Quantitative synchronization criteria for flow boiling in parallel microchannels: coupling-coefficient thresholds, design mapping, measurement, and validated actuation

Flow boiling in parallel microchannels enables high-heat-flux cooling but suffers flow maldistribution and self-sustained flow oscillations. Jin, Wen and Narayanan (2020) showed, with a lumped two-channel model and experiments, that flow oscillations of sufficient amplitude synchronize the temperatures of parallel channels coupled thermally (heat conduction through the shared wall, coefficient $U_t$) and/or hydraulically (common inlet/outlet manifolds, coefficient $U_p$), turning oscillation into a maldistribution-mitigation tool — leaving open the quantitative problem: determine the threshold values and ranges of $U_t$ and $U_p$ that yield synchronization for given evaporator geometry and operating conditions; derive explicit mappings from channel geometry, wall thickness, wall thermal conductivity, manifold design and local convective heat transfer to $U_t$ and $U_p$; measure per-channel instantaneous mass flow rates during oscillatory operation; extend the one-zone, channel-averaged model to capture spatial variations; and validate valve/pump actuation strategies.

Background

Microchannel evaporators remove very high heat fluxes, such as those of power electronics and high-performance computing chips, by flow boiling: a pumped liquid coolant vaporizes inside many parallel channels of sub-millimeter scale, absorbing latent heat. Boiling in parallel channels produces two characteristic pathologies. First, flow maldistribution: because the pressure-drop-versus-flow curve of a boiling channel can be non-monotonic, channels sharing inlet and outlet manifolds can settle into asymmetric steady states in which some channels are starved of liquid — the classical static Ledinegg instability (flow excursion). Second, dynamic instabilities: when the loop contains a compressible volume (an accumulator or trapped vapor), the boiling channel can interact with it in a self-sustained limit cycle (a periodic oscillation the system maintains without external forcing) known as pressure-drop oscillation, a large-amplitude, low-frequency oscillation of flow rate, pressure and wall temperature. Both produce uneven wall temperatures, premature dryout (early loss of liquid contact with the channel wall), and degraded cooling.

Two neighboring literatures address these instabilities. For the static problem, models of two-phase flow distribution that include lateral heat conduction through the solid walls between channels — the thermal coupling between channels — show that wall conduction redistributes heat away from flow-starved channels and suppresses maldistribution: Van Oevelen, Weibel and Garimella (2018) predicted, and Miglani, Weibel and Garimella (2021) confirmed experimentally with per-channel flow-rate measurement, that thermally coupled parallel channels maldistribute only beyond a critical heat flux that rises with increasing lateral wall conductance. For the dynamic problem, an experimental line (Rafałko et al., 2020-2026), using high-speed imaging and recurrence analysis (a nonlinear time-series method that detects whether two fluctuating signals visit shared dynamical states), documented that boiling in neighboring minichannels can synchronize — flow patterns and pressure fluctuations lock into common dynamics — with the degree of synchronization depending on the prevailing flow pattern.

Jin, Wen and Narayanan (2020) connected these threads. They studied an evaporator of two parallel channels that interact (i) thermally, by heat conduction through the shared wall between them, summarized by an effective thermal-coupling coefficient $U_t$, and (ii) hydraulically, through the common inlet and outlet manifolds, summarized by a flow-coupling coefficient $U_p$. Operationally, $U_t$ measures the strength of the thermal interaction — the heat exchanged between neighboring channels through the shared wall per unit temperature difference between them — and $U_p$ measures the strength of the flow interaction — the extent to which a flow disturbance in one channel, such as a sudden change in flow rate or pressure, is transmitted to the other through the common manifolds; both are effective parameters of the channel-averaged description and are, in general, state-dependent during oscillations rather than fixed constants. A lumped, one-zone model — meaning each channel is treated as a single well-mixed control volume with properties averaged over its length, with vapor quality (the vapor mass fraction of the two-phase mixture) treated as a channel average — predicts when the two channels' self-sustained boiling oscillations synchronize. Their experiments on thermally coupled versus decoupled channel pairs showed that flow oscillations of sufficient amplitude synchronize the channel temperatures, in the sense of reducing the wall temperature difference between channels, when the channels are thermally coupled, flow coupled, or both; they proposed deliberately allowing controllable flow oscillations as a simple strategy to mitigate maldistribution.

