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Chaotic advection and chaos theory are two very related topics that have been put on a pedestal among the chemical engineering community, as applications of the idea are in high demand in fields such microfluidics and packed media. The aim of this research project is to continue the good work done in the first semester where simulation of upscaled models was completed and studied. Completing the circle for this semester is the undertaking of such experiments in the lab to understand the differences between a simulated model and an experimental model. Since a simulated model that was done last semester was done through computational methods, real-life variables were not applied. As such, variables that come into play during the undertaking of experiments in the lab should be recorded and studied.
The originator of the term “chaotic advection” is Hassan Aref, who termed it in 1984 when his aim was to search for the median between laminar and turbulent flow. Fundamentally, chaotic advection is the motion of particles sensitive to initial conditions. Any minor changes to the initial conditions should result in bifurcating trajectories. In a nutshell, integrable advection equations implies that the flow is steady in two dimensions and just regular advection. However, non-integrable advection equations implies that the flow is unsteady in two dimensions and thus is the very definition of chaotic advection. Hassan credits his discovery to a few predecessors before him, select researchers and scientists that encountered some foundations of chaotic advection long before Hassan himself.
This article provides an insight into the literature of chaotic advection with a complete methodology section on how to perform the experiment as well as a thorough discussion on the results obtained from the experiments conducted. The conclusion section highlights the results and discussion obtained throughout the research project.
Keywords: chaos, advection, chaos theory, nonlinear dynamics, reaction rate, mixing rate, chaotic flow.
Chaotic advection and chaos theory are two intertwined topics that has been of major interest to the chemical engineering fields lately (Aref 2002). As most fluid flows in the industry are turbulent, predicting the rate of reaction has always been an arduous task (Arratia & Gollub 2006). The idea of mixing laminar flows efficiently can help alleviate that problem thus the reason for such heightened interest in the topic of chaos theory and chaotic advection (Aref 1994). As a continuation to previous semester’s research topic of chaos theory and chaotic advection, the main idea and aim of this semester’s research project is to complete the circle of research and perform experimentally similar studies of flow characteristics and chaotic advection.
The purpose of this research project is as follows;
- Achieve a deeper understanding of the concept of chaotic advection and chaos theory.
- Personal experimentation in the form of lab work.
- Compare computational simulation data from the first semester and actual lab experimentation.
- Interpreting the differences between a simulated model and an experimental model.
- Recognising how the results of the research can help further the field of chemical engineering.
This research can help improve the many fields of chemical engineering such as microfluidics and packed media where the application of chaotic advection is highly sought after. Comparing the computational simulated model with a lab experimental model can also help understand what the differences between them are and what variables are in play when implementing such research into the industry.
The word “chaos” means disorder or confusion but in chaos theory the definition of chaos is much more precise. In chaos theory, it is more defined as complex, intricate yet with no hint of randomness. In simpler terms, chaos is the sensitivity to initial conditions, widely recognized as the “butterfly effect” (Layek 2015). For example, rounding errors in a numerical calculation can result in hugely deviated outcomes (Boeing 2016). Thus, the behaviour of unpredictability of these systems, termed by Edward Lorenz, is called deterministic chaos (Doherty & Ottino 1988). Chaos theory is mainly part of mathematics and physics, but as of late, has been much more recognised in the chemical engineering community as there is major interest on its application towards chemical engineering fields (Layek 2015).
Chaotic advection, on the other hand, is a mixture of chaos theory and fluid dynamics (Aref 2002). The term “chaotic advection” was first coined by Hassan Aref in 1984. The idea of chaotic advection is basically finding the middle ground between turbulent and laminar advection (Aref 2002). Chaotic advection can also be described as particle motion sensitive to initial conditions; trajectories bifurcate exponentially even with extremely minor changes to initial conditions are an indication that the flow is chaotic (Liang 2006). Turbulent flows have always mixed well, but the velocity is too chaotic while laminar flows are non-mixing, but the flow is smooth. Thus, the idea that laminar flow can mix well is essentially the concept behind chaotic advection (Aref 1994).
Rossi et al. (2012) showed that the basis of how laminar flow can mix quickly can be explained through Baker’s flow. When a material flow is continuously stretched and folded which increases the material flow exponentially, by 2N. This creates a lamination with thin material lines stacking over each other which greatly improves mixing.
