Thermodynamics, kinetics, and computational fluid dynamics modeling of Escherichia coli and Salmonella Typhi inactivation during the thermosonication process of celery juice

Highlights

• •Thermosinoculation was performed to inactivate E.coli and S. Typhi in celery juice.
• •Mathematical and thermodynamic models were performed on the inactivation data.
• •As the temperature increased, the sensitivity to sonication increased.
• •The inactivation process for microorganisms was not spontaneous.

Abstract

In this study, thermosonication (37 KHz, 300 W; 50, 60, and 70 °C) of celery juice was performed to inactivate Escherichia coli and Salmonella Typhi in 6 min. The inactivation of pathogens and the process were modeled using mathematical, thermodynamic, and computational fluid dynamics models. The findings indicated that the distribution of power dissipation density was not uniform across the entire domain, including the beaker area, with a maximum value of 27.8 × 10³ W/m³. At lower temperatures, E. coli showed a 9.4 % higher resistance to sonication, while at higher temperatures, S. Typhi had a 5.4 % higher durability than E. coli. Increasing the temperature decreased the maximum inactivation rate of both S. Typhi and E. coli by 15.5 % and 20.5 % respectively, while increasing the thermal level by 20 °C reduced the log time to achieve the maximum inactivation rate by 20.3 % and 34.9 % for S. Typhi and E. coli respectively, highlighting the stronger effect of sonication at higher temperatures. According to the results, the positive magnitudes of ΔG were observed in both E. coli and S. Typhi, indicating a similar range of variations. Additionally, the magnitude of ΔG increased by approximately 5.2 to 5.5 % for both microorganisms which suggested the inactivation process was not spontaneous.

1. Introduction

Fruit and vegetable juices are in great demand all over the world, and as non-alcoholic drinks that are full of nutrients; they can play a big role in the health of consumers. In addition to nutritional value, fruit and vegetable juices can have functional properties such as antioxidant, antimicrobial, anti-inflammatory, and anti-cancer properties [1], [2], [3]. Food safety is a global issue and diseases caused by food pathogens play a very important role in this issue. Escherichia coli and Salmonella play an important role in foodborne diseases and have been abundantly isolated from different types of food products [4]. As a result, the control of these pathogens is necessary to ensure the health of the consumers. One of the methods that are widely used today to destroy pathogens in food is the use of thermal processes [5]. Although these traditional methods can destroy pathogens, they can have adverse effects, including damage to nutrients. For this reason, the uses of alternative methods that have both the potential to eliminate pathogens and minimize damage to nutrients have become very popular [6]. Ultrasound is a technology that uses sound waves that are not in the human hearing threshold. Today, the use of ultrasound in food processing has attracted the attention of many researchers. With its efficiency, ultrasound can be used in various fields of food production and processing [7], [8]. Inactivation of microorganisms can be one of the attractive functions of ultrasound, which can be taken a step forward in its industrialization by examining and obtaining accurate and practical information in this case [9], [10]. Computational fluid dynamics (CFD) is a valuable tool that can be used to investigate and optimize the sonication inactivation of microorganisms [11]. It allows researchers to study the effects of various parameters such as frequency, amplitude, duration, and positioning of the ultrasonic probe on the distribution of acoustic energy and fluid flow patterns. By simulating the fluid dynamics using CFD, researchers can better understand the mechanisms involved in the sonication process. It can be used to visualize the flow patterns, pressure distributions, and acoustic energy distribution within the sonication chamber or container [12], [13]. This information can help identify regions of high-intensity acoustic energy and areas with low energy, which are critical for the effective inactivation of microorganisms [10].

