# Measurement of heat transfer and mixing enhancement by chaotic advection: Experimental methods

Nouhaila El Hani, Tom Lacassagne, Souria Hamidouche, S. Amir Bahrani

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Nouhaila El Hani, Tom Lacassagne, Souria Hamidouche, S. Amir Bahrani. Measurement of heat transfer and mixing enhancement by chaotic advection: Experimental methods. Review of Scientific Instruments, 2025, 96, pp.061501. 10.1063/5.0254063. hal-05114940

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# Measurement of heat transfer and mixing enhancement by chaotic advection: Experimental methods

Nouhaila El Hani,¹ Tom Lacassagne,¹ Souria Hamidouche,¹ and S. Amir Bahrani¹

IMT Nord Europe, Institut Mines Télécom, Univ. Lille, Center for Energy and Environment, Lille, 59000, France

(*Electronic mail: amir.bahrani@imt-nord-europe.fr)

(Dated: 16 May 2025)

The enhancement of heat and mass transfer is the key to design energy-efficient industrial devices with the lowest environment impact, a challenge for sustainable development and energy transition. Several approaches can be used, either by extending the transfer surface and/or by acting on the fluid flow to increase the heat transfer rate, in passive or active manners. In this review, the focus is placed on one type of passive technique, known as chaotic advection, which generates unsteady and irregular fluid motion at relatively low flow rate, with negligible additional energy input, thus enhancing mixing and mass transfer. Chaotic advection is used in a wide range of applications and fields, including fluid mixing, heat exchangers and microfluidics. In this context, experimental methods offer a direct means of observing, quantifying and unravelling the complexity of fluid dynamics coupled with heat transfer, and ultimately finding effective strategies for intensifying exchanges. The main purpose of this review paper is to summarize experimental techniques used in the literature, both local and global, enabling an in-depth examination of the interplay between chaotic advection, heat transfer and mixing efficiency. To this end, a compilation of methods for visualizing and quantifying transfer mechanisms is detailed in this review. They give access to local velocity and scalar fields, and help depict chaotic transfer mechanisms. Several global parameters and performance coefficients are also defined, providing valuable information on overall thermohydraulic efficiency. The advantages and drawbacks of the different experimental techniques presented are discussed. Finally, this review also discusses some promising experimental techniques that have not yet been used in the literature, in the context of chaotic exchangers, but hold the potential to deepen our understanding of both external and internal mixing and thermal behavior of such systems.

## I. INTRODUCTION

Many industrial processes rely heavily on two essential pieces of equipment: heat exchangers and mixers. Heat exchangers are designed to transfer heat between two or more fluids, either by direct or indirect contact. In direct contact heat exchangers, the fluids physically mix or come into direct contact with each other, while indirect heat exchangers use a separating wall to prevent mixing¹. Heat exchangers are used in a variety of applications, including food and pharmaceuticals production², automotive³, air conditioning⁴ and solar energy systems⁵. Mixers, on the other hand, are devices that enable the uniform distribution of different substances by combining them into a single phase or mixture. They are essential in industries such as food and beverage production⁶⁻⁹ and pharmaceuticals and cosmetics⁶,⁷,¹⁰.

The efficiency of industrial processes strongly depends on how mixers and exchangers are able to promote efficient heat transfer and the creation of homogeneous mixtures essential to achieving the desired product properties and quality. To further improve productivity and efficiency, process intensification techniques are employed in both types of equipment. These techniques aim to improve heat transfer rates and mixing efficiency, resulting in significant advances in productivity, product quality and reduced energy consumption. In heat exchangers, intensification methods often focus on increasing the surface area available for heat transfer. This can be achieved by various means, such as microchannel designs, the implementation of finned surfaces and different types of inserts such as twisted tapes¹¹,¹². These improvements can

significantly increase heat transfer rates and, in some cases, reduce the size and cost of the heat exchanger. For mixers, process intensification often involves the use of high-speed impellers, static mixing elements or various configurations of inserts and baffles¹³⁻¹⁵. These approaches are designed to optimize the mixing process, promoting a more homogenous distribution of components and improving overall operational efficiency. In heat and mass exchange systems, two important factors affect how well they work: the convective exchange coefficient and the surface area available for exchanging heat or mass. The challenge, here, is to decide whether to improve how efficiently the exchange happens in a given area or to increase the total area where substances can interact. This choice depends on the specific requirements and constraints of each system. For example, when improving a chemical reactor, the focus might be on getting the best exchange rate in a small space, while for large industrial heat exchangers, the goal might be to increase the overall surface area. Finding the right balance between these two factors is key to achieving efficiency. This balance is important for ensuring energy efficiency, speeding up processes, and maintaining product quality. Some industrial processes face unique heat transfer and mixing challenges, particularly in situations involving highly viscous fluids or where conservation of energy requires lower flow rates. These conditions often result in laminar flow regimes, which can limit the efficiency of heat exchange and mixing. To overcome these limitations, research has focused on developing innovative intensification strategies for low Reynolds number regime. One promising approach is to apply the principles of chaotic advection. This method involves introducing specially designed geomet-
ric features or patterns into heat exchangers and mixers. The general principles of chaotic advection in curved tubes has been extensively discussed in the literature<sup>16-18</sup>; in brief, carefully designed bended structures induce complex and chaotic inner flows, even under laminar flow conditions. By promoting these complex fluid motions, these designs can significantly improve heat transfer and mixing efficiency, offering a new solution to long-standing process engineering challenges. Recent research has highlighted the effectiveness of chaotic advection techniques in improving the performance of heat exchangers<sup>10,19-26</sup> and mixers<sup>27-33</sup>. Yet, and despite experimental evidence of the efficiency of chaotic advection<sup>34,35</sup>, the complex mechanisms governing it remain to be fully understood if one wants to model, predict, and optimize the performances of chaotic advection heat exchangers and mixers, a prerequisite for their popularization in the industry. This gap in understanding highlights the need for experimental methods, as they provide a concrete means of observing and analyzing the complex fluid dynamics and transport phenomena associated with chaotic advection.

In order to characterize the heat transfer and mixing enhancement, many local or global experimental techniques have been developed. These techniques involve the measurement of various parameters such as temperature, pressure and fluid velocity to assess the effectiveness induced by chaotic advection. Under favorable conditions, that is to say most of the time in lab-scale prototype exchangers, optical measurement techniques are useful to characterize chaotic advection due to their non-intrusive nature and their high temporal and spatial resolution. They can provide precision information about the velocity gradients and other flow characteristics which are necessary to model fluid behavior and understand many aspects of fluid dynamics, including chaotic advection. Optical techniques provide local experimental data that can be used to validate and verify computational Fluid Dynamics (CFD) models on well-chosen cases and then use the models for an optimization to identify the optimal exchange conditions. The following sections of this paper aim at providing a comprehensive and detailed review of the experimental techniques used to characterize heat and mass transfer enhancement by chaotic advection. The review is structured as follows: local and global characterization techniques are described in sections II and III, respectively; section IV is devoted to a discussion of potentially interesting experimental techniques, not used in the literature, and the conclusion in section V includes a summary of the insights drawn from the previous sections.

