Supercritical Carbon Dioxide Density And Viscosity Correlations m1r5g

EXECUTIVE SUMMARY

Use of supercritical carbon dioxide have become prevalent in many industries due to its unique properties, readily realized critical conditions (Vesovic et al. 1990) and environmentally friendly nature. Thermodynamic properties of supercritical carbon dioxide are of great importance in engineering designs and are substantial to be estimated accurately. The properties of interest in this research are density and viscosity. Accurate densities are substantial for use in custody-transfer calculations and can also provide a stringent test for theoretical models such as descriptions of molecular interactions. Likewise, viscosity plays a crucial role in describing transport processes involving supercritical carbon dioxide. In recent years, different thermodynamic models have been proposed to predict the density and viscosity of supercritical carbon dioxide but many of these models are designed for limited ranges of temperature and pressure. Enhanced correlations to predict both density and viscosity of supercritical carbon dioxide for a wider range of applications is yet to be found. This progress report aims to identify the research gap needed for development of new empirical correlations that better predict the density and viscosity of supercritical carbon dioxide. Prior to development of new empirical correlations, comparative study had been conducted upon existing models to analyse their weaknesses and shortcomings so that the research gap for new models development can be identified. Initial comparative study was done by comparing the predicted properties of CO2 from the existing models with their counterparts from NIST carbon dioxide database. According to the comparative study, for density prediction, modified Redlich and Kwong equation of state had the largest valid range of temperature and pressure, from 268 K to 1100 K and 0.1 MPa to 330 MPa respectively. It accurately predicted the density of CO2 within 4.83% absolute average residual deviation from NIST data. The downside of this equation is that it is more complex than empirical correlations. In of accuracy, Liang-Biao Ouyang’s density correlation was the most accurate with %AARD of only 1.25%. Nonetheless, it only worked at temperatures between 313 K and 373 K, and pressures between 7 MPa and 60 MPa. Bahadori’s density model worked for wider ranges compared to Liang-Biao Ouyang’s correlation but it had four times less accuracy. As for viscosity prediction, Jossi et al’s viscosity correlation could predict the viscosity of CO2 at temperatures between 268 K and 1100K and pressures between 0.1 MPa 1|Page

and 330 MPa within 2.41% average deviation from NIST data. Liang-Biao Ouyang’s viscosity correlation was the most accurate with %AARD of only 0.86%. However, it only worked at temperatures between 313 K and 373 K, and pressures between 7 MPa and 60 MPa. Bahadori’s viscosity model worked at wider range than Liang-Biao’s correlation but the accuracy was lower (%AARD of 1.67%). In conclusion, the models that worked for smaller ranges of temperature and pressure had better accuracy while the short ranges of validity was the major downside. Meanwhile, the models that work for wide ranges of temperature and pressure gave unsatisfactory accuracy. Therefore, this calls for the need to develop new empirical models which better predict the density and viscosity of supercritical carbon dioxide. Keywords: SC: Supercritical, CO2: Carbon dioxide, density, viscosity, empirical correlations, equations of state

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TABLE OF CONTENTS

CHAPTER 1 .............................................................................................................................. 8 1.1

Background ................................................................................................................. 8

1.2

Objectives .................................................................................................................. 13

1.3

Significances ............................................................................................................. 13

1.3.1

Theoretical Significance .................................................................................... 13

1.3.2

Practical Significance......................................................................................... 13

1.4

Scope of the study ..................................................................................................... 13

1.5

Layout of the report ................................................................................................... 14

CHAPTER 2 ............................................................................................................................ 15 2.1

Existing Models ............................................................................................................ 15 2.1.1

Bahadori’s CO2 Density Correlation ................................................................. 15

2.1.2

Modified Redlich-Kwong equation of state for density of CO2 ........................ 18

2.1.3

Liang-Biao Ouyang’s density correlation .......................................................... 20

2.1.4

Bahadori’s CO2 viscosity correlation................................................................. 22

2.1.5

Jossi’s CO2 viscosity prediction model.............................................................. 25

2.1.6

Liang-Biao Ouyang’s viscosity correlation ....................................................... 26

CHAPTER 3 ............................................................................................................................ 29 3.1

General Description................................................................................................... 29

3.2

Detailed Description for each stage .......................................................................... 30

3.2.1

Stage 1: Literature Review on Existing Models ................................................ 30

3.2.2

Stage 2: Collection of data ................................................................................. 30

3.2.3

Stage 3: Performance testing on each model ..................................................... 30

3.2.4

Stage 4 and 5: Modelling ................................................................................... 35

3.3

Facilities and Resources ............................................................................................ 36

CHAPTER 4 ............................................................................................................................ 37 4.1

Results ....................................................................................................................... 37

4.2

Graphical Comparisons ............................................................................................. 38

4.2.1

Bahadori’s density correlation ........................................................................... 38

4.2.2

Modified Redlich-Kwong equation of state for density of CO2 ........................ 40

4.2.3

Liang-Biao Ouyang’s density correlation .......................................................... 42

4.2.4

Bahadori’s viscosity correlation ........................................................................ 44 3|Page

4.2.5

Jossi’s CO2 viscosity prediction model.............................................................. 46

4.2.6

Liang-Biao Ouyang’s viscosity correlation ....................................................... 48

4.3

Discussion ................................................................................................................. 50

CHAPTER 5 ............................................................................................................................ 52 5.1

Conclusion .................................................................................................................... 52

5.2

Recommendations ......................................................................................................... 53

5.3

Workplan for research project 412 ............................................................................... 53

REFERENCES ........................................................................................................................ 55 APPENDIX A: NIST database ................................................................................................ 58 APPENDIX B: Predicted database .......................................................................................... 64

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LIST OF FIGURES Figure 1.1: Density-pressure isotherms for carbon dioxide (A.A. Clifford 1991) .................... 9 Figure 1.2: Isotherms for the kinematic viscosity,

,for carbon dioxide (A.A. Clifford

1991) ........................................................................................................................................ 11 Figure 3.1: Research Methodology .......................................................................................... 29 Figure 4.1: Density vs Temperature plot of Bahadori's Density Model (Literature Range) ... 39 Figure 4.2: Bahdori Density versus NIST Density .................................................................. 39 Figure 4.3: Density vs temperature plot of Bahadori's Density Model (wider range) ............. 40 Figure 4.4: Density vs temperature plot of modified Redlich-Kwong Equation of State........ 41 Figure 4.5: Modified Redlich-Kwong density vs NIST density .............................................. 41 Figure 4.6: Density vs temperature plot of Liang-Biao Ouyang's model (Literature Range) . 42 Figure 4.7: Liang-Biao Ouyang’s model density vs NIST density .......................................... 43 Figure 4.8: Density vs Temperature plot of Liang-Biao Ouyang’s model (wider range)........ 43 Figure 4.9: Viscosity vs pressure plot of Bahadori’s viscosity model (260K -340 K) ............ 44 Figure 4.10: Viscosity vs pressure plot of Bahadori's viscosity model (340K-450K) ............ 45 Figure 4.11: Bahadori's Model Viscosity vs NIST Viscosity .................................................. 45 Figure 4.12: Viscosity vs pressure plot for Bahadori's viscosity model (wider range) ........... 46 Figure 4.13: Viscosity vs temperature plot for Jossi's Model (300K-600K) ........................... 47 Figure 4.14: Viscosity vs temperature plot for Jossi's model (700K-1000K) ......................... 47 Figure 4.15: Jossi's Model Viscosity vs NIST viscosity .......................................................... 48 Figure 4.16: Viscosity vs pressure plot for Liang-Biao Ouyang' s model (313K-343K) ........ 48 Figure 4.17: Viscosity vs pressure plot for Liang-Biao Ouyang's model (353K-373K) ........ 49 Figure 4.18: Liang-Biao Ouyang's model viscosity vs NIST viscosity ................................... 49 Figure 4.19: Viscosity vs pressure plot for Liang-Biao Ouyang's viscosity model (wider range) .................................................................................................................................................. 50

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LIST OF TABLES Table 2.1: Tuned coefficients used in equation 15-18 ............................................................. 17 Table 2.2: Tuned Coefficients used in Equation (23) .............................................................. 19 Table 2.3 Values of Coefficients for Equation (32) ................................................................. 21 Table 2.4: Tuned coefficients for Equation (46) ...................................................................... 24 Table 2.5: Values of coefficients for Equation (59) ................................................................ 28 Table 3.1: Matlab parameters used for Bahadori's Density Model .......................................... 31 Table 3.2: Matlab parameters for Modified Redlich-Kwong Equation of State ..................... 32 Table 3.3: Matlab parameters for Liang-Biao Ouyang's density correlation ........................... 33 Table 3.4: Matlab parameters for Bahadori’s viscosity model ................................................ 34 Table 3.5: Matlab parameters for Jossi's CO2 viscosity prediction model .............................. 34 Table 3.6: Matlab parameters for Ouyang's Viscosity Correlation ......................................... 35 Table 4.1: Summary of results for density models .................................................................. 37 Table 4.2: Summary of results for viscosity models................................................................ 38 Table 5.1: Workplan for Research Project 412 ........................................................................ 54 Table A.1: NIST density data .................................................................................................. 58 Table A.2: NIST viscosity data ................................................................................................ 61 Table B.1: Bahador's Density Correlation Predicted Results .................................................. 64 Table B.2: Modified Redlich-Kwong's equation of state predicted results ............................. 65 Table B.3: Liang-Biao Ouyang's density correlation predicted results ................................... 66 Table B.4: Bahadori's viscosity correlatin predicted results .................................................... 67 Table B.5: Jossi's viscosity model predicted results ................................................................ 68 Table B.6: Liang-Biao Ouyang's viscosity correlation predicted results ................................. 69

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NOMENCLATURES Ai (i=1 to 4)

Coefficient

Bi (i=1 to 4)

Coefficient

Ci (i=1 to 4)

Coefficient

Di (i=1 to 4)

Coefficient

P

Pressure

T

Temperature

PC

Critical pressure

TC

Critical temperature

Pr

Reduced pressure

Tr

Reduced temperature

R

Ideal gas constant

M

Molecular weight

a

Energy Parameter for Modified Redlich-Kwong’s EOS

b

Modified Parameter for Modified Redlich-Kwong’s EOS

a, b, c, d

Coefficient/Exponent

e

Exponent

f

Exponent

m

Exponent

n

Exponent

V

Specific volume

Z

Compressibility factor

α

Coefficient

β

Coefficient

γ

Coefficient

θ

Coefficient

ρ

Density Residual property Viscosity Viscosity at low pressures Viscosity parameter

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CHAPTER 1 INTRODUCTION

1.1

Background Supercritical fluids are gases which are highly compressed in order to exhibit

combined properties of both gases and liquids (Poliakoff 1997). The liquid-like density of supercritical fluid solvent contributes to high solvent power while the gas-like viscosity and diffusivity, with zero surface tension, grant outstanding transport properties to the supercritical fluid solvent. Due to their unique properties, they are widely applied in different fields of chemical industries namely purification, extraction, separation, crystal growth, reaction and fractionation of different compounds (Mukhopadhyay 2000). Supercritical carbon dioxide (scCO2) being non-toxic, non-flammable, and available in abundance at high purity in nature is a very much preferred solvent candidate for the described industries (Amooey 2014). In addition to that, another factor of choosing supercritical CO2 over other supercritical fluids is its convenient critical temperature, which is relatively low at only 304 K (Caude and Thiebaut 1999). This means that supercritical CO2 can be used as a solvent with materials that decomposes at high temperatures. Ultimately, supercritical CO2 is gaining popularity due to its great potential for high recoverability and superior-grade natural extracts from a variety of biomaterials (Mukhopadhyay 2000). Thermodynamic properties, such as viscosity, density and thermal conductivity of supercritical CO2 are important for the most efficient engineering design of many processes in biotechnological, oil and chemical industries. They portray the response of supercritical CO2 to changes in temperature and pressure. Thermodynamically speaking, supercritical fluid is a state where the pressure and temperature are beyond the critical point values and these thermodynamic properties changes rapidly near the critical point. In addition to that, the properties of supercritical CO2 are significantly different from other common fluids transported by pipeline (Bahadori and Vuthaluru 2010). Therefore, accurate predictions for these properties is crucial in determining the actual flow behaviour in many applications of supercritical CO2 (Yamamoto, Furusawa and Matsuzawa 2011).

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The first thermodynamic property of interest in this research proposal is density of supercritical CO2. The density-pressure isotherms for CO2 is as illustrated in Figure 1. It illustrates three density-pressure isotherms with the lowest temperature of 6 K above the critical temperature.

