. The number of chemically independent components is two because the charge

of the cations has to be balanced with an equivalent charge of the anions. The number

of cations is therefore depended on the number of anions. The solution is therefore

Phase diagrams

23

considered a binary solution, a solution of two chemically independent components

(Atkins 2002).

4.2 The H2S system

The understanding of the pure component system H2S is of importance for

understanding the more complex systems H2S-H2O, H2S-H2O-NH3, and H2S-H2O-CO2.

For the pure component system H2S there is only one component, which yields the

following phase rule for the H2S system.

F = C-P+2 = 3-P (4-2)

From equation () it is seen that the invariant point (F=0) is a triple point, since there are

three co-existing phases (P=3) when the degrees of freedom is zero. The three co-

existing phases are L-S-V (V is vapor, S is solid and L is liquid hydrogen sulfide). For the

system, where only two phases exist, there is one degree of freedom and the loci are S-

V, L-S and L-V. For the system with one degree of freedom there is only one phase

which is solid, liquid or vapor.

Table 4-1. The degrees of freedom, number of phases and loci for the pure component system H2S. L is the liquid phase, S is the solid phase and V is the vapor phase.

Degrees of freedom Number of phases Loci

0 (Triple point) 3 L-S-V

1 2 S-V

L-S

L-V

2 1 S

V

L

The vapor pressure of pure H2S was measured by Cardoso (1921), Reamer (1950),

Clarke (1970), and Klemenc (1932). There is good agreement between the vapor

pressure obtained by Cardoso (1921), Reamer (1950) and Clarke (1970). Klemenc

(1932) measured the vapor pressure at a lower temperature than Cardoso (1921),

Phase diagrams

24

Reamer (1950) and Clarke (1970), so direct comparison is not possible. The vapor

pressure of pure H2S is seen in

Figure 4-2.

0

10

20

30

40

50

60

70

80

90

150 200 250 300 350 400

Temperature [K]

Pre

ssur

e [a

tm.]

0

0,2

0,4

0,6

0,8

1

1,2

Pre

ssur

e [a

tm.]

Cardoso (1921) Clarke (1970)

Reamer(1950) Klemenc (1932)

→

←

Figure 4-2. Vapor pressure as a function of the temperature for pure H2S. The vapor pressure measured by Cardoso (1921), Reamer (1950) and Clarke (1970) are represented by the left axis (bigger scale) and the vapor pressure from Klemenc (1932) are represented by the right axis (smaller scale).

4.3 The H2S-H2O system

The H2S-H2O system has been studied extensively due to its importance in the chemical

and petrochemical sector. A review of the most important discoveries for the H2S-H2O

system is presented in this section.

Hydrogen sulfide is a weak diprotic acid that dissociates in two steps as shown in the

following reactions together with the ionic equilibrium of water.

Phase diagrams

25

( ) ( ) ( )1

+ - 7

2 aH S aq H aq +HS aq K 10 (25 )C−= �

� (4-1)

( ) ( ) ( )2

- + 2- 19

aHS aq H aq +S aq K 10 (25 )C−≈ �

� (4-2)

+ - 14

2 wH O H +OH 10 (25 )K C−= �

� (4-3)

From the small 2aK value it is seen that sulfide ion 2-S has a high affinity for protons. In

an acidic solution where the concentration of protons is high the concentration of sulfide

ion will be relatively small, since under these conditions the dissociation equilibrium will

lie far to the left. On the contrary, in basic solutions the sulfide ion will be relatively large,

since the small proton concentration will be responsible for producing relatively large

amounts of sulfide ions (Zumdahl 2002).

In Figure 4-3 the phase diagram for hydrogen sulfide-water is seen. It must be noticed

that the diagram is not scaled and several of the three-phase loci are speculation by the

authors (Carroll 1991) and included for completeness.

The pure component two-phase loci, denoted by the solid lines, are well established for

hydrogen sulfide and water (Carroll 1991). Of the three-phase loci the LA-H-V (Aqueous

liquid-Hydrate-Vapor), LS(Hydrogen sulfide rich liquid)-H-V, LA-LS-H, H-I-V and LA-LS-V

loci have been determined.

The first and second quadruple points have been well determined, while the third and

fourth quadruple points have not and it is not sure they exist.

Phase diagrams

26

Figure 4-3. Pressure-temperature diagram for the H2S-H2O system. The diagram is not scaled. The pure component loci are represented by the solid lines. The three-phase loci are represented by the dotted lines. The three-phase loci LA-H-V, LS-H-V, LA-LS-H, H-I-V, and LA-LS-V have been determined while the others are speculation. The quadruple points Q1 and Q2 have been determined while Q3 and Q4 are believed to exist but not determined (Carroll 1991).

For the H2S-H2O system there are two components, which yield the following phase rule

for the system.

F = C-P+2 = 4-P (4-4)

From the phase rule it is seen that the invariant points (F=0) are quadruple points, since

there are four co-existing phases when there is zero degrees of freedom. For the system

where only three phases exist there is one degree of freedom. For the system where two

phases exist there are two degrees of freedom and for the system with one phase there

are three degrees of freedom.

Phase diagrams

27

Table 4-2. The degrees of freedom, number of existing phases, the three and four phase loci and the physical occurrence for the quadruple points for the H2S-H2O system (Carroll 1991).

Degrees of freedom Number of phases Loci Physical occurrence

0 (Quadruple point) 4 LA-LS-H-V

H-I-V- LA

H-I-S-V*

H-S-LS-V*

29.4°C and 2.24 MPa

-0.4°C and 93.1 kPa

1

3

LA-H-V

LS-H-V

LA-LS-H

H-I-V

LA-LS-V

S-I-V*

S-I-H*

H-S-V*

LS-S-V*

LA-I-V*

LS-H-S*

LA-I-H*

2 2

1 3 * Locus not measured, but believed to exist (Carroll 1991).

4.3.1.1 The LA-H-V locus

Several people have determined the LA-H-V locus. Carroll (1991) have collected data

from several sources and correlated this locus from a Clausius-Clapeyron type equation

( ) ( )2788.88

ln 26.8952 0.15139 3.5786lnP T TT

= − + + −

(4-5)

where P is the total pressure in kPa and T is the temperature in Kelvin.

Phase diagrams

28

The data points for the LA-H-V locus from the different sources used to make the

Clausius-Clayperon type correlation (equation (4-5)) are seen in Figure 4-4.

Figure 4-4. The aqueous liquid-hydrate-vapor locus for the H2S-H2O system. Equation (2) on the figure corresponds to equation (4-5) in this work. Data from several authors are included: Forcrand (1882a,b), Cailletet (1882), Wright (1932), Bond (1949), Selleck (1951), Carroll (1990). The figure is taken from Carroll (1991).

The composition of the co-existing phases along the LA-H-V locus with varying

temperature and pressure is seen in Table 4-3.

Table 4-3. Compositions of the co-existing phases along the LA-H-V locus for the system hydrogen sulfide-water (Carroll 1991).

Mole Percent Hydrogen Sulfide Temp [°C] Pressure

[kPa](1) Aqueous(2) Hydrate(3) Vapor(2)

0 98.6 0.366 14.3 99.373 5 163.9 0.511 14.4 99.458 10 274.7 0.725 14.5 99.539 15 463.6 1.053 14.6 99.613 20 787.9 1.525 14.6 99.676 (1) – from equation (4-6). (2) – from the model presented by Carroll (1988c). (3) – from a modified van der Waals-Platteeuw (1959) model using the parameters from Munck (1988) and the hypothesis of Cady (1981).

Phase diagrams

29

4.3.1.2 The Ls-H-V locus

The Ls-H-V locus was correlated as a Clausius-Clapeyron type equation by Carroll

(1991) with experimental data points from several sources (see also Figure 4-5)

( )2061.05

ln 14.5229PT

= −

(4-6)

where P is the total pressure in kPa and T the temperature in Kelvin.

The vapor pressure for the Ls-H-V locus for the H2S-H2O system is seen in Figure 4-5

and the vapor pressure for pure H2S is also presented by Goodwin (1983). After a

thorough study of Goodwin’s article (1983) I concluded that the vapor presented by

Goodwin (1983) is doubtful and it is therefore advised to be used with caution. The major

concern regarding Goodwin’s article (1983) is whether he actually measured the vapor

pressure or calculated it. This is not presented clearly.

Figure 4-5. The Ls-H-V locus for the H2S-H2O system. The solid line is the correlation expressed in equation (4-6). The data points are from Carroll (1990), Scheffer (1913),

and

Selleck (1951). The dotted line is the vapor pressure of pure H2S presented by Goodwin (1983)

*. Q1 represents the quadruple point at 29.4°C and 2.24 kPa. The figure is taken from

Carroll (1991).

* Argued in section 4.3.1.4 that the vapor pressure presented by Goodwin (1983) is doubtful.

Phase diagrams

30

4.3.1.3 The LA-Ls-H locus

Only very limited data was available for the LA-Ls-H locus. Two data points from Scheffer

(1913) and three data points from Selleck (1951) are found in the literature. From the

data points Carroll (1991) obtained the following correlation,

11.083 3352.515P T= − (4-7)

where P is the total pressure in MPa and T is the temperature in Kelvin. A graphical presentation of the experimental data points and the correlation (Equation

(4-7)) for the LA-Ls-H locus are seen in Figure 4-6.

Figure 4-6. The Ls-La-H locus for the H2S-H2O system. Equation 4 on the figure corresponds to the correlation given in equation (4-7) with data points from Scheffer (1913) and Selleck (1951). Q1 represents the quadruple point at 29.4°C and 2.24 kPa. The figure is taken from Carroll (1991).

4.3.1.4 The H-I-V locus

Experimental data points from Scheffer (1913) and Meyer (1919a, b) were correlated by

Carroll (1991). The following Clausius-Clapeyron type equation was obtained

Phase diagrams

31

( )3070.13

ln 15.8059PT

= − (4-8)

where P is the total in pressure kPa and T is the temperature in Kelvin.

A graphical presentation of H-I-V locus is seen in Figure 4-7 and the respective

correlation (Equation (4-8)).

Figure 4-7. The H-I-V locus for the H2S-H2O system. Equation 5 on the figure corresponds to the correlation given in equation (4-8). Q2 represents the quadruple point at -0.4°C and 93.1 kPa. The figure was taken from Carroll (1991).

4.3.1.5 The LA-Ls-V locus

The LA-Ls-V locus was determined by Carroll (1989a). From the experimental data points

a Clausius-Clapeyron-type equation was obtained

( )2156.9

ln 7.928PT

= − (4-9)

Phase diagrams

32

where P is the total pressure in kPa and T is the temperature in Kelvin.

A graphical presentation of the experimental data points for the LA-LS-V locus and the

correlation (equation (4-9)) is seen in Figure 4-8. The vapor pressure of pure H2S is also

presented by Goodwin (1983), but it was earlier concluded that correctness of the vapor

pressure is doubtful (see section 4.3.1.2).

