Page 281 - The Combined Finite-Discrete Element Method

264 FLUID COUPLING

and after substitution into (8.30)
b−1
aρ c
b−1 (ρ/ρ c ) b−1 −(k−1) [1−(ρ/ρ c ) b−1 ]
p/p c = (ρ/ρ c ) k 1 + aρ c e b − 1 (8.32)
b−1
1 + aρ c
By substituting ρ = 1/v
b−1
a/v c
b−1 )(v c /v) b−1 −(k−1) [1−(v c /v) b−1 ]
1 + (a/v c
p/p c = (v c /v) k e b − 1 (8.33)
b−1
1 + a/v c
It also follows that
e
p v c a/v c b−1 (b−1) ln(v c /v) 1 + a/v c b−1 (b−1) ln(v c /v)
ln = k ln − (k − 1) [1 − e ] + ln b−1
p c v b − 1 1 + a/v c
(8.34)
Again, as a result of a conveniently assumed equation of state, an analytical solution to
the gas expansion has been obtained.

8.2.4 Detonation gas expansion in a partially filled
non-rigid chamber

If the detonation gas is placed in a thermally insulating chamber, the walls of which
are able to move, and initially the chamber is only partially filled with detonation gas
(i.e. at initial density the volume of the gas is smaller than the volume of the chamber),
the expansion of the gas is coupled with the motion of the chamber walls. The initial
expansion of the detonation gas until the chamber is filled also involves pressure being
exerted onto the chamber walls and mechanical work being done by the expanding gas.
Thus, in contrast to the rigid chamber, the process would not be irreversible adiabatic.
This situation occurs in rock blasting when the borehole is only partially filled with
explosive. After initiation of the explosive charge, the detonation gas first fills the borehole,
exerting the pressure onto the borehole walls and gradually transferring energy onto the
rock in the form of mechanical work.
In the initial stages, even if the chamber walls are free to move, their motion is much
restricted by inertia. In real applications the mass of the fracturing solid is much larger
than the mass of explosive charge used to break the solid. Thus, the acceleration and
motion of the solid is relatively slow in comparison to the rate at which any transient
motion of the detonation gas yields thermodynamic equilibrium.
Consequently, at every stage of the motion of the chamber walls, it is assumed that
the detonation gas is in a state of thermodynamic equilibrium, and that its expansion in
a non-rigid chamber takes place in two stages:
• The first stage involves no mechanical work, and is completed when the chamber is
completely filled with detonation gas.
• The second stage involves expansion of the detonation gas in which the chamber walls
move together with the detonation gas and the internal energy of the detonation gas is
transferred into mechanical energy of the chamber walls.