Energy Engineering - Energy Conversion A

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C OURSE OF E NERGY C ONVERSION GLOSSARY These class notes are for the students o f the course "Energy Conversion " at Politecnico di Milano. Anyone who find s inaccuracies or , anyhow, wishes to send comments to improve them is invited to the lecturer ( [email protected] ), who thanks in advance. Gl ossary - Energy Conversion – V1.1 2 of 9 English Italian absolute velocity velocità assoluta adiabatic flame temperatur e temperatura adiabatica di fiamma aero derivative aeroderivativa adapted nozzle ugello adattato air -cooled condenser condensatore ad aria alpha particle particella alpha angular molecule molecola angolare degree of partial arc grado di ammissione ash collection hopper tramoggia di raccolta delle ceneri atom atomo atomic number numero atomico attemperation attemperamento axial chord corda assiale balance constant costante di equilibrio bending stress sforzo di flessione bent molecule molecola angolare beta particle particella beta binding energy energia di legame blade paletta blade base base di pala blade chord corda di pala blade height altezza di pala blade row palettatura, schiera blade spacing passo di pala blade throat gola di pala blade tip apice di pala blade to blade plane piano interpalare bled spillato bleed spillamento (to) bleed spillare boundary layer strato limite boundary layer separation distacco di vena branched molecule molecola ramificata breeding autofertilizzante burn -out crisi termica Carnot's theorem enunciato di Carnot "(to) Carnotize" "Carnotizzare" case cassa centrifugal stress sforzo centrifugo chain reaction reazione a catena characteristic curve curva caratteristica chocked flow blocco sonico cladding rivestimento Clausius -Clapeyron relation equazione di Clausius -Clapeyron clearance gioco closed cycle ciclo chiuso coefficient of performance coefficiente di prestazione cold sink pozzo freddo Gl ossary - Energy Conversion – V1.1 3 of 9 English Italian combustion air aria comburente combustion chamber camera di combustione combustor combustore compressibility factor fattore di comprimibilità compression ratio rapporto di compressione (to) collide collidere condenser condensatore continuous reactor reattore fluente control rod barr a di controllo convection heat transfer coefficient coefficient e di scambio termico conve ttivo convergent -divergent nozzle ugello convergente -divergente coolant fluido di raffreddamento corner vortex vortice di spigolo correction term termine di correzione critical heat flux flusso di calore critico critical pressure pressione critica day ahead market Mercato del Giorno Prima (to) decay decadere dead state punto morto, stato morto deaerator degasatore degree of freedom grado di libertà degree of reaction grado di reazione deflection angle deflessione angolare (to) desuperheat desurriscaldare deuterium deuterio degradation degradazione diatomic biatomico diffuser diffusore direct heat exchanger scambiatore a miscela direct preheater preriscaldatore a miscela discharge flow angle angol o di scarico (to) downgrade decadere , degradare driving machine macchina motrice drum tamburo effectiveness efficacia efficiency rendimento efficiency loss perdita di rendimento efflux speed velocità di efflusso eigenmode modo di vibrare ejector eiettore elastic collision urto elastico electron elettrone endothermic endotermico energy balance bilancio energetico energy level stato energetico energy performance prestazione energetica engine motore enthalpy entalpia Gl ossary - Energy Conversion – V1.1 4 of 9 English Italian enthalpy change salto entalpico enthalpy drop salto entalpico enthalpy of condensation calore di condensazione enthalpy of reaction calore di reazione enthalpy of vaporization calore di evaporazione entropy entropia entropy balance bilancio entropico entropy of mixing entropia di miscelamento Equation of State (EOS) equazione di stato Eulerian work lavoro euleriano evaporator tube tubo EVA exergy exergia exhaust gas gas combusto exhaust pipe condotto di scarico exothermic esotermico expander espansore expansion ratio rapporto di espansione fast neutron neutrone veloce FE parameter parametro FE feedwater acqua di alimento fertile isotope isotopo fertile film boiling ebollizione a film final velocity velocità finale finned tube tubo alettato fissile fissile fission fissione flaring angle angolo di apertura della palettatura flow coefficient coefficiente di portata flue gas fumi fluid machine macchina a fluido fluid machinery macchine a fluido fluid dynamics fluidodinamica forced convection convezione forzata fossil fuel combustibile fossile fouling sporcamento free vortex method metodo del vortice libero fuel combustibile fully turbulent regime regime pienamente turbolento fusion fusione gamma particle particella gamma Gibbs equation relazione di Gibbs Gibbs free energy energia libera di Gibbs groove scanalatura gross power potenza lorda half -life time tempo di dimezzamento halogenated hydrocarbon idrocarburo alogenato harmonic oscillator oscillatore armonico head coefficient coefficiente di carico Gl ossary - Energy Conversion – V1.1 5 of 9 English Italian heat capacity capacità termica heat pump pompa di calore heat sink pozzo caldo heat transfer scambio termico heating value potere calorifico heavy water acqua pesante Helmholtz free energy energia libera di Helmholtz higher heating value potere calorifico superiore hoop cerchiatura horseshoe vortex vortice a ferro di cavallo hot source sorgente calda hub mozzo hydrocarbon idrocarburi hydrocyclon ciclone ideal cycle ciclo ideale ideal gas gas perfetto ideal liquid liquido perfetto impulse stage stadio ad azione indirect preheater preriscaldatore a superficie initial velocity velocità iniziale inlet temperature temperatura di ingresso integrand funzione integranda intercooled interraffreddato