Prior design for tomographic volume fraction estimation in pneumatic conveying systems from capacitive data

Abstract

Pneumatic conveying systems have become a standard technique for the transport of bulk materials such as powdery or granulates. The spatial dependence of the material density and the stream velocity in such transport systems require a volumetric measurement principle for flow measurement. In this paper we analyse the capability to estimate the volume fraction from capacitive sensing data using electrical capacitance tomography (ECT). In particular, we investigate the capability of back-projection type imaging algorithms. The ill-posed nature of the imaging problem of ECT require the incorporation of prior knowledge in the design of the estimator. We analyse the different flow profiles in pneumatic conveying in order to generate specific sample-based prior information to improve the estimation performance and robustness. We discuss the construction of different linear image reconstruction algorithms and present a framework, which allows a detailed statistical analysis of the estimator performance. Simulation studies show the estimation behaviour of different algorithms with respect to the incorporated prior information. We demonstrate, that the incorporation of specific prior knowledge leads to an improved estimator behaviour; for example, reduced variance and unbiased estimates. We implemented laboratory experiments in order to analyse the presented approach for the application in real pneumatic conveying processes. We demonstrate the improved robust estimation behaviour by means of comparative reconstruction results obtained with different algorithms and priors. Furthermore, the uncertainty of the estimated volume fraction is analysed in steady state conveying processes. Hereby, it is demonstrated, that appropriate prior information improves the estimation performance also for measurements coming from real pneumatic conveying processes, making ECT a suitable tool for the volume fraction estimation in such transport systems.

Keywords

Pneumatic conveying electrical capacitance tomography inverse problem prior design flow measurement

Introduction

Pneumatic conveying is a standard technique for the transport of particulate and granular materials (Klinzing et al., 2011). It is used in various industries including agriculture and food industries; for example, the transport of wheat or flour – or heavy and chemical industries; for example, the transportation of cement, and so forth (Kraume, 2012). In pneumatic conveying, the material is transported by means of a gas stream through a pipe system. The transport process is characterized by means of the spatial material distribution of the particulate material within the transport pipe. Figure 1 sketches different flow regimes for pneumatic conveying and the relation for pressure drop and gas velocity (Jama et al., 2000). For high stream velocities of the transport gas, the material is uniformly distributed over the cross section of the transport pipe. The density of the material stream is low, the flow regime is referred to as dilute or dispersed flow. For decreased gas velocities the flow regime shows a distinct material layer moving at the bottom of the pipe with a dilute phase above. Flow profiles showing this distinct bottom layer are generally referred to as stationary flows. Due to the decreased pressure drop these flow regimes are favourable because of the reduced energy demands. For further decreased gas velocities the material moves by means of a concentrated material bulk.

The efficient operation of a pneumatic conveying systems requires the continuous determination of certain flow parameters such as the volume flow or the mass flow. The volume flow $\overset{\cdot}{V}$ in $m^{3} / s$ in pneumatic conveying systems is determined by (Jaworski and Dyakowski, 2001)

\overset{\cdot}{V} (t) = \int_{Ω_{ROI}} R (x, y, t) v (x, y, t) d Γ,

(1)

where $v (x, y)$ is the spatial velocity in m/s and the dimensionless quantity $R (x, y, t)$ is referred to as density function. $x$ and $y$ are the Cartesian coordinates, the cross section of the transport pipe is referred to as region of interest $Ω_{ROI}$ . The mass flow $\overset{\cdot}{m}$ in $kg / s$ can be obtained by multiplying the integrand of equation (1) with the density of the material $ρ$ in kg/m ³. Following equation (1), the determination of the volume flow requires the simultaneous determination of the velocity and the density function, which represents the flow regime. The spatial dependency of the indivitual quantities and the properties of pneumatic conveying systems are making the measurement of the volume flow $\overset{\cdot}{V}$ and the mass flow $\overset{\cdot}{m}$ a challenging task. Due to the nature of the different flow regimes, scale-based systems can only be applied at the end of the system if possible. Other mechanical measurement principles for inline flow measurements – for example, impact plates (Pugh, 2003) – are causing an additional pressure drop in the system making the conveying process more likely to clog especially for dense flow regimes. A further issue is the highly abrasive nature of particulate flows disabling the use of invasive measurement priciples (Yan and Stewart, 2001).

