Derivation of a Revised Tsiolkovsky Rocket Equation That Predicts Combustion Oscillations

Our study identifies a subtle deviation from Newton’s third law in the derivation of the ideal rocket equation, also known as the Tsiolkovsky Rocket Equation (TRE). TRE can be derived using a 1D elastic collision model of the momentum exchange between the differential propellant mass element (dm) and the rocket final mass (m1), in which dm initially travels forward to collide with m1 and rebounds to exit through the exhaust nozzle with a velocity that is known as the effective exhaust velocity ve. We observe that such a model does not explain how dm was able to acquire its initial forward velocity without the support of a reactive mass traveling in the opposite direction. We show instead that the initial kinetic energy of dm is generated from dm itself by a process of self-combustion and expansion. In our ideal rocket with a single particle dm confined inside a hollow tube with one closed end, we show that the process of self-combustion and expansion of dm will result in a pair of differential particles each with a mass dm/2, and each traveling away from one another along the tube axis, from the center of combustion. These two identical particles represent the active and reactive sub-components of dm, co-generated in compliance with Newton’s third law of equal action and reaction. Building on this model, we derive a linear momentum ODE of the system, the solution of which yields what we call the Revised Tsiolkovsky Rocket Equation (RTRE). We show that RTRE has a mathematical form that is similar to TRE, with the exception of the effective exhaust velocity (ve) term. The ve term in TRE is replaced in RTRE by the average of two distinct exhaust velocities that we refer to as fast-jet, vx1, and slow-jet, vx2. These two velocities correspond, respectively, to the velocities of the detonation pressure wave that is vectored directly towards the exhaust nozzle, and the retonation wave that is initially vectored in the direction of rocket propagation, but subsequently becomes reflected from the thrust surface of the combustion chamber to exit through the exhaust nozzle with a time lag behind the detonation wave. The detonation-retonation phenomenon is supported by experimental evidence in the published literature. Finally, we use a convolution model to simulate the composite exhaust pressure wave, highlighting the frequency spectrum of the pressure perturbations that are generated by the mutual interference between the fast-jet and slow-jet components. Our analysis offers insights into the origin of combustion oscillations in rocket engines, with possible extensions beyond rocket engineering into other fields of combustion engineering.

A New Methodology for Early Detection of Thermoacoustic Combustion Oscillations Based on Permutation Entropy

We carry out a series of experimental investigations in a model combustor to detect a precursor of thermoacoustic combustion oscillations based on permutation entropy,which can amplify the subtle changes effected in the time sequence to identify the anomaly.By changing the flame’s location or the fuel flow to a value,an abrupt switch from aperiodic small-amplitude oscillations to periodic large-amplitude oscillations would occur in pressure fluctuations.The characteristic frequency of combustion oscillation is obtained by spectral analysis,with which a modified algorithm of the permutation entropy is proposed.The impact evaluation on key parameters such as moving step sizes and window sizes reveals that the moving data permutation entropy has strong robustness,and can accurately detect the onset of thermoacoustic oscillations.Further nonlinear analysis exhibits peculiar dynamics of the combustion system,which result in specific patterns in the time series and provide a theoretical basis for anomaly detection.Our results suggest that the permutation entropy has a certain potential in early warning and detection of combustion oscillations.

Experimental study on combustion pressure oscillation of soybean bio diesel oil

The principal objectives of this study were to examine in-cylinder combustion pressure oscillation characteristics of soybean biodiesel in time domain and time-frequency domain,and their influences on the control and operational parameters,such as injection timing,exhaust gas recirculation(EGR)ratio,engine load and engine speed.In this study,the combustion pressure oscillation characteristics of biodiesel engine for various injection timing,EGR ratio and engine speed were investigated.The corresponding relation of pressure characteristics in the time domain and frequency domain were obtained.The results showed that the pressure oscillation and peak pressure rise acceleration occurred mainly in the diffusion combustion,and the peak pressure rise rate located in the premixed combustion.The in-cylinder pressure level curve can be divided into three stages.The pressure levels of stage 1,stage 2 and stage 3 represent the peak in-cylinder pressure,the maximum amplitude of pressure rise rate and pressure rise acceleration,respectively.As the injection timing retards,the pressure levels of stage 1 and stage 3 decrease gradually.The pressure level curve of stage 3 with 25°before top dead center(BTDC)is the highest and the oscillation is the most significant.It is worth noting that the location of each stage with various operate conditions is not fixed.At 0.41 MPa indicated mean effective pressure(IMEP),with the increase of EGR rate,the pressure levels of stage 1 and stage 2 decrease gradually.The pressure level curve of stage 3 and the maximum amplitude of pressure rise acceleration with 0%EGR rate are the highest.The oscillation with 0%EGR rate is the most significant at 0.41 MPa IMEP.Compared to 0.41 MPa IMEP,the frequency bands of stage 1 and stage 2 at 1.1 MPa IMEP are relatively low due to the soft combustion in the cylinder.As EGR rate increases,the pressure level of stage 1 decreases,and those of stage 2 and stage 3 increase gradually.The oscillation with 30%EGR rate is the most significant.With the increase of engine speed,the pressure levels of stage 1 and stage 2 decrease,and move to the low frequency.The pressure level in the high frequency domain at 1600 r/min is less than that at 1100 r/min,and the combustion process is smooth.