## II. LOCAL CHARACTERIZATION TECHNIQUES

### A. Overview

Local characterization techniques for fluid mechanics (36,37) allow investigation of the flow behavior in geometries of most of the time fundamental, as well as of sometimes industrial, interest. Among these techniques we find laser Doppler velocimetry (LDV), particle image velocimetry (PIV)

and laser induced fluorescence (LIF) that were used in several experimental works in the literature. Those optical techniques are minimally intrusive, as they only require seeding the fluid with either particles (LDV, PIV) or a dye (LIF), at low concentrations, and no insertion of physical probes in the flow. They provide local information on the heat and mass transfer, by giving access to velocity, temperature or concentration fields or profiles, without unduly affecting the studied process. In most cases, velocity measurements are performed using either LDV as a point measurement technique or PIV as a whole field technique. For the measurement of local scalar fields in the liquid, such as passive scalar concentration or temperature at high spatial resolution, LIF is the most common and appropriate technique. The selection of a technique depends on its ability to provide measurement with high resolution in both time and space domains. The use of these optical techniques is limited to use in transparent geometries. These methods, indeed, depend on the transmission and detection of light, making them effective only in scenarios where light can pass through the material under study. Those optical techniques are among the most accurate flow measurement techniques. However, they are subject to potential source of errors<sup>36,38</sup>, as discussed below. For an exhaustive explanation of experimental methods, the reader is advised to consult these few reference books:<sup>36,37</sup>

### B. Visualization, colorimetry and fluorescence imaging

Flow visualization is a set of techniques used to observe and study the flow of fluids in a system. The main asset of flow visualization methods is that they allow understanding the flow pattern and identify vortical structures and to figure out how it contributes to the mixing process. One common principle used for flow visualization is the use of dyes or particles added to the fluid. These tracers are typically small and highly visible, making it possible to track the flow patterns and observe any mixing or separation. The movement of the tracers can be captured using imaging techniques such as photography, video, or microscopy, and analyzed to quantify the flow properties.

Schonfeld et al.<sup>39</sup> used transmitted light microscopy for flow-pattern visualization, enabling the examination and imaging of dyed solutions to evaluate mixing performance of the optimized split and recombine (SAR) micro-mixer. They validated both numerically and experimentally the SAR mixing concept. The study of Bringer et al.<sup>40-42</sup> used a microfluidic device that relies on chaotic advection, in order to mix multiple reagents in droplets. The geometry caused the droplets to undergo repeated stretching and folding, leading to chaotic advection and enhanced mixing. The droplets were labeled with a fluorescent dye to enable visualization of the mixing process, and the mixing was characterized by analyzing the intensity distribution of the fluorescent signal. Flow visualization is a powerful tool for evaluating micromixer performance, as it enables direct observation of the mixing process and the identification of regions with incomplete mixing. The work of Sudarsan et al.<sup>43</sup> describes an experimental study of fluid
mixing in a planar spiral channel, which is a type of microfluidic device that consists of a spiral-shaped channel etched onto a flat surface. They investigated five spiral designs by varying channel lengths. The mixing was characterized by analyzing the mixing intensity of the green color that was generated when the two streams (blue and yellow dyes) mixed. At higher flow rates, secondary Dean effects become active and help to increase mixing levels. As the fluid moves downstream within the spiral contours, it experiences an increase in the magnitude of centrifugal forces accompanied by a corresponding improvement in mixing performance. For flow visualization, fluorescence microscopy is used by⁴⁴ in order to visualize the mixing process and quantitatively evaluate the mixing performance of the device, which is in the form of a variable pattern of grooves on the floor of the channel. The grooves generate a transverse flow and counter-rotating elliptical trajectories. The mixture improves after each cycle. The study of Tekin et al.⁴⁵ presents a microfluidic mixing device that uses source-sink flows to generate chaotic advection and enhance mixing efficiency. Several protocols of operation of source-sink pairs were studied and the mixing performance was characterized by analyzing the intensity profiles of the chamber from the obtained images. In the optimized protocol, they found that the mixing is realized in just a sub-second time interval, thanks to the implementation of chaotic advection. Liu et al.⁴⁶ have studied the mixing in a three-dimensional serpentine microchannel design with a “C-shaped” repeating unit. The mixing ability of this mixer is quantified by optically determining how much of the pH indicator (phenolphthalein) used changes color during the mixing process. Ohkawa et al.⁴⁷ have investigated the flow and the mixing characteristics of a static mixer composed of σ-shaped elements. Mixing was visualized by using the decolorizing reaction of iodine with sodium. They found that at Re > 10, secondary flows are generated due to the two- and three-dimensional curves of the mixer, which distort the systematic splitting and recombination. They contribute to accelerate the mixing progress by deforming and elongating the interfaces between the fluids to be mixed.

### C. Velocity measurements

Velocimetry methods, such as Laser Doppler velocimetry (LDV) or particle image velocimetry (PIV), the principle of which are illustrated in figure 3(a) and (b), offer the advantage of providing instantaneous velocity measurements, providing valuable information on fluid behavior and on the strength or speed of vortical structures responsible for mixing.

The LDV is a technique based on the Doppler effect (the change in frequency of a wave under the effect of the speed)⁴⁸. It measures the flow velocity at a given point with a high temporal resolution, it has the limitation that it cannot characterize the spatial structure of the flow field. The intersection of two laser beams ($\lambda$) forms a volume where an interference fringe is formed ($d_f$). When a particle passes through this interference, the intensity that it reflects fluctuates according to scattered light received by the photo-detector (figure 3(a)).

The photo-detector records a signal burst whose amplitude is modulated by the fringe pattern. The frequency ($\Delta t$) of this variation is equivalent to the difference between the incident and scattered light frequencies (Doppler shift) and is related to the component of particle velocity. Thus, the local velocity of the fluid can be determined. LDV data represent irregular velocity time sampling, as they depend on the moments at which particles pass through the measurement volume. The accuracy of this technique enables the identification of flow instabilities and to characterize complex turbulent flow. However, there are several source of errors, for LDV, associated to frequency measurements; one common error is related to the alignment of the optical system, specifically the dimension of the fringes. If the fringe dimension is not correctly adjusted, it can lead to imprecision in the velocity measurements. Another error is caused by the presence of a velocity gradient within the measuring control volume. Additionally, the sensitivity of the electronic signal processing system can lead to errors into LDV measurements. Electronic noise and the need to adjust the high voltage level of the system are also factors that can affect the accuracy of LDV velocity measurements³⁸. LDV was used by Castelain et al.⁴⁹ in order to probe the Eulerian nature of the flow by measuring the axial velocity across the square cross-section duct width at several Dean numbers (figure 1(a))(De = $Re\sqrt{\frac{r}{R_c}}$ where Re is the Reynolds number, r is the radius of the circular section of the pipe and $R_c$ is the radius of curvature of the pipe). The use of a transparent duct allowed velocity measurements to be performed in all bends of the chaotic exchanger. In Mokrani et al.⁵⁰ however, the use of non transparent material imposed the measurement of velocity profiles only at the exit, where a straight plexiglass tube with flat outer walls was positioned at the exit of the stainless steel chaotic heat exchanger. The profiles of the axial velocity display a parabolic shape (figure 1(b)). The study of Timité et al.⁵¹ was performed on unsteady Dean flow for three radial positions in the transparent cross-section. LDV measurements showed that the flow pulsation gives rise to pure sinusoidal instantaneous average velocities. In order to determine the inlet conditions upstream of the geometry studied, Zambaux et al.⁵² used the LDV to measure the longitudinal velocity profiles. LDV can also be used to locally validate other whole field velocity measurement methods, such as PIV, which is presented in the next paragraph.

