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CALCULATION OF PROCESSES IN THE COMBUSTION CHAMBER AND
NOZZLE OF A LIQUID-PROPELLANT ROCKET ENGINE
By A. V. Bolgarskiy
October 1959
116 Pages
PREPARED BY -
LIAISON OFFICE
.TECHNICAL INFORMATION CENTER
MC LTD ?
WRIGHT-PATTERSON AIR FORCE BASE. OHIO
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V
A. V. Bolgarskiy
Raschet Protsessov V Kamere Sgoraniya-
i sople Zhidkostnogo Raketnogo Dvigatelya
Gosudarstvennoe Izdatellstvo Oboronnoy Promyshlennost'i
Mciskva 1957
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2
This book is devoted to questions of combustion and discharge at high tempera-
tures in a liquid-propellant rocket engine. It gives the detailed technique for
the thermodynamic calculation of the working process, illustrated by the solution
of a series of practical problems. This book offers the bases for the possibility
of applying a graphical method for calculating the parameters of the liquid-
propellant rocket engine, which greatly facilitates this calculation. It gives a
detailed exposition of the technique of constructing such nomograms. Two nomograms
for two types of propellant)constructed by the author, are appended at the end of
the book.
This book is intended for students of higher institutes of aviation who are
studying the theory of combustion and exhaust: but it may also be useful to engi-
neers working in this field.
Critice: Instructor YU.K.Zastela,
Professor A.V.Kvasnikov, Doctor of Technical Sciences .
Editor: Engineer G.Yu.Yanovskiy
Supervisor of Editorial Staff, Engineer A.I.Sokolov
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It
PREFACE
In this book we give an exposition of the method developed by the author for
calculating the processes of fuel combustion and the discharge of the combustion -
products from liquid-propellant rocket engines. The entire calculation is based on
the use of the conventional equation of energy, which is known to the student from
his course in aerodynamics. The description of the proposed method is accompanied
by a detailed exposition of the computational procedure, and by examples.
The diagrams at the end of the book (Appendixes V, VI, and VII) considerably
accelerate the calculations and may be useful, especially in the preliminary rough
calculations. If these diagrams do prove useful and find practical application, it
will be possible to organize the calculation and construction of similar diagrams
for all available fuels of liquid-propellant rocket engines.
This book will be useful not only to students in studying the theory and de-
sign of liquid-propellant rocket engines, but also to engineers working in this
field.
19,
The author expresses his thanks to Instructor V.Ye.Alemasov, Candidate in
Technical Sciences, for his help in the preparation of the manuscript, and for his
permission to use the Tables (Appendixes I, II, III and IV) which have been care-
fully recalculated and checked by him.
The author also notes the work by L.V.Ignatyyeva, laboratory assistant of the
Department of Heat Engines, for arrangement of the computational diagrams and in
illustrating this book, and expresses his thanks to her.
,
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I.
The author Id11 be very grateful for any comments and suggestions, addressed to
. OBORONGIZ Moscow, I-51, Petrovka, 24. -
-
???
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a
INTRODUCTION
The appearance of a new type of thermal engine, the liquid-propellant rocket
engine, demanded the development of new methods of thermal calculation since the
processes of conversion of chemical energy into thermal energy and then into kinetic
energy, in this engine, have certain peculiarities which complicate the calculations.
The theoretical combustion temperatures in these engines are considerably high-
er than in other types of thermal engines, since the oxidizers used are liquids
richer in free oxygen than atmospheric air. Indeed, with the theoretically neces-
sary quantity of oxidizer, the weight of the combustion products of 1 kg of kerosene
71W-iri round numbers, as follows: with atmospheric air as oxidizer, 16 kg; with
hydrogen peroxide, 8.5 kg; with nitric acid, 6.3 kg; and with liquid oxygen, 4.5 kg.
And yet the quantity of heat liberated on combustion is almost the same with these
different oxidizers. It is entirely understandable that, with.a smaller quantity of
combustion products, they must be heated to a higher temperature.
The high theoretical combustion temperature causes a more extensive dissocia-
tion in the combustion chamber and a partial recombination of the molecules as the
gases flow into the exhaust nozzle where, on expansion, the temperature of the gases
drops. The need to take these phenomena into account makes it more difficult to
. handle the thermal calculation.
