Combustion, Explosion, and Shock Waves, Vol. 39, No. 1, pp. 68–74, 2003

Relation between Gunpowder Burning-Rate Responses

to Oscillating Pressure and an Oscillating Radiant Heat Flux

UDC 536.46

B. V. Novozhilov,

1

M. Kono,

2

and T. Morita

3

Translated from Fizika Goreniya i Vzryva, Vol. 39, No. 1, pp. 79–86, January–February, 2003.
Original article submitted March 27, 2002.

In the linear approximation of the Zel’dovich–Novozhilov theory, we found an ana-
lytical relation between the response function to oscillating pressure, determined at
a certain initial temperature, and the response function to oscillating radiant heat
flux, determined at the same pressure but at another, lower, initial temperature. The
difference between the initial temperatures satisfies the equality condition between
the steady-state burning rates in the absence and in the presence of the radiant heat
flux and is proportional to the latter.

Key words:

gunpowder, unsteady combustion, Zel’dovich–Novozhilov theory,

burning-rate response, oscillating perturbation.

INTRODUCTION

The idea of using the gunpowder burning-rate re-

sponse function to an oscillating radiant heat flux for
gaining information about the response function to os-
cillating pressure, which plays a decisive role in studying
unsteady operating regimes of solid-propellant rocket
motors, was put forward about thirty years ago [1].
Since then, there have been several attempts to analyti-
cally relate these two functions (see, for instance, [2–5]),
but these attempts were not successful. One function
was expressed in terms of the other either in the triv-
ial case of extremely weak radiant heat fluxes and zero
mean free path of radiation in a condensed matter [2–4]
or for a dsubstantially restricted range of propellants [5].

The reason for this failure is quite obvious. The

steady component of the radiant heat flux changes the
steady-state temperature profile inside the condensed
phase. In this case, the steady-state burning rate and
steady-state surface temperature turn out to be higher
than those in the absence of the heat flux. For this rea-
son, the burning-rate oscillations caused by the fluctuat-
ing radiant-flux component occur on a different steady-
state background, compared to the burning-rate oscil-
lations arosed by oscillating pressure.

1

Semenov Institute of Chemical Physics, Russian Academy
of Sciences, Moscow 117977; [e-mail].

2

Institute of Outer Space and Astronautics,
Sagamihara, Japan.

3

Tokai University, Hiratsuka, Japan.

Formally, in the mathematical treatment of the

problem, this circumstance is manifested in the fact that
the response functions to oscillating pressure and an os-
cillating radiant heat flux, found at identical values of
base pressure and initial temperature, turn out to be
dependent on different parameters.

It is well known

that, in the linear approximation of the Zel’dovich–
Novozhilov theory (ZN-theory) [6, 7], any unsteady pro-
cess is characterized by first four partial derivatives of
the laws of steady combustion, namely, by the deriva-
tives of the gunpowder burning rate and gunpowder sur-
face temperature with respect to pressure and initial
temperature. Without a radiant heat flux, the deriva-
tives with respect to initial temperature are calculated
at the true value of this temperature. As was shown
in [8], with a radiant heat flux, these derivatives should
be found at a higher initial temperature determined by
the steady radiation-flux component and by the new
value of the steady-state burning rate. As a result, the
linear theory of any unsteady process in the presence
of a heat flux includes gunpowder parameters that dif-
fer from the governing parameters of the same problem
in the absence of an external heat flux. In addition,
in studying oscillatory regimes in the presence of radi-
ant heat fluxes, a new dimensionless frequency, related
to another steady-state burning rate, appears. Appar-
ently, in the general case, it is impossible to derive an
analytical relation between two functions of different ar-
guments, being characterized by different parameters.

68

0010-5082/03/3901-0068 $25.00 c

2003

Plenum Publishing Corporation

Relation between Gunpowder Burning-Rate Responses to Pressure and Radiant Heat Flux

69

In the present work, we show that, nonetheless, it

is possible to find the response function to oscillating
pressure at given values of the initial temperature T

a

and pressure from the response function to an oscillating
heat flux at the same pressure but at a lower initial tem-
perature T

e

. The new initial temperature T

e

should be

chosen such that to ensure equality between the steady-
state burning rate under permanent irradiation at the
temperature T

e

and the steady-state burning rate with-

out irradiation at the temperature T

a

. In this case, the

response function to an oscillating radiant heat flux at
the temperature T

e

depends on the same parameters as

the response function to the oscillating pressure at the
higher initial temperature T

a

. Naturally, now these two

functions can be analytically related to each other.

