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Principles of Energy Conversion Jeffrey S. Allen Part 6. Review of Engineering Thermodynamic September 15, 2016 Copyright © 2016 by J. S. Allen 1 What is Thermodynamics? 1.1 Typical Engineering Thermodynamic Definitions . . . . . . . . . . . . . . . 1.2 Our Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 2 2 Thermodynamic Energy Types & Classifications 2.1 Closed and Open Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Efficiency and the First Law of Thermodynamics . . . . . . . . . . . . . . . 2.3 Energy Flux Through A Surface . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 5 5 3 Heat Engines 3.1 Newcomen’s Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Watt’s Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Evans’ Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6 6 6 4 Work from Steam 4.1 Measuring Pressure and Volume - Engine Indicators . . . . . . . . . . . . . 4.2 Boundary Motion Work & Polytropic Processes . . . . . . . . . . . . . . . . 7 8 9 5 Steam from Heat 5.1 Caloric Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 10 6 Energy to/from a Fluid 6.1 Measures of Mass . . . . . . . . . . . . . . 6.2 But what is ∆e? . . . . . . . . . . . . . . 6.3 Internal Energy (U, u) . . . . . . . . . . . 6.3.1 Joule’s Experiment . . . . . . . . . 6.3.2 Definition of Internal Energy . . . 6.4 Enthalpy . . . . . . . . . . . . . . . . . . . 6.4.1 Flow Potential vs Boundary Work 6.4.2 Definition of Enthalpy . . . . . . . . . . . . . . . 12 12 12 14 14 14 15 15 16 7 Measuring Changes in Energy 7.1 Specific Heats (c, cv , cp ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Measurable Properties and Changes in Energy . . . . . . . . . . . . . . . . . 7.2.1 State Property Relationships . . . . . . . . . . . . . . . . . . . . . . 17 17 21 21 References 23 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What is Thermodynamics? “Thermodynamics is a funny subject. The first time you go through it, you don’t understand it at all. The second time you go through it, you think you understand it, except for one or two points. The third time you go through it you know you don’t understand, but by that time you are so used to the subject, it doesn’t bother you any more.” – Arnold Sommerfield “In lecturing on any subject, it seems to be the natural course to begin with a clear explanation of the nature, purpose, and scope of the subject. But in answer to the question ‘what is thermo-dynamics?’ I feel tempted to reply ‘It is a very difficult subject, nearly, if not quite, unfit for a lecture.” – Osborn Reynolds On the General Theory of Thermo-dynamics, November, 1883 “Thermodynamics” was first used in a publication by Lord Kelvin in 1849. The word thermodynamics comes from the Greek language: therme ≡ heat } “power of heat” dynamis ≡ power Typical Engineering Thermodynamic Definitions → the science of the relationship between heat and work and the substances related to the interaction of heat and work → the science of energy and entropy → “Thermodynamics is the science and changes in state of physical systems and the interactions between systems which may accompany changes in state. . . . Classical thermodynamics is concerned with changes between equilibrium states.” Kestin Our Definition → conversion of transitional thermal and mechanical energy via energy exchange with a fluid; a subset of the field of energy conversion 2 Thermodynamic Energy Types & Classifications Energy Classification Energy Type Transitional Stored mechanical work inertial pressure gravitational thermal heat internal energy sensible heat latent heat Closed and Open Systems Closed System – no mass flow in or out of system work heat energy state 1 time = t1 energy state 2 time = t2 t1 < t < t2 Open System – mass flow in and/or out of system h e a t flu x , Q m 1 e (1 ) (2 ) 1 m 2 e 2 p o w e r, W With a Closed System, we are analyzing a fixed mass. An Open System, however, is a volume in space with mass passing through. Conservation Laws are only strictly applicable to a Closed System (fixed mass, inertial frame of reference). Therefore, Conservation of Energy on an Open System has an extra energy term associated with mass flow in and out of the volume to compensate for the non-fixed mass. The extra energy term will be referred to as flow energy and it looks the same as boundary work. Though not identical, the two are equivalent mathematically. 