Page 277 - The Combined Finite-Discrete Element Method

260 FLUID COUPLING

the chamber; the detonation gas expands until the rigid chamber is completely filled
with the detonation gas. Because the chamber is rigid, no mechanical work is done by
the expanding gas. The chamber walls being impermeable to the flow of heat (adiabatic
walls) means that no heat transfer between the chamber walls and gas takes place. The
detonation gas expands into the chamber, and when all the transient motion has ceased
the gas reaches equilibrium. In this process the gas has done no mechanical work, and
no energy in the form of heat has been exchanged. As a result, no change in the internal
energy of detonation gas occurs.
In a state of thermodynamic equilibrium, the internal energy of the unit mass of deto-
nation gas, u, depends only upon the thermodynamic state of the gas, which in this case
is completely described by the specific volume v and temperature T . Consequently, any
change in u is given by

∂u ∂u
du = dT + dv (8.8)
∂T v ∂v T
where any changes in u depend only upon the initial and final states, and not on the path
(for instance, changing v at constant T and then changing T at constant v yields the same
u as changing T at constant v and then changing v at constant T ).
With an additional assumption that the internal energy per unit mass of the detonation
gas at a given temperature does not depend upon the specific volume, any change in
internal energy can be given in terms of the temperature only:
∂u

du = dT = c v dT (8.9)
∂T
v
where c v is the heat capacity per unit mass of the detonation gas at constant volume.
The heat capacity per unit mass of the detonation gas is in general a function of the
temperature of the detonation gas. However, for the sake of simplicity it is assumed that
c v is constant and calculated in such a way that the internal energy of the detonation gas
at maximum temperature is accurately represented. In other words, c v is such that the
specific explosive energy of the explosive is accurately represented.
No change in internal energy implies no change in temperature, and consequently the
temperature of the detonation gas after the expansion is completed and equilibrium state
is reached is equal to the temperature of the detonation gas before the expansion, i.e.
expansion is also isothermal and

u c = u 0 ; T c = T 0 (8.10)

where subscripts o and c indicate equilibrium states before and after the expansion.
From the equation of state, it follows that

p o p c
T 0 = = T c = (8.11)
b
b
R(ρ o + aρ o ) R(ρ c + aρ c )
which yields the expression for the pressure of the detonation gas after expansion, p c ,as
function of the density of detonation gas after expansion:
b b−1 b−1 b−1
p c ρ c + aρ c ρ c 1 + aρ c ρ c 1 + aρ o (ρ c /ρ o )
= = = (8.12)
p o ρ o + aρ o b ρ o 1 + aρ o b−1 ρ o 1 + aρ o b−1