# Development of a numerical model for measuring the electrical conductivity (EC) of a cake batter

Mamadou Lamine Niane, Olivier Rouaud, Anthony Ogé, Delphine Quéveau, Alain Le-Bail, Patricia Le-Bail

► To cite this version:

Mamadou Lamine Niane, Olivier Rouaud, Anthony Ogé, Delphine Quéveau, Alain Le-Bail, et al.  
Development of a numerical model for measuring the electrical conductivity (EC) of a cake batter.  
Journal of Food Engineering, 2026, 402, pp.112687. 10.1016/j.jfoodeng.2025.112687. hal-05171364

HAL Id: hal-05171364

https://hal.science/hal-05171364v1

Submitted on 5 Aug 2025

**HAL** is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

# Development of a numerical model for measuring the electrical conductivity (EC) of a cake batter

Mamadou Lamine Niane<sup>a, c, ●</sup>, Olivier Rouaud<sup>a, c, ●</sup>, Anthony Ogé<sup>a, c</sup>, Delphine Quéveau<sup>a, c</sup>, Alain Le-Bail<sup>a, c</sup>, Patricia Le-Bail<sup>b, c</sup>

## ARTICLE INFO

### Keywords:

Electrical conductivity measurement

Cake batter

Numerical simulation

Temperature non-uniformity

Starch gelatinization

## ABSTRACT

The electrical conductivity (EC) of materials represents their ability to conduct electrical current and determines the power dissipated within the material. This parameter can be temperature and electric field dependent. When EC is measured in the electric fields used for ohmic heating, the increase in temperature results in heating non-uniformity in the measuring cell. As a result, the relationship between EC and temperature cannot be accurately determined. In addition, for cake batter, starch gelatinization occurs during the measurement, leading to more complex EC curves. To address these issues, this study proposes a numerical method for determining EC that accounts for temperature non-uniformity. This model is coupled with a starch gelatinization model. The principle is based on the estimation of the EC using the method of least squares between the experimental temperature and the numerical one. The estimation of the EC of the cake batter consists of two steps: first, the device was characterized with xanthan and potassium chloride solutions of known electrical conductivities. A conversion efficiency of 0.77 was found. This efficiency was used to estimate the EC of the cake batter as a function of temperature and for different electric fields. Results showed that EC became independent of the electric field from 34.57 V/cm.

## 1. Introduction

Electrical conductivity (EC) is the ability of a material to conduct an electric current. EC can be considered as the most important influencing parameter in an ohmic heating (OH) process (de Alwis and Fryer, 1992; Mitchell and De Alwis, 1989; Palaniappan and Sastry, 1991a). As reported by

Zareifard et al. (2013), the success of OH depends on the EC, the rate of heat generation, the electric field strength and the residence time. Therefore, proper knowledge of the EC is essential for the design of an OH process. EC is a complex parameter, as it depends on temperature, frequency, electrolyte concentration, electrical potential gradient and starch gelatinization for bakery products (Assiry et al., 2006; Icier, 2012). Several authors have shown that the EC of liquids, formulated products and vegetables increases linearly with temperature between 20 and 80 °C (Fryer et al., 1993; Qihua et al., 1993). According to Knirsch et al. (2010), the increase in EC is due to the increase in the ionic mobility. For fluids with a high concentration of dissolved ions, EC is a

strong function of the frequency (Marcotte et al., 1998). This result was confirmed by Imai et al. (1995) in the OH of Japanese white radish. Depending on the frequency used, polarization phenomena may occur, leading to measurement errors. The extent of polarization is a factor to control during the measurement (Zareifard et al., 2013). According to Cummings and Torrance (1985), the effect of frequency is particularly important for the range of measured ECs. For commercial applications, the choice of frequency is usually a compromise between avoiding polarization problems without introducing capacitive errors into the measurements (Fryer et al., 1993; Mitchell and De Alwis, 1989).

According to several authors, the electric field has an increasing effect on EC (Filiz and Coscan, 2004; Palaniappan and Sastry, 1991a; Ramaswamy et al., 2014). This effect is due to electroosmotic effects (Halden et al., 1990; Palaniappan and Sastry, 1991b), and the osmotic pressure across a capillary medium is directly proportional to the electric field. However, Marcotte (1999) investigated the effect of electric fields on EC for four types of hydrocolloid solutions containing 0.5 % salt, using a Teflon cell at two electrode distances (14.33 and 20 cm) and

<footnote><sup>a</sup> Oniris, Nantes Université, CNRS, GEPEA, UMR 6144, F-44000, Nantes, France</footnote>

<footnote><sup>b</sup> INRAE, Biopolymères Interactions Assemblages, rue de la Géraudière, BP 71627, cedex 3, F-44316, Nantes, France</footnote>

<footnote><sup>c</sup> USC 1498 INRAE-TRANSFORM Department and GEPEA UMR CNRS 6144, France</footnote>

<footnote>* Corresponding author. Oniris, Nantes Université, CNRS, GEPEA, UMR 6144, F-44000, Nantes, France.</footnote>

<footnote>E-mail address: olivier.rouaud@oniris-nantes.fr (O. Rouaud).</footnote>
three electrical potentials (100, 250 and 400 V). The results indicated that the electric field had no effect on EC. Palaniappan and Sastry (1991b) had the same result while heating orange juice for electric field in the range of 30–60 V/cm. Yongsawatdigul et al. (1995) and Ramaswamy et al. (2014) found that a high salt concentration (3–4 %) in surimi was necessary to observe the effect of the electric field on EC.

While the influence of the electric field is not obvious, the starch gelatinization in bakery products has an effect on the EC. Li et al. (2004) investigated the ohmic heating of starch suspensions at 90 °C with an electric field of 10 V/cm and a frequency of 50 Hz. The results showed that the EC evolved linearly with temperature, except in the gelatinization range. Gally et al. (2016) obtained similar results on their work on unfermented bread batter by OH. Three phases were distinguished on the EC curve: (1) EC increased with temperature, (2) EC slightly decreased or became almost constant (plateau), and (3) EC increased again with temperature. The second phase was the gelatinization phase, during which structural changes occur with an increase in bound water (Kaur and Singh, 2016; Wang and Sastry, 1997). According to Li et al. (2004) and Kaur and Singh (2016), the main reason for the plateau (phase 2) is probably due to the fact that, during gelatinization, the swelling of the starch granules reduces the area of movement of the charged particles and increases the resistance to movement of the swollen particles.

