Colloidal State Machines as Smart Tracers for Chemical Reactor Analysis

A widely utilized tool in reactor analysis is passive tracers that report the residence time distribution, allowing estimation of the conversion and other properties of the system. Recently, advances in microrobotics have introduced powered and functional entities with sizes comparable to some traditional tracers. This has motivated the concept of Smart Tracers that could record the local chemical concentrations, temperature, or other conditions as they progress through reactors. Herein, the design constraints and advantages of Smart Tracers by simulating their operation in a laminar flow reactor model conducting chemical reactions of various orders are analyzed. It is noted that far fewer particles are necessary to completely map even the most complex concentration gradients compared with their conventional counterparts. Design criteria explored herein include sampling frequency, memory storage capacity, and ensemble number necessary to achieve the required accuracy to inform a reactor model. Cases of severe particle diffusion and sensor noise appear to bind the functional upper limit of such probes and require consideration for future design. The results of the study provide a starting framework for applying the new technology of microrobotics to the broad and impactful set of problems classified as chemical reactor analysis.

In the past decade, the continuous improvement of microfabrication technologies helps to push the size of microrobots down to tens of micrometers. [58,59,61]Various functions have been demonstrated by researchers around the world. [62,63,65,68,69,72,73]][76] Using optical communication, researchers can read out the sensor status and send instructions to these micromachines in real time. [77,78]Innovations in functional materials enabled researchers to add legs onto these electronic microparticles, turning them into microrobots that can actuate. [65,69,79]Further integration with onboard digital timers and more complex logic circuits has made fully autonomous microscale robots a reality. [70]In particular, cell-sized machines that can be dispersed in a solution or aerosolized have been created, to access confined spaces such as brain tissue and soil matrix with minimum disturbance. [64,66,67][102][103] With higher intelligence, electronic microrobots have the potential to conduct far more complicated tasks than simple microparticles, such as glucose-responsive in vivo insulin control [72] and neural activity monitoring, [77,78] among many others.Combining the dispersibility of microparticles and the complex functionality of electronic devices, these colloidal microrobots have equipped us with a new tool to study the interior of chemical reactors noninvasively.
In this work, we analyze the design constraints of generic Smart Tracers by connecting the data that they collect locally to the global state of a complex chemical flow reactor operating under various conditions.Equipped with sensors, an onboard digital timer, memory units, and some simple control circuits, these smart tracers will be able to record local information as they travel through the reactors.Via numerical simulation, we find that the collective information logged in a small group of smart tracer particles (50 or less) with limited data storage capability (less than 20 data points per particle) is sufficient to reconstruct the concentration profiles inside reactors with unprecedented details.Various concentration fields within laminar flow reactors have been successfully recreated using this method.We notice a conflict between increasing the fitting resolution and maintaining a long enough working time, especially when the particles have very limited space for data storage.This is solved by combining the information from smart tracers with different sampling frequencies.Lastly, we investigate the impact of particle dispersion and cross-stream migration on the accuracy of profile reconstruction and estimate the maximum length of reactors for the smart tracer method to be effective.The results should advance the application of this new technology to the broad class of existing reactor analysis problems at all scales.

Problem Statement
The problem addressed in this work is how to use microrobotic technology to significantly improve upon the prediction of chemical reactor conversion of conventional approaches.Passive tracers rely on the information collected at the exit of the reactor; combined with knowledge about the reaction kinetics and fluid mixing (Figure 1a), engineers have been able to establish fairly accurate models of common chemical reactors (Figure 1b).However, these conventional approaches cannot distinguish between systems with degenerate outcomes. [104]To illustrate this, three simulated reactors are compared in Figure 1c; the concentration profiles at the outlet are almost identical for these reactors, despite them having active regions (in blue) at different locations along the tube (Figure 1d).Without inserting many sensors along the tube axis, it is impossible to determine the position of active regions with any measurements at the exit of reactors.Similar problems arise in many relevant scenarios, such as leakage in an oil or water pipe, [105][106][107][108] cancer cells around a blood vessel, [109][110][111][112][113][114] and natural resources or pollutants in a geological tunnel. [26,28,115,116]In most of these systems, the interior of the "reactor" volume is considered inaccessible in that it is either too disruptive, expensive, or logistically impossible to outfit sensors throughout the interior of the control volume.Hence, in all of these cases, the metrology of the problem is constrained to take place at the reactor exit.