In the present work, the narration is organized as

follows.

First of all, we give definitions of the gun-

powder burning-rate response functions to oscillating
pressure and an oscillating radiant heat flux, only lin-
ear approximation with respect to the amplitudes of
time-dependent quantities being considered. Then, we
give the well-known expression [9] for the gunpowder
burning-rate response function to oscillating pressure.
Next, we discuss in detail the linear time-dependent
theory in the absence and in the presence of a radiant
heat flux and substantiate the procedure for choosing
the new initial temperature. Simultaneously, we point
to the fact that the adopted approach is devoid of the
difficulty related to the necessity of determining the ra-
diation flux in the depth of the gunpowder. Generally,
the radiant heat flux can be found from the source inten-
sity, taking into account only absorption of radiation in
the gas phase and its reflection by the burning surface.
In our approach, the radiant heat flux can easily be ex-
pressed through the difference between the initial tem-
peratures T

a

and T

e

, which can readily be determined

experimentally; in fact, the final result is independent
on the radiant flux. In final sections of the paper, we
derive an expression for the gunpowder burning-rate re-
sponse to a harmonically oscillating radiant flux and
establish a relation between the two response functions.

GUNPOWDER BURNING-RATE
RESPONSES TO HARMONICALLY
OSCILLATING PRESSURE AND
OSCILLATING RADIANT HEAT FLUX

If the surface pressure p of burning gunpowder ex-

erts low-amplitude oscillations with an amplitude

p = p

0

+ p

1

cos Ωt

(p

1

p

0

),

then, the linear burning rate of the gunpowder oscillates
with the same frequency but with a certain phase shift

with respect to the pressure oscillations:

u = u

0

+ u

1

cos(Ωt + ψ),

u

1

u

0

.

The superscript 0 everywhere refers to the steady com-
bustion mode.

In the linear approximation, we may conveniently

use the method of dimensionless complex amplitudes.
In this case,

η = 1 + [η

1

exp(iΩt) + c.c.],

v = 1 + [v

1

exp(iΩt) + c.c.],

where

η =

p

p

0

, v =

u

u

0

, η

1

=

p

1

2p

0

, v

1

=

u

1

2u

0

exp(iψ)

(c.c. denotes a complex-conjugate quantity for the first
terms in the brackets).

The complex quantity

U

p

= ν

1

/η

1

is called the linear burning-rate response function to
oscillating pressure.

In a similar manner, we define the response func-

tion to an oscillating radiant heat flux:

I = I

0

+ I

1

cos Ωt,

I

1

I

0

.

Here, the burning rate is

u

r

= u

0
r

+ u

r1

cos(Ωt + ψ

r

),

u

r1

u

0
r

,

where the subscript “r” indicates the difference between
the burning rates in the absence and in the presence of
the radiant heat flux. Using the method of complex
amplitudes, we obtain

I = I

0

+

h

I

1

2

exp(iΩt) + c.c.

i

,

ν

r

= 1 + [ν

r1

exp(iΩt) + c.c.],

ν

r1

=

u

r1

2u

0

r

exp(iψ

r

).

The complex quantity

U

r

=

ν

r1

I

1

/2I

0

is called the gunpowder burning-rate response function
to a harmonically oscillating heat flux.

Subsequently, we will use expressions for gunpow-

der response functions at various initial temperatures.
To indicate the initial temperature at which these func-
tions are taken, we mark the symbols of these functions
with proper superscripts. For instance, if the response
function to oscillating pressure is calculated at a tem-

perature T

a

, this function is designated as U

(a)

p

. The

response functions to an oscillating radiant heat flux,
calculated at initial temperatures T

a

and T

e

, are desig-

nated as U

(a)

r

and U

(e)

r

.

70

Novozhilov, Kono, and Morita

GUNPOWDER BURNING-RATE RESPONSE
TO HARMONICALLY OSCILLATING
PRESSURE

In the ZN-theory, used in the present study, of pri-

mary significance are steady-state dependences of the
burning rate and surface temperature on pressure and
initial temperature:

u

0

= F (p, T

a

),

T

0

s

= Φ(p, T

a

).

