ISSN 0012-5008, Doklady Chemistry, 2006, Vol. 410, Part 1, pp. 158–164. © Pleiades Publishing, Inc., 2006.Original Russian Text © V.V. Klyucharev, 2006, published in Doklady Akademii Nauk, 2006, Vol. 410, No. 3, pp. 347–353.

158

Self-propagating processes are currently considerednot only as an energy source but also as a procedure ofchemical synthesis [1, 2]. However, the nature of thephenomena underlying this procedure is still unclear[3–5].

Let us recall the essence of the problem by analyz-ing the law of energy conservation for a self-propagat-ing chemical reaction wave. In the one-dimensionalcase, in the absence of heat losses to the environment,the change in the heat flux

q

passing through an ele-mentary layer

dx

of density

ρ

is equivalent to thechange in the enthalpy of this layer

dH

occurring overthe time of thermal relaxation

t

:

(1)

Inasmuch as the quantity

dH

includes, at least, theenthalpy change of the reacting system due to the heat-ing of its components

dh

and the enthalpy change of thereacting system due to a chemical reaction

d

(

α∆

)

,where

α

is the degree of conversion of a unit mass ofthe reagents and

∆

is the standard enthalpy ofreaction, we can write

(2)

In the general case,

h

=

h

[

x

(

t

),

θ

(

t

),

t

]

and

α

=

α

[

x

(

t

),

L

(

t

),

t

]

, where

θ

(

t

)

is the thermal relaxation functionand

L

(

t

)

is the thermochemical kinetic function. If thethermal and thermochemical relaxation times are takento be infinitesimal and the pressure in the reaction zoneis taken to be constant, we may assume that

where

T

is the temperature,

C

p

(

T

)

is the heat capacity atconstant pressure, and

K

(

T

)

is the chemical reaction

∂q∂x------ ρdH

dt-------.=

Hreact°

Hreact°

dH dh ∆Hreact° dα.+=

θ t( ) Cp T( ) and L t( ) K T( ),= =

rate constant; therefore, formula (1) can be rearrangedto the form

(3)

or written in the form

(4)

where

λ

is the heat conductivity coefficient,

u

=

is

the normal flame propagation velocity, and

Q

R

=

−∆

is the standard heat of chemical transforma-tion.

As is known, Eq. (4) allows for several statements ofthe problem of the critical conditions of combustion [3,4]. One of them was developed by Zel’dovich andFrank-Kamenetskii [6]. For a zero-order reaction withArrhenius kinetics, this statement of the problem can beformulated as

(5)

where

A

and

E

are, respectively, the pre-exponent andthe activation energy of chemical transformation. Thephysical meaning of simplification (5) becomes clear ifwe note that it follows from Eq. (4) when two condi-tions are met. The first one,

∂q∂x------ ρCp

∂T∂t------⎝ ⎠

⎛ ⎞x

dxdt------ρCp

∂T∂x------⎝ ⎠

⎛ ⎞t

+=

+ ρ∆Hreact° ∂α∂t-------⎝ ⎠

⎛ ⎞x

dxdt------ρ∆Hreact° ∂α

∂x-------⎝ ⎠

⎛ ⎞t

+

∂∂x------ λ∂T

∂x------⎝ ⎠

⎛ ⎞ ρ Cp∂T∂t------⎝ ⎠

⎛ ⎞x

QR∂α∂t-------⎝ ⎠

⎛ ⎞x

–=

+ uρ Cp∂T∂x------⎝ ⎠

⎛ ⎞t

QR∂α∂x-------⎝ ⎠

⎛ ⎞t

– ,

dxdt------

Hreact°

∂∂x------ λ∂T

∂x------⎝ ⎠

⎛ ⎞ uρCp∂T∂x------⎝ ⎠

⎛ ⎞t

–

+ ρQR∂T∂t------⎝ ⎠

⎛ ⎞x

AeE

RT-------–

0,=

Dimensions of Becoming of Self-Propagating Chemical Processes

V. V. KlyucharevPresented by Academician Yu.D. Tret’yakov April 28, 2006

Received May 12, 2006

DOI: 10.1134/S0012500806090047

Institute of Problems of Chemical Physics, Russian Academy of Sciences, Institutskii pr. 18, Chernogolovka, Moscow oblast, 142432 Russia