Figure 4-8. The liquid-liquid-vapor locus for the system hydrogen sulfide-water. K is the three-phase critical end point. The vapor pressure of pure H2S is from Goodwin (1983). The correlation denoted by a solid line corresponds to equation (4-9). Q1 represents the quadruple point at 29.4°C and 2.24 kPa. The figure was taken from Carroll (1991).

The composition given as mole fraction of H2S of the co-existing phases along LA-Ls-V

locus is seen in Table 4-4.

Phase diagrams

33

Table 4-4. The composition given as mole fraction of H2S of the co-existing phases for the L-L-V locus. The + in the first rows under the vapor compositions denotes that the composition could not be determined more accurately than within two decimals (Carroll 1989a).

Mole fraction of H2S

Temp [°C] Aqueous liquid H2S-Rich liquid Vapor

40 0.0335 0.987 0.99+

60 0.0341 0.973 0.99+

80 0.0385 0.965 0.987

100 0.0440 0.951 0.975

105 0.0451 0.955 0.973

4.3.2 Summary of the three-phase loci

The three-phase loci for the complex H2S-H2O system are summarized in this section. In

Table 4-3 the earlier presented correlations and the loci that are believed to exist, but not

determined are presented.

Table 4-5. Correlations for 5 known three-phase loci for the H2O- H2S system. Three-phase loci that are believed to exist, but not determined are also included (Carroll and Mather 1991, 1989a).

Locus Correlation LA-H-V

( ) ( )2788.88

ln 26.8952 0.15139 3.5786lnP T TT

= − + + −

LS-H-V ( )

2061.05ln 14.5229P

T= −

LA-LS-H 11.083 3352.515P T= − H-I-V

( )3070.13

ln 15.8059PT

= −

LA-LS-V ( )

2156.9ln 7.928P

T= −

S-I-V Not determined S-I-H Not determined H-S-V Not determined LS-S-V Not determined LA-I-V Not determined LS-H-S Not determined LA-I-H Not determined

Phase diagrams

34

A thorough review of the literature regarding the pure H2S system and the binary system

H2S-H2O was presented in this section. It is concluded that much research has been

carried out and that the phase behavior for several loci are well described, but still much

research must be carried out in order to increase the current knowledge of the important

systems H2S and H2S-H2O, which is of great importance in the petrochemical and

chemical sector.

H2S-H2O-weak electrolyte systems

35

5 H2S-H2O-weak electrolyte systems

A review of the most important earlier works from the literature describing the H2S-H2O-

NH3 and H2S-H2O-CO2 systems together with the important equilibrium in the ternary

system is presented in this section.

In the literature very scarce information concerning the SLVE for the H2S-NH3-H2O and

H2S-CO2-H2O systems where the solid is ice or gas hydrates were found.

The Extended UNIQUAC model has successfully been applied for the H2O-NH3-CO2

system by Thomsen (1999). The work presented here can be seen as a continuation

and the Extended UNIQUAC model in this work is applied to the ternary system H2S-

H2O-NH3. The system H2S-H2O-CO2 is also described, but the lack of experimental data

made it impossible to estimate the parameters.

5.1 Solid-liquid-vapor equilibrium

The solubility of volatile weak electrolytes in water results from two equilibrium: Vapor-

liquid and ionization. In the case of hydrogen sulfide, for instance the following

equilibrium exists.

( ) ( ) ( ) ( )VLE Ionization

- +

2 2H S g H S aq HS aq +H aq� � (5-1)

A more general form for the VLE and ionization for electrolyte systems for single solute

systems are seen in Figure 5-1. The weak electrolyte in the liquid phase exists in two

forms which are the molecular (VLE) and ionic form. The equilibrium depends on

pressure and temperature.

H2S-H2O-weak electrolyte systems

36

Figure 5-1. Equilibrium for vapor-liquid and ionization for a single-solute system (Edwards et al. 1975).

In the systems H2S-NH3-H2O and H2S-CO2-H2O several equilibrium exist. Due to the

chemical reactions in the liquid phase and a strong deviation from ideality, correlating

and predicting the thermodynamic properties of aqueous systems containing ammonia

and sour gases is an extremely difficult task (Beatier 1978 and Rumpf 1998).

Figure 5-2. The temperature dependence for the first dissociation equilibrium constant for ammonia, carbon dioxide and hydrogen sulfide in water (Edwards et al. 1978).

For systems with only a single weak electrolyte the only chemical reaction occurring is

ionic dissociation. For very dilute concentrations this effect is appreciable. At more

H2S-H2O-weak electrolyte systems

37

moderate concentrations the effect is less appreciable and only a relatively small fraction

of the weak electrolyte is in ionic form. For example, for the single solute system

hydrogen sulfide, the vast majority of electrolyte in the solution exists in molecular form

and not ionic. However, by for example adding ammonia to the solution the

concentration of the molecular form decreases significantly and the concentration of the

ionic form increases significantly. Depending on the concentration of the weak acid and

weak base in the solution, the fraction of weak electrolyte in molecular form may be

greatly reduced. Since it is the molecular species that are in equilibrium with the vapor

phase, the partial pressure of the weak electrolyte in the vapor phase may also be

greatly reduced (Edwards et al. 1975; Edwards et al. 1978; Beutier 1978).

5.1.1 The H2S-NH3-H2O system

In this section the equilibrium of importance, which are later used to estimate the

Extended UNIQUAC parameters, are presented for the H2S-NH3-H2O system. The

following VLE, LLV and SLE are considered for the system.

Vapor-liquid equilibrium:

( ) ( )3 3NH g NH aq� (5-2)

( ) ( )2 2H S g H S aq� (5-3)

( ) ( )2 2H O g H O l� (5-4)

Speciation equilbria (Ionization):

( ) ( ) + -

3 2 4NH aq +H O l NH +OH� (5-5)

( ) + -

2H S aq H +HS� (4-1)

+ -

2H O H +OH� (5-6)

Solid-liquid equilibrium:

- + 2-HS H +S� (4-2)

H2S-H2O-weak electrolyte systems

38

( ) ( )2 2H S aq + H O s Gas hydrate� (5-13)

( ) ( )2 2H O s H O l� (5-7)

At sufficiently low temperatures solid phases of water and ammonia exist where x is a

not specified stoichiometric coefficient (Thomsen, not in the reference list). The

equilibrium condition 5-8 does not have importance in this work, since the temperature

used to estimate the Extended UNIQUAC parameters is not sufficiently low to make the

equilibrium have any influence, but presented here for completeness.

( )2 3 2 3H O l NH (aq) H O XNH ( )s+ ⋅� (5-8)

5.1.2 The H2S-CO2-H2O System

In this section important VLE, LLV and SLE equilibrium for the H2S-CO2-H2O system are

presented. The behavior of the solubility of carbon dioxide is very similar to that of

hydrogen sulfide in many systems. Carbon dioxide also forms a weak diprotic acid when

dissolved in water. Many of the H2S observations can be directly translated to the

behavior of CO2 although the results will be different for the two gases, but the

qualitative phenomena are often the same (Carroll 1998).

Vapor-liquid equilibrium:

( ) ( )2 2CO g CO aq� (5-9)

( ) ( )2 2H S g H S aq� (5-3)

( ) ( )2 2H O g H O l� (5-4)

Speciation equilibrium:

( ) ( ) + -

2 2 3CO aq +H O l H +HCO� (5-10)

H2S-H2O-weak electrolyte systems

39

( ) + -

2H S aq H +HS� (4-1)

( ) + -

2H O l H +OH� (4-3)

Solid-liquid equilibrium:

( )2 2H S aq + H O(s) Gas hydrate� (5-12)

( )2 2CO aq +H O(s) Gas hydrate� (5-13)

( )2 2H O l H O(s)� (5-14)

5.1.3 Vapor-liquid equilibrium models

In the literature a number of different models have been applied to describe the VLE for

the H2S-NH3-H2O system and the NH3-CO2-H2O system. The Pitzer equation (Pitzer

1973) has been widely applied for these systems and the authors Edwards et al. (1975;

1978), Beutier (1978) and Rumpf (1999) use it directly or in a modified form. Models

published by Krevelen (1949), Wilson (1980), Leyko (1964a), Leyko (1964b) and

Ginzburg (1965) are widely empirical. Wilson (1980) and Krevelen (1949) base their

models on equilibrium constants and Henry’s law coefficients with out taking the ionic

activity into account. Daumn (1986) have developed an EoS for weak electrolyte VLE

behavior. All of the mentioned methods are quite complex and require a lot of

information.

The most applied models used to describe the vapor phase for the systems H2S-NH3-

H2O and H2S-CO2-H2O are the SRK, Peng-Robinson (PR), Nothnagel et al. (1973) and

Nakamura et al. (1976).

In the literature no information concerning the H2S-NH3-H2O and H2S-CO2-H2O systems

modeled with the Extended UNIQUAC was found.

- + 2-

3 3HCO H +CO� (5-11)

- + 2-HS H +S� (4-2)

H2S-H2O-weak electrolyte systems

40

A minor review of the article published by Edwards et al. (1978) is presented here,

because of its importance in describing the H2S-NH3-H2O and NH3-CO2-H2O systems.

For more details the reader is referred to the literature.

5.1.4 Vapor-liquid equilibrium model by Edwards

The model describing the VLE based on the Pitzer equation used by Edwards et al.

(1975; 1978) is presented in this section.

The liquid phase activity is calculated from the Pitzer equation and the vapor phase

activity is calculated by an extended Henry’s law. The Pitzer equation is a semi-empirical

virial type model.

The Pitzer equation is used to calculate the activity coefficients of both neutral and ionic

species and the excess Gibbs energy of an aqueous electrolyte mixture is

( ) ( ) ( ) ( )( )( )( )

Bin,0 Bin,1

1 2

, , ,

E

i j ij ij i j k ijk

i j w i j k ww w

Gf I m m f I m m m

RTn Mβ β τ

≠ ≠

= + + +∑ ∑

(5-15)

where 1f and 2f are functions of ionic strength I, ( )Bin,0

,i jβ , ( )Bin, 1

,i jβ and , ,i j kτ are

respectively binary and ternary interaction parameters, m is the concentration in molality

of other species than water, R is the gas constant, T the temperature in Kelvin, w

M the

molecular weight of water, and w

n the number of moles of water.