The EC of foodstuffs is generally measured using an ohmic device consisting of 2 electrodes of surface area S facing each other, separated by a distance d and between which the product is placed. The EC is then calculated using Eq. (1).

$$ \sigma = \frac{d I}{S V} \qquad (1) $$

where I is the electrical current and V is the electrical potential. Knowing the dimensions of the device (S and d), it is theoretically possible to calculate the EC. This method is not recommended because it requires knowledge of the exact geometry of the sample, which is difficult to obtain for small geometries. Since most of the authors develop their own devices (Marcotte, 1999; Mitchell and De Alwis, 1989; Palaniappan and Sastry, 1991a), these geometries are not optimized and can lead to measurement uncertainties. The common practice is to calibrate the cell by estimating the d/S parameter with solutions whose ECs are precisely known at a reference temperature (Zareifard et al., 2013). This is the most common method for calculating the EC as a function of temperature (Gally et al., 2016; Khodeir et al., 2021; Marcotte, 1999; Marcotte et al., 1998). When high electric fields, such as those found in OH, are used during the measurement, the temperature rises and causes gradients near the electrodes. According to Ito et al. (2014), the electrodes are responsible for this temperature gradient. Due to their low electrical resistivity, they do not heat up under the effect of the electric field, making them cold spots. Conversely, their thermal conductivity is high, which has the effect of easily dissipating heat from the product (Gally et al., 2016). The coldest point in an ohmic cell is at the corner between the electrodes and the ohmic device wall, due to the combination of the heat transfer from the wall to the outside and the heat dissipation from the electrode (Ito et al., 2014). To minimize these gradients, Zell et al. (2011) found that the temperature at the electrode surface could be increased by using thinner electrodes. However, temperature gradients are never eliminated. In this case, the standard method of EC measurement is unsuitable to obtain the temperature-dependent EC used in OH processes.

The aim of this work was to accurately estimate the EC of a cake batter during baking (i.e. at high electric field). This knowledge is absolutely necessary to improve the robustness of numerical models used to reliably predict solidification due to starch gelatinization as a function of temperature during ohmic baking. As shown in (Niane et al., 2024), it is a key physical phenomenon for applications such as 3D food printing where precise control of solidification is required to guarantee

the quality and accuracy of printed shapes. In addition to its intended use for cake batter, the method can also be used for other products.

To achieve this objective, a numerical method for estimating EC that takes into account temperature gradients has been proposed in this study. After a numerical calibration of the measurement cell with known solutions, the method was used to develop a model of EC for a cake batter. In this model, the effect of the starch gelatinization and the influence of the electric field were taken into account in the calculations. The frequency of the electric field was set at 50 Hz. This method consisted of estimating the EC from the measured temperature profiles at varying positions within the ohmic device. The Nelder-Mead Simplex Method was employed to align the numerical and experimental temperature profiles. The entire problem was solved using the Comsol Multiphysics® 5.6 software. The method consisted of two steps.

i) Characterization of the ohmic device: the aim was to determine the conversion efficiency ($η_{conv}$) of the device, defined as the proportion of electrical energy ($V_0 \times I, W$) converted into heat ($Q_{abs}, W$) as shown in Eq. (2). For a conversion efficiency $η_{conv}$ of 1, all the electrical energy is converted into heat. However, in practice, the efficiency is less than 1 due to faradic reactions (Samaranayake and Sastry, 2014).

$$ \eta_{\text{conv}} = \frac{Q_{\text{abs}}}{V_0 I} \qquad (2) $$

$V_0$ (V) is the initial electrical potential, and I (A) is the electrical current circulating between the electrodes. Two solutions of known ECs were used: xanthan and potassium chloride solutions. Xanthan solutions were used to determine the efficiency $η_{conv}$ while potassium chloride was used to validate this efficiency estimation.

ii) Estimation of the batter's EC: the efficiency $η_{conv}$ was then used to estimate the EC of the batter.

## 2. Ohmic cell characterization

### 2.1. Material and methods

#### 2.1.1. Ohmic device

The device used in this study is presented in Fig. 1. It was developed in previous projects (Gally, 2017; Khodeir, 2020). The length of the device (1) is 60.5 mm, and it comprises two electrodes (2) in titanium, with a distance of 40.5 mm between them. The electrodes have a thickness of 2 mm and a diameter of 29 mm. Each electrode is inserted into a polyacetal plug of a total thickness of 20 mm, with 10 mm within the device and 10 mm outside. The plugs are perforated to facilitate electrical connection to the electrodes. The device exhibits inner and outer diameters of 29 and 42 mm, respectively. When the electrical potential is applied to the electrodes via a generator (3), the values of electrical current (I) and the temperature are recorded over time via a DATALOG (AOIP-France) program on a computer (4) connected to a data acquisition unit (5). Two temperature probes were used: one positioned centrally within the product (T₁) and another situated 2 mm from the device wall and the electrode (T₂). This device was used for all 3 products (xanthan, KCl and batter).

#### 2.1.2. Preparation of xanthan solutions

The xanthan used was of 99 % purity. The xanthan solutions were prepared using the method proposed by Marcotte et al. (1998), which involves dispersing the xanthan powder in a distilled water heated to 80 °C and blended at 700 rpm for 10 min. The solutions were subjected to centrifugation at 2000 rpm for a period of 5 min, with the objective of removing any air bubbles that may have formed. The solutions were prepared at three concentrations: S1 (0.4 % w/w), S2 (0.65 % w/w), and S3 (1 % w/w).
<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Left: experimental system: (1) OH device, (2) Electrodes and plugs, (3) Generator, (4) computer, (5) Data acquisition unit. Right; details of the OH device with its electrodes.</figcaption></figure>

### 2.1.3. Model development

The electrical potential field within the heating section was calculated by solving the Laplace equation (Eq. (3)).

$$ \nabla(\sigma\nabla V) = 0 \qquad (3) $$

where $\sigma$ (S.m$^{-1}$) is the EC, V is the electrical potential and $\nabla$ is the Nabla operator.

The heat equation for the heat transfer through convection and conduction in fluids was employed (Eq. (4)).

$$ \rho C_p \frac{\partial T}{\partial t} + \rho C_p u \nabla T = \nabla \cdot (\lambda \nabla T) + \dot{Q}_{abs} \qquad (4) $$

where $C_p$ (J kg$^{-1}$ K$^{-1}$) is the heat capacity, $\lambda$ (W m$^{-1}$ K$^{-1}$) is the thermal conductivity, $\rho$ (kg m$^{-3}$) is the density and $\dot{Q}_{abs}$ is the volumetric heating power (Eq. (5)).

$$ \dot{Q}_{abs} = \eta_{conv} \sigma |\nabla V|^2 \qquad (5) $$

$\nabla V$ is calculated numerically by the software from Eq. (3).

As xanthan solutions are gels, there is no natural convection ($u = 0$), and heat transfer is by conduction. The boundary conditions are summarized in Table 1.

