The CRC Handbook of Mechanical Engineering. Chapter 2. Engineering Thermodynamics (776125), страница 11
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An energy balance reduces to give Q˙ cv / m˙ = W˙ cv / m˙ = –135.3 kJ/kg. (c) For an isentropiccompression, Q̇cv = 0 and an energy rate balance reduces to give W˙ cv / m˙ = –(h2s – h1), where 2s denotesthe exit state. With Equation 2.59a and pr data, h2s = 464.8 kJ/kg (T2s = 463K). Then W˙ cv / m˙ = –(464.8– 293.2) = –171.6 kJ/kg. Area 1-2s-a-b on Figure 2.11 represents the magnitude of the work required,per unit of mass of air flowing.Ideal Gas MixturesWhen applied to an ideal gas mixture, the additive pressure rule (Section 2.3, p-v-T Relations) is knownas the Dalton model.
According to this model, each gas in the mixture acts as if it exists separately atthe volume and temperature of the mixture. Applying the ideal gas equation of state to the mixture asa whole and to each component i, pV = nRT, piV = ni RT , where pi, the partial pressure of componenti, is the pressure that component i would exert if ni moles occupied the full volume V at the temperatureT. Forming a ratio, the partial pressure of component i ispi =nip = yi pn(2.60)where yi is the mole fraction of component i. The sum of the partial pressures equals the mixture pressure.The internal energy, enthalpy, and entropy of the mixture can be determined as the sum of the respectiveproperties of the component gases, provided that the contribution from each gas is evaluated at thecondition at which the gas exists in the mixture.
On a molar basis,jU=∑n ui ij∑y uor u =i ii =1jH=∑jni hi or h =i =1∑∑y hi i(2.61b)i =1jS=(2.61a)i =1jni si or s =i =1∑y si i(2.61c)i =1The specific heats cv and c p for an ideal gas mixture in terms of the corresponding specific heats of thecomponents are expressed similarly:jcv =∑y ci vii =1© 1999 by CRC Press LLC(2.61d)2-50Section 2jcp =∑y c(2.61e)i pii =1When working on a mass basis, expressions similar in form to Equations 2.61 can be written using massand mass fractions in place of moles and mole fractions, respectively, and using u, h, s, cp, and cv inplace of u , h , s , c p , and cV , respectively.The internal energy and enthalpy of an ideal gas depend only on temperature, and thus the ui andhi terms appearing in Equations 2.61 are evaluated at the temperature of the mixture. Since entropydepends on two independent properties, the si terms are evaluated either at the temperature and thepartial pressure pi of component i, or at the temperature and volume of the mixture.
In the former casejS=∑ n s (T , p )i iii =1(2.62)j=∑ n s (T , x p)i iii =1Inserting the expressions for H and S given by Equations 2.61b and 2.61c into the Gibbs function,G = H – TS,jG=∑ni hi (T ) − Ti =1j∑ n s (T , p )i iii =1(2.63)j=∑ n g (T , p )i iii =1where the molar-specific Gibbs function of component i is gi(T, pi) = hi(T) – Tsi(T, pi). The Gibbs functionof i can be expressed alternatively asgi (T , pi ) = gi (T , p ′) + RT ln( pi p ′)= gi (T , p ′) + RT ln( xi p p ′)(2.64)were p′ is some specified pressure.
Equation 2.64 is obtained by integrating Equation 2.32d at fixedtemperature T from pressure p′ to pi.Moist AirAn ideal gas mixture of particular interest for many practical applications is moist air. Moist air refersto a mixture of dry air and water vapor in which the dry air is treated as if it were a pure component.Ideal gas mixture principles usually apply to moist air. In particular, the Dalton model is applicable, andso the mixture pressure p is the sum of the partial pressures pa and pv of the dry air and water vapor,respectively.Saturated air is a mixture of dry air and saturated water vapor.