The activity coefficient in Pitzer’s theory describes the molecule-molecule, molecule-ion

and ion-ion interaction. The unsymmetrical activity coefficient i

γ ∗ for solute i (ion or

molecule in molecular form) derived by Pitzer’s theory (Pitzer 1973) is

H2S-H2O-weak electrolyte systems

41

( ) ( )

( )( )

( ) ( )

( ) ( ) ( ){ }

2

Bin,1

Bin ,0

2Bin,1

2

2ln ln 1 1.2

1.21 1.2

2 1 1 1.2 exp 22

1 1 2 2 exp 24

i i

ij

j ij

j w

ij k jk

j w k w

IA z I

I

m I II

zm m I I I

I

φγ

ββ

β

∗

≠

≠ ≠

= − + +

+

+ + − + −

− − + + −

∑

∑∑

(5-16)

where Aφ is a Debye-Hückel parameter and I is the ionic strength.

The activity of water is determined from the Gibbs-Duhem equation.

( ) ( ) ( ) ( )3 2

Bin,0 Bin,12ln exp 2

1 1.2w w i j ij ij w i

i w j w i w

A Ia M m m I M m

I

φ β β≠ ≠ ≠

= − + − − + ∑∑ ∑

(5-17)

An extended version of Henry’s law is used to describe the solubility of the weak

electrolytes. Henry’s law states that for ideal solutions that the vapor pressure of solute

i is proportional to the mole fraction of component i in the liquid phase. The

proportionality factor is an experimental constant named Henry’s constant H.

i i ip x H= (5-18)

Henry’s law is modified and the non-ideality is taking into account by introducing the

activity coefficient for the liquid phase and the fugacity coefficient for the gas phase. The

equilibrium for the molecular solute between the vapor and liquid phase is then

i i i i iy P m Hφ γ= (5-19)

where P is the total pressure,

iφ is the fugacity coefficient for component i , and

im is

the molality of solute i .

Henry’s constant is a strong function of temperature and a weak function of pressure.

The pressure dependence is given by the following equation

H2S-H2O-weak electrolyte systems

42

( )( ) ( ) ( )wln ln

satw

sat

p aPP P

H HRT

υ∞ − = +

(5-20)

where a

υ ∞ is the partial molar volume of molecular solute at infinite dilution andw

satP is

the saturation pressure of water at system temperature.

The temperature dependence of Henry’s constant is given by the following empirical

function

( ) ( )21 3 4ln ln

HH H H T H T

T= + + +

(5-21)

where 1 2 3, ,H H H , and 4H are empirical constants.

The equilibrium constants are calculated from the following empirical correlation

( ) ( )21 3 4ln ln

CK C C T C T

T= + + +

(5-22)

where 1 2 3, ,C C C , and 4C are empirical constant.

From the thermodynamic models presented in this section Edwards et al. (1978)

describe the H2S-NH3-H2O and NH3-CO2-H2O systems in the temperature range 0°C to

170°C. The composition range, depending on the extent ionization, may be as high as

10 to 20 molal.

Rumpf (1999) successfully describe the H2S-NH3-H2O system in the temperature range

40°C to 120°C and a total pressure up to 0.7 MPa.

Concentration calculations for the H2S-H2O system

43

6 Concentration calculations for the H2S-H2O system

Ideal concentration calculations for the H2S-H2O system are carried out in this section.

This is done in order to get an overview of the concentrations of the considered species

for different amounts of total dissolved hydrogen sulfide, which is useful when estimating

the Extended UNIQUAC parameters. The species H2S(aq), HS-, S2-, H+, OH- and H2O(l)

are considered for the H2S-H2O system.

The concentrations of the 6 species depend on the total amount of H2S dissolved in the

solution. There are 6 species and 6 independent equations must therefore be specified

to determine the concentration of the species. The 6 equations consist of three

equilibrium conditions, one electro-neutrality and two mass balances (sulfide and

oxygen).

The molality m, which is defined as mol pr. kg solvent, is used in the equilibrium

calculations. In order to determine the molalities of the species 1 kg water is considered.

The concentrations are relatively small and the assumption of ideality is therefore good.

The three chemical equilibrium considered are the first dissociation equilibrium constant

for H2S 1K , the second dissociation equilibrium constant for H2S 2K , and the equilibrium

constant for self-ionization of water wK . The equilibrium constants were determined

from the standard Gibbs energy of formation at 25°C. The data for the species are from

the database NIST. The equilibrium constants were calculated from the following well-

known thermodynamic relation.

( )RT lnG K∆ = − (6-1)

Concentration calculations for the H2S-H2O system

44

Table 6-1. The standard Gibbs energy of formation at 25°C for the 6 species considered in the H2S-H2O system. The data are from the database NIST.

Specie ( )25 CfG∆ � [ KJ Mol ]

H2S(aq) -27.83000

HS- 12.08000

S2- 85.80000

H2O(l) -237.1290

H+ 0

OH- -157.2481

The 6 conditions setup to determine the concentrations in the H2S-H2O system are seen

below. The 3 calculated equilibrium constants K1, K2 and Kw are seen first, then the

electro neutrality and then the sulfide and oxygen mass balance.

2

-7HS H1

H S

1.019 10m m

Km

− +⋅= = ⋅

(6-2)

-13S H2

HS

1.216 10m m

Km

− +⋅= = ⋅

(6-3)

-

-14

w OH H1.013 10 K m m += ⋅ = ⋅ (6-4)

There is no net charge in the solution and an electro neutrality condition is specified.

2H OH HS S2m m m m+ − − −= + + ⋅ (6-5)

Two mass balances are set up, which are for elemental sulfur and elemental oxygen

22 2

Total

H S H S HS Sm m m m− −= + + (6-6)

where Total

Sm is the total amount of sulfide dissolved in the solution.

Concentration calculations for the H2S-H2O system

45

2 2

Total

H O H O OHm m m −= + (6-7)

The molality of water was included, even though it is usually neglected, since it occurs in

the mass balance for oxygen.

The three equilibrium equations, the electro neutrality and the two mass balances are

solved, and the concentrations of the 6 species in the system are determined. In Table

6-2 the calculated concentrations are seen for several total amounts of dissolved

hydrogen sulfide. The equations were solved by the use of the math program Maple and

the calculation for 2

Total

H S 1m = is as an example seen in appendix 13.1.1.

Table 6-2. Calculated concentrations for total amounts of dissolved hydrogen sulfide. The molality of water is also included, since it was used in the mass balance for oxygen.

2

Total

H Sm 2H Sm

HSm − 2S

m − 2H Om

OHm −

Hm +

0.001 49.900 10−⋅ 51.004 10−⋅ 131.216 10−⋅ 55.508 91.008 10−⋅ 51.004 10−⋅

0.010 39.968 10−⋅ 53.187 10−⋅ 131.217 10−⋅ 55.508 103.178 10−⋅ 53.187 10−⋅

0.100 0.100 41.008 10−⋅ 131.217 10−⋅ 55.508 101.004 10−⋅ 41.009 10−⋅

1.000 0.9997 43.191 10−⋅ 131.217 10−⋅ 55.508 113.173 10−⋅ 43.191 10−⋅

2.000 1.9995 44.513 10−⋅ 131.217 10−⋅ 55.508 112.244 10−⋅ 44.513 10−⋅

3.000 2.999 45.528 10−⋅ 131.217 10−⋅ 55.508 111.832 10−⋅ 45.528 10−⋅

From Table 6-2 it is seen that the molality of S-2 is very small and it is therefore not

considered when estimating the Extended UNIQUAC parameters.

Parameter estimation and data description

46

7 Parameter estimation and data description

The general method used to estimate the parameters in the Extended UNIQUAC model

and the gas hydrate model by Munck (1988) is presented in this section. The algorithm

used in the parameter estimation, the uncertainty of the parameters and a review of the

experimental data is also presented.

The Extended UNIQUAC parameters (r, q, u0 and ut) for H2S and HS- are important for

the systems H2S-H2O, H2S-NH3-H2O and H2S-CO2-H2O. The Langmuir constants (A and

B) in the gas hydrate model for H2S are important for the systems H2S-H2O, H2S-NH3-

H2O and H2S-CO2-H2O.

In order to estimate the Extended UNIQUAC parameters and the Langmuir constants, a

FORTRAN code with the Extended UNIQUAC model, SRK EoS and the gas hydrate

model was used. Kaj Thomsen (supervisor) had programmed the general code and only

minor changes were applied to the existing FORTRAN code, such as adding the

hydrogen sulfide species and adding the experimental data points found in the literature.

The task of estimating the parameters is usually quite tedious, although in theory it is

only necessary to add the data to the existing program and run the simulation until it

converges. For several reasons such as, for example, poor data and poor start guesses,

the process of estimating the parameters is often time consuming, since several start

guesses must be tried in order to obtain good parameter values. If the model for some

reason did not describe some experimental values, then I tried in the best possible way

to compare them with other experimental data from the literature in order to verify the

reliability of the data.

7.1 Minimization

In this work the Levenberg-Marquardt (LM) algorithm and the Nelder-Mead simplex

search method (NM) algorithm are combined, in order to locate the global minimum of

the specified objective function, and thereby find the best estimate of the investigated

parameters. The objective function includes the squared summation of the difference

between the experimental and calculated values. Different weighting factors can be

included in this summation in order to give different weight (importance) to the different

types of data points (for example VLE and SLVE).

Parameter estimation and data description

47

The reason for combining the LM and the NM algorithms is because they have proven to

work well together, since they can compensate for each others drawbacks (Kaj

Thomsen, not in the reference list). A short summary of the LM and NM algorithm is

presented in the two following subsections, but for more specific information the reader

should consult the literature.

7.1.1 Levenberg-Marquardt Algorithm

The levenberg-Marquardt (LM) algorithm is an iterative technique that locates the

minimum of a multivariate function that is expressed as the sum of squares of non-linear

real valued functions. It has become a standard technique for non-linear least-squares

problems and can be thought of as a combination of steepest descent and the Gauss-

Newton method. It is sure to converge, since it uses a gradient method, but with risk of

getting trapped in local minimums. It is therefore maybe not able to find the global

minimum without very good initial guess for the parameter values (Lourakis, 2005).

7.1.2 Nelder-Mead Algorithm

The NM algorithm locates minimums with out using the gradient method, but has

difficulty in converging, which is a serious drawback. The advantage of the LM algorithm

is that it avoids getting trapped in local minimums and therefore a good method to find

the global minimum in a difficult terrain (Nelder and Mead 1965 ; Barker and Conway

2007).

7.2 Confidence limit for estimated parameters

The estimated parameter values were determined with a confidence limit of two standard

deviations (st. dev. commonly denoted σ). The following probability statement for the

confidence limit for the parameter is used to determine the accuracy of the parameter

estimation

( )P L Parameter U 1 α≤ ≤ = − (7-1)

Parameter estimation and data description

48

where P is the probability, L is the lower confidence limit, U is the upper confidence limit,

and α is the probability of rejecting the parameter value when it is true (in common

statistics called probability of type I error).