Here, we propose a concept that enables model-free determination of the entire concentration profile in a laminar flow reactor by using colloidal microrobots as smart tracers.The tracer particles are equipped with a sensor for measurement, a clock circuit to keep track of time, and a memory array for recording information (Figure 2a). Figure 2 shows the schematics of the workflow: smart tracer microparticles are injected into a simulated reactor at the inlet, travel passively with the fluid and record the local concentration at a fixed frequency (Figure 2b), and are collected at the exit of the reactor.The recorded data in each particle may be read out electrically or optically; alternatively, the particles can transmit the data once they leave the reactors.Combining the recordings from multiple tracers with the velocity function (Figure 2c,d), the concentration profile inside the tubular reactor can be mapped out (Figure 2e).By fitting with a 2D piecewise bilinear function, we eliminate the requirement of any prior knowledge or models for the reactor.Even the velocity profile vðrÞ can be deconvoluted from the RTD of tracers (Figure S1 and Note S2, Supporting Information) as long as it is a monotonic function of radial position.

Effects of Sampling Frequency and Number of Tracers Under Ideal Conditions
In the first section, we explore the effectiveness of the smart tracer method under ideal conditions, where the particles follow the streamline (no diffusion), take accurate measurements, and have unlimited storage capacity.For each combination of sampling frequency (f τ) and the number of particles (n), we perform fitting on eight batches of different particles and report the average and standard deviation of the deviation from ground truth defined in Equation (3).The dimensionless sampling frequency and the number of particles determine the resolution of the collected dataset in the axial and radial directions, respectively.As the top row in Figure 3 shows, the deviation from ground truth decreases with the increasing number of particles and sampling frequencies for all reactors due to the increase in spatial resolution.The curves reach a plateau at a high number of particles (n > 30).Two factors are limiting the further improvement of fitting accuracy here.When the sampling frequency is low (f τ ¼ 4), the fitting accuracy is limited by the axial resolution of the dataset because a lot of the particles are only able to collect 3-4 data points before being flushed out of the reactor.When the sampling frequency is higher (f τ ≥ 20), the accuracy is likely limited by the number of grids in our fitting function, as we only used an 8-by-8 matrix in the fitting.This is also supported by the convergence of curves with different sampling frequencies (f τ ¼ 20 and f τ ¼ 100), which suggests that the axial resolution of the dataset is not the bottleneck.In addition to the average of deviation from ground truth, we noticed that the variance of deviation from ground truth between different batches also shows a decreasing trend with the number of particles in each batch.This agrees with our expectation that a larger batch of particles can be more evenly distributed on the cross-section of the reactor, hence improving the certainty of the fitting.
The deviation from ground truth also provides a quantitative measure of the overall accuracy of the fitting.It is also inherently normalized because all the concentration data are already normalized.The smart tracer approach provides reliable fitting accuracy for all concentration profiles we studied (top row of Figure 3 and S2, Supporting Information), with a deviation from ground truth no more than 0.2 for the worst case (f τ ¼ 4, n ¼ 10) and less than 0.02 for the best case (f τ ¼ 100, n ¼ 50).
