Page 1
Entropy 2005, 7[2], 134-147 134
Entropy
ISSN 1099-4300
www.mdpi.org/entropy/
On the Entropy Production Due to Explosion in Seawater
R.P.Yadav1, P.K.Agarwal2 and Atul Sharma3, *
1. Department of Physics, Govt. P.G. College, Bisalpur (Pilibhit), U.P., India
2. Department of Physics, D.A.V. College, Muzaffarnagar, U.P., India
3. Department of Physics, V.S.P. Govt. College, Kairana (Muzaffarnagar), U.P., India
* Email addresses of authors:
rpyadav93PPhysics@yahoo.com (R.P.Yadav), pkaphysics@yahoo.com (P.K.Agarwal) and
attulsharma@yahoo.co.in (Atul Sharma)
Received: 7 March 2005 / Accepted: 10 May 2005 / Published: 12 May 2005
__________________________________________________________________________________
Abstract
The change in entropy is calculated due to propagation of blast waves, produced by the explosion of
spherical charge in sea water, using the energy hypothesis of Thomas. The release of energy is
considered as instantaneous and the gravitation of earth is taken into account, assuming the earth to be
a sphere of uniform density. For the sake of simplicity, effect of rotation of the earth is not considered.
The explosion is considered at different depths. It has been found that the change in entropy of water
decreases at different radial points, as the shock moves away from the point of explosion. Explosion
occurred at larger depths, produces a smaller change in entropy of water, then the explosion of same
energy, at smaller depths. Directional dependence of entropy production and the motion of the shock
are also studied. It has been found that, entropy production is larger in upward motion of the
underwater shock. However the shock velocity increases in downward direction.
Keywords: explosion, entropy, underwater shock, Tsunami waves
Nomenclature
R0= radius of spherical charge (m)
Z d =depth of explosion (m) Superscripts
p = pressure of fluid (seawater) * = at the explosive boundary
g = defined in equation (4)
gs= acceleration due to gravity at
the surface of earth (m/s2) Subscripts
Re=radius of earth (m) z = unshocked state at a depth z
r = radial distance from the point of explosion 0 = state at zero pressure
z =depth of any point from the water surface (m) 1= state at the water surface
u= radial component of fluid velocity (m/s) 2=state just behind the shock front
v= transverse component of fluid velocity (m/s) D= detonation
U =shock velocity (m/s)
R=shock radius (m)
Page 2
Entropy 2005, 7[2] 135
R’=nondimensional shock radius (=R/R0)
n =a constant for water Greek letters
B(s)= slowly varying function of entropy, ρ =density of fluid (seawater)
normally considered as constant (kb) θ = angle measured from vertical direction
T’=energy released during explosion (J) δ = compression ratio (ρ2/ρz)
UD= detonation velocity (m/s) ε = total energy /unit mass (sum of
cp =specific heat of water (kJ/kg-K) internal and kinetic energies for
T=absolute temperature (K) unit mass = E+½ u2)
E=internal energy/unit mass (J/kg) α =constant defined in equation (22)
V=specific volume (m3/kg)
W=work, defined in equation (29)
s =specific entropy (entropy /unit mass)(kJ/kg-K)
∆Q= heat flow / unit mass (J/kg)
Introduction
During and after Second World War, many scientists have shown their interest in analyzing
theoretically and experimentally the explosions in different types of media. Recently, Tsunami
waves had produced a disastrous effect on the human lives, in the coastal areas of south east Asia,
which are caused by shock waves due to submarine earthquake. Underwater explosions and shock
waves have been studied earlier mainly by Kirkwood and Bethe [12], Penny and Dasgupta [14],
Kirwood and Brinkley [13]. Later, Hunter [11], Berger and Holt [1], Bhatnagar et al. [2], Ranga Rav
et al. [16], Singh et al. [21,22], Singh [18,19,20] and Vishvakarma et al. [25] have studied shock
propagation in the water. Many methods are developed for the study of shock waves. Thomas [24]
used ‘Energy hypothesis’ for spherical blast waves. This hypothesis was successfully applied by
Bhutani [3] to cylindrical blast waves in Magnetogasdynamics. A theoretical study of explosion in
water (without considering the effect of gravity) is carried out by Singh et al. [21] and Singh [18]
using Energy hypothesis. Experimental verification of Energy hypothesis is given by Singh et al. [22]
and Singh [20]. Vishvakarma et al. [25] used Energy hypothesis to explosion problem in sea water
considering the effect of gravity.