Particle image velocimetry (PIV) measures the instantaneous velocity field in a flow³⁶. The flow is seeded with tracer particles (figure 3(b)) chosen carefully in a way that their concentration and size must be low enough to not disturb the flow but high enough to be detected⁵⁵. A plane light sheet within the flow is illuminated by a laser system. The light scattered by the particles, at two successive times, is recorded by the camera. By calculating the displacements of the particles, the velocity field of the flow is determined. The camera settings, used to capture the images of the tracer particles, is an important factor as well as the duration of the illumination light (laser pulse). For example, if the duration of the illumination light pulse is too long, it can lead to motion blur and make it difficult to accurately follow the motion of the particles. Jarrahi et al.⁵⁶ used the PIV technique to investigate the sec-
<figure><img src="image_2.png" /><figcaption>FIG. 1. Axial velocity profiles measured on the symmetry plane of the bend: (a) of the third bend (Figure inspired by$^{53,54}$, (b) of the last bend of the chaotic configuration (Reproduced with permission from Int. J. Heat Mass Transf. 40, (1997). Copyright 1997 Elsevier Science Ltd.$^{50}$)</figcaption></figure>

ondary flow velocity fields in steady and pulsating alternating Dean flow. In the steady case, structure of the secondary flow is always composed of two counter-rotation cells close to the convex wall in all bends (figure 2(a)). Nevertheless, the number of bends and also the Reynolds number have an impact on the symmetry of the flow structure and the positions of the cell centers. In the work of Younes et al.$^{57}$, the PIV technique was employed to characterize the flow in a transparent channel with three co or counter rotating cylinders placed vertically along the entire height of the channel. The velocity fields were measured at different times during the rotation period of the cylinders. They observed different features of chaotic flow. They found that during the first half of the period, the fluid elements undergo strong recirculation in the elliptical regions near the cylinders (figure 2(b)), while during the second half, they become folded.

In microscale flows, PIV may face some limitations such as small dimensions of the measuring range and signal-to-noise ratio due to the smaller size of the particles and the flow features. To overcome these limitations, a variation of PIV called micro Particle Image Velocimetry (μPIV) has been developed. μPIV designed for microscale flow applications and specialized optics to achieve higher spatial resolution and signal-to-noise ratio in microfluidic applications. μPIV can provide valuable insights into the behavior of microfluidic devices, such as micromixers, and enables the visualization and quantification of complex flow phenomena, including chaotic mixing. However, μPIV presents an illumination complexity that stems from the need to efficiently illuminate a volume, which can be difficult to achieve as it requires the generation of well-defined laser planes, and access to these measurement planes depends on the microscope used. Hsieh et al.$^{58}$ used μPIV to study the flow field and mixing performance in three different type of chaotic helical micromixers. They found that secondary flow could be a contributing factor in generating lateral chaotic mixing. Although the increase in Dean number is relatively small in this study, as it does increase, chaotic mixing becomes more apparent.

<figure><img src="image_3.png" /><figcaption>FIG. 2. Examples of velocity fields measured by the PIV technique:(a) in steady alternating Dean flow at the sixth bend for Re = 420 (Reproduced with permission from Chem. Eng. Process. 74, (2013). Copyright 2013 Elsevier Science Ltd.$^{56}$), (b) at the period $t = T + 3T/4$ of rotation of the cylinders (Reproduced with permission from Int. J. Heat Mass Transf. 187, (2022). Copyright 2022 Elsevier Science Ltd.$^{57}$).</figcaption></figure>

## D. Temperature and scalar field measurements

The most common and suitable technique used to measure local scalar fields in liquids (passive scalar concentration, temperature...), at high spatial resolution, is Laser-Induced Fluorescence (LIF). This technique involves the visualization of the luminescence light emitted by a fluorescent molecule by an imaging system (e.g. a camera) and thus obtaining a 2D map of the concentration of this molecule. The fluorescent dye is most of the time excited with a laser source shaped as a beam or as a plane, and emits a fluorescence intensity which depends on its local concentration (see Figure 3 (c)), the laser excitation intensity, and other environmental factors such as the pH, dissolved oxygen concentration, or temperature. Indeed, a thermo-sensitive variant of LIF allows, under homogeneous marker concentration, the measurement of local temperature fields, as the fluorescence intensity also depends on temperature. The most conventional thermo-sensitive LIF setups involve the recording of the fluorescence of a single dye, and a calibration of temperature measurements based on the total fluoresced light intensity emitted by this dye at
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0254063

<figure><img src="image_4.png" /><figcaption>FIG. 3. Schematic illustration of common optical experimental techniques used to characterize chaotic advection flows in heat exchangers/reactors. (a) Laser Doppler velocimetry (LDV) is used to measure fluid velocity locally with a high temporal resolution, (b) Particle image velocimetry (PIV) is used to measure flow velocity, (c) Laser induced fluorescence (LIF) can be either used to measure the temperature or the concentration with a high spatial resolution</figcaption></figure>

all wavelengths⁵⁹. More advanced variants of this thermo-sensitive LIF have been developed from the 2000's, to account for large flow fields and or heterogeneity in the laser excitation patterns. Those require the recording of the intensity of fluoresced light at two distinct wavelengths, one sensitive to temperature and one insensitive, and the temperature measurement is calibrated over the ratio of the two intensities which is itself insensitive to any other factors (such as laser reflection or absorption)⁶⁰⁻⁶². The two wavelengths can be emitted by either the same dye, or two dyes. The latter case is referred to as the two-color/two-dye variant⁶³. This was used by Shi et al.⁶⁴, who measured the temperature field in a millimetric zigzag channel using two different dyes with different fluorescence spectra, one of which was used to measure the temperature field, while the other one was used to visualize the flow.

In the following, a non-exhaustive list of examples from the literature where LIF was employed to provide quantitative information of scalar fields in the fluid flow is proposed. For example, in the study of Liu et al.⁶⁵, they used hanging droplets within a micro-channel, enabling mixing to take place. To observe the dynamic mixing process within each droplet, the LIF technique was employed. By increasing the flow rate, the inner chaotic flows within the droplets contributed to the acceleration and efficiency of the mixing process. The study of Arratia et al.⁶⁶ LIF technique to ac-

curately measure the concentration of dye within a smooth and continuous three-dimensional flow in a stirred tank reactor. The reactor is stirred by three impellers. The primary objective of this method was to provide insights into the spatial arrangement of the developing mixing patterns and their associated statistical characteristics. By conducting a statistical analysis of concentration fields of the tracer used, they discovered that the distribution of concentrations does not follow a Gaussian pattern for chaotic flow conditions. Additionally, they observed an exponential decay in the variability of the tracer. They found that the concentration pattern exhibited self-similarity throughout its evolution, despite the fact that the concentration amplitude decreased as the number of flow periods increased. This implies that the intricate structures formed during the initial mixing process tend to maintain their characteristics within this three-dimensional flow. Noroozi et al.⁶⁷ have developed a method for achieving microfluidic mixing on a centrifugal platform. This approach involves inducing localized chaotic advection within the fluid flow through the use of various forces, namely the Coriolis force, the centrifugal force, reciprocating flow, and oscillatory volume contractions. By programming the temporal trajectory of the spinning disk or cartridge and manipulating the angular frequency of the rotor, they were able to induce a reciprocal motion of the liquid, resulting in highly efficient mixing through the generation of local chaotic flow, even at
<figure><img src="image_5.png" /></figure>

This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0254063

low Reynolds numbers. The process of mixing was effectively monitored using the LIF technique. The application of the LIF technique, in the study of Mangi et al.⁶⁸ allowed the examination of passive scalar concentration fields within a developing laminar pulsating flow occurring in a 90° curved pipe. They obtained detailed insights into the secondary flow patterns, revealing that the presence of pulsating flow in a curved pipe significantly enhances the process of mixing. Wang et al.⁶⁹ conducted an extensive investigation of the mixing process of miscible liquid-liquid systems within micro channels. The study focused on comprehensively examining the liquid mixing process specifically in T-shaped micro-channels, employing the micro Laser-Induced Fluorescence technique (μLIF), which is a variant of the laser induced fluorescence. By visualizing the mixing processes in two dimensions, they aimed to enhance the understanding of the mixing phenomena, particularly when dealing with miscible liquids. The findings revealed that superior mixing performance can be attained at both low and high Reynolds numbers compared to the performance observed at intermediate Reynolds numbers in the micro-channels.