Assuming a uniform velocity distribution with velocity $\bar{v} (t)$ , equation (1) can be simplified to (Yan and Stewart, 2001)

\overset{\cdot}{V} (t) = \bar{β} (t) \bar{v} (t) A_{Ω_{ROI}},

(2)

where $\bar{β} (t)$ is the normalized integral of the density function $R (x, y, t)$

\bar{β} (t) = \frac{1}{A_{Ω_{ROI}}} \int_{Ω_{ROI}} R (x, y, t) d Γ,

(3)

and $A_{Ω_{ROI}}$ is the cross sectional area. $\bar{β} (t)$ is termed the volume fraction. Note that this simplification is not valid for all flow profiles; for example, for the stationary flow profiles the dilute phase and the dense phase typically have different velocities. However, due to the large ratio between $ρ_{dense}$ and $ρ_{dilute}$ the error made by the simplification given by equation (2) can be considered small.

In this paper, we focus on the determination of $\bar{β} (t)$ from electrical sensing principles. Due to the shortcomings of mechanical measurement systems for inline flow measurements in pneumatic conveying, electrical measurement systems have seen extensive research. Electrical systems maintain sensor effects like electrostatics, attenuation of microwave signals, or capacitive sensing principles to gain information about the material distribution and subsequently about the parameter $\bar{β} (t)$ (Yan and Stewart, 2001). Due to the nonlinear nature of the sensing principles, the determination of $\bar{β} (t)$ from the sensing signals forms a hard estimation problem. Electrical capacitance tomography (ECT) is a tomographic imaging principle that maintains capacitive measurements to provide images for the spatial material distributions (Brandstätter, 2003a; Holder, 2005; Neumayer et al., 2011b), making it a suitable tool for flow measurements in processes with a spatial dependency of the quantities of interest (Plaskowski et al., 1987). It has been suggested by several authors for estimating the material distribution and subsequently the parameter $\bar{β} (t)$ from capacitance measurements (Hunt, 2014; Sun et al., 2008; White, 2002).

The comparatively low permittivity values of the materials used in pneumatic conveying processes (Kraume, 2012) enable the application of linear reconstruction algorithms (Neumayer et al., 2011b). For this reason and since computational costs are an immanent issue for the continuous determination of flow parameter by means of ECT, we investigate the estimation of $\bar{β} (t)$ using different linear back-projection algorithms. We discuss the different flow regimes in pneumatic conveying and show the construction of prior distributions for a robust parameter estimation by means of a sample-based approach. A framework for the evaluation of estimation algorithms for $\bar{β} (t)$ based on tomographic principles is presented. The performance of different estimators is analysed with respect to the incorporated prior information. We analyse the estimation behaviour for sample-based prior distributions, which were suggested for the reconstruction of arbitrary material distributions (Zangl et al., 2007).This results are compared with the estimation performance achieved with prior distributions, which we specifically designed for the reconstruction of flow patterns in pneumatic conveying systems.

This work is a technically extended version of the conference paper (Suppan et al., 2018) the authors presented at the $9^{th}$ World Congress on Industrial Process Tomography, where comprehensive statistical analyses of different estimators were given. The results of these simulations are summarized briefly. The technical novelty of the research presented in this paper is given by the following points:

Extension of the sample-based prior presented in Suppan et al. (2018) to improve the robustness of the volume fraction estimation approach in real pneumatic conveying processes by means of the implementation of robust priors.

Demonstration of the improved estimation behaviour by means of comparative reconstruction results from lab experiments obtained with different priors.

Uncertainty analysis of the estimation result for the application of the presented approach in real pneumatic conveying processes. Hereby, steady state flow conditions are generated on a test rig.

For the application in real pneumatic conveying systems, the presented sample-based prior approach is extended in order to improve the robustness of the estimates with respect to variations in the flow profiles. The reconstruction behaviour of different estimators and priors is analysed by means of a time varying horizontal conveying process. The necessity of a proper robust prior is addressed when it comes to the reconstruction of material distributions in real conveying processes. The uncertainty of different estimators and different priors is analysed in order to show the improved estimation performance of the presented approach. This analysis is implemented for steady state conveying processes, where the material stream has a constant mass flow rate with a constant velocity and therefore also a constant volume fraction. For the analysis of the uncertainty the standard deviation of the estimated volume fraction is calculated for stationary flow conditions. These experimental results are compared with the findings of the simulation study.

This paper is structured as follows. In Section II, we will briefly discuss ECT. In Section III, we will present the design of the sample-based prior and demonstrate the behaviour of different prior distributions in simulation studies. The laboratory experiments are presented in Section IV.