In most of the underwater shock research stated above, entropy change at the shock front is considered
negligible. This approximation of negligible entropy change during shock propagation in water is also
supported by the fact that the equations for conservation of mass and momentum, across a shock (the
mechanical conditions) together with the equation of state for the fluid are sufficient to determine the
shock process. However a definite amount of mechanical energy of the shock wave is converted into
heat, producing entropy. In case of strong shocks, it cannot be neglected [11]. To calculate it, equation
for conservation of energy is also needed [7].
In the present paper, an attempt has been made to study the change in entropy of seawater due to the
propagation of blast waves generated by strong explosion. The energy released during explosion is
considered instantaneous and its distribution is uniform in the shocked field. It is also assumed that the
whole energy liberated during explosion is used in the propagation of the shock. The basic equations
of Vishvakarma et al. [25] are considered, in which Energy hypothesis deviced by Thomas [24] is
used. This hypothesis is well suited to express, at least to a first approximation, the basic property of
energy behavior, within the blast caused by the explosion.
Page 3
Entropy 2005, 7[2] 136
()()()
0
cot
21
r
u
=++
∂
∂
θ
+
∂
∂
+
∂
∂
∂
v
rr
u
v
u
r
∂
t
ρ
θ
ρ
ρ
ρ
ρ
0 cos
1
ρ
=+
∂
∂
+
∂
∂
+
∂
+
∂
θ
θ
g
r
p
r
v
r
u
u
t
u
0 sin
1
=−
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
θ
θρθ
g
p
r
v
r
v
r
v
u
t
v
θ
ρ
1
cos
1
g
r
∂
p
−=
∂
∂
θ
θ∂ρ
sin
g
p
r
=
θ
cos
Rzz
d−=
Basic formulation
Let the explosion take place due to a spherical charge, at a time t=0 and at a depth ‘Zd’ from the
surface of water. This point is taken as origin. This explosion produces spherical blast waves (fig.1).
Figure 1 – Shock front in the seawater
The partial differential equations representing the conservation of mass and momentum of the flow are
given by:
Assuming the earth to be a homogeneous sphere (uniform density), `g’ varies only with the depth from
the surface of the earth, not due to non uniform density [17].The variation of acceleration due to
gravity ‘g’ at a depth ‘z’ below the earth surface is given by:
R
e
In the stationary conditions u=o and v=o we have:
The depth ‘z’ at any point on the shock front[r=R(t)] is (figure 1):
−=
z
g
g
s1
Where ‘θ’ is the angle between vertical direction and the shock radius (R) corresponding to the point.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Page 4
Entropy 2005, 7[2] 137
Equation of state for sea water is:
From (8), pz is given by:
From (8) and (9) we have:
Similarly, with pressure and density at water surface, we get:
Solving equation (4), (5), (7) and (11), we get the values of ρz and pz as:
As the shock travels forward, the spherical front changes its shape [6]. It is assumed that the deviation
of shock front from its spherical shape is negligible. Hence the transverse component v of particle
velocity, at the shock front is always tangent to it and it is unchanged during shock transition.
Taking unshocked water as stationary, the Rankine –Hugoniot conditions relating the fluid parameters,
in front and behind the shock front are given by:
(
U
u
U
z
ρρ
=−
2
2
(15)
u
U
pp
z
z
2
2
ρ
=−
(16)
u
p
EE
z
2
ρ
With compression ratio δ = (ρ2/ρz), we have the jump conditions as follows:
(
pp
B
p
zz
+−+=
1
2
δ
(18)
(
(
1
ρ
z
()
−
+=−
1
1
11
n
pBpp
ρ
ρ
()
()
e
s
R
g
pBn
n
K
1
1
1
where
+
−
=
ρ
()
[]() 1
−
1
11
−+=
n
ez
zzRK
ρρ
()(){}()
1
1
1
11
pzzRKpBp
n
n
−
ez
+
−−++=
−
=
1)(
0
n
sBp
ρ
ρ
( )
s
−
=
1
0
n
z
z
B
p
ρ
ρ
()
−
+=−
1
n
z
zz
p
B
p
p
ρ
ρ
)
u
U
z
2
2
2
2
2
1
−=−
(17)
)[]
n
)(
δ
)
δ
)
2
1
1
−
−+
=
δ
n
zp
B
U
(19)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
Page 5
Entropy 2005, 7[2] 138
(
2
ρ
z
These are four relations [(from (17) to (20)] in five unknown p2,u2,U,ρ2 and E2. To solve these, another
fifth relation is needed.
