# Investigation of mixing/segregation patterns of two dissimilar granular solids in fluidized beds by capacitance probes

Laura Molignano, Maurizio Troiano, Roberto Solimene, Sina Tebianian, Jean-François Joly, Piero Salatino

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## To cite this version:

Laura Molignano, Maurizio Troiano, Roberto Solimene, Sina Tebianian, Jean-François Joly, et al. Investigation of mixing/segregation patterns of two dissimilar granular solids in fluidized beds by capacitance probes. Chemical Engineering Science, 2026, 322, pp.123047. 10.1016/j.ces.2025.123047. hal-05454470

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https://ifp.hal.science/hal-05454470v1

Submitted on 12 Jan 2026

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# Investigation of mixing/segregation patterns of two dissimilar granular solids in fluidized beds by capacitance probes

Laura Molignano<sup>a,b</sup>, Maurizio Troiano<sup>a,c</sup>, Roberto Solimene<sup>c,*</sup>, Sina Tebianian<sup>b</sup>, Jean-François Joly<sup>b</sup>, Piero Salatino<sup>a</sup>

## ARTICLE INFO

### Keywords:

Gas-solid fluidization

Flotsam/jetsam solids

Mixing/segregation

In-situ diagnostics

High-temperature capacitance probe

Time series statistical analysis

## ABSTRACT

The present study addresses mixing/segregation phenomena relevant to several chemical processes in bubbling fluidized bed reactors. Mixing between two dissimilar solids belonging to group B and D of Geldart classification is investigated in a lab-scale fluidized bed at high temperature. Capacitance probes are selected and tested innovatively to assess the transient phase of the mixing process between the two solids and the measurement of steady local mixture composition thanks to dissimilar dielectric properties of gas and solids and of the two chosen solids. A new procedure to gather quantitative results has been used considering expansion of the emulsion phase voidage and validated based on experimental data measured at 500 °C in the bubbling fluidized bed regime. The tests in transient conditions comprise the determination of two axial dispersion coefficients, $\mathcal{D}_{s,macro}^a$, on average, $3.1 \times 10^{-2}m^2/s$, and $\mathcal{D}_s^a$, ranging between 1.8 and $3.1 \times 10^{-3}m^2/s$, referred to the tracers batch crossing the bed vertically without mixing with the rest of the bed and to the tracer batch disaggregating and reaching local steady-state concentrations in the dense bed, respectively. New insights are mainly revealed with reference to the phenomenology at steady-state conditions. In particular, axial and radial segregation patterns could be determined. Critical issues and corresponding improvements of the technique are highlighted and are mainly related to the properties of the two dissimilar solids and the capacitance probes amplification system.

## 1. Introduction

Fluidization technology commonly involves the use of inert solids acting as dispersing medium of active solids and gases. The inert bed material is never truly monodisperse. Active solids are, generally, dissimilar with respect to the inert bed in terms of size, shape, density, and chemical composition, and such properties undergo an evolution throughout the process. This is the case, for example, of solid fuels dispersed in an inert bed of sand. A third different solid material could be present, acting as a catalyst or sorbent material (Leckner, 2022). Therefore, one or more types of active particles are commonly involved in a process held in a fluidized bed (FB) unit. Mixing/segregation conditions between such solids strongly influence the performance of the process, as they regulate particle-particle interactions and gas-solid mass transfer, with consequences on heat transfer and reactants conversion. Even in circulating FBs, solids mixing/segregation in the bottom

bed zone is determinant for the overall process (Schlichthaerle and Werther, 2001).

Requirements about the level of mixing/segregation of active solids in the dense FB, in both axial and lateral directions, depend on the type of process. Effective mixing conditions of different solids are typically necessary in most of the FB applications, although in some cases segregation is to be emphasized. Conventional combustion of solid fuels relies on relatively intense solids mixing to ensure complete burn out of fuel, minimize the number of feeding ports, and favour an even distribution of heat and gas released from the fuel solids (Pallarès and Johnsson, 2008). In dual FB gasifiers, characterized by two interconnected reactors, the aim is to control the solids mixing to ensure sufficient fuel particle residence time within the gasifier. Too high lateral dispersion results in a loss of char from the gasifier to the connected combustor (Pröll et al., 2007). In CLC fuel reactors, in contrast to direct combustion, moderate levels of lateral mixing increase the fuel residence time, thereby minimizing losses of unconverted char to the air

<footnote><sup>a</sup> Dipartimento di Ingegneria Chimica, dei Materiali e della Produzione Industriale, Università degli Studi di Napoli Federico II, Piazzale Tecchio 80, 80125 Napoli, Italy</footnote>

<footnote><sup>b</sup> Direction Conception Modélisation Procédés, IFP Energies Nouvelles, Rond-point de l'échangeur de Solaize 69360, Solaize, France</footnote>

<footnote><sup>c</sup> Istituto di Scienze e Tecnologie per l'Energia e la Mobilità Sostenibili, Consiglio Nazionale delle Ricerche, Piazzale Tecchio 80, 80125 Napoli, Italy</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: roberto.solimene@cnr.it (R. Solimene).</footnote>
**Nomenclature**

**Notation**

<table><tr><td>A</td><td>area of the PDF [-]</td></tr><tr><td>C</td><td>capacitance [F]</td></tr><tr><td>d</td><td>diameter [m]</td></tr><tr><td>~d</td><td>mean diameter [m]</td></tr><tr><td>D</td><td>column diameter [m]</td></tr><tr><td>D̃<sub>s</sub></td><td>solid dispersion coefficient [m<sup>2</sup>/s]</td></tr><tr><td>f</td><td>solid volume fraction (fluid-free basis) [-]</td></tr><tr><td>f̃</td><td>spatially-averaged solid volume fraction (fluid-free basis) [-]</td></tr><tr><td>Gain</td><td>gain factor of the amplifier for capacitance probes [pF V]</td></tr><tr><td>H</td><td>height [m]</td></tr><tr><td>r</td><td>radius [m]</td></tr><tr><td>R</td><td>column radius [m]</td></tr><tr><td>S</td><td>column cross section [m<sup>2</sup>]</td></tr><tr><td>t</td><td>time [s]</td></tr><tr><td>T</td><td>temperature [°C]</td></tr><tr><td>U</td><td>superficial gas velocity [m/s]</td></tr><tr><td>v</td><td>volume [m<sup>3</sup>]</td></tr><tr><td>V</td><td>voltage [V]</td></tr><tr><td>V̄</td><td>representative modal voltage value [V]</td></tr><tr><td>z</td><td>axial coordinate [m]</td></tr></table>

**Greek symbols**

<table><tr><td>ε</td><td>voidage [-]</td></tr><tr><td>ε<sub>0</sub></td><td>void dielectric constant [pF/m]</td></tr><tr><td>κ</td><td>relative dielectric constant [-]</td></tr><tr><td>ρ</td><td>density [kg/m<sup>3</sup>]</td></tr><tr><td>ρ̃</td><td>mean density [kg/m<sup>3</sup>]</td></tr><tr><td>φ</td><td>particle sphericity [-]</td></tr></table>

<table><tr><td>Σ</td><td>capacitance probe geometric characteristic [m]</td></tr></table>

**Subscripts**

<table><tr><td>bed</td><td>bed material</td></tr><tr><td>e</td><td>emulsion/dense phase</td></tr><tr><td>H<sub>1</sub></td><td>axial position in the FB</td></tr><tr><td>H<sub>3</sub></td><td>axial position in the FB</td></tr><tr><td>i</td><td>subsection number</td></tr><tr><td>LV</td><td>LV-phase</td></tr><tr><td>HV</td><td>HV-phase</td></tr><tr><td>macro</td><td>macromixing</td></tr><tr><td>mf</td><td>minimum/incipient fluidization conditions</td></tr><tr><td>mb</td><td>minimum bubbling</td></tr><tr><td>p</td><td>particle</td></tr><tr><td>pack</td><td>packed</td></tr><tr><td>r</td><td>radius</td></tr><tr><td>s</td><td>solid</td></tr><tr><td>t</td><td>terminal</td></tr><tr><td>tot</td><td>total</td></tr><tr><td>tr</td><td>tracer material</td></tr><tr><td>0</td><td>air/void</td></tr><tr><td>1</td><td>solid 1 in a binary mixture</td></tr><tr><td>2</td><td>solid 2 in a binary mixture</td></tr><tr><td>γ</td><td>γ-alumina</td></tr></table>

**Superscripts**

<table><tr><td>a</td><td>axial</td></tr><tr><td>bed</td><td>bed material</td></tr><tr><td>mode</td><td>mode</td></tr><tr><td>r</td><td>radial</td></tr><tr><td>tr</td><td>tracer material</td></tr></table>

reactor (Lyngfelt, 2020). In calcium looping applied for thermochemical energy storage, by choosing appropriate conditions, segregation phenomena could be favored over mixing, enabling the separation of the more carbonated material from the material which remains unconverted (Di Lauro et al., 2023) thanks to the addition of a separation unit downward the carbonator (Sarrión et al., 2016; Sarrión et al., 2018; Pascual et al., 2021; Tregambi et al., 2023). This last aspect would increase the efficiency of closed CO₂ cycles for power generation. The level of solids mixing/segregation has a key role also in the performance of industrial FB dryers, which mixing regimes may lie between the back-mixed and plug flow model (Daud, 2008).

