n
INVESTIGATIONS INTO
THE SELF4GNITION HAZARD
OF NITRATE ESTER PROPELLANTS
TECHNOLOGICAL LABORATORY RVO - TNO RIJSWIJK (Z.-H.) - POSTBOX 4545 - THE NETHERLANDS
INVESTIGATIONS INTO
THE SELF4GNITION HAZARD
OF NITRATE ESTER PROPELLANTS
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS DR. IR. C. J. D. M. VERHAGEN, HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUURKUNDE,
VOOR EEN COMMISSIE UIT DE SENAAT T E VERDEDIGEN OP WOENSDAG 12 FEBRUARI 1969 T E 14.00 UUR
DOOR
JOHANNES LEONARDUS CORNELIS VAN GEEL
SCHEIKUNDIG INGENIEUR GEBOREN TE EINDHOVEN
DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOREN PROF. DR. IR. C. BOELHOUWER EN PROF. DR. IR. W. J. BEEK
to mg parents
to all who contributed to the accomplishment of this thesis
THE INVESTIGATIONS DESCRIBED IN THIS THESIS HAVE BEEN CARRIED OUT
IN THE TECHNOLOGICAL LABORATORY OF THE NATIONAL DEFENCE RESEARCH ORGANIZATION TNO,
C O N T E N T S CHAPTER 1 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. CHAPTER 2 2.1. 2.2. 2.3. 2.4. CHAPTER 3 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. LIST O F SYMBOLS I N T R O D U C T I O N Nitrate ester propellants Introduction
History
Classification of nitrate ester propellants .. Propellant additives
Degradation of nitrate ester propellants .. 1.5.1. Denitration...
1.5.2. Influence of free oxygen on the degra dation
1.5.3. Influence of the temperature on the degradation
Stability tests
Theory of self-heating Introduction
Theory of thermal explosion 2.2.1. Heat balance
2.2.2. Criteria of self-ignition
2.2.3. Temperature distribution as a func tion of time
2.2.4. Critical state
Rate of heat generation as a function of temperature
Discussion
The isothermal heat-generation meter Principle of operation Heat-flow meter ... Temperature control Compensation Sample vessel Calibration Accuracy Page 9 13
CHAPTER 4 4.1. 4.2. 4.3. 4.4. 4.5.
Heat generation of nitrate ester propellants Introduction
Experiments
Theory of iso-Q-lines
Analysis of the thermograms 4.4.1. Apparent activation energy 4.4.2. Heat-generation factor
Rate of heat generation as a function of time 4.5.1. Heat generation and degree of
degra-dation
4.5.2. Free oxygen depletion and pressure variation
4.5.3. Increase in moisture content 4.5.4. Stabilizer depletion 4.6. Discussion Page 45 45 45 50 53 53 58 59 59 59 64 65 67 CHAPTER 5 5.1. 5.2. 5.3. 5.4. 5.5. Appendix I Appendix II Appendix III Appendix I V
Temperature distribution as a function of
time 68 Theory ... 68
Apparatus for the measurement of the
tem-perature distribution as a function of time 71
Experiments 72 Calculated and measured temperature
distribution 11 General discussion and further investigations 79
Time constant of heat-generation meter ... 85
Specific heat 87 Tables 91 Analytical procedures 97 S U M M A R Y 99 S A M E N V A T T I N G 101 R E F E R E N C E S 103 A C K N O W L E D G E M E N T S 105 C U R R I C U L U M V I T A E 107 S T E L L I N G E N 109
LIST O F SYMBOLS D . A = constant. B = constant. C = heat-generation factor. C„ = calibration constant. Dg = dimensionless group, D„ =/?(5. Dn = dimensionless group, tang D^ =
-a — I
E = apparent activation energy.
E„ = apparent activation energy corresponding to a distinct degree of degradation, n = I, II, III etc.
F ( Q ) = heat-generation factor, a function of Q. H = heat capacity of the measuring unit. M = mass.
P = pressure.
Q = amount of heat generated per unit of mass.
R = gas constant.
S„ = standard deviation of the output of zero-line.
T = absolute temperature.
ATjn = temperature fluctuation of the heating-liquid in the ultra-thermostate.
'^'^'(,„4 = temperature fluctuation of the internal room of the heat-generation meter.
U = output of the heat-generation meter. U' = maximum error of U.
$ = rate of heat generation.
b = constant, specific for the degree of con-version, involwed.
c = specific heat at constant volume. d = thickness.
f(Q) = function of Q.
h = heat-transfer coefficient.
i = dimensionless number, i = 1, 2, 3 m = dimensionless number, m = 0 for a slab,
m = 1 for a cylinder, m = 2 for a sphere.
[mol/kg sec] [sec~'] [ W / k g ] [ W / V ] [KJ/mol] [KJ/mol] [ W / k g ] [J] [kg] [N/m2] [J/kg] [J/mol °C] [ V ] [°K] [°C] [°C] ,[V] [V] [ W ] [I/kg] [J/kg] [m] [sec] [W/m2 °C]
n q r s t V X « N o X = dimensionless number, n = 1, 2, 3
= rate of heat generation per unit of mass. = characteristic dimension of a symmetrical
body; half thickness of a slab, the radius of a cylinder and a sphere.
= surface. = time.
volume.
= distance coordinate.
X
= reduced distance coordinate, z = .
= dimensionless group of Biot, a = r hr dimensionless number, for a sphere: /J= 1-f 0.369 ( 0 J , + 0 . 0 9 0 8 ( 0 J , ^ ratio of the volume of the free oxygen in the container to the volume of the propellant in the container.
ratio of the volume of the free nitrogen in the container to the volume of the propellant in the container.
ratio of the volume of the air in the con-tainer to the volume of the propellant in the container.
Q
, degree of conversion. Q
distinct degree of degradation, n = I. II, III etc.
dimensionless group of Kamenetskii, r2 E g(, E C e x p ( - - — ) . 2 R T / = heat conductivity = mass density. = bulk density. = reduced time, T = R T . [ W / k g ] [m2] [sec] [m3] [m] [ W / m °C] [kg/m3] [kg/m3] CQb^
Tg = time constant of the heat-generation meter. [sec]
E(T-TJ
0 = reduced temperature, 0 =RT,
2 ©0 At OJ = = = Subscripts a b c expl ig ind i m r s sa z reduced temperatu time interval. frequency. ambient bulk critical explosion ignition induction initial measurement reference material steady state sample location z [sec] [rad/sec]I N T R O D U C T I O N
Until the end of the 19th century gunpowder was used as a pro-pellant. It consists of a mixture of potassium nitrate, sulphur and charcoal. Gunpowder has a rather low calorific value and about half of its reaction products is solid. This involves moderate ballistic properties, serious fouling of the weapon and smoke pro-duction at firing. Smoke is a serious handicap for military use since it reveals the location of the fire-arm to the enemy and prevents the marksman from observing the target.
In the course of the 19th century gunpowder was gradually re-placed by nitrate ester propellants. T h e main components of this new type of propellant are cellulose nitrate and glycerol trinitrate. Nitrate ester propellants show much better propulsion properties than gunpowder. Their reaction products are almost completely gaseous so that fouling of the weapon and smoke production are greatly reduced. However, many violent explosions showed that the chemical stability of these nitrate ester propellants was much less than that of gunpowder i) 2). In 1847 the first factory of cellulose nitrate at Faversham, England, was destroyed by an ex-plosion. The laboratory investigations of glycerol trinitrate were also accompanied by several explosions. In 1905 the Japanese war-ship Mikara was destroyed due to the explosion of the ammunition magazine and about 600 peoples lost their lives. Many similar disasters followed, such as the explosion of the French war-ship Liberté in 1911. It became more and more clear that most of these accidents were the result of self-heating, due to the exothermic degradation process of the nitrate ester propellants.
The degradation process of the nitrate ester propellants involves denitration reactions due to the thermal decomposition of the R O - N O 2 bond and hydrolysis by the moisture present in the propellant.
