Plug Regime Flow Velocity Measurement Problem Based on Correlability Notion and Twin Plane Electrical Capacitance Tomography: Use Case - MDPI

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Plug Regime Flow Velocity Measurement Problem Based on Correlability Notion and Twin Plane Electrical Capacitance Tomography: Use Case - MDPI
sensors
Article
Plug Regime Flow Velocity Measurement Problem Based on
Correlability Notion and Twin Plane Electrical Capacitance
Tomography: Use Case
Volodymyr Mosorov * , Grzegorz Rybak                          and Dominik Sankowski

                                          Institute of Applied Computer Science, Lodz University of Technology, 90-537 Łódź, Poland;
                                          grzegorz.rybak@p.lodz.pl (G.R.); dsan@kis.p.lodz.pl (D.S.)
                                          * Correspondence: volodymyr.mosorov@p.lodz.pl

                                          Abstract: In this paper, the authors present the flow velocity measurement based on twin plane
                                          sensor electrical capacitance tomography and the cross-correlation method. It is shown that such a
                                          technique has a significant restriction for its use, particularly for the plug regime of a flow. The major
                                          issue is with the irregular regime of the flow when portions of propagated material appear in different
                                          time moments. Thus, the requirement of correlability of analyzed input signal patterns should be
                                          met. Therefore, the checking of the correlability should be considered by such a technique. The article
                                          presents a study of the efficiency of the original algorithm of automatic extraction of the suitable
                                          signal patterns which has been recently proposed, to calculate flow velocity. The obtained results
                                          allow for choosing in practice the required parameters of the algorithm to correct the extraction of
         
                                   signal patterns in a proper and accurate way. Various examples of the application of the discussed
                                          algorithm were presented, along with the analysis of the influence of the parameters used on the
Citation: Mosorov, V.; Rybak, G.;
Sankowski, D. Plug Regime Flow
                                          quality of plugs identification and determination of material flow.
Velocity Measurement Problem Based
on Correlability Notion and Twin          Keywords: electrical capacitance tomography; pattern; correlability; multiphase flow
Plane Electrical Capacitance
Tomography: Use Case. Sensors 2021,
21, 2189. https://doi.org/10.3390/
s21062189                                 1. Introduction
                                                The flow velocity measurements are often applied to many industrial processes,
Academic Editors: Kenneth Loh and
                                          such as food, mineral, chemical, and pharmaceutical production [1–5]. For one phase
Ruben Specogna
                                          flow, velocity measurement is a relatively simple task. However, it is still challenged
                                          in multiphase flow applications when the velocity measurement of the chosen phase is
Received: 22 January 2021
                                          required. One of the concrete applications is plastic bottle production, where a pneumatic
Accepted: 16 March 2021
                                          transport system transports granular material. One of the basic efficiency criteria for such
Published: 21 March 2021
                                          a transport system is the maintenance of low air pressure during material propagation.
                                          Thus, continuous measurement of solid material velocity is necessary for estimating the
Publisher’s Note: MDPI stays neutral
                                          transportation efficiency of such a system. Nowadays, in practice, different techniques for
with regard to jurisdictional claims in
published maps and institutional affil-
                                          the flow velocity measurement are used, although they are still not perfect. Sometimes,
iations.
                                          invasive techniques are used. They require sensors inserted into the pipe, and as a result,
                                          they disturb the material propagation and are exposed to abrasion over time. There are
                                          also laser methods and other optical methods, but they have a limited scope of application
                                          related to its radiation issues.
                                                One of the techniques that are not invasive is process tomography. The electrical
Copyright: © 2021 by the authors.
                                          capacitance tomography (ECT) technique is successfully applied when flow phases are not
Licensee MDPI, Basel, Switzerland.
                                          electrically conductive. In the mentioned pneumatic transport, this situation occurs. Such a
This article is an open access article
distributed under the terms and
                                          technique is relatively low-cost and safe, especially its measurement sensor part. The classi-
conditions of the Creative Commons
                                          cal tomographic image velocimeter (TIV) includes twin sensor planes mounted around an
Attribution (CC BY) license (https://
                                          investigated pipe/vessel/column. Each sensor consists of electrodes (typically, the number
creativecommons.org/licenses/by/          of the electrodes is 8, 12 rarely) and it is used to measure the electrical capacitance between
4.0/).                                    any pairs. Based on these capacitance measurements, an image reconstruction algorithm is

Sensors 2021, 21, 2189. https://doi.org/10.3390/s21062189                                                    https://www.mdpi.com/journal/sensors
Plug Regime Flow Velocity Measurement Problem Based on Correlability Notion and Twin Plane Electrical Capacitance Tomography: Use Case - MDPI
Sensors 2021, 21, 2189                                                                                            2 of 13

                         next utilized to reconstruct the material distribution inside the investigated volume of inter-
                         est. To calculate the material velocity between the sensors’ planes, a series of reconstructed
                         images are required.
                               The correctness and reliability of a particle velocity measurement of multiphase
                         flow are still a key challenge, despite the fact that currently there are many published
                         elaborations related to this research scope [6–8]. Excluding the problems concerning the
                         time stability of the measured capacitances and calibration limitations, the main problem is
                         how to correctly calculate flow velocity based on reconstructed images.
                               Although different techniques are proposed, for instance, the method based on the
                         computation of the sensitivity matrix spatial gradient [9,10], the main method of velocity
                         calculation is still a classical cross-correlation one [11–16]. To estimate flow velocity in
                         the chosen pixel of the volume of interest, first, the cross-correlation function of signals
                         representing material distribution changes within the pixel of the first and second plane of
                         the tomographic unit is calculated. It is assumed that the global maximum of the calculated
                         function corresponds to a transit time of material propagation between two sensor planes.
                         Moreover, it is important to mention that the transit time is calculated for a time window.
                         Therefore, this calculated velocity corresponds to the average material velocity between
                         two sensor planes.
                               The signal correlation function is based on the indicated time intervals, in which the
                         range of measurements used in the calculations is determined. The indicated time intervals
                         constitute time windows for the measurement data. Most often, such a window is set as a
                         constant value before performing the calculations and its scope does not change during the
                         analysis. Unfortunately, there are no hints or information on what basis the parameters
                         were determined [16–20]. On the other hand, it turns out that the window width parameter
                         is important in the context of ensuring the effectiveness of the algorithm. Due to this fact,
                         it is obvious that the pattern should be computed based on the signal that is processed.
                         Warsito and Fan 2008 [21] proposed the technique of tracing flow structures but without
                         indicating the mechanism of time window selection. Fuchs et al. [22] described a method
                         based on determining the length of a plug, but without explaining how to determine the
                         length of time windows for the flow velocity estimation algorithm. The problem is still
                         omitted by the researchers, for instance, in the articles published in the last years [4,23,24].
                               The existing gap has recently been filled up by developing a new concept for an
                         appropriate algorithm allowing the automatic determination of the time window (Mosorov
                         et al. 2020) [11]. The authors propose automatically detecting signal patterns to measure
                         the particle velocity based on the obtained series of tomographic images. However, the
                         proposed algorithm uses arbitrary chosen parameters, and their choices have not been
                         justified. There is no study presenting the dependence between the accuracy of flow
                         velocity measurement and parameter values. Therefore, the article has several aims:
                         •    Justification of the choice of parameters to detect signal patterns to measure flow
                              velocity.
                         •    Determination of the problems for utilization of the proposed approach in real-case-
                              studies.
                              The authors suppose that the obtained results of the conducted study can be success-
                         fully utilized in many applications that use the cross-correlation method to calculate time
                         parameters of flow measurements, where the plug regime occurs.