Although chaotic advection is introduced by Hassan Aref in 1984, Hassan himself attributes the idea of chaotic advection to a few progenitors before him where the concept was hinted in hindsight (Aref 2002). Among the first precursors to chaotic advection was Carl Eckart in 1948, who made a case that mixing, and stirring are two distinct physical processes where stirring is the result of advection and mixing is the result of diffusion (Aref 2002). Another progenitor was recognized as Pierre Welander, who showed that the slow diffusion of a blob of dye ends in a highly chaotic pattern with fine streamlines all around. Hassan identified that the underlying flow was in fact not of turbulent flows (Aref 2002). The third precursor was by V.I. Arnol’d and M. Hénon in 1960s where they proposed some form of the nonintegrability of the advection equations used in the present, but in Beltrami flow, where vorticity and velocity are equivalent all over (Aref 2002).
The advection equations are an idea, that if a particle has negligible weight, is inactive as well as acquiescent to the flow of the fluid thus immersing itself all over the fluid, therefore the equation for the particles motion are the advection equations itself which are a system of ordinary differential equations (ODEs) (Aref 1994). The advection equations follow the Lagrangrian dynamics of fluid motions more than the Eulerian dynamics (Aref 2002).
Technically, three ODEs, which means 3-dimensional flows are in play, are more than ample for the production of chaotic dynamics, or in simpler terms, non-integrable equations (Aref 1984). Steady flows in 3-D are sufficient to create chaos without being time-dependent (Aref 2002). However, in 2-D steady flows are integrable thus leading to just regular advection instead of chaotic, because of the existence of a stream function which defines the velocity of the flow (Aref 1994). Therefore, producing chaotic particle motion in 2-D, incompressible flow requires time-dependent flow (Aref 2002).
By deriving velocity through a stream function, ψ, and combining them with the advection equations
results in Hamilton’s equations (Doherty & Ottino 1988). Any dynamical system is called a Hamiltonian system with one degree of freedom when it has a structure similar to the equations above (Aref 2002).
As the mixing regions and quality cannot exactly be seen clearly by the naked eye, Poincaré sections are used to discern between mixing and non-mixing regions (Moura et al. 2012). A Poincaré section is essentially a snapshot of particle locations at each t = nT which is then superimposed providing a clear picture of the mixing regions of the flow (Metcalfe et al. 2012).
According to Aref (2002), the established concept of chaotic advection was conveyed in a simple model which is called a “batch stirring device”, now recognised as the “blinking vortex model”. The model is essentially a fluid that is restricted to a circular disk with two agitators that can be switched “on” or “off” as a fixed point for mixing, with the flow being two-dimensional, inviscid and incompressible. The two agitators are switched on and off alternatively at a time, t = T. By putting a blob of particles in the blinking vortex model and initiating the process, the dispersion of the particles is chaotic which can be seen clearly through a Poincaré section. With the fluid flow incompressible and bounded to a circular domain, the mixing and dispersion of the particles suggest that the material lines are stretched continuously. This indicates repeated stretching and folding which is the basis of how laminar flow can mix quickly.
Figure 6 : Poincaré sections of a blinking vortex system. Evolution of Poincaré sections as the value of the parameter µ is increased. Number of iterations were 1000.
µ value: (a) = 0.01, (b) = 0.15, (c) = 0.25, (d) = 0.3, (e) = 0.4, (f) = 0.5
Moura et al (2012) points out that fixed points, otherwise known as periodic points, can easily be classified. Periodic points fundamentally control the mixing of the particles, classified as either elliptic or hyperbolic periodic points. Elliptic periodic points are easily identifiable through the appearance of circular orbits of non-mixing particles called “camtori” surrounding stable islands called “KAM islands” which are considered the non-mixing points. Hyperbolic periodic points, on the other hand, are mixing regions that show good mixing which are also called the “chaotic sea” that surrounds the “KAM islands”.
Materials for the experiment include glycerol, fluorescein, sodium hydroxide 0.1M, hydrochloric acid 0.1M, phenolphthalein. Equipment includes 1L tank, beakers, measuring cylinder, pipettes, fume hood, overhead stirrer and camera with digital video capture.
In the first experiment, 400ml of glycerol was poured into a 1L tank. Fluorescein dye was produced in a beaker where a small amount of water was added to red fluorescein powder and small pellets of sodium hydroxide were also added to allow the fluorescein powder to dissolve in water. All three ingredients were then stirred using a magnetic stirrer to allow the dye to form. A few drops of the produced fluorescein dye was added in the tank containing glycerol. The overhead stirrer was set at a speed of 10 rpm, and the overhead camera was set up to digital video capture while the drop of fluorescein dye gets stirred in the glycerol. The above steps were repeated with differing stirrer speeds which were set at 30, 50, 70 and 90 rpm respectively. The flow characteristics were then studied through the captured video and snapshots.