3 Steady Energy Balance For a steady-state, open system: ∆Q − ∆W = ∆E = (Ep + Ek + Ei + Ef )∣out − (Ep + Ek + Ei + Ef )∣in ∆Q: ∆W : ∆E : heat into system work produced by system change in system energy potential energy = mzg /gc kinetic energy = mV 2 /2gc internal energy = mu flow energy = P∀ = mP/ρ Ep : Ek : Ei : Ef : This is the first law of thermodynamics, which is really: change in transitional energy ⇐⇒ change in stored energy Steady implies that there is no energy or mass accumulation within the system. That is different than the exchange of energy in the mass which passes through the system. The mass which passes through may accumulate energy. First Law of Thermodynamics Steady, rate form of the 1st Law:1 ∑ Q̇ − ∑ Ẇ = ∑ ṁe − ∑ ṁe out m 2 e 2 (2 ) (3 ) m 1 e 1 (1) in (1 ) h e a t flu x , Q m 3 e 3 p o w e r, W Heat (Q̇) & Work (Ẇ ) ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ Not the same as useful heat and work The 1st Law of Thermodynamics applies to the fluid! 1 The sign convention on heat (Q) and work (W ) may change. For example, some thermodynamic textbooks will write the first law of thermodynamics (equation 1) as ∆Q + ∆W , in which case ∆W is the work into the system. The choice of sign convention will vary between engineering disciplines as well. It is not important which sign convention you use as long as you are consistent. The meaning of a positive or negative work can always be ascertained by examining how the stored energy term (∆E ) changes. 4 Efficiency and the First Law of Thermodynamics Conservation of Energy, and by corollary 1st Law of Thermodynamics, demands that the total change in stored energy is balanced by the transitional energies to/from the energy converter of interest. In practice, an imbalance in heat, work and energy change is nearly always observed. Some of the energy put into the system is converted into something other than useful work. The ratio defines efficiency. η= energy sought actual power output Ẇuseful = = energy cost ideal power output ∆Ė Energy Flux Through A Surface The sign convention for ∆E is based on the definition of flux through a surface. 1 Q̇2 − 1 Ẇ2 = +Ėout − Ėin 5 Heat Engines Newcomen’s Engine – 1707 W o s c illa tin g b e a m P Work from piston: p is to n & c o n d e n se r b o ile r m∆e = Wout + {Qout + Wlost } Q a tm Q o u t c o ld w a te r in d ra in to p u m p Watt’s Engine – 1764 W o s c illa tin g b e a m P b o ile r Work from piston: a tm p is to n c o ld w a te r m∆e = Wout + {Wlost } Q to p u m p in Q c o n d e n se r o u t d ra in Evans’ Engine – 1805 W for the cycle: P ∆e = 0 ∑ Q = ∑ Wout + ∑ Wlost b o ile r p is to n Work from piston: m∆e = Wout + {Wlost } Q a tm p u m p p re b o ile r in 6 to v a c u u m p u m p Q o u t c o n d e n se r Work from Steam Work is the expansion (or contraction) of a pressurized gas or vapor. From the perspective of a steam cylinder, work is related to pressure P and volume ∀. Work = F ∆x = (Pnet Apiston ) ( ∆∀ ) = Pnet ∆∀ Apiston Pnet is the net pressure difference across the piston. For the atmospheric engines, the pressure difference across the slow moving cylinders was equal to Patmospheric − Psaturation which was essentially constant. Psaturation is the pressure at which the steam condenses (or evaporates). Work (1 ) D x (2 ) = (Patm − Psat ) ∆∀ = (Patm − Psat ) Apiston ∆x v a p o r The work extracted from the atmosphere could be measured by the force F exerted from the piston and the piston displacement ∆x during the power stroke. After Evan’s engine development, the work was extracted from pressurized steam instead of the atmosphere. Thus, the net pressure difference, Psteam − Patm was not constant during the power stroke and the work during a power stroke required integration of the pressure and volume. W =∫ 2 Pnet d∀ 1 As engine technology improved, the speed of engines increased; becoming too fast to watch a pressure gauge or manometer and record the pressures as a function of piston position. Some sort of device was needed to measure pressure and volume and correlate the two values. Such a device was developed in the late 1700’s. The theory was of an indicator diagram, P versus ∀, was proposed by Davies Gilbert in 1792. In 1796, John Southern, who was an associate of Watt, invents the engine indicator. [? ] 7 Measuring Pressure and Volume - Engine Indicators Steam engine indicators from Peabody [1]. Indicator Plots The indicator plot is a pressurevolume plot with pressure on the vertical axis and volume on the horizontal axis. The work is found by integrating over the area swept by the indicator. W = ∫ Pd∀ The procedure was to measure height, P, of thin, d∀, vertical bands and sum lengths. The proper scaling had to be applied to convert the indicator lengths to pressure and volume. The steam pressure becomes less than back pressure from 6 to 10. Peabody [1] Modern Engines Steam engine indicators are only suitable for low-rpm engines because of the inertia of the indicator components, such as the oscillating drum. As the engine runs faster, the indicator does not accurately reflect pressure and volume in the cylinder. Today, we use crank angle sensors and dynamic pressure transducers to measure pressure and volume. The sensor signals are captured via high speed data acquisition hardware. 