1 Figure 1.1: Density-pressure isotherms for carbon dioxide (A.A. Clifford 1991)

The density changes rapidly at around the critical pressure, 74 bar (Caude and Thiebaut 1999). Therefore, it is hard to control the density near the critical temperature and pressure which leads to difficulty in controlling experiments and processes since many properties are correlated with the density (Caude and Thiebaut 1999). The solvent capacity is one of these density dependent properties which varies significantly for small changes in either temperature or pressure. Thus, by adjusting the pressure and temperature of supercritical CO2, it is possible to indirectly control the density, which in turn controls the solvent power of the fluid. This enables fine tuning of solvent power for selective separation of many active constituents from feed materials (Mukhopadhyay 2000). Therefore, it is possible to create a wide spectrum of solvent properties in supercritical CO2 just by slightly altering the temperature and pressure (thus density). Hence, it is of great importance to be able to predict the density of supercritical CO2 for a wide range of temperature and pressure. The behaviour of density and other thermodynamic properties, as a function of pressure and temperature can be estimated by using equation of sates (EOS) (Burce E. Poling, John M. Prausnitz and O'Connell 2011). The first EOS to calculate density of CO2 was created by Van Der Waals based on attractive and repulsive (Ding-Yu. Peng and Robinson 1976). It was later improved by Redlich and Kwong by introducing temperature dependence for the attractive term (Heidaryan and Jarrahian 2013). Only recently, Heidaryan and Jarrahian (2013) further refined Redlich and Kwong’s EOS to predict density of 9|Page

supercritical CO2 with absolute error of only 1.63% compared to literature data. Amongst all these EOS, Peng and Robinson’s cubic EOS is the most well-known analytical form of EOS used to calculate density of supercritical CO2 (Caude and Thiebaut 1999). However, the best equations in of accuracy are complex numerical forms obtained through intelligent fitting of a wide range of thermodynamic data. These models are based on simple error minimization using least squares method and most of them do not require the use of physicochemical properties. An example would be the EOS predicting the density of CO2 developed and standardized by International Union of Pure and Applied Chemistry (IUPAC) (Yamamoto, Furusawa and Matsuzawa 2011). Similarly, Ely, Haynes, and Bain (1989) developed a 32-term equation of state for pure CO2 which covers a wide range of densities from gaseous state to supercritical fluid state. It is valid for temperature between 250 K and 330 K, and pressure up to 350 bar. Ely et al’s model having 32 makes it complicated to be solved and may result in computational overkill. Better than Ely’s model, Holste et al. (1987) derived second and third virial coefficients based on wider range of experimental density measurements made at temperatures from 217.01K to 448.15K. With progressing computational technology, Bahadori, Vuthaluru and Mokhatab (2009) proposed an easy-touse correlation, which is simpler than available models involving many parameters and complicated computations, to accurately predict the CO2 density. It is valid for pressure between 25 and 700 bar and temperatures between 293 and 433K (Bahadori, Vuthaluru and Mokhatab 2009). This correlation was developed to calculate density of supercritical CO2 first, which will then be used to calculate solubility of different solutes in it. Likewise, Jouyban, Chan, and Foster (2002) also obtained an equation empirically from correlating density of pure supercritical CO2 data at 80-450 bar and 300-450 K. This equation, like Bahdori’s, was also developed for the purpose of predicting density of supercritical CO2 which will be used to calculate solubility of solutes in it. Next, Ouyang (2011) came up with a CO2 density empirical correlation which works very well for pressures in the range of 7-62 MPa and temperatures in the range of 40-100 C. Just recently, Haghbakhsh et al (2013) proposed a supercritical CO2 density correlation which they claimed to have better performance (lower AARD%) and higher density prediction ability for wider range of pressure and temperature compared to Bahdori’s model and Jouyban’s model. They used sensitivity analysis to model the density of supercritical CO2 based on 1240 data sets (Haghbakhsh et al. 2013). The major problem is that most of these empirical correlations were developed to predict the density in somewhat limited range of pressure and temperature. Furthermore, researchers who developed these equations did not bother to check the accuracy 10 | P a g e

of the correlation predicting the density and more importantly did not consider the applicability of the said correlations in wider ranges of temperature and pressure since their interest of research is solely in the solubility of solutes in supercritical CO2. Therefore, a modified correlation which is specifically designed to predict the density of supercritical CO2 for a wide range of temperature and pressure is yet to be found. The second property of interest is the viscosity of supercritical CO2. Viscosity of the supercritical CO2 solvent is one of the crucial key factors in deciding the time required for extraction and the sizes of the critical components of the process plant as it can be used to describe the momentum transport related to the pressure difference. Hence successful design and development of supercritical fluid extraction process depends heavily upon accurate prediction of viscosity (Mukhopadhyay 2000). Isotherms for the kinematic viscosity of CO2 are shown in Figure 2. Kinematic viscosity decreases in low pressure region until the critical density and it rises back slightly (Caude and Thiebaut 1999).

2 Figure 1.2: Isotherms for the kinematic viscosity, 1991)

,for carbon dioxide (A.A. Clifford

The dynamic viscosity value of supercritical fluids is normally close to that of their gas counterparts then to that of liquid. This gives an advantage as pressure drop across the chromatographic columns in supercritical extraction processes will be much less compared to that of liquid transport. In order to fully utilize this advantage in industries, ability to accurately predict the viscosity plays an extremely crucial role. Relying on commercial soft-wares and using equations of state to predict properties is convenient and easy-to-use, but, unlike density, they do not perform equally well for all properties; especially for viscosity calculation (Bahadori 2008). Jossi, Stiel, and Thodos (1962) created the CO2 viscosity correlation which was valid in the reduced density range of 0.1-3, and it was accurate within the range of 5-10%. Later, Ely and Hanley (1981) embraced

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the principles of corresponding states for estimation of viscosity of supercritical CO2 in the reduced temperature range of 0.5-2.0. This method requires critical constants and acentric factors of pure components which makes it rather complicated to solve. However, it has an accuracy within 8%. Next, Lucas et al. (1987) came up with an empirical expression for supercritical CO2 viscosity within 3% accuracy give or take. Suring them all, Vesovic et al. (1990) published a comprehensive correlation to predict the viscosity of CO2 in the temperature range of 200K to 1500K and density less than 1200 kg m-3. It is the best working model so far compared to all the models mentioned. However, Vesovic’s model is extremely difficult to be used as a quick way to estimate the viscosity because his model was correlated based on the density of carbon dioxide, which may not be readily available in some situations. Recently, simpler model to predict viscosity with less computational time and parameters was proposed by Bahadori and Vuthaluru (2010). It works well for temperature between 260K and 450K and pressure between 10 MPa and 70 MPa. However, the pressure range for Bahadori and Vuthaluru’s correlation is only in the range of pressure for CO2 sequestration. In 2011, Ouyang came up with viscosity correlation which performs well for pressures between 7 MPa and 62 MPa, and temperatures between 40 C to 100 C. Ouyang’s viscosity correlation was also designed for CO2 sequestration range and therefore, its applicability is limited. A correlation that predicts the viscosity of supercritical CO2 more accurately than Vesovic’s and has wider application than Bahadori and Vuthaluru’s is yet to be found. According to this initial assessment upon literature, despite having so many correlations to predict the density of supercritical CO2, most of them do not accurately predict the density in the vapour and critical region. Those that do accurately predict the density in these regions are only limited to a certain range of thermodynamic state of interest. This calls for the need for the establishment of a new simplified correlation that accurately predicts the density of supercritical CO2 for a wider range of applications, which this research proposal is going to address. On the other hand, models that predict the viscosity of the supercritical CO2 are mostly outdated and have plenty of room for improvement. To date, no viscosity correlation has ever sured the performance of Vesovic’s model (Mukhopadhyay 2000). Even one of the latest and accurate correlations such as Bahadori’s model is limited to CO2 sequestration pressure range only. Therefore, this research aims to propose new models to better predict both density and viscosity of supercritical CO2.

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1.2

Objectives 1. To review and to perform a comparative study on existing models used to predict density and viscosity of supercritical carbon dioxide. 2. To propose improved correlations used to predict density and viscosity of supercritical carbon dioxide.

1.3

Significances

1.3.1 Theoretical Significance 1. The end results of this research project being the simpler and better correlations to predict density and viscosity of supercritical carbon dioxide is intended to be used by the scientific community in general. 2. The new correlations are also expected to be of great significance for any researchers studying the supercritical carbon dioxide as they can use these correlations to quickly determine the properties of scCO2. 1.3.2 Practical Significance 1. They can be a significant practical value for engineers to have a quick check on properties of supercritical CO2 at different temperatures and pressures without having to resort to experimental measurements. 2. The correlations involving less parameters and requiring less computational time is of significant importance for process engineers who are dealing with the conceptual design stage and FEED stage as they can easily calculate the properties without relying on intensive computational abilities and at the same time, data with high accuracy can be achieved.

1.4

Scope of the study Existing correlations to predict density and viscosity of scCO2 are either only

applicable for small ranges of pressures and temperature or the prediction accuracy is unsatisfactory. Therefore, new correlations that can better predict these properties for wider ranges of temperature and pressure are needed to be found. The scope of this study is confined to studying two properties, namely viscosity and density of supercritical carbon dioxide. The scope of this study is limited to conducting comparative study upon the existing models to analyse their performances and developing improved correlations that better 13 | P a g e

predict the density and viscosity of CO2 in supercritical region. This study has utilized the carbon dioxide database from National Institute of Standards and Technology (NIST) for comparative study. The database constructed has a temperature range of 220K-1100K and pressure range of 0.1MPa-300MPa, covering a total of 40969 data points for each property. The properties of scCO2 outside this range will neither be considered in comparative study nor in new models development. NIST employed the equations of states to predict the densities of CO2 for which at pressures up to 30 MPa and temperatures up to 523 K, the estimated uncertainty ranges from 0.03% to 0.05% (R Span and Wagner 1996). The uncertainty in supercritical density ranges is from 0.05% in the isotherms near critical point up to 2% at high pressures. The uncertainty in viscosity ranges from 0.3% in the dilute gas near room temperature and increases up to 5% at high pressures (E.W. Lemmon, McLinden. and Friend 2011). 1.5

Layout of the report The second chapter of this report covers the extensive literature review discussing the

existing models that predict the density and viscosity of supercritical carbon dioxide. The second chapter is divided into six sections where each section discusses about each existing model in depth and detail. Next, the third chapter of this report describes the methodology employed in this research project. The first section of this chapter describes the methodology generally. The second section discusses the methodology used for each model in detail. The third section describes the equipment and facilities used in the research project. As for the fourth chapter, it was divided into three sections, namely results, graphical comparison and discussion. The results section includes the summary of results whereas the graphical comparison includes graphs produced from model. Comparisons are made in this section of the chapter. Afterwards, in the discussion section, the overall results and comparisons are discussed. Next, the fifth chapter includes the conclusion, recommendation and workplans for Research Project 412. After the fifth chapter, there are references and appendix.

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CHAPTER 2 LITERATURE REVIEW

2.1

Existing Models Density of supercritical carbon dioxide can be predicted by equations of state, a

black oil model or empirical correlations. Black oil models have been developed to calculate density of carbon dioxide in oil and gas industries and they tend to have large errors when applied to determine the density of pure carbon dioxide (Ouyang 2011). The same is true for equations of state. Moreover, the equations of state may result in computational overkill or they have long computational time under certain circumstances due to their complexity (Hassanzadeh et al. 2008). The empirical correlations are usually simpler and easier to use compared to equations of state. Nevertheless, they have limited ranges of temperature and pressure where they can be applied and the poor accuracy is also a concern. Similarly, there are equations of state and empirical correlations to calculate the viscosity of scCO2. In this section of the report, detailed literature review for each existing models predicting the density and viscosity of CO2 has been conducted for further understanding. Their literature reviews are as follows; 2.1.1 Bahadori’s CO2 Density Correlation Bahadori, Vuthaluru, and Mokhatab (2009) developed an easy-to-use empirical model that predicts the density of scCO2 as part of the research to predict the solubility of CO2 in different solutions for the temperature range of 293-433 K and pressure range of 25 bar to 700 bar. Their objective was to identify a polynomial equation that is able to correlate the density of CO2 with the influential properties namely temperature and pressure. They assumed that the best-fit polynomial equation predicting the density is the one that produces the minimal sum of deviations squared from a given set of data. ∑

(

)

(1)

For given equation, the coefficients α, β, γ and θ are all unknowns whereas its respective

and

are taken from literature. To acquire the least square error, these

coefficients has to produce zero first derivatives: 15 | P a g e

(

∑

)

(2)

∑

(

)

∑

(

)

∑

(

)

(3)

(4)

(5)

Expansion and conversion of above equations to linear form results in the following: ∑

∑

∑

∑

∑

∑

∑

∑

(6) ∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

(7)

(8)

(9)

Knowing that the coefficients are a function of pressure, the previously tuned coefficients are needed to be made as a function of pressure. Therefore, ∑

(

)

∑

(

)

∑

(

)

∑

(

)

(10)

(11)

(12)

(13)

The tuned coefficients for equation (10-13) are derived by applying the same procedure for Equation (1-9). The required coefficients are provided in Table 2.1.

