Unformatted text preview: does not change with specific volume. That is, cv is not a function of specific volume either. Therefore we conclude that the internal energy of an ideal gas is a function of temperature only (Fig. 12–11). (b) For an incompressible substance, v from Eq. 12–49, cp cv c since a b Then Eq. 12–29 reduces to constant and thus dv 0. Also 0 for incompressible substances.
u = u(T ) cv = cv (T ) cp = cp(T ) AIR du c dT Again we need to show that the specific heat c depends on temperature only and not on pressure or specific volume. This is done with the help of Eq. 12–43: u = u(T ) c = c(T ) LAKE a 0 cp 0P b T Ta 0 2v b 0T 2 P 0 since v constant. Therefore, we conclude that the internal energy of a truly incompressible substance depends on temperature only. FIGURE 12–11 The internal energies and specific heats of ideal gases and incompressible substances depend on temperature only. EXAMPLE 12–9
Show that cp cv The Specific Heat Difference of an Ideal Gas
R for an ideal gas. Solution It is to be shown that the specific heat difference for an ideal gas
is equal to its gas constant. Analysis This relation is easily proved by showing that the right-hand side of Eq. 12–46 is equivalent to the gas constant R of the ideal gas: cp P v
Substituting, cv Ta RT v2 a 0v 2 0P ba b 0T P 0v T P v RT 0P Sa b v 0v T RT 0v 2 Sa b P 0T P 0v 2 0P ba b 0T P 0v T cp cv R2 b P R2 ba P P b v Ta
Therefore, Ta R R cen84959_ch12.qxd 4/5/05 3:58 PM Page 668 668 |
T1 = 20°C Thermodynamics
> T2 = 20°C < P2 = 200 kPa 12–5 ■ THE JOULE-THOMSON COEFFICIENT P1 = 800 kPa FIGURE 12–12 The temperature of a fluid may increase, decrease, or remain constant during a throttling process.
T P2, T2 (varied) P1, T1 (fixed) When a fluid passes through a restriction such as a porous plug, a capillary tube, or an ordinary valve, its pressure decreases. As we have shown in Chap. 5, the enthalpy of the fluid remains approximately constant during such a throttling process. You will remember that a fluid may experience a large drop in its temperature as a result of throttling, which forms the basis of operation for refrigerators and air conditioners. This is not always the case, however. The temperature of the fluid may remain unchanged, or it may even increase during a throttling process (Fig. 12–12). The temperature behavior of a fluid during a throttling (h constant) process is described by the Joule-Thomson coefficient, defined as
m a 0T b 0P h
(12–51) Exit states 2 2 Thus the Joule-Thomson coefficient is a measure of the change in temperature with pressure during a constant-enthalpy process. Notice that if
m JT • 60 0 70 temperature increases temperature remains constant temperature decreases 2 2 2 Inlet state h = constant line 1 P1 P FIGURE 12–13 The development of an h line on a P-T diagram. constant T Maximum inversion temperature µJT > 0 µJT < 0
h = const. Inversion line P FIGURE 12–14 Constant-enthalpy lines of a substance on a T-P diagram. during a throttling process. A careful...
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