intercooling interraffreddamento interblade channel canale interpalare internal energy energia interna irreversibility irreversibilità isenthalpic isoentalpico isentropic isoentropico isentropic head coefficient coefficiente di carico isoentropico isobar isobara isobaric expansion coefficient coefficiente di espansione isobaro isochore isocora isotherm isoterma isothermobaric isotermobarico isotope isotopo Joule's experiment esperienza di Joule Junker ’s gas calorimeter calorimetro di Junker kinetic energy energia cinetica law of equipartition of energy principio di equipartizione dell'energia law of thermodynamics principio di termodinamica leading edge bordo di attacco leakage trafilamento levers leveraggi leakage vortex vortice di trafilamento liberated energy energia liberata line segment segmento load ramp rampa di carico Gl ossary - Energy Conversion – V1.1 6 of 9 English Italian lower heating value potere calorifico inferiore Mach number numero di Mach Mahler bomb calorimeter bomba calorimetrica di Mahler mass balance bilancio di massa mass defect difetto di massa mass flow rate portata massica mass fraction frazione massica mass number numero di massa mass unit unità di massa mass -basis massico Maxwell ’s relation equazione di Maxwell mean log temperatura temperatura media logaritmica mean log temperatura difference differenza di temperatura media logaritmica mean temperature temperatura media mean temperature difference differenza di temperatura media membrane watertube tubo membranato meridian plane piano meridiano method of least squares metodo dei minimi quadrati mixture miscela moderator moderatore molar mass massa molare molar -basis molare mole fraction frazione molare molecular weight massa molecolare momentum quantità di moto monatomic monoatomico natural convection convezione naturale network code codice di rete nucleate boiling ebollizione nucleata normal shock wave onda d'urto normale nozzle ugello nucleus nucleo nucleon number numero di massa nuclide nuclide objective function funzione obiettivo oblique compression shock wave onda d'urto obliqua di espansione oblique expansion shock wave onda d'urto obliqua di compressione oblique shock wave onda d'urto obliqua opening angle angolo di apertura operating machine macchina operatrice organic fluid fluido organico outlet temperature temperature di uscita partial arc admission ammis sione parzializzata pebble bed reactor reattore tipo “pebble bed” peripheral velocity velocità periferica phase diagram diagramma di stato plant impianto plutonium plutonio Gl ossary - Energy Conversion – V1.1 7 of 9 English Italian polyatomic poliatomico polytropic efficiency efficienza politropica power cycle ciclo di potenza power island isola di potenza Prandtl -Meyer ’s expansion fan onde (o moto) di espansione di Prandtl -Meyer preheating preriscaldamento pressure drop perdita di carico pressure side ventre (di una pala) principle of corresponding states legge degli stati corrispondenti proton protone protium idrogeno, prozio pulse impulso purge spurgo quantity grandezza radial clearance gioco radiale radial seal tenuta radiale radiation radiazione radioactive radioattivo reacting system sistema reagente reaction stage stadio a reazione reagent reagente real fluid gas reale, fluido reale recirculation pump pompa di ricircolo recuperated recuperativo recuperator recuperatore reduced pressure pressione ridotta reduced temperature temperatura ridotta regenerator rigeneratore reheater risurriscaldatore reheating risurriscaldamento relation relazione relative velocity velocità relativa residual enthalpy correzione di entalpia residual entropy correzione di entropia Reynolds number numero di Reynolds rotational speed velocità di rotazione root -mean -square translational speed velocità di traslazione quadratica media rotor rotore rotor disk disco rotorico saturated vapor curve curva limite superiore saturation dome campana di saturazione saturation pressure tensione di vapore saturation pressure curve curva di tensione di vapore second -law analysis analisi entropica segment segmento shock wave onda d'urto siloxane silossano similitude theory teoria della similitudine Gl ossary - Energy Conversion – V1.1 8 of 9 English Italian size parameter parametro (dimensionale) di taglia spacing passo specific angular velocity velocità angolare specifica specific heat calore specifico speed of sound velocità del suono stage stadio starting engine motore di partenza state function funzione di stato state quantity grandezza di stato state variable variabile di stato stator statore stator disk disco statorico steam generator generatore di vapore strain sforzo stress sforzo subcooling sottoraffreddamento subsonic subsonico suction side dorso (di una pala) supercritical power plant centrale ipercritica superheated surriscaldato supersonic supersonico supply duct condotto di alimentazione survey casistica tapered rastremata temperature change salto di temperatura thermal conductivity conduttività termica thermal energy energia termica thermal load carico termico thermal loss perdita termica thermal motion agitazione termica thermal neutron neutrone termico thermal power potenza termica thorium torio throttling laminazione total enthalpy entalpia totale traction stress sforzo di trazione trailing edge bordo di uscita transition boiling ebollizione di transizione translatory traslazionale trapezoidal cycle ciclo trapezoidale trend andamento triangular cycle ciclo triangolare tritium trizio turbine cylinder corpo di turbina turbine section corpo di turbina turbomachine turbomacchina turbomachinery turbomacchine turbopump turbopompa Gl ossary - Energy Conversion – V1.1 9 of 9 English Italian twisted blade pala svergolata ultra -supercritical (USC) ultra -super critico ( USC ) unit weight massa unitaria uranium uranio useful work lavoro utile Van der Walls cubic equation equazione cubica di Van der Walls vapor vapore velocity field campo di moto vibrational frequency frequenza di vibrazione vibrational mode modo di vibrare volumetric flow rate portata volumetrica waste heat cascami termici wasted power potenza persa wasted work lavoro perso water -cooled condenser condensatore ad acqua wheel chamber camera ruota work of friction lavoro di attrito working fluid fluido di lavoro C OURSE OF E NERGY C ONVERSION NOMENCLATURE These class notes are for the students o f the course "Energy conversion " at Politecnico di Milano. Anyone who find s inaccuracies or , anyhow, wishes to send comments to improve them is invited to the lecturer ( [email protected] ), who thanks in advance. Nomenclature - Energy Conversion – V1. 