Millimeter-Wave Interferometry for Opaque Particle-Laden Flows

A novel method to measure the concentration of particles in optically opaque particle-laden flows is presented. This method is based on the principle of millimeter wave interferometry, using a fully-integrated frequency modulated continuous-wave (FMCW) radar operating between 77 and 81 GHz to measure path-integrated particle concentrations between the radar and a reflector. The instrument is capable of quantitative, high-speed (20 kHz) path-integrated concentration measurements in dispersed multiphase flows with concentrations one to two orders of magnitude higher than those at reach with state-of-the-art optical methods. The interferometer was demonstrated and calibrated for path-integrated number concentrations up to <inline-formula> <tex-math notation="LaTeX">$(4.36 \pm 0.24) \times 10^8 \mathrm{~m}^{-2}$ </tex-math></inline-formula> using glass microspheres with a mean diameter of <inline-formula> <tex-math notation="LaTeX">$109.2 \mu \mathrm {m}$ </tex-math></inline-formula>. Two independent measurements of particle size distribution (PSD) were performed using X-ray microtomography and dry sieving. The calibration setup relied on high-resolution particle shadowgraphy applied to individual thin particle streams and used multistreams superposition to reproduce large optical depths in a controlled particle-air mixture. The instrument exhibited excellent linearity and low error during the calibration, with a phase shift-to-number concentration slope of <inline-formula> <tex-math notation="LaTeX">$(1.378 \pm 0.043) \times 10^{-7} \mathrm{~m}^2$ </tex-math></inline-formula>, validating the measurement concept and paving the way for practical applications. The leading uncertainties are discussed, providing guidelines for exploiting the measurement concept without necessarily performing a direct calibration.


I. INTRODUCTION
P ARTICLE-LADEN flows, also known as dispersed multi- phase flows, are defined by the presence of small discrete particles (bubbles, droplets, or grains) suspended in a continuous fluid [1].They are important to a wide range of natural and man-made environments, including, among others, fluidized bed reactors [2], [3] and sediment transport [4], [5].Particleladen flows are the direct cause of a number of hazards with significant human and material costs.Dust explosions caused 119 deaths and hundreds of millions of dollars in property The authors are with the Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (e-mail: rasmont2@illinois.edu).
Color versions of one or more figures in this article are available at https://doi.org/10.1109/TMTT.2023.3277527.
Digital Object Identifier 10.1109/TMTT.2023.3277527damage in the United States between 1980 and 2005 [6].Aeolian soil erosion is a significant contributor to desertification and loss of arable land [7].Dispersed multiphase flows are also of high importance during the landing of aerospace vehicles on unprepared surfaces.Dust lifted by the downwash of a rotorcraft can lead to a complete or partial loss of visibility during landing, a dangerous condition called brownout.Spatial disorientation due to brownout has been the cause of 400 rotorcraft losses at a cost of 152 lives and one billion United States Dollar (USD) in damage in the U.S. Army alone between 2002 and 2015 [8].The interactions between rocket plumes and granular planetary surfaces (plume-surface interactions or PSI) during the landing of a human-class spacecraft on the Moon or Mars give rise to a number of challenges.The erosion of the surface by the jet generates multiple tons of supersonic, highly abrasive ejecta capable of damaging the spacecraft and any infrastructure nearby [9], [10], and lead to the formation of a crater deep enough to topple or bury the lander [11], [12], [13].The ejecta cloud itself can blind pilots and landing sensors during terminal descent [14].
Improving understanding of particle-laden flows is critical for industrial and aerospace safety, environmental preservation, and a wide range of other fields.Theoretical and numerical descriptions of dispersed multiphase flows are challenged by the coupling between a discrete inertial particle phase and the continuum flow phase, generally turbulent.Despite significant advances in the last 30 years from experiments and simulations, numerical and theoretical models of multiphase flows are still not suitable for predictive capabilities in many realistic applications.Current state-of-the-art point-particle Lagrangian models coupled with an Eulerian flow description can be used to simulate thousands to millions of particles at the expense of computational power [15], [16], [17], but these numbers are still far from practical real-life situations involving billions or trillions of particles [18], [19].In addition, the fidelity of those simulations depends on the range of particle-fluid nondimensional parameters that dominate the two-way coupled interactions being modeled.The computational cost of Eulerian-Eulerian models on the other hand is more permissive for large simulations but they necessitate constitutive models for interparticle and interphase dynamics [20], [21].
Experimental research in particle-laden flows has been much benefited from the rapid development of high-speed and highresolution cameras in the last two decades.While essential for dilute conditions, quantitative concentration measurements via imagining are challenged as concentration increases due to apparent particle overlap and ghost particles [16].Laser and phase Doppler anemometry (PDA) and laser diffraction techniques are also commonly used to measure particle concentrations, velocities or sizes [22], [23], [24].They are most suitable for dilute conditions, with measurement uncertainties that increase with concentration.At volume fractions above 0.1%-0.01%,depending on the particle size and material properties, optical diagnostics fail due to the opacity of the particlefluid mixture [25], [26].Measurements in particle-laden flows also face other challenges absent in single-phase flows as particles can coat or damage experimental hardware through erosion, impact, mechanical jamming, or triboelectric charging [27], hindering the use of classical intrusive instrumentation.Some nonintrusive, nonoptical diagnostics are applicable to opaque flows.X-ray transmission computed tomography (CT) [28], X-ray diffraction tomography [29], gamma-ray transmission CT [30], neutron transmission CT [31], and positron emission CT [32], use ionizing radiations to characterize opaque flows with extremely high volume fractions, theoretically up to 100%, at high spatial resolution.However, they suffer from high cost, low temporal resolution, and bring a range of health and safety risks.Other nonionizing methods include nuclear magnetic resonance (NMR) imaging [33], ultrasound tomography [34], microwave tomography [35], and a set of related electrical tomography methods: capacitance tomography (ECT) [36], resistivity tomography (ERT) [37], and impedance tomography (EIT) [38].They typically suffer from high attenuation, low acquisition frequency, and, with the exception of NMR imaging, have relatively low resolution, on the order of centimeters.Most of those nonoptical diagnostics require the experiment to fit within the instrument, restricting their applicability for fluid-driven experiments.
This work presents a novel method to measure the volume fraction in particle-laden flows that overcomes some of the limitations of existing diagnostics in high concentration environments.Originally developed for studying plume-surface interactions in planetary landing experiments [39], it is amenable to the study of opaque gas-particle mixtures where optical diagnostics fail due to excessive attenuation.The method is based on the principle of millimeterwave (mm-Wave) interferometry, which has been extensively investigated for medical [40], [41] and industrial [42], [43] purposes.mm-Wave interferometric techniques are also used to measure electron densities in plasma devices [44], [45].In the current study, a fully integrated, frequency-modulated continuous-wave (FMCW) radar operating between 77 and 81 GHz illuminates the area-under-test, tracking the phase shift of a wave bouncing off a reflector located at a known, fixed position.The FMCW mode of operation allows the instrument to discriminate between the reflector echo and clutter without requiring the complex quasi-optical beam-forming system of continuous-wave interferometers [46].The phase of the electromagnetic wave provides the concentration of particles integrated along the line of sight between the radar and the reflector at a high repetition rate, independently of the size distribution of the material, and for a range of concentrations at least one order of magnitude higher than optical techniques.
Sampling rates of 20 kHz were achieved in this work, but several MHz are possible using a dedicated, optimized radar system.The sensor compares favorably in terms of cost, power, size, and mass with conventional nonintrusive particle concentration diagnostics and unlike most of them, can be extended to field measurements.
This article is an extension of the works presented in [47] and [48].Focusing on the specific use case of an ongoing plume-surface interaction experiment, they served to demonstrate the potential of mm-Wave interferometry for rapidly varying optically opaque environments.This manuscript presents a refined version of a mm-Wave interferometer for particle concentrations that is application agnostic.The theoretical foundations of the technique are revisited in detail in Section II.The instrument and the experiment specifically designed for its demonstration and calibration are described in Section III, with their results presented in Section IV.This work also extends on detailed propagation of uncertainties not previously explored, essential to improve the accuracy of the calibrated instrument and to evaluate the validity of the theoretical models.The second objective is achieved using Monte-Carlo (MC) simulations fed with probability distributions for the unknown parameters in the models derived for the Maxwell-Garnett equation.The analysis in Section V suggests that calibration-free measurements are possible and provides a framework to identify which particle properties need to be more accurately known to minimize measurement uncertainties.Published experimental data in particle-laden jets is used to evaluate the potential of the new instrument to extend the range of particle concentrations that can be quantified.The strength and limits of mm-Wave interferometry are also discussed, together with recommendations for future developments.

