The CRC Handbook of Mechanical Engineering. Chapter 2. Engineering Thermodynamics (776125), страница 8
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and Dugan, R.E. 1996. Engineering Thermodynamics, Prentice-Hall,Englewood Cliffs, NJ, based on data and formulations from Haar, L., Gallagher, J.S., and Kell, G.S. 1984. NBS/NRC Steam Tables. Hemisphere,Washington, D.C.)2-31© 1999 by CRC Press LLC2-32Section 23600100.10.31.01.00.03700°C2100.10υ=0.001 m 3/kg38003003004000600°C900°C500°C0.01800°C3400400°C3200300°C700°C00330000.200°C01m.0=0υ=1260000T=100°C600°CM3Pa/kg2800px=90%2400%802200%8020004567s, kJ/kg⋅K8910FIGURE 2.8 Enthalpy-entropy (Mollier) diagram for water.
(Source: Jones, J.B. and Dugan, R.E. 1996. EngineeringThermodynamics. Prentice-Hall, Englewood Cliffs, NJ, based on data and formulations from Haar, L., Gallagher,J.S., and Kell, G.S. 1984. NBS/NRC Steam Tables. Hemisphere, Washington, D.C.)pR =p,pcTR =T,Tcv R′ =(vRTc pc)(2.40)In these expressions, R is the universal gas constant and pc and Tc denote the critical pressure andtemperature, respectively.
Values of pc and Tc are given for several substances in Table A.9. The reducedisotherms of Figure 2.10 represent the best curves fitted to the data of several gases. For the 30 gasesused in developing the chart, the deviation of observed values from those of the chart is at most on theorder of 5% and for most ranges is much less.*Figure 2.10 gives a common value of about 0.27 for the compressibility factor at the critical point.As the critical compressibility factor for different substances actually varies from 0.23 to 0.33, the chartis inaccurate in the vicinity of the critical point.
This source of inaccuracy can be removed by restrictingthe correlation to substances having essentially the same Zc values. which is equivalent to including thecritical compressibility factor as an independent variable: Z = f (TR, pR, Zc). To achieve greater accuracy* To determine Z for hydrogen, helium, and neon above a T of 5, the reduced temperature and pressure shouldRbe calculated using TR = T/(Tc + 8) and PR = p/(pc + 8), where temperatures are in K and pressures are in atm.© 1999 by CRC Press LLC325 m /kgν=0.001nJoule-Thomson inversio5.00.5 1.02001.52.02.53.03.54.05.54.5700°C1100°C800°C1000°C900°C10080600°C406.5600°C100°C200°C300°C0.0020.001500°C7.0400°C20p, MPaT=1200°C6.07.510864Engineering Thermodynamics1000800600400300°C8.028.5200°C10.80.60.49.0x=10%20%30%40%50%60%70%80%x=909.50.2T=100°C0.10.080.0610.00.0410.50.02s=11.0 kJ/kg⋅K0.01010002000300040005000h, kJ/kg© 1999 by CRC Press LLC2-33FIGURE 2.9 Pressure-enthalpy diagram for water.
(Source: Jones, J.B. and Dugan, R.E. 1996. Engineering Thermodynamics. Prentice-Hall, EnglewoodCliffs, NJ, based on data and formulations from Haar, L., Gallagher, J.S., and Kell, G.S. 1984. NBS/NRC Steam Tables. Hemisphere, Washington, D.C.)2-34© 1998 by CRC Press LLCSection 2FIGURE 2.10 Generalized compressibility chart (TR = T/TC, pR = p/pC, v R¢ = vpC / RTC ) for pR £ 10. (Source: Obert, E.F. 1960 Concepts of Thermodynamics.McGraw-Hill, New York.)2-35Engineering Thermodynamicsvariables other than Zc have been proposed as a third parameter — for example, the acentric factor (see,e.g., Reid and Sherwood, 1966).Generalized compressibility data are also available in tabular form (see, e.g., Reid and Sherwood,1966) and in equation form (see, e.g., Reynolds, 1979).
The use of generalized data in any form (graphical,tabular, or equation) allows p, v, and T for gases to be evaluated simply and with reasonable accuracy.When accuracy is an essential consideration, generalized compressibility data should not be used as asubstitute for p-v-T data for a given substance as provided by computer software, a table, or an equationof state.Equations of StateConsidering the isotherms of Figure 2.10, it is plausible that the variation of the compressibility factormight be expressed as an equation, at least for certain intervals of p and T. Two expressions can bewritten that enjoy a theoretical basis. One gives the compressibility factor as an infinite series expansionin pressure,Z = 1 + Bˆ (T ) p + Cˆ (T ) p 2 + Dˆ (T ) p 3 + Kand the other is a series in 1/ v ,Z = 1+B(T ) C(T ) D(T )+ 2 + 3 +KvvvThese expressions are known as virial expansions, and the coefficients Bˆ , Cˆ Dˆ , … and B, C, D … arecalled virial coefficients.
In principle, the virial coefficients can be calculated using expressions fromstatistical mechanics derived from consideration of the force fields around the molecules. Thus far onlythe first few coefficients have been calculated and only for gases consisting of relatively simple molecules.The coefficients also can be found, in principle, by fitting p-v-T data in particular realms of interest.Only the first few coefficients can be found accurately this way, however, and the result is a truncatedequation valid only at certain states.Over 100 equations of state have been developed in an attempt to portray accurately the p-v-T behaviorof substances and yet avoid the complexities inherent in a full virial series.