Another relevant aspect is the characteristic mixing/segregation time of active solids in the dense fluidized bed. The relationship between the characteristic times of mixing/segregation and competing processes determines the effectiveness of the overall process. For example, operational experience of FB-combustors of high-volatile biomasses has shown that fuel particles rapidly tend to segregate at the bed surface during the devolatilization stage, leading to “stratified” conversion patterns (Salatino and Solimene, 2017).

Overall, the success of any process involving a variety of different types of solids depends on a thorough understanding of the key factors impacting mixing. To properly design and operate FB reactors, the dynamics of polydisperse mixtures and their response to changes in operating conditions need to be deeply understood and predicted.

Mechanisms of mixing and segregation in dense FBs mainly rely on the phenomenon of “bubbling”. Solids can be transported toward the bed surface within the wake and the upward drift that generates around the vertical axis of the bubble (mechanisms of wake and drift (Clift et al., 1985)). These patterns have their closure in the splash and grid zone, where solids motion takes place along the horizontal direction. The

described bubble-induced gross solids circulation, known as gulf-streaming motion, occurs at the scale of the bed height. Internal circulation of solids, instead, occurs at the scale of bubble diameter (Norouzi et al., 2011). Internal circulation mechanisms include solids sinking in the emulsion phase to fill the gap created by the rising bubble and unstable circulation motion of particles within bubble wake (wake shedding mechanism). In this last mechanism, solids are laterally released in the surrounding emulsion phase and replaced. Bubbles coalescence is another mechanism occurring at the bubble-diameter scale, which contributes to lateral solids mixing (Norouzi et al., 2011).

Literature on gas-fluidized bed units of polydisperse solids can be resumed into two major categories. The first category of FBs of polydisperse solids can be referred as mixtures of bed materials. The bed is characterized by the presence of two or more dissimilar solids in fairly equal or moderate proportion with one another. The solid species generally differ from each other in size and/or density in a limited extent. The inert bed of sand as adopted in industrial practices can be conceived as polydisperse bed as it is generally composed of a wide size range of fluidized particles having the same density. The second category of FBs polydisperse mixtures refer to solids species differing in size and density by orders of magnitude. The former solid species is present in small percentages with respect to bed material solids in a way that, in a perfectly mixed state, it is surrounded only by bed material solids. Referring to examples of practical interest, the solid species which is present in smaller amounts is generally larger and lighter than bed material solids. It is the case of biogenic large solid fuels (Köhler et al., 2021), representing a small percentage of the bed inventory in a thermochemical process. In both cases, the presence of *flotsam* and *jetsam* components can be recognized. The terminology refers to *flotsam* the particles that tend to accumulate at the top of the bed and to *jetsam* those
that sink at the bottom of the bed (Rowe et al., 1972).

Most of the numerical and experimental literature studies are focused on the fluidization of binary systems of solids. Such systems are easier to investigate, and results can be extrapolated with some confidence to more complex cases. Rarely, literature studies on ternary or polydisperse mixtures are reported (Wang and Chou, 1995). A fluidized bed of two dissimilar solids can be characterized by determining a series of properties. Among them, the initial fluidization velocity and the final fluidization velocity serve to identify the gas velocity interval along which fluidization of the binary mixture takes place (Formisani et al., 2011). Besides, the interest in most of the studies on binary mixtures lies in determining the equilibrium condition between the mixing and the segregating tendencies of particles under the fluidization state. Mixing and segregation occur concurrently, and their dynamic equilibrium yields axial/radial profiles of particle concentrations. The mixing or segregation index (Rowe et al., 1972; Lacey, 1954; Marzocchella et al., 2000; Goldschmidt et al., 2003; Zhu et al., 2025) represents the best way to assess such equilibrium based on solids properties and operating conditions. This approach can reveal the dominant property in the establishment of segregation patterns (size-segregating (Formisani et al., 2001), density segregating (Formisani et al., 2008)). As for the description of the dynamics of mixing/segregation patterns, some studies addressed the unsteady state of fluidization that may establish at superficial gas velocity below the threshold for complete and uniform fluidization of solid mixtures and the dynamics of particle segregation expressed through the drift flux velocity of mixture components (Marzocchella et al., 2000; Olivieri et al., 2004). Among mathematical models, a completely predictive one (Di Maio et al., 2012), without adjustable parameters, is able to foresee the segregation tendency/direction of solid species in unconventional binary mixtures of dissimilar materials in which the role of jetsam and flotsam cannot be established a priori. The characterization methods above are generally applied to binary mixtures studied in small-scale units.

In the industrial field, where large-scale fluidized beds are operated, multiple mixing cells (Pallarès et al., 2007; Olsson et al., 2012) establish along the vertical or transversal direction depending on whether deep or shallow beds are utilized. In this context, the dispersion model represents the most reliable and feasible option to assess mixing in the solid phase at macroscopic level. The flux of particles is expressed as a solid dispersion coefficient, $\bar{D}_s$, multiplied by the gradient of particle concentration and is substantially referred to solids mass transfer along multiple adjacent mixing cells. Utilization of the macroscopic dispersion model to small apparatuses or units in which one or very few mixing cells can be identified is questionable. In this kind of apparatuses mixing can be addressed at a meso-to-micro scale level (Abanades and Grasa, 2001). Given that the mechanisms of solids mixing along the horizontal direction result, overall, to be less effective than those active in the vertical direction (Solimene, 2017), research has been quite focused on the investigation of lateral dispersion of solids. Lateral dispersion coefficients evaluated in 3D medium-to-large scale units with bed aspect ratios less than unity, span over a very broad range of four orders of magnitude ($10^{-4} - 10^{-1} \text{ m}^2/\text{s}$) (Olsson et al., 2012; Bellgardt and Werther, 1986; Berruti et al., 1986; Borodulya et al., 1982; Highley and Merrick, 1971; Mostoufi and Chaouki, 2001; Niklasson et al., 2002; Qian et al., 1987; Sette et al., 2014; Sette et al., 2014; Sette et al., 2016; Sette et al., 2015; Guío-Pérez et al., 2023). These results have been interpreted over years in the light of the dependance of mixing phenomena on several operating parameters. Correlations proposed in the literature for the prediction of the horizontal dispersion coefficient are comprehensively resumed in the review of Solimene (Solimene, 2017).

The main diagnostics employed in literature to characterize mixing and segregation of bidisperse solids in dense gas-fluidized beds are summarized as follows. Eulerian techniques locally measure a tracer concentration, not giving information on the trajectory of single tracer particles. In many cases such diagnostic techniques are intrusive, as the measuring probes have to access different points of the dense bed. When

possible, tracer concentrations measured along the axial and/or radial coordinate of the dense bed can be fitted with a dispersion model to derive a solid dispersion coefficient (Niklasson et al., 2002). Lagrangian techniques track the patterns of a single or multiple tracers, and are mostly non-intrusive with respect to Eulerian methods. After measuring tracer particle trajectories, a dispersion coefficient can be calculated using the Einstein's equation for Brownian motions (Guío-Pérez et al., 2023; Einstein, 1926).