This made synchronization a potential design tool — but an empirical one. What remains missing to make it quantitative: threshold values and ranges of $U_t$ and $U_p$ required for synchronization for given evaporator geometries and operating conditions; explicit relations mapping channel geometry, wall thickness, wall thermal conductivity, manifold design and local convective heat transfer onto the effective coupling coefficients used in the lumped model; direct measurement of the instantaneous mass flow rate of each channel during oscillatory operation (their testbed inferred flow redistribution only indirectly, from temperatures and pressures); extension of the model beyond the one-zone channel average to capture spatial variations along the channel, including the subcooled inlet region (liquid cooled below its saturation temperature) and transient local variations of the heat transfer coefficient; and actuation strategies (valve or pump modulation) validated to produce desirable oscillation amplitudes and frequencies without unacceptable reliability or performance penalties.

Problem Statement

The problem arising from the results of Jin, Wen and Narayanan (2020), for a microchannel evaporator with parallel boiling channels, is to convert the observed temperature-synchronization phenomenon into quantitative, design-usable knowledge, so that synchronization can be predicted from geometry and operating conditions and actuated beneficially. Its linked components are:

(1) Determine the threshold values and ranges of the thermal-coupling coefficient $U_t$ and the flow-coupling coefficient $U_p$ within which the flow-boiling oscillations of the parallel channels synchronize so as to equalize channel temperatures — that is, establish the synchronization boundary in the ($U_t$, $U_p$) plane and how it depends on evaporator geometry and operating conditions (heat load, total flow rate, inlet subcooling). The $U_t$ and $U_p$ of this boundary are the effective scalar coefficients of the one-zone, channel-averaged model that characterize the two coupled channels at the stated operating conditions — their nominal, cycle-averaged values at that operating point — so that the threshold is a well-defined locus in the plane. The instantaneous, state-dependent variation of these coefficients along an oscillation cycle is the subject of the model extension in component (4), not of the boundary itself.

(2) Derive explicit relations that map the physical design parameters — channel geometry, wall thickness, wall thermal conductivity, manifold design, and local convective heat transfer — onto the effective coupling coefficients $U_t$ and $U_p$ used in the lumped model, so that the thresholds from (1) become predictable design quantities rather than fitted constants.

(3) Measure, experimentally, the instantaneous mass flow rate of each channel during oscillatory operation, in order to directly quantify how the flow redistributes on average as synchronization develops, replacing the present indirect inference from temperatures and pressures.

(4) Extend the theoretical model beyond the one-zone, channel-averaged approximation so that it captures spatial variations along the channels — including the subcooled inlet region and local variations of the heat transfer coefficient during transients — and thereby improves quantitative agreement with the observed synchronization dynamics.

(5) Design and validate controlled actuation strategies (valve or pump modulation) that generate oscillations of desirable amplitude and frequency, as predicted by the criteria of (1)-(2), without introducing unacceptable reliability or performance penalties.

These components are complementary and serve a single objective — turning the observed synchronization phenomenon into robust design and control prescriptions for practical microchannel evaporators. Components (1)-(2), together with the model extension of (4), form the quantitative core: the synchronization boundary and its geometry-to-coefficient mapping can be established theoretically or by simulation, and a complete answer to them resolves the stated objective. Components (3) and (5) require physical experiment and validated hardware; they are the experimental realization and the design validation that the quantitative core is meant to enable, and they do not gate resolution of a theoretical or simulation answer.