                         2. Theoretical Considerations
                               An ECT tomography system to determine a velocity profile in the region of interest
                         should consist of at least two plane sensors mounted around a pipe (see Figure 1). To ensure
                         fulfilling the assumption on the laminar character of the investigated flow, a distance
                         between the two planes should be chosen, as short as possible. Such an ECT tomographic
                         velocimeter provides the constant speed of imagining, representing the distribution of the
                         material concentrations in cross-sections of both planes.
Plug Regime Flow Velocity Measurement Problem Based on Correlability Notion and Twin Plane Electrical Capacitance Tomography: Use Case - MDPI
ensure fulfilling the assumption on the laminar character of the investigated flow, a dis-
                tance between the two planes should be chosen, as short as possible. Such an ECT tomo-
Sensors 2021, 21, 2189                                                                                                                            3 of 13
                graphic velocimeter provides the constant speed of imagining, representing the distribu-
                tion of the material concentrations in cross-sections of both planes.

              Diagram
    Figure 1.Figure   1. of flow velocity
                         Diagram          measurement
                                    of flow             realized by a two-plane-sensor
                                            velocity measurement                       electrical capacitance
                                                                      realized by a two-plane-sensor          tomography
                                                                                                         electrical capaci-(ECT)
             tanced is
    tomography,        the distance (ECT)
                    tomography      between  the two planes.
                                           tomography,    d is the distance between the two planes.

                                         Material concentrations are represented as images reconstructed based on raw mea-
                   Material concentrations are represented as images reconstructed based on raw meas-
                              surement. The typical image resolution is 32 × 32 pixels. To calculate the flow velocity
            urement. The typical      image resolution is 32 × 32 pixels. To calculate the flow velocity pro-
                              profile, the cross-correlation method is utilized. First, the cross-correlation function is
            file, the cross-correlation
                              calculated methodbetween is       utilized.
                                                            changes            First,
                                                                           in the      the cross-correlation
                                                                                   corresponding       pixel values     function     is calcu-
                                                                                                                           of the images,      and then,
            lated between changes
                              the maximumin the of   corresponding
                                                         the cross-correlationpixel values   of isthe
                                                                                      function          images, and
                                                                                                    determined.         Suchthen,   the max-
                                                                                                                               a founded      maximum
            imum of the cross-correlation
                              determines the time       function
                                                            delay of   is material
                                                                           determined.      Such abetween
                                                                                      propagation       founded      themaximum
                                                                                                                          two planes.deter-
                                    Let    x
            mines the time delay of material i,j ( nT ) and   y     (
                                                           propagation
                                                                i,j   nT ) define  the  material    distribution
                                                                                between the two planes.                changes    for (i, j) pixel of the
                   Let , ( nth) and            ( ) defineimage
                                   I × J , tomographic                       (see Figure
                                                                     the material          1) captured
                                                                                       distribution           with the
                                                                                                           changes            ( , ) rate
                                                                                                                         forframe     pixelresolution
                                                                                                                                              of         T
                              obtained      from     planes    X    and   Y  of the twin  plane
            the th I × J tomographic image (see Figure 1) captured with the frame rate resolution  tomograph,         respectively.
                                    As mentioned before, it is assumed that flow has a laminar character. Therefore,
            obtained from planes              and         of the twin plane tomograph, respectively.
                              a strong relationship between the material distribution changes coming from the
                   As mentioned     before,        it isand
                                                          assumed         that flow has      a laminar         character.      Therefore,     a X plane
                              and Y   one     exists,        the    cross-correlation    function    R i,j ( kT ) of the  two  time   series   xi,j (nT )
            strong relationship
                              and between
                                    
                                      yi,j (nT ) ,the  n =material
                                                           N, . . . N +distribution
                                                                             M − 1 can bechanges
                                                                                            applied: coming from the X plane
            and Y one exists, and the cross-correlation function , ( ) of the two time series
                , (  ) and       , (     ) , n = N, … N +1MN +            − M1−can
                                                                                1   be applied:
                                                    Ri,j (kT ) =
                                                                   M     ∑      xi,j (nT )yi,j ((n − k ) T ), k = . . . , −1, 0, 1 . . .              (1)
                                                  1           −1         n= N
                                    ,(      )=        ∑   =         ,(      ) , (( − ) ),              = ⋯ , −1,0,1 …                      (1)
                            where M is the number of frames determining the time window considered in the calculations.
            where M is the number     of frames
                                 The founded       determining
                                                maximum     of thethe  time window
                                                                    cross-correlation    considered
                                                                                       Rmax           inthe
                                                                                             determines  thetime
                                                                                                             calcu-
                                                                                                                 delay τ0 in
            lations.        a frame rate between the two signals representing material change distributions. Then,
                 The foundeda velocity
                                maximumVi,j of
                                            of the
                                                theflow  in (i, j) pixel of
                                                    cross-correlation        thedetermines
                                                                          Rmax    cross-section calculated
                                                                                              the          in timeinwindow
                                                                                                  time delay
                            [N, N +the
            a frame rate between    M] two
                                        is given  by the
                                             signals     formula:
                                                       representing    material change distributions. Then, a
                                                                                 d
            velocity , of the flow in (i, j) pixel of the cross-section Vi,j =calculated in time window [N, N            (2)
                                                                                 τ0
            + M] is given by the formula:
                                  It is worth nothing that it is an average velocity of material propagation in the time
                                                                , =
                             window, for instance, in the frame        [N, N + M].                                       (2)
                                  The changes in the concentration of the pneumatic flow material measured in the
                 It is worth vertical
                              nothingpipethatsection
                                               it is anhave  a characteristic
                                                         average     velocity ofshape,  namely
                                                                                   material        quasi-regular
                                                                                               propagation          pulses
                                                                                                                in the timesimilar to
                             rectangular    ones.   In terms
            window, for instance, in the frame [N, N + M].     of  fluid  mechanics,   plug   flow  is a basic   model  describing  a
                             material  flowing    in a pipe.  In  plug  flow, the velocity  of the
                 The changes in the concentration of the pneumatic flow material measured in the   fluid is  assumed  to be  constant
                             across any cross-section of the pipe perpendicular to the axis of the pipe. Such a plug
            vertical pipe section have a characteristic shape, namely quasi-regular pulses similar to
                             flow model assumes that there is no boundary layer adjacent to the inner wall of the pipe.
            rectangular ones.   In terms of fluid mechanics, plug flow is a basic model describing a ma-
                             For instance, Figure 2 shows the chosen time interval of typical normalized concentration
            terial flowing inchanges
                               a pipe.inIntheplug
                                              chosenflow,   the
                                                        pixel of velocity   of the fluid
                                                                  the tomographic    images isfor
                                                                                               assumed      to beconveying,
                                                                                                  a pneumatic      constant where 0
            across any cross-section
                             corresponds of the
                                            to anpipe  perpendicular
                                                   empty   pipe and 1 meansto the
                                                                                theaxis of the
                                                                                    highest      pipe. concentration
                                                                                              material  Such a plug flow
                                                                                                                       (to normalize
            model assumes material
                              that there   is no boundary
                                        concentration,           layer adjacent
                                                          the calibration          to the
                                                                            procedure       inner wall of the pipe. For
                                                                                        is required).
            instance, Figure 2 shows the chosen time interval of typical normalized concentration
            changes in the chosen pixel of the tomographic images for a pneumatic conveying, where
Plug Regime Flow Velocity Measurement Problem Based on Correlability Notion and Twin Plane Electrical Capacitance Tomography: Use Case - MDPI
signal patterns. Hence, the velocity calculation should consider choosing the appropriate
                         signal patterns of time series , ( ) and , ( ), i.e., determination of a time window.
                              Since the plug propagation time is not constant, the choice of a time window with a
                         fixed length M is not possible. In addition, plugs appear irregularly, therefore, there is a
Sensors 2021, 21, 2189   problem: How to automatically determine the time of appearance of the plug and its4 fin-of 13
                         ishing. Hence, the algorithm for determining the velocity based on Equation (1) should
                         automatically detect the moments of plug occurrence.

