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European Journal of Applied Sciences – Vol. 10, No. 6
Publication Date: December 25, 2022
DOI:10.14738/aivp.106.13564. Partom, Y. (2022). Detonation Velocity as Function of Curvature from Diameter Effect Data. European Journal of Applied Sciences,
10(6). 434-443.
Services for Science and Education – United Kingdom
Detonation Velocity as Function of Curvature from Diameter
Effect Data
Y. Partom
Retired, 18 HaBanim, Zikhron Ya'akov 3094017, ISRAEL
ABSTRACT
Dn(k) (normal detonation velocity as function of curvature) is usually calibrated
from rate stick breakout data. Using this approach, it is often found that sections of
the Dn(k) curve, obtained from sticks of different diameters, do not agree at
overlapping k ranges. We therefore ask if it is possible to calibrate Dn(k) from
diameter effect data, and show that this is indeed possible. We use for that an
indirect calibration procedure: 1) assume a Dn(k) curve; 2) for different values of
the steady rate stick detonation velocity Ds, integrate the two front shape ODEs until
the boundary angle is reached, and record the resulting diameter d; and 3) compare
the resulting Ds,d pairs to the experimental diameter effect curve. We show here
that by appropriately correcting the assumed Dn(k) curve, we can easily get
agreement with the experimental diameter effect curve, thereby completing the
calibration procedure. As a byproduct we show that the Dn(k) curve must have a
limit curvature, or otherwise the failure stick diameter is not reproduced. Finally,
we apply the same procedure to predict the size effect (like diameter effect, but for
nonstick geometries) and the breakout curves for other steady detonations.
INTRODUCTION
The model called DSD (= Detonation Shock Dynamics) makes it possible to calculate
propagation of a steady or quasi-steady diverging detonation front, independent of the
hydrodynamics behind it [1,2]. According to DSD, what determines the normal detonation
velocity (Dn) at a point on the detonation front is the mean curvature (k) of the front at that
point. To apply the DSD model, we therefore need to measure or calibrate the relation Dn(k) for
the considered explosive. In addition, what determines the front shape as a whole is also the
boundary angle qb (= angle between the front and the normal to the boundary), specific to the
explosive and the inert material behind the boundary. Many times, DSD is applied to situations
that are not quite steady-state. When this is the case, normal detonation velocity at a point on
the front is still determined by Dn(k), but the boundary angle condition is treated as a limit
condition: qb3qL (= Limit angle).
The standard practice of calibrating D(k) and qL is from steady detonation in cylinders (or rate
sticks). In a typical test one measures the steady axial detonation velocity Ds, the breakout curve
from the far end of the stick t(r), and the boundary angle qb. Tests like this are performed for
several stick diameters (d), and each of them covers a certain range of the curvature k with
some overlap. The procedure for extracting Dn(k) and qb from such rate stick test data is as
follows: 1) convert the breakout curve t(r) to a front shape curve zex(r), using the measured
axial detonation velocity Ds; 2) curve fit zex(r) to obtain z(r). Such curve fitting is needed because
we have to differentiate z(r) twice to obtain the curvature k, and the end result is quite sensitive
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Partom, Y. (2022). Detonation Velocity as Function of Curvature from Diameter Effect Data. European Journal of Applied Sciences, 10(6). 434-443.
URL: http://dx.doi.org/10.14738/aivp.106.13564
to experimental noise and to the curve fit function; 3) differentiate z(r) once to get q, Dn=Dscosq,
and the perpendicular curvature component k2=sinq/r; 4) differentiate twice to get the
curvature component k1 and the mean curvature k=(k1+k2)/2. We call this kind of approach
Differential Calibration (= DIFC). Using DIFC we encounter three problems: 1) results are
sensitive to the curve fit function; 2) Dn(k) sections extracted from tests with different stick
diameters do not connect to a single curve [3,4]; 3) the slope of z(r) changes rapidly near the
boundary, and the error in extracting qb is large.
On this background we ask the following: 1) is it possible to assume Dn(k) and qb and calculate
from them the diameter effect Ds(d)?; 2) if yes, can we go further and calibrate Dn(k) to match
the measured diameter effect. We call this calibration procedure Integral Calibration (= INTC).
In [5] we’ve already demonstrated that INTC might work.
In what follows we first perform integral calibration to match diameter effect data from [3,6].
We then apply the results to predict the diameter effect (or size effect) and breakout curves for
other steady detonations.
INTEGRAL CALIBRATION
As outlined above, to perform integral calibration we need the diameter effect curve Ds(d), and
the boundary angle qb. For Ds(d) we use data from [3,6], as shown in Fig. 1. The value to use for
qb is somewhat uncertain, as demonstrated in [3]. We performed reactive flow calculations of
detonation in sticks (not shown here), and extracted from them the boundary angle. We
obtained qb=0.5 radian, which is at the lower end of the range given in [3], and this is what we
use here. We show later, that instead of using qb we can use the boundary front lag Δz, which is
determined from rate stick tests with less uncertainty. Also, as qb changes rapidly near the
boundary, integral calibration is not sensitive to it.
Figure 1. Diameter effect data from [3,6].
7.4
7.45
7.5
7.55
7.6
7.65
7.7
0 0.02 0.04 0.06 0.08 0.1 0.12
Axial detonation velocity (km/s)
Inverse diameter (1/mm)
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European Journal of Applied Sciences (EJAS) Vol. 10, Issue 6, December-2022
Services for Science and Education – United Kingdom
The procedure to calculate Ds(d) from Dn(k) and qb is: 1) choose a value for Ds. In fact, we go
through a Ds range in steps ΔD, starting from DCJ; 2) integrate the detonation front equations
along the front, starting from the axis. These equations are:
(1)
where:
(2)
Eqs. (1) are a system of two first order ODEs, and we integrate them numerically from q=0 to
q=qb, where r=a and d=2a. The pairs (Ds,d) thus obtained are on the diameter effect curve
corresponding to the Dn(k) (or k(Dn)) relation assumed.
Initially we assumed for Dn(k) a straight line Dn=DCJ(1-bk), with b=1mm. In Fig. 2 we compare
the Ds(d) relation obtained to the data points from Fig. 1. We see that: 1) the calculated curve is
convex upwards, has a horizontal tangent and goes above the data points near 1/d=0; 2) at high
1/d values the calculated curve goes below the data points. We conclude that Dn(k) needs to
have a higher slope initially and a lower slope later on. The easiest way to achieve that is by
using a bi-linear curve. After some trials we ended up with the following curve:
(3)
In Fig. 3 we show the agreement obtained with the curve defined by Eqs. (3).
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