It was found that some of the degradation products formed, namely nitrogen-dioxide, nitric acid and nitrous acid cause conse-cutive reactions and catalyse the hydrolysis of the nitrate esters. Due to the exothermic degradation process the temperature of a stored propellant mass will rise. This temperature rise is accompanied
by an increase in the rate of degradation and thus by an increase in heat generation. This in turn accelerates the temperature rise, etc. W h e n the rate of heat generation remains higher than the rate of heat loss to the surroundings, a thermal explosion occurs after a certain induction periode.
T o prevent the undesirable effect of the degradation products formed, stabilizers were added to the propellant to bind the men-tioned degradation products. To-day diphenylaraine and diethyl-diphenylurea are widely used as stabilizers.
But although the stability of the propellants is markedly improved by these stabilizers, careful stability control is still necessary. This was experienced 3) rather recently from several disastrous explo-sions of propellants manufactured during World W a r II.
So far, the chemical stability of nitrate ester propellants is con-trolled by means of several stability tests, which generally involve an artificial aging at an elevated temperature. The tests are based on the detection of distinct degradation products or phenomena as for instance the decrease in stabilizer content, the loss in weight, the evolution of nitrous gases and other gaseous degradations pro-ducts. T h e results are compared with those of reference propellants of which it is known from experience that the stability is sufficient for safe storage.
However, problems arise when the stability of new propellant compositions has to be examined for which no reference propellant is available.
From the foregoing it follows that the mentioned stability tests give only indirect information on the hazards of self-ignition.
It is noteworthy that in he United Kingdom around 1900 the so-called "silvered vessel test" has been developed. This test in-volves the measurement of the temperature rise as a function of time of about 20 gram of propellant stored in a spherical vacuum jacketted flask, located in a stove, which is kept at a constant tem-perature of 80 °C'. T h e temtem-perature of the sample is measured in the center of the sphere. W h e n the temperature has gone up to 2 °C above the stove temperature the test is discontinued.
Although with this test more direct information is obtained with respect to the liability of self-heating, even in England the silvered vessel test has not been applied widely for safety inspection. T h e disadvantages of the test are that the amount of heat loss to the
surroundings depends on the design of the test apparatus and that the risk of self-ignition at testing is great.
Also at the Technological Laboratory of the National Defence Research Organization in the Netherlands research work has been done to get more direct information on the self-ignition hazard of heat-generating substances. Lindeyer and Heykoop developed an apparatus in which a sample of propellant could be kept in an approximately adiabatic condition. This apparatus was improved by Pasman et al"^), resulting in the so-called adiabatic storage test. T h e investigations described in this thesis fit in with these studies. They aim at getting quantitative information on the factors govern-ing the self-heatgovern-ing process of nitrate ester propellants durgovern-ing storage viz.
— the kinetics and heat effects of the degradation,
— the physical properties of the propellant, such as the specific heat and the heat conductivity,
— the storage conditions, such as the temperature of the surround-ings, the heat-transfer coefficient to the surroundings and the shape and size of the propellant mass.
Arrangement of this thesis
Chapter 1 gives a general description of the chemical and physical properties of nitrate ester propellants.
In Chapter 2 a literature review is given of the factors governing the self-heating process.
T o investigate the influence of the temperature on the rate of heat generation and its variation with time a sensitive heat-genera-tion meter has been constructed. T h e meter enables the measure-ment of the rate of heat generation as a function of time under isothermal conditions at different temperatures. T h e heat-generation meter is described in Chapter 3.
T h e results of the measurements of four different types of nitrate ester propellants are described and discussed in Chapter 4.
Based on these results, in Chapter 5 a method is developed by which the temperature distribution as a function of time can be calculated in a stored propellant mass. T h e calculation method has been verified with experiments.
CHAPTER 1
N I T R A T E E S T E R P R O P E L L A N T S
1.1. Introduction
Nitrate ester propellants belong to the class of explosive substances. An explosive substance is capable of sustaining a propagating auto-combustion process.
Explosives are divided in deflagrating explosives (low explosives) and detonating explosives (high explosives). W h e n an explosive deflagrates the velocity of the reaction zone is governed by the rate of heat transfer to the adjacent layer by conduction, convection and radiation. The velocity with which the reaction zone proceeds through the substance is of the order of centimeters per second. In detonating explosives the reaction zone proceeds much faster through the substance; the velocity is of the order of kilometers per second. The auto-combustion process of a detonating explosive proceeds through the substance due to energy transfer to the adjacent layer by a shock wave. T h e chemical energy evolved sustains the shock wave, thus providing a steady state detonation.
Propellants belong to the group of deflagrating explosives. The application of propellants is based upon the property that they can deflagrate with strong gas evolution. The formation of gas is essential since the purpose of a propellant is to accelerate a projectile or to propulse a rocket by the combustion gases. At present nitrate ester propellants are the main propulsion substances for fire-arms. In rockets not only nitrate ester propellants but also composite pro-pellants are used. A composite propellant consists basically of an inorganic oxidizer (for instance ammonium perchlorate) embedded in an organic substance as fuel (polyurethane, polyester, poly buta-diene).
Experiments have shown that composite propellants may be less dangerous with respect to self-heating than nitrate ester propel-lants 5). In this study only nitrate ester propelpropel-lants have been in-vestigated.
In this chapter, after a short historical review of the development of nitrate ester propellants, the classification and different types of
additives will be dealt with. Also the degradation of nitrate ester propellants and some stability tests will be discussed. A detailed description of the chemical and physical properties of nitrate ester propellants has been published recently by U r b a n s k i i ) . Hence in this thesis only a general discussion on the properties of nitrate ester propellants is given.
1.2. History
The first nitrate ester propellant was made in 1864 by Major Schultze, a Prussian Army Officer. Schultze's propellant consisted of cellulose nitrate mixed with sodium nitrate and potassium nitrate. However, the new propellant burned too fast and not regular enough for use in rifles. Moreover, cellulose nitrate in its fibrous form can be brought rather easy to detonation.
In 1884 a French engineer, Paul Vielle, found that cellulose nitrate could be brought in colloidal form using a mixture of ether and alcohol. After removing the greater part of the solvents, the cellulose nitrate could be kneaded into a stiff jellied mass, which could be rolled out into thin sheets and cut to small squares. It turn-ed out that the burning properties of the gelatinizturn-ed cellulose nitrate were much better and that the hazard of detonation was reduced to a safe level.
Around 1888 Nobel combined cellulose nitrate with glycerol tri-nitrate, the latter having a gelatinizing action on cellulose nitrate. T h e presence of glycerol trinitrate gives the propellant a higher calorific value, which markedly influences the ballistic properties.
1.3. Classification of nitrate ester propellants
Nitrate ester propellants are classified according to their composition.
A. Single base propellants
These propellants contain cellulose nitrate as their main ingredient. They are also called nitrocellulose powders.
B. Double base propellants
These propellants contain two nitrate esters as their main ingre-dients. Besides cellulose nitrate generally glycerol trinitrate is added. Such a propellant is also called nitroglycerine powder. In stead of glycerol trinitrate sometimes diethyleneglycol dinitrate is used as the second nitrate ester.
W h e n a double base propellant is used for rocket propulsion the propellant is also called homogeneous rocket propellant.
The physical structure of these propellants is homogeneous as contrasted with a second type of rocket propellant: the composite propellants.
C. Triple base propellants
These propellants contain besides cellulose nitrate and glycerol trinitrate also nitroguanidine. Owing to the presence of nitro-guanidine the burning gases have a high nitrogen content and a relatively low temperature. This results in a reduced chance of a muzzle flash and in a decrease in the erosion of the bore.
Triple base propellants are mainly used for artillery ammunition. In Table 1.1 examples of the three types of nitrate ester propel-lants are given.
