Internal ballistics and C ? computation of a test solid rocket motor - Technical report - Group “MaxQ” Tommaso Mauriello 10803889 - 968434 Tommaso Zagatti 10581563 - 964800 Peter Achanccaray 10794333 - 975457 Course of Space Propulsion MSc in Space Engineering A.Y. 2020/21 Politecnico di Milano 1 Experimental setup A BATES-like motor (BAllistic Test Evaluation System), whose geometrical features are shown in Fig.1, has been used for the testing of 27 nominally identical propellant grains. The 27 pressure traces are organized in 9 batches, each characterized by three pressure curves (low, medium, high). Figure 1: Scheme of the experimental setup. The sides of the propellant grain are also burning in order to obtain a constant burning area and thus a constant combustion pressure (the Vieille law fit performed afterward requires in fact steady state conditions). The different pressure levels are obtained by changing the nozzle throat area: Pressure level Throat diameter [mm] High 21.81 Medium 25.25 Low 28.80 Table 1: Throat diameters and pressure levels. The propellant has the following composition: Name Formula Density [g/ cm 3 ] Std. enth. of formation [kJ/kmol] Amm. perchlorate N H 4 ClO 4 1.95 -295.77 Aluminum Al 2.7 0 HTPB C 7 075 H 10 65 O 0 223 N 0 063 0.92 -58 Table 2: Composition of the propellant grain. 2 Analysis method The scope of the analysis is to evaluate: • the coefficients a and n to fit the experimental ballistic data with the Vieille law r b = aP n ; • the experimental C ? ( C ? exp ); • the C ? efficiency η C ? = C ? exp C ? ideal with respect to the ideal C ? computed with the CEA code. The effective pressure of each pressure trace is computed with a modified Bayern-Chemie method (PoliMi-3 method), considering the 5% of P max to compute P A and P D . Fig. 2 shows one of the 1 Figure 2: Example of pressure trace and relevant pressure points. 27 experimental pressure traces, together with the relevant pressure points used in the analysis methodology. These points, together with the action time t act and the burning time t b , are computed as follows: P A = P D = 5% P max , t act = t D − t A , P B = P C = P ref = ∫ t D t A P ( t ) dt 2 t act t b = t C − t B , P ef f = ∫ t C t B P ( t ) dt t b The experimental C ? can be computed as: C ?exp = A t ∫ t C t B P ( t ) dt M tot where A t is the nozzle throat area. The total mass of propellant M tot can be computed knowing the volume of the cylindrical grain and the average propellant density, which is: ρ = AP % + Al % + HT P B % ρ AP AP % + ρ Al Al % + ρ HT P B HT P B % (1) The burning rate r b is computed as the ratio between the web thickness r ext − r int and the burning time t b . Finally, the ideal C ? is computed as: C ?ideal = 1 Γ( γ ) √ R M m T cc , Γ( γ ) = √ γ ( 2 γ + 1 ) γ +1 γ − 1 (2) where the ratio of the specific heats γ has been obtained from the values of the molar mass and the specific heat at constant pressure of the exhaust gases for an ideal expansion, computed using the CEA code (imposing a frozen equilibrium model, a storage temperature of the propellant of 298 K and a combustion pressure equal to the P ef f computed for each trace). 3 Results From the 27 pairs P ef f - r b a Vieille-law fit is computed using the function Uncertainty.m , obtaining: a = 1 7629 mm · s − 1 · bar − 1 ( uncert. = 0 0184 mm · s − 1 · bar − 1 ) , n = 0 3821 ( uncert. = 0 0027) as it is shown in Fig.3. 2 Figure 3: Vieille fitting of the experimental data on pressure and burning rate. The full trend of the ideal and experimental C ? with respect to the pressure is shown in Fig.4. Figure 4: Trend of experimental and ideal values of C ? as a function of effective pressure. Table 3 shows the mean value ( μ ) and standard deviation ( σ ) of the experimental C ? and of the C ? efficiency ( η ) for the three pressure ranges and for the whole set of measurements. P low P medium P high Total μ C ? [m/s] 1521.8 1518.3 1497.1 1512.4 σ C ? [m/s] 3.4 4.8 3.2 11.7 η C ? [m/s] 94.9% 94.3% 92.7% 93.97% Table 3: Results of the statistical analysis on the experimental C ? It’s observed that both the experimental C ? and the C ? efficiency decrease with pressure. This may be explained considering that at higher pressures the burning rate, the mass flow rate and thus the erosive burning are all increased. The increment in burning rate produced by the erosive burning can potentially cause an early burnout of the web with consequent loss of of structural integrity, resulting in the production of pieces of unburned propellant that are ejected through the nozzle. Furthermore, as the residence time of the molecules inside the combustion chamber is decreased, less dissociated molecules will be able to reassociate and release the stored enthalpy. Finally, since the sides of propellant grain are also burning, these regions may be affected by recirculating flow and non-ideal processes, whose importance grows with the flow velocity (and thus with the mass flow rate, which is increased at higher pressures), effectively consuming part of the enthalpy released by the combustion. All these phenomena may explain why the combustion efficiency, measured by the C ? , decreases as the combustion pressure grows. 3