THEORY OF COMBUSTION OF UNMIXED GASES

NASA TECHNICAL TRANSLATION

NASA TT F-I4,360

THEORY OF COMBUSTION OF UNMIXED GASES
Ya. B. Zel'dovich

Translation of: "K Teorii Goreniya
Neperemeshannykh Grazov," Zhuvnal Tekhnicheskoy Fiziki,
Vol. 19, No. 10, 1949, pp. 1199-1210.

, HC $3.00

G3/33

N73-15956

Onclas
62615

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
WASHINGTON, D. C. 20546 JANUARY 1973

NASA TT F- 14, 360
THEORY OF COMBUSTION OF UNMIXED GASES
Ya. B, Zel'dovich

INTRODUCTION

The author will consider the chemical reaction o£ two substances (a fuel /UgQ"
and oxygen) accompanying the formation of new substances which are the products
of combustion and the liberation of heat.

The author will examine the stationary process with continuous supply of
raw materials and removal of products. The special characteristic of the case
in question consists in that the fuel and oxygen, or air, are fed separately,
i.e., they are not mixed beforehand. Therefore, even in the case when the
constant of the reaction rate of oxygen with the fuel is high, the intensity of
combustion will not exceed a certain level which depends upon the rate of
mixing of the fuel and the oxygen.

Note that the fact of combustion itself significantly changes the dis-
tribution of the concentrations relative to the distribution of concentrations
when the same gases are mixed without combustion.

It has been known for a very long time, since the time of Faraday, if
not earlier, that there is a basic qualitative concept which states that the
surface of a flame separates a region in which there is oxygen and no fuel
(oxidizing region) from the region in which there is no oxygen but there is
fuel (reduction region) .

Burke and Schi^mann [1] calculated the shape of the surface of a flame

in a very specific case of combustion in parallel concentric laminar flows of

fuel and oxygen or air. They failed to take into account the details of the

phenomena that occur in the zone of the flame.

The recently published work by Shvab [2], which was performed in Leningrad
back in 1940, is the most complete to data. This paper contains a detailed

*Numbers in the margin indicate pagination in the foreign text.

description o£ the turbulent tongue of flame, both as a case with feed of pure
burning gas and a case of feed of a gas mixture with an insufficient quantity
of air, which the author has not taken into account. Shvab finds the
relationships between the fields of concentration of gas, oxygen and combustion
products, temperature and the velocity field of the gas which the author did
not take into account. A number of the results obtained by Shvab (in particular,
the constancy of the concentration of combustion products and temperature) at
the surface of the flame are given in this article for the sake of completeness.

The detailed discussion of the zone and the kinetics of the chemical
reaction are essentially new (§5).

In the case of laminar combustion it is possible to go further on the
basis of this investigation and to determine the limits of possible intensifi-
cation of the combustion in view of the fact that at a high rate of feed of
fuel and oxygen to the surface of the flame and insufficient rate of the
chemical reaction results.

§ 1. General Equations /12£0

One should consider the region in which the gas is moving at a rate —
the density of the gas is p, the weight concentration of the component in which
we are interested is a, the coefficient of diffusion is D, the thermal
conductivity is X, the temperature conductivity is the fraction ^ = X, the
temperature is T, with all of the above values variable (depending upon the
coordinates). The flow of component a is given by the formula

9a = pa « — f £> grad a.

(1)

The vector q gives the direction of the flow and its value in grams per
centimeter squared per second at a given point. The general equation of
diffusion will have the form

i divl = L{a)=-^{fa)-*-F,. ^ (2)

On the left-hand side is the divisibility of the flow, i.e., the difference
between the quantity, of the substance brought in by the flow and the amount
carried away by it, based on a unit volume; on the right,?side is the change in

A

amount of substance a per unit volume ^^' and the amount of substance a
which is formed per unit volume as a result of the chemical reaction.

L(a) is an abbreviated designation for the differential operator

In a stationary flow

L (a) = div( pa«| — div (p£> grad a).
|-(pa) = 0. ,| = -div(p«) = 0.

(3)

(4)
(5)

j[;,(o) = fCUg^ada — div (p£) grade).

In a stationary process

The equation of thermal conductivity has an analogous form^ to that of the

heat flux . . - . — - -.

