Acoustic and Thermal Analysis of Food

Abstract

Exploring the food acoustic features can help to understand and effectively apply some preservation treatments that extend their expiration date. The food composition and properties are crucial issues in their acoustic behavior when stimulated with acoustic waves. If these waves are varied in frequency and intensity, the temperature of food could be affected facilitating the moisture removal or degrading its nutritional condition. Therefore, we presented a guide to determine and apply the most influential spectral component of ultrasound waves on apple and tomato when dehydrated in an ultrasound-assisted dehydration system. In this guide, applying the finite element method, we study, simulate, and analyze the acoustic and thermic behavior of apple and tomato inside a chamber when radiated with acoustic waves at (1 Hz, 1 MHz) by using up to three piezoelectric transducers. From the physical parameters defined in the simulation environment for apple and tomato, we find the relevant spectral components that can produce temperature changes in each food sample considering the radiation time and the food sample location. This work represents an analysis guide that allows for determining the best conditions for the acoustic radiation of foods, avoiding their structural and nutritional damage, and seeking the design of energy-efficient processes.

1. Introduction

Commercially available food must have an expiration date. After that date, we should consider that food has lost a significant nutrient content, or it has been degraded by contamination to detrimental levels to human health. Different methods have been developed for food preservation. For example, dehydration, refrigeration/freezing, fermentation, canning, pasteurization, and incorporation of chemical additives. Food dehydration is a reliable method commonly used for food preservation, based on food moisture removal, it must be performed by using energy-, cost-, and time-efficient technologies, and it must use practical methods that do not interfere with time spent on personal and professional daily activities. In this sense, it should be considered that, according to Kumar et al. [1] and Merone et al. [2], food dehydration implies energy-intensive processes approximately using 25% of the total energy consumed by the food industry. In addition, food dehydration may be a practical process with hygiene and temperature conditions. Therefore, under the demands of today’s life, food dehydration must be an automatically controlled process ensuring that the food structure, content, and quality are preserved on time. In general, food dehydration systems have low efficiency, and consequently, the development of efficient systems is a relevant scientific and technological challenge [3, 4, 5]. Hence, when developing or selecting a food dehydration system, its efficiency must be evaluated considering removed moisture content and the moisture removal rate. On the other hand, energy consumption and dehydrated food quality are two critical requirements for the design, implementation, or selection of a dehydration system [6]. In this sense, ultrasound-assisted dehydration systems can be a promising alternative [7], because they can help to reduce energy consumption [8, 9, 10]. However, we must keep in mind that the internal microstructure of food could change drastically when it is immersed in ultrasound waves, reducing its resistance to water diffusion, and increasing its temperature. But, an ultrasound-assisted dehydration system could have higher energy, cost, and time efficiency than simple convection dehydration systems [11, 12, 13]. Additionally, they have been demonstrated to be reproducible processes that avoid further wastewater treatment and additional energy use [14, 15].

An overview of ultrasound waves application in other technologies for food processing can be revised in the works published in 2021 by Singla and Sit [16] and Khadhraoui et al. [17]. In conventional dehydration technologies combined with ultrasound systems, it is worth noting that the ultrasound-assisted dehydration systems can increase the dehydration rates or decrease the dehydration temperature since the ultrasound waves strongly accelerate mass transfer maintaining food quality. For example, in 2017, Fei et al. showed that ultrasound osmotic dehydration produces samples with reduced sugar, ascorbic acid, and soluble protein content at significantly higher rates than osmotic dehydration, and the food samples showed a better texture and microstructure. The ultrasound osmotic dehydration process not only retained the nutrient composition and flavor material more effectively but also improved the texture and efficiency of osmosis-treated mushrooms [18]. Complementary to the application of ultrasound waves in food dehydration systems, we can mention that, in 2016, Başlar et al. [19] reviewed different ultrasound-assisted dehydration processes, including convective, osmotic, vacuum, and freeze dehydration applications, as well as the various types of ultrasonic equipment used. They summarized the mechanisms, applications, advantages, disadvantages, and recent investigations of ultrasound-assisted dehydration concluding that ultrasound treatments can potentially provide significant improvement in food dehydration, as the dehydration process simultaneously accelerates heat and mass transfers in the system. In general, they showed that the dehydration methods combined with ultrasound-assisted dehydration resulted in less dehydration temperature and duration. Finally, they suggested that these two advantages in ultrasound-assisted dehydration systems may avoid the reduction effect of food quality in comparison with other dehydration techniques.

Therefore, considering the work of Başlar et al. [19], we have compared different dehydration systems against ultrasound-assisted versions of the same systems. In Table 1, we have included convective, osmotic, vacuum, and freeze dehydration systems considering that the ultrasound-assisted dehydration systems offer higher dehydration rates or lower food dehydration temperatures than conventional systems.

