THERMODYNAMIC RESEARCH BY THE METHOD OF EXPOLOSION AND COMPUTATION OF COMBUSTION PROCESSES

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• Original Classification: K
• Document Page Count: 189
• Document Creation Date: December 23, 2016
• Document Release Date: June 12, 2014
• Sequence Number: 11
• Publication Date: September 1, 1959
• Content Type: REPORT

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TRELIIT1011
THERMODYNAMIC RESEARCH BY THE METHOD OF EXPLOSION
AND COMPUTATION OF COMBUSTION PROCESSES
By A. M. Gurvich,- Yu. Kh. Shaulov
September 1959
181 Pages
STAT .
PREPARED BY
LIAISON OFFICE
TECHNICAL INFORMATION CENTER
NC LTD
WRIGHT-PATTERSON AIR FORCE BASE. OHIO
__STAT
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TERMODIN.AMICHESUYE ISSLEDOVANIYA METODOM VZRYVA
Izdatel'stvo Moskovskogo Universiteta
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,
Published by resolution of the
Council of Editors and Publishers
of Moscow Un.iversity
,??
Dedicated to the unfading memory of
Professor Andrey Vladimirovich Frost
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Foreword
Chapter I
Chapter II
Chapter III
Chapter IV
Chapter V
Chapter VI
Chapter VII
Chapter VIII
Chapter IX
Conclusions
Appendix I
.Appendix II
Appendix III
Appendix IV
Bibliography
TABLE OF CONTENTS -
_
Introduction
Experimental Determination of Maximum Explosion Pressure 4
Heat Losses and the Degi.ee to which Equilibrium is Achieved
during Explosions in Spherical Vessels with Central Ignition 15'
General Principles of Thermodynamic Computation OfCom-
bustion Processes 23
Calculation of Combustion ProcesSes at Constant Volume 29
Computation of Combustion Processes at Constant Pressure
Direct Method of Calculation
Computation of Combustion Processes at Constant Piessure
Approximation Method of Calculation /16/
On the Methods of Composition Calculation for Mixtures
of Combustion Products
Thermodynamic Research by the Method of Explosion
Elementary Information on Statistical Methods of Compu-
tation of Thermodynamic Quantities
MOD =I
ONO ??11
Entropy Calculation of the Mixture of Combustion Products
at Constant Pressure
58
69
89
105
135
137
160
Method of Rapid Computation of Square Roots on the Basis
of Their Approximate Values with the Help of the Calcu-
lating Machine 167
Initial Data for Thermodynamic Computations 170
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FOREWORD
This book is closely related to the previously published books Statisticheskiye
Metody rascheta termodinamicheskikh velichin (Statistical Methods of Computation of
Thermodynamic Quantities) by V. M. Gryaznuv and A. V. Frost, and Svobodnve energii
organicheskikh soyedineniv (The Free Energies of Organic Compounds) by V. V. Korobov
and A. V. Frost. It is a result of the great theoretical and pedagogic work con-
ducted in'the Department of Physical Chemistry of MO (Moscow-State University)
under the guidance of Professor A. V. Frost. Regrettably, the untimely death of
A. V. Frost made it impossible for him to participate in the preparation of this
book which was written after his death. The book contains an outline of the methods
of computation of the characteristics of combustion processes on the basis of thermo-
dynamic functions and equilibrium constants, the computation procedures for which
were analyzed in the two above-mentioned books. At the same time, the book is so
compiled that the knowledge acquired within the scope of the physical chemistry- cur-
ricula for university students in departments of chemistry is sufficient for its
understanding and no preliminary perusal is required of-the books by Gryaznov and
Frost or Korobov and Frost. For this purpose, in particular, Appendix I was incor-
porated at the end of this book; there we find a brief review of the statistical
methods of computation for thermodynamic quantities (to the approximation of har-
monic oscillator - - rigid rotator), as well as a calculation of equilibrium con-
stants. .
The fundamental principles of the computation methods outlined in the book are
'described in the monographs by Zel,dovich and Polyarny, Vanichev, and Lewis and Elbe.
The authors introduced a series of improvements into these methods, consolidated the
computation procedures for constant volume and constant pressure, and made an effort
to present them formally and at the same time sufficiently graphically. The deriva-
tions of a considerable number of formulas recorded in the book constitute something
iii
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original. Several simple formulas were developed to facilitate a Check on the ac-
curacy of the initial stages of calculation, in the course of which -- as it is
evidenced in practice -- errors are often committed Which nullity all the subsequent
,
highly laborious work. . ChapterSt contains the methods developed by the authors for
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the calculation of the correction .pertaining to the maximum explosion pressure for
the temperature gradient'In the products of CimbustiOn,' and for the "amputation of
temperature distribution in the mixture of combustion products depending on the dis-
tance from the center of the exploiion vessel...,
. .,,. . ., .
A simple method is also given for the computation of -thermal capacities of the
combustion products according to experimental values of maximum explosion pressure.
All calculations related to thermodynamic research by method of explosion are
ve't
brought into conformance with the consolidated methods of computation of combustion
at constant volume and at constant pressure, as outlined , :hapters V and VI.
The examples of thermodynamic research by the method of explosion and the calcu-
lations connected therewith -- which are analyzed in Chapter IX on the basis Of all
the material contained in the preceding chapters and in the appendixes -- are de-
signed, apart from their direct object, to facilitate a more critical approach by
the reader to the evaluation of the accuracy of calculations required in every given
case, in particular, with respect to the required precision for calculation of the
thermodynamic functions of substances by statistical methods, which is amatter of
special importance in view of the labor-consuming character of this operation.
The methods and computation procedures incorporated in the book are illustrated
by means of a considerable number of examples. The reader will easily- perceive a
definite system in the selection of these examples and the relationship of some of
them to each other.. Ail examples are computed by the authors.
The methods of calculation are presented in such a manner as to facilitate
their utilization by a vide circle of research workers and professional engineers
engaged in work involving combustion processes. This fact constitutes the great
practical significance of this book.
,?
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A
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Ta. I. Gerasimov
Corresponding Member
Academy of Sciences, USSR
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Chapter I
INTRODUCTION
Very high temperatures develop as a result of combustion of gaseous explosive
mixtures. It is natural to assume that, in view of the great progress rate of re-
actions under these conditions, the system arrives at a state of chemical equilibrium
by the moment combustion is terminated. Owing to the speed of burning, the heat
losses into the ambient space are, as a rule, small. These circumstances make pos-
sible the application of thermodynamics to calculations of such quantities as maxim.
mum pressure of explosion, degree of expansion, combustion temperature, and so on.