1 2 of 9 1 Acronyms ................................ ................................ ................................ ................................ ...... 3 2 Greek symbols ................................ ................................ ................................ ............................... 4 3 Roman symbols ................................ ................................ ................................ ............................. 5 4 Superscripts ................................ ................................ ................................ ................................ ... 7 5 Subscripts ................................ ................................ ................................ ................................ ...... 7 6 Accents ................................ ................................ ................................ ................................ .......... 9 Nomenclature - Energy Conversion – V1. 1 3 of 9 1 ACRONYMS Acronym Meaning ��� Benedict -Webb -Rubin ���� Benedict -Webb -Rubin -Starling ��� Carrying Charge Factor ��� Coefficient of Performance ��� Combustor Outlet Temperature ��� Economizer ��� Emission Monitoring System �� Gas Turbine ��� Higher Heating Value �� Higher Pressure ���� Heat Recovery Steam Generator ��� Inlet Guide Vanes ��� International Standard Organization ��� Lower Heating Values �� Low Pressure ���� Modified Benedict -Webb -Rubin �� Middle Pressure �� −���� Perturbed Chain Statistical Association Fluid Theory �� Peng -Robinson �� Reheating ��� Revolutions per minute ��� Revolutions per second �� Superheating �� Size parameter ��� Soave -Redlich -Kwong ��� Turbine Inlet Temperature ��� Turbine Outlet Temperature �� Total to Static �� Total to Total ��� Ultra -super -critical ��� Variable Guide Vanes Nomenclature - Energy Conversion – V1. 1 4 of 9 2 GR EEK SYMBOLS Symbol Description Dimensions Typical units � Expansion coefficient Θ -1 1/K � Air -to-fuel ratio - - � Absolute v elocity angle - rad, ° � Pressure ratio - - � Relative velocity angle - rad, ° � Heat capacity ratio - - � Reduced density - - � Shock wave angle - rad, ° �� Radial clearance L mm � Angular opening - rad, ° � Effectiveness - -, % � Partial admission arc factor - - � Efficiency - -, % � Temperature ratio - - � Threshhold temperature Θ K, °C � �= (�−1)�−1 - - � Combustion ratio - - � Chemical potential L2MN-1T-2 J/mol, kJ /mol � Temperature ratio - - � Loss coefficient - - � Generic specific property Π Generic extensive property � Density L-3M kg/m 3, g/cm 3 ������ Stress L-1MT -2 kPa, MPa ������ Inverse of reduced temperature - - ������ Flow coefficient - - � Diffuser efficiency - -, % � Degree of reaction (static -to-static) - - � Generic phase fraction - - � Head coefficient (static -to-static) - - � Rotational speed T-1 rad/s Nomenclature - Energy Conversion – V1. 1 5 of 9 3 ROMAN SYMBOLS Symbol Description Dimensions Typical units � Area L2 m2, cm 2 � Helmholtz free energy L2MT -2 J �̃ Helmholtz free energy on mole basis L2MT -2N-1 J/kmol � Helmholtz free energy on mass basis L2T-2 J/kg � Inter -blade channel width L cm, mm � Polynomial coefficients for specific heat Various Various � Speed of sound LT -1 m/s � Exergy L2MT -2 J �̃ Exergy on mole basis L2MT -2N-1 J/kmol � Exergy on mass basis L2T-2 J/kg � Axial chord length for blade rows L m, cm � Heat capacity L2MT -2Θ-1 J/K, kJ/K � Axial chord length for insulated profiles L m, cm � Specific heat L2T-2Θ-1 J/(kg K), kJ/(kg K) � Diameter L m, cm � Energy L2MT -2 J �̃ Energy on mole basis L2MT -2N-1 J/kmol � Energy on mass basis L2T-2 J/kg � Excess - - � Generic function - - � Degrees of freedom for specific heat - - � Gibbs free energy L2MT -2 J �̃ Gibbs free energy on mole basis L2MT -2N-1 J/kmol � Gibbs free energy on mass basis L2T-2 J/kg � Gravitational constant LT -2 9.81 m/s 2 ℎ Planck constant L2MT 6.626 * 10 -34 J s � Enthalpy L2MT -2 J ℎ̃ Specific e nthalpy on mole basis L2MT -2N-1 J/kmol ℎ Specific e nthalpy on mass basis L2T-2 J/kg � Stress incrementation parameter - - � Compressibility coefficient L1M-1T2 1/bar, 1/MPa � Boltzmann constant L2MT -2Θ-1 1.38065 * 10 -23J/K � Head coefficient (total -to-static) - - �� Surface finish L µm, nm � Specific work L2T-2 J/kg, kJ/kg � Mach number - - � Moment L2MT -2 N m � Mass M kg, g �� Molar mass MN -1 kg/kmol, g/mol � Rotational velocity T-1 RPM � Moles N kmol, mol � Blade channel throath L m, cm � Power L2MT -3 W, kW, MW � Pressure L-1MT -2 Pa, kPa, MPa, bar, atm � Thermal energy L2MT -2 J/m 2, J � Thermal flux MT -2 J/m2 � Heat transfer resistance M-1T3 m2/W � Radius L m, cm �� Reynolds number - - ������� Specific gas constant L2T-2Θ-1 J/(kg K), kJ/(kg K) ������� Universal gas constant L2MN-1T-2Θ-1 8314 J/(kmol K) Nomenclature - Energy Conversion – V1. 