Troels Dyhr Pedersen.indd - Solid Mechanics

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12.3.8 Determination of the Chapman Jouget velocity
The derivations leading to the expressions in this section are too extensive to be
explained, so the expressions are given as final results of these derivations. Please refer to
Glassman [25] for explanations and derivations.
Due to the convenience of built in functions for determination of all properties required
(internal energy, density and specific heats) as well as being a numerical solver, EES was
considered an ideal tool for this problem. The following equations constitute the basis for
solving the detonation problem. The entire set of equations implemented in the EES
model is found in appendix.
The initial conditions required are the pressure, temperature, density and internal energy
of the gas:
p = 2400000 Pa
( 9)
T
( 10)
1
1
= 726
ρ =
1
e
1
=
[ K]
xi
xi
[ ]
3
( ρ ) MW [ kg / m ]
MW
e
1
i
i
1
[ kJ / kg]
xi is the mole fraction of the individual species with density i (kmol/m 3 ) in the unburned
gas. MW1 is the average molar weight of the gas. Hence 1 is the average density, and e1 is
the average internal energy.
The model is setup assuming a non-reacted gas. The initial conditions may however
change if the gas is partly reacted. In that case the mole fractions must be set according to
the composition assumed, and the temperature and pressure adjusted as well. EES does
not have the capability to look up internal energies for radicals, so if the gas composition
has a large amount of radicals then other means of determining the internal energy must
be used.
The post detonation conditions require simultaneous solution of the average internal
energy e2 in the burned gas and the temperature T2, since these depend on each other. The
two equations required are 11 and 12. The specific heat ratio after the detonation is also
required and is calculated in eq. 13:
( 11)
( 12)
( 13)
e
e
γ
2
2
2
− e
=
=
1
R ⋅T2
= ½
γ
⋅ MW
x e
i
MW
i
i
2
i
x cp
i
x cv
2
i
2
[ kJ / kg]
[ kJ / kg]