When microorganisms are exposed to sonication, several mechanical phenomena occur that contribute to their inactivation including cavitation, shear stress, acoustic microstreaming, and enhanced mass transfer. Sonication generates bubbles in the medium that rapidly expand and collapse, releasing energy in the form of shockwaves and extreme temperatures [14]. This disrupts microorganisms' cell walls or membranes (cavitation). On the other hand, sonication vibrations cause deformation and disruption of microorganisms' cellular structures. High-frequency energy also intensifies fluid flow near the cells, leading to further shear stress and cell damage (shear stress). Moreover, sonication produces small-scale circulation patterns, aiding in the uniform distribution of energy throughout the liquid. This ensures maximum inactivation of targeted microorganisms (acoustic microstreaming). Finally, sonication removes protective layers or biofilms from microorganisms, exposing them to antimicrobial agents in the surrounding medium [15]. This improves the effectiveness of chemical inactivation agents or disinfectants in combination with sonication (enhanced mass transfer). In this research, thermosonication was used to inactivate E. coli and S. Typhi in celery juice, and then the mechanism of this process was investigated using mathematical and thermodynamic models.

2. Materials and methods

2.1. Bacterial strain

Escherichia coli PTCC 1789 and Salmonella Typhi PTCC 1609 were purchased from the Persian Type Culture Collection (IROST; Tehran, Iran). Pathogenic bacteria were activated in a nutrient broth medium at 37 °C for 24 h.

2.2. Samples treatments

Celery was purchased from a vegetable shop in Shiraz City (Fars province, Iran), and after separating the undesirable parts of the vegetable, it was washed well with sterile water. Then it was placed at room temperature for 2 h to dry its surface moisture. In the next part, the vegetable was divided into different parts and juiced by a home juicer (Pars Khazar, JBG-610P). After filtering the juice with a cleaning cloth, it was passed through a 0.22-µm PVDF filter. Microorganisms were inoculated separately into vegetable juice (∼8.1 log CFU/mL) and control samples were without microorganisms. Celery juice samples (50 mL) were poured into a 100 mL beaker and then treated by an ultrasonic bath (Elmasonic S 300H; 37 KHz, 300 W; Germany) at temperatures of 50, 60, and 70 °C for 6 min.

2.3. Enumeration of bacteria

To count the bacteria, at first proper dilution was done and then different dilutions were cultured in the plate. MacConkey agar medium was used for S. Typhi and nutrient agar medium was used for E. coli. Finally, colonies were counted after incubation at 37 °C for 24 h.

2.4. Numerical modeling

2.4.1. Inactivation kinetics

The first-order kinetics is one of the simplest and most widely used models to describe the inactivation of microorganisms [16], [17]. This model is based on the existence of a linear relation between the logarithmic quantity of survived microorganisms and time. The first-order kinetics can be formulated as:

$$\mathit{log}\frac{N_t}{N_0}=-\frac{t}{D_T}$$ (1)

where in the above equation $N_t$, $N_0$, $t$ and are the number of survived and initial bacteria’s, time (min), and decimal reaction time (min), respectively. The parameter is defined as the required time to inactivate 90 % of the initial microorganism’s population. Besides the first-order kinetics, the decimal reaction rate can be substituted with the reaction rate constant $k$ (min⁻¹) as:

$$\frac{\mathit{dN}}{N}=-k.dt$$ (2)

The first-order kinetics is suitable for inactivation processes for which a log-linear relation between time and the number of survived bacteria is occurred. However, for more complex inactivation trends, several more sophisticated mathematical models are available such as Weibull and log-logistic models.

The Weibull model is a mathematical extension of first-order kinetics [18]:

$$\mathit{log}\frac{N_t}{N_0}=-b{t}^n$$ (3)

where and $n$ are the scale parameters of characteristic slope (min^-n) and dimensionless shape factor. The curvature of the inactivation trend is altered as the shape factor varies from lower to upper unity values. The successive increase or decrease of bacteria resistance to the inactivation process can be well described by n < 1 and n > 1, respectively.

Finally, the log-logistic model can be used as a non-log-linear model with the formulation of [19]

$$\mathit{log}\frac{N_t}{N_0}=\frac{A}{1+e^{4\sigma (\tau -\log \left(t\right))/A}}+\frac{A}{1+e^{4\sigma (\tau -\log \left(t_0\right))/A}}$$ (4)

where in the above equation, $A$ the difference between two asymptotes (log(CFU/mL)), $\sigma$ is the maximum of reaction rate log(CFU/mL)/log(min), $\tau$ is the required time (log(min)) to achieve the mentioned maximum inactivation rate and is set to 10⁻⁶ in the calculation [20], [21].

