That is psat f tsat therefore the partial derivative

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Unformatted text preview: associated with a phase change (such as the enthalpy of vaporization hfg) from a knowledge of P, v, and T data alone. Consider the third Maxwell relation, Eq. 12–18: a 0P b 0T v a 0s b 0v T During a phase-change process, the pressure is the saturation pressure, which depends on the temperature only and is independent of the specific cen84959_ch12.qxd 4/5/05 3:58 PM Page 659 Chapter 12 volume. That is, Psat f (Tsat). Therefore, the partial derivative ( P/ T )v can be expressed as a total derivative (dP/dT )sat, which is the slope of the saturation curve on a P-T diagram at a specified saturation state (Fig. 12–9). This slope is independent of the specific volume, and thus it can be treated as a constant during the integration of Eq. 12–18 between two saturation states at the same temperature. For an isothermal liquid–vapor phase-change process, for example, the integration yields sg sf dP a b 1v dT sat g sfg vfg vf 2 (12–20) P | 659 LIQUID SOLID P (∂–– ) ∂T VAPOR T = const. sat or dP a b dT sat (12–21) T During this process the pressure also remains constant. Therefore, from Eq. 12–11, dh T ds 0 v dP → S f g g FIGURE 12–9 The slope of the saturation curve on a P-T diagram is constant at a constant T or P. dh f T ds S h fg Tsfg Substituting this result into Eq. 12–21, we obtain a dP b dT sat hfg Tvfg (12–22) which is called the Clapeyron equation after the French engineer and physicist E. Clapeyron (1799–1864). This is an important thermodynamic relation since it enables us to determine the enthalpy of vaporization hfg at a given temperature by simply measuring the slope of the saturation curve on a P-T diagram and the specific volume of saturated liquid and saturated vapor at the given temperature. The Clapeyron equation is applicable to any phase-change process that occurs at constant temperature and pressure. It can be expressed in a general form as a dP b dT sat h12 Tv12 (12–23) where the subscripts 1 and 2 indicate the two phases. EXAMPLE 12–5 Evaluating the h fg of a Substance from t he P - v - T Data Using the Clapeyron equation, estimate the value of the enthalpy of vaporization of refrigerant-134a at 20°C, and compare it with the tabulated value. Solution The hfg of refrigerant-134a is to be determined using the Clapeyron equation. Analysis From Eq. 12–22, hfg Tvfg a dP b dT sat cen84959_ch12.qxd 4/5/05 3:58 PM Page 660 660 | Thermodynamics where, from Table A–11, vfg a dP b d T sat,20°C 1 vg a vf 2 @ 20°C 0.035969 Psat @ 24°C 24°C 0.0008161 Psat @ 16°C 16°C 0.035153 m3> kg ¢P b ¢ T sat,20°C 646.18 since T (°C) 504.58 kPa 8°C 17.70 kPa> K 1 kJ b 1 kPa # m3 T (K). Substituting, we get h fg 1 293.15 K 2 1 0.035153 m3> kg 2 1 17.70 kPa> K 2 a 182.40 kJ/kg The tabulated value of hfg at 20°C is 182.27 kJ/kg. The small difference between the two values is due to the approximation used in determining the slope of the saturation curve at 20°C. The Clapeyron equation can be simplified for liquid–vapor and solid–vapor vf , phase changes by utilizing some approximations. At low pressures vg and thus vfg vg. By treating the vapor as an ideal gas, we have vg RT/P. Substituting...
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