System	Advantages	Disadvantages
Convective dehydration	Easy operation. It prevents microbial growth. It reduces water activity in food. It prevents enzymatic activity. It allows easy packaging and transport of food.	High energy consumption. Low dehydration rate. Low product quality. High dehydration temperature. Long dehydration time.
Ultrasound-assisted convective dehydration	It reduces dehydration time. It reduces dehydration temperature operation. It increases the moisture transfer rate. It minimizes quality losses in the final products. Low-energy consumption	The operating frequency must be selected and adjusted. An acoustic decoupling is produced between the media on which the ultrasound signals are propagated.
Osmotic dehydration	It minimizes the flavoring compound losses. It minimizes the aromatic substance’s losses. It minimizes color losses. The dehydrated products have a better appearance and nutritional properties.	Mass transfer is limited. Low-mass transfer. Mass transfer completely stops after a certain time.
Ultrasound-assisted osmotic dehydration	It enhances mass transfer during osmotic dehydration. It improves the fluxes of osmotic solute to the intercellular of the dewatered food. It reduces the dehydration time and cost.	The choice of appropriate parameters in the ultrasound generator is a considerable issue. The ultrasound signal effects are reduced at high temperatures.
Vacuum dehydration	Low temperature. It prevents food oxidation. It is an energy-efficient dehydration process. Low dehydration time. It retains the organoleptic and nutritional properties of food. Risk of occupational accidents.	It does not perform a continuous dehydration process. It is an expensive process due to high maintenance costs. It does not remove sufficient moisture from the center to the surface in food samples. Low-heat transfer. It requires pretreatment or combination with other methods to improve the dehydration process efficiency.
Ultrasound-assisted vacuum dehydration	It has a near-perfect rehydration rate. Fast food dehydration. It enhances food quality and appearance. It accelerates mass transfer from the food center. Dehydration rate increases without increasing the temperature in the dehydration chamber. It improves diffusion and raises the convective mass transfer rate. It has a reduced overall processing time. Low ultrasound power.	Specific pressure and temperature parameters are required for the dehydration process of each food.
Freeze dehydration	Preservation of the activity of food nutraceuticals and pharmaceuticals. Preservation of food flavor and aroma. Fast water penetration and recovery of the original characteristics during a rehydration process.	Long dehydration times (hours or days). High operating costs. Limited industrial application. To reduce its dehydration time, the dehydration system must be assisted by other processes.
Ultrasound-assisted freeze dehydration	It increases the freeze-dehydration efficiency. It increases the dehydration rate. It reduces operating costs. Dehydration times less than 1 day. It does not deteriorate food quality.	The food sample under dehydration must be small. The process requires specific values of ultrasound signal frequency and power, as well as temperature and airflow in the dehydration chamber for each food.

Table 1.

Advantages and disadvantages of conventional and conventional ultrasound-assisted dehydration systems.

Regarding the applicability status of dehydration technology, we must say that ultrasound-assisted dehydration is mainly done at the laboratory level, which is used to study the dehydration kinetics and physicochemical, microbiological, structural, and rehydration characteristics of some foods [20]. Ultrasound-assisted dehydration systems are based on conventional dehydration techniques that are assisted by ultrasound waves resulting in different methods, such as ultrasound-assisted convective dehydration [21], ultrasound-assisted osmotic dehydration [22], ultrasound-assisted vacuum dehydration [23], and ultrasound-assisted freeze dehydration [24]. In ultrasound-assisted dehydration systems, ultrasound waves are used to increase the dehydration rates or decrease the dehydration temperature, since these waves strongly accelerate mass transfer maintaining food quality.

On the other hand, with respect to commercial alternatives, it can be stated that some ultrasound-assisted systems are beginning to emerge, such as the one used for dehydration of heat-sensitive biological materials and food processing. Some examples of these commercial systems include the following:

1. The Center for Advanced Research in Drying (CARD) is devoted to research on the drying of moist and porous materials, such as food, agricultural products, chemical products, and biopharmaceuticals. It is investigating the use of high-power ultrasound as a nonthermal, faster, and less energy-intensive method for protein dehydration [25].

2. Dehydration Assisted by Ultrasound by PUSONICS SL is a technology that offers power ultrasonic plate-transducers capable of transmitting sonic and ultrasonic energy through the air for drying heat-sensitive materials, where micro-vibrations could prevent spoilage due to damaging high-temperature levels [26].

3. Ultrasound enhanced osmotic dehydrating method and apparatus is an infiltration dehydrating method with ultrasonic strengthen and correlated device applicable to ripe fresh fruits with smooth epidermises, such as apples and pears [27].

4. Ultrasound applied to food processing as yogurt fermentation, extraction of flavors and bioactive compounds, milk homogenization, sugar crystallization for confectionery, edible oils hydrogenation, honey liquefaction, juices stabilization, wine and liquor aging, accelerated ice cream freezing, batter aeration, chocolate crystallization and conching, and meat tenderization [28, 29].

Therefore, in this work, we focus on sound–assisted convection dehydration systems, which could reduce dehydration time and cost by at least 30% compared to conventional methods [30] for food dehydration. Furthermore, it should be noted that these dehydration systems are not harmful to humans as microwave, gamma radiation, or electromagnetic field pulses systems; therefore, they can be applied in the food industry for safety reasons [31, 32]. Thereby, it is important to study, analyze, and understand the food acoustic-thermal behavior when dehydrated by using an ultrasound-assisted convection dehydration system. In this regard, we presented a guide to determine and apply the most influential spectral components of ultrasound waves on food under dehydration in an ultrasound-assisted dehydration system for increasing the temperature of food under dehydration. For this purpose, we have defined two case studies; in the first one, we study and analyze the thermo-acoustic behavior of apple samples, and in the second study case, we consider tomato samples. In both case studies, food samples are immersed in ultrasound waves and they are radiated using one and three transducers. With these case studies, after identifying the intensity and frequency of the ultrasound waves that have the greatest influence on the dehydration process of each food, we intended to observe the temperature changes experienced by food samples while they are being dehydrated. In this way, this work allows us to find the best conditions for food acoustic radiation avoiding structural and nutritional damage to the food. Also, in this guide, we show how to apply the finite element method to simulate and analyze the thermal and acoustic behavior of foods under dehydration inside the dehydration chamber of an ultrasound-assisted system.

The rest of the chapter is organized as follows. Section 2 describes some scientific works related to the analysis of ultrasound-assisted food dehydration systems. Section 3 describes the two case studies for which the thermo-acoustic analysis was performed, the used configurations for the dehydration chamber, the food composition and properties, a brief overview of transducers and acoustic waves, and the selected simulation environment. Section 4 provides the acoustic and thermal mathematical model used for the thermo-acoustic analysis by the finite element method, the spatial behavior of the ultrasound waves inside the dehydration chamber, and the temporal behavior of the temperature change in the food samples. Section 5 shows the opportunities for future work when ultrasound–assisted convection systems are used in food dehydration. Finally, Section 6 is devoted to conclusions and work perspectives.