CHEMISTRY

DOKLADY CHEMISTRY Vol. 410 Part 1 2006

DIMENSIONS OF BECOMING OF SELF-PROPAGATING CHEMICAL PROCESSES 159

(6)

reflects a nonuniform stationary temperature distribu-tion in the heating zone. The second condition,

(7)

(8)

uρCp∂T∂x------⎝ ⎠

⎛ ⎞t

0; ρCp∂T∂t------⎝ ⎠

⎛ ⎞x

≠ 0=

uρQR∂T∂x------⎝ ⎠

⎛ ⎞t

AeE

RT-------–

0,=

ρQR∂T∂t------⎝ ⎠

⎛ ⎞x

AeE

RT-------–

0≠

refers to a homogeneous nonstationary chemical reac-tion. Hence, Eq. (5) describes the coexistence of twoprocesses, a chemical reaction and heat absorption [7],which is limited by the thermal properties of themedium in the preheating zone, while the self-ignitiontemperature, according to Eq. (7), coincides with theexact upper boundary of the temperature field in theself-propagating process wave

(9)

The alternative statement of the problem of the crit-ical conditions of combustion according to Eqs. (6),(10), and (11),

TB sup T{ }shp.=

DTlog9log3----------- 2= =

(‡) (b) (c)

(d) (e) (f)

(g) (h) (i)

DTlog8log3----------- 1.8928≈= DT

log7log3----------- 1.7712≈=

DTlog6log3----------- 1.6309≈= DT

log5log3----------- 1.4650≈= DT

log4log3----------- 1.2619≈=

DTlog3log3----------- 1= = DT

log1log3----------- 0= =DT

log2log3----------- 0.6309≈=

Fig. 1. Dimensions of becoming upon the threefold scaling of a square.

160

DOKLADY CHEMISTRY Vol. 410 Part 1 2006

KLYUCHAREV

(10)

(11)

allows for the existence of the stationary self-propagat-ing reaction wave and the ignition temperature

(12)

which, due to inequality (10), does not coincide withthe maximum temperature of combustion. This idea isreflected by the Daniell equation [8] written in the form

(13)

The strategy for elucidation of the combustionmechanism suggested by Zel’dovich and Frank-Kamenetskii [6] has formed the basis of many success-ful applications, and it is quite reasonable that it hasappeared in textbooks. At the same time, Daniell’sapproach is only occasionally mentioned in the litera-ture as a kind of riddle that specialists attempt to solvein one way or another [3–5].

Physical interpretation of the stationary combustionwave according to conditions (6), (10), and (11) pre-sents difficulties, which are caused by the fact that, inthe 20th century, an object whose reflection is the non-zero partial derivative of the conversion with respect to

the coordinate (10) was perceived either as a

category of formal logic [9] or as an instantaneous jumpfrom the preceding state to the subsequent state [10,11]. As a result, the question of the nature of the motion

uρQR∂T∂x------⎝ ⎠

⎛ ⎞t

AeE

RT-------–

0,≠

ρQR∂T∂t------⎝ ⎠

⎛ ⎞x

AeE

RT-------–

0=

TC inf T{ }shp,=

∂∂x------ λ∂T

∂x------⎝ ⎠

⎛ ⎞ uρCP∂T∂x------⎝ ⎠

⎛ ⎞t

– uρQR∂T∂x------⎝ ⎠

⎛ ⎞t

AeE

RT-------–

+ 0.=

∂α∂x-------⎝ ⎠

⎛ ⎞t

at the stage of becoming of a new substance in the self-propagating reaction wave remains unanswered.Zel’dovich and Frank-Kamenetskii’s equation (5),

which is derived from Eq. (4) at = 0, has only

obscured this fact. The aim of the present work is tooffer new tools that will allow one to elucidate the kine-matics of becoming of a new substance in a self-propa-gating reaction wave and, on this basis, to clearly differ-entiate burning, combustion, and explosion as phenom-ena with a radically different organization of chemicaltransformation.