In the second experiment, a measuring cylinder was used to pour 50ml of sodium hydroxide 0.1M into a beaker. A drop of phenolphthalein was added into the beaker. Another measuring cylinder was then used to pour 50ml of hydrochloric acid 0.1M into the beaker containing the 50ml sodium hydroxide 0.1M and the phenolphthalein (The previous few steps involving acid and basic chemicals were done under a fume hood). The overhead stirrer was set at a speed of 10 and similar to the previous experiment, the camera was set up as an overhead camera and digital video capture while the acid-base reaction and the drop of phenolphthalein gets stirred. The above steps were again repeated with differing stirrer speeds which were set at 30, 50, 70 and 90 rpm respectively. The flow characteristics were then studied through the captured video and snapshots. This experiment however, does not correlate with the first experiment due to unforeseen circumstances but the results section will discuss this in further detail.
Known safety issues from the above experiments include the corrosivity of the sodium hydroxide 0.1M and hydrochloric acid 0.1M solutions as well as its toxicity. Associated risks for the handling of such chemicals include respiratory irritation, skin burns and eye damage. The controls for such risks include the use of fume hood when handling or using such chemicals for the experiments while also ensuring that the nitrile gloves are worn at all times when handling such chemicals. While there is also the risk of slip and trip hazards, by ensuring everything is run cautiously and carefully the risk can be eliminated.
RESULTS AND DISCUSSION
Results of first experiment
The first set of experiments was done 5 times, with each set having a different impeller speed (rpm) as a method to manipulate the Peclet number. Each set was run at a set time of about 5 minutes to allow for any changes to the mixing pattern. The conducted sets and its respective stirrer speeds are as follows:
Table 1: Experimental sets
To calibrate the concentration variance over time of each set, the snapshot obtained from the digital overhead video camera was converted to grayscale and the pixel intensity was used to calculate to concentration variance. Each set produces a differing graph that will be further discussed below.
- SET 1
By running the overhead stirrer at a speed of 10 rpm, the video and snapshot of the pattern (as shown below in figure 8) was analysed and used to calibrate the concentration variance over time.
Figure 8: Grayscale snapshot and graph of Set 1
- SET 2
By running the overhead stirrer at a speed of 30 rpm, the video and snapshot of the pattern (as shown below in figure 9) was analysed and used to calibrate the concentration variance over time.
Figure 9: Grayscale snapshot and graph of Set 2
- SET 3
By running the overhead stirrer at a speed of 50 rpm, the video and snapshot of the pattern (as shown below in figure 10) was analysed and used to calibrate the concentration variance over time.
Figure 10: Grayscale snapshot and graph of Set 3
- SET 4
By running the overhead stirrer at a speed of 70 rpm, the video and snapshot of the pattern (as shown below in figure 11) was analysed and used to calibrate the concentration variance over time.
Figure 11: Grayscale snapshot and graph of Set 4
- SET 5
By running the overhead stirrer at a speed of 90 rpm, the video and snapshot of the pattern (as shown below in figure 12) was analysed and used to calibrate the concentration variance over time.
Figure 12: Grayscale snapshot and graph of Set 5
Discussion of experimental results
By comparing all five sets, a unified graph was created to further highlight the difference.
Figure 13: Graphical comparison of all sets
From the graph, it can be seen that the concentration variance in set 1 changes much more over time than in set 5. This can be explained through the snapshots of the experiments. From figure 8 in set 1, the dye is much more concentrated towards the centre with long tails of dye protruding from the centre. This results in the concentration variance altering much more through time. From figure 12 in set 5, the dye is spread almost throughout the entire tank, with only small parts unoccupied by the dye. This results in the concentration variance altering at a much reduced rate as the dye has almost completely diffused in the glycerol.
Results of second experiment
The results of the second set of experiments unfortunately were of no correlation to the first set of experiments due to various unexpected circumstances. With the lab technician often not in for continuous weeks in the semester and miscommunication between the student, lab technician and supervisor, the conducted second experiment was not related to the first experiment as the medium used was water, when it should have been glycerol. Furthermore, the acid was continuously dropped into the basic solution with phenolphthalein when it should have been a single, large drop instead of small, continuous drops of acid.
The results from the second experiment produced will still however be discussed, however not in detail. The experiment was done in five sets, similar to the first experiment. Set 1 results showed that the pink colour of the phenolphthalein in alkaline solution turned colourless after the longest period of time when compared to the other sets. Set 5 results showed the quickest time for the pink colour of the phenolphthalein to turn colourless after adding drops of acid. From the results, it can be inferred that the faster the stirrer speed and mixing speed, the quicker the reaction of the acid-base mixture.