2. Related works

There are some works reporting analysis of ultrasound-assisted food dehydration systems. For example, in 2006, García-Pérez et al. proposed an ultrasound-assisted convection dehydration system consisted of an aluminum–cylinder drying chamber able to create a high-intensity ultrasound field at 21.8 kHz [8]. In 2011, Khmelev et al. [9] also proposed an ultrasound-assisted dehydration system consisted of a resonant drying chamber to amplify the ultrasound waves. They determined that the efficiency of their dehydration system was 20% higher than the efficiency of a pure convection system. In 2015, Fernandes et al. examined the influence of ultrasound waves on apple dehydration and estimated the effective moisture diffusivity [33]. Lastly, Sabarez et al. developed and tested a high-intensity ultrasound system to assist a conventional dehydration system, which was more efficient between 46 and 57% than a conventional dehydration system [10]. Also, there are other works related to the energy and environmental analysis of ultrasound-assisted food dehydration. For example, in 2022 Chavan et al. simulated dehydration at an industrial scale for some vegetables [34]. On the other hand, there are works related to parametric studies about ultrasound-assisted food dehydration systems. For example, in 2020, Huang et al. considered the air flow rate, ultrasonic power, and mass loading in the analysis of a food ultrasound-assisted dehydration system [35].

Considering this variety of scientific works, we decided to include in this chapter the study and simulation analysis of the thermo-acoustic behavior of apple and tomato samples when they are dehydrated in an ultrasound-assisted convection system. We performed this simulation analysis by using the finite element method (FEM) implemented in COMSOL Multiphysics™. In this way, we analyzed the temporal and spectral behavior of the dehydration system, which we estimated, as a function of the radiation time and the ultrasound waves frequency, and the temperature changes of the 5 mm-size food plates (apple or tomato). This simulation analysis also identifies changes in food temperature based on the location of the samples within the dehydration chamber.

3. Case studies

In order to study the ultrasound waves behavior and their thermal effects on apple and tomato samples, we proposed an ultrasound-assisted dehydration system based on the forced convection dehydration system shown in Figure 1 considering two case studies. In the first case study, represented by Figure 2a, we used one piezoelectric transducer of size (0.5 cm × 4 cm). In the second case study, represented by Figure 2b, we use three piezoelectric transducers as the first case study. Considering a transversal cut at (x, z)-plane, note in Figure 2 that the dehydration system has a (27 cm × 20 cm × 20 cm)-chamber composed of five gridded trays spaced 3 cm from each other, an electronic module to regulate the dehydration temperature, an ultrasound waves generator, and up to three piezoelectric transducers placed at the dehydration chamber base.

Figure 1.

Ultrasonic dehydration chamber.

Figure 2.

Proposed geometry for the thermo-acoustic simulation when apples and tomatoes were dehydrated using: (a) one PZT, and (b) three PZT’s.

In both case studies, we consider 0.5 cm-thick apple and tomato plates placed in the five gridded trays and assumed that the dehydration chamber is in equilibrium conditions at a temperature of 60°C. From the geometries proposed in Figure 2, and considering that the simplified geometry of the dehydration chamber derived by the transversal cut at (x, z)-plane can help to simplify the thermo-acoustic analysis of the proposed system, we show an example of the FEM simulation results for both case studies considering the effects of the ultrasound waves on apple and tomato when a conventional convection dehydration system is used.

3.1 Food composition and properties

In order to perform a thermal and acoustic FEM analysis for apple and tomato inside an ultrasonic-assisted convection dehydrator, in Table 2, we summarize the physical parameters of apple and tomato, such as density, sound propagation velocity, heat capacity, thermal conductivity, and sound absorption coefficient [36, 37, 38, 39].

Parameter	Apple	Tomato	Units
Density ($\rho$)	840	560	[kg/m^3]
Sound speed (c)	49.89	231.45	[m/s]
Thermal capacity ($C_p$)	3810	4080	[J/(kgK)]
Thermal conductivity (k)	0.418	0.66	[W/(mK)]
Acoustic absorption coefficient ($\alpha$)	0.22	0.35	[dB/m]

Table 2.

Thermo–acoustic parameters of apple and tomato.

For the acoustic and thermal analysis for the apple and tomato, we defined the parameters shown in Table 2 in order to determine the dynamics produced by the drying chamber in the apple and tomato. We should emphasize that if any of these parameters are omitted, the FEM analysis cannot be performed properly and the spatial and temporal dynamics will have a significant error since acoustic and thermal phenomena are completely interrelated.

3.2 Acoustic waves and piezoelectric transducers

Firstly, a wave can be defined as the energy and momentum transfer from one point in a medium to another point in the same medium without net matter transport between the two points [40]. When the waves require a medium for their propagation are called mechanical or elastic waves [41]. In this case, the medium particles perform a periodic motion around a mean position as the wave propagates through the medium. A mechanical wave is produced when a particle is perturbed in the propagating medium and interacts with the neighboring particle and its energy is transmitted to the next particle (due to the inertia of the medium) [42]. The perturbed particles return to equilibrium due to the medium elasticity after a finite time. Thus, when the mechanical wave motion is produced the following parameters must be considered: frequency, propagation velocity, period, phase, and wavelength.

