Coefficient of Isothermal Oil Compressibility for Oil Well No

Total Page:16

File Type:pdf, Size:1020Kb

Coefficient of Isothermal Oil Compressibility for Oil Well No COEFFICIENT OF ISOTHERMAL OIL COMPRESSIBILITY FOR RESERVOIR FLUIDS BY CUBIC EQUATION-OF-STATE by OLAOLUWA OPEOLUWA ADEPOJU, B.Tech. Chem Engr. A THESIS IN PETROLEUM ENGINEERING Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE IN PETROLEUM ENGINEERING Approved Lloyd Heinze Chairperson of the Committee Shameem Siddiqui Accepted John Borrelli Dean of the Graduate School December, 2006 ACKNOWLEDGEMENTS I extend my profound gratitude to Dr. Akanni Lawal for inspiring me into phase behavior and into this research. Special thanks to Dr. Lloyd Heinze, the chair of my Masters committee and Dr. Shameem Siddiqui for their support. I would like to acknowledge the African Development Bank for awarding me the 2005- 2006 ADB/Japan Fellowship award. I extend my profound appreciation to my family for their support and encouragement. I also acknowledge my colleagues in Texas Tech University for their support. To God be the glory. ii TABLE OF CONTENTS ACKNOWLEDGEMENTS ii ABSTRACT v LIST OF TABLES vi LIST OF FIGURES viii LIST OF ABBREVIATIONS x CHAPTER 1 INTRODUCTION 1.1 Background Information 1 1.2 Importance of isothermal Oil Compressibility 4 1.3 Scope of the Project 9 1.4 Objectives of the Project 9 2 COEFFIECIENT OF ISOTHERMAL OIL COMPRESSIBILITY 2.1 Defining Equations for Isothermal Oil Compressibility 10 2.2 Isothermal Oil Compressibility Correlation and Computation Methods 24 3 DESIGN OF CUBIC EQUATION OF STATE 3.1 Generalized Cubic Equation of State 35 3.2 Characterization of Heavy Petroleum Fractions 43 3.3 Cubic EOS Based Isothermal Compressibility Equation 43 4 ANALYSIS OF PREDICTION RESULTS 4.1 Computing Isothermal Oil Compressibility from Reservoir Fluid Study Report 56 4.2 Predicted Molar Volume from Cubic Equation of State 56 4.3 Predicted Coefficient of Isothermal Oil Compressibility from Cubic iii Equation of State 66 4.4 Discussion of Results 74 5 CONCLUSIONS AND RECOMMENDATIONS 5.1 Conclusions 77 5.2 Recommendations 78 REFERENCES 79 APPENDIX A. DERIVATION OF PR EOS COMPRESSIBILITY FACTOR AND MOLAR VOLUME EQUATION. 83 B. DERIVATION OF PR EOS BASED ISOTHERMAL COMPRESSIBILITY EQUATION. 86 C. DERIVATION OF THE PR EOS ISOTHERMAL GAS COMPRESSIBILITY EQUATION. 88 D. DERIVATION OF THE FLASH CALCULATION ALGORITHM 90 E. MATHEMATICAL CONSISITENCY OF ISOTHERMAL COMPRESSIBILITY ADDITION BELOW BUBBLE POINT PRESSURE. 94 F. RESERVOIR FLUID STUDY REPORT AND ANALYSIS 97 G. RESERVOIR FLUIDS CRITICAL PROPERTIES TABLE 108 H. AVERAGE ABSOLUTE PERCENT DEVIATION (AAPD) 109 I. FORTRAN CODE DOCUMENTATION 116 J. DEVELOPED FORTRAN CODE 120 K. VITA 162 iv ABSTRACT Calculations of reservoir performance for petroleum reservoirs require accurate knowledge of the volumetric behavior of hydrocarbon mixtures, both liquid and gaseous. Coefficient of Isothermal oil compressibility is required in transient fluid flow problems, extension of fluid properties from values at the bubble point pressure to higher pressures of interest and in material balance calculations38, 35. Coefficient of Isothermal oil compressibility is a measure of the fractional change in volume