*1.1. Flow Patterns in PHPs*

Whilst PHPs are drawing the attention of a growing number of research groups, including both experimental and numerical approaches, the industrialization of such technology is still in its preliminary phase, and examples of off-the-shelf PHPs are not yet common and limited to specific applications. The complex interplay of evaporation/condensation phenomena, surface tension, and inertial effects has been the object of several numerical investigations with the aim of developing a robust modeling tool. Nikolayev [24] developed one of the first models able to describe the chaotic self-sustained oscillations in a PHP with an arbitrary number of branches and arbitrary number of bubbles. Further improvements of the same model led to the implementation of the effect of the tube conductivity on the start-up phase [25] and the impact of the PHP orientation on the overall performance [26].

The operation of a PHP is strongly linked to the existence of a dominant slug/plug flow throughout the required range of operating conditions. Due to the variation of flow direction, pressure drop, and liquid film thickness in a PHP, several flow patterns have been observed [27], showing transitions between slug/plug, semi-annular, and annular flow (Figure 1). For a given geometry, the flow pattern is highly influenced by filling ratio and power input [28], due to the effect on the vapor quality, showing a higher ratio of bubble length over tube diameter [29]. As a result, the slug/plug flow pattern can transition into an annular flow, which in the long run can lead to a reduction in thermal performance and a stoppage of the oscillation due to critical drying out of the evaporator. The flow pattern has been extensively investigated in flow boiling in millimeter-scale channels, and it is the result of the interaction of interfacial, inertial, viscous, and gravitational forces. Without an exhaustive knowledge of the flow pattern, the correct thermal and hydraulic design parameters cannot be calculated properly. Despite the crucial role played, the majority of the available flow boiling pressure drop correlations have been formulated without reference to the flow pattern condition they covered [30]. It is also known that the available heat transfer correlations are very sensitive to the flow pattern condition [31]. Frequently, the expected flow pattern is roughly linked to the dimension of the channel. A rough classification proposed by Kandlikar [32] fixed 3 mm as the transition limit between macro-channels and micro-channels, not considering fluid properties, inertial effects, and gravity levels. In varying gravity conditions, a transition from a thermosyphon mode (semiannular dominant) to PHP mode (slug/plug dominant) impacts the thermal performance, operating range, and start-up power [33].

**Figure 1.** Flow patterns observed in the adiabatic section of a 1.6 mm ID PHP filled with ethanol during a series of experiments [9]. S: slug/plug; SA: semi-annular; A: annular.

One of the main limitations of the available numerical tools is the inability to define the dominant flow pattern given a set of operating conditions. The flow pattern is assumed a priori, and only static criteria are considered. Bond number (*Bo* = *ρgd*2/*σ*), or its relevant form considering the wettability through the contact angle (*θ* < 90◦) (*Bo* = *ρgd*2/(*σcosθ*)), and the confinement number (*Co* = 1/*Bo*1/2) are implemented to establish whether the initial existence condition for slug/plug flow are met. Once the motion is activated, there is no real control of the transitions of the flow pattern, mainly ignoring inertial effects. Break-ups and coalescence events were reviewed in a numerical investigation from Andredaki et al. [34]. An approach to the development of flow pattern maps for oscillating flows based on dimensionless numbers was proposed by Pietrasanta et al. [29], drawing the attention to break-up and coalescence phenomena in a simplified PHP loop, suggesting the effective use of the actual bubble acceleration rather than the static, nominal *g* value (i.e., gravitational acceleration) and the actual bubble velocity to describe the transition between slug/plug and semi-annular flow. This last methodology, even if much more accurate than the use of Bond number and other dimensionless numbers such as Weber or Reynolds number, has the disadvantage that it cannot be used for design purposes, but only for a posteriori validation of numerical codes.

Therefore, despite the great effort shown so far, the development of comprehensive design tools, validated over a wide range of operating conditions and able to assist thermal engineers, is not yet complete.
