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European Journal of Applied Sciences – Vol. 10, No. 2
Publication Date: April 25, 2022
DOI:10.14738/aivp.102.12260. Partom, Y. (2022). TDRR Model Equations with Bulk Reaction Between the Burn Surfaces. European Journal of Applied Sciences,
10(2). 588-595.
Services for Science and Education – United Kingdom
TDRR Model Equations with Bulk Reaction Between the Burn
Surfaces
Yehuda Partom
Retired from RAFAEL, P.O. Box 2250, Haifa, Israel
ABSTRACT
The reaction rate of our reactive flow model (TDRR) is determined by the moving
burn fronts emanating from hot spots and any bulk reaction between those fronts
is ignored. Our TDRR reaction rate depends on the reactant temperature, and has a
multiplier called Burn Topology Function (BTF), that from lack of specific
knowledge we’re not changing for different situations. Recent shock initiation tests
with the explosive PBX9502 show that our TDRR reaction rate is somewhat too slow
towards the end of the reaction. To account for this small discrepancy, we check the
assumption that what increases the reaction rate in a computational cell is the bulk
reaction between the reaction fronts towards the end of the reaction. This calls for
expanding TDRR to include this bulk reaction. We find that including the bulk
reaction has very little influence on the end result. We then check whether by
changing somewhat our BTF can move our computational results closer to the test
results. We find that by increasing somewhat the initial value of our BTF, we’re able
to adjust our reactant burn rate to get a better agreement with test results.
INTRODUCTION
We’re checking here the validity of one of the assumptions of our reactive flow model called
TDRR (= reactant Temperature Dependent Reaction Rate) [1-3]. With the equations of TDRR
we assume that on the mesoscale, the explosive reaction develops by means of burn surfaces
that spread out of hot spots, and we refer to this mechanism as surface burn. The speed of the
burn surfaces increases with the temperature of the reactant in front of them, thereby making
the reaction rate to increase with the reactant temperature. The reaction rate equation includes
a multiplier that we call Burn Topology Function (BTF) denoted by y(W), where W is the
reaction parameter. The function y(W) starts from a low but finite value, increases to a
maximum (normalized to unity), and decreases to zero at W=1. Until recently there were no
data to calibrate the BTF for any specific explosive. We therefore assumed for it an artificial
function that fulfils the demands stated above. We refer to this function as our original BTF. It
is composed of two parabolas with a common tangent at W=W1=0.48 and y=1. When we
calibrate a reaction rate for some explosive, we always use our original BTF. But we promised
ourselves that whenever new data related to the BTF of some explosive becomes available, we’ll
use it to recalibrate an improved BTF for that explosive. Such new data became available for
PBX9502 through the tests of Gustavsen [4]. But before moving away from our original BTF, we
check another assumption that we always make. This assumption has to do with what happens
in the yet unreacted volumes of the considered cell, that are between the progressing reaction
fronts, at the final stage of the cell reaction. At this stage the temperature there may be quite
high, and bulk reaction [5] should be going on there. As a result, the temperature between the
expanding reaction fronts is probably higher than what we calculate when we ignore the bulk
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Partom, Y. (2022). TDRR Model Equations with Bulk Reaction Between the Burn Surfaces. European Journal of Applied Sciences, 10(2). 588-595.
URL: http://dx.doi.org/10.14738/aivp.102.12260
reaction, and so are the speed of the reaction fronts and the reaction rate. It is therefore
possible, that the slight disagreement between the test results of [4] and our predictions, are
not the outcome of our BTF but follow from ignoring the bulk reaction between the moving
reaction fronts. In what follows we check this possibility. To do this we upgrade our TDRR
model to consider simultaneously two reactions: 1. Surface burn at the reaction fronts coming
out of hot spots, and 2) bulk reactions between those reaction fronts. We then make
computations with the upgraded model to see if inclusion of the bulk reaction makes a change
that accounts for the small discrepancy between test data and TDRR predictions.
RATE EQUATIONS
The reaction rate equations that we develop and use here are for the special case of
hydrodynamic equilibrium, which means that all the different components considered on the
mesoscale have the same pressure. Applying this assumption of pressure equilibrium or
hydrodynamic equilibrium is permissible whenever the time scale of pressure equalization is
shorter than the time scale of the reactions considered. For instance, in both our surface burn
model and our volume reaction model, hydrodynamic equilibrium means:
(1)
where g denotes products (gas) and s denotes reactant (solids). For simultaneous surface burn
and bulk reaction of the explosive volumes between the expanding burn surfaces,
hydrodynamic equilibrium means:
(2)
where the index f is for the expanding reaction fronts, and the index b is for the reactant- products mixture between the fronts, where bulk reaction takes place. For a general
hydrodynamic equilibrium case with several variables, the macroscopic rate equation for any
of the variables x is of the form:
(3)
where Dx and Exi are coefficients derived from the state variables and from the equations of
state, and are the reaction rates of the various reactions. When there is only a single
reaction rate, we get:
(4)
and when there are two reaction rates, as in the case discussed here (surface burn plus bulk
reaction), we get:
(5)
To develop rate equations for a given mesoscopic situation we therefore need to derive the
coefficients Dx and Exi from the equations of state of the various components and from the mix- rules between them.
In what follows we develop the rate equations for the combined case, where we have surface
burn at the burn fronts expanding from hot spots, and bulk reaction of the volumes of explosive
between those fronts.
PPP
g s = =
PP P PP f gb sb b = = ==
n
x xi i
i 1
x DP EW =
! = +! ! å
Wi
!
x x x DP EW = +! ! !
x xb xf x D P E W FW =+ + !! ! !
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European Journal of Applied Sciences (EJAS) Vol. 10, Issue 2, April-2022
Services for Science and Education – United Kingdom
RATE EQUATIONS FOR THE COMBINED CASE OF BURN FRONTS OUT OF HOT SPOTS AND
BULK REACTION BETWEEN THOSE FRONTS
We start with bulk reaction between the burning fronts. The rate equations are the same as
those derived in [5].
(6)
(7)
where Q is the heat of detonation.
Solving equations (6) and (7) we get:
(8)
and also:
(9)
( )[ ]
( )
( )( )
sb sb gb gb pb wb b
sb b sb sb
gb b gb gb
pb b sb b gb
wb b
b sb gb sb gb
AV AV AP A W
A 1W E V Pq
A W E V Pq
A 1W E PWE P
A F
F QE E PqV V
+ =+
= - ¶ ¶ + +
= é ù ¶ ¶ + + ë û
= - - ¶ ¶ é ù + ¶ ¶ ë û
=
= + - + + -
! ! !!
sb sb gb gb pb
sb sb sb v b s0 s0
gb gb gb v b g gb
pb sb gb
BV BV BP
B E V P CT V
B E V P CT V
B EPE P
+ =
= ¶ ¶ + - G
= - ¶ ¶ é ù + - G ë û
= - ¶ ¶ -¶ ¶ é ù ë û
!!!
sb vsb vsb
gb vgb vgb b
b sb gb gb sb
vsb pb gb gb pb b
vsb wb gb b
vgb sb pb pb sb b
vgb wb sb b
V DP EW
VD P E W
AB AB
D AB AB
E AB
D AB AB
E AB
= +
+
D = -
= é ù - D ë û
= D
= - D
= - D
! !!
!! !
( )
b tb tb b
tb gb vgb gb v
tb vgb gb v
T DP EW
D E PDB C
E EB C
= +
= ¶ ¶ -
= -
! !!