as pressure is changed at constant temperature29. Coefficients of isothermal oil compressibility are usually obtained from reservoir fluid analysis. Reservoir fluid analysis is an expensive and time consuming operation that is not always available when the volumetric properties of reservoir fluids are needed. For this reason correlations have been developed and are being developed for predicting fluid properties including the coefficient of isothermal oil compressibility. This project developed a mathematical model for predicting the coefficient of isothermal oil compressibility based on Peng-Robinson Equation of State (PR EOS). A computer program was developed to predict the coefficient of isothermal compressibility using the developed model. The predicted coefficient of isothermal oil compressibility closely matches the experimentally derived coefficient of isothermal compressibility. v LIST OF TABLES 2.1 Coefficients for the co Correlations 34 4.1 Predicted Molar Volume for Oil Well No. 4, Good Oil Company, Samson, Texas, Bubble Point = 2619.7 psia 57 4.2 Predicted molar volume for Jehlicka 1A, Wilshire Oil Co. of Texas, Beaver, Oklahoma. Bubble Point = 2666.7 psia 60 4.3 Predicted molar volume for Jacques Unit #5603, Rough Ride Field, A.C.T. Operating Company, Fisher County, Texas. Bubble Point = 1689.7 psia 63 4.4 Predicted Coefficient of Isothermal Oil Compressibility for Oil Well No. 4, Good Oil Company. Samson, Texas. Bubble Point = 2619.7 psia 66 4.5 Predicted Coefficient of Isothermal Oil Compressibility for Jehlicka 1A, Wilshire Oil Co. of Texas, Beaver, Oklahoma. Bubble Point = 2666.7 psia. 69 4.6 Predicted Coefficient of Isothermal Oil Compressibility for Jacques Unit #5603, Rough Ride Field, A.C.T. Operating Company, Fisher County, Texas. Bubble Point = 1689.7 psia 72 F.1 Reservoir Fluid Composition for Oil Well No. 4 98 F.2 Molar Volume Determination from Pressure-Volume Relations for Oil Well No. 4 99 F.3 Differential Vaporization data for Oil Well No. 4 100 F.4 Reservoir Fluid Composition for Jehlicka 1A 101 F.5 Molar Volume Determination from Pressure-Volume Relations for Jehlicka 1A 102 F.6 Differential Vaporization data for Jehlicka 1A 103 F.7 Reservoir Fluid Composition for Jacques Unit #5603 104 F.8 Molar Volume Determination from Pressure-Volume Relations for Jacques Unit #5603 105 F.9 Differential Vaporization data for Jacques Unit #5603 106 G.1 Reservoir Fluids Critical Properties 107 vi H.1 The AAPD for the Predicted Molar Volume for Oil Well No. 4, Good Oil Company. Samson, Texas 108 H.2 The AAPD for the Predicted Coefficient of Isothermal Oil Compressibility For Oil Well No. 4, Good Oil Company. Samson, Texas. 109 H.3 The AAPD for the Predicted Molar Volume for Jacques Unit #5603, Rough Ride Field, A.C.T. Operating Company, Fisher County, Texas. 110 H.4 The AAPD for the Predicted Coefficient of Isothermal Oil Compressibility for Jacques Unit #5603, Rough Ride Field, A.C.T. Operating Company, Fisher County, Texas (Using 11 Components). 111 H.5 The AAPD for the Predicted Molar Volume for Jehlicka 1A, Wilshire Oil Co. of Texas, Beaver, Oklahoma. 112 H.6 The AAPD for the Predicted Coefficient of Isothermal Oil Compressibility for Jehlicka 1A, Wilshire Oil Co. of Texas, Beaver, Oklahoma. 