II. THEORY OF OPERATION A. Fundamentals
For the purpose of modeling electromagnetic wave propagation, particle-laden flows with randomly distributed particles can be assumed to behave as a homogeneous, isotropic material with an effective dielectric constant ε eff .The Maxwell-Garnett equation shown in (1) [49] provides ε eff as a function of the volume fraction of particles δ (ratio of the volume of particles to total volume), the dielectric constant of the particle material ε i , and the dielectric constant of the host fluid The Maxwell-Garnett equation is valid for particles smaller than the wavelength by at least an order of magnitude and considers the particles as random spherical inclusions in the medium.(1) can be simplified if |ε eff − ε h | ≪ ε h , or equivalently δ ≪ 1, to obtain a closed-form expression for ε eff .This simplification is valid for dispersed multiphase flows of interest for the instrument.Multiple scattering effects are usually neglected in the Maxwell-Garnett equation in dilute particle mixtures (δ ≪ 1) due to the very low scattering efficiencies of particles smaller than the wavelength, as described in Section II-B.Even in high particle concentration mixtures for which multiple scattering could be significant, experiments and simulation have shown that the Maxwell-Garnett equation stays valid [50], [51] The time of flight (TOF) τ of an electromagnetic wave propagating in the fluid-particle mixture between an emitter (TX) and a receiver (RX) is given by (3) as a function of the speed of light in vacuum c 0 and the effective dielectric constant ε eff along the propagation path of the wave l TX,RX The time delay τ between a wave propagating in a particle-laden medium and a wave propagating in an unladen medium can be measured as a phase shift φ by an interferometer.This approach eliminates any offset time due to instrument components (i.e., antennas, transmission lines, etc.) Combining ( 2)-( 4) provides the phase shift caused by the presence of the particles as a function of the path-integrated particle volume fraction between emitter and receiver, and electromagnetic wave, particle and continuum host medium properties, to obtain the measurement equation for the instrument Two major benefits of interferometry for measuring the concentration of the dispersed phase are evidenced in (5): the measured phase shift is directly proportional to the path-integrated volume fraction of particles, and it is independent of the particle size distribution (PSD).The proportionality between phase shift and path-integrated volume fraction is advantageous to many realistic applications in which the dispersed particle phase is poorly characterized.The PSD is nonetheless required if the desired output quantity is the number concentration of particles.

