One of the major concerns in the operability of power generation systems is their susceptibility to combustion instabilities. In this work, we explore whether a heat exchanger, an integral component of a domestic boiler, can be made to act as a passive controller that suppresses combustion instabilities. The combustor is modelled as a quarter-wave resonator (1-D, open at one end, closed at the other) with a compact heat source inside, which is modelled by a time-lag law. The heat exchanger is modelled as an array of tubes with bias flow and is placed near the closed end of the resonator, causing it to behave like a cavity-backed slit plate: an effective acoustic absorber. For simplicity and ease of analysis, we treat the physical processes of heat transfer and acoustic scattering occurring at the heat exchanger as two individual processes separated by an infinitesimal distance. The aeroacoustic response of the tube array is modelled using a quasi-steady approach and the heat transfer across the heat exchanger is modelled by assuming it to be a heat sink. Unsteady numerical simulations were carried out to obtain the heat exchanger transfer function, which is the response of the heat transfer at heat exchanger to upstream velocity perturbations. Combining the aeroacoustic response and the heat exchanger transfer function, in the limit of the distance between these processes tending to zero, gives the net influence of the heat exchanger. Other parameters of interest are the heat source location and the cavity length (the distance between the tube array and the closed end). We then construct stability maps for the first resonant mode of the aforementioned combustor configuration, for various parameter combinations. Our model predicts that stability can be achieved for a wide range of parameters.