ECT

In this section, we briefly discuss the sensing principle and the simulation model of ECT and the inverse problem for image reconstruction. Figure 2 depicts the scheme of an ECT sensor. The sensor consists of several electrodes, a non-conductive pipe and a shield. The electrodes are placed on the outer circumference of the pipe. The capacitances between the electrodes are influenced by the dielectric properties of the materials inside the pipe (Brandstätter et al., 2003b; Holder, 2005; Neumayer et al., 2011b). An ECT measurement system determines the capacitances between the electrodes using suitable circuitry (Wegleitner et al., 2005, 2008). The shield guards the sensor from external influences.

The inverse problem of ECT refers to the estimation of the spatial dielectric material distribution from the capacitance measurements. This step requires a simulation model of the sensor. To simulate the sensor, the governing partial differential equations have to be solved by an appropriate numerical technique. Electric fields inside the ECT sensor are governed by the potential equation $\nabla \cdot (ε_{0} ε_{r} \nabla V) = 0$ (Brandstätter et al., 2003b), where $V$ is the electric scalar potential, $ε_{0}$ denotes the permittivity of vacuum in As/Vm and $ε_{r}$ is the relative permittivity. Following the measurement process, Dirichlet type boundary conditions are applied to the electrodes and the capacitances are evaluated. For further discussion, we summarize these computations by the forward map $y = F (x)$ , where the vector $y \in R^{M}$ holds the simulated capacitances and the state vector $x \in R^{N}$ holds the discretized spatial dielectric material distribution. We use a finite element simulation for the sensor (Polycarpou, 2006). Subsequently, we assemble the state vector $x$ from the dielectric material values of the finite elements inside the pipe of the sensor model.

Inverse problem and prior information for pneumatic conveying

The inverse problem of ECT is of severe ill-posed nature (Hansen, 1998). To obtain a stable result specific measures have to be applied. Within the Bayesian framework or statistical inversion theory, prior information is incorporated into the estimator design (Kaipio and Somersalo, 2005). Statistical inversion theory considers any variables to be random variables. The characterization is based on probability density functions (pdfs) $π (\cdot)$ . Prior information about the material distribution is incorporated by the prior distribution $π (x)$ . The formulation of a prior distribution $π (x)$ is again a complex problem. A simple approach are so-called sample-based priors and summary statistics (Kaipio and Somersalo, 2005). Figures 3 and 4 depict samples from prior distributions for arbitrary material patterns. Figure 3 depicts samples from a distribution, which we refer to as rod-type distribution. Hereby, a random number of circular inclusions are placed on randomly drawn positions inside the pipe. Also, the permittivity of the inclusions is generated randomly. Figure 4 depicts exemplary samples from a distribution, which we refer to as Gaussian-type distribution. Hereby, Gaussian shaped objects are used instead of rods. Both distributions have been suggested for the assembly of sample-based priors for the reconstruction of arbitrary material distributions (Zangl et al., 2007). For the incorporation of the sample based prior to an actual reconstruction algorithm it is common to compute a summary statistic from an ensemble of samples (Neumayer et al., 2018a). A reasonable approach is the use of a Gaussian Summary statistic, that is, $π (x) = N (μ_{X}, Σ_{X})$ , where $μ_{X}$ and $Σ_{X}$ are the mean vector and the covariance matrix of a multivariate Gaussian distribution, respectively.

State reduction

A further technique to incorporate prior information to the solution of the inverse problem is given by the approach of a state reduction (Neumayer et al., 2019). Given an ensemble of prior samples, a state reduction of form $x = P_{N_{R}} x_{R}$ can be constructed (Neumayer et al., 2017), where $x_{R}$ is the reduced state vector and $P_{N_{R}}$ is a projection matrix. The vector $x_{R} \in R^{N_{R}}$ is of significantly lower dimension, that is, $N_{R} << N$ , with respect to the state vector $x$ . Therefore, this approach not only incorporates prior information to the solution of the inverse problem but also reduces the computational costs of the estimation task. To compute the projection matrix, a principal component analysis (Jolliffe, 2002) by means of a singular value decomposition $U Σ V^{T} = X_{Ensemble}$ is performed for the matrix $X_{Ensemble}$ , which holds an ensemble of prior samples. Then the matrix $P_{N_{R}}$ can be constructed by $P_{N_{R}} = [\begin{matrix} 1 & u_{1} & u_{2} & \dots & u_{N_{R} - 1} \end{matrix}]$ where $u_{i}$ denotes the column vectors of the unitary matrix $U$ . The state reduction can incorporate prior knowledge due to the intrinsic information of the basis vectors. The solution space is limited by the column vectors of the projection matrix.