113 H.7 The AAPD for the Predicted Coefficient of Isothermal Oil Compressibility for Jacques Unit #5603, Rough Ride Field, A.C.T. Operating Company, Fisher County, Texas (Using 11 Components). 114 vii LIST OF FIGURES 1.1 Pressure-Volume/Compressibility Relationship 7 2.1 Typical Shape of the Isothermal Gas Compressibility of Gas as a Function of Pressure at Constant Reservoir Temperature 11 2.2 Standing and Katz compressibility factors chart 15 2.3 Variation of Reduced Compressibility with Reduced Pressures for Various Fixed values of Reduced Temperature for Natural Gases (0.1< cr <1.0) 18 2.4 Variation of Reduced Compressibility with Reduced Pressures for Various Fixed values of Reduced Temperature for Natural Gases (0.01< cr < 0.1) 18 2.5 Typical Shape of the Isothermal Oil Compressibility as a Function of Reservoir Pressure at Constant Temperatures at Pressures above the Bubble Point Pressure 20 2.6 Typical Shape of the Coefficient of Isothermal Oil Compressibility as a Function of Pressure at Constant Reservoir Temperature 24 2.7 Trube’s Pseudo Reduced Compressibility of Undersaturated Crude Oils 25 2.8 Coefficient of Isothermal Compressibility of Undersaturated Black Oils 27 2.9 Coefficient of Isothermal Compressibility of Saturated Black Oils 29 3.1 Volumetric Behavior of Pure Compounds by Van der Waal Cubic EOS 37 3.2 Reservoir Oil at Pressure above the Bubble Point Pressure 44 3.3 Reservoir Oil at Pressure below the Bubble Point Pressure 47 3.4 Schematic diagram of Flash Liberation Experiment 49 4.1 Predicted Molar Volume for Oil Well No- 4, Good Oil Company. Samson, Texas (Model Based on PR EOS). 58 4.2 Predicted Molar Volume for Oil Well No- 4, Good Oil Company. Samson, Texas (Model Based on MPR EOS). 59 4.3 Predicted Molar Volume for Jehlicka 1A, Wilshire Oil Co. of Texas, Beaver, Oklahoma. (Model Based on PR EOS). 61 viii 4.4 Predicted Molar Volume for Jehlicka 1A, Wilshire Oil Co. of Texas, Beaver, Oklahoma. (Model Based on MPR EOS). 62 4.5 Predicted Molar Volume for Jacques Unit #5603, Rough Ride Field, Fisher County, Texas. (Model Based on PR EOS). 64 4.6 Predicted Molar Volume for Jacques Unit #5603, Rough Ride Field, Fisher County, Texas. (Model Based on MPR EOS). 65 4.7 Predicted Coefficient of Isothermal Oil Compressibility for Oil Well No. 4, Good Oil Company. Samson, Texas. (Model Based on PR EOS) 67 4.8 Predicted Coefficient of Isothermal Oil Compressibility for Oil Well No. 4, Oil Company. Samson, Texas. (Model Based on MPR EOS). 68 4.9 Predicted Coefficient of Isothermal Oil Compressibility for Jehlicka 1A, Wilshire Oil Co. of Texas, Beaver, Oklahoma. (Model Based on PR EOS). 70 4.10 Predicted Coefficient of Isothermal Oil Compressibility for Jehlicka 1A, Wilshire Oil Co. of Texas, Beaver, Oklahoma. (Model Based on MPR EOS) 71 4.11 Predicted Coefficient of Isothermal Oil Compressibility for Jacques Unit #5603, Rough Ride Field, A.C.T. Operating Company, Fisher County, Texas. (Model Based on PR EOS). 73 4.12 Predicted Coefficient of Isothermal Oil Compressibility for Jacques Unit #5603, Rough Ride Field, A.C.T. Operating Company, Fisher County, Texas. (Model Based on MPR EOS). 74
Recommended publications
  • How Compressible Are Innovation Processes?
