## 2 Application of 1-D EIS and 2-D ERT as a PAT in Crystallization Processes  
### Application of Electrical Impedance Spectroscopy to the Crystallization Process  
Impedance is dependent on the electrical conductivity of the solution. Electrical conductivity is calculated from the measurement of the corresponding material property such as resistance [41]. Fluid may have many dielectric mechanisms or polarization effects that contribute to its overall permittivity, such as ionic, dipolar, atomic, or electronic, depending on the frequency applied. Equation (1) gives the general Ohms law for calculating impedance \(Z\) and its dependence on the conductivity in phasor notation. Here sinusoidal signals at angular frequency \(\omega\) are considered where \(V\) denotes the electric voltage phasor applied to the sensor (or sensing cell) and 1 the resulting electric current phasor. The impedance \(Z\) is a complex number and has two components, namely magnitude and phase, as given in Equation (2). The equation of impedance in phasor form is  
\[Z(\omega)=\frac{V(\omega)}{I(\omega)}=A(\omega)e^{j\varphi(\omega)} \tag{1}\]  
where \(A\) is the magnitude (impedance module) and \(\varphi\) is the phase of the evaluated impedance. As a complex quantity, the impedance can be expressed in the rectangular form as follows:  
\[Z(\omega)=R(\omega)+jX(\omega) \tag{2}\]  
where \(R\) is the resistance and \(X\) is the reactance of the evaluated impedance. Impedance spectroscopy aims at measuring the impedances of the circuit, material, or any other component by applying the alternating voltage or current stimulus at different frequencies and measuring the response at these frequencies. These evaluations of the electrochemical behaviour of electrolyte materials are usually performed using two identical electrodes facing the sample inside a cylindrical cell [42]. In the field of electrochemical impedance spectroscopy [43], electrochemistry, instrumentation, process models, and statistical interpretation strategies have been discussed. Electrohydrodynamic impedances were evaluated to form a generalized transfer function of the non-electrical quantities. Multiple interpretation strategies for impedance representations and graphical methods were studied in detail. Due to the advancement in modern impedance acquisition equipment and the development of signal processing capabilities, it is now faster to acquire and process the measurement data [44; 45]. Several low and high accuracy methods for determining the conductivity of different types of liquids were studied along with fringe electric fields and interfacial effects due to the electrode area [46]. Impedance spectroscopy has been used from monitoring ions in aqueous solutions [47] to studying the cooling crystallization of ice [48]. In industrial processes, EIS has been used in batch crystallization processes as a 1-D PAT tool in order to characterize the amount of the precipitate in the mixture of the batch reactor. It is utilized to monitor the production and control of the quality of the product produced. Studies of the electrical behaviour of glucose using impedance spectroscopy [49] and impedance spectrum characterization after the addition of Na\({}_{2}\)CO\({}_{3}\) have been performed [30]. The study of impedance signatures as a PAT tool for sucrose crystallization was recently done by [31]. During the crystallization of sugar using ethanol as an antisolvent, there is a mixture of three components in the solution. The impedances of all these three individual components are different. The study of the impedance signatures of sucrose solutions at various concentrations and at various ethanol to water ratios, and to model them into a transfer function is our interest of study in this module. Figure 1a shows the schematic of the setup made for the antisolvent crystallization. Figure 1b shows the sucrose-water phase equilibrium.