Untitled - Aerobib - Universidad Politécnica de Madrid

70 CHAPTER 3. GENERAL EQUATIONS
Energy Equation
ρ Du
Dt = −p∇ · ¯v + Φ + ∇ · (λ∇T ) − ∇ · (
ρ ∑ i
Y i h i¯v di
)
. (3.34)
State Equation
p
ρ = R mT. (3.39)
Moreover the laws of variation of the thermodynamic functions, of the transport coefficients
and of the reaction rates as functions of the state and composition of the
mixture must be known.
3.6 Entropy variation
It is interesting to bring forth the entropy variation and therewith the causes of the
irreversibility of the process. Moreover, the entropy variations have a great influence
upon motion, for example, producing vortexes. 13
Thermodynamics teaches 14 that the variation dS of the entropy of a reactant
system corresponding to the variations dU, dV and dm i of its internal energy. volume
and masses of the chemical species respectively, is given by the expression
T dS = dU + p dV − ∑ i
µ i dm i , (3.40)
where µ i is the chemical potential of species A i .
If one applies this expression to the mass contained in the volume element V
of Fig. 3.1 following the motion, and bearing in mind that for this volume
S = ρV s, U = ρV u, m i = ρV Y i ,
1
V
DV
Dt
= ∇ · ¯v, (3.41)
where s is the entropy of the mixture per unit mass, the following expression is obtained
for the time variation of the entropy of the mass contained in the unit volume
(
ρ Ds
Dt = 1 ρ Du
T Dt + p∇ · ¯v − ρ ∑ i
µ i
DY i
Dt
)
. (3.42)
13 See chapter 9.
14 See chapter 1.