1 6 of 9 Symbol Description Dimensions Typical units �∗ Isentropic degree of reaction (total -to-static) - - � Entropy L2MT -2Θ-1 J/K �̃ Specific entropy on mole basis L2MN-1T-2Θ-1 J/(kmol K) � Specific entropy on mass basis L2T-2Θ-1 J/(kg K) � Blade spacing L m, cm � Temperature Θ K, °C � Thickness L mm ������� Traling edge thickness L mm � Heat transfer coefficient MT -3 W/m 2 � Internal energy L2MT -2 J �̃ Specific internal energy on mole basis L2MT -2N-1 J/kmol � Specific internal energy on mass basis L2T-2 J/kg � Activation of vibrational modes parameter - - � Peripheral velocity LT -1 m/s � Velocity LT -1 m/s � Volume L3 m3, lt �̃ Molar specific volume L3N-1 m3/kmol � Specific volume L3M-1 m3/kg � Proper oscillation frequency T-1 Hz, kHz, MHz � Section modulus L3 m3, cm 3 � Relative velocity LT -1 m/s � Work L2MT-2 J � Specific work L2T-2 J/kg � Generic property Various Various � Liquid fraction - - � Vapor fraction - - � Compressibility - - � Height L m � Number of blades - - Nomenclature - Energy Conversion – V1. 1 7 of 9 4 SUPERSCRIPTS Symbol Meaning ° Standard 0 Ideal gas �ℎ Chemical � Gas � Liquid �ℎ Physical ������� Solid of type i ��� Saturated � Vapor 5 SUBSCRIPTS Symbol Meaning 0 Reference state 0 Stator inlet (in expan sion stage) 1 Rotor inlet 2 Rotor outlet 3 Stator outlet (in compression stage) 2� Two -dimensional 3� Three -dimensional � Axial �� Adiabatic ��� Admittable �� Adiabatic Flame ��� Ambient �� Approach ��� Auxiliaries � Cold ����� Centrifugal ��� Chiller ��� Compressor ��� Combustor ���� Combustion ��� Condenser ���� Condensation �� Critical �� (Perfect) crystal transformation ��� Desuperheater ���� Diffuser ��������� Dissipated ��� Economizer �� Electric �� Equivalent �� Eulerian ��� Evaporator ��� Evaporation �� Exit �� ℎ Exhaust gas ��� Expander ��� External � Final � Formation Nomenclature - Energy Conversion – V1. 1 8 of 9 Symbol Meaning ��� Fluid ���� Flexural ���� Fouling ��� Fusion ��� Geometric ℎ Hydraulic ℎ Hot �� Heat pump � Initial � First law of thermodynamics �� Second law of thermodynamics �������� Impulse ������� Inlet �������� Internal �������� Irreversible ������� Isentropic �� Leading edge �������� Limit � Mean ��� Maximum �� Mechanical �������� Minimum �������� Mixing ��� Mean logarithmic ��� Optimal ��� Outlet � Products � Constant pressure �� Pinch point �� Pressure Side � Reactants � Reduced ���� Reaction ��� Recuperated ��� Regenerated ��� Reversible ��� Rotor ��� Rotational � Static � Specific �� Subcooling ��� Secondary �� Source �� Suction side ���� Stator ��� Stage ���������� Stoichiometric � Constant temperature � Total ��� Turbine � Translational ���� Tangential �� Trailing edge �ℎ� Throat �� Triple Nomenclature - Energy Conversion – V1. 1 9 of 9 Symbol Meaning ���� Transformer ��������� Trilateral � Constant volume � Vibrational �� Velocity triangle � Wall � Wasted � Polytropic 6 ACCENTS Symbol Meaning ̅ Average ̃ Mole basis ̇ Time derivative ̂ Volume basis C OURSE OF E NERGY C ONVERSION T HERMODYNAMIC P ROPERTIES OF F LUIDS These class notes are for the students of the course "Energy Conversion " at Politecnico di Milano. Anyone who find s inaccuracies or , anyhow, wishes to send comments to improve them is invited to the lecturer ( [email protected] ), who thanks in advance. Thermodynamic properties of fluids - Energy Conversion 2 of 109 Why this classnotes ................................ ................................ ................................ ............................... 4 1 Introduction ................................ ................................ ................................ ................................ ...... 5 2 Review of thermodynamics fundamental relations ................................ ................................ .......... 8 3 The ideal gas model ................................ ................................ ................................ ........................ 14 3.1 Definition of pure fluid ................................ ................................ ................................ ........... 14 3.2 Thermodynamic behavior of ideal gases ................................ ................................ ................. 15 3.3 Thermodynamic properties of ideal gases calculation ................................ ............................ 17 3.4 Specific heat of ideal gases calculation ................................ ................................ ................... 18 3.4.1 Monoatomic molecules ................................ ................................ ............................... 22 3.4.2 Diatomic molecules ................................ ................................ ................................ ..... 23 3.4. 3 Polyatomic molecules ................................ ................................ ................................ .. 24 3.4.4 Practical examples ................................ ................................ ................................ ....... 25 3.5 Thermodynamic diagrams and transformations ................................ ................................ ...... 25 3.5.1 Trend of the isobars of ideal gas in the T -s diagram ................................ ................... 25 3.5.2 Temperature rise in an isentropic compression ................................ ........................... 27 3.5.3 Optimum compression ratio in a closed cycle ................................ ............................. 29 3.5.4 Molecular complexity effect on the isentropic compression ratio .............................. 30 3.5.5 Molecular complexity and mol ecular weight effect on isentropic enthalpy change ... 31 3.5.6 Molecular complexity effect on the volumetric flow rate in a heat exch anger ........... 31 4 The ideal liquid model ................................ ................................ ................................ .................... 33 4.1 Thermodynamic properties of an ideal liquid calculation ................................ ....................... 34 4.1.1 Internal energy ................................ ................................ ................................ ............. 34 4.1.2 Enthalpy ................................ ................................ ................................ ....................... 