Steady-State Behavior of a Three-Dimensional Pool-Boiling System

[+] Author and Article Information
Michel Speetjens

Energy Technology Laboratory, Mechanical Engineering Department, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlandsm.f.m.speetjens@tue.nl

J. Electron. Packag 130(4), 041102 (Nov 13, 2008) (7 pages) doi:10.1115/1.2993147 History: Received August 31, 2007; Revised March 29, 2008; Published November 13, 2008

Pool-boiling serves as the physical model problem for electronics cooling by means of phase-change heat-transfer. The key for optimal and reliable cooling capacity is better understanding of the conditions that determine the critical heat-flux (CHF). Exceeding CHF results in the transition from efficient nucleate-boiling to inefficient film-boiling. This transition is intimately related to the formation and stability of multiple (steady) states on the fluid-heater interface. To this end, the steady-state behavior of a three-dimensional pool-boiling system has been studied in terms of a representative mathematical model problem. This model problem involves only the temperature field within the heater and models the heat exchange with the boiling medium via a nonlinear boundary condition imposed on the fluid-heater interface. The steady-state behavior is investigated via a bifurcation analysis with a continuation algorithm based on the treatment of the model with the method of separation of variables and a Fourier-collocation method. This revealed that steady-state solutions with homogeneous interface temperatures may undergo bifurcations that result in multiple solutions with essentially heterogeneous interface temperatures. These heterogeneous states phenomenologically correspond with vapor patches (“dry spots”) on the interface that characterize transition conditions. The findings on the model problem are consistent with laboratory experiments.

Copyright © 2008 by American Society of Mechanical Engineers
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Figure 1

Typical phase-change cooling system for electronics devices. The actual cooling section (bottom part) is in essence a pool-boiling system (from Ref. 1).

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Figure 2

Nondimensional model problem: heater configuration (a) and heat-flux function (b) for W=1 and Π1=4. The stars indicate the local maximum (critical heat-flux) and local minimum (Leidenfrost point). The dashed line represents the normalized heat supply (q¯H/Π2=Π2−1).

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Figure 3

Introduction of nonlinearity to the heat-flux function via the nonlinearity parameter λ. Shown is the transition from linear (λ=0) to physical (λ=1) profile. The arrow indicates increasing λ.

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Figure 4

Homogeneous solutions of the nonlinear system as a function of the nonlinearity parameter λ. Transition from the single-solution state (a) to the triple-solution state (b) takes place via the tangent bifurcation occurring at λ=λ∗ (not shown).

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Figure 5

Pairwise emergence of typical two-cluster (TF,TF∗,1) of heterogeneous solutions from the bifurcating homogeneous solution TF(2). Shown is the state of the interface temperatures at three consecutive λ-values in the direct proximity of the associated pitchfork bifurcation at λ≈0.9761. More pronounced crests and troughs correspond with increasing λ. The solutions relate via TF∗,1(x,y)=TF(x+1/2,y)=TF(x,y+1/2).

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Figure 6

Physically meaningful (λ=1) state of the two-cluster of heterogeneous solutions shown in Fig. 5

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Figure 7

Physically meaningful state (λ=1) of a typical four-cluster of heterogeneous solutions. Parent and dual solutions relate via symmetry relations 16.

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Figure 8

Physically meaningful (λ=1) state of the parent solutions TF of other four-clusters of heterogeneous solutions. Dual solutions corresponding with TF (not shown) are for the top and bottom rows given by symmetry relations 17,18, respectively.



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