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Table 2.1: Tuned coefficients used in equation 15-18 Coefficient A1 B1 C1 D1 A2 B2 C2 D2 A3 B3 C3 D3 A4 B4 C4 D4

25 bar < P < 100 bar 2.089800972761597×105 -1.456286332143609×104 2.885813588280259×102 -1.597103845187521 -1.675182353338921×103 1.16799554255704×102 -2.31558333122805 1.284012022012305×10-2 4.450600950630782 -3.10430147581379×10-1 6.157718845508209×10-3 -3.420339567335051×10-5 -3.919844561756813×10-3 2.734973744483903×10-4 -5.428007373890436×10-6 3.019572090945029×10-8

100 bar < P < 700 bar 1.053293651041897×105 -9.396448507019846×102 2.397414334181339 -1.819046028481314×10-3 -8.253383504614545×102 7.618125848567747 -1.963563757655062×10-2 1.497658394413360×10-5 2.135712083402950 -2.023128850373911×10-2 5.272125417813041×10-5 -4.043564072108339×10-8 -1.827956524285481×10-3 1.768297712855951×10-5 -4.653377143658811×10-8 3.586708189749551×10-11

The correlation that predicts the density (ρ, kg/m3) of CO2 as a function of temperature (T, K) and pressure (P, bar) is as follows: (14) Where, (15) (16) (17) (18)

Equation (14) was only designed to capture the variation of density of CO2 in CO2 sequestration process and enhance oil recovery process. Therefore, it has limited applicable ranges of temperature and pressure.

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2.1.2 Modified Redlich-Kwong equation of state for density of CO2 Many equations of state that predict the density of supercritical carbon dioxide has high accuracy, but they are often not implemented in many applications due to their complexity. Generally, the cubic equations of state are simpler, flexible and able to produce accurate results for different practical applications (Heidaryan and Jarrahian 2013). The semi-empirical cubic equations of state are widely used in engineering applications due to their simplicity despite lacking the sound theoretical basis. The main concern for these semi-empirical cubic equations of state is the poor accuracy of properties predicted by them. To solve this problem, Heidaryan and Jarrahian (2013) introduced a modified Redlich-Kwong semi-empirical cubic equation of state to calculate the density of carbon dioxide in the supercritical region as a function of reduced temperature and reduced pressure. They used a total of 3742 experimental density data which was spread out over the temperature range of 304 K to 1273 K and pressure range of 7381.427 kPa to 452,500 kPa. The original Redlich-Kwong equation of state is as shown (Redlich and Kwong 1949): (19) (

)

This was modified to become: (20) (

)

Where, Energy Parameter,

(

)

(21) Co-Volume Parameter, Coefficient,

(

)

(22)

( ) ( )

( (

)

)

(23) (

)

Reduced Temperature,

(24)

Reduced Pressure,

(25)

The dimensionless constants, β1- β6, are listed in Table 2.2.

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Table 2.2: Tuned Coefficients used in Equation (23) Constant

Determined Constant

β1

1.00257031×1000

β2

-3.60123418×10-1

β3

-8.19348589×10-2

β4

4.45472775×10-1

β5

-1.92850236×10-1

β6

2.40911796×10-2

The critical pressure and temperature of CO2 is 7377.3 kPa and 304.128 K respectively. R, the ideal gas constant is 8.3144621 JK-1mol-1. In order to find the density, Equation (19) has been rearranged into a cubic form of compressibility factor as follows: (

)

(26)

Where

(27) (

) (28)

Roots of Equation (26) can be calculated by employing Cardano’s method. The compressibility Z obtained can be converted to density by the following equation: (29) Where M, the molecular weight of carbon dioxide is 44.01 kg/kmol. This modified Redlich-Kwong equation of state is much simpler to use compared to other older equations of state and it possesses high accuracy. However, Heidaryan and Jarrahian do not recommend this equation to be used for very low temperature and very high pressure region.

19 | P a g e

2.1.3 Liang-Biao Ouyang’s density correlation Similar to Bahadori’s density correlation, Ouyang (2011) introduced a simple and explicit empirical correlation to predict the density of CO2 under operational conditions expected for carbon capture and sequestration project. The model is designed for pressure of 7-62 MPa and temperature of 40-100 K. Ouyang extracted the carbon dioxide properties under the temperature and pressure ranges anticipated for carbon sequestration operations from NIST database. He then applied the least square approach just like Bahadori to develop a simple correlation for prediction of carbon dioxide density. The correlation formulae and the relevant correlation coefficients were adjusted in such a way that it fulfilled the following criteria: ∑(

)

∑(

(

)

)

(30)

The final density correlation is shown below as a function of pressure: (31) Where density, ρ is in kg/m3 and pressure, P is in psia. The coefficients A0 to A4 as a function of temperature (0C) can be found from the following equation: (32) (i=0 to 4) Ouyang (2011)’s research indicated that his new correlation performs much better than Bahadori’s correlation. However, the temperature and pressure ranges where this correlation is applicable is limited as his interest was only to predict the density of CO 2 in CO2 sequestration conditions.

20 | P a g e

The correlation coefficients values bi0 to bi4 are listed in Table 2.3. Table 2.3 Values of Coefficients for Equation (32) bi0

bi1

bi2

bi3

bi4

Value of bij Coefficients used in Equation (32) for Pressure < 3000 Psia i=0

-2.148322085348E+05

1.168116599408E+04

-2.302236659392E+02

1.967428940167E+00

-6.184842764145E-03

i=1

4.757146002428E+02

-2.619250287624E+01

5.215134206837E-01

-4.494511089838E-03

1.423058795982E-05

i=2

-3.713900186613E-01

2.072488876536E-02

-4.169082831078E-04

3.622975674137E-06

-1.155050860329E-08

i=3

1.228907393482E-04

-6.930063746226E-06

1.406317206628E-07

-1.230995287169E-09

3.948417428040E-12

i=4

-1.466408011784E-08

8.338008651366E-10

-1.704242447194E-11

1.500878861807E-13

-4.838826574173E-16

Value of bij Coefficients used in Equation (32) for Pressure > 3000 Psia i=0

6.897382693936E+02

2.730479206931E+00

-2.254102364542E-02

-4.651196146917E-03

3.439702234956E-05

i=1

2.213692462613E-01

6.547268255814E-03

5.982258882656E-05

2.274997412526E-06

-1.888361337660E-08

i=2

-5.118724890479E-05

2.019697017603E-06

-2.311332097185E-08

-4.079557404679E-10

3.893599641874E-12

i=3

5.517971126745E-09

2.415814703211E-10

3.121603486524E-12

3.171271084870E-14

-3.560785550401E-16

i=4

-2.184152941323E-13

1.010703706059E-14

-1.406620681883E-16

-8.957731136447E-19

1.215810469539E-20

21 | P a g e

2.1.4 Bahadori’s CO2 viscosity correlation Applications of equations of state and commercial soft-wares to predict properties is often easy and convenient for many engineers. However, they do not perform well for all properties, especially when predicting the viscosity of supercritical carbon dioxide (Bahadori 2008). There are many theoretical and semi-theoretical correlations of viscosity which often include other parameters like the density (Sastri and Rao 1999). Hence, data or correlations for these parameter are required before utilizing the viscosity correlations. As a result, this affects the accuracy of correlations predicting the viscosity due to uncertainties involved in the correlations predicting these other parameters. Moreover, this creates an inconvenience to find data or predict data using correlations for such parameters before finding the property that is wanted. These problems can be solved by directly correlating the viscosity of scCO2 as a function of temperature and pressure (Civan 2007). Bahadori and Vuthaluru (2010) proposed a simple correlation for prediction of CO2 viscosity as a function of pressure and temperature which works well between the temperature range of 260K-450K and pressure range of 10-70MPa. The objective of the adopted methodology for developing the new correlation is to identify a way to correlate the transport properties of CO2 with the two influential properties, namely temperature and pressure. They assumed that the best governing equation is the one that gives the minimal sum of deviations squared from the set of literature viscosity data. ∑, ( )

*

( )

( )

( ) +-

(33)

The coefficients, ‘a’,’b’,’c’ and ‘d’ for objective equation (33), can be calculated if the best correlated viscosity data, ‘ln (μ i)’, has the minimum error from ‘(1/P)i’. Therefore, these coefficients must yield zero for first derivatives of objective function (equation (33)). ∑, ( )

∑( ) , ( )

*

( )

*

( )

( )

( )

( ) +-

( ) +-

(34)

(35)

22 | P a g e

∑( ) , ( )

*

( )

( )

( ) +-

∑( ) , ( )

*

( )

( )

( ) +-

(36)

(37)

Expansion and conversion of above equations to linear form results in the following: ∑ ( )

∑ ( ) ( )

∑( )

∑( )

∑( )

∑( )

∑( )

∑( )

(38) ∑( )

∑ ( ) ( )

∑( )

∑( )

∑( )

∑( )

∑ ( ) ( )

∑( )

∑( )

∑( )

∑( )

(39)

(40)

(41)

The coefficients, ‘a’, ‘b’, ‘c’ and ‘d’ are obtained through solving equation (38-39). Knowing that these coefficients are a function of temperature, the previously tuned coefficients are needed to be made as a function of temperature. Therefore, ∑

(

( )

( )

( ) )

∑

(

( )

( )

( ) )

∑

(

( )

( )

( ) )

∑

(

( )

( )

( ) )

(42)

(43)

(44)

(45)

The tuned coefficients in equations (42-43) are derived by using the same procedure for equations (33-41). The required tuned coefficients are provided in Table 2.4.

23 | P a g e

Table 2.4: Tuned coefficients for Equation (46) Coefficient A1 B1 C1 D1 A2 B2 C2 D2 A3 B3 C3 D3 A4 B4 C4 D4

T<340 K -8.381727231932328×101 7.170262916398216×104 -2.088352606491789×107 2.035238087953347×109 7.688274861237018×103 -6.832908603727831×106 2.00319868619153×109 -1.94536522596535×1011 -1.967260059076993×105 1.732142393454871×108 -5.049067845006425×1010 4.882358762211981×1012 1.3529778432466×106 -1.19567721576674×109 3.498814034450212×1011 -3.395109635057981×1013

T>340 K -6.304360942940384×101 7.089412819202834×104 -2.729618206187531×107 3.491954145885637×109 5.392507286567643×103 -6.48675327864201×106 2.543938513422521×109 -3.281228975928387×1011 -1.182481836340281×105 1.438608961427429×108 -5.738803284656972×1010 7.535042772730154×1012 6.947087585578619×105 -8.506349304338924×108 3.424312685872325×1011 -4.542379235870166×1013

The correlation that predicts the viscosity (μ, mPa S) of CO2 as a function of temperature (T, K) and pressure (P, MPa) is as follows: ( )

(46)

Where, (47) (48) (49) (50)

Equation (46) was only designed to capture the variation of viscosity of CO2 in CO2 sequestration process condition. Therefore, it has limited applicable ranges of temperature and pressure

24 | P a g e

2.1.5 Jossi’s CO2 viscosity prediction model In 1962, Jossi, Stiel and Thodos correlated viscosity of carbon dioxide with reduced density by the use of dimensional analysis and the Abas-zade expression for the residual viscosity to deliver a single generalized equation which can be presented analytically. The properties needed for the viscosity calculation from this equation are the molecular weight, the critical constants, and the density of carbon dioxide at the temperature and pressure of interest (Jossi, Stiel and Thodos 1962). Prior to that, Stiel and Thodos used a dimensional analysis approach and literature viscosity data to develop a correlation which could easily be used to predict the viscosity of pure CO2 at 0.1 to 5 atm. According to them, carbon dioxide being a non-polar gas have the viscosity correlations as follows (Stiel and Thodos 1961): (51)

for

(52)

for Where

(53) Viscosity values from equations (51) and (52) for CO2 are very close to their experimental counterparts. However, these equations are valid for low pressures of 0.1 to 5 atm only. Hence, a similar approach was to be utilized to develop correlations for prediction of viscosity of CO2 at high pressures, especially in supercritical region. Researchers later discovered that the residual viscosity, μ-μ* could be a function of density and dimensional constants which are specific to CO2 (note: μ*=viscosity at normal pressures). Assuming these constants are molecular weight and the critical constants of CO2, Jossi, Stiel, and Thodos (1962) applied the Rayleigh method of dimensional analysis to the variables to obtain the following equation, (54) By using exponents, equation (54) was reduced to (

)

(55)

Using experimental CO2 viscosity data reported in literature and experimental PVT data, the constant

and the exponents m and n of equation (55) were identified. Since the

exponent n in equation (54) is not a constant and the (

) dependence on

is

25 | P a g e

complex but continuous function, it was found that fourth degree polynomial was necessary to accurately predict the viscosity of CO2 over the entire range of reduced densities. The coefficients of first, second, third and fourth degree polynomials were calculated by the method of least squares and the following equation was formed as a result: (

(56)

)

Knowing that viscosity parameter of CO2 is 0.0224, the viscosity of CO2 in centipoise for a wide range of temperature and pressure can be calculated using equation (56) and (51)/(52). Nonetheless, the major downside of this correlation is that it requires the density of CO2 to find the viscosity. Using other correlations to find the density can be an inconvenience and may involve uncertainties depending on which correlation has been used.