Hsieh et al.⁷⁰ conducted a study to investigate the enhancement of mixing in five T-shaped micro-channels through various surface modifications. These modifications included rib-positioned surfaces, wavy surfaces, and cross-section expansions or contractions for each micro-channel. They employed μLIF to track the evolution of a dye and obtain detailed concentration distributions for the six different configurations under different flow conditions. The flow within these micro-channels exhibited three-dimensional characteristics, with secondary flows generated across the channel cross-section in addition to the main flow along the channel axis. Among the five channels, the alternating wavy surface with rib-positioned channels demonstrated superior mixing performance, as indicated by the shortest mixing length, highest mixing efficiency, and largest stretching rate. The study revealed that the system exhibited chaotic behavior due to periodic folding and stretching of the streamlines, with flow periodically splitting and recombining as it approached the ribs, influenced by the rib geometry.

Hsieh et al.⁵⁸ used the same geometry as mentioned in the previous section above IIC where the mixing performance was measured using μLIF technique at extremely low Reynolds numbers. Three variations of the helical type micromixers were designed and fabricated, with differing numbers of strands (three, four, and six) and channel widths. The mixing performance was evaluated by analyzing the color evolution along the downstream region, which provided insights into the detailed concentration distribution. Among the three variations, the four-strand micromixer demonstrated the most effective mixing, achieving a high mixing efficiency of 83%. In the experimental study of Castelain et al.⁴⁹, the LIF technique was used for the purpose of visualizing and quantifying the mixing and transport processes in the twisted duct flow. The authors were able to obtain a tracer map of the flow in order to study the behavior of a passive scalar, which is the dye used, and to determine how it was mixed and transported by the chaotic velocity field (figure 4(a)). Timité et

al.⁵¹,⁷¹ also used the LIF technique in their experiment in order to see the impact of pulsation on the development of secondary flow in a curved pipe (figure 4(b)). Chagny et al.⁷² used LIF in order to understand the flow structure, in two heat exchangers configurations (chaotic and helical), under heating conditions. The laser sheet was generated at the outlet of the heat exchanger. They found that, after a number of bends, the Dean roll-cells were not completely destroyed between two bends and thus can influence the flow structure in the following bends. The upper Dean roll-cell was folded inside the lower one, under the influence of the previous cells, which were not totally destroyed (figure 4(c)). The tracer was better distributed and uniform in the chaotic configuration than in the helical one, which means that mixing is better in this configuration.

One common issue for LIF technique is crosstalk, which can occur when fluorescence from one species overlaps with that of another, making it difficult to distinguish between them. This can be mitigated by selecting an appropriate excitation wavelength and using filters to block unwanted fluorescence. Another issue is quenching, which can occur when a molecule is exposed to high-energy radiation or collisions with other molecules, reducing the amount of fluorescence emitted. The concentration of the sample should be optimized, and quenching inhibitors can be added to reduce the effects of quenching. Spectral overlap is another issue, and multiple excitation wavelengths can be used to distinguish between fluorescent species. Photobleaching, which is the loss of fluorescence intensity over time due to the destruction of the fluorescent molecule, can be mitigated by using lower laser power or reducing the exposure time. Finally, changes in the propensity of the dye fluorescence under a given excitation level are another potential source of error. For example, Rhodamine 6G is pH and temperature dependent. These quantities must be held constant during an experiment that doesn't require a thermal condition. These factors should be carefully controlled and monitored, and appropriate calibration and controls should be used to ensure accurate results.

This section has highlighted ways of measuring both local velocity and local scalars, enabling a better understanding of flow mechanisms. The possible coupling of these two methods, as illustrated in figure 3, to obtain scalar flows and observe the influence of flow on mixing and heat transfer is not yet explored for chaotic advection purpose. Although such coupling has been applied in various flow configurations, its application in chaotic heat exchangers, as we shall see in section IV, is rare, if non-existent. The local experimental data provided in this context are an invaluable resource for understanding phenomena, validating simulations and numerical models⁷³, and elucidating overall heat exchanger behavior. The following section focuses on characterizing the overall heat transfer performance of heat exchangers and mixers.

### III. GLOBAL CHARACTERIZATION METHODS

In this section, we will explore global methods, which encompass experimental approaches aimed at evaluating heat
<figure><img src="image_6.png" /><figcaption>FIG. 4. Examples of flow visualizations using LIF in chaotic configurations: (a) Flow visualization in fourth bend at $D_n = 354$ (Figure inspired by$^{53,54}$, (b) Tracer dye deformation at the exit of the sixth bend at $Re_{st} = 430$ (Reproduced with permission from Int. J. Heat Fluid Flow. 31, (2010). Copyright 2010 Elsevier Science Ltd.$^{51}$), (c) Flow visualizations for the exit of the chaotic configuration at $Re = 50$ (Reproduced with permission from Appl. Therm. Eng. 20, (2000). Copyright 2000 Elsevier Science Ltd.$^{72}$)</figcaption></figure>

exchanger performance while avoiding detailed examinations of local behavior. The main objectives are to understand the quantification of key parameters such as the residence time of fluid particles in the system, the total energy supplied to the fluid as it passes through the heat exchanger, variations in mixing efficiency between inlet and outlet of mixer, and energy dissipation or consumption during heat exchange processes. In addition, we aim to show all the performance coefficients that have been used in the literature to evaluate overall system efficiency, in order to provide an overview of the performance of heat exchangers/reactors based on chaotic advection.

## A. Residence time distribution

The study of particle residence time distributions (RTD) provides a wealth of information on flow behavior$^{74-76}$, making it possible to predict the average time particles spend in the system, and the presence of any preferential pathways or regions. The RTD is a method that describes the distribution of the time that fluid particles spend within a system, as heat exchanger or reactor, and therefore the quality of mixing. In the context of heat exchangers and reactors based on chaotic advection, determining the RTD can be particularly useful for assessing the performance and understanding the overall behavior of these devices. Castelain et al.$^{75,77}$, conducted both experimental and numerical investigations to analyze the RTD in two distinct types of twisted pipe flows: helically coiled and chaotic systems. The resulting RTD obtained from these configurations were then compared to those observed in a straight pipe flow. To obtain the RTD experimentally, they employed a two-measurement-point conductimetric method. This method allowed them to measure the concentration of a non-reactive tracer injected into the mixer, and analyzing the response at the outlet. As the Reynolds number increased, they observed a consistent decrease in the axial dispersion coefficient. This decrease in dispersion was attributed to two specific factors in chaotic twisted pipe flow: the emergence of secondary flow patterns and the initiation of chaotic trajectories within the fluid flow. The results showed that the twisted geometry of the pipe can significantly impact the residence time distribution, leading to longer residence times and greater mixing of fluid particles. To explore the stretching and folding process