    1 How Compressible are Innovation Processes? Hamid Ghourchian, Arash Amini, Senior Member, IEEE, and Amin Gohari, Senior Member, IEEE Abstract The sparsity and compressibility of finite-dimensional signals are of great interest in fields such as compressed sensing. The notion of compressibility is also extended to infinite sequences of i.i.d. or ergodic random variables based on the observed error in their nonlinear k-term approximation. In this work, we use the entropy measure to study the compressibility of continuous-domain innovation processes (alternatively known as white noise). Specifically, we define such a measure as the entropy limit of the doubly quantized (time and amplitude) process. This provides a tool to compare the compressibility of various innovation processes. It also allows us to identify an analogue of the concept of “entropy dimension” which was originally defined by R´enyi for random variables. Particular attention is given to stable and impulsive Poisson innovation processes. Here, our results recognize Poisson innovations as the more compressible ones with an entropy measure far below that of stable innovations. While this result departs from the previous knowledge regarding the compressibility of fat-tailed distributions, our entropy measure ranks stable innovations according to their tail decay. Index Terms Compressibility, entropy, impulsive Poisson process, stable innovation, white L´evy noise. I. INTRODUCTION The compressible signal models have been extensively used to represent or approximate various types of data such as audio, image and video signals. The concept of compressibility has been separately studied in information theory and signal processing societies. In information theory, this concept is usually studied via the well-known entropy measure and its variants.
    [Show full text]
  • Chapter 3 3.4-2 the Compressibility Factor Equation of State
    Chapter 3 3.4-2 The Compressibility Factor Equation of State The dimensionless compressibility factor, Z, for a gaseous species is defined as the ratio pv Z = (3.4-1) RT If the gas behaves ideally Z = 1. The extent to which Z differs from 1 is a measure of the extent to which the gas is behaving nonideally. The compressibility can be determined from experimental data where Z is plotted versus a dimensionless reduced pressure pR and reduced temperature TR, defined as pR = p/pc and TR = T/Tc In these expressions, pc and Tc denote the critical pressure and temperature, respectively. A generalized compressibility chart of the form Z = f(pR, TR) is shown in Figure 3.4-1 for 10 different gases. The solid lines represent the best curves fitted to the data. Figure 3.4-1 Generalized compressibility chart for various gases10. It can be seen from Figure 3.4-1 that the value of Z tends to unity for all temperatures as pressure approach zero and Z also approaches unity for all pressure at very high temperature. If the p, v, and T data are available in table format or computer software then you should not use the generalized compressibility chart to evaluate p, v, and T since using Z is just another approximation to the real data. 10 Moran, M. J. and Shapiro H. N., Fundamentals of Engineering Thermodynamics, Wiley, 2008, pg. 112 3-19 Example 3.4-2 ---------------------------------------------------------------------------------- A closed, rigid tank filled with water vapor, initially at 20 MPa, 520oC, is cooled until its temperature reaches 400oC.
    [Show full text]
  • A Thermodynamic Theory of Economics
    ________________________________________________________________________________________________ A Thermodynamic Theory of Economics Final Post Review Version ________________________________________________________________________________________________ John Bryant VOCAT International Ltd, 10 Falconers Field, Harpenden, AL5 3ES, United Kingdom. E-mail: [email protected] Abstract: An analogy between thermodynamic and economic theories and processes is developed further, following a previous paper published by the author in 1982. Economic equivalents are set out concerning the ideal gas equation, the gas constant, pressure, temperature, entropy, work done, specific heat and the 1st and 2nd Laws of Thermodynamics. The law of diminishing marginal utility was derived from thermodynamic first principles. Conditions are set out concerning the relationship of economic processes to entropic gain. A link between the Le Chatelier principle and economic processes is developed, culminating in a derivation of an equation similar in format to that of Cobb Douglas production function, but with an equilibrium constant and a disequilibrium function added to it. A trade cycle is constructed, utilising thermodynamic processes, and equations are derived for cycle efficiency, growth and entropy gain. A thermodynamic model of a money system is set out, and an attempt is made to relate interest rates, the rate of return, money demand and the velocity of circulation to entropy gain. Aspects concerning the measurement of economic value in thermodynamic terms are discussed. Keywords: Thermodynamics, economics, Le Chatelier, entropy, utility, money, equilibrium, value, energy. Publisher/Journal: This paper is the final post review version of a paper submitted to the International Journal of Exergy, published by Inderscience Publishers. Reference: Bryant, J. (2007) ‘A thermodynamic theory of economics’, Int. J. Exergy, Vol 4, No.
