Three-dimensional electrical capacitance tomography

Electrical Capacitance Volume Tomography (ECVT) is a non-invasive 3D imaging technology applied primarily to multiphase flows. It was first introduced by W. Warsito, Q. Marashdeh, and L.S. Fan as an extension of the conventional Electrical Capacitance Tomography (ECT). In conventional ECT, sensor plates are distributed around a surface of interest. Measured capacitance between plate combinations is used to reconstruct 2D images (tomograms) of material distribution. In ECT, the fringing field from the edges of the plates is viewed as a source of distortion to the final reconstructed image and is thus mitigated by guard electrodes. ECVT exploits this fringing field and expands it through 3D sensor designs that deliberately establish an electric field variation in all three dimensions. The image reconstruction algorithms are similar in nature to ECT; nevertheless, the reconstruction problem in ECVT is more complicated. The sensitivity matrix of an ECVT sensor is more ill-conditioned and the overall reconstruction problem is more ill-posed compared to ECT. The ECVT approach to sensor design allows direct 3D imaging of the outrounded geometry. This is different than 3D-ECT that relies on stacking images from individual ECT sensors. 3D-ECT can also be accomplished by stacking frames from a sequence of time intervals of ECT measurements. Because the ECT sensor plates are required to have lengths on the order of the domain cross-section, 3D-ECT does not provide the required resolution in the axial dimension. ECVT solves this problem by going directly to the image reconstruction and avoiding the stacking approach. This is accomplished by using a sensor that is inherently three-dimensional.

Principles

Capacitance and Field Equations in ECVT

Two metal electrodes held at different electric potential $V$ and separated by a finite distance will induce an electric field $E$ in the region between and surrounding them. The field distribution is determined by the geometry of the problem and the constitutive medium properties such as permittivity $\varepsilon$ and conductivity $\sigma$. Assuming a static or quasi-static regime and the presence of a lossless dielectric medium, such as a perfect insulator, in the region between the plates, the field obeys the following equation:

$\nabla .(\varepsilon \nabla \varphi )=0$

where $\varphi$ denotes the electric potential distribution. In a homogeneous medium with uniform $\varepsilon$, this equation reduces to the Laplace equation. In a lossy medium with finite conductivity, such as water, the field obeys the generalized Ampere equation,

$\nabla \times H=\sigma E+j\omega \varepsilon E$

By taking divergence of this equation and using the fact that $E=-\nabla \varphi$, it follows:

$\nabla .((\sigma +j\omega \varepsilon )\nabla \varphi )=0$

when the plates are excited by a time-harmonic voltage potential with frequency $\omega$.

The capacitance $C$ is a measure of electric energy $W$ stored in the medium, which can be quantified via the following relation:

$W={\frac {1}{2}}\int _{}^{}\varepsilon E^{2}\,dv={\frac {1}{2}}CV^{2}$

where $E^{2}$ is the square magnitude of the electric field. The capacitance changes as a nonlinear function of the dielectric permittivity $\varepsilon$ because the electric field distribution in the above integral is also a function of $\varepsilon$.

Soft-Field Tomography

File:MRI ECT Comparison.png

Comparison of image resolution of voidage in a gas-fluidized bed between ECVT (soft-field) and MRI (hard-field)

Soft-field tomography refers to a set of imaging modalities such as electrical capacitance tomography (ECT), electrical impedance tomography (EIT), electrical resistivity tomography (ERT), etc., wherein electric (or magnetic) field lines undergo changes in the presence of a perturbation in the medium. This is in contrast to hard-field tomography, such as X-ray CT, where the electric field lines do not change in the presence of a test subject. A fundamental characteristic of soft-field tomography is its ill-posedness. This contributes for making the reconstruction more challenging to achieve good spatial resolution in soft-field tomography as compared to hard-field tomography. A number of techniques, such Tikhonov regularization, can be used to alleviate the ill-posed problem. The figure at the right shows a comparison in image resolution between ECVT and MRI.