The interpretation of equation (7-1) is that the interval the parameter lies in will be true

( )100 1 %α− of the times that the statement is made. 1.96 standard deviations

correspond to a 95% confidence interval and the two standard deviations that are used

as upper and lower confidence limit, therefore correspond to a slightly higher confidence

interval than 95% (any standard book of statistics, for example Applied Statistics and

Probability for Engineers by Douglas C. Montgomery).

The estimated parameter values in the following sections are presented with many

significant digits. This means that the parameter value is often presented with more

digits than the confidence limit (for example 0.05 ± 0.06). Many significant digits are

presented because this allows other people to easily reproduce this work.

In general it is desired to have as few as possible parameters in the model because the

significant parameters in this way are estimated more precisely (e.g. smaller confidence

limit).

7.3 Collection and review of data

The open literature was used to collect experimental data points in order to estimate

parameters in the Extended UNIQUAC model and the gas hydrate model. The process

of collecting and reviewing experimental data is time consuming, since it takes time to

find articles with experimental data and a careful examination of the experimental data

must be carried out. From articles, especially older, it is often unclear how the

experiments were carried out. Important information is sometimes missing in the articles

and assumptions must be made. For example, often only the partial pressures were

presented and the vapor pressure of water therefore had to be added in order to obtain

the total pressure. Sometimes authors refer to data in the literature and investigation of

the original article shows that the referred data had been misinterpreted. Therefore all

the experimental data used are from the original articles except from Carroll (1977). By

using the original articles misprints from other articles are also eliminated. It was not

possible to obtain the original data from Carroll (1977) and the data published by Fogg

(1988) was therefore used.

H2S and HS- parameter estimation

49

8 H2S and HS- parameter estimation

The estimated Extended UNIQUAC parameters, which are the surface (q), volume (r)

and interaction (u0 and ut) parameters, for H2S and HS- are presented in this section.

The Extended UNIQUAC model with the estimated parameters from this work and the

SRK EoS are used to correlate the experimental data points for the pure H2S system

and the binary H2S-H2O system.

The S2- specie has very low concentrations (see section 6) in the concentration range

given by the data and it was therefore decided not to estimate any parameters for this

specie. 462 experimental VLE data points from the systems H2S and H2S-H2O were

used to estimate the Extended UNIQUAC parameters.

8.1.1 Objective function

In order to obtain the best parameter estimate the following objective function was

minimized

( )

2

calc exp

VLEdata exp

P -PSSQ

0.05 P 0.01

=

+ ∑

(8-1)

where Pcalc is the calculated total pressure and Pexp is the experimental total pressure in

bars.

0.01 is added to the experimental pressure in the denominator to prevent giving too

much weight to the data points at very low pressures. The data points with small

pressures and relatively big pressure differences between the calculated and

experimental pressures might otherwise result in a very big contribution to the total sum

of squares compared to the data points with big pressures. However, if the factor is too

big the data points at small pressures will not have significant influence in the parameter

estimation and a value of 0.01 was therefore found appropriate. The factor 0.05 is only

important when other equilibriums than VLE exist and are used to simultaneously

estimate the parameters. The factor 0.05 is a weighting factor and can be changed in

order to decrease or increase the weight of the VLE data by respectively increase or

decrease the value of the factor.

H2S and HS- parameter estimation

50

8.1.2 Vapor-liquid equilibrium data

Pure component data for the H2S system and data from the H2S-H2O system were used

to estimate the Extended UNIQUAC parameters. In Table 8-1 the range of temperature,

pressure and molality, the sources, and the number of data points used to estimate the

Extended UNIQUAC parameters are seen.

By inspection of the data from West (1948) it was seen that the data points are identical

to the earlier published data by Cardoso (1921), with the exception of 3 extra data points

published by West (1948). West (1948) does not refer to Cardoso (1921) and due to

experimental error the only reasonable conclusion is that West (1948) copied the data

from Cardoso (1921). Only data from the original article except for Lee (1977) has been

used in order to avoid misprints and wrong interpretation of the data. It was not possible

to obtain the original data, since they are in the National Depository of Unpublished

Data, Ottowa, Canada†. The data from Lee (1977) was taken from a compendium by

Fogg (1998). By comparing other experimental data presented in the compendium by

Fogg (1998) and the original articles available in the open literature there was found no

wrong presentation of the data by Fogg (1998). The data presented by Fogg (1998) is

therefore believed to be correct.

† National Depository of Unpublished Data, National Science Library, National Research Council,

Ottowa, Ontario, K1A OS2, Canada.

H2S and HS- parameter estimation

51

Table 8-1. The references, the number of data points and the respective temperature, pressure and molality (liquid phase) range that were used to estimate the Extended UNIQUAC parameters for H2S and HS

-. In the last row the total range of temperature,

pressure and total number of data points is included.

T, [K] P, [bar] Molality of H2S Source NP

273 – 298 1 0.10 – 0.21 Kiss (1937) 3

278 – 333 c0.4 – 5 0.036 – 0.39 Wright (1932) 52

273 – 323 c0.5 – 1 0.027 – 0.17 Clarke (1970) 36

313 5 – 25 0.39 – 1.72 Kuranov (1996) 9

311 – 378 c7 – 69 0.43 – 2.70 Selleck (1952) 22

278 – 374 12 – 90 a0.62 Reamer (1950) 9

273 – 370 12 – 85 a 0.62 bWest (1948) 19

273 – 374 10 –89 a 0.62 Cardoso (1921) 16

245 – 303 4 – 23 a 0.62 Clarke (1970) 20

298 1 0.10 Kapustinskii (1941) 1

283 – 393 1.6 – 67 0.070 – 2.27 Lee (1977) 275

273 – 393 0.4 – 89 0.027 – 2.70 All the above sources 462 aPure hydrogen sulfide data. bThe data published by West J. R. are identical to the data earlier published from Cardoso E.,

except from 3 extra data points. cPartial pressures or vapor phase composition measured.

8.1.3 Estimated UNIQUAC parameters

The estimated surface and volume UNIQUAC parameters for H2S and HS- are seen in

Table 8-2. The volume parameter for H2S has a relatively small confidence limit and the

estimate of the parameter therefore seems to be fairly accurate. A small confidence limit

for model parameters in general indicates that the parameter is very significant in the

model. From experience the confidence limit does not always mean this in the program

used in this work. By experience, parameters that are very important for the model often

have big confidence limits, although they are very important for the model to describe

the experimental data well. A measure of how important the parameters are in this

context means that the correlation decreases significantly by changing the parameter

value slightly. The confidence limit presented in the following tables with the parameter

H2S and HS- parameter estimation

52

values should therefore not be taken as an exact estimate of how significant (important)

the parameter is.

The surface and the volume parameters for HS- have relatively small confidence limits

and the estimate of these parameters therefore seem to be estimated accurately. By

changing the parameter values of the HS- ion and the H2S specie it was discovered that

the ion does have significant influence when correlating the experimental data with the

Extended UNIQUAC model, but that the H2S specie is far more important. The H2S

specie is more important than the HS- ion, because a small change of the H2S parameter

values decreases the correlation with the experimental data more than the HS- ion does.

The confidence limit for the H2S and HS- do not show this, but as previously mentioned

the confidence limit is not necessarily a measure of how important the parameter is.

The HS- ion is probably less important than the H2S specie because of the relatively low

concentration of the HS- ion compared to the H2S specie in the H2S-H2O system. The

HS- ion probably has more importance when the concentration is higher as in for

example systems with more species (for example H2S-H2O-NH3)

More data points might give a better estimate with smaller confidence limits for the

parameters of the HS- ion and the H2S specie. Usually the surface and the volume

parameters are within the range of 0-15 and the estimated parameters are also in this

range, but since no published parameter values are available it is not possible to

compare the parameters values with the literature.

Table 8-2. Estimated surface and volume UNIQUAC parameters for the H2S and HS-

species and the confidence limit. The confidence limit assigned as ± is two times the standard deviation.

Species Volume parameter (r) Surface parameter (q)

H2S b0.6205 ± 0.06 b0.05128 ± 0.03

HS- b0.6645 ± 0.4 b1.0245 ± 0.006

H2O a0.92 a1.4

OH- a9.3973 a8.8171

H+ a0.13779 a 1610− aParameter value from literature (Thomsen 1999) bEstimated parameter (this work)

H2S and HS- parameter estimation

53

In Table 8-3 the estimated Extended UNIQUAC interaction parameters from this work at

reference temperature 298.15 K ( 0u ) are seen and the temperature dependent

parameters ( Tu ) are seen in Table 8-4. The interaction parameters between the

aqueous species (OH-, H+ and H2O) have previously been determined, but also

presented here for completeness (Thomsen 1999).

The most important parameters were estimated by minimizing the sum of squares of the

objective function (equation (8-1)). Parameters that have no significant influence were

given a very high value for temperature independent parameter and a value of 0 for the

temperature dependent parameter. Other parameters that had only little influence in the

model were given a fixed value. The fixed value was found by trying different values and

afterwards fix the parameter value when a satisfactory correlation was obtained.

It was discovered that the HS- own interaction does not have very much influence in the

model. This is probably due to the very low concentration in the H2S-H2O system. A

good correlation was obtained by assigning the interaction parameter between HS- and

H2S the value 1300. The interaction parameter between HS- and water was assigned the

value of 1100, which gives a good correlation. The true value of the interaction between

H2S and water could be investigated further, because there is some uncertainty of the

value.

Table 8-3. The temperature independent parameters in the Extended UNIQUAC model

(0 0u uki ik

= ) at reference temperature (298.15 K) for the system H2S-H2O system

( ( )0 Tu u u T 298.15ki kiki = + − .

H2S HS- H2O OH- H+

H2S b-1439 ± 815

HS- d1300 d1000

H2O b-585 ± 120 d1100 c0

OH- a 1010 a 1010 c600.5 c1562.9

H+ a 1010 a 1010 c 410 a 1010 c0

aThe parameter was assigned this value, because it does not have significant influence in the model. bEstimated parameter value (this work). cParameter value from literature (Thomsen 1999). dThe parameter was assigned (not estimated) this value in order to obtain the best correlation.

H2S and HS- parameter estimation

54

The estimated temperature dependent parameters (uT) in the Extended UNIQUAC

model for H2S and HS- are seen in Table 8-4. The interaction parameters between H2S

and H2S, H2S and HS-, and HS- and water were fixed. The final parameter values were

fixed when a relatively low sum of squares in the objective was obtained.

Table 8-4. The temperature dependent interaction parameters in the Extended UNIQUAC

model (T T

ij jiu u= ) for the system H2S-H2O system ( ( )0 Tu u u T 298.15ki kiki = + − .

H2S HS- H2O OH- H+

H2S d5

HS- d-15 a0

H2O b1.99 ± 0.8 d12 0

OH- a1 a1 c8.5455 c5.6169

H+ c1 a1 c0.50922 c1 c0 aThe parameter was assigned this value, because it does not have significant influence in the

model. bEstimated parameter value (this work). cParameter value from literature (Thomsen 1999). dThe pameter was assigned (not estimated) this value.