■Jy^fTcw — X?rad7'=prc« — xcpgradr, ^^^

where c is the specific thermal capacity of a unit mass which the author con-
siders to be constant. From this it follows

div^r = cL(r)=--cy,(pr)-+- Q,

where Q is the volume velocity of liberation of heat.
In a stationary process ^ '

(8)

(9)

If the coefficients of diffusion of various substances are originally a,
b and the combustion products are g, h and the coefficient's of thermal conductivity
are all equal to each other

i z>.=A=/>*=*^»=''=Av J (10)

the operators L(a), L(b),..., 1(1) will also coincide in a formula of the form
(6), written for various substances and in formula (9) for the temperature.

In a chemical reaction involving combustion, the amount of raw materials
entering into the reaction, the amount of combustion products formed and the

/1201

^Here.the author disregards the heat loss through radiation (see below).

amount of heat i liberated come within certain strictiy constant jrelationships.
Using the F to designate the volume velocity of the reaction, the author uses
it to express all of the va.lues _ _

_ /-. Q _ F

O — „ » ^i> — ft » ^9 — r ' * »i ♦ c

(11)

by means of constant and positive coefficients a,..., ir. The signs in (11)
indicate that a, b are used up and g, h and heat are given off in the reaction.
The coefficients a,..., x are placed in the denominator for convenience of
further calculation. For example:

cal

, = CH*i 6=0,; ^ = CO,j A = H,0; c=0.5-^^ ;

the heat of the reaction, is 192,000

cal

mole CH,

Expressing F as the consumption of all substances entering into the
reaction in g/cm • sec, one obtains

a = 5; p = 1.25; Y = l-82; *i=2.22; t = -^ = 2.08 • 10-*rC)-^

By means of these coefficients, all of the differential equations of
diffusion of various substances at equations of thermal conductivity in a
chemical reaction will take on a completely identical form, including L and F
in all formulas ;■ .

L(^a) = _F; Lm^-F\ L('{g)^F; L{r>h) = F; L(Tr^ =F. (12)

(12)

However, it is still not possible to conclude from this that the fields

of all of the values a,..., T of interest are the same,, inasmuch as the field

of each value depends not only on the differential equation which this value

satisfies but also oh boundary conditions.

A combustible gas is fed through a pipe (I); accordingly, within the pipe
the following will be valid

Ir a=ao; ,b = g = h = Qi T=:Ti

o>

(13)

where a_ is the concentration of the fuel in the gas reaching the combustion
point. This gas may be diluted, for example , by nitrogen.

Through another tube (II) air is supplied

II. 6=V a = g = h=^0; T=To,

(14)

where b is the concentration of oxygen in the air. During burning of the
flame in the open atmosphere condition II refers not to a pipe containing air
but to the concentrations in the atmosphere at an infinite distance from the
flame.

On the surfaces of the pipes, on the surface of the combustion chamber,
etc., the boundary conditions consist in the fact that the flow of any substance
through the material surface is equal to zero, so that the component of the
corresponding flow vector which is normal , to the surface is equal to zero.
These conditions are the same for all substances. The boundary conditions for
the temperature will be the same as the boundary conditions for a,..., h, if
one does not remove any heat from the flame, i.e., if only heat-insulated and
not heat-radiating surfaces are employed or if the walls in general with
temperature T_ are located only where the gas temperature is equal to T^.

One assumes that these conditions are satisfied so that the boundary
conditions for all of the substances and the temperatures on the walls are the
same.
§ 2. Analysis of Equations. Equation For the Surface of the Flame

Now let us proceed to an analysis of the equations. The principal
difficulty in a direct solution of these equations consist in the fact that the
rate of the reaction E is highly dependent upon the very values which the
author is trying to find, a, b and T.

Calculating the first from the second one has

Z,(aa — Pfc) = /.(p) = 0,where|^=cta-^^, | (15)

with boundary conditions „„^

(16)

I I) p = OLa„ II)p = -3V

Hence, the difference between the concentrations of the fuel and oxygen,
taken with corresponding stoichiometric coefficients, obeys the equation of
diffusion in which the rate of the reaction F is not involved. This is the
equation which was discussed by Burke and Schumann to determine the shape of
the flame, proceeding on the basis of the fact that the fuel may be viewed as
negative oxygen. If we are dealing not with a slow reaction but with combustion,

5

this means that the function F with simultaneous a =/= 0, b ?^ is very high.
Inasmuch as the total amount o£ substance which is burned per unit time is
limited by the amount of fuel which is supplied, the equation F(a, b, T) with
a, b and T given means that in reality the width of the reaction zone in which
a T^ and b =^ is decreasing in the flame and the values a and b are decreasing
in this zone.