                                                                     (a)                                                  (b)
                         Figure
                          Figure2.2.Normalized
                                    Normalized concentration
                                                concentration changes within aa chosen
                                                              changes within    chosenpixel
                                                                                       pixel(15,15)
                                                                                             (15,15)ofofthe
                                                                                                         thetomographic
                                                                                                             tomographicimages
                                                                                                                         im-
                         ages (a) and the image captured by a high speed camera (#220–#320) (b); s0—some threshold level
                          (a) and the image captured by a high speed camera (#220–#320) (b); s0 —some threshold level and
                         and m0—the arbitrarily chosen number of frames.
                          m0 —the arbitrarily chosen number of frames.

                                Applying    simple
                                 As mentioned        thresholding
                                                 in Mosorov     et al. to determine
                                                                        2020 [11], thethe  time moments
                                                                                       estimation            is not
                                                                                                   of the time      possible
                                                                                                                 delay betweenduetwo
                                                                                                                                  to
                         the  fact  that  there can  be a  high  probability     of short pulses which    are   not suitable
                          time signals requires fulfilling the condition of their correlability. The notion of correlability for de-
                         termining     the material
                          of input signals    means velocity
                                                      that the by   the correlation
                                                               calculated              method due
                                                                              cross-correlation     to their
                                                                                                function   willshort duration.
                                                                                                                 allow         Such
                                                                                                                        determining
                         aacase   can be described
                            dependency      between as theimpulsive
                                                            analyzednoise      in terms
                                                                         signals.        of signal processing
                                                                                   The correlability  is strictly notions.
                                                                                                                  connected with a
                                In Mosorov
                          problem             et al. 2020
                                      determining          [11],window
                                                     the time    the algorithm      to determine
                                                                            to the calculation of the
                                                                                                   the time   interval is proposed.
                                                                                                        cross-correlation   function.
                         ItThis
                            includes    the following   steps:
                                 means that the choice of the time window depends on the considered signal patterns.
                         1.Hence,   the velocity
                               Choose     arbitrarilycalculation     shouldvalue
                                                         the threshold          consider    choosing the appropriate
                                                                                      s0 experimentally                    signal
                                                                                                             and an interval        patterns of
                                                                                                                                of confidence
                          timemseries    xi,j (nTm)0 and
                                  0T, where                yi,j (arbitrarily
                                                      is the      nT ), i.e., determination
                                                                                chosen number     of aoftime  window.
                                                                                                          frames.  It is assumed that the
                                Since the values
                               threshold       plug propagation
                                                        must be above    time    is known
                                                                               the  not constant,     the choice of a time window with
                                                                                             noise level.
                          a fixed   length    M   is
                         2. Waiting when the signal ,not  possible.     (In   )
                                                                            addition,
                                                                                 or ,    (    )
                                                                                          plugs   appear
                                                                                                is is below irregularly,  therefore,
                                                                                                              the threshold     value sthere
                                                                                                                                         0.   is
                         3. If the signal value , ( ) or , ( ) exceeds the threshold value s0 during the con-
                          a  problem:    How       to automatically        determine       the  time   of  appearance    of the  plug    and its
                          finishing.
                               fidence Hence,
                                          intervalthe[Nalgorithm
                                                          bT, NbT +for     determining
                                                                        m0T],    then moment the velocity
                                                                                                   NbT − mbased     on Equation
                                                                                                             0T determines           (1) should
                                                                                                                               the beginning
                          automatically
                               of the signal  detect   the moments of plug occurrence.
                                                  pattern.
                         4. If the signal value thresholding
                                Applying       simple
                                                             , (    )          to determine
                                                                              , (    ) is below  the
                                                                                                   thetime  moments
                                                                                                         threshold      is not
                                                                                                                    values       possible due
                                                                                                                            0 during the in-
                          to the
                               terval of confidence [NeT, NeT + m0T], then moment NeT + m0T is chosen assuitable
                                   fact that    there  can   be   a high    probability     of  short   pulses  which   are  not    the endfor
                                                                                                                                             of
                          determining      the
                               the signal pattern.material    velocity     by   the  correlation    method     due to  their  short   duration.
                         5.SuchThe
                                 a case  cancorrelation
                                     cross       be described      as impulsive
                                                              function                noise as
                                                                           is calculated     in follows:
                                                                                                 terms of signal processing notions.
                                In Mosorov et al. 2020 [11], the algorithm to determine the time interval is proposed.
                          It includes the following steps:
                         1.     Choose arbitrarily the threshold value s0 experimentally and an interval of confidence
                                m0 T, where m0 is the arbitrarily chosen number of frames. It is assumed that the
                                threshold values must be above the known noise level.
                         2.     Waiting when the signal xi,j (nT ) or yi,j (nT ) is below the threshold value s0 .
                         3.     If the signal value xi,j (nT ) or yi,j (nT ) exceeds the threshold value s0 during the
                                confidence interval [Nb T, Nb T + m0 T], then moment Nb T − m0 T determines the
                                beginning of the signal pattern.
                         4.     If the signal value xi,j (nT ) or yi,j (nT ) is below the threshold value s0 during the
                                interval of confidence [Ne T, Ne T + m0 T], then moment Ne T + m0 T is chosen as the end
                                of the signal pattern.
                         5.     The cross correlation function is calculated as follows:

                                                                        Ne + M0
                                                       1
                                  Ri,j (kT ) =
                                                 Ne − Nb + 2M0             ∑       xi,j (nT )yi,j ((n − k) T ), k = . . . , −1, 0, 1 . . . (3)
                                                                      n= Nb − M0
Plug Regime Flow Velocity Measurement Problem Based on Correlability Notion and Twin Plane Electrical Capacitance Tomography: Use Case - MDPI
,   (   )=             ∑               ,   (   )   ,   (( − ) ),    = ⋯ , −1,0,1 …        (3)

                         6.    Finally, the velocity of the flow in (I,j) pixel is calculated according to Equation (2),
                               during the interval [NeT, NeT + m0T]:
Sensors 2021, 21, 2189                                                                                                           5 of 13
                                                                      , |[   ,           ]   =                                      (4)