On the other hand, the piezoelectric transducer is an important component of ultrasound instrumentation systems [43]. Piezoelectric transducers convert electrical waves into mechanical vibrations and mechanical vibrations into electrical waves [44]. These devices are mainly used to generate waves in the ultrasound range (frequencies higher than 20 kHz) at low, medium, and high intensity. Piezoelectric transducers can be produced by using ceramics, quartz, Rochelle salts, and metal alloys to be used in ultrasound wave generators applicable in multiple industrial areas [45]. For example, drying, ultrasonic cleaning, fuel oil injection into burners, and medical treatments, among others.

3.3 Simulation environment

To describe the acoustic and thermal behavior of apple and tomato in the proposed dehydrator, COMSOL Multiphysics™ has been chosen as the simulation environment, which is a software tool for finite element analysis useful in various physics and engineering applications, especially for couple or multiphysics phenomena [46]. The possibility of analyzing different physical phenomena integrated into COMSOL Multiphysics™ allows the user to model and analyze scenarios involving multiple interacting physical phenomena. The phenomena that can be modeled in COMSOL Multiphysics™ are related to acoustics, electromagnetism, micro-electromechanical systems (MEMS), microwaves, radio frequency components, semiconductor devices, and wave propagation, among many others. In this work, the acoustics and heat transfer modules have been used to perform the acoustic and thermal analysis of the proposed system.

The finite element method is based on dividing the body, structure, or domain over which the equations characterizing the phenomena physical behavior are defined into subdomains called finite elements [47]. The finite elements set forms a domain partition also called discretization or mesh. Due to the subdivisions generated in the geometry, the mathematical equations that govern the phenomena physical behavior could not be solved in an exact way, but in an approximate way, since the solution that results in the simulation environment depend on the nodes and elements number, as well as of the elements size and type defined in the mesh [48]. From the approximation provided by the solution of the equations describing the desired physical phenomenon from the generated finite elements, it is possible to describe the desired system behavior.

Using a COMSOL Multiphysics™ simulation environment and the FEM, the acoustic and thermal analysis of the proposed system for apple and tomato has been performed from the geometry shown in Figure 2. Additionally, we have used the mathematical models developed in Subsection 4.1, and we have shown the obtained results in Subsections 4.2 and 4.3.

4. Analysis methodology

In order to perform the thermo-acoustic analysis when apple and tomato are dehydrated in the ultrasound-assisted convection dehydrator, we define the mathematical models that describe the behavior of the ultrasound waves inside the dehydration chamber and within the food samples, as well as the temperature change of the food samples. Subsection 4.1 shows the model describing the ultrasound waves propagation in the system and the thermodynamic model that describes the temperature change inside the food. Subsequently, Subsection 4.2 shows the system acoustic analysis and Subsection 4.3 shows the thermic analysis of food samples.

4.1 Mathematical model

The mathematical model used to describe the ultrasound waves propagation is derived from the reduction of the mass, momentum, energy, and state balance equations. According to Blackstock and Everest in [49, 50], mass, momentum, energy, and states are defined by Eqs. (1)–(4), respectively.

$$\frac{\mathit{D\rho }}{\mathit{Dt}}+\rho \frac{\partial u}{\partial x}=0,$$E1

$$\rho \frac{\mathit{Du}}{\mathit{Dt}}+\frac{\partial P}{\partial x}=0,$$E2

$$\rho \frac{\mathit{D\epsilon }}{\mathit{Dt}}+P\frac{\partial u}{\partial x}=-\frac{\partial q}{\partial x},$$E3

$$P=c_0^2\mathit{\delta \rho }\left[1+\frac{B}{2!A}\frac{\mathit{\delta \rho }}{{\rho }_0}+\frac{C}{3!A}{\left(\frac{\mathit{\delta \rho }}{{\rho }_0}\right)}^2+\dots \right],$$E4

where $\frac{D}{\mathit{Dt}}$ is the Stokes derivative of the variable studied, $\rho$ is the medium’s density, P is the medium’s pressure, $\epsilon$ is the medium’s internal energy, q is the heat flow inside the dehydration chamber, $c_0$ is the sound speed, ${\delta }_{\rho }$ is the density excess ($\mathit{\delta \rho }=\rho -{\rho }_0$), and A, B, and C are the coefficients of the Taylor series for P.

Also, according to Kinsler [51], the wave equation defined by Eq. (5) is used considering that it idealizes many types of wave motion produced in an isolated medium that does not exchange energy, momentum, or mass with its environment.

$$c^2{\nabla }^{2}u-\frac{{\partial }^{2}u}{{\partial }^{2}t}=0,$$E5

where u represents acoustic waves, ${\nabla }^2$ represents the Laplacian applied to u, c is the wave speed, and t is time.

It should be noted that the mathematical model given by Eq. (5) is not enough when describing the waves behavior in a fluid with losses. Then, in order to include those losses, it is considered that u depends on the energy dissipation in a three-dimensional viscous medium [52]. Now, considering that the complex wave number $k=\beta +\mathit{j\alpha }$ is used to calculate a solution by a harmonic time, from Eq. (5) the expression shown in Eq. (6) is obtained,

$$u=u_0{e}^{-\mathit{\alpha x}}{e}^{j\left(\mathit{wt}-\mathit{\beta x}\right)},$$E6

where $u_0$ is the wave amplitude in $t=0$, $\alpha$ is the absorption coefficient, and $\beta$ is the wave cycles number per distance unit.

In a similar way, solving Eq. (5) for P, Eq. (7) is obtained.

$$P=P_0{e}^{-\mathit{\alpha x}}{e}^{j\left(\mathit{wt}-\mathit{\beta x}\right)},$$E7

where $P_0$ is the wave pressure in $t=0$.

Based on the geometry proposed for the FEM simulation, an approximation of the thermodynamic model for food has been made. For this purpose, the model has been developed considering the equilibrium equation that describes the thermal system considering as a particular case the first law of thermodynamics, where it is considered that the heat transfer system does not generate work, and the system dynamics is a function of heat flow and temperature. The system variables description to be considered taking into account the heat flows, temperatures, thermal capacitances, and thermal resistances are shown in Figure 3.

