The CRC Handbook of Mechanical Engineering. Chapter 2. Engineering Thermodynamics (776125), страница 7

Просмтор этого файла доступен только зарегистрированным пользователям. Но у нас супер быстрая регистрация: достаточно только электронной почты!

Текст из файла (страница 7)

Theycan also be determined macroscopically through exacting property measurements. Specific heat data forcommon gases, liquids, and solids are provided by the handbooks and property reference volumes listedamong the Chapter 2 references. Specific heats are also considered in Section 2.3 as a part of thediscussions of the incompressible model and the ideal gas model. Figure 2.4 shows how cp for watervapor varies as a function of temperature and pressure. Other gases exhibit similar behavior. The figurealso gives the variation of cp with temperature in the limit as pressure tends to zero (the ideal gas limit).In this limit cp increases with increasing temperature, which is a characteristic exhibited by other gasesas wellThe following two equations are often convenient for establishing relations among properties: ∂x   ∂y     = 1 ∂y  z ∂x z(2.36a) ∂y   ∂z   ∂x       = −1 ∂z  x  ∂x  y  ∂y  z(2.36b)Their use is illustrated in Example 5.Example 5Obtain Equations 2 and 11 of Table 2.4 from Equation 1.Solution.

Identifying x, y, z with s, T, and v, respectively, Equation 2.36b reads ∂T   ∂v   ∂s       = −1 ∂v  s  ∂s  T  ∂T  vApplying Equation 2.36a to each of (∂T/∂v)s and (∂v/∂s)T ,© 1999 by CRC Press LLC71.661.4cp Btu/lb ⋅ °R54015320 2590807060105,01.050301.2101004d vapor1.8Saturate8d vapor2.0Saturatecp kJ/kg ⋅ K2-269100.810200120201500006000500040003000,080000000lbf/in. 25000.6200Zero pressure limit0Zero pressure limit100200300400500°C6007008000.4 2004006008001000120014001600°F© 1999 by CRC Press LLCSection 2FIGURE 2.4 cp of water vapor as a function of temperature and pressure.

(Adapted from Keenan, J.H., Keyes, F.G., Hill, P.G., and Moore, J.G. 1969 and 1978. SteamTables — S.I. Units (English Units). John Wiley & Sons, New York.)2-27Engineering ThermodynamicsTABLE 2.4 Specific Heat Relationsa ∂u  ∂s cv =   = T   ∂T  v ∂T  v ∂p   ∂v = −T     ∂T  v  ∂T  s ∂h  ∂s cp =   = T  ∂T  p ∂T  p ∂v   ∂p = T    ∂T  p  ∂T  s ∂p   ∂v c p − cv = T     ∂T  v  ∂T  p(1)(2)(3)(4)(5)2 ∂v   ∂p = −T     ∂T  p  ∂v  T=cp =a(7)1   ∂v T   − v µ J  ∂T p(8)1   ∂p T   − pη   ∂T  v(9) ∂v   ∂p =    ∂p  T  ∂v  s10)cv = −k=Tvβ 2κ(6)cpcv ∂2 p  ∂cv  = T 2  ∂v  T ∂T  v(11) ∂c p  ∂2v  ∂p  = −T  ∂T 2 Tp(12)See, for example, Moran, M.J.

andShapiro, H.N. 1995. Fundamentals ofEngineering Thermodynamics, 3rd ed.Wiley, New York, chap. 11.1 ∂s  ∂v   ∂s = −     =− ∂T  v(∂T ∂v)s (∂v ∂s)T  ∂T  s  ∂v  TIntroducing the Maxwell relation from Table 2.2 corresponding to ψ(T, v), ∂s  ∂v   ∂p   = −    ∂T  v ∂T  s  ∂T  vWith this, Equation 2 of Table 2.4 is obtained from Equation 1, which in turn is obtained in Example6. Equation 11 of Table 2.4 can be obtained by differentiating Equation 1 with repect to specific volumeat fixed temperature, and again using the Maxwell relation corresponding to ψ.© 1999 by CRC Press LLC2-28Section 2P-v-T RelationsConsiderable pressure, specific volume, and temperature data have been accumulated for industriallyimportant gases and liquids.

These data can be represented in the form p = f (v, T ), called an equationof state. Equations of state can be expressed in tabular, graphical, and analytical forms.P-v-T SurfaceThe graph of a function p = f (v, T) is a surface in three-dimensional space. Figure 2.5 shows the p-vT relationship for water. Figure 2.5b shows the projection of the surface onto the pressure-temperatureplane, called the phase diagram.

The projection onto the p–v plane is shown in Figure 2.5c.FIGURE 2.5 Pressure-specific volume-temperature surface and projections for water (not to scale).Figure 2.5 has three regions labeled solid, liquid, and vapor where the substance exists only in a singlephase. Between the single phase regions lie two-phase regions, where two phases coexist in equilibrium.The lines separating the single-phase regions from the two-phase regions are saturation lines. Any staterepresented by a point on a saturation line is a saturation state.