At the limit, with an infinitely rapid reaction, a and b tend toward zero

in the reaction zone, so that nowhere (except for an infinitely narrow zone)

can a and b differ simultaneously from zero. Thus, in the case of combustion,

finding the distribution p in space from the solution of the linear equation

(15) with conditions (16), one can readily find the field a and the field b as

well

with| p>0, a = -7' * =

wi^th] p<0, a = 0, b — — ^

(17)

The condition p = also constitutes an equation for the surface of the flame.
It is easy to see that the flows of substance a reaching the surface from one
side and substance b reaching the same surface from the other side are immediately
in a stoichiometric relationship. To prove this, note that on the surface of

"-> -* j^

the flame p = a = b = 0, so that the convective parts of the flows pan, nbrn^ ?p(^
are also equal to zero. Therefore at the surface

|o = P^?fada = 7P^ff'adp; qi = fDgTadb = j^DgTaipv (17a)

and the value grad p has no special characteristics for the surface of the flame,
wherever p = 0, so that on this surface F is large when used in the equation for
a and b but not in the equation for p.

The surface of the flame (p = 0) was found by Burke and Schumann through
integration of (15) for the simplest case of concentric flows of fuel and air,
moving at the same velocity; their conclusions are in satisfactory agreement
with experimental data.

Let us turn now to the equations for the temperature and the combustion
products and set up problems for expressing these values through a and b.

Considering the equation for one specific product of the reaction

and comparing it with the equation for a and b

L(aa) = -/', Lm = — F, \
one can see that it is possible to exclude F from the equation in different
ways. However, if one wishes also to obtain the simplest boundary conditions,
it will be necessary to select a new variable in a completely determined
fashion _ _ . -

I

'0''

(18)

it is easy to see that under these conditions

L(r) = 0. I) r = l, U)r = l. 1 ^g-,

§ 3. Distribution of Reaction Products and Temperature

It is clear from equation (18) and conditions (19) that in the entire
region r satisfies the equation of diffusion and convective transfer without
sources or flows (inasmuch as the first part of the equation is equal to zero) .
Then r is selected so that r = 1 in all flows which enter the region in question
both in a flow of gas (1) and in a flow of air or in an atmosphere (11) . It
is clear that if a certain substance (r) is contained in a certain concentration
which is the same in two mixing flows, precisely the same constant concentration
of this substance will be found all through the entire mixing area. This is
expressed mathematically as follows: r s 1 is then the solution of (18) - (19) .
From this one can find the expression g through a and b

aft agba — ahn — aob [ . .

^ = Y aoo-H36o I "-^"-'

and completely analogously for h or T obtained by substituting y for n or t.

Thus, the problem which the author set himself has been solved.

^Any combination z.-='n.(,Yg + maaj+ (1 - m)6bi, where n and ra are any constants
gives L = n[F - m.F - (1 - m)Fj = 0. However, in order for z to have -the same
values with a = aQ',',b = 6 and; a -= 0, b r.^Q> i^ is necessary for nmaa^ = n(l - m)

gb =1, which gives the expression r written above [formula (18)].

In the case of rapid combustion, when (18) is satisfied, one obtains at
the front (p = 0, a = b = 0) . . _ . .

goo— V (xoo-+-W(i (21)

This result has a very simple and clear physical significance: the values

a and b characterize the concentrations of active substances in the burning

gases. In order to obtain the stoichiometric mixture in which 1 gram would be

subject to reaction, it is necessary to take — grams of substance a and — of

substance b; this will give us — of substance g. Taking into account the fact

that substances a and b in the original gases are dilute, it is necessary to

take of one gas and -rv— of the other. After combustion, the amount - of

^% ebQ Y

substance g will be contained in the total amount of combustion products which
is equal to ■ ■ — i i

which gives us a concentration g_. which satisfied formula (21) .

Hence, in the case of rapid combustion of unmixed gases the reaction zone
will contain precisely the same concentration of combustion products as if one
had mixed burning gases in a stoichiometric ratio and had performed a chemical
reaction invoiving combustion without any diffusion volume.