8 Boundary Motion Work & Polytropic Processes Engineers found that when the pressure-volume traces were replotted on log-log axis, that the power and compression strokes were linear. The power and compression strokes being the portion of the P-∀ trace which represented work to or from the fluid. log P = −a log ∀ + b ← linear relationship on log-log plot P∀a = e b (constant) The value of the polytropic coefficient, a, is a measure of the efficiency the process. Often we approximate actual expansion processes with idealizations by setting the polytropic coefficient a to specific values. For example, if the expansion process can be assumed to occur at constant temperature, then a = 1. We will discuss the implications of various idealized polytropic processes later. 1 W2 =∫ ∀2 ∀1 Pd∀ For P∀a = constant, integration results in an analytical expression for work based on the starting and ending fluid states. P2 ∀2 − P1 ∀1 1−a a ≠ 1, 1 W2 = a = 1, 1 W2 = P1 ∀1 ln ( ∀2 ) ∀1 Typical pressure-volume trace for an Otto cycle; spark-ignition internal combustion engine [2]. work process polytropic coefficent work constant pressure (isobaric) 0 P(∀2 − ∀1 ) constant volume (isochoric) ∞ 0 constant temperature (isothermal) 1 P1 ∀1 ln(∀2 /∀1 ) constant entropy (isentropic) k ≡ cp /cv (P2 ∀2 − P1 ∀1 )/(k − 1) 9 Steam from Heat {Heat} = {Change in Fluid Energy} v a p o r Q = ∆E p h a se c h a n g e liq u id h e a t The change in fluid energy, ∆E , was considered different than that which occurred during the work on the piston. The concept of energy and the relationship between heat and work took a long time to become established. Caloric Theory Up until the mid-1800’s, the relationship between heat, work, and energy exchange with a fluid (steam or gas) was not understood. Work was known to be related to pressure and volume, but heat added to the fluid from which work was extracted was a mystery. Two theories were originally put forth to explain heat. 1. Live Force: Heat is a level of randomness of atoms and molecules in a substance; heat is a form of kinetic energy of very small particles. In other words, heat is manifestation of motion at the molecular level. 2. Caloric: Heat is a fluid-like substance that is massless, colorless, oderless, and tasteless. Caloric was a substance that had to be conserved; as such, an important principle of caloric theory is that work can not be converted into heat because work is not a substance. Caloric theory was proposed by French chemist Antoine Lavosier in 1789 and was the prevailing theory until the mid-1800’s. Caloric is a substance and can be “poured” from one body to another. Since it is a substance, caloric is subject to conservation laws; i.e., caloric is conserved. When caloric is added to a body, the temperature of that body increases. Remove caloric and the body’s temperature decreases. “When a body could not contain any more caloric, much the same way as when a glass of water could not dissolve any more salt or sugar, the body was said to be saturated with caloric. The interpretation gives rise to the terms saturated liquid, saturated vapor, and heat flow that are still in use today.” [? ] There were many challenges to caloric theory, • American Engineer Benjamin Thompson (Count Rumsford, 1754–1814) – continuous heating due to friction during boring of cannons “demonstrated categorically” that caloric theory was not valid – science community remained unconvinced • English Scientist James P. Joule (1818–1889) – published experimental results in 1843 that “definitively” proved that heat is not a substance subject to laws of conservation 10 – science community remained unconvinced • German Physician Robert Mayer (1814–1878) – in 1840’s, proposed heat and work were different forms of the same thing – science community remained unconvinced • Joule and Kelvin performed a series of experiments between 1850 and 1860 which quantified the relationship between heat and work – took another 50 years for caloric theory to pass 11 Energy to/from a Fluid The piston work is related to energy transfer from the steam (fluid). Heat + Work ∼ change in fluid energy, E 1Q2 − 1W2 = E2 − E1 The amount of energy is mass dependent (extrinsic). At a given temperature, the total energy in a glass of water depends on the mass