Linear image reconstruction algorithms

In this section, we briefly address the design of estimators, or reconstruction algorithms within the Bayesian famework (Cui et al., 2016; Watzenig and Fox, 2009). Most materials of particulate flows have a comparatively low relative permittivity (Kraume, 2012). Due to this the forward map can be approximated by $F (x_{0} + d x) \approx F (x_{0}) + J d x$ , where $J$ is the $M \times N$ Jacobian matrix and $x_{0}$ denotes the linearization point. Detailed information about the determination of the Jacobian can be found in Brandstätter (2003a). We assume an additive Gaussian noise model $\tilde{d} = F (x) + v$ . Hereby, $\tilde{d}$ denotes the measurements corrupted with noise and $v$ denotes the measurement noise with $V \propto N (0, Σ_{V})$ , where $Σ_{V}$ denotes the covariance matrix of the noise. Then the linearized maximum a posteriori (MAP) estimator (Kay, 1993) is given by

{\hat{x}}_{1} = μ_{X} + {(J^{T} Σ_{V}^{- 1} J + Σ_{X}^{- 1})}^{- 1} (J^{T} Σ_{V}^{- 1} (\tilde{d} - F (μ_{X}))) .

(4)

In combination with the state reduction, the estimator becomes (Neumayer et al., 2018a)

\begin{matrix} {\hat{x}}_{2} = & μ_{X} + P_{N_{R}} {(P_{N_{R}}^{T} J^{T} Σ_{V}^{- 1} J P_{N_{R}} + P_{N_{R}}^{T} Σ_{X}^{- 1} P_{N_{R}})}^{- 1} \cdot \\ (P_{N_{R}}^{T} J^{T} Σ_{V}^{- 1} (\tilde{d} - F (μ_{X}))) . \end{matrix}

(5)

Both estimators fall into the class of linear back projection algorithms, as the computation requires a single matrix vector multiplication.

It has been demonstrated, that the incorporation of a state reduction incorporates prior information to algorithms due to the intrinsic information of the basis vectors. This involves a stabilization of the ill-posed inverse problem without the necessity of an explicit prior distribution $π (x)$ (Neumayer et al., 2019). Therefore, it is possible to skip $π (x)$ for estimator 5 leading do a maximum likelihood (ML) type estimator (Kay, 1993) for the inverse problem

{\hat{x}}_{3} = μ_{X} + P_{N_{R}} {(P_{N_{R}}^{T} J^{T} Σ_{V}^{- 1} J P_{N_{R}})}^{- 1} (P_{N_{R}}^{T} J^{T} Σ_{V}^{- 1} (\tilde{d} - F (μ_{X}))) .

(6)

Further, algorithms (4), (5) and (6) can be extended by state constraints to avoid infeasible estimation results; for example, $ε_{r} < 1$ . Therefore, the estimators are recasted into quadratic programs and solved with subject to $x_{\min} \leq x \leq x_{\max}$ , where $x_{\min}$ and $x_{\max}$ denotes a lower and an upper bound of $x$ , respectively. The estimation problems can be solved reasonably fast when using dedicated numerical libraries (Neumayer et al., 2011a).

A further back projection type reconstruction algorithm of form ${\hat{x}}_{4} = P {\tilde{d}}_{a}$ can be constructed directly from the ensemble of prior samples (Zangl et al., 2007). ${\tilde{d}}_{a}$ is referred to as augmented data vector. In this work, we use ${\tilde{d}}_{a} = {[\begin{matrix} 1 & {\tilde{d}}^{T} & {({\tilde{d}}^{T})}^{2} \end{matrix}]}^{T}$ where $({\tilde{d}}^{T})^{2}$ refers to a vector with element wise squared entries. The reconstruction matrix $P$ can be obtained by solving the equation system $H P^{T} = X_{Ensemble}^{T}$ in a least squares sense. The matrix $H$ holds the augmented data vectors in a column wise sense. This estimator is referred to as optimal second order approximation (OSOA).

Estimation of $\bar{β}$

Given an estimate $\hat{x}$ for the material distribution, the parameter $\bar{β}$ can be estimated by

{\bar{β}}_{est .} = \frac{1}{A_{Ω_{ROI}} χ_{prior}} \int_{A_{Ω_{ROI}}} \hat{χ} (x, y) d Γ

(7)

Hereby, $χ$ denotes the dielectric susceptibility, which is given by $χ = ε_{r} - 1$ . This is due to the relative permittivity of air, being $ε_{r, air} = 1$ . Using the finite element method, the integral can be approximated by a weighted sum using the area of the finite elements. $χ_{prior}$ in equation (7) is a normalization constant. It is determined from the relative permittivity of the transported material in a dense state. Using this approach, the relation $0 \leq \bar{β} \leq 1$ holds.