within a chaotic system and to emphasize the sensitivity of such a system to initial conditions, Younes et al.$^{57}$ conducted a study in which they calculated the time taken for each particle to pass through the mixer, knowing the position of each particle at each time step and the time spent between two consecutive time steps. By examining the residence time distribution of these particles, they gained valuable insights into the flow behavior and were able to make predictions regarding the quality of mixing. By tracking the position of each particle at each time step and calculating the time spent between consecutive time steps, they were able to determine the time taken by each particle to pass through the mixer. Subsequently, the authors evaluated the residence time distribution of 1000 particles initially distributed along the inlet section. The obtained residence time distributions revealed that under favorable controlling conditions, the fluid particles spent more time within the mixer, indicating improved mixing efficiency. Saxena et al.$^{74}$ conducted a study that aimed at optimizing the coiled configuration to achieve a narrower distribution of RTD. The study investigated several parameters that could impact the RTD in coiled flow inverters, including the number of bends, spacing between bends, angle between different arms of the helix, and Dean number. They found that the narrowest distribution of RTD was obtained when the bends were set at a 90-degree angle and equally spaced. Increasing the number of bends significantly reduced the RTD. Additionally, when the Dean number exceeded 1.5, the first mechanism no longer influenced the RTD, but the secondary flow facilitated more efficient shifting at the bends, resulting in further narrowing of the RTD. The study also revealed that bent coils were more effective than coiled coils in reducing the spread of residence time. Habchi et al.$^{27}$ developed a mechanistic model in which they used a Lagrangian analysis of the flow to explain the observed smaller drop size under chaotic advection flow in water/oil dispersions. The trajectories of fluid particles after 25 bends in helically coiled and chaotic advection flows were analyzed to elucidate the superior dispersive performance of the chaotic advection flow. They established residence time distributions by computing trajectories for a large number of particles, ≈ 6000, uniformly distributed in the inlet plane section. The analysis revealed that the residence time dispersion was narrower in the chaotic advection configuration, indicating enhanced radial transfer (figure 5(a)). Interestingly, even for
laminar flows with very small Dean numbers, the chaotic advection flow exhibited a residence time distribution as narrow as an axially dispersed plug flow, whereas the helically coiled tube flow demonstrated a residence time distribution similar to that of a straight tube flow highlighting the efficiency of chaotic advection. Boskovic et al.⁷⁸ conducted a study to investigate and compare the RTD characteristics of three microreactors with different passive mixing structures. The microreactors examined included a three-dimensional serpentine structure, a split-and-recombine (SAR) structure, and a staggered herringbone structure. The study employed a flexible input response to assess and compare their performance. The split-and-recombine structure showed slightly better RTD behavior across a wider range of flow rates compared to the three-dimensional serpentine structure (figure 5 (b)). This improved performance can be attributed to the mixing structure of the split and recombine, which induces additional serial multilamination and positively impacts the RTD.

## B. Global performance factors

Global performance factors, commonly found in the literature, are used in order to assess the overall efficiency of heat exchangers, considering key parameters such as heat transfer coefficients, pressure drops and temperature uniformity. They also simplify the comparison of different heat exchanger designs or configurations. As discussed earlier, they have to take into account the balance between optimizing heat transfer, which ensures thermal effectiveness in performance, and minimizing pressure drop which reduces energy consumption and operating costs.

Consequently, there are a number of coefficients used in the literature to assess the efficiency of these heat exchangers, which we have chosen to summarize in table I. One of those factors is the thermal enhancement factor $\eta$ also known as the thermohydraulic performance enhancement criterion (PEC)⁷⁹,⁸⁰ and is defined as follows:

$$ PEC = \frac{Nu/Nu_0}{(f/f_0)^{1/3}} \quad (1) $$

where the index 0 represents the reference geometry. When it comes to assessing the global efficiency of a heat exchanger, there are two main parameters to consider: heat transfer coefficient ($h$) which is represented by Nusselt number ($Nu$) and pressure drop ($\Delta p$) which transposes to the non-dimensional Darcy coefficient $f$. The Nusselt number is a critical indicator of this efficiency, quantifying the ratio of convective to conductive heat transfer in the system, such as: $Nu = \frac{hD_h}{k}$ with $k$ the thermal conductivity of the flow, $D_h$ the hydraulic diameter, $h$ the heat transfer coefficient calculated from the wall heat flux density $Q$ and the DTLM, such that $h = \frac{Q}{A \times DTLM}$, where $A$ is the exchange area and DTLM is the logarithmic mean temperature difference between the inlet and outlet of a channel: $DTLM = \frac{T_o - T_i}{\log\left(\frac{T_o - T_i}{T_i - T_o}\right)}$. Where $T_i$ is the fluid inlet temperature, $T_o$ the fluid outlet temperature and $T_w$ is the heat exchanger

wall temperature. Meanwhile, the friction coefficient $f$ quantifies the resistance to fluid flow through the geometry affecting the energy required for fluid transport and is expressed as $f = \frac{2D_h\Delta p}{\rho l v^2}$ with $\Delta p$ the pressure drop, $l$ the total length of the pipe and $\rho$ the fluid density.

In the literature, PEC quantifications are more frequently encountered in numerical than in experimental studies. Numerical simulations offer the capacity to examine various parameters, explore a wide range of conditions and geometries, and carry out cost-effective parametric studies efficiently. They can also be used, in other cases, to extend experimental studies carried out beforehand. Aldor et al.²¹ conducted a numerical study on a heat exchanger, sine-helical, employing a channel design that melds elements of both helical coil and sine-wave configuration. The investigation focused on laminar flow conditions, with Reynolds numbers spanning from 100 to 1400. The outcome of this study revealed that the PEC (eq.1) increases from 5.5% to 20.7% in the sine-helical flow in comparison to the helical channel setup. Bahrani et al.⁸¹ conducted an experimental study on Chaotic Laminar Mixers (CLMs), which belong to the category of split-and-recombine (SAR) mixers characterized by utilizing chaotic advection principles inspired by the baker's transformation. The investigation highlighted that within the Reynolds number range examined (100–2000), CLMs exhibit twice the heat transfer efficiency of helical heat exchangers. However, when considering the impact of pressure losses, helical exchangers demonstrate a global energy efficiency comparable to that of CLMs. A comprehensive comparison between the developed CLMs and helical exchangers was undertaken, aiming to determine the potential applicability of CLMs within a broader energy framework, using the PEC (eq.1). CLMs are especially suitable in applications prioritizing efficient heat transfer, even though they require higher pumping power.

Xia et al.⁸² employed an experimental method to investigate the temperature distribution and pressure drops within the corrugated microchannel heat sinks (CMCHS). The performance is carried out through the use of the thermal enhancement factor PEC (eq.1). In comparison to an equivalently shaped rectangular microchannel heat sink, the average and maximum temperatures are notably lowered, accompanied by a uniform temperature distribution. However, this is achieved at the expense of higher pressure losses for flow rates exceeding 100 ml/min. The enhancement factor exceeds 1 when the Reynolds number is between 300 and 600 (figure 6), and continues to increase as the Reynolds number rises. This suggests that the overall performance of the CMCHS improves significantly once the Reynolds number is between 300 and 600. Specifically, the thermal enhancement factor can reach up to 1.24 at a Reynolds number of 611, leading to 19% reduction in the total thermal resistance of the CMCHS under the given pumping power of 0.446 K/W. Zambaux et al.⁸³ studied a configuration involving consecutive alternating deformations applied to both the external and internal walls of a coaxial tube. Their investigation, carried out through numerical analysis, aimed to demonstrate the effect of all deformations on the flow structures. They studied the influence
<figure><img src="image_7.png" /><figcaption>FIG. 5. The representation of dimensionless residence time distribution in the literature: (a) in the study of Habchi et al.²⁷ in 25 bends for Dn=100 (Reproduced with permission from Int. J. Multiph. Flow. 35, (2009). Copyright 2009 Elsevier Science Ltd.²⁷), (b) as a function of normalized density distribution function E(θ) in the experimental study of Boskovic et al.⁷⁸ in the split-and-recombine reactor (SAR) (Reproduced with permission from Chem. Eng. Technol. 34, (2011). Copyright 2011 Wiley.⁷⁸)</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>FIG. 6. Comparison between some studies using the Performance Evaluation Criteria 'PEC'.</figcaption></figure>