Nitrate ester propellants are manufactured in many forms (e.g. flakes, strips, tubes, pellets, sheets and cords) to adjust the rate of energy release at burning. By suitable choice of the geometric form and size of the grain one propellant composition can meet the performance requirements for a wide range of weapons.
TABLE 1.1.
Composition in weight % of some nitrate ester propellants
Ingredients Single base Double base propellants Triple base propellant propellant
1 2 3 4 Cellulose nitrate Glycerol trinitrate Nitroguadinine Trinitrotoluene Dibutylphthalate Diphenylaraine N, N'-diethyl-N, N'-di-phenylurea N, N'-dimethyl-N, N'-di-phenylurea Vaseline Potassium cryolite Potassium nitrate Potassium sulphate Graphite Moisture Solvent Calcium carbonate 91.0 6.2 0.6 0.5 0.05 1.2 0.45 55.4 40.7 0.2 1.8 0.4 0.8 0.2 0.5 82.1 9.3 5.3 0.6 0.8 0.5 0.5 0.15 0.75 19.8 20.0 56.0 3.6 0.3 0.2 0.1 1.4. Propellant additives Stabilizers
An important group of propellant additives are the stabilizers. As has already been discussed in the introduction the function of the stabilizer is to bind the evolved nitrogen dioxide and its acids and thus to prevent the acceleration of the degradation process.
2nitrodiphenylamine, N, N'diethylN, N'diphenylurea (centralite I) and N, N ' -dimethyl-N, N'-diphenylurea (centralite II). In the next section, which deals with the degradation process of nitrate ester propel-lants, the action of the stabilizers is further discussed.
Plasticizers
For the gelatination of cellulose nitrate, originally only solvents (acetone or an ether-alcohol mixture) were used. To-day also non-volatile plasticizers are used such as diethylphthalate, dibutyl-phthalate, diamyldibutyl-phthalate, dinitrotoluene and centralite. It has been discussed before that centralite is also used as a stabilizer. Many propellant additives have raore than one function. These plasticizers reraain in the propellant. In general they also reduce the hygro-scopicity of the propellant. The gelatinizing effect of glycerol tri-nitrate on cellulose tri-nitrate has already been raentioned before.
Coatings
T h e burning properties of the nitrate ester propellants can be adjusted by coating the propellant with a moderator, e.g. dinitro-toluene or centralite. W h e n applied to the surface of the propellant these substances lower the linear burning rate at the outside of the grain as compared with that of the main body of the propellant.
Nitrate ester propellants are often coated with graphite. It gives the propellant grain a smooth surface which facilitates the auto-matic loading of the cartridge since it permits the propellant to flow freely through the loading machines.
It prevents also the formation of electric charges at blending and loading.
Anti-flash agents
In order to reduce the chance of a flash, anti-flash agents are often added. These agents, normally potassium salts, are mixed with the propellant or added separately to the propellant loading. Potassium sulphate, potassium bitartrate, potassium cryolite and potassium nitrate are commonly used.
1.5. Degradation of nitrate ester propellants
1.5.1. Denitration
The knowledge of the slow degradation process of nitrate ester propellants stored under the circumstances in practice is scarce. T h e degradation is complex since it involves a great number of siraultaneous and consecutive reactions.
It is generally assumed that the denitration of the nitrate ester at storage is due to hydrolysis and thermal decompositions) i s ) . Both types of reactions result in the release of the R - N O 2 bond. This denitration reaction results in the formation of nitrogen dioxide and its acids, and in the formation of organic fragments. T h e hydro-lysis of the nitrate ester propellants is autocatalyzed by the acids. Moreover, the nitrogen dioxide and its acids oxidize the nitrate esters and their organic fragments. In these consecutive reactions nitrogen dioxide is reduced to nitric oxide, nitrous oxide or even to nitrogen. As long as gaseous oxygen is present nitric oxide is oxi-dized to nitrogen dioxide and the oxidation-reduction sequence continues until nitrogen dioxide is bound by the stabilizer.
T h e stabilizer is converted into nitro- en nitroso derivatives i^) 20). W h a t sort of derivatives are formed depends on different factors as for instance the extent of degradation and the amount of stabilizer originally present in the propellant.
1.5.2. Influence of free oxygen on the degradation
W h e n a propellant is stored in a gastight container the amount of free oxygen available to the propellant (often in granular form) is limited. It is dependent on the free space in the container; thus also on the degree of filling of the container.
The role played by the free oxygen in the degradation process of the nitrate ester propellants has been investigated by Boyars and G o u g h 2 i ) . These authors observed the variation in pressure, re-sulting from the evolution of gaseous degradation products. T h e aging of the propellant was carried out at 110 °C in a gastight container. T h e confining atmosphere was either nitrogen, air or oxygen. In Figure 1.1 the results of Boyars and Gough are pre-sented for a double base propellant.
^ 1 2 0 -E E Q9 0 6 0 30 0 -B / I / n i t r o g e n / / / oxygen / consumed / 1 1 2 t ~~~~ 6 /TL / a i r
1
Stabiliser consumed / a / 1 10 JUL /oxygen / 12 t ( h r ) Figure 1.1.Pressure variation o[ a double base propellant containing 1 % centralite I, stored at 110 °C in a gastight container under nitrogen, air and oxygen,
according to Boyars and Cough ^^)
F o r t h e interpretation of their results B o y a r s a n d G o u g h a s s u m e d t h e following simplified reaction scheme:
INITIAL DENITRATION
nitrate ester - ^ d e c o m p o s e d n i t r a t e ester -i- N O 2 ( I )
SUBSEQUENT OXIDATION REACTION
N O 2 + decomposed n i t r a t e ester ->- N O , N g O , N o , C O , C O g ,
H 2 , H 2 O a n d organic fragments (II)
STABILIZER REACTION
N O 2 + stabilizer - > n i t r o - a n d nitroso-derivatives ( I I I )
OXYDATION OF NITRIC OXYDE BY OXYGEN
If the rate of gas removal by reaction (III) and (IV) exceeds the rate of gas evolution due to reaction (I) and (II), a decrease in pressure is the result. This is the case as long as free oxygen and stabilizer are present.
If no free oxygen is present, the pressure increases from the be-ginning as is shown by curve I. If sufficient stabilizer is present the pressure decreases until the free oxygen is consumed (see curve I I ) . However, when sufficient oxygen is present (see curve III) the pressure decreases until the stabilizer is depleted.
1.5.3. Influence of the temperature on the degradation
Recently Frey2.2) studied the influence of the temperature on the rate of degradation of several nitrate ester propellants. T h e author measured the rate of stabilizer depletion between 5 0 - 100 °C. T h e aging was carried out in stoppered weighing bottles.
It turned out that the rate of stabilizer depletion is independent of the stabilizer content and obeys the equation for a zero order reaction:
d (stabilizer) E
= A exp ( ) (1.1.) dt '^ '^ R T ^
where t = time, A = constant, E = apparent activation energy of the stabilizer depletion reaction and T = absolute temperature.
Frey also measured the time to reach a distinct degree of con-version at different temperatures. Loss of weight, the appearance of nitrogen oxides and the depletion of stabilizer were used as the indications for the degree of conversion.
Frey concluded from his experiments that the time (t) to reach a certain degree of conversion may be described by
E
t - i = B e x p ( ) (1.2.) ^ R T ^
where B is a constant depending on the degree of conversion in-volved.
For the investigated nitrate ester propellants, Frey found E values between 27 - 35 kcal/mol.
Equation (1.1.) and (1.2.) are in line with Frey's proposition that the initial denitration reaction (I) is the rate determining step of all consecutive reactions II, III and IV.
1.6. Stability tests
A great number of tests has been developed to check the chemical stability of the nitrate ester propellants.
A survey of the stability tests is given by Beach 23). Stability tests generally involve an accelerated aging of the propellant and the examination of distinct degradation phenomena or products. The accelerated aging is achieved by storing the propellant at an elevated temperature. Aging temperatures differ for the different stability tests. T h e majority of tests are carried out at temperatures between 60 and 140 °C. Most stability tests performed nowadays are based on the assessment of one or raore of the following phenomena: — the rate of stabilizer depletion
— the rate of gas evolution — the loss in weight
— the lapse of time until the appearance of the nitrogen oxide gases is detected.