(1)

These dependences can be found from experiments on
steady combustion or from some theoretical model of
gunpowder.

The burning-rate response function to oscillating

pressure was found in [9]. This function has the form

U

(a)

p

=

ν + δ(z

− 1)

1 + (z

− 1)(r − k/z)

,

(2)

where

k = (T

0

s

− T

a

)

∂lnF

∂T

a

(a)

p

0

,

r =

∂Φ

∂T

a

(a)

p

0

,

ν =

∂lnF

∂ ln p

0

(a)

T

a

, µ =

1

T

0

s

− T

a

∂Φ

∂ ln p

0

(a)

T

a

,

(3)

δ = νr

− µk, z =

1

2

(1 +

√

1 + 4iω), ω =

æΩ

(u

0

)

2

.

Here æ is the thermal diffusivity in the condensed phase,
and all derivatives are calculated at the initial temper-
ature T

a

.

It should be noted that the symbol T

a

has two

meanings. First, it is used as an argument in station-
ary combustion laws (1) with respect to which differ-
entiation can be performed. Second, it gives the initial
temperature at which the response function to oscilla-
tion pressure is found. Thus, alongside with the deriva-
tives with respect to the initial temperature, which are
given by (3), derivatives with respect to this argument
at other values of the initial temperature are also mean-

ingful. For instance, (∂F/∂T

a

)

(e)
p

0

is the derivative of

the steady-state burning rate with respect to the initial
temperature, taken at a fixed pressure and at the initial
temperature T

e

.

LINEAR NONSTATIONARY MODEL
IN THE ABSENCE AND IN THE PRESENCE
OF A RADIANT HEAT FLUX

Without a radiant flux, the linear approximation

of the theory of any unsteady process includes four

derivatives, namely, derivatives of gunpowder burning
rate and gunpowder surface temperature with respect
to pressure and initial temperature (k, r, ν, and µ),
which are given by expressions (3).

A constant radiant heat flux alters the condensed-

phase temperature profile and gunpowder burning rate.
In particular, the steady-state burning rate u

0

r

and

steady-state surface temperature T

0

s,r

in this case be-

come higher than in the absence of the radiant heat
flux: u

0

r

> u

0

and T

0

s,r

> T

0

s

.

The method for finding the steady-state burning

rate and steady-state surface temperature is reproduced
here following [8].

In the stationary case with no radiant heat flux,

the heat-conduction equation for the condensed phase
and the boundary conditions have the form

u

0
r

dT

0

r

dx

=

d

dx

æ

dT

0

r

dx

+

I

0

ρc

exp(αx)

,

x =

−∞: T

0

r

= T

a

;

x = 0:

T

0

r

= T

0

s,r

.

Here α is the linear absorption coefficient [we assume
that the mean free path of radiation (α

−1

) is much

greater than the thickness of the condensed-phase re-
acting layer], and the coordinate system is attached to
the interface between the phases (x = 0). Integrating
this equation over the entire volume of the condensed
phase, we find the surface-temperature gradient:

f

0

r

=

u

0

r

æ

T

0

s,r

− T

a

−

I

0

ρcu

0

r

.

(4)

In the absence of the external radiant heat flux, we ob-
tain the following well-known relation for this quantity:

f

0

=

u

0

æ

(T

0

s

− T

a

).

Using the latter relation, we can reduce stationary com-
bustion laws (1) to the form

u

0

= F

p, T

0

s

−

æf

0

u

0

,

(5)

T

0

s

= Φ

p, T

0

s

−

æf

0

u

0

.

The dependences of the burning rate and surface tem-
perature on the pressure and surface-temperature gradi-
ents are universal: relations (5) also hold for conditions
with the radiant heat flux. Substituting (4) into (5), we
obtain

u

0
r

= F

p, T

a

+

I

0

ρcu

0

r

,

(6)

T

0

s,r

= Φ

p, T

a

+

I

0

ρcu

0

r

.

Relation between Gunpowder Burning-Rate Responses to Pressure and Radiant Heat Flux

71

Thus, the burning rate and surface temperature during
combustion under the action of a constant external heat
flux can be found from the stationary combustion laws
(1) in which, instead of the true initial temperature, a
higher temperature determined by the radiant heat flux
should be taken:

u

0
r

= F (p, T

I

),

T

0

s,r

= Φ(p, T

I

).

Here

T

I

= T

a

+ I

0

/ρcu

0
r

.