Design of specific prior information for flow profiles and analysis of the estimation behaviour

The concept of sample-based priors can be adapted for flow regimes in pneumatic conveying systems (Neumayer et al., 2018b). In this section, we address the design of specific sample-based prior information for flow patterns occurring in pneumatic conveying systems and we analyse and compare the estimation behaviour of different algorithms and different priors.

Prior design

The aim of this subsection is the construction of specific prior information for pneumatic conveying systems. In order to achieve this, the flow regimes depicted in Figure 1 are analysed. The different flow regimes can be summarized by three main cases. The first case is given by dilute flow regimes where the particles are distributed over the whole cross section of the pipe with a low density. In the second case a distinct bottom layer of a dense phase and a dilute phase in the upper domain of the pipe is present. The third case is given by the slug flow regime, where the whole cross section of the pipe is filled with dense bulk material. A sample-based prior for these material distributions is given by random samples of a two-dimensional cross sectional representation of the different flow regimes.

Horizontal flow samples with even boundaries between the phases

In the simplest cases, the random samples are generated by the parametrized material distribution depicted in Figure 5. The material distribution is given by a dense bottom layer with a certain height $h$ and the material value $ε_{r 1}$ and a dilute upper layer with the material value $ε_{r 2}$ . Hereby, $ε_{r 1} > ε_{r 2}$ holds due to the higher density of the bottom layer. In the first step of the generation of random samples for flow type material distributions the flow regime is selected randomly. Depending on the selected case, the following steps are performed. In order to generate random samples for dilute flow regimes, the height $h$ is set to zero and the whole cross section is filled with the randomly generated material value $ε_{r 2}$ . For the flow regimes with a distinct bottom layer, all three parameters – $h$ , $ε_{r 1}$ and $ε_{r 2}$ – are generated randomly. For the dense slug flow regime, the whole cross section of the pipe is filled with the random material value $ε_{r 1}$ . Prior information generated by samples for these different cases is referred to as flow-type prior.

Horizontal flow samples with uneven and smooth boundaries between the phases

In order to generate a prior, which is robust against deviations in the flow profiles, a more complex representation of the different flow regimes is given by the parametrized material distribution depicted in Figure 6(a). Hereby, the boundary between the two phases is uneven and has smooth behaviour. In Guan et al. (2011) it is demonstrated that this kind of flow profiles can occur in horizontal pneumatic conveying processes. The dilute flow regime and the dense slug flow regimes are generated in the same way as discussed previously. The flow regimes with a distinct bottom layer are in this case parametrized by three heights – $h_{1}$ , $h_{2}$ and $h_{3}$ – at certain grid points and the two material values – $ε_{r 1}$ and $ε_{r 2}$ . Between these grid points the boundary is interpolated by means of a quadratic polynomial function. Figure 6(b) depicts an ensemble of randomly generated boundaries. These ensemble illustrates, that the boundaries can have convex as well as concave behaviour. Using this kind of samples should allow the reconstruction of more complex material distributions within the ECT sensor and therefore this sample-based prior should improve the robustness against deviations in the flow profiles. We refer to prior information generated by this kind of samples as robust-flow-type prior.

Simulation study to analyse the behaviour of different priors

In this subsection, we analyse the estimation behaviour of different algorithms with respect to the prior information. We compare the rod-type and Gaussian-type prior for arbitrary material distribution with specific prior information for flow patterns. In order to analyse the estimation performance, we use the evaluation framework depicted in Figure 7. The evaluation maintains two sets of ensembles, which are evaluated on different simulation models for the sensor. For the generation of simulated data and the determination of the true parameter ${\bar{β}}_{true}$ , a finite element mesh with a fine discretization is used. On this fine mesh, random samples for flow type material distributions are generated. The data are determined from the first ensemble. The reconstruction and computation of ${\bar{β}}_{est .}$ is performed on a coarse mesh suitable for fast image reconstruction. Prior knowledge from the second ensemble is used to create the reconstruction algorithm. The second ensemble is coming from prior distributions for arbitrary material patterns as well as from prior distributions for specific flow type material distributions. Exemplary samples for arbitrary rod-type and Gaussian-type priors are depicted in Figure 3 and Figure 4, respectively. The material patterns for specific flow prior information are depicted in Figure 5 for flow-type prior and in Figure 6(a) for robust-flow-type prior. The approach of using different simulation models reduces the impact of an inverse crime (Kaipio and Somersalo, 2007). Prior to the reconstruction, the data from ensemble one is corrupted by additive measurement noise and calibrated by means of an offset-gain simulation between the fine and the coarse mesh (Neumayer et al., 2011b).

