The CRC Handbook of Mechanical Engineering. Chapter 2. Engineering Thermodynamics (776125), страница 6

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For such devices Ẇcv = 0 and Equation 2.27c reducesfurther to readhe ≅ hi(throttling process)(2.27d)That is, upstream and downstream of the throttling device, the specific enthalpies are equal.A nozzle is a flow passage of varying cross-sectional area in which the velocity of a gas or liquidincreases in the direction of flow. In a diffuser, the gas or liquid decelerates in the direction of flow. Forsuch devices, Ẇcv = 0. The heat transfer and potential energy change are also generally negligible. ThenEquation 2.27b reduces to0 = hi − he +© 1999 by CRC Press LLCv i2 − v e22(2.27e)2-19Engineering ThermodynamicsSolving for the outlet velocityv e = v i2 + 2(hi − he )(nozzle, diffuser)(2.27f)Further discussion of the flow-through nozzles and diffusers is provided in Chapter 3.The mass, energy, and entropy rate balances, Equations 2.26, can be applied to control volumes withmultiple inlets and/or outlets, as, for example, cases involving heat-recovery steam generators, feedwaterheaters, and counterflow and crossflow heat exchangers.

Transient (or unsteady) analyses can be conducted with Equations 2.19, 2.24, and 2.25. Illustrations of all such applications are provided by Moranand Shapiro (1995).Example 1A turbine receives steam at 7 MPa, 440°C and exhausts at 0.2 MPa for subsequent process heating duty.If heat transfer and kinetic/potential energy effects are negligible, determine the steam mass flow rate,in kg/hr, for a turbine power output of 30 MW when (a) the steam quality at the turbine outlet is 95%,(b) the turbine expansion is internally reversible.Solution. With the indicated idealizations, Equation 2.27c is appropriate. Solving, m˙ = W˙ cv /(hi − he ).Steam table data (Table A.5) at the inlet condition are hi = 3261.7 kJ/kg, si = 6.6022 kJ/kg · K.(a) At 0.2 MPa and x = 0.95, he = 2596.5 kJ/kg.

Thenm˙ = 10 3 kJ sec   3600 sec 30 MW(3261.7 − 2596.5) kJ kg  1 MW   1 hr = 162, 357 kg hr(b) For an internally reversible expansion, Equation 2.28b reduces to give se = si. For this case, he =2499.6 kJ/kg (x = 0.906), and ṁ = 141,714 kg/hr.Example 2Air at 500°F, 150 lbf/in.2, and 10 ft/sec expands adiabatically through a nozzle and exits at 60°F, 15lbf/in.2. For a mass flow rate of 5 lb/sec determine the exit area, in in.2. Repeat for an isentropic expansionto 15 lbf/in.2.

Model the air as an ideal gas (Section 2.3, Ideal Gas Model) with specific heat cp = 0.24Btu/lb · °R (k = 1.4).Solution. The nozle exit area can be evaluated using Equation 2.20, together with the ideal gas equation,v = RT/p:Ae =m˙ ν e m˙ ( RTe pe )=veveThe exit velocity required by this expression is obtained using Equation 2.27f′ of Table 2.1,v e = v i2 + 2c p (Ti − Te )2 32.174 lb ⋅ ft sec 2 10 ft Btu   778.17 ft ⋅ lbf = + 2 0.24440°R) ( s lb ⋅ R  1 Btu1 lbf= 2299.5 ft sec© 1999 by CRC Press LLC2-20Section 2FIGURE 2.2 Open feedwater heater.Finally, with R = R /M = 53.33 ft · lbf/lb · °R, 5 lb   53.3 ft ⋅ lbf  520°R() sec  lb ⋅ °R Ae == 4.02 in.2 2299.5 ft  15 lbf sec   in.2 Using Equation 2.27f″ in Table 2.1 for the isentropic expansion,ve =(10)2 + 2(0.24)(778.17)(960)(32.174) 1 − 15 150 0.4 1.4= 2358.3 ft secThen Ae = 3.92 in.2.Example 3Figure 2.2 provides steady-state operating data for an open feedwater heater.