?
On the other hand, there are no fundamentally new phenomena in the thermal
processes in liquid-propellant rocket engines, and consequently, the entire thermal
calculatibh shoUld be founded on the Propositions generally adopted in thermal
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I.
C,
e -
technology.
The book by A.V.Bolgarskiy and V.K.ShchUkii(tfilorking Processes in Liquid-
Propellant Rocket Engines", Oborongiz, 1953) gives the principles of a method of
calculating the thermal processes for liquid-propellant rocket engines, based on the
general propositions and concepts of thermal technology and of thermal engines.
The present work represents an expanded and detailed exposition of the calcula-
tion method which is a logical development of general thermal calculations extended
to a new type of thermal engine.
On the basis of the fact that all the processes taking place in the combustion
chamber and nozzle of the liquid-propellant rocket engine are conversions of energy
from one form to another (chemical - thermal - kinetic), the author bases his entire
technique on the usual equation of energy (cf. Chapter 11); in this equation, from
the total quantity of heat Q he separates the heat obtained on combustion of the
fuel.
Only the simplest propellants, in widest use, are considered in this book.
They are chemical compOUnds of the systems C, H, 0, and N.
The application of the proposed calculation method to fuels of other types, and
the construction of computational diagrams for other fuels to be used, is a task for
the future.
The calculation per mole of fuel has been taken as the basis of exposition of
the method.
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Ho = 100%,
Ro
(2)
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a
CHAPTER I
PROPFT ANT CALCULATIONS
1. Chemical Ccmposition of the Components
The fuel for liquid-propellant rocket engines consists of a combustible and an
oxidizer. The combustibles ordinarily employed are either hydrocarbons or alcohols;
kerosene, ethyl and methyl alcohols are most often used. The usual oxidizers are
nitric acid, liquid oxygen, or hydrogen peroxide.
Hereafter we shall take CnHm0p as the general chemical formula for the com-
bustible and HtNuOvCq for the oxidizer.
If the components are specified by chemical formulas, then their molecular
weight is calculated by the formulas
tic =12n -I-m-1-16p;
110. td- 14a+ 16v+ 12q.
(1)
The conversion to elementary Composition by weight is performed according to
the formulas
Cc_ I2n 10014,
Pc
Fic.7-2?: 100%,
Pc
?Oc ?16p 100%
and by analogy, for the oxidizer,
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Ni0==i4u100915,
Ro
00 . 16v
ILo
100%,
Co --12q 10056.
(3)
If the components are specified by elementary composition by weight,. i.e.:
Combustible - Cc%, Hc%, Oc%,
Oxidizer -
e- n cc, v po%, N0%,
nes vo,
then the calculation for finding the chemical formula is performed as follows: The
number of atoms of hydrogen are taken as equal to its percentage content in the com-
ponent, and then the number of atoms of the remaining elements in the component are
Cc 0
calculated. Thus, the combustible contains atoms of carbon' c atoms of
12 16
No co '
oxygen, while the oxidizer contains atoms of nitrogen, atoms of carbon
14 12
0
and ?16 atoms of oxygen; we then get the following arbitrary formulas:
Combustible ? CCe Hl-ic 00e ?
12 16
(4)
Oxidizer --fiHoliN001100Cco.
14 16 12
In this case the arbitrary molecular weight is taken as equal to 100.
Very often the components of a propellant do not constitute 100% of the sub-
stance, but have been diluted with water (for instance ethyl and methyl alcohols,
nitric acid, hydrogen peroxide). We allow for the presence of water in the com-
ponents by their concentrations, expressing the percentage content of the pure sub-
stance in a mixture with water; for example, if the concentration of hydrogen per-
oxide is 80%, this means that 1 kg of this component contains 0.8 kg of pure hydro-
gen peroxide and 0.2 kg of water.
If the component is to be expressed by a chemical formula, then it is more con-
venient to express the quantity of water by the number of moles of water to 1 mole
of the substance.
