Bogdan FLORCZAK* – Institute of Industrial Organic Chemistry, Warsaw, Poland; Ewelina
BEDNARCZYK, Andrzej MARANDA – Military University of Technology, Warsaw, Poland
Please cite as: CHEMIK 2015, 69, 3, 136–145
Introduction
The initial stage of the technological process of manufacture of
solid heterogeneous rocket propellants involves a mixture of liquid and
solid chemical substances of semi-solid consistency, which as a result
of mixing in specific time and defined temperature becomes dense
liquid mass that can be classified as a highly-filled suspension of nonNewtonian fluid properties. Viscosity of such a suspension depends
on a number of factors including time, temperature and shear rate.
This means that under given experimental conditions only its apparent
viscosity (ηz) can be determined. This property depends on liquid
phase viscosity (ηc) and structural viscosity (ηs) [1].
ηz = ηc+ ηs
(1)
ηs(Φ) = ηc[2.5Φ+ 10.5Φ2+ exp(a+ bΦ)]
(2)
ηs(Φ) = ηcKΦ/(1/Φ – 1/Φm)
(3)
which gives:
ηz = ηc[1+ 2.5Φ+ 10.5 Φ2+ exp(a+ bΦ)]
(4)
ηz = ηc[1+ KΦ/(1/Φ -1/Φm)]
(5)
η(t)= η0ekηt
(8)
where:
η – viscosity, mPas
η0 – viscosity for t = 0, mPas
t – time, s
kη – rate coefficent for viscosity increase (consistency multiplier).
Relationship between viscosity and rotational speed in model
representing viscosity change (η) with time (t) can be obtained
by measurement of propellant suspension viscosity as a function
of time [3].
Relationship of viscosity change versus time and temperature can
be described using the following formula [4]:
lnη(t, T) = lna(T)+ b(T)t
(9)
where a, b – constants determined experimentally.
where:
Φ – relative volume fraction of solid phase in suspension
Φm – maximum relative volume fraction of solid phase in suspension
a, b, K – constants determined experimentally.
The most important factors affecting rheological properties of
rocket propeller suspension are following: time, temperature and
shear rate. Dependence of shear stress on shear rate is usually
described by means of Ostwald-de Vaele rheological model in the
form of power law [2]:
τ = kDn
where:
η – viscosity, mPa s
k – constistency multiplier
n – flow index
R – rotational speed, rpm
and formula [3]:
Slope b(t) and lna(T) change linearly with the inverse of temperature
(1/T), i.e. lna(t) = ma/T+ ca and b(T) = mb/T+ cb (ma, mb and ca and cb are
slopes and free terms of these functions, respectively) [4]. Substitution
of the formulas of these functions in the equation (9) gives:
lnη(t,T) = ma/T+ ca+ (mb/T+ cb)t
(10)
Differentiation of the equation (10) with respect to time and
temperature gives, respectively:
(6)
where:
τ – shear stress, N/m2
k – constistency coefficient
n – flow index
D – shear rate, 1/s.
Consistency coefficient in the formula (6) is a measure of
apparent viscosity, while flow index is a dimensionless parameter that
is a measure of fluid deviation from non-Newtonian fluid. In order
to describe rheological properties of the rocket propeller suspension,
the following formula is used [2]:
(11)
(12)
The presented relationships (11), (12) allow determination of
temperature at which viscosity does not change (by maintaining this
suspension temperature cross-linking reaction is stopped, which
increases the life-time of the suspension).
η = kRn
(13)
and
(14)
(7)
Corresponding author:
Bogdan FLORCZAK – Ph.D., Eng., Associate Professor, e-mail: [email protected]
nr 3/2015 • tom 69
as well as time after which suspension loses its usefulness for
processing:
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Studies of rheological properties of suspension
of heterogeneous rocket propellant based
on HTPB rubber
science • technique
(15)
and then:
(16)
Based on the experimental data from viscosity measurements
of rocket propellant suspension one can determine also its
thermodynamic parameters, i.e. activation energy (Ea), entropy
(∆Sη*), enthalpy (∆Hη*) and free enthalpy (∆Gη*) from Arrhenius
and Eyring equations [4]:
Preparation of propellant suspension
Initial semi-liquid propellant mass without curing agent DDI was
produced in ZPS „GAMRAT” Sp. z o. o. Such a prepared mass was
dosed into NETZSCH planetary mixer (Fig. 1) of 0.5 dm3 capacity. After
introducing appropriate amount of DDI (ratio of equivalent weights
for HTPB and cross-linking agent DDI was 0.9), the mass was mixed
subsequently for 30 min. under atmospheric pressure and for 30 min.
under reduced pressure. The propellant suspension obtained in such
a manner was poured in vacuo (Fig. 1) into the viscosity measurement
vessel of approx. 115 cm3 capacity.