of water present. Here, extrinsic properties are denoted by capital letters such as energy E , internal energy U, and volume ∀. Mass independent properties (intrinsic) are denoted by lower case letters, e.g. specific energy e, specific internal energy u, and specific volume v . Measures of Mass There are several means to quantify mass during energy conversion processes. For closed process, mass m is often used directly. The total mass in a closed system is proportional to the volume of the system, m ∼ ∀. The constant of proportionality is density ρ, which has units of mass per volume [kg/m3 ]. The inverse of density is specific volume v with units of [m3 /kg]. For an open system (mass flow in and out), density is more convenient when considering momentum exchange. Thus, in fluid mechanics, density is a preferred measure of mass. When considering energy exchange, specific volume is more convenient since a key factor is the pressurevolume product. Thus, in thermodynamics with open systems, specific volume is the preferred measure of mass. Other commonly used measures of mass are specific gravity, SG = ρ/ρH20 , which is the fluid density normalized by the density of water at room temperature and specific weight, γ = ρg , where g is acceleration due to gravity. But what is ∆e? The term energy was first used by Thomas Young in 1807 to describe capillary phenomena. Energy, in a thermodynamic context, was explained in 1840 by William Thomson (a.k.a. Lord Kelvin). Modern engineering thermodynamics expand ∆E to include internal energy and a few other forms of stored energy. • Stored Thermal Energy – sensible heat ≡ internal energy, U,2 • Stored Mechanical Energy – potential (gravitational), mgz, – kinetic (inertial), mV 2 /2, 2 Internal energy is often described as sensible heat, which is a stored form of thermal energy. Strictly speaking, however, internal energy encompasses more than just the stored thermal energy component. We will attempt to separate the other components of internal energy, such as the chemical and electrical, from the thermal portion. Internal energy is not exactly a stored form of thermal energy, but it is convenient to categorize it as such for the purposes of this discussion. 12 – flow work potential –or– flow energy, P∀, • Stored Chemical Energy (only known form) – chemical potential – enthalpy of formation Mechanical energy is easy to conceptualize and measure. We require measures of mass, velocity, elevation, pressure and volume. Thermal energy, however, is not easy to conceptualize nor measure. Thermal energy can only be defined by a change that occurs during a process. 13 Internal Energy (U, u) The first law of thermodynamics is based on the principle that energy is a conservative property. As such, the first law is what defines a thermodynamic property of energy, or more correctly, a change in a thermodynamic property. Internal energy3 , denoted as U, is the intrinsic energy of a substance neglecting other forms of potential energy (gravitational, inertial, flow). Thus, for a closed system in which the mass is fixed, the energy change ∆E is ∆U. 1 Q2 − 1 W2 = U2 − U1 = m(u2 − u1 ) The concept of internal energy is somewhat intuitive. However, how do we measure internal energy? What other measurable properties is internal energy dependent upon? Joule’s Experiment Joule determined for gases that internal energy only depended upon temperature. His experiment consisted of two vessels in an insulated water bath. The vessels were left in the water bath for a long time so that the system was isothermal. The vessels were connected, but isolated from one another by a stopcock (c). One vessel (a) was filled with air at 2 atmospheres and the second vessel (b) was under vacuum. (c ) (a ) a ir a t 2 a tm (b ) v a c u u m in s u la te d w a te r b a th Joule’s experiment Joule opened the stopcock allowing the air to move from vessel (a) into the vessel (b) until the pressure equilibrated at 1 atmosphere. The water bath did not change temperature, but pressure and volume changed. There was no heat or work transfer, so the first law for this experiment is: 1Q2 − 1W2 = 0 = U2 − U1 Since pressure and volume changed and the temperature did not, his conclusion was that internal energy (for a gas) is only dependent upon temperature. u = f (T ) only! (ideal gas only) Definition of Internal Energy The first law applied to a closed system defines the change in internal energy, ∆U. The value of internal energy, U or u, at a given temperature is the change relative to some arbitrary state at which U = 0. 3 The term internal energy, coined by Clausius and Rankine, replaced terms such as inner work, internal work, and intrinsic energy. 