In precisely the same way, in the absence of heat loss through radiation
and cooling of the surfaces in the flame, and with equality of the coefficient
of temperature conductivity and diffusion, it can be shown that the temperature

is also equal

to the temperature of combustion at a constant pressure of the stoichiometric
mixture of the gases in question.

§ 4. Comparison of the Temperature of the Flame With Experiment

The conclusion which was reached earlier, namely that the temperature of
the unmixed flame is equal to the temperature of combustion of a stoichiometric
mixture is contradictory with respect to experience: it is well known from
daily laboratory experience that in the combustion of a given amount of
illuminating gas in a Bunsen burner with the openings used for air intake at the

\<L^ On b^

in the combustion zone of the diffusion flame T ^ ~ T 'xa^-t-^bn

bottom closed, the flame temperature is lower than when the same gas is burning
with' the openings open, so that a prepared mixture of gas and air reaches the
flame. ^

However, this contradiction with respect to experience is explained not.
by an error in the calculations but by the fact that the condition for usability
of the calculation was not fulfilled earlier; in reality, in the thermal
balance of a laboratory burner it is impossible to disregard the amount of heat
which is given off by the form of radiation.

With an equal amount of burning gas, i.e., with equal liberation of heat,
without feeding air, the size of the flame is much larger than when air is fed,
so that the radiant surface is larger and the liberation of chemical energy
per unit surface is less. In addition, the brightness of the flame when air is
not supplied is greater due to the appearance of tiny particles of carbon in it
which come from the decomposition of the hydrocarbons in the fuel; when air is
supplied, the carbon (lamp black) disappears. The presence of the soot in the
flame of unmixed gas is also quite natural; let us examine the point A in the
reducing area (Figure 1), i.e., within the surface of the flame. Let us say
that this point which is located near the surface of the flame; in this instance,
the temperature at point A will be high, the gas will already have been diluted
strongly by the products of combustion and nitrogen, but there will be no
oxygen in it. Combustion in the absence of oxygen will also lead to the
formation of soot.

In the flame of a mixture of a gas with air, the gas is also heated ahead
of the flame front, but this heating takes place in the presence of oxygen,
and those accumulations of hydrocarbon molecules which could become starting
poipts for the formation of soot particles, are immediately oxidized. As a
result, the .thermal radiation of the flame of the mixture is much less, while
the temperature is much higher than in a flame of an unmixed gas, although the

%ere and below, it is assumed that the stoichiometric amount of air is drawn..
in. It is possible to consider (although one shall not do so here) the case of
an insufficient quantity, when two "cones" of flame are formed, the inner
(compressing mixture) and outer (burning in the surrounding air).

initial "ideal" theoretical value o£ the temperature of combustion in both
cases is the same. As has been already pointed out, this theoretical value
constitutes the temperature which must be reached in combustion in the absence
of losses through radiation and side reactions, but with complete consideration
of the conductive and convective heat exchange of the flame with the gas and
air.

P'0,o-0,b'0 Consideration of the conductive and

convective heat exchange is theoretically
necessary, inasmuch as this heat exchange
is unavoidably bound up with the processes
which feed the fuel and oxygen to the
flame. Regardless of the ratio between the
present and supplied amounts of air and gas,
in the case when they are fed separately
the flame is always located in the same
position, such that the fuel and oxygen- reach the surface in a stoichiometric
ratio. When the coefficients of diffusion and temperature conductivity are
equal (especially in a turbulent motion, which ensures such equality), the
concentration of combustion products and temperature in the flame correspond
precisely to the combustion of a stoichiometric mixture (with equal losses to
radiation) , as the result of this calculation.

§ 5. Combustion Limit of Unmixed Gases

The methods described above makes it possible to calculate the position of
a surface of a flame when any quantity of gas and air is fed at a caloric
content of the gas which can be as small as desired] this calculation is based
on the assumption of a high rate of chemical reaction at the surface of the
flame (and at combustion temperature) , which leads to a narrow thickness of the
zone in which the chemical reaction takes place and to the possibility of
viewing the flame as a geometric surface.