(17)
(18)
where:
T- temperature, K
R – universal gas constant, 8.314 J·mol-1·K-1
k – reaction rate constant
h – Avogadro constant, 6.02x1023
N – Planck constant, 6.62x10‒34
∆Sη* – entropy, J/K
∆Hη* – activation enthalpy, J.
The values of enthalpy and entropy can be calculated from curve
slope and free term of equation (18), respectively, while free enthalpy
can be determined using the following relationship:
∆Gη*= ∆Hη*- T·∆Sη*
(19)
Experimental part
The propellant of the following compositions was studied (Tab. 1).
Table 1
Fig. 1. Station for propellant pouring under reduced pressure (on the
left) and NETZSCH mixer (on the right) (equipment of the Institute
of Industrial Organic Chemistry in Warsaw)
Preparation of cross-linking systems
For the measurements of viscosity of studied rocket propellant
suspensions new generation Brookfield DV-II + Pro Viscometer with
T-D spindle and TC-550 thermostat (Fig. 2) were used. The viscometer
is a rotational measuring instrument and the measurement is performed
by rotation of a measuring tip (spindle) immersed in suspension (Fig. 2).
The spindle is coupled with calibrated spring. Drag force resulting from
the viscosity of studied material retards the rotating spindle and causes
deformation of the spring, which is measured electronically.
Composition of rocket propellant suspension
Component
Content, %
Ammonium chlorate(VII)
70.0
Aluminum dust
15.0
BEFP
0.2
Liquid components (HTPB R45M + DDI + DOA) + additives
14.8
The propellant suspension was prepared using rubber R45M
(manufacturer IPI), as well as the following substances: dioctyl
adipate (DOA) (manufacturer Boryszew Erg S.A.) as a plasticizer,
ρ= 0.925 g/cm3, boiling point: 690 K, acid value: 0.008 mg KOH/g,
volatile substance content at 373 K of not more than 0.08%,
water content of not more than 0.05%, 2-heptyl-3,4-bis(9isocyanatononyl)-1-pentylcyclohexane (DDI) (manufacturer IPI) as
a curing agent, % NCO = 13.79, 2,2’-bis(ethylferrocenyl) propane
(BEFP) (manufacturer Neo Organics) Fe content approx. 23%, H2O
content 0.03%, insolubility in chloroform below 0.02%, ammonium
chlorate(VII) (manufacturer IPI), above 0.4 mm 2%, above 0.25 mm
38%, above 0.125 mm 55%, above 0.071 mm 3%, below 0.071 mm
2%, aluminum dust (manufacturer Benda Lutz), D10 = 3.14 μm,
D50= 6.30 μm, D90= 12.72 μm.
142 •
Fig. 2. Equipment for viscosity measurement
Viscometer was mounted on Helipath drive. The viscometer with
appropriately mounted T-shaped measuring spindle, thanks to such
a solution, can be slowly lowered and lifted during the measurement.
This allows spindle motion (while maintaining its rotation) along a spiral
path inside the studied sample (Fig. 3). This eliminates the error related
to the formation of a kind of “channel” along the spindle [6].
The sample of studied propellant suspension was placed in the
teflon measuring vessel of 115 cm3 capacity (height 70 mm, diameter
nr 3/2015 • tom 69
Moreover, average values of viscosity were determined for each
thirty-minute measurement period (15th minute of each thirty-minute
measurement period). They are presented in Figure 5, while values of
constants η0 and kη are shown in Table 3.
Table 3
Values of η0 and kη versus temperature of the studied propellant
T, K
η0, mPas
kη, h-1
R2
313
245012
0.0989
0.99
318
215287
0.1086
0.99
323
198043
0.1245
0.99
328
142959
0.1854
0.99
333
147826
0.2018
0.99
338
111885
0.277
0.99
Fig. 3. Scheme of spindle motion during the measurement using
Helipath drive [6]
Determination of cross-linking reaction rate constant
The viscosity of propellant suspension during cross-linking was
measured at equal time intervals at six different temperatures (313
K, 318 K, 323 K, 328 K, 333 K and 338 K) with spindle rotational
speed of 4 rpm. Experimental data was approximated by exponential
function (8) η(t) = η0·ekηt by means of least square method with
determination of constants η0 and kη. Relation viscosity vs time for
different temperatures is presented in Figure 4, while values of
constants are shown in Table 2.
Fig. 5. Graph of averaged viscosity change versus time
Determination of thermodynamic parameters
The value of activation energy (Ea) was determined from Arrhenius
equation (15), which describes relationship of reaction rate and
temperature. The activation energy was determined from the curve
slope of the relationship ln kη= f(1/T) presented in Figure 6.
