Empirical Modeling of Chemoviscosity of Hydroxy Terminated Polybutadiene Based Solid Composite Propellant Slurry

Available online at www.fkkksa.utm.my/mpj Empirical Modeling of Chemoviscosity of Hydroxy Terminated Polybutadiene Based Solid Composite Propellant Slurry Mahanta Abhay K. 1, Goyal Monika 1 and Pathak Devendra D. 2 * 1 Defence Research & Development Organization, SF Complex, Jagdalpur-494001, Chhattisgarh, India. 2 Department of Applied Chemistry, Indian School of Mines University, Dhanbad-826004, Jharkhand, India ABSTRACT: In this study, rheological properties of hydroxyl-terminated polybutadiene based solid composite propellant slurry, after addition of toluene diisocyanate, have been determined at different time interval and temperatures by using Brookfield rotational viscometer. The time dependency of viscosity is described by exponential function. The flow behavior index (n) and consistency coefficient (K) values were obtained by fitting the rotational speed (rpm) versus apparent viscosity (η) data to a power law model of the form η = Kγ n-1. The slurry was found to exhibit non-newtonian, shear thinning behavior at all temperatures with a yield stress. The flow behavior index and the consistency coefficient, both parameters were significantly affected by temperature variation. The change of flow behavior indexes are found to be more or less independent of the cure time up to three hours while with temperature follows an exponential trend. The relation between temperature and K was successfully described by an Arrhenius-type function. Casson equation was utilized to assess the variation of yield stress of the propellant slurry at various time interval and temperature. A mathematical model was formulated to determine the combined effect of temperature, cure time and shear rate on apparent viscosity of the propellant slurry. The model equation is found to be satisfying the experimental data within ± 20 %. The proposed model equations can be used in designing equipment for the slurries, in quality control, process control applications and sensory evaluation of the product. Keywords: apparent viscosity, chemorheology, pseudoplastic behavior, propellant slurry, yield point, flow behavior index, consistency coefficient 1.0 INTRODUCTION Solid propellant rocket technology makes use of polyurethane propellant based on hydroxyl-terminated polybutadiene as binder due to high solid loading capability, convenient viscosity of slurry for casting and high fuel value. Solid loading level in modern system goes as high as 86-90% by weight in the liquid polymer [1]. Manufacturing of solid propellant involves preparation of premix consisting of liquid ingredients, incorporation of ammonium perchlorate, aluminum powder and then, the desired quantity of diisocyanate is added and mixed thoroughly just before casting the propellant. Casting time is mainly influenced by viscosity at end of mixing and viscosity build-up of the slurry. Viscosity build up is basically affected by the particulate matter and binder chemistry. Binder chemistry involves chemical reaction, which is basically condensation reaction between isocyanate groups of curator and hydroxyl groups of the binder. The main reaction between the hydroxy-terminated polybutadiene (HTPB) and toluene diisocyanate (TDI) is given in Scheme 1. Corresponding author: Pathak Devendra D, E-mail: ddpathak@yahoo.com 1

HO OH + n CH 3 NCO OOCHN CH 3 CH 3 NHCOO NCO NHCOO OCONH n Scheme 1: HTPB-TDI polyurethane reaction. The propellant slurry being non-newtonian does not show a characteristic viscosity over the range of test conditions. Instead, only an apparent viscosity can be measured, and its value depends on the various conditions under which it is measured. In a reactive polymeric system such as the propellant slurry, after addition of curing agent, rheology depends on a number of variables e.g. time, temperature, deformation rate, filler concentration, chemical formulation, reaction kinetics etc, which makes rheological characterization very complex. A good understanding of the rheology of the propellant slurry is very important for process design and control as well as quality control and performances evaluation of the propellant slurry. A number of studies have been conducted on the kinetics of the polyurethane cure reaction and a variety of kinetic models have been propounded to co-relate the rate of the chemical reaction with time, temperature and extent of cure [2-6]. However, a few studies are reported on modeling the chemoviscosity of HTPB based propellant slurry [7]. Osgood [8] describes the non-newtonian nature of propellant slurry in terms of change of pseudoplasticity index against time with progress of the curing reaction. The propellant slurry has been characterized as pseudoplastic material by power law equation and the optimum temperature for casting of the slurry has been determined by studying its behavior at four different temperatures [9]. The present investigation was aimed at studying chemorheology of the propellant slurry and developing a mathematical model to predict the combined effect of temperature, cure time and shear rate on apparent viscosity of the propellant slurry. The study was composed of mainly three sections. In the first section, the effect of temperature variation and curing time on the chemo-viscosity of the propellant slurry has been studied and subsequently, the rate constants (k η ) have been calculated. In the second part, the flow behavior of the slurry at varying shear rate and temperature were studied by analyzing the viscosity data against the shear rate (rpm). Rheological parameters like yield point, consistency coefficient and flow behavior index of the slurry have been calculated. Third section covered the development of a generalized correlation consisting of the combined effect of temperature, cure time and shear rate on apparent viscosity of the propellant slurry. The developed empirical model equations would be useful in proper designing of processing equipments for the propellant slurries, in quality control, process control applications as well as sensory evaluation of the product. 2

