Untitled - Aerobib - Universidad Politécnica de Madrid

5.2. WAVE EQUATIONS 107
of diffusion, this degree of advancement of the combustion differs from the mass
fraction Y (x) of the burnt gases that defines the mixture composition at point x,
that is, the fraction Y (x) of burnt gases that would be obtained by analyzing the
composition of a gas sample taken from the said point.
The simplifying assumptions previously stated do not limit the extent of the
study that follows, which is of a qualitative nature. On the other hand, these assumptions
simplify calculations.
The wave equations are obtained by particularizing the general equations of
continuity, momentum and energy 1 to the case of a one-dimensional stationary motion
(∂/∂y = ∂/∂z = ∂/∂t = 0). When this is done, and in addition to the preceding
assumptions are taken into account, the following system of equations is obtained:
a) Continuity equation.
that is
d (ρv)
dx
ρv = m,
= 0, (5.1)
(5.1.a)
where m is the mass flow normal to the wave, per unit surface.
b) Continuity equation for the burnt gases. Since the mass flow per unit surface is
ρv and the burnt fraction is ε, the mass flow of the burnt gases at section x is
ρvε. Its variation with x is due to chemical reactions. The equation that gives
this variation is
d (ρvε)
= w, (5.2)
dx
where w is the reaction rate. In virtue of (5.1.a), equation (5.2) can also be
written in the form
c) Momentum equation.
m dε
dx = w.
ρv dv
dx = − dp
dx + 4 3
(5.2.a)
(
d
µ dv )
. (5.3)
dx dx
This equation can be immediately integrated by taking into account (5.1.a), giving
p + mv − 4 3 µ dv
dx = i,
(5.3.a)
where i is an integration constant that must be determined by the boundary conditions.
2
1 See chapter 3.
2 In equations (5.3) and (5.4) it has been assumed that the second viscosity coefficient of the mixture
(see chapter 2) is zero. When not, it is sufficient to substitute µ for `µ + 3 4 µ′´ in (5.3) and (5.4).