Feedback stabilisation of a pool-boiling system R.W. van Gils *,1,2 , M.F.M. Speetjens 2 and H. Nijmeijer 1 Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven Department of Mechanical Engineering, 1 Dynamics and Control, 2 Energy Technology Email: * r.w.v.gils@tue.nl 1 Introduction: Electronics cooling Cutting-edge technologies increasingly require the ability for massive heat removal. Pool boiling affords cooling ca- pacities substantially beyond that of conventional methods, and is as a result emerging as novel cooling technique. Pool boiling refers to boiling heat transfer by natural convec- tion and admits two stable modes: nucleate and film boil- ing, for low and high temperatures, respectively. Nucleate boiling is the desired state in cooling applications. How- ever, it transits into undesired film boiling if the heat gen- eration exceeds the so-called “critical heat flux” (CHF) [1]. This causes collapse of the cooling capacity. Hence, optimal cooling performance is a trade-off between efficiency (close to CHF) and low risk (safety margin to CHF). High uncertainty in predicting CHF and the inability to ac- tively respond to fluctuating cooling conditions result in large safety margins for current applications. Objective is diminishing this margin by stabilisation of the unstable equi- libria in the highly unstable transition regime. 2 Pool boiling model description The model considered, presented in [1], involves only the temperature distribution within the heater, Figure 1 (left). It models the heat exchange with the boiling medium via a nonlinear boundary condition imposed at the fluid-heater in- terface, which is given by the boiling curve, Figure 1 (right). 1 D ∂T ∂t =κ∇ 2 T boiling liquid: Λ ∂T ∂y = - Π2q F (T) y x 0 heat supply: Λ ∂T ∂y =q h + u(t) - Figure 1 Two-dimensional rectangular heater (left) and boiling curve (heavy) with the constant heat supply (dotted) (right). This model is spatially discretised using a Fourier-cosine and a Chebyshev-tau expansion in x- and y-direction, re- spectively [2]. This yields the following nonlinear ODE for T, which represents the temperature profile in the heater, ˙ T = A nl T + B nl v (u, T F ) , T F (x)= T (x, D)= CT, (1) with constant matrices and a nonlinear vector v(u, T F ). Be- sides a stable homogeneous (i.e. uniform T F , T F (x)= c) equilibrium in the nucleate and film boiling region, one ho- mogeneous and several heterogeneous unstable equilibria exist in the transition boiling regime [1]. Linearisation of (1) around the equilibrium T ∞ considered, allows analysis of the nonlinear model using standard control techniques. 3 Stabilisation of the system First the unstable homogeneous equilibrium is considered for which the system can be reduced to a one-dimensional (1D) (i.e. x-independent) system. The compact model is found fully controllable and observable and thus a feedback controller in combination with a linear identity observer is introduced. This linear controller-observer pair effectively stabilises the nonlinear system by regulating the heat sup- ply as a function of the internal state. In Figure 2 the inter- face temperature T F is given for small (left) and large (right) initial perturbations. Local asymptotic stability is proven (using Lyapunov), while global asymptotic stability is sug- gested by the large number of simulations with large initial perturbations (see Figure 2). Furthermore, these simulations put forth the state-feedback controller as a viable option for the rapid respond to fluctuating cooling conditions. 0 0.2 0.4 0.6 0.8 1 -20 -15 -10 -5 0 5 x 10 -3 t T F - T F, ∞ nonlinear model linearised model 0 0.2 0.4 0.6 0.8 1 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 t T F - T F, ∞ nonlinear model linearised model Figure 2 Fluid-heater interface temperature for small (left) and large (right) initial conditions. 4 Future Work Analyses reveal that some of the unstable equilibria of the pool-boiling system can be stabilised. Therefore future work should comprise: (i) finalisation of the analysis of the 2D system, (ii) extension to 3D pool-boiling system and (iii) experimental validation. References [1] M. SPEETJENS, et al. Steady-state solutions in a nonlinear pool-boiling model, Comm. Nonlin. Sci. Numer. Simul., 13 (2008), pp. 1475-1494. [2] R.W. VAN GILS, et al. 2008 Feedback stabilisation of a pool-boiling system. Submitted