                         7.   Return to step 2 where n = Ne, i.e., starting from the moment of the interval when the
                              signal , (         )      , (       ) is below the threshold value s0.
                         6.   Finally, the velocity of       the flow in (I,j) pixel is calculated according to Equation (2),
                              In practice,
                              during         it is unnecessary
                                      the interval     [Ne T, Ne Tto  +m combine
                                                                            0 T]:
                                                                                        two signal values , or , and it has to
                         choose one of them. In the article, such a basic signal is , . However, there are no studies
                         showing how the rightness of the velocity     Vi,j|[ Ne T,calculation     willd depend on the choice of s0 and
                                                                                    Ne T + M0 T ] =                                  (4)
                         m0 parameters. Therefore, the next section describes the carried            τ0      out studies regarding the
                         estimation of influence of these parameters on the rightness of velocity calculation and
                         7.   Return
                         finding      to step
                                 the best        2 where
                                            values   for the n =calculation
                                                                 Ne , i.e., starting      from the moment of the interval when the
                                                                                  procedure.
                              signal xi,j ( Ne T ) or yi,j ( Ne T ) is below the threshold value s0 .
                         3. Experiments
                              In practice,and    unnecessary to combine two signal values xi,j or yi,j and it has to
                                           it isDiscussion
                         choose
                              The measurementthe
                                 one of them.   In    article,
                                                   is made     such
                                                            with  ana electrical
                                                                      basic signal  is xi,j . However,
                                                                                 capacitance           there are
                                                                                                tomography   unitno
                                                                                                                  bystudies
                                                                                                                      meas-
                         showing how the rightness of the velocity calculation will depend on the choice of s0 and
                         uring the value of the electric capacity between successive pairs of electrodes Nel. Elec-
                         m0 parameters. Therefore, the next section describes the carried out studies regarding the
                         trodes are placed around the pipe at an equal distance, as shown in Figure 3. The meas-
                         estimation of influence of these parameters on the rightness of velocity calculation and
                         urement output has the form of the vector where its elements stand for consecutive elec-
                         finding the best values for the calculation procedure.
                         trode pairs:
                         3. Experiments and Discussion V = [ 1.0073 … 1.0114].
                              The measurement is made with an electrical capacitance tomography unit by measuring
                              The number of electrode pairs Nel can be calculated as follows:
                         the value of the electric capacity between successive pairs of electrodes Nel . Electrodes are
                                                                             (      )
                         placed around the pipe at an equal distance, =as shown      in Figure 3. The measurement output(5)
                         has the form of the vector where its elements stand for consecutive electrode pairs:
                         where ne is the number of electrodes.                            
                                                        V = of1.0073
                              In our experiment, the number               . . .ne is
                                                                 electrodes       1.0114     .
                                                                                     8 and hence,  the number of electrode
                         pairs Nel is 28.

                                                       (a)                                                   (b)
                         Figure 3.
                         Figure 3. Electrical
                                   Electrical capacitance
                                              capacitance tomography
                                                           tomographymeasurement
                                                                         measurementunitunitand
                                                                                             andsensor
                                                                                                 sensorplacing
                                                                                                         placingon
                                                                                                                 onthe
                                                                                                                    thepipe
                                                                                                                        pipe(a).
                                                                                                                             (a).
                         On the right hand,   the cross-section of the pipe is presented with electrodes (e1, e2 …) mounted
                         On the right hand, the cross-section of the pipe is presented with electrodes (e1, e2 . . . ) mounted
                         around the pipe (b).
                         around the pipe (b).

                               The number
                              The  process of
                                           of obtaining  a tomogram,
                                              electrode pairs           thecalculated
                                                               Nel can be    graphicalasrepresentation
                                                                                          follows:     of reconstructed
                         tomographic data requires several stages, which include: Data acquisition from tomo-
                         graphic sensors, device calibration (determining     − 1range
                                                                       ne (ne the )     of the minimum and maximum
                                                               Nel =                                                  (5)
                         value for the process, normalization of    measurement
                                                                            2        data, preparation of the sensitivity
                         matrix, and reconstruction of the tomographic image).
                         where ne is the number of electrodes.
                               One of the ECT data processing stages is to perform an image reconstruction algo-
                              In our experiment, the number of electrodes ne is 8 and hence, the number of electrode
                         rithm that allows preparing the pipe cross-section       view. There are many such methods.
                         pairs Nel is 28.
                         However, despite the lower quality, the most frequently used method in real-time systems
                              The process of obtaining a tomogram, the graphical representation of reconstructed to-
                         mographic data requires several stages, which include: Data acquisition from tomographic
                         sensors, device calibration (determining the range of the minimum and maximum value
                         for the process, normalization of measurement data, preparation of the sensitivity matrix,
                         and reconstruction of the tomographic image).
                              One of the ECT data processing stages is to perform an image reconstruction algorithm
                         that allows preparing the pipe cross-section view. There are many such methods. However,
                         despite the lower quality, the most frequently used method in real-time systems is the linear
SensorsSensors
        2021, 21, x FOR
               2021, 21, x PEER REVIEW
                           FOR PEER REVIEW                                                                                                                 6 of 13 6 of 13
      Sensors 2021, 21, 2189                                                                                                                              6 of 13

                                        is the
                                    is the     linear
                                            linear     backprojection
                                                     back     projection (LBP)
                                                                           (LBP) method
                                                                                    method[25,26]
                                                                                               [25,26]due
                                                                                                        dueto to
                                                                                                              thethelowlow
                                                                                                                         costcost
                                                                                                                              of computing.
                                                                                                                                   of computing.The The
                                       back
                                        methodprojection
                                                  of linear(LBP)
                                                             back  method
                                                                    projection[25,26]
                                                                                 is a  due to the
                                                                                      method        low costtoofthe
                                                                                                 belonging         computing.
                                                                                                                      group   of  The method
                                                                                                                                 inverse         of
                                                                                                                                           problem
                                    method of linear back projection is a method belonging to the group of inverse problem
                                       linear  back
                                        solution     projection
                                                  [27],  allowing is to
                                                                      a method
                                                                        calculatebelonging    to the
                                                                                    more result      group
                                                                                                   data  thanofthe
                                                                                                                 inverse
                                                                                                                     amountproblem   solution
                                                                                                                              of input   data. [27],
                                                                                                                                               This
                                    solution [27], allowing to calculate more result data than the amount of input data. This
                                       allowing   to calculate
                                        method includes         more result
                                                             calculating       data than
                                                                           all points      thetomogram
                                                                                       of the   amount ofusing
                                                                                                            input thedata. This method
                                                                                                                        so-called         includes
                                                                                                                                   sensitivity  ma-
                                    method     includes
                                       calculating   all   calculating
                                                         points  of  the   all pointsusing
                                                                         tomogram       of the
                                                                                             thetomogram
                                                                                                  so-called   using thematrix.
                                                                                                             sensitivity    so-called
                                                                                                                                    Thissensitivity
                                                                                                                                         structure   ma-
                                        trix. This structure is based on a predefined N-element image table, where each element
                                    trix.
                                       is  This structure
                                          based   on a to
                                        corresponds
                                                              is based
                                                        predefined        on ofa predefined
                                                                       N-element
                                                          the distribution
                                                                                                 N-element
                                                                                     imagecapacitance
                                                                                  electric  table,  where   eachimage
                                                                                                          between  elementtable, where
                                                                                                                             corresponds
                                                                                                                     two sensor
                                                                                                                                          each
                                                                                                                                  electrodes.
                                                                                                                                                 element
                                                                                                                                             to Part
                                                                                                                                                the
                                    corresponds
                                       distribution  to  the  distribution
                                                       of electric
                                        of the sensitivity   matrix capacitanceof electric
                                                                       with examplebetween  capacitance
                                                                                       pairs two            between
                                                                                                   sensor electrodes.
                                                                                              of electrodes              two
                                                                                                              are shownPart   sensor   electrodes.
                                                                                                                                       sensitivity Part
                                                                                                                                of the 4.
                                                                                                                            in Figure
                                    of matrix
                                       the sensitivity     matrix
                                                with example     pairswith  example pairs
                                                                         of electrodes          of electrodes
                                                                                         are shown    in Figure 4.are shown in Figure 4.