34 4.1.3 Entropy ................................ ................................ ................................ ........................ 36 5 The real fluid properties, Single -phase region ................................ ................................ ............... 37 5.1 Phenomenology at the microscopic level ................................ ................................ ................ 37 5.2 The compressibility factor ................................ ................................ ................................ ....... 38 5.3 The Corresponding States Principle ................................ ................................ ........................ 40 5.4 A brief history of the equations of State ................................ ................................ ................. 43 5.5 Calculation programs ................................ ................................ ................................ .............. 49 5.6 Thermodynamic properties of a real fluid in actual coordinates ................................ ............. 50 5.6.1 T-v diagram ................................ ................................ ................................ ................. 51 5.6.2 Specific heat residual ................................ ................................ ................................ ... 52 5.6.3 Residual enthalpy ................................ ................................ ................................ ........ 56 5.6.4 Residual entropy ................................ ................................ ................................ .......... 60 5.7 Effects on the T -s diagram ................................ ................................ ................................ ...... 62 5.7.1 Trend of the isobar curves ................................ ................................ ........................... 62 5.7.2 Trend of the isenthalpic curves ................................ ................................ ................... 63 5.8 Thermodynamic properties in reduced coordinates ................................ ................................ 64 5.8.1 Enthalpy difference between two thermodynamic states calculation ......................... 67 5.8.2 Specific heat on a molar -basis variation between saturated liquid and vapor ............. 68 6 The real fluid properties, Phase -change region ................................ ................................ .............. 70 6.1 Clapeyron and Clausius -Clapeyron equations ................................ ................................ ........ 70 6.2 The saturation pressure curve ................................ ................................ ................................ .. 73 6.3 Approximated estimate of enthalpy of evaporation ................................ ................................ 78 6.4 The acentric factor ................................ ................................ ................................ ................... 79 6.5 Molecular complexity effect of the fluid on the shape of the T -s diagram ............................. 80 7 The real liquid behavior ................................ ................................ ................................ ................. 83 7.1 Ideal compression work of a pump ................................ ................................ ......................... 83 7.2 Evaluation of the heating caused by an isentropic compression ................................ ............. 85 8 Solutions and Mixtures ................................ ................................ ................................ ................... 88 Thermodynamic properties of fluids - Energy Conversion 3 of 109 8.1 Introduction to solutions ................................ ................................ ................................ ......... 88 8.2 Ideal mixtures ................................ ................................ ................................ .......................... 89 8.3 Mixtures in combustion ................................ ................................ ................................ ........... 91 9 Fossil and renewable fuels ................................ ................................ ................................ ............. 93 9.1 Stoichiometry in combustion reactions ................................ ................................ ................... 93 9.2 Enthalpies of formation and enthalpy balance of combustion ................................ ................ 95 9.3 Adiabatic flame temperature ................................ ................................ ................................ ... 98 9.4 Heating value of the fuel ................................ ................................ ................................ ......... 98 9.5 Recuperative preheating ................................ ................................ ................................ ........ 101 9.6 Entropy balance for a reacting system ................................ ................................ .................. 102 10 Appendices ................................ ................................ ................................ ................................ ... 105 10.1 Pressure of radiation: a practical example ................................ ................................ ............ 105 10.2 Thermodynamic square ................................ ................................ ................................ ......... 105 10.3 Application of the kinetic theory of gases ................................ ................................ ............. 