B. Extinction Effects
Extinction is a significant challenge for optical diagnostics operating in densely loaded multiphase flows and one of the motivations for the development of the current mm-Wave interferometric instrument.A brief review of extinction theory is useful in explaining how radar interferometry can improve the measurement range of optical methods by several orders of magnitude.An electromagnetic wave propagating through a fluid-particle mixture is scattered and absorbed by the particles, leading to a reduction in flux intensity with distance.This phenomenon is described by the Beer-Lambert law [52] in (6), relating the flux intensity of an electromagnetic wave I RX measured by a receiver (RX) after propagating through a diffusive material with an initial flux intensity of I TX where µ(l) is the linear attenuation coefficient along the path between (TX) and (RX).In the case of solid particles suspended in a transparent medium, µ(l) is given by (7) as a function of the particle number distribution N p (D), number concentration n p , particle diameters D, and extinction efficiency The extinction efficiency Q ext (D, f 0 ) can be analytically derived using Mie theory [53], assuming spherical particles with a known dielectric constant ε or refractive index √ ε as a function of the frequency f 0 (or wavelength λ 0 ) of the incoming electromagnetic wave and particle diameter D.
When λ 0 ≪ D, scattering is governed by the propagation of light rays inside the particle according to geometric optics.The extinction efficiency in this configuration is constant at Q ext = 2, independent of material properties, frequency, or particle size.This is the situation encountered by most optical diagnostics in particle-laden flows, with particles in the 10-1000 µm range and wavelengths in the 100-1000 nm range.For λ 0 ≈ D, Mie scattering is encountered.The amplitude of the extinction efficiency oscillates with respect to the size parameter, D/λ 0 .While this behavior can be used to determine particle sizes and constituent material properties, it is associated with strong extinctions efficiencies and is hard to access for particles in the 10-1000 µm range, as effective wave generation and measurement technologies are not yet available in this section of the electromagnetic spectrum (the "terahertz gap").When λ 0 ≫ D, Rayleigh scattering happens.The extinction efficiency is proportional to the fourth power of the frequency (Q ext ∝ f 4 0 ), until absorption becomes dominant over scattering (Q ext ∝ f 0 ), depending on the loss tangent of the material.Extinction efficiencies are several orders of magnitude lower in the Rayleigh regime than in the geometric regime.
This means that for particles in the 10-1000 µm range, an instrument operating in the microwave range, λ 0 = 1-100 mm, will be able to perform measurements at particle concentrations much higher than an optical instrument of comparable power and dynamic range.Fig. 1 shows the extinction efficiency predicted by Mie theory for 105 µm diameter glass particles, very similar to those used in the experiments presented in this article, for different wavelengths and frequencies.The material is assumed to have a dielectric constant 4.5 + j6.75 × 10 −2 at 77 GHz (3.89 mm) and a refractive index of 1.51 at 650 nm, within the expected range for glass material [54], [55].The particles have an extinction efficiency 20 000 times lower when illuminated by a 77 GHz mm-Wave radar, than by a visible 650 nm laser.
Equations ( 5) and ( 6) can be used to compare the performance of conventional optical diagnostics and mm-Wave interferometry for 105 µm glass particles.Fig. 2 shows the  ] and must therefore be unwrapped when values exceed this threshold.The maximum phase up to which unwrapping can be reliably performed is an important parameter to determine the measurement range of the interferometer.It depends on the time derivative of the particle volume fraction, on the acquisition frequency, and on the noise of the instrument.If restricted to the unambiguous [0 • , 360 • ] angular range, mm-Wave interferometry is capable of measuring particle concentrations one order of magnitude higher than a laser instrument with a 60 dB dynamic range.The proof-of-concept interferometer developed in this study has been demonstrated to measure phase up to 20π radians [48].Such a capability extends its measurement range by another order of magnitude, while the signal attenuation stays at a manageable level.
The ability to measure high particle concentrations is a fundamental advantage of radar interferometry over optical attenuation or scattering-based techniques.Indeed, one could imagine to compensate for excessive extinction by simply increasing the sensitivity of an optical instrument, pushing the dynamic range higher and higher by using more illumination power or a better light sensor.However, this approach is foiled by the exponential nature of light extinction, which results in dramatically diminishing returns in measurement range with increases in sensitivity.For 105 µm particles, increasing the measurement range of an optical attenuation sensor from a path-integrated volume fraction of 2 × 10 −2 m-2 × 10 −1 m, well within the capability of a mm-Wave interferometer, requires 20 orders of magnitude (200 dB) increase in sensitivity, impossible to achieve.Another benefit of radar interferometry is that it can be tailored to a concentration range of interest: lower frequencies are well-suited for measuring high concentrations, as they are less attenuated and have a higher unambiguous measurement range, while higher frequencies are more sensitive to lower concentrations.