Ignoring heat transfer andkinetic/potential energy effects, determine the ratio of mass flow rates, m˙ 1 / m˙ 2 .Solution. For this case Equations 2.26a and 2.26b reduce to read, respectively,m˙ 1 + m˙ 2 = m˙ 30 = m˙ 1h1 + m˙ 2 h2 − m˙ 3 h3Combining and solving for the ratio m˙ 1 / m˙ 2 ,m˙ 1 h2 − h3=m˙ 2 h3 − h1Inserting steam table data, in kJ/kg, from Table A.5,m˙ 1 2844.8 − 697.2== 4.06m˙ 2697.2 − 167.6Internally Reversible Heat Transfer and WorkFor one-inlet, one-outlet control volumes at steady state, the following expressions give the heat transferrate and power in the absence of internal irreversibilities:© 1999 by CRC Press LLC2-21Engineering Thermodynamics Q˙ cv  m˙  int =rev W˙ cv  m˙  int = −rev∫2νdp +1∫ Tds2(2.29)1v12 − v 22+ g( z1 − z2 )2(2.30a)(see, e.g., Moran and Shapiro, 1995).If there is no significant change in kinetic or potential energy from inlet to outlet, Equation 2.30a reads2 W˙ cv =−νdp m˙  int1rev∫(∆ke = ∆pe = 0)(2.30b)The specific volume remains approximately constant in many applications with liquids.

Then Equation30b becomes W˙ cv  m˙  int = − v( p2 − p1 )(v = constant)(2.30c)revWhen the states visited by a unit of mass flowing without irreversibilities from inlet to outlet aredescribed by a continuous curve on a plot of temperature vs. specific entropy, Equation 2.29 impliesthat the area under the curve is the magnitude of the heat transfer per unit of mass flowing.

When suchan ideal process is described by a curve on a plot of pressure vs. specific volume, as shown in Figure2.3, the magnitude of the integral ∫vdp of Equations 2.30a and 2.30b is represented by the area a-b-c-dbehind the curve. The area a-b-c′-d′ under the curve is identified with the magnitude of the integral ∫pdvof Equation 2.10.FIGURE 2.3 Internally reversible process on p–v coordinates.© 1999 by CRC Press LLC2-22Section 22.3 Property Relations and DataPressure, temperature, volume, and mass can be found experimentally.

The relationships between thespecific heats cv and cp and temperature at relatively low pressure are also accessible experimentally, asare certain other property data. Specific internal energy, enthalpy, and entropy are among those propertiesthat are not so readily obtained in the laboratory. Values for such properties are calculated usingexperimental data of properties that are more amenable to measurement, together with appropriateproperty relations derived using the principles of thermodynamics. In this section property relations anddata sources are considered for simple compressible systems, which include a wide range of industriallyimportant substances.Property data are provided in the publications of the National Institute of Standards and Technology(formerly the U.S.

Bureau of Standards), of professional groups such as the American Society ofMechanical Engineering (ASME), the American Society of Heating. Refrigerating, and Air ConditioningEngineers (ASHRAE), and the American Chemical Society, and of corporate entities such as Dupont andDow Chemical. Handbooks and property reference volumes such as included in the list of referencesfor this chapter are readily accessed sources of data. Property data are also retrievable from variouscommercial online data bases.

Computer software is increasingly available for this purpose as well.Basic Relations for Pure SubstancesAn energy balance in differential form for a closed system undergoing an internally reversible processin the absence of overall system motion and the effect of gravity readsdU = (δQ)int − (δW )intrevrevFrom Equation 2.14b, (δQ)int= TdS.