2.0 EXPERIMENTAL 2.1 Materials Malaysian Polymer Journal, Vol. 5, No. 1, p 1-16, 2010 Hydroxy-terminated polybutadiene manufactured by free radical polymerization with hydroxyl value = 40 mg KOH/g, polydispersity = 1.80, viscosity at 30 C = 6000 cp, viscosity at 60 C = 1448 cp, cis / trans / vinyl (%) = 25/59/16 was procured from trade. Ammonium perchlorate with purity > 99 per cent was used in bimodal distributions having average particle size 280 m and 49 m respectively. Dioctyl adipate (saponification value=300 mg KOH/g), toluene diisocyanate (purity > 99 percent and a mixture of 2, 4 and 2, 6-isomers in 80:20 ratio), aluminum powder (mean diameter=33.51 µm) used as plasticizer, curator and fuel, respectively, were procured from trade. Butanediol with hydroxyl value = 1232 mg KOH/g, trimethylol propane with hydroxyl value = 1227 mg KOH/g were used as chain extender and cross-linker respectively. Particle size of ammonium perchlorate (AP) and aluminum (Al) powder were measured by CILAS particle size analyzer-1180 model. 2.2 Methods 2.2.1 Preparation of Propellant Slurry Propellant mixing was carried out with 86 % solid loading in which 68 % is bimodal ammonium perchlorate with a coarse to fines ratio 3:1 and 18 % is aluminum powder. The stochiometric ratio of the curator to binder was fixed at 0.8 for this formulation. The mixing was carried out in two phases. In the first phase, all the ingredients except the curing agent were premixed thoroughly for about 3 h at 38 2 0 C. Hot water was circulated through the jacket of the mixture bowl to keep a constant temperature throughout the mixing cycle. A homogeneous test of the slurry was carried out after completion of the premix to confirm the uniform dispersion of AP and Al powder. In the second phase of mixing, a curing agent was added to the premixed slurry and mixed for 40 minutes at 40 1 0 C. In the present investigation, two propellant mixes with the identical raw materials were carried out. The rheological data of first mix was utilized for developing the empirical model equation and the applicability of the model equation was verified by comparing the predicted values with the actual viscosity values of the second mix. 2.2.2 Measurement of Viscosity Three slurry samples from the final mix (after addition of TDI), were withdrawn in 500 ml container of 84 mm diameter and 110 mm in length and kept in thermostatic water bath (Brookfield) separately at 40 0, 50 0 and 60 0 C. The slurry viscosity was measured at a regular interval of time up to five hours from end of mixing by Brookfield viscometer model HADV-II + equipped with a motorized stand (helipath stand). The function of the helipath stand is to slowly raise and lower the viscometer at the rate of 7/8-inch per minute during viscosity measurement and to eliminate the channeling effect due to the spindle rotation, giving high consistency in measurement. The flow behavior of the slurry is usually characterized by measuring viscosity at low shear rates. Therefore, rotational speeds from 0.5 to 10 rpm were selected. This selection was based on requirement of the applied percentage torque that is proportional to the applied shear stress, which is to be 10 % of the maximum measurable range for the spindle and speed combination of Brookfield viscometer. This requirement could be fulfilled by measuring viscosity at 0.5 to 10 rpm by T-E spindle of HADVII+ viscometer. Viscosity of all the slurry samples was measured at 0.5, 1, 2.5, 5 and 10 rpm by T-E spindle. For measurement of viscosity, the spindle was attached to the lower shaft of the viscometer and was centered in the propellant slurry. Initially, it was immersed about 7 mm into the slurry by using the helipath stand. The 3

