**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.

$$\frac{D\rho}{Dt} + \rho \frac{\partial u}{\partial \mathbf{x}} = \mathbf{0},\tag{1}$$

$$
\rho \frac{Du}{Dt} + \frac{\partial P}{\partial \mathbf{x}} = \mathbf{0},
\tag{2}
$$

$$
\rho \frac{D\varepsilon}{Dt} + P \frac{\partial u}{\partial \mathbf{x}} = -\frac{\partial q}{\partial \mathbf{x}},\tag{3}
$$

$$P = c\_0^2 \delta \rho \left[ \mathbf{1} + \frac{B}{2!A} \frac{\delta \rho}{\rho\_0} + \frac{C}{3!A} \left( \frac{\delta \rho}{\rho\_0} \right)^2 + \dots \right],\tag{4}$$

where *<sup>D</sup> Dt* is the Stokes derivative of the variable studied, *ρ* is the medium's density, *P* is the medium's pressure, *ε* is the medium's internal energy, *q* is the heat flow inside the dehydration chamber, *c*<sup>0</sup> is the sound speed, *δρ* is the density excess (*δρ* ¼ *ρ* � *ρ*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.

$$
\omega^2 \nabla^2 u - \frac{\partial^2 u}{\partial^2 t} = \mathbf{0},
\tag{5}
$$

where *u* represents acoustic waves, ∇<sup>2</sup> 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 threedimensional viscous medium [52]. Now, considering that the complex wave number *k* ¼ *β* þ *jα* is used to calculate a solution by a harmonic time, from Eq. (5) the expression shown in Eq. (6) is obtained,

$$
\mu = \mu\_0 e^{-\alpha \mathbf{x}} e^{j(wt - \beta \mathbf{x})},\tag{6}
$$

where *u*<sup>0</sup> is the wave amplitude in *t* ¼ 0, *α* is the absorption coefficient, and *β* is the wave cycles number per distance unit.

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

$$P = P\_0 e^{-\alpha \mathbf{x}} e^{j(wt - \beta \mathbf{x})},\tag{7}$$

where *P*<sup>0</sup> is the wave pressure in *t* ¼ 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**.

where, *Qu* is the heat flux generated by the ultrasound wave, *Q*1, *Q*2, and *Q*<sup>3</sup> are the heat fluxes leaving the system, *Rf* 1, *Rf* 2, and *Rf* <sup>3</sup> are the thermal resistances of the sample walls, *Cf* is the food thermal capacitance, *Tf* is the food temperature, and *Tdc* 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).

**Figure 3.** *System variables for thermodynamic modeling.*

*Acoustic and Thermal Analysis of Food DOI: http://dx.doi.org/10.5772/intechopen.108007*

$$Q\_1 R\_{f1} = T\_f - T\_{dc},\tag{8}$$

$$
\mathcal{Q}\_2 \mathcal{R}\_{f2} = T\_f - T\_{dc}, \tag{9}
$$

$$Q\_3 R\_{\hat{f}3} = T\_{\hat{f}} - T\_{dc} \,. \tag{10}$$

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).

$$C\_f \frac{\mathbf{d}T\_f}{\mathbf{d}t} = \sum Q\_{input} - \sum Q\_{output},\tag{11}$$

where *Qinput* are the incoming heat fluxes, and *Qoutput* 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 *Qu* 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).

$$\frac{\text{d}T\_f}{\text{d}t} = \frac{\text{Q}\_u}{\text{C}\_f} - \frac{\text{3}}{\text{C}\_f \text{R}\_{fT}} T\_f - \frac{\text{3}}{\text{C}\_f \text{R}\_{fT}} T\_{dc},\tag{12}$$

where *Tf* is the food temperature, *t* is the time, *Qu* is the heat flux produced by the ultrasound, *Cf* is the food heat capacity, *RfT* is the total thermal resistance generated by the apple walls, and *Tdc* is the drying chamber temperature.

In this case, it is contemplated that *Qu* ¼ 2*αI*, where *α* is the local acoustic absorption coefficient of the food, and *I* is the local sound intensity. In the same way, *Cf* ¼ *ρfCpf A* where *ρ<sup>f</sup>* is the food density, *Cpf* is the food-specific heat, and *A* is the food transverse area. Thus, Eq. (12) is also expressed by Eq. (13).

$$\frac{\mathbf{d}T\_f}{\mathbf{d}t} = \frac{2aI}{\rho\_f \mathbf{C}\_{pf}A} - \frac{3}{\rho\_f \mathbf{C}\_{pf}A \mathbf{R}\_{fT}} T\_f - \frac{3}{\rho\_f \mathbf{C}\_{pf}A \mathbf{R}\_{fT}} T\_{dc} \tag{13}$$

If Eq. (13) is lumped and considering that *RfT* <sup>¼</sup> <sup>Δ</sup>*<sup>x</sup> kf* , where Δ*x* is the thickness of the sample, and *kf* is the food thermal conductivity, then, it would be written as:

$$\frac{\text{d}T\_f}{\text{d}t} = \frac{2aI}{\rho\_f \text{C}\_{pf}A} - \frac{3k\_f}{\rho\_f \text{C}\_{pf}A} \frac{\Delta T\_f}{\Delta \infty} \tag{14}$$

where <sup>Δ</sup>*Tf* <sup>¼</sup> *Tf* � *Tdc*, and *<sup>I</sup>* <sup>¼</sup> <sup>ℝ</sup> <sup>1</sup> <sup>2</sup> *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).

$$v = \frac{\partial \overline{e}}{\partial t} \tag{15}$$

where *ε* is expressed as is shown by Eq. (16).

$$
\overline{\boldsymbol{\varepsilon}} = \frac{\partial \overline{\boldsymbol{\varepsilon}}}{\partial \boldsymbol{\varepsilon}} \boldsymbol{\hat{x}} + \frac{\partial \overline{\boldsymbol{\varepsilon}}}{\partial \boldsymbol{\gamma}} \boldsymbol{\hat{y}} + \frac{\partial \overline{\boldsymbol{\varepsilon}}}{\partial \boldsymbol{z}} \boldsymbol{\hat{z}} \tag{16}
$$

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).

$$\frac{\partial T\_f}{\partial t} = \frac{aP}{\rho\_f \mathbf{C}\_{pf} A} \frac{\partial \overline{\mathbf{c}}}{\partial \mathbf{x}} - \frac{\mathbf{3}k\_f}{\rho\_f \mathbf{C}\_{pf} A} \nabla T \tag{17}$$

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:


pressure ! select geometries corresponding to air ! domain ! acoustic pressure ! select geometries corresponding to food ! contour ! pressure ! select geometries corresponding to piezoelectric transducers ! enter test pressure value (2 Pa)! initial values ! 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.


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 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.*

**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.

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

#### **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.*

### *Acoustic and Thermal Analysis of Food DOI: http://dx.doi.org/10.5772/intechopen.108007*

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.

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.