2.1.6 Liang-Biao Ouyang’s viscosity correlation Similar to predicting density, carbon dioxide viscosity can be predicted through equations of state, semi-equations of states and also from empirical correlations. Equations of state and semi-equations of state models require complex numerical computations which is often unwanted in many engineering applications. Meanwhile, as mentioned previously, some correlations require density as a dependent parameter to predict the viscosity. Therefore, Ouyang proposed a simple and new viscosity correlation for conditions under carbon capture and sequestration applications. Ouyang extracted the carbon dioxide properties under the temperature and pressure ranges anticipated for carbon sequestration operations from NIST database. He then applied the least square approach just like Bahadori to develop a simple correlation for prediction of carbon dioxide viscosity. The correlation formulae and the relevant correlation coefficients were adjusted in such a way that it fulfilled the following criteria: ∑(

)

∑(

(

)

)

(57)

The final viscosity correlation is shown below as a function of pressure: 26 | P a g e

(58) Where viscosity,

is in centipoises and pressure, P is in psia. The coefficients C0 to C4 as

a function of temperature (0C) can be found from the equation as follows: (59) (i=0 to 4) The correlation coefficients values di0 to di4 are listed in Table 2.5. Ouyang (2011)’s research indicated that his new correlation performs much better than Bahadori’s viscosity prediction correlation. However, the temperature and pressure ranges where this correlation is applicable is limited as his interest was only to predict the viscosity of CO2 in CO2 sequestration conditions.

27 | P a g e

Table 2.5: Values of coefficients for Equation (59) di0

di1

di2

di3

di4

Value of dij Coefficients used in Equation (59) for Pressure < 3000 Psia i=0

-1.958098980443E+01

1.123243298270E+00

-2.320378874100E-02

2.067060943050E-04

-6.740205984528E-07

i=1

4.187280585109E-02

-2.425666731623E-03

5.051177210444E-05

-4.527585394282E-07

1.483580144144E-09

i=2

-3.164424775231E-05

1.853493293079E-06

-3.892243662924E-08

3.511599795831E-10

-1.156613338683E-12

i=3

1.018084854204E-08

-6.013995738056E-10

1.271924622771E-11

-1.154170663233E-13

3.819260251596E-16

i=4

-1.185834697489E-12

7.052301533772E-14

-1.500321307714E-15

1.368104294236E-17

-4.545472651918E-20

Value of dij Coefficients used in Equation (59) for Pressure > 3000 Psia i=0

1.856798626054E-02

3.083186834281E-03

-1.004022090988E-04

8.331453343531E-07

-1.824126204417E-09

i=1

6.519276827948E-05

-3.174897980949E-06

7.524167185714E-08

-6.141534284471E-10

1.463896995503E-12

i=2

-1.310632653461E-08

7.702474418324E-10

-1.830098887313E-11

1.530419648245E-13

3.852361658746E-16

i=3

1.335772487425E-12

-8.113168443709E-14

1.921794651400E-15

-1.632868926659E-17

4.257160059035E-20

i=4

-5.047795395464E-17

3.115707980951E-18

-7.370406590957E-20

6.333570782917E-22

-1.691344581198E-24

28 | P a g e

CHAPTER 3 RESEARCH METHODOLOGY / MODELLING

3.1

General Description Thermodynamic and transport properties of CO2 are readily available from a variety

of literature sources. Numerical correlations are only to be developed through empirical modelling approach using the database at hand. Therefore, no experiment is required. The modelling of correlations will be done in the context of computer software such as Matlab and Microsoft Excel. The step-by-step method of how the research is to be done is as illustrated in Figure 3.1.

• Literature review on existing models • This will be done by accessing journals, articles and books from university Stage 1 library and the internet. • Collection of data • Physical properties of supercritical carbon dioxide and their respective and pressure will be collected from a variety of literature soruces Stage 2 temperature to construct a databse in a spreadsheet • Performance testing on current models • Using Matlab and Excel, all models will be tested against the database to Stage 3 analyse the performance.

Stage 4

• Modelling • Based on the database created in stage 2, emipirical models will be developed using Matlab and Excel to predict the density and viscosity of CO2

Stage 5

• Peformance testing on new models • Prediction ability of new correlations found from stage 4 will be tested with the property database and compared to existing models.

Figure 3.1: Research Methodology

29 | P a g e

3.2

Detailed Description for each stage 3.2.1 Stage 1: Literature Review on Existing Models Literature review was done by finding journals, articles, and books which are related to developing empirical correlations or equations of state to predict the density and viscosity of scCO2. Detailed analysis of each research material was done as described in previous chapter of this report.

3.2.2 Stage 2: Collection of data The physical and transport properties of supercritical carbon dioxide were easily found from NIST database. The National Institute of Standards and Technology (NIST) delivers a high quality copy of the database and verifies that the data contained in it have been chosen on the basis of sound scientific judgement (E.W. Lemmon, McLinden. and Friend 2011). The density (kg/m3) and viscosity of carbon dioxide (μPa s) with their corresponding temperature and pressure were copied into a Microsoft Excel Spreadsheet. The sets of data copied has a pressure range from 0.1 MPa to 300 MPa and temperature range from 220K to 1100K, covering a total 40969 data points for each property. The database is stored in a personal computer and several backups have been stored in two different external hard drives and Google drive storage. The condensed form of the database can be found in Table A.1 and Table A.2 in Appendix A.

3.2.3 Stage 3: Performance testing on each model The empirical correlations and equations of state found from the literature review were used in Matlab to predict the properties of scCO2 and the results were compared to NIST standard database. Description of how each model has been tested is described as follow: 3.2.3.1 Bahadori’s density correlation Bahadori’s density model is a well-defined empirical model as seen from equations (14-18) and the tuned coefficients are given in Table 2.1. Therefore, in Matlab, equation (14-18) and coefficient values were easily replicated. Then, the density equation was programmed to predict the density values of CO2 in the ranges

30 | P a g e

of temperature and pressure stated in the literature. The summary of the parameters used in Matlab are stated in Table 3.1. Table 3.1: Matlab parameters used for Bahadori's Density Model

Parameter

Min

Max

Interval

No. of Points

Temperature

293 K

433 K

1K

141

Pressure

25 bar

700 bar

Increasing

14

Density to be 31.5571 predicted

1019.1

kg/m3

1974

kg/m3

The predicted data was stored in a matrix. The NIST density standard database was duplicated and resized according to the parameters in Table 3.1. Then, read function in Matlab was used to copy the NIST density values from the resized database into a second matrix. Then, the predicted density values from the first matrix were compared to their corresponding NIST density values from the second matrix by calculating the percentage absolute average residual deviation (%AARD). Equation of %AARD is stated as follows: ∑

|

| (60)

Where , density predicted from correlation , density from NIST number of data points. Afterwards, the model was tested for the much larger pressure and temperature ranges of the NIST database prepared, where pressure range is from 0.1MPa to 300 MPa while temperature range is from 268K to 1100K. Next, the %AARD of the Bahadori’s model in the wider ranges of temperature and pressure was calculated using equation (60). Thereafter, the density versus temperature graphs were plotted in isobaric lines

31 | P a g e

to conduct visual comparison between predicted density and NIST density. In the end, graphs of predicted density versus NIST density was plotted for overall comparison. 3.2.3.2 Modified Redlich-Kwong equation of state for density of CO2 The modified Redlich-Kwong equation of state given in the literature was not as straightforward as Bahadori’s density correlation. The equation of state given had pressure as a function of temperature and specific volume as seen in equation (20). This was rearranged into cubic form of compressibility as shown in equation (26). From the compressibility factor, the density of CO2 was calculated for the temperature and pressure ranges given in Table 3.2 and stored in a matrix. Table 3.2: Matlab parameters for Modified Redlich-Kwong Equation of State Parameter

Min

Max

Interval

No. of Points

Temperature

268K

1100 K

1K

833

Pressure

0.1 MPa

300 MPa

Increasing

47

Density to be 0.4811 predicted

kg/m3

1249.4

39151

kg/m3

Then, the density values from NIST database were copied into a new Matlab matrix and compared to their corresponding predicted values by calculating %AARD with equation (60). Thereafter, the density versus temperature graphs were plotted in isobaric lines to conduct visual comparison between predicted density and NIST density. In the end, graphs of predicted density versus NIST density was plotted for overall comparison.

3.2.3.3 Liang-Biao Ouyang’s density correlation The approach used for Ouyang’s density correlation in Matlab is similar to the approach used for Bahadori’s density correlation. For initial assessment, Ouyang’s equation was tested for reported ranges of temperature and pressure in literature and predicted values were stored in a matrix. The summary of the parameters used in Matlab are stated in Table 3.3.

32 | P a g e

Table 3.3: Matlab parameters for Liang-Biao Ouyang's density correlation Parameter

Min

Max

Interval

No. of Points

Temperature

313 K

373 K

1K

63

Pressure

7 MPa

60 MPa

Increasing

17

Density to be 50.1659 predicted

kg/m3

1020.5

1071

kg/m3

Then, the density values from NIST database were copied into a new Matlab matrix and compared to their corresponding predicted values by calculating %AARD with equation (60). Afterwards, the model was tested for larger pressure and temperature, where pressure range is from 0.1 MPa to 300 MPa while temperature range is from 268K to 1100K. Then, the %AARD of this model in the wider ranges of temperature and pressure was calculated using equation (60).

Thereafter, the density versus

temperature graphs were plotted in isobaric lines to conduct visual comparison between predicted density and NIST density. In the end, graphs of predicted density versus NIST density was plotted for overall comparison.

3.2.3.4 Bahadori’s viscosity correlation The approach used for Bahdori’s viscosity correlation in Matlab is similar to the approach used for Bahadori’s density correlation. For initial assessment, the equation was tested for reported ranges of temperature and pressure in literature and predicted values were stored in a matrix. The summary of the parameters used in Matlab for initial assessment are stated in Table 3.4.

33 | P a g e

Table 3.4: Matlab parameters for Bahadori’s viscosity model

Parameter

Min

Max

Interval

No. of Points

Temperature

260 K

450 K

1K

63

Pressure

10 MPa

70 MPa

Increasing

17

Viscosity to 21.6226 be predicted

μPa

225.2907

2483

μPa

The rest of the procedure is the same as the procedure for Bahadori’s density correlation. However, for the graphs, the viscosities were plotted against the pressure in of isothermal lines.

3.2.3.5 Jossi’s CO2 viscosity prediction model For Jossi’s CO2 viscosity prediction model, in order to calculate the viscosity, the density of the CO2 at the required temperature and pressure is needed to be known. Therefore, the CO2 density data was copied from the database into an array in Matlab first. Afterwards, equations (51, 52 and 56) were used to predict the viscosity of CO2 by substituting in the temperatures, pressures and corresponding densities, and the predicted viscosities were stored in an array. The summary of the parameters used in Matlab are stated in Table 3.5 Table 3.5: Matlab parameters for Jossi's CO2 viscosity prediction model Parameter

Min

Max

Interval

No. of Points

Temperature

268 K

1100 K

1K

833

Pressure

0.1 MPa

300 MPa

Increasing

47

Viscosity to 13.9217

541.9899

be predicted

μPa

39151

μPa

Then, the viscosity values from NIST database were copied into a new Matlab matrix and compared to their corresponding predicted values by calculating %AARD with 34 | P a g e

equation (60). Thereafter, the viscosity versus pressure graphs were plotted in isothermal lines to conduct visual comparison between predicted viscosity and NIST density. In the end, graphs of predicted viscosity versus NIST viscosity was plotted for overall comparison.

3.2.3.6 Liang-Biao Ouyang’s viscosity correlation The approach used for Liang-Biao’s viscosity correlation in Matlab is similar to the approach used for Bahadori’s viscosity correlation. For initial assessment, the equation was tested for reported ranges of temperature and pressure in literature and predicted values were stored in a matrix. The summary of the parameters used in Matlab for initial assessment are stated in Table 3.4. Table 3.6: Matlab parameters for Ouyang's Viscosity Correlation Parameter

Min

Max

Interval

No. of Points

Temperature

313 K

373 K

1K

63

Pressure

7 MPa

60 MPa

Increasing

17

Viscosity to 7.8192 μPa

128.4066

be predicted

μPa

1071

The rest of the procedure is the same as the procedure for Bahadori’s viscosity correlation. As for the graphs, the viscosities were plotted against the pressure in of isothermal lines.

