Different frying models have been proposed in the literature (

Figure 2).

Physical models are built based on universal physical laws and the underlying mechanisms behind frying process, so their predictions are usually more precise and based on fundamental physical phenomena.

Observational models are built based on the fitting of experimental data, so they are also known as data-driven models.

Kinetic models are built to describe the rates of chemical reactions relevant to some universal physical laws by fitting experimental data into the model. Therefore, kinetic models are classified under both physical and observational models [

73]. In this review, selected frying models will be briefly discussed and then evaluated regarding their suitability to be used in describing the influence of PEF treatment on the frying process of plant-based foods.

#### 5.1. Physical Models to Describe Heat, Mass and Momentum Transfers

Physical frying models require equations that describe changes in mass, heat and momentum transfers, and simultaneously considering the phase changes and physicochemical changes of plant-based foods during frying. They can provide an in-depth understanding of the physical process of frying, and they are usually more precise because the models are built based on the universal laws. Most physical models for frying are macroscopic continuum models of heat, mass and momentum transfers [

73]. However, the coordinate systems, suitable equations and boundary conditions may vary between different frying conditions. Because of the complexity of frying process, different types of physical models have been built (

Table 2). These can be divided into

simple diffusion-based,

crust-core moving boundary and

multiphase porous media models.

The procedures of building physical models for the frying process are straightforward (

Figure 2). The first step is to define the purpose and describe the questions to be solved with this model. Then, some assumptions (about shapes, geometrical dimensions, mass or heat transfer coefficients, material properties and volume changes) are usually made to simplify the complex real-life situation [

74]. The governing equations for heat, mass and momentum transfers are the core of theoretical models and the typical equations vary between different types of models (

Table 3).

Heat transfer is usually modelled by conservation of heat equation and Fourier’s equation,

mass transfer is generally modelled by conservation of mass equation and Fick’s law of diffusion and

momentum transfer is always described by conservation of momentum equation and Navier–Stokes equation [

74]. Each governing equation has its boundary conditions, which reflect the interaction between the material being fried and surroundings, in order to describe the frying process accurately. Knowledge of boundary conditions occurring during the frying process are required to solve these equations numerically. The commonly used methods to obtain solutions for the boundary conditions include the finite difference, finite element, finite volume, boundary elements, lattice gas cellular automata and lattice Boltzman methods, and many of them rely on the commercial computational software, such as the computational fluid dynamics and computer aided food process engineering. Then input parameters such as density, specific heat capacity, thermal conductivity and permeability are introduced to the model. After building the theoretical model, experimental data from frying process is used to verify the model.

The

simple diffusion-based frying model is the simplest physical model to describe frying. It is built based on the simple heat conduction and moisture diffusion processes while ignoring the oil absorption and water evaporation altogether in the fried material. Rice and Gamble [

75] attempted to build a one-dimensional water diffusion model combining both Fick’s first law and Arrhenius relationship to predict the moisture loss during the frying process of potato slices and successfully proved that the model made valid predictions regarding the early stage of frying (within the first 180 s). Likewise, Pedreschi et al. [

76] applied Fick’s law of diffusion with constant and variable effective diffusion coefficients in order to model the water loss during frying. They found that a simple diffusion-based frying model that considered the change of diffusivity coefficient value with frying time was relatively more precise (better fit to experimental data) in predicting the moisture content of potato slices compared to a classic model with a constant effective moisture diffusion coefficient. However, it is important to note that simple diffusion-based frying models are only able to provide a limited understanding of the frying process because the complex pressure driven flow is simplified to effective diffusion and empirical parameters, which influences their accuracy and restricts the application of these models for different plant materials and frying conditions [

77].