It is obvious that when the rate of the chemical reaction is insufficient
one can expect deviations from this picture. By analogy with other phenomena
of combustion and explosion, one can expect that a decrease in the rate of

10

reaction with all other conditions equal will cause first o£ all a slight
quantitative change -- expansion of the reaction zone and then after achieving
a certain critical value, damping of the flame; combustion becomes impossible
and ceases, and instead of combustion there will be mixing of cold gas and air
without, any reaction. In the following the author will attempt to discuss the
critical conditions for damping in a simple schematic case.

In 1940 the author discussed [3] the conditions for- the possibility of

combustion (propagation of a flame) in a prepared mixture of gases. In this

case, the limits depended upon a drop in temperature of combustion due to heat

-,,ȣ. loss by the side] walls of the tube and heat loss through radiation. The

decrease in the temperature of combustion in turn led to a drop in the rate of

flame propagation, i.e., to a decrease in the heat emission per second. With a

decrease in the speed of the flame the relative thermal losses increase, etc.

Therefore, the critical condition for the possibility of combustion of a prepared

mixture can be written so that the decrease in the combustion temperature from /1206

2
the action of thermal losses must not exceed a certain low limit (RT /A where

A is the heat of activation of the combustion reaction and T is the combustion

temperature) .

However, this theory of the effect of external experimental losses cannot
be applied to combustion of unmixed gases. As a matter of fact, the decrease in
the combustion temperature in this case does not lead to a change in the
quantity of gas which is burned per unit surface of the flame, inasmuch as the
rate of combustion is governed here exclusively by the rate of diffusion feed
of oxygen and fuel to the surface of the flame, and not by the rate of propagation
(which depends upon the rate, of the reaction), as. was the case for the prepared
mixture.

The combustion limit of unmixed gases is set by the decrease in temperature
which depends upon the finite velocity of the chemical reaction.

Let us examine the distribution of the concentrations and the temperature ,
in the zone of the reaction. If the rate of the reaction were infinite, the
distribution would be given by Figure 2. The dashed line shows the position of
the zone in which a = 0, b = 0. Using it as the origin for calculating the

11

coordinate x^ one directs the x-axis perpendicular to the surface of the flame.
If the total amount of material which reacts per unit surface in a unit time is
represented by M, the diffusion flows a and b will be equal (respectively)'*

(22)

r\ da t M ^ »M r\ ^l> "■ »t

p

so that. the distribution of the reacting components near the zone is given by
formulas i

4r>0, a = 0, A = p4^' |

By means of formula (20) and (21), rewritten for T, one obtains the
corresponding distribution of temperature. In the general case

(23)

Substituting expression a and b in (23), one has

'^^'^^FS

rAT

pO « -^ &

Oj» Jr<0

The corresponding curve is also shown in Figure 2.

(24)

(25).

Figure 2.

How do the curves change. when the
reaction is not instantaneous?

It is obvious that with an equal
amount of substance burning per unit
surface M, the concentration gradient and
the entire distribution of a and b does
not change far from the zone. However,
now curves a and b cannot undergo sharp
curvature (corresponding to an

^Inasmuch as the author is interested in the vicinity of the zone of the reaction
in these formulas and below p and D will be taken at a temperature in the
reaction zone which is rounded off to T,

^0^0

OO"
+ T-. In the following, one shall consider

'More exactly, T^^ = — ^^^ ^ ^^^ ■ . ^.

the initial temperature T to be small in comparison with the combustion
temperature and will disregard T everywhere.

12

instantaneous reaction) at the origin o£ the coordinates, as in Figure 2^yThey
will cross, as shown in Figure I, asymptotically approaching zero in the region
occupied by the second component. The dashed line in this curve shows the
distribution for an instantaneous reaction.

In order to be able to determine
precisely the curves for the distrbution
of concentration in Figure 3, it would be
necessary to integrate the equations for
the diffusion of the form (12) , substitu-
ting a specific expression for the rate of

the chemical reaction, for example

F=^abKe

(26)

Figure 3.

Inasmuch as F depends on at least 3
values a, b and T, it is necessary to view
the system of 3 equations of the second
order. However, thanks to the concepts developed above, one can find first of
all p, which links a and b, and then express T through a and b and thereby
reduce the problem to a single equation of the second order, for example for
a, in which F will be expressed by a and a known function p(x).

However, for our purposes such an approach is too complicated, and the
conclusions in which the author is interested in will be obtained (admittedly
with an accuracy up to numerical factors which the author does not know) by
methods of analysis of dimensionality and the theory of similarity.

