V. A.
THE FOUNDERMIKHELSON

OF THE PHYSICS OF COMBUSTION
IN RUSSIATO CELEBRATETHE OCCASION

OF HIS HUNDREDTH

BIRTHDAYA. S. PR~VOD~TELEV
Department of Molecular Physics, Moscow State University, U.S.S.R.
(Receilled 30 October 1961)

THK ~ND~DTH
bi~hday of the prominent
Russian scientist V. A. Mikhelson was celebrated on 27 June. His most outstanding
investigation was the study of the inflammationof detonating gas mixtures. In spite of the factthat this problem was the object of numerous
investigations before V. A. Mikhelson, nevertheless it is true to state that this is the work
which made him very famous, and made him
the founder of a vast branch of science inRussia known as the physics of combustion andexplosions.
Mi~elson’s works on the physics of combustion and explosions were compiled into an
extensive monograph which was submitted to
the Moscow University as a thesis for a master’sdegree. Numerous explosions in coal pits, with
their attendant heavy toll on humanity, initiatedthe investigation of processes of combustion andexplosions. This mono~aph
is full of ideas
which still preserve their freshness. Someof Mikhelson’s ideas may be easily developed
and are capable of expansion into forms ofconsiderable use nowadays; this is the aim of thepresent work.
The classical memoirs of Mallyar and Le
Chatelier published in 1883 open the historyof this problem. However, besides the worksof these scientists there were importantinvestigations carried out by Davy, Bunsen,
Schlesing and Fonseca, as a result of which theso-called heat theory of fire travel was outlined.
Proceeding from the equation of energy balance
in a combustion process, Mallyar and LeY435

Chatelier for the first time made an attempt togive a formula for the rate of fire travel in acombustible mixture. The problem investigated by Mallyar and Le Chatelier was thedetermination of a functional relationship between the propagation rate of a front, physicalproperties of a gas mixture, and chemical
kinetics of combustion. Mallyar and Le Chatelier
did not completely manage to achieve theirobjective, but they described the qualitative
aspect of the phenomenon correctly.
The historical memoirs of Mallyar and I..e
Chatelier had already been published five years
when Mikhelson’s works on the physics ofcombustion appeared. An experimental part ofhis work was carried out at the Physical Instituteof the Berlin University, where Mikhelson had
been sent in order to expand his scientificeducation. Investigations were sanctioned by
Helmholtz and Kundt.
In his thesis, besides giving striking experimental results on the rate of propagation of acombustion front in various combustible gas
mixtures, Mikhelson gave a theoretical statement of the problem in the form which is still
believed to be the only true one.
Even now the problem of fire propagation’isone of great controversy, and, approaching thisquestion quite objectively, one may con&m thatthe theories of all the investigations carried out
in the Soviet Union and abroad do not possess
any advantages over the statement of theproblem proposed by Mikhelson.
Mikhelson as well as Mallyar and Le Chatelier

436

A.

S. PREDVODITELEV

considered that fire propagation over a combustible mixture should be investigated inrelation to the heat propagation over a noncombustible gas mixture. This means that theconduction equation should be used as thebasis of the theory, but this is not all; the heatconduction equation differs in its character fromthe wave equation, which in its di~erential form
contains the idea of a wave being propagatedat a definite rate. As is known, the sohttionsof the wave equation are always as follows:
Q, = @(r rt gt)
Here, the direction of the ray is designatedthrough r and the velocity of wave propagation
through g.
The differential heat conduction equation by
itself does not have such solutions. The questionof why the front propagates with a definite
rate is in fact the case with combustion.
Mikhelson understood quite clearly that sucha possibility may appear in all cases where there
are conjugated processes similar to heat distribution in chemically reacting media. The joint
consideration of the differential heat conduction equation with the equation of kinetics
of chemical reactions may yield solutions
similar to those of the wave equation. That is
the reason why, when searching for the functional
relationship between the rate of fire travel andthat of a chemical reaction, he transformed theone-~mensional heat conduction equation to avariable :.f = x - gt.

