TITLE:
Derivation of a Revised Tsiolkovsky Rocket Equation That Predicts Combustion Oscillations
AUTHORS:
Zaki Harari
KEYWORDS:
Tsiolkovsky Rocket Equation, Ideal Rocket Equation, Rocket Propulsion, Newton’s Third Law, Combustion Oscillations, Combustion Instability
JOURNAL NAME:
Advances in Aerospace Science and Technology,
Vol.9 No.1,
March
14,
2024
ABSTRACT: 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.