Figure 3.

System variables for thermodynamic modeling.

where, $Q_u$ is the heat flux generated by the ultrasound wave, $Q_1$, $Q_2$, and $Q_3$ are the heat fluxes leaving the system, $R_{f1}$, $R_{f2}$, and $R_{f3}$ are the thermal resistances of the sample walls, $C_f$ is the food thermal capacitance, $T_f$ is the food temperature, and $T_{\mathit{dc}}$ is the drying chamber temperature.

Considering heat fluxes, thermal resistances, food temperature, and environment temperature, equations for each thermal element that composes the food (walls of the food describing the heat flow through them) are written according to Eqs. (8)–(10).

$$Q_1{R}_{f1}=T_f-T_{\mathit{dc}},$$E8

$$Q_2{R}_{f2}=T_f-T_{\mathit{dc}},$$E9

$$Q_3{R}_{f3}=T_f-T_{\mathit{dc}}.$$E10

Once the element equations are defined, the equilibrium equation is established, starting from the first law of thermodynamics, which will define the system behavior from the incoming and outgoing system flows considering that there is only heat transfer, and no work is generated. This equilibrium equation can be written by Eq. (11).

$$C_f\frac{\mathrm{d}{T}_f}{\mathrm{d}t}=\sum Q_{\mathit{\text{input}}}-\sum Q_{\mathit{\text{output}}},$$E11

where $Q_{\mathit{\text{input}}}$ are the incoming heat fluxes, and $Q_{\mathit{\text{output}}}$ are the outgoing heat fluxes.

Since the equilibrium equation is a function of the incoming and outgoing heat fluxes, each of these must be defined considering the system elements. In the system, the inflow will only be given by $Q_u$ and the outflows can be calculated from Eqs. (8)–(10). Substituting the heat fluxes in the equilibrium equation and taking as output the rate of food temperature change as a time function, the system dynamics are described by Eq. (12).

$$\frac{\mathrm{d}{T}_f}{\mathrm{d}t}=\frac{Q_u}{C_f}-\frac{3}{C_f{R}_{\mathit{\text{fT}}}}{T}_f-\frac{3}{C_f{R}_{\mathit{\text{fT}}}}{T}_{\mathit{dc}},$$E12

where $T_f$ is the food temperature, t is the time, $Q_u$ is the heat flux produced by the ultrasound, $C_f$ is the food heat capacity, $R_{\mathit{\text{fT}}}$ is the total thermal resistance generated by the apple walls, and $T_{\mathit{dc}}$ is the drying chamber temperature.

In this case, it is contemplated that $Q_u=2\mathit{\alpha I}$, where $\alpha$ is the local acoustic absorption coefficient of the food, and I is the local sound intensity. In the same way, $C_f={\rho }_f{C}_{\mathit{pf}}A$ where ${\rho }_f$ is the food density, $C_{\mathit{pf}}$ is the food-specific heat, and A is the food transverse area. Thus, Eq. (12) is also expressed by Eq. (13).

$$\frac{\mathrm{d}{T}_f}{\mathrm{d}t}=\frac{2\mathit{\alpha I}}{{\rho }_f{C}_{\mathit{pf}}A}-\frac{3}{{\rho }_f{C}_{\mathit{pf}}{\mathit{AR}}_{\mathit{\text{fT}}}}{T}_f-\frac{3}{{\rho }_f{C}_{\mathit{pf}}{\mathit{AR}}_{\mathit{\text{fT}}}}{T}_{\mathit{dc}}$$E13

If Eq. (13) is lumped and considering that $R_{\mathit{\text{fT}}}=\frac{\mathrm{\Delta }x}{k_f}$, where $\mathrm{\Delta }x$ is the thickness of the sample, and $k_f$ is the food thermal conductivity, then, it would be written as:

$$\frac{\mathrm{d}{T}_f}{\mathrm{d}t}=\frac{2\mathit{\alpha I}}{{\rho }_f{C}_{\mathit{pf}}A}-\frac{3k_f}{{\rho }_f{C}_{\mathit{pf}}A}\frac{\mathrm{\Delta }{T}_f}{\mathrm{\Delta }x}$$E14

where $\mathrm{\Delta }{T}_f=T_f-T_{\mathit{dc}}$, and $I=\mathrm{ℝ}\frac{1}{2}\mathit{Pv}$, where P is the pressure generated by the sound wave and $v$ is the particle velocity.

Therefore, $v$ can be written as shown in Eq. (15).

$$v=\frac{\partial \overline{\epsilon }}{\partial t}$$E15

where $\epsilon$ is expressed as is shown by Eq. (16).

$$\overline{\epsilon }=\frac{\partial \overline{\epsilon }}{\partial x}\widehat{x}+\frac{\partial \overline{\epsilon }}{\partial y}\widehat{y}+\frac{\partial \overline{\epsilon }}{\partial z}\widehat{z}$$E16

By substituting the Eqs. (15) and (16) in we obtain that the rate of change of temperature with respect to time of the food sample is described by Eq. (17).

$$\frac{\partial T_f}{\partial t}=\frac{\mathit{\alpha P}}{{\rho }_f{C}_{\mathit{pf}}A}\frac{\partial \overline{\epsilon }}{\partial x}-\frac{3k_f}{{\rho }_f{C}_{\mathit{pf}}A}\nabla T$$E17

In order to describe the acoustic and thermal behavior of the apple and the tomato in the spatial and temporal domain, the FEM will be used to solve the solutions to Eqs. (6) and (17). The description of the analysis performed for each of the cases is shown in Subsections 4.2 and 4.3.