The middle row of Figure 3 demonstrates the fitted concentration maps with 50 particles and a dimensionless frequency of 100.Compared with the actual profiles in the bottom row, the fitted profiles are able to reproduce the real concentration field with very high fidelity, matching not only the trend but also many details.Due to the limited number of fitting parameters, some detailed features of the profile may not be fully captured, such as the extremely sharp gradient near r = 1 and z = 0 in Figure 3c.This contributes to the higher deviation from ground truth in the case of the complex reaction network (Figure 3c) compared to other systems.Overall, the method works well for all kinds of reaction systems, from bulk reaction (Figure 3a) to surface reaction (Figure 3b), from simple first-order kinetics (Figure 3a), to complex reaction networks (Figure 3c).Most notably, we can easily figure out the location and size of active regions in a reactor by directly observing the fitted profiles (Figure 3d and additional profiles in Figure S2, Supporting Information).As a result, we can visually tell apart the location of active regions (blue area) in the top two subplots of Figure 3d, which is impossible to do if we only have information at the exits of reactors.When there are two smaller active regions in a reactor, the smart tracer method can easily resolve them if they are far apart (bottom subplot of Figure 3d).To resolve features that are closer  together spatially, the number of grid points in the fitting function can be increased, as Figure S2c, Supporting Information, shows.Overall, we have successfully demonstrated the most significant advantage of the smart tracer approach over the conventional chemical engineering approach, which motivated us to do this study in the first place (Figure 1c,d).
To analyze the requirements and performance of smart tracer fitting, we will focus on a specific reactor system, which involves a first-order surface reaction on the wall with one active region from z = 2 to z = 6 (Figure 4a). Figure 4b shows the worst fitting that we have studied, with only 10 particles and a dimensionless frequency of 4. Surprisingly, the fitted profile is still able to correctly reveal the position of the active region (z = 2 to z = 6), although the magnitude of the minimum concentration is quite far from the real value (Figure 4d).When we use 50 particles with the same sampling frequency for fitting, we are able to get both the position and magnitude of the active region with high accuracy (Figure 4c).The line plots of the fitted concentration along the tube wall (Figure 4d,e) also show that only a small number of particles (n ¼ 10) and a low sampling frequency (f τ ¼ 4) are needed to capture the location of the active region.It is notable that the fitted curves in Figure 4d,e, and S3, Supporting Information, do not capture the smooth curvature of the actual profiles (red dashed line).This is caused by the bilinear nature of the fitting function, which means the function is piecewise linear in r and z directions, respectively, although the overall function is quadratic.To obtain a smooth curve, a C 1 continuous interpolation method, such as cubic interpolation, can be used in either the drawing or the fitting itself (Figure S4, Supporting Information).
Conversion is an important metric in conventional chemical reactor design, which could be calculated from the fitted maps of the smart tracer method.The predicted conversion as a function of axial position matches quite well with the actual conversion curve (Figure 4f,g) for any frequency greater than 10 and particle number greater than 10.The change in the slope of conversion curves ( dX dz ) near the active regions is clearly predicted as well in Figure 4f,g and S5, Supporting Information.At the lowest frequency f τ ¼ 4, however, the predicted conversion curves do not match the actual conversion even if we use a large number (n ¼ 50) of particles, as Figure 4g shows.To analyze the origin of this large error at the low sampling frequency, we plot the absolute value of the difference between fitted and actual concentration profiles in Figure S6, Supporting Information.A comparison between Figure S6a,b, Supporting Information (or S6c,d, Supporting Information) reveals that a low sampling frequency leads to a large discrepancy on the order of 0.1 spreading across the whole reactor.On the other hand, when the sampling frequency is high, the error is smaller and is concentrated around the active regions.A very low sampling frequency not only reduces the axial resolution but also worsens the radial resolution as a result of large uncertainties in the residence time of particles.
To sum up, the results under ideal conditions (no diffusion, unlimited storage) provide guidance on the minimum requirements of the smart tracer fitting method.A dimensionless sampling frequency of 20 (f τ ¼ 20) is determined to be enough for the profiles studied in this work, and 50 particles (n ¼ 50) for each fitting is more than sufficient.Hereafter, we will use these two conditions to study the effects of other nonideal conditions.In practice, the number of grid points and the sampling frequency should both satisfy the necessary axial resolution for the specific problem.The radial resolution is less likely to be a problem, and 20-30 particles are usually enough.