The estimated parameters for H2S and HS- in the Extended UNIQUAC model are seen

to include a wide range of temperature (273 – 393 K), pressure (0.4 – 89 bar) and

molality (0.027 – 2.70 mol/kg). A total number of 462 experimental data points were

used from 11 different sources. In the following figures it is seen that the Extended

UNIQUAC model with the estimated parameters from this work and the SRK EoS

correlate the experimental total vapor pressure very well for the pure H2S system and

the binary H2S-H2O system.

In Figure 8-1 the experimental total pressure and the correlation of the vapor pressure of

pure hydrogen sulfide by the Extended UNIQUAC model and the SRK EoS are seen.

H2S and HS- parameter estimation

55

0

20

40

60

80

100

240 260 280 300 320 340 360 380

Temperature [Kelvin]

Pre

ssu

re [

Bar]

Reamer (1950) Cardoso (1921) Clarke (1970) Model

Figure 8-1. The vapor pressure correlated by the Extended UNIQUAC model and the SRK EoS are seen (Reamer 1950; Cardoso 1921); Clarke 1970).

In Figure 8-2 and Figure 8-3 the experimental values and the model correlation of the

total pressure of the H2S-H2O system are seen. It is seen that the Extended UNIQUAC

model with the estimated parameters from this work and the SRK EoS correlate the H2S-

H2O system very well. In Figure 8-3 a close up of Figure 8-2 is seen for the region at low

pressure and low molality of H2S.

H2S and HS- parameter estimation

56

0

10

20

30

40

50

60

70

0 0,5 1 1,5 2 2,5

Molality [mol/kg]

Exp. 0°CMod. 0°CExp. 4.96°CMod. 4.96°CExp. 5°CMod. 5°CExp. 10°CMod. 10°CExp. 15°CMod. 15°CExp. 20°CMod. 20°CExp. 25°CMod. 25°CExp. 30°CMod. 30°CExp. 37.78°CMod. 37.78°CExp. 40°CMod. 40°CExp. 50°CMod. 50°CExp. 60°CMod. 60°CExp. 71°CMod. 71°CExp. 71.11°CMod. 71.11°CExp. 90°CMod. 90°CExp. 104.44°CMod. 104.44°CExp. 120°CMod. 120°C

Figure 8-2. The total vapor pressure for the system H2S-H2O with different molalities for H2S at isotherms from 0°C to 120°C. The model (Extended UNIQUAC and SRK EoS) with the estimated parameters for H2S and HS

- is represented by the solid line and the

experimental data points are represented by dots.

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

2

2,2

0,02 0,07 0,12 0,17 0,22 0,27

Molality [mol/kg water]

To

tal p

ressu

re [

Bar]

Exp. 0°CMod. 0°CExp. 4.96°CMod. 4.96°CExp. 5°CMod. 5°CExp. 10°CMod. 10°CExp. 15°CMod. 15°CExp. 20°CMod. 20°CExp. 25°CMod. 25°CExp. 30°CMod. 30°CExp. 37.78°CMod. 37.78°CExp. 40°CMod. 40°CExp. 50°CMod. 50°CExp. 60°CMod. 60°C

Figure 8-3. The total vapor pressure for the H2S-H2O system with different molalities for H2S at isotherms from 0°C to 60°C. The model (Extended UNIQUAC and SRK EoS) with the estimated parameters for H2S and HS

- is represented by the solid line and the experimental

data points are represented by dots.

H2S and HS- parameter estimation

57

The mole fraction of H2S in the gas phase as a function of the H2S molality is seen for

several isotherms in Figure 8-4 and Figure 8-5. It is seen that the prediction of the

Extended UNIQUAC model with the estimated parameters from this work for H2S

together with the SRK EoS correlate the vapor composition very well in a molality range

from 0 to 2 mol/kg water and a temperature range between 0°C and 104.4°C.

0,76

0,81

0,86

0,91

0,96

0,02 0,07 0,12 0,17

Molality (Liquid phase) [kg/mol]

Mo

le f

racti

on

(G

as p

hase)

0°C Exp0°C Mod4.96°C Exp4.96°C Mod10°C Exp10°C Mod15°C Exp15°C Mod20°C Exp20°C Mod 25°C Exp25°C Mod30°C Exp30°C Mod40°C Exp40°C Mod50°C Exp50°C Mod

Figure 8-4. The H2S mole fraction of the vapor phase and the H2S molality is seen for several isotherms for the binary system H2S-H2O. The experimental data points are from Clarke (1970).

H2S and HS- parameter estimation

58

0,89

0,91

0,93

0,95

0,97

0,99

0,1 0,6 1,1 1,6 2,1 2,6

H2S molality [mol/kg water]

Mo

lefr

acti

on

of

H2S

(G

as p

hase)

37.8 C Mod

37.8 C Exp

71.1 C Mod

71.1 C Exp

104.4 C Mod

104.4 C Exp

Figure 8-5. The correlation (Extended UNIQUAC and SRK EoS) and experimental data of the H2S mole fraction in the vapor phase. The H2S molality is seen for 3 isotherms for the binary system H2S-H2O. The experimental data points are from Selleck (1952).

For several isotherms the partial pressure of H2S as a function of the molality is seen in

Figure 8-6. It is seen that partial pressure is well described by the Extended UNIQUAC

model with the estimated parameters for H2S and the SRK EoS. The partial pressure is

used in Figure 8-6 because Wright (1932) presents the experimental data as the partial

pressure of H2S. In Figure 8-4 and Figure 8-5 the mole fraction is presented because

Selleck (1952) and Clarke (1970) present the experimental data points as the mole

fraction of H2S.

H2S and HS- parameter estimation

59

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

5

0 0,1 0,2 0,3 0,4

Molality [mol/kg water]

Pa

rtia

l p

res

su

re [

ba

r]

5°C Exp

5°C Mod

10°C Exp.

10 °C Mod

15°C Exp

15°C Mod

20°C Exp.

20°C Mod

25°C Exp.

25°C Mod

30°C Exp

30°C Mod

40°C Exp.

40°C Mod

50°C Exp

50°C Mod

60°C Exp

60°C Mod

Figure 8-6. The partial pressure of H2S and the molality for the binary system H2S- H2O is seen for several isotherms. The experimental data points are from Wright R. H. (1932).

As a conclusion of this section it is stated that the Extended UNIQUAC model and the

SRK EoS, with the estimated Extended UNIQUAC parameters from this work for H2S

and HS- successfully correlate the VLE for pure H2S and H2S-H2O. Some uncertainty

exists for the interaction parameters between HS- and water, but a good estimate of

these interaction parameters can probably be obtained by using SLE experimental data.

For industrial purposes the Extended UNIQUAC ought to be sufficiently good in most

cases to predict the VLE of pure H2S and the H2S-H2O system in a wide range of

temperature (273 – 393 K), pressure (0.4 – 89 bar) and molality (0.027 – 2.70 mol/kg).

Gas hydrate parameter estimation

60

9 Gas hydrate parameter estimation

In this section the estimated Langmuir parameters in the gas hydrate model used to

describe the phase behavior of the system H2S-H2O with gas hydrates are presented.

The gas hydrate model is presented in details in section 3.3.

The estimated parameters in the gas hydrate model are the Langmuir constants AKi and

BKi for hydrogen sulfide in structure I. The Extended UNIQUAC parameters estimated in

section 8 are used to calculate the activity in the liquid phase and the SRK EoS is used

to calculate the fugacity in the gas phase.

The parameter estimation was carried out by minimizing the sum of squares of the

objective function. The objective function seen below is the squared sum of all the

pressure differences between the calculated and experimental pressures in bars.

2

Calc Exp

i

ssq = P P − ∑

(9-1)

In Table 9-1 the sources, the number of data points, the temperature and pressure range

used to estimate the Langmuir constants for hydrogen sulfide in structure I are

presented.

Table 9-1. The references, the number of data points and the respective temperature and pressure range that were used to estimate the Langmuir parameters for hydrogen sulfide in SLE gas hydrate model. In the last row the total range of temperature, pressure and total number of data points is included.

T, [K] P, [bar] Source NP

273 – 302 0.9 – 22 Selleck F. T. (1952) 13

289 – 302 5 – 22 Scheffer F. E. (1913) 10

273 – 302 0.9 – 22 All the above sources 23

The gas hydrate model used to estimate the Langmuir constants AKi and BKi are seen in

equation (9-2). For structure I, as in the H2S-H2O system, there are 2 small cavities and

6 large cavities per unit cell (i

υ ).

Gas hydrate parameter estimation

61

( )

0

Ki Ki

0 p 00 0

2i K ji ji 0

j

j

A Bexp

V T TP dT ln 1

A BR1 exp

T T

T k

i

T

fH C T T

RT RTTf

µυ

∆ + ∆ −∆ ∆ = − − −

+

∑ ∑∫∑

(9-2)

The pressure P, calculated by eq. (9-2), was used to estimate the Langmuir constants so

that the sum of squares for the objective function was minimized.

During the estimation process of the Langmuir constants it was found that both of the B

(large and small cavity) Langmuir constants changed less than 1% and it was therefore

decided to keep them at the initial value, as presented by Munck (1988), and only fit the

A parameters.

During the estimation process of the A parameters it was found that best estimate (e.g.

smallest ssq.) of the As (s represents the small cavity) parameter value was obtained

with a small negative value of As. A negative value of As does not have any physical

significance, since the probability of cave Ys being occupied by a hydrogen sulfide

molecule then would be negative according to the gas hydrate model.

The value of As was therefore assigned the very low value of 10-6. The reason for the

negative As value without constrains can be investigated further, but it is beyond the

scope of this work. A possible reason for the negative As value could be poor

experimental data or model flaws.

The constant Al (l represent the large cavity) for hydrogen sulfide was estimated to a

value of 0.025339 ± 0.05 (two standard deviations). It is seen that the parameter value

is approximately two times smaller than the standard deviation and a big uncertainty is

therefore connected to parameter value AKl. Possible reasons for the big uncertainty

could be the use of only 23 data points or that the parameter is not very important in the

model.

The estimated Langmuir constants from this work and by Munck (1988) for the small and

large cavity for hydrogen sulfide are seen in Table 9-2. It is seen that parameter values

differ significantly for the As parameter compared to Munck (1988), but as previously

mentioned some uncertainty is connected to the estimated parameter value As. The

parameter value AL differ significantly from the value by Munck (1988), but the estimated

AL value is still within 2 standard deviations. Munck (1988) uses 10 fewer experimental

data points than in this work (23). In this work two sources were used (Scheffer (1913)

and Selleck (1952)) while Munck only uses one source (Selleck (1952)). The values in

Gas hydrate parameter estimation

62

this work therefore ought to correlate the system H2S-H2O where gas hydrates are

present better than the work by Munck (1988).