Where **n** is the normal vector, J (A m$^{-2}$) is the current density, h (W m$^{-2}$ K$^{-1}$) is the convection coefficient. It was directly calculated by COMSOL using the standard correlations integrated into the software. It was observed that its value varies along the wall between 3.2 and 4.6 W m$^{-2}$ K$^{-1}$. $\sigma_{bz}$ (1.380649 × 10$^{-23}$ J K$^{-1}$) is Boltzmann's constant. The emissivity $\epsilon$ was considered at 0.9. The initial temperature was 24 °C. The EC of each solution was calculated from the data of Marcotte et al. (1998) and was expressed by a linear equation in the range 20 °C–100 °C as shown in Table 2.

The heat capacity (Cp) and thermal conductivity ($\lambda$) were determined using data from Zhang et al. (2020). Specific heat values were extracted at 25, 35, 45, and 55 °C for xanthan concentrations between 0.05 % and 0.35 %, while thermal conductivity data were obtained for concentrations from 0.1 % to 0.5 %. Quadratic regressions allowed extrapolation to higher concentrations up to 1 % and equations are given in Table 3.

<table><caption>**Table 1**
Boundary and initial conditions.</caption><thead><tr><th>Boundary</th><th>Description</th><th>Value</th></tr></thead><tbody><tr><td><strong>Electrostatic</strong></td><td></td><td></td></tr><tr><td>Plugs, wall</td><td>Electric insulation</td><td>nJ = 0</td></tr><tr><td>Electrode 1</td><td>Electric potential</td><td>V<sub>0</sub> (Table 5)</td></tr><tr><td>Electrode 2</td><td>Ground</td><td>V = 0</td></tr><tr><td><strong>Heat transfer</strong></td><td></td><td></td></tr><tr><td>Plugs, wall</td><td>Natural convection and radiation</td><td>nλ∇T = h(T<sub>amb</sub> - T) + εσ<sub>bz</sub>(T<sup>4</sup><sub>amb</sub> - T<sup>4</sup>)</td></tr><tr><td><strong>Initial condition</strong></td><td>Temperature</td><td>24 °C</td></tr></tbody></table>

<table><caption>**Table 2**
ECs of xanthan solutions (Marcotte et al., 1998).</caption><thead><tr><th>Solution</th><th>σ (T)</th></tr></thead><tbody><tr><td>S1 (0.4 %)</td><td>σ = 0.05303 + 0.00142 (T - 25)</td></tr><tr><td>S2 (0.65 %)</td><td>σ = 0.0868 + 0.0023 (T - 25)</td></tr><tr><td>S3 (1 %)</td><td>σ = 0.1394 + 0.0041 (T - 25)</td></tr></tbody></table>

The properties of the solid materials were found in the COMSOL library (Table 4).

The operating conditions are presented in Table 5. The experiments were replicated six times.

Comsol Multiphysics® 5.6 software was used to solve the entire problem on a Dell Precision Tower 7810 workstation (2 × 2 GHz Intel Xeon processors, 131.072 GB (56 CPUs) of RAM, Windows 10 Professional, 64 bit). To optimize computation time, a 2D axisymmetric was used. The mesh consisted of 3648 triangles, 508 boundary elements, 23,274 degrees of freedom and the average mesh quality was 0.812. The calculation time required was 20s with the following modules selected in Comsol Multiphysics® 5.6: Heat Transfer, Electric Current, Laminar Flow for KCl where natural convection occurs and General Optimization.

### 2.2. Conversion efficiency of the device

Fig. 2 shows how temperatures and electrical currents change over

<table><caption>**Table 3**
Thermophysical properties of xanthan solutions.</caption><thead><tr><th colspan="4">Density</th></tr><tr><th>T(°C)</th><th>S1 (0.4 %)</th><th>S2 (0.65 %)</th><th>S3 (1 %)</th></tr></thead><tbody><tr><td>25</td><td>998.68</td><td>997.49</td><td>999.79</td></tr><tr><td>30</td><td>999.60</td><td>996.36</td><td>996.18</td></tr><tr><td>40</td><td>979.35</td><td>989.65</td><td>986.89</td></tr><tr><td>50</td><td>959.67</td><td>977.02</td><td>974.84</td></tr></tbody><tbody><tr><th colspan="4">Thermal properties rowhead</th></tr><tr><td>Cp (J kg<sup>-1</sup> K<sup>-1</sup>)</td><td>(-6.5 10<sup>-3</sup>T<sup>2</sup> - 0.31T - 1124.9)%<sub>xanthan</sub> + (-3.35 10<sup>-2</sup>T<sup>2</sup> + 2.96T + 4263.3)</td></tr><tr><td>λ (W m<sup>-1</sup> K<sup>-1</sup>)</td><td>(-9 10<sup>-6</sup>T<sup>2</sup> + 5.7 10<sup>-4</sup>T + 6.38 10<sup>-3</sup>)%<sub>xanthan</sub> + (-1.7 10<sup>-5</sup>T<sup>2</sup> + 2.32 10<sup>-3</sup>T + 0.55082)</td></tr></tbody></table>

where T is the temperature (°C) and %xanthan is the concentration of xanthan (0.4, 0.65 or 1).
<table><caption>**Table 4**
Physical properties.</caption><thead><tr><th>Properties</th><th>Polyacetal</th><th>Titanium</th></tr></thead><tbody><tr><td>ρ (kg m<sup>-3</sup>)</td><td>1410</td><td>4940</td></tr><tr><td>σ (S m<sup>-1</sup>)</td><td>10<sup>-12</sup></td><td>7.407,10<sup>5</sup></td></tr><tr><td>λ (W m<sup>-1</sup> K<sup>-1</sup>)</td><td>0.39</td><td>7.5</td></tr><tr><td>Cp (J kg<sup>-1</sup> K<sup>-1</sup>)</td><td>1400</td><td>710</td></tr></tbody></table>

<table><caption>**Table 5**
Operating conditions for temperature measurements.</caption><thead><tr><th>Xanthan solution (% w/w)</th><th>Electrical potential (V)</th></tr></thead><tbody><tr><td>S1(0.4 %)</td><td>120; 150</td></tr><tr><td>S2(0.65 %)</td><td>120; 150</td></tr><tr><td>S3(1 %)</td><td>80; 120</td></tr></tbody></table>

time. The presentation of a single curve is justified by the high degree of reproducibility observed in the measurements. Fig. 2a and b and c illustrate the temperatures given by thermocouple number 1 (T₁) and number 2 (T₂), the impact of electrical potential, and the weight concentration on the temperature profile, respectively. As shown in Fig. 2a, the temperature at the centre (T₁) increases at a faster rate than the temperature near the wall (T₂). This serves to confirm the non-uniformity of the heating process. Fig. 2b and c demonstrate that the electrical potential and the concentration of xanthan increase the heating rate. This is because an increase in electrical potential results in an increase in the volumetric heating power (Q̇<sub>gen</sub>), which consequently leads to an increase in the electrical current (Fig. 2d). The weight concentration is proportional to the concentration of dissolved ions, which facilitate the circulation of the electrical current (Fig. 2e).

