Thermodynamically Consistent Modeling and Simulation of Multi-Component Two-Phase Flow Model with Partial Miscibility
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Thermodynamic Potentials and Natural Variables
Revista Brasileira de Ensino de Física, vol. 42, e20190127 (2020) Articles www.scielo.br/rbef cb DOI: http://dx.doi.org/10.1590/1806-9126-RBEF-2019-0127 Licença Creative Commons Thermodynamic Potentials and Natural Variables M. Amaku1,2, F. A. B. Coutinho*1, L. N. Oliveira3 1Universidade de São Paulo, Faculdade de Medicina, São Paulo, SP, Brasil 2Universidade de São Paulo, Faculdade de Medicina Veterinária e Zootecnia, São Paulo, SP, Brasil 3Universidade de São Paulo, Instituto de Física de São Carlos, São Carlos, SP, Brasil Received on May 30, 2019. Revised on September 13, 2018. Accepted on October 4, 2019. Most books on Thermodynamics explain what thermodynamic potentials are and how conveniently they describe the properties of physical systems. Certain books add that, to be useful, the thermodynamic potentials must be expressed in their “natural variables”. Here we show that, given a set of physical variables, an appropriate thermodynamic potential can always be defined, which contains all the thermodynamic information about the system. We adopt various perspectives to discuss this point, which to the best of our knowledge has not been clearly presented in the literature. Keywords: Thermodynamic Potentials, natural variables, Legendre transforms. 1. Introduction same statement cannot be applied to the temperature. In real fluids, even in simple ones, the proportionality Basic concepts are most easily understood when we dis- to T is washed out, and the Internal Energy is more cuss simple systems. Consider an ideal gas in a cylinder. conveniently expressed as a function of the entropy and The cylinder is closed, its walls are conducting, and a volume: U = U(S, V ). -
Thermodynamics Notes
Thermodynamics Notes Steven K. Krueger Department of Atmospheric Sciences, University of Utah August 2020 Contents 1 Introduction 1 1.1 What is thermodynamics? . .1 1.2 The atmosphere . .1 2 The Equation of State 1 2.1 State variables . .1 2.2 Charles' Law and absolute temperature . .2 2.3 Boyle's Law . .3 2.4 Equation of state of an ideal gas . .3 2.5 Mixtures of gases . .4 2.6 Ideal gas law: molecular viewpoint . .6 3 Conservation of Energy 8 3.1 Conservation of energy in mechanics . .8 3.2 Conservation of energy: A system of point masses . .8 3.3 Kinetic energy exchange in molecular collisions . .9 3.4 Working and Heating . .9 4 The Principles of Thermodynamics 11 4.1 Conservation of energy and the first law of thermodynamics . 11 4.1.1 Conservation of energy . 11 4.1.2 The first law of thermodynamics . 11 4.1.3 Work . 12 4.1.4 Energy transferred by heating . 13 4.2 Quantity of energy transferred by heating . 14 4.3 The first law of thermodynamics for an ideal gas . 15 4.4 Applications of the first law . 16 4.4.1 Isothermal process . 16 4.4.2 Isobaric process . 17 4.4.3 Isosteric process . 18 4.5 Adiabatic processes . 18 5 The Thermodynamics of Water Vapor and Moist Air 21 5.1 Thermal properties of water substance . 21 5.2 Equation of state of moist air . 21 5.3 Mixing ratio . 22 5.4 Moisture variables . 22 5.5 Changes of phase and latent heats . -
PDF Version of Helmholtz Free Energy
Free energy Free Energy at Constant T and V Starting with the First Law dU = δw + δq At constant temperature and volume we have δw = 0 and dU = δq Free Energy at Constant T and V Starting with the First Law dU = δw + δq At constant temperature and volume we have δw = 0 and dU = δq Recall that dS ≥ δq/T so we have dU ≤ TdS Free Energy at Constant T and V Starting with the First Law dU = δw + δq At constant temperature and volume we have δw = 0 and dU = δq Recall that dS ≥ δq/T so we have dU ≤ TdS which leads to dU - TdS ≤ 0 Since T and V are constant we can write this as d(U - TS) ≤ 0 The quantity in parentheses is a measure of the spontaneity of the system that depends on known state functions. Definition of Helmholtz Free Energy We define a new state function: A = U -TS such that dA ≤ 0. We call A the