Nonlinear analysis of tilted toroidal thermosyphon models Arturo Pacheco-Vega a , Walfre Franco a , Hsueh-Chia Chang b , Mihir Sen a, * a Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556-5637, USA b Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA Received 27 December 2000; received in revised form 10 August 2001 Abstract We analyze one-dimensional models for single-phase tilted toroidal thermosyphons for three different heating conditions: known heat flux, known wall temperature and mixed heating. For the first two the governing equations lend themselves to exact reduction to a set of three ordinary differential equations, while for the third the equations remain coupled as an infinite set. For all three cases, the tilt angle is stabilizing while the heat rate is a destabilizer. A nonlinear analysis is carried out using center manifold theory and normal form analysis. The known heat flux solutions lose stability through a supercritical Hopf bifurcation, while for the other two heating conditions the Hopf bifurcation is supercritical under some conditions and subcritical under others. Stable limit-cycle oscillations exist only for the supercritical cases, otherwise instability leads directly to chaos. Analysis also provides an estimate for the amplitude of oscillation for the supercritical conditions. Numerical experiments have confirmed the theoretical predictions quali- tatively and quantitatively. Ó 2002 Elsevier Science Ltd. All rights reserved. 1. Introduction Single-phase natural convective loops or thermosy- phons are used in a variety of engineering applications, suchasnuclearreactorcooling,solarcollectors,etc.[1–4]. They have also been studied because they provide an ex- cellent theoretical introduction to convection in more complex geometries. Experiments have been carried out with toroidal [5–7] and rectangular [8] loops. Of these [6] and [8] worked with variable loop inclinations. Most an- alytical models are one-dimensional in the sense that the fluid velocity and temperature are averaged over a cross- section. These models have been especially significant since it has been found that, for certain geometries and undercertainthermalconditions,thegoverningequations can be reduced to a set of three ordinary differential equations that can exhibit chaotic behavior [9–11]. Due to the variety of geometries that can be used and thermal conditions that can be applied, comparison between theory and experiment is approximate unless the same conditions are considered for both. It is sometimes assumed that all thermal conditions and loop geometries give similar results; this may perhaps be true for overall qualitative behavior, but certainly not in the details. The occurrence of chaos in these mathematical models has been an aspect that has attracted the atten- tion of many researchers, and certainly all of them, under specific conditions, do that. The focus on chaos has obscured the fact that the ge- ometry and heating condition significantly affect the de- tailsofsystembehaviorandthereissomeconfusioninthe literature in this regard. For example, some experiments [5,8] have shown the presence of stable oscillations while the most commonly used model based on the Lorenz equations does not show the existence of stable limit cy- cles [9]. The existence or not of oscillations is important for the design of convection loops in actual applications. In this work we will clarify the situation by choosing the simplest geometry possible, the torus, and apply the commonly used heating conditions. In each case, we will be interested in the effect of nondimensional parameters corresponding to heating and tilt angle. Each of the heating conditions will be examined separately though in a very similar manner. The steady states will be found and their linear stability determined. International Journal of Heat and Mass Transfer 45 (2002) 1379–1391 www.elsevier.com/locate/ijhmt * Corresponding author. Tel.: +1-219-631-5975; fax: +1-219- 631-8341. E-mail address: mihir.sen.1@nd.edu (M. Sen). 0017-9310/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII:S0017-9310(01)00265-4