Dimensions of the ECT sensor and simulation setup

Figure 8(a) depicts a photograph of a lab test rig with an ECT sensor. It is used to perform flow experiments with particulate materials and simultaneously acquire ECT measurements. The dimensions of the ECT sensor, which is used in the simulation study is the same as the sensor of the lab test rig. The inner radius of the process pipe is given by $r_{i} = 16$ mm and the outer radius is given by $r_{o} = 20$ mm. The shield has a radius of $r_{s} = 31.5$ mm. The sensor has eight electrodes, each electrode has an angular width of $l_{elec} = 10$ mm and the electrodes are equidistant arranged around the circumference of the pipe.

The number of eight electrodes for the lab sensor and the study was selected by the authors, as we are interested in monitoring of particulate flows with capacitive sensors, while keeping the instrumentation effort low. The capability for flow monitoring with a reduced number of electrodes has been presented in Neumayer and Bretterklieber (2014), where an ECT sensor with only five electrodes is used. For arbitrary material distributions, various articles suggesting 12 or 16 electrodes have been presented. The capability to reconstruct flow profiles with an ECT sensor with a lower number of electrodes is due to the specific structure of flows, which is addressed by the prior design. A fewer number of electrodes allow a sensor design with larger electrodes and therefore increased coupling capacitances between the electrodes, which generally leads to improved signal to noise ratios (SNR).

Figure 8(b) depicts an SNR measurement of the lab ECT sensor. The SNR is indicated with respect to the total signal change between an empty measurement and a measurement for the sensor filled with particulate polyethylene pellets with $ε_{r} = 2.4$ . The SNR achieved with the lab measurement system is used as noise level in order to generate the simulated data. The relative permittivity of the polyethylene pellets is used as maximum $ε_{r}$ value for the simulation of the data as well as for the prior samples. The reconstruction is performed on a coarse finite element mesh with approximately 500 finite elements within the interior of the pipe.

Statistical evaluation of the estimation error

For the evaluation of the reconstruction algorithms we analyse the estimation error $e$ given by

e = {\bar{β}}_{est .} - {\bar{β}}_{true} .

(8)

Figures 9 and 10 depict the simulation results obtained with the OSOA algorithm and the linearized MAP estimator (LMAP), respectively. The plots show the mean $μ_{e}$ and the standard deviation $σ_{e}$ of the estimation error $e$ over the range of ${\bar{β}}_{true}$ . The simulated data was generated by means of the material distribution depicted in Figure 6(a). Both estimators show a significant variance as well as a bias error when using the rod-type prior. Considering the results of the linearized MAP estimator depicted in Figure 10, the incorporation of a Gaussian-type prior provides estimation results of similar quality compared to the result achieved with the rod-type prior. The results obtained with the OSOA algorithm depicted in Figure 9 show in this case a reduced uncertainty compared to the rod-type prior. This can be explained by the smooth behaviour of this kind of prior samples and the soft field nature of ECT, as ECT images suffer from blurring (Neumayer et al., 2011b), disabling the reconstruction of sharp boundaries. Both estimators exhibit a reduced bias error for incorporated flow-type priors.

For the linearized MAP estimator also, the variance is decreased over the whole range of ${\bar{β}}_{true}$ . For the variance achieved with the OSOA algorithm, the flow-type prior only shows an improvement for larger values of ${\bar{β}}_{true}$ compared with the Gaussian-type prior. Finally, the incorporation of the robust-flow-type prior improves the estimation behaviour with respect to the bias error as well as to the variances for both demonstrated estimators. In this case, the prior samples are corresponding to the samples used to generate the data. Comparing the two different estimators, a decreased bias error is evident for the OSOA algorithm. The linearized MAP estimator shows an increased bias error for low values of ${\bar{β}}_{true}$ , but remains generally smaller with respect to the other priors. This is explained due to the linearization in the point $μ_{X}$ . For an overall comparison between the different algorithms and priors, Figure 11 depicts the mean ${\bar{μ}}_{| e |}$ and the standard deviation ${\bar{σ}}_{| e |}$ of the absolute value of the estimation error $| e |$ over all simulated samples. This representation of the simulation results demonstrate the improved overall estimation performance of the presented prior information with respect to the bias error and the uncertainty of the estimates. In Suppan et al. (2018) detailed studies are shown for different implementations of the estimators; for example, constraint, non-constraint and for an incorporated state reduction as well as for the volume fraction in vertical pneumatic conveying systems.