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Let the concentration of the substance be 0%, and its molecular weight be 4;
then the calculation is conducted as follows: 1 kg of component contains 100 kg
100 -
of the pure substance and akg of water; and it contains the following quan-
tity of water per 1 kg of pure substance:
There is 100 - a
0
100?a a
100 100 a
100 ? a
KvIC5r.
LI kg of water per mole of pure substance; expreesing this
quantity of water in moles and remembering that the molecular weight of water is
?w = 18, we get:
100 -- a 100--0
m-=
ap? 187
mote/mole
(5)
Table 1 gives the values of in for components most frequently encountered in
concentrations below 100%.
Table 1
Type of
Component
Molecular
Weight
Concentration in %
95
90
85
80 1
75
70
Ethyl alcohol
46
0,135
0,284
0,451
0,6391 0,852
1,095
Methyl alcohol
32
0,094
0,198
0,314
A AAK
A KO'l
A 'MI
Nitric acid
63
0,184
0,389
0,618
0,875
1,167
1,500
Hydrogen peroxide
34
0,099
0,209
0,333
0,472
0,630
0,809
In the symbols used above, the chemical formulas of components with the concen-
tration ac% and aC% will be of the form
C,1-1#101, ? mci-120 and HEN.OvCq? MOHO,
where 0
100 ? a,
P. I
me =
1!3ac
5
100?a.
Mo? Ro?
18a0
(6)
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The weight of Imola of the component, of a concentration 4, will hereafter be
termed the molecular weight of the component and will be designated by 41. At any
concentration, this weight will be determined as follows:
la? 8nt = + 1814 106 ? a 100v.
18t: ?
Kg/mole
Table 2 gives the values of pf for the most frequently used components.
(7)
Table 2
i ?
Component
Chemical
Formula
Molecular Weight at Concentration 0%:
100
95
90
85
80
I 75
I 70
Ethyl iilcohOl
C2Hs0
46
48,4
51,1
54,1
57,5
61,3
65,7
Methyl alcohol
CH40
32
33,6
35,5
37,7
40,0
42,6
45,7
Nitric acid
HNOs
63
66,3
70,0
74,1
78,8
84,0
90,0
Hydrogen peroxide
H202
34
35,7
37,7
40,0
42,5
45,3
48,6
For a component expressed by a chemical formula, at a concentration below 100%,
its elementary composition by weight is calculated as follows.
Let a combustible be specified by the chemical formula CnNm0p ? m020.
The number of atoms of the various elements per mole of combustible and their
weights will be determined as follows:
Carbon - number of atoms - n weight 12 n
Hydrogen - ff tt M + 2mc " in 2mc
Oxygen- if ft -p + mc " 16(p 4' mc)
Total weight of 1 mole of combustible 12n + in + 16p + 18%.
.e On the basis of eqs.(1) and (7) we find
12n+ m+ 18inc=itc+18Mcw=K.
Consequently, the elementary composition of the coMbustible by weight
? (of. eq.(2)] will be expressed by the following relations:
4
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Cc' =r 12n, 10()%,
Pc
m +,2m` 1.00%,
== 16 (p m ) 100%.
c
imilarly, we get for the oxidizer:
12q
, 100%,
0
110
H,0 t + Fin? 100%,
0'0 =-- 16 (v + mo) 100%,
14u
= --7- 100%.
0
v0
?.(8)
Example 1. Calculate the elementary composition, by weight, of ethyl alcohol,
at concentrations of 60, 80, and 100%.
The chemical formula of pure ethyl alcohol is as follows:
C2H5 (OH) .----C2H60.
Consequently, its molecular weight will be
!le= 12.2+1.6+161=46.
The number of molecules of water per mole of the pure substance at the speci?
fied concentratio& is calculated on the basis of eq.(5) and will be
10D18?:ca,== 10D--ac 46
=2,555 1?437
--
18 a
ac
At the specified concentrations, this formula yields
m100 =_? 0,
m? =O,639,
m6?. 1,703.
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The weight of 1 mole of the component, of a concentration 4, will hereafter be
termed the molecular weight of the component and will be designated by pl. At any
concentration, this weight will be determined as follows:
/1?'14+18fft=p,.+.1814 100?a .1_00 Kg /mole
18o a
Table 2 gives the values of pt for the most frequently used components.