3.2.4 Stage 4 and 5: Modelling This stage is to be completed as part of Research Project 412. The modelling approaches are to be adopted from some models in literature review and models are to be developed in Matlab and Microsoft Excel based on the NIST database prepared. The correlations obtained will be compared against the NIST database using the same approach used to compare existing models to NIST.

35 | P a g e

3.3

Facilities and Resources Since this research project is aimed to develop empirical correlations based on literature

data, the whole research project can be completed in a personal computer with the right software. The specifications of the computers and the software utilized are described as follows:

1. ICT Lab 1 of Curtin University Sarawak Malaysia 

Dell Desktop Computer with the following specifications and software o Intel Core i5 2.1 GHz o 2 Gb RAM o 320 Gb HDD o Windows 7



Matlab R2008a



Microsoft Excel 2010

2. Personal Computer 

Lenovo Laptop Computer with the following specifications and software o Intel Core i7 2.2 GHz o 6 Gb RAM o 720 Gb HDD o Windows 8



Matlab R2001a



Microsoft Excel 2013

36 | P a g e

CHAPTER 4 RESULTS AND DISCUSSIONS

4.1

Results Using the methods described in the previous chapter of this report, the densities and

viscosities of carbon dioxide in both supercritical and normal phase were determined for each model. Afterwards, the performances of each model were determined by calculating the %AARD using equation (60). The condense form of predicted densities and viscosities are tabulated and shown in Appendix B for all six models. The performances of all the density models were summarized and tabulated in Table 4.1. Likewise, the performances of all the viscosity models were summarized and tabulated in Table 4.2. Table 4.1: Summary of results for density models No Model

Temperature Pressure Range, K Range, MPa

1

293- 433

2.5-70

No of data points used 1974

268 - 1100

0.1-300

39151

Bahadori

Performa nce, %AARD

Remarks

4.39

Moderate accuracy for small ranges of T and P Moderate accuracy for wide ranges of T and P High accuracy for small ranges of T and P

2

Modified RedlichKwong

268-1100

0.1-300

39151

4.83

3

Liang-Biao Ouyang

313- 373

7-60

1071

1.25

268-1100

0.1-300

39151

37 | P a g e

Table 4.2: Summary of results for viscosity models No Model

Temperature Pressure Range, K Range, MPa

1

Bahadori

260-450

2

Jossi

3

Liang-Biao Ouyang

4.2

Performa nce, %AARD

Remarks

10-70

No of data points used 2483

1.67

268-1100 268-1100

0.1-300 0.1-300

39151 39151

Infinity 2.41

313-373

7-60

1071

0.86

268-1100

0.1-300

39151

High accuracy for small ranges of T and P High accuracy for wide ranges of T and P Very high accuracy for small ranges of T and P

Graphical Comparisons From the results obtained, graphical comparisons were made between all model

predictions with the NIST data obtained to further understand the performance and weaknesses of each model. 4.2.1 Bahadori’s density correlation The predicted density values of Bahadori’s model were plotted against temperature as seen in Figure 4.1. Figure 4.1 indicates that Bahadori’s density correlation performs fairly well for low pressure ranges and predicted values started to deviate from the NIST database at about 400 bar (40 MPa). The isobaric lines for pressure below 400 bar were overlapping with the NIST data lines for most of the temperature and therefore, the model performs very well in the temperature range given in the literature.

38 | P a g e

Figure 4.1: Density vs Temperature plot of Bahadori's Density Model (Literature Range) As seen in Figure 4.2, many of the predicted density values from this model were in agreement with the NIST data as most of them lie in close proximity to the straight line ing through zero. However, there was a significant amount of predicted density which deviated away from the NIST. This had probably caused the %AARD in Table 4.1 to be relatively higher.

Figure 4.2: Bahdori Density versus NIST Density

39 | P a g e

Figure 4.3: Density vs temperature plot of Bahadori's Density Model (wider range) Figure 4.3 consists of isobaric lines at pressure and temperature ranges reported in the literature as well as isobaric lines that are outside the literature working range. As seen in Figure 4.3, the predicted isobaric line at 10 bar, which is below the minimum pressure stated in the literature, did not conform to the trend of NIST density data at 10 bar. Likewise, the predicted isobaric line at 800 bar, where the pressure is higher than the maximum pressure stated in literature, did not follow the NIST density data trend. Next, the predicted isobaric line at 250 bar is in the literature pressure range and therefore it followed the corresponding NIST density trend. However, at temperature above 433 K, it started to deviate from the NIST density trend. Hence, it can be justified that this model does not function at all at temperatures and pressures which are out of the ranges stated in the literature. This explains why the %AARD of this model in Table 4.1 is so large when the model had been applied to wider ranges of temperature and pressure.

4.2.2 Modified Redlich-Kwong equation of state for density of CO2 The predicted density values of modified Redlich-Kwong equation of state were plotted against temperature as seen in Figure 4.4.

40 | P a g e

Figure 4.4: Density vs temperature plot of modified Redlich-Kwong Equation of State According to Figure 4.4, the predicted density values were in closer proximity to their NIST counterpart at low pressure and high temperature. The deviation of predicted data from NIST data became significant starting from 10 MPa at low temperatures and remained constant with increasing pressure. However, the deviation decreased with increasing temperature. Figure 4.5 illustrates how far the predicted density values are from the NIST data. As seen in Figure 4.5, density values were in close proximity to their NIST counterparts for low and high densities.

Figure 4.5: Modified Redlich-Kwong density vs NIST density 41 | P a g e

There were quite a number of density values deviating away from the NIST as seen in the figure. However, compared to the large amount of data that were close to their corresponding NIST values, they can be considered negligible. Therefore, the overall %AARD is only at 4.83% as seen in Table 4.1.

4.2.3 Liang-Biao Ouyang’s density correlation The predicted density values of modified Liang-Biao Ouyang’s model were plotted against temperature as seen in Figure 4.6.

Figure 4.6: Density vs temperature plot of Liang-Biao Ouyang's model (Literature Range) According to Figure 4.6, the predicted density values were mostly similar to their corresponding NIST density values except for the isobaric line at 7 MPa; there were some deviations at low temperature. Figure 4.7 indicates that most of the predicted densities were more or less the same as the NIST densities except for a few densities below 300kg/m3. Hence, the %AARD in the reported range of temperature and pressure is only 1.25%.

42 | P a g e

Figure 4.7: Liang-Biao Ouyang’s model density vs NIST density Figure 4.8 was plotted to study the performance of Liang-Biao Ouyang’s model at the conditions outside the reported ranges of temperature and pressure.

Figure 4.8: Density vs Temperature plot of Liang-Biao Ouyang’s model (wider range)

According to Figure 4.8, the predicted isobaric line at 6 MPa, which is below the minimum pressure stated in the literature for this model, did not conform to the trend of its NIST counterpart. The same was true for predicted density isobaric line at 80 MPa, which is above the maximum pressure stated in the literature for this model. The predicted isobaric lines with pressure values that lie in the literature reported range followed the NIST density trend lines within the temperature range stated in the literature for this model. Therefore, it can be deduced that the model does not function at all outside the 43 | P a g e

literature ranges of temperature and pressure, which explains the extremely large %AARD when this model was applied outside its literature ranges of temperature and pressure.

4.2.4 Bahadori’s viscosity correlation The predicted viscosity values of Bahadori’s model were plotted against pressure as seen in Figure 4.9 and Figure 4.10. According to Figure 4.9 and Figure 4.10, the predicted viscosities in of isothermal lines were more or less the same as their corresponding NIST viscosities for the whole range of pressure reported in the literature. The deviations were small and negligible. This verifies that the model works perfectly well within the reported ranges of temperature and pressure.

Figure 4.9: Viscosity vs pressure plot of Bahadori’s viscosity model (260K -340 K)

44 | P a g e

Figure 4.10: Viscosity vs pressure plot of Bahadori's viscosity model (340K-450K)

Figure 4.11 illustrates how far the predicted viscosity values are from the NIST data. As seen in Figure 4.11, most viscosity values were in close proximity to their NIST counterparts. Therefore, the %AARD is only 1.67% as seen in Table 4.2 for the predicted values of this model in reported ranges of temperature and pressure.

Figure 4.11: Bahadori's Model Viscosity vs NIST Viscosity

Figure 4.12 was plotted to study the performance of Bahadori’s viscosity model at the conditions outside the reported ranges of temperature and pressure. 45 | P a g e

Figure 4.12: Viscosity vs pressure plot for Bahadori's viscosity model (wider range) As seen from Figure 4.12, below 10 MPa, which is the minimum pressure stated in the literature for this model, all predicted viscosities were different from corresponding NIST viscosities. Likewise, the predicted viscosities start to deviate significantly from NIST viscosities for most isotherms above 70 MPa, which is the maximum pressure for this model. Therefore, this confirmed that the model is only applicable for the ranges of temperature and pressure stated in the literature. This also explains the high %AARD when the model was tested outside the ranges specified in the literature.

4.2.5 Jossi’s CO2 viscosity prediction model The predicted viscosity values of Jossi’s model were plotted against pressure as seen in Figure 4.13 and Figure 4.14. As seen in Figure 4.13 and 4.14, the predicted viscosities were fairly close to their NIST counterparts except for 300 K and 700 K isotherms. Therefore, this model performs very well throughout a wide range of temperatures and pressures.

46 | P a g e

Figure 4.13: Viscosity vs temperature plot for Jossi's Model (300K-600K)

Figure 4.14: Viscosity vs temperature plot for Jossi's model (700K-1000K)

Figure 4.15 illustrates how far the predicted viscosity values are from the NIST data. As seen in Figure 4.15, most viscosity values were in close proximity to their NIST counterparts. Therefore, the %AARD is only 2.41% as seen in Table 4.2 for the predicted values of this model for a wide range of temperature and pressure.

47 | P a g e

Figure 4.15: Jossi's Model Viscosity vs NIST viscosity

4.2.6 Liang-Biao Ouyang’s viscosity correlation The predicted viscosity values of Jossi’s model were plotted against pressure as seen in Figure 4.16 and Figure 4.17.

Figure 4.16: Viscosity vs pressure plot for Liang-Biao Ouyang' s model (313K-343K)

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Figure 4.17: Viscosity vs pressure plot for Liang-Biao Ouyang's model (353K-373K)

As seen from Figure 4.16 and 4.17, the predicted viscosities of this model were in an excellent agreement with the NIST viscosities when the model was tested between the temperature and pressure ranges reported in the literature. Figure 4.18 confirmed the accuracy of this model by demonstrating a nearly perfect straight line through zero. Therefore, the %AARD was only 0.86% as reported in Table 4.2.

Figure 4.18: Liang-Biao Ouyang's model viscosity vs NIST viscosity

49 | P a g e

Figure 4.19 was plotted to study the performance of Liang-Biao Ouyang’s viscosity model at the conditions outside the reported ranges of temperature and pressure. According to Figure 4.19, below 7 MPa (1015 bar), which is the minimum pressure for this model, the predicted viscosities were inaccurately calculated as the values become negative. This indicated that the model will not perform outside the ranges of temperature and pressure specified in the literature. This explains the extremely high %AARD as seen in Table 4.2 when the model was tested outside its valid range.

Figure 4.19: Viscosity vs pressure plot for Liang-Biao Ouyang's viscosity model (wider range)

4.3

Discussion According to Table 4.1, Liang-Biao Ouyang’s density correlation had the most

accurate prediction of the density of CO2 in supercritical region. The major short coming for this equation is the small ranges of temperature and pressure in which the equation works. The %AARD in Table 4.1 and Figure 4.8 had confirmed the fact this model does not work at all outside its literature ranges. Compared to Liang-Biao Ouyang’s model, Bahadori’s density model had a slightly wider range of operability. However, there was a trade-off in accuracy where Bahadori’s %AARD was nearly four times Liang-Biao Ouyang’s model. Furthermore, according to Figure 4.1, the isobaric trends at higher pressures for Bahadori’s model differ significantly from NIST density trends even though they were still in the literature 50 | P a g e

temperature and pressure range. In of operable ranges, modified Redlich-Kwong’s equation of state was far more superior compared to Liang-Biao Ouyang’s and Bahadori’s models. The %AARD of modified Redlich-Kwong’s is decent for a model that is valid for high temperature and pressure ranges. The downside for this model is that it requires more extensive programming compared to Bahadori’s and Liang-Biao Ouyang’s correlations since it is a semi-equation of state. However, it was much simpler and easier to use compared to other equations of state. As for viscosity, amongst the models compared, Liang-Biao Ouyang’s viscosity correlation had the most accurate prediction of the viscosity of CO2 in supercritical region. Again, the shortcoming of this model was that it had very small ranges of temperature and pressure where the model worked. Bahadori’s viscosity model had wider ranges of temperature and pressure in which the model worked with less accuracy (two times the %AARD of Liang-Biao Ouyang’s model). Jossi’s model, however, had the best prediction for a very wide range of temperature and pressure with a very decent accuracy. However, the major disadvantage of this equation was that it required density values. The accuracy was high probably due to the fact that NIST densities were substituted inside to predict the viscosity. If there were no CO2 densities available and other correlations were to be used to predict the density first, this model would not have performed well. From this comparative study, it was found that models that had high accuracy to predict the density and viscosity of scCO2 had lower ranges of applicability whereas those with high ranges of applicability had unsatisfactory accuracy. Moreover, some models like modified Redlich-Kwong’s equation of state and Jossi’s viscosity correlation are inconvenient to be used to predict the properties due to their complexity. Judging from the shortcomings of existing models, there is an opportunity to create a relatively simpler and easier-to-use equations that can better predict the densities and viscosities of CO2 in supercritical region.

