Feedback stabilisation of pool-boiling systems : for application in thermal management schemes

Abstract

The research scope of this thesis is the stabilisation of unstable states in a pool-boiling system. Thereto, a compact mathematical model is employed. Pool-boiling systems serve as physical model for practical applications of boiling heat transfer in industry. Boiling has advantages over conventional heat-transfer methods based on air- or single-phase liquids by enabling extremely high heat-transfer rates at isothermal conditions. This o?ers solutions to thermal issues emerging in cutting-edge technologies as semi-conductor manufacturing and electric vehicles (EVs). Continuous miniaturisation in micro-electronics is pushing heat-?ux densities beyond the limits of standard cooling schemes and growing architecture complexity makes thermal uniformity during chip manufacturing increasingly critical. Further development of EVs may bene?t equally from boiling heat transfer by its utilisation for actuator cooling and thermal conditioning of battery packs. A pool-boiling system consists of a heater that is submerged in a pool of boiling liquid. The theater is the to-be-cooled device (or a thermally conducting element between the device and the boiling liquid) and is heated at its bottom. On top of the heater, heat is extracted by the boiling liquid. In order to exploit boiling to its fullest e?ciency, unstable modes need to be stabilised to avoid the formation of a thermally-insulating vapour ?lm on the heater that causes collapse of the cooling capacity and that heralds a dangerous and ine?cient mode of boiling. The pool-boiling model comprises a partial di?erential equation (PDE), i.e. the well- known heat equation, and corresponding boundary conditions that represent adiabatic sidewalls, a uniform heat supply at the bottom, and a nonuniform and nonlinear heat extraction at the heater top. This nonlinear boundary condition renders the entire model nonlinear, resulting in multiple equilibria and complex and exciting dynamics. Restriction to uniform temperature distributions within the heater admits description by a model of one spatial dimension (1D). The 1D model is investigated mathematically and the results are compared with those found by the analyses of spatial-discretisations of the model. Two spatial-discretisation schemes, based on a ?nite-di?erence method and a spectral method, are investigated. The latter shows far better convergence properties than the former. Moreover, application of full state feedback of the spectral modes (modal control) results in signi?cantly better properties than by regulation via standard P-control. In practical applications, the heater temperature can only be measured at the heater top. Consequently, an observer is implemented that estimates the spectral modes of the temperature within the heater, which are subsequently used in the feedback-law. The e?ciency and performance of this controller-observer combination is examined by numerical simulations. A pool-boiling system with an electrically heated wire as heater can be represented by the model as described above, but now with two spatial dimensions (2D). The 2D model can be analysed mathematically only for uniform equilibria, i.e. the equilibria that exist also for the 1D system. For nonuniform equilibria, the mathematical analysis becomes too complex and a spatial discretisation is required to obtain results. A 1D characteristic equation on the ?uid-heater interface can be obtained by analytical reduction of the 2D eigenvalue problem using the method of separation of variables. The system poles follow from spatially discretising this equation. Because of its outstanding performance for the 1D model, the 2D model is again stabilised by a modal controller (full state feedback) in combination with an observer. Simulations are again performed to determine the e?ciency of the controller-observer combination. If a thermally conducting foil is considered as heater, the three-dimensional (3D) form of the model must be investigated. This involves essentially the same methodology as described above, resulting in a 2D characteristic equation on the ?uid-heater interface. However, spatial discretisation of this equation yields large system matrices and requires excessive calculation times. Hence, the 3D system is analysed only at moderate discretisation orders. The above modal control strategy is, as before, applied in combination with an observer to stabilise unstable equilibria and the evolution of the nonlinear system is again investigated and demonstrated by way of simulations. Finally, a series of exploratory experiments, to investigate the application of pool-boiling to thermally condition battery cells in EVs, is considered. Experiments are performed to investigate the ability for thermal homogenisation of the boiling process and the ability to manipulate the boiling process via the pressure in the boiling chamber. Furthermore, the application of pool-boiling to overcome thermal issues in high-end technologies is investigated by numerical simulations.