The

crust-core moving boundary model is built based on the core and the crust regions formed in the food material during the frying process, taking into account the moving boundary, where the interface between the core and crust regions moves [

31]. This model is expected to be more precise than the simple diffusion-based frying model because it considers the diffusional and pressure driven transports as well as the distributed evaporation. For example, a good agreement was found between the experimental data (water content, centre temperature, surface temperature and crust thickness) and predicted values using a one-dimensional moving boundary model that included pressure-driven flow, albeit ignoring diffusion flow in the crust region [

78]. Other types of crust-core moving boundary models have been built dealing with different conditions. Acknowledging the temperature difference at different positions of a fried food, Southern et al. [

79] developed a moving boundary model using the Fourier’s law and energy balance equation to describe heat transfer in both the core and crust regions and significantly improved the theoretical prediction of the experimental temperature-time values in different locations of potato crisps during frying. Moreover, van Koerten et al. [

80] built a crust-core moving boundary model based on a Nusselt correlation connecting heat transfer coefficient and water evaporation rate, which was demonstrated to be a simple but effective model for predicting water evaporation and temperature profile in potato cylinders of different diameters (8.5, 10.5 and 14 mm). To allow the models to be applied more widely for different conditions, Farid and Kizilel [

42] developed a unified moving boundary model, by defining a parameter which could reflect the extent of mass diffusion relative to thermal diffusion, to predict the temperature and moisture distribution in a food material, which can be applied in any drying and frying processes. The unified model successfully described the temperature and moisture distributions during the frying and air-drying process for thick (25.4 mm) and thin (2–3 mm) potato slices. Thus, crust-core moving boundary model can be a suitable model to describe the frying process regardless of the dimensions and positions of the fried materials. However, the rate of heat transfer during frying process is highly dependent on the food properties, such as the thermal conductivity and water diffusivity and hence, they should be considered in the mathematical model [

81]. In addition, analytical solutions for complicated equations and boundary conditions are sometimes unavailable. Application of PEF to plant materials is expected to alter the microstructure and physicochemical properties (e.g., thermal conductivity and water diffusivity), hence influencing the moving boundary of crust-core regions. Therefore, crust-core moving boundary model could be considered in future research to describe the differences of frying process between untreated and PEF-treated plant-based foods in a simple yet precise manner.

The

multiphase porous media model is built based on the simultaneous heat, mass and momentum transfers. Multiphase transport in a porous media can be due to three underlying mechanisms including the molecular diffusion for gases, capillary diffusion for liquids and convection (pressure driven or Darcy flow for liquids or gases) [

32]. The multiphase porous media model is considered more realistic, comprehensive and can provide better insight into frying process because it includes the temporal and spatial profiles of temperature and the transport mechanisms of water (liquid and vapour) and air inside the fried materials [

82,

83]. Researchers have applied the multiphase porous media model for different frying conditions, and hence, are able to explain the frying process from multiple perspectives. For example, Ni and Datta [

84] developed a multiphase porous media model to predict the temperature, moisture, oil uptake and crust thickness of potato slices that took into consideration the pressure driven flow for the oil, vapour and air phase in the porous medium. This model shows that there is a pseudo-steady state region in the dry crust and a transient diffusion-like profile in the wet core but it becomes spatially uniform with frying time. Similarly, Halder et al. [

77] have also developed a multiphase porous media model for potato frying and postfrying cooling process based on the nonequilibrium equation for evaporation. The estimated heat and mass transfer coefficients accurately reflect the process of different phases including the nonboiling phase and surface boiling and falling rate stages in the boiling phase. As a result, there is a reasonably good agreement between the experimental data and predicted values of quality parameters, such as the oil content, crust thickness and acrylamide content and this model can be applied to describe baking, meat cooking and drying processes with minor modifications. Multiphase porous media model can be applied to vacuum frying, where Warning et al. [

85] have developed a model of potato chips by modifying the Darcy’s law to account for the Klinkenberg effect. It works well to predict the moisture, temperature, pressure, oil content and acrylamide content during vacuum frying and it implies that the core pressure is approximately 40 kPa higher than the surface pressure of the potato chips. While it is clear that multiphase porous media models are capable of describing the multifaceted physics behind frying process, some of them are difficult to implement in real-life scenario due to the complexity of calculating the evaporation rate. Application of PEF to plant materials is expected to influence the porosity of raw materials. Therefore, multiphase porous media model could be suitable to describe the frying process of PEF-treated plant materials since the model can describe the heat and mass transfers inside a porous material in conjunction with the air flow outside the porous material, and phase changes such as evaporation and condensation [

86].

