A Method For Setting The Nozzle Expansion-Ratio In A Boost-Sustain Rocket Motor

Abstract

In a boost-sustain rocket motor with a fixed throat area and a relatively high boost-tosustain thrust ratio (or thrust turndown), the question periodically arises regarding how best to set the nozzle expansion ratio. To provide an answer to this question, a closed-form solution was derived for the nozzle expansion ratio that will minimize the total propellant mass required to deliver a specified thrust turndown and boost-to-sustain impulse ratio in a boost-sustain rocket motor. This nozzle expansion ratio was referred to as either e e e emm or the mass-minimizing expansion ratio. The fundamental equations that govern the performance of an ideal rocket motor and the standard methods from calculus for determining the extrema of a one-variable function formed the basis of the derivation. No approximations were made in the derivation, and the closed-form solution that was obtained for emm is exact. In addition, the closed-form solution is completely general in that the predicted value of e e e emm is a function of the ratio of the ambient pressure to the boost chamber pressure, the thrust turndown, the boost-to-sustain impulse ratio, and the specific heat ratio of the motor’s propellant. The derivation illustrated that the minimization of the total propellant mass in a boost-sustain rocket motor is equivalent to maximizing the mass-weighted average thrust coefficient. After the solution for e e e emm was derived, the manner in which e e e emm varies as a function of several independent variables was investigated for a wide range of scenarios. For all of the scenarios that were investigated it was shown that the mass-minimizing expansion ratio was unique (i.e. there were never two different expansion ratio values that could be chosen to minimize the total propellant mass). It was also shown that for motors with impulse ratios not much greater than 1.0, for any thrust turndown the mass-minimizing expansion ratio is usually less than the expansion ratio at which flow separation will typically occur in the sustain phase. In addition, it was shown that the mass-minimizing expansion ratio can be a relatively small value; for scenarios where the impulse ratio is not much greater than 1.0, the thrust turndown is greater than 15.0, and the ratio of the boost chamber pressure to the ambient pressure is less than 200, the value of emm is usually less than 5.0. The benefits of using the nozzle expansion ratio given by emm in a boost-sustain motor were also quantified relative to a hypothetical, “ideal” boost-sustain nozzle that could deliver optimal expansion (exit pressure equal to ambient pressure) during both the boost phase and the sustain phase. Ultimately, it is suggested that the closed-form solution derived for the mass-minimizing expansion ratio should prove to be useful for a wide range of design problems involving boost-sustain motors that use conventional nozzles.