Among Eulerian techniques, solid sampling is used to measure the local concentration of a tracer in the dense bed by withdrawing solid samples from it. It is a direct method as one can directly measure the concentration of the tracer in the retrieved sample. Tracer is distinguished from bed solids by size (Qian et al., 1987; Sette et al., 2014; Liu and Chen, 2011; Lardier et al., 2016; Zhang et al., 2009), density (Lardier et al., 2016), color (Fan and Shl, 1984; Jin et al., 2009), chemical composition (Berruti et al., 1986; Chesonis et al., 1990; Kato et al., 1984), magnetic (Fu et al., 2020; Beeckmans and Stahl, 1987; Gabor, 1964) or radioactive (Highley and Merrick, 1971) properties. Solid sampling can be realized either on-line (Berruti et al., 1986; Sette et al., 2014; Lardier et al., 2016; Fan and Shl, 1984; Chesonis et al., 1990) or after de-fluidization of the bed (frozen bed method) (Qian et al., 1987; Liu and Chen, 2011; Lardier et al., 2016; Zhang et al., 2009; Jin et al., 2009; Kato et al., 1984; Fu et al., 2020) to measure the tracer concentration at steady state conditions of the fluidized bed or in a transient phase of the mixing/segregation phenomenon. On-line samples collection out from the reactor is a drawback and can be limited in frequency if compared to the solids mixing/segregation characteristic times. Bed-collapse method can cause solids segregation and alter the measurement. Most of the cited authors adopted solid sampling at ambient temperature; only Lardier et al. (Lardier et al., 2016) performed experiments of solid sampling at temperature conditions typical for biomass gasification, whereas Sette et al. (Sette et al., 2014) carried out the experiments in a cold-flow unit adopting fluid-dynamic scaling laws. Thermal tracing consists in assessing mixing by means of heated (Borodulya et al., 1982; Liu and Chen, 2012; Westphalen and Glicksman, 1995; Valenzuela and Glicksman, 1984; Salam et al., 1987; Shen and Zhang, 1991) or cooled tracers (Schlichthaerle and Werther, 2001; Bellgardt and Werther, 1986; Glicksman et al., 2008). It requires the bed material to be at a different temperature than the injected sample of tracer solids. It is an indirect method as solid tracer concentration is gathered passing through the measurement of local temperature inside the system. High-frequency fast sensors are required to obtain reliable measures in transient conditions. The method loses its effectiveness when temperature becomes even inside the bed. Therefore, it is important to understand which of the two mechanisms, solid mixing and heat transfer, is the controlling step in the operation of the fluidized bed. In studies at steady state conditions, the dispersion coefficient is obtained by measuring the heat transfer rates of bed solids when electric heating and water cooling sections are placed at different sections of the bed (Lee and Kim, 1990). The method is based on the assumption that the heat capacity ratio between the solids and air is so high that the transported sensible heat in the bed primarily results from the mixing of heated particles. Another option consists in using hot air as tracer instead of heated/cooled solid tracers (Subbarao et al., 1985). The major drawback of the method is that the heat transfer coefficients do not necessarily coincide with dispersion coefficients of fluidized tracer particles, thus an accurate heat transfer model is needed to correctly interpret the experimental data. Thermal tracing could also be implemented non-intrusively by measuring temperature with an infrared camera at the surface of the dense fluidized bed or to 2D units (Martinez Castilla et al., 2020). Gas sampling (Niklasson et al., 2002; Chirone et al., 2004; Winaya et al., 2007; Xiao et al., 1998) enables the study of time-resolved solid mixing in dense beds. By sampling and analyzing the gas above the bed coming from tracer particles, it is possible to indirectly measure the gas-emitting tracer concentration in the bed. The non-intrusive usage of this Eulerian technique implies uncertainties for
inferring the local tracer concentration from gas analysis, especially if evaporation, desorption, gasification or reaction rates are too low to have an accurate on-line measurement. Moreover, inference on mixing/segregation mechanisms along the bed axial coordinate can hardly be accomplished. Thus, such a technique could be considered if coupled with other ones, such as in-bed temperature measurements when frozen CO₂ tracers are used (Schlichthaerle and Werther, 2001; Bellgardt and Werther, 1986). The use of gas emitting tracers can simulate the formation of endogenous bubbles inside the bed and the method allows low contamination of the bed if tracers are progressively worn out through evaporation or chemical reaction. When axial mixing/segregation is to be addressed in very deep beds, multiple pressure probes can be fitted to the column and bed-slices composition can be measured by determining the local minimum fluidization velocity of the mixture (Turrado et al., 2019). Capacitance probes can be used to indirectly measure the concentration of a tracer in a dense fluidized bed of bidisperse solids (Huang et al., 2017; Wan et al., 2020; Li et al., 2021; Song et al., 2024; Wan et al., 2024). Time-resolved voltage signal associated with the emulsion phase of the bed depends on void fraction and on the relative amount of the two solids. Tracer relative dielectric constant must be sufficiently different from that of bed material. An assumption on the emulsion-phase voidage is required to postprocess the emulsion-phase signal. The technique has been tested only with bidisperse mixtures of solids having the same size. The method can be applied to high-temperature 3D large-scale systems, although this has not been tested yet. Optical probes with phosphorescent tracers can be effectively used to measure the axial and radial solids mixing behavior (Du et al., 2002; Du and Wei, 2002). The phosphorescent material becomes fluorescent for a short period of time after being excited. When tracer particles pass by the detector, the light emitted by the tracers is received by the optical fiber and transmitted to the photomultiplier where tracer concentration is quantified. The advantages of this technique include fast on-line detection of particles. The method has not been tested in high temperature units. Electrical resistance probes are able to measure local solid mixing rates (Hayakawa et al., 1964). The probes consist of two platinum rings mounted on a ceramic rod and are submerged in the FB. Connecting wires pass through the holes in the rod. The voltage drop occurring as a result of current flow via the conducting bed between the electrodes is recorded when a DC voltage is applied to the probe electrodes in series with a fixed resistor. The recorded voltage changes as a result of changes in the resistance properties of the bed caused by mixing of dissimilar materials, and fluctuates as a result of changes in bed properties caused by the bubble flow.

Some Lagrangian techniques consist in video recording and subsequent digital image analysis of particle trajectories. A fast camera is put on the roof of the reactor or outside a transparent 2D apparatus. The tracer has to be clearly distinguishable from the bed material. Phosphorescent tracers (Olsson et al., 2012; Pallarès and Johnsson, 2006) are preferred over pigmented/dyed particles (Wei et al., 2006; Vandewalle et al., 2022; Shen et al., 2007) although phosphorescence is a time-limited phenomenon. Light emission from tracers is accomplished also by other chemical compounds like strontium aluminate coatings (Soria-Verdugo et al., 2011), or fluorescent paints, illuminated with soft ultraviolet lights (Sanderson and Rhodes, 2003). Hot tracers can also be adopted (Sette et al., 2016; Errigo et al., 2023). Single or multiple tracers can be used. The technique is non-intrusive and allows time-transient and steady-state studies limited to the domain above the dense bed surface. X-rays, γ-rays (Seo and Gidaspow, 1987), single/multiple Positron Emission Particle Tracking (PEPT) (Yang et al., 2007; Stein et al., 2000) and electrical capacitance (Du et al., 2002) are tomography-tracking techniques more suited for 3D systems if compared to camera-based ones. Most of these techniques require substantial safety measures. In magnetic particle tracing (Halow et al., 2012), magnetoresistive sensors track a single magnetic tracer particle, whose properties can be tuned to have the required particle density and diameter.

Generally, there are limitations in the maximum operating temperature depending on the Curie point of the magnet. Moreover, the insulation and the heating system can represent an obstacle for the effectiveness of the method. The operation of the fluidized bed in fluid-dynamic scaled conditions represents the solution to the stated problem (Sette et al., 2014). In this case, the use of large apparatuses is still advised if operation of large-scale units is to be represented, to avoid that measurements could be affected by wall effects (Guío-Pérez et al., 2023).

A summary of the experimental techniques available in literature for the investigation of solids mixing/segregation in dense FB is presented in Table 1.

In the framework of the above-listed diagnostic techniques, capacitance probes represent a powerful tool to have multiple measurements at the same time, such as bed voidage distribution, bubble hold-up, bubbles distribution, velocity measurements, fluidization regimes characterization, assessment of fluidization quality. Moreover, they present favorable features as compared with alternative diagnostics in terms of ease of operation and signal interpretation, temporal resolution, cost, applicability in severe high temperature environments, and applicability in industrial units. Furthermore, needle capacitance probes are usually utilized in fluidized beds since the miniaturized design causes very little disturbances of the flow (Huang et al., 2017; Li et al., 2021; Wan et al., 2020; Huang et al., 2022; Zhang et al., 2025).

Recently, a new method based on capacitance probes has been proposed and tested for the characterization of solids mixing in a gas-fluidized 2D and 3D units operated at ambient temperature and in bubbling regime with binary mixtures of solids having the same size and different density, belonging to the groups B and D of Geldart classification (Huang et al., 2017; Li et al., 2021; Wan et al., 2020; Huang et al., 2022; Zhang et al., 2025). The output voltage signal of the capacitance probe depends on the combination of the fluidizing gas and the two different solids, which together cross the sensing volume of the probe. This signal has been postprocessed to get the local solid tracer concentration in the emulsion phase of the dense FB resolved over time. It has been assumed that the probe output signal changes linearly with the volume fraction of each component in the sensing field. Therefore, bubbles have been filtered-out and the emulsion-phase signal has been elaborated considering that, according to the *two-phase theory*, the contribution of the gas is constant with the superficial gas velocity and equal to that at minimum fluidization conditions ($ε_{mf}$). However, very few data are available in the literature in 3D units and at high temperature. Furthermore, mixing has been evaluated considering that the *two-phase theory* holds.

In the present study, capacitance probes have been employed to study the solid mixing phenomena of binary mixtures of dissimilar solids materials having different size and density in a 3D lab-scale FB apparatus at high temperature. Furthermore, the operating conditions under investigation do not allow to assume that the *two-phase theory* holds true. For this reason, preliminary calibration tests in fixed bed conditions have been carried out to the validation of a new relationship, that would correlate the relative dielectric constant of the medium composed by air and two different solids in certain proportions with the relative dielectric constant of the different elements taken separately according to their relative amounts in the sensing volume of the capacitance probe. Then, mixing tests in a 3D laboratory-FB unit operated at 500 °C in the bubbling regime have been performed, in which one of the two solid species acting as tracer was rapidly injected in the fluidizing medium to investigate both the transient and the steady-state conditions of solids mixing. An effective signal postprocessing procedure has been proposed in the light of the deviations of the system from the *two-phase theory* and results successfully validated. The potential and the limitations of the diagnostic technique based on capacitance probes have been discussed.
<table><caption>**Table 1**