of longitudinal phase-shifting between the external and internal deformations. The thermohydraulic performances were evaluated using the PEC (eq.1). All the examined configurations showed enhanced performance when compared to a smooth annular tube as reference. Significantly, through adjustments in longitudinal phase-shifting, the PEC exhibited enhanced performance. The changes in the PEC were influenced by modifications to the placements of the internal and external tubes. Yahiat et al.³³ conducted a numerical investigation involving configurations that incorporated wall macrodeformations, in the form of external sinusoidal wall deformations, alongside an internal swirled core. They examined the impact of swirl pitch on thermohydraulic performance of the heat exchanger. To assess overall performance, they, also, employed the PEC (eq.1) across the specified range of Reynolds numbers (figure 6). The Reynolds numbers were ranging from 200 to 1000. They found that for a Reynolds number of 1000,

the PEC value was of up to 3.10 for the configuration featuring the smallest pitch. Tohidi et al.⁸⁴ investigated numerically two configurations: a developed chaotic design and the conventional helical one. Their study focused on the thermal performance of these heat exchanger configurations, evaluated through the calculation of the Nusselt number. A notable improvement in the chaotic model's performance is showed when compared to the standard design. Regarding pressure loss, the reorientation of flow within the chaotic configuration led to higher pressure losses compared to the normal helical coil. The evaluation of the chaotic heat exchanger's performance was conducted by determining the ratio of the Nusselt number to the friction factor. In this work, Nusselt number was expressed as $Nu = \frac{q_w d}{T_b - T_w}$ where $q_w$ is the heat flux, $T_b$ is the temperature of mean bulk fluid over the cross-sectional area of the coil and $T_w$ is the wall temperature and the friction factor was expressed as $f = \frac{\tau_w}{\frac{1}{2}\rho v_m^2}$ where $\tau_w$ is the peripherally averaged shear stress at the wall and $v_m$ is the mean velocity. There was a remarkable enhancement of 4% to 26% in heat transfer, while the increase in pressure drop was ranging from 5% to 8%. However, the increase in the friction factor outweighed the improvement in heat transfer, resulting in the coil heat exchanger demonstrating slightly better performance, around 1%. Nevertheless, as Reynolds numbers increased, the chaotic heat exchanger exhibited impressive performance enhancement. By combining the Nusselt number curve and the coefficient of friction as a function of Reynolds number presented in the paper, we have calculated the PEC factor and presented it in figure 6. El hani et al.³⁴ conducted an experimental investigation of a chaotic helix exchanger (CHE) and compared its performance to that of a helical heat exchanger (HHE), both having the same surface area. The heat exchanger under testing was submerged in a cylindrical tank filled with water maintained at a controlled temperature. They found that the CHE consistently outperformed HHE for all Reynolds number values considered. The performance enhancement of CHE was notable, with PEC values consistently exceeding 3 and increasing with Reynolds
<figure><img src="image_9.png" /></figure>

number (Figure 6). In contrast, the PEC value for the HHE configuration remained constant at 3, which is the minimum value observed for CHE, beyond a Reynolds number of 600. Within the range of 800 <Re<1700, the PEC factor for CHE demonstrated an increase between 52% and 136%.

Following the same concept of the PEC coefficient, i.e., comparing thermal performance with energy expenditure. Chagny et al.⁷² used the L criterion (table I), that takes into account thermal performance and energy expenditure and is expressed as $L = \frac{j_H}{C_f}$ where $j_H$ is the Colburn coefficient which is defined as $j_H = \frac{Nu}{RePr^{1/3}}$ and $C_f$ is the friction factor. They made an experimental comparison over a large range of Reynolds numbers from 30 to 30,000 for two different heat exchanger configurations: helical and chaotic geometries having the same heat transfer area. The comparison was made by using the criterion L. For low Re (30-150), the L criterion was found to be higher for the chaotic configuration than the helical one. For the mid-range (Re from 150 to 3000), the performance of the chaotic heat exchanger is higher than that of the helical one at low Re without increased energy expenditure. When the Reynolds number increases, the dynamical effect of the Dean roll-cells increases and then the performances of the two heat exchangers approach each other.

Creyssels et al.¹⁹ studied a multi-level laminating mixer (MLLM), a static mixing device that derives from a three-dimensional interpretation of the baker's transformation. They presented both numerical and experimental investigations. The authors found that the Nusselt number, averaged across each fundamental component composing the MLLM structure, remained nearly constant, irrespective of the specific location of these components. They exhibited that the Nusselt number associated with each component experienced an asymptotic increase proportionate to the square root of the Reynolds number. The work of Mokrani et al.⁵⁰,⁸⁵ examined, experimentally, the effects of chaotic advection on heat transfer at low Reynolds numbers (from 60 to 200). To evaluate the potential of heat transfer improvement through chaotic advection, a comparison was conducted between a helical heat exchanger and a chaotic heat exchanger, possessing identical heat transfer surface areas and tube lengths. Analysis of temperature profiles within the coil cross-sections illustrated that chaotic advection played an important role in homogenizing and facilitating heating effects. When considering overall heat transfer efficiency, the results highlighted that the chaotic heat exchanger outperformed its helical counterpart, exhibiting an efficiency enhancement ranging from 13% to 27%. The calculation of the heat exchanger's efficiency was grounded in the ratio $\epsilon = \frac{T_o - T_i}{T_{is} - T_i}$ where $T_o$ and $T_i$ are respectively the outlet and inlet temperatures of the geometry and $T_{is}$ is the inlet temperature of the shell where the heat exchanger is immersed. Kumar et al.'s⁸⁶ research involved an experimental study on a coiled flow inverter (CFI) with Reynolds numbers spanning from 1000 to 16000. The heat exchanger was composed of coils and 90° bends integrated within a closed shell. The study focused on investigating the influence of flow inversion on overall heat transfer efficiency. The quantification of efficiency involved the ratio of two temperature differences.

Notably, at lower Reynolds numbers (1000-7000), the heat exchanger's efficiency approached 1, indicating highly efficient heat exchange. However, as the Reynolds number increased, the efficiency gradually decreased. Also, in comparison to conventional coiled tubes, heat transfer was 25% greater at lower Reynolds numbers. The study of Acharya et al.²⁰ centered around a geometry where the coil's axis undergoes a periodic 90-degree rotation, allowing for a comparison with a coil without this change in axis. To predict particle trajectories, the study employed Dean's classical perturbation solution to analyze secondary flows. The experimental results showed a substantial improvement of 6 to 8% in the heat transfer coefficient within the tube. This enhancement was observed alongside a moderate increase in pressure drop, ranging between 1.5% and 2.5%, over a Reynolds number interval between 300 and 10000. Jarrahi et al.⁸⁷ studied two different SAR configurations. They used numerical simulations to predict temperature fields under the same experimental conditions. The evaluation of temperature uniformity involved employing H parameter, which was derived from the standard deviation within the temperature field. Specifically, H is the ratio of standard deviations of temperature at the inlet and outlet cross-sections of both SAR geometries and the straight tube. The results showed that both SAR1 and SAR2 configurations offer a more uniform temperature field in comparison to a conventional straight tube. Anxionnaz-Minvielle et al.⁸⁸ explored the performance of three SAR geometries incorporated into three Heat Exchanger (HEX) reactors. The assessment of HEX reactors performance involved a wide range of Reynolds numbers from 0.1 to 10,000. Pressure drops were measured using a differential pressure sensor between the inlet and the outlet of the HEX reactors. Additionally, temperature sensors were integrated into the HEX reactors to characterize temperature profiles across different flow regimes. Temperature measurements were also taken between the inlet and outlet of each row. In order to evaluate the thermal behavior of the HEX reactors, a thermal intensification factor, denoted as UA/V (kW/K/m³), was employed. U represents the overall heat transfer coefficient, A represents the heat exchanger area, and V denotes the volume of fluid. The thermal intensification factor was directly derived from temperature measurements, instead of relying on heat transfer area. It was calculated as the ratio between $P_{th}$ (W), the exchanged power between the process fluid and the cooling fluid, and DTML, the logarithmic mean temperature. The study also involved a comparison of the heat transfer capacity of each HEX reactor for both SAR geometries and a 2D corrugated design, with a focus on the energy dissipation rate $\alpha$, a parameter defining the required pumping power which is defined as $\alpha = \frac{F_p \Delta p}{\rho V}$ where $F_p$ is the process flow rate and V is the volume of the fluid. The SAR design showed higher energy efficiency compared to the corrugated design, especially when dealing with more viscous fluids. Remarkably, the advantages in energy efficiency associated with SAR heat exchangers/reactors became apparent at Reynolds numbers below 50.

