fluid_mechanics

11.1 Ideal Gas Relationships 581
Thus,
T 2
ǔ 2 ǔ 1 c v dT
(11.4)
Equation 11.4 is useful because it allows us to evaluate the change in internal energy, ǔ 2 ǔ 1 , associated
with ideal gas flow from section 112 to section 122 in a flow. For simplicity, we can assume
that c v is constant for a particular ideal gas and obtain from Eq. 11.4
ǔ 2 ǔ 1 c v 1T 2 T 1 2
(11.5)
Actually, c v for a particular gas varies with temperature 1see Ref. 22. However, for moderate changes
in temperature, the constant c v assumption is reasonable.
The fluid property enthalpy, ȟ, is defined as
T 1
ȟ ǔ p r
(11.6)
It combines internal energy, ǔ, and pressure energy, pr, and is useful when dealing with the energy
equation 1Eq. 5.692. For an ideal gas, we have already stated that
From the equation of state 1Eq. 11.12
Thus, it follows that
ǔ ǔ1T2
p
r RT
ȟ ȟ1T2
Since for an ideal gas, enthalpy is a function of temperature only, the ideal gas specific heat at constant
pressure, c p , can be expressed as
c p a 0ȟ
0T b dȟ
p dT
(11.7)
where the subscript p on the partial derivative refers to differentiation at constant pressure, and
is a function of temperature only. The rearrangement of Eq. 11.7 leads to
and
dȟ c p dT
c p
T 2
ȟ 2 ȟ 1 c p dT
T 1
(11.8)
Equation 11.8 is useful because it allows us to evaluate the change in enthalpy, ȟ 2 ȟ 1 , associated
with ideal gas flow from section 112 to section 122 in a flow. For simplicity, we can assume
that c p is constant for a specific ideal gas and obtain from Eq. 11.8
ȟ 2 ȟ 1 c p 1T 2 T 1 2
(11.9)
For moderate temperature
changes,
specific heat values
can be considered
constant.
As is true for c v , the value of c p for a given gas varies with temperature. Nevertheless, for moderate
changes in temperature, the constant c p assumption is reasonable.
From Eqs. 11.5 and 11.9 we see that changes in internal energy and enthalpy are related
to changes in temperature by values of c v and c p . We turn our attention now to developing useful
relationships for determining c v and c p . Combining Eqs. 11.6 and 11.1 we get
ȟ ǔ RT
(11.10)