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:

${\displaystyle }$ $\nabla .(\varepsilon \nabla \varphi )=0$

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

Figure 1: 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. Figure 1 shows a comparison in image resolution between ECVT and MRI.

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