A non-linear dynamic framework is presented for the modelling of a novel two-phase heat engine termed 'Up-THERM', which features a single solid moving-part (piston). When applied across the device, a constant temperature difference between an external (low- to medium-grade) heat source and an external heat sink is converted into sustained and persistent oscillations of pressure and volumetric fluid displacement. These oscillations are transformed in a load arrangement into a unidirectional flow from which power is extracted by a hydraulic motor. The Up-THERM engine is modelled using a system of first order differential equations that describe the dominant thermal fluid processes in each component of the device. For certain components where the deviations from a linear approximation are non negligible (gas spring in the displacer cylinder, check valves and piston valve, and heat exchangers), a non-linear description is employed. A comparison between the linear and non-linear descriptions of the gas spring at the top of the displacer cylinder reveals that the non-linear description results in more realistic predictions of the oscillation frequency compared to experimental data from a similar device. Furthermore, the shape of the temperature profile over the heat-exchanger surfaces is modelled as following a hyperbolic tangent function, based on findings from an experimental investigation. Following the validation of these important device components, a parametric study is performed on the Up-THERM engine model with the aforementioned non-linear component descriptions, aimed at investigating the effects of important geometric parameters and of the heat-source temperature on key performance indicators, namely the oscillation frequency, power output and exergy efficiency of the engine. The results indicate that the geometric design of the displacer cylinder, including the height of the gas spring at the top of the cylinder, and the heat-source temperature have the most significant influence on the performance of the engine. A maximum exergy efficiency of 2.8% and a maximum power output of 175 W are observed at the proposed operating temperature of 450 degrees C for a nominal Up-THERM design (based on the physical dimensions of a device prototype and water as the working fluid; the role of the working fluid is explored in follow-up paper Ref. [1].) but with shorter displacer cylinder gas-spring lengths relative to a nominal design. The results and insight can assist the further development of this technology, in particular as a prime mover in combined heat and power applications. (C) 2016 The Authors. Published by Elsevier Ltd.