Table 2
(7)
Molecular Weight at Concentration a%:
Component
Chemical
Formula
100
95
90
85
80
I 75
1 70
Ethyl alcohdl
C.21-40
46
48,4
51,1
54,1
57,5
61,3
65,7
Methyl alcohol
CH40
32
33,6
35,5
37,7
40,0
42.6
45,7
Nitric acid
HNOs
63
66,3
70,0
74,1
78,8
84,0
90,0
Hydrogen peroxide
H202
34
35,7
37,7
40,0
42,5
45,3
48,6
For a component expressed by a chemical formula, at a concentration below 100%,
its elementary composition by weight is calculated as follows.
Let a combustible be specified by the chemical formula Cn Nm 0p ? m_1120.
The
The number of atoms of the various elements per mole of combustible and their
weights will be determined as follows:
carbon - number of atoms - n weight 12 n
Hydrogen -
Oxygen -
tt
11 11
? m * 2mc
P 'no
11 m 2mc
n 16(p +m0)
Total weight of Imola of combustible 12n + m + 16p + 18%.
e On the basis of eqe.(1) and (7) we find
12n+m-1-16p+18tnc=pc+18mc=K.
Consequently, the elementary composition of the combustible by weight
[cf. eq.(2)] will be expressed by the following relations:
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The corresponding chemical formulas will have the form
at
ac == 10)
--- C.21160,
.
at
=80%
? C2H 60 ?
0,639 H20,
at
ac = 60 %
? C2H60 ?
1,703H20.
On the basis of eq.(7), the molecular weight of alcohol is
at a .100%
at
ac = 80 %
at
= 46,
? 100.46
= 80? 575'
? 100.46? 76,67.
c 60
We recall that the term molecular weight of a substance of lower concentration
(o 1, then the
heat value of the propellant, after reaching a :efinite value at a = 1, varies only
slightly thereafter (just like the chemical energy) exclusively as a function of the
heat value of the oxidizers somewhat decreasing for a negative heat value of the oxi-
dizer and increasing for a positive value. We present a more exact definition of
the concept of the heat value of a propellant for a liquid rocket engine: The heat
value of a propellant is the quantity of heat liberated on full utilization of one
of the components during the combustion process.
In accordance with the above, the heat value of a propellant at a 1
Hp,. (1 027 003 ? 8,5 .14 905) "II 900 400a
900400a
Ha. 100 563,7a
fill,==-X and 14==.1c.
2. 95% ethyl alcohol (chemical formulas C2H60 ? 0.135 H20, Hile 290,580 kcal/kg,
= 42.4) and 80% hydrogen peroxide (chemical formula H202 ? 0.472 H20, 1-4t =
= 7240 kcal/mole, = 42.5); ko = 6.
The corresponding computational formulas will have the form
X.290 580 + 43 4402, 14= 33334400202,
... 290 580 4- 43 440a
48,4+ 255a ?' t= 48,4 -IF 2.55a
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0
CHAPTER III
CALCULATIONS OF PROCESSES IN LIQUID-PROPELLANT ROCKET ENGINE
1. The Principal Processes in a Liouid-Pronellant Rocket Engine
For the further calculations the following notation is adopted for the princi-
pal cross sections in liquid-propellant rocket engines (Fig.2).
At/. the section c - c the atomized mixture of the propellant components arrives,
and the combustion process begins; in the section z - z the process of combustion
is substantially concluded, and the combustion products enter the exhaust nozzle;
the passage of the combustion products
Fig.2 - For
Calculating
through the nozzle is accompanied by further
combustion of the propellant, which has not
been completed in the combustion chamber,
and by a substantial recombination of the
molecules, due to the decreased tempera-
the Processes
ture during efflux.
in a Liquid-Propellant Rocket Engine
The velocity of the chemical reaction
of combustion of the propellant evidently varies for different rocket propellants,
just as is the case for the duration of the preparatory period; besides the form of
the propellant, a considerable number of other factors affect these quantities: the
combustion temperature, the weight ratio of the components, the temperature of the
arriving components, the degree of atomization and mixing, etc.