The conversion efficiency was estimated from the experimental temperature profiles measured at the centre (T₁) and near the wall (T₂), obtained under different electrical voltages and for each xanthan solution. The simulation of the non-linear coupled equations (3)–(5) allows

for numerically solving the problem and obtaining the temperatures as a function of time. The Nelder-Mead Simplex Solver, available in the COMSOL general Optimization module, was used to minimize the root mean square difference between the experimental and numerical temperatures, thereby obtaining an optimized value of the efficiency. Fig. 3a and b and c illustrate the discrepancy between the experimental and numerical data of the temperature (T₁ and T₂) for solutions 0.4 %w/w, 0.65 %w/w and 1 %w/w, respectively. Fig. 3d shows the conversion efficiency obtained from each solution, with the 0.4 %w/w solution represented in blue, the 0.65 %w/w solution in grey, and the 1 % w/w solution in green. Additionally, Fig. 3d illustrates the comparison between the numerical and experimental current. The correlation factor of the mathematical equation that links the numerical and experimental temperatures is approximately 1, indicating that the discrepancy between the two is minimal. The efficiency of each solution exhibits a slight variation in response to changes in electrical potential. It can thus be concluded that the effect of the electric field is negligible. The efficiency declines in direct proportion to the concentration of xanthan. The respective values for S1, S2, and S3 are 0.77 ± 0.04, 0.62 ± 0.03, and 0.48 ± 0.03. As the weight concentration is correlated to the EC, the efficiency is observed to decrease with an increase in EC (high ion concentration). As the ion concentration increases, the model’s ability to accurately predict the electric current decreases. The energy efficiency of ohmic heating depends on many factors: the type of current used, the material of the electrodes, the conductivity of the fluid, the control of the power applied, the thermal non-uniformity and the device design. The nature and frequency of the power supply also play a decisive role. At low frequencies (50–60 Hz), electrochemical reactions such as electrode corrosion and fluid electrolysis can occur, causing energy losses and resulting in efficiency drops of up to 67 %. However, by using specific electrodes, such as dimensionally stable anodes (DSA), or high-frequency currents, these phenomena can be limited and thermal efficiencies of over 90 % can be achieved. The geometry of the cell and the distribution of the electric field determine the thermal homogeneity of the fluid in ohmic heating. A uniform geometry and a well-distributed

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Experimental data as a function of time a) temperatures given by thermocouples number 1 (T₁) and number 2 (T₂) for S1 at 120V, b) temperature T₁ for S2 at 120 and 150 V, c) temperature T₁ for S1, S2 and S3 at 120 V d) electrical current for S2 at 120 and 150 V, e) electrical current for S1, S2 and S3 at 120 V.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 3. (a), (b) and (c): Experimental vs numerical temperatures (T₁ and T₂) for 0.4 %, 0.65 % and 1 % w/w solutions respectively, (d) Efficiency, (e) Comparison between the numerical and electrical current.</figcaption></figure>

field are essential to avoid temperature gradients and limit overheating or underheating zones, which have an impact on efficiency. In addition, non-thermal losses, such as electrode fouling, can reduce efficiency, particularly with complex solid-liquid products (Ramaswamy et al., 2014). Another phenomenon that can occur in ohmic heating processes is polarization. Phenomenon of polarization can occur at low-frequency and results in electrical current losses. The extent of polarization depends on several factors, the most important are the nature of the electrode surface and the frequency of the alternating current (Cummings et al., 2003). In our case where a low frequency (50Hz) was used, the efficiency decreases with the concentration that would be correlated to the speed of electrochemical reactions and polarization phenomena.

These phenomena should be less significant for solution 1 and then differences between numerical and experimental current are less important. As these phenomena are not considered in the model, solution 1 gives a better estimate of the efficiency. In addition, solution 1 (0.4 %w/w) covers an EC range closer to that of the batter used by Khodeir et al. (2021) (0.05–0.21 S/m between 20 °C and 100 °C). The only difference with the cake batter used in this study was the addition of baking powder, which could increase porosity and consequently decrease EC. As a result, the efficiency conversion of the ohmic device was 0.77 and has been retained for the rest of the study. The following step was to validate the efficiency (ηₐ₅ₐ) using KCl solution having an EC within the EC range of the 0.4 %w/w xanthan solution.

### 3. Validation of the efficiency with potassium chloride solution (KCl)

Efficiency validation involved a determination of whether it was possible to estimate a known EC of a KCl solution, assuming the ohmic device efficiency to be 0.77.

#### 3.1. Material and methods

##### 3.1.1. KCl solution

A 0.03 % w/w KCl solution was prepared with distilled water. The EC range of this solution (0.05–0.13 S/m between 20 and 100 °C) calculated from Eq. (6) is included in that of the 0.4 % xanthan solution. This equation was determined using the data cited in Gally (2017).

$$ \sigma = \sigma_{25} (1 + m_T (T - 25)) \sigma_{25} = 0.0631 \text{ S m}^{-1} \text{ and } m_T = 0.0174 \text{ K}^{-1} \quad (6) $$

##### 3.1.2. Model development

The potassium chloride solution is liquid and therefore potentially subject to natural convection. Natural convection is caused by buoyancy forces acting on the fluid in which a density gradient exists (Bergman et al., 2011) and was therefore considered for the KCl solution (u ≠ 0). It is described by the Navier-Stokes equations (Eq. (7)).

$$ \frac{\partial \rho u}{\partial t} + \nabla \cdot (\rho u \cdot u) = - \nabla p + \nabla \cdot \bar{\tau} + \rho g \quad (7) $$

$\bar{\tau}$ is the viscous stress tensor (Pa), g acceleration of gravity, p is the
hydraulic pressure. Eq. (7), Laplace equation (Eq. (3)) and temperature (Eq. (4)) were then solved. Boundary conditions for heat transfer and electrostatics presented in Table 1 remained unchanged. For the natural convection, the initial velocity was zero and the no-slip conditions at the wall were considered. The heat capacity was calculated from the formula below:

$$C_p^{\text{solution}} = x_{\text{KCl}}C_p^{\text{KCl}} + x_{\text{water}}C_p^{\text{water}} \quad (8)$$

where $x_{KCl}$ and $x_{water}$ are mass fractions of KCl and water respectively, $C_{pKCl}$ (5.2 T(°C)+2260.38)

J kg⁻¹ K⁻¹) is the heat capacity of the cristal KCl (El-Sharkawy et al., 1984) and $C_{pwater}$ (4181.4)

J kg⁻¹ K⁻¹) the heat capacity of water (Engineering Toolbox, 2004). The thermal conductivity (0.64 W m⁻¹ K⁻¹) and the viscosity ((-1E-05 T (°C) + 0.0013 Pa s) were calculated from the data found in the literature (Kestin et al., 1981; Ramires & Nieto De Castro, 2000). The density ((-0.7268 T(°C)+1016.9) kg m⁻³) was measured at different temperatures between 22 °C and 50 °C.