rotational speed was adjusted at 0.5 rpm. For each experiment, data collection was accomplished after one complete revolution. Samples were sheared using five different rotational speeds at an increasing order. For each successive revolution, total 20 readings, each at an interval of one-second were recorded at the set rotational speed by using Wingather software. The data were then transferred to excel worksheet. The average viscosity value was calculated and used for data analysis and modeling. 3.0 RESULTS AND DISCUSSIONS 3.1 Time Modeling of Propellant Slurry The variation of viscosity during the cure process has been studied by several workers [10-12]. Tajima and Crozier [13] have suggested semi empirical model to predict the variation of viscosity during the cure process. The model representing the changes of viscosity (η) with reaction time (t) has the following form: η (t) = η 0 e kη t (1) Where η 0 is the viscosity at t = 0 and k η is the rate constant for viscosity build up. Taking the logarithm we get: ln η (t) = ln η 0 + k η t (2) Therefore, plotting ln η against t should yield a straight line and the slope of which is the rate constant for the viscosity build up. A typical plot of ln η vs. t of the slurry at 40 0, 50 0 and 60 0 C (η at 2.5 rpm) is shown in Figure 1. A good linearity of the plots shows that the experimental data are well fitted with the exponential model. In other words, the viscosity build up can be described as a first order process with respect to the viscosity of curing mixture at any given time. From the slope of the regressed line, the values of rate constant and from the intercept, the value of η 0 was determined. These values of η 0, k η at all the studied temperatures are given in Table 1. By plotting ln k η versus 1/T (K), the activation energy of the propellant slurry was found to be 25.46 kj/mol. Table 1: Values of constant (η 0 ) and rate constant (k η ) of the propellant slurry at three different temperatures. Temperature ( 0 C) η 0 (Poise) k η x10 3 (min -1 ) 40 8706 1.91 50 6890 2.49 60 5347 3.44 4

Table 2: Power law parameters of the propellant slurry at three different temperatures with varying time interval. Cure time (min) 40 0 C 50 0 C 60 0 C n K r 2 n K r 2 n K r 2 30 0.73 11014 0.987 0.68 9569 0.995 0.65 8100 0.999 60 0.73 11519 0.992 0.68 10161 0.995 0.66 8860 0.999 90 0.74 12047 0.996 0.68 10789 0.995 0.66 9692 0.998 120 0.74 12600 0.993 0.68 11457 0.995 0.67 10601 0.995 150 0.74 13178 0.985 0.68 12165 0.995 0.68 11596 0.991 180 0.75 13782 0.972 0.68 12917 0.995 0.69 12684 0.985 Table 3: Casson parameters of the propellant slurry at three different temperatures with varying time interval. Cure time (min) 40 0 C 50 0 C 60 0 C τ 0 r 2 τ 0 r 2 τ 0 r 2 30 1179 4692 0.950 1362 3452 0.961 1423 2535 0.983 60 1233 4922 0.969 1446 3665 0.961 1516 2837 0.986 90 1292 5164 0.983 1536 3893 0.961 1613 3174 0.989 120 1353 5418 0.992 1631 4133 0.961 1715 3552 0.992 150 1417 5687 0.997 1732 4388 0.961 1822 3975 0.993 180 1484 5968 0.996 1839 4659 0.961 1934 4449 0.994 5

3.2 Shear Rate Modeling of the Propellant Slurry In order to characterize the chemorheology of the propellant slurry, the power law model which has been extensively used in theoretical analysis and in practical engineering calculations was employed. Mathematically, the power law equation can be expressed as n τ = Kγ (3) Where, τ is the shear stress (dynes/cm 2 ), K is the consistency coefficient (dynes/cm 2.sec n ) and n is the flow behavior index. From the power law model, the apparent viscosity (η) can be defined as n-1 η = Kγ (4) As the shear rate is approximately a linear function of viscometer speed (rpm) over the narrow shear rate range available with the Brookfield instrument, the rotational speed (rpm) of the spindle used in viscosity measurement was taken as the shear rate in the present investigation for rheological analysis. Therefore, from Equation 4, it can be inferred that the logarithmic plot of apparent viscosity versus shear rate (rpm) would be a straight line. The slope and intercept of the straight line would be the measure of flow behavior index (n) and consistency coefficient (K) respectively. The experimental viscosity data were fitted to an exponential function and the viscosities of the propellant slurry at the desired time intervals were computed and plotted against the shear rate (rpm). Figure 2 shows a typical plot of ln η vs. ln γ the slurry at 40 0 C. A good linearity of the plots shows that the viscosities data are well fitted with the power law model. From the slope of the regressed line, the flow behavior index (n) and from the intercept, consistency coefficient (K) was determined. These values of n, K along with the correlation coefficient (r 2 ) at all the studied temperatures at different time intervals are summarized in Table 2. The model appeared to be suitable for describing the rheological behavior of the propellant slurry as the correlation coefficient ranges from 0.972 to 0.999 at all the studied temperatures. The consistency coefficient (K) ranged from 11014-13782 Poise at 40 0 C. The values of flow behavior index varied between 0.65 and 0.75 indicating shear-thinning (pseudoplastic) behavior since figures were smaller than unity (n<1).the degree of pseudoplasticity can be measured by the flow behavior index which is a measure of deviation from Newtonian behaviour. As n increases, pseudoplasticity decreases. 6