2.4.2. Thermodynamics relations

To evaluate the thermodynamic aspects of the process, the Arrhenius model in addition to inactivation heat (ΔH), entropy (ΔS), and Gibbs energy (ΔG) was implemented. The Arrhenius equation determines the reaction rate as a function of temperature and activation energy. Lower activation energy barrier and higher thermal levels aid the reaction to be accomplished in lower time intervals. The Arrhenius model can be stated in de Oliveira et al. and Peleg et al. [22], [23]

$$k=A{e}^{-\frac{E_a}{\mathit{RT}}}$$ (5)

where is the pre-exponent coefficient, $E_a$ is the activation energy (J/mol), R is the universal gas constant (8.314 J/mol.K) and T is the temperature (K). To simplify the linear curve fitting procedure for the determination of unknown parameters of A and E_a, the natural logarithm operator is applied to both sides of the Arrhenius model as:

$$\mathrm{l}\mathrm{n}(k)=\mathrm{l}\mathrm{n}(A)-\frac{E_a}{\mathit{RT}}$$ (6)

The thermodynamic properties of the inactivation process can be evaluated based on the obtained kinetics data:

$${\mathrm{\Delta }H=E}_a-RT$$ (7)

$${\mathrm{\Delta }G=E}_a+RTln\left(\frac{K_{b}T}{\mathit{hA}}\right)$$ (8)

$$\mathrm{\Delta }S=\frac{\mathrm{\Delta }H-\mathrm{\Delta }G}{T}$$ (9)

In Eq. (7), (8), (9), $K_b$ is the Boltzmann constant (1.381 × 10⁻²³ J/K) and h is the Plank constant (6.63 × 10⁻³⁴ J.s).

The microbial inactivation process involves the breaking of chemical bonds within microorganisms, leading to their destruction. Two important factors in this process are enthalpy (ΔH) and entropy (ΔS), which influence the overall change in Gibbs energy (ΔG). Enthalpy represents the heat energy transferred during the process and is determined by whether energy is absorbed (positive ΔH) or released (negative ΔH). On the other hand, entropy represents the molecular disorder in the system, with positive ΔS indicating increased disorder and negative ΔS indicating increased order.

The goal of the microbial inactivation process is to achieve a negative ΔG, indicating a spontaneous process. The relationship between ΔH, ΔS, and temperature (T) determines whether this is achieved. If ΔH and ΔS have opposite signs, the temperature plays a crucial role in determining spontaneity. At high temperatures, the TΔS term can overcome the ΔH term, resulting in a negative ΔG and a spontaneous process. However, if ΔH and ΔS have the same sign, both temperature and the magnitude of ΔH and ΔS influence the spontaneity, and the temperature at which ΔG becomes negative or positive can vary depending on the system.

2.4.3. Statistical model assessment

To evaluate the performance of the applied mathematical models several assessment parameters including the root mean square error (RMSE), the coefficient of determination (R²), and the adjusted coefficient of determination are implemented (Adj-R²). These parameters are calculated as:

$$\mathit{RMSE}=\sqrt{\frac{\sum {(predictedvalue-observedvalue)}^2}{i-n}}$$ (10)

$$R^2=\frac{\sum {(predictedvalue-average)}^2}{\sum {(observedvalue-average)}^2}$$ (11)

In the above equations, the numbers of observations are represented by i and the number of parameters is denoted by n.