Zheng et al.⁸⁹ and Shi et al.⁹⁰ used the same heat enhancement factor $e_{Nu}$ defined as $e_{Nu} = \frac{Nu_m}{Nu_0}$ where $Nu_m$ is the average Nusselt number over channel length and $Nu_0$ the Nusselt
number in a straight channel. Zheng et al.⁸⁹ used numerical simulations to investigate steady laminar flow and heat transfer within periodic zigzag channels characterized by square cross-sections, while considering the influence of chaotic advection. The computational domain encompassed up to 14 repeating zigzag units, seamlessly connected at the inlet and outlet sections. Within this framework, the heat transfer enhancement ($e_{Nu}$) for each unit was evaluated. For Reynolds numbers ($Re$) less than 200, the heat transfer enhancements for each unit remained consistent, except for the initial few units still experiencing the effects of developing flow near the inlet section. However, as $Re$ exceeded 200, $e_{Nu}$ exhibited variability from the 1st unit to the 14th unit. The $e_{Nu}$ for the first two units consistently exceeded the average values due to the pronounced entrance effects. The numerical findings revealed that following the short length of developing flow near the inlet section, the flow field demonstrated spatial periodicity for $Re$ less than 200. In contrast, at higher Reynolds numbers, the flow lost its periodic nature and evolved into a more chaotic state as $Re$ increased. The presence of secondary flow patterns, such as Dean vortices, and the impact of chaotic advection, had significantly influenced heat transfer enhancement and led to an increase in pressure-drop penalties. Shi et al.⁹⁰ conducted a simulation of heat transfer within millimetric channels featuring a zigzag configuration with square cross-sections. They evaluated heat transfer enhancement using the factor $e_{Nu}$, as expressed above, alongside a thermohydraulic performance factor ($PF$) that serves as a valuable criterion for comparing heat transfer and pressure drop. $PF$ is calculated as the ratio of heat exchange capacity (global heat transfer coefficient $h_m$ multiplied by heat exchange area $A$) within the technology under consideration to that in a straight tube, divided by the ratio of flow friction power in the technology ($P$) to that in a straight tube ($P_0$), expressed as $PF = (hA)/(h_0A_0)$. $PF$ increased while increasing Reynolds numbers, with particularly significant increase at lower $Re$ values. The zigzag channel improved heat transfer while having a relatively smaller impact on pressure drop compared to a straight channel. The ratio of the Nusselt number in zigzag channels to that in straight channels consistently exceeded one, rising with increasing Reynolds numbers, reaching values as high as 6.4 in the studied cases. Even when considering the pressure drop penalty in the performance factor, an enhancement was still evident, with a factor of up to 2.5.

Cui et al.⁹¹ conducted a numerical study on a circle-arc-wavy helically coiled pipe heat exchanger (CAWC). Their study focused on evaluating the overall performance of the CAWC using two key performance criteria. The first, denoted as $PEC\_h$, represents the ratio of the convective heat transfer coefficient of the CAWC to that of a conventional helically coiled heat exchanger (CC) and is expressed as $PEC\_h = \frac{h}{h_{CC}}$ when operating under the same pumping power and heat transfer area. The second factor, $PEC\_Q$, represents the ratio of the total heat transfer rate of the CAWC to that of the CC under the same pumping power and heat transfer area and is expressed as $PEC\_Q = \frac{Q}{Q_{CC}}$. Their findings revealed that when subjected to identical pumping power and heat exchange

area conditions, the CAWC demonstrated a 40% increase in the total convective heat coefficient compared to the conventional coiled heat exchanger.

## IV. PERSPECTIVES

In this section, we will explore some interesting techniques that have not yet been used in the framework of chaotic advection but hold the potential to deepen our understanding of chaotic advection and how it influences mixing, species distribution, and thermal behavior both internally, in wall bounded forced convection conditions, and externally in free convection.

### A. Extension to 3D and high temporal resolutions

Conventional PIV as frequently used in the literature mainly provides two-dimensional velocity information in a single plane, making it suitable for examining flow patterns at a specific fluid cross-section. Yet, as highlighted in this review, chaotic flows by nature tend to be highly three-dimensional. Tomographic Particle Image Velocimetry (Tomo PIV)⁹² extends the technique into three dimensions, allowing the measurement of velocity components in a volume. While achieving temporal resolution with Tomo-PIV proves to be quite experimentally expensive, Particle Tracking Velocimetry in Four Dimensions (PTV-4D) can be an alternative⁹³. Tracking particles, in three dimensions yields time-resolved trajectories and enables a better understanding of Lagrangian chaotic motion and the study of transitional and unstable flow phenomena, offering at the same time the possibility of a subsequent reconstruction of three-dimensional flow velocity fields. These techniques can be very useful in the study of complex 3D flows such as chaotic advection.

### B. Method coupling and flux measurements

PIV and LIF give access to local velocity and temperature/concentration fields, respectively, whereas measurement and understanding of local heat fluxes requires simultaneous probing of both. This can be achieved by a coupling of optical methods. When these optical techniques are coupled together, they can deepen our fundamental understanding of the mechanisms governing heat and mass transfer. This combination has been used in many different applications, the heat-exchanger and mixer systems. A non exhaustive list of examples includes the study of:

- turbulent superstructures in Rayleigh-Bénard convection⁹⁴,

- the flow and heat transfer characteristics of nucleating vapour bubbles⁹⁵,

- the turbulent mixing performance of a scaled-up multi-inlet vortex reactor⁹⁶,
<table><caption>TABLE I. Summary of factors used to evaluate the performance of heat exchangers based on chaotic advection</caption><thead><tr><th colspan="10">Parameters expression</th></tr><tr><th></th><th>f = 2D<sub>h</sub>Δp / ρl<sup>2</sup></th><th>C<sub>f</sub> = 2D<sub>h</sub>Δp / ρl<sup>2</sup></th><th>-</th><th>-</th><th>f = 2D<sub>h</sub>Δp / ρl<sup>2</sup></th><th>-</th><th>P = ΔpQ</th><th>-</th><th>f = π<sub>o</sub> / (1/2ρv<sub>h</sub><sup>2</sup>)</th></tr></thead><tbody><tr><td>Friction factor</td><td>Nu = hD<sub>h</sub> / k</td><td>j<sub>H</sub> = Nu / RePr<sup>1/3</sup></td><td>T<sub>o</sub> - T<sub>i</sub></td><td>T<sub>i</sub>, T<sub>o</sub></td><td>-</td><td>Nu<sub>m</sub></td><td>h<sub>m</sub></td><td>h, Q</td><td>Nu = q<sub>u</sub> / (T<sub>h</sub> - T<sub>o</sub>)</td></tr><tr><td>Internal heat transfer</td><td>-</td><td>-</td><td>T<sub>is</sub> - T<sub>i</sub></td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>External heat transfer</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Heat transfer ratio</td><td>-</td><td>-</td><td>ε = (T<sub>o</sub> - T<sub>i</sub>) / (T<sub>o</sub> - T<sub>i</sub>)</td><td>H = T<sub>o</sub> / T<sub>i</sub></td><td>-</td><td>e<sub>Nu</sub> = Nu<sub>m</sub> / Nu<sub>0</sub></td><td>-</td><td>PEC_h = h / h<sub>CC</sub>, PEC_Q = Q / Q<sub>CC</sub></td><td>-</td></tr><tr><td>Global performance ratio</td><td>PEC = Nu/Nu<sub>0</sub> / ((f/f<sub>0</sub>)<sup>1/3</sup>)</td><td>L = j<sub>u</sub> / C<sub>f</sub></td><td>-</td><td>-</td><td>α = F<sub>p</sub>Δp / ρV</td><td>-</td><td>PF = (hA) / (h<sub>0</sub>A<sub>0</sub>) / P / P<sub>0</sub></td><td>-</td><td>Nu/f</td></tr><tr><td>References</td><td>21,33,81–83</td><td>72</td><td>50,86</td><td>87</td><td>88</td><td>89,90</td><td>90</td><td>91</td><td>84</td></tr></tbody></table>