(7)

For this reason, the linear theory of any unsteady

process under irradiation includes parameters differing
from those given by expressions (3). For the first time,
this fact was established in [8], where it was shown that,
in the linear approximation, instead of the parameters
k and r, the unsteady-combustion theory includes new
quantities

k

r

= (T

0

s,r

− T

I

)

∂lnF

∂T

a

(I)

p

0

,

(8)

r

r

=

∂Φ

∂T

a

(I)

p

0

,

which differ from k and r by the fact that the derivatives
with respect to the initial temperature should be calcu-
lated not at the actual initial temperature of gunpow-
der T

a

but at a higher temperature T

I

. That is why the

derivatives in (8) are marked with a superscript pointing
to this temperature. Moreover, in studying oscillatory
regimes, there arises a new dimensionless frequency re-
lated to another value of the steady-state burning rate:

ω

r

=

æΩ

(u

0

r

)

2

.

(9)

Thus, the response functions defined above depend

on different parameters of burning gunpowder. Namely,

— the function U

(a)

p

is determined by the param-

eters k and r and depends on the dimensionless fre-
quency ω;

— the function U

(a)

r

is determined by the param-

eters k

r

and r

r

and depends on the dimensionless fre-

quency ω

r

.

Apparently, in the general case, it is impossible to

establish an analytical relation between two functions
that depend on different arguments and are character-
ized by different parameters.

We show now that it is, nevertheless, possible to

find an analytical relation between the response func-

tion to oscillating pressure U

(a)

p

, measured at the initial

temperature T

a

, and the response function to the os-

cillating radiant heat flux U

(e)

r

, found at a lower initial

temperature T

e

.

Let us choose the temperature T

e

so that the

steady-state burning rate at this initial temperature
under a constant radiant heat flux I

0

be equal to the

steady-state burning rate at the initial temperature T

a

without the radiant flux, i.e.,

u

0
r

(p, T

e

, I

0

) = u

0

(p, T

a

).

(10)

From the first relation in (6), we have

F

p, T

e

+

I

0

ρcu

0

= F (p, T

a

);

therefore, equality (10) holds if

T

e

= T

a

− I

0

/ρcu

0

.

(11)

At the same time, in view of the second relation in (6),
the equality between the surface temperatures in the
regimes under consideration holds:

T

0

s,r

(p, T

e

, I

0

) = T

0

s

(p, T

a

).

In addition, it follows from (8) and (9) that

k

r

= k,

r

r

= r,

ω

r

= ω.

Thus, the response function to the oscillating radiant

heat flux U

(e)

r

found at the temperature T

e

depends on

the same parameters as the response function to oscil-

lating pressure U

(a)

p

, found at a higher temperature T

a

.

Naturally, the two functions can be analytically related
to each other.

One more important point is worth noting. The

theory of condensed-system combustion under irradia-
tion includes the value of the radiant heat flux in the
depth of gunpowder (under the interface between the
phases). With a known mean free path of radiation, the
radiant heat flux at each point of the condensed phase
can easily be found from the flux penetrating through
the interface. An estimate of the latter flux from the
radiation-source intensity is hampered by radiation ab-
sorption in the gas phase and its reflection from the
burning surface. The approach under consideration is
devoid of this difficulty. Indeed, in the stationary case,
the radiant heat flux at the surface can easily be ex-
pressed through the difference between the initial tem-
peratures T

a

and T

e

, which can be found experimentally,

and through the steady-state burning rate u

0

:

I

0

= ρcu

0

(T

a

− T

e

).

The theory dealing with unsteady burning processes
also includes the modulation length of radiation I

1

/I

0

,

which coincides with the modulation length of radia-
tion emitted by the source (under the obviously valid
assumption about similarity of absorption and reflec-
tion processes for constant and variable components of
the total radiant heat flux).

72

Novozhilov, Kono, and Morita

GUNPOWDER BURNING-RATE RESPONSE
TO A HARMONICALLY OSCILLATING
RADIANT HEAT FLUX

Within the framework of the ZN-theory, the gun-

powder burning-rate response function to an oscillating
radiant heat flux was found in [4]. Let us find it now for
the case in which the initial temperature T

e

is related

to the steady component of the radiant heat flux by the
formula

T

e

= T

a

− I

0

/ρcu

0

,

where u

0

is the steady-state burning rate in the absence

of radiation at the initial temperature T

a

or the burning

rate in the presence of radiation at the initial tempera-
ture T

e

.