4.2 Acoustic behavior analysis

To perform the acoustic analysis in the spatial domain by means of the FEM in COMSOL Multiphysics™, the following programming sequence is proposed:

1. Programming environment startup: the programming of the simulation environment in COMSOL Multiphysics${}^{\mathit{TM}}$ should be as follows: open COMSOL Multiphysics${}^{\mathit{TM}}$ $\to$ new document $\to$ model wizard $\to$ space dimension selection $\to$ 2D $\to$ physics selection $\to$ acoustics $\to$ acoustic pressure $\to$ frequency (acpr) $\to$ click in add $\to$study $\to$ frequency domain $\to$ done. This programming sequence will open the simulation environment already configured for the acoustic analysis.

2. Geometry construction: to generate the geometry according to the dimensions given in Figure 2, follow the following path: geometry $\to$ units $\to$ cm $\to$ geometry $\to$ primitives $\to$ rectangle $\to$ width (enter value in cm of the rectangle to be constructed) $\to$ height (enter value in cm of the rectangle to be constructed) $\to$ location (enter coordinates according to Figure 2) $\to$ construct all. This step must be performed for each of the rectangles of the geometry proposed in Figure 2.

3. Assignment of materials: for the selection of materials corresponding to each geometry the following path must be followed: materials $\to$ add material $\to$ fluids $\to$ gases $\to$ air $\to$ add material $\to$ materials $\to$ blank material $\to$ material properties $\to$ select properties (Parameters of Table 2) $\to$ add properties $\to$ name material (tomato or apple as the case may be) $\to$ select air material $\to$ select geometric entities that compose it $\to$ select food material $\to$ select geometric entities that compose it. In this way, you will have assigned the corresponding materials to each of the geometries that represent the dehydration chamber and the food samples.

4. Physics configuration: to configure the physics (acoustics), the following sequence must be followed: acoustic pressure, frequency (acpr) $\to$ domain $\to$ acoustic pressure $\to$ select geometries corresponding to air $\to$ domain $\to$ acoustic pressure $\to$ select geometries corresponding to food $\to$ contour $\to$ pressure $\to$ select geometries corresponding to piezoelectric transducers $\to$ enter test pressure value (2 Pa)$\to$ initial values $\to$ pressure value and temperature (2 Pa at 60°C). From this sequence, the physics will be configured to perform the acoustic analysis of the food samples.

5. Mesh construction: to build the mesh follow the following path: mesh $\to$ select the geometries that will be interacting with the acoustic field $\to$ sequence type $\to$ physics controlled $\to$ element size $\to$ extra fine $\to$ build mesh. With these steps, a mesh will be generated with enough elements to have a good approximation of the interaction of the acoustic field in the dehydration chamber.

6. Study configuration: the following sequence is used to set up the study: study $\to$ frequency $\to$ frequency units $\to$ Hertz $\to$ study frequencies $\to$ enter desired frequencies (1 Hz - 1 MHz) $\to$ physics selection $\to$ acoustic pressure $\to$ frequency $\to$ mesh $\to$ mesh 1 $\to$ value of dependent variables $\to$ defined by physics $\to$ compute. From this sequence, the simulation environment will perform the calculation of the approximation of the acoustic field behavior and the frequency spectrum of the food samples.

7. Results: for the visualization of the obtained results, just select the results option and the frequency spectrum, sound pressure, and sound pressure level graphs will be displayed. In this way, the acoustic analysis will be finished.

From these steps, it is possible to analyze the acoustic and spatial behaviors of the ultrasound waves at (1 Hz, 1 MHz) inside the dehydration chamber. Then, we can determine the optimal operating frequencies at which the dehydration system can perform the most efficient dehydration on each test food (apple and tomato) considering the average system pressure. Thus, in Figure 4, we can identify the spectral component the most influential spectral component for each food under consideration; that is, we selected the frequency band with the highest average pressure. This frequency band is centered around 34 kHz (see Figure 4a) when apple samples are considered and around 70 kHz (see Figure 4b) for tomato samples. Consequently, Figure 5 shows the spatial behavior of the ultrasound waves when apple samples are radiated by a single piezoelectric transducer (see Figure 5a) and three piezoelectric transducers (see Figure 5b) at 34 kHz and 2 Pa of pressure.

Figure 4.

Apple and tomato frequency spectrum: (a) apple and (b) tomato.

Figure 5.

Acoustic field when apple samples are radiated by ultrasound waves with fundamental frequency at 34 kHz: (a) one piezoelectric transducer and (b) three piezoelectric transducers.

Also, Figure 5 shows that the moisture removal at the surface level or internally in the food is a function of the intensity and frequency of the ultrasound waves applied. It should be considered that in this study a sweep of frequencies between 1 Hz and 1 MHz was performed and it was noticed that there are frequencies with greater influence than others. On the other hand, also trying not to exceed a temperature of 70°C in the foods under dehydration to avoid their structural and nutritional damage, we determined the most appropriate pressure that should be exerted on the food. Note that when three piezoelectric transducers are used, the ultrasound waves have a more homogeneous spatial distribution than a single piezoelectric transducer is considered (see Figure 5b). In addition, the apple sample closest to the piezoelectric transducer has the greatest influence by ultrasound waves. Therefore, the acoustic field distribution also influences the apple samples, and it depends on the transducer number.

In a similar way to apple samples, Figure 6 shows the spatial behavior of the acoustic field when the tomato samples are radiated by ultrasound waves by a single transducer (see Figure 6a) and three piezoelectric transducers (Figure 6b) at 70 kHz and 2 Pa. Note that the acoustic field generated by the ultrasound waves is distributed more homogeneously when the tomato samples are radiated by three piezoelectric transducers, while when they are radiated with only one piezoelectric transducer the acoustic field has more influence on the tomato samples that are closer to the piezoelectric transducer.