Energy hypothesis [24] provides this fifth relation. According to it, change in total energy per unit
mass of water at a distance R from the centre of explosion due to the propagation of spherical shock is
given by:
T
z
3
2
4
ρ
π
α is a constant given by [18]:
(
εε
α
z
RR
T
′
3
From equation (17) and (21) we get:
T
U
z
2
4
ρ
π
ρ
Putting the values of p2,u2 and U from (18), (19) and (20) in (23) we get:
(
R
4
π
At the boundary of spherical charge (at R=R0), we have δ= δ*, then:
(
R
0
4
π
Dividing (24) by (25) we get:
(
(
R
pp
B
zz
11
δδ
δ* is calculated using the mismatch method due to Butchanan and James [5].According to it, at the
explosive boundary:
U
p
z
D
D
D
+
ρρ
The values of ρD ,pD and UD are available from detonation data for a known quantity of a particular
explosive. Then for a particular direction θ, the fluid parameters δ, p2,, u2, and U are calculated at
different R′ values, using equations (26), (18), (19) and (20) [9].
)()()
2
1
11
−−+
=
δ
δ
δ
n
zp
B
u
(20)
R
2
3
α
εε
′
=−
(21)
)
ρ
π
4
R
−=
→
2
2
3
lim
0
(22)
R
u
p
3
2
2
3
α
′
=
(23)
)(){}(
z
)
T
pp
B
n
z
3
3
11
α
δ
δ
′
=−+−+
(24)
)()
{}()
T
pp
B
z
n
z
3
3
11
α
δδ
′
=−+−+
∗
∗
∗
∗
(25)
)(
)(
)
)
{
{
}(
}(
)
)
pp
B
n
z
∗
n
z
′
=
−+−+
−+−+
∗∗
∗
3
1
11
δ
δ
(26)
UU
p
z
∗
∗
∗
2
=
ρ
2
(27)
Page 6
Entropy 2005, 7[2] 139
Now according to the first law of thermodynamics, heat flow during an irreversible process is [26]:
(
EE
Q
−=∆
2
(28)
Work done W, by the shock wave on the unit mass of seawater in compressing it is given by:
2
dVpW
(29)
) W
z−
∫ −=
z
Using equation of state (8), (29) and (28), we get:
δ
ρ
z
The well known thermodynamic relation is:
V
T dT
c
dQ
p
∂
At the shock front, both temperature and pressure are changing. However in case of solid and liquids,
an increase of pressure produces only a small temperature change. Also experiments show that cp
hardly changes even for an increase 10,000 Bars [26]. So for unit mass of seawater, the heat flow can
be approximated as:
dT
c
dQ
p
≅
(32)
From Second law of Thermodynamics:
dsT dQ ≅
(33)
From (32) and (33), for the states above and behind the shock front, we get:
T
TT
z
z
.Also from (32), we have:
(
TT
c
Q
z
p
−=∆
2
(35)
From (34), (35) and (30) we get finally the ‘rate of specific entropy production’{It must be
remembered that R-H conditions (15), (16) and (17) are defined for unit area of shock wave per unit
time} across the shock front is given as:
(
(
z
n
TcTc
(
(
)
)
()
{
δ
}
−
−
+
+
−−∆=∆
−
1
1
1
1
1
ρ
n
z
z
n
p
B
B
EQ
(30)
dp
T
p
∂
−=
(31)
=
∫
=
∫
z
=∆
T
c
T
dT
c
dss
pp
2
2
ln
2
(34)
)
)
)
()
(
δ
)
−
−
+
+
−−
∆
+=∆
−
1
1
1
1
δ
1
1 ln
1
ρρ
n
z
z
zpzp
p
p
B
BE
c
s
(36)
Page 7
Entropy 2005, 7[2] 140
Results
The typical variation of specific entropy production ∆s with the radial distance R’ in upward direction
(θ=00) at the depth of explosion Zd=4 km is shown in figure 2.The corresponding variation of shock
velocity is shown in figure 3.
4.50
Figure 2-Entropy variation with radial distance
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
05 1015 20
Figure 3- Shock velocity variation with radial distance
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
5500
6000
6500
7000
05 1015 20
∆s (kJ/kg-K)
R’
U (m/s)
R’
Page 8
Entropy 2005, 7[2] 141
Entropy variation also depends on the depth of explosion. Figure 4 shows the variation in entropy
production with radial distance at different depths of explosions (Zd), as the shock moves upwards
(θ=00).
0.14
Figure 4- Effect of depth of explosion on entropy variation
Variation of entropy production with radial distance in different directions at the depth of explosion
Zd=7000 m is shown in figure 5.
0.00
0.02
0.04
0.06
0.08
0.10
0.12
3.54 4.55
1000m
4000m
7000m
R’
Figure 5- Typical variation of entropy with radial distance in different directions
0.966489633
0.966489833
0.966490033
0.966490233
0.966490433
0.966490633
1.7999995 1.81.8000005
at 0 deg.
at 60 deg.
at 90 deg.
at 120 deg.
at 180 deg.