Temperature profiles were measured at 100 V. The experiments were replicated four times, and the parameters of the numerical EC ($\sigma_{25}$ and $m_T$) were estimated.

## 3.2. Validation of the conversion efficiency

Fig. 4a and b displays the comparison between the numerical and experimental temperature profiles in the centre ($T_1$) and near the wall ($T_2$) respectively for one experiment among the four. The coefficient of

correlation $R^2$ was calculated from Eq. (9) to assess the accuracy of the model. The results obtained were 0.9987 for $T_1$ and 0.9970 for $T_2$, indicating good agreement between the numerical and experimental data.

$$R^2 = 1 - \frac{\sum (T_{\text{experimental}} - T_{\text{numeric}})^2}{(T_{\text{experimental}} - \bar{T}_{\text{experimental}})^2} \quad (9)$$

Fig. 4c and d illustrate the ECs and the comparison between numerical and experimental current respectively. The average numerical values of the parameters estimated from the different temperature profiles are 0.0602 ± 0.0038 S/m for $\sigma_{25}$ and 0.0176 ± 0.0006 S m⁻¹ K⁻¹ for $m_T$. The model predictions are in the interval ±5 % of the value of the EC from the literature. For the electrical current, there is a difference of 20 mA between the numerical and experimental current. The model predictions are in ±20 % of the value of the experimental current. We can be satisfied of this result, given the uncertainties of experimental measurements on the efficiency and the uncertainties that can result from properties taken from the literature. The efficiency of 0.77 was then validated. It was considered that the efficiency is independent of the product in this EC values range. Therefore, it is possible to find the EC of other products with EC ranges close to this solution, including wheat flour - based batter.

<figure><img src="image_6.png" /><figcaption>Fig. 4. (a) Experimental vs numerical temperature $T_1$, (b) Experimental vs numerical temperature $T_2$, (c) numerical (blue) and experimental (pink) conductivity vs temperature of KCl solution and (d) Comparison between experimental and numerical current.</figcaption></figure>
# 4. Estimation of the EC of the cake batter

## 4.1. Material and methods

### 4.1.1. Cake batter recipe

The product used in this study was based on a cake batter recipe (29.5 % of wheat flour, 25 % of sugar, 20 % of fat, 25 % of egg and 0.5 % of salt) used in previous study (Khodeir et al., 2021). The water content in the recipe was measured and found to be 22 % w/w. A baking powder (mixture of Sodium Bicarbonate and Sodium Pyrophosphate Acid) was added to this formula to obtain a honeycomb cake structure.

### 4.1.2. Model development

Due to the high viscosity of the batter and as the cake batter solidifies with temperature, it was assumed that there was no mass transfer and natural convection in the batter. Laplace and temperature equations (Eqs. (3) and (4) respectively) were solved. The convection term in Eq. (4) is zero (no natural convection). The boundary conditions presented in Table 1 were used also for the batter. The density, thermal conductivity and heat capacity were determined in the study of Niane et al. (2024). The density was 943 ± 7 kg m⁻³. Fig. 5 shows the thermal conductivity and heat capacity that are temperature-dependent, which make the equations of the model non-linear.

To our knowledge, there is no mathematical equation of the EC of a batter considering the starch gelatinization. However, Marcotte et al. (1998), using a statistical analysis, found an equation of the EC of hydrocolloids solutions over temperature and solid rate (Sₜ) which have a negative effect on the EC (Eq. (10)).

$$ \sigma = \sigma_{25} + K_T (T - 25) + K_{TS_r} (T - 25) S_r \quad (10) $$

Where σ₂₅, Kₜ and KₜSₜ are constants. By analogy, a model of EC of the cake batter as a function of temperature and gelatinization rate α (Eq. (11)) was proposed in this study. The first two terms on the right-hand side of Eq. (11) represent the linear relationship between EC and temperature. The last term is the interaction effect.

$$ \sigma = \sigma_{25} + K_T (T - 25) + K_{T\alpha} (T - 25)\alpha \quad (11) $$

Mathematically, the degree of starch gelatinization (α) follows first-order kinetics (Eq. (12)) where the reaction rate constant k (Eq. (13)) varies with temperature according to the Arrhenius equation (Chhanwal and Anandharamakrishnan, 2014; Pérez-Santos et al., 2016; Zanoni et al., 1995; Zanoni et al., 1995).

$$ (1 - \frac{\alpha}{\alpha_0}) = \exp(-k t) \quad (12) $$

$$ k = k_0 \exp \left( -\frac{E_a}{RT} \right) \quad (13) $$

where α₀ is the maximum gelatinization rate at a given temperature, t (s) is the time, k (s⁻¹) is the reaction rate constant, k₀ (s⁻¹) is the pre-exponential factor, Eₐ (J mol⁻¹) is the activation energy and R (J mol⁻¹ K⁻¹) is the ideal gas constant.

The parameters α₀, Eₐ and k₀ were estimated from experimental data on the rate of starch gelatinization with temperature of Asamoah (2023) who worked on baking the same cake batter by OH under a temperature rise of 3.5 °C/min. Gelatinization rates were determined by Differential Scanning Calorimetry (DSC) which measures energy variations associated with phase transitions batter-cake. The rate of partial gelatinization was measured on the cake batter (ΔHₐbatter) in excess of water and on the cake after baking (ΔHₐsample) also in excess of water. The gelatinization rate was calculated according to the equation below:

$$ \alpha = \frac{\Delta H_{\text{batter}} - \Delta H_{\text{sample}}}{\Delta H_{\text{batter}}} \quad (14) $$

Fig. 6 shows the experimental gelatinization rate and the one obtained from the model. The parameters α₀, Eₐ and k₀ estimated from the experimental data were 0.6, 144,455 J mol⁻¹ and 5.36 10¹⁸ s⁻¹ respectively. The parameters Eₐ and k₀ are comparable with the ones found in the literature of bakery product (Eₐ = 138 kJ mol⁻¹, k₀ = 2.8 10¹⁸ s⁻¹ in Zanoni et al. (1995)). We observed that starch began to gelatinize at around 60 °C, the onset temperature for most types of starch (Pérez-Santos et al., 2016; Pineda-Gómez et al., 2010). A rapid increase in gelatinization rate was observed, reaching 0.6 at 82 °C before

<figure><img src="image_7.png" /><figcaption>Fig. 6. Gelatinization rate over temperature for a rise of 3.5 °C/min during OH; experimental values from Asamoah (2023).</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 5. a) Heat capacity b) thermal conductivity of the cake batter (Niane et al., 2024).</figcaption></figure>
becoming constant. The other parameters to be estimated were $\sigma_{25}$, $K_T$ and $K_{T\alpha}$.

