3.13.1 Du, Dh, c𝝊, and cp Relations
For a gas obeying the ideal gas model, specific internal energy depends only on tem- perature. Hence, the specific heat c𝜐, defined by Eq. 3.8, is also a function of temperature alone. That is,
c𝜐(T)5 du
dT (ideal gas) (3.38)
This is expressed as an ordinary derivative because u depends only on T.
By separating variables in Eq. 3.38
du5c𝜐(T) dT (3.39)
118 Chapter 3 Evaluating Properties
On integration, the change in specific internal energy is u(T2)2u(T1)5 #TT2
1
c𝜐(T) dT (ideal gas) (3.40)
Similarly, for a gas obeying the ideal gas model, the specific enthalpy depends only on temperature, so the specific heat cp, defined by Eq. 3.9, is also a function of tem- perature alone. That is
cp(T)5 dh
dT (ideal gas) (3.41)
Separating variables in Eq. 3.41
dh5cp(T) dT (3.42)
On integration, the change in specific enthalpy is h(T2)2h(T1)5 #TT2
1
cp(T) dT (ideal gas) (3.43)
An important relationship between the ideal gas specific heats can be developed by differentiating Eq. 3.37 with respect to temperature
dh dT⫽ du
dT⫹R and introducing Eqs. 3.38 and 3.41 to obtain
cp(T)5c𝜐(T)1R (ideal gas) (3.44) On a molar basis, this is written as
cp(T)5c𝜐(T)1R (ideal gas) (3.45) Although each of the two ideal gas specific heats is a function of temperature, Eqs. 3.44 and 3.45 show that the specific heats differ by just a constant: the gas constant. Knowl- edge of either specific heat for a particular gas allows the other to be calculated by using only the gas constant. The above equations also show that cp . c𝜐 and cp . c𝜐, respectively.
For an ideal gas, the specific heat ratio, k, is also a function of temperature only k5 cp(T)
c𝜐(T) (ideal gas) (3.46)
Since cp . c𝜐, it follows that k . 1. Combining Eqs. 3.44 and 3.46 results in cp(T)5 kR
k21 (3.47a)
(ideal gas)
c𝜐(T)5 R
k21 (3.47b)
Similar expressions can be written for the specific heats on a molar basis, with R being replaced by R.
3.13 Internal Energy, Enthalpy, and Specific Heats of Ideal Gases 119
3.13.2 Using Specific Heat Functions
The foregoing expressions require the ideal gas specific heats as functions of temperature. These functions are available for gases of practical interest in vari- ous forms, including graphs, tables, and equations. Figure 3.13 illustrates the vari- ation of cp (molar basis) with temperature for a number of common gases. In the range of temperature shown, cp increases with temperature for all gases, except for the monatonic gases Ar, Ne, and He. For these, cp is constant at the value predicted by kinetic theory: cp⫽52R. Tabular specific heat data for selected gases are presented versus temperature in Tables A-20. Specific heats are also available in equation form. Several alternative forms of such equations are found in the engineering literature. An equation that is relatively easy to integrate is the poly- nomial form
cp
R 5𝛼1𝛽T1𝛾T21𝛿T3 1𝜀T4 (3.48) Values of the constants α, β, δ, and ε are listed in Tables A-21 for several gases in the temperature range 300 to 1000 K.
to illustrate the use of Eq. 3.48, let us evaluate the change in spe- cific enthalpy, in kJ/kg, of air modeled as an ideal gas from a state where T1 5 400 K to a state where T2 5 900 K. Inserting the expression for cp(T) given by Eq. 3.48 into Eq. 3.43 and integrating with respect to temperature
h22h15 R M#TT2
1
(𝛼1𝛽T1𝛾T2 1𝛿T3 1𝜀T4)dT
5 R
Mc𝛼(T22T1)1 𝛽
2(T222T21)1 𝛾
3(T322T31)1 𝛿
4(T422T41)1 𝜀
5(T522T51)d
7
6
5
4
3
0 1000 2000 3000
cp
R
Temperature, K Ar, Ne, He
CO2 H2O
O2 CO H2 Air
Fig. 3.13 Variation of –cp/–
R with temperature for a number of gases modeled as ideal gases.
120 Chapter 3 Evaluating Properties
where the molecular weight M has been introduced to obtain the result on a unit mass basis. With values for the constants from Table A-21
h22h15 8.314
28.97e3.653(9002400)2 1.337
2(10)3[(900)22 (400)2] 1 3.294
3(10)6[(900)32(400)3]2 1.913
4(10)9[(900)42(400)4] 1 0.2763
5(10)12[(900)52 (400)5]f 5531.69 kJ/kg b b b b b
Specific heat functions cʋ(T) and cp(T) are also available in IT: Interactive Ther- modynamics in the PROPERTIES menu. These functions can be integrated using the integral function of the program to calculate Du and Dh, respectively.
let us repeat the immediately preceding example using IT. For air, the IT code is
cp 5 cp_T (“Air” ,T) delh 5 Integral(cp,T)
Pushing SOLVE and sweeping T from 400 K to 900 K, the change in specific enthalpy is delh 5 531.7 kJ/kg, which agrees closely with the value obtained by integrating the specific heat function from Table A-21, as illustrated above. b b b b b
The source of ideal gas specific heat data is experiment. Specific heats can be determined macroscopically from painstaking property measurements. In the limit as pressure tends to zero, the properties of a gas tend to merge into those of its ideal gas model, so macroscopically determined specific heats of a gas extrapolated to very low pressures may be called either zero-pressure specific heats or ideal gas specific heats. Although zero-pressure specific heats can be obtained by extrapolating macro- scopically determined experimental data, this is rarely done nowadays because ideal gas specific heats can be readily calculated with expressions from statistical mechan- ics by using spectral data, which can be obtained experimentally with precision. The determination of ideal gas specific heats is one of the important areas where the microscopic approach contributes significantly to the application of engineering thermodynamics.