The line separating the liquid phase and© 1999 by CRC Press LLC2-29Engineering Thermodynamicsthe two-phase liquid-vapor region is the saturated liquid line. The state denoted by f is a saturated liquidstate. The saturated vapor line separates the vapor region and the two-phase liquid-vapor region. Thestate denoted by g is a saturated vapor state. The saturated liquid line and the saturated vapor line meetat the critical point.

At the critical point, the pressure is the critical pressure pc, and the temperature isthe critical temperature Tc . Three phases can coexist in equilibrium along the line labeled triple line.The triple line projects onto a point on the phase diagram. The triple point of water is used in definingthe Kelvin temperature scale (Section 2.1, Basic Concepts and Definitions; The Second Law of Thermodynamics, Entropy).When a phase change occurs during constant pressure heating or cooling, the temperature remainsconstant as long as both phases are present. Accordingly, in the two-phase liquid-vapor region, a line ofconstant pressure is also a line of constant temperature. For a specified pressure, the correspondingtemperature is called the saturation temperature.

For a specified temperature, the corresponding pressureis called the saturation pressure. The region to the right of the saturated vapor line is known as thesuperheated vapor region because the vapor exists at a temperature greater than the saturation temperaturefor its pressure. The region to the left of the saturated liquid line is known as the compressed liquidregion because the liquid is at a pressure higher than the saturation pressure for its temperature.When a mixture of liquid and vapor coexists in equilibrium, the liquid phase is a saturated liquid andthe vapor phase is a saturated vapor.

The total volume of any such mixture is V = Vf + Vg; or, alternatively,mv = mfvf + mgvg, where m and v denote mass and specific volume, respectively. Dividing by the totalmass of the mixture m and letting the mass fraction of the vapor in the mixture, mg /m, be symbolizedby x, called the quality, the apparent specific volume v of the mixture isv = (1 − x )vf + xvg(2.37a)= vf + xvfgwhere vfg = vg – vf.

Expressions similar in form can be written for internal energy, enthalpy, and entropy:u = (1 − x )uf + xug(2.37b)= uf + xufgh = (1 − x )hf + xhg(2.37c)= hf + xhfgs = (1 − x )sf + xsg(2.37d)= sf + xsfgFor the case of water, Figure 2.6 illustrates the phase change from solid to liquid (melting): a-b-c;from solid to vapor (sublimation): a′-b′-c′; and from liquid to vapor (vaporization): a″-b″-c″. Duringany such phase change the temperature and pressure remain constant and thus are not independentproperties. The Clapeyron equation allows the change in enthalpy during a phase change at fixedtemperature to be evaluated from p-v-T data pertaining to the phase change.

For vaporization, theClapeyron equation reads dp  = hg − hf dT  sat T v − vgf(© 1999 by CRC Press LLC)(2.38)2-30Section 2FIGURE 2.6 Phase diagram for water (not to scale).where (dp/dT)sat is the slope of the saturation pressure-temperature curve at the point determined by thetemperature held constant during the phase change.

Expressions similar in form to Equation 2.38 canbe written for sublimation and melting.The Clapeyron equation shows that the slope of a saturation line on a phase diagram depends on thesigns of the specific volume and enthalpy changes accompanying the phase change. In most cases, whena phase change takes place with an increase in specific enthalpy, the specific volume also increases, and(dp/dT)sat is positive. However, in the case of the melting of ice and a few other substances, the specificvolume decreases on melting. The slope of the saturated solid-liquid curve for these few substances isnegative, as illustrated for water in Figure 2.6.Graphical RepresentationsThe intensive states of a pure, simple compressible system can be represented graphically with any twoindependent intensive properties as the coordinates, excluding properties associated with motion andgravity.

While any such pair may be used, there are several selections that are conventionally employed.These include the p-T and p-v diagrams of Figure 2.5, the T-s diagram of Figure 2.7, the h-s (Mollier)diagram of Figure 2.8, and the p-h diagram of Figure 2.9. The compressibility charts considered nextuse the compressibility factor as one of the coordinates.Compressibility ChartsThe p-v-T relation for a wide range of common gases is illustrated by the generalized compressibilitychart of Figure 2.10. In this chart, the compressibility factor, Z, is plotted vs.

the reduced pressure, pR,reduced temperature, TR, and pseudoreduced specific volume, v R′ , whereZ=and© 1999 by CRC Press LLCpvRT(2.39)100.030.10.31.01.02100.10300300g1 m 3/k3600700°C600°C900°Cυ=0.003800Engineering Thermodynamics4000500°C0.01800°C3400400°C3200300°C700°C00330000.200°C01m.0=0υ=1260000T=100°C600°CM3Pa/kg2800px=90%240080%220080%200045678910FIGURE 2.7 Temperature-entropy diagram for water. (Source: Jones, J.B.

Характеристики

Список файлов книги