Test rig measurements and lab demonstration

This section contains a demonstration of the improved estimation behaviour of the presented approach with measurements coming from the lab test rig. A photograph of the lab test rig setup is depicted in Figure 8(a). It contains a hopper filled with particulate polyethylene pellets with a relative permittivity of $ε_{r} = 2.4$ . The hopper is connected to a pipe system on which the ECT sensor is mounted. A valve is situated at the end of the pipe system, which is used to initiate the flow experiments. The ECT sensor is connected to the ECT measurement unit by means of coaxial cables. The whole test rig is tiltable enabling the generation of different flows states reaching from vertical outflow experiments to horizontal pneumatic conveying experiments, which are driven by pressurized air. Hereby, the velocity of the material stream can be influenced by means of the tilt angle, the valve position and the air pressure. The test rig features an additional scale signal to provide information about the mass flow.

Reconstruction experiments

In this subsection, we demonstrate the reconstruction behaviour of linear back projection type estimators with different priors for measurements coming from lab experiments. Additionally, we analyse the required reconstruction times of different algorithms. Hereby, a horizontal pneumatic conveying flow is generated where the material stream decreases with time. After initiating the flow by the opening of the valve the hopper is gradually emptied due to the gravitational flow of the particulate material. At a certain time instant pressurized air is injected in the pipe system. This gas injection causes increased velocities of the particles and a simultaneous decrease of the volume fraction. Figure 12 depicts camera recordings of the flow experiment at different time instants. Figure 12(a) shows the flow state before the pressurized air is injected. Here, the whole cross section of the pipe is filled with bulk material. Figure 12(b) depicts the flow state at the beginning of the gas injection. The gas stream creates a dilute upper phase. The photograph shown in Figure 12(c) depicts the flow state at a later time instant where the hopper is emptied.

Figure 13 depicts exemplary sequences of reconstruction results obtained by the linearized MAP estimator with different priors. Figure 13(a) depicts the reconstructed images obtained by the incorporation of rod-type prior information. These images show distinct artefacts in the dense phase as well as in the dilute phase. The reconstruction with rod-type prior information does not allow the representation of homogeneous and constant material distributions. Figure 13(b) depicts the images obtained with Gaussian-type prior information. These results suffer from the same issues than the results obtained with the rod-type prior. The artefacts however are less distinct and the material distribution shows a smoother behaviour. Figure 13(c) depicts the reconstruction results obtained by means of the flow-type prior. The reconstructed images show distinct minima in the dense flow phase as well as artefacts in the dilute flow phase. This result suggests that the flow-type prior depicted in Figure 5 is too simplistic to provide meaningful reconstruction results for real pneumatic conveying processes. Other linear back projection type estimators – for example, the OSOA algorithm – show similar behaviour when it comes to the reconstruction of real flow experiments. Figure 13(d) depicts the sequential reconstruction results achieved with the linearized MAP estimator and the robust-flow-type prior. The use of this prior information involves a smoother behaviour of the reconstruction results with a more homogeneous distribution of the permittivity in the dense phase as well as in the dilute phase. Also, the appearance of artefacts in the reconstructed images is significantly reduced. This demonstrates the advantages of the incorporation of appropriate prior information to the estimation task.

Computation times

Regarding computation times, reconstruction rates of about 150 Frames/s for the back-projection algorithms like the OSOA and the linearized MAP estimator were achieved. When applying state constrains the frame rate is reduced to 10 Frames/s. The results were computed with MATLAB2017 on an Intel Core i7-4510u CPU with 2x2 GHz clock rate.

Uncertainty analysis by means of steady state conveying experiments

In this subsection, we present a measurement-based analysis of the uncertainty of different reconstruction algorithms and prior distributions. The test rig is used to produce horizontal steady state flows with a constant particle velocity $\bar{v}$ and a constant volume fraction $\bar{β}$ . In order to achieve this steady state flow the test rig is run with constant air pressure and constant valve position at constant tilt angles $α$ in the range of $20 ° \leq α \leq 60 °$ . To quantify the uncertainty, we use the standard deviation of the estimated volume fraction $σ_{\bar{β}}$ within time intervals with constant flow states. Figure 14(a) depicts the scale-based mass signal for an exemplary flow experiment. The experiment starts at approximately 18 s by opening a valve at the end of the pipe system. The intervals for the uncertainty analysis are chosen to correspond to the time range where the mass signal exhibits a constant slope. Figure 14(b) depicts the exemplary estimation results obtained with the OSOA algorithm and different prior distributions. The prior distributions used for this experiment are given by the rod-type distribution, the Gaussian-type distribution and the robust-flow-type prior. Due to the shortcomings of the flow-type prior demonstrated in the previous subsection this prior is not considered for this analysis. The estimator provides a constant volume fraction ${\bar{β}}_{est .}$ within the interval with a constant mass flow rate. However, the different priors show differences with respect to their expectation and their variances. The estimators provide volume fractions in the range of $0.8$ before the start of the flow experiment. The valve, which is used to initiate the flow is situated after the ECT sensor and therefore the sensor is filled with bulk material before the start of the experiment. Since no velocity information is available it is not possible to make conclusions about the true volume fraction by means of the scale-based mass signal and therefore also no statements about the expectations of the estimators are currently possible. Due to this reason and because of the circumstance that a bias of the estimates can be handled by means of calibration schemes, the focus of the following analysis lies on the variance of the estimation results. The experiments for the uncertainty analysis are performed for large fillings of the pipe and therefore large values of ${\bar{β}}_{true}$ . The test rig currently allows only the generation of stationary flow with a comparatively large volume fraction. This is also due to the particle size of the polyethylene pellets, which makes it difficult to generate steady state flow conditions with low volume fractions. For this reason, a comparison of the lab experiments with the simulation results cannot cover the whole range of $\bar{β}$ .