As has already been discussed in the introduction the interpretation of the results of these stability tests is based on the comparison of the degradation phenomena observed with those of reference propellants. If a too large deviation in the results of the stability test is observed, the propellant is considered insufficiently stable and will not be accepted.
CHAPTER 2
T H E O R Y O F S E L F - H E A T I N G
2.1. Introduction
Already in 1884 V a n 't Hof f 24) postulated that a thermal explosion will occur when the rate of heat generation in a heat-generating substance continues to exceed the rate of heat loss to the surround-ings. This postulation was first described in raatheraatical form by Frank-Kamenetzkii 25) v/ho presented the criteria for self-ignition of substances with a heat generation according to
q = C e x p ( - ^ ^ ^ ) (2.1.) where q = rate of heat generation per unit of mass, C = constant, E =
activation energy, R = gas constant, and T = absolute temperature. In literature 26-33) jt is often assumed that Equation (2.1.) holds for nitrate ester propellants. The theory of Frank-Kamenetzkii has been extended by many investigators e.g. Thomas 3*) and Kim-bara and Akita 35). For a good understanding of the factors which govern the self-heating process, the theories of the authors mention-ed will be discussmention-ed in this chapter. Also a description is given of the experiments presented in literature which deal with the deter-mination of the quantities of C and E of Equation (2.1.) for dif-ferent types of propellants.
2.2. Theory of thermal explosion
2.2.1. Heat balance
If it is assumed that the heat transport in a homogeneous, isotropic heat-generating substance takes place by conduction only, the tem-perature distribution as a function of time is described by the equation:
3T
Q^c—— = X V 2 T + Q,q (2.2.)
o t
self heating heat loss heat generation Q^, = bulk density of the heat-generating substance
c = specific heat of the heat-generating substance
X = heat conductivity of the heat-generating substance
V 2 = Laplacian operator
The differential Equation (2.2.) is of the second order in space co-ordinate, which accounts for two integration constants in the general solution. These constants are determined by the boundary conditions which are derived from the storage conditions under con-sideration.
2.2.2. Criteria of self-ignition of Frank-Kamenetzkii
In 1939 Frank-Kamenetzkii 25) was the first who described the cri-teria for self-ignition mathematically. He assumed the following sim-plifications.
a. The physical quantities of the homogeneous heat-generating substance, Q^^, C, X, are independent of teraperature and degree of conversion. The heat transport takes place by conduction only.
b. T h e heat generation of the substance is due to a zero order
reaction. Thus Frank-Kamenetzkii considered a substance with a heat generation in accordance with Equation (2.1.).
c. The heat-generating substance has a symmetrical shape: a sphere, an infinite long cylinder and an infinite large slab.
d. At t = 0 the temperature of the substance is homogeneous and
equal to the ambient temperature.
e. T h e temperature of the surface is constant and equal to the sur-roundings.
W i t h respect to assumption c, differential equations, equivalent to Equation (2.3.) are
3 T 3 2 T m 3 T E
py,c = X ( + • ) + Pb C exp ( ) (2.4.) ^ 3 t ^ 3 x 2 X 3 x ^ ^ ' ^ ^ R T ^
with m 1= 0 for a slab, ra = 1 for a cylinder, m = 2 for a sphere, X = distance co-ordinate frora the center.
T h e boundary condition regarding assumptions d is
T = T^ for 0 ^ X ^ r at t = 0 (2.4.1.) T h e next boundary condition is given by assumption e
T = T^ at x = r (2.4.2.) where r = characteristic dimension of the substance, the half
thick-ness of the slab, the radius of a cylinder or a sphere.
Frank-Kamenetzkii introduced an approximation with respect to the heat-generation term making Equation (2.4.) more tractable.
E
H e developed the argument of the exponential term into a T - T ,
Taylor series of powers of .
T - T , ( T - T J 2
+-R T +-R T , T 2
T - T ,
Frank-Kamenetzkii ommitted powers of greater than one ^ a
since T - T^ « T„, as can be noticed later, and thus
(T - T J
c
R T R T ^ RT^2 Hence,
exp ( - - ^ ) ^ exp ( - ^ ) exp ( ( T - T j ^ } (2.5.) Introducing this approximation in Equation (2.4.) gives
3 T X 3 2 T m 3 T E E ( T - T J
= — ( + — • ) + C exp ( ) exp ( \ (2.6.) 3 t Ob ^ 3x2 X 3 x ^ ^^ R T / ^ ^ R T , 2 f ^
Frank-Kamenetzkii introduced the dimensionless variables E 0 = ( T - T J - — ^ (2.7. (2.8.) (2.9.) =
(T
c z- T J
/ (?br2 X RT^2 • t and r2E^,, E d = ^ C exp ( ) (2.10.) /IRT,2 "^ ^ R T , ^ Then Equation (2.5.) becomes3 0 32 0 m 3 0
= 4- + (5 exp 0 (2.11.) 3 T 3 z2 Z 3 z
with the boundary conditions in accordance with (2.4.1.)
0 = 0 at T = 0 for 0 ^ z ^ 1 (2.11.1.) and (2.4.2.)
0 = 0 at z = 1 (2.11.2.) Frank-Kamenetzkii considered all stable situations where the
tem-perature distribution of the substance does not vary with time any
3 0
more. In that stationary state situation the term = 0 and
Equa-3 T
tion (2.11.) becomes
d2 0 m d 0
H- (5 exp 0 = 0 (2.12.) d z2 z d z
Equation (2.12.) describes the temperature distribution when a stationary state has been reached. In that situation the total heat generation in the substance is equal to the heat loss to the sur-roundings.
Frank-Kamenetzkii showed that a solution of Equation (2.12.) is only possible if 3 is smaller than or equal to a critical value d^,
i.e. the criterium for the possibility of reaching a stationary state is
If (5 > Sc, Equation (2.12.) cannot be solved and no stationary state is possible, hence the temperature of the substance continues to rise and a thermal explosion will be the result. For ra = 0 (a slab), Frank-Kamenetzkii calculated d,. analytically from the Equations (2.12.), (2.11.1.) and (2.11.2.). This resulted in d, = 0.88. He calculated also lor the critical state the corresponding value of the reduced temperature in the center (z = 0 ) , which gives (0o)e = LIO. For m = 1 (a cylinder) and m = 2 (a sphere) Frank-Kamenetzkii
cal-culated óp and (0o)c numerically. T h e calcal-culated values of 3^. and (0o)c are listed in Table 2.1.
TABLE 2.1.
Critical values of d and @Q for several shapes
slab cylinder sphere <3, = 0.88 d, = 2.00 Ó, = 3.32 (0o)c = 1-10 (0o)o = 1-38 (0o)c = 1-61
2.2.3. Temperature distribution as a function of time
In 1960 Kimbara and Akita 35) described an approximate solution of Equation (2.3.) taking into account that Frank-Kamenetzkii's assumption of an infinite heat-transfer coefficient h for the heat transport from the surface of the substance to the surroundings generally does not agree with the storage conditions in practice. This implicates that the boundary condition given by Equation
(2.4.2.) has to be modified into 3 T
h ( T - T J = - X at X = r (2.4.3.) 3 x
In the same way as Frank-Kamenetzkii has done, Kimbara and Akita transformed Equation (2.4.), (2.4.1.) and (2.4.3.) into a di-mensionless form, resulting in Equation (2.11.), the boundary con-dition (2.11.1.) and
3 0
a@ + = 0 at z = 1 (2.11.3.)
3 z respectively.