Figure 6.

Acoustic field when tomato samples are radiated by ultrasound waves with fundamental frequency at 70 kHz: (a) one piezoelectric transducer, and (b) three piezoelectric transducers.

It should be noted that in Figures 5 and 6 the sound intensity level is not uniform in food samples, which implies that the moisture removal is different at the surface level than inside the food. In this way, the moisture removal is a function of the intensity and frequency of the ultrasound waves applied. In addition, it should be remembered that in this study we made a sweep of frequencies between 1 Hz and 1 MHz and we noticed that there are frequencies with greater influence than others. On the other hand, also trying not to exceed a temperature of 70°C in the foods to be dehydrated, to avoid their structural and nutritional damage.

Now, from these results, Subsection 4.3 shows the ultrasound waves influence at 34 kHz and 70 kHz on the temperature change for apple and tomato samples, respectively.

4.3 Thermic behavior analysis

In order to perform the thermal analysis on the apple and tomato samples, a multiphysical analysis is performed in which the energy produced in the food samples by the ultrasound waves is used to obtain the temperature change in each of them during a dehydration time of 40,000 seconds. To perform the thermal analysis of the ultrasound-assisted convection dehydration system, the following steps are established:

1. Programming simulation environment: the construction of the geometry and the integration of the materials will be maintained, as the thermal analysis will be carried out only as an add-on segment to the acoustic analysis. To configure the simulation environment for the thermal analysis we must follow the following sequence within the same file used for the thermal analysis: physics $\to$ add physics $\to$ heat transfer $\to$ study $\to$ add study $\to$ time domain. With this sequence, the simulation environment for thermal analysis is now configured.

2. Physics configuration: to configure the heat transfer physics, follow the following path: heat transfer $\to$ domain $\to$ heat source $\to$ select the geometries corresponding to the chamber and the food samples $\to$ contour $\to$ temperature $\to$ enter initial temperature (60°C) $\to$ initial conditions (initial temperature 60°C) $\to$ heat source $\to$ enter the command acpr.Qpw (indicates the heat produced in the system by the acoustic source). With these steps, the physics will already be configured.

3. Mesh construction: for the thermal analysis, a mesh must be constructed independently of the one generated for the acoustic analysis, for this the following sequence is performed: mesh $\to$ new mesh $\to$ add mesh $\to$ sequence size $\to$ controlled by physics $\to$ element size $\to$ fine. This mesh will be the one that will help to give the approximation of the thermal analysis in the function of the acoustic analysis.

4. Study configuration: to set up the study, follow the following sequence: study $\to$ time domain $\to$ time units $\to$ seconds $\to$ times $\to$ select time range (0–40,000 seconds) $\to$ select physics and variables $\to$ heat transfer $\to$ dependent variable values $\to$ settings $\to$ user-controlled $\to$ method $\to$ solution $\to$ study $\to$ study in frequency $\to$ frequency parameter value $\to$ selected according to the optimum operating frequency for each food $\to$ compute. In this way, the simulation environment will be able to give the approximation of the temperature change in the food samples as a function of the frequency that is radiating them.

5. Results: to visualize the results obtained, simply select the results option and the temperature contour graphs of the dehydration system and the temperature rise curve for each food sample will be displayed. Thus, the thermal analysis according to the optimum operating frequency of the dehydration chamber is finished.

Using the above steps, we can determine the temperature change in the dehydration system. Note that the food mass should not exceed 70°C. This temperature limit is given according to Michelice and Ohaco [53] in their guide to food dehydration and drying, and as experimentally proven by Tao et al. [54] in their tests carried out in the dehydration of blackberries. This temperature has been established since the use of higher temperatures may cause structural and nutritional damage to foods. For this purpose, simulation runs must be performed by varying the ultrasound waves pressure until the food samples temperature do not exceed 70°C.

For the first case study, Figure 7 shows the thermal analysis performed on the five levels of apple samples at 34 kHz, 80 kPa, and an initial temperature of 60°C. Figure 7a shows the isothermal curves generated inside the dehydration system after 40,000 seconds. Figure 7a shows the effect of ultrasound waves radiation on food samples. Note that the food samples closest to the piezoelectric transducer reaches a maximum temperature of 70.8°C, while the other samples have minimal increases in temperature. The temperature change in the apple samples is shown in Figure 7b. Note that the first food samples have the greatest influence on their temperature change. It should be noted that in Figure 7b each food sample is denoted with its Cartesian coordinates (x,z) within the system, being so for the first sample (0, 6.7), the second (0, 9.7), the third (0, 12.7), the fourth (0, 15.7), and the fifth (0, 18.7).

Figure 7.

Temperature of the apple samples radiated with a piezoelectric transducer.

Similarly, Figure 8 shows the temperature changes of five tomato samples radiated with ultrasound waves using a piezoelectric transducer at 70 kHz and 22 kPa for 40,000 seconds. Figure 8a shows the isothermal curves inside the dehydration system, in which it is observed that the first and second samples have a significant increase in temperature reaching a maximum of 70.1°C, while the other samples have smaller temperature changes. In Figure 8b, these temperature changes can be better observed from the curves plotted as a function of time.

Figure 8.

Temperature of the tomato samples radiated with a piezoelectric transducer.

However, Figure 9 shows the temperature change of the apple samples radiated with ultrasound waves using three piezoelectric transducers at 34 kHz, 52 kPa, for 40,000 seconds. Note that the maximum temperature reached was 70.2°C. Figure 9a shows the isothermal curves generated after 400,000 seconds in apple samples located at the five levels are more homogeneous compared to the isothermal curves in Figure 7a. These temperature changes are more visible in Figure 9. These curves show that all the apple samples have a significant temperature change due to the ultrasound waves radiation and that the difference between each of them is not so large compared to those obtained for a single transducer in Figure 7.