#### 5.2. Observational Models for Quality Prediction of PEF-Treated Fried Foods

Observational models are useful in building a relationship between the input and output parameters, especially when the practical situations are too complex to understand and building a physical model becomes unrealistic. When building observational models, assumptions and physical interpretations are not necessary since only experimental data are needed to fit a mathematical equation. Commonly used observational models includes

classical statistical,

artificial neural network,

genetic algorithm,

fuzzy and

fractal analysis models [

73]. Observational models are particularly useful to help optimise the frying process in order to obtain high-quality fried products. For example, statistical and two-stage fuzzy models have been built to optimize the blanching and frying parameters (e.g., oil temperature, thermal power) resulting in quality improvements of fried foods and efficiency of frying process [

92,

93]. Artificial neural network models have become popular in recent years and are considered to provide an accurate quality prediction. These models are built by selecting proper network structures and tuning model parameters such as weight, connections and threshold values until the fit to the experimental data is maximised. Mohebbi et al. [

91] built a genetic algorithm-artificial neural network model which could predict the moisture and oil content of fried mushroom accurately. However, the predictive results from observational models can be significantly affected when the physical properties or environmental conditions are altered [

77]. To study the influence of PEF on some reactions (e.g., Maillard reaction, starch gelatinisation, etc.) and quality parameters (colour, acrylamide content, etc.) of fried foods during frying process, observational models would be suitable because they can predict the results directly without understanding the complex reaction process.

#### 5.3. Kinetic Models for Predicting the Rate of Frying Reactions after PEF Treatment

The quality of fried foods changes with frying time owing to chemical and physical reactions during the frying process. The changes of food composition and quality parameters are always described by the kinetic rate of frying reactions, which may also influence the heat and mass transfers [

81]. The chemical, physical and biochemical changes such as degradation and formation of substances (amino acid, sugar, acrylamide, etc.), texture degradation and starch gelatinisation can be evaluated by kinetic models [

94,

95]. A clearer understanding of molecular level changes during the frying process can be obtained from kinetic models because they contain characteristic kinetic parameters such as rate constants and activation energies [

96]. The frying reactions for PEF-treated plant materials can be influenced by the treatment intensity, and accordingly, alter the kinetic parameters of the chemical, physical and biochemical reactions occurring during frying.

The basis of a kinetic model for predicting quality changes is as follows:

P: quality parameters,

k: rate constant,

t: time,

n: reaction order.

Normally, reaction orders can be determined from the characteristics of a chemical reaction. The reaction orders may vary under different circumstances due to the complicated nature of food systems. As reported in the literature, the time dependency of chemical reactions and quality changes of texture, colour and nutrient content usually follows a zero or first order of reaction [

97,

98].

The temperature dependency of the rate constant (

k) can be modelled using Arrhenius equation and Eyring’s absolute reaction rate theory model. The Arrhenius model is very common in the food industry for quality prediction, which can be written as:

A: pre-exponential factor,

E_{a}: activation energy,

R: ideal gas constant (8.3136 J/molK),

T: absolute temperature (K).

As for the Eyring’s absolute reaction rate theory, it is formed based on the transition state theory:

A and

B: molecules,

AC+: activated complex or transition state,

C: product.

The Eyring equation is as follows:

Δ

S^{+}: activation entropy, Δ

H^{+}: activation enthalpy

k_{b}: Boltzmann’s constant (1.381 × 10

^{−23} J/K),

h: Planck’s constant (6.626 × 10

^{−34} Js).

The application of Eyring’s absolute reaction rate theory model has not been widely used to describe frying process, but it is possible to describe some underlying mechanisms during this process. For example, Moyano and Zúñiga [

99] used the enthalpy–entropy compensation approach based on Eyring’s absolute reaction rate theory to study the colour kinetics of French fries during frying. The results indicated that the activation entropy decreased during frying because of the limited space for molecular movement after drying.