Let us introduce the effective values -- the width of zone x^^, the
temperature in the zone of the reaction T^, concentrations a^^ and b^^. The
total amount of substance which reacts in the entire zone is expressed by these
effective values

Let us express all of the values on the right-hand side by x . In order to
link them, the author notes that one is given a relationship a and b upon x
at distances from the zone which are large in comparison with the wide zone of

13

^

the chemical reaction x , and this relationship is linear, i.e., it is

8a

3b

characterized by certain values set externally, by gradients J^ ' '^ °^
relationships -^ or - [see formulas (23)^]. It is clear from the dimensionality

X X

that relationship a , b and x must be given by similar formulas

Af

:x» Aj =

(28)

The relationship of T with a and b as given by formula (24) from which /12Q8
by substituting (28) we can have

Ti = Too (1 — ?Mjf , ), wherejp = -^ -^-

a

a3

1

(29)

pZ? a -+- P a -«- P pDx 7*00
Substituting in (27) , one finally obtains the equation which links M and x^

"""■Ke "'"x,^e

ifMz,S
"Bfi^

(30)

~"(a-+-/?)2(pO)2

In the second line of (30) the author expanded the value in the exponent
in a series using the Frank-Kamenetskiy method [4] . The value on the right-hand
side, depending upon x , has the form x e' and consequently passes through a
maximum at a certain critical value x = x .

l:p.

M..

i__ «P ( RTq^ y

Ke

RTm

(31)

(32)

The significance of the maximum M as a function of x consists in !the fact
that when x is small the covered areas a and b are small and the reaction zone
is narrow so that the concentrations of reacting substances in it are small.

a b
^The ratios — and — change' slightly only at distances which are greater than

X X

X , but less than the size of the flame. At distances which are comparable to

the dimensions of the flame it is not permissible to disregard the convective
terms in the diffusion equation.

14

When X is small and the given temperature is T , the temperature T will

1 T 'Z ^^

differ slightly from T q, M ~ x^, a^ ~ b^ ~ x^ ~ 7M , in accordance with the
known results for flames which show good. diffusion in a high vacuum [5]. When
X is large, the concentrations of reacting substances are high, so that the
temperature decreases leading to a drop in the total amount of reacting substance.

By intensifying the combustion, increasing the feed of reacting substances
a and b to the zone, one simultaneously intensify the cooling of the reaction
zone. Until the rate of the reaction is sufficiently high, x^ is less than
maximum, the temperature practically does not decrease. However, at a certain
■critical value M [formula (32)] a decrease occurs in the temperature of the
zone which leads to a further decrease in the rate of the reaction and a further
drop in the temperature; combustion is interrupted, and instead of combustion

there is a mixing of cold gases. The maximum possible temperature drop prior

3RT 2
to separation is the fraction 00 . Note the curious similarity of expression

E

M (32) with the expression for the amount of substance burnt per unit time in

the flame propagating according to a previously prepared stoichiometric mixture

of combustible gas and air, whose combustion in unmixed forms was discussed

above.

For the propagation of a flame in a stoichiometric mixture, according to
the work of Frank-Kamenetskiy and the author [6] , one obtains in the designations

U ■*■*;] (33)

which differs from (32) only in the factor "^^ which does not have a numerator
inasmuch as when (32) is derived the author discarded the numerical factors.
This coincidence is very interesting from the theoretical standpoint inasmuch as
it shows that the maximum intensity of combustion of the mixture even of unmixed
gases, if the mixing is sufficiently intensified, is of the same order.

In the theory of combustion of an explosive mixture, the author has shown
that the chemical reaction proceeds in a zone in which the concentration of the

reacting gas (that which is inadequate in the mixture or of both in the case of

RT
a stoichiometric mixture) is very small -- on the order of 00 of the initial

E

15

concentration. As the calculations above indicate, in the combustion of unmixed
gases the concentrations of both reacting substances (fuel and oxygen) in the
reaction zone is very small. These concentrations depend upon the intensity of
combustion: in contrast to an explosive mixture, in which there is a
characteristic value for the intensity of combustion (speed of the flame) , the
intensity of combustion of a flame of unmixed gases M depends upon the external
conditions. However, at the maximum possible M, at the point of separation of
the flame, the concentrations in the reaction zone do not exceed the concen-
trations in the stoichiometric mixture in terms of the order of magnitude of

RT
the amount 00 .