References

1. Qi Jin, John T. Wen, Shankar Narayanan, 'Temperature synchronization across parallel microchannels during flow boiling', International Journal of Thermal Sciences 156 (2020) 106476. DOI: 10.1016/j.ijthermalsci.2020.106476. https://doi.org/10.1016/j.ijthermalsci.2020.106476
2. Qi Jin, John T. Wen, Shankar Narayan, 'Oscillatory valve effect on temperature synchronization in microchannel cooling systems', Applied Thermal Engineering 204 (2022) 117999. DOI: 10.1016/j.applthermaleng.2021.117999. https://doi.org/10.1016/j.applthermaleng.2021.117999
3. Qi Jin, 'Limit cycle analysis and maldistribution mitigation for multi-channel cooling system', Applied Thermal Engineering 204 (2022) 117982. DOI: 10.1016/j.applthermaleng.2021.117982. https://doi.org/10.1016/j.applthermaleng.2021.117982
4. Qi Jin, Yanshun Yu, Yaobiao Xia, 'Oscillatory valve effect on pressure drop oscillation in microchannel cooling system', International Journal of Heat and Mass Transfer 206 (2023) 123931. DOI: 10.1016/j.ijheatmasstransfer.2023.123931. https://doi.org/10.1016/j.ijheatmasstransfer.2023.123931
5. Qi Jin, John T. Wen, Shankar Narayanan, 'The analysis and prediction of pressure drop oscillation in phase-change cooling systems', International Journal of Heat and Mass Transfer 165 (2021) 120621. DOI: 10.1016/j.ijheatmasstransfer.2020.120621. https://doi.org/10.1016/j.ijheatmasstransfer.2020.120621
6. Qi Jin, John T. Wen, Shankar Narayanan, 'Experimental study and mitigation of pressure drop oscillation using active control', Journal of Electronic Packaging 143 (4) (2021) 041102. DOI: 10.1115/1.4051942. https://doi.org/10.1115/1.4051942
7. Qi Jin, Charles C. Okaeme, John T. Wen, Shankar Narayanan, 'A comparison of finite element and lumped modeling techniques to analyze flow boiling in microchannels', IEEE Transactions on Components, Packaging and Manufacturing Technology 11 (10) (2021) 1655-1667. DOI: 10.1109/TCPMT.2021.3095148. https://doi.org/10.1109/TCPMT.2021.3095148
8. Tijs Van Oevelen, Justin A. Weibel, Suresh V. Garimella, 'The effect of lateral thermal coupling between parallel microchannels on two-phase flow distribution', International Journal of Heat and Mass Transfer 124 (2018) 769-781. DOI: 10.1016/j.ijheatmasstransfer.2018.03.073. https://doi.org/10.1016/j.ijheatmasstransfer.2018.03.073
9. Ankur Miglani, Justin A. Weibel, Suresh V. Garimella, 'An experimental investigation of the effect of thermal coupling between parallel microchannels undergoing boiling on the Ledinegg instability-induced flow maldistribution', International Journal of Multiphase Flow 139 (2021) 103536. DOI: 10.1016/j.ijmultiphaseflow.2020.103536. https://doi.org/10.1016/j.ijmultiphaseflow.2020.103536
10. Ankur Miglani, Justin A. Weibel, Suresh V. Garimella, 'Measurement of flow maldistribution induced by the Ledinegg instability during boiling in thermally isolated parallel microchannels', International Journal of Multiphase Flow 139 (2021) 103644. DOI: 10.1016/j.ijmultiphaseflow.2021.103644. https://doi.org/10.1016/j.ijmultiphaseflow.2021.103644
11. Gabriela Rafałko, Iwona Zaborowska, Hubert Grzybowski, Romuald Mosdorf, 'Boiling synchronization in two parallel minichannels—image analysis', Energies 13 (2020) 1409. DOI: 10.3390/en13061409. https://doi.org/10.3390/en13061409
12. Gabriela Rafałko, Hubert Grzybowski, Paweł Dzienis, Iwona Zaborowska, Romuald Mosdorf, Grzegorz Litak, 'Recurrence analysis of phase distribution changes during boiling flow in parallel minichannels', The European Physical Journal Special Topics 232 (2023) 201-207 (published online December 2022). DOI: 10.1140/epjs/s11734-022-00741-0. https://doi.org/10.1140/epjs/s11734-022-00741-0
13. Gabriela Rafałko, Hubert Grzybowski, Paweł Dzienis, Romuald Mosdorf, Iwona Zaborowska, Grzegorz Litak, 'Multichannel flow boiling maldistribution and synchronization', Experimental and Computational Multiphase Flow (2026). DOI: 10.1007/s42757-026-0283-4. https://doi.org/10.1007/s42757-026-0283-4
14. Luiz A. Silva, Fernando Lizarralde, 'Synchronization on Van der Pol oscillators and application in micro-thermal-fluid cooling systems', Proceedings do Simpósio Brasileiro de Sistemas Elétricos (2023). DOI: 10.20906/sbai-sbse-2023/3775. https://doi.org/10.20906/sbai-sbse-2023/3775
15. Luiz A. Silva, Fernando Lizarralde, Alessandro J. Peixoto, 'Smooth sliding control of Van der Pol oscillators with a single input: application to micro-thermal-fluid cooling systems', IEEE Control Systems Letters 7 (2023) 3932-3937. DOI: 10.1109/LCSYS.2023.3343432. https://doi.org/10.1109/LCSYS.2023.3343432
16. Toochukwu Aka, Shankar Narayan, 'Using entropy balance to determine multiphase flow distribution in thermally decoupled parallel channels with shared inlet and outlet headers', Physics of Fluids 36 (2024) 054118. DOI: 10.1063/5.0207373. https://doi.org/10.1063/5.0207373
17. Joseph Rendall, Tugba Turnaoglu, Viral K. Patel, 'Experimental results of a magnetically coupled piezoelectric actuator to relieve microchannel heat exchanger maldistribution', International Communications in Heat and Mass Transfer 133 (2022) 105944. DOI: 10.1016/j.icheatmasstransfer.2022.105944. https://doi.org/10.1016/j.icheatmasstransfer.2022.105944