Helmholtz free energy. At constant T and V the Helmholtz free energy will decrease until all possible spontaneous processes have occurred. At that point the system will be in equilibrium. The condition for equilibrium is dA = 0. A time Definition of Helmholtz Free Energy Expressing the change in the Helmholtz free energy we have ∆A = ∆U – T∆S for an isothermal change from one state to another. The condition for spontaneous change is that ∆A is less than zero and the condition for equilibrium is that ∆A = 0. We write ∆A = ∆U – T∆S ≤ 0 (at constant T and V) Definition of Helmholtz Free Energy Expressing the change in the Helmholtz free energy we have ∆A = ∆U – T∆S for an isothermal change from one state to another. -
Thermodynamics
ME346A Introduction to Statistical Mechanics { Wei Cai { Stanford University { Win 2011 Handout 6. Thermodynamics January 26, 2011 Contents 1 Laws of thermodynamics 2 1.1 The zeroth law . .3 1.2 The first law . .4 1.3 The second law . .5 1.3.1 Efficiency of Carnot engine . .5 1.3.2 Alternative statements of the second law . .7 1.4 The third law . .8 2 Mathematics of thermodynamics 9 2.1 Equation of state . .9 2.2 Gibbs-Duhem relation . 11 2.2.1 Homogeneous function . 11 2.2.2 Virial theorem / Euler theorem . 12 2.3 Maxwell relations . 13 2.4 Legendre transform . 15 2.5 Thermodynamic potentials . 16 3 Worked examples 21 3.1 Thermodynamic potentials and Maxwell's relation . 21 3.2 Properties of ideal gas . 24 3.3 Gas expansion . 28 4 Irreversible processes 32 4.1 Entropy and irreversibility . 32 4.2 Variational statement of second law . 32 1 In the 1st lecture, we will discuss the concepts of thermodynamics, namely its 4 laws. The most important concepts are the second law and the notion of Entropy. (reading assignment: Reif x 3.10, 3.11) In the 2nd lecture, We will discuss the mathematics of thermodynamics, i.e. the machinery to make quantitative predictions. We will deal with partial derivatives and Legendre transforms. (reading assignment: Reif x 4.1-4.7, 5.1-5.12) 1 Laws of thermodynamics Thermodynamics is a branch of science connected with the nature of heat and its conver- sion to mechanical, electrical and chemical energy. (The Webster pocket dictionary defines, Thermodynamics: physics of heat.) Historically, it grew out of efforts to construct more efficient heat engines | devices for ex- tracting useful work from expanding hot gases (http://www.answers.com/thermodynamics). -
Thermodynamic Analysis Based on the Second-Order Variations of Thermodynamic Potentials
Theoret. Appl. Mech., Vol.35, No.1-3, pp. 215{234, Belgrade 2008 Thermodynamic analysis based on the second-order variations of thermodynamic potentials Vlado A. Lubarda ¤ Abstract An analysis of the Gibbs conditions of stable thermodynamic equi- librium, based on the constrained minimization of the four fundamen- tal thermodynamic potentials, is presented with a particular attention given to the previously unexplored connections between the second- order variations of thermodynamic potentials. These connections are used to establish the convexity properties of all potentials in relation to each other, which systematically deliver thermodynamic relationships between the speci¯c heats, and the isentropic and isothermal bulk mod- uli and compressibilities. The comparison with the classical derivation is then given. Keywords: Gibbs conditions, internal energy, second-order variations, speci¯c heats, thermodynamic potentials 1 Introduction The Gibbs conditions of thermodynamic equilibrium are of great importance in the analysis of the equilibrium and stability of homogeneous and heterogeneous thermodynamic systems [1]. The system is in a thermodynamic equilibrium if its state variables do not spontaneously change with time. As a consequence of the second law of thermodynamics, the equilibrium state of an isolated sys- tem at constant volume and internal energy is the state with the maximum value of the total entropy. Alternatively, among all neighboring states with the ¤Department of Mechanical and Aerospace Engineering, University of California, San Diego; La Jolla, CA 92093-0411, USA and Montenegrin Academy of Sciences and Arts, Rista Stijovi¶ca5, 81000 Podgorica, Montenegro, e-mail: [email protected]; [email protected] 215 216 Vlado A. Lubarda same volume and total entropy, the equilibrium state is one with the lowest total internal energy. -