Summary of diagnostic techniques for the investigation of solids mixing/segregation in fluidized beds.</caption><thead><tr><th>Technique</th><th>Solid sampling</th><th>Thermal tracing</th><th>Gas sampling</th><th>Pressure measurement</th><th>Capacitance probes</th><th>Optical probes</th><th>Electrical resistance probes</th><th>DIA/Image analysis</th><th>X-/γ-rays, PEPT</th><th>ECT</th><th>Magnetic</th></tr></thead><tbody><tr><td>Operating principle</td><td>Suction probes retrieve solid samples. Tracer distinguishability through different size, density, color, composition, magnetic/radioactive properties.</td><td>Heated/ cooled solid/ gas tracers, T profiles along the bed give mixing information.</td><td>It is possible to get gas-emitting tracer concentration in the bed through suction probes.</td><td>Curves of pressure vs. velocity at different bed heights indicate the level of mixing/segregation of the bed.</td><td>Based on different dielectric properties of bubbles and solids, and of the two adopted solids.</td><td>Particle concentration is measured either by the degree of reflection or by the degree of transmission of a light bundle.</td><td>The recorded voltage changes as a result of changes in the resistance properties of the bed caused by mixing of dissimilar materials.</td><td>It requires a camera to be put on the roof of the reactor or a transparent reactor that allows external observations.</td><td>Measure of how much the emitted radiation, from one or several emitters, passes through the bed to a receiver.</td><td>Plane capacitance sensors are coupled with suited electronics and a computer system for images reconstruction.</td><td>Magneto resistive sensors track a single magnetic tracer particle.</td></tr><tr><td>Unit</td><td>2D-3D</td><td>2D-3D</td><td>2D-3D</td><td>2D-3D</td><td>2D-3D</td><td>2D-3D</td><td>2D-3D</td><td>2D-limited in 3D</td><td>2D-3D with tomographic tools</td><td>3D</td><td>2D-3D</td></tr><tr><td>Temperature</td><td>No limitations</td><td>Limitations are dictated by the temperature sensor</td><td>No limitations</td><td>No limitations</td><td>No particular limitations</td><td>No limitations</td><td>Not reported</td><td>Some limitations at high T</td><td>No limitations</td><td>Applications at high temperature not found</td><td>Limits in the maximum T depending on the magnet Curie point</td></tr><tr><td>Analyzed phase</td><td>Dense/Diluted</td><td>Dense/Diluted</td><td>Gas in dense phase or freeboard</td><td>Dense</td><td>Dense/Diluted</td><td>Dense/Diluted</td><td>Dense/Diluted</td><td>Dense/Diluted</td><td>Dense/Diluted</td><td>Dense/Diluted</td><td>Dense/Diluted</td></tr><tr><td>Bed coordinate Intrusive</td><td>Radial/Axial</td><td>Radial/Axial</td><td>Radial/Axial</td><td>Axial</td><td>Radial/Axial</td><td>Radial/Axial</td><td>Radial/Axial</td><td>Radial/Axial</td><td>Radial/Axial</td><td>Radial/Axial</td><td>Radial/Axial</td></tr><tr><td>On-/Off-line Self/mutual dispersion</td><td>Yes</td><td>Yes</td><td>Yes, if performed in the dense bed</td><td>No</td><td>Yes</td><td>Yes</td><td>Yes</td><td>No</td><td>No</td><td>No</td><td>No</td></tr><tr><td>Eulerian/ Lagrangian Measurement Frequency</td><td>Both</td><td>On-line</td><td>On-line</td><td>On-line</td><td>On-line</td><td>On-line</td><td>On-line</td><td>Both</td><td>On-line</td><td>On-line</td><td>On-line</td></tr><tr><td>Other comments</td><td>Both</td><td>Both</td><td>Both</td><td>Mutual</td><td>Both</td><td>Both</td><td>Both</td><td>Both</td><td>Both</td><td>Both</td><td>Both</td></tr><tr><td></td><td>Eulerian</td><td>Eulerian</td><td>Eulerian</td><td>Eulerian</td><td>Eulerian</td><td>Eulerian</td><td>Eulerian</td><td>Lagrangian in 2D beds</td><td>Lagrangian</td><td>Lagrangian</td><td>Lagrangian</td></tr><tr><td></td><td>Low</td><td>High</td><td>High</td><td>High</td><td>High</td><td>High</td><td>High</td><td>High</td><td>High</td><td>High</td><td>High</td></tr><tr><td></td><td>Alteration of bed inventory after sampling.</td><td>Very fast response sensors required, difficulty in inferring solids dispersion from heat transfer models.</td><td>Uncertainties for inferring local tracer concentration if the probe is put in the freeboard. Simulation of the formation of endogenous bubbles. Low contamination of the bed.</td><td>Mixing study in transient and stationary conditions. Need for sufficient pressure ports.</td><td>Mixing study in transient and stationary conditions. The needle-type probe guarantees a small footprint. Assumption on the emulsion-phase voidage required to postprocess the signal.</td><td>Sufficiently small optical fiber can detect one particle at a time. Tracers with distinctive optical properties could be used for mixing experiments.</td><td>Only one old study in literature witnesses the applicability of such method.</td><td>Time-transient and steady-state studies. Tracer has to be clearly distinguishable from the bed material.</td><td>Relatively High Expensive if compared to others, they require substantial safety measures. Limited to not too big diameters and thin walls columns.</td><td>Application in high-pressure and/or high-temperature units, and with metallic vessels is often problematic.</td><td>Fluid-dynamic scaling overcomes temperature limitations.</td></tr></tbody></table>
## 2. Experimental

### 2.1. Apparatus

The experimental apparatus is based on a stainless steel 3" diameter (schedule 40: ID 77.92 mm) and 1300 mm high column operated as atmospheric bubbling fluidized bed (Fig. 1). The fluidization gas distributor belongs to the low pressure drop type. It is a 2 mm triangular pitch perforated plate placed under two stainless steel nets (mesh size of 40 µm), which provide the major contribution to the distributor pressure drop. The average pressure drop across the gas distributor is about 173 Pa, and its ratio to the bed pressure drop ranges between 8 to 11 %, depending on gas superficial velocity. These data refer to a bed inventory of 900 g and a superficial gas velocity range of 0.17–0.80 m/s at 500 °C. The windbox is a 225 mm high stainless-steel column filled with ceramic rings to equalize the gas flow. The fluidization column is thermally insulated and heated by two semi-cylindrical 2.4 kW ceramic ovens, while the bed temperature is kept at the set point by a PID control unit. Gases are fed to the column via high-precision digital mass flowmeters/controllers. Entrained dust due to abrasion of dense bed particles during testing is removed via a fume hood. The two ports located under and right above the gas distributor, respectively, are 4 mm ID tubes exclusively used for pressure and temperature measurements. The second pressure tap is located at 0.01 m above the grid. The column presents other six 3/8" diameter (schedule 40: ID 12.4 mm) tubes, mounted flush to the column wall, that provide access to capacitance probes and for pressure/temperature measurements. These lateral accesses are placed at different vertical positions along the column, at distances above the distributor plate of 0.06, 0.1, 0.14, 0.18, 0.34 and 0.95 m. In Fig. 1 only the ports which have been used for the experiments are represented. Each measurement probe is connected to its data acquisition module and signals are acquired simultaneously on a Personal Computer via a program developed in LabVIEW. Additional details about the experimental unit are reported by Molignano et al. (Molignano et al., 2024; Molignano et al., 2025).

### 2.2. Materials

Silicon carbide (SiC) and γ-alumina have been chosen as they present sufficiently different relative dielectric constants (Table 2), which do not depend much on the temperature level. Thus, they can potentially be

<table><caption>**Table 2**

Properties of adopted materials.</caption><thead><tr><th>Material</th><th>Silicon carbide</th><th>γ-Alumina</th></tr></thead><tbody><tr><td>Sieving range, d<sub>p</sub> [µm]</td><td>420–590<br>505</td><td>Monosize<br>1000</td></tr><tr><td>φ<sub>[-]</sub></td><td>0.63</td><td>1</td></tr><tr><td>ρ<sub>p</sub> [kg/m<sup>3</sup>]</td><td>3253</td><td>1209</td></tr><tr><td>ρ<sub>pack</sub>(20 °C) [kg/m<sup>3</sup>]</td><td>1764</td><td>779</td></tr><tr><td>Geldart group</td><td>B/D</td><td>D</td></tr><tr><td>U<sub>mf</sub>(20 °C) [m/s] (experimental)</td><td>0.305</td><td>0.411</td></tr><tr><td>U<sub>mf</sub>(500 °C) [m/s] (experimental)</td><td>0.190</td><td>0.300</td></tr><tr><td>U<sub>t</sub>(20 °C) [m/s] (Haider and Levenspiel, 1989) (Kunii and Levenspiel, 1991)</td><td>2.72</td><td>4.72</td></tr><tr><td>U<sub>t</sub>(500 °C) [m/s] (Haider and Levenspiel, 1989) (Kunii and Levenspiel, 1991)</td><td>3.58</td><td>5.97</td></tr><tr><td>ε<sub>pack</sub>(20 °C) [-]</td><td>0.46</td><td>0.33</td></tr><tr><td>ε<sub>mf</sub>(20 °C) [-]</td><td>0.5</td><td>0.43</td></tr><tr><td>ε<sub>mf</sub>(500 °C) [-] (Ergun, 1952) (Kunii and Levenspiel, 1991)</td><td>0.52</td><td>0.43</td></tr><tr><td>κ<sub>s</sub>(20 °C) [-] (experimental)</td><td>12.7</td><td>3.3</td></tr><tr><td>κ<sub>s</sub>(500 °C) [-] (experimental)</td><td>11.8</td><td>2.7</td></tr></tbody></table>

well distinguished one from the other by the capacitance probe. The adopted samples of SiC and γ-alumina are dissimilar in size and density but not by orders of magnitude. γ-alumina is lighter than SiC, with a particle density ratio of 0.37, and larger, with a particle diameter ratio of 2. In addition, γ-alumina particles are perfectly spherical, whereas SiC has an irregular shape. Additional details about the adopted samples are reported in the study of Molignano et al. (Molignano et al., 2024), in which the two materials have been tested separately by capacitance probes during experiments of hydrodynamic characterization.

