Page 285 - The Combined Finite-Discrete Element Method

268 FLUID COUPLING

where k = c p /c v is the specific heat ratio, f is an averaged friction factor along the length
of the duct, A is the area of the duct and P is the perimeter of the duct.
Step 3: Assuming isentropic flow close to the entrance of the constant area duct, calculate:
(a) Pressure at the entrance of the duct, p 1 :

k/(k−1)
1
1 2
p 1 = p b 2 (8.36)
1 + [(k − 1)/2]Ma
1
where p b is the pressure of the detonation gas inside the borehole. This formula comes
from the assumption that the gas flow at the entrance to the crack can be described
by the flow of ideal gas through a converging no-friction duct.
(b) Temperature at the entrance of the duct, T 1 :
1
T 1 = T b 2 (8.37)
1 + [(k − 1)/2]Ma
1
where T b is the temperature of the detonation gas inside the borehole. This formula
also comes from the assumption that the gas flow at the entrance to the crack can be
described by the flow of ideal gas through a converging no-friction duct.
(c) Density at the entrance of the duct, ρ 1 is obtained

1 2 1/(k−1)
1
ρ 1 = ρ b 2 (8.38)
1 + [(k − 1)/2]Ma
1
where ρ b is the density of the detonation gas inside the borehole. This formula also
comes from the assumption that the gas flow at the entrance to the crack can be
described by the flow of ideal gas through a converging no-friction duct.
Step 4: Assuming nonisentropic flow with friction for the rest of the constant area duct,
calculate pressure at the exit of the duct, p 2 :
2 1/2
1 2
Ma 1 1 + [(k − 1)/2]Ma 1
p 2 = p 1 2 (8.39)
Ma 2 1 + [(k − 1)/2]Ma
2
and if the pressure at the exit of the duct is different from atmospheric pressure p a ,and
the assumed Mach number at the exit of the duct is smaller than 1,gotothe step 1.
Step 5: At the end of iteration, calculate the velocity at the entrance of the constant
area duct:

#
v 1 = Ma 1 kRT 1 (8.40)
and gas flow rate
dm
q = = v 1 Aρ 1 (8.41)
dt
where m is the total remaining mass of the detonation gas.