Calculation of Both High Velocity Detonation and Low Velocity Detonation
Processes Using a Unified Model
Shaoming Hu and F. Richard Yu
Department of Systems and Computer Engineering, Carleton University, Ottawa, ON, Canada
Email: [email protected], [email protected]
Abstract. Explosives detonate at two stable velocities: high velocity detonation (HVD)
and low velocity detonation (LVD). But the ZND model, the classical detonation theory,
only has a single mathematical solution. Taking into account the transport effects and
complex multidimensional movement neglected by the ZND model, a new detonation
model, named Least Action Detonation Model (LADM), has four mathematical solutions.
In this paper, we use two of these solutions to calculate both HVD and LVD. Explosion
heat Q measured by calorimeter is used in the calculation. When γi, the parameters in
equation of state (EOS), are near to the theoretical values, the calculated detonation
velocities of four kinds of explosives coincide with experimental data well. Moreover, the
other two solutions in the LADM have the potential for studying other
detonation/combustion phenomena.
Introduction
Since discovered by Stettbacker in 1919, low
velocity detonation has been spurring detonation
scientists for a long time [1]. The well-known
ZND model [2] only has a single mathematical
solution for high velocity detonation (HVD), and
no other independent mathematical solution is
available for low velocity detonation (LVD).
During the early days of LVD study, some
factors that cause LVD have been studied, such as
density, diameter of charge, air bubbles in
explosive, shell materials, particle size, charging
conditions, etc. [3].
Recently, researchers have investigated other
factors, such as two-step chemical reaction [4],
collision of a shock wave with roughness [5],
diffusion of pressure [6], chemical–acoustic
interactions [7], etc.
Efforts have also been made in theoretical
study: Manzhalei [8] and Grebenkin [9] proposed
models for low velocity detonation; Eyring [10],
Fickett [11] and Schall [12] proposed models that
have two detonation velocities, etc.
Although these excellent works have been
done, to the best of our knowledge, very few
theoretical models have been proposed that can
summarize these factors and calculate LVD
processes, let alone a unified theoretical model for
both HVD and LVD processes.
Recently,
a
multi-solution
theoretical
detonation model, named Least Action Detonation
Model (LADM), has been proposed [13] [14], and
has been verified by experiments [14] (See
Appendix 1). Taking into account the transport
effects and complex multidimensional movement
neglected by the ZND model, the LADM has four
mathematical solutions that correspond to a variety
of phenomena related to detonation and
combustion [15].
In this paper, both HVD and LVD are
calculated with two solutions of this unified
LADM. Detonation velocities of nitroglycerine
(NG), trinitrotoluene (TNT), nitromethane (NM)
and hexogen (RDX) are calculated. Calculation
results show that, when the parameters γi are near
to the theoretical values, the calculated detonation
velocities coincide with the experimental data well.
Moreover, the other two solutions in the LADM
have the potential for studying other
detonation/combustion phenomena.
For the convenience to reproduce and check
the calculation results, all of the original data and
the intermediate calculating steps are presented in
this paper. In addition, the executable program is
also available upon request (please send us an
email for it). Besides these four explosives
calculated in this paper, this program can calculate
other explosives as well (γi of specified explosive
should be determined by experimental data before
using this program).
The rest of this paper is organized as follows.
Section 1 presents the LADM; Section 2 and
Section 3 detail the calculation of HVD and LVD;
Section 4 discusses the potential of the LADM.
Finally, we conclude this study in Section 5.
Appendix 1 describes the verification of the
LADM. Appendix 2 describes that the LADM
settles the entropy argument.
1. The Least Action Detonation Model (LADM)
Being a simple theory, the ZND model [2]
neglects transport effects and assumes that the
detonation products movement is one-dimensional.
But, experiments show that the transport effects do
exist and complex multidimensional movement
play an important role in the detonation process [2],
which should be taken into account by more
sophisticated models.
The LADM uses the entropy principle to
specify the result of transport effects, and
Hamilton’s principle to describe the complex
multidimensional movement of detonation
products particles, and to determine the reaction
path from explosive to detonation products [13]
[14].
Similar to the ZND model, the detonation
wave of the LADM is also composed of three parts:
the front, the reaction zone and the final point. The
front is the same in both models. In the LADM,
because the transport effects and complex
multidimensional movement play an important
role, the reaction zone and the final point are
different from those in the ZND model.
Fig. 1 shows the wave configuration and p-v
plane of HVD in the LADM.
H2
H1
p
Reaction
path
p
1
1
front
2
Reaction path
R
20
0
0
distance x
v0
Configuration
p-v
Fig. 1. High velocity detonation in the LADM [15].
1.1. Shock Front (line "0" → "1")
Similar to the ZND model, the shock front of
the LADM is described by conservation laws:
1 ( D  u1 )   0 ( D  u0 )
(1)
p1  1 ( D  u1 ) 2  p0   0 ( D  u0 ) 2
( 2)
1
1
e1  p1v1  ( D  u1 ) 2  e0  p0 v 0  ( D  u0 ) 2 (3)
2
2
where D is detonation velocity, u is particles’
velocity, p is pressure, v is specific volume, ρ is
density, and e is internal energy. Subscription "0"
is explosive, "1" is the front, and "2" is the final
point of detonation process.
1.2 Final Point (Point "2")
According to the Second Thermodynamic Law,
the entropy at the final point of any thermal
process must be the maximum, so do the
detonation process:
S2  max .
( 4)
where S is entropy.
According to the First Thermodynamic Law,
the internal energy at final point "2" of a
detonation process is:
e 2  e0  Q
(5)
where Q is the explosion heat measured with
calorimeter.
The Entropy at point "2" is the maximum
instead of the minimum at the CJ point, which
settles the entropy argument in detonation science
(see Appendix 2).
1.3 Reaction Zone (Path "1"→"2")
It is difficult to describe the transport effects
and the complex multidimensional movement by
conventional method such as Conservation Laws.
Therefore, the LADM uses the least action
principle to describe them.
The least action principle is one of the basic
principles in nature, from which nearly all of the
physical equations can be derived, including the
equations of Newton’s mechanics, relativistic
mechanics, Maxwell equations, electrodynamics,
Fermat’s principle in optics, Schrodinger equations
in quantum mechanics, etc. Hamilton principle is
the least action principle in mechanics, which is:
2