A. Instrument
The mm-Wave interferometer used in this work was made of two evaluation boards manufactured by Texas Instruments, the IWR1443BOOST and DCA1000EVM.The IWR1443BOOST carries a IWR1443 radar chip, which integrates a complete 3TX, 4RX 77-81 GHz frequency-modulated continuous wave radar transceiver chip, analog-to-digital converters (ADCs), and a microcontroller in a 10.4 × 10.4 mm package.The DCA1000EVM is an interfacing board that captures the raw ADC data from the radar and streams it over a 1-GBps Ethernet link to a computer running the mmWaveStudio software.Previous studies by the authors [47] were conducted using the lower frequency IWR6843 as part of a proof-of-concept instrument.It was found to suffer from an excessive imbalance between its in-phase and quadrature channels that leads to high noise.The IWR1443 board does not present this problem.The entire radar system fits in a 130 mm by 130 mm package, weights less 300g (100g for the electronic components themselves) with a peak power consumption of 5 W. The specific radar used emits chirps at a repetition rate of 20 kHz, which defines the temporal resolution of the instrument, similar to conventional high-speed cameras.The chirps have a slope of 100 MHz/µs for a chirp time of 40 µs and idle time between chirps of 10 µs.These parameters correspond to the maximum chirp bandwidth of 4 GHz available to the radar system, which helps minimize clutter and multipath-interference.Fig. 3 shows the radar interferometer in the subscale experiment that motivated the development of this instrument.The setup reproduces the plume-surface interaction phenomenology of a rocket landing on a planetary surface [39].A Mach 5 jet impinging on a granular surface leads to a dense cloud Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. of ejecta and a crater that evolves in time.The radar system setup is identical to that used in this work for demonstrating and calibrating the instrument.A 31.75 mm diameter copper disk reflector defines the line-of-sight with the radar antenna, and thus the measurement chord, which is 58 cm long.The spatial resolution and the measurement volume of the instrument is defined by the apparent angular size of the reflector seen by the radar antenna, 3.1 • for the reflector and antenna cross sections and distance between them in this work.A planar reflector is used instead of a more conventional corner cube because its highly directional cross section reduces the effect of multipath-interference.The IWR1443BOOST and DCA1000EVM boards are protected from the particles by a 3-D printed PLA and acrylic enclosure including a PTFE window transparent to mm-Waves.The signal echo corresponding to the reflector is extracted from the radar range data, and the phase of the range bin corresponding to the reflector tracked between each chirp, providing a time-resolved interferometric phase shift measurement.

B. Instrument Calibration
The theoretical framework presented in Section II suggests a linear relationship between the phase shift measured by the instrument and the path-integrated volume particle concentration.The constant of proportionality depends on the radar, host medium, and dispersed phase properties.Provided (5) holds exact, for a given radar it would suffice to know accurately the dielectric constants of the host medium and particles, which could be achieved using a vector network analyzer (VNA) in the mm-Wave range.However, a number of simplifying assumptions are involved in the Maxwell-Garnett theory that although widely used to predict the dielectric constant of mixtures, may render the proportionality factor, or even the linear relation, not accurate.A calibration approach that would directly serve two purposes was preferred: verifying the linearity between phase shift and path-integrated volume fraction, and directly obtaining the relation between them without the need of measuring dielectric constants, thus avoiding the need for a mm-Wave VNA, an expensive and very specific instrument not widely available.The calibration process is based on the superposition of thin curtains of falling particles, each of which can be individually characterized using an optical counting method.When combined, the curtains provide an optically thick medium with a known path-integrated particle concentration within the measurement range of the instrument.Using this calibration procedure, the instrument measures the travel delay through the same mass of air, laden and unladen, at an interval of a few seconds, which also eliminates any variability due to the host medium.This is in contrast with solid matrix calibration methods [56], which require tight control on the matrix dimensions and physical properties as they directly influence the propagation time of the waves through the mixture.The calibration was implemented using a funnel with individually addressable slots to generate thin curtains of falling particles.The path-integrated concentration of each curtain was measured by a shadowgraphic optical particle-counting method illustrated in Fig. 4(a).Once the particle curtains generated by each slot were characterized, a particle cloud with a known path-integrated concentration was generated by simultaneously opening multiple slots, as shown in Fig. 4(b).By measuring the phase shift for an increasing number of open slots, a calibration curve for phase shift as a function of the number particle concentration is obtained.The calibration set-up covered up to 15% of the unambiguous (0 • -360 • ) measurement range of the instrument.The relationship between these parameters is assumed to remain linear, i.e., the calibration can be extrapolated to higher concentrations.
This methodology requires particle curtains to be identical when individually or jointly operated, which is not the case at ambient conditions due to aerodynamic entertainment.The interaction between the gravity-driven particle flow and the otherwise quiescent surrounding air leads to the generation of a turbulent flow.When multiple particle streams in close proximity are operated, the cross section of a given stream, particle velocities, and thus local particle concentration are modified, leading to a path-averaged concentration that is different from the sum of the contributions of each individual curtain.To avoid these undesired effects the calibration was performed in a chamber in a reduced atmosphere The particles used as regolith simulant were Ballotini solid glass microspheres with a quoted size range of 90-150 µm and a density of 2500 kg.m −3 .A 1.5 W white LED collimated by a biconvex lens with a focal length of 100 mm provided the back-illumination for the shadowgraphy particle counting.A 1.3 Mpixel Chronos 1.4 high-speed camera equipped with a Canon EF 100 mm macro lens was used for imaging the particle stream at 1057 frames/s.An exposure of 5 µs was used to prevent motion blur of the falling particles.With an image pixel size of 13.23 µm/pixel, the glass particles were resolved by 53 pixels on average.The following procedure was used to count the particles.First, images were segmented from the background and a sample of well-resolved single particles was selected using a combination of size and roundness thresholds.The total area of all particles in the image was then divided by the average particle area of the selected sample, and a correction factor was applied to account for overlapping.The correction factor was derived by comparing the number of particles counted by the pipeline n o against synthetic images with a known number of particles n g .The synthetic images used are representative of real background-subtracted images, with dark circular particles against a white background.Particles are uniformly distributed and their diameter follows the size distribution measured by CT, described in Section III-C.For each value of n g , ten synthetic images were generated to allow