Figure 9.

Temperature of the apple samples radiated with three piezoelectric transducers.

Now, the temperature change in the tomato samples radiated by ultrasound waves using three piezoelectric transducers at 70 kHz and 5 kPa during 40,000 seconds is shown in Figure 10. Note in Figure 10a that the isothermal curves for the tomato samples show homogeneous temperature changes reaching a maximum temperature of 70.8°C. These temperature changes in the tomato samples are more evident in Figure 10b. Note the temperature increase on five levels of food samples occurs to a greater extent when radiated with three piezoelectric transducers than if the samples are radiated with a single transducer as shown in Figure 9b.

Figure 10.

Temperature change of tomato when radiated with three piezoelectric transducers.

From the thermal-acoustic analysis performed, it is possible to highlight different aspects that are relevant at the moment of designing a convection dehydration system assisted by ultrasound waves: (a) the distribution of the acoustic field produced by the ultrasound waves depends on the frequency, the number of transducers and the dimensions of the dehydration chamber, (b) the increase in the temperature of the food samples is a function of the frequency and pressure with which the samples are radiated, and (c) it is possible to radiate more than one sample simultaneously within the same dehydration chamber placed at different distances from the radiation source. Thus, in this way, it is possible to give an approximation of the thermo-acoustic behavior of different foods within the processes of dehydration by convection assisted by ultrasound waves.

5. Discussion and work perspectives

The performed analysis describes the thermo-acoustic behavior of ultrasound waves and food samples in an ultrasound-assisted convection dehydration system considering two case studies. In the first one, we use separately apple and tomato samples placed in five gridded trays spaced 3 cm from each other and ultrasound–radiated inside the dehydration chamber by using one piezoelectric transducer. In the second case study, we use the same conditions of the spatial distribution of food samples but using three piezoelectric transducers. From the thermo-acoustic analysis, the optimal operating frequencies were identified for apple (around 34 kHz) and tomato (around 70 kHz) samples. Also, from a temperature control on food samples, we identified the operating pressures that avoid the structural and nutritional damage of the food samples under consideration in the dehydration process. By obtaining the acoustic and thermal behavior of apple and tomato samples in an ultrasound-assisted convection dehydration system, we can design an ultrasound-assisted dehydration system that improves the quality, time, cost, and energy of small– and large-scale dehydration processes. The development of new technologies focused on ultrasound-assisted convection dehydrator systems for food dehydration remains a challenge for the scientific community when considering acoustic, thermal, mechanical, and biochemical approaches in the design, purchase, or implementation of dehydration systems applied to human consumption foods. Moreover, from these new technologies, it is possible to obtain dehydrated products with high nutritional content and little structural damage that, without leaving behind a good taste and texture, may allow the human being to store food and its subsequent consumption in a long-term time. Finally, it is necessary to emphasize that although there are several commercially available ultrasound–assisted dehydration systems, there is still a gap to be filled so that ultrasound-assisted systems can be applied in multiple food fields at an industrial level, and at the same time, that these systems tend to be used on a daily basis at affordable prices in households around the world.

6. Conclusions

From the thermo-acoustic analysis for 0.5 cm-plate apple and tomato samples inside an ultrasound-assisted convection dehydrator assisted by ultrasound waves, and considering the physical dimensioning of the dehydration chamber, the physical parameters of the apple and tomato, and the wave and heat equations, we can determine the spectral behavior of ultrasound waves inside food and the temperature changes on the apple and tomato samples. Based on this analysis, we identify the optimum operating frequencies from the average pressure applied in the dehydration system for apple and tomato samples using ultrasound waves at 34 kHz and 70 kHz, respectively. Using these operating frequencies, we performed a spatial analysis when the ultrasound-assisted convection dehydrator was implemented with one and three piezoelectric transducers radiating a test acoustic field at 2 Pa. However, the spectral analysis showed that using ultrasound waves at 34 kHz for apples, the acoustic field was more uniformly distributed on apple samples located at five levels inside the dehydration chamber using three piezoelectric transducers. The same phenomenon occurred when the tomato was radiated at 70 kHz with three piezoelectric transducers. In addition, considering the distribution of the acoustic field in the two case studies, the temporal analysis to obtain the temperature changes inside the apple and tomato samples at the test frequencies was performed. The results obtained showed that when the five racks were used to place the apple and tomato samples and they were radiated using one piezoelectric transducer, only the temperature of food sample closest to the transducer was increased, while the rest of the food samples remained without significant temperature changes. Note that for the case of a single piezoelectric transducer, the pressure levels ranged between 80 kPa for apples and 22 kPa for the tomato to reach a temperature close to 70°C, which avoids structural and nutritional damage to the foods. When three piezoelectric transducers were used, uniform temperature changes were observed, and a similar temperature increase was observed in food samples of the five racks. Under these conditions, the apple and tomato samples reached temperatures between 67 and 70°C. When apple and tomato samples were radiated with three piezoelectric transducers using pressure between 52 kPa and 5 kPa respectively, less energy was required than when the food samples were radiated with one transducer. Thus, from the FEM analysis, it was possible to determine the optimal operating conditions at which an ultrasound-assisted convection dehydrator for apple and tomato samples can operate most efficiently based on its spectral and thermodynamic behavior.

Acknowledgments

The authors thank Instituto Politécnico Nacional (IPN-México) for financial support under grant numbers SIP–20220531 (Rubén Vázquez–Medina), SIP–20220572 (Omar Jiménez-Ramírez), and SIP-20220933 (Gonzalo Alonso Ramos-López). Daniel Aguilar-Torres (CVU-829790) thanks for the scholarship provided by Consejo Nacional de Ciencia y Tecnología (Mexico).

Conflict of interest

“The authors declare no conflict of interest.”