Untitled - Aerobib - Universidad Politécnica de Madrid

3.6. ENTROPY VARIATION 71
Taking into this expression the variations of u and Y i given by Eqs. (3.34) and
(3.7) respectively, one obtains
ρT Ds
Dt =Φ + ∇ · (λ∇T ) − ∇ · (ρ ∑ i
− ∑ i
µ i w i + ∑ i
Y i h i¯v di )
µ i ∇ · (ρY i¯v di ).
(3.43)
As can easily be verified, this equation can be written in the form
[
]
ρ Ds
Dt =∇ · λ ∇T
T
− ρ ∑
Y i (h i − µ i )¯v di + 1 (∇T )2
[Φ + λ
T
T T
− ρ ∇T
T
i
∑
Y i (h i − µ i )¯v di − ρ ∑
i
i
Y i¯v di · ∇µ i − ∑ i
µ i w i
] (3.44)
Then we have 15 µ i = h i − T s i , (3.45)
where h i and s i are, respectively, the enthalpy and the entropy per unit mass of species
A i at the partial pressure p i of the species and at the temperature T of the mixture.
When taking this expression into Eq. (3.44) one obtains
[
ρ Ds
Dt =∇ · λ ∇T
T
− ρ ∑ ]
Y i s i¯v di + 1 (∇T )2
[Φ + λ
T T
i
− ρ∇T ∑ Y i s i¯v di − ρ ∑ Y i¯v di · ∇µ i − ∑ ]
µ i w i .
i
i
i
(3.46)
This equation admits a simple interpretation. In fact the first term of the right hand
side of this equation gives the reversible entropy flux across the boundary of the unit
volume. This flux originates from: a) heat transport through conduction, and b) diffusion
of the species. The second and third terms originate from the entropy generated
in the gas per unit volume and unit time, due to irreversibilities of the process. Such
irreversibilities, shown in Eq. (3.46), arise from: a) viscosity, b) heat conductivity,
c) diffusion, and d) chemical reactions. Making use of Eq. (3.11) the term ∑ µ i w i
i
corresponding to d) may be written in the form
∑
µ i w i = − ∑ r j α j , (3.47)
i
j
where α j is the chemical affinity corresponding to reaction j. 16 All contributions to
the variation of the entropy from the irreversibilities of the process must be positive. 17
15 See chapter 1.
16 See chapter 1.
17 For further information, see Refs. [4] or [5].