Through a corresponding experimental setup, heat losses, as a matter of fact,
may be reduced to insignificantly small values. In this manner, it becomes possible
under definite conditions to obtain good agreement between the experimental values'
of the combustion characteristics and their values calculated on the basis of the
application of thermodynamic laws. But in this case the experiment in tarn makes
it possible to calculate the unknown thermodynamic quantities included in the calcu-
lation scheme of this or that quantity observed experimentally. For this purpose,
use is ordinarily made of the method of explosion in a closed spherical vessel
(bomb) with central ignition. In such a vessel, the flame comes into contact with
the walls of the bomb simultaneously over their entire surface as a result of total
combustion of the explosive mixture. Up to this moment, which coincides with maxi-
mum pressure in the vessel, the heat losses for rapidly burning mixtures are negli
tibia.' The maximum explosion pressure is measured by means of special pressure indi-
cators. This method, initially-suggested by Bunsen, was employed after a number of
?
technical difficulties were overcome by PAer (52, 53) --by a number of researchers
for the determination of the mean specific heat of gases and for other purposes. It
is possible to record.a considerable number of examples in which this method was
successfully used for the determination of thermodynamic quantities at high tempera-
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tures (61, 47), to define more precisely the oat* on the structure of molecules (46),
for the study of the combustion process itself (63, 21. 62) and the determination of
the possible degree of useful utilization of the energy contained in explosive mix-
tures. Other experimental methods of thermodynamic research, though sometimes even
more accurate than the method of explosion, do not permit research to be conducted
at high temperatures on the order of the combustion temperature of explosive gas
Mixtures..
We refrain entirely from dealing here with the matter of the feasibility of
quantitative thermodynamic investigations based on the comparison of the measured
detonation rates with those calculated theoretically. Such investigations were car-
ried out by Zel'dovich and Ratner (17) and a few other researchers. Computation in
this case is similar to that for adiabatic explosion in a closed vessel. This meth-
od allows us to study the thermodynamic properties of gaset, at even higher tempera-
tures than this is possible by the method of explosion in a bomb.
For the study of combustion at constant pressure, a method is applied which
roughly consists of the following. An explosive mixture is ignited by 'a spark in
the center of a soap bubble (57) or a spherical film of thin transparent rubber (54).
The flame, while travelling in such a "constant pressure bomb", retains its spheri-
cal form with the change of diameter being registered on a moving photographic film.
In this manner one can determine the degree of expansion, i.e., the ratio be-
tween the ultimate volume of combustion products at constant pressure and the initial
volume of the original mixture.
The most reliable method of the flame temperature measurement is the reversal
of spectral lines (21). The light from an incandescent wire passes through the
flame, colored by traces of sodium. With the rise of the wire temperature, the
bright lines of sodium radiation change into black absorption lines at the moment
when the temperature of the wire becomes equal to that of the flame.
The satisfactory coincidence of the experimental values of combustion character-
istics obtained, by the mentioned methods, with the theoretical values confirms the
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validity of the basic postulates and the initial data supporting thermodynamic cal-
culations. Thus, we have a fairly reliable method of computation of combustion
processes for industrial purposes. Selection of fuel (37) and material for the
manufacture of equipment (3) evaluation of the effect which the composition of the
initial mixture may produce on these by other characteristics of various processes
(37, 26), determination of the possible degree of utilization of the chemical energy
of fuel for the performance of certain work (26), calculation of the technical char-
acteristics of all types of engines (16, 2, 41, 42), solution of various problems in
metallurgy (23), in the field of welding and in processes of Awl i,asificaticm
??? ???
All this constitutes a far from complete list of the fields 1)f possible application
of thermodynamic computation of combustion processes.
For these calculations, use is made of thermodynamic quantities computed t, the
statistical method on the basis of data on molecular and atomic structure obtaihed
mainly by means of spectroscopic and electron diffraction studies. It is in this
manner that a "mathematical bridge" is built from experiments pertaining to the de-
termination of microscopic quantities to experiments intended to define the =zero-
scopic quantities providing direct characteristics of the real processes.
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chapter II
EXPERIMENTAL DETERMINATION OF MAXIMUM EXPLOSION PRESSURE,
The problem of measuring the maximum explosion pressure with accuracy satis-
factory for scientific purposes is quite complex. The experimental equipment used
will be described briefly here in order to furnish a real physical basis for the
thermodynamic calculations of combustion processes, the methods for which are ana-
lyzed in detail below.
Figure la. Diagram of spherical bomb and auxiliary equipment.
(1) To oscillograph; (2) gas from flask or
gasmeter; (3) to vacuum pump.
Such an assembly (Figure la) consists of an explosion vessel -- a bomb with
equipment for its evacuation, inlet of gases and vapors, and the ignition of the
mixtures -- and a diaphragm pressure indicator * to register the change in pressure
during explosion in relation to time. The unit may be also provided with other de-
vices and apparatuses in accordance with the research objectives.
* Indicators of the piston type cannot be used for accurate measurements because of
high inertia.
The history of pressure indicator development for thermodynamic research by the
method of explosion in a bomb is to be found in the book by Bone and Townend(27).
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se.
Section 1. The Bomb
The bomb (Figure lb) constitutes a steel vessel whose inner hollow space has a
strictly spherical shape. It consists of two sections fastened together by means
of massive bolts and a gasket of soft metal. This arrangement assures the possi-
bility of inspection and cleaning of the bomb and at the same time provides for
sufficiently tight sealing.
The spark electrodes are so mounted as to make sure that the gap between them
is right in the center of the sphere.
. Figure lb. Cross-section of the spherical bomb.
(1) two-point spark plug; (2) quartz window; (3) gasket;
(4) condenser pickup; (5) bomb body; (6) window for
photographing of flame propagation.
To permit photographing of the flame propagation?which, in particular, is im-
portant to check that the flame reaches the walls of the vessel simultaneously at
all points -- the bomb is provided with a window in the form of a narrow slit cov-
ered by thick glass. The flame is photographed on a motion picture film fixed on a
rotating drum whose axis runs parallel to the slit. A typical recording of the
flame propagatioi is shown in Figure 2.
Different researchers used bombs with volumes ranging from 3.5 to 35 liters
(53, 47). Bombs of small size are disadvantageous because the eccentricity of the
spark gap in them may be relQtively large (47). Besides, the larger the bomb, the
smaller is the surface-voIume ratio and the smaller, therefore, are the heat losses
through the walls. However, an increase of the explosion vessel radius facilitates
5
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detonation, which entails a narrowing of the range of the initial mixture concentra:-
tions that may be utilized for thermodynamic research by the method of explosion in
a bomb.