∆s (kJ/kg-K)
R’
∆s (kJ/kg-K)
Page 9
Entropy 2005, 7[2] 142
Figure 6 shows the variation of entropy production with the direction, at a fixed radial distance
(R’=3.0), at a depth of explosion of 7000 m. In it, ∆s axis is plotted at θ=900, because for θ=900, z=Zd
i.e. shock moves horizontally. So the directions <900 signifies the upward motion of the shock,
whereas the directions >900 signifies the downward motion of the shock.
0
Figure 6- Variation in entropy production with direction (at R’=3.0)
Shock velocity variation (Corresponding to figure 6) is shown in figure 7.
30
60
90
120
150
180
0.17423700.17423710.1742372 0.17423730.1742374 0.17423750.1742376 0.17423770.17423780.1742379 0.1742380
Figure 7- Variation in shock velocity with direction (at R’=3.0)
Radial plots in figure 8, shows the shock velocity and entropy profiles at different radial positions
(R’=1.4, 3.0 and 5.0) and at Zd =7000m. Radial axes show the different directions and the shaded area
gives the magnitude of ∆s (and U) in corresponding profiles.
0
30
60
90
120
150
180
2714.434 2714.4345 2714.435 2714.43552714.436 2714.43652714.4372714.4375 2714.4382714.4385
∆s (kJ/kg-K)
Direction (θ)
U (m/s)
Direction (θ)
Page 10
Entropy 2005, 7[2] 143
1.9533405
1.9533410
1.9533415
1.9533420
1.9533425
0
30
45
60
90
120
135
150
180
210
225
240
270
300
315
330
at R’=1.4
4654.654
4654.655
4654.656
4654.657
4654.658
4654.659
0
30
45
60
90
120
135
150
180
210
225
240
270
300
315
330
0.1742366
0.1742368
0.1742370
0.1742372
0.1742374
0.1742376
0.1742378
0
30
45
60
90
120
135
150
180
210
225
240
270
300
315
330
at R’=3.0
2714.432
2714.433
2714.434
2714.435
2714.436
2714.437
2714.438
0
30
45
60
90
120
135
150
180
210
225
240
270
300
315
330
0.0233684
0.0233685
0.0233686
0.0233687
0.0233688
0.0233689
330
0
30
45
60
90
120
135
150
180
210
225
240
270
300
315
at R’=5.0
2127.31
2127.312
2127.314
2127.316
2127.318
2127.32
2127.322
330
0
30
45
60
90
120
135
150
180
210
225
240
270
300
315
Figure 8- Entropy profiles and corresponding shock velocity profiles at different radial
Positions, as the shock moves away from the point of explosion
ENTROPY
PROFILES (kJ/kg-K)
SHOCK VELOCITY
PROFILES (m/s)
Page 11
Entropy 2005, 7[2] 144
Analysis of results
Characteristics curves for blast waves:
Figure 2 shows that as the shock moves away from the point of explosion, the entropy production is
large near the point of explosion and it decreases very rapidly towards negligible value. At a distance
R’=5.0, it is almost negligible. This is in agreement with figure 3 (between shock velocity and radial
distance R’). As R’ increases, U decreases and approaches the value of local sound velocity (=1580.28
m/s).These curves presents the basic nature of blast waves.
Propagation of shock in any medium produces a disordered motion of medium particles, which
depends on the strength of the shock. An increase in the disorder of a medium is described by the
increase in entropy [8,26].As the shock moves away from the point of explosion its velocity decreases
producing less disorder (Entropy) than its previous position. In this manner, the blast waves produce a
non-isentropic flow field behind it, in which entropy is maximum at the explosive boundary.
Effect of depth of explosion:
Figure 4 shows that Explosion occurred at larger depths, produces a smaller change in entropy of
water, then the explosion of same energy, at smaller depths. The corresponding dependence of shock
velocity on depth of explosion is given in table 1.
Table 1- Shock velocity variation with depth of explosion
Table 1 show that shock moves with a larger velocity at larger depths. An increment of 1000m of
depth produces a decrement in ∆s of the order of 10-3 kJ/kg-K and a large increment in U of the order
of 14m/s.
This behavior can be explained well on the basis of static pressure pz. As the depth increases, static
pressure increases sharply. The large static pressure causes the shock front to produce lesser
disordered motion of water molecules. In other words, entropy production decreases as the depth of
explosion increases.
Directional dependence of entropy production:
Figure 5 shows that, in general when the shock moves upward (θ<900), entropy production is larger
than the entropy production in downward direction (θ>900).