Fig. 4. Graph of relation viscosity (η) vs time (t)
Table 2
Values of η0 and kη as a function of temperature of the studied
propellant
T, K
η0, mPas
kη, h-1
R2
313
244569
0.0966
0.99
318
214188
0.1088
0.90
323
196811
0.1246
0.93
328
141779
0.1848
0.95
333
146597
0.2008
0.94
338
110671
0.2748
0.97
nr 3/2015 • tom 69
Fig. 6. Graph of functions ln kη= f(1/T) and ln kηav= f(1/T)
The value of enthalpy (∆Hη*) and entropy (∆Sη*) of the crosslinking of rocket propellant was determined using Eyring formula
(16) based on slope and free term of the curve representing fuction
ln kη/T= f(1/T) (Fig. 7), respectively.
Using the equation (17) the values of free enthalpy at measurement
temperatures were determined. The values of determined
thermodynamic parameters are presented in Tables 4 and 5.
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46 mm) placed in a smaller thermostat equipped with temperature
sensor. The smaller thermostat was connected to a bigger one using
silicone hoses. The measurements were carried out with Brookfield’s
Rheocalc software and T-D measuring spindle. The Helipath drive
was lifting and lowering viscometer with speed of 2.6 cm/min. Initially,
the measuring spindle was immersed in the propellant suspension
at depth of 8 mm. The studies of rheological properties of the
rocket propellant suspension involved recording value of viscosity as
a function of time and rotational speed of the spindle at the following
temperatures: 313 K, 318 K, 323 K, 328 K, 333 K and 338 K.
science • technique
Table 6
Values of lna, b and coefficient R2
for η
T, K
Fig. 7. Graph of linear functions ln kη/T= f(1/T) and ln kηav/T= f(1/T)
Table 4
Values of determined thermodynamic parameters
Ea, kJ/mol
∆Hη*, kJ/mol
∆Sη*, J/(K∙mol)
for η
36,15 ±4,84
33,44 ±4,83
-158,92 ±14,84
for ηav
37,11 ±4,03
34,41 ±4,02
-155,44 ±12,37
Table 5
Values of free enthalpy
T, K
313
318
323
328
333
338
∆G for η, kJ/mol
83.18
83.98
84.78
85.57
86.36
87.16
∆G for ηav, kJ/mol
83.06
83.84
84.62
85.39
86.17
86.95
Determination of cross-linking inhibition and maximum
technological usefulness time
Based on the slope and free term of curve lnη = f(t) obtained after
logarithmization of the equation (8), the values of lna = lnη0 b = kη
were determined. The presented relations (Fig. 8) of these parameters
and temperature lna = f(1/T) and b = f(1/T) allowed determination of
the following coefficients ma, mb, ca and cb.
for ηav
ln a
B
R2
ln a
b
R2
313
12.407
0.0966
0.99
12.409
0.0989
0.99
318
12.275
0.1088
0.90
12.28
0.1086
0.99
323
12.19
0.1246
0.93
12.196
0.1245
0.99
328
11.862
0.1848
0.95
11.904
0.2018
0.99
333
11.895
0.2008
0.94
11.87
0.1854
0.99
338
11.614
0.2748
0.97
11.625
0.277
0.99
Table 7
Values of coefficient and cross-linking inhibition temperature, as well
as maximum time of technological usefulness
ma
ca
mb
cb
T, K
t, h
for ηav
3284.4
3284.4
-720.73
-720.73
302.5
4.6
for η
3280.6
3280.6
-737.91
-737.91
303
4.4
Effect of shear rate on the viscosity of rocket propellant
suspension
Measurements aiming to verify if the propellant suspension has
thixotropic or anti-thixotropic properties were also carried out.
To this end, relationship of viscosity vs spindle rotation speed changing
in range of 4.5–9 rpm, with 0.5 rpm step, was studied. The results of
measurements are presented in Figures 9–14.
Fig. 9. Graph of η changes as a function of increase and decrease in R
at T = 313 K
Fig. 8. Graphs showing linear relationship lna and b vs inverse
temperature (x = 1/T)
Substitution of numerical values corresponding to ma and mb
gives the value of time t=4.4 h, while for the average viscosity this
gives a result of t=4.6 h. After this time the viscosity increases
rapidly and the suspension is no longer useful for processing.
By proceeding in a similar manner, substitutions of values mb and
cb in equation (14) give the value of temperature T = 303 K, while
for the averaged viscosity it is T=302.5 K. This is a temperature at
which cross-linking reaction in propellant is inhibited. The detailed
calculated values are presented in Tables 6 and 7.
144 •
Fig. 10. Graph of η changes as a function of increase and decrease in
R at T = 318 K
nr 3/2015 • tom 69
Fig. 11. Graph of η changes as a function of increase and decrease in
R at T = 323 K
Fig. 12. Graph of η changes as a function of increase and decrease in
R at T = 328 K
Conclusions
The study leads to the following conclusions:
1. The cross-linking reaction of the studied rocket propellant
suspension is weakly time-dependent. This is shown by the low
values of reaction rate constants (kη), which vary in range from
0.0963 h-1 up to 0.2570 h-1, depending on temperature.
