  In ducts with varying cross-sectional area and sustaining a subsonic
non-isentropic mean flow, the axially varying flow conditions affect the
acoustic energy balance of the system. This is significant in understanding and
controlling thermo-acoustic phenomena, particularly in combustors. This work
aims at quantifying the acoustic energy change in such configurations, using
the acoustic absorption coefficient, $\Delta$. The acoustic response of the
duct to acoustic forcing is determined using an analytical model, neglecting
the effect of entropy fluctuations on the acoustic field, and subsequently,
$\Delta$ is estimated. The model predictions of $\Delta$ are validated using a
linearised Euler equations (LEEs) solver. The model was found to be accurate
for Mach numbers below $0.25$, provided the lower frequency limit set by the
analytical solution is satisfied. For conically varying area ducts with linear
mean temperature gradient, it was observed that $\Delta$ showed very little
dependence on frequency, and that the absolute value of $\Delta$ tended to be
maximised when the upstream boundary was anechoic rather than non-anechoic.
More importantly, $\Delta$ was also observed to show stronger dependence on the
mean temperature gradient than area gradient variation for such configurations.
Further parametric and optimisation studies for $\Delta$ revealed a crucial
finding that a positive mean temperature gradient, representing a heated duct
caused acoustic energy absorption. Similarly, a negative mean temperature
gradient, representing a cooled duct caused acoustic energy generation -- a key
result of this analysis. This behaviour was shown to be consistent with a
simplified analysis of the acoustic energy balance. Based on this finding, a
linearly proportional reduction in acoustic energy generation was achieved by
changing the mean temperature gradient.