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CHAPTER 5 CONCLUSIONS, RECOMMENDATIONS AND WORKPLAN FOR RESEARCH PROJECT 412

5.1

Conclusion In conclusion, the ability to accurately predict the densities and viscosities of

supercritical CO2 has become very crucial due to the fact that supercritical CO2 has been widely used in many chemical industries, especially in the extraction processes. They can be predicted through a variety of empirical correlations and equations of state which are reported in the literature. There are also semi-equations of state which are a hybrid of the empirical correlations and equations of state and they normally are simpler than the pure equations of state. To determine the performance of existing models, a comparative study had been conducted in this research project on 6 models, namely Bahadori’s density correlation, modified Redlich-Kwong’s equation of state, Liang-Biao Ouyang’s density correlation, Bahadroi’s viscosity correlation, Jossi’s viscosity correlation and Liang-Biao Ouyang’s viscosity correlation. The predicted values from all these models were compared to the density and viscosities from NIST database, which is a high quality database used by many researchers. According to the results, Liang-Biao Ouyang’s density correlation and LiangBiao Ouyang’s viscosity correlation had the highest accuracy, with only 1.25% and 0.86% percentage absolute average residual deviation (%AARD) respectively compared to NIST database. Nevertheless, both of them could only perform in very small ranges of temperature and pressure, which are 7 MPa-60 MPa and 313 K –373 K respectively. Bahadori’s density correlation worked between pressure of 2.5 MPa-70 MPa and temperature of 293 K – 433 K with %AARD of 4.39%. Modified Redlich-Kwong’s equation of state for density prediction worked well for a large range of temperature between 268 K – 1100 K and pressure between 0.1 MPa – 300 MPa. The accuracy of this model could be considered decent as the %AARD was only 4.83%. Next, Bahadori’s viscosity correlation worked well at pressures between 10 MPa-70 MPa and temperatures between 260 K -450 K with %AARD of only 1.67%. On the other hand, Jossi’s viscosity correlation worked for a large range of temperature between 268 K and 1100 K, and pressure between 0.1 MPa and 300 MPa, with %AARD of only 2.41%. However, this equation requires the densities of CO2 beforehand which makes it rather 52 | P a g e

complicated to use. The comparative study results indicated that there is still room for improvements in existing models because some of them had unsatisfactory accuracy in predictions while the others had very short ranges of validity. Therefore, new correlations which are simple, easy-to-use, and at the same time applicable for wider ranges of temperature and pressure with high accuracy are to be modelled in the next part of this research project. These improved correlations would be useful for engineers and the scientific community in general and they could be used as a quick and explicit way to determine the properties of supercritical CO2.

5.2

Recommendations The results of the comparative study are summarized in Table 4.1 and 4.2 as seen in

Chapter 4 of this report. If any of the existing models are to be used, these two tables could be referred to determine the ranges of temperature and pressure where these models are valid and their corresponding accuracies. The comparative study could have been more thorough and comprehensive if the ranges of temperature and pressure were broken down into subranges and the models were tested in these sub-ranges. This is to identify in which region of temperature and pressure the models work best and where the models perform weakly. Next, the results of the comparative study highlighted the weaknesses of these models and knowing these weaknesses could be crucial in developing new and better models. These weaknesses must be addressed in the development of new models and these new models are recommended to be free from these shortcomings of existing models.

5.3

Workplan for research project 412 In Research Project 412, the primary goal is to develop two entirely new empirical

correlations; one correlation to predict the density of supercritical CO2 and the other predicting the viscosity of CO2. Workplan on how this goal is going to be achieved is as described in Table 5.1.

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Table 5.1: Workplan for Research Project 412 Jul

Aug

Sep

Oct

Nov

Literature review on modelling Modelling of density correlation Modelling of viscosity correlation Completion of Modelling

Milestone 1

Testing of new correlations Conducting comparative study of new correlations and existing models Completion of comparative study

Milestone 2

Writing of draft Thesis Finalizing Thesis Completion of Research project

Milestone 3

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Haghbakhsh, Reza, Hossein Hayer, Majid Saidi, Simin Keshtkari, and Feridun Esmaeilzadeh. 2013. "Density Estimation of Pure Carbon Dioxide at Supercritical Region and Estimation Solubility of Solid Compounds in Supercritical Carbon Dioxide: Correlation Approach Based on Sensitivity Analysis." Fluid Phase Equilibria 342 (0): 31-41. doi: http://dx.doi.org/10.1016/j.fluid.2012.12.029. Hassanzadeh, Hassan, Mehran Pooladi-Darvish, Adel M. Elsharkawy, David W. Keith, and Yuri Leonenko. 2008. "Predicting Pvt Data for Co2–Brine Mixtures for Black-Oil Simulation of Co2 Geological Storage." International Journal of Greenhouse Gas Control 2 (1): 65-77. doi: http://dx.doi.org/10.1016/S1750-5836(07)00010-2. Heidaryan, Ehsan, and Azad Jarrahian. 2013. "Modified Redlich⿿Kwong Equation of State for Supercritical Carbon Dioxide." The Journal of Supercritical Fluids 81 (0): 92-98. doi: http://dx.doi.org/10.1016/j.supflu.2013.05.009. Jossi, John A., Leonard I. Stiel, and George Thodos. 1962. "The Viscosity of Pure Substances in the Dense Gaseous and Liquid Phases." AIChE Journal 8 (1): 59-63. doi: 10.1002/aic.690080116. Jouyban, Abolghasem, Hak-Kim Chan, and Neil R. Foster. 2002. "Mathematical Representation of Solute Solubility in Supercritical Carbon Dioxide Using Empirical Expressions."

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Redlich, Otto, and J. N. S. Kwong. 1949. "On the Thermodynamics of Solutions. V. An Equation of State. Fugacities of Gaseous Solutions." Chemical Reviews 44 (1): 233244. doi: 10.1021/cr60137a013. Sastri, S. R. S., and K. K. Rao. 1999. "A New Temperature–Thermal Conductivity Relationship for Predicting Saturated Liquid Thermal Conductivity." Chemical Engineering Journal 74 (3): 161-169. doi: http://dx.doi.org/10.1016/S13858947(99)00046-7. Stiel, Leonard I., and George Thodos. 1961. "The Viscosity of Nonpolar Gases at Normal Pressures." AIChE Journal 7 (4): 611-615. doi: 10.1002/aic.690070416. Vesovic, V., A. W. Wakeham, G. A. Olchowy, J. V. Sengers, J. T. R. Watson, and J. Millat. 1990. "The Transport Properties of Carbon Dioxide." Journal of Physical and Chemical Reference Data 19 (3). Yamamoto, Satoru, Takashi Furusawa, and Ryo Matsuzawa. 2011. "Numerical Simulation of Supercritical Carbon Dioxide Flows across Critical Point." International Journal of Heat

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57 | P a g e

APPENDIX A: NIST database Table A.1: NIST density data P (Mpa) T (K) 220 250 300 350 400 450 500 550 600 650 700 750 800 850 900 1000 1050 1100

0.1

0.5

1.0

1.5

2.0

2.5

2.4394 2.1363 1.773 1.5167 1.3257 1.1776 1.0594 0.96283 0.88242 0.81444 0.75619 0.70572 0.66158 0.62264 0.58803 0.52921 0.504 0.48109

12.974 11.097 9.0456 7.6736 6.677 5.9154 5.3126 4.8229 4.4168 4.0743 3.7814 3.5281 3.3067 3.1116 2.9383 2.6439 2.5179 2.4033

1167 23.435 18.579 15.581 13.477 11.899 10.664 9.6675 8.8449 8.1535 7.5638 7.0546 6.6102 6.219 5.8718 5.2826 5.0304 4.8014

1168.1 37.509 28.674 23.739 20.405 17.952 16.054 14.533 13.284 12.237 11.347 10.579 9.9104 9.322 8.8004 7.9159 7.5376 7.1941

1169.2 1046.9 39.42 32.165 27.464 24.075 21.482 19.42 17.733 16.325 15.13 14.102 13.207 12.421 11.724 10.544 10.039 9.5815

1170.3 1049 50.936 40.878 34.658 30.268 26.948 24.326 22.193 20.417 18.914 17.622 16.5 15.515 14.643 13.166 12.536 11.964

3.0 3.5 Density (kg/m3) 1171.4 1172.5 1051 1053 63.376 76.952 49.901 59.258 41.992 49.469 36.531 42.865 32.451 37.99 29.252 34.197 26.661 31.139 24.513 28.611 22.697 26.48 21.14 24.656 19.789 23.074 18.604 21.689 17.556 20.464 15.783 18.395 15.027 17.512 14.34 16.711

4.0

4.5

5.0

5.5

6.0

6.5

1173.5 1055 91.965 68.976 57.094 49.27 43.566 39.16 35.625 32.712 30.262 28.168 26.355 24.769 23.367 21.001 19.992 19.077

1174.6 1057 108.87 79.085 64.871 55.747 49.178 44.141 40.12 36.816 34.043 31.678 29.632 27.843 26.265 23.601 22.466 21.438

1175.6 1058.9 128.4 89.619 72.804 62.295 54.826 49.14 44.621 40.921 37.823 35.184 32.904 30.913 29.157 26.196 24.935 23.793

1176.7 1060.7 151.89 100.62 80.897 68.914 60.507 54.155 49.13 45.028 41.601 38.686 36.172 33.977 32.043 28.785 27.398 26.142

1177.7 1062.6 182.31 112.12 89.155 75.605 66.223 59.186 53.645 49.136 45.377 42.185 39.435 37.036 34.924 31.368 29.856 28.486

1178.7 1064.4 229.58 124.17 97.581 82.367 71.972 64.233 58.167 53.245 49.151 45.68 42.692 40.09 37.799 33.946 32.308 30.825

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P (MPa) T (K) 220 250 300 350 400 450 500 550 600 650 700 750 800 850 900 1000 1050 1100

7.0

7.5

10.0

12.5

15.0

17.5

1179.7 1066.2 706.06 136.82 106.18 89.2 77.754 69.294 62.694 57.354 52.922 49.171 45.945 43.138 40.668 36.517 34.754 33.158

1180.7 1068 733.9 150.13 114.95 96.102 83.566 74.369 67.225 61.464 56.691 52.657 49.193 46.18 43.532 39.083 37.195 35.485

1185.6 1076.4 801.62 228.8 161.53 131.63 113.07 99.93 89.941 81.999 75.486 70.019 65.349 61.304 57.759 51.825 49.312 47.04

1190.4 1084.3 838.84 332.37 212.58 168.72 143.2 125.72 112.7 102.48 94.173 87.246 81.356 76.273 71.831 64.417 61.284 58.455

1195 1091.8 865.82 449.2 267.42 207.01 173.76 151.63 135.43 122.86 112.72 104.32 97.199 91.077 85.74 76.856 73.11 69.73

1098.8 887.39 545.7 324.27 246.03 204.53 177.54 158.06 143.09 131.1 121.21 112.86 105.71 99.48 89.14 84.787 80.864

20.0 22.5 Density (kg/m3) 1105.5 905.57 614.18 380.5 285.14 235.24 203.3 180.5 163.12 149.27 137.9 128.34 120.15 113.04 101.27 96.315 91.857

1111.8 921.37 663.92 433.49 323.68 265.64 228.79 202.69 182.9 167.22 154.37 143.6 134.4 126.43 113.23 107.69 102.71

25.0

27.5

30.0

35.0

40.0

45.0

1117.9 935.42 702.22 481.55 361.01 295.46 253.87 224.54 202.4 184.9 170.61 158.65 148.45 139.62 125.04 118.92 113.42

1123.7 948.11 733.13 524.13 396.64 324.48 278.43 245.99 221.56 202.3 186.59 173.47 162.3 152.63 136.68 129.99 123.98

1129.3 959.7 758.98 561.5 430.25 352.51 302.37 266.99 240.36 219.39 202.31 188.05 175.92 165.44 148.15 140.91 134.4

1139.8 980.32 800.68 623.24 490.88 405.16 348.09 307.43 276.74 252.56 232.88 216.47 202.51 190.46 170.6 162.29 154.82