The experiments were replicated six times at 65V corresponding to a temperature rise of 3.5 °C/min the same used for the determination of the activation energy (E_a) and the pre-expansion factor (k₀) in the previous work of Asamoah (2023).

## 4.2. Results and discussion

The parameters $\sigma_{25}$, $K_T$ and $K_{T\alpha}$ were estimated by minimizing the temperature difference between experimental and numerical values. Parametric estimation was carried out in two steps.

- The parameter $K_{T\alpha}$, which characterizes the negative effect of starch gelatinization on EC, was initially estimated. A series of negative values were evaluated, and the value that resulted in a plateau (gelatinization plateau) on the EC curve was selected. The value of $K_{T\alpha}$ was found to be: $-0.0018 \pm 0.0001 \text{ S m}^{-1} \text{ K}^{-1}$.

- The parameters $\sigma_{25}$ and $K_T$ were estimated by applying the Nelder-Mead Simplex Method to the experimental and numerical temperature profiles. The estimated parameters were $0.03352 \pm 0.0031 \text{ S/m}$ for $\sigma_{25}$ and $0.003501 \pm 0.0001 \text{ S m}^{-1} \text{ K}^{-1}$ for $K_T$. Fig. 7 shows the evolution of experimental and numerical temperature (a), electrical current (b) and EC. The difference between experimental and numerical temperature are in the range of 0.6–5 °C. For the electrical current, the differences between the experimental and numerical results are in the range of 0–30 mA. These differences were more pronounced during the gelatinization phase, wherein the numerical electrical current assumes a greater significance than the experimental one. These differences could be due to the decrease of the efficiency during the gelatinization phase. As mentioned by Gally et al. (2016) and Li et al. (2004) with starch solutions, the EC curve presents three phases in the range 0 °C–100 °C:

- phase 1 (23–65 °C): the EC demonstrates a linear increase ($R^2 = 0.993$) with temperature up to the onset temperature of 65 °C, exhibiting a slope of $0.0039 \text{ S m}^{-1} \text{ K}^{-1}$.

- phase 2 (65–82 °C): the EC continues to increase at a slower rate due to the negative value of the $K_{T\alpha}$ parameter ($-0.0018 \pm 0.00010 \text{ S m}^{-1} \text{ C}^{-1}$). This is the gelatinization phase. Starch gelatinization leads to changes in structure and properties. Gelatinization is accompanied by the absorption of water by the starch granules, followed by their swelling, which modifies the rheology of the batter. Held together by associative bonds, the swollen granules form a continuous gel-like system (Leach, 1965), giving a three-dimensional network in which water is both bound and entrapped (Ott and Hester, 1965). It therefore reduces significantly water mobility (Collison and McDonald, 1960). This matrix structure, the decreasing unbound water

and the change of the rheology affect the current flow and thus the EC.

- phase 3: from 82 °C, the EC increases linearly ($R^2 = 0.999$) with a slope of $0.0014 \text{ S m}^{-1} \text{ C}^{-1}$. All the starch has been gelatinized at the end of phase 2. The current flow is no longer affected, and the EC increases again. As mentioned by Li et al. (2004), the slopes observed before and after gelatinization differ and the one obtain after gelatinization is less important. According to Gally (2017), the completion of starch gelatinization would put an end to its negative impact on EC, thereby allowing the positive effect of temperature to fully take place and leading to a further increase in EC. However, structural changes in the batter, such as changes in rheology and the reduction of the liquid phase between starch granules, would mitigate this thermal effect, thus limiting the rate at which EC increases.

This EC was obtained at 65 V, which is equivalent to an electric field intensity of 16.05 V/cm. Additionally, the EC was also calculated for other electrical potentials. The electric fields were calculated for 100, 120, 140, and 160 V, which correspond to the respective values of 24.69, 29.63, 34.57, and 39.51 V/cm. The gelatinization parameters identified in the preceding analysis were considered to be independent of the electric field and were therefore incorporated into the simulations. Fig. 8 shows the ECs of the batter for each electrical potential. For each voltage, a single curve is presented, given the high degree of repeatability observed in the measurements. The results show that the EC increased with increasing voltage, reaching a maximum at 120–140 V. According to Halden et al. (1990) and (Palaniappan and Sastry, 1991a),

<figure><img src="image_9.png" /><figcaption>Fig. 8. EC of cake batter at different electrical potentials.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 7. (a) Numerical vs experimental temperature, (b) numerical vs experimental current, (c) EC of the batter.</figcaption></figure>
this increase is mainly due to electro-osmotic effects. Furthermore, it has been shown that the osmotic pressure across a straight capillary media was directly proportional to the voltage gradient (Crow, 1988 as cited in Ramaswamy et al., 2014).

The influence of the electric field was particularly pronounced during the starch gelatinization process and in the subsequent post-gelatinization phase. It is noteworthy that the gelatinization plateau was observed to shift to higher temperatures as the electric field increased. Furthermore, from 34.57 V/m (140 V), the EC appeared to become independent of the electric field.

## 5. Conclusion

The aim of this study was to develop a numerical method for estimating the EC of a cake batter as a function of temperature in the context of high electric fields, such as those observed in ohmic heating processes. The estimation was carried out using the inverse method, specifically the least squares method (Nelder-Mead simplex method), which was applied to the numerical and experimental temperature profiles. The estimation was done in two steps. First, a characterization of the device was conducted using xanthan and KCl solutions, with the objective of determining the efficiency conversion in the EC range corresponding to the product under study (cake batter). A conversion efficiency of 0.77 enabled the validation of a model for estimating the EC of both solutions in the presence of temperature non-uniformity. Subsequently, an estimation of the EC of the cake batter was performed. Laplace and heat transfer equations, including a model for the starch gelatinization process, were solved using Comsol Multiphysics® 5.6. The EC of the cake batter was then studied as a function of the electric field. Increasing the electrical field led to an increase in EC, which stabilized at 34.57 V/cm (140 V). This EC can thus be used as an input parameter of OH processes

dedicated to cake baking. Moreover, the methodology developed for EC measurement can now be applied to other liquid and solid food materials.

## CRediT authorship contribution statement

**Mamadou Lamine Niane:** Writing – original draft, Methodology, Investigation. **Olivier Rouaud:** Writing – review & editing, Supervision, Conceptualization. **Anthony Ogé:** Methodology. **Delphine Quéveau:** Methodology. **Alain Le-Bail:** Writing – review & editing, Project administration, Funding acquisition. **Patricia Le-Bail:** Writing – review & editing, Supervision.

## Research ethics

Not applicable.

## Research funding

The present paper/communication has been co-funded by ONIRIS-GEPEA and by the FEDER European funds Ref 2020/FEDER/n°PL0019794 within the international Chair “CONT-E-FOOD” on continuous food processes under electrical disturbances. Disclaimer: the content found on this contribution reflects only the author’s view. The EU commission is not responsible for any use that may be made of the information it contains.