The basis for solution of this problem is the conceptthat the space of the transformation in the state ofbecoming is formed by appearing and disappearing ele-ments of the old and new wholenesses [11, 12]. Hence,the dimension of the state can be defined as

(14)

where N is the number of new equal unit elements of acovering that retain their quality upon scaling of thestate in accord with the similarity coefficient m (Fig. 1).By the definition of DT, the condition

(15)

is always valid (DL is the topological dimension of theplace occupied by the object of transformation),whereas the inequality

(16)

is always true for the fractal dimension [13].Let us explain this discrepancy by the example of

the Koch curve (Fig. 2). This curve is constructed asfollows: the initial segment is divided into three equalsections, and the middle one is turned at an angle of 60°to the initial direction and a fourth segment is added(shown by the dashed line). According to Mandelbrot[13], the transformation result is described by the frac-

tal dimension DF = ≈ 1.2619. This means that the

development of the one-dimensional object takes placein a space with a topological dimension DL > 1. How-ever, if the “act of creation” is considered, it is easy tosee (curve b) that the object comprises four equal sub-objects, one of them being subjected to transformation.Hence, according to Eq. (14), the becoming of the Kochcurve is characterized by the fractional dimension DT =

≈ 0.7925. The physical meaning of inequality

(15) is that the transformation as a state cannot gobeyond the boundaries of the region where this trans-formation occurs.

It is worth noting that the dimension of becomingcan also be DT = DL, which is forbidden for the fractaldimension by condition (16). This equality takes place

∂α∂x-------⎝ ⎠

⎛ ⎞t

DTNlogmlog

-------------,=

DT DL≤

DF DL.>

4log3log

-----------

3log4log

-----------

c

b

‡

Fig. 2. Koch curve: the (a) zeroth, (b) first, and (c) secondgenerations.

DOKLADY CHEMISTRY Vol. 410 Part 1 2006

DIMENSIONS OF BECOMING OF SELF-PROPAGATING CHEMICAL PROCESSES 161

if the conversion of an object consists in division of awhole into its parts or combination of parts into awhole, displacement of the parts of a whole, expansion,

or contraction. For clarity (Fig. 3), at DT = ≈

1.8928, a Lebesgue point can transform to the Sierpin-ski carpet (Figs. 3a–3c), whereas, at DT = 0, a Lebesguepoint remains an object that requires one minimal cov-ering for its mapping in space (Figs. 3d–3f).

Assessing the becoming of the chemically trans-forming bodies with the use of adequate dimensions, allself-propagating processes can be easily classified intothree groups (Fig. 4). For the first group (Figs. 4a, 4d),it suffices that the reaction products concentrated in avolume of scale δk initiate the reaction wave in the adja-cent layer of the initial reagents of scale δi meeting thecondition

(17)

For the second group (Figs. 4c, 4f), the condition

(18)

is met. The boundary between these groups (Figs. 4b,4e) is the equality

(19)

8log3log

-----------

δi δk.<

δi δk>

δi δk,=

which corresponds to stationary self-development. Thedimensions of becoming corresponding to conditions(17), (18), and (19) are described by Eqs. (20)–(22)(Figs. 4a, 4c, and 4b) for one-dimensional systems, byEqs. (23)–(25) (Figs. 4d, 4f, and 4e) for two-dimen-sional systems, and by Eqs. (26)–(28) (Figs. 4d, 4f, and4e) for three-dimensional systems.

(20)

(21)

(22)

(23)

(24)

(25)

DT2 m m 1–( )–[ ]log

mlog---------------------------------------------

2logmlog

------------ 0,= =m → ∞

DT4 2–( )log

2log------------------------- 2log

2log----------- 1,= = =

DT2 m 1–( )log

mlog------------------------------ 1,= m → ∞

DTm2 m 1–( )2–[ ]log

mlog----------------------------------------------

2m 1–( )logmlog

------------------------------ 1,= = m → ∞

DT4 1–( )log

2log-------------------------

3log2log

----------- 1.5850,≈= =

DTm2 1–( )log

mlog---------------------------- 2,= m → ∞

(‡) (b) (c)

(d) (e) (f)

Fig. 3. Self-similar transformation of an object with the dimension (a–c) DT = ≈ 1.8928 and (d–f) DT = = 0.8log3log

----------- 1log3log

-----------

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DOKLADY CHEMISTRY Vol. 410 Part 1 2006

KLYUCHAREV

(26)

(27)

(28)

The physical meaning of this classification becomesclear if we elucidate the correlation between the formsof motion described by Eqs. (20)–(28) and the energycorresponding to the critical conditions of their exist-ence. For a chemical reaction wave caused by heattransfer from the hot layer to the neighboring cold oneorganized so that limit (20), (23), or (26) is fulfilled, theself-propagating process will be sustained if the calo-rific value of combustible cells is no less than

