Modern Engineering Thermodynamics

446 CHAPTER 12: Mixtures of Gases and Vapors
replacing T DB with T WB . The wet bulb temperature can now be
used as an input parameter. Prompt for inputs of p m , T DB , and
either T WB , ϕ, orω. Return to the screen properly formatted values
(with appropriate units) of p m , T DB , T WB , T DP , ω, ϕ, h # , and v a .
Can we then define Ti to be the partial specific temperature of gas
i in the mixture and define T i = m i^T i to be the partial temperature
of gas i in the mixture (Figure 12.16)? Does this lead us to a
fourth composition measure, the temperature fraction?
Special Problems
67. The implicit function theorem from calculus tells us that if
MV m /MT m ≠ 0 in Eq. (12.5), then we can write the temperature
T m of a mixture of N gases as a function of the mixture total
volume V m , mixture total pressure p m , and the mass
composition of the mixture m 1 , m 2 , … , m N as
Is this the partial
temperature of gas i
T i ????
Mixture
pressure
p m
T m = T m ðV m , p m , m 1 , m 2 , … , m N Þ
However, when the total volume and pressure of the mixture are
constant, this generally is not a homogenous function of the first
degree in the masses m i . The temperature usually varies inversely
with the system mass in most equations of state for gases and
vapors. Consequently, when the mixture masses are multiplied
by an arbitrary constant λ, the mixture temperature is multiplied
by 1/λ, or
ð1=λÞT m = T m ðp m , Vm, λm 1 , λm 2 , ::.,λm N Þ
Show that differentiating this equation with respect to λ while
holding the mixture pressure and volume constant gives
∂ðT m /λÞ
= − T m
λ pm,Vm λ 2
= ∂T m
j pm,V m
∂λm1
∂λm 1 ∂λ
= ∂T m
∂λm 1
m 1 + … +
and setting λ = 1 gives
T m j pm,Vm = −
∂T
m
m 1 − …−
∂m 1
where
+ … + ∂T m
∂λm N
∂T m
∂λm N
m N
∂T m
m N =∑ N
∂m N
^T i = − ∂T
m
∂m i pm,Vm,mj
∂λmN
∂λ
m 1^T i =∑ N
T i
i = 1 i = 1
FIGURE 12.16
Problem 67.
Gas i at the mixture volume
V m
Is this the temperature fraction of gas i in a mixture of gases?
τ i = T i =T m
Also, in the case of a mixture of ideal gases, can the following
interpretation be developed similar to the Dalton and Amagat law
ideal gas partial pressure and partial volume? Can the partial
temperature of ideal gas i (T i ) be defined as the temperature
exhibited by ideal gas i when it alone occupies the volume of the
mixture V m at the pressure of the mixture p m ?Or,
T i = p mV m
m i R i