    [Show full text]
  • Real Gases – As Opposed to a Perfect Or Ideal Gas – Exhibit Properties That Cannot Be Explained Entirely Using the Ideal Gas Law
    Basic principle II Second class Dr. Arkan Jasim Hadi 1. Real gas Real gases – as opposed to a perfect or ideal gas – exhibit properties that cannot be explained entirely using the ideal gas law. To understand the behavior of real gases, the following must be taken into account: compressibility effects; variable specific heat capacity; van der Waals forces; non-equilibrium thermodynamic effects; Issues with molecular dissociation and elementary reactions with variable composition. Critical state and Reduced conditions Critical point: The point at highest temp. (Tc) and Pressure (Pc) at which a pure chemical species can exist in vapour/liquid equilibrium. The point critical is the point at which the liquid and vapour phases are not distinguishable; because of the liquid and vapour having same properties. Reduced properties of a fluid are a set of state variables normalized by the fluid's state properties at its critical point. These dimensionless thermodynamic coordinates, taken together with a substance's compressibility factor, provide the basis for the simplest form of the theorem of corresponding states The reduced pressure is defined as its actual pressure divided by its critical pressure : The reduced temperature of a fluid is its actual temperature, divided by its critical temperature: The reduced specific volume ") of a fluid is computed from the ideal gas law at the substance's critical pressure and temperature: This property is useful when the specific volume and either temperature or pressure are known, in which case the missing third property can be computed directly. 1 Basic principle II Second class Dr. Arkan Jasim Hadi In Kay's method, pseudocritical values for mixtures of gases are calculated on the assumption that each component in the mixture contributes to the pseudocritical value in the same proportion as the mol fraction of that component in the gas.
    [Show full text]
  • Thermodynamics As a Substantive and Formal Theory for the Analysis of Economic and Biological Systems
    Thermodynamics as a Substantive and Formal Theory for the Analysis of Economic and Biological Systems The research presented in this thesis was carried out at the Department of Theoretical Life Sciences, Vrije Universiteit Amsterdam, The Netherlands and at the Environment and Energy Section, Instituto Superior T´ecnico, Lisbon, Portugal. VRIJE UNIVERSITEIT THERMODYNAMICS AS A SUBSTANTIVE AND FORMAL THEORY FOR THE ANALYSIS OF ECONOMIC AND BIOLOGICAL SYSTEMS ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Vrije Universiteit Amsterdam, op gezag van de rector magnificus prof.dr. L.M. Bouter, in het openbaar te verdedigen ten overstaan van de promotiecommissie van de faculteit der Aard- en Levenswetenschappen op dinsdag 6 februari 2007 om 10.00 uur in de aula van de Instituto Superior T´ecnico Av. Rovisco Pais, n◦1 1049-001 Lisboa door Taniaˆ Alexandra dos Santos Costa e Sousa geboren te Beira, Mozambique promotor: prof.dr. S.A.L.M. Kooijman Contents 1 Introduction 1 2 IsNeoclassicalEconomicsFormallyValid? 9 Published in Ecological Economics 58 (2006): 160-169 2.1 Introduction................................. 10 2.2 IstheFormalismofNeoclassicalEconomicswrong? . ...... 12 2.3 A Unified Formalism for Neoclassical Economics and Equilibrium Ther- modynamics................................. 14 2.4 DiscussingtheFormalism. 19 2.5 Conclusions................................. 21 3 Equilibrium Econophysics 29 Published in Physica A 371 (2006): 492-512 3.1 Introduction................................. 30 3.2 FundamentalEquationandtheEquilibriumState . ...... 31 3.3 DualityFormulation............................. 33 3.4 Reversible, Irreversible and Impossible Processes . .......... 35 3.5 Many-stepProcesses ............................ 36 3.6 LegendreTransforms ............................ 38 3.7 Elasticities.................................. 40 3.8 MaxwellRelations ............................. 43 3.9 StabilityConditions and the Le Chatelier Principle . ......... 45 3.10 EquationsofStateandIntegrability . ..... 47 3.11 FirstOrderPhaseTransitions .