C. Granular Material Characterization
The phase shift measured by the radar interferometer is proportional to path-integrated volume fraction and is agnostic to the size distribution of the particles.However, because it was calibrated against an optical method providing the path-integrated number concentration of a particle cloud, the conversion of path-integrated number concentration into path-integrated volume fraction requires knowledge of the PSD of the material.While this partially negates one of the advantages of mm-Wave interferometry, it arises from the need of verifying the theoretical measuring principle using robust optical methods for particle-counting in an air-particle mixture environment similar to that of the target experiments.Knowledge of the PSD is not required if the desired output is number concentration of particles and the instrument is calibrated against that quantity, as in the present work.The relation between path-integrated number concentration n p and path-integrated volume fraction δ is given by the mean of the volume-weighted PSD δ = (1/6)πn p (D m,v ) 3 .
The size distribution of the particles was measured via X-ray microtomography, in addition to the results from planar shadowgraphy, due to the higher accuracy of CT particle sizing.Uncompacted particles samples were imaged by an Xradia MicroXCT-400 system.The tomographic images, with a voxel size of 2.61 µm/voxel, resolved each particle by 39 000 voxels on average.The images were segmented using a watershed algorithm, resulting in a set of individually-labeled particles whose diameter was measured and binned to determine the sample PSD.The outcome of the segmentation is presented in Fig. 6.For comparison, the particle PSD was also measured by a dry sieve shaker with size breaks at 53, 63, 90, 106, 125, 150, 180, 250, and 350 µm.Results are reported in Section IV-A as volume distributions.
The are two main error contributors to the PSD as derived from micro-CT measurements: error on the particle diameters due to the image resolution, σ D , and error on the diameter Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
binning made by the algorithm σ V .The image resolution error can be estimated as r CT / √ 2, with r CT the voxel size.The binning error σ V is due to particles that have escaped detection by the watershed segmentation algorithm, usually due to poor convexity.The upper and lower bounds of the volume-weighted mean particle diameter D m,v due to those missing particles were computed as follows.First, the fraction of particles volume missed by the algorithm was estimated by comparing the volume of the raw, unsegmented particles to the volume of segmented particles.Then, two particle volume distributions were generated: V l (D), in which all the unaccounted volume was assigned to the fifth percentile bin of the distribution, and V u (D) in which all the unaccounted volume was assigned to the 95th percentile bin.For each distribution, the corresponding mean particle diameter D m,V l and D m,V u are computed.While those distributions are highly unlikely, they serve to define upper and lower bounds of the true value of the mean diameter D m,v , for which the error can then be defined as The final value of the microCT mean diameter error was 6.7 µm, or 6.1% of D m,v .A similar procedure was used to estimate the error on the mean particle diameter obtained by sieving, yielding an error of 9.7 µm, 8.8% of the mean.The mean particle diameter from the planar shadowgraphy images is affected by image resolution and also by the particle identification and postprocessing methods.Shadowgraphy measurements were focused on particle number counting and not optimized for size characterization, and have the largest error on the volume weighted mean diameter.Only the resolution error is provided here as a lowest bound estimate, r pix / √ 2 = 9.4 µm, or 7.8% of the mean.

A. Particle Size Measurements
Micro-CT imaging of particle samples provided the particle volume distribution used to convert number to volume concentrations in this work, shown in Fig. 7. Results are compared with those obtained by sieving and direct optical imaging during the calibration.The mean, median, mode, and fifth and 95th percentile diameters are reported in Table I together with the measurement error of each technique.There is a close agreement between the mean diameter as measured via micro-CT and sieving, to within 1%.The median, fifth percentile, and 95th percentiles are also within 5% of each other.The larger gap in the mode diameter of 12% can be explained by the lower resolution of the sieving distribution.The volume distribution obtained through optical imaging appears to be a version of the micro-CT distribution skewed toward higher values by approximately 11 µm.This is considered an artifact due to the thresholding method used in the particle segmentation, affected by the lower resolution of optical imaging.