Effects of Limited Storage Space
Although a high sampling frequency seems to have no disadvantages under ideal conditions, it encounters certain limitations in practice.The tracers have to be small enough compared to the reactors to follow the fluid motion and to minimize any perturbations to the system under investigation.The size constraints will put a certain cap on the capabilities of each smart tracer particle, such as the number of electronic components and the amount of energy it can store. [117]Therefore, it is of great importance for design to figure out how the capabilities of each particle affect the overall fitting.In particular, we focus on the effects of data storage space, as it is often linearly correlated with the physical size of particles.
The data storage space on each particle is quantified by the number of storage bits on it.Note that in the most general case, the Smart Tracer need not have digital memory but could store information in analog form, as shown recently in aerosolizable systems. [67]Each data storage unit can keep one measurement result (one data point).As Figure 5a shows, the particles at the center of a reactor move faster than the particles near the wall due to the quadratic velocity profile of the laminar flow.This can create a dilemma when the particles have a limited number of data storage units: the particles at the center need a high sampling frequency because they are moving fast, while the particles near the wall require a lower frequency; otherwise, they will run out of memory half-way through the reactor.As shown in the top row of Figure 5c-e, decreasing the data storage space significantly deteriorates the fitting accuracy for high-frequency particles (f τ ¼ 20), while it has little effect on low-frequency particles (f τ ¼ 4).The result of these two different trends is a crossover between the orange and violet fitting accuracy curves at about 50 data storage units per particle.The middle row of Figure 5c-e shows the fitting results for high-frequency particles with very little data storage space (15 bits per particle), which are far from the actual profiles as expected.Although low-frequency particles have much better accuracy under this limited data storage space, the conversion and the profile near the center are not well predicted, as we discussed in Figure 4c,g.
Combining the results of both the high-and low-frequency particles provides a simple way to achieve high accuracy fitting with limited storage space.Both types of particles will be released at the entrance with a uniform areal density, but the particles need to be selected after being collected.For the center of the reactor, only the information from high-frequency particles will be used, while only low-frequency particles will be used near the wall, resulting in a distribution shown in Figure 5b.Across a range from 15 to 100 data storage units per particle, the combined frequency strategy consistently allows for higher accuracy than either single frequency fitting approach, demonstrated by We choose a combination of high-frequency particles at the center (red) and low-frequency particles near the wall (blue) to give the best fitting.c,d) Fitting results for two concentration profiles.Top row: deviation from ground truth versus the number of data storage units (bits) per particle.Middle row: fitted concentration profiles (2D) as a function of radial and axial positions, with 15 bits per particle, 50 particles, and a dimensionless frequency of 20.Bottom row: Fitted concentration profiles (2D) with 15 bits per particle, 25 high-frequency particles (f τ ¼ 20), and 25 low-frequency particles (f τ ¼ 4).(c) Firstorder irreversible bulk reaction.(d) Mixed order series-parallel reaction network (van de Vusse system).e) Fitted concentration profiles (2D) of three reactors with degenerate outlet conditions using combined information from high and low-frequency particles.
the blue curves in the top row of Figure 5c,d.With 40 bits per particle, the combined approach already reaches the optimal accuracy (ε < 0.02) that requires 100 bits per particle in the high-frequency case and could never be achieved with lowfrequency particles alone.The better fitting accuracy is more clearly visualized by the profiles in the bottom row of Figure 5c,d, forming a sharp contrast with the profiles in the middle row.Additional results on other reactor systems also support the same conclusion (Figure S7, Supporting Information).The combined frequency approach also accurately captures the location of active regions in reactors with degenerate outlet conditions (Figure 5e), while high-frequency particles alone cannot (Figure S7, Supporting Information, left column).For the concentration along the wall, the combined frequency approach has the same accuracy as the low-frequency method, much better than the high-frequency particles (Figure S8, Supporting Information).The combined frequency approach has significant advantages over single frequency approaches in accurately predicting the conversion with limited data storage space, as Figure S9, Supporting Information shows.However, there is one tradeoff for achieving better fitting with the combined approach, which is the cost of throwing away the information from half of the particles.

Effects of Particle Diffusion and Measurement Error
So far, we have assumed ideal behaviors for the tracer particles.In this section, we will evaluate how the nonideal behaviors of particles, namely diffusion and measurement error, influence the fitting accuracy.For the fitting in this section, we will be using the combined frequency strategy with 25 high-frequency particles (f τ ¼ 20) and 25 low-frequency particles (f τ ¼ 4), and 40 data storage units per particle.