As a result of such a transformation and thesolution of the heat conduction equation he getsthe following formula for the rate of displacement of a flat combustion front:g = _ ” lg LX5
E Tl- T,(1)

where T, is the initial gas temperature; Tl is theapron
tempera~re;a is the so-called
thermal diffusivity of the gas mixture.
In order not to have indefinite values of thevariables 6 and T in the formula mentioned, werelate it to that surface behind the front where
the temperature reaches its limiting value,

designated by Tf. If the origin and time reading
are connected with the front head, then in thiscase the co-ordinate 6 by its absolute value, will
be equal to the front thickness d and will havethe inverse sign. Thus, Mikhelson’s formula forthe rate of front motion may be written as
follows :nT, - _-.
r,
ST= 2 k Tl_..___ T,’

But, on the other hand, the quantity of a burnt
gas in a combustion zone is equal to some finitevalue 4. Consequently, the mass flow throughthe front surface required for combustion is:

This flow should be equal to the following value
if the rate of a chemical reaction is designatedthrough dy/dt:

Therefore, we have.JZg = jsd
ord z=:_+&g&
Substituting this expression forMikhelson formula, we obtain :(2)d intotheTheequation
given
above
essentiallycompletes this problem, It is interesting to notethat all the existing formulae obtained to dateby Soviet and foreign scientists differ only slightly
in form from that in which Mikhelson’s formula
was presented. No experiments carried out up to
now have disproved Mikhelson’s concept, or new
definitions of it introduced anything essential. At
present the object of discussions is only temperatures T, and T,, i.e. those conditions whichshould be assumed on the surfaces restricting
the zone of the comb~tion front. But theseconditions are highly restricted in their variety.
Proceeding from Mikhelson’s concept it isalso easy to obtain another type of the formulafor the velocity of displacement of a normal
combustion front. For this purpose it is assumed

V. A.

MIKHELSON:

THEFOUNDER

that the velocity of the temperature change at themoment of front formation has always a finiteand constant value.
The integral of heat conduction equation (1)in a non-logarithmic form is written as follows:
T =(Tl_To) ,-@i@z.&zin)t

Determine the first-timefunction T. It will be:derivative= ; (Tl - To),THE

PHYSICSTo.

ofthe(4)The time required for the front formation isdesignated through 7. Let the maximum front
temperature be equal to Tf. Therefore, thevelocity of the temperature change at the front
formation may be expressed in the following
form :(Tf - TI)/T.
But according to the hypothesis given above
this expression must be equal to relation (3).
Consequently, such an equality may be written:

OFTfJTaking this into account, equation (5a) may be
given in the form:

The formula obtained, as is easy to see, is very
close in its physical content to the formula ofLewis and Elbe as well as to the fommla obtainedby our Soviet investigators Zeldovich andFrank-Kamenetsky.
Experimental results described by Mikhelson
in his thesis are also excellent. He established
two essential facts. The investigation on thedependence of the normal rate of front displacement upon the mixture composition led
to the discovery of limiting points, i.e. the rateof front displacement does not become zero atany change of mixture concentration, it always
has lower and upper limits.
It means that in Mikhelson’s formula (3) for
the thermal diffusivity coefhcient the multiplierequal to.flg;i&1

But from equation (2) which is also valid for thegiven conditions, we have:&d- =7=-?-.
8Y
Now formula (5) may be presented in thefollowing form :T, - Tl

g2=aT,--T,2.

r’(W4371dy= T, - Tl Ip1x-t dT.

g2(T, - 7-0)/a = (T, - T,)/q.
HenceCOMBUSTION

The average reaction rate during the timenecessary for the formation of a combustion
zone may also be written in the form of thefollowing integral:-7+Upon referring this expression to the origin ofco-ordinates and initial moment it will give:f==OOF0never becomes zero. From the relation mentioned
above it is seen that at equalities Tf = T,
Tl = To the logarithm of the relation becomes
zero, but the value & is equal to zero as well.
Another phenomenon observed by Mikhelson
and closely connected with the one described
is, the rate of normal front displacement at its
maximum is dependant on the mixture composition. Thus, the curve of the dependence of thefront displacement rate on the composition has
the form of a bell resting upon the limiting
points. These facts are of paramount importancefor an understanding of the true kinetics ofthe chemical substance conversion.

438

A. S. PREDVODITELEV

The last but one chapter of the thesis deals
with the theory of the Bunsen burner. The author
gave such thorough development of the problem
that even now text-books and monographs
contain it almost in the same form as Mikhelsongave it.

We see that Mikhelson’s scientific achievements in the field of the physics of combustion
are so considerable that we may consider him
by right the founder of this teaching. Soviet
scientists must strengthen and develop scientificconcepts proposed by Mikhelson.