The fuel mixture is prepared either in the bomb itself (53, 62) by means of
alternate introduction of gases, or in special mixing containers
(37) sufficiently
large to permit the preparation of a mixture for a number of experiments. On their
way to the bomb, the gases pass through a drying system. To assure a complete mix-
ing of the mixture components, and in order to communicate to them the temperature
of the atmosphere surrounding the bomb, a certain period of time must be permitted
to elapse between the filling of the bomb and the explosion. This period depends on
the density of the mixture gases and the order of their introduction into the bomb
or the mixing container.
?
Figure 2. Photorecording of flame propagation in a spherical
vessel with central ignition (Fiok and King).
To introduce water vapor into the explosion vessel, a corresponding amount of
water is vaporized in a special test tube (37), or, alternatively, prior to the in-
let of gases into the bomb, the gases are saturated by water vapor at a definite
temperature. If experiments are to be conducted with a considerable content of H20
in the initial mixture, then use is made of an explosion vessel with a steam jacket
(37, 31). Since at constant volume and constant temperature, the molar cencentra-
.
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,tions of gases are proportional to their partial pressures, it suffices; for the
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preparation of a mixture of a given composition, to measure tial?presnres
,
during the alternate introduction of the mixture components: For :'the 'measurement of
..
pressure, use is made of a mercury manometer with cathetometer. Whenever the initi-
al pressure exceeds 1 atm, a reference manometer of the 02 class may be utilized.
For gases with a relatively high critical temperature, such as RCI or C12 the
deviation from the ideal gas state should be calculated While determination is being
made of their initial quantities on the basis of partial pressures. From the Van,.
der Waals equation we derive:
pv3?(bp ?RT)v2?Fav?ab =0, ?
where v is the volume of one mole of gas and p is its partial pressure,
Tic 27 /21 ? 71
=T3-14; Pk?
(II.2)
In working out equation ar.ly by the approximation method with application of
graphicinterpolatiaa(22), we find the volume v of one mole of the given gas.
From the Clapeyron equation, we calculate the partial gas pressure p* Which it
should have had, had it conformed to the law.of ideal gases
In summing up the partial pressures of the initial mixture components reduced to the
state of ideal gases, we obtain a corrected initial pressure of the mixture P.
Example. The partial pressure of chlorine p = 663,9 mm Hg, the temperature of
the mixture T = 291? K, the critical temperature of chlorine Ter = 417,2? K, the
critical pressure per = 76,1 atm.
From equation (II.2), we determine
b.= 0,05623 a = 6,496
1-atm
By substituting the numerical values of p (in atm), a,h,R (in ) and Tin equa?
aeg.
tion (II.1), we find
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0.8738 v3 - 23,9257 v + 6,496 v - 0,37 = 0
On solving this equation by the approximation method, we determine that
v = 27.114 lit
whence
p* = 0,8806 atm = 669;3 mm
For p = 241,0 mm, we find p* = 241,7 mm by similar'procedure".-
Analogous considerations exist also for water vapor. If a considerable quan-
tity of water vapor is to be introduced into the mixture, then experimental data on
its compressibility should be utilized.
Section 2. Pressure Indicators
The pressure indicators consist of two basic parts: the sensitive element
(pickup) receiving pressure, and the recording unit. The existing indicators may
be divided into three groups: mechanical, optical, and electric indicators. The two
latter groups are most suitable for precise measurements.
In optical and electric indicators, the main pickup element is the diaphragm
which responds directly to pressure. The deflection of the optical indicator dia-
phragm causes the mirror, which is connected thereto, to turn, entailing a deviation
of the reflected light beam which is registered on a moving photofilm. In electric
indicators, the shifting of the diaphragm is translated into a change of one or an-
other parameter of the electric circuit registered by the oscillograph.
Of a number of different types (4, 14) of electric indicators, we shall con-
sider the two most suitable ones for accurate measurement: the condenser and piezo-
electric indicators.
A pressure indicator used for thermodynamic investigations by the method of ex-
plosion must record rapid changes in pressure with the greatest possible precision
and the least possible delay. For this purpose, it should be characterized by small
inertia and small hysteresis, while being sufficiently stable in operation and de-
pending as little as possible on temperature in its recordings. To obviate the
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possibility of a resonance with the fluctuations of the measured pressure, the dia-
phragm must have its own high frequency. The indicator diaphragm must be at one
level with the inner surface of the bomb.
Illatical Pressure Indicators
This type of indicator is widely used in ther'dynamic research by the explosion
/ method, since it is relatively simple and assures considerable accuracy of meas-
nrement. The diaphragm of the optical indicator is connected to the mirror by means
of a flexible system. A voltaic arc serves as the source of illumination. The beam
of light reflected by the mirror is focused by the objective of the photographic re-
corder onto a sensitive film fixed to a rotating drum, and shifts proportionally to
the magnitude.of the diaphragm deflection (if it is not too large), While tracing
the pressure-time curve, i.e., the indicator diagram (Figure 9a). This gives us
more than 1,000-fold magnification. By measuring the height of the sharp maximum on
the indicator diagram, the corresponding pressure incrementaP is determined (data
on the calibration of indicators will be found below). By adding AP to the initial
pressure Pi, we get the maximum value of explosion pressure Pe (experiment) = Pi 4.
4. P.
Figure 3 illustrates the diaphragm indicator proposed by Lewis and Elbe(45).
The characteristic feature of this indicator is that the diaphragm is not clamped in
the supporting frame (as this is the case with most diaphragm indicators), but con-
stitutes one solid piece with it. This insures the stability of the diaphragm's
zero position and the absence of an apparent hysteresis caused by the inelastic
nature of its attachment. The measurement accuracy exceeds 0.1% of the magnitude
of maximum pressure (26). Optical indicators used by other investigators showed an
accuracy of the order of 0,1 - 0,2% (61). Discrepancies between the results of
individual measurements range within the limits of 0,2 - 0,8% (46, 47, 61).
9
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Pickup of optical pressure indicator developed
by Lewis and Elbe.