Table 9-2. The estimated Langmuir constants A and B, from this work and the work by Munck (1988), in the gas hydrate model for hydrogen sulfide in aqueous solution in small (S) and large cavities (L) are seen.

Parameter Estimated Value (This work) Value from Munck (1988)

AL b 0.025339 ± 0.05 0.01634

BL a3737 3737

AS c10-6 52.5 10−⋅

BS a4568 4568 aThese parameter values were estimated very close to the values of Munck and therefore kept

equal to estimated parameter values of Munck (1988). bEstimated parameter value cThe parameter was assigned a fixed value, because it otherwise gave negative probabilities of

H2S being trapped in the ice structure (gas hydrate).

From a statistical analysis it was found that the four Langmuir constants for hydrogen

sulfide are highly correlated. The statistical analysis is based on the Durbin-Watson test,

which is also often referred to as D-test. The D-test is very complex and it is therefore

beyond the scope this report to treat it in details. The reader can for example consult the

following relevant literature regarding the D-test (Durbin and Watson 1950; Durbin and

Watson 1951). Due to the high correlation it is justified to keep BKl and BKs constant and

only fit the AKl and AKs. The high correlation might be because of the few data points or

also have a physical explanation. A physical explanation could be that probability of the

small and the large cavity being occupied by hydrogen sulfide is constantly proportional

and does not depended on for example pressure and temperature. However, this is

doubtful speculation and should be investigated further to make better conclusions.

In Figure 9-1 it is seen that the gas hydrate model with the estimated Langmuir gas

hydrate parameters from this work, the Extended UNIQUAC parameters presented in

section 8, and the SRK EoS correlate the experimental data points very well.

Gas hydrate parameter estimation

63

0

5

10

15

20

25

270 280 290 300

Temperature [K]

Pre

ss

ure

[B

ar]

Experiment Model

Figure 9-1. The experimental data points correlated by the gas hydrate model, with estimated Langmuir parameters from this work, the SRK EoS, and the Extended UNIQUAC model, with the parameters estimated in section 8 (Selleck (1952) ; Scheffer (1913)).

In Table 9-3 Langmuir constants estimated by Parrish (1972) for structure I for hydrogen

sulfide are seen. The physical properties used in the model are seen in Table 3-4.

Parrish (1972) uses different values for the physical properties and he uses the Kihara

potential to calculate the adsorption coefficient KiC where as in this work a simple two

parameter approach is used. It is seen by comparing the estimated parameter values

that there is relatively good agreement for the A parameters, while the B parameters

differ significantly. Due to the use of different physical properties the parameter values

from this work is not completely comparable to the ones obtained by Parrish (1972).

Table 9-3. Estimated Langmuir constants between 260 K and 300 K by Parrish (1972) for hydrogen sulfide in structure I.

Parameter Value

AL 0.01674

BL 3610.9

AS 33.0343 10−⋅

BS 3736.0

The physical properties used in the gas hydrate model by Munck (1988) are not treated

in the article and these properties could also be investigated further in order to obtain a

Gas hydrate parameter estimation

64

better model. The physical properties presented by Munck (1988) and several other

authors are seen in Table 3-4 and significant differences are seen. Especially does the

molar temperature dependent enthalpy difference between the empty hydrate lattice and

the liquid state ( p (liq)C∆ ) differ and the correct value of the physical properties is

therefore uncertain. The difference in the physical properties would be of interest to

investigate in order to determine what the correct values of the physical properties are,

but it is beyond the scope of this project.

As a conclusion for this section it is stated there is some uncertainty of the estimated

Langmuir parameters A. More data points might help to estimate these parameters

better. The estimated B parameters are in very good agreement with Munck (1988). The

estimated parameters also agree fairly with Parrish (1972), although they use other

values for the physical properties and some other experimental data points.

From Figure 9-1 it is seen that the 23 data points are correlated very well by the gas

hydrate model with the estimated Langmuir parameters from this work, the Extended

UNIQUAC model with the parameters from section 8, and the SRK EoS. The good

correlation of the experimental data points could therefore be of interest, to for example

the industry, when designing process equipment and preventing plugging of pipelines

when water and hydrogen sulfide are present at relatively low temperatures and high

pressures.

Parameter estimation for ternary systems

65

10 Parameter estimation for ternary systems

In this section the estimated Extended UNIQUAC parameters for H2S and HS- in the

ternary systems H2S-NH3-H2O are presented. In section 8 the surface (q) and volume (r)

parameters were estimated for H2S and HS- and the interaction energy between H2S-

H2S and H2S-H2O were also estimated. In this section these parameters are estimated

again with all the experimental data collected (639 data points), which include the

systems H2S, H2S-H2O, H2S-NH3-H2O, and H2S-H2O-gas hydrates. The interaction

parameters in the ternary system H2S-NH3-H2O are also estimated. Due to the lack of

good experimental data it was not possible to estimate the Extended UNIQUAC

interaction parameters for H2S and HS- in the ternary system H2S-CO2-H2O.

10.1 Description of the H2S-CO2-H2O system

The important ternary system H2S-CO2-H2O is often encountered as a part of the natural

gas systems (Guo et al. (2003) and exhaustion gas of coal-fired power plants and

predictive methods to describe the phase behavior is therefore of importance when for

example designing processing equipment and preventing plugging of pipelines.

Despite the importance of this system experimental data is extremely scarce in the open

literature and only a single author (Golutvin (1958)) was found to present experimental

data for the H2S-CO2-H2O system. The data could not be used to estimate the Extended

UNIQUAC parameters in the system, since after reviewing the data it was concluded

that the data is very inconsistent with well established experimental data from the binary

systems H2S-H2O and H2O-CO2.

In Table 10-1 3 experimental data points, for the H2S-CO2-H2O system, measured by

Golutvin (1958) is presented together with experimental data points for the H2S-H2O

system from Wright (1932) and Clarke (1970). The experimental data points chosen for

the H2S-H2O system have almost the same molality of H2S as in the H2S-CO2-H2O

system. This is done in order to compare the H2S-CO2-H2O system and the H2S-H2O

system. The difference in vapor pressure between the systems H2S-CO2-H2O and H2S-

H2O should mainly be due CO2, if the data is correct.

In Table 10-2 the H2S molality difference is seen for the different experimental data sets

and the experimental total pressure given in Table 10-1. For example is operation a-d

Parameter estimation for ternary systems

66

equal to the H2S molality difference between the data sets a and d, and likewise for the

total pressure difference. For example, for operation a subtracted by d (a-d) in Table

10-2 it is seen that the pressure difference is approximately 0.53 bar. This means that

the very small concentration of CO2 of 0.00686 mol/kg should contribute with

approximately 0.53 bar. Because CO2 in pure water at 20°C does not even closely

contribute to a partial pressure of 0.5 bar (Novak 1961) and the partial pressure of water

is negligible it is together with analog observations for the other experimental data points

concluded that the data set from Golutvin (1958) does not satisfactorily describe vapor

pressure of the H2S-CO2-H2O system.

Table 10-1. A sample of experimental data from Golutvin (1958), Wright (1932) and Clarke (1970). The data sample is used to justify that the experimental data Golutvin (1958) are incorrect. All data points are measured at 20°C.

Molality CO2 [mol/kg] Molality H2S [mol/kg] Exp. Pres. [Bar] Source a0.00686 0.05265 1.01325 Golutvin (1958) b0.00229 0.08263 1.01325 Golutvin (1958) c0.00150 0.09171 1.01325 Golutvin (1958) d 0.05304 0.48369 Wright (1932) e 0.08036 0.72623 Clarke (1970) f 0.10617 0.95362 Clarke (1970) g 0.10807 0.96525 Wright (1932)

In Table 10-2 the H2S molality difference is seen for the different experimental data set

and the experimental total pressure is given in Table 10-1.

Table 10-2. The difference in H2S molality and experimental pressure refers to the difference between the data sets from Table 10-1. All data points are measured at 20°C.

∆H2S Molality

[mol/kg]

∆Exp. pres. [Bar] CO2 Molality

[mol/kg]

Operation

-0.00039 0.5488 0.00686 a-d

0.00227 0.30627 0.00229 b-e

-0.01447 0.0789 0.00150 c-f

-0.01636 0.0673 0.00150 c-g

In the literature a wide range of experimental data for the system H2O-CO2 is available.

For the H2O-CO2 system Novak (1961) measured a pressure of 0.52 bars with a

Parameter estimation for ternary systems

67

molality of 0.04199 mol/kg at 20°C. By comparing operation a-d in Table 10-2 with the

measured pressure from Novak (1961) it is seen the low molality of 0.00686 can not give

a pressure of approximately 0.55 bars. The molality should be in the order of 0.04 mol/kg

in order to give a pressure of approximately 0.55 bars. The pressure measured by

Golutvin (1958) is therefore too low. The trend with too low pressure was also observed

for the other experimental data points by Golutvin (1958) and comparison to the

literature. It is therefore safe to conclude the experimental data from Golutvin (1958) is

not correct.

To summarize, it was not possible to estimate the Extended UNIQUAC interaction

parameters between H2S and CO2 due to the very scarce experimental data. It was

argued that the only found experimental data set describing the H2S-CO2-H2O system is

incorrect and therefore could not be used to estimate the Extended UNIQUAC

parameters.

10.2 Parameter estimation of the H2S-NH3-H2O system

In this section the Extended UNIQUAC model parameter estimation for the ternary

system H2S-NH3-H2O are presented. The collected data for the systems H2S, H2O-H2S,

H2S-NH3-H2O and H2S-H2O-gas hydrates were used to estimate the parameters for H2S

and HS-. A total number of 639 experimental data points were used.

In order to determine which interaction parameters that are of possible importance in the

H2S-NH3-H2O system a Bjerrum diagram was made for the system. The equilibrium

constants were calculated as described in section 6 with Gibbs energy of formation seen

in Table 10-3.

Table 10-3. The standard Gibbs energy of formation at 25°C for the ammonia species considered in the H2S-H2O-NH3 system. The other standard Gibbs energy of formation are seen in Table 6-1.

Specie ( )25 CfG∆ � [ KJ Mol ]

NH3(aq) -26.5

+

4NH -79.31

It is seen in Figure 10-1 that the interaction parameters for H2S and HS- that might have

big importance in the Extended UNIQUAC model are H2S-HS-, H2S- +

4NH , H2S-NH3, HS--

Parameter estimation for ternary systems

68

+

4NH , and HS--NH3. The S2- ion is not considered, because it is believed not to be

present at the conditions specified for the collected data.

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

3 5 7 9 11 13 15

pH

Mo

lefr

ac

tio

n

NH3 H2S HS-

3NH

HS− 2S −2H S

+

4NH

Figure 10-1. Bjerrum diagram for the system H2S-NH3-H2O at 25°C. The diagram can be used to exclude interaction parameters with no influence.