As a general discussion, we assume that the motion of a particle can be decomposed into a random walk in the cross-sectional area and a piecewise uniform linear motion in the axial direction following the local fluid velocity.Diffusion along the axial direction is neglected because convection should dominate the motion.Although we refer to the cross-sectional motion as "diffusion", it is not limited to the passive Brownian motion but can also include active motions as long as they are isotropic.The extent of this diffusive motion is quantified by the dimensionless diffusion length (L D =R) defined in Equation (7), which is inversely proportional to the square root of the Péclet number commonly used by chemical engineers.This type of motion leads to tortuous trajectories in the extreme case where the (dimensional) diffusion length of the particle is comparable to the tube radius (Figure 6a), while the trajectories are fairly straight when the diffusion length is one order of magnitude smaller (Figure 6b).The problem is hard to treat because the particles no longer have a constant velocity, which prevents us from accurately determining their radial and axial position.We have to use the average velocity of each particle to estimate its position, introducing a large error in the case of high diffusivity.
The top row of Figure 6c,d indicates that the accuracy of profile reconstruction is pretty much unaffected by the particle diffusion when the dimensionless diffusion length is small.Interestingly, a turning point at a dimensionless diffusion length of about 0.1 is present in each of these graphs, above which the deviation from ground truth grows much faster.The fitted profiles under these high diffusivity conditions are only able to capture the rough trend and the magnitude of the actual concentration, as the middle row of Figure 6c,d illustrates.Most of the important details, such as the location of the active regions, are lost due to inaccurate information about the particle positions (Figure 6e).To resolve these level of details, the dimensionless diffusion length should not exceed the turning point at 0.1, as the profiles in the bottom row show.In all the reactor systems that we simulated, we observed the same turning point and the loss of resolution at high diffusivity (Figure S10, Supporting Information).As the dimensionless diffusion length is proportional to the square root of diffusivity and residence time, this sets a limit on the length of reactors for a given diffusivity Beyond this limit, the information collected by the particles is blurred by the diffusion so much that we lose the essential features of the concentration profile.A more complicated algorithm is needed to improve the resolution of fitting under these high diffusion conditions, [118] which can be important for long tubes or very slow flow.
Another significant source of nonideality is the measurement error (noise), which can come from many components of the particle, such as the sensor, the data storage unit, the clock, or the postcollection information retrieval.All these error sources are lumped together as one stochastic noise in the measured concentration, which has a magnitude proportional to the actual concentration and a uniform distribution over the possible range.The resulting deviation from ground truth is roughly linear with the measurement error for all reactor systems we have studied (Figure S11, Supporting Information).Although quantitatively the deviation from ground truth at high noise levels ( ΔC C ¼ 0.6) was not much smaller than that at high diffusivity ( L D R > 1), the fitted profiles reveal much more details in the former case (middle column of Figure S11, Supporting Information).In particular, we are able to resolve the active regions clearly in Figure S11d-f, Supporting Information.This resistance to noise is partially due to the concentration-dependent noise magnitude and partially due to the self-averaging nature of our method that uses tens of particles.

Discussion
In this work, we propose for the first time using micrometerscale, colloidal robots to map the concentration profiles in tubular, laminar flow reactors.We formulate the computational method and illustrate the feasibility of this new approach by examining the influence of various design parameters as well as some practical constraints.This new framework, which we term the smart tracer approach, has enabled us to tell apart reactor systems that are indistinguishable to other noninvasive methods.More impressively, detailed mapping of the interior of reactors can be achieved without any prior knowledge or model.This method works well for a large variety of concentration profiles, and its accuracy is not affected by the kinetics of reactions.
For example, the profile reconstruction works equally well for second and 0th-order reactions as for first-order reactions (Figure S12, Supporting Information), and different reactors with degenerate outlet conditions can be easily distinguished from the reconstructed profile.