1
 Ld  0
( 6)
where δ is variational calculus, τ is time, L= T-V
is Lagrangian function, T is kinetic energy, and V
is potential energy.
Eq. (6) means that, for a conservative system,
the integral of the Lagrangian function has a
stationary value for the actual curve compared
with all other theoretical curves having the same
start and end points that performed in the same
time.
In mechanics, Hamilton’s principle and
Newton’s law are equivalent. Hamilton’s principle
use energy relations; while Newton’s law in
classical detonation theory uses force relations.
However, the same result will be achieved.
Studying energies instead of forces,
Hamilton’s principle bypasses the difficulty to
describe
transport
effects
and
complex
multidimensional movement in detonation process.
From Eq. (6), we have:
2

1
 (T  V )d  0
(7)
In order to apply Hamilton’s principle,
Lagrangian function including kinetic T and
potential energies V shall be specified at first. The
order energies, kinetic energy and chemical energy
dissipate into disorder thermal energy in
detonation process. Lagrangian function correlates
these energies. The kinetic energy T is [13]:
D 2v 2
(8)
T
2 v 02
where β is a function of p and v, i.e., β=β(p,v),
which is a coefficient that characterizes the
complex multidimensional movement [13].
When all the particles of detonation products
move towards the same direction of the detonation
wave, β = 1; when particles of detonation products
move in different directions, β < 1.
In analytical mechanics, the potentials V are
mechanics ones. But in detonation systems, only
e+pv and Q exist, which are not mechanics
potentials.
According to thermodynamics,
e  pv  H
( 9)
where H is enthalpy.
Q is the reaction heat transformed from
chemical energy, which is the energy that
maintains detonation wave propagation. According
to Hess’s Law, chemical reaction heat is the
difference between the enthalpy of detonation
products H2 and the enthalpy of explosives H0, i.e.,
Q  H2  H0
(10)
Glasdorff and Prigogine [16] pointed out that
the thermodynamic state function, enthalpy H, is
the thermodynamic potentials.
Therefore, in detonation processes, the
potential energy is:
V  e  pv  Q  H  Q
(11)
where λ is reaction degree.
Although the kinetic and potential energies
vary in an adiabatic system (no heat and mass
exchange with the surroundings), their sum
remains constant, i.e., it is conservative. For all
sections in the reaction zone, the sum of kinetic
energy and potential energy conserve.
D 2v 2
 H  Q  Const.
(12)
2 v02
Then, for detonation processes, Lagrangian
function L is specified as:
D 2v 2
 H  Q
(13)
L  T V 
2 v 02
Eq. (13) means that the order energies, kinetic
energy D 2 v 2 2 v 02
and chemical energy Q,
transform into disorder thermal energy H, i.e., the
complex movement and transport effects play roles
and have been taken into account by the LADM.
Substituting Eq. (13) into Hamilton’s principle
Eq. (6):
2
D 2v 2

