Rockets, Missiles and Computers

Transcription

1 Rockets, Missiles and Computers Burn Rate Analysis Method INTRODUCTION As has been mentioned several times throughout this book, SFA, ARMAC and ROMANS-I all use a unique means of determining web-burn-time. Having realized early in the career that no good method of determining web-burn-time existed, he searched diligently for a good method to determine web-burn-time. For many years, the author, like his cohorts, believed the tangent bisector method of determining web-burntime was the best that the solid propellant rocket industry could do. Unlike the overwhelming majority of his cohorts, he believed it was not good enough. Eventually his search paid off in the development of the method to be described herein. The solid propellant rocket motor industry also occasionally used a method called the balance That method was crude, at best, and yielded a high degree of bias along with a large coefficient of variation vector (mass balance). The inability to accurately define web-burn-time and the lack of understanding and properly defining nozzle throat efficiency (the reciprocal of C-star efficiency) was the major contributor to lack of accuracy in determining the mass balance propellant burn rate. It is axiomatic that the solid propellant grain in a rocket motor can have only one burn rate (excluding bidirectional and other anomalous conditions) at any given point in time or when averaged over the web-burn-time. Therefore, the instantaneous and web averaged burn rate must have the same value, whether web-over-time or mass balance determined. Being in a position to evaluate the accuracy of the mass balance burn rate, by use of the web-over-time burn rate, allows one to accurately determine the natural logarithm of burn rate as a function of the natural logarithm of rocket motor chamber pressure. Under the correct conditions, the burn rate over a significant pressure range can be determined, allowing one to obtain the burn rate equation pressure exponent from a single motor static firing. Page 1

2 DISCUSSION Mass Balance Burn Rate Calculation Depending on the situation, in some instances the propellant burn rate is calculated by one of the various web-over-time methods. In other instances the burn rate is calculated using the so called mass balance equation. As can be observed in Figure 1, the mass balance equation sets the mass flow rate of propellant gases generated equal to those stored plus those that flow through the nozzle. However, the mass of gas storage rate is so small, compared to that generated under quasi steady state, it is Figure 1 - Mass Balance Equation Development ignored. That leaves the mass flow rate generated equal to the mass flow rate out through the nozzle throat. The mass flow rate generated is equal to the product of the burning surface area times the burning rate, at operating pressure, and propellant solid density. The mass flow rate out through the nozzle is equal to the product of the chamber pressure times the nozzle throat area times the standard gravity constant times the nozzle throat efficiency, all divided by the characteristic exhaust velocity. The nozzle throat efficiency is rarely ever correctly determined, if at all, or never included in the equation. The burn rate at operating pressure can be obtained by solving the equation at the bottom of Figure 1. As noted above, historically, all web-over-time burn rate calculations have directly Page 2

3 or indirectly assumed that the first lit point on the propellant grain is the first burn out point. That assumption is emphatically not the case in nearly all solid propellant rocket motor firings, as repeatedly demonstrated by the author. A mass balance determined burn rate rarely ever matches a web-over-time burn rate when the web-over-time burn rate has been calculated by any of the standard methods used, such as the tangent bisector method. Both the web-over-time and the mass balance determined burn rates must match because the propellant can have only one correct burn rate except in instances where there is bidirectional burning. Page 3

4 Web-Over-Time Historically, the burn rate of the propellant in a solid propellant rocket motor was calculated by what is called the tangent bisection method. There have been many other methods proposed but the tangent bisector was the most prevalent. The tangent bisector method is, by far, the better method, compared to the other fraction of maximum or fraction of average pressure points often used. However, the tangent bisector method was quite non-reproducible. From Equation 1, the burning surface area of the propellant grain, as seen by the rocket motor, is Figure 2 - Initial Surface Traces calculated, using the mass balance equation, after taking into account the nozzle throat efficiency (same as the reciprocal of the C* efficiency). Then the burning surface area, as calculated from the propellant grain, is shown in Equation Figure 2 shows, schematically, how these two traces would look if Figure 3 - Next Surface Traces plotted together. The burning surface area trace calculated from the propellant grain is noted as the geometric trace. The burning surface calculated from Equation 1 is noted as the mass balance trace. The geometric trace is simply a square wave trace, showing no influence of ignition rise or tail-off decay. The mass balance trace shows both ignition rise and tail-off. Figure 4 - Matched Surface Traces Using a fraction of the pressure integral at ignition, the computer shifts the geometric burning surface area trace to the right until its initial ignition rise slope closely matches the mass balance ignition Page 4