ECVT Measurement Acquisition Systems

The hardware of ECVT systems consists of sensing electrode plates, the data acquisition circuitry, and the computer to control the overall system and process the data. ECVT is a non-intrusive and non-invasive imaging modality due to its contactless operation. Prior to the actual measurements, a calibration and normalization procedure is necessary to cancel out the effects of stray capacitance and any insulating wall between the electrodes and the region of interest to be imaged. After calibration and normalization, the measurements can be divided into a sequence of acquisitions where two separate electrodes are involved: one electrode (TX) is excited with AC voltage source in the quasi-electrostatic regime, typically below 10 MHz, while a second electrode (RX) is placed at the ground potential used for measuring the resultant current. The remaining electrodes are also placed at ground potential.

This process is repeated for all possible electrode pairs. Note that reversing the roles of TX and RX electrodes would result in the same mutual capacitance due to the reciprocity. As a result, for ECVT systems with N number of plates, the number of independent measurement is equal to N(N-1)/2. This process is typically automated through data acquisition circuitry. Depending on the operation frequency, number of plates and frame rate per second of the measurement system, one full measurement cycle can vary; however, this is in the order of few seconds or less. One of the most critical parts of ECVT systems is sensor design. As the previous discussion suggests, increasing the number of electrodes also increases the amount of independent information about the region of interests. However this results in smaller electrode sizes which in turn results in low signal to noise ratio . Increasing the electrode size, on the other hand, does not result in non-uniform charge distribution over the plates, which may exacerbate the ill-posedness of the problem. The sensor dimension is also limited by the gaps between the sensing electrodes. These are important due to fringe effects. The use of guard plates between electrodes have been shown to reduce these effects. Based on the intended application, ECVT sensors can be composed of single or more layers along the axial direction. The volume tomography with ECVT is not obtained from merging of 2D scans but rather from 3D discretized voxels sensitivities.

The design of the electrodes is also dictated by the shape of the domain under investigation. Some domains can be relatively simple geometries (cylindrical, rectangular prism, etc.) where symmetrical electrode placement can be used. However, complex geometries (corner joints, T-shaped domains, etc.) require specially designed electrodes to properly surround the domain. The flexibility of ECVT makes it very useful for field applications where the sensing plates cannot be placed symmetrically. It should also be noted that since the Laplace equation lacks a characteristic length (such as the wavelength in Helmholtz equation), the fundamental physics of the ECVT problem is scalable in size as long as quasi-static regime properties are preserved.

Image Reconstruction Methods for ECVT

Reconstruction methods address the inverse problem of ECVT imaging, i.e. to determine the volumetric permittivity distribution form the mutual capacitance measurements. Traditionally, the inverse problem is handled through the linearization of the (nonlinear) relationship between the capacitance and the material permittivity equation using the Born approximation. Typically, this approximation is only valid for small permittivity contrasts. For other cases, the nonlinear nature of the electric field distribution poses a challenge for both 2D and 3D image reconstruction, making the reconstruction methods an active research area for better image quality. Reconstruction methods for ECVT/ECT can be categorized as iterative and non-iterative (single step) methods. The examples of non-iterative methods are linear back projection (LBP), and direct method based on singular value decomposition and Tikhonov regularization. These algorithms are computationally inexpensive; however, their tradeoff is less accurate images without quantitative information. Iterative methods can be roughly classified into projection-based and optimization-based methods. Some of the linear projection iterative algorithms used for ECVT include Newton-Raphson, Landweber iteration and steepest descent algebraic reconstruction and simultaneous reconstruction techniques, and model-based iteration. Similar to single step methods, these algorithms also use linearized sensitivity matrix for the projections to obtain the permittivity distribution inside the domain. Projection-based iterative methods typically provide better images than non-iterative algorithms yet require more computational resources. The second type of iterative reconstruction methods are optimization-based reconstruction algorithms such as neural network optimization. These methods need more computational resources than the previously mentioned methods along with added complexity for the implementation. Optimization reconstruction methods employ multiple objective functions and use iterative process to minimize them. The resultant images contain less artifacts from the nonlinear nature and tend to be more reliable for quantitative applications.