107 Thermodynamic properties of fluids - Energy Conversion 4 of 109 W HY THIS CLASSNOTES Fluids are one of the three cornerstones of energy conversion systems together with cycles and equipment. Models of the thermodynamic properties of fluids are necessa ry to study properly energy conversion systems from both the cycle and the equipment perspectives . In this context , it is fundamental to understand when it is possible to apply one model depending upon the required accuracy. Specifically, a distinction bet ween ideal and real fluids and mixtures shall be considered. The scope of this class note is to provide the students of Energy Conversion with the knowledge and competence to understand and apply thermodynamic modelling of fluids . The methodology adopted here is based on a review of fundamental thermodynamic relations first for pure fluids, developed first of ideal pure fluids and then for real pure fluids. Similarly, fundamentals relations for solutions are only outlined and detailed f or ideal mixtures . The structure of the present document is as follows. • Chapter 1 introduces the topic of thermodynamic properties of fluids and their modelling • Ch apter 2 reviews the thermodynamic fundamental relations • Chapter 3 deals with the ideal gas model • Chapter 4 deals with the ideal liquid model • Chapter 5 deals with real fluid properties in the single -phase region • Chapter 6 deals with real fluid properties in the phase -change region • Chapter 7 deals with the real liquid model • Chapter 8 deals with solution and mixture properties Thermodynamic properties of fluids - Energy Conversion 5 of 109 1 INTRODUCTION The thermodynamic state of a fluid in equilibrium can be described by a set of parameters called state variables . These state variables represent the effect of the behavior of atoms and m olecules at the microscopic scale (typical of the Kinetic theory of gases) on the macroscopic scal e (of Classical thermodynamic s). The approach here is investigating mathematically matter at the macroscopic scale, providing the relations of Classic thermodynamics, and analyzing qualitatively at the microscopic scale, recalling the main outcomes of the Kinetic theory of gases. A pure fluid is a fluid made of a single chemical component (or species); in contrast, solutions and mixtures are made of two or more species , where a mixtures is a solution in the gaseous phase. P ure fluid s and solution can exist in one or more phase s, where a phase is a quantity of matter that is homogeneous throughout in both physical structure and chemical composition. More phases can exist at equilibrium at one time; for instance, vapor -liquid equilibria are conditions in whic h a vapor phase and a liquid phase coexist. Vapor -liquid equilibria can be des cribed mathematically in two different manners: 1. direct approach , in which both vapor and liquid phases are describe d by an equation of state, introduces soon after; 2. indirect approach , in which only the vapor phase is described by an equation of state, while the liquid phase by activity coefficients. For a pure fluid , a thermodynamic state of equilibrium is completely defined when two independent state variables are k nown. For example, temperature of a stable equilibrium state can be calculated as a function of pressure and specific volume. The relation between these three state variables is called “volumetric Equation of State (EOS)” and can be expressed explicitly in pressure as: ⥍(⥗⏬⥝⏬⥁)= ╽❧ ⥗ = ⥍(⥁⏬⥝) (1.1) It is possible to derive any other state variable and fully characterize the properties of the fluid from this equation (or from any other Equation of State that relates three thermodynamic state variables) through differentiation and integration operations of the Equation of Sta te itself . The Equation of State can take on particularly simple analytical expressions in the case of ideal gases or ideal liquids , while in other cases its formulation may require a large number of terms. The choice of the Equation of State that models the properties of the fluid of interests most accurately is mainly a function of the fluid itself , its thermodynamic conditions and the desired accuracy . The most widely used fluids in the field of power generation plants are certainly water in steam power plants , air and exhaust gases in gas turbines. While air and - within certain limits - exhaust gases can be treated as ideal gases with results of reason able accuracy in most applications , in the case of water it is usually necessary to adopt formulations that refer to the real fluid behavior . Many other fluids, such as hydrocarbons, can b e used in specific applications; on top of these, if inverse cycles also are considered for refrigerant systems , the database becomes even broader. Since the se fluids cannot be considered ideal gases, the availability of accurate Equations of State is essential to model any process and to design, optimize and ultimately ma nufacture every single plant component with a good degree of confidence . Thus, the need arises to have a sufficiently precise calculation method for determining the thermodynamic properties of fluid s via an Equations of State able to describe the real flui d behavior. Thermodynamic properties of fluids - Energy Conversion 6 of 109 In case of pure fluids, a real fluid property is calculated adding a residual property to the fluid property in the hypothetical ideal gas state . The residual property is sometimes called also departure property because it indicates the deviat ion from the ideal gas behavior. In their turn, also solution and mixture can be ideal and real. Ideal solution properties can be simply computed from the real fluid properties of the pure components making the solution. Ultimately, a real solution property is calculated adding the solution excess property to the solution property in the hypothetical ideal solution state. As a first example of the importance of real fluid properties , c onsider an ideal turbine that expands a fluid through an isentropic process from an initial state 0 (characterized by ⥁ⴿ and ⥗ⴿ) to a final state 1 (characterized by ⥗ⵀ and ⥚ⵀ). The work obt ainable via an adiabatic