### 2.3. Capacitance probe diagnostics

House-made needle-type uncooled capacitance probes have been used as main diagnostics for the investigation of mixing/segregation of two dissimilar solids in fluidized beds. A detailed description of the probes design and of their operating principle is provided by Molignano et al. (Molignano et al., 2023). Fig. 2 shows the layout of the needle-type cylindrical capacitance probe. The ground tube outer diameter is 6.75 mm, the needle length and diameter are 5.7 mm and 1.6 mm,

<figure><img src="image_5.png" /><figcaption>**Fig. 1.** Layout of the lab-scale experimental apparatus equipped for the experimental campaign.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 2. Outline of the uncooled capacitance probe (needle-type cylindric design) electrically wired to the amplifier by a low-noise coaxial cable.</figcaption></figure>

respectively. The volume of the measured domain can be estimated around 130 mm³, calculated as mean value of the volume associated to a cylinder and to a cone. The choice of the probe volume, defined by the needle length, outer diameter and the ground electrode outer diameter is coherent with other designs presented in literature, as it guarantees the representation of continuous phases minimizing disturbances to the flows (Tebianian et al., 2015). The connection of the central and guard electrodes to the customized amplifier (Accumeasure™ System 9000, MTI Instrument Inc.) via a low noise BNC cable limits any disturbances on the measured signal.

The voltage signal, V, shows inverse proportionality with the capacitance, C, of the medium which crosses the sensing volume through the gain factor of the amplifier, Gain (Fig. 3):

$$V = \frac{Gain}{C} = \frac{Gain}{\kappa\epsilon_0\Sigma} \qquad (1)$$

where $\epsilon_0$ is the void dielectric constant, $\kappa$ is the relative dielectric constant of the medium, and $\Sigma$ is the geometric characteristic of the probe sensing volume. When the capacitance probe is immersed in a fluidized bed, the relative dielectric constant of the medium is generally expressed as:

$$\kappa = \kappa_s(1 - \varepsilon) + \varepsilon \qquad (2)$$

a linear correlation first validated by Brereton (Brereton, 1987), in which $\kappa_s$ is the relative dielectric constant of the solid material and $\varepsilon$ represents the void fraction. The same relationship can be further generalized to a system composed by gas and a mixture of two dissimilar solids, with one of the two considered as tracer:

$$\kappa = (\kappa_s^{tr} \bullet f_{tr} + \kappa_s^{bed} \bullet (1 - f_{tr})) \bullet (1 - \varepsilon) + \varepsilon \qquad (3)$$

<figure><img src="image_7.png" /><figcaption>Fig. 3. Relationship between output voltage V and relative dielectric constant $\kappa$ for the applied capacitance measuring instrument ($V_0 = 7.6$ V, $\Sigma = 0.3$, $Gain = 21$ pF V).</figcaption></figure>

$f_{tr}$ represents on a fluid-free basis the tracer volume fraction in the analyzed medium or, which is the same, an expression of the mixing ratio of particles. Given the emulsion-phase voidage, $\varepsilon$, $1 - \varepsilon$ represents the fraction of volume occupied by the two solids and $f_{tr}$ the fraction of one of the two particles in this volume. $f_{tr}$ can be expressed accordingly as:

$$f_{tr} = \frac{v_{tr}}{v_{tr} + v_{bed}} \qquad (4)$$

Validity of the relationship expressed by Eq. (3) is subject to verification that the dielectric constants of the two solids do not influence each other in the mixture.

## 2.4. Experimental procedure

Calibration tests with mixtures of SiC and $\gamma$-alumina in fixed bed conditions have been performed in the graduated cylinder of 500 ml to validate the correlation proposed for the relative dielectric constant of a system of two different solids, as expressed by Eq. (3). Nine perfectly-mixed mixtures of the two materials at known volume fraction of one of the two components, varying between 0.1 and 0.9, were prepared in a vessel and poured in the graduated cylinder avoiding any segregation of the two solids and in a way that a loose fixed bed could be obtained. The capacitance probe was immersed in the fixed bed to acquire the voltage signal. The same procedure was repeated 10 times for each composition of the mixture SiC/$\gamma$-alumina.

The mixing experiments in the fluidized column represented in Fig. 1, operated at 500 °C, consisted in the rapid injection of a batch of tracer particles in the fluidized bed of monodisperse solids. A stainless-steel vessel, thermally insulated, has been used to pour the tracer material in the dense bed. Thermal insulation was adopted as tracer solids were pre-heated at the temperature of the fluidized bed before being injected. The transient phases of tracer penetration in the dense bed were followed until the establishment of the mixing/segregation equilibrium between the two materials. Finally, the local tracer concentration in the emulsion phase at steady-state conditions was measured while varying the superficial gas velocity in the range 0.34–0.87 m/s. Once injected, tracer solids arranged homogeneously along the column cross section in their downward motion toward the FB surface. This condition was previously checked by performing the rapid injection of tracer solids in a column having the same ID of the FB, and in absence of fluidization so that the distribution of tracer solids on the horizontal plane could be easily assessed. During the test, it was ensured that the injection point had the same distance from the floor as there was between the injection and the dense FB surface in the experiment.

Two experiments have been performed: in a first test, SiC was used as bed material and $\gamma$-alumina as tracer; in the second experiment, $\gamma$-alumina has been used as bed material. The amount of bed material loaded in the column was the same as in the experiments described in
the study of Molignano et al. (Molignano et al., 2024), namely 1074 g of SiC and 486 g of γ-alumina when they were used as bed solids, respectively. The batch of tracer solids corresponds to a volume concentration of 13 % – defined on a fluid-free basis as $f_{tr} = \frac{v_e}{v_e+v_{bed}}$, v being the volume occupied by the particles – in perfectly-mixed conditions of the bed. The choice of a 13 % volume fraction is the result of an optimization having as objectives the minimization of the injection time and of the increase in bed height, and the maximization of readability of the capacitance probe signal. The tracer batch was preheated at 500 °C before being injected in the system. The capacitance signal was recorded at three different axial positions inside the bed, $H_1$, $H_2$ and $H_3$, at a distance from the distributor plate of 0.06, 0.1 and 0.14 m (Fig. 1). At each bed height, capacitance probes were placed at three radial positions, namely bed center, mid-radius and wall, corresponding to r/R ratios of 0, 1/2 and 7/9, respectively. Before recording capacitance probe signals at steady-state conditions and different superficial gas velocities, the system was given sufficient time to establish the mixing/segregation equilibrium between the two solids. Signal acquisition time, recorded at a frequency of 1 kHz, was set at 300 s for all the experimental conditions, widely sufficient to make reliable statistical evaluations.

## 2.5. Data postprocessing

Data postprocessing in the transient phase of the experiment was mainly based on detailed time-resolved analysis of the voltage signals, aided by Matlab signal-processing tools able to detect signal *changepoints* with a parametric global method (Lavielle, 2005; Killick et al., 2012). A *changepoint* is a sample or time instant at which a statistical property of a signal changes abruptly. In the present case, the property under consideration was the mean of the signal. The function:

1. Chooses signal points to divide the signal into two or more sections.

2. Computes the mean signal value for each section.

3. At each point within a section, measures how much it deviates from the mean and makes the sum of the deviations for all points within the section.

4. Makes the sum of the deviations section-to-section to find the total residual error.

5. Varies the location of the division points between each section until the total residual error attains a minimum.

A result of the implementation of the method presented above is shown in Fig. 4.

As for data analysis at steady-state condition, the volumetric fraction of tracer particles, $f_{tr}$, in the emulsion phase was calculated at each of the investigated axial/radial coordinates in the dense bed and superficial gas velocities. This is possible when the emulsion phase crosses the sensing volume of the capacitance probe. Therefore, only parts of the signal associated with the dense phase have to be considered.

On the basis of previous studies on fluidized beds of monodisperse solids (Molignano et al., 2024; Molignano et al., 2023), both bubble and emulsion phases cannot be identified with a single voidage value. Instead, a Probability Distribution Function (PDF) of voidages exists for each phase, and reliable threshold values must be determined to correctly separate the emulsion from the bubble phase. Once extracted from the overall time-resolved voidage signal, the signal associated with the emulsion phase of a monodisperse bed, therefore, fluctuates between minimum and maximum values around a characteristic void fraction, which corresponds to the mode of the emulsion-phase PDF. It has been observed that the modal value associated with the emulsion phase does not necessarily corresponds to that at minimum fluidization conditions, $\epsilon_{mf}$, and that non-negligible expansion of the dense phase in the bubbling regime can occur (Molignano et al., 2024). There are some specific experimental conditions at which two modal values associated with the emulsion phase have been observed, a lower voidage LV-phase and a higher voidage HV-phase (Molignano et al., 2024).

The addition of a second solid material in the FB, different by size and density from the bed material, implies that the relative dielectric constant of the emulsion phase that crosses the sensing volume of the capacitance probe has to be treated with Eq. (3), in which $\epsilon = \epsilon_e$. However, if the emulsion-phase characteristic voidage does not keep constant due to expansion and compression patterns, as previously pointed out, not only $f_{tr}$ is unknown, but also $\epsilon_e$. This aspect implies that $f_{tr}$ cannot be tracked in the time domain, but only on a statistical basis.

