Troels Dyhr Pedersen.indd - Solid Mechanics

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In the expressions above, e2 is the internal energy, R is the universal gas constant, 8314
J/kmol-K, T2 is the temperature, 2 is the specific heat ratio, MW2 the average molar mass,
cp is the heat capacity at constant pressure and cv is the heat capacity at constant volume.
Subscript 2 refers to post detonation conditions.
The internal energy e2 depends on the chemical composition of the combustion products.
A first guess on chemical composition can be made by finding the chemical equilibrium
for a constant volume explosion in CHEMKIN.
The CHEMKIN reactor is initialized with conditions that match the conditions in the
cylinder at a compression ratio of 10. An adiabatic compression results in a pressure of
24 MPa and a temperature of 726 K. The initial gas composition is calculated from an
equivalence ratio of 0.33.
The resulting temperature from the CHEMKIN run is around 1800 K and the pressure
around 6 MPa. This temperature results in negligible dissociation, hence complete
combustion products are assumed. For higher combustion temperatures dissociation will
however be more significant, and the full range of equilibrium products must be included
in the calculations of internal energy.
With proper guess on the mass fractions of species (in this case complete combustion
products) after the detonation, the internal energy and specific heat ratio may be
determined. The temperature T2 is now found by iteration in EES. It is required that the
guess for T2 is close to the final value for the solution to converge. It was found that the
best way to find the correct temperature was to insert a temporary error variable in
equation (5) and then find its minimum through parametric variation of T2.
T2 is found to be approx. 1989 K. The pressure is then determined from the following
expression:
( 14)
The constant μ is defined as:
p2 MW1
T2
= μ p1
MW2
T1
( 15)
γ 2 + 1
μ =
γ
2
[ Pa]
2 was found previously. μ has a value close to 1.8 if the detonation takes place in air.
Solving equation 14 gives a pressure of approx. 12 MPa.
It is required to evaluate the chemical composition once again at the new temperature and
pressure, since dissociation may have become significant. The new equilibrium
calculation is performed as a constant volume reaction in CHEMKIN. The reactor is
initialized with the new temperature T2, pressure p2 and the complete combustion
products. At equivalence ratio of 0.33 it is found that it still reasonable to neglect