**2. Basic principle of ERT**

The principle of operation of electrical resistance tomography (ERT) is to inject a known electrical current into a pair of the (usually) 16 electrodes in a plane equispaced around a vessel or pipe wall and measure the voltages present on the remaining electrodes. The process is cycled through the remaining possible pairs of injection current locations (15 pairs) to give an extensive data set for the measurement plane. The complete set of measured voltages is then used to reconstruct an image of the conductivity in the vessel's cross section from which the solids concentration or gas fraction can be obtained in principle, if the liquid- and solid-phase conductivities are known (gas and most solids in slurries would have zero conductivity). A basic schematic of the measurement and reconstruction process is shown in **Figure 1**.

ERT requires an inverse problem algorithm to calculate the conductivity distribution from the voltage measurements. If the conductivity distribution is known from such a calculation, the volume concentration of the nonconductive solids or gas can be calculated from the measured conductivity via the reduced Maxwell equation:

$$\phi = \frac{2\left(\sigma\_1 - \sigma\_{mc}\right)}{\sigma\_{mc} + 2\sigma\_1} \tag{1}$$

where.

σ1 = continuous phase or background conductivity (typically water or another homogenous supernatant such as a carrier fluid or fine particle slurry).

σmc = measured reconstructed conductivity.

ϕ = volume fraction of the nonconductive phase.

Note that the aforementioned equation is applicable for zero conductivity solids only with a more complex version applicable if the solids exhibit some level of conductivity. The equation is also applicable for a gas phase which can also be considered having zero conductivity. The volume fraction may be readily converted to other measures of concentration if the solids and liquids densities are known.

Areas of an image with a lower conductivity indicate the presence of solids to be higher as the particles present in many slurries are nonconductive. The data acquisition process is repeated across multiple planes in the vessel or pipe; however,

**Figure 1.** *Basic operation principle of ERT. Differential reconstruction assumed.*

usually only one plane can be measured at any one time with most commercial hardware and software.