5 rise slope, as shown in Figure 3. Then, assuring no change in propellant grain volume, the tail-off decay of the geometric burning surface area trace during tail-off is matched to the mass balance tail-off trace, as shown in Figure 4. Using Mass Balance Burn Rate to Obtain a Burn Rate Graph The web distance burned and the time to burn that distance is, then used to determine the web-over-time propellant grain burn rate. Recognizing the need to ensure that both methods produce the same value resulted in a Figure 5 - Log-rate versus log-pressure modification to the computer code to create a data table that could be used to calculate and graph a ratio of the mass balance burn rate divided by the web-over-time burn rate as a function of web or time. It was soon realized that, since the mass balance burn rate table existed within the computer output data, the logarithm of it could be plotted against the logarithm of the pressure, as shown in Figure 5. Figure 5 consists of six sub scale rocket motor firings where all motors were of identical configurations and of the same propellant batch. Note that there were two rocket motors fired at the high pressure end of the curve and two low pressure motors at the low end of the curve. There were two motors having ablative nozzles on them. One was designed to ignite at the high pressure end of the matrix and one designed to ignite at the mid point of the spectrum. Page 5

6 Burn Rate Influence Factors Figure 6 shows the relative influences of a number of factors on web-over-time and mass balance burn rates. Some of these influence factors are mixed blessings. Those factors which have an influence may reduce the accuracy of determining the true burn rate but may also provide a method of detecting measurement errors. When a group of similar motors, containing propellant grains from the same batch of propellant are statically fired, a comparison of the burn rates among the motors may reveal data measurement errors. Figure 6 - Burn Rate Determination Influence Factors Page 6

7 ROMANS-I Objectives As noted in Figure 7, the primary objective of ROMANS-I, with respect to sub-scale burn rate motor analysis has been the most accurate burn rate possible. That objective was essentially achieved on the ASRM program when a burn rate coefficient of variation of 0.23 percent was achieved. A secondary objective of ROMANS-I was the isolation and elimination of data measurement errors in sub-scale burn rate determination motors. Those objectives were obtained on a number of programs, including the ASRM program. Figure 7 - ROMANS-I Objectives A tertiary objective was the reduction of costs to obtain burn rate and other data from sub-scale motors. The use of the mass balance burn rate method to obtain a log-rate versus log-pressure graph which automatically provides the propellant burn rate pressure exponent achieved that goal. Page 7

8 ROMANS-I Bonus Data While Providing Burn Rate As is shown in Figure 8, determining burn rate using the ROMANS-I computer code to analyze sub-scale rocket motor data provides many more pieces of data. It also provides the most accurate propellant burn rate data in the industry at a much reduced cost. With groups of motors from the same propellant batch, it provides an excellent way to ensure that all measurements are correct. As a failure investigation tool for full scale motors, it is unequaled. In addition, it overcomes the false concept that the first lit point Figure 8 - Bonuses Accrued by obtaining Burn Rate with ROMANS-I is the first burn out point. Page 8

9 Burn Rate Reproducibility Comparison Figure 9 presents some comparisons between the coefficients of variation of burn rate, as obtained by the tangent bisector method and those obtained by using the ARMAC code (ARMAC is technically identical to ROMANS-I). The same identical rocket motor firings were first analyzed by using the tangent bisector method to obtain burn rate and then ARMAC was used to obtain the burn rate. It is worth noting that since these were exactly the same motors, the propellant is identically the same propellant and the firing traces were exactly the same firings. Since the observed coefficients of variation include bias, uncertainty and actual Figure 9 - Comparison of Coefficients of Variation for Burn Rate burn rate variability, the variability of propellant burn rate remains unchanged. The improvement in the coefficients of variation strictly represents an improvement in technique. Page 9