E
The limit found above to the intensity of combustion of unmixed gases

explains at least qualitatively the fact that when a rapid stream emerges from

a tube the flame breaks down completely some distance from the outlet cross-

-section of the tube, so that at the outlet, where mixing of the reacting

components is most intensive, the flame separates. Previously the author did

not. take into account the thermal losses due to radiation. When they are

taken into account, for example, in formulas (29-32) instead of the theoretical

value T - which was calculated on the basis of thermal capacity, one must use

the maximum possible temperature for an instantaneous reaction, but with

consideration of radiation -- the temperature T '. This T ' is less than T

even in the infinitely thin zone where x^ ->- 0, due to the radiation of heated

gases on the left and right of the reaction zone. The temperature T^ ' decreases

as M decreases, so that at small M less heat is given out and the radiant zone

is wider. When the radiation is taken into account, when M decreases due to the

decrease in T_^' M will also decrease according to (32) and with very small

M a second, lower limit M will arise, with separation of the flame at too low

an intensity of combustion. Finally, with low caloric content of the gas, the

upper and lower limits of M may coincide, and combustion will become completely

impossible. Qualitatively speaking, the picture is similar to the simpler case

of exothermal reaction in a flow with consideration of the heat loss considered

by Zysin and the author [7] .

The -author will not take up in this paper the practically significant but
more complex problem of the limit of intensification of turbulent combustion

16

of unmixed gases: the complexity of this problem has to do with the fact that
in the presence of turbulence we cannot correctly link the average velocity of
the reaction with the average temperature. Apparently it will be necessary to
determine experimentally the limits of possible combustion conditions.

Finally, it should be pointed out that by means of equations (512) with a
concrete form of function F [for example (26)] it is also possible to solve the
very interesting problem of the diffusion flashback of the fuel through the
zone of the flame; as was shown in Figure 3, the concentration of the mutually
penetrating reacting substances during the transition through the reaction
zone decreases sharply but does not return to zero. Inasmuch as the temperature
and the rate of the reaction also drop on both sides of the reaction zone, the
concentration of the fuel which has already reached a certain distance from
the flame in the oxidizing zone will not change further. However our approxi- /1210
mate method will not do for solving this problem; it is necessary to solve
equations in the manner described in the text after formula (26) .

Conclusions

The distribution of the concentration products and temperature is discussed
in the combustion of mixed gases. It is shown that in the simplest conditions
these concentrations and temperature at the surface of a flame are the same as
in the combustion of previously prepared stoichiometric mixtures of the same
gases.

A possible limit was found to the intensification of combustion of unmixed
gases, depending upon the limiting rate of the chemical reaction. In order of
magnitude, this limit is close to the rate of combustion of stoichiometric
mixtures.

17

REFERENCES

1. Burke and Schumann, Ind. Eng. Chem,,, Vol. 20, p. 998, 1928.

2. Shvab. V. A., "The Relationship Between Temperature and Velocity Fields of

A Gas Flame," in the collection: Issledovaniye Protsessov Goreniya
Natural'nogo Topliva [Investigation of the Combustion Processes of Natural
Fuel], Moscow-Leningrad, Cos. Energ. Izd. Press, 1943.

3. Zel'dovich, Ya, B., ZhETF, Vol. 11, p. 159, 1941.. _

4. Frank-Kamenetskiy, D, A., Diffuziya i TeplodtdaoHa v Khirmoheskoy Kvnet%ke
'^'[ Diffusion "and Heat Loss In Chemical Kinetics] ,' USSR Academy of Sciences

Publishing House, 1947.

^ . 5. Bentler and Polanyi, Zs f.]Phys. Chem. , Vol. 18, p. 1, 1928.

^ ^ 6. Zel'dovich, Ya. B. and D'. A. Frank-Kamenetskiy, ZhFKh, Vol. 12, p. 100, 1938,

7. Zel'dovich, Ya. B. and Yu. A. Zysin, ZhTF, Vol. 11, p. 501, 1941.

Translated for the National Aeronautics and Space Administration under contract
no. NASw-2485 by Techtran Corporation, P. 0. Box 729, Glen Burnie, Maryland,
21061, Translator, William J. Grimes, M.I.L.

18