To translate the methods and results of this work into practical applications, several hurdles remain to be overcome.Experimentally, there will be significant challenges regarding the fabrication of reliable devices, the integration of many components, the effective retrieval of collected information, and so on.But we believe those challenges can be solved in the coming years with the advancement of photolithography techniques.Key components such as sensors, [63,67,68] memory devices, [66,72] and clock circuits [72,117] have already been demonstrated individually on other microrobots.Reliable methods for releasing and collecting smart tracer particles are also needed to ensure a uniform sampling over the cross-sectional area of reactors.For some application scenarios such as human body and natural reservoirs, biocompatibility, health, and environmental impact of these particles should be considered as well.
On the theory side, this work has been restricted to Hagen-Poiseuille flow in tubular reactors to make the problem tractable.However, the exact same method can be applied to laminar flow between two parallel plates or a very thin rectangular channel in practice, where the flow is essentially 1D, and the concentration field is 2D.In the extreme case of complete turbulent flow (Re > 10000), the velocity profile approaches a plug flow.Although this precludes us from determining the radial position of particles, the mixing is so strong that there is very little gradient in the radial direction, and therefore the radial position does not matter.The only exception is the region in close proximity to the wall, where particles may enter and leave as they travel down the tube.A detailed study on the behavior of particles in this complicated hydrodynamic environment is needed to figure out whether we can apply the smart tracer method to turbulent flow.A concentration profile that is not axisymmetric may also create some confusion; the fitted profile will average over the angular direction since the azimuthal position of particles is unknown.For reactor systems with more complex flow fields, such as packed bed reactors and fluidized bed reactors, the smart tracer method cannot be applied yet.As an intermediate step, it would be interesting to generalize the method in this article toward reactor networks with multiple branches in parallel, which resemble human circulatory systems.
As we aim at establishing a generic method, we haven't specified the dimensions, shape, materials, and other properties of the smart tracer particles.A good rule of thumb is to make particles as small as possible, with a density as close to the fluid in the reactors as possible, so as to reduce the disturbance to the concentration field.Smaller particles tend to have higher diffusivity, which can be a disadvantage.But the diffusion coefficient due to Brownian motion is well below 10 À13 m s s À1 for particles with diameters on the order of several micrometers. [119]Such a lowdiffusivity guarantees that our method can be effective for channels up to 200 m long with a diameter of 2 mm and an average flow velocity of 1 cm s À1 , based on Equation (1).Hydrodynamic forces in a laminar shear flow are shown to focus microparticles to a specific radius, [120][121][122] preventing uniform sampling over the cross-section.The focusing length is inversely proportional to the square of the size ratio between the rigid particle and the channel, [123,124] so a smaller particle size is desired to delay the focus (see Note S3, Supporting Information).However, the inertial focusing behavior can be used to our advantage if it is inevitable.[127][128] Moreover, the focusing of particles essentially eliminates the problem of diffusion, which is the biggest obstacle in the way toward high-accuracy fitting.

Experimental Section
To start with, different concentration profiles in tubular reactors (radius R, length L) are first simulated numerically using the finite difference method, which serves as the "real" profile (C real ) that we want to reconstruct.Note S1, Supporting Information, gives a detailed description of all the reactor models used in this study.Next, a certain number of tracer particles (400 for most of this study) are released at the entrance of these reactors (z ¼ 0), with random initial locations following a uniform distribution within the circular cross-section.All the particles then begin to move and record the local concentration at a fixed frequency f.The motion of the particles is separated into the cross-sectional and axial components.In the axial direction, the particles move at the same velocity vðrÞ as the local fluid.In the radial and azimuthal directions, the particles perform a confined random walk with a fixed step length proportional to the diffusion length between consecutive sampling events dx ¼ 2 ffiffi ffi D f q .A higher diffusivity (D) means that the particles are more likely to switch to a streamline that is far away from its original one.The recording of a particular particle will stop as soon as it exits the reactor (z ≥ L).Particles starting at different radii will spend different amounts of time in the reactor and provide different lengths of a recorded sequence of concentration data.After collecting and examining particles at the exit of a reactor, we obtain a raw dataset of the form fC i,j ðt i,j Þg, where i ¼ 1, : : : , n is the index of particles and j ¼ 1, : : : , m i is the index of the time step for each particle.The fitting starts with determining the radial and axial position of each particle at each time step.The radial position r i of particle i can be determined from its residence time and the velocity function vðrÞ.As the particles only take measurements at discrete time steps, the residence time is also a multiple of the time step size.Here we assume the fluid flow is unidirectional and the particle follows a single streamline.We will see later how this assumption affects the fitting accuracy when particles do not exactly follow the streamlines.Using the estimated velocity of each particle, each time step can also be correlated to an axial position by z i,j ¼ t i,j vðr i Þ.As a result, we now have a processed dataset of the form fr i , z i,j , C i,j g.