B. Optical Characterization of the Calibration Set-up
Particle concentrations measured for each of the 25 individual funnel slots as described in Section III-B are presented in Fig. 8.Each series, labeled as A-C, corresponds to an independent measurement.Since the spread of the data for each slot is comparable to the total standard deviation of all data points independent of the slot index, each measurement is treated as a single realization of a random variable.The average particle concentration and measurement error for all slots in the

C. Radar Calibration
Radar measurements were conducted for increasing the number of slots open simultaneously.The sequence followed for opening the slots did not have any impact on the calibration regression.Measured phase shift as a function of path-integrated number concentration is shown in Fig. 9. Error bars represent the uncertainty on phase shift and path-integrated number concentration measurements.Due to the presence of error on both parameters, the calibration curve was determined using a zero-intercept Deming regression.MC simulations were used to propagate the combination of systematic and random errors for both phase and integrated number concentration measurements, and to derive a 95% confidence interval fit.The MC probability of a data point being located at a given number density and phase is given by the greyscale contour.The main error source on the radar phase measurement is local clutter, while the optical measurements have error terms related to the output of the slot funnel and to the particle counting algorithm.
The slope of the calibration curve is There is an excellent linearity between phase shift and path-integrated particle concentration, with a generalized coefficient of determination r 2 g = 0.9964.This linearity partially verifies the validity of the assumptions involved in the Maxwell-Garnett equation and the theoretical model in (5).The proposed calibration method provides the functional relation between phase shift and path-integrated concentration.Therefore, even if the relation had not been linear, once calibrated the proposed mm-Wave interformeter would still enable path-integrated concentration measurements.

A. Comparison Between Experimental and Theoretical Results
In this section, the experimental results are compared to the theoretical framework presented in Section II.While the linearity of the experimental calibration curve is in agreement with the theory, the aim is to compare the measured slope to its theoretical prediction.It should be noted that the theoretical relation is not needed if a calibration such as the one previously described can be performed, which provides the true relation between phase shift and path-integrated particle number density for a given radar and particle-medium mixture.However, verification of the theoretical relation is important for calibration-free methods that would allow knowing the factor of proportionality between path-integrated concentrations and phase shift provided the medium and particle dielectric constants are known with the required measurement accuracy.For this purpose, a statistical model of the calibration experiment based on (5) was used to obtain a probability distribution of possible calibration slope.The uncertainty of our theoretical slope prediction is due to the measured or estimated uncertainties of the parameters involved in the theoretical model.A p-value hypothesis test was conducted between the experimental slope and the theoretical distribution of slopes to quantify their agreement, with a 5% significance level.A sensitivity analysis was conducted to identify the contribution of each parameter to the spread of the model.Finally, the values of the model parameters most in agreement with the experimental slope were derived using maximum likelihood estimation.The theoretical model of the experiment is presented in (9).It combines (5), which predicts the phase shift as a function of path-integrated volume fraction, with the path-integrated volume fraction generated by each particle stream.The use of a passive reflector means that the mm-Waves travel through the particles twice, effectively doubling the signal for a given concentration The parameters involved in the model were modeled as probability distributions according to experimental measurements or a priori estimates from available data in the literature.The radar operating frequency f 0 , the number of open slots N s , and the dielectric constant of the unladen propagation medium (air or vacuum), ε h , were considered known, not requiring probabilistic modeling.The mean of the volume-weighted particle distribution D m,v and the path-integrated particle number concentration per slot n s were modeled as Gaussian variables with mean and standard deviation provided by the experimental results described in Sections IV-A and IV-B, respectively.Similarly, a noise term was applied to the radar phase shift φ to account for the measurement error reported in Section IV-C.The dielectric constant of the particle material, ε p , not measured in the current work, was modeled by a uniform probability distribution in the range of 4-7, consistent with values drawn from literature for glass at mm-Wave range [54], [57].A calibration curve made of 25 data points was generated by varying the number of open slots, N s .The slope of the calibration curve was calculated using a Deming regression.Given the number of uncertain parameters and the covariance between individual data points due to systematic error terms, MC methods were used for uncertainty propagation with 15 000 repetitions instead of analytical error propagation.The MC probability of a measured phase shift for a given path-integrated number density is represented in greyscale in Fig. 10 along with experimental data for comparison.The mean and 95% confidence interval of the slope distribution predicted by the model are indicated by solid and dashed lines as 1.163±0.316× 10 −7 m 2 .The experimental data is in agreement with the numerical model, as it is within the 95% confidence interval, with a two-tailed p-value of 18.6%.
Model uncertainties obscure potential deviations between the experimental data the and theoretical model.A model with high uncertainty on its slope predictions has a correspondingly high risk of false positive.That is, any experimental slope will have a high p-value, even if it does not actually follow the model's base assumption.Efforts to reduce the uncertainty on the slope prediction should be guided by a prior identification of the parameters with a leading impact on the global model uncertainty.For each variable parameter independently ( φ, ε p , D m,s , n s ), 3000 MC replications were made by randomly choosing values from its probability distribution while keeping all other parameters constant at their mean value.The 95% confidence interval of the calibration slope distributions associated with each parameter is reported in Table II as % of the mean MC slope.The main contributor to the spread of the model is the mean particle diameter of the volume distribution D m,v , followed by ε p , n s , and φ.The leading effect of the mean volume-weighted diameter was expected given that it participates in the functional relation as cubed parameter.That motivated efforts on accurately measuring PSD.Using mono-dispersed particles together with higher resolution micro-CT measurements would significantly reduce the uncertainty on model parameters for the purpose of more accurately verifying the theoretical underlying foundation of mm-Wave radar interferometry.
Using the simulated MC data based on the experimental model and parameter distributions, it is possible to find the set of parameters that maximize the likelihood of the model distribution.Those maximum likelihood parameters provide a useful reference for comparison to future parameter measurements.If the theoretical assumptions of the model agree with the calibration measurements, as previously shown within the model uncertainty, the measured value of each parameter converges toward the maximum likelihood value as the accuracy of its measurement increases.The maximum likelihood values for ε p , D m,v , and n s were derived by extracting every slope within the experimental margin of error of β n from the MC dataset.The model parameters realizing those slopes were sorted and binned into probability distributions.The mode of each parameter distribution is the value that maximizes the likelihood of the experimental slope.Maximum likelihood values are reported in Table III along with the mean parameter values in the overall MC distribution for reference.It must be remarked that the maximum likelihood value of each parameter is in the upper range of their distribution.The maximum likelihood dielectric constant, in particular, is close to the edge of the chosen range.These results also suggest that the volume-weighted mean particle diameter could be slightly underestimated by current measurement techniques, possibly due to segmentation thresholds used in the particle sizing and optical particle counting algorithms.
This discussion focused on the role of particle parameter uncertainties in the assessment of theoretical models, or for the inverse determination of particle parameters from the theoretical models combined with experimental data.However, specific knowledge of those parameters or robustness of the theoretical framework is only needed for a calibration-free measurement of concentration from phase shift measurements.If the instrument is calibrated against a gas-particle mixture with a known concentration, using the calibration procedure described or another methodology, the measurement uncertainty is dictated by the calibration.The calibration of the mm-Wave interferometer in this work yielded a path-integrated concentration measurement uncertainty of ±3.12%, see Section IV-C.