    [Show full text]
  • Compressibility of the Fluid
    EPJ Web of Conferences 67, 02048 (2014) DOI: 10.1051/epjconf/20146702048 C Owned by the authors, published by EDP Sciences, 2014 Compressibility of the fluid Jana Jablonská1,a 1 Department of Hydromechanics and Hydraulic Equipment, Faculty of Mechanical Engineering, VŠB-Technical University of Ostrava, 17. listopadu 15, 708 33 Ostrava, Czech republic Abstract. The presence of air in the liquid causes the dynamic system behaviour. When solve to issue of the dynamics we often meet problems of cavitation. Cavitation is an undesirable phenomenon, since it causes a disruption of the surrounding material and material destruction. Cavitation is accompanied by loud sound effects and reduces the efficiency of such pumps, etc. Therefore, it is desirable to model systems in which the cavitation might occur. A typical example is a solution of water hammer. 1 Introduction Fluid density after compression is given by m m In solving dynamics of hydraulic systems we often 0 4 4 (4) encounter the problem with cavitation. Cavitation is an V0 V 1 p 1 p undesirable phenomenon that we are trying to prevent. Cavitation depends on the liquid used in the hydraulic Thereof system. Liquids - hydraulic fluids are typical of its air content. The air in the fluid occurs in two forms - C p S D1 T (5) dissolved and undissolved. Dissolved air according to the 0 E K U literature [4] ia controled by Henry's law, its release is complex and difficult to describe action. If we consider the fluid flow with undissolved air it is a flow of mixture - water – air. 3 Determining the modulus of elasticity of liquid 2 Compressibility of the fluid The modulus of elasticity of water is generally in the literature determines the value of 2.1·109 Pa under normal Compressibility of the fluid is character to shrink in conditions, which corresponds to theory 1 (see Figure 3).
    [Show full text]
  • Thermodynamic Properties of Molecular Oxygen
    NATIONAL BUREAU OF STANDARDS REPORT 2611 THERMODYNAMIC PROPERTIES OF MOLECULAR OXYGEN tv Harold ¥, Woolley Themodynamics Section Division of Heat and Power U. S. DEPARTMENT OF COMMERCE NATIONAL BUREAU OF STANDARDS U. S. DEPARTMENT OF COMMERCE Sinclair Weeks, Secretary NATIONAL BUREAU OF STANDARDS A. V. Astin, Director THE NATIONAL BUREAU OF STANDARDS The scope of activities of the National Bureau of Standards is suggested in the following listing of the divisions and sections engaged in technical work. In general, each section is engaged in special- ized research, development, and engineering in the field indicated by its title. A brief description of the activities, and of the resultant reports and publications, appears on the inside of the back cover of this report. Electricity. Resistance Measurements. Inductance and Capacitance. Electrical Instruments. Magnetic Measurements. Applied Electricity. Electrochemistry. Optics and Metrology. Photometry and Colorimetry. Optical Instruments. Photographic Technology. Length. Gage. Heat and Power. Temperature Measurements. Thermodynamics. Cryogenics. Engines and Lubrication. Engine Fuels. Cryogenic Engineering. Atomic and Radiation Physics. Spectroscopy. Radiometry. Mass Spectrometry. Solid State Physics. Electron Physics. Atomic Physics. Neutron Measurements. Infrared Spectroscopy. Nuclear Physics. Radioactivity. X-Rays. Betatron. Nucleonic Instrumentation. Radio- logical Equipment. Atomic Energy Commission Instruments Branch. Chemistry. Organic Coatings. Surface Cheipistry. Organic Chemistry.