The heat-conduction equation for the condensed

phase with a source of heat induced by radiation ab-
sorption and the boundary conditions have the form

∂T

r

∂t

= æ

∂

2

T

r

∂x

2

− u

r

∂T

r

∂x

+

αI(t)

ρc

exp (αx),

−∞ < x 6 0,

x

→ −∞: T

r

= T

e

;

x = 0: T

r

= T

s,r

(t).

In what follows, we use the dimensionless variables

ξ =

u

0

x

æ

,

τ =

(u

0

)

2

t

æ

,

v =

u

r

u

0

,

θ =

T

r

− T

e

T

0

s

− T

a

,

ϑ =

T

s,r

− T

e

T

0

s

− T

a

,

ϕ =

f

r

f

0

r

,

l =

u

0

αæ

,

S(τ ) =

I(t)

ρcu

0

(T

0

s

− T

a

)

,

where f

0

r

is the stationary temperature gradient at the

interface between the phases given by formula (4).

The heat-conduction equation and the boundary

conditions expressed in terms of these variables have
the form

∂θ

∂τ

=

∂

2

θ

∂ξ

2

− v

∂θ

∂ξ

+

S(τ )

l

exp

ξ

l

,

(12)

ξ

→ −∞: θ = 0;

ξ = 0:

θ = ϑ(τ );

in the stationary regime, we have

d

2

θ

0

dξ

2

−

dθ

0

dξ

+

S

0

l

exp

ξ

l

= 0,

ξ

→ −∞: θ

0

= 0;

ξ = 0:

θ

0

= 1 + S

0

with the solution

θ

0

=

1

−

S

0

l

− 1

exp ξ +

lS

0

l

− 1

exp

ξ

l

,

(13)

S

0

=

I

0

ρcu

0

(T

0

s

− T

a

)

.

In the case of low-amplitude oscillations of the ra-

diant heat flux

S = S

0

+ [S

1

exp(iωτ ) + c.c.]

(S

1

S

0

),

all time-dependent quantities in the linear approxima-
tion can be represented as sums of a certain mean value
and a small harmonic component:

v = 1 + [v

1

exp(iωτ ) + c.c.],

ϕ = 1 + [ϕ

1

exp(iωτ ) + c.c.],

(14)

θ(ξ, τ ) = θ

0

(ξ) + [θ

1

(ξ) exp(iωτ ) + c.c.],

ϑ = 1 + S

0

+ [ϑ

1

exp(iωτ ) + c.c.].

Substitution

of

these

expressions

into

the

heat-

conduction equation (12) yields

θ

00

1

− θ

0

1

− iωθ

1

= v

1

dθ

0

dξ

−

S

1

l

exp

ξ

l

.

Taking into account the stationary solution (13), we
arrive at the equation for weak perturbations

θ

00

1

− θ

0

1

− iωθ

1

= v

1

1

−

S

0

l

− 1

exp ξ

+

v

1

S

0

l

− 1

−

S

1

l

exp

ξ

l

with the boundary condition

ξ

→ −∞: θ

0

= 0.

The solution of this equation has the form

θ

1

(ξ) = C exp (zξ)

−

v

1

z(z

− 1)

1

−

S

0

l

− 1

exp ξ

+

l

2

(1

− lz)[1 + l(z − 1)]

v

1

S

0

l

− 1

−

S

1

l

exp

ξ

l

,

where C is the constant of integration and the function
z = z(ω) is given by expression (3). At ξ = 0, the latter
expression yields corrections to the steady-state surface
temperature and its gradient:

ϑ

1

= C

−

v

1

z(z

− 1)

1

−

S

0

l

− 1

+

l

2

(1

− lz)[1 + l(z − 1)]

v

1

S

0

l

− 1

−

S

1

l

,

ϕ

1

= Cz

−

v

1

z(z

− 1)

1

−

S

0

l

− 1

+

l

(1

− lz)[1 + l(z − 1)]

v

1

S

0

l

− 1

−

S

1

l

.

Relation between Gunpowder Burning-Rate Responses to Pressure and Radiant Heat Flux

73

Eliminating the constant of integration from these ex-
pressions, we obtain a linear relation between the cor-
rections to the burning rate and temperature gradient
on the surface:

zϑ

1

− ϕ

1

+

v

1

z

h

1 +

S

0

1 + l(z

− 1)

i

=

S

1

1 + l(z

− 1)

.

