The field of engines with periodic workflow includes several different types of engines. We can first note the internal combustion engines, Stirling engines (external combustion) and among them, one of the famous types of engines is the pulse jet engine (Fig. 1). Research has shown that, due to the low cycle pressure, pulse jet is inferior to other types of engines in terms of specific parameters. But this engine can be useful for exploring various processes that are similar to all other periodic workflow engines.
Fig. 1. Valved (right) and valveless (center and left) pulsejet engines.
The reason was the pulsejet's advantages have not disappeared anywhere, this is the utmost simplicity and accessibility, extremely low cost and reliability due to the absence of rotating parts. It is precisely because of the understandable advantages that interest in this engine type still exists, as evidenced by ongoing research, the number of publications on this topic.
At the same time, despite certain successes in research, the theoretical models traditionally used in the calculations, have drawbacks. So, not all of them are clear enough and allow applying the results obtained to engines of different dimension. But most importantly, in some cases, it is difficult to verify theoretically the stability of the pulsating cycle for a given engine geometry.
For this reason, in the study, a simple piston analogy method (or gas piston method, see Fig. 2) was chosen and applied to a simple valved pulsejet engine to obtain general patterns.
Fig. 2. The "piston" (mechanical) analogy adopted for the model.
The essence of the method is that the engine combustion chamber is represented in a zero-dimensional formulation in the same way as it is done in the thermodynamic description of the in-cylinder process of an internal combustion engine, that is, with instantaneous gas parameters uniformly distributed by the volume. The gas flow in the resonance tube in the 1st approximation is considered as an oscillatory motion of the gas column. In other words, the engine is not represented as a pipe (as in the usual methods), but as a Helmholtz resonator or a mechanical oscillatory system.
These assumptions make it possible to compose a mathematical model from continuity, momentum and energy equations. This is a system (1) of the 1st order differential equations, for the instantaneous gas parameters (as functions of time τ): pressure pτ, temperature Tτ in the combustion chamber and gas velocity in the resonance pipe va, taking into account the formed flow zones:
Numerical solution of the system (1) with initial conditions makes it possible to obtain not only various parameters, but also to check the absence of geometric errors and the engine performance. The main condition is that 6-8 cycles after the start-up cycle, a stable operating cycle of auto-oscillations should be obtained, when the differences in the parameters of each subsequent cycle from the previous one do not exceed 1% (Fig. 3).
When developing the model, it was also found that if the derivation of the calculation equations is performed using dimensionless variables (relative to atmospheric pressure p0, temperature T0, sound speed a0, etc.), some previously unknown regularities can be revealed. As a result, dimensionless similarity criteria for the pulsejet engine were obtained, including the complex parameter Λ=Fa L/V (relative pipe volume) and the area factor Φ=Fe⁄Fa (relative inlet area). The obtained criterion dependencies (2) were tested against the data of the known engines and gave satisfactory convergence in the wide range of their sizes in terms of dimensionless thrust T and cycle frequency f, specific fuel consumption TSFC (Fig. 3).
Fig. 3. Calculation results: instantaneous values of gas dimensionless parameters in cycle (right) and criterion dependence for the integral cycle parameters (left)
Thus, the calculation results show that, in contrast to well-known models, the proposed method for calculating pulse jet engines based on criterion dependencies and piston analogy has a fairly high accuracy in the widest range of parameters. That means the proposed method has prospects for further research and development.
References
1.Khrulev A., Saraieva І., Vorobiov O., Sokhin A. Evaluation of the possibility of using mathematical models for expert research of car engine damage. Vehicle and electronics. Innovative technologies, Vol. 21, 2022, pp. 79-86. DOI: https://doi.org/10.30977/VЕІТ.2022.21.0.06
2.Ismail R.S., Jailani A., Muhammad A.H. Kadenancy Effect, Acoustical Resonance Effect Valveless Pulse Jet Engine. 3rd Electronic and Green Materials International Conference, 2017 (EGM 2017). AIP Conf. Proc., 1885, pp. 020036-1–020036-8; DOI: https://doi.org/10.1063/1.5002230.
3.Van Heerbeek P.A. Mathematical Modelling of a Pulse Combustor of the Helmholtz-type. A thesis submitted to the Delft Institute of Applied Mathematics for the degree Master of Science in Applied Mathematics. Delft, 2008. 146 p.
4.Anand V., Jodele J., Gutmark E., Prisell E., Lyrsell O. Dynamic Features of Internal and External Flowfields of Pulsejet Engines. AIAAJ Aeronautics and Astronautics, 2020. Volume 58, Number 10. 8 p. DOI: https://doi.org/10.2514/ 1.J059685.
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Науковий керівник: Сараєв Олексій Вікторович, доктор технічних наук, професор
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