(29)

where T0 is the initial temperature of the combustiblemixture, Tign is the self-ignition temperature describedby Eq. (9) or (12), and CK is the heat capacity of thereaction products. This is due to the fact that the initial

DTm3 m 1–( )3–[ ]log

mlog----------------------------------------------=

= 3m2 3m– 1+( )log

mlog---------------------------------------------- 2,m → ∞

DT8 1–( )log

2log-------------------------

7log2log

----------- 2.8074,≈= =

DTm3 1–( )log

mlog---------------------------- 3.=

m → ∞

Q CK T ,d

T0

T ign

∫=

substance to be heated is point in the one-dimensionalcase (20), line in the two-dimensional case (23), andsurface in the three-dimensional case (26). If the reac-tion is organazed so that limiting condition (21), (22),(24), (25), (27), or (28) is fulfilled, for the self-propa-gating process to be sustained, the calorific value ofcombustible cells should be no less than

(30)

where CI is the heat capacity of the initial reagents, QI

is the heat of phase transitions and chemical transfor-mations in the zone of heating from T0 to Tign, and n isthe ratio of the spatial fraction of the initial reagents tothe spatial fraction of the reaction products. This is dueto the fact that, in this case, the n value cannot be takenas negligible.

The thermodynamic vulnerability of self-develop-ment as compared to adaptation, which follows fromEqs. (29) and (30), is of special interest since dimension(21), (24), or (27) can be calculated at the same m = 2according to both condition (20), (23), or (26) and con-dition (22), (25), or (28). Therefore, it may seem thatthe adiabatic limit of the self-propagating reaction isalways condition (29) and, hence, these transforma-tions, from the chemical standpoint, do not differ fromthe processes induced by external heating in a furnace [2].

Q nQI CK nCI+( ) T ,d

T0

T ign

∫+=

(‡) (b) (c)

(f)(e)(d)

DTlog2log4----------- 0.5= = DT

log2log2----------- 1= = DT

log6log4------------ 1.2925≈=

DTlog15log

-------------- 1.9534≈=

DTlog63log4-------------- 2.9886≈=DT

log7log2----------- 2.8074≈=

DTlog3log2----------- 1.5850≈=DT

log7log4----------- 1.4037≈=

DTlog37log4-------------- 2.6047≈=

Two-dimensional burning

One-dimensional burning

Three-dimensional burning

One-dimensional combustion

Two-dimensional combustion

Three-dimensional combustion

One-dimensional explosionis impossible

Two-dimensional explosion

Three-dimensional explosion

Fig. 4. Dimension of becoming of self-propagating processes.

DOKLADY CHEMISTRY Vol. 410 Part 1 2006

DIMENSIONS OF BECOMING OF SELF-PROPAGATING CHEMICAL PROCESSES 163

As a result, the existence of limit (30) should be verifiedboth logically and experimentally. This did not meetwith success in the 20th century.

From the standpoint of the classical mechanics ofchemical transformation, principles of which are pre-sented in this work, one-dimensional combustion is themost suitable object for solving this problem. In thiscase, there is no need to answer the question of how theobjects of fractional dimension that emerge at the stageof becoming convert to Euclidean objects since DT = DL

(Fig. 4b). In addition, condition (22) has a physicalmeaning only at m = 2 since the value of function (22)exceeds unity at any m > 2, i.e., DT > DL, which isimpossible because of condition (15).

It is rather simple to accomplish one-dimensionalcombustion if a stoichiometric fuel diluted with achemically inert binding agent is used. In this case, theformation of two- or three-dimensional burning sites isinefficient due to a local deficiency in the componentsof the combustible mixture.

As an example, let us consider the combustion limitsobtained by dilution of the stoichiometric fuel Mg–CaO2 with inert filler. The initial components—a mag-nesium powder with a particle size of 65–100 µm, cal-cium peroxide (95.7 wt % CaO2) with a particle size of20–50 µm, and sodium and potassium chlorides(99.8 wt % NaCl or KCl) with a particle size of 65–100 µm dried at 250°ë—were mixed and pressed intoblocks 30.2 mm in diameter and 50–60 mm high with adensity of 1.7–1.8 g/cm3. The self-propagating reactionwave was initiated by an electrically heated Nichromewire at one end of the block. In Tables 1 and 2, the frac-tions of NaCl and KCl at which the ignited sampleceased to completely burn up and the adiabatic combus-tion limits calculated by Eq. (30) in accord with (12) atTign = TC, T0 = 20°C, and n = 1 taking into account thedissociation of CaO2 in the preheating zone are com-pared.