(15)

We obtain two additional linear relations between

the corrections to the burning rate, the surface tem-
perature, and the temperature gradient on the surface,
considering unsteady combustion laws. Steady combus-
tion laws in the form (5) are also valid in unsteady cases,
i.e.,

u

r

= F

p, T

s,r

−

æf

r

u

r

,

(16)

T

s,r

= Φ

p, T

s,r

−

æf

r

u

r

.

In the linear approximation, insertion of expansions (14)
into these relations yields two other linear relations:

v

1

= k(ϑ

1

− ϕ

1

+ v

1

),

ϑ

1

= r(ϑ

1

− ϕ

1

+ v

1

)

or

ϑ

1

=

r

k

v

1

,

ϕ

1

=

k + r

− 1

k

v

1

.

(17)

It should be emphasized once again that the parameters
k and r represent the same quantities that enter the ex-
pression for the response function to oscillating pressure
(2). Indeed, in decomposing functions (16) near steady-
state values, there arise derivatives with respect to the
initial temperature, which should be calculated at the
initial temperature T

e

+ I

0

/ρcu

0

. The latter temper-

ature is T

a

exactly. It is just these derivatives that,

according to (3), enter the definitions of the parameters
k and r.

The response function to an oscillating radiant heat

flux

U

(e)

r

=

v

1

S

1

/S

0

can be derived from (15) and (17):

U

(e)

r

=

kS

0

[1 + l(z

− 1)]D + kS

0

/z

,

(18)

where

D = 1 + (z

− 1)

r

−

k

z

.

The product kS

0

can be expressed in terms of the

burning-rate temperature coefficient at the temperature
T

a

and the difference in the initial temperatures T

a

−T

e

.

Indeed, from (3), (11), and (13), we have

kS

0

= β∆, β =

∂ ln u

0

∂T

a

(a)

p

0

, ∆ = T

a

− T

e

.

RELATION BETWEEN THE FUNCTIONS
U

(a)

p

AND U

(e)

r

Thus, the burning-rate response function to an os-

cillating radiant heat flux, found at the initial temper-
ature T

e

(18), can be written in the form

U

(e)

r

=

β∆

[1 + l(z

− 1)]D + β∆/z

.

(19)

Instead of the quantity S

0

, which is hard to determine,

this expression contains the difference between the tem-
peratures at which the burning-rate response function
to oscillating pressure or an oscillating radiant heat flux
are measured.

Comparing (2) and (19), we can ex-

press the response to oscillating pressure through the
response to an oscillating radiant heat flux:

U

(a)

p

=

[ν + δ(z

− 1)][1 + l(z − 1)]

β∆(1/U

(e)

r

− 1/z)

.

(20)

A more symmetrical form of the relation between the
two functions is

ν + δ(z

− 1)

U

(a)

p

=

β∆

[1 + l(z

− 1)]

1

U

(e)

r

−

1

z

.

(21)

It should be noted that application of (20) or (21)

requires the mean free path of radiation in the con-
densed phase, its thermal diffusivity, the steady-burning
parameters of gunpowder ν and δ, and also the burning-
rate temperature coefficient β to be preliminarily deter-
mined.

The difference between the initial temperatures

should be set in the experiment, considering the mea-
surement capability of the experimental setup. The sim-
plest way to making the proper choice of this parameter
includes the following steps:

— the steady-state burning rate u

0

under a given

pressure p

0

and initial temperature T

a

in the absence of

a radiant heat flux should be found;

— under the same pressure, the temperature of the

specimen under study should be decreased to a certain
new initial temperature T

e

;

— the burning rate at this new initial temperature

and in the presence of a constant external radiant heat
flux u

0

r

should be measured, and the source intensity

such that the equality u

0

r

= u

0

be satisfied should be

chosen from (11).

The present-day radiation sources furnish radiation

intensities of about one megawatt per square meter. De-
pending on pressure, at normal densities and heat ca-
pacity of gunpowders, the difference between the initial
temperatures in actual measurements ranges from sev-
eral tens to two hundreds of degrees.

This work was supported by the Russian Foun-

dation for Fundamental Research (Grant No. 02–03–
32077).

74

Novozhilov, Kono, and Morita

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