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.
History
Electrical Capacitance Volume Tomography was first introduced by W. Warsito and L.-S. Fan in a presentation at the 3rd World Congress in Process Tomography in Banff Canada in 2003. The term was coined in 2005 in a patent filed by W. Warsito, Q. Marashdeh, and L.S. Fan with the emphasis on volume to distinguish the technology from earlier and ongoing development of a form called 3D-ECT where 2D tomograms are stacked on top of each other to create a pseudo 3D image. This traditional 3D-ECT approach limited the use of 3D imaging as the significant length of ECT electrodes introduced a major penalty on axial resolution of such 3D images. ECVT emerged as a solution to this limitation. ECVT provides direct 3D imaging by exploiting the X, Y, and Z components of the electric field, which are a function of sensor design. The original presentation in 2003 was followed by a publication by Q. Marashdeh and F. Teixeira in 2004 where they introduced a method to build a sensitivity matrix for these new sensors. This new emerging form of the technology was referred to as 3D-ECT up until the filing of the patent in 2005, where it was distinguished as ECVT. A journal paper was later published, in 2007, detailing the scientific background of the technology, and the chronological order of the development of ECVT was also published in a journal publication that same year.
Principles
Capacitance and Field Equations in ECVT
Two metal electrodes held at different electric potential and separated by a finite distance will induce an electric field 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 and conductivity . 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:
where denotes the electric potential distribution. In a homogeneous medium with uniform , this equation reduces to the Laplace equation. In a lossy medium with finite conductivity, such as water, the field obeys the generalized Ampere equation,
By taking divergence of this equation and using the fact that , it follows:
when the plates are excited by a time-harmonic voltage potential with frequency .
The capacitance is a measure of electric energy stored in the medium, which can be quantified via the following relation:
where is the square magnitude of the electric field. The capacitance changes as a nonlinear function of the dielectric permittivity because the electric field distribution in the above integral is also a function of .