(2) Condenser Pressure Indicators
- The diaphragm of a condenser indicator is one of the plates of the adjustable
capacitor-condenser pickup (Figure 4), whose second plate is insulated from the in-
dicator body and connected with an amplifier_rT---
11
114
MOM
4.03A 4
12 , r
15
16
Et.E.E.1. Condenser pickup;
1- diaphragm; 2- mica gasket; 3- fixed lining; 4- rod;
5- washer; 6- nut; 7- conductor; 8- terminal block;
9 - conductor seal-in; 10- shaped nut; 11- packing block;
12- insulating bush; 13- housing; 14- thread for housing
fitting; 15- ebonite sleeve; 16- support bush.
The small capacitance variations of the pickup resulting from diaphragm deform-
ation under the influence of pressure cause almost linear changes of the voltage
supplied after amplification to the vertical scanning plates of the cathode-ray
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II
oscillograph. Saw-tooth voltage is delivered to the plates of horizontal scanning.
During explosion, the luminous point on the electron-ray tube screen traces a pres-
sure-time curve Which may be photographed.
The response of the pickup to pressure variations increases as the superficial
area of the diaphragm (and the fixed lining) increases; it decreases as the distance
between the pickup linings decreases. At the same time, this distance must be large
in comparison to its changes under the influence of pressure, so as to assure the
linearity of the pickup characteristics*. This may be achieved through the use of
a relatively thick diaphragm, characteristic for small values of deflection, which
in turn enables the pickup linings to be placed close to each other. A thick mem-
brane is more advantageous than a thin diaphragm, in that its zero position has
greater stability and that it has a smaller hysteresis (60). The diaphragm is made
of stainless steel or special alloys (elinvar, mnimonica).
- -
Elgure 5. Single-contact pickup.
1- separator, lower; 2- separator, upper; 3- packing block;
4- perforated insulator; 5- packing block; 6- gas inlet tube;
7- contact guide; 8- cap Ilut; 9- contact washer; 10- insulat-
ing washer; 11- upper housing; 12- water inlet pipe; 13- lower
housing; 14- packing sleeve; 15- regulated contact; 16- water
jacket; 17 - packing washer; 18- diaphragm.
* It should be borne in mind that condenser pressure indicators
terized by a linear relationship between the point displacement
the pressure only over a relatively narrow range (49).
11
are usually charac-
on the diagram and
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For the determination of the indicator diagram scale ?.provided the indicator
characteristic, which is checked by calibration, is linear -- it is sufficient to
have two points on the curve for which the corresponding pressure values are known.
One of these points is the point of ignition -- the basis of the rising branch of
pressure curve. This point corresponds to the initial pressure of the mixture. The
second calibration point may be plotted on the indicator diagram with the aid of a
single-contact pickup (Figure 5), which, when a definite pressure is reached in the
bomb, produces a pip in the form of a bright point on the steep left-hand branch of
the diagram. The single-contact pickup (26) has a diaphragm and a contact screw be-
tween which there is a small gap, if pressure on both sides of the diaphragm is
equal. The difference in pressure required to press the membrane to the contact is
being measured before each test ("correction for back pressure" "P back). Fri"
to the explosion, a definite pressure ("back pressure" - n
back) is applied to the
pickup diagram from the side of the contact screw. When the pressure in the bomb,
affecting the inner face of the diaphragm, exceeds back pressure by the quantity
"
back' the circuit closes and a bright point appears on the indicator diagram.
Time is plotted on the diagram by means of a special devA. e. The entire system
of instruments is synchronized, ignition of the mixture, reg tration of pressure,
and, whenever needed, the photographing of flame propagation, are affected by a
single depression of the push-button.
(3) Piezoelectric Pressure Indicators (5).
The diaphragm of a piezoelectric indicator activates a quartz crystal on whose
surface there appears an electric charge during compression (this is known as the
piezoelectric effect). The charge magnitude is proportional to crystal deformation.
Through an electric converter, the pickup is connected with the oscillograph, by
means of which the pressure-time variation is registered.
The advantages of the piezoelectric indicator are the absence of hysteresis and
independence of registration from temperature. The short-comings of this type e
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indicators, as of any electric pressure indicators in general, are the relative com-
plexity of their construction and of their o-peration.
(4) Calibration of Pressure Indicators.
There are two types of pressure-indicator calibration. We have static calibra-
tion, in the process of Which a known pressure is applied to the diaphragm gradually,
and we have dynamic calibration, when the diaphragm is subjected to rapid loading.
Dynamic calibration corresponds to the unit's operating conditions during explosion
and should therefore be given preference. However, in case of total absence of
hysteresis, both methods should produce coinciding results.
1
1. 8xot7 za3a
It' friannatempy
. _
Figure 6. Instrument for dynamic calibration of optical pressure
indicator (Lewis and Elbe).
(1) Gas inlet; (2) to manometer.
During static calibration (37, 59) the bomb is filled with air or any other gas,
for instance, nitrogen or carbon dioxide, up to different values of pressure meas-
ured by manometer so as to assure the greatest possible precision.
For dynamic calibration, use is made of special cylinders or flasks filled with
gas. One of such units is shown in Figure 6 (45). The calibration cylinder is
placed inside the bomb. Between it and the diaphragm of the pickup, screwed into
the opening in the explosion vessel, there is a small space isolated from the re-
maining bomb space by a rubber ring. Pressure is applied to the diaphragm by means
of a swift turn of the screw. The time of pressure growth comprises approximately
0,01 sec. Si-AT
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On the basis of the indicator calibration data, a diagram is constructed for
the dependence of the height of the indicator diagram on pressure. If the character-
istic of the indicator is linear, the graph has the form of a straight line.
In summing up this short description of the equipment for the measurement of
maximum explosion pressure, it should be noted that these measurements may be suf-
ficiently reliable for the determination of thermodynamic quantities by the method
of explosion (47) --'if use is made of carefully calibrated pressure indicators of
good design and capable of satisfying the above mentioned requirements; if the inner
surface of the bomb has a strictly spherical shape and there are no "dead spaces"
outside of this Surface; if the bomb volume is sufficiently large to assure a pre-
cise central position of the spark gap; and if the experimental unit permits us to
reduce the heat losses to a minimum before the flame has reached the walls of the
bomb. If all these requirements are duly observed when use is made of rapidly burn-
ing mixtures, indicator diagrams with a sharp maximum may be produced. This maximum
corresponds to the moment when the spherical flame makes contact with the inner sur-
face of the bomb.