B. Comparison With Optical Techniques
The ability of the radar interferometer to provide concentration data in particle-laden flows over a wide range of particle loadings is shown in the context of previous works on particle-laden jets.Particle concentration, velocity, and flux are important outputs needed to characterize turbulent mixing Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.and particle dispersion.Experimental studies with particle diameters and bulk densities similar to the present work were selected, with the goal of identifying the capabilities of the mm-Wave within the envelope of concentration conditions relevant and previously explored in particle-laden flows.They are summarized in Table IV.The measurement techniques used include laser Doppler anemometry (LDA), PDA, planar nephelometry (PN), and laser diffraction method (LDM).A challenge present in all optical methods is light extinction, which impedes signal propagation at high particle concentrations.LDA and PDA can provide absolute values of local concentration or flux, but absolute quantities are only accurate in dilute conditions without multiple particles at once within the measurement volume [22], [23].Therefore, results are most often reported as relative quantities with respect to centerline values or with respect to estimates based on inlet particle mass loading ratios.For the purpose of comparison, the relative concentration results in the reference studies are converted to absolute path-integrated number concentrations.Equation (10) was used to convert the experimental data to path-integrated volume concentrations, where m is the jet mass loading ratio, ρ p and ρ g are the particles bulk and gas densities, respectively, and d is the nozzle exit diameter, considered equal to the jet diameter near the nozzle exit where the particle concentration is highest.This conversion assumes equal average particle and gas velocities, a reasonable estimate for most of the particle Stokes numbers in those works.The mean particle diameter D m,n was used to translate path-integrated volume fractions δdl to path-integrated number concentrations n Each experimental condition reported is documented in Fig. 11 and corresponds to a path-integrated number concentration and particle size.Data points corresponding to the calibration results presented in this work are included, as well as the range of path-integrated concentrations measured with the mm-Wave interferometer in demonstration experiments for plume-surface interactions [48].Data points are overlaid on a map of optical absorbance A, (11), calculated using the Beer-Lambert Law using an extinction efficiency Q ext of 2 for geometric scattering.The optical absorbance associated with each experiment is a measure of the signal extinction that an optical diagnostic encounters for that particle size and pathintegrated concentration.Optical techniques are expected to fail once absorbance becomes sufficient to attenuate the signal received by the instrument beyond practical use Most of the experiments reported in the literature are in the dilute regime, with correspondingly low absorbance values (A < 1).Only Barlow and Morrisson's work [70] is at a regime where the estimated optical absorbance is above 1 (A = 1.75).It is likely that this value is close to the absolute limit for LDA measurements, with authors reporting that higher mass loading ratios were trialed but only a partial signal was recovered by the instrument.
The measurement range of the instrument demonstrated during the calibration significantly overlaps with optical methods at moderate path-integrated concentrations Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.(10 7  − 10 8 m −2 ).The experiments reported by Gillant et al. [58], Shuen et al. [61], Mostafa et al. [65] and Lau and Nathan [66] used particles of similar size to those in the present work (±10 µm), making them useful for one-to-one comparison.Mostafa et al. [65] and Lau and Nathan [66] experiments had similar path-integrated concentrations to those achieved during calibration, while Gillant et al. [58] and Shuen et al.'s citeShuen1984 path-integrated concentrations were lower than the noise floor of the present radar instrument.This indicates that radar interferometry cannot compete with optical methods for very dilute particle-laden flows, the most ideal regime for optical techniques.However, the concentration range during the instrument calibration extends one order of magnitude above those reference experiments, into regions with high optical absorbance.The maximum optical absorbance for the calibration range is 3.6, more than double the absorbance of Barlow and Morrisson's experiment and one to two orders of magnitude higher than the rest of the experiments analyzed.Even higher concentration values were measured using the same instrument during plume-surface interaction demonstration experiments [48], reaching up to 5.71 × 10 9 m −2 .This corresponds to an optical absorbance of 50, far above the capability of any optical system.
Altogether, the new mm-Wave interferometer is able to operate in a wide range of concentrations and particle sizes, complementing optical diagnostics at low to moderate particle loadings, and extending the range of loading conditions measurable experimentally into otherwise inaccessible optically thick flows.The instrument has a number of additional desirable characteristics such as a high sampling rate, low power requirements, and low size, mass, and cost.It provides absolute concentrations of particles, in contrast with most optical techniques other than for very dilute conditions.In addition, for high dielectric constant materials (e.g., titanium dioxide, barium titanate, water, ethanol, as well as most polar liquids), calibration-free measurements are possible, as (7) simplifies into φ ≈ (3π f 0 /c 0 ) RX TX δdl, with φ independent of the particle dielectric constant.For low dielectric constant material (ε < 15), a calibration procedure is necessary, either following the procedure described in this work, or by measuring the dielectric constant of the material.The spatial resolution of the instrument, of the order of the angular size of the reflector seen by the radar antenna, cannot be compared to that of camera-based optical techniques or point measurements.Higher resolutions are possible at the expense of signal strength (smaller passive reflectors) or complexity (active frequency-coded reflectors).