            Figure 4. Visualization of sensitivity matrix parts, an example. The red color stands for amplified values between pairs of
            electrodes.
           Figure
      Figure       4. Visualization
              4. Visualization       of sensitivity
                                 of sensitivity     matrix
                                                 matrix    parts,an
                                                         parts,   anexample.
                                                                     example. The
                                                                              The red
                                                                                  red color
                                                                                      color stands
                                                                                             standsfor
                                                                                                    foramplified
                                                                                                        amplifiedvalues between
                                                                                                                   values       pairs
                                                                                                                           between  pairs of
           of electrodes.
      electrodes.                              The sensitivity matrix is a graphical representation of the electric capacitance distri-
                                        bution between the electrodes, which results in the amplification effect of the values be-
                                               The sensitivity matrix is a graphical representation of the electric capacitance dis-
                                            The the
                                        tween     sensitivity     matrix
                                                        electrodes,   whileisweakening
                                                                                a graphicaltherepresentation
                                                                                                    signal’s influence    of theon electric
                                                                                                                                     the central capacitance
                                                                                                                                                     area of thedistri-
                                       tribution between the electrodes, which results in the amplification effect of the values
                                        analyzed
                                    bution     betweenplane.theTheelectrodes,
                                                                    size of eachwhichsensitivity
                                                                                             resultsmatrix
                                                                                                        in    image
                                                                                                            the         is equal to 32
                                                                                                                   amplification             × 32 of
                                                                                                                                          effect   pixels
                                                                                                                                                       theofof  the
                                                                                                                                                             values
                                       between the electrodes, while weakening the signal’s influence on the central area                                      the be-
                                        same
                                    tween        dimensions,
                                               the electrodes,    as  a  result,
                                                                     while        a reconstructed
                                                                               weakening         the    image.
                                                                                                      signal’s      The    output
                                                                                                                    influence          is obtained
                                                                                                                                     on32the           thanks
                                                                                                                                                 central    area  to
                                       analyzed      plane. The size      of each   sensitivity    matrix    image     is equal to          × 32   pixels of   theof the
                                        Equation
                                    analyzed          (6):
                                                   plane. Theassize       of each    sensitivity matrix
                                       same dimensions,              a result,   a reconstructed       image.image          is equal
                                                                                                                   The output             to 32 × 32
                                                                                                                                      is obtained         pixels
                                                                                                                                                      thanks     to of the
                                    same      dimensions,
                                       Equation      (6):        as a result, a reconstructed      = × image. ∗        The output is obtained thanks             (6)     to
                                    Equation (6):                                  I MG Mx1 = S−      1
                                                                                                      ×M    ∗  C Lx1                                            (6)
                                        where M is the ECT image resolution, L is theLvector                    size of one ECT measurement, CLx1 is
                                       where
                                        the ECT  M is  the ECT image
                                                     measurement          resolution,
                                                                        vector,        L is
                                                                                 SL×M is     thesensitivity
                                                                                           the    vector
                                                                                                     = × ∗of one ECT
                                                                                                           size matrix,    and measurement,           CLx1 is the
                                                                                                                                   IMG is the tomographic               (6)
                                       ECT
                                        image. measurement vector, SL×M is the sensitivity matrix, and IMG is the tomographic image.
                                    where ThereM   is the
                                               There     areECT
                                                       are  many
                                                             many image
                                                                   numerical resolution,
                                                                     numerical   problems
                                                                                   problems  Lassociated
                                                                                               isassociated
                                                                                                   the vectorwith  size
                                                                                                                    thethe
                                                                                                                 with     ofsensitivity
                                                                                                                               one ECT
                                                                                                                          sensitivity        measurement,
                                                                                                                                          matrix.   This
                                                                                                                                              matrix.          ma-CLx1 is
                                                                                                                                                           matrix
                                                                                                                                                        This
                                    theistrix
                                           notisanot
                                          ECT     square    matrix.
                                                  measurement        Its  dimensions
                                                                       vector,   S      depend
                                                                                        is  the     on the
                                                                                                  sensitivitynumber      of
                                                                                                                   matrix,   independent
                                                       a square matrix. Its dimensions depend on the number of independent meas-
                                                                                   L×M                                           and    IMG     measurements
                                                                                                                                                is the   tomographic
                                       and
                                    image.    from aand
                                        urements        significantly   greater number
                                                            from a significantly      greaterof image
                                                                                                 number  points.
                                                                                                            of imageHence,     it is impossible
                                                                                                                          points.     Hence, it isto    calculate
                                                                                                                                                     impossible
                                       the   inverse
                                        to There
                                            calculate    of
                                                     arethe matrices
                                                              inverse
                                                           many         and   in various
                                                                        of matrices
                                                                    numerical               known
                                                                                       and in various
                                                                                   problems           methods
                                                                                                  associatedknown    the
                                                                                                                    with  pseudo-reverse
                                                                                                                        methods                   of the
                                                                                                                                       the pseudo-reverse
                                                                                                                            the sensitivity        matrix. image  of ma-
                                                                                                                                                               This
                                       reconstruction
                                        the   image         is used.
                                                       reconstruction The  isproblem
                                                                              used.    is
                                                                                     The   considered
                                                                                           problem    is  as   ill-conditioned.
                                                                                                          considered       as          In the
                                                                                                                                ill-conditioned.LBP  algorithm,
                                                                                                                                                      In  the  LBP
                                    trix is not a square matrix. Its dimensions depend on the number of independent meas-
                                       the   pseudo-inverse
                                        algorithm,      the       sensitivity matrix
                                                             pseudo-inverse               is computedisthrough
                                                                                 sensitivity                             its transpositions.
                                    urements   After
                                                    and   from    a significantly     greatermatrix
                                                                                                  number      computed
                                                                                                                of image      through
                                                                                                                               points.     its transpositions.
                                                                                                                                          Hence,      it is impossible
                                               After performing the image reconstruction algorithm, the cross-section image
                                                       performing     the   image   reconstruction       algorithm,       the   cross-section      image is  is ob-
                                                                                                                                                                ob-
                                    to tained
                                       calculate       the inverse
                                                 (Figure    5a).       of matrices and in various known methods the pseudo-reverse of
                                        tained (Figure 5a).
                                    the image reconstruction is used. The problem is considered as ill-conditioned. In the LBP
                                    algorithm, the pseudo-inverse sensitivity matrix is computed through its transpositions.
                                         After performing the image reconstruction algorithm, the cross-section image is ob-
                                    tained (Figure 5a).

                                                            (a)                                                (b)
                                       Figure 5.
                                       Figure   5. Reconstructed
                                                   Reconstructed ECT
                                                                 ECT image
                                                                      image (a)
                                                                             (a) and
                                                                                 and the
                                                                                      the photography
                                                                                          photographythat
                                                                                                        thatpresents
                                                                                                             presentsthe
                                                                                                                      thematerial
                                                                                                                          materialinside
                                                                                                                                   insidethe
                                                                                                                                          the
                                       pipe. (b) The red color shows the material inside the pipe, and the black color represents an empty
                                       pipe. (b) The red color shows the material inside the pipe, and the black color represents an empty
                                       part of the pipe. On the right hand, (b) the image of the pipe section is presented.
                                       part of the pipe. On the right hand, (b) the image of the pipe section is presented.
                                                         (a)                                                      (b)
                                    Figure 5. Reconstructed ECT image (a) and the photography that presents the material inside the
                                    pipe. (b) The red color shows the material inside the pipe, and the black color represents an empty
                                    part of the pipe. On the right hand, (b) the image of the pipe section is presented.
stances in the transport route or the distribution of material concentration in the cross-
                         section of the tank during industrial processes. Despite the low quality, the accuracy of
                         concentration changes is determined at the level of about 3%.
                               The research was based on a two-phase pneumatic gas-bulk material flow. The
Sensors 2021, 21, 2189   equipment configuration includes two 100 L tanks placed one below the other,                            7 of 13a rotary
                         feeder, a pump, piping in the horizontal section (2 × 7 m ) and vertical (approx. 3 m), as
                                                                                                      2