At steady-state conditions of the FB with the two dissimilar solids, the statistical approach based on the computation of the PDF of voltage signal led to the identification of the characteristic voltage (mode) of the mixture in the emulsion phase (Fig. 5). With reference to those cases in which two modal components have been observed in the PDF associated with the emulsion phase of the FB, a single modal value was considered. The representative modal voltage value for the emulsion phase was obtained by the following Equation:

<figure><img src="image_8.png" /><figcaption>Fig. 4. Identification of signal *changepoints* based on the evaluation of signal mean within each section. Operating conditions: bed axis, $H_1$, T = 500 °C, U = 0.44 m/s.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Fig. 5. PDF of a voltage signal recorded by the capacitance probe. Example of bimodality in the distribution associated with the emulsion phase. Solid curve: experimental; dotted curve: smoothing of the experimental curve, smoothing factor: 0.999. Sensing volume placed at bed axis and H₂. Operating conditions: T = 500 °C, U = 0.87 m/s, fluidized bed of SiC used as bed material and γ-alumina as solid tracer.</figcaption></figure>

$$ \bar{V}_e = A_{LV}V_{LV}^{mode} + A_{HV}V_{HV}^{mode} \qquad (5) $$

With A representing the areas of the two monomodal overlapping distributions resulting from the deconvolution of the PDF according to the method proposed in the study of Molignano et al. (Molignano et al., 2024). It was observed that the same representative modal voltage can be obtained by smoothing the PDF using a smoothing factor of 0.999, as represented in Fig. 5. Then, the corresponding relative permittivity of the mixture at steady-state condition has been derived from Eq. (1), in which the previously chosen mode of V was used.

Calculation of $f_{tr}$ in the emulsion phase crossing the sensing volume of the probe depends on the void fraction, $\epsilon_e$, as expressed in Eq. (3). Given the superficial gas velocity and the axial/radial bed coordinate, there will be a statistically relevant emulsion-phase voidage, $\epsilon_e$, referred

to γ-alumina and silicon carbide, respectively. These two characteristic modal values do not necessarily coincide with each other, because the two materials have different shape factors, mean particle diameters and densities. A representative value of $\epsilon_e$ for Eq. (3) associated to a mixture of the two solids can be obtained by using a linear correlation:

$$ \epsilon_e = \epsilon_{tr}f_{tr} + \epsilon_{bed}(1 - f_{tr}) \qquad (6) $$

Conversely, the assumption that $\epsilon_e$ in the mixture remains equal to that of the solid species present in larger amount would determine major errors in case of marked segregation of the tracer in one zone of the bed.

Local values of $\epsilon_e$ for the two materials were obtained recording probe signals in FBs of the two solids fluidized separately at the same superficial gas velocities of the mixing experiments. Modal values were extracted from the PDFs of voltage signals referred to the dense phase; the corresponding relative permittivities were, consequently, calculated from Eq. (1), and $\epsilon_e$, in turn, from Eq. (2). The overall steady-state data postprocessing procedure is reported in the detailed step-by-step flow-chart of Fig. 6.

The correlation, as expressed by Eq. (6), has been successfully validated for the adopted couple of materials during the calibration tests with mixtures of SiC and γ-alumina in fixed bed conditions reported in the paragraph 2.4. The mixture bed voidage was obtained at each investigated composition with the relationship expressed as:

$$ \epsilon = \frac{v_{tot} - v_1 - v_2}{v_{tot}} \qquad (7) $$

where $v_{tot}$ is the total volume occupied by the batch in the graduated cylinder, $v_1$ and $v_2$ are the known values of volumes occupied by the two solids, easily calculated from the weight and the particle density of each material. The results of calibration tests with mixtures of SiC and γ-alumina in fixed bed conditions are showed in Fig. 7. The experimental points represent mean voltages measured by the capacitance probe immersed in the mixture in fixed bed conditions and are reported with the standard deviation bars calculated over ten measurements. The dotted line represents the predicted voltage obtained from Eq. (1) after calculating the mixture relative dielectric constant from Eq. (3). Mixture voidage, $\epsilon$, has been measured in the same graduated cylinder (Eq. (7)). f symbol in Fig. 7 refers to the volume fraction of γ-alumina. Results demonstrate that the correlation expressed by Eq. (3) holds true, meaning that the dielectric constants of the two adopted solids do not

<figure><img src="image_10.png" /><figcaption>Fig. 6. Flowchart with the step-by-step postprocessing procedure used to determine $f_{tr}$ from the experiments at steady-state conditions.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Fig. 7. Capacitance probe voltage as a function of solid volume fraction (fluid-free basis) of γ-alumina. The experimental points refer to mixtures at known homogeneous composition of γ-alumina and silicon carbide (properties reported in Table 2). The dotted line represents the predicted voltages.</figcaption></figure>

vary in mixture.

It was assumed that the linear correlation, expressed by Eq. (6), and obtained for loose fixed beds of the mixtures SiC/γ-alumina holds true in the emulsion phase of the bubbling fluidized bed even when simplifications from the two-phase theory cannot be applied. The linear trend for the void fraction in fixed loose conditions is reasonable as the size ratio between the two adopted materials does not approaches 0, a condition that would lead small particles to fill the interstices of large ones (Yang, 2003), resulting in a concave voidage profile with a minimum voidage value at a certain mixture composition. Besides, as in the present case the large particles are spherical, the high level of packing of the spheres may probably prevent even more SiC particles from filling the interstices created by γ-alumina. The final calculation of $f_{ir}$ values was achieved through Eq. (3) by using an objective function tool.

A validation of the whole adopted postprocessing procedure, including the Eq. (6) already validated under fixed bed conditions, was carried out through the computation of the balance on tracer volume fraction over the fluidized bed (Eq. (8)). The experimental domain has been discretized, under axisymmetric assumption, in three subdomains, which concentric cross-sections have the center of the probe needle equally spaced with respect to their edges. Local tracer concentration values measured at the same axial coordinate and belonging to each subdomain have been averaged over bed cross section, as following:

$$ \langle f_{ir} \rangle_r = \frac{\sum_{i=1}^{3} f_{ir} S_i}{S} \qquad (8) $$

with $S_i$ being the generic concentric cross-subsectional area, and $i$ the subsection number. Afterwards, the arithmetic mean of the three resulting values of $\langle f_{ir} \rangle_r$ has been computed and a mean concentration value referred to the whole FB has been gathered.

### 3. Results and discussion

#### 3.1. Study of the transient phase of mixing

Experiments aimed at the assessment of the transient state of mixing between the batch of tracer solids and bed material have been performed at a superficial gas velocity of 0.44 m/s and 500 °C. Injected tracer solids arranged homogeneously along the column cross section in their downward motion toward the dense FB and reached together the bed surface. As matter of fact, the time-resolved pressure signal measured at 0.01 m above the gas distributor exhibits a one-step shift at a pressure level corresponding to the increased weight of the bed at about 120 s

after the beginning of signals acquisition. The time instant at which pressure signal abruptly increases was taken as reference time for the starting of mixing between the tracer particles batch and the FB of different granular solid. Right after, tracer solids penetrated the FB and this was witnessed by modifications in the capacitance probes signal, which sensing volume was interested by the passage of a solid material different from that of the bed. The passage of tracer solids through the sensing volume of probes placed at different bed axial levels is represented in Fig. 8 as example. It can be clearly observed that the batch of tracer solids enters the bed by, first, crossing the sensing volume of the upper probes and, finally, reaching the sensing volume of the probe placed at $H_1$. The described pattern represented in Fig. 8 refers to a system in which SiC has been injected in the bed of γ-alumina solids. The time instant at which the batch passes through the sensing volume of the probes could be easily identified as the signal baseline immediately deviates toward lower signal levels, given the higher dielectric constant of SiC with respect to γ-alumina. In the “complementary” experiment, in which γ-alumina was used as tracer material, the passage of γ-alumina is identified by an increase of the signal baseline characteristic value. However, signal modification is much smoother in this last case, and the reason will be discussed more in detail in the following sections. Apart from differences due to instruments response times, $t_{H_1}$ basically coincides with the mixing on-time identified by the pressure signal (Fig. 8). The difference between the time defined by the pressure probe and $t_{H_1}$, derived from the capacitance probe placed at $H_1$, can be considered as an axial macro-mixing time of the tracer, namely the time required for the batch to vertically cross the entire bed height without being truly mixed with the rest of the bed yet. Once defined this characteristic time of the system, it was observed that it lasted, on average, less than 1 s (0.71 ± 0.46 s). The reported value results from the mean over six experiment replicates, in which the capacitance probes were placed two times at bed center, two times at mid-radius and two times at column wall. Concerning the time required for the tracer to reach steady-state concentrations in the dense bed, as established by the changepoints procedure (Fig. 4), it was one or two orders of magnitude higher, being in the average range 7–12 s. The edges of this range constitute, respectively, the minimum and maximum value of the time required for the adopted tracer to reach steady-state concentrations in the dense bed at a superficial gas velocity of 0.44 m/s. These values derive from the mean of time values computed over six experiment replicates. During

<figure><img src="image_12.png" /><figcaption>Fig. 8. Time-resolved voltage signals recorded at bed axis with the probes sensing volume placed at $H_1$ (bottom) and $H_3$ (top). Operating conditions: T = 500 °C, U = 0.44 m/s; bed of γ-alumina solids (0.486 kg), tracer batch of SiC solids (0.196 kg). Injection performed at about 120 s. $t_{H_1}$ and $t_{H_3}$ represent the times at which the batch crosses compactly the sensing volume of the two probes, respectively. $t_{H_1} > t_{H_3}$.</figcaption></figure>
each experiment, two probes were fitted in the fluidized bed at the same
radial coordinate and different axial levels, therefore twelve values were
used for the estimate of the mean.