2.4.4. The mass conservation equation

The sonication process leads to the creation of micro/nano vapor bubbles within the flow domain. So, the fluid domain should be treated as a mixture. The continuity (mass conservation) equation can be presented by Roohi and Hashemi [15]:

$$\frac{\partial }{\partial \mathrm{t}}\left({\rho }_{\mathit{mix}}\right)+\mathrm{\nabla }.\left({\rho }_{\mathit{mix}}{\overrightarrow{V}}_{\mathit{mix}}\right)=0$$ (12)

In Eq. (1) the density and velocity vector are represented as $\rho$ and $\overrightarrow{V}$. The mixture properties are mentioned using the subscript mix. These properties are determined based on the properties of liquid (liq) and vapor (vap) phases using the vapor volume fraction ($\alpha$) as the weight function:

$${\rho }_{\mathit{mix}}=\alpha {\rho }_{\mathit{vap}}+(1-\alpha ){\rho }_{\mathit{liq}}$$ (13)

It should be noted that the simulation of vapor bubble transition from nucleation to growth, transport, and collapse is a complicated process for which one could refer to Abedi et al., 2020 for more information [11].

2.4.5. The momentum conservation equation

To determine the overall momentum equation, the sum of momentum equations for each phase can be implemented. Using the mentioned procedure, the interaction terms in the momentum equation of each phase can be eliminated as:

$$\frac{\partial }{\partial \mathrm{t}}\left(\sum _{j=1}^n{\alpha }_j{\rho }_j{\overrightarrow{V}}_j\right)+\mathrm{\nabla }.\left(\sum _{j=1}^n{\alpha }_j{\rho }_j{\overrightarrow{V}}_j{\overrightarrow{V}}_j\right)=-\sum _{j=1}^n{\alpha }_j\mathrm{\nabla }{p}_j+\mathrm{\nabla }.\sum _{j=1}^n{\alpha }_j({\tau }_j+{\tau }_{\mathit{Tj}})+\sum _{j=1}^n{\alpha }_j{\rho }_j\overrightarrow{g}+\sum _{j=1}^n{\overrightarrow{S}}_j$$ (14)

p denotes the pressure, $\tau$ stands for shear stress, and the force vector source term is represented by $\overrightarrow{S}$. Besides, the summation over each phase is denoted by the j index and the turbulent shear stress is pointed out using the T index.

2.4.6. The energy conservation equation

The determination of the thermal field and the impact of ultrasonic waves on the transferred energy can be performed using the summation of the energy equation of liquid and vapor phases:

$$\frac{\partial }{\partial \mathrm{t}}\left(\sum _{j=1}^n{\alpha }_j{\rho }_j{\mathrm{\Psi }}_j\right)+\mathrm{\nabla }.\left(\sum _{j=1}^n{\alpha }_j{{\overrightarrow{V}}_j(\rho }_j{\mathrm{\Psi }}_j+p\right)=\mathrm{\nabla }.\left(\left(\sum _{j=1}^n{\alpha }_j(K_j+K_T\right)\mathrm{\nabla }T\right)+E_{\mathit{source}}$$ (15)

In the above equation, the thermal conductivity is denoted by K and total heat generation via all of the involved mechanisms including the sonication process is denoted by $E_{\mathit{source}}$. Besides, the energy content of each phase can be pointed as the sum of internal energy, pressure work component, and the kinetic energy term as:

$$E_j=h_j-\frac{p}{{\rho }_j}+\frac{{{\overrightarrow{u}}_j}^2}{2}$$ (16)

In Eq. (16) the enthalpy of phase j is represented by $h_j$.

2.4.7. Acoustic pressure field

The acoustic pressure field is generated as a result of high-frequency oscillations of the transducers. The time-dependent propagation of the pressure waves can be formulated based on the Helmholtz equation [24]:

$$\mathrm{\nabla }\left(\frac{1}{\rho }\mathrm{\nabla }P\left(\overrightarrow{X},t\right)\right)-\frac{1}{\rho {S_{\mathit{sound}}}^2}\frac{{\partial }^{2}P\left(\overrightarrow{X},t\right)}{\partial t^2}=0$$ (17)

In the above equation is the magnitude of pressure created by sound waves and is the location vector. Moreover, the sound speed in the domain is represented by $S_{\mathit{sound}}$.