Figure 15 depicts an analysis of the standard deviation of different reconstruction experiments, where each trend represents the results for a single flow experiment and each sub-figure represents a different estimator. We maintain the same representation of the uncertainty as depicted in Figure 11(b). The results show the improvement in the reduction of the standard deviation, due to the use of an appropriate prior. The evaluated standard deviations for the measurements are yet higher than the simulated standard deviations; for example, the results depicted in Figures 9, 10 or Figure 11(b). We consider this increase to be caused by variations inside the transport process of the measurement experiment – for example, local density variations – which are yet not modeled by the priors.

Figure 16 depicts the mean standard deviation $σ_{\bar{β}}$ obtained with different algorithms and priors for the flow experiments. Hereby, the incorporated prior information is stated on the axis of abscissa. The axis of ordinates displays the standard deviation $σ_{\bar{β}}$ of ${\bar{β}}_{est .}$ . The abbreviation OSOA in the legend refers to the optimal second order approximation algorithm and LMAP refers to the linearized MAP estimator, respectively. The abbreviation PCA30 refers to the application of a state reduction with 30 basis vectors and LMAPc refers to a linearized MAP estimator with the incorporated state constraints $ε_{r} \geq 1$ . The result depicted in Figure 16 again shows that the estimators with rod-type prior information achieved the highest standard deviation $σ_{\bar{β}}$ and the estimators with incorporated robust-flow-type prior achieved the lowest uncertainties for every algorithm. This demonstrates the improved estimation behaviour of the presented approach.

Conclusion

In this paper, we presented the estimation of the volume fraction in pneumatic conveying systems from capacitive sensor data using ECT. We addressed the construction of prior information and the modification of different tomographic reconstructions methods. We presented a framework for the systematic analysis of the different algorithms and demonstrate the advantages of incorporating specific prior information for flow profiles in pneumatic conveying processes; for example, unbiased estimates and reduced variances. The presented approach was also applied to measurements coming from a lab test rig in order to demonstrate the improved and robust estimation behaviour also for data coming from real measurements. Comparative reconstruction results show an improved image quality and representation of the true material distribution within the conveying process, when applying specific prior information. The results achieved in the lab experiments also demonstrate the improved estimation behaviour regarding the uncertainties of the estimates when applying the demonstrated approach. The achieved results show the capability of tomographic signal processing methods for flow parameter estimation in pneumatic conveying systems.

Footnotes

Acknowledgements

The authors would like to thank Mr Thomas Wiener for his support during the lab experiments.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is funded by the FFG Project (Bridge 1) TomoFlow under the FFG Project Number 6833795 in cooperation with voestalpine Stahl GmbH.

ORCID iD

Markus Neumayer

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Prior design for tomographic volume fraction estimation in pneumatic conveying systems from capacitive data

Abstract

Keywords

Introduction

ECT

Inverse problem and prior information for pneumatic conveying

State reduction

Linear image reconstruction algorithms

Estimation of β ¯

Design of specific prior information for flow profiles and analysis of the estimation behaviour

Prior design

Horizontal flow samples with even boundaries between the phases

Horizontal flow samples with uneven and smooth boundaries between the phases

Simulation study to analyse the behaviour of different priors

Dimensions of the ECT sensor and simulation setup

Statistical evaluation of the estimation error

Test rig measurements and lab demonstration

Reconstruction experiments

Computation times

Uncertainty analysis by means of steady state conveying experiments

Conclusion

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

ORCID iD

References

Estimation of $\bar{β}$