Here a is a dimensionless parameter hr
Equation (2.11.) with (2.11.1.) and (2.11.3.) has a solution of the form © = f (z, Ó, a, T ) . Kimbara and Akita have shown that for 0 ^ 0c this function can be calculated analytically by substituting the term I + fi& for the term exp 0 in Equation (2.11.), where
/3 = 1 + a, ( 0 o ) , + ao (0o),2 + ag (0o),3 (2.15.) and Zi, 32 are constants, depending on the shape of the heat-gene-rating substanc and (0o)s is the reduced temperature in the center for the stationary state.
Kimbara and Akita calculated 35) the values a^ and a2 for a sphere, using the results of Frank-Kamenetzkii with respect to the values
d^ and (0o)c for a sphere, and obtained:
)8 = 1 + 0.3691 (0o)s + 0.0908 (0o),^ (2.16.) Introducing the term Ï+/3Q for exp 0 in Equation (2.11.) gives for a sphere (m = 2 ) :
3 0 3 2 0 2 3 0
d T o z-^ z 0 z
S t a r t i n g from t h e condition 0 = 0 at T = 0, Kimbara a n d A k i t a calculated from E q u a t i o n s ( 2 . 1 7 . ) , (2.11.1.), (2.11.3.) a n d (2.16.) t h e t e m p e r a t u r e distribution as function of time, d a n d z.
T h i s resulted in 1
0
P
a sin Do z
z [Do cos Do -I- ( a - I) sinDo]
4aDo2 X p °o sin Do s i n D „ z X 2 exp (Do2 - D„2) r (2.18.) „ = , D , ( 2 D , - s i n 2 D J ( D „ 2 _ D o 2 ) 2 where D , = V('5/Ö) (2.18.1.) and t a n g D „ = ^ ; n = 1 2 ( D „ _ i < D J (2.18.2.) a — 1
In accordance with Equation (2.18.), the temperature distribu-tion in the stadistribu-tionary state (t = oo and d ^ è,.) is presented in Figure 2.1, for several values of d and a.
For t = oo Equation (2.18.) becomes
0 =
P
a sin DQ Z
z [Do cos Do + ( a - 1) sinDo] (2.19.)
Figure 2.1.
Reduced temperature (&) as a function of the reduced distance co-ordinate (z), in the stationary state of a spheri-cal heat-generating substance, for several values of a and S,
according to Kimbara and Akita ^^). Solid curves a = oo, dashed curves a = 15.
W i t h Equation (2.18.) Kimbara and Akita calculated also the re-duced center temperature 0o as a function of T for several valus of
d and a.
For z = 0 Equation (2.18.) becomes
©„ = « D p Do cos Do + (a - 1) sin Do - 1 4 a D o 2 00 • 2 sin D„
P
( 2 D „ - s i n 2 D J ( D , 2 - D o 2 ) exp (D„2 - D„2) r (2.20.)The results are shown in Figure 2.2.
Figure 2.2.
Reduced temperature of the center (BQ) as a function of the reduced time (T) for a spherical heat-generating substance for several values of S and a,
according to Kimbara and Akita ^^). Solid curves a = oc, dashed curves a = 15.
2.2.4. Critical state
Figure 2.2. shows that up to a distinct maximum value of ó (<5c), a stationary state of the temperature distribution is possible. The value of df. for a = oo refers to the 3^ value obtained by Frank-Kamenetzkii. It can be noted from figure 2.2. that the value of 3^ decreases for lower values of a.
Thomas 34) calculated analytically for a slab, a cylinder and a sphere, 3^ as a function of a. The results are presented in Figure 2.3.
Each curve divides the graph in two regions: above the curve (Ó > <5e) a super critical region and under the curve a sub critical region (ó < óp). A critical situation exists when 3 has values lying on the curve {3 = 3^).
6c c J O -2.0 J l.C -B| / c ^--^ ^ super _ ^ _ _ ^_ critical ( 6>6c) c r i t i c a l ( 6 = 6 c 2 _ _ _ - —
^_,^^^rrr
c r i t r c a l ( 6 = 6c' criticat(Ö = 6c) a — - oo sphere sub critfcaH 5<6c) a — - oo 6 c — * 2.0 " cylinder a m. oo 6c—*088 slab Figure 2.3.Critical value of the dimensionless group of Kamenetzkii (Se) as a function of the dimensionless group of Biot (a), for a heat-generating substance in the form of a sphere, a cylinder and a slab, according to Thomas ^^).
In Figure 2.3 point A represents a sub critical storage situation for a spherical heat-generating substance since 3 < d^..
W h e n the spherical substance is supposed to be covered with an insulating material, so that the heat-transfer coefficient (h) is de-creased for instance with a factor one fifth, then a storage situation arises corresponding to point B, since the value of 3 remains constant
{3 is not a function of the heat-transfer coefficient h) and a is
decreased with a factor one fifth (a •= ) . From Figure 2.3 it
follows that the supposed decrease in h results in a super critical state. If it is supposed that the teraperature of the surroundings (T^) is lowered to such an extent that the value of 3 is halved
(c
r 2 E
e\>
; . R T 2 C e x p . - R T , then a storage situation exists corresponding to point C and again a sub critical state is obtained. Doubling the quantity of the spherical propellant results in an in-crease of the radius with a factor 2''^. This corresponds with point D since 3 increases to the square with r, and a varies linearly with r. Thus in the example considered, doubling of the quantity results in a super critical state.
2.3. Rate of heat generation as a function of temperature
In literature only a few experiments have been described dealing with the influence of the temperature on the rate of heat generation of nitrate ester propellants.
Gross and Amster ^) investigated a double base propellant. They measured the rate of temperature rise of the propellant stored under adiabatic conditions. During the measurement the gaseous degra-dation products could escape out of the sample holder. Gross and Amster plotted the logarithm of the rate of temperature increase
d T
(log ) as a function of the reciprocal value of the absolute d t
1
temperature ( ).
As a straight line was obtained for the investigated temperature range of 1 2 5 - 1 5 5 °C, it was concluded that
q (= c ) = C exp ( - ) (2.21.)
^ d t ^ ^ R T ' Gross and Amster calculated from their measurements:
C = 1.72 X 1022 W / k g and E = 38.8 kcal/mol.
Adiabatic experiments have also been carried out by Smith and Velicky36). They determined the C and E value of a composite propellant with cellulose nitrate as a binder. T w o experiments were carried out. In the first experiment the initial teraperature of the bath and the assembled container with sample was approximately the same and amounted to 103 °C. Thirty-one minutes were required to bring the system to thermal equilibrium. After an additional period
d T of 88 minutes a temperature rise was noticed. A plot of
d t 1
against resulted in a straight line for the investigated terape-rature range of 1 0 3 - 1 6 0 °C.
W i t h an estimated value for c= 1.46x10-3 J / k g ° C , Smith and Velicky calculated that C - 2 4 0 x IO22 W / k g and E = 3 8 . 0 kcal/mol. In the second experiment the initial temperature of the bath and the assembled bomb was 25 °C. T h e thermal equilibrium at 108 °C was reached after 33 minutes and after an additional 40 minutes
the temperature rise set in. Again a straight line was obtained. The authors calculated C = 103 x 1022 W / k g and E = 34.75 kcal/mol. In both experiments the propellant sample was positioned in a non-gastight container.
Gross and Robertson 37) calculated from their adiabatic experi-ments between 1 3 5 - 170 °C the C and E values of gelatinized cellulose nitrate. They obtained C = 840xl022 W / k g and E = 42.0 kcal/mol.
2.4. Discussion
Frora the foregoing it follows that the self-heating process is not exclusively governed by the chemical stability of the propellant (C, E) but is also dependent on its physical properties (c, X, Q<a) and on the storage conditions (r, h, T^, t ) .
T o what extent the discussed mathematics on the self-heating process can be applied to examine the self-ignition hazard of nitrate ester propellants is one of the subjects of this thesis. In this respect special attention will be given to the validity of Equation (2.1.).
CHAPTER 3
T H E I S O T H E R M A L H E A T - G E N E R A T I O N M E T E R
3.1. Principle of operation
The heat-generation meter (Fig. 3.1.) enables the measurement of the rate of heat generation of a sample (1) stored under isothermal conditions.
