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62 CHAPTER 3. GENERAL EQUATIONS
Therefore, the continuity equation for species A i is
Dρ i
Dt = −ρ i∇ · ¯v − ∇ · (ρ i¯v di ) + w i . (3.2)
If there are l different species A i , there is an equation similar to Eq. (3.2) for
each of them. 5
Adding the l equations corresponding to the l different species and making use
of the obvious relation
and of
∑
ρ i = ρ (3.3)
i
∑
w i = 0, (3.4)
i
which expresses the condition that the chemical reactions do not change the mass, and
of
∑
ρ i¯v di = ¯0, (3.5)
which was deduced in chapter 2, the following expression is obtained
i
Dρ
+ ρ∇ · ¯v = 0, (3.6)
Dt
which is the continuity equation for the mixture and is identical to the continuity
equation of Gas Dynamics.
Equation (3.6) allows simplification in the form of (3.2). For this purpose it is
convenient to express Eq. (3.2) as a function of the mass fraction Y i = ρ i /ρ of species
5 The detailed deduction of Eq. (3.2) is as follows.
The mass m i of species A i contained in volume V is
ZZZ
m i = ρ i dV.
V
Its time variation is (see Appendix)
dm i
= d ZZZ ZZZ
ρ i dV =
dt dt V
V
ZZZ » –
∂ρ i
∂t
ZZΣ
dV + ∂ρi
ρ i (¯v · ¯n) dσ =
V ∂t + ∇ · (ρ i¯v) dV.
The mass diffusing from V through Σ per unit time, is
Z
ZZZ
ρ i¯v di · ¯n dσ = ∇ · (ρ i¯v di ) dV.
Σ
V
The mass production in V by the chemical reaction, per unit time, is
ZZZ
w i dV.
V
Then when establishing the balance and expressing that this condition must be satisfied for all V , we obtain
Eq. (3.2).
In all the other formulae that follow the argumentation is identical, and for this reason the expansion of
the complete calculation is not considered necessary.