In Table 10-4 the experimental data points that were collected for the ternary systems

H2S-NH3-H2O with the temperature, pressure and molality range together with the

sources and the number of data points are seen. 5 data points from Terres (1957) were

excluded from the collected data because the Extended UNIQUAC model gave a very

bad correlation of these data points. In general very good correlations were obtained

with the Extended UNIQUAC model and a large amount data was used. The reason for

these data points giving a bad correlation could be that the data is not very good.

Parameter estimation for ternary systems

69

Table 10-4. Experimental data points collected for the H2S-NH3-H2O system from several different sources, the temperature, pressure, molality range and the number of data points.

Molality [mole/kg water]

T [K] P [bar] H2S NH3 Source NP 293 – 333 a 0.48 – 3.40 3.40 - 0.42 0.48 – 6.8 cTerres

(1957) 16

353 – 373 a 1.67 – 4.60 2.4 7.8 Leyko (1964b )

4

313 – 393 b 0.16 – 6.85 0.22 – 5.40 3.0 – 6.0 Rumpf (1999)

64

293 – 323 a 0.044 – 0.78

1.1 – 9.1 2.3 – 19.1 Leyko (1959)

14

313 – 373 b 0.9 – 19.3 1.14 7.9 – 53.3 Stimming (1985)

14

273 – 364 a 1.01 0.27 – 3.3 0.21 –7.3 Ginzburg (1963)

27

353 – 393 a 1.06 – 16.9 1.1– 7.9 0.11 – 22.6 Miles (1974)

19

273-393 0.16 – 16.9 0.22 – 9.1 0.11– 53.3 All sources 158 a) All partial pressures or mole vapor fractions were measured b) Only the total pressure was measured

c) 5 data points were not used in the parameter estimate since they deviate significantly from the

model correlation.

The estimated UNIQUAC parameters from this work and the parameters used for the

ammonia species are seen in Table 10-5.

Table 10-5. The UNIQUAC parameters estimated in this work. Data from the systems H2S, H2S-H2O, H2S-H2O-NH3, and H2S-H2O-gas hydrates were used.

Species Volume parameter (r) Surface parameter (q)

H2S a3.951 ± 0.9 a 1.290 ± 0.7

HS- a7.023 ± 3 a 15 ± 4

NH3 b1.435 b2.092

+

4NH b4.815 b4.603

aParameters estimated in this work bParameters from the literature (Thomsen 1999)

Parameter estimation for ternary systems

70

The estimated temperature independent Extended UNIQUAC parameters are seen in

Table 10-6. Some parameters were assigned a fixed value, which was done in order to

estimate other parameters better. If too many parameters are estimated in the same

parameter estimation the best correlation is not always obtained. By fixing some

parameters often better correlations are obtained than if the fixed parameters are also

estimated. Some parameters were therefore fixed. The parameters that were given a

fixed value were not of big importance in the model (e.g. influence on the ssq. in the

objective function). The most important parameters were estimated.

Table 10-6. The temperature independent Extended UNIQUAC parameters with reference temperature 298.15 K. Data from the systems H2S, H2S-H2O, H2S-H2O-NH3, and H2S-H2O-gas hydrates were used.

H2S HS- +

4NH NH3 H2O

OH- H+

H2S b-730 ± 186

HS- d1100 d1000

+

4NH b-1113 ±

142

d-15 c0

NH3 d200 b522 ± 68 c359.9 c1140.2

H2O b-614 ± 106 b292 ± 67 c52.730 c371.60 c0

OH- a 1010 a 1010 1877.9 c359.9 c600.5 c1562.9

H+ a 1010 a 1010 a 1010 a 1010 c 410 a 1010 c0

aThe parameter was assigned this high value, because it does not have significant influence in the model. bEstimated parameter value (this work). cParameter value from literature (Thomsen 1999). dThe parameter was assigned (not estimated) this value in order to obtain a good correlation. The estimated temperature dependent Extended UNIQUAC parameters are seen in

Table 10-7. Some parameters were assigned a fixed value, which was done in order to

estimate other parameters better. The parameters that were given a fixed value were not

of big importance. The most important parameters were estimated.

Parameter estimation for ternary systems

71

Table 10-7. The temperature dependent Extended UNIQUAC parameters with reference temperature is 298.15 K. Data from the systems H2S, H2S-H2O, H2S-H2O-NH3, and H2S-H2O-gas hydrates were used.

H2S HS- +

4NH NH3 H2O

OH- H+

H2S b3.80 ± 3

HS- d1 d0

+

4NH b-0.956 ± 1 d1 c0

NH3 d-4.5 b6.18 ± 1 c6.54 c4.02

H2O b-1.26 ± 0.3 b1.51 ± 0.2 c0.509 c6.19 c0

OH- a1 a1 a1 c0.0904 c8.55 c5.62

H+ c0 a1 c0 c0 c0 a0 c0 aThe parameter was assigned this value, because it does not have significant influence in the model. bEstimated parameter value (this work). cParameter value from literature (Thomsen 1999). dThe parameter was assigned (not estimated) this value in order to obtain a good correlation.

The correlation of the experimental data with the Extended UNIQUAC model, with the

parameters in Table 10-5, Table 10-6 and Table 10-7, the SRK EoS, and gas hydrate

model are seen in the following figures. The correlation of the ternary system H2S-H2O-

NH3, the pure component H2S, and the H2O-H2S system including gas hydrates are

presented.

In Figure 10-2 the total vapor pressure for the ternary system H2S-H2O-NH3 is seen for

isotherms at 40°C, 70°C and 100°C with a constant H2S concentration of 1.137 mol/kg

water. It is seen that the correlation is good, although the models correlate the vapor

pressure a little bit too low.

Parameter estimation for ternary systems

72

.

Molality H2S=1.137 m

0

5

10

15

20

5 15 25 35 45 55

Molality of NH3 [mol/kg water]

Pre

ssu

re [

bar]

40°C Mod

40°C Exp

70°C Mod

70°C Exp

100°C Mod

100°C Exp

Figure 10-2. The total vapor pressure for the ternary system H2S-H2O-NH3 for three isoterms with a constant H2S molality of 1.137 m (Stimming 1985).

In Figure 10-3 the vapor pressure for the ternary system H2S-H2O-NH3 is seen for

isotherms at 20°C, 30°C and 40°C with different with molality ratios of 3 2NH H S . The

models correlate the experimental well, except for the data at 40°C and a ratio of 2.

Parameter estimation for ternary systems

73

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0 5 10 15 20

Molality NH3 [mol/kg water]

Pre

ssu

re [

bar]

20°C NH3/H2S=2.1 Mod 20°C NH3/H2S=2.1 Exp20°C NH3/H2S=1.0 Mod 20°C NH3/H2S=1.0 Exp30°C NH3/H2S=2.1 Mod 30°C NH3/H2S=2.1 Exp40°C NH3/H2S=2.0 Mod 40°C NH3/H2S=2.0 Exp40°C NH3/H2S=2.1 Mod 40°C NH3/H2S=2.1 Exp

Figure 10-3. The vapor pressure for the ternary system H2S-H2O-NH3 for isotherms at 20°C,

30°C and 40°C with different with molality ratios of 3 2NH H S (Terres 1957; Leyko 1959).

In Figure 10-4 the vapor pressure is seen for the ternary system H2S-H2O-NH3 for

isotherms at 40°C, 80°C and 120°C with constant ammonia concentrations of 3.2, 5, 5.8

and 6 molality and varying concentration of H2S. The models correlate the experimental

data very well.

Parameter estimation for ternary systems

74

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5

Molality of H2S [mol/kg water]

Pre

ssu

re [

bar]

40°C NH3=3.2 Mod 40°C NH3=3.2 Exp 40°C NH3=5 Mod

40°C NH3=5 Exp 80°C NH3=3.2 Mod 80°C NH3=3.2 Exp

80°C NH3=6 Exp 80°C NH3=6 Mod 120°C NH3=3.2 Mod

120°C NH3=3.2 Exp 120°C NH3=5.8 Mod 120°C NH3=5.8 Exp

Figure 10-4. The vapor pressure for the ternary system H2S-H2O-NH3 with isotherms at 40°C and 80°C with constant ammonia concentrations of 3.2, 5, 5.8 and 6 molality and varying concentration of H2S (Rumpf 1999).

In Figure 10-5 the vapor pressure for the ternary system H2S-H2O-NH3 with isotherms at

50°C and 60°C with different molality ratios of 3 2NH H S is seen. The models correlate

the experimental data very well.

Parameter estimation for ternary systems

75

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1 3 5 7 9 11

Molality of NH3 [mol/kg water]

Pre

ss

ure

[b

ar]

50°C NH3/H2S=2.1 Mod

50°C NH3/H2S=2.1 Exp

60°C NH3/H2S=2.0 Mod

60°C NH3/H2S=2.0 Exp

Figure 10-5. The vapor pressure for isotherms at 50°C and 60°C with different molality ratios (Terres 1957; Leyko 1959).

In Figure 10-2, Figure 10-3, Figure 10-4 and Figure 10-5 it is seen that the Extended

UNIQUAC model, with the estimated parameters for H2S and HS-, and the SRK EoS

correlate the experimental data very well. Few data points are not correlated very well

and this can be due to model flaws or poor experimental data. Some variation of the

experimental data must be expected and it is therefore safe to conclude that the

Extended UNIQUAC model and the SRK EoS used together are able to correlate the

experimental data well for the ternary system H2S-H2O-NH3.

The gas hydrate data from the system H2S-H2O was also used when estimating the

parameters in the ternary system. A good correlation of the gas hydrate data is again, as

in section 9, obtained. It is seen that the gas hydrate model, with the already estimated

Langmuir constants, the SRK EoS and the Extended UNIQUAC model in general

correlate the vapor pressure well. The vapor pressure is correlated a little bit too low

from approximately 297-303 K.

Parameter estimation for ternary systems

76

0

5

10

15

20

25

270 275 280 285 290 295 300 305

Temperature [K]

Pre

ssu

re [

bar]

Model

Experimental value

Figure 10-6. The vapor pressure of the system H2S-H2O where gas hydrates are present correlated with the estimated Extended UNIQUAC parameters from all the collected data (Scheffer 1913; Selleck 1952).

The vapor pressure of pure H2S correlated with the estimated Extended UNIQUAC

parameters where all the collected data is used is seen in Figure 10-7. It is seen that

correlation of the Extended UNIQUAC model and the SRK EoS correlate the

experimental data very well.

0

1020

30

4050

60

708090

100

230 250 270 290 310 330 350 370 390

Temperature [K]

Pre

ss

ure

[b

ar]

Model

Experimental value

Figure 10-7 The vapor pressure of pure H2S correlated with the estimated Extended UNIQUAC parameters with all the collected data used.