1149.7 998.35 833.72 672.08 542.98 452.94 390.67 345.6 311.36 284.29 262.24 243.83 228.18 214.66 192.37 183.05 174.67

1158.9 1014.4 861.19 711.97 587.64 495.9 430 381.4 344.13 314.54 290.35 270.13 252.91 238.02 213.47 203.19 193.95

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P (MPa) T (K) 220 250 300 350 400 450 500 550 600 650 700 750 800 850 900 1000 1050 1100

50.0

55.0

60.0

65.0

70.0

75.0

1167.6 1028.9 884.76 745.45 626.2 534.42 466.18 414.84 375.06 343.27 317.2 295.34 276.69 260.55 233.89 222.71 212.66

1175.8 1042.2 905.45 774.19 659.87 568.99 499.42 446.04 404.19 370.53 342.79 319.47 299.53 282.25 253.64 241.63 230.82

1183.6 1054.5 923.92 799.32 689.59 600.13 529.96 475.12 431.61 396.36 367.18 342.56 321.46 303.13 272.73 259.95 248.43

1191.1 1065.9 940.63 821.66 716.1 628.32 558.07 502.23 457.43 420.84 390.4 364.64 342.5 323.22 291.18 277.68 265.51

1198.2 1076.5 955.9 841.76 739.96 654 584.01 527.55 481.74 444.05 412.53 385.75 362.68 342.55 309.01 294.85 282.06

1205 1086.6 969.98 860.03 761.62 677.52 608.02 551.22 504.66 466.06 433.62 405.95 382.04 361.14 326.23 311.46 298.11

80.0 85.0 Density (kg/m3) 1211.6 1096 983.05 876.8 781.42 699.16 630.31 573.4 526.29 486.96 453.73 425.28 400.64 379.04 342.88 327.54 313.67

1217.9 1105 995.26 892.29 799.66 719.18 651.09 594.22 546.74 506.83 472.93 443.8 418.5 396.27 358.97 343.11 328.75

90.0

95.0

100.0

150.0

200.0

300.0

1224 1113.5 1006.7 906.7 816.55 737.78 670.52 613.82 566.11 525.73 491.28 461.56 435.66 412.88 374.52 358.2 343.38

1229.9 1121.7 1017.5 920.16 832.28 755.12 688.73 632.3 584.47 543.75 508.82 478.6 452.18 428.88 389.58 372.81 357.58

1235.6 1129.4 1027.7 932.81 847 771.37 705.86 649.77 601.91 560.93 525.63 494.96 468.08 444.32 404.15 386.97 371.36

1284.4 1193.4 1108.3 1029.8 958.22 893.75 836.09 784.69 738.93 698.17 661.81 629.28 600.07 573.72 528.11 508.24 489.99

1241.6 1166.3 1097 1033.7 976.16 923.96 876.69 833.86 795 759.66 727.46 698.04 671.1 623.57 602.51 583

1314.4 1250.5 1192.2 1138.7 1089.7 1044.6 1003.2 964.92 929.62 896.96 866.67 838.51 812.27 764.83 743.33 723.14

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Table A.2: NIST viscosity data P (Mpa) T(K) 220 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1050 1100

0.1

0.5

1.0

1.5

2.0

2.5

11.063 12.565 15.021 17.402 19.696 21.901 24.017 26.048 27.999 29.873 31.676 33.413 35.088 36.707 38.273 39.79 41.262 42.692 44.084

11.118 12.61 15.057 17.431 19.721 21.922 24.036 26.066 28.014 29.887 31.689 33.425 35.1 36.718 38.283 39.799 41.271 42.701 44.092

242.81 12.691 15.114 17.476 19.758 21.954 24.064 26.09 28.037 29.907 31.708 33.442 35.116 36.732 38.297 39.812 41.283 42.712 44.103

243.81 12.81 15.19 17.533 19.803 21.992 24.097 26.119 28.062 29.93 31.728 33.461 35.133 36.749 38.312 39.826 41.296 42.725 44.115

244.8 147.18 15.286 17.601 19.856 22.036 24.133 26.151 28.09 29.955 31.751 33.482 35.152 36.766 38.328 39.842 41.31 42.738 44.127

245.79 148.2 15.408 17.682 19.918 22.085 24.175 26.186 28.121 29.983 31.776 33.504 35.172 36.785 38.345 39.858 41.326 42.752 44.141

3.0 3.5 Viscosity (μPa s) 246.78 247.76 149.21 150.21 15.561 15.753 17.778 17.889 19.988 20.068 22.141 22.203 24.221 24.271 26.225 26.268 28.155 28.192 30.013 30.045 31.802 31.831 33.528 33.554 35.194 35.218 36.805 36.827 38.364 38.384 39.875 39.893 41.342 41.359 42.767 42.783 44.155 44.17

4.0

4.5

5.0

5.5

6.0

6.5

248.74 151.2 15.997 18.019 20.157 22.271 24.327 26.314 28.231 30.08 31.862 33.582 35.243 36.849 38.405 39.913 41.377 42.8 44.186

249.72 152.18 16.31 18.168 20.257 22.346 24.387 26.364 28.274 30.117 31.895 33.611 35.27 36.874 38.427 39.933 41.396 42.818 44.202

250.69 153.15 16.723 18.338 20.368 22.429 24.452 26.418 28.32 30.157 31.93 33.642 35.298 36.899 38.45 39.955 41.416 42.836 44.219

251.66 154.12 17.293 18.534 20.491 22.518 24.523 26.476 28.369 30.199 31.967 33.675 35.327 36.926 38.475 39.977 41.436 42.855 44.237

252.62 155.07 18.149 18.757 20.625 22.615 24.598 26.537 28.421 30.243 32.006 33.71 35.358 36.954 38.5 40.001 41.458 42.876 44.256

253.58 156.02 19.745 19.011 20.773 22.719 24.679 26.603 28.475 30.29 32.047 33.746 35.391 36.983 38.527 40.025 41.481 42.897 44.276

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P (Mpa) T (K) 220 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1050 1100

7.0

7.5

10.0

12.5

15.0

17.5

254.54 156.96 56.519 19.3 20.934 22.832 24.764 26.672 28.533 30.34 32.09 33.784 35.425 37.014 38.555 40.051 41.504 42.918 44.296

255.49 157.9 60.336 19.629 21.11 22.952 24.856 26.745 28.594 30.392 32.135 33.824 35.46 37.046 38.584 40.077 41.529 42.941 44.317

260.22 162.47 71.03 22.097 22.226 23.679 25.393 27.17 28.943 30.687 32.39 34.048 35.659 37.224 38.746 40.225 41.664 43.066 44.432

264.88 166.89 77.952 26.715 23.809 24.632 26.072 27.694 29.368 31.043 32.695 34.313 35.893 37.434 38.934 40.396 41.82 43.21 44.566

269.47 171.18 83.54 33.852 25.929 25.824 26.893 28.317 29.867 31.456 33.046 34.618 36.161 37.672 39.149 40.59 41.998 43.372 44.716

175.37 88.406 41.429 28.589 27.256 27.857 29.036 30.437 31.926 33.444 34.961 36.462 37.939 39.388 40.806 42.194 43.553 44.882

20.0 22.5 Viscosity (μPa s) 179.46 92.808 47.891 31.687 28.912 28.954 29.847 31.075 32.449 33.885 35.34 36.794 38.233 39.65 41.043 42.41 43.75 45.063

183.46 96.885 53.271 35.045 30.762 30.175 30.744 31.777 33.022 34.366 35.754 37.155 38.552 39.936 41.3 42.643 43.963 45.259

25.0

27.5

30.0

35.0

40.0

45.0

187.38 100.72 57.883 38.473 32.762 31.502 31.718 32.539 33.642 34.887 36.2 37.543 38.895 40.242 41.576 42.894 44.192 45.469

191.23 104.36 61.957 41.833 34.864 32.918 32.76 33.353 34.305 35.442 36.676 37.958 39.261 40.569 41.87 43.161 44.435 45.693

195.02 107.85 65.64 45.05 37.023 34.403 33.86 34.214 35.007 36.031 37.18 38.397 39.648 40.914 42.181 43.443 44.693 45.929

202.42 114.46 72.195 50.98 41.367 37.507 36.192 36.053 36.51 37.293 38.262 39.34 40.481 41.657 42.85 44.049 45.246 46.437

209.63 120.71 78.025 56.302 45.591 40.688 38.641 38.007 38.118 38.65 39.429 40.358 41.381 42.461 43.575 44.707 45.847 46.989

216.66 126.67 83.376 61.141 49.61 43.857 41.144 40.034 39.802 40.078 40.662 41.439 42.339 43.318 44.347 45.409 46.489 47.579

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P (Mpa) T (K) 220 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1050 1100

50.0

55.0

60.0

65.0

70.0

75.0

223.55 132.41 88.389 65.607 53.411 46.961 43.658 42.101 41.535 41.559 41.947 42.568 43.343 44.218 45.161 46.15 47.167 48.203

230.3 137.98 93.151 69.786 57.012 49.973 46.153 44.181 43.297 43.075 43.27 43.736 44.384 45.155 46.01 46.923 47.877 48.857

236.94 143.39 97.719 73.74 60.438 52.885 48.608 46.258 45.073 44.614 44.619 44.933 45.455 46.121 46.887 47.725 48.613 49.536

243.47 148.68 102.14 77.518 63.714 55.699 51.013 48.317 46.85 46.164 45.987 46.15 46.548 47.11 47.788 48.55 49.372 50.238

249.92 153.86 106.43 81.156 66.863 58.423 53.364 50.351 48.621 47.718 47.364 47.382 47.658 48.117 48.707 49.393 50.15 50.958

256.27 158.96 110.62 84.683 69.903 61.064 55.662 52.355 50.377 49.27 48.746 48.622 48.779 49.137 49.641 50.252 50.943 51.694

80.0 85.0 Viscosity (μPa s) 262.56 163.97 114.73 88.118 72.852 63.632 57.907 54.326 52.117 50.814 50.127 49.867 49.908 50.167 50.586 51.123 51.75 52.443

268.77 168.91 118.77 91.478 75.724 66.135 60.103 56.264 53.837 52.349 51.505 51.113 51.041 51.204 51.539 52.004 52.566 53.203

90.0

95.0

100.0

150.0

200.0

300.0

274.92 173.8 122.74 94.777 78.532 68.58 62.255 58.17 55.535 53.871 52.878 52.357 52.177 52.245 52.499 52.892 53.391 53.972

281.01 178.62 126.67 98.024 81.287 70.974 64.365 60.046 57.212 55.38 54.242 53.598 53.312 53.288 53.462 53.785 54.222 54.749

287.05 183.4 130.55 101.23 83.996 73.325 66.439 61.892 58.868 56.874 55.597 54.834 54.445 54.332 54.428 54.682 55.058 55.53

345.26 229.42 167.91 131.99 109.79 95.472 85.87 79.215 74.488 71.078 68.6 66.803 65.514 64.613 64.013 63.648 63.473 63.449

273.72 204.19 162.08 135.04 116.98 104.52 95.68 89.257 84.493 80.907 78.179 76.095 74.504 73.297 72.395 71.737 71.278

360.97 276.94 223.51 187.33 161.85 143.42 129.8 119.54 111.7 105.61 100.83 97.029 93.988 91.538 89.557 87.951 86.651

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APPENDIX B: Predicted database Table B.1: Bahador's Density Correlation Predicted Results P (Mpa) T (K) 293 298 300 310 320 330 340 350 360 370 380 390 400 410 420 430 431 432 433

2.5

5.0

10

20

25

52.591 51.292 50.789 48.417 46.265 44.316 42.550 40.949 39.494 38.167 36.949 35.822 34.766 33.763 32.795 31.843 31.747 31.652 31.557

140.393 133.307 130.638 118.636 108.689 100.553 93.985 88.743 84.583 81.262 78.537 76.164 73.901 71.504 68.730 65.336 64.953 64.562 64.161

916.559 813.600 775.191 605.605 470.565 366.145 288.417 233.455 197.333 176.123 165.898 162.733 162.700 161.872 156.322 142.125 140.078 137.902 135.592

947.707 916.980 904.611 842.371 780.060 718.484 658.449 600.759 546.221 495.638 449.818 409.565 375.684 348.981 330.261 320.329 319.851 319.469 319.184

962.468 948.188 941.907 906.117 864.103 817.331 767.272 715.395 663.169 612.063 563.546 519.087 480.155 448.219 424.749 411.214 410.463 409.828 409.310