The sample vessel (2) is positioned in a cylindrical holder (3) which is placed on a disc (4). The greater part of the heat generated by the sample passes through the disc and causes a temperature difference between the upper surface and the base of the disc.
This temperature difference is measured by means of a thermo-pile which consists of a large number of thermo-couples, which are connected electrically in series.
T h e thermo-electrical output (5) is amplified (6) and recorded (7).
Figure 3.1. Heat-generation meter
15. Inert material
3.2. Heat-flow disc
T h e heat-flow discs (4) used in the heat-generation raeters are de-veloped and manufactured b y the Technical Physical Department T N O - T U * ) . The discs are fixed (with silicone rubber) between the cylindrical holder of the measuring unit (3) and the aluminium block (9).
In principle the heat-flow disc consists of a spiral of constantan wire wound on a pliable flat core of teflon. The windings are partially silvered, each winding thus forming a thermo-couple, (Figure 3.2.).
The teflon core with the partially silvered wire is spiralized simul-taneously with an insulating teflon tape without wire. In this way a disc (see Fig. 3.3.) is formed which is kept together by a teflon ring.
Figure 33.
Spiralized core with thermo-couples.
For electrical insulation the upper surface and the base of the disc are covered with thin sheets of teflon.
For detailed information on the construction and operation of the heat-flow disc, reference is made to De Jong 38) and Van Ooyen 39).
Technical data of the heat-flow disc are listed in Table 3.1. TABLE 3.1.
Technical data of the heat-f low disc
Outer diameter of teflon ring Inner diameter of teflon ring Height of spiralized core Total number of thermo-couples Diameter of constantan wire
Overall heat conductivity of heat-flow disc Electric resistance
Maximura operation temperature
56 mm 53 mm 2 ram 2100 60 X 10-6 m 0.285 W / m ° C 1050 a 200 ° C 3.3. Temperature control
The thermostated cylindrical roora (8) is enclosed in an aluminium block (9) (see Fig. 3.1.). This block is surrounded by a layer of polyurethane foam (10) which is enclosed in a steel house (11). The temperature of this house is kept constant by circulating a liquid (12) of constant temperature through bifilar tubes.
In order to reduce the heat loss of the heating-liquid to the surroundings, the heat-generation meter is insulated with glass wool (14). T h e heating-hquid is pumped (1 1/min) from an ultra thermostate. The temperature of the heating-liquid rises from mi-nimum to maximura (about 0.1 ° C ) in about 20 seconds. Then, in about the same time, the teraperature falls off to its minimum. These temperature fluctuations are effectively attenuated in the internal room due to both the heat capacity of the aluminium block and the low heat conductivity of the foam layer.
T h e temperature of the internal room is measured by a thermo-couple (13), positioned between both heat-flow discs.
T h e time constant ( T ^ ) of the heat-generation meter amounts to about 13000 seconds (3.6 hours) * ) .
Information on the fluctuation of the temperature in the internal
room of the heat-generation meter (AT^m) due to the fluctuation of the temperature of the heating liquid ( A T J ^ ) can be obtained by
(ATo„J
calculating the amplitude ratio of the temperature disturb-( A T J J
ance, assuming a sinosiodal temperature variation of the heating-liquid applied to a first order system. Then for the amplitude ratio holds 40).
ATo„t _ 1 ATi„ " l / ( l + c o 2 r G 2 ) '
where w is the frequency of the temperature fluctuation of the heating-liquid.
2ji
From the foregoing it follows (ATJ^ = 0.1 °C, (x» = rad s e c - i , TQ = 1 3 0 0 0 seconds) that ATo„t = 5 x 10-5 °C.
It should be noted that the slow temperature variations of the ambient temperature will not be attenuated effectively in the internal room. In general the temperature of the surroundings varies too slowly with respect to the value of the time constant of the heat-generation meter. Moreover, these fluctuations are often quite large. 3.4. Compensation
Slow temperature variations of the ambient temperature interfere with the output of the heat-generation meter, since each variation of the ambient temperature is accompanied by a heat flux through the heat-flow disc of the measuring unit. T o solve this problem two identical measuring units are fitted in the heat-generation meter.
One is the measuring unit containing the sample (1), the other contains an inert material (15). If it is assumed that the thermal diffusivity and heat conductivity of both measuring units are equal the internal temperature fluctuations effect both measuring units simultaneously and equally. As both heat-flow discs are connected electrically in counter phase, the effect of the variation of the ambient temperature on the output of the heat-generation meter is
compen-sated.
As can be noticed from Figure 3.4 the measuring unit for the sample and the measuring unit for the inert material are located sym-metrically to the center of the bottom of the internal room. This is done to achieve an equal effect of the variation of the ambient tem-perature on both measuring units.
h A £ : ^
Figure 3.4.
Measuring units ^or the sample and the inert material in the internal room of the heat-generation meter.
3.5. Sample vessel
T h e sample vessel (and the vessel for the inert material) have a volume of 60 cm3 (diameter 5.1 cm). T o assure gastightness the vessel is covered with a teflon sheet and metal foil, which are secured by means of a gland and bolts (see Fig. 3.5.). The thickness of the metal foil is chosen in such a way that it can act as a bursting plate. The teflon foil (located under the metal foil) protects the
Figure 3.5.
bursting plate from corrosion and acts as a sealing foil). In each vessel two concentric stainless steel cylinders are positioned. These cylinders reduce the v/arming-up time after insertion the vessel in the heat-generation meter and the self-heating of the sample during the measuring period, as they increase the rate of heat transport
of the saraplc to the sample vessel. T h e cylinders have slits to fit in the grooves located in the bottom of the vessel.
3.6. Calibration
T o calibrate the heat-generation meter a calibration vessel has been constructed: around the inner cylinder of a sample vessel an insulated constantan wire has been wound (diameter 90 ^m, electric resistance 432 n ) .
For calibration the heat-generation meter, the calibration vessel (filled v^ith glass spheres) is brought into the measuring unit for the sample and the ends of the calibration wire are connected with gilded contact pins to the electric circuit (5) of the heat-generation meter
(see Fig. 3.4.).
By means of a direct electric current through the calibration wind-ing a known quantity of heat can be introduced. It has been found that there is a linear relation between the heat introduced ( $ ) and the output (U) of the heat-generation meter. So is a constant, which is called the "calibration constant".
The calibration constants of a number of heat-generation meters used in the experiments are presented in Table 3.2.
TABLE 3.2.
Calibration constants of the heat-generation meters at different operation temperatures
Heat-generation meter 1 2 3 4 5 6
Operation
temperature ( ° C ) 50 60 70 70 80 90 100 110 Calibration constant
3.7. Accuracy
The normal procedure for measuring the heat generation of a sample is to put two vessels simultaneously in the heat-generation meter. One vessel is filled with the sample, the other vessel with glass spheres (diameter 100 ,«). Before the vessels are positioned in the heat-generation meter the vessels are heated up to 30 ° C . After insertion, the vessels are simultaneously warmed up to the operation teraperature of thci heat-generation meter ( 5 0 - 1 1 0 ° C ) . During the insertion procedure the cover of the heat-generation meter is open and the meter is out of therraal balance. Therefore a short time after the insertion procedure the output of the heat-gene-ration meter does not correspond exactly with the rate of heat generation of the sample in spite of the compensation system. Infor-raation on the error of the output of the heat-generation meter during this initial period is obtained by measuring the output of the meter after insertion of two vessels, both filled with glass spheres. The output of the heat-generation meter after insertion is called the "zero line". Both vessels were conditioned before insertion at 30 °C.
Zero lines were measured at several operating temperatures. T h e results of these measurements are gathered in Table 3.3.
From the measurements it can be noticed that the standard devia-tion (Su) of the output decreases with increasing measuring time. T h e low reproducibility of the zero line in the initial period of the measureraents is due to the poor reproducibility of the insertion procedure.