STAT
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Chapter III
HEAT LOSSES AND THE DEGREE TO WHICH EQUILIBRIUM IS ACHIETED DURING
EXPLOSIONS IN SPHERICAL VESSXLS WITH CENTRAL IGNITION
1. Phenomena Arisiagjaat_gomb during Explosion
Daring ignition of an explosive mixture of gases in the bomb, the flame travels
from the spark gap in all directions. The transformation of the initial mixture in-
t) combustion products takes place in the reaction sone which constitaLys a thin
spherical 'Ayer whose surface is called the flame front. The reaction sone sepa..
rates the dark region of the unburned mixture from the luminous region of oombustion
products which are in a state of chemical equilibrium. The spherical Shape of the
vessel with central ignition assures the preservation of the sphericity of the flame
front and its simultaneous reaching Of the bomb walls at all points, provided Um+
rate of combustion is sufficiently high (see Figure 7).
4-
Figure 7. Instantaneous photos of the flame propagating in
a spherical glass vessel with central ignition
(Ellis).
At the expense of heat released during combustion, the transformation products
remaining behind the reaction zone expand due to strong heating and compress the un-
burned gas. As a result of this, pressure increases continuously. Moreover, in
view of the fact that flame velocity is low as compared to the speed of sound, it is
possible to consider, without too much of an error, that pressure is even within the
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whole volume of the vessel at any moment in time, with the exception of those cases
when the combustion rate is very high (29, 56). At the same time, the flame fails
to heat up the unburned mixture in any .appreciable way, so that, before it reaches
the walls of the bomb, the latter are in contact only with the gases faintly heated
in consequence of compression. This assures the almost totally adiabatic character
of combustion.
Experiments show (40, 34) that at the moment when the maximum explosion pres--
MIMis reached, the temperature at the center of the bomb, near the ignition point,
exceeds the gas temperature at its walls by several hundred degrees. This fact can
be explained as follows (13, 18, 21, 38, 40, 50). Let the volume of the vessel be
mentally divided into a namber of thin spherical layers with a common center at the
ignition point. In practice, each such elementary layer, whose volume comprises but
an insignificant part of the total gas volume, burns up at constant pressure. .Thus,
the volume element situated at the center burns at initial pressure P. whereby the
pressure performs the work of expansion. Later on, it is compressed almost to its
initialvolume*bypressurewhichrisescontinuouslyfrom.tomaximum pressure Pe
Pi
grad, i.e., on an average, by a pressure considerably greater than Pi, as a result
of which fact it initially releases less energy than is reabsorbed by it at a later
time. As to the volume element which burns the last, i.e., practically at Pe grad.
pressure, the situation is reversed. Consequently, at the moment when the maximum
explosion pressure is reached, the temperature at the center of the vessel turns
out to be higher than at the walls.
Since the specific heat of gases grows with rising temperature, it is reason-
able to expect that, during temperature. equalization without heat losses, the pres-
sure of the mixture of combustion products must increase. Let us assume, for ex-
ample, that the vessel is divided into two parts, containing equal masses of gas at
* At the same time we must bear in mind that the described effect constitutes a
secondary effect as compared with the thermal effect of combustion which is the
primary factor responsible for the general increase of temperature.
Let it be noted that, in the qualitative explanations provided herein, the dis-
sociation of combustion products has so far not been taken into account.
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different temperatures. With the establinhment of an even temperature distribution,
one part would have absorbed the same amount of energy as was lost by the other, but
Its temperature, at the same time, would have grown in a higher measure than would
have been the drop of temperature in the other part. In the absence of dissociation
of combustion products, pressure is directly proportional to temperature. Hence, in
the first part of the vessel it would also have grown to a greater extent than it
would have slackened in the second part. Consequently, during its equalization, a
higher pressure would have been registered than that which corresponds to the
initial, uneven distribution of temperature.
The dissociation of gases, which increases with rising temperature, accelerates
the growth of pressure with temperature. Therefore, in its presence the mentioned
effect of pressure growth with temperature equalization will be somewhat leveled
down.
In addition to this, dissociation as an endothermic process should reduce the
temperature gradient (in comparison with what it would have been in the absence of
dissociation). Figure 8 shows the distribution of temperature and pressure in the
bomb for three time instants: before the commencement of burning, at moment ti,
when the flame attains the sphere with radius 1.1, and at the moment when combustion
ends te. Throughout the entire process of combustion, the temperature at the ig-
nition point is at the maximum. In the reaction zone it drops sharply. In the re-
gion of unburned mixture, it is uniform though rising somewhat with time.
The increase in steepness of the rising branch of the indicator diagram (Figure
9a) as the maximum is approached evidences the growth of the rate of transformation.
However, since the flame surface grows proportionally to the square of the radius,
. the speed at which the flame front moves in the direction of the normal to its sur-
face (the so-called, spatial flame velocity), not only fails to increase as the
flame approaches the walls of the bomb, but even diminishes somewhat. This may be
seen from the generatrix of the cone on photographs of flame propagation in spheri-
cal vessels (Figure 2).
17
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r
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4
3. Padatic?...-
Figure 8. Variation of pressure and temperature during explosion
in a spherical vessel
(1) Temperature; (2) pressure; (3) radius;
After the flame has come in contact with the bomb walls, the cooling of the
mixture of combustion products begins, as a result of which a drop in pressure is
to be registered. Minor cambers, which sometimes may be observed on the drooping
branch of the pressure diagram (Figure 9b), are to be attributed to temporary re-
tardation of cooling in consequence of heat influx from the bomb center (21, 47).
In the case of rapidly burning mixtures, pressure sometimes fails to equalize
and compression waves -- shock waves -- are being formed. This provokes powerful
diaphragm vibrations which usually arise toward the end of the combustion process
when the transformation rate reaches the minimum. The indicator diagrams resulting
therefrom are unsuitable for purposes of thermodynamic research. Consequently, such
mixtures are being exposed to flegmatization through dilution by one of their prin-
ciple components taking part in the reaction, or by some inert gas. Dilution with
inert gas allows us to maintain the required proportion among the reacting sub-
stances, and at the same time permits us to shift the region of experiments, capable
of being processed, toward higher temperatures than those registered in a case where
we have an excess of one of the principal components (63).
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7
fi
.3% 4
So.+.
lel 0 C,0 0112 403 404 Pa 406
2.;8Pemil 8 ceic
? ? a
0,07 408
CI /Pow
Figure 9. (a) Indicator diagram. (b) Effect of the temperature gradient
on the shape of the curve of cooling (Lewis and Elbe).
(1) Absolute pressure in atm; (2)time in sec; (3) pressure;
(4) time.
Apart from compression waves, several researchers were able to observe also
other types of vibrations in rapidly burning mixtures, for example, fluctuations in
certain slowly burning mixtures arising long before the maximum pressure is attained,
irregular fluctuations in hydrogen-oxygen mixtures at low temperatures (63), and
other types. The causes of these phenomena are not in all cases sufficiently firmly
established.