,
Thus the coefficient Of heat liberation at the end of the combustion chamber,
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of
410
&inc z* may have various values governed by the above factors. It is most natural
that this coefficient should not be amenable to theoretical determination, meaning
that'it must be established on the basis of practical considerations and data for
existing liquid-propellant rocket engines. In the same way, without data on the
completeness of the afterburning of the propellant during the exhaust process, we
must confine ourselves to practical and experimental data on the values of the co-
efficient of heat liberation in a cross section of the nozzle,alue
&inc a The value
of
of inc a also depends on the residence time of the combustion products in the
nozzle, i.e., on the rate of flow of the gases and on the length of the nozzle.
The process of efflux is complicated not only by the afterburning of the pro-
pellant but also by the recombination of the molecules due to the drop in the tem-
perature of the combustion products. There is no reliable assurance that the re-
combination of the molecules proceeds so rapidly that the combustion products are
always in a state of chemical equilibrium, nor that the oscill tory energy of the
atoms in the molecules (which is restored in any case more slowly than the rotation-
al and translational energy of the molecules) may vary so rapidly that the combus-
tion products are always in energetic equilibrium. There are no definite data as to
how completely chemical and energetic equilibrium is restored. This is even more
emphasized by the fact that these quantities also depend on the residence time of
the combustion products in the nozzle and thus on the velocity of flow and on the
length of the nozzle.
Theoretically, one might imagine two extreme hypotheses about the efflux of the
combustion products:
1. Extreme non-equilibrium exhaust conditions, when it is assumed that the
chemical and energetic equilibriums are restored very slowly; it is assumed in this
case that the chemical composition of the combustion products does not vary at all,
since there is no afterburning of the fuel, and the oscillatory energy of the atoms
has remained constant. Thus, under this stipulation it is assumed that no additional
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processes other than the conversion of part of the enthalpy of the combustion
products into kinetic energy takes place, i.e., that in this case the process of
efflux of the combustion product8. is adiabatic.
2. Complete equilibrium efflux, when it is assumed that both the chemical and
energetic equilibriums are very rapidly restored; under such conditions it may be
assumed that the chemical composition of the combustion products, like the oscilla?
tory energy of the atoms, corresponds at all times to the temperature; moreover,
there is also an additional partial afterburning of the propellant, so that the co?
efficient of heat liberation at emergence from the nozzle &inc a is higher than the
same coefficient at the exit of the combustion chamber, &inc z. Thus, in this case,
the process of efflux takes place with the addition of a considerable quantity of
heat, due both to the afterburning of part of the propellant and to recombination of
the molecules; consequently, this process must be regarded as a polytropic process .
with an index of polytropy n staller than k.
The actual process of efflux must be somewhere between these two extremes, and
the longer the nozzle and the longer the time of exhaust, the closer will the
process come to complete equilibrium of efflux. Recent research has shown that the
actual process of efflux takes place in a manner very close to the equilibrium state.
It is therefore more rational to calculate precisely this equilibrium discharge,
and then to apply practical coefficients to allow for the deviation of the process
from equilibrium.
2. Composition of the Combustion Products
The principal quantities determining the composition of the combustion products
are as follows: the coefficient of excess oxidizer a, the coefficient of libera?
tion of heat with respect to completeness of mixing Cinc z' and the actual combus?
tion temperature T. The composition of the combustion products is very complex,
owing to the considerable degree of dissociation. If we neglect the presence of
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(
;
several gases contained in very small amounts in the combustion products, we may
assume that in the general case the combustion products consist of CO2, CO, H20, 1-12,
OH, N2, 02, NO2, H, and O.
Calculations and experiments show that, at lower dissociation temperatures,
there is more H2 than OH, while at higher temperature3 the quantity of OH consider-
ably exceeds that of H2. The monatomic gases H and 0 appear in appreciable quanti-
ties only at temperatures over 24000 abs.
In the most generally adopted case, the dissociation of the combustion products
is determined by the following chemical equations:
Co + 0,502 7*---3. CO2;
H2 + 01502 "E? H10;
OH + 0,5H2 E-? H20;
2H Hs;
20 4-2: 02;
NO