For saturated air, the partial pressureof the water vapor equals psat(T), which is the saturation pressure of water corresponding to the dry-bulb(mixture) temperature T. The makeup of moist air can be described in terms of the humidity ratio (specifichumidity) and the relative humidity. The bulb of a wet-bulb thermometer is covered with a wick saturatedwith liquid water, and the wet-bulb temperature of an air-water vapor mixture is the temperature indicatedby such a thermometer exposed to the mixture.© 1999 by CRC Press LLC2-51Engineering ThermodynamicsWhen a sample of moist air is cooled at constant pressure, the temperature at which the samplebecomes saturated is called the dew point temperature. Cooling below the dew point temperature resultsin the condensation of some of the water vapor initially present.
When cooled to a final equilibriumstate at a temperature below the dew point temperature, the original sample would consist of a gas phaseof dry air and saturated water vapor in equilibrium with a liquid water phase.Psychrometric charts are plotted with various moist air parameters, including the dry-bulb and wetbulb temperatures, the humidity ratio, and the relative humidity, usually for a specified value of themixture pressure such as 1 atm. Further discussion of moist air and related psychrometric principles andapplications is provided in Chapter 9.Generalized Charts for Enthalpy, Entropy, and FugacityThe changes in enthalpy and entropy between two states can be determined in principle by correctingthe respective property change determined using the ideal gas model.
The corrections can be obtained,at least approximately, by inspection of the generalized enthalpy correction and entropy correction charts,Figures 2.12 and 2.13, respectively. Such data are also available in tabular form (see, e.g., Reid andSherwood, 1966) and calculable using a generalized equation for the compressibility factor (Reynolds,1979).
Using the superscript * to identify ideal gas property values, the changes in specific enthalpy andspecific entropy between states 1 and 2 are h * − h h* − h h2 − h1 = h2* − h1* − RTc − RTc 2 RTc 1 (2.65a) s * − s s* − s s2 − s1 = s2* − s1* − R − R 2 R 1 (2.65b)The first underlined term on the right side of each expression represents the respective property changeassuming ideal gas behavior. The second underlined term is the correction that must be applied to theideal gas value to obtain the actual value.
The quantities (h * − h ) / RTc and (s * − s ) / R at state 1 wouldbe read from the respective correction chart or table or calculated, using the reduced temperature TR1and reduced pressure pR1 corresponding to the temperature T1 and pressure p1 at state 1, respectively.Similarly, (h * − h ) / RTc and (s * − s ) / R at state 2 would be obtained using TR2 and pR2. Mixture valuesfor Tc and pc determined by applying Kay’s rule or some other mixture rule also can be used to enterthe generalized enthalpy correction and entropy correction charts.Figure 2.14 gives the fugacity coefficient, f/p, as a function of reduced pressure and reduced temperature. The fugacity f plays a similar role in determining the specific Gibbs function for a real gas aspressure plays for the ideal gas.
To develop this, consider the variation of the specific Gibbs functionwith pressure at fixed temperature (from Table 2.2)∂g =v∂p TFor an ideal gas, integration at fixed temperature givesg * = RT ln p + C(T )where C(T) is a function of integration. To evaluate g for a real gas, fugacity replaces pressure,© 1999 by CRC Press LLC2-52Section 20.507.0TR0.550.600.656.00.70Saturatedliquid0.750.800.905.00.850.920.940.960.981.004.01.020.750.800.850.901.04h* - hRTc0.921.151.080.941.201.060.961.101.253.00.981.301.401.502.01.601.701.801.902.201.02.002.402.60Saturatedvapor2.803.004.000-1.00.10.20.3 0.4 0.51.02.03.0 4.0 5.0102030Reduced pressure, pRFIGURE 2.12 Generalized enthalpy correction chart.