This comparison shows that the self-ignition tem-perature is determined by the melting of chloride andthe combustion limits are specified by formula (30),which includes the power spent for heating of the initialcomponents. This is clearly seen from the paradoxicaldecrease in the allowable fraction of the inert diluent atthe combustion limit when a component with a rela-tively high melting point and enthalpy of melting(NaCl) is exchanged for a component with a relativelylow melting point and enthalpy of melting (KCl). Itturns out that sodium chloride should be dissolved inthe layer of the products and should only be heated tothe melting point in the layer of the initial reagents.Potassium chloride should be melted in both the prod-uct layer and the initial reagent layer.

The data obtained unambiguously indicate that theprinciples of chemical mechanics presented in thiswork make it possible to clearly differentiate burning,combustion, and explosion as phenomena with a radi-

cally different organization of chemical transformation.The first of them is characterized by the dimension

DT ∈ (0, 1) in the one-dimensional case, DT ∈ 0,

in the two-dimensional case, and DT ∈ 0,

in the three-dimensional case. The stationary

self-development is characterized by the dimension

invariants DT = 1, DT = ≈ 1.5850, and DT = ≈

2.8074, whence it follows [14] that, on the scale of thebecoming, two- and three-dimensional combustionhave a fractal nature. Explosion is characterized by the

dimension DT ∈ , 2 in the two-dimensional

case and DT ∈ , 3 in the three-dimensional

case. If the explosion velocity is lower than the speed of

-⎝⎛

3log2log

-----------⎠⎞ -⎝

⎛

7log2log

-----------⎠⎞

3log2log

----------- 7log2log

-----------

3log2log

-----------⎝⎛

⎠⎞

7log2log

-----------⎝⎛

⎠⎞

Table 1. Adiabatic limits and the experimentally found steady-step combustion limit in the system 1 M Mg–1 M CaO2–x MNaCl

Tign, °C Combustionlimit, x M NaCl Note

651 5.84 Without melting of Mgin the initial layer

5.71 With melting of Mg in the initial layer

801 4.15 Without melting of NaCl3.14 Without melting of NaCl

in the initial layer2.53 With melting of NaCl

in the initial layer3.05 ± 0.10 Experiment

Table 2. Adiabatic limits and the experimentally found steady-step combustion limit in the system 1 M Mg–1 M CaO2–x MKCl

Tign, °C Combustionlimit, x M KCl Note

651 5.85 Without melting of Mgin the initial layer

5.72 With melting of Mgin the initial layer

771 4.40 Without melting of KCl3.36 Without melting of KCl

in the initial layer2.72 With melting of NaCl

in the initial layer2.80 ± 0.10 Experiment

164

DOKLADY CHEMISTRY Vol. 410 Part 1 2006

KLYUCHAREV

sound in a combustible material, deflagration takesplace; when the explosion velocity exceeds the speed ofsound in a combustible material, detonation takesplace.

The above classification makes it possible to consol-idate combustion chemistry and explosion chemistry asparticular areas of general chemistry. Until recently,these phenomena have been considered mainly fromthe standpoint of chemical kinetics since chemicalmechanics in the 20th century was focused on studyingatoms, molecules, and molecular intermediates [15].The use of a mole, which includes not only the struc-tures of molecules and supramolecular associates butalso the organization of chemical motion at the levelcorresponding to the substance in its classical sense, isa new step in understanding chemism. As a result, itbecomes clear that self-organization of chemical pro-cesses is possible not only in the adaptation mode,dominated by energy and matter exchange with theenvironment [6], by also in the self-development modeunder the action of the store of energy and matter in thereaction system [8]. Therefore, as yet unwritten chap-ters devoted to combustion and explosion chemistryshould be added to general chemistry in the 21st cen-tury.

ACKNOWLEDGMENTS

I am grateful to Academician Yu.D. Tret’yakov andProfessor L.P. Kholpanov for helpful discussions.

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