The pressure waves are propagated as a harmonic wave. Therefore, considering the very small period of oscillations in comparison to the time constant of fluid field formation, it is reasonable to transform the Helmholtz equation into the frequency domain and utilize the obtained data as a predefined domain for calculations of the velocity field via the solution of the Navier-Stokes equations. The mentioned transformation from time to frequency domain can be applied where $\mathrm{\Omega }$ is the ultrasonic frequency [25]. So, the Helmholtz equation will have the following form:

$$\mathrm{\nabla }\left(\frac{1}{\rho }\mathrm{\nabla }P\left(\overrightarrow{X}\right)\right)-\frac{{\mathrm{\Omega }}^2}{\rho {S_{\mathit{sound}}}^2}P\left(\overrightarrow{X}\right)=0$$ (18)

2.5. Domain boundary condition schematic and computational grids

Proper boundary conditions should be applied to determine the acoustic pressure field. Three types of boundaries were present in the sonication simulation namely, the transducer's surface, the far field boundaries, and other surfaces. At the transducer’s surfaces, a specified amplitude with the frequency of the simulated aperture was applied. The two bottom transducer amplitude was determine based on the device power, the material properties, as well as the frequency. The peripheral walls which were referred to as far field zones were modeled using the perfectly matched layer (PML) condition which assumed the sound wave could perfectly pass through them without any reflections. The other boundaries were simulated using the hard wall assumption.

The determination of the velocity field was performed using the CFD simulation. The appropriate boundary conditions for the described problem included the moving and stationary walls in addition to the constant pressure surface. At the transducers’ walls, a normal velocity profile with sinusoidal time behavior was assigned. The local distribution of the velocity profile was based on the experimental data proposed by Rahimi, et al. [26]. According to this profile, the velocity reaches its peak value at the center of the transducer and is gradually reduced toward the edges. Additionally, all of the other boundaries were assigned with the no-slip boundary condition.

The development of the computational grids was performed in two steps. First, for the calculations of the acoustic pressure field, the grid size must be reduced between λ/10 to λ/6 for near transducer and farther locations, respectively (λ is the wavelength of the sonic waves). According to this procedure and by utilization of the grid study routine, the optimal number of computational cells for the acoustic field was determined to be 193,457. The schematic of the CFD and acoustic domain with applied boundary conditions and grid cells are represented in Fig. 1.

Fig. 1.

The schematic of a) CFD computational domain, b) acoustic domain and boundary conditions, c) acoustic domain grids and d) the zoomed view of acoustic domain grids near the transducer.

To perform CFD calculations that accurately simulate the boundary layer, it is crucial to meet certain requirements such as high grid density in regions with steep variations in variables. However, this comes at the cost of increased computational resources. After careful analysis and evaluation, it has been determined that the ideal number of cells required for these calculations is 78,391. This specific value strikes a balance between capturing the intricacies of the boundary layer and managing computational costs effectively. For the implemented fluid properties and well as the acoustic operating conditions one could refer to Hashemi and Roohi [10].

2.6. Numerical simulation specifics

The RNG turbulent model was implemented due to its capability to simulate the curvature of streamlines and intense strain rates. The coupled acoustic and CFD simulations were performed using the COMSOL Multiphysics^TM 6.2. The frequency and time domains were selected to solve the ultrasonic and flow field domains, respectively [16], [27]. The computations were performed via a workstation with the specification of CPU: Intel® Xeon® Processor E6540 (18 M Cache), RAM: 16 GB DDR5. The computational time varied between 48 and 51 h.

In this study, we utilized OriginPro 9.8.0.200 software developed by OriginLab Corp. in Massachusetts, USA. This software was employed for various data analyses, including evaluating variance, conducting linear and non-linear analysis, and creating graphical representations. To ensure the accuracy of our findings, the data were collected and analyzed in triplicates. The analysis was performed using the analysis of variance (ANOVA) method. The results were expressed as the average data to provide a measure of the precision of the data. To determine the significant differences between the means of different groups, we used the post-hoc Turkey test. The level of significance chosen for this analysis was p < 0.05, indicating that any observed differences with a probability lower than 5 % were considered statistically