Table 4: Comparison of predicted model results with experimental data of the propellant slurry at 40 0 C and 50 0 C. Temp. ( 0 C) Cure time (min) η (Poise) @ 2.5 rpm η (Poise) @ 5 rpm Model equation Experimental Model equation Experimental 40 49 9976 9560 8485 7998 108 10738 10485 9136 8658 160 11458 13838 9751 9478 210 12196 14200 10383 10900 240 12662 14900 10781 11600 50 63 8937 8212 7498 6704 110 9792 9109 8214 7101 170 11005 10282 9227 8600 215 12012 11800 10068 9956 245 12735 12756 10672 10785 3.3 Determination of Yield Stress Since the propellant slurry behaves much like a solid at zero shear rates i.e. it doesn t flow until a certain amount of force is applied. This force is called the yield value and measuring it is often worthwhile. Yield value can help in determining, whether the vacuum level has sufficient power to make the slurry flow during casting, and often correlated with the flow behavior of the propellant slurry. In other words, the pour ability of propellant slurry is directly related to its yield value. Casson [14] proposed an equation to relate the shear stress to shear rate of pigment-oil suspension of the printing ink type. Asbeck [15] has subsequently modified this equation as: 1/2 = 1/2 + τ 0 1/2 γ -1/2 (5) Where is the viscosity, is the infinite shear rate viscosity, τ 0 is the yield point and γ is the shear rate. The plot of 1/2 against γ -1/2 should, therefore, yield a straight line. The slope and intercept of the line gives the value of τ 0 1/2 and 1/2. The two parameters and τ 0 can be related to the product performance and composition variations [16]. Figure 3 7

shows the plots of square root of viscosity ( 1/2 ) vs. square root of reciprocal shear rate in rpm (γ -1/2 ) of the propellant slurry at 40 0 C. Similar plots were obtained for slurry samples maintained at different temperatures. The values τ 0 and were evaluated from the slopes and intercepts of the regressed lines thus obtained, and presented in Table 3. The viscosities of the slurry at any given time (t) could be related to the shear rate (rpm) through Casson equation as the correlation coefficient (r 2 ) varied in the range 0.950 to 0.997. 3.4 Development of Empirical Model: Combined Effect of Shear Rate, Time and Temperature The experimental results indicate that viscosity of the propellant slurry is a function of cure time (t), temperature (T) and shear rate (γ ). Mathematically, we can write, η = f (γ, T, t) (6) The rheological parameters of the propellant slurry like yield points (τ 0 ), consistency coefficient (K), flow behavior index (n), are found to be function of cure time and temperature. A simple equation describing the combined effect of temperature, time and shear rate on the viscosity of propellant slurry is a very useful tool for engineering application. For this purpose, the following model equations were investigated. τ 0 = τ 0 (T, t) (7) K = K (T, t) (8) n = n (T, t) (9) η = τ 0 (T, t)+k(t,t)( γ ) n(t, t) (10) 3.4.1 Evaluation of τ 0 = τ 0 (T, t) Figure 4 shows the semi-logarithmic plots of yield point versus reciprocal of temperature at various cure time of the propellant slurry. Using least square linear regression, the plots were fitted with a regression line and the values of slope (A 1 ) and intercept (A 0 ) at various cure time were determined. The slopes and intercepts are found to be the linear function of cure time, as can be seen in Figures 5 and 6, respectively. Thus we can write: ln τ 0 =A o +A 1 /T (11) A o = m o t + c 0 and A 1 = m 1 t + c 1 (12) The values of m o, m 1, c 0 and c 1 have been evaluated to be 0.01, -2.665, 9.962 and -916.2 respectively. So the final empirical relation corresponding to Equation 7 becomes: τ 0 = e {(0.01t + 9.962) - (2.665t + 916.2)/T} (13) 3.4.2 Evaluation of K=K (T, t) Figure 7 shows the semi-logarithmic plots of consistency coefficient (K) versus reciprocal of temperature at different cure time of the propellant slurry. Using least square linear regression, the plots were fitted with a regression line and the values of slope (S k ) and intercept (I k ) at various cure time were determined. The slopes and intercepts are found to be the linear function of cure time, as can be seen in Figures 8 and 9 respectively. Thus we can write: 8

