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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)

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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.

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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)

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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)

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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)