14 Enthalpy Flow Potential vs Boundary Work w o r k c lo s e d s y s te m The first law of thermodynamics only applies to a fixed mass; i.e., a closed system. If we track a fixed mass through a turbine (modeled as an open system), there is boundary work associated with the compressed, high pressure gas moving to an uncompressed, low pressure region. A tu r b in e o p e n sy ste m A Another simple conceptualization is to examine the total state at time 1 and then the total state at time 2 as shown, where total state includes the gas volume. w o r k e n e r g y sta te 1 tim e = t 1 e n e r g y sta te 2 t1 < t < t tim e = t 2 2 w o r k A tu r b in e tu r b in e A Recall from the engine indicators, that the total work output of the fluid passing through the turbine is related to the change in P∀. −1 W2 = ∫ Pd∀ = E2 − E1 , where E2 −E1 is the change in stored energy of the fluid. For an atmospheric engine, ∫ Pd∀ = P2 ∀2 − P1 ∀1 Often, there is also an accompanying change in temperature related to a heat transfer to or from the fluid. Therefore, the total change in energy is becomes (from the 1st Law): Q − W = E2 − E1 = (U2 − U1 ) + (P2 ∀2 − P1 ∀1 ) 15 For continuous flow, the transitional energies become transitional power (thermal Q̇ and mechanical Ẇ ) and with conservation of mass applied to a single inlet and outlet the energy balance becomes Q̇ − Ẇ = ṁ [(u2 − u1 ) + (P2 v2 − P1 v1 )] Definition of Enthalpy In open systems, the combination of internal energy, pressure and volume always appear in the energy balance in the form of U + P∀. For convenience, engineers defined a new energy, commonly called a heat, which included internal energy plus flow potential energy. H = U + P∀ [kJ, BTU] and on a per mass basis h = u + Pv [kJ/kg, BTU/lbm] This property simplified looking up fluid energy values by reducing the three properties to a single property. This new property was called by many names including total heat, heat content, and heat of liquid. In the July 1936 issue of Mechanical Engineering, published by the American Society of Mechanical Engineers (ASME), the term enthalpy was selected to designate the thermodynamic function defined as internal energy plus the pressure volume product of any working substance. Enthalpy is from the Greek word “enthalpo”, which means “heat within”. The original symbol used to designate enthalpy was I , i. The use of I is still common nomenclature in the heating, ventilation and air conditioning (HVAC) industry. 16 Measuring Changes in Energy The first law of thermodynamics relates changes in transitional energies heat and work to stored energies of a fluid. For both open and closed systems, the measure of stored thermal energy in a fluid is related to the internal energy, U. Unfortunately, there is no direct measure of thermal fluid energy. We must infer changes in fluid energy from measurable parameters that include pressure P, temperature T , volume ∀ and mass m. Somehow we must relate these four measures to other measures of fluid energy: u internal energy, h enthalpy, and µ̂ chemical potential. Specific Heats (c, cv , cp ) In order to relate changes in thermal energy to something measurable, we must use Conservation of Energy and Mass. Beginning with simple compressible substances such as air or an ideal gas such as Helium, an experiment can be designed such that Conservation of Energy reduces to: Q − W = ∆Ũ = m∆ũ , ´¹¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ transitional (2) stored in gas where only work and heat for the transitional energies and only stored thermal energy are considered. An example of a simple experiment is shown in Figure 1. Heat is applied to a constant volume container holding a gas. Temperature of the gas is measured while heat is applied. The subsequent data shows a linear relationship between the amount of heat added and the change in temperature. qin = Qin = C ∆T , mgas where C is a proportionality constant, which is simply the slope of the experimental data. For this experiment, no work is applied. Combining the experimental observation with the reduced Conservation of Energy (equation 2) results in a relationship between internal energy and temperature. qin = ∆ũ = C ∆T For infinitesimal changes, δqin = d ũ = CdT For this scenario, the proportionality constant C is known as the specific heat at constant volume and labeled cv . A common misconception is that cv can only be used when considering constant volume processes. Specific heats relate measurable property changes, in this case temperature, to changes in stored energy. Thus, cv relates temperature changes 17 h e a t a p p lie d , q te m p e ra tu re g a s T 1 h e a t 0 o C te m p e ra tu re Figure 1. Conceptual experiment for determining relationship between stored thermal energy and temperature. (measurable) to changes in internal energy (not measurable) regardless of the process being considered. 