In Figure 10-8 the vapor pressure for the binary system H2S-H2O is correlated by the

Extended UNIQUAC model with the estimated parameters for H2S and HS- and the SRK

Parameter estimation for ternary systems

77

EoS. The parameters estimated are with use of all the experimental data collected from

all the system investigated in this work. In Figure 10-9 a close up of Figure 10-8 is seen

at the region with low pressure and low molality of H2S. By comparing the correlation

obtained in section 8 it is seen that the correlation presented in this section (9) is not as

good as in section 8. The correlation is in general good, but the isotherms 90°C,

104.4°C, and 120°C are not correlated well. The other isotherms at temperatures lower

than 90°C are correlated well.

0

10

20

30

40

50

60

70

0 0,5 1 1,5 2

Molality of H2S [mol/kg water]

Pre

ssu

re [

bar]

0°C Mod0°C Exp5°C Mod5°C Exp10°C Mod10°C Exp15°C Mod15°C Exp20°C Mod20°C Exp25°C Mod25°C Exp30°C Mod30°C Exp37.8°C Mod37.8°C Exp40°C Mod40°C Exp50°C Mod50°C Exp60°C Mod60°C Exp71°C Mod71°C Exp71.1°C Mod71.1°C Exp90°C Mod90°C Exp104.4°C Mod104.4°C Exp120°C Mod

Figure 10-8. The vapor pressure for the binary system H2S-H2O correlated by the Extended UNIQUAC parameters with the estimated parameters where all the collected was used.

In Figure 10-9 the vapor pressure for the binary system H2S-H2O is correlated by the

Extended UNIQUAC model with the estimated parameters for H2S and HS- and the SRK

EoS. The parameters estimated in this section are with use of all the experimental data

collected from all the system investigated in this work.

Parameter estimation for ternary systems

78

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

2

2,2

0,02 0,07 0,12 0,17 0,22 0,27

Molality of H2S [mol/kg water]

Pre

ssu

re [

bar]

0°C Mod0°C Exp5°C Mod5°C Exp10°C Mod10°C Exp15°C Mod15°C Exp20°C Mod20°C Exp25°C Mod25°C Exp30°C Mod30°C Exp37.8°C Mod37.8°C Exp40°C Mod40°C Exp50°C Mod50°C Exp60°C Mod60°C Exp

Figure 10-9. The vapor pressure for the binary system H2S-H2O correlated by the Extended UNIQUAC parameters with the estimated parameters where all the collected was used.

As a conclusion of this section it is concluded that the Extended UNIQUAC model with

the estimated parameters for H2S and HS-, the SRK EoS and the gas hydrate model

correlate the systems H2S, H2S-H2O, H2S-H2O-gas hydrates, and H2S-H2O-NH3 well.

The ternary system H2S-H2O-NH3, the pure H2S system and the H2S-H2O-gas hydrate

system are correlated very well. The H2S-H2O system is in general correlated well, but

the isotherms at 90°C, 104.4°C, and 120°C are not correlated well. The best correlation

for this system is obtained by using the parameters presented in 8.

Conclusion

79

11 Conclusion

Parameters for the Extended UNIQUAC model have successfully been estimated for the

hydrogen sulfide species H2S and HS-. Interaction parameters for H2S and HS- in the

systems H2S-H2O and H2S-H2O-NH3 were successfully estimated. The parameters were

estimated from a large amount of collected experimental data (639 data points). The

Extended UNIQUAC model with the estimated parameters for H2S and HS- and the SRK

EoS correlate the pure H2S system, the binary H2S-H2O system, and the ternary H2S-

H2O-NH3 systems very well. The Extended UNIQUAC model could therefore prove

useful when designing process equipment for these systems.

Langmuir parameters for hydrogen sulfide in the system H2S-H2O were estimated by

using the gas hydrate model presented by Munck (1988). A minor number of

experimental data points (23) were collected and some uncertainties of the parameters

exist. However, the gas hydrate model, the Extended UNIQUAC and the SRK EoS

correlate the experimental data very well with the estimated Langmuir parameters and

the estimated Extended UNIQUAC parameters.

In this work a comprehensive review of the literature for the systems H2S, H2S-H2O (with

and without gas hydrates), H2S-CO2-H2O and H2S-NH3-H2O was carried out. The review

included collection of a large number of VLE and a minor number of SLVE (gas hydrate

data) experimental data. The most important VLE models were presented and the gas

hydrate model presented by Munck (1988) was presented and discussed. Some

uncertainty of the physical properties in the gas hydrate model was discovered.

It was argued that the very scarce experimental data found in the literature for the

system H2S-CO2-H2O was wrong. The data could therefore not be used to estimate the

parameters in the Extended UNIQUAC model.

The successful correlation of the VLE for the systems H2S, H2S-H2O and H2S-CO2-H2O

by use of the Extended UNIQUAC model and the SRK EoS can be used to describe the

phase behavior in a wide range of temperature, pressure and molality. This means that

the model most likely can be useful when designing separation equipment and

separation processes in order to minimize the outlet of these undesirable compounds

from for example the exhaust gas of coal-fired power plants.

In this work a successful correlation of the phase behavior for the system H2S-H2O

where gas hydrates are present was obtained. The work presented here can therefore

Conclusion

80

prove useful when for example preventing plugging of transmission lines and designing

process equipment.

Future work

81

12 Future work

The parameters in the Extended UNIQUAC model could most likely be estimated more

accurately resulting in an even better description of the phase behavior of the treated

systems. Especially would it be of interest to estimate the HS- ion more accurately. A

better estimate of the HS- ion could most likely be obtained by using SLE experimental

data from, for example, the systems NaOH-H2S-H2O and KOH-H2S-H2O.

Due to the lack of experimental data for the H2S-CO2-H2O system experimental data

could be obtained in order to estimate the interaction parameters between H2S species

and CO2 species in the Extended UNIQUAC model. A natural continuation of this work

would be to estimate the interaction parameters in the Extended UNIQUAC model

between the H2S species and the species SO2 and HCN, since these components are

often present in the process of gasification of coal.

The estimated Langmuir constants for H2S could be investigated further, since there is a

relatively big uncertainty connected to these parameters. Experimental data for the

system H2S-H2O at gas hydrate conditions could be obtained and thereby estimate the

Langmuir parameters better for H2S in structure I.

The quaternary system H2S-H2O-NH3-CO2 could be investigated.

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Mahadev, N. M. and P. R. Bishnoi, “Equilibrium Conditions for the Hydrogen Sulfide Hydrate Formation in the Presence of Electrolytes and Methanol” Can. J. Chem. Eng. Vol. 67, p.718-722 (1999). Maurer, G., and Prausnitz J.M., “On the derivation and Extension of the UNIQUAC Equation”, Fluid Phase Equilibrium, Vol. 2, p. 91-99, (1978). Montgomery, D. C., G. C. Runger, “Applied Statistics and Probability for Engineers”, John Wiley & Sons, Inc., 3rd ed., (2002). Munck, J., S. Skjold-Jørgensen and P. Rasmussen, “ Computations of the formation of gas hydrates”, Chem. Eng. Sci., Vol. 43, p.2661-2672, (1988). Nelder, J. A., R. Mead, “A simplex method for function minimization”, Computer J., Vol. 7, p. 308-313, (1965). Novak, J. V. Fried Pick, J. “Löslichkeit des Kohlendiosyds in Wasser bei Verschiedenen Drücken und Temperaturen", Czech. Chem. Commun., Vol. 26, p. 2266-2270, (1961). Parrish, W. R., P. B. Dharmawardhana, E. D. Sloan, “Experimental Thermodynamic Parameters for the Prediction of Natural Gas Hydrate Dissocation Conditions”, Ind. Eng. Chem. Fundam. Vol. 19, p. 410-414, (1980). Parrish, W. R., J. M. Prausnitz, “Dissociation Pressures of Gas Hydrates Formed by Gas Mixtures”, Ind. Eng. Chem. Process Des. Dev., Vol. 11, No. 1, p. 26-35, (1972). Pitzer, K.S., J. Phys. Chem., Vol. 77, p. 268-277, (1973). Thomsen, K., P. Rasmussen, “Modeling of vapor-liquid-solid equilibrium in gas-aqueous electrolyte systems”, Chem. Eng. Sci., Vol. 54, p. 1787-1802, (1999). Thomsen, K., “Thermodynamics of Electrolyte Solutions”, Personal notes not published, (2005). Reamer, H. H., B. H. Sage and W. N. Lacey, “Volumetric Behavior of Hydrogen Sulfide”, J. Industrial and Engineering Chemstry, Vol. 42, p. 140-143, (1950). Rumpf, B., Á. Pérez-Salado Kamps, R. Sing, G. Maurer, “Simultaneous solubility of ammonia and hydrogen sulfide in water at temperatures from 313 K to 393 K., Fluid Phase Equilibrium Vol. 158, No.160, p. 923-932. (1999) Sander, B., “Extended UNIFAC/UNIQUAC Models for (1) Gas Solubilities and (2) Electrolyte Solutions”. Ph.D. Thesis, Department of Chemical Engineering, Technical University of Denmark, Dk-2800 Lyngby Denmark, (1984).

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Appendices

i

13 Appendices

13.1 Appendix A

13.1.1 Concentration calculations for the H2O-H2S system

In this appendix the equations used to estimate the concentrations of hydrogen

sulfide species with the assumption that the system is ideal is presented.

The equations used to determine the concentrations of the species in the H2S-

H2O system were set up and solved in the math program Maple 11. The Maple

output that is presented further down corresponds to a total molality of 3 for

hydrogen sulfide. The program gave four solutions (separated by the curly

brackets), but only the first one is valid since the other solutions have no physical

meaning (negative molality).

The first three equations (1, 2 and 3) are equilibrium equations. Equation 4 is a

mass balance for sulfide. Equation 5 is the electro neutrality condition that

implies no net charge in the solution. Equation 6 is a mass balance for oxygen.

The molality of water is the reciprocal of the molar mass of water (eq. 6). 1 kg of

water is considered in the calculations.

The value of respectively the first and second dissociation equilibrium constant

for hydrogen sulfide and equilibrium constant for water are as follows.

k1=1.01877132e-7 k2=1.21569950e-13 kw=1.012707941965e-14

> eq1 := 1.01877132e-7 = (HS*H)/(H2S);

> eq2 := 1.21569950e-13 = (S*H)/HS;

> eq3 := 1.012707941965e-14 = OH*H;

Appendices

ii

> eq4 := 3=H2S+HS+S; > eq5 := H=OH+HS+2*S;

> eq6 := 1/0.01801528 = H2O+OH;

Determination of concentrations (solving equations) and the output is seen. > solve( [eq1, eq2, eq3, eq4, eq5, eq6], [H2S, HS, S, OH,

H, H2O] );