According to the amount of bed material poured in the column, the
height of the fluidized bed at minimum fluidization conditions is such
that the aspect ratio, H/D, is higher than unity. Consequently, the
displacement of the tracer solids along the column section can be
neglected over that along the bed height, namely, it can be considered
instantaneous with respect to the mechanisms which develop across the
height of the FB in transient conditions. Although the operation at
relatively high temperature in a steel column does not allow the
observation of the bubbles patterns, the characteristic dimensions of the
FB-system under investigation do not suggest the possible identification
of multiple mixing cells along the axial coordinate. Therefore, a
macroscopic dispersion model can hardly be used to describe and pre-
dict our data. Nevertheless, with reference to the above-mentioned
experimental characteristic mixing times, two axial tracer dispersion
coefficients can be calculated: a macro dispersion one, $\mathcal{D}_{s,macro}^a$, referred
to the tracers batch crossing the bed vertically without mixing with the
rest of the bed; an axial dispersion coefficient, $\mathcal{D}_s^a$, referred to the tracer
batch disaggregating and reaching local steady-state concentrations in
the dense bed. $\mathcal{D}_{s,macro}^a$ and $\mathcal{D}_s^a$ have been calculated as a squared space
$\Delta z$ divided by a characteristic time $\Delta t$. 0.71 s and 7–12 s were assumed

as characteristic times for macromixing and complete mixing, respec-
tively. As $\Delta z$, the $\gamma$-alumina bed height at minimum fluidization condi-
tions has been used, equal to 0.148 m. It resulted that $\mathcal{D}_{s,macro}^a$ is, on
average, $3.1 \times 10^{-2}m^2/s$, while $\mathcal{D}_s^a$ ranges between 1.8 and $3.1 \times
10^{-3}m^2/s$. These values are slightly underestimated because they are
obtained using the bed height information referred to minimum fluid-
ization conditions; bed expansion due to the higher superficial gas ve-
locity is not taken into account. In light of the above, the calculated $\mathcal{D}_s^a$
coefficient is coeherent with the axial diffusion coefficient of Mostoufi
and Chaouki (Mostoufi and Chaouki, 2001) obtained on a larger
laboratory-scale unit, in which the static bed height is comparable with
that of the present experimental study. Moreover, $\mathcal{D}_s^a$ is, as expected,
one order of magnitude higher than the radial dispersion coefficient
measured in some literature studies on wide laboratory-scale fluidized
beds (Berruti et al., 1986; Kunii and Levenspiel, 1969).

3.2. Steady-state profiles of solid tracer concentration in the dense phase of the fluidized bed

With reference to the nomenclature proposed by Rowe (Rowe et al.,
1972), the dependence of the gas drag force on the projected particle
surface generally determines that larger/smaller solids tend to act as
jetsam/flotsam, respectively. Denser solids tend to sink to the bottom

<figure><img src="image_13.png" /><figcaption>**Fig. 9.** Concentration profiles of silicon carbide (left) and γ-alumina (right) in the emulsion phase of the fluidized bed of γ-alumina and silicon carbide, respectively. Tracer volume fraction is represented versus superficial gas velocity, and parametric in the axial coordinate.</figcaption></figure>
while less dense solids are typically segregated at the bed surface. In this context, higher-velocity fluidized beds tend to reach a more uniform, mixed state. The properties of the two materials adopted in the present study make the basic question of which component does act as flotsam in the mixture not trivial at all. Indeed, it can happen that the combination of properties driving towards opposed tendencies leads to an unpredictable behavior.

In Fig. 9, tracer concentration profiles are reported for the two complementary experiments. Tracer volume fraction is reported versus superficial gas velocity and parametric in the axial coordinate of the bed. Each graph refers to a different radial coordinate, namely column center, mid-radius, and wall. A continuous black line at a tracer volume fraction, $f_{tr}$, of 13 % is reported as reference for perfectly-mixed conditions. When the measurement has been repeated more than one time, mean tracer volume fraction is obtained after averaging results from the independent experiments and the scatter around reported mean values is displayed as error bar having minimum and maximum measured values as ends. In Fig. 9 on the left, the volume fraction of silicon carbide acting as tracer in the fluidized bed of γ-alumina is reported on y-axis versus U: at a superficial gas velocity of 0.44 m/s the FB results well mixed along the axial coordinate. As superficial gas velocity is increased, the system exhibits a vertical segregation with silicon carbide moving towards the top of the bed, resulting in a progressive pauperization of silicon carbide in the lower and intermediate zone of the bed. These patterns are particularly evident at bed axis and mid-radius, whereas the bed seems to be exempt from segregation mechanisms at the column wall. Looking at concentration values at different radial positions, at a superficial gas velocity of 0.44 m/s, the fluidized bed results well mixed along the radial coordinate of the bed, as well. As superficial gas velocity is increased, radial segregation patterns develop especially at $H_1$ and $H_2$. Silicon carbide tends to segregate towards bed walls. $H_3$ is less subject to segregation mechanisms, although the slight radial segregation drives silicon carbide to concentrate towards the axis of the column, as opposite to what observed at $H_1$ and $H_2$. In summary, superficial gas velocity intensifies the axial and lateral segregation of the mixture. Vertical segregation with superficial gas velocity is more pronounced at bed axis becoming progressively less intense towards bed walls. Radial segregation with superficial gas velocity is much emphasized at $H_1$ and the slight tracer segregation identified at $H_3$ occurs in the opposite direction with respect to $H_1$ and $H_2$.

In Fig. 9 on the right, results from the complementary experiment, in which γ-alumina has been used as tracer material, are presented: at a superficial gas velocity of 0.44 m/s the fluidized bed results moderately mixed along the axial coordinate. As superficial gas velocity is increased, the system exhibits a vertical segregation with γ-alumina moving towards the bottom and the intermediate zone of the bed, resulting in a progressive pauperization of γ-alumina at the top of the bed. These patterns are particularly evident at bed axis and mid-radius, whereas mechanisms at the column wall are not well understood. Looking at concentration values at different radial positions, at a superficial gas velocity of 0.44 m/s, the fluidized bed results fairly mixed along the radial coordinate of the bed. As superficial gas velocity is increased, radial segregation patterns develop at $H_1$ and $H_2$, where γ-alumina tends to segregate towards bed axis. At $H_3$, the radial segregation drives γ-alumina to concentrate towards the walls of the column, although, from $U = 0.49$ m/s, $f_{tr}$ at $H_3$ decreases independently from the radial coordinate. Also in these experiments, with silicon carbide as bed material, superficial gas velocity is found to emphasize the axial and lateral segregation of the binary mixture. Some mechanisms at bed walls and $H_3$ are less clear.

### 3.3. Final validation

The final validation of the procedure which led to the steady-state profiles of solid tracer concentration in the dense phase of the fluidized bed comes from the balance on tracer volume fraction over the

whole bed, as explained in paragraph 2.5. Results for the two mixtures are shown in Fig. 10. A continuous black line at a tracer volume fraction, $f_{tr}$, of 13 % is reported as reference for perfectly-mixed conditions. The balance on tracer volume fraction can be considered fulfilled in both cases. The mean tracer concentration in the graph (a), represented by the dotted line, is a bit lower that the reference value for perfectly-mixed conditions. The trends in Fig. 10 suggested that SiC tends to move towards bed walls. The radial coordinate to which the terminology Wall is associated, actually corresponds to 7/9 of the column radius, with 0 being the value for column axis. Therefore, it is expected that even higher tracer volume fractions can be found at the real walls of the column. Along the same path, as SiC tends to concentrate at the top of the bed, it is likely for a layer of silicon carbide to establish above the capacitance probe placed at $H_3$. The discretization of the bed domain seems to be, instead, more favorable in the case (b), as demonstrated by the balance on tracer volume fraction. Overall, experiments on the two complementary mixtures seem really to show coherent results, even if more acquisition points along the radial direction can be helpful for a more comprehensive characterization of the tracer behaviour and for an even better closure of the balance on the tracer volume fraction.

### 3.4. Conceptualization

Semi-quantitative 2D-maps of tracer concentration referred to the dense phase of the fluidized bed have been derived from the profiles of Fig. 9, following the domain discretization along the radial coordinate, as explained in paragraph 2.5. As for the axial coordinate, three subdomains of equal extent were considered. Results with the FB of γ-alumina are represented in Fig. 11.