Section 2. De ree of uilibrium Attained durin losions.
Ds ?
In computations of combustion processes connected with thermodynamic research
by means of explosion and in other cases, as well as in general calculations of
parameters characterizing any physical phenomena, it is customary to proceed from
some definite simplifying assumptions which are apt to idealize somewhat the true
conditions of experimentation. To assure correct deductions made on the basis of
these calculations and comparisons of the computed quantities with those observed,
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it is important to know to what extent do these assumptions correspond to reality.
First of all, our calculations will be based on the supposition that combustion
leads to the establishment of equilibrium in the mixture of combustion products be-
hind the reaction zone. The high development rate of reactions thermodynamicallY
possible in gases at high temperatures provides a basis for such assumptions. The
fact that combustion, in case of sufficiently rapidly burning mixtures, stops instan-
taneouayand completely is evidenced by a sharp break in the indicator graph at the
point of transition from the pressure growth curve to the curve of cooling. Had the
chemical changes not been termintaed by that moment, the maximum of the pressure
diagram would have been rounded off. Some investigators (26, 34, 35) are inclined
to interpret the secondary glow at the ignition point -- observed by a number of?
authors at the end of the combustion process -- as a proof that the processes do
not reach equilibrium in the reaction zone. However, this point of view runs into
a number of objections (18, 21, 47). In particular, it is believed that this phe-
nomenon may be attributed to the existence of a temperature gradient.
The degree to which the postulates, on which calculations are based, corre-
smulto the actual conditions of experimentation is often checked by comparison of
the observed data on maximum explosion pressure with those calculated on the basis
of thermochemical, spectroscopic, and thermodynamic constants whose certainty is
beyond question. The underrated experimental values of maximum explosion pressure
as compared with those obtained by calculation are usually attributed to heat losses,
whose sources will be discussed in Section 3 (some authors (26) explain them by the
incompleteness of chemical changes; see above). The (sometimes by a few percent)
overrated experimental values of maximum pressure -- which are to be observed, for
instance, in mixtures of hydrocarbons and H2 with oxygen, with excess of the latter
-- are ascribed to a delay in excitation of vibrational energy levels of certain
molecules (21, 63). As a result of this, the energy initially is distributed mostly
according to the progressive and rotary degrees of freedom, and the thus arising
excess of forward-motion energy leads to a rise in pressure as compared to that
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which would have been registered in case of equilibrium distribution of energy ac-
cording to the degrees of freedom. It should be observed here that this is not an
unquestionable explanation.
Section 3. Heat Losses in Explosions in Bombs with Central
Ignition
Since it is impossible to determine the precise magnitude of heat losses, ex-
periments are carried out linder such conditions when the losses of heat during the
period of pressure rise are insignificantly small.
The most important source of heat losses, which may not be disregarded in case
of slowly burning mixtures, is constituted by the convective upward motion of the
hot combustion products (Figure 10). As a result of this, the flame initially comes
in contact with the upper section of the bomb mall, which results in strong losses
of heat even before the maximum explosion pressure has been reached. In this case,
heat losses make the burning of the last portion of gas ineffective and the maximum
of the indicator diagram therefore becomes smoother.
Figure 10. Convective
upward motion of a
slaw flame (Ellis).
Heat losses through the wall, resulting from the heating of unburned mixture
owing toadiabatic compression, are minor -- first of all, because the rise of tem-
perature of gases at the wall is consequently small (see Table 17, Chapter IX), and,
second, because it occurs mainly at the end of tho process of combustion (37).
During the period of pressure growth, the surface of the spark electrodes (of
the plug) -- which right from the moment of ignition remains in direct contact with
the burning gases -- may become a substantial source of heat losses. Consequently,
the ignition device should have the least possible surface and be manufactured from
21
AF-WP-O-OCT 59 40
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*ft
a material with the least possible heat conductivity. It should be mentioned in
passing that the margin of the spark energy is too small to produce a substantial
effect on the maximum pressure of explosion.
, _ -
Finally, radiation constitutes an important source of heat losses. order,to
reduce heat losses caused by radiation to the minimum, the internalsurfice of the
. .
bomb should possess a good reflecting power. 'Wohl and Elbe (62) explain the reduc-
tion in heat losses -- observable after the introduction of a small quantity (
?ri
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o cr-I
04
00 1t.1 g,
11 ?
? +3
40 r
H c0
n-1 II
O +1 0
Q 00.
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_
WI 100
V 70
so 40 m
M JO
M 20
47
4J
1,0 1111
47
45 0,7
0,5
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7 7 7
5 t 5
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09017 MX MO MOO 21003000?5X 102 zar poo2m43133)
03 03 IL
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SOO Z707 7137 MX re
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hip
car
Figure 16. Equilibrium composition of the products of coal gas
combustion in ambient atmosphere, in air enriched
up to 60% by oxygen, and in pure oxygen ( o< = 1)
(Fehling)
(1) Pure oxygen; (2) 60% oxygen; (3) air.
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Chapter IX
THERMODYNAMIC RESEARCH BY THE METHOD OF EXPLOSION
Section 1. General Principles
The comparison of the experimental data on maximum explosion pressure with the
results of theoretical computations enables us to draw a conclusion on the accuracy
of the data taken as a basis for the calculation; it also permits us to determine
individual unknown quantities. The explosion method used to be applied most fre-
quently in the past and is still used in the present (see, for instance,(59D to de-
termine the specific heats and the heats of dissociation of the products of combus-
tion, although the progress of statistical physics and experimental research in the
field of molecular structures has somewhat changed the character of this method's
application. At the same time this has given deeper insight into the nature of the
phenomena occuring during explosion in a sealed vessel and it has permitted us to
specify the conditions necessary for the application of the method of explosion; as
a result of this, a nuaber of errors committed by reasearchers in the past can now
be successfully avoided.
It is a matter of primary importance to make sure that the postulates on which
the computation is based (see Chapter (IV) should sufficiently accurately correspond
to the actual progress of the process. The degree of this correspondence may be
established by comparing the experimental and theoretical values of the maximum
pressure of explosion in conditions in which it is a priori possible to dis-
regard the effect of the unknown quantities (heat capacities, heats of dissociation)
upon the magnitude of the explosion pressure; the degree of correspondence may also
be established by means of analysis of indicator diagrams.
The composition of the initial mixture is determined by the problems facing the
researcher. At the same time due consideration should be given to the limits of
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ignitability and the rate of burning, which should neither be too high, nor too low.