30 35 40 45 50 60 70 3 Density (kg/m ) 975.653 987.204 997.065 1005.182 1011.499 1018.506 1017.642 970.035 984.246 992.548 996.666 998.326 1001.176 1014.905 967.070 982.452 990.400 993.265 993.397 994.856 1013.576 946.620 968.639 976.933 976.262 971.386 968.056 1005.020 917.925 947.634 959.339 959.147 953.166 948.269 993.509 882.528 920.629 938.191 941.769 937.919 934.157 979.347 841.973 888.814 914.061 923.976 924.826 924.381 962.841 797.804 853.380 887.519 905.616 913.066 917.605 944.297 751.565 815.519 859.139 886.537 901.821 912.489 924.021 704.801 776.421 829.493 866.587 890.272 907.695 902.317 659.053 737.277 799.151 845.614 877.599 901.884 879.494 615.868 699.278 768.687 823.466 862.984 893.720 855.855 576.787 663.615 738.672 799.992 845.607 881.863 831.708 543.356 631.480 709.678 775.039 824.650 864.975 807.358 517.119 604.063 682.277 748.455 799.292 841.718 783.112 499.618 582.555 657.041 720.089 768.715 810.754 759.274 498.407 580.775 654.658 717.149 765.339 807.182 756.925 497.302 579.067 652.304 714.189 761.901 803.519 754.583 496.302 577.433 649.978 711.209 758.402 799.762 752.248

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Table B.2: Modified Redlich-Kwong's equation of state predicted results P (Mpa) T (K) 268 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1050 1100

0.1

3.0

25.0

50.0

1.987 102.366 1031.271 1050.350 1.772 67.465 882.231 940.024 1.516 50.991 613.069 772.916 1.325 42.375 419.778 637.264 1.177 36.673 325.828 539.594 1.059 32.499 273.186 470.191 0.963 29.263 238.613 419.086 0.882 26.658 213.569 379.824 0.814 24.505 194.264 348.533 0.756 22.690 178.744 322.852 0.706 21.135 165.888 301.279 0.662 19.786 154.997 282.819 0.623 18.604 145.612 266.784 0.588 17.558 137.413 252.684 0.557 16.626 130.169 240.159 0.529 15.789 123.711 228.937 0.504 15.034 117.907 218.809 0.481 14.348 112.657 209.610

75.0 100.0 150.0 200.0 250.0 300.0 3 Density (kg/m ) 1078.415 1106.335 1154.811 1193.396 1224.283 1249.422 990.931 1033.463 1099.595 1148.595 1186.433 1216.577 863.100 927.841 1019.347 1083.007 1130.631 1167.858 755.185 836.452 947.701 1023.280 1079.104 1122.406 669.087 759.926 884.676 969.333 1031.769 1080.149 601.278 696.421 829.486 920.734 988.357 1040.905 547.278 643.469 781.083 876.922 948.523 1004.450 503.423 598.836 738.431 837.320 911.914 970.548 467.073 560.737 700.618 801.396 878.189 938.967 436.378 527.808 666.875 768.677 847.038 909.494 410.037 499.023 636.568 738.756 818.182 881.931 387.122 473.603 609.183 711.283 791.379 856.101 366.956 450.953 584.298 685.962 766.413 831.845 349.033 430.613 561.567 662.540 743.097 809.022 332.969 412.222 540.708 640.802 721.267 787.505 318.467 395.491 521.483 620.563 700.780 767.182 305.291 380.191 503.697 601.666 681.511 747.953 293.255 366.131 487.183 583.974 663.349 729.728

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Table B.3: Liang-Biao Ouyang's density correlation predicted results P (MPa) T (K) 313 320 325 330 335 340 350 355 360 365 370 375 380

7.0

10.0

15.0

20.0

25.0

50.166 119.884 148.172 162.862 167.361 164.700 148.155 138.462 129.993 123.907 120.989 121.649 125.924

609.233 449.684 372.900 319.108 282.289 257.367 227.610 217.325 208.035 199.367 191.888 187.105 187.465

778.972 739.063 693.376 640.691 586.188 533.889 446.175 412.995 386.488 364.869 345.193 323.352 294.079

843.778 799.244 769.096 739.307 709.259 678.602 615.384 583.446 552.147 522.461 495.627 473.152 456.804

879.942 848.445 825.177 801.291 776.846 751.936 701.260 675.848 650.678 626.012 602.145 579.405 558.155

30.0 35.0 Density (kg/m3) 910.486 935.474 883.015 910.710 863.021 892.818 842.721 874.767 822.135 856.569 801.298 838.243 759.081 801.311 737.844 782.771 716.640 764.234 695.577 745.746 674.776 727.360 654.375 709.133 634.524 691.128

40.0

45.0

956.578 933.710 917.238 900.668 884.018 867.308 833.797 817.045 800.333 783.690 767.148 750.742 734.507

975.038 953.620 938.221 922.760 907.255 891.727 860.693 845.232 829.842 814.549 799.380 784.362 769.526

50.0

55.0

60.0

991.662 1006.825 1020.473 971.463 987.684 1002.150 956.983 974.011 989.066 942.475 960.344 975.994 927.953 946.691 962.945 913.434 933.057 949.926 884.463 905.878 924.018 870.046 892.347 911.145 855.697 878.864 898.339 841.437 865.437 885.607 827.282 852.073 872.957 813.255 838.780 860.397 799.374 825.565 847.937

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Table B.4: Bahadori's viscosity correlatin predicted results P (MPa) T (K) 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450

10

15

20

25

30

142.251 135.826 112.914 86.809 64.395 47.470 35.485 27.262 21.713 21.839 21.645 21.639 21.761 21.966 22.223 22.503 22.787 23.059 23.305 23.516

153.692 133.555 116.291 100.629 86.104 72.649 60.365 49.381 39.786 34.103 30.617 28.441 27.097 26.300 25.863 25.662 25.606 25.629 25.680 25.720

159.435 139.822 122.847 107.825 94.388 82.329 71.526 61.890 53.346 46.483 41.557 37.937 35.246 33.232 31.724 30.604 29.786 29.207 28.820 28.591

167.354 148.315 130.945 115.514 102.030 90.363 80.323 71.704 64.307 57.338 51.689 47.142 43.488 40.561 38.230 36.392 34.969 33.896 33.125 32.618

175.868 156.807 138.991 123.156 109.494 97.917 88.209 80.121 73.408 66.556 60.555 55.454 51.172 47.618 44.697 42.327 40.435 38.958 37.847 37.059

35

40 45 Viscosity (μPa s) 184.126 191.811 198.836 164.651 171.704 177.991 146.450 153.193 159.232 130.347 136.947 142.937 116.555 123.093 129.082 104.977 111.492 117.460 95.381 101.893 107.795 87.500 94.020 99.808 81.076 87.614 93.244 74.348 80.962 86.619 68.194 74.770 80.452 62.792 69.231 74.885 58.133 64.372 69.946 54.174 60.171 65.621 50.849 56.582 61.871 48.092 53.549 58.647 45.838 51.016 55.899 44.027 48.926 53.576 42.609 47.233 51.634 41.542 45.893 50.031

50

55

60

65

70

205.211 183.587 164.631 148.353 134.542 122.907 113.144 104.970 98.137 91.499 85.392 79.866 74.931 70.564 66.728 63.377 60.465 57.946 55.779 53.925

210.985 188.576 169.465 153.248 139.513 127.875 117.997 109.594 102.425 95.743 89.715 84.275 79.398 75.048 71.181 67.751 64.715 62.030 59.659 57.568

216.218 193.040 173.803 157.679 144.041 132.409 122.410 113.754 106.210 99.463 93.525 88.196 83.415 79.123 75.265 71.792 68.662 65.836 63.280 60.964

220.968 197.049 177.712 161.700 148.174 136.554 126.432 117.512 109.575 102.748 96.902 91.703 87.041 82.834 79.013 75.526 72.327 69.380 66.656 64.128

225.291 200.664 181.246 165.359 151.954 140.352 130.108 120.921 112.584 105.667 99.916 94.853 90.326 86.223 82.461 78.978 75.729 72.679 69.801 67.074

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Table B.5: Jossi's viscosity model predicted results P (MPa) T (K) 268 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1050 1100

0.1 13.922 15.427 17.761 20.076 22.373 24.402 26.282 28.083 29.817 31.491 33.113 34.688 36.221 37.715 39.174 40.601 41.998 43.368

3.0

25.0

50.0

75.0

100.0

15.077 128.237 160.029 191.208 223.256 16.253 91.884 118.769 143.599 168.446 18.370 58.602 83.900 102.729 120.671 20.568 40.606 65.850 81.894 95.948 22.790 34.547 55.445 70.117 82.069 24.764 32.927 49.211 62.612 73.371 26.603 32.839 45.639 57.631 67.527 28.371 33.415 43.740 54.326 63.465 30.079 34.311 42.861 52.178 60.597 31.731 35.375 42.616 50.841 58.579 33.335 36.533 42.778 50.084 57.190 34.894 37.742 43.211 49.743 56.276 36.413 38.979 43.828 49.708 55.725 37.895 40.229 44.574 49.902 55.456 39.344 41.483 45.410 50.266 55.408 40.761 42.736 46.312 50.761 55.536 42.150 43.982 47.261 51.358 55.804 43.512 45.220 48.244 52.033 56.184

125.0 150.0 Viscosity (μPa s) 256.444 291.169 194.200 221.076 138.889 157.708 109.703 123.750 93.211 104.291 82.977 92.222 76.184 84.272 71.452 78.777 68.044 74.823 65.550 71.900 63.722 69.708 62.399 68.061 61.467 66.833 60.844 65.934 60.470 65.302 60.297 64.887 60.287 64.651 60.412 64.565

175.0 327.740 249.286 177.477 138.373 115.656 101.503 92.200 85.803 81.225 77.842 75.292 73.346 71.856 70.725 69.876 69.261 68.836 68.573

200.0 366.232 279.026 198.376 153.699 127.489 111.050 100.212 92.773 87.467 83.563 80.618 78.362 76.616 75.263 74.220 73.426 72.837 72.420

225.0 406.889 310.384 220.409 169.905 139.958 120.995 108.465 99.849 93.712 89.206 85.816 83.217 81.197 79.617 78.379 77.415 76.674 76.116

250.0 449.543 343.582 243.719 187.077 153.127 131.449 117.066 107.144 100.071 94.883 90.986 87.998 85.676 83.849 82.409 81.271 80.377 79.683

275.0

300.0

494.719 378.727 268.330 205.236 167.016 142.478 126.095 114.739 106.632 100.679 96.208 92.784 90.119 88.022 86.360 85.036 83.984 83.150

541.990 415.617 294.407 224.521 181.741 154.169 135.594 122.717 113.456 106.659 101.549 97.634 94.587 92.187 90.278 88.751 87.527 86.550

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Table B.6: Liang-Biao Ouyang's viscosity correlation predicted results P (MPa) T (K) 313 315 320 325 330 335 340 345 350 355 360 365 370 375 380

7.0

10.0

15.0

20.0

22.5

25.0

7.819 10.964 16.538 19.565 20.835 21.010 20.621 20.073 19.640 19.467 19.571 19.840 20.031 19.776 18.574

46.338 41.889 33.447 28.068 24.907 23.250 22.516 22.255 22.151 22.018 21.804 21.586 21.576 22.116 23.683

68.242 66.239 60.772 55.041 49.457 44.340 39.912 36.303 33.547 31.586 30.266 29.340 28.465 27.206 25.032

78.514 76.359 71.311 66.682 62.406 58.430 54.721 51.260 48.044 45.090 42.427 40.103 38.181 36.742 35.883

82.809 80.849 76.142 71.700 67.517 63.587 59.907 56.477 53.297 50.370 47.702 45.300 43.173 41.333 39.794

86.750 84.812 80.175 75.819 71.730 67.895 64.305 60.952 57.830 54.935 52.269 49.831 47.625 45.659 43.941

27.5 30.0 Viscosity (μPa s) 90.463 93.979 88.538 92.059 83.942 87.483 79.639 83.208 75.610 79.211 71.837 75.473 68.306 71.975 65.004 68.704 61.921 65.644 59.050 62.786 56.385 60.122 53.923 57.645 51.665 55.351 49.611 53.238 47.766 51.308

35.0

40.0

45.0

50.0

55.0

60.0

100.536 106.619 112.376 117.902 123.247 128.407 98.610 104.673 110.402 115.897 121.208 126.331 94.026 100.039 105.699 111.119 116.350 121.383 89.749 95.717 101.310 106.658 111.813 116.758 85.758 91.683 97.213 102.491 107.574 112.434 82.029 87.916 93.387 98.599 103.611 108.388 78.544 84.397 89.812 94.959 99.904 104.601 75.285 81.108 86.471 91.556 96.432 101.053 72.235 78.032 83.346 88.370 93.179 97.727 69.380 75.153 80.422 85.387 90.128 94.607 66.709 72.459 77.686 82.592 87.265 91.678 64.211 69.936 75.123 79.972 84.576 88.927 61.877 67.574 72.723 77.515 82.051 86.343 59.701 65.364 70.476 75.212 79.679 83.915 57.678 63.299 68.371 73.052 77.452 81.635

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