In general, these equationsexhibit little in the way of fundamental physical significance and are mainly empirical in character. Mostare developed for gases, but some describe the p-v-T behavior of the liquid phase, at least qualitatively.Every equation of state is restricted to particular states. The realm of applicability is often indicated bygiving an interval of pressure, or density, where the equation can be expected to represent the p-v-Tbehavior faithfully. When it is not stated, the realm of applicability often may be approximated byexpressing the equation in terms of the compressibility factor Z and the reduced properties, and superimposing the result on a generalized compressibility chart or comparing with compressibility data fromthe literature.Equations of state can be classified by the number of adjustable constants they involve.
The RedlichKwong equation is considered by many to be the best of the two-constant equations of state. It givespressure as a function of temperature and specific volume and thus is explicit in pressure:p=RTa−v − b v (v + b)T 1 2(2.41)This equation is primarily empirical in nature, with no rigorous justification in terms of moleculararguments.
Values for the Redlich-Kwong constants for several substances are provided in Table A.9.Modified forms of the equation have been proposed with the aim of achieving better accuracy.© 1999 by CRC Press LLC2-36Section 2Although the two-constant Redlich-Kwong equation performs better than some equations of statehaving several adjustable constants, two-constant equations tend to be limited in accuracy as pressure(or density) increases.
Increased accuracy normally requires a greater number of adjustable constants.For example, the Benedict-Webb-Rubin equation, which involves eight adjustable constants, has beensuccessful in predicting the p-v-T behavior of light hydrocarbons. The Benedict-Webb-Rubin equationis also explicit in pressure,p=()bRT − aγγRT C 1aαc+ BRT − A − 2 2 ++ 6 + 3 2 1 + 2 exp − 2 v vT vv3vv T v (2.42)Values of the Benedict-Webb-Rubin constants for various gases are provided in the literature (see, e.g.,Cooper and Goldfrank, 1967). A modification of the Benedict-Webb-Rubin equation involving 12constants is discussed by Lee and Kessler, 1975. Many multiconstant equations can be found in theengineering literature, and with the advent of high speed computers, equations having 50 or moreconstants have been developed for representing the p-v-T behavior of different substances.Gas MixturesSince an unlimited variety of mixtures can be formed from a given set of pure components by varyingthe relative amounts present, the properties of mixtures are reported only in special cases such as air.Means are available for predicting the properties of mixtures, however.
Most techniques for predictingmixture properties are empirical in character and are not derived from fundamental physical principles.The realm of validity of any particular technique can be established by comparing predicted propertyvalues with empirical data. In this section, methods for evaluating the p-v-T relations for pure componentsare adapted to obtain plausible estimates for gas mixtures. The case of ideal gas mixtures is discussedin Section 2.3, Ideal Gas Model.
In Section 2.3, Multicomponent Systems, some general aspects ofproperty evaluation for multicomponent systems are presented.The total number of moles of mixture, n, is the sum of the number of moles of the components, ni:jn = n1 + n2 + Kn j =∑n(2.43)ii =1The relative amounts of the components present can be described in terms of mole fractions. The molefraction yi of component i is yi = ni /n.
The sum of the mole fractions of all components present is equalto unity. The apparent molecular weight M is the mole fraction average of the component molecularweights, such thatjM =∑y Mi(2.44)ii =1The relative amounts of the components present also can be described in terms of mass fractions: mi /m,where mi is the mass of component i and m is the total mass of mixture.The p-v-T relation for a gas mixture can be estimated by applying an equation of state to the overallmixture.
The constants appearing in the equation of state are mixture values determined with empiricalcombining rules developed for the equation. For example, mixture values of the constants a and b foruse in the Redlich-Kwong equation are obtained using relations of the forma=© 1999 by CRC Press LLCj∑i =12ya ,12i ijb=∑y bi ii =1(2.45)2-37Engineering Thermodynamicswhere ai and bi are the values of the constants for component i.
Combination rules for obtaining mixturevalues for the constants in other equations of state are also found in the literature.Another approach is to regard the mixture as if it were a single pure component having criticalproperties calculated by one of several mixture rules. Kay’s rule is perhaps the simplest of these, requiringonly the determination of a mole fraction averaged critical temperature Tc and critical pressure pc :jTc =∑jyi Tc,i ,pc =i =1∑y pic,i(2.46)i =1where Tc,i and pc,i are the critical temperature and critical pressure of component i, respectively. UsingTc and pc, the mixture compressibility factor Z is obtained as for a single pure component. The unkownquantity from among the pressure p, volume V, temperature T, and total number of moles n of the gasmixture can then be obtained by solving Z = pV/n R T.Additional means for predicting the p-v-T relation of a mixture are provided by empirical mixturerules.
Several are found in the engineering literature. According to the additive pressure rule, the pressureof a gas mixture is expressible as a sum of pressures exerted by the individual components:]p = p1 + p2 + p3 K T ,V(2.47a)where the pressures p1, p2, etc. are evaluated by considering the respective components to be at thevolume V and temperature T of the mixture.
The additive pressure rule can be expressed alternatively asjZ=∑ y Z i(2.47b)ii =1T ,Vwhere Z is the compressibility factor of the mixture and the compressibility factors Zi are determinedassuming that component i occupies the entire volume of the mixture at the temperature T.The additive volume rule postulates that the volume V of a gas mixture is expressible as the sum ofvolumes occupied by the individual components:]V = V1 + V2 + V3 Kp,T(2.48a)where the volumes V1, V2, etc.
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