                         shown in Figure 6. Stainless steel piping was used with a diameter of Ø1 = 57 mm (inner).
                         In the  uppertopart
                               Thanks        ECT,ofusing
                                                    the installation,     there itisisa possible
                                                            the LBP method,              filter that
                                                                                                   to separates
                                                                                                      determine the  the phase
                                                                                                                          number  ofofthe loose
                         material   from    the  gaseous     medium.     A  polymer       granulate     with   a size
                          substances in the transport route or the distribution of material concentration in the cross-of  3 mm     × 3 mm × 1
                         mm    and
                          section of athe
                                        dielectric
                                           tank duringpermittivity
                                                          industrial of   approximately
                                                                       processes.   Despite the 2.1 low
                                                                                                     wasquality,
                                                                                                            adopted the as   the transported
                                                                                                                         accuracy     of
                         substance
                          concentration(Jaworski
                                           changesand     Dyakowski
                                                     is determined        2001)
                                                                      at the     [28].
                                                                             level  of about 3%.
                               The  research
                               As seen          was based
                                           in Figure         on tomographic
                                                        6, the  a two-phase pneumatic
                                                                                 sensors gas-bulk
                                                                                              have been  material  flow.inThe
                                                                                                             installed       theequip-
                                                                                                                                  vertical sec-
                          mentEach
                         tion.  configuration
                                       of them includes
                                                  consiststwo     100 Lelectrodes
                                                              of eight   tanks placed      one below
                                                                                        placed          the other,
                                                                                                  at equal           a rotary
                                                                                                              intervals        feeder,
                                                                                                                          on the     outer wall
                          a pump,  piping    in the horizontal   section (2 ×  7 m  2 ) and vertical (approx. 3 m), as shown in
                         (electrode width 30 mm). The distance between them is constant and was d = 130 mm. The
                          Figure 6. Stainless steel piping was used with a diameter of Ø1 = 57 mm (inner). In the
                         measurement was made using the mentioned tomographic unit with a sampling fre-
                          upper part of the installation, there is a filter that separates the phase of the loose material
                         quency
                          from the of   100 Hz.
                                   gaseous         The gas
                                               medium.         velocity
                                                          A polymer       (for anwith
                                                                       granulate     empty      pipe)
                                                                                           a size       can ×vary
                                                                                                  of 3 mm       3 mm from    1.0 to
                                                                                                                        × 1 mm     and 5 m/s by
                         inserting   three    different   nozzles   and   the  pressure       control.   It follows
                          a dielectric permittivity of approximately 2.1 was adopted as the transported substance      from   the   above   that
                         the  highest
                         (Jaworski   and achievable
                                           Dyakowskimaterial       velocity is 5 m/s.
                                                         2001) [28].

                         Figure  6. Pneumatic
                         Figure 6.  Pneumaticmaterial
                                              material  conveying
                                                      conveying    system
                                                                system    schema
                                                                       schema withwith tomography
                                                                                  tomography       sensors
                                                                                             sensors       andcam-
                                                                                                     and video video cam-
                         era
                         era placement.
                             placement.

                              As seen in Figure 6, the tomographic sensors have been installed in the vertical
                         section. Each of them consists of eight electrodes placed at equal intervals on the outer
                         wall (electrode width 30 mm). The distance between them is constant and was d = 130 mm.
                         The measurement was made using the mentioned tomographic unit with a sampling
                         frequency of 100 Hz. The gas velocity (for an empty pipe) can vary from 1.0 to 5 m/s by
Sensors
Sensors       21, x21,
        2021,2021,  FOR2189
                          PEER REVIEW                                                                                                       8 of 13 8 of 13

                                       inserting
                                           As part  three  different
                                                       of the         nozzles the
                                                               experiment,       andacquisition
                                                                                       the pressureofcontrol.     It followsvalues
                                                                                                          measurement          from thewasabove  that
                                                                                                                                            performed    us-
                                       the highest achievable material velocity is 5 m/s.
                                    ing a tomographic sensor from two planes, and then the data were processed according
                                             As part of the experiment, the acquisition of measurement values was performed
                                    tousing
                                        the steps    shown in sensor
                                                a tomographic     Figurefrom
                                                                           1. The twomain    parts
                                                                                        planes,  andofthen
                                                                                                         thethe
                                                                                                              algorithm
                                                                                                                 data were  are   the plugs
                                                                                                                               processed     identification
                                                                                                                                           according
                                    section     and the
                                       to the steps        cross-correlation
                                                       shown   in Figure 1. The module.
                                                                                    main partsWhen        the measurement
                                                                                                  of the algorithm     are the plugssamples   arrive at the
                                                                                                                                       identification
                                    plug    identification     module,    the   s
                                       section and the cross-correlation module. When the measurement samples arrive at the Then,
                                                                                  0 threshold    is used    to determine      if the  plug  appears.
                                    if plug
                                       the higher      material
                                              identification       concentration
                                                               module,  the s0 thresholdis recorded
                                                                                              is used in   the periodif greater
                                                                                                        to determine      the plug than     m0 (confidence
                                                                                                                                      appears.  Then,
                                       if the higher
                                    threshold),      thematerial  concentration
                                                          plug beginning             is recorded
                                                                               is indicated.     Thein the
                                                                                                       opposite      greater thanismperformed
                                                                                                            periodprocedure            0 (confidencefor plug
                                       threshold),    the plug   beginning   is  indicated.   The   opposite    procedure
                                    end determination. Figure 7 presents example graphs showing changes in the concentra-    is  performed   for plug
                                       endvalue
                                    tion     determination.
                                                    in the pipe Figure 7 presents example
                                                                   cross-section       based on graphs
                                                                                                    a 15showing
                                                                                                           × 15 pixelchanges    in the concentration
                                                                                                                         of tomographic       images from
                                       value in the pipe cross-section based on a 15 × 15 pixel of tomographic images from two
                                    two planes and the result of the discussed plug identification algorithm to parameters s0,
                                       planes and the result of the discussed plug identification algorithm to parameters s0 , m0 .
                                    m0As. As   can be seen, the pattern of the time range (dotted red line) covers the time range of
                                           can be seen, the pattern of the time range (dotted red line) covers the time range of plug
                                    plug    appearance
                                       appearance      (blue(blue   line) completely.
                                                             line) completely.

     Figure 7. Plug
         Figure      identification
                7. Plug  identification algorithm
                                          algorithmoutput
                                                      output for severalcases.
                                                             for several cases.The
                                                                                Thesignal
                                                                                     signal  from
                                                                                          from  thethe second
                                                                                                    second planeplane   is presented
                                                                                                                  is presented         with an
                                                                                                                                 with an
     orange color.
         orange    TheThe
                color.   dotted
                            dotted lines
                                     linespresent
                                           presentranges   of computed
                                                    ranges of computedpatterns:
                                                                          patterns:
                                                                                 (a)(a) Shows
                                                                                     Shows      patterns
                                                                                             patterns    where
                                                                                                      where       s0 =and
                                                                                                            s0 = 0.1   0.1mand   m0 = 20; (b–d)
                                                                                                                            0 = 20; (b–d)
     shows  patterns
         shows        with
                patterns    s0 =s00.6
                          with        and
                                   = 0.6 andm0m=0 30.
                                                  = 30.