By looking at the phenomenology along the height of the fluidized bed, it may in some extent resemble what in liquid fluidization has been called layer inversion phenomenon. It occurs with mixtures of two particle species, species 1 being larger in size and with lower density than species 2. An idealized and over-simplified description of the layer inversion phenomenon can be given as follows: at very low liquid velocity, the two species form distinct layers, with the denser smaller particles (species 2) at the bottom, and the larger lower-density particles (species 1) at the top. At high velocities, two distinct layers are again visible, but the layers are inverted, species 1 and 2 being at the bottom and top, respectively. The liquid velocity at which the bed inverts corresponds to homogeneous mixing of the two species (Epstein and LeClair, 1985). Layer inversion has been extensively studied in the field of liquid fluidization (Epstein and LeClair, 1985; Di Felice, 1995; Moritomi et al., 1982; Gibilaro et al., 1986; Escudié et al., 2006). Rasul et al. (Rasul et al., 1999) showed how the theory of layer inversion can also be applied to gas fluidized beds. The authors operated under constrained conditions, choosing small particles that could give inversion between $U_{mf}$ and $U_{mb}$, to prevent bubbles from interfering with the observation of the phenomenon. Afterwards, van Wachem et al. (van Wachem et al., 2001) reproduced the layer inversion phenomenon through a CFD model developed for gas-solid fluidized beds containing a mixture of two particle species. They claimed that the observed phenomenon can be explained taking into account the different mechanisms promoting mixing and segregation patterns: at low velocity the controlling step is mostly related to the difference in upward drag and gravity force acting on the two solid species, whereas at high velocity the different behaviour at granular temperature and pressure gradients plays a role. It can be speculated that the results of the present study could be coherent with the described phenomenology. The superficial gas velocity of 0.44 m/s may represent the velocity at which the bed starts displaying axial inversion patterns from a homogeneously mixed condition. At higher gas velocity, the high level of “turbulence” induced by the intense bubbling and the high voidage enhance the role of random motion and the mobility of the particles rather than the classical mechanisms of bubble wake and drift on the mixing/segregation patterns. Under these
<figure><img src="image_14.png" /><figcaption>**Fig. 10.** Tracer concentration in the emulsion phase of the FB averaged over the column cross section represented versus superficial gas velocity and parametric in the axial coordinate. (a) silicon carbide used as tracer; (b) γ-alumina used as tracer. Dotted lines represent tracer concentration profile averaged over the whole bed.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Fig. 11.** Semi-quantitative phenomenological maps of solid tracer concentration in the dense phase at steady-state conditions of the fluidized bed of γ-alumina and SiC used as tracer.</figcaption></figure>

conditions finer SiC particles are more prone to be moved in the upper part of the fluidized bed than larger alumina particles inducing axial segregation. The same mechanism can be evocated for the radial segregation at wall column; again, the more extensive mobility of finer SiC particles than alumina ones allows to reach the typically lower voidage region of the bed. This effect could be emphasized at high fluidization velocity by the distribution of bubbles that are typically concentrated in the middle of the fluidized bed. However, further investigations at superficial gas velocities lower than 0.44 m/s are needed to confirm the described phenomenology as well as at different fluidization column scale to exclude wall effects derived by the laboratory-scale apparatus.

Overall, it is worth to note that the hydrodynamic heterogeneities related to the bubbles phase and the bubble “turbulence” can determine the conditions of perfect mixing for the two solid species of particles in the fluidized bed volume, but at the same time can interact differently on each solid species determining an uneven distribution of solid species in the fluidized bed volume (Troiano et al., 2022; Köhler et al., 2017; Köhler et al., 2020). Thus, it can be considered a solid species perfectly mixed in the fluidized bed volume, but non-perfectly mixed with the other solid species present in the same control volume.

### 3.5. Improving the capacitance probes diagnostics for binary mixtures of dissimilar solids in dense FBs

A new method, based on capacitance probe diagnostics, has been proposed for the characterization of solids mixing in a gas-FB of two dissimilar solids in the bubbling regime. Silicon carbide (SiC) and γ-alumina have been chosen as they present sufficiently different relative dielectric constants (Table 2). Thus, they can potentially be well distinguished one from the other by the capacitance probe method. At steady-state conditions, the comparison between capacitance probe signals associated with mixtures γ-alumina/SiC of different composition and with the two beds of monodisperse solids can be made by a statistical approach. PDFs of voltage signals corresponding to different local

volume fractions of γ-alumina in the emulsion phase, $f_γ$, recorded at steady-state conditions of the fluidized bed, are, thus, reported in Fig. 12. All the curves refer to systems operated at a superficial gas velocity of 0.44 m/s, corresponding to $U - U_{mf}$ values of 0.25 and 0.17 m/s with reference to SiC and γ-alumina, respectively. The green curves correspond to the fluidized bed of γ-alumina, the blue curves to bed of SiC particles. The yellow curves result from the injection of a batch of SiC in the γ-alumina bed, the red curves result from the complementary experiment. It is interesting to note that the blue curves and red curves of each figure are very similar one from the other. In the red curves, the part of the PDF referred to the emulsion phase of the fluidized bed is slightly shifted toward higher voltage values, indicating the presence of small amounts of γ-alumina. By considering comparable amounts of SiC in the bed of γ-alumina (yellow curves), the shift of the PDF with respect to that associated to the bed of monodisperse γ-alumina (green curves) is much more evident. The extent of the shift of the red curve from the blue one risks being of the same order of magnitude of experimental random errors. This aspect has caused more repetitions of the measurement to be required for this type of system.

The reason for such different results depends on the non-linearity in the characteristic curve of the capacitance measuring instrument (capacitance probe + amplifier). Non-linearity has consequences on the instrument sensitivity, which will depend on where the relative dielectric constant of the medium is positioned on the curve. This can be better understood by the observation of Fig. 13, in which the characteristic curve of the instrument is re-proposed. The highest voltage value corresponds to the case in which the sensing volume of the probe is immersed in free air ($V_0$). Measured voltage values decrease as the relative dielectric constant of the medium in the sensing volume increases. The same change in relative dielectric constant, $\Delta\kappa$, as shown by the distance along the x-axis between the first pair of magenta and light-blue points on the left and the second pair on the right, produces a completely different difference in the output voltage. Therefore, non-linearity between the relative dielectric constant of the medium and the output voltage penalizes measurements carried out in the bed of SiC,
<figure><img src="image_16.png" /><figcaption>**Fig. 12.** PDFs of voltage signals recorded by capacitance probe with the sensing volume placed at bed axis and, from left to right, $H_1$, $H_2$, and $H_3$. Operating conditions: T = 500 °C, U = 0.44 m/s. $f_γ$ is the local volume fraction of γ-alumina in the emulsion phase of the fluidized bed.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Fig. 13.** Characteristic curve of the capacitance measuring instrument (relationship between output voltage V and relative dielectric constant $κ$). Different effects of the same relative dielectric constant variation ($Δκ$ = 1) on the measured voltage.</figcaption></figure>

in which γ-alumina is injected as tracer. This aspect also clarifies why, when γ-alumina was used as tracer material, signal modification due to passage of γ-alumina resulted undetectable in transient conditions (paragraph 3.2). The present issue can be solved by using an amplifier with a linear characteristic curve between the output voltage and the solids relative dielectric constant. Moreover, we have learnt that the difference in terms of dielectric constants between bed material and tracer material has not to be necessarily maximized. Instead, difference has to be optimized with respect to the instrument curve available. The above-listed adjustments would surely improve the flexibility of the technique towards more combinations of solid materials and increase the intelligibility of signals in the time-domain, with less repetitions needed for a single measurement.

## 4. Conclusions

Capacitance probes have been employed innovatively to characterize fluidized beds of bidisperse solids. A correlation which predicts the relative dielectric constant of a binary mixture has been proposed and successfully validated. The implementation of the correlation to the dense fluidized bed of two dissimilar solids requires the voidage of the dense phase to be known at the operating conditions of interest.

The effectiveness of the method to quantitatively characterize gas-fluidized beds of dissimilar solids strongly depends on the difference

between the dielectric properties of the two materials and on the characteristic instrument curve. Some drawbacks are found in the nonlinearity of the measurement instrument curve: this issue can be easily overcome by using a different amplification system.

More serious issues concern the difficulty in following the time-series of local tracer concentration due to the unknown temporal evolution of the emulsion-phase voidage. The voltage signal from a bed of two dissimilar solids is composed, in fact, of two unknowns, namely bed voidage and solid composition. This is a consequence of the fact that deviations from the *two-phase theory of fluidization* can, in some cases, hardly be neglected. Therefore, in the present study, tracer concentration profiles have been gathered only at steady-state condition by a statistical approach and evaluation of transient phases have been accomplished only by evaluation of the voltage signal, which does not give any information on the evolution of tracer concentration with time. The axial dispersion coefficient calculated during studies in transient conditions is coherent with results from literature and it is in the range 1.8–3.1 x 10⁻³ m²/s.

Results on local concentrations of solid tracers in the emulsion phase of the fluidized bed at steady-state conditions are satisfactory and encouraging. The adopted binary mixtures exhibit the *layer inversion* phenomenon, well known in liquid fluidized beds. This means that, when size and density ratios of the two solid species are on opposite sides of unity, a superficial gas velocity range exists in which particle size plays a major role over density in the phenomenology of segregation.

### CRediT authorship contribution statement

**Laura Molignano:** Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Conceptualization. **Maurizio Troiano:** Writing – review & editing, Supervision, Conceptualization. **Roberto Solimene:** Writing – review & editing, Supervision, Resources, Project administration, Funding acquisition, Conceptualization. **Sina Tebianian:** Writing – review & editing, Supervision, Conceptualization. **Jean-François Joly:** Funding acquisition. **Piero Salatino:** Writing – review & editing, Funding acquisition, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgements

The authors gratefully acknowledge Mr. Antonio Cammarota for the set-up of the experimental apparatus, and express their gratitude to Ms.
Cristiana Rocca and Mr. Mario Mobiglia, University of Naples, for the experimental support.

## Data availability

Data will be made available on request.

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