Module P7.4 Specific Heat, Latent Heat and Entropy
FLEXIBLE LEARNING APPROACH TO PHYSICS Module P7.4 Specific heat, latent heat and entropy 1 Opening items 4 PVT-surfaces and changes of phase 1.1 Module introduction 4.1 The critical point 1.2 Fast track questions 4.2 The triple point 1.3 Ready to study? 4.3 The Clausius–Clapeyron equation 2 Heating solids and liquids 5 Entropy and the second law of thermodynamics 2.1 Heat, work and internal energy 5.1 The second law of thermodynamics 2.2 Changes of temperature: specific heat 5.2 Entropy: a function of state 2.3 Changes of phase: latent heat 5.3 The principle of entropy increase 2.4 Measuring specific heats and latent heats 5.4 The irreversibility of nature 3 Heating gases 6 Closing items 3.1 Ideal gases 6.1 Module summary 3.2 Principal specific heats: monatomic ideal gases 6.2 Achievements 3.3 Principal specific heats: other gases 6.3 Exit test 3.4 Isothermal and adiabatic processes Exit module FLAP P7.4 Specific heat, latent heat and entropy COPYRIGHT © 1998 THE OPEN UNIVERSITY S570 V1.1 1 Opening items 1.1 Module introduction What happens when a substance is heated? Its temperature may rise; it may melt or evaporate; it may expand and do work1—1the net effect of the heating depends on the conditions under which the heating takes place. In this module we discuss the heating of solids, liquids and gases under a variety of conditions. We also look more generally at the problem of converting heat into useful work, and the related issue of the irreversibility of many natural processes. -
Thermodynamic Potential of Free Energy for Thermo-Elastic-Plastic Body
Continuum Mech. Thermodyn. (2018) 30:221–232 https://doi.org/10.1007/s00161-017-0597-3 ORIGINAL ARTICLE Z. Sloderbach´ · J. Paja˛k Thermodynamic potential of free energy for thermo-elastic-plastic body Received: 22 January 2017 / Accepted: 11 September 2017 / Published online: 22 September 2017 © The Author(s) 2017. This article is an open access publication Abstract The procedure of derivation of thermodynamic potential of free energy (Helmholtz free energy) for a thermo-elastic-plastic body is presented. This procedure concerns a special thermodynamic model of a thermo-elastic-plastic body with isotropic hardening characteristics. The classical thermodynamics of irre- versible processes for material characterized by macroscopic internal parameters is used in the derivation. Thermodynamic potential of free energy may be used for practical determination of the level of stored energy accumulated in material during plastic processing applied, e.g., for industry components and other machinery parts received by plastic deformation processing. In this paper the stored energy for the simple stretching of austenitic steel will be presented. Keywords Free energy · Thermo-elastic-plastic body · Enthalpy · Exergy · Stored energy of plastic deformations · Mechanical energy dissipation · Thermodynamic reference state 1 Introduction The derivation of thermodynamic potential of free energy for a model of the thermo-elastic-plastic body with the isotropic hardening is presented. The additive form of the thermodynamic potential is assumed, see, e.g., [1–4]. Such assumption causes that the influence of plastic strains on thermo-elastic properties of the bodies is not taken into account, meaning the lack of “elastic–plastic” coupling effects, see Sloderbach´ [4–10]. -
Thermodynamic Potentials