The rate of heat generation per unit of raass (M) of sample is calculated with
U • C^
q = - — - ^ (3.1.) where q = rate of heat generation per unit of mass
U = output of heat-generation meter
Cn = calibration constant of heat-generation meter n Msa = mass of sample
Frora Equation (3.1.) it can be noticed that the maximum error of q (in % ) is the summation of the maximum error (in % ) of U, C„ and M,^.
TABLE 3.3.
Output U (juV) and standard deviation S^^ (juV) of heat-generation meters 1-6, in the period after insertion dummy samples (zero-lines)
Heat-generation raeter Operation teraperature (°C) Time (hrs) after insertion 4 6 8 12 18 24 36 48 1 50 u S„ - 8 8 - 4 4 - 3 3 - 2 2 - 0 . 4 0.3 - 0 . 2 0.3 - 0 . 1 0.3 - 0.2 0.2 2 60 u S„ - 9 10 - 4 4 - 1 3 - 0 . 9 1 - 0 . 3 0.3 - 0.2 0.3 - 0 . 1 0.3 - 0 . 2 0.3 3 70 u S„ - 1 3 8 - 6 5 - 3 2 - 1 1 - 0.6 0.4 - 0.4 0.3 - 0.4 0.3 - 0.4 0.3 4 70 U S„ - 1 6 7 - 7 3 - 4 2 - 2 1 - 1.2 0.5 - 0.4 0.4 - 0.3 0.3 - 0.3 0.3 5 6
1
80 90 U S„ - 1 8 10 - 4 6 - 2.5 2 - 0.9 1 - 0.5 0.6 - 0.4 0.4 - 0.3 0.4 - 0.3 0.4 u S„ - 1 8 23 - 8 7 - 3 4 - 2 2 - 1 1 - 0.3 0.4 - 0.3 0.4 - 0.3 0.4 100 U S„ - 2 0 30 - 1 1 13 - 6 8 - 4 4 - 2 1 - 0.6 0.5 - 0.4 0.5 - 0.4 0.5 110 U S„ - 3 0 40 - 1 2 20 - 9 8 - 4 3 - 1.1 0.7 - 0.5 0.5 - 0.3 0.5 - 0.4 0.5 lj>.The maximum error (in %) of the output is equal to the summation
of the reading error (assumed to be 1 % ) plus the maximum error of the zero line in %, which is a function of the measuring time. It is assumed that the raaxiraum error of the zero line (U') is three times the standard deviation (S,,) viz. U ' = 3 S^. So the raaximum error of U is
U '
(1 + 100) % (3.2.) U
As already has been indicated in Tabic 3.2, the maximum error 0.1
of the calibration constant is ± O.I, or • 100 % ^ 0.7 %.
15
T h e mass of the sample is measured with an accuracy of ± 1 rag. Since the sample mass is more than 10 gram the maximum error of
the mass is less than 0.01 %
From the foregoing it follows that the maximum error of q is U '
CHAPTER 4
H E A T G E N E R A T I O N
O F N I T R A T E E S T E R P R O P E L L A N T S
4.1. Introduction
In Chapter 2 a mathematical description has been given of the self-heating process of substances which generate heat according to
q = C e x p ( - ^ ^ ) (4.1.)
In this Chapter the validity of Equation (4.1.) is verified for four nitrate ester propellants by measuring the rate of heat generation as a function of time under isothermal conditions at different tempera-tures. These heat generation/time curves are called "thermograms" The measurements have been carried out with the heat-generation meters described in Chapter 3. If Equation (4.1.) holds for the nitrate ester propellants the rate of heat generation (q) as a function of time under isothermal conditions must be constant.
In this Chapter also experiments are described concerning the investigation of the factors governing the rate of heat generation of nitrate ester propellants.
4.2. Experiments
The chemical composition of the four investigated propellants is given in Table 1.1. The investigated propellants have a granular form. In Table 4.1. some chemical and physical properties of the investigated propellants are listed.
TABLE 4.1.
Chemical and physical properties of the propellants 1-4.
Propellant
Specific heat ( l O M A g )
Specific gravity pg (kg/m^)
Bulk density pb (kg/m^) Calorimetric value Qig (103 J/kg) Heat-conductivity coefficient X ( W / m °C) Granular form 1 12.2 ± 0 . 2 1562 ± 5 963 ± 5 3750 ± 10 88 ± 4 tube 2 14.9 ± 0 . 2 1691 ± 5 740 ± 4 5030 ± 10 78 ± 3 flake 3 13.2 ± 0 . 2 1629 ± 5 983 ± 5 3760 ± 10 94 ± 4 sphere 4 14.1 ± 0 . 2 1678 ± 5 899 ± 5 3710 ± 10 98 ± 4 irregular Reference *) «) 4 1 ) 4 1 ) 4 2 ) Dimensions (lO--* m) Outer diameter 0.84 Inner diameter 0.15 Length 0.15 Thickness 0.22 Breadth 1.41 Length 1.50 Diameter between 0,35 and 0.71 Cut into fragments passing a sieve with holes of 0.6 mm diameter and retained by a sieve with holes of 0.4 mm diameter
*) The specific heat has been measured with the isothermal heat-generation meter. The measure-ments are described in Appendix II.
T h e rate of heat generation as a function of time has been measured under isothermal conditions at temperatures between 6 0 - 110 °C. All measurements were carried out in duplicate. T h e rate of heat generation at 80 °C and higher was measured con-tinuously during the whole degradation process.
T h e measureraents below 80 °C were carried out continuously only during the first ten 'days. After this period the rate of heat generation was measured discontinuously. During the intervals between the measurements the samples were stored in stoves, which were kept at the same temperature as the operating terape-rature of the heat-generation meter.
T h e rate of heat generation was measured after about every tenth day of storage in the stoves.
T o raeasure the rate of heat generation after each interval of storage in the stoves, the saraple vessel was inserted in the heat-generation raeter, and after about one day, the output of the meter was measured. As can be noticed from Table 3.3 thermal equilibrium had then been attained.
T h e thermograms obtained are shown in Fig. 4.1 - 4.4. Numerical data of these measureraents are listed in Table 1 - 4 of Appen-dix III. The first conclusion that can be drawn from the thermo-grams is:
T h e rate of heat generation is not constant with time for the investigated propellants, stored under the pertaining experi-mental conditions.
Therefore Equation (4.1.) does not hold for the investigated propellants stored in completely filled gastight containers under isothermal conditions.
As will be demonstrated in section 4.4, analysis of the thermograms of Fig. 4.1 - 4.4 leads to a modification of Equation (4.1.):
q = f(r) exp i--:^) K2.)
where i (y) is a function of the degree of conversion {y).Q
The degree of conversion can be denoted by 7 = ,where
t =°
Q = I q d t, the total amount of heat generated after time t, 0
and
00
Qj^ = I q d t, the total amount of heat generated when degra-0
dation is completed.
From the foregoing it follows that f (7) can be written as F ( Q ) , where F ( Q ) is a function of Q.
Figure 4.1.
Rate of heat generation (q) as a function of time (t) of propellant 1, stored in a completely filled gastight container
under isothermal conditions at different temperatures.
t (days)
Figure 4.2.
Rate of heat generation (q) as a [unction of time (t) o{ propellant 2, stored in a completely filled gastight containe:
I(dais)
Figure 4.3.
Rate of heat generation (q) as a function of time (t) of propellant 3, stored in a completely filled gastight container
under isothermal conditions at different temperatures.
Figure 4.4.
Rate of heat generation (q) as a function of time (t) of propellant 4, stored in a completely filled gastight container
Thus Equation (4.2) may be written as E
q = F ( Q ) exp ( - ^ ^ ) (4.3.) where F ( Q ) is called the "heat-generation factor".
Before discussing the validity of Equation (4.3.) for the nitrate ester propellants, some theoretical considerations will be given.