- dispersion of passive scalar downstream of the multi-scale turbulence generator⁹⁷,

- turbulent mass transfer near gas-liquid interfaces in water and shear-thinning dilute polymer solution⁹⁸.

The simultaneous use of PIV and LIF not only provides a comprehensive understanding of the interaction between fluid dynamics and heat transfer, but also characterizes the local heat or mass fluxes, as mentioned earlier, which can be interesting for investigating chaotic advection in heat exchangers or reactors. Coupled measurements can also be a way to bridge the gap in level of detail between advanced numerical simulations and experiments by providing reference data with high spatial and temporal resolution for the development and validation of accurate and reliable computing tools.

### C. Measurement in non-transparent systems

Most of the local methods mentioned in section II rely on a good optical access to the flow and transparency of the fluid. Although these techniques are very effective for single-phase flows, the presence of a dispersed phase even at a minimal volume fraction of only 0.5% can make these techniques useless⁹⁹,¹⁰⁰. Another important limitation of these optical techniques is that they only apply to transparent geometries, restricting their utility in some industrial applications. Ultrasonic based velocimetry, known under various names as ultrasound imaging velocimetry (UIV) or echo-PIV, can be an alternative technique. It was used for example to study particle-laden flows¹⁰¹ or to investigate single-phase Taylor-Couette flow¹⁰². Images are formed by emitting sound waves and interpreting the echoes received, generated by changes in the local acoustic impedance of the material. While using a different imaging approach, UIV preserves the main advantage of PIV, namely, its ability to generate immediate, two-dimensional, two-component velocity fields¹⁰³,¹⁰⁴.

As for the micro-devices, experimental methods based on the nuclear magnetic resonance (NMR), e.g. MRI (magnetic resonance imaging), allow non-invasive analysis of flows¹⁰⁵,¹⁰⁶. They provide quantitative 3D measurements of the flow field, even in opaque devices inaccessible to optical velocimetry techniques which can be interesting for micromixers based on chaotic advection.

### D. Measuring external convection

The measurement of external convection around a heat exchanger (see figure 7) is interesting because of its direct influence on the performance and efficiency of the heat exchanger, enabling its design to be optimized. Although previous work has effectively measured external convection using traditional methods such as thermocouples²⁰,⁵⁰,⁷², these techniques often offer limited spatial and temporal resolution. Optical techniques, such as PIV or LIF for example, mentioned above but on a larger measurement scale¹⁰⁷⁻¹⁰⁹, can be used for this purpose, as they provide a comprehensive and visual approach to capture and analyze fluid velocity and temperature fields. Another interesting technique is infrared thermography (IR), based on the emission of temperature-dependent thermal radiation from an object. IR is an important technique for determining the influence of convective heat flows on surface temperatures¹¹⁰,¹¹¹. Schlieren and Shadowgraphy techniques are also interesting options, as they are seedless (not relying on the seeding of the flow with dyes or particles) and applicable on large flow scales techniques¹¹²⁻¹¹⁴. The schlieren technique is linked to refractive index gradients, density and temperature variations in a fluid flow. Shadowgraphy, on the other hand, relies on variations in light intensity to capture fluid density gradients, providing valuable information on convective heat transfer and the characteristics of the flow field around the heat exchanger. Figure 7 shows an illustration of the above-mentioned techniques.

### V. CONCLUSION

The present review article has focused on different experimental methods used, in the literature, to characterize mixing and heat transfer by chaotic advection. The main elements in this review can be listed as follows:

- Chaotic advection can enhance mixing and heat transfer in laminar flow, which is particularly beneficial in scenarios where efficient and rapid heat or mass transfer is challenging, such as in highly viscous fluids or small-scale microfluidic geometries. This can be achieved through geometries designed to promote chaotic advection.

- The experimental techniques used to characterize chaotic advection can be divided into two parts: local
<figure><img src="image_10.png" /><figcaption>FIG. 7. Schematic representation of perspective techniques that can be used to measure the external convection of a heat exchanger based on chaotic advection, whether in the open air, shown on the right, or immersed in a liquid-filled tank, shown on the left.</figcaption></figure>

characterization techniques and global characterization
techniques. Local characterization techniques focus on
specific points or regions in the flow field. They provide
a better understanding of chaotic advection phenomena,
including parameters such as velocity and concentra-
tion, at specific points or regions. Conversely, global
characterization techniques assess the overall efficiency
of chaotic advection across the entire system.

- Flow visualization, such as dye tracing or tracer particles, enables the complex flow structures associated with chaotic advection to be observed and studied directly. This visual information is fundamental for qualitative analysis but it does not provide quantitative measurements of flow parameters such as concentration fields.

- PIV and LDV provide quantitative data by measur-
  ing velocity fields at different points within the flow.
  PIV provides measurements over the whole-field, while
  LDV provides precise point measurements. These tech-
  niques enable flow parameters such as velocity gradi-
  ents and vorticity to be quantified, facilitating the iden-
  tification of chaotic flow regions. As for local scalar
  fields (passive scalar concentration, temperature, etc.),
  LIF is the most common technique for measuring them
  at high spatial resolution. These optical metrology tech-
  niques listed above also have their drawbacks, in partic-
  ular their use only on transparent geometries and they
  can in some cases be costly as they require large, ex-
  pensive equipment such as lasers, cameras, etc.

- Global performance factors (PEC, L, ε, etc) are a set of criteria for evaluating the efficiency of an industrial device (heat exchangers, mixers) based on chaotic advection. These criteria can include dimensionless numbers or factors such as Nusselt number or friction factor, which indicate the efficiency of heat transfer within the system.

Although extensive studies have been carried out into the
use of PIV, LIF and LDV for the characterization of chaotic
advection, there remain promising but relatively unexplored
techniques that could be the subject of further research. These
include Tomo PIV and PTV-4D, which offer high temporal
resolution insight into complex 3D flows, coupling PIV and
LIF to improve understanding of local heat fluxes, using PIV
and NMR to measure the flow field in non-transparent geome-
tries, and examining external convection to assess how heat
exchangers interact with the external environment. These un-
explored paths are likely to expand our knowledge in the field
of chaotic advection.

ACKNOWLEDGMENTS

This work is supported by the Corenstock Industrial Chair
ANR-20-CHIN-0004-01. Funded by the French National Re-
search Agency (ANR).

AUTHOR DECLARATIONS

Conflict of Interest

The author has no conflicts to disclose.

Author Contributions

.....

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PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0254063

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