Figure 6 presents a relative difference of ∆s, in different directions at a particular position of the shock
front. This change in entropy is of the order 10-7 kJ/kg-K. Figure 7 shows a relative difference of U in
R'
1
1.2
1.4
1.6
1.8
2
2.5
3
U at different Zd's in the direction
θ=00 (upward direction)
7000m 4000m
6247.47 6206.10
5307.25 5265.28
4654.66 4612.42
4177.20 4134.86
3813.98 3771.62
3529.23 3486.89
3032.16 2989.93
2714.43 2672.31
1000m
6162.46
5221.10
4568.04
4090.44
3727.24
3442.58
2945.85
2628.45
Page 12
Entropy 2005, 7[2] 145
different directions at a particular position of the shock front. This difference is of the order 10-4m/s.
These plots can be explained on the same ground as figure 4.
In figure 8, as R’ increases, shock velocity U decays and correspondingly ∆S also decrease. But in
general, decay of shock in upward direction (θ<900) is faster than its decay in downward direction
(θ>900). Whereas entropy production in upward directions is higher than downward directions of
shock motion. It can also be seen that initially the profiles are not smooth. Velocity profiles shows
fluctuations along 300 and 450. However entropy profile shows fluctuations along 300,450 and 1350
directions. As the radial distance increases, these fluctuation decreases and profiles shows smoothness
in all directions.
Conclusion
The results of this paper show that the entropy production in seawater depends on the direction of
shock propagation as well as on the depth of explosion. The results also shows that as the shock moves
away from the point of explosion, the entropy production is larger near the point of explosion and it
decreases very rapidly towards negligible value. At a distance of 5 radii of charge, entropy production
is almost negligible.
The results obtained here are agreed with those given by Kirwood and Montroll [6]. However the
estimation given in [6] was only for pure water at 200C, up to a pressure of 50 kb. The effect of the
depth of explosion and directional dependence was not considered in their estimation.
For the sake of simplicity, effect of the rotation of the earth is not considered here. Rotation of the
earth produces centrifugal and Coriolis forces. Effect of centrifugal force is very small in comparison
to gravitational force for the case of earth. Whereas horizontal component of Coriolis force produces
ocean currents. We expect to include it in our next paper. Presence of the gas bubble at the point of
explosion is also not considered. This gas bubble certainly affects the shock motion.
Shock propagation is an irreversible process with non-equilibrium states above and behind it. However
from entropy point of view, a Local Thermo dynamical Equilibrium (LTE) is considered across the
shock. The increase in entropy across the shock is an essential part of shock process and it is also a
necessary condition for the stability of the shock [10]. The results of this paper also prove that
irreversible process taking place inside a thermodynamic system always lowers the value of entropy
production per unit time [15].
The most important aspect of the study of entropy production in seawater lies in the fact that, it
presents the behavior of the medium particle (seawater) for the shock waves. Generation of entropy in
this process causes the conversion of explosion energy into unavailable form. So less energy is
available for shock propagation in the direction in which change in entropy is larger. In other words,
this analysis provides the nonuniform distribution of explosion energy, available for shock
propagation in different directions and at different depths.
These results are important in the modeling of Tsunami waves, caused by shock waves due to
submarine earthquakes, landslides or volcanic eruptions. In this paper we have studied only a very
small amount of explosive which liberates a small amount of explosion energy. However, a very large
amount of energy is released in seismic and volcanic activities. In such cases, the effect of these
results will be of considerable amount. Predictions and analysis of Tsunami waves can be improved
using these results, which needs a further study in this field.
Page 13
Entropy 2005, 7[2] 146
Data used:
1. Re =6371230 m 2. gs = 9.81 m/s2
3. B(s) = 2.94 kb [4,11] 4. n = 7.25 [4,11]
5. p1 = 1 b 6. ρ1 = 1027 kg/m3 [23]
7. Tz (average temperature in deep sea) =276 K [23]
8. cp = 4.186 kJ/kg-K
9. Detonation data for the explosive RDX/TNT (60:40), [18]:-
• R0 =0.0375 m • Mass of the charge =0.365 kg
• ρD =1680 kg/m3 • UD = 7800 m/s
• pD = 255.528 kb • T’ =2252.11 kJ
References
1. Berger, S.A.; Holt, M. Implosive phase of spherical explosion in sea water. Phy. Fluids 1962,
5, no.4, 426-431.
2. Bhatnagar, P.L.; Sachdev, P.L.; Prasad, P. Spherical piston problem in water. J. Fluid Mech.
1969, 39, 587-600.
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