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European Journal of Applied Sciences – Vol. 10, No. 1
Publication Date: February 25, 2022
DOI:10.14738/aivp.101.11599. Partom, Y. (2022). Similarity Deviation of Diameter and Thickness Effects. European Journal of Applied Sciences, 10(1). 161-167.
Services for Science and Education – United Kingdom
Similarity Deviation of Diameter and Thickness Effects
Yehuda Partom
Retired from RAFAEL, P.O. Box 2250, Haifa, Israel
ABSTRACT
One way to characterize the similarity of detonations (in the sense of their reaction
rate) is through size effect tests. Using cylinders (rods), they’re called diameter
effect tests, and using slabs, they’re called thickness effect tests. From the steady
state equations of motion of these configurations it follows that they scale as
r(D)=h(D), where r=rod radius, h=slab thickness, and D=steady detonation velocity.
But tests show deviations from such scaling. Here we propose an explanation to
these deviations. The explanation is based on a previous work in which we show
that a boundary layer of partial reaction usually forms when a detonation wave is
grazing along a free boundary. We suggest and show via computer simulations that
those scaling deviations are related to this partially reacted boundary layer effect.
INTRODUCTION
It is customary to characterize sensitivity or insensitivity (in the sense of reaction rate in a
detonation wave) of explosives in terms of a size effect test. When the test explosive is a long
cylinder, it is known as the diameter effect test. In this test we shock initiate the explosive
cylinder (which is unconfined) at one end, and measure the steady detonation velocity after the
detonation wave has travelled a distance of several diameters. As is well known, the steady
detonation velocity (D) is lower for cylinders with a smaller diameter (d), and there exists a
failure diameter (df) under which the detonation wave decays and steady detonation is not
reached. Usually, we perform detonation velocity tests with cylinders of different diameters,
and plot the results as a D(1/d) curve called the diameter effect curve. We can then extrapolate
to 1/d=0 which represents a large diameter cylinder, and find the detonation velocity D0 which
is a reasonable approximation to the plane wave detonation velocity of the explosive (DCJ).
When the diameter effect curve of an explosive is flat (detonation velocity doesn’t change much
when the diameter increases), the explosive is regarded as ideal. This would happen for
explosives with high reaction rate and a narrow reaction zone. A good example of an ideal
explosive is a pressed HMX based mixture like PBX9501, with a reaction zone thickness of a few
tenths of a mm. An explosive with a more steeply descending diameter effect curve is regarded
as semi-ideal, and a common example for that are TATB based explosives, with a reaction zone
thickness of 1-2mm. And there are also explosives with a rather low reaction rate, regarded as
non-ideal. An example of such an explosive is ANFO = Ammonium nitrate + fuel oil.
In addition to the diameter effect test for characterizing the ideality measure or the sensitivity
measure of explosives, researchers sometimes also use the thickness effect test. In this test the
explosive sample is a wide slab. Similar to the diameter effect test we measure in this test the
steady detonation velocity (D) as function of the slab thickness (h), and the thickness effect
curve is D(1/h).
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European Journal of Applied Sciences (EJAS) Vol. 10, Issue 1, February-2022
Services for Science and Education – United Kingdom
Next we show that from the equations of motion in both geometries it follows that:
(1)
which means that for steady detonation in both geometries, the flow fields and detonation
velocities are identical when r=h.
Using cartesian coordinates, and adding -D to these equations we get for cylindrical symmetry:
(2)
where u,v,w are the x,y,z particle velocity components, and r=density.
but as the x and y directions are equivalent, we get:
(3)
For slab symmetry we get:
(4)
It follows that:
(5)
From momentum conservation we get for cyl. Symmetry:
(6)
but u=v, , so that . It follows that:
(7)
( )
( )
( )
( )
dD rD
2 or 1
hD hD = =
0 u v w for cylindrical symmetry ( ) ( ) ( ) tx y z
¶r ¶ ¶ ¶ = = r + r + r ¶¶ ¶ ¶
0 2 u w for cyl. symmetry ( ) ( ) t xz
¶r ¶ ¶ = = r + r ¶¶¶
0 u w for slab symmetry ( ) ( ) tx z
¶r ¶ ¶ = = r + r ¶¶ ¶
( ) ( )
h
2
2 1
x cyl. x slab
2 1 or r h or d 2h
r
= D D
\ = = =
u u uP uvw 0
x y zx
æ ö ¶ ¶ ¶¶
r + + += ç ÷ è ø ¶ ¶ ¶¶
x y
¶ ¶
¶ ¶ = u u u v x y
¶ ¶
¶ ¶ =
u uP 2u w 0 for cyl. symmetry
x zx
u uP u w 0 for slab symmetry
x zx
æ ö ¶ ¶¶
rç ÷ + += è ø ¶ ¶¶
æ ö ¶ ¶¶
rç ÷ + += è ø ¶ ¶¶
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Partom, Y. (2022). Similarity Deviation of Diameter and Thickness Effects. European Journal of Applied Sciences, 10(1). 161-167.
URL: http://dx.doi.org/10.14738/aivp.101.11599
which leads to:
(8)
which by using Eq. (5) leads to:
(9)
From Eqs. (5) and (9) it follows that a steady detonation wave in a rod of radius r (or diameter
d=2r), should be the same as a steady detonation wave in a slab of thickness h=r.
But tests [1] show that this kind of equivalence exists only approximately for ideal explosives,
and almost not at all for non-ideal explosives. To show this they measured in tests the diameter
effect curve D(1/d) for rods, and the thickness effect curve D(1/t) for slabs (where t=slab
thickness=h). In Figs. 1,2 and 3 we show the results obtained for these curves in [1] for the
explosives: PBX9501 (ideal), PBX9502 (semi-ideal) and ANFO (Ammonium nitrate + fuel oil,
non-ideal).
Figure 1. Diameter effect curve and thickness effect curve for PBX9501 (ideal explosive).
D=detonation velocity, d=rod diameter, t=slab thickness
( )
( )
( )
( )
u cyl. u slab 2
x cyl. x slab = D D
u cyl. u slab at r cyl. x slab ( ) ( ) ( ) ( )
where x 0 and x h are the slab boundaries.
= =
= =