18 19 20 Measurable Properties and Changes in Energy As a general rule, we can only measure changes in energy. There is no method by which we cam measure the absolute energy; we can only measure energy relative to another state or reference, and then only through inference. An example is the closed system heat addition to an ideal gas discussed in §7.1. The total energy transferred to the tank containing the ideal gas can be calculated by knowing the heat transfer rate, or power, and the duration of applying that power. Qin = Q̇in ∆t An example would be heat addition via an electric heater in which the electrical power is measured. From the First Law of Thermodynamics (Conservation of Energy), the heat input is equal to the change in the energy of the ideal gas: Qin = U2 − U1 = mgas (u2 − u1 ) As heat is added, the measured temperature of the gas increases in proportion to the quantity of heat. Qin ∼ T2 − T1 = c (T2 − T1 ) , where c is a constant of proportionality. This particular constant of proportionality between heat addition and temperature increase is known as specific heat. Thus, by carefully designing and controlling the process of energy change, which in this case is a constant volume (isochoric) process, the change in energy as expressed using the first law of thermodynamics can be related to a change in the measurable property temperature. State Property Relationships The primary properties that can be measured are pressure, temperature, volume and mass. Other useful properties can be derived from these four such as density (ρ = mass/volume) and specific volume (v = volume/mass). When there is no change in energy occurring, a state known as equilibrium, then the measurable properties can be related to one another. An example is the Ideal Gas Law, where pressure, volume, and mass are found to be proportional to temperature for simple compressible gases. P∀ ∼ T = RT , m where R is a constant of proportionality known as the gas constant. The ideal gas law is but one example of numerous relationships between properties of a substance in an equilibrium state, which again means that there is no change in energy occurring. Another example of such a property relationship is the Non-Ideal Gas Law. P∀ = Z mRT , where Z is an experimentally determined factor that accounts for non-ideal compression of more complex gases such as butane. Relationships between properties 21 measured during a state of equilibrium are collectively known as equations of state. A simple equation of state for an incompressible fluid (liquid) is m ∼ ∀ = ρ∀. In this instance ρ is a constant of proportionality that does not change with temperature, pressure, etc. In other words, this particular equation of state indicates that the fluid density is independent of all other properties. There are many equations of state for gases, liquids and solids. When a substance is near the transition between vapor-liquid, liquid-solid, or solid-gas, then defining an equation of state becomes extremely difficult. In these cases, where the equilibrium state is near a phase transition, the relationship between properties is experimentally measured and tabulated. Examples where tabulated property relationships are common include steam, refrigerants and gases that are near the boiling point such as hydrogen at 21 K. Steam tables include three sets of relationships: (a) vapor in equilibrium, known as superheated vapor which does not behave as an ideal gas; (b) vapor and liquid in equilibrium with one another, known saturated liquid/vapor; and (c) liquid in equilibrium, known as subcooled or compressed liquid which does not necessarily behave as an incompressible liquid. In contrast to an ideal gas, when a saturated liquid is heated there is a change in the internal energy, but that energy change is not accompanied by an increase temperature. . . . least is known about subcooled liquid finish section with process of determining energy changes The relationship between measurable properties and energies are developed using a combination of: 1. fluid property relationships 2. first law of thermodynamics 3. heat transfer and fluid mechanics relationships (pitot-static measurement, hot-wire anemometry, . . . ) 22 References [1] Cecil H. Peabody. Manual of the Steam-Engine Indicator. John Wiley & Sons, Inc., 1914. [2] Edward F. Obert. Internal Combustion Engines and Air Pollution. Harper & Row, Publishers, 1973. 23