Global Continuity and the Continuity of the Species

Continuity equation: If M is the total mass in the system then N = M and for the system DN/Dt = DM/Dt = 0. In addition, since g = M/N = 1 Eq. 2.27 reads

(2.28)

Using the divergence theorem, the second term at the right hand side of Eq. 2.28 reads as (Hildebrand 1976),

JJJV.(qU)dV = фp (~ . dA) (2.29)

The new form of Eq. 2.28 becomes

JJJJp dV + JJJ V. (qA)dV = JJJ (jjp + V. (pU)^dV = 0 (2.30)

In Eq. 2.30, the control volume does not change with time therefore, the time derivative can be taken into inside of the first term without causing any alteration. Since the volume element dV is arbitrary and different from 0, to satisfy Eq. 2.30 the integrand must be zero to give the differential form, strong form, of the continuity equation.

OP + V.(p~) = 0 (2.31)

At high temperatures when the real gas effects take place, the air starts to disassociate and chemical reactions create new species. Because of this, we may need to write continuity of the species for each specie separately. If we consider specie i whose density is pt and its production rate is W in a control volume, then we have to have a source term at the left hand side of Eq. 2.27.

Iff WdV = OiIff Р^+ ff Pi (Ai-dA) (2.32)

Here, the velocity V, is the mass velocity of specie i. The differential form of Eq. 2.32 reads as

JPi +V. (Pi-Ui) = Wi (2.33)

Defining the mass fraction or the concentration of a specie with c, = p/p, the total density then becomes p = Rc;p;. The mass velocity Vi of a specie in a mixture is related with the global velocity as follows: V = Rc;V,. A mass velocity of a specie is found with adding its diffusion velocity Ui to the global velocity V i. e., Vi = V + Uj. According to the Ficks law of diffusion, the diffusion speed of a specie is proportional with its concentration. If we denote the proportionality constant with Dmi the diffusion velocity of i reads

U i ciDmiV ci

If we combine Eq. 2.34 with 2.31 and use it in Eq. 2.31, we obtain the conti­nuity of the species in terms of their concentrations as follows (Anderson 1989),

Dci ~ . .

q— = v • p (DmiVc^ + Wi