[
(14)
1 2v02  H  Q ]  0
E  T V 
The mathematical solutions that make the
integral of Eq. (14) be equal to zero are the real
paths in the reaction zone of a detonation wave.
2. Calculation of High Velocity Detonation
Using the LADM
By variational calculus, we have obtained four
solutions [15] that correspond to different
detonation/combustion phenomena, among which
Solution One and Solution Two are used for
calculating HVD and LVD, respectively.
2.1. Solution One: Isothermal Curve
Using variational calculus to Eq. (14), we can
derive the first solution. When:
t  Const.
(15)
where t is temperature, and

D 2 v 2  D 2 v v H 



Q  0 (16)
2  2 v 02 p v 02 p p p
Eqs. (15) and (16) that make Eq. (14) be equal to
zero are the first solution.
Fig. 1 is the wave configuration of high
velocity detonation of the LADM. Shocked by the
front, the state point moves from the explosive
point "0" up the Rayleigh line to the Neumann
spike point "1", then, chemical reaction takes place
and the state point moves down the theoretical
reaction path "1" → "2" to point "2".
As Fig.1 shows, this solution has a sharp
pressure PEAK, therefore, the obtained velocity is
higher than the one driven by the pressure plateau
in LVD as shown in Fig. 2.
Eq. (16) describes the profile of the reaction
zone.
2.2 Original Data
The original data used in the calculation are
shown in Table 1:
Table 1. Original data of explosives
No.
NG
TNT
NM
ρ
1590
1000
1128
Q 6445000 4185000 5150000
cv
942
1372
1720
ρ: density of explosive (kg/m3)
Q: explosion heat (J/kg)
cv : specific heat capacity (J/kg· K)
RDX
1800
5450000
1172
Please note that Q is the heat measured by
calorimeter. For comparison, in the ZND model
the QCJ is an intermediate datum in calculation,
which cannot be measured and is not always
published in the papers concerning detonation
velocity calculation.
2.3. The Equation of State
In this paper, γ- EOS is used in calculation:
ei 
pi vi
(17)
i 1
According to thermodynamics, the theoretical
value γ of multi-atom molecule is:
c
9
  p   1.286
cv 7
and the theoretical value γ of double-atom
molecule is:
c
7
  p   1.40
cv 5
where cp is constant pressure specific heat, cv is
constant volume specific heat.
Fitted with experimental data, the values of γi
used in calculation are listed in Table II.
Table 2. Values of γi used in calculation
No.
NG
TNT
NM
RDX
γ1
1.312
1.254
1.239
1.260
γ2
1.480
1.480
1.480
1.480
γ1: adiabatic index of explosives
γ2: adiabatic index of detonation products
It is interesting that the γ1 is near to the
theoretical value of multi-atom gases 1.286; and
the γ2 is near to the theoretical value of doubleatom molecular gases 1.40.
In the ZND model, γ ≈ 3, which is far from the
theoretical value [2].
2.4. Calculation of Pressure and Temperature at
Point "2"
From Eq. (5), we have:
t 2  t0  Q
(18)
cv
From Eq. (17), we have:
Q ( 2  1)
p2  p0 
v2
(19)
The pressure and temperature at the final point
"2" are calculated as shown in Table 3.
Table 3. Pressure and temperature at point “2”
No.
NG
TNTρ=1000
NM
RDX
p2 4.92·109 2.01·109 2.79·109 4.71·109
t2
7115
3323
3267
4923
H1 H2
p
Reaction
path
p
front
2
2.5 HVD Calculation Result
According to thermodynamics, for adiabatic
processes:

 1
 t1    p1 
 t   p 
0
 0 
We can get p1 as:
( 20)
1
 1
t
p1  p0  1  1
 t0 
From Eq. (15), we have:
(21)
t1  t 2
( 22)
Substituting Eq. (22) into Eq. (21), then:
1
 1
p1  p0  t 2  1
( 23)
t
0