fluid machine , neglecting the kinetic and potential energ y terms , is: ⥞ = ⥏ⴿ(⥁ⴿ⏬⥗ⴿ)⽑ ⥏ⵀ(⥗ⵀ⏬⥚ⴿ(⥁ⴿ⏬⥗ⴿ)) (1.2) As shown in Section 3, in the case of ideal gas, enthalpy depends only on temperature. For a real fluid, enthalpy is instead a function of both temperature and pressure. Underestimating the dependency of enthalpy on pressure can lead in general to severe errors for some fluids in certain conditions . Therefore, an accurate calculation of the work obtainable from a fluid machine cannot disregard an accurate calculation of the thermodynamic properties of the working f luid . Likewise , when using sophisticated design methods for turbomachines based on fluid dynamics, adopt ing accurate thermodynamic properties to determine the velocity field of the fluid, the presence of supersonic flows, as well as other phenomena, is a crucial step for reliable results . The second example , regarding gas cycle s, highlight s the importance of an accurate calculation of the thermodynamic properties of the working fluid. Consider the regeneration in a Brayton closed cycle ope rating respectively with an ideal gas and a real fluid (cycles with helium, He, and carbon dioxide, CO 2, respectively in Fig. 1.1 left and right ). Fig. 1.1 - Closed gas cycle operating between the same extreme temperatures: in the Helium cycle (a) , the fluid can be considered an ideal gas at each point, but in the CO 2 cycle (b) the fluid close to the saturatio n dome shows important effects of real fluid . He CO 2 1 2 3 4 5 6 1 2 3 4 5 6 a) b) Thermodynamic properties of fluids - Energy Conversion 7 of 109 In the cycle with helium ( Fig. 1.1a), which can be considered an ideal gas in these conditions because it operates at temperatures much higher than its critical temperature, an infinite surface regenerator would give rise to a reversible process , resulting in heat transfer under infinitesimal temperature differences , leading to ⥁ⵁ= ⥁ⵄ and ⥁ⵅ= ⥁ⵃ. On the contrary, in the CO 2 cycle (Fig. 1.1b) an ideal regeneration involves an irreversibility in heat transmission , hence ⥁ⵁ= ⥁ⵄ but ⥁ⵅ< ⥁ⵃ, because of the real fluid effect of pressure on the specific heat along the high pressure isobar, as explained later . In the following chapters , the equations for calculating the properties of t he ideal pure gases as well as ideal pure liquid are described ; then , the corrections due to the real fluid behavior are analyzed. Subsequently , the P rinciple of Corresponding States and a brief history of the evolution of the Equations of State is discussed , in addition to the theoretical description of the trend of the thermodynamic quantities in the ⥁ ⽑ ⥚ diagram . Ultimately, concepts for solutions of fluids are provided, focusing specifically on idea l mi xtures. Thermodynamic properties of fluids - Energy Conversion 8 of 109 2 REVIEW OF THERMODYNAMICS FUNDAMENTAL RELATIONS This chapter deals with a review of the thermodynamics fundamental relations for the calculation of fluid properties. For a single -phase multicomponent homogeneous system in stable equilibrium , the internal energy ⥂ can be expressed as follows : ⥂ = ⥂(⥀⏬⥃⏬⪟) (2.1) which is c alled the fundamental relation in the energy form and where ⥀ is the entropy of the system, ⥃ is the volume and ⪟ is the vector of the number of moles of each component . By differentiating it: ⥋⥂ = (⧽⥂ ⧽⥀ ) ⷚ⏬ⷬ⻟ ⥋⥀ ⽐ (⧽⥂ ⧽⥃ ) ⷗⏬ⷬ⻟ ⥋⥃ ⽐ ⿘ (⧽⥂ ⧽⥕ⷧ ) ⷗⏬ⷚ⏬ⷬ⻠⺺⻟ ⷧ ⥋⥕ⷧ (2.2) where ⥕ⷧ is the number of moles of the i-th component and in which the partial derivatives have a physical meaning: (⧽⥂ ⧽⥀ ) ⷚ⏬ⷬ⻟ ⠫ ⥁ (⊡⊒⊚⊝⊒⊟⊎⊡⊢⊟⊒) (2.3) ⽑ (⧽⥂ ⧽⥃ ) ⷗⏬ⷬ⻟ ⠫ ⥗ (⊝⊟⊒⊠⊠⊢⊟⊒) (2.4) (⧽⥂ ⧽⥕ⷧ ) ⷗⏬ⷚ⏬ⷬ⻠⺺⻟ ⠫ ⧯ⷧ (⊐⊕⊒⊚⊖⊐⊎⊙ ⊝⊜⊡⊒⊛⊡⊖⊎⊙) (2.5) Consequently, Eq. (2.2) can be rewritten as : ⥋⥂ = ⥁⥋⥀ ⽑ ⥗⥋⥃ ⽐ ⿘ ⧯ⷧ ⷧ ⥋⥕ⷧ (2.6) which in the case of a closed -flow system turns to be : ⥋⥂ = ⥁⥋⥀ ⽑ ⥗⥋⥃ ⥁⥋⥀ = ⥋⥂ ⽐ ⥗⥋⥃ (2.7) Eq. (2.7) is called the Gibbs equation and defines the relations between states of stable equilibrium : evolving from one condition of stable equilibrium to another of stable equilibrium, the state variables (⥂, ⥀, ⥃) must satisfy the Gibbs equation. Otherwise, the final state is no longer of stable equilibrium. The same result can also be obtained by rearrangin g the energy balance of a closed -flow system following a reversible process . In fact, consider a cylinder -piston system in which a gas at temperature ⥁ and pressure ⥗ receives reversibly energy by heat interaction at constant temperature and expands moving the piston against an environment at same pressure (Fig. 2.1a). Thermodynamic properties of fluids - Energy Conversion 9 of 109 Fig. 2.1 - Reference system formed by a frictionless piston in which both the expansion and the introduction of energy by heat interaction take place reversibly (a) and the case in which the expansion is not reversible (b) . The energy balance for the closed -flow system is: ⥋⤾ = ⥋⥂ ⽐ ⥋⥄ (2.8) where ⤾ is the energy transfer by heat interation (commonly said exchanged heat ) and ⥄ is the energy transfer by work interaction ( exchanged work ). Moreover, assuming a reversible process , entropy can be computed as : ⥋⥀ = ⥋⤾ ⥁ (2.9) while , under the reversibility assumption, the energy transfer by work interaction is : ⥋⥄ = ⥗⥋⥃ (2.10 ) Replacing Eqs. (2.9) and (2.10 ) in Eq. (2.8) and referring to a unit weight: ⥋⥜ = ⥁⥋⥚ ⽑ ⥗⥋⥝ (2.11 ) This relation may seem to be valid only along reversible transformations because of the way it has been derive d here . Its validity applies to the general case, as it relates ultimately change s in state quantities. This statement may also be proven by examining the terms of an irreversible transformation, considering a pressure inside the cylinder higher than the external one , ⥗ > ⥗⷟ⷫⷠ . In this case, t he energy balance can thus be formulated as follows: ⧧⥘ = ⥋⥜ ⽐ ⧧⥞ = ⥋⥜ ⽐ ⥗⷟ⷫⷠ ⥋⥝ (2.12 ) in which the work actually exchanged with the outside environment is expressed as