When consideration is limited to simple compressible systems:revsystems for which the only significant work in an internally reversible process is associated with volumechange, (δW )int= pdV, the following equation is obtained:revdU = TdS − pdV(2.31a)Introducing enthalpy, H = U + pV, the Helmholtz function, Ψ = U – TS, and the Gibbs function, G = H– TS, three additional expressions are obtained:dH = TdS + Vdp(2.31b)dΨ = − pdV − SdT(2.31c)dG = Vdp − SdT(2.31d)Equations 2.31 can be expressed on a per-unit-mass basis as© 1999 by CRC Press LLCdu = Tds − pdv(2.32a)dh = Tds + vdp(2.32b)dψ = − pdv − sdT(2.32c)dg = vdp − sdT(2.32d)Engineering Thermodynamics2-23Similar expressions can be written on a per-mole basis.Maxwell RelationsSince only properties are involved, each of the four differential expressions given by Equations 2.32 isan exact differential exhibiting the general form dz = M(x, y)dx + N(x, y)dy, where the second mixedpartial derivatives are equal: (∂M/∂y) = (∂N/∂x).

Underlying these exact differentials are, respectively,functions of the form u(s, v), h(s, p), ψ(v, T), and g(T, p). From such considerations the Maxwell relationsgiven in Table 2.2 can be established.Example 4Derive the Maxwell relation following from Equation 2.32a.TABLE 2.2 Relations from Exact Differentials© 1999 by CRC Press LLC2-24Section 2Solution. The differential of the function u = u(s, v) is ∂u  ∂u du =   ds +   dv ∂v  s ∂s  vBy comparison with Equation 2.32a, ∂u T =  , ∂s  v ∂u −p=  ∂v  sIn Equation 2.32a, T plays the role of M and –p plays the role of N, so the equality of second mixedpartial derivatives gives the Maxwell relation, ∂T  ∂p   = −  ∂v  s ∂s  vSince each of the properties T, p, v, and s appears on the right side of two of the eight coefficients ofTable 2.2, four additional property relations can be obtained by equating such expressions: ∂u  ∂h   =  , ∂s  v  ∂s  p ∂u  ∂ψ   = ∂v  s  ∂v  T ∂g  ∂h   =  , ∂p  s  ∂p  T ∂ψ  ∂g  =  ∂T  v  ∂T  pThese four relations are identified in Table 2.2 by brackets.

As any three of Equations 2.32 can beobtained from the fourth simply by manipulation, the 16 property relations of Table 2.2 also can beregarded as following from this single differential expression. Several additional first-derivative propertyrelations can be derived; see, e.g., Zemansky, 1972.Specific Heats and Other PropertiesEngineering thermodynamics uses a wide assortment of thermodynamic properties and relations amongthese properties. Table 2.3 lists several commonly encountered properties.Among the entries of Table 2.3 are the specific heats cv and cp. These intensive properties are oftenrequired for thermodynamic analysis, and are defined as partial derivations of the functions u(T, v) andh(T, p), respectively, ∂u cv =   ∂T  v(2.33) ∂h cp =   ∂T  p(2.34)Since u and h can be expressed either on a unit mass basis or a per-mole basis, values of the specificheats can be similarly expressed. Table 2.4 summarizes relations involving cv and cp.

The property k,the specific heat ratio, isk=© 1999 by CRC Press LLCcpcv(2.35)2-25Engineering ThermodynamicsTABLE 2.3 Symbols and Definitions for Selected PropertiesPropertySymbolDefinitionPropertySymbolDefinition(∂u ∂T ) v(∂h ∂T ) pPressurepSpecific heat, constant volumecvTemperatureTSpecific heat, constant pressurecpSpecific volumevVolume expansivityβSpecific internal energyuIsothermal compressivityκ−Specific entropysIsentropic compressibilityα−Specific enthalpyhu + pvIsothermal bulk modulusB1(∂v ∂p) sv− v(∂p ∂v) TSpecific Helmholtz functionψu – TsIsentropic bulk modulusBs− v(∂p ∂v) sSpecific Gibbs functiongh – TsJoule-Thomson coefficientµJCompressibility factorZpv/RTJoule coefficientηSpecific heat ratiokcp /cvVelocity of soundc1(∂v ∂T ) pv1(∂v ∂p) Tv(∂T ∂p) h(∂T ∂v)u− v 2 (∂p ∂v) sValues for cv and cp can be obtained via statistical mechanics using spectroscopic measurements.

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