4.3. Theory of iso-Q-lines
A heat-generating substance is considered to obey Equation (4.3.). For such a substance holds
E
log q = - — — l o g e + log F ( Q ) (4.4.) H 1
From Equation (4.4.) it follows that log q as a function of T is a straight line, if the relevant q values correspond to a same value of Q, i.e. to a same degree of conversion.
Fig. 4.5 shows a log q versus log t graph with three thermograms at T i , T2 and T3 °K of, a hypothetical substance.
In order to find in Figure 4.5 points of equal degrees of conversion (or equal Q-values), the following reasoning is hold.
d Q
For isothermal conditions q = , a n d ' E q u a t i o n (4.3.) can be d t
written as
d Q E
= exp ( ) dt (4.5.) F ( Q ) ^ R T ^
Integration of Equation (4.5.) between Q = 0 at t = 0 and Q = Q at t = t gives where f ( Q ) = t exp (--1^) (4.6.) Q
r'
dQ
f ( Q ) =J
F(Q)
0Equation (4.6.) implicates that a heat-generating substance, obey-ing Equation (4.3.) and stored under isothermal condition at diffe-rent temperatures, has generated an equal amount of heat ( Q ) , when
E
the product t • exp ( — ) has the same value. So for two
heat-R r
generation/time curves measured at T^ and T2 (see Fig. 4.5.) a same amount of heat (Q^ = Q2) has been generated when
t , . e x p ( - ^ ) = t 2 . e x p ( - - ^ ) (4.7.) v/here
tj = reaction time to generate an amount of heat Q j under iso-thermal conditions at T^.
t2 = reaction time to generate an amount of heat Q2 (= Q i ) under isothermal conditions at T2.
Figure 4.5.
Rate of heat generation (q) as a function of time (t) at different temperatures (T) of a hypothetical substance,
Denoting the rate of heat generation at T^, after time tx as qi and the rate of heat generation at Tg after time t2 as qg, then it follows from Equation (4.3.) that
q, '""^-^^
qa E x exp ( J
"^ "^ R T 2
since the times t^ and t2 are so chosen that Q^ = Q2. Combining Equation (4.7.) with (4.8.) gives
q i t2 qa t i
(4.8.)
(4.9.)
So one may state that for the thermograms at different tempera-tures holds that
q • t = constant (= b ) , (4.10.) when the relevant values q and t correspond to a same degree of
conversion.
b is a constant, which value is specific for each degree of con-version.
Equation (4.10.) can be written as
log q = - l o g t + b (4.11.) Equation (4.11) implicates that in a log q versus log t graph
(Fig. 4.5.) points, representing a same degree of conversion (or an equal amount of heat generated) Ue on straight lines. These lines are called "iso-conversion lines" or "iso-Q-lines". From the foregoing it •, follows for example that in Fig. 4.5. the points q^ , t^ ; q^ , t^ and q^ , t^ correspond to the same degree of conversion.
In Fig. 4.6, the relevant q values of the intersections of the heat generation/time curves with each of the iso-Q-lines drawn in Fig.
1
Figure 4.6.
Rate of heat generation (q) as a function of the reciprocal value of the absolute tem-perature ( ) for the intersections of the
T
heat-generation/time curves with the tso»Q-Unes A, B and C of Figure 4.5.
In accordance with Equation (4.4.), the slope (a') of these lines is given by
E
t g « ' =
-R log e (4.12.)
4.4. Analysis of the thennograms 4.4.1. Apparent activation energy
T h e validity of Equation (4.2.) or (4.3.) will be demonstrated for the investigated propellants, using the thermograms presented in Fig. 4.1 - 4.4. In these figures n straight lines have been drawn in accordance with Equation (4.11.). Each line corresponds to a distinct degree of degradation y„, n = I, II, III, etc.
T h e q values of the intersections of the thermograms with each of the straight lines of Fig. 4.1 - 4.4 are plotted logarithmically as a
1
function of in Fig. 4.7 - 4.10 respectively. The numerical values of q and T corresponding with these intersections are listed in Table 5.8 of Appendix III.
As can be concluded from Fig. 4.7-4.10, for the intersections of the thermograms with each straight line n holds that the relevant
1
log q value plotted as a function of is a straigt line. Frora the slope of each line* the E value has been calculated according to Equation (4.12.). T h e E value corresponding to the intersections of the therraogram with straight line number n is denoted as E„.
T h e E„ values thus obtained are listed in Table 4.2.
In the last column of Table 4.2, for each propellant the average value of the corresponding E^ values is listed. From the data of Table 4.2. it can be noted that for each propellant the differences between the E^ and E values are within the maximum error of the experiment. This corresponds with Equation (4.3.), where the appa-rent activation energy E is independent of the degree of conversion.
Figure 4.7.
Rate of heat generation (q) of propellant 1 as a function
of the reciprocal absolute
temperature ( ) for the intersection points of the thermograms with the straight lines number n = I, II, III etc., as presented in Fig. 4.1.
Figure 4.8. Rate of heat generation (q) of propellant 2 as a function of the reciprocal absolute temperature ( ) for the
T
intersection points of the thermograms with the straight lines number n =
/ , II, III etc., as presented
Figure 4.9.
Rate of heat generation (q) of propellant 3 as a function of the reciprocal absolute temperature { ) for the
T
intersection points of the thermograms with the straight lines number n = I, II, III etc., as presented in Fig. 4.3.
Figure 4.10. Rate of heat generation (q) of propellant 4 as a function of the reciprocal absolute temperature ( ) for the
T
intersection points of the thermograms with the straight lines number n = I, II, III etc., as presented
E„ values of propellants 1-4, TABLE 4.2. Propellant 1 2 3 4 Figure 4.7. 4.8. 4.9. 4.10. n = I KJ/mol kcal/mol 1 4 4 ± 6 34 ± 2 1 3 6 ± 7 a ± 2 1 3 4 ± 7 33 ± 2 1 5 3 ± 7 36 + 2 n = II Kl/mol kcal/mol 139 ± 4 33 ± 1 1 4 0 ± 2 33.4 ±0.6 1 4 0 ± 2 33.4 ±0.6 1 4 8 ± 3 35.4 ±0.7 n = III KJ/mol kcal/mol 1 4 2 ± : 2 33.8 ±0.4 141 ± 1 33.7 ±0.4 1 3 9 ± 2 33.2 ±0.5 1 4 9 ± 2 35.6 ± 0 . 5 n = IV KJ/mol kcal/mol 141 ± 1 33.7 ±0.3 141 ± 1 33.6 ±0.3 1 4 0 ± 1 33.5 ± 0 . 3 1 4 9 ± 1 35.6 ± 0 . 3 n = V KJ/mol kc^l/mol 1 4 2 ± 1 33.9+0.3 1 4 0 ± 1 33.4 + 0.3 1 3 9 ± : 1 33.3 + 0.2 1 5 0 ± 1 35.7 + 0.2
Average value of E,,
KJ/mol kcal/mol 1 4 2 ± 1 33.8 ±0.2 1 4 0 ± 1 33.5 ±0.2 1 4 0 +- 1 33.4 ±0.2 1 5 0 ± 1 35-7 ±0.2 U l
4.4.2. Heat-generation factor
Knowing the E value of the propellant, the heat-generation factor as a function of Q can be obtained by measuring the heat generation as a function of time under isothermal conditions at temperature T and plotting
'^ ( ^ F ( Q U a . s a f u n c t i o n o f f q d t ( = Q )
xp (
-R T )
This is shown for the investigated propellants in Fig. 4.11. The curves presented in Fig. 4.11 are obtained frora the measured rate of heat generation as a function of time at 80 °C, as shown
S 5 propellant 3 / / / p r o p e l l a n t 2 propellant U propellant \ —T" SO — 1 — 100 " T — ISO —r-200 Q t 1 0 ^ j / k g ) Figure 4.11.
Heat-generation factor F (Q) as a function of the total amount of heat generated (Q) of propellants 1-4.
