Troels Dyhr Pedersen.indd - Solid Mechanics

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12.3.5 Conservation equations
In the theoretical treatment of the wave it is regarded as a control volume. What happens
in the shock or detonation wave is not of interest in the theoretical treatment, only the
states before and after.
The shock wave is now regarded as being stationary within a control volume as defined
by figure 24. The conservation laws for the control volume are as follows:
Conservation of mass:
Conservation of momentum:
Conservation of energy:
The enthalpy is defined as:
( 1)
ρ u = ρ u
1
1
1
2
1u1
( 2)
P + ρ = P + ρ u
1
2
1
( 3)
h + ½u
= h + ½u
( 4)
h P
2
2
2
2
≡ c T + h°
where h° is the standard enthalpy of formation. In case of changes in the chemical
composition due to chemical reactions the heat release is defined as:
( 5)
q = h1°
− h2°
q is zero in a shock wave without heat release or dissociation, which means that the
enthalpy can be calculated from temperature alone.
12.3.6 The Rayleigh relation
By combining the equations for mass and momentum a relation for P2 as function of the
specific volume v2 is formed:
2
1
2 2
1 u1
2
2
2
2
2
( v )
( 6)
p = p − ρ − v
This relation is called the Rayleigh relation. It is universally applicable to both reacting
and non-reacting flows since the equations do not hold any term for energy. Given a set
of initial conditions (P1, v1 and u1) the relation can be plotted as a straight line in a p-v
plot as in figure 26.
2
1