When the explosions are highly intensive, pressure often cannot be determined from
the indicator diagram due to the ensuing irregular diaphragm vibrations. In order
to reduce the intensity of the explosion, the initial mixture is being diluted by
one of the principal substances (participating in the reaction) or by inert gases,
if it is desired to maintain the ratio between the basic mixture components un-
changed. Since this reduces the upper limit of the temperature range over which the
thermodynamic properties of the combustion products may be determined, it is desir-
able that other methods for the prevention of irregular fluctuations of gas be de-
vised. Moreover, these new methods should not diminish the precision in the deter-
mination of the maximum explosion pressure due to increased heat losses, for ex-
ample. Certain steps have already been taken in this direction. In particular,
positive results were produced through replacement of oxygen as oxidizer by N20, the
decrease of initial pressure (63), and the reduction of the distance between the
ignition point and the walls of the explosion vessel.
In slow explosions, we can notice convection which leads to considerable heat
losses. Attempts were made to calculate these heat losses (30,37); however, no
sufficiently reliable method for their determination is available so far. Moreover,
it has been demonstrated by experimental data that, in cases of low-velocity explo-
sions, the postulates -- which form the basis for maximum explosion pressure calcu-
lations and the pertinent corrections for temperature gradient (see Section 2 of
this chapter) -- no longer correspond to the actual development of the process,
which in this case turns out to be irregular (37, 50). Thermodynamic investigations
by method of explosion should therefore be conducted in such conditions when the
explosion rate is sufficiently high to permit us to ignore the convection of com-
bustion products. An increase in combustion velocity may be achieved by adding
small quantities of substances which contribute to the formation of radicals assuring
the rapid progress of the chain reaction. Thus, for example, explosions of mixtures
of carbon monoxide with oxygen are accelerated by the addition of small quantities
?%.
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of hydrogen (37,63).
Apart from convection, other sources of heat losses referred to in Chapter III
should also be taken into account. A rigorous centering of the spark-gap is impera-
tive. The explosion vessel should not be too small, lest it become difficult to
avoid a considerable eccentricity of the spark-gap (47). The surface of ignition
electrodes must be as small as possible (63). In certain mixtures -- for instance,
in mixtures of hydrogen with oxygen containing excess hydrogen -- heat losses may
be reduced to nil through addition of a small proportion of water vapors. A rise in
starting pressure of the initial mixture also contributes to a reduction of heat
losses.
A serious complication is constituted by the transcendence of the experimental
values of maximum explosion pressure over those obtained theoretically under the
postulates referred to above. For example, in the case of oxyacetylene mixture
with an excess of 02, this phenomenon attains a value of 5% (21). In view of the
fact that this phenomenon is little known and, at any rate, cannot be quantitatively
accounted for or regulated in any possible way, one should avoid making experiments
under conditions in which this phenomenon may occur, for example, in case of 02
excess in oxyhydrogen, methane-oxygen, and oxyacetylene mixtures.
In the calculation of maximum explosion pressure in closed volume, recorded in
Chapter V, it was assumed that there is an even distribution of temperature in com-
bustion products at the moment when the flame reaches the walls of the vessel. This
simplifying premise is fundamentally wrong, but in most cases it does not lead to
considerable errors. When these errors assume a substantial significance, a cor-
rection for the temperature gradient in combustion products should be introduced in-
'
to the value of maximum explosion pressure. The methods of computation of this cor-
rection will be discussed in the next section.
The initial mixture composition and other conditions of the experiment are
selected in conformity with the research objectives. If the aim is to determine a
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certain quantity, the experimental conditions must be such as to assure that the
proportion of the unknown quantity in the total heat contents of the mixture be
sufficiently large. For example, if the value of the thermal effect is to be found,
then efforts should be made to make the experiment in such a manner as to contribute
to the maximum development of the corresponding process. For the heat-capacity de-
termination of one or another reaction product, it is necessary that its concentra-
tion at the moment of realization of the maximum explosion pressure be large, while
the degree of its dissociation and the degree of reaction development between this
and the other components of the mixture be small enough. These requirements will be
defined below in a more concrete form during the discussion of various examples of
thermoolynamic research by the method of explosion.
Section 2. Calculation of the Maximum Explosion-Pressure Correction for
Irregularity of Temperature Distribution in
Combustion Products
According to Lewis-Elbe's data, the presence of a temperature gradient reduces
the pressure in the system by 0.1-1.0% in comparison to that which would have been
registered in it had temperature distribution been uniform (47, 48). In the tests
conducted by Fenning and Whiffin, this quantity happened to reach 1.3%. This also
is approximately the order of the greatest deviation of the measured maximum explo-
sion-pressure values from the average measurement figures. Nevertheless, in a
number ofcases, particularly during heat-capacity determinations, the introduction
of a correction for the temperature gradient seems to be advisable. The following
considerations speak in favor of this recommendation. First of all, the error in
an average figure is smaller than the error in an individual measurement figure.
However, when the results of tests -- say, the experimental values of heat capacity
of this or that combustion product -- are represented in the form of functions of
temperature, then these results are subjected to further averaging by the method of
least squares, for instance. At the same time, in consequence of the change of the
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qiamtlimtive composition of the initial mixture and the corresponding change of com-
bustion temperature, the magnitude of the correction for temperature gradient changes
considerably (37, 43) and this may substantially affect the shape of the experimental
curve. If this correction is not calculated, then -- according to Fenning and
Whiffen -- absurd results may ensue in a number of cases. The specific heats of in-
ert gases might grow strongly with temperature, the thermal capacity of carbon monox-
ide might prove to be almost constant, and so on. The following fact also speaks in
favor of the conclusion that calculation of the correction for the temperature gra-
dient is unquestionably helpful. The variation of this correction with the variation
of the combustion temperature constitutes a considerable percentage of the corre-
sponding growth or drop of the specific heat which occurs mostly at the expense of .
the vibrational component. Thus the introduction of this correction may prove to be
very important for a more precise definition of the data which form the basis for the
calculation of thermodynamic functions by statistical methods.