ln K(T,t) =I k +S k /T (14) I k = m i t + c i and S k = m s t + c s (15) The values of m i, m s, c i and c s have been evaluated to be 0.026, -7.758, 3.415 and 1831 respectively. So the final empirical relation corresponding to Equation 8 becomes K= e {(0.026t + 3.415) - (7.758t - 1831)/T} (16) 3.4.3 Evaluation of n = n (T, t) As it can be seen from Table 2, the flow behavior index (n) is more or less independent of cure time up to three hours. Hence, the first three hours of n values were averaged and plotted against temperature. The data are well fitted to an exponential function as shown in Figure 10. n = 3.47197e (-0.00498T) (17) 3.4.4 Generalized Empirical Modeling The final empirical model equation having the combined effect of temperature, time and shear rate on viscosity of the propellant slurry up to the first three hours of cure time is obtained as: η = e {(0.01t + 9.962) - (2.665t + 916.2)/T} + e {(0.026t + 3.415) - (7.758t - 1831)/T} x (γ ) 3.47197e (-0.00498T) (18) The viscosity calculated using this model equation for different rates of shear, times and temperatures have been given against the experimental values in Table 4. This equation is found to satisfy all the experimental data within ± 20%. 9

Figure 1: Plots of ln η vs. cure time for the propellant slurry at three different temperatures. Figure 2: Plots of ln η vs. ln at 40 C for the propellant slurry at different cure time. 10

Figure 3: Casson plots of η 1/2 vs. -1/2 at 40ºC for the propellant slurry at various cure time. Figure 4: Plots of ln σ 0 vs. 1/T for the propellant slurry at various cure time. 11

-800-900 -1000 A 1-1100 -1200 y = -2.665x - 916.2 R² = 0.997-1300 -1400-1500 30 60 90 120 150 180 Curing time (min) Figure 5: Variation of A 1 vs. cure time for the propellant slurry. 12 11.8 11.6 11.4 A 0 11.2 11 10.8 10.6 10.4 10.2 y = 0.010x + 9.962 R² = 0.998 10 30 60 90 120 150 180 Cure time (min) Figure 6: Variation of A o with cure time for the propellant slurry. 12

Figure 7: Isochrones of ln K vs. 1/T for the propellant slurry. Figure 8: Variation of S k with cure time for the propellant slurry. 13

Figure 9: Variation of I k with cure time for the propellant slurry. Flow behaviour Index (n) 0.75 0.74 0.73 0.72 0.71 0.7 0.69 0.68 0.67 0.66 y = 3.47197e -0.00498x R² = 0.87604 0.65 310 315 320 325 330 335 Temperature (K) Figure 10: Variation of flow behavior index with temperature for the propellant slurry. 14

