Untitled - Aerobib - Universidad Politécnica de Madrid

78 CHAPTER 3. GENERAL EQUATIONS
reduces to
Y v d1
D 12
System (3.69) reduces in this case to the single equation
m = dε = w. (3.76)
dx
Equation of motion (3.72) remains the same and the energy equation (3.74)
m
(h 2 + (h 1 − h 2 ) ε + 1 )
2 v2 − λ dT
dx − 4 dv
µv = e. (3.77)
3 dx
Finally, the system of diffusion equations (3.75) reduces to the single equation
(
+ dY
dx + (M2 − M 1 ) [ ])
(M 2 − M 1 ) Y + M 1 Y (1 − Y ) dp
= 0, (3.78)
M 1 M 2 p dx
which making use of Eq. (3.68) that in this case reduces to
can be written as
mε = ρY (v + v d1 ) = mY + ρY v d1 , (3.79)
dY
dx + (M 2 − M 1 ) [ ]
(M 2 − M 1 ) Y + M 1 Y (1 − Y ) dp
M 1 M 2 p dx = m (Y − ε). (3.80)
ρD 12
The unknown of the problem are in this case v, p, ρ, T , Y and ε. The six equation
which determine their values are the following: continuity (3.65) and (3.76), motion
(3.72), energy (3.77), diffusion (3.80) and state (3.39). The system thus formed
will be studied in detail in chapters 5 and 6 in discussing the structure of the combustion
waves, specially under the simplified form studied in the following. Let us
assume, in particular, the following two conditions:
1) The heat capacities at constant pressure of both species are equal
c p1 = c p2 = c p . (3.81)
2) The molar masses of both species are equal, or else the gradient of pressure is
very small.
By virtue of the first assumption, and taking into account (3.60), one has
h 2 − h 1 = h 20 − h 10 = q, (3.82)
where q is the heat of reaction per unit mass of the mixture.
Due to (3.82), equation (3.77) reduces to
m
(h 2 − qε + 1 )
2 v2 − λ dT
dx − 4 dv
µv = e. (3.83)
3 dx