As a model-free approach, we use a piecewise bilinear function interpolated from an 8-by-8 matrix as the fitting function C fit ðr, zjC grid Þ, where C grid is the 8-by-8 matrix representing the concentration values on the fixed grid points.This function, therefore, has 64 fitting parameters, and the concentration at each point is predicted via bilinear interpolation of the values at four neighboring grid points.We proceed by minimizing the following objective function where the first term is the least-square error between the model prediction and the actual recording of particle i at the jth time step, and the second term λΨðC grid Þ is a small regularization term that penalizes excessive gradients.The problem is treated using the nonlinear least square fitting method (lsqnonlin) in MATLAB.All the concentration data are normalized to the maximum concentration.The fitting parameters C grid are constrained between 0 and 1, and a uniform initial guess of 0.5 is selected for all parameters.
To quantify the accuracy of the fitting results, we use the following root mean square error as a generic metric In addition to the deviation from ground truth and the concentration profile itself, we also report the conversion of reactants that is sometimes of interest for chemical engineers The conversion is a function of axial position and is averaged over the cross-sectional area.
The velocity profile is critical knowledge that enables us to extract spatial information from the time sequence recorded by smart tracers.In this study, we assume that all reactors have a laminar flow of Newtonian fluid (Hagen-Poiseuille flow) with the typical quadratic velocity distribution This assumption is convenient and is close to reality in most application scenarios, but it is not necessary.Our model can work with other types of velocity profiles as long as the fluid velocity is a monotonic function of radius. [129,130]Moreover, the velocity function can be derived from the modified RTD of passive tracers, removing the only requirement of prior knowledge and making the method truly model-free.The iterative method for velocity deconvolution is described in detail in Note S2 and Figure S1, Supporting Information.
The relevant time scale for a reactor is the average residence time The sampling frequency can be nondimensionalized as f τ.
The dimensionless diffusion/dispersion length is defined as follows Other dimensionless variables studied in this work include sensor noise ΔC C and number of storage units per particle n bit for recording information.

Figure 1 .
Figure 1.Problem statement using a 1D flow chemical reactor illustrating the limits of passive tracers.a) Schematic of the conventional approach that combines RTD, mixing models, and kinetics to predict conversion and concentration at the exit.b) Three typical concentration profiles in LFRs that can be distinguished by conventional approaches.From left to right, the profiles represent 1) a first-order irreversible bulk reaction, 2) a series-parallel reaction network (van de Vusse system) with bulk reactions of different orders, and 3) a constant flux from the tube wall with no bulk reactions.c) Concentration versus radial position at the exit of three reactors, showing degenerate outlet profiles.The legend indicates the position of active regions along the z-axis of the tubular reactors.d) Concentration profiles corresponding to the three curves in (c).The wall of these tubular reactors consumes the solute in the fluid with first-order kinetics.The active regions shown in blue have a surface rate constant 11 times as high as other parts of the wall.Details of all the reactor models are provided in Note S1, Supporting Information.

Figure 2 .
Figure 2. Concept and workflow of the smart tracer approach for mapping concentration profiles in laminar flow reactors.a) Schematics of the essential components of a microelectronic tracer particle.b) Schematics of the smart tracers traveling through a tubular reactor.c) The data recorded by one tracer (concentration versus time) are converted to concentration versus z position at a fixed radial position.d) Compiling data from all tracers, we get concentration versus z position at multiple radial positions.e) An example of the fitted concentration profile as a function or r and z.