THERMODYNAMIC POTENTIALS, PHASE TRANSITION AND LOW TEMPERATURE PHYSICS Syllabus: Unit II - Thermodynamic potentials : Internal Energy; Enthalpy; Helmholtz free energy; Gibbs free energy and their significance; Maxwell's thermodynamic relations (using thermodynamic potentials) and their significance; TdS relations; Energy equations and Heat Capacity equations; Third law of thermodynamics (Nernst Heat theorem) Thermodynamics deals with the conversion of heat energy to other forms of energy or vice versa in general. A thermodynamic system is the quantity of matter under study which is in general macroscopic in nature. Examples: Gas, vapour, vapour in contact with liquid etc.. Thermodynamic stateor condition of a system is one which is described by the macroscopic physical quantities like pressure (P), volume (V), temperature (T) and entropy (S). The physical quantities like P, V, T and S are called thermodynamic variables. Any two of these variables are independent variables and the rest are dependent variables. A general relation between the thermodynamic variables is called equation of state. The relation between the variables that describe a thermodynamic system are given by first and second law of thermodynamics. According to first law of thermodynamics, when a substance absorbs an amount of heat dQ at constant pressure P, its internal energy increases by dU and the substance does work dW by increase in its volume by dV. Mathematically it is represented by 풅푸 = 풅푼 + 풅푾 = 풅푼 + 푷 풅푽….(1) If a substance absorbs an amount of heat dQ at a temperature T and if all changes that take place are perfectly reversible, then the change in entropy from the second 풅푸 law of thermodynamics is 풅푺 = or 풅푸 = 푻 풅푺….(2) 푻 From equations (1) and (2), 푻 풅푺 = 풅푼 + 푷 풅푽 This is the basic equation that connects the first and second laws of thermodynamics. -
Thermodynamic Potentials and Thermodynamic Relations In
arXiv:1004.0337 Thermodynamic potentials and Thermodynamic Relations in Nonextensive Thermodynamics Guo Lina, Du Jiulin Department of Physics, School of Science, Tianjin University, Tianjin 300072, China Abstract The generalized Gibbs free energy and enthalpy is derived in the framework of nonextensive thermodynamics by using the so-called physical temperature and the physical pressure. Some thermodynamical relations are studied by considering the difference between the physical temperature and the inverse of Lagrange multiplier. The thermodynamical relation between the heat capacities at a constant volume and at a constant pressure is obtained using the generalized thermodynamical potential, which is found to be different from the traditional one in Gibbs thermodynamics. But, the expressions for the heat capacities using the generalized thermodynamical potentials are unchanged. PACS: 05.70.-a; 05.20.-y; 05.90.+m Keywords: Nonextensive thermodynamics; The generalized thermodynamical relations 1. Introduction In the traditional thermodynamics, there are several fundamental thermodynamic potentials, such as internal energy U , Helmholtz free energy F , enthalpy H , Gibbs free energyG . Each of them is a function of temperature T , pressure P and volume V . They and their thermodynamical relations constitute the basis of classical thermodynamics. Recently, nonextensive thermo-statitstics has attracted significent interests and has obtained wide applications to so many interesting fields, such as astrophysics [2, 1 3], real gases [4], plasma [5], nuclear reactions [6] and so on. Especially, one has been studying the problems whether the thermodynamic potentials and their thermo- dynamic relations in nonextensive thermodynamics are the same as those in the classical thermodynamics [7, 8]. In this paper, under the framework of nonextensive thermodynamics, we study the generalized Gibbs free energy Gq in section 2, the heat capacity at constant volume CVq and heat capacity at constant pressure CPq in section 3, and the generalized enthalpy H q in Sec.4. -
Equation of State - Wikipedia, the Free Encyclopedia 頁 1 / 8
