## 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
Application of a Voltage Injected-Current Detected (V-C) Based Low Conductivity Sensitive Electrical Resistance Tomography (ERT)  
Although the EIS provides fast acquisition of data, this information is obtained in one dimension. This presents limitations for locating the crystal density distribution inside a reactor. Electrical resistance tomography is an imaging modality to visualize the binary or ternary phase flows and mixing in reactors based on the conductivity distribution. The conductivity distribution is visualized using electrical voltage or current measurements acquired from the boundary of the object. The electrical conductivity \(\sigma\) and electrical potential distribution \(\Phi\); are related to [PERSON]'s equation in the sensing field \(\Omega\). Assuming a source-free region, in the case of ERT it can be simplified in the form of Equation (3) [51]. \[\
abla(\sigma\
abla\Phi)=0\text{ in }\Omega \tag{3}\]  
The forward problem is shown in Equation (4), where potential differences V between the electrode pair of a given injection is evaluated using the knowledge of \(\sigma\). Usually, the finite element method (FEM) is used to govern Equation (3), to compute the numerical solution [51; 52]. \[V=F(\sigma) \tag{4}\]  
Figure 1: (**a**) Schematic showing antisolvent crystallization; (**b**) sucrose–water phase equilibrium diagram, Reproduced with permission from [50], Elsevier, 2016. In the inverse problem, the Jacobian matrix \(\mathsf{J}\) is evaluated using current \(\mathsf{I}\) in Equation (6), where the conductivity distribution \(\sigma\) is considered as piecewise constant and is expressed as in Equation (5) [53]. \[\sigma=\sum_{n=1}^{N}\sigma_{n}X_{n}, \tag{5}\]  
where \(X_{n}\) is the characteristic function, and \(X_{n}=1\) in the \(n^{th}\) element, and \(\Omega_{n}\) and \(X_{n}=0\) otherwise. \[\mathsf{J}=\frac{\partial I}{\partial\sigma_{n}} \tag{6}\]  
Novel reconstruction methods based on apriori information were developed by [PERSON] [54]. Various algorithms for ERT reconstruction have been discussed and reviewed in [55; 56]. Image reconstructions using voltage-current methods were discussed in [53]. The ERT reconstruction methods involve deterministic and probabilistic approaches amongst others. Bayesian reconstruction is a probabilistic method where the regularization technique is connected to the covariance of the prior density, which is different from the regularization practices involving a regularization parameter. Probabilistic methods can be used even in the case of systems which do not have probabilistic behaviour. In these types of methods, the problems are formulated based on the probability theory. All the probability densities are considered as continuous. The joint density of the random vector \(x=(x_{1},\ldots\ x_{N})^{T}\) is \(p(x)\). The joint density of the components of two random vectors \(x\) and \(y\) are denoted by \(p(x,y)\). The cross-covariance of \(x\) and \(y\) is defined as [54];  
\[C_{xy}=E\big{\{}xx^{T}\big{\}}-\eta_{x}\eta_{y}^{T} \tag{7}\]  
where, \(E\big{\{}xx^{T}\big{\}}\) is the correlation matrix of random vector \(x\) and \(\eta_{x}\) is the mean or expected value given by;  
\[\eta_{x}=\int_{-\infty}^{\infty}xp(x)dx \tag{8}\]  
The conditional density of \(x\) given \(y\) is defined as;  
\[p(x|y)=\frac{p(x,y)}{p(y)} \tag{9}\]  
Whenever \(p(y)\
eq 0\) and \(p(x|y)=0\) otherwise. Clearly, it is also;  
\[Z(\omega)=R(\omega)+jX(\omega) \tag{2}\]0  
Which helps in arriving at the [PERSON]'s theorem which is given as;  
\[Z(\omega)=R(\omega)+jX(\omega) \tag{2}\]1  
Mathematical models for ERT and models for forward and inverse problem using Bayesian theorems along with a comparison of simulated and experimental data were performed by [51]. In this work, the commercially available software solutions from Rocsole Inc. was used to implement Bayesian reconstruction. To complement the PAT evaluations in low conductivity ternary solutions, the low conductivity ERT monitoring data acquisition and sensor was designed and tested.