Comparison with previous semester results
The results from the computational method in the previous semester also produced graphs of concentration variance over time. The results from the previous semester were compared to current experimental results below, which show that the results show a very striking difference.
Figure 14: Sample results from previous semester
Figure 15: Graphical comparison of Experiment 1 results
The results obtained from the previous semester showed that the concentration variance reduces slowly over time, while the current experimental results show a graph that decreases, increases then decreases again over time. The results achieved from the experiment conducted was much different due to the fact the concentration variance calibrated experimentally was done using a two-dimensional snapshot which reduces or eliminates concentration gradients and variants while the computationally produced result was done in three-dimensions. In simple terms, the graph produced from the experimental results would not be possible if the calibration was done in three-dimensions.
Besides that, when comparing the pattern produced from mixing the fluorescein dye in glycerol to the first semester’s results of dyetrace plotting, the images were largely dissimilar besides the final set where the both the computational and experimental results show that the dye has diffused almost entirely in the medium solvent. The dissimilarity between the two results may be because of the differing density of the solvent medium as glycerol is a non-Newtonian fluid.
Prediction of reaction rates
Using the graph from figure 15, trendlines were used to find the fitted value of λ, of each set. The results are tabled below.
Table 2: Fitted value of λ
From the λ values obtained above, the rate of reaction was predicted using the Mathematica codes from the previous semester by inputting the found fitted values of λ into the equation from the theory which yielded rate of reaction values tabled as follows:
Table 3: Predicted reaction rates
- Chaotic advection is a very intriguing yet very challenging idea that may require further research and experiments as it is relatively new in concept.
- The concentration variance alters at a much reduced rate when the stirrer speed is higher as the fluorescein dye has almost entirely diffused in glycerol.
- The pattern of dye produced when stirred in a solvent medium highly depends on properties of the dye and solvent.
- The graph of concentration variance against time does not continuously decrease over time as it should in the experimental results due to the fact that the calibration was done using a two-dimensional snapshot. The result obtained should not be possible when done in three dimensions.
- Reaction rates are predicted to be higher when stirrer speeds are slower. Thus, reaction rates are harder to predict as the stirrer speed increases as the value would encroach closer and closer to zero as the speed increases.
- The results of the second experiment could have been tied in and correlated with the first set of experiments but failed due to various circumstances. The second experiment results would have given a clearer picture of predicting the reaction rates had it been done properly.
- Research and experimental results of chaotic advection can be implemented into chemical reactors and distillation columns in chemical engineering plants where the complex flow systems makes it difficult to deduce the rate of reaction.
- Arratia, PE & Gollub, JP 2006, “Predicting the Progress of Diffusively Limited Chemical Reactions in the Presence on Chaotic Advection”, Physical Review Letters, vol. 96, no. 2.
- Aref, H 1984, “Stirring by chaotic advection”, Journal of Fluid Mechanics, vol. 143, pp. 1-21.
- Aref, H 1994, “Chaotic Advection in Perspective”, Chaos, Solitons & Fractals, vol. 4, no. 6, pp. 745-748.
- Aref, H 2002, “The development of chaotic advection”, Physics of Fluids, vol. 14, no. 4, pp. 1315-1325.
- Boeing, G 2016, “Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction”, Systems, vol. 4, no. 4, pp. 37-54.
- Doherty, MF & Ottino, JM 1988, “Chaos in Deterministic Systems: Strange Attractors, Turbulence, and Applications in Chemical Engineering”, Chemical Engineering Science, vol. 43, no. 2, pp. 139-189.
- Layek, GC 2015, An Introduction to Dynamical Systems & Chaos, Springer, New Delhi, India, viewed 15 April 2018, ResearchGate database.
- Liang, Q 2006, Chaotic Advection, updated 28 March 2006, viewed 15 April 2018, <https://www.staff.ncl.ac.uk/qiuhua.liang/Research/Chaotic_advection.html>.
- Metcalfe, G, Speetjens, MFM, Lester, DR & Clercx, HJH 2012, “Beyond Passive: Chaotic Transport in Stirred Fluids”, Advances in Applied Mechanics, vol. 45, pp. 109-188.
- Moura, A, Feudel, U & Gouillart, E 2012, “Mixing & Chaos in Open Flows”, Advances in Applied Mechanics, vol. 45, pp. 1-50.
- Prants, SV 2014, “Chaotic Lagrangrian transport and mixing in the ocean”, European Physical Journal-Special Topics, vol. 223, no. 13, pp. 2723-2743.
- Rossi, L, Doorly, D & Kustrin, D 2012, “Lamination and mixing in laminar flows driven by Lorentz body forces”, Europhysics Letters, vol. 97, no .1.
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