## Declaration of competing interest

The authors state no conflict of interest.

## Nomenclature

<table><tr><td>Cp</td><td>Heat capacity</td><td>J kg<sup>-1</sup> K<sup>-1</sup></td></tr><tr><td>d</td><td>Distance between electrodes</td><td>m</td></tr><tr><td>E<sub>a</sub></td><td>Activation energy</td><td>J mol<sup>-1</sup></td></tr><tr><td>g</td><td>Acceleration of gravity</td><td>m.s<sup>-2</sup></td></tr><tr><td>ΔH</td><td>Enthalpy difference from DSC curves</td><td>J kg<sup>-1</sup></td></tr><tr><td>I</td><td>Current</td><td>A</td></tr><tr><td>k</td><td>Reaction rate constant</td><td>s<sup>-1</sup></td></tr><tr><td>k<sub>0</sub></td><td>Pre-exponential factor</td><td>s<sup>-1</sup></td></tr><tr><td>Q<sub>abs</sub></td><td>Source term</td><td>W m<sup>-3</sup></td></tr><tr><td>R</td><td>Ideal gas constant</td><td>J mol<sup>-1</sup> K<sup>-1</sup></td></tr><tr><td>S</td><td>Electrodes surface</td><td>m<sup>2</sup></td></tr><tr><td>T</td><td>Temperature</td><td>°C or K</td></tr><tr><td>t</td><td>Time</td><td>s</td></tr><tr><td>V</td><td>Electrical potential</td><td>V</td></tr><tr><td>V<sub>0</sub></td><td>Potential difference across the electrodes</td><td>V</td></tr><tr><td>η<sub>conv</sub></td><td>Conversion efficiency</td><td>–</td></tr><tr><td>α</td><td>Gelatinization rate</td><td>–</td></tr><tr><td>ρ</td><td>Density</td><td>kg m<sup>-3</sup></td></tr><tr><td>λ</td><td>Thermal conductivity</td><td>W K<sup>-1</sup> m<sup>-1</sup></td></tr><tr><td>ε</td><td>Emissivity</td><td>–</td></tr><tr><td>μ</td><td>Dynamic viscosity</td><td>Pa s</td></tr></table>

## Data availability

Data will be made available on request.

## References

Asamoah, E., 2023. E-OHM CAKE cuisson de cake par chauffage ohmique en procédé batch et continu, impact sur la structureet le produit. In: Thèse de doctorat. Nantes Université, France, p. 275.

Assiry, A.M., Sastry, S.K., Samaranayake, C.P., 2006. Influence of temperature, electrical conductivity, power and pH on ascorbic acid degradation kinetics during ohmic heating using stainless steel electrodes. Bioelectrochemistry 68 (1), 7–13. https://doi.org/10.1016/j.bioelechem.2005.02.005.

Bergman, T.L., Lavine, A.S., Incropera, F.P., Dewitt, D.P., 2011. Fundamentals of Heat and Mass Transfer (I. John Wiley & Sons.

Chhanwal, N., Anandharamakrishnan, C., 2014. Temperature- and moisture-based modeling for prediction of starch gelatinization and crumb softness during bread-baking process. J. Texture Stud. 45 (6), 462–476. https://doi.org/10.1111/jtxs.12097.
Collison, R., McDonald, M.P., 1960. Broadening of the water proton lint in high-resolution nuclear magnetic resonance spectra of starch gels. Nature 186, 548-549.

Cummings, W.G., Torrance, K., 1985. Chemical analysis—electrochemical techniques. In: *Jones Instrument Technology. Measurement of Temperature and Chemical Composition* (4th Edition).

Cummings, W.G., Torrance, K., Verhappen, I., 2003. Chemical analysis: electrochemical techniques. Instrumentation Reference Book. Third Edition, pp. 343–381. https://doi.org/10.1016/B978-075067123-1/50021-1.

de Alwis, A.A.P., Fryer, P.J., 1992. Operability of the ohmic heating process: electrical conductivity effects. J. Food Eng. 15 (1), 21–48. https://doi.org/10.1016/0260-8774(92)90038-8.

El-Sharkawy, A.A., Rashed, I.H., Zaghloul, M.S., Ghoniem, M.H., 1984. Study of thermal properties of polycrystalline KCl, KBr, and KI in the temperature bange 300 to 600 K. Phys. Status Solidi 85 (2), 429–434. https://doi.org/10.1002/pssa.2210850212.

Engineering Toolbox, 2004. Water - specific heat vs. Temperature. https://www.engi neeringtoolbox.com/specific-heat-capacity-water-d_660.html. (Accessed 13 March 2021).

Filiz, I., Coscan, I., 2004. Sour cherry juice concentrates during ohmic heating. J. Food Process 3884000, 159-180.

Fryer, P.J., de Alwis, A.A.P., Koury, E., Stapley, A.G.F., Zhang, L., 1993. Ohmic processing of solid-liquid mixtures: heat generation and convection effects. J. Food Eng. 18 (2), 101–125. https://doi.org/10.1016/0260-8774(93)90031-E.

Gally, T., 2017. Etudes expérimentales et numériques du procédé de chauffage ohmique appliqué à la panification, 238p. Thèse de doctorat, Université Nantes, France.

Gally, T., Rouaud, O., Jury, V., Le-Bail, A., 2016. Bread baking using ohmic heating technology; a comprehensive study based on experiments and modelling. J. Food Eng. 190, 176-184. https://doi.org/10.1016/j.jfoodeng.2016.06.029.

Halden, K., De Alwis, A.A.P., Fryer, P.J., 1990. Changes in the electrical conductivity of foods during ohmic heating. Int. J. Food Sci. Technol. 25 (1), 9-25. https://doi.org/10.1111/j.1365-2621.1990.tb01055.x.

Icier, F., 2012. Ohmic heating of fluid foods. In: Novel Thermal and Non-Thermal Technologies for Fluid Foods. Elsevier Inc. https://doi.org/10.1016/B978-0-12-381470-8.00011-6.

Imai, T., Uemura, K., Ishida, N., Yoshizaki, S., Noguchi, A., 1995. Ohmic heating of Japanese white radish. Int. J. Food Sci. Technol. 30, 461-472.

Ito, R., Fukuoka, M., Hamada-Sato, N., 2014. Innovative food processing technology using ohmic heating and aseptic packaging for meat. Meat Sci. 96 (2), 675-681. https://doi.org/10.1016/j.meatsci.2013.10.012.

Kaur, N., Singh, A.K., 2016. Ohmic heating: concept and applications—a review. Crit. Rev. Food Sci. Nutr. 56 (14), 2338-2351. https://doi.org/10.1080/10408398.2013.835303.