                                            Thediscussed
                                          The   discussed algorithm,
                                                            algorithm, together withwith
                                                                         together    arbitrarily set parameters,
                                                                                          arbitrarily            apparently
                                                                                                        set parameters,     indicates indi-
                                                                                                                         apparently
                                      the correct  speed  estimates. However,   there are  situations  when  the obtained
                                    cates the correct speed estimates. However, there are situations when the obtained    values  arevalues
                                      inconsistent with reality. One of the reasons for the occurrence of inappropriate estimates
                                    are inconsistent with reality. One of the reasons for the occurrence of inappropriate esti-
                                      is the reliance on incorrect algorithm arguments. The analysis of the plug registration
                                    mates   is the reliance on incorrect algorithm arguments. The analysis of the plug registra-
                                      threshold s0 impact on the plugs identification algorithm effectiveness was performed.
                                    tion
                                      Thethreshold     s0 impactinon
                                           results are presented        the 8a,b.
                                                                     Figure  plugs identification algorithm effectiveness was per-
                                    formed. Using the parameters s0 = 0.35inand
                                               The  results are  presented      Figure
                                                                                    m0 =8a,b.
                                                                                          5, an additional wrong plug is detected
                                      (Figure 8a). It results from the occurrence of situations where the instantaneous fluctua-
                                      tions of the concentration level exceed the indicated threshold and remain above m0 = 5
                                      units. In such a situation, a signal pattern is generated, where the maximum correlation
                                      reaches the value of n = 21 (wrong velocity value eq. 0.62 m/s). The plug identification
                                      algorithm requires proper parameter setting that would enable the correct speed estima-
                                      tion. The authors conducted a series of experiments that allow indicating the right values.
                                      The results are shown in Figure 9.
SensorsSensors 2021,
            2021, 21,    21, 2189
                      x FOR    PEER REVIEW                                                                                                             9 of 139 of 13

                                                            (a)

                                                                                 (a)

                                                            (b)
Material velocity estimation chart in relation to the parameter s0 for each plug. The different colors represent
 0: (a) Full data, (b) removed data for s0 = 0.35 m, and plug number 9.

                          Using the parameters s0 = 0.35 and m0 = 5, an additional wrong plug is detected (Fig-
                   ure 8a). It results from the occurrence of situations where the instantaneous fluctuations
                   of the concentration level exceed the indicated threshold and remain above m0 = 5 units.
                   In such a situation, a signal pattern is generated, where the maximum correlation reaches
                   the value of n = 21 (wrong velocity value eq.                    (b)0.62 m/s). The plug identification algorithm
                 8.requires
         FigureFigure
                    Material     proper
                                 velocity
                       8. Material
                                            parameter
                                           estimation
                                     velocity   estimation
                                                           setting
                                                         chart
                                                           chart
                                                                       that would
                                                                ininrelation
                                                                      relation toto the
                                                                                          enable sthe
                                                                                        parameter
                                                                                    the parameter  0s0for
                                                                                                         correct
                                                                                                        for   each
                                                                                                           each
                                                                                                                    speed
                                                                                                                   plug.
                                                                                                                 plug.
                                                                                                                            estimation.
                                                                                                                       TheThe  different
                                                                                                                           different
                                                                                                                                             Therepresent
                                                                                                                                         colors
                                                                                                                                     colors represent
         parameter authors
              parameter         conducted
                     s0: (a)s0Full
                              : (a) data,data,
                                    Full  (b) (b)a removed
                                                   seriesdata
                                               removed     ofdata
                                                               experiments
                                                                  fors0s0= =0.35
                                                                for           0.35m,that
                                                                                   m,and  allow
                                                                                      and plug   indicating
                                                                                               number
                                                                                          plug number    9.9.     the right values. The re-
                   sults are shown in Figure 9.
                                                  Using the parameters s0 = 0.35 and m0 = 5, an additional wrong plug is detected (Fig-
                                           ure 8a). It results from the occurrence of situations where the instantaneous fluctuations
                                           of the concentration level exceed the indicated threshold and remain above m0 = 5 units.
                                           In such a situation, a signal pattern is generated, where the maximum correlation reaches
                                           the value of n = 21 (wrong velocity value eq. 0.62 m/s). The plug identification algorithm
                                           requires proper parameter setting that would enable the correct speed estimation. The
                                           authors conducted a series of experiments that allow indicating the right values. The re-
                                           sults are shown in Figure 9.

                                                  (a)                                                             (b)
                    Figure
                Figure  9. (a)9.Analysis
                                 (a) Analysis      of the influence
                                           of the influence           of parameters
                                                            of parameters  s0 , m0 on thes0number
                                                                                           , m0 on of
                                                                                                    theidentifiable
                                                                                                         number of     identifiable
                                                                                                                    plugs             plugs (red
                                                                                                                           (red line—expected    value);
                    line—expected
                (b) absolute              value);
                              error for the         (b) absolute
                                              correlation function error for thetocorrelation
                                                                   with respect     parameters s0function    withof
                                                                                                  , m0 . The color  respect   to parameters
                                                                                                                      circles indicates         s0,of m0
                                                                                                                                        the value
                    m0. The(the
                parameter      color
                                  scaleofiscircles
                                            shown indicates
                                                    on the rightthe value of m0 parameter (the scale is shown on the right hand).
                                                                 hand).

                                                                        (a)                                                          (b)
                                          Figure 9. (a) Analysis of the influence of parameters s0, m0 on the number of identifiable plugs (red
                                          line—expected value); (b) absolute error for the correlation function with respect to parameters s0,
Sensors 2021, 21, 2189                                                                                           10 of 13

                              The conducted experiments show unequivocally that the use of the right algorithm
                         parameters is crucial when assessing its effectiveness. For the real data series, which shows
                         the passage of 20 plugs (Figure 9a red line), it is noticeable that for the threshold s0 ≥ 0.75
                         the number of identified plugs does not reach the expected value and additionally shows
                         an extreme influence of the m0 parameter. Moreover, for a value of s0 ≤ 1, a similar
                         situation exists.
                              On the other hand, if there is a higher s0 threshold, then a lower m0 parameter should
                         be set to return the required number of plugs. Additionally, reducing the m0 allows
                         identifying more plugs.
                              The second experiment was performed in order to visually estimate the signal time
                         shifts (p) for successive plugs. Then, results were taken as reference values to calculate the
                         root of mean square error (RMSE); Equation (7)) for the algorithm with respect to s0 , m0 .
                                                                   s
                                                                                         2
                                                                       ∑nPn=1 ( p̂ − p(n))
                                                        RMSE =                             ,                         (7)
                                                                                 Pn