Displacement-Current Phase Tomography (DCPT)

Displacement-Current Phase Tomography is an imaging modality that relies on the same hardware as ECVT. ECVT does not make use of the real part (conductance component) of the obtained mutual admittance measurements. This component of the measurement is related with the material losses in the region of interest (conductivity and/or dielectric losses). DCPT utilizes the full admittance information by means of the small angle phase component of this complex valued data. DCPT can only be used when the electrodes are excited with AC voltage. It applies only to domains that include material losses, otherwise the measured phase will be zero (real part of the admittance will be zero). DCPT is designed to be used with the same reconstruction algorithms designed for ECVT. Therefore, DCPT can be used simultaneously with ECVT to image the spatial tangent loss distribution of the medium along with its spatial relative permittivity distribution from ECT.

Multi-Frequency ECVT Operation

Multiphase flows are invariably complex. Advanced measuring techniques are required to monitor and quantify phase hold ups in such multiphase flows. Due to their relatively fast speed of acquisition and non-intrusive characteristics, ECT and ECVT are widely used in industries for flow monitoring. However, the flow decomposition and monitoring capabilities of ECT/ECVT for multiphase flow containing three or more phases (e.g., a combination of oil, air, and water) is somewhat limited. Multi-frequency excitations and measurements has have been exploited and successfully used in ECT image reconstruction in those cases. Multi-frequency measurements allow the exploitation of the Maxwell-Wagner-Sillars (MWS) effect on the response of the measured data (e.g., admittance, capacitance, and etc.) as a function of excitation frequency. This effect was first discovered by Maxwell in 1982 and later studied by Wagner and Silliars. The MWS effect is a consequence of surface migration polarization at the interface between materials when at least one of them is conducting. Typically a dielectric material presents a Debye-type relaxation effect at microwave frequencies. However, due to the presence of the MWS effect (or the MWS polarization) a mixture containing at least one conducting phase will exhibit this relaxation at much lower frequencies. The MWS effect depends on several factors such as volume fraction of each phase, phase orientation, conductivity and other mixture parameters. Wagner formula for dilute mixture and Bruggeman formula for dense mixtures are among the most notable formulation of effective dielectric constant. Hanai’s formulation of complex dielectric constant, an extension of Bruggeman formula of effective dielectric constant, is instrumental in analyzing MWS effect for complex dielectric constant. Hanai’s formula for complex dielectric writes as ((ϵ_1^*-ϵ_2^*)/(ϵ_1^*-ϵ^* ))^3 ϵ^*/(ϵ_2^* )=1/(1-ϕ)^3 , where ϵ_1^*,ϵ_2^*, and ϵ^* are the complex effective permittivity of the dispersed phase, continuous phase, and mixture, respectively. ϕ is the volume fraction of the dispersed phase. Knowing that a mixture will exhibit dielectric relaxation due to the MWS effect, this additional measuring dimension can be exploited to decompose multiphase flows when at least one of the phases is conducting.

Advantages

The ECVT technology provides low profile and flexibility of capacitance sensors, an increase in the number of imaging frames per second, and relatively low cost. These features moved the technology to the top of the list as a tool that could be used in industrial imaging.

Applications

ECVT is applied for multi-phase flow systems measurements. The ECVT would provide Measurements in real-time in a three-dimensional components of multi-phase reactors. The ECVT could be applied to gas-liquid and gas, gas-solid fluidized bed, gas-solid circulating fluidized beds, liquid-solid bubble columns, and multi-phase flows in vessels with complex geometries.

See also

• Electrical capacitance tomography
• Electrical impedance tomography
• Electrical resistivity tomography
• Process tomography