the product of the external pressure , not internal! , by the volume increase. Furthermore, t he differential entropy for an irreversible transformation can be rewritten as: ⥋⥚ = ⧧⥘ ⥁ ⽐ ⥋⥚ ⷧⷰⷰ (2.13 ) which indicates that the change in entropy of the system comprises two terms: the first term takes into account the change in entropy due to the heat interaction , ⧧⥘ ⥁▯ ; the second term corresponds to the internal generation of entropy due to the irreversibility , ⥋⥚ ⷧⷰⷰ . Rearranging : Thermodynamic properties of fluids - Energy Conversion 10 of 109 ⧧⥘ = ⥁(⥋⥚ ⽑ ⥋⥚ ⷧⷰⷰ ) (2.14 ) that replaced in (2.12 ) becomes : ⥁(⥋⥚ ⽑ ⥋⥚ ⷧⷰⷰ )= ⥋⥜ ⽐ ⥗⷟ⷫⷠ ⥋⥝ (2.15 ) or : ⥋⥜ = ⥁⥋⥚ ⽑ (⥁⥋⥚ ⷧⷰⷰ ⽐ ⥗⷟ⷫⷠ ⥋⥝ ) (2.16 ) Now let u s calculate the entropy generation , ⥋⥚ ⷧⷰⷰ . The wasted work in the irreversible transformation (a class notes is dedicated to this topic, comprising irreversibility and wasted work) is the difference between the internal work of the expansion , ⥗⥋⥝ , and the work that the external environment can receive, ⥗⥋⥝ . In general, this difference will t ransform into kinetic energy of the particles of the fluid. In accordance with the fact that the transformation is irreversible, the kinetic energy degrades into internal energy. The dissipation leads to an increase in entropy given by: ⥋⥚ ⷧⷰⷰ = (⥗⥋⥝ ⽑ ⥗⷟ⷫⷠ ⥋⥝ ) ⥁ (2.17 ) from which: ⥁⥋⥚ ⷧⷰⷰ = (⥗⥋⥝ ⽑ ⥗⷟ⷫⷠ ⥋⥝ ) (2.18 ) that replaced in (2.16 ) still supplies the (2.11 ) whose overall validity is proven . Therefore, it can be considered a thermodynamic identity valid for a general case . ⥋⥜ = ⥁⥋⥚ ⽑ ⥗⥋⥝ (2.19 ) Let us introduce the thermodynamic state variable specific enthalpy , ⥏, and its differential, ⥋⥏: ⥏ = ⥜⽐ ⥗⥝ (2.20 ) ⥋⥏ = ⥋⥜ ⽐ ⥗⥋⥝ ⽐ ⥝⥋⥗ (2.21 ) Replacing in (2.11 ): ⥋⥏ = ⥁⥋⥚ ⽑ ⥗⥋⥝ ⽐ ⥗⥋⥝ ⽐ ⥝⥋⥗ = ⥁⥋⥚ ⽐ ⥝⥋⥗ (2.22 ) Now it is possible to introduce the following definitions: ⥊ⷴ⠨ (⧽⥜ ⧽⥁ ) ⷴ ⊠⊝⊒⊐⊖⊓⊖⊐ ⊕⊒⊎⊡ ⊎⊡ ⊐⊜⊛⊠⊡⊎⊛⊡ ⊣⊜⊙⊢⊚⊒ (2.23 ) ⥊ⷮ⠨ (⧽⥏ ⧽⥁ ) ⷮ ⊠⊝⊒⊐⊖⊓⊖⊐ ⊕⊒⊎⊡ ⊎⊡ ⊐⊜⊛⊠⊡⊎⊛⊡ ⊝⊟⊒⊠⊠⊢⊟⊒ (2.24 ) ⧤ⷮ⠨ ╾ ⥝(⧽⥝ ⧽⥁ ) ⷮ ⊖⊠⊜⊏⊎⊟⊖⊐ ⊒⊥⊝⊎⊛⊠⊖⊜⊛ ⊐⊜⊒⊓⊓⊖⊐⊖⊒⊛⊡ (2.25 ) ⥒ⷘ⠨ ⽑ ╾ ⥝(⧽⥝ ⧽⥗ ) ⷘ ⊖⊠⊜⊡⊕⊒⊟⊚⊎⊙ ⊐⊜⊚⊝⊟⊒⊠⊠⊖⊏⊖⊙⊖⊡⊦ ⊐⊜⊒⊓⊓⊖⊐⊖⊒⊛⊡ (2.26 ) Thermodynamic properties of fluids - Energy Conversion 11 of 109 It is also possible to define two other state variables, i.e. the specific Helmholtz free energy ⥈ and the Gibbs free energy . ⥈ = ⥜⽑ ⥁⥚ (2.27 ) ⥎ = ⥏⽑ ⥁⥚ (2.28 ) Upon differenti ati ng , ⥈ and ⥎ become : ⥋⥈ = ⥁⥋⥚ ⽑ ⥗⥋⥝ ⽑ ⥁⥋⥚ ⽑ ⥚⥋⥁ = ⽑⥗⥋⥝ ⽑ ⥚⥋⥁ (2.29 ) ⥋⥎ = ⥁⥋⥚ ⽐ ⥝⥋⥗ ⽑ ⥁⥋⥚ ⽑ ⥚⥋⥁ = ⥝⥋⥗ ⽑ ⥚⥋⥁ (2.30 ) which are thermodynamic identities that will be widely used in the following discussion. It is now possible to derive the first two Maxwell ’s relations , which relate the thermal behavior to the volumetric behavior of the fluid , from the derivatives of ⥈ and ⥎. Helmholtz free energy ⥈ = ⥜⽑ ⥁⥚ By deriving the expression of ⥈: ⥋⥈ = ⽑⥗⥋⥝ ⽑ ⥚⥋⥁ Hence, considering a transformation at constant volume : (⧽⥈ ⧽⥁ ) ⷴ = ⽑⥚ (2.31 ) and a transformation at constant temperature : (⧽⥈ ⧽⥝ ) ⷘ = ⽑⥗ (2.32 ) Again deriving (2.31 ) and (2.32 ): ( ⧽ⵁ⥈ ⧽⥁⧽⥝ ) ⷴ⏬ⷘ = ⽑ (⧽⥚ ⧽⥝ ) ⷘ (2.33 ) ( ⧽ⵁ⥈ ⧽⥝⧽⥁ ) ⷘ⏬ⷴ = ⽑ (⧽⥗ ⧽⥁ ) ⷴ (2.34 ) Since ⥈ is a state function, Schwarz’s theorem on the symmetry of second order derivatives applie s, making (2.33 ) equal to (2.34 ): (⧽⥚ ⧽⥝ ) ⷘ = (⧽⥗ ⧽⥁ ) ⷴ (2.35 ) which is the first Maxwell ’s relation . Gibbs free energy ⥎ = ⥏⽑ ⥁⥚ By deriving the expression of ⤴: ⥋⥎ = ⥝⥋⥗ ⽑ ⥚⥋⥁ Hence, considering a transformation at constant pressure : (⧽⥎ ⧽⥁ ) ⷮ = ⽑⥚ (2.36 ) and a transformation at constant temperature : (⧽⥎ ⧽⥗ ) ⷘ = ⥝ (2.37 ) Again deriving (2.36 ) and (2.37 ): ( ⧽ⵁ⥎ ⧽⥁⧽⥗ ) ⷮ⏬ⷘ = ⽑ (⧽⥚ ⧽⥗ ) ⷘ (2.38 ) ( ⧽ⵁ⥎ ⧽⥗⧽⥁ ) ⷘ⏬ⷮ = (⧽⥝ ⧽⥁ ) ⷮ (2.39 ) Since ⥎ is a state function, Schwarz’s theorem on the symmetry of second order derivatives applie s, making (2.38 ) equal to (2.39 ): ⽑ (⧽⥚ ⧽⥗ ) ⷘ = (⧽⥝ ⧽⥁ ) ⷮ (2.40 ) which is the second Maxwell ’s relation . Thermodynamic properties of fluids - Energy Conversion 12 of 109 These relations are of fundamental importance for the analysis of properties of real fluids . They are useful to get the general definition of the state fu nction differentials , such as specific volume, enthalpy and entropy as a function of measurable quantity differential ⥋⥁ , ⥋⥗ and volumetric behavior of the fluid. One practical example is shown in Appendix 10.1 . The differential of specific volume as a function of temperature and pressure, ⥝ = ⥝(⥁⏬⥗), is : ⥋⥝ = (⧽⥝ ⧽⥁ ) ⷮ ⥋⥁ ⽐ (⧽⥝ ⧽⥗ ) ⷘ ⥋⥗ (2.41 ) or: ⥋⥝ = ⥝⧤ⷮ⥋⥁ ⽑ ⥝⥒ⷘ⥋⥗ (2.42 ) From the formula of the differential of ⥝ – see Eq. (2.42 ) – by setting ⥋⥝ = ╽ it is possible to obtain: (⧽⥗ ⧽⥁ ) ⷴ = ⧤ⷮ ⥒ⷘ (2.43 ) Moreover, the differential of entropy as a function of temperature and pressure, ⥚= ⥚(⥁⏬⥝), is : ⥋⥚ = (⧽⥚ ⧽⥁ ) ⷴ ⥋⥁ ⽐ (⧽⥚ ⧽⥝ ) ⷘ ⥋⥝ (2.44 ) that substitut ed into the thermodynamic relation ⥋⥜ = ⥁⥋⥚ ⽑ ⥗⥋⥝ gives : ⥋⥜ = ⥁(⧽⥚ ⧽⥁ ) ⷴ ⥋⥁ ⽐ ⥁(⧽⥚ ⧽⥝ ) ⷘ ⥋⥝ ⽑ ⥗⥋⥝ = ⥁(⧽⥚ ⧽⥁ ) ⷴ ⥋⥁ ⽐ [⥁(⧽⥚ ⧽⥝ ) ⷘ ⽑ ⥗]⥋⥝ (2.45 ) where by replacing the first Maxwell relation (⸬ⷱ ⸬ⷴ )ⷘ= (⸬ⷮ ⸬ⷘ )ⷴ and considering that ⥁(⸬ⷱ ⸬ⷘ )ⷴ= ⥊ⷴ: ⥋⥜ = ⥊ⷴ⥋⥁ ⽐ [⥁(⧽⥗ ⧽⥁ ) ⷴ ⽑ ⥗]⥋⥝ (2.46 ) Replacing Eq. (2.43 ) into Eq. (2.46 ): ⥋⥜ = ⥊ⷴ⥋⥁ ⽐ ⥝(⥁⧤ⷮ ⥒ⷘ ⽑ ⥗)(⧤ⷮ⥋⥁ ⽑ ⥒ⷘ⥋⥗ ) = [⥊ⷴ⽑ ⥝⧤ⷮ ⥒ⷘ (⥗⥒ⷘ⽑ ⥁⧤ⷮ)]⥋⥁ ⽐ ⥝(⥗⥒ⷘ⽑ ⥁⧤ⷮ)⥋⥗ (2.47 ) that is the differential of the state function ⥜(⥁⏬⥗). A similar procedure can be followed for enthalpy , ⥋⥏ = ⥁⥋⥚ ⽐ ⥝⥋⥗ , differentiating ⥚(⥁⏬⥗): ⥋⥏ = ⥁(⧽⥚ ⧽⥁ ) ⷮ ⥋⥁ ⽐ [⥁(⧽⥚ ⧽⥗ ) ⷘ ⽐ ⥝]⥋⥗ (2.48 ) Thermodynamic properties of fluids - Energy Conversion 13 of 109 For the second Maxwell relation ⽑ (⸬ⷱ ⸬ⷮ )ⷘ= (⸬ⷴ ⸬ⷘ )ⷮ and considering that ⥁(⸬ⷱ ⸬ⷘ)ⷮ= ⥊⥗: ⥋⥏ = ⥊ⷮ⥋⥁ ⽐ [⽑⥁(⧽⥝ ⧽⥁ ) ⷮ ⽐ ⥝]⥋⥗ (2.49 ) Recalling that the isobaric expansion coefficient is ⧤ⷮ= ⵀ ⷴ(⧽⥝ ⧽⥁)⥗ leads to: ⥋⥏ = ⥊ⷮ⥋⥁ ⽐ ⥝[╾⽑ ⥁⧤ⷮ]⥋⥗ (2.50 ) that is the differential of enthalpy as function of temperature and pressure ⥏(⥁⏬⥗). From this, by recalling the thermodynamic Tds relation ⥋⥏ = ⥁⥋⥚ ⽐ ⥝⥋⥗ , a formulation for the differential of entropy ⥋⥚ is obtained: ⥋⥚ = ⥋⥏ ⥁ ⽑ ⥝⥋⥗ ⥁ = ⥊ⷮ⥋⥁ ⥁ ⽐ ⥝⥋⥗ ⥁ ⽑ ⥝⥁ ⑈ⷮ⥋⥗ ⥁ ⽑ ⥝⥋⥗ ⥁ = ⥊ⷮ⥋