Stability Analysis of Two-Phase Cooling Systems

Effective and efficient removal of heat is of critical importance in a wide range of applications, from human comfort to high power electronic devices and high efficiency photovoltaic arrays. This paper considers the stability and control of a cooling system operating in the two-phase regime. Two-phase cooling uses boiling to transfer heat from the source to the working fluid. Compared with the usual single-phase operation which relies on the temperature differential for heat transfer, boiling has superior efficiency by taking advantage of the latent heat of vaporization. However, it is known that two-phase operation is susceptible to various types of instability. We consider a cooling cycle consisting of two heat exchangers, an evaporator to extract heat from the source and a condenser to dissipate the heat to the ambient environment, and a pump and valves to regulate the flow. The heat exchangers are modeled by one-dimensional partial differential equations based on mass balance, momentum balance and energy balance. By using a novel Lyapunov function and impose the condition that the thermal subsystem time constant (in terms of enthalpy) is much slower than the fluid subsystem (in terms of pressure and mass flow rate), we obtain explicit stability condition in terms of the pressure demand curves (pressure drop as a function of the mass flow rate) of the heat exchangers. Using this Lypunov funciton, we derive passivity conditions for the fluid system which may in turn be used for flow stabilization. The general Lyapunov framework presented here may be extended to the stability analysis and control design of more complex thermodynamic systems.