VI. CONCLUSION
This work presented the development of a mm-Wave radar interferometer capable of measuring absolute values of path-integrated concentrations in optically opaque gas-particle mixtures.The instrument was calibrated against particle number concentration measurements by optical particle-counting, demonstrating a high degree of linearity between measured phase shift and path-integrated concentration.MC simulations were used to evaluate the agreement the experimental results and theoretical models of wave-particle interactions based on the Maxwell-Garnett theory of effective medium.The experimental results were found to be within the upper margin of error of the theoretical predictions, given in terms of path-integrated volume concentration.While this confirms the validity of the theoretical models, it also indicates a potential bias toward an underestimation of some experimental parameters, the PSD and dielectric constant in particular.Improving the accuracy of those parameters will allow smaller deviations of experimental results from theoretical predictions to be detected, improving theoretical understanding of the technique and paving the way toward simpler calibration procedures based on dielectric constant measurements, or even waiving the need for a calibration entirely in flows involving high dielectric constant material.Nonetheless, if a calibration such as the one proposed in this work is performed, knowledge of the specific particle properties is not required for accurate measurements of path-integrated particle concentrations.The mm-Wave interferometer extends the achievable higher end of the measurement range by at least one order of magnitude with respect to state-of-the-art optical concentration techniques.Enabling absolute measurements of path-integrated concentration, this instrument is particularly relevant for moderate to a high concentration particle-gas mixtures where existing techniques provide at most relative values of local particle densities.Local values can be estimated from path-integrated measurements in mixtures with small concentration gradients.They can also be directly resolved, in time and space, by extending this concept into a tomographic system, currently under development by our group.Other extensions of this work using a mm-Wave radar to measure particle velocities are also envisioned.The instrument can be used in many multiphase flow applications beyond the specific plume-surface interaction problem that motivated its development.Glass particles in the 80-140 µm range were used in this work, but the same principles apply to other particle size and material, such as liquid droplets or soil particles.With a low weight, low cost, compact design, and high frequency resolution (above 20 kHz for the specific radar presented), it is well suited for laboratory and field applications where more expensive, fragile, and bulkier optical equipment is not practical.

Manuscript received 15
February 2023; revised 6 April 2023; accepted 21 April 2023.Date of publication 31 May 2023; date of current version 6 November 2023.This work was supported in part by the National Aeronautics and Space Administration (NASA) issued through the Early Stage Innovation Program under Grant 80NSSC20K0304 and in part by the Future Investigator in NASA Space Science and Technology (FINESST) Fellowship under Grant 80NSSC22K1332.(Corresponding author: Nicolas Rasmont.)

Fig. 1 .
Fig. 1.Extinction efficiency of 105 µm glass particles as a function of the frequency and wavelength of the electromagnetic wave using Mie theory.

Fig. 2 .
Fig. 2. Comparison of the attenuation and phase shift of 650 nm laser and 3.89 mm radar (77 GHz) beams traveling through monodisperse glass regolith simulant with a particle diameter of 105 µm.

Fig. 5 .
Fig. 5. Piston-actuated slot funnel used for the calibration of the instrument.

Fig. 6 .
Fig. 6.Cross section view of a CT scan of the particle sample.(a) Raw scan.(b) Segmented particles detected in the sample.

Fig. 7 .
Fig. 7.Comparison between the volume PSD measured by micro-CT scanning, optical imaging, and dry sieving.

Fig. 9 .
Fig. 9. calibration, phase shift versus path-integrated number concentration, with MC probability distribution and associated Deming regression.

Fig. 11 .
Fig. 11.of optical absorbance as function of path-integrated number concentration and particle diameter.Symbols correspond to published experimental conditions and measurements by mm-Wave interferometer.

TABLE IV PARTICLE
-LADEN JET EXPERIMENTS REPORTED IN THE LITERATURE.THE NOZZLE DIAMETER, MASS LOADING RATIOS, PARTICLE SIZE, AND BULK DENSITY ARE USED TO ESTIMATE THE PATH-INTEGRATED CONCENTRATION