    [Show full text]
  • Entropy and Compressibility of Symbol Sequences
    PhysComp96 Entropy and Compressibility of Symbol Sequences Full paper Draft, February 23, 1997 Werner Ebeling [email protected] Thorsten PÄoschel [email protected] Alexander Neiman [email protected] Institute of Physics, Humboldt University Invalidenstr. 110, D-10115 Berlin, Germany The purpose of this paper is to investigate long-range correlations in symbol sequences using meth- ods of statistical physics and nonlinear dynamics. Beside the principal interest in the analysis of correlations and fluctuations comprising many letters, our main aim is related here to the problem of sequence compression. In spite of the great progress in this ¯eld achieved in the work of Shannon, Fano, Hu®man, Lempel, Ziv and others [1] many questions still remain open. In particular one must note that since the basic work by Lempel and Ziv the improvement of the standard compression algorithms is rather slow not exceeding a few percent per decade. One the other hand several experts expressed the idee that the long range correlations, which clearly exist in texts, computer programs etc. are not su±ciently taken into account by the standard algorithms [1]. Thus, our interest in compressibility is twofold: (i) We would like to explore how far compressibility is able to measure correlations. In particular we apply the standard algorithms to model sequences with known correlations such as, for instance, repeat sequences and symbolic sequences generated by maps. (ii) We aim to detect correlations which are not yet exploited in the standard compression algorithms and belong therefore to potential reservoirs for compression algorithms.
    [Show full text]
  • Ideal and Real Gases 1 the Ideal Gas Law 2 Virial Equations
    Ideal and Real Gases Ramu Ramachandran 1 The ideal gas law The ideal gas law can be derived from the two empirical gas laws, namely, Boyle’s law, pV = k1, at constant T, (1) and Charles’ law, V = k2T, at constant p, (2) where k1 and k2 are constants. From Eq. (1) and (2), we may assume that V is a function of p and T. Therefore, µ ¶ µ ¶ ∂V ∂V dV (p, T )= dp + dT. (3) ∂p T ∂T p 2 Substituting from above, we get (∂V/∂p)T = k1/p = V/p and (∂V/∂T )p = k2 = V/T.Thus, − − V V dV = dP + dT, or − p T dV dp dT + = . V p T Integrating this equation, we get ln V +lnp =lnT + c (4) where c is the constant of integration. Assigning (quite arbitrarily) ec = R,weget ln V +lnp =lnT +lnR,or pV = RT. (5) The constant R is, of course, the gas constant, whose value has been determined experimentally. 2 Virial equations The compressibility factor of a gas Z is defined as Z = pV/(nRT )=pVm/(RT), where the subscript on V indicates that this is a molar quantity. Obviously, for an ideal gas, Z =1always. For real gases, additional corrections have to be introduced. Virial equations are expressions in which such corrections can be systematically incorporated. For example, B(T ) C(T ) Z =1+ + 2 + ... (6) Vm Vm 1 We may substitute for Vm in terms of the ideal gas law and get an expression in terms of pressure, which is often more convenient to use: B C Z =1+ p + p2 + ...
    [Show full text]
  • Thermodynamics.Pdf
    1 Statistical Thermodynamics Professor Dmitry Garanin Thermodynamics February 24, 2021 I. PREFACE The course of Statistical Thermodynamics consist of two parts: Thermodynamics and Statistical Physics. These both branches of physics deal with systems of a large number of particles (atoms, molecules, etc.) at equilibrium. 3 19 One cm of an ideal gas under normal conditions contains NL = 2:69×10 atoms, the so-called Loschmidt number. Although one may describe the motion of the atoms with the help of Newton's equations, direct solution of such a large number of differential equations is impossible. On the other hand, one does not need the too detailed information about the motion of the individual particles, the microscopic behavior of the system. One is rather interested in the macroscopic quantities, such as the pressure P . Pressure in gases is due to the bombardment of the walls of the container by the flying atoms of the contained gas. It does not exist if there are only a few gas molecules. Macroscopic quantities such as pressure arise only in systems of a large number of particles. Both thermodynamics and statistical physics study macroscopic quantities and relations between them. Some macroscopics quantities, such as temperature and entropy, are non-mechanical. Equilibruim, or thermodynamic equilibrium, is the state of the system that is achieved after some time after time-dependent forces acting on the system have been switched off. One can say that the system approaches the equilibrium, if undisturbed. Again, thermodynamic equilibrium arises solely in macroscopic systems. There is no thermodynamic equilibrium in a system of a few particles that are moving according to the Newton's law.