(Source: Adapted from Van Wylen, G. J. and Sonntag, R. E.1986. Fundamentals of Classical Thermodynamics, 3rd ed., English/SI. Wiley, New York.)g = RT ln f + C(T )In terms of the fugacity coefficient the departure of the real gas value from the ideal gas value at fixedtemperature is then© 1999 by CRC Press LLC2-53Engineering Thermodynamics10.09.08.00.50TR0.557.0Saturatedliquid0.600.656.00.70S* - SR0.750.805.00.850.750.800.850.900.900.920.924.00.960.940.980.941.001.021.040.963.00.981.061.081.101.152.01.201.301.401.501.0Saturatedvapor01.601.802.000.952.500.903.000.10.20.3 0.4 0.51.02.03.0 4.05.0102030Reduced pressure, pRFIGURE 2.13 Generalized entropy correction chart. (Source: Adapted from Van Wylen, G. J.
and Sonntag, R. E.1986. Fundamentals of Classical Thermodynamics, 3rd ed., English/SI. Wiley, New York.)g − g * = RT lnfp(2.66)As pressure is reduced at fixed temperature, f/p tends to unity, and the specific Gibbs function is givenby the ideal gas value.© 1999 by CRC Press LLC2-54Section 21.51.41.31.21.115.001.0Fugacity coefficient, f/p10.00TR6.002.000.9Saturation1.901.80line1.700.81.601.500.71.450.850.800.60.901.401.351.300.51.250.751.200.41.151.101.081.061.041.021.000.980.960.940.920.900.30.700.20.650.10.850.800.6000.10.20.750.30.4 0.51.02.03.04.0 5.00.70100.602030Reduced pressure, pRFIGURE 2.14 Generalized fugacity coefficient chart. (Source: Van Wylen, G. J. and Sonntag, R.
E. 1986. Fundamentals of Classical Thermodynamics, 3rd ed., English/SI. Wiley, New York.)Multicomponent SystemsIn this section are presented some general aspects of the properties of multicomponent systems consistingof nonreacting mixtures. For a single phase multicomponent system consisting of j components, anextensive property X may be regarded as a function of temperature, pressure, and the number of molesof each component present in the mixture: X = X(T, p, n1, n2, … nj). Since X is mathematicallyhomogeneous of degree one in the n’s, the function is expressible as© 1999 by CRC Press LLC2-55Engineering ThermodynamicsjX=∑n Xi(2.67)ii =1where the partial molar property Xi is by definitionXi =∂X ∂ni T , p,n(2.68)land the subscript n, denotes that all n’s except ni are held fixed during differentiation.
As Xi dependsin general on temperature, pressure, and mixture composition: Xi (T, p, n1, n2, … nj), the partial molalproperty Xi is an intensive property of the mixture and not simply a property of the ith component.Selecting the extensive property X to be volume, internal energy, enthalpy, entropy, and the Gibbsfunction, respectively, givesjV=∑jU=ni Vi ,i =1iii =1jH=∑n Uj∑n H ,iS=ii =1∑n Si i(2.69)i =1jG=∑n Giii =1where Vi , Ui , Hi , Si , and Gi denote the respective partial molal properties.When pure components, each initially at the same temperature and pressure, are mixed, the changesin volume, internal energy, enthalpy, and entropy on mixing are given byj∆Vmixing =∑ n (V − v )iii(2.70a)i =1j∆U mixing =∑ n (U − u )iii(2.70b)i =1j∆H mixing =∑ n (H − h )iii(2.70c)i =1j∆Smixing =∑ n (S − s )iii(2.70d)i =1where vi , ui , hi , and si denote the molar-specific volume, internal energy, enthalpy, and entropy ofpure component i.Chemical PotentialThe partial molal Gibbs function of the ith component of a multicomponent system is the chemicalpotential, µi,© 1999 by CRC Press LLC2-56Section 2∂G ∂ni T , p,nµ i = Gi =(2.71)lLike temperature and pressure, the chemical potential, µi is an intensive property.When written in terms of chemical potentials, Equation 2.67 for the Gibbs function readsjG=∑n µi(2.72)ii =1For a single component sysrem, Equation 2.72 reduces to G = nµ; that is, the chemical potential equalsthe molar Gibbs function.
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