The design of rocket nozzles aims to expand the combustion gases until the exit pressure matches the ambient pressure, thereby maximizing thrust. However, ambient pressuredecreases as the rocket ascends, posing a challenge for conventional nozzles, which operate efficiently only at a specific altitude [ 30 ]. By adjusting their flow characteristics to the varying atmospheric pressure, aerospike nozzles offer a significant advantage in this regard.
To quantify this impact, the thrust coefficient () is used, a dimensionless parameter that compares the actual nozzle thrust to the ideal thrust that would be achieved if the gases expanded to zero ambient pressure [ 38 ]:whereis the thrust,is the chamber pressure,is the throat area,is the exit pressure,is the ambient pressure,is the specific heat ratio, andis the exit area. Equation ( 1 ) illustrates the three possible scenarios:
Perfectly expanded nozzle ( P e = P a ): The exit pressure matches the ambient pressure. The pressure thrust term is nullified, maximizing C f for a given area ratio. This condition represents the ideal scenario for a nozzle operating at a fixed altitude.
Underexpanded nozzle ( P e > P a ): The exit pressure is higher than the ambient pressure. The pressure thrust term is positive, increasing the total thrust but at the cost of incomplete flow expansion. Expansion waves form outside the nozzle, indicating that thermal energy is not fully converted into kinetic energy.
Overexpanded nozzle ( P e < P a ): The exit pressure is lower than the ambient pressure. The pressure thrust term is negative, reducing the total thrust. Shock waves can form inside the nozzle, negatively affecting performance and potentially causing structural damage.
Due to their design, aerospike nozzles maintain an almost perfectly expanded flow throughout the ascent (see Figure 2 ), adapting to the varying atmospheric pressure and optimizingacross the entire altitude range. The base pressure of the aerospike naturally equalizes with ambient pressure, nullifying the pressure thrust term, unlike conventional nozzles. This adaptability makes aerospike nozzles ideal for single-stage-to-orbit (SSTO) vehicles, opening new possibilities for space transportation.
In the combustion chamber, the chemical reaction between fuel and oxidizer generates gases at high pressure (on the order of hundreds of atmospheres) and high temperature (around 3000 K or more). The composition of these gases depends on the propellants used. As they expand through the nozzle, the gases accelerate to supersonic speeds and experience reductions in pressure and temperature, as well as possible changes in their chemical composition.
To predict the thermodynamic behavior in the nozzle, simplified models are employed for preliminary analysis without capturing the full complexity of the real process. The frozen flow model assumes that composition remains constant during expansion, ignoring additional reactions that could release extra energy. Conversely, the chemical equilibrium flow (CEQ) model assumes that species react instantaneously, reaching a local equilibrium that minimizes Gibbs free energy at every point [ 40 ].
Although these models are useful for initial analysis, they have limitations. The frozen flow model tends to underestimate performance by neglecting additional reactions, while the CEQ model may overestimate performance by assuming instantaneous reactions without considering finite reaction kinetics [ 41 ]. Moreover, gas–wall friction and heat losses prevent achieving ideal isentropic expansion, reducing flow efficiency.
For more realistic predictions, two-dimensional simulations combining computational fluid dynamics (CFDs) with flow chemistry models are used. Specialized tools such as the TDK (Two-Dimensional Kinetics) code allow for the simulation of reactive flows in nozzles, considering finite chemical kinetics, friction, and heat transfer, providing a more accurate analysis of real engine behavior [ 42 ].
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