4.0 CONCLUSIONS Malaysian Polymer Journal, Vol. 5, No. 1, p 1-16, 2010 The viscosity buildup during the cure of solid propellant slurry has been studied at various temperature and shear rate. Flow behavior and rheological properties of the slurries were determined by applying linear regression method via Microsoft Excel Software. Model equations and correlation coefficient were reported. The experimental results indicate that at a particular temperature and cure time the propellant slurries shows yield stress with pseudoplastic behavior. The time modeling of the viscosity data followed an exponential curve. Apparent viscosity-shear rate (rpm) relationship of the propellant slurry can be successfully represented by power-law model. The change of flow behavior indexes are found to be more or less independent of cure time up to three hours while with temperature follows an exponential trend. The consistency coefficient and temperature relationship can successfully be described by an Arrhenius-type function. The combined effect of temperature, time and shear rate on viscosity of the slurry can be described by the empirical model equations i.e. η = e {(0.01t + 9.962) - (2.665t + 916.2)/T} + e {(0.026t + 3.415) - (7.758t - 1831)/T} x (γ ) 3.47197e (-0.00498T). This equation is found to be satisfying the experimental data within ± 20 %. The chemorheological model equation can be a useful tool in process design and control of chemical unit opearations as well as quality control and sensory evaluation of the product. The methodology for development of the rheological model described in this article can be applied not only to the propellant slurry but also to any type of slurry under investigation. Acknowledgement The authors are thankful to Shri T. Mohan Reddy, General Manager, SF Complex, Jagdalpur for his kind permission to publish this research article. References [1] Muthiah R.M., Krishnamurthy V.N., Gupta B.R.: Rheology of HTPB propellant. 1. Effect of solid loading, oxidizer particle size, and aluminum content, J. Appl. Polym. Sci., 44:2043-52(1992). [2] Singh M., Kanungo B.K., Bansal T.K.: Kinetic studies on curing of hydroxylterminated polybutadiene prepolymer-based polyurethane networks, J. Appl.Polym.Sci., 85: 842-46(2002). [3] Sekkar V., Krishnamurthy V.N., Jain S.R.: Kinetics of copolyurethane network formation, J. Appl.Polym.Sci., 66: 1795-1801(1997). [4] Abraham V., ScariahK.J., Bera S.C., Rama Rao M., Sastri K.S.: Processibility charecteristics of hydroxy terminated polybutadienes, Eur.Polym.J. 32(1): 79-83(1996). [5] Kothandaraman K., Sultan Nasar A.: The kinetics of the polymerization reaction of toluene diisocyanate with HTPB prepolymer, J. Appl.Polym.Sci., 50: 1611-17(1993). 15

[6] Coutinho F.M.B., Rezende L.C., Quijada R.C.P.: Kinetic study of the reaction between hydroxylated polybutadienes and isocyanate. 1. Reaction with tolylene diisocyanate (TDI), J Polym Sci: Part A: Polym Chem., 24: 727-35(1986). [7] Lakshmi K., Athithan S.K.: An empirical model for the viscosity build up of hydroxyl terminated polybutadiene based solid propellant slurry, Polym. Composites, 20 (3) :346-56(1999). [8] Osgood A.A.: Rheological characterization of non-newtonian propellant for casting optimization, AIAA 5 th Prop Jt Sp Conf Colorado, A69-32665 (1969). [9] Mahanta A.K., Dharmsaktu I., Pattnayak P.K.: Rheological behavior of HTPB based composite propellant: Effect of temperature and pot-life on casting rate, Def. Sci. J., 57 (4): 581-88(2007). [10] Yousefi A., Lafleur P.G.: Kinetic studies of thermoset cure reaction: A review, Polym. Composites, 18(2):157-168(1997). [11] White R.P.Jr.: Time-temperature superpositioning of viscosity-time profiles of three high temperature epoxy resins, Polym. Eng. Sci., 14(1): 50-57 (1974). [12] Roller M.B.: Rheology of curing thermosets: A review, Polym. Eng. Sci., 26(6): 432-440 (1986). [13] Tajima Y.A., Crozier D.: Thermokinetic modeling of an epoxy resin 1.Chemoviscosity, Polym. Eng. Sci., 23(4): 186-190 (1983). [14] Casson N.: Rheology of disperse systems, C.C.Mills, Editor, Pergamon Press, New York 84 (1959). [15] Asbeck W.K.:Official Digest, 33, No.432: 65(1961). [16] Pierce P.E.: Measurement of rheology of thixotropic organic coatings and resins with the Brookfield viscometer, J. of Paint Tech., 43 No.557: 35-43 (1971). NOMENCLATURE η = Apparent viscosity (Poise) η 0 = Apparent viscosity (Poise) at t=0 k η = Reaction rate constant (min -1 ) t = Cure time (min) τ = Shear stress K = Consistency coefficient γ = Shear rate (rpm) n = Flow behavior index τ 0 = Yield point = Infinite shear viscosity (Poise) T = Temperature (K) A 0, A 1, m o, m 1, c 0, c 1, m i, m s, c i, = Various constants in the model equation c s, I k,s k 16