Figure 3 .
Figure 3. Effects of sampling frequency and the number of tracers on the accuracy of profile reconstruction (ideal conditions).Top row: Deviation from ground truth versus the number of particles with different sampling frequencies.Middle row: Fitted concentration profiles (2D) as a function of radial and axial positions, with 50 particles and a dimensionless frequency of 100.Bottom row: Actual concentration profiles.Each column corresponds to one reactor system.a) First-order irreversible bulk reaction.b) Constant flux from the tube wall with no bulk reactions.c) Mixed order series-parallel reaction network (van de Vusse system).d) Fitted concentration profiles (2D) of three reactors with degenerate outlet conditions using 50 particles and a dimensionless frequency of 100.First-order surface reaction happens in different active regions for the three plots.

Figure 4 .
Figure 4. Detailed analysis of the effects of sampling frequency and number of tracers on the accuracy of profile reconstruction (ideal conditions).a) 3D profile of the actual concentration in an LFR with a first-order surface reaction on the wall and one active region from z = 2 to z = 6.b) Fitted concentration profile (2D) with 10 particles and a dimensionless frequency of 4. c) Fitted concentration profile (2D) with 50 particles and a dimensionless frequency of 4. d) Concentration versus axial position (z) curves at the wall of the reactor (r ¼ 1), extracted from fitting results with 10 particles and various frequencies.e) Concentration versus axial position (z) curves at the wall of the reactor (r ¼ 1), extracted from fitting results with 50 particles and various frequencies.f ) Conversion versus axial position (z) curves calculated from fitting results with 10 particles and various frequencies.g) Conversion versus axial position (z) curves calculated from fitting results with 50 particles and various frequencies.

Figure 5 .
Figure 5. Effects of limited data storage on the accuracy of profile reconstruction (no diffusion).a) Schematics of a typical velocity profile and the different traveling speeds for particles at the center (red) or near the wall (blue).b) Cross-sectional positions of tracer particles with different sampling frequencies.We choose a combination of high-frequency particles at the center (red) and low-frequency particles near the wall (blue) to give the best fitting.c,d) Fitting results for two concentration profiles.Top row: deviation from ground truth versus the number of data storage units (bits) per particle.Middle row: fitted concentration profiles (2D) as a function of radial and axial positions, with 15 bits per particle, 50 particles, and a dimensionless frequency of 20.Bottom row: Fitted concentration profiles (2D) with 15 bits per particle, 25 high-frequency particles (f τ ¼ 20), and 25 low-frequency particles (f τ ¼ 4).(c) Firstorder irreversible bulk reaction.(d) Mixed order series-parallel reaction network (van de Vusse system).e) Fitted concentration profiles (2D) of three reactors with degenerate outlet conditions using combined information from high and low-frequency particles.

Figure 6 .
Figure 6.Effects of particle diffusion and measurement error.a) Trajectories (magenta) of smart tracer particles with high diffusivity ( L D R ¼ 0.89) in a reactor.b) Trajectories (magenta) of smart tracer particles with low diffusivity ( L D R ¼ 0.09) in a reactor.c,d) Fitting results for two concentration profiles.Top row: Deviation from ground truth versus the dimensionless diffusion length of particles.Middle row: Fitted concentration profiles (2D) as a function of radial and axial positions, using 50 particles with high diffusivity and a dimensionless frequency of 20.Bottom row: Fitted concentration profiles (2D) using 50 particles with low diffusivity and a dimensionless frequency of 20.Each column corresponds to one reactor system.(c) First-order irreversible bulk reaction.(d) Mixed-order series-parallel reaction network (van de Vusse system).e) Fitted concentration profiles (2D) of reactors with degenerate outlet conditions.Top: Fitted profile under high particle diffusion.Middle and Bottom: Fitted profile under low particle diffusion.The locations of the active regions are marked in the plots.