## 3 Electrical Tomography Principles and Applications
### Fundamentals of Electrical Tomography
#### 3.2.1 Capacitance Tomography  
The purpose of ECT it to obtain a reconstruction of the dielectric characteristics of a system, using measures of capacitance taken between all existing electrode pairs. In Figure 1, a schematic of a cross-section representation of an ECT system possessing eight electrodes can be seen. If the pipe where the suspension is flowing is made of a conductive material, then the electrodes are internally mounted. If, conversely, the pipe is made of an insulating material, then the electrodes will be located externally. External electrodes are easily conceived, assembled and maintained; additionally, once they have been directly exposed to extreme turbulence, pressures and temperatures, their electrical behaviour remains unchanged for a longer time (contamination by the materials flowing in the tube is not encountered, either). Their greatest disadvantage is that they possess non-linear characteristics; however, the use of adequate correction factors allows them to display almost linear characteristics. The design of internal electrodes is harder and more intricate, because they are exposed as described above to extreme conditions, and may also suffer from corrosion; however, the capacitance changes observed with internal electrodes can be assumed to be directly proportional to the permittivity changes occurring inside the pipe. In ECT, all electrodes are stimulated, one at a time, and all pairs of capacitances between the stimulated electrode and the remaining ones are measured; for an \(N\) electrodes system, this produces \(N\)(\(N\)-1)/2 independent measurements, once capacitance \(C_{ij}=C_{j,i}\) and \(C_{i,i}\), i.e., the self-capacitance, are all ignored. In summary, the measuring protocol operates in a similar way to the source-detector movement in computerised tomography in medical imaging; in ECT, the electrical field rotates around the pipe cross-section in steps, with a step angle of 360\({}^{\circ}\)/\(N\). The number \(N\) of electrodes is chosen based on a balance between the desired spatial resolution and acquisition rate of the images (acquisition rates around 100 frames per second are frequent). Maxwell equations make it possible to establish a relationship between the measured capacitances and the actual spatial distribution of the permittivity. In ECT, only one electrode is stimulated at one time, while the remaining are at a virtual earth potential. Consequently, over tall electrode surfaces, the total electric flux is equal to zero. The capacitance measures can be easily influenced by external parasite capacitances (usually greater than the measured ones), such as earth capacitance, because the inter-electrode capacitance is quite small. To prevent this perturbation, electrode shielding is implemented. With respect to the excitation signal frequency, a value of around 1 MHz is common; for this order of magnitude, the signal wavelength is of some hundreds of meters, greatly exceeding the sensor size by many orders of magnitude. In summary, electrostatic field theory can be described by the electrical potential distribution inside the medium under inspection.  
ERT sensing arrays are less complex than those required by ECT, and the difficulty in their conception arises when the domains present high conductivity [63; 64]. Consequently, it can be stated that it is quite adequate for the inspection of non-conductive mixtures containing diverse permittivity's. ECT produces low-resolution images, but, when dealing with disperse system flows, offers good accuracy with respect to volume fraction estimation [65]. If in ECT a dual plane system is used, then cross correlating the cross-section average time series enables the evaluation of a particle's velocity [65].