    [Show full text]
  • This Working Paper Was Published in the Context of INSEAD's Centre For
    "WASTE POTENTIAL ENTROPY: THE ULTIMATE ECOTOXIC?" by R. U. AYRES and K. MARTINAS 94/36/EPS This working paper was published in the context of INSEAD's Centre for the Management of Environmental Resources, an R&D partnership sponsored by Ciba-Geigy, Danfoss, Otto Group, Royal Dutch/Shell and Sandoz AG. * Professor of Environmental Economics, Sandoz Chair in Management and the Environment, at INSEAD, Boulevard de Constance, Fontainebleau 77305 Cedex, France. ** Research Fellow, at INSEAD, Boulevard de Constance, Fontainebleau 77305 Cedex, France. A working paper in the INSEAD Working Paper Series is intended as a means whereby a faculty researcher's thoughts and findings may be communicated to interested readers. The paper should be considered preliminary in nature and may require revision. Printed at INSEAD, Fontainebleau, France WASTE POTENTIAL ENTROPY: THE ULTIMATE ECOTOXIC? Robert U. Ayres Katalin Martinas CMER, INSEAD Fontainebleau, France July 1994 Abstract We explore some economic implications of three simple ideas. These are as follows: (1) all resource flows and all economic goods and services can be characterized as stocks or flows of "useful" embodied information, (2) the economy is an information processor, in the sense that large quantities of low grade "physical" information are converted, by intention, into smaller quantities of higher grade "morphological" and "symbolic" information and (3) the most general pollutant is the physical information in the waste. Production, in the economic system, is the conversion of low economic value information-content raw materials into high economic value information-content goods and services. We point out in this paper that entropy generated in the creation of economic goods or services is not a priori damaging to the environment.
    [Show full text]
  • Effects of Compressibility on the Performance of a Wave-Energy Conversion Device with an Impulse Turbine Using a Numerical Simulation Technique
    International Journal of Rotating Machinery, 9: 443–450, 2003 Copyright c Taylor & Francis Inc. ISSN: 1023-621X print DOI: 10.1080/10236210390253418 Effects of Compressibility on the Performance of a Wave-Energy Conversion Device with an Impulse Turbine Using a Numerical Simulation Technique A. Thakker and T. S. Dhanasekaran Wave Energy Research Team, Department of Mechanical and Aeronautical Engineering, University of Limerick, Limerick, Ireland M. Takao Department of Control Engineering, Matsue National College of Technology, Matsue-shi, Shimane, Japan T. Setoguchi Department of Mechanical Engineering, Saga University, Saga-shi, Saga, Japan data used is one of the important factors in the simulation This article presents work carried out to predict the be- technique. havior of a 0.6 m impulse turbine with fixed guide vanes as compared with that of a 0.6 hub-to-tip ratio turbine un- Keywords Impulse turbine, Irregular waves, Numerical simulation, der real sea conditions. In order to predict the true perfor- Wave energy conversion mance of the actual oscillating water column (OWC), the numerical technique was fine-tuned by incorporating the compressibility effect. Water surface elevation versus time Over the past two decades, scientists have been investigating history was used as the input data for this purpose. The effect and defining various methods of extracting power from wave of compressibility inside the air chamber and the turbine’s motion. Some of these devices utilize the principle of an os- performance under unsteady and irregular flow conditions cillating water column (OWC). The OWC-based wave-energy were analyzed numerically. Considering the quasi-steady as- power plants convert wave energy into low-pressure pneumatic sumptions, the unidirectional steady-flow experimental data power in the form of bidirectional airflow.
    [Show full text]