2. The technological time limit for processing of propellant suspension
determined based on experimental data is quite long (t = 4.4 h),
which indicates its good technological usefulness.
3. Experimental data was also used to determine temperature, at
which curing of the propellant is almost inhibited is equal to approx.
303 K. This value compared with the literature value T = 264 K
[4] is preferable.
Research project funded from science funds for the years 2010–2013 as
a development project.
Literature
Fig. 13. Graph of η changes as a function of increase and decrease in
R at T = 333 K
1. Florczak B., Stokowski P., Maranda A.: Badania właściwości wytypowanych
lepiszczy stałych paliw rakietowych niejednorodnych. Przemysł Chemiczny
2013, 92, 6, 957–961.
2. Brookfield DV-II+ Pro Viscometer. Manual, http://www.brookfieldengineering.
com/download/files/DV2Pro_Manual.pdf (21.01.2014).
3. Mahanta A. K., Goyal M., Pathak D. D.: Empirical modeling of chemoviscosity
of hydroxy terminated polybutadiene based solid composite propellant slurry.
Malaysian Polymer Journal, 2010, 5, 1, 1–16.
4. Mahanta A. K., Goyal M., Pathak D. D.: Rheokinetic analysis of hydroxy
terminated polybutadiene based solid propellant slurry. E-Journal of Chemistry,
2010, 7, 1, 171–179.
5. Muthiah R. M., Manjari R., Krishnamurthy V. N., Gupta B. R.: Rheology of
HTPB propellant: effect of mixing speed and mixing time. Defence Science
Journal, 1993, 43, 2, 167–172.
6. Viscosity, texture, powder, Labo Plus Sp. z o.o., 2012, http://www.laboplus.
pl/ (24.01.2014).
Fig. 14. Graph of η changes as a function of increase and decrease
in R at T = 338 K
*Bogdan FLORCZAK – Ph.D., Eng., Associate Professor of the Institute
of Industrial Organic Chemistry has graduated from the Faculty of Chemistry
and Technical Physics of the Military University of Technology (1976). He has
obtained his Ph.D. from the Faculty of Chemistry and Technical Physics of the
Military University of Technology (1990). Currently he works at the Institute of
Industrial Organic Chemistry. Scientific interests: chemistry and technology of
energetic materials, especially solid rocket propellants; materials science and
engineering. He has authored or co-authored 70 papers in scientific and technical journals, as well as 60 oral presentations and posters at national and international conferences. He has co-authored 26 patents and 9 patent applications.
e-mail: [email protected], phone: +48 609 819 698
The hysteresis loops recorded during the measurement indicate
that the studied propellant suspension shows thixotropic properties.
The graphs showing relationship viscosity vs time prove that as the
temperature increases the suspension viscosity decreases for a given
time interval. Moreover, the increase in the value of this viscosity with
time is observed, which shows the progress of cross-linking reaction
in rocket propellant.
The reaction rate constants calculated using experimental results
are increasing with temperature. This shows the obvious dependence
of this reaction on temperature. The constants determined for averaged
viscosity and ones obtained directly from experimental results show
small differences.
Based on the conducted measurement, the values of the following
thermodynamic parameters were determined: activation, enthalpy,
entropy and free enthalpy. The obtained activation energy value for
nr 3/2015 • tom 69
Ewelina BEDNARCZYK - has completed (2014) full-time B.Sc. course at
the Faculty of Advanced Technologies and Chemistry of the Military University
of Technology with specialization in dangerous materials and chemical rescue.
Currently she is a student of the M.Sc. course at the aforementioned faculty.
Andrzej MARANDA – Professor (Sc.D., Eng) is a graduate of the Faculty
of Chemistry, Warsaw University of Technology (1971). Currently, he works
for the Military University of Technology and the Institute of Industrial Organic
Chemistry. Research interests: chemistry, technology and the use of explosives, protection of the environment. He is the author of 5 monographs, 20
patents, over 500 articles, papers and posters at national and international
conferences.
e-mail: [email protected], phone: +48 22 683 75 41
• 145
science • technique
cross-linking reaction for the studied propellant suspension in 36
kJ/mol. The calculated positive value of free enthalpy (∆Gη) at all
temperatures of measurements proves that the cross-linking reaction
cannot proceed spontaneously. While the positive value of enthalpy
confirm the endothermic nature of cross-linking reaction. Moreover,
the measurements allowed also calculation of the temperature at
which the reaction is effectively inhibited, so the life-time of such
a propellant suspension can be prolonged. The measurements allowed
also to determine the time after rapid increase of viscosity is observed
and the suspension stops being technologically useful.