We shall compute the quantity Pe/Pegrad, i.e., the relationship of maximum ex-
plosion pressure -- which corresponds to uniform temperature distribution -- to the
pressure, which corresponds to its true distribution at the moment when the flame
contacts the walls of the bomb. Let us mentally split the gas enclosed within the
spherical vessel into a series of small concentric spherical layers, with the center
at the ignition point. In calculating the ratio Pe/P egrad it is assumed; (1) that
convection, as well as heat exchange, between the neighboring layers is insignifi-
cantly small; (2) that the growth of pressure in the bomb is proportional to the
mixture mass burned; (3) that the compression of the unit spherical layers up to
their combustion proceeds adiabatically; (4) that they burn practically at constant
pressure; (5) that subsequent compression of the spherical layers to maximum explo-
sion pressure develops adiabatically; and (6) that by the moment the flame reaches
the walls of the bomb, complete chemical equilibrium and equilibrium energy dis-
tribution according to the degrees of freedom are achieved in all the layers.
109 STAT
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?
?
?, ?
If c< is the proportion of burned gas in relation to the initial mixture as a
whole and Poc is the pressure formed in the bomb by that time, then, according to
postulate (2):
Although the approximative character of this relationship is obvious, a detailed
analysis (18,21) shows that its accuracy is totally acceptable. As to the other as-
sumptions, their validity for cases of not overly slow burning is confirmed by the
fact that calculations made on the basis of these postulates produce a temperature-
drop value of the same order of magnitude as those of the experimentally determined
temperature gradients.
On the basis of these assumptions, one may determine the state of the combus-
tion products in a number of elementary layers; now the ratio Pe/Pegrad is calcu-
lated according to the thus determined temperature distribution, concentrations
(total number of moles), and specific intrinsic energy.'
The computation of the state of combustion products in the elementary spherical
layer do( at the boundary of the sphere, within which a proportion Kof the gas mix-
ture is contained, is composed of three stages: (1) calculation of temperature TU,
up to which the mixture is heated in consequence of adiabatic compression prior to
combustion, (2) calculation of temperature Tb' reached by the mixture as a result of
burning at constant pressure Po< , (3) calculation of equilibrium temperature Te,
corresponding to pressure P grad?
First stage. From the known thermodynamic equation
1-y
TP 7 = const
it follows that
--1 Pc.
10.11==ig7'rA-214----Ig7r, (1.3)
Tv
where Poc is determined from equation (IXA), and..-
4i
is determined by the following
u
formula:
H? LmijifTsti?ElniitiTii
Un? Ui EntijUrui? EntiPTii
-
110-
? ??,i-t ???!,
(IX.4)
STAT
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^
(here, to simplify calculations, we use the average specific heats*). Hence,
lu -- 1
Einliffrui '"-'12M131-1T ?
R TO Erni
(ILO)
Equation (IX.3) may be worked out by trial-and-error method, which does not present
any difficulty, since the right-hand part of the equation depends relatively little
on Tu.
Second slam:. The state of the combustion products of a mixture enclosed in
layer dc( at constant pressure may be determined by the previously described direct
or approximate methods of calculation for P = const. However, a considerably
simpler and more accurate procedure is the following one (13), which is based on the
Zeltdovich-Polyarpy- method of approximation outlined in Chapter VII. Let us suppose
that the first to burn at constant pressure is the layer do c whereby pressure P'
is being attained. Thereafter, reversible and isothermic compression or expansion
to pressure Po< is assumed to occur. In accordance with this assumption, one should
first determine the state of all the combustion products of layer dei( for constant
volume. The initial state of the layer immediately before the commencement of burn-
ing is the state which corresponds to Tu calculated on the basis of equation (IX.3).
We have
T.
Pui= P-fi,iii 9
(Th.5)
The material balance equations for the -e-th element at initial temperatures Ti and
Tle respectively, are:
(P)rri = (1 lip07.
* TO attain a greater degree of accuracy in the first stage of the Pe/Pegrad com-
putation, this stage could be broken up into a series of substages in each of which
one could analyze the effect of the small increase of pressure as compared with
that of the preceding substage.
111 STAT
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?
wheretj is the number of atoms Of thele-th element in the j-th component of the
initial mixture. Substituting the value of puj from equation (I1,5) in equation
3-
This determines the following order of calculation. On the basis
) ;
?
(IX.8)
of the scheme
recorded in Chapter V, we calculate the composition of the mixture of combustion
products and its pressure P' for a series of temperatures T, located both above and
,
? -
below Te which was determined in the assumption that temperature distribution is
.. ?
"
uniform. In addition to this for these T values we calculate enthalpy H' on the
basis of formula (VI.18), lEml according to formula*.
GP %
Ent' ?
ZPkilk
k
(compare with VI.21), ri from equation (VII.39), and 42 by formula
(compare with V11.59)-..--..-4-
Cf = E-Mf I
tm
(I1.9)
-
For one of the temperatures, approximately in the middle of the interval in
question, lie determine the same quantities for three values of Pi, for example, for
the given value, for half that value, and for *one-quarter that value. The average
values- of nH and nt can be found from formulas (VII.62) and (VII.63).
Further, on the basis of a formula similar to (VII.64), we calculate for each
dc c layer the enthalpy of combustion products at constant pressure, Hb, for three
temperatures in such a manner as to assure that enthalpy Hu of the layer in the
initial state with respect to burning, which is to be found from expression
should
should appear between the two extreme values of Hb. Thereupon, by interpolation
* Here we have assigned the ' (prime) mark to quantities which have a conditional -
significance for this calculation.
Z.7
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nAnMeses....
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from condition
(IX.12)
we can find the values of combustion temperature Tb of the layer. After having
calculated the magnitude of gb for the same three temperatures on the basis of a
formula similar to (VII.65), we determine, by means of interpolation the value of
corresponding to the known value of Tb. Knowing ,t;b, from an equation similar
to (VII.67) we finary then Ub=Hb 7 amORTb.
Thus, the above simplified method of Tb and Imb calculation calls for no com-
putation of the mixture composition of combustion products rok each of the con-
sidered do( layers, which must be no less than five in number. In accordance with
the statements made in Section 1, Chapter III, it should be emphasized here that,
in calculating the ratio Pe/P egrad, one should take the dissociation of combustion
products into account to the fullest extent possible.
Third stage. During adiabatic compression of the mixture of combustion prod-
ucts in layer de{, the temperature of the mixture rises to the value of Te which is
determinable by formula
P
? Pmai
1 Ig Ig To+ Ig , .
This formula is analogous to formula (IX.3). But the third stage differs from the
first in that here account must be taken of the degree of dissociation of combustion
products. For this purpose, in calculating Tb and 2;mb for the layer contiguous to
the wall of the bomb, i.e., fore( = 1, the quantities Hb, Ub and2mb must be com-
puted for four to five temperature values. The highest of them must not be lower
than Te for the elementary volume located in the center of the vessel, i.e., for
i