Equation of state - Wikipedia, the free encyclopedia 頁 1 / 8 Equation of state From Wikipedia, the free encyclopedia In physics and thermodynamics, an equation of state is a relation between state variables.[1] More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions. It is a constitutive equation which provides a mathematical relationship between two or more state functions associated with the matter, such as its temperature, pressure, volume, or internal energy. Equations of state are useful in describing the properties of fluids, mixtures of fluids, solids, and even the interior of stars. Thermodynamics Contents ■ 1 Overview ■ 2Historical ■ 2.1 Boyle's law (1662) ■ 2.2 Charles's law or Law of Charles and Gay-Lussac (1787) Branches ■ 2.3 Dalton's law of partial pressures (1801) ■ 2.4 The ideal gas law (1834) Classical · Statistical · Chemical ■ 2.5 Van der Waals equation of state (1873) Equilibrium / Non-equilibrium ■ 3 Major equations of state Thermofluids ■ 3.1 Classical ideal gas law Laws ■ 4 Cubic equations of state Zeroth · First · Second · Third ■ 4.1 Van der Waals equation of state ■ 4.2 Redlich–Kwong equation of state Systems ■ 4.3 Soave modification of Redlich-Kwong State: ■ 4.4 Peng–Robinson equation of state Equation of state ■ 4.5 Peng-Robinson-Stryjek-Vera equations of state Ideal gas · Real gas ■ 4.5.1 PRSV1 Phase of matter · Equilibrium ■ 4.5.2 PRSV2 Control volume · Instruments ■ 4.6 Elliott, Suresh, Donohue equation of state Processes: -
Use of Equations of State and Equation of State Software Packages
USE OF EQUATIONS OF STATE AND EQUATION OF STATE SOFTWARE PACKAGES Adam G. Hawley Darin L. George Southwest Research Institute 6220 Culebra Road San Antonio, TX 78238 Introduction Peng-Robinson (PR) Determination of fluid properties and phase The Peng-Robison (PR) EOS (Peng and Robinson, conditions of hydrocarbon mixtures is critical 1976) is referred to as a cubic equation of state, for accurate hydrocarbon measurement, because the basic equations can be rewritten as cubic representative sampling, and overall pipeline polynomials in specific volume. The Peng Robison operation. Fluid properties such as EOS is derived from the basic ideal gas law along with other corrections, to account for the behavior of compressibility and density are critical for flow a “real” gas. The Peng Robison EOS is very measurement and determination of the versatile and can be used to determine properties hydrocarbon due point is important to verify such as density, compressibility, and sound speed. that heavier hydrocarbons will not condense out The Peng Robison EOS can also be used to of a gas mixture in changing process conditions. determine phase boundaries and the phase conditions of hydrocarbon mixtures. In the oil and gas industry, equations of state (EOS) are typically used to determine the Soave-Redlich-Kwong (SRK) properties and the phase conditions of hydrocarbon mixtures. Equations of state are The Soave-Redlich-Kwaon (SRK) EOS (Soave, 1972) is a cubic equation of state, similar to the Peng mathematical correlations that relate properties Robison EOS. The main difference between the of hydrocarbons to pressure, temperature, and SRK and Peng Robison EOS is the different sets of fluid composition. -
PHYS 521: Statistical Mechanics Homework #1
PHYS 521: Statistical Mechanics Homework #1 Prakash Gautam April 17, 2018 1. A particular system obeys two equations of state 3As2 As3 T = (thermal equation of state);P = (mechanical equation of state): v v2 Where A is a constant. (a) Find µ as a function of s and v, and then find the fundamental equation. Solution: Given P and T the differential of each of them can be calculated as 6As 3As2 6As2 3As3 dT = ds − dv ) sdT = ds − dv v v2 v v2 3As2 2As3 3As2 2As3 dP = ds − dv ) vdP = ds − dv v2 v3 v v2 The Gibbs-Duhem relation in energy representation allows to calculate the value of µ. dµ = vdP − sdT 3As2 2As3 6As2 3As3 = ds − dv − ds + dv v[ v2 ] v ( v)2 3As2 As3 As3 = − ds − dv = −d v v2 v As3 This can be identified as the total derivative of v so the rel ( ) As3 As3 dµ = −d ) µ = − + k v v Where k is arbitrary constant. We can plug this back to Euler relation to find the fundamental equation as. u = T s − P v + µ 3As3 As3 As3 As3 = − − + k = + k v v v v As3 □ So the fundamental equation of the system is v + k. (b) Find the fundamental equation of this system by direct integration of the molar form of the equation. Solution: The differential form of internal energy is du = T ds − P dv 3As2 As3 = ds − dv v v2 1 As3 As before this is just the total differential of v so the relation leads to ( ) As3 As3 du = d ) u = + k v2 v This k should be the same arbitrary constant that we got in the previous problem.