Kestin, J., Khalifa, H.E., Correia, R.J., 1981. Tables of the dynamic and kinematic viscosity of aqueous KCl solutions in the temperature range 25–150°C and the pressure range 0.1–35 MPa. J. Phys. Chem. Ref. Data 10 (1), 57–70. https://doi.org/10.1063/1.555640.

Khodeir, M., 2020. *Conception et développement d'un système innovant de cuisson en continu de matrices céréalières (type cake) par chauffage ohmique visant une application 3D.*

*Thèse de doctorat. Université de Nantes, France, p. 245p.*

Khodeir, M., Rouaud, O., Ogé, A., Jury, V., Le-Bail, P., Le-Bail, A., 2021. Study of continuous cake pre-baking in a rectangular channel using ohmic heating. Innov. Food Sci. Emerg. Technol. 67 (October 2020). https://doi.org/10.1016/j.ifset.2020.102580.

Knirsch, M.C., Alves dos Santos, C., Martins de Oliveira Soares Vicente, A.A., Vessoni Penna, T.C., 2010. Ohmic heating - a review. Trends Food Sci. Technol. 21 (9), 436-441. https://doi.org/10.1016/j.tifs.2010.06.003.

Leach, H.W., 1965. Gelatinization of starch. In: Whistler, R.L., Paschall, E.F. (Eds.), *Sturch: Chemistry and Technology, Vol1. Fundamental Aspects*. Academic Press, pp. 289-307.

Li, F. De, Li, L. Te, Li, Z., Tatsumi, E., 2004. Determination of starch gelatinization temperature by ohmic heating. J. Food Eng. 62 (2), 113–120. https://doi.org/10.1016/S0260-8774(03)00199-7.

Marcotte, M., 1999. Ohmic Heating of Viscous Liquid Foods. University Sainte-Anne-de-Belevue, Québec, Canada.

Marcotte, M., Piette, J.P.G., Ramaswamy, H.S., 1998. Electrical conductivities of hydrocolloid solutions. J. Food Process. Eng. 21 (6), 503–520. https://doi.org/10.1111/j.1745-4530.1998.tb00466.x.

Mitchell, F.R.G., De Alwis, A.A.P., 1989. Electrical conductivity meter for food samples. J. Phys. E Sci. Instrum. 22 (8), 554–556. https://doi.org/10.1088/0022-3735/22/8/004.

Niane, M.L., Rouaud, O., Ogé, A., Quéveau, D., Le-bail, A., 2024. Optimization of the temperature profile of cake batter in an ohmic heating - assisted printing nozzle for 3D food printing applications. Int. J. Food Eng. 1–11. https://doi.org/10.1515/ijfeco-2023-0028.

Ott, M., Hester, E.E., 1965. *Gel formation as related to concentration of amylose and degree of starch swelling. Cereal Chem.* 42, 476-484.

Palaniappan, S., Sastry, S.K., 1991a. Electrical conductivities of selected solid foods during ohmic heating. J. Food Process. Eng. 14 (3), 221–236. https://doi.org/10.1111/j.1745-4530.1991.tb00093.x.

Palaniappan, S., Sastry, S.K., 1991b. Electrical conductivity of selected juices: influences of temperature, solids content, applied voltage, and particle size. J. Food Process. Eng. 14 (4), 247–260. https://doi.org/10.1111/j.1745-4530.1991.tb00135.x.

Pérez-Santos, D.M., Velazquez, G., Canonico-Franco, M., Morales-Sanchez, E., Gaytan-Martínez, M., Yanez-Limon, J.M., Herrera-Gomez, A., 2016. Modeling the limited degree of starch gelatinization. Starch/Staerke 68 (7-8), 727-733. https://doi.org/10.1002/star.201500220.

Pineda-Gómez, P., Coral, D.F., Arciniegas, M.L., Rorales Rivera, A., Rodríguez García, M. E., 2010. Papel del agua en la gelatinización delalmidón de maíz: estudio por calorimetría diferencial de barrido. Ingeniería y Ciencia - Ing.Cienc 6 (11), 129-141. http://publicaciones.eafit.edu.co/index.php/ingciencia/article/view/252.

Qihua, T., Jindal, V.K., van Winden, J., 1993. Design and performance evaluation of an ohmic heating unit for liquid foods. Comput. Electron. Agric. 9 (3), 243–253. https://doi.org/10.1016/0168-1699(93)90042-Y.

Ramaswamy, H.S., Marcotte, M., Sastry, S., Abdelrahim, K., 2014. Ohmic heating in food processing. In CRC press inc (ed.). J. Agric. Food Chem. 3 (6) (1st Editio).

Ramires, M.L.V., Nieto De Castro, C.A., 2000. Thermal conductivity of aqueous potassium chloride solutions. Int. J. Thermophys. 21 (3), 671-679. https://doi.org/10.1023/A:1006628419636.

Samaranayake, C.P., Sastry, S.K., 2014. Electrochemical reactions during ohmic heating and moderate electric field processing. In: Ohmic Heating in Food Processing. CRC Press. https://doi.org/10.1201/B16605-16.

Wang, W.C., Sastry, S.K., 1997. Starch gelatinization in ohmic heating. J. Food Eng. 34 (3), 225-242. https://doi.org/10.1016/S0260-8774(97)00085-X.

Yongsawatdigul, J., Park, J.W., Kolbe, E., 1995. Electrical conductivity of pacific whiting surimi paste during ohmic heating. J. Food Sci. 60 (5), 922-925. https://doi.org/10.1111/j.1365-2621.1995.tb06262.x.

Zanoni, B., Peri, C., Bruno, D., 1995a. Modelling of starch gelatinization kinetics of bread crumb during baking. Lebensm.-Wiss. Technol. 28 (3), 314-318. https://doi.org/10.1016/S0023-6438(95)94458-3.

Zanoni, B., Schiraldi, A., Simonetta, R., 1995b. A naive model of starch gelatinization kinetics. J. Food Eng. 24 (1), 25–33. https://doi.org/10.1016/0260-8774(94)P1605-W.

Zareifard, M.R., Marcotte, M., Ramaswamy, H.S., Karimi, Y., 2013. Ohmic heating in food processing: electrical conductivity importance and methods of measurement. In Taylor & Francis group. In: Nature, 531. CRC Press. https://doi.org/10.1038/531139a. Issue 7593.

Zell, M., Lyng, J.G., Morgan, D.J., Cronin, D.A., 2011. Minimising heat losses during batch ohmic heating of solid food. Food Bioprod. Process. 89, 128-134.

Zhang, Y., Zhou, F., Kang, J., 2020. Flow and heat transfer in drag-reducing polymer solution flow through the corrugated tube and circular tube. Appl. Therm. Eng. 174 (February), 115185. https://doi.org/10.1016/j.applthermaleng.2020.115185.