                         where Pn is the plug count; n is the plug number; p(n) is the time shift for n-th plug
                         between planes; p̂ is the expected time shift (visual analysis).
                               As a result of the analysis, the diagram (see Figure 9b) was obtained. The reference
                         data (visual analysis) may be inaccurate. Therefore, the values on the graph fluctuate
                         around RMSE = 1. It is noticeable that for low m0 and high s0 values, the estimation error
                         is lower. Based on the above analyses, it can be assumed that the best parameters of the
                         discussed algorithm are s0 = 0.6 and m0 = 10.
                               Another aspect of the algorithm’s efficiency is the speed estimation error based on the
                         correlation result resolution. The maximum speed that can be obtained with the indicated
                         measurement parameters of the system equals n1 = 13 m/s (where n = 1 indicates the
                         number of units of shifting the signals between two measurement planes). The follow-
                         ing values are: n1 = 13 m/s, n2 = 6.5 m/s, n3 = 4.3 m/s, n4 = 3.25 m/s, n5 = 2.6 m/s,
                         n6 = 2.17 m/s, n7 = 1.86 m/s, etc. Hence, the speed estimation error resulting from the
                         erroneous calculation of the maximum value of the correlation increases with the increase
                         in speed, reaching the maximum absolute value equal to v > 13 m/s (100%). In the current
                         experiment, where the velocity of the medium is known, it is expected to obtain the plug
                         velocity v < 6.5 m/s, which gives an error of 51%, and for the value of v = 2.17 m/s (n = 6),
                         respectively 17%.
                               The authors conducted an algorithm accuracy analysis. The first step was to check
                         the changes in the estimated velocity to the parameter s0 . It was expected to show that
                         different velocity estimates were obtained for different threshold values. In the second
                         step, an error resulting from the estimation using of the considered algorithm to visual
                         measurements was demonstrated.
                               The results of the experiments are presented in Figure 10. The top chart (a) shows the
                         relative error from the elaborated algorithm.
                               The analysis proves that for a low threshold (smaller than 0.4) and higher than 0.7
                         results are not stable. Contrary, the center values give constant velocity estimation that
                         allows us to assume this range as correct. On the lower plot, the error rate was calculated
                         based on the reference measurement performed as a visual analysis. Same as above,
                         a threshold greater than 0.4 gives better results. For the discussed case, the average error
                         oscillates around 13.3% and the maximum error reaches up to 33.3%.
The authors conducted an algorithm accuracy analysis. The first step was to check
                                  the changes in the estimated velocity to the parameter s0. It was expected to show that
                                  different velocity estimates were obtained for different threshold values. In the second
                                  step, an error resulting from the estimation using of the considered algorithm to visual
   Sensors 2021, 21, 2189         measurements was demonstrated.                                                      11 of 13
                                       The results of the experiments are presented in Figure 10. The top chart (a) shows the
                                  relative error from the elaborated algorithm.

Sensors 2021, 21, x FOR PEER REVIEW                                                                                                           11 of 13

                                                                      (a)

                                                                       (b)
     Figure   10.10.
         Figure    Plug identification
                     Plug  identificationalgorithm
                                          algorithmthreshold
                                                      thresholdvalue
                                                                value impact  on the
                                                                      impact on  the velocity
                                                                                     velocityestimation
                                                                                               estimationaccuracy.
                                                                                                           accuracy.  The
                                                                                                                    The    plot
                                                                                                                        plot onon
                                                                                                                                thethe
                                                                                                                                    toptop
     (a) (a)
          shows   the relative error  for s0 = 0.7 as a reference (expected value). The relative error based on the visual analysis as a
             shows the relative error for s0 = 0.7 as a reference (expected value). The relative error based on the visual analysis as a
     reference   is shown beneath (b).
         reference is shown beneath (b).

                                         The analysis proves that for a low threshold (smaller than 0.4) and higher than 0.7
                                     4. Conclusions
                                   results   are not stable. Contrary, the center values give constant velocity estimation that
                                   allowsAlthough
                                            us to assumean ECTthistomograph
                                                                    range as correct.
                                                                                 system Oncanthe  lower plot, the
                                                                                              be successfully          errortorate
                                                                                                                 utilized           was calculated
                                                                                                                                determine     a ve-
                                     locityon
                                   based     profile in multiphase
                                                the reference           flow, however,
                                                                   measurement             there still
                                                                                     performed      as is a lack of
                                                                                                       a visual       the efficient
                                                                                                                   analysis.    Same algorithms
                                                                                                                                        as above, a
                                     implemented
                                   threshold          in such
                                                greater    thana 0.4
                                                                  system
                                                                      givesto better
                                                                              calculate  the velocity
                                                                                      results.  For the profile in the right
                                                                                                           discussed     case, way.  It is shown
                                                                                                                                the average      error
                                     that  the  key  factor   is the  choice   of  appropriate   time
                                   oscillates around 13.3% and the maximum error reaches up to 33.3%.   intervals   for  velocity   calculations.
                                     This carried out research concerns the study of dependence between the threshold level
                                   4.parameters
                                      Conclusions  of signal pattern detection and rightness of the calculated velocity values. It is
                                     shown that these threshold level parameters have a major influence on the appropriate
                                     plugAlthough
                                            detection, anasECT
                                                             welltomograph       system can be successfully
                                                                   as on the cross-correlation                     utilized
                                                                                                     result that stands     fortoaccurate
                                                                                                                                  determine      a ve-
                                                                                                                                             veloc-
                                   locity  profile  in  multiphase      flow,   however,    there  still is a lack   of the  efficient
                                     ity estimation. In this article, a use case is presented which proves the relation between          algorithms
                                   implemented      in such
                                     the s0 threshold     and athesystem   to calculate
                                                                     m0 parameter         thein
                                                                                      given   velocity   profile inalgorithm.
                                                                                                 the elaborated       the right way.     It is shown
                                                                                                                                   The larger    s0 ,
                                   that the
                                     then     key factor
                                           smaller  m0 mustis thebechoice   of appropriate
                                                                    to obtain   a valid counttime    intervals
                                                                                               of plugs.        for velocity
                                                                                                           Additionally,    the calculations.
                                                                                                                                 correlation out- This
                                   carried   out determined
                                     put is also  research concerns        the study
                                                                  by a threshold        of For
                                                                                    value.  dependence
                                                                                                the range between
                                                                                                            of s0 = ,
                                                                                                                              threshold     level pa-
                                                                                                                                  the algorithm
                                   rameters
                                     gives theoflowest
                                                  signalerror
                                                            pattern   detection
                                                                 referring  to theand    rightness
                                                                                    visual  analysis.of the calculated velocity values. It is
                                   shownAlthough
                                             that thesethethreshold
                                                           conducted level
                                                                        study parameters
                                                                                was obtainedhave
                                                                                               for gravity  flowinfluence
                                                                                                      a major    experiments  onand
                                                                                                                                  the pneumatic
                                                                                                                                       appropriate
                                     conveying,
                                   plug           otherasstudies
                                          detection,       well as can
                                                                     onalso
                                                                          thebecross-correlation
                                                                                 valuable for flowsresult
                                                                                                      with different   tomographic
                                                                                                             that stands              modalities.
                                                                                                                            for accurate     velocity
                                   estimation. In this article, a use case is presented which proves the relation between the s0
                                     Author Contributions: Investigation, V.M. and G.R.; project administration, D.S.; resources, V.M.;
                                   threshold    and the m0 parameter given in the elaborated algorithm. The larger s0, then
                                     writing—original draft, V.M. and G.R. All authors have read and agreed to the published version of
                                   smaller   m 0 must be to obtain a valid count of plugs. Additionally, the correlation output
                                     the manuscript.
                                   is also determined by a threshold value. For the range of s0 = , the algorithm gives
                                        lowestThis
                                     Funding:
                                   the               work
                                                 error    was partly
                                                       referring     funded
                                                                 to the      by the
                                                                         visual     Intelligent Development Operational Program 2014–
                                                                                 analysis.
                                     2020 co-financed by the European Regional Development Fund, project POIR.04.01.02-00-0089/17-00.
                                         Although the conducted study was obtained for gravity flow experiments and pneu-
                                     Institutional
                                   matic conveying,Review  Board
                                                        other     Statement:
                                                               studies       Not applicable.
                                                                        can also  be valuable for flows with different tomographic
                                   modalities.
                                     Informed Consent Statement: Not applicable.

                                   Author Contributions: Investigation, V.M. and G.R.; project administration, D.S.; resources, V.M.;
                                   writing—original draft, V.M. and G.R. All authors have read and agreed to the published version
                                   of the manuscript.
                                   Funding: This work was partly funded by the Intelligent Development Operational Program 2014-
Sensors 2021, 21, 2189                                                                                                              12 of 13

                                   Data Availability Statement: Not applicable.
                                   Conflicts of Interest: The authors declare no conflict of interest.

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