Troels Dyhr Pedersen.indd - Solid Mechanics

3,0 x 10 6
2,5 x 10 6
2,0 x 10 6
1,5 x 10 6
1,0 x 10 6
5,0 x 10 5
P2 [pa]
Hugoniot kurve
3
2
- 64 - -
Rayleigh linje for U1=2000 m/s
Rayleigh linje for U1=1000 m/s
0,0
0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
x10 0
v2 [m^3/kg]
Figure 26: Rayleigh lines for shock waves at 1000 and 2000 m/s intersecting the Hugoniot line
The slope of the line also indicates the speed of the wave, since it may be written as:
2
2
2 1 1
2
( 7)
2 1
1
2
p − p u u
= −
= −
v − v v v
12.3.7 The Hugoniot relation
The third conservation requirement may be rewritten to include the pressure:
( 8)
h
2
− h
1
= ( p
2
− p
1
)
( v + v )
This expression is called the Hugoniot relation. It includes the change in enthalpy, which
in case of heat release determines the vertical position of the plot in the p-v diagram.
In non-reacting shock waves with no change in standard enthalpy the relation is referred
to as the Shock adiabat.
A solution to the Hugoniot relation has been plotted in figure 26, using the same
reference point (1) as the Rayleigh lines. The intersections between the Rayleigh lines
and the Hugoniot curve mark the specific solutions to the two velocities. It is noted that a
shock velocity of 1000 m/s results in a pressure increase five times the original pressure,
whereas 2000 m/s results in a pressure increase 20 times the original pressure.
Figure 27 illustrates the Rayleigh and Hugoniot relations as well as the various solution
regions I - V. The plot has been constructed in EES using typical conditions for HCCI.
2
2
1
1