The internal energy e1 is:
pv
( 24)
e1  1 1
 1 1
The detonation velocity D of a HVD process
is obtained by solving the group of Eqs. (1), (2),
(3), (23) and (24). The calculation result is shown
in Table 4.
Table 4. Experimental and
velocity detonation
No.
NG
TNTρ=1000
Exp.
8140
5010
Calc.
8091
5073
calculated High
NM
6290
6203
RDX
8754
8760
Reaction path
R
20
1
0
0
distance x
Configuration
v0
p-v
Fig. 2. Low velocity detonation in the LADM [15].
Fig. 2 is the wave configuration of low
velocity detonation of the LADM. Shocked by the
front, the state point moves from the explosive
point "0" up the Rayleigh line to the Neumann
spike point "1", then, chemical reaction takes place
and the state point moves along the horizontal
reaction path "1" → "2" to point "2".
Some factors, such as those studied in [3]-[7],
level the pressure peak of detonation wave to a
pressure PLATEAU, as shown in Fig.2. Therefore,
the obtained velocity is lower than the one of HVD
driven by the pressure peak, as shown in Fig.1.
3.2. The Original Data
They are the same as those used in calculating
HVD.
3.3. The Equation of State (EOS)
It is the same as that used in calculating HVD.
3.4. LVD Calculation Result
From Eq. (25), we have:
p1  p 2
( 27)
3. Calculation of Low Velocity Detonation
Process Using the LADM
3.1. Solution Two: Isobaric Curve
Using variational calculus to Eq. (14), we can
derive the second solution. When
p  Const.
( 25)
Substitute Eq.(27) into Eq.(17), then:
pv
e1  2 1
( 28)
 1 1
The detonation velocity D of LVD process is
obtained by solving group of Eqs. (1), (2), (3), (27)
and (28). Table 5 dhows the calculation result.
and
Table 5. Experimental and calculated low velocity
detonation
No.
NG
TNTρ=1000 NM
RDX
Exp. 1740<2000
1600—
2030
1700
Calc. 1891
1505
1664
1719
D 2 v 2  D 2 v H 




Q  0 ( 26)
2  2 v02 v v02 v v
Eqs. (25) and (26) that make Eq. (14) be equal to
zero are the second solution.
Eq. (26) describes the profile of the reaction
zone.
4. Potential Applications of the LADM
Besides the above two solutions applied to
HVD and LVD, the LADM has other two
solutions that have the potential for studying other
detonation/combustion phenomena..
Using variational calculus to Eq. (14), we can
derive the fourth solution. When

D 2 v 2  D 2 v v H 



Q  0 (33)
2  2 v 02 p v02 p p p

D 2 v 2  D 2 v H 



Q0
2  2 v02 v v02 v v
and
4.1. Solution Three: Isochoric Curve
Using variational calculus to Eq. (14), we can
derive the third solution. When:
v  Const.
( 29)
and

D 2 v 2  D 2 v v H 



Q  0 (30)
2  2 v 02 p v 02 p p p
Based on Eq. (29), we have
D 2 v v
0
v02 p
(31)
Then, Eq. (30) becomes:

D 2 v 2  H 


Q0
2  2 v 02 p p p
(32)
Eqs. (29) and (32) that make Eq. (14) be equal
to zero are the third solution.
Fig. 3 is the wave configuration of this solution.
While chemical reaction is initiated, temperature
of explosive at point "0" begins to rise, and state
point goes up the constant volume line vertically
along path "1" → "2" to the final point "2". Eq.
(32) describes the profile of the reaction zone.
There is NOT SHOCK in this solution.
This process is likely controlled by other
factors, such as heat transfer, which might be a
combustion phenomenon.
H1
p
H2
p
Reaction path
Reaction
path
2
20
0
0
distance x
Configuration
v0
p-v
Fig. 3 Isochoric solution [15].
4.2. Solution Four: Multivariable Curve
In this solution, all of p, v and t are variables.
(34)
Eqs. (33) and (34) that make Eq. (14) be equal to
zero are the fourth solution.
The wave configuration and physical meaning
of this solution are still unknown.
4.3. The LADM Includes More Information
Substituting the obtained velocity D into Eqs.
(16) and ((26), we can obtain β=β (p,v), which
contains information of the complex structure in
the reaction zone.
With different boundary conditions, the
differential equation solutions of the LADM shall
describe the reaction zone under these boundary
conditions.
5. Conclusion
In this paper, using a unified Least Action
Detonation Model (LADM), we showed that both
high velocity detonation (HVD) and low velocity
detonation (LVD) processes can be calculated.
Corresponding to the complex and varied
detonation phenomena, the LADM has four
mathematical solutions that provide much more
information than the single solution of the ZND
model. We showed that, when coefficients γi are
near to the theoretical values, two solutions of the
LADM can calculate detonation velocities of both
HVD and LVD. We demonstrated the
effectiveness of the proposed approach using
numerical examples of four explosives, NG, TNT,
NM, and RDX. All of the original data and the
intermediate calculating steps were presented.
The LADM has four sets of solutions, two of
which have been used to calculate HVD and LVD
processes in this paper. The other two sets of
solutions have the potential for studying other
detonation/combustion phenomena.
Appendix 1: Verification of the LADM
Because the LADM takes transport effects and
complex multidimensional movement into account,
the detonation wave configuration described by the
LADM differs from that described by the ZND
model: the flow after the reaction zone in the ZND
model is the Taylor rarefaction in which
detonation particles always move forward [2],
whereas it is a stationary state in the LADM.
MOVE or NOT? The flow state of detonation
products can serve as the criterion for these two
models.
A1.1. Stationary Titanium Foil in Detonation
Products
From the displacement of titanium foils
embodied in the explosive charge, the movement
state of detonation products particles can be
known [14].
Fig. 4 shows the position of the titanium foil at
0 μs, 3.6 μs,_and 5.6 μs, sequentially [14]. Flashes
are taken twice that create two images at one photo:
the mark “before wave” means the original
position of foil in explosive; “behind wave” means
the instant position of foil in detonation products.
Shocked by the detonation front, the titanium
foil at first moves 1-2 mm because of the moving
particles in the reaction zone; then stops moving,
which coincides with the prediction of the LADM.
It is clear that during the time interval from 3.6
μs_to 5.6 μs, the front moves down a certain
distance, but the foil remains at the same position,
which means that the detonation products is in
stationary state in the after-flow of detonation
products. They do NOT move!
A1.2 The LADM has been Verified Long Time
Ago by Predecessors
Concerning above experiment, the most
frequently asked question is: “Why has the
stationary state of detonation products never been
observed before?” In fact, much evidence that
detonation particles are in a stationary state has
already appeared in the literature on detonation in
the form of data, graphs and photographs. Because
such evidence contradicts with the ZND model,
the stationary state has not been addressed.
Moreover, it has never been considered as the
essence of detonation.
For example, in the 5th Int’l Symp. on
Detonation (Pasadena, 1970), Rivard, Venable,
Fickett and Davis from Los Alamos Scientific Lab.
published the following work [17].
before
wave
behind
wave
front
3.6μs
5.6μs
Fig. 4 Position of titanium foil in detonation products at different times [14]
The flow field behind a plane detonation
wave in Composition B-3 was examined by
radiographically observing the motion of metal
foils placed between slabs of explosives [17] [18].
The schematic of typical experimental assembly is
shown by Fig. 5 [18].
That is, at a certain distance apart from the front,
particles velocity is zero. They do NOT move!
This experiment carried out in LANL by Rivard
et al. [17] [18] has already observed the stationary
state, which has proved the LADM.
Δh
foils
Plane wave generator
detonator
Fig.5 Schematic of typical experimental assembly
by Rivard and Venable [18].
Table 6 shows the interval Δh between foils
measured before and after detonation wave. The
results are calculated by us based on the original
data taken from Shot NW431, Table 2-I [18].
Table 6. Space between foils before and after the
detonation wave [18]
No.
Δh0
Δh2
1-2
6.35
6.28
2-3
6.33
6.40
3-4
6.34
6.11
4-5
6.35
6.01
5-6
6.36
5.22
6-7
6.38
5.86
7-8
6.38
5.17
8-9
6.34
4.80
where: Δh0 is space between two foils before
detonation; Δh2 is space between two foils after
detonation.
Near the front, where foils "8"and "9" locate:
i.e.,
products is equal to the specific volume of the
explosives. i.e.,
v2  v0
(38)
Substituting Eq. (38) and u0=0, the particles
velocity of explosive, into mass conservation Eq.
(36), we have:
u2  0
(39)
h2  h0
v2  v0
(35)
Substituting it into mass conservation
( D  u2 ) / v 2  ( D  u0 ) / v0
(36)
then we have:
u2  0
That means in the reaction zone near the front, the
detonation products move forward.
In the after flow, where foils "1", "2" and "3"
locate:
h2  h0
That means, here the specific volume of detonation
A 1.3 The “NOT Moving State” has been Used by
the Gurney Model Since 1940s
In explosion engineering, the most popular
model is the Gurney model, which is simple and
accurate, and has been extensively used since
1940s [19].
Gurney model is established on the assumption
that the detonation products have a uniform
density without longitudinal movement, i.e., after
the reaction zone the particles of detonation
products do NOT move towards the direction of
detonation propagation, which is exactly the
stationary state point "2"" of the LADM.
The success of the Gurney model proves its
assumption, so proves the LADM.
A1.4 The “NOT Moving State” is the Most
Probable State of Detonation Products
The definition of entropy given by Boltzmann
is:
S  k  ln 
(40)
where Ω is the probability of macro states; k is
Boltzmann coefficient.
According to Eq. (4), the entropy at point “2”,
the stationary state, is the maximum, therefore:
 2  max .
Ω2 is the maximum means the probability of the
stationary state is the most probable one. We will
show how large Ω2 is by comparing it with ΩCJ, the
probability of the CJ state.
The difference of entropies at point “2” and
point CJ is [14]:
S2  SCJ  10 R
where R is gas constant. From Eq. (40) we have:
S2  SCJ
24
24
2
 e k  e6.02310  102.6210
( 41)
CJ
In our daily life, the dimension of the universe
is the largest and the dimension of atom is the
smallest, then their diameters’ ratio is:
d univ . 1026 m

 1036
( 42)
d atom 1010 m
1036 is an extremely large astronomy figure. But,
24
comparing it with 102.6210 in Eq. (41), we find:
2
d
 univ .
 CJ
d atom
( 43)
Eq. (43) means Ω2, the probability of the
stationary state, is so large that this “NOT moving
state” is the inevitable result of detonation process;
and ΩCJ, the probability of the moving CJ state, is
so trivial that it is much scarcer than a single atom
in the whole universe!
Appendix 2: The LADM Settles the Entropy
Argument
According to the Second Thermodynamic Law,
the entropy at the end of all of thermal processes
must be the maximum; but, derived by
Conservation Laws, the entropy at the CJ state is
the minimum [2]. Von Neumann, one of the
founders of the ZND model, commented it as “the
apparent conflict between the conservation laws
for energy and entropy” [20].
A 2.1. Entropy Argument
Minimum entropy of detonation products has
been argued since the very beginning of detonation
science.
The identification of the CJ point is an end
state characterized by a relative minimum in
entropy production has been noted by early
investigators. Chapman [21] hoped that the
entropy at the CJ point would be the maximum.
Jouguet [22] pointed out that the entropy at the CJ
point is the minimum instead of the maximum as
Chapman expected. Zeldovich [23] noted that the
minimum entropy at the CJ point cannot be
applied in any way to explain the choice of the
detonation velocity according to the principle of
least increasing entropy. Courant [24] noted that
person might have expected a detonation for which
the entropy is a maximum, and this expectation
has misled some writers into claiming that entropy
has a maximum at the CJ point.
In order to explain the minimum entropy of
detonation products, Scorah [25] proposed with
the “principle of degradation of energy”, Duffey
[26] proposed with the ”principle of minimum
entropy production”, and Brown [27] proposed
with the ”entropy of effective reaction” in the 12th
Int’l Symp. on Detonation, etc.
Recently, the minimum entropy of detonation
products is still recognized as of interest in
detonation propulsion. Wintenberger and Shepherd
[28] noted that the entropy minimum associated
with the CJ detonation and its potential
implications on the thermal efficiency of these
systems has been one of the main motivations to
explore detonation applications to propulsion.
Some authors have taken it as the formal basis for
the superiority of detonation-based power
generation or propulsion.
A 2.2. Ending the Entropy Argument
The entropy argument originates from the
assumptions of the ZND model.
In detonation processes, three factors increase
entropy: chemical reactions, transport effects and
complex multidimensional movement. The ZND
model only takes chemical reaction into account,
but neglects transport effects and complex
multidimensional movement, therefore, the
entropy at the CJ state becomes the minimum,
which contradicts with the Second Law and causes
the entropy argument.
When all of these three factors are taken into
account by the LADM, entropy at the end of
detonation process is the maximum, which
coincides with the Second Law. No any
explanation is needed; therefore, the LADM ends
the entropy argument.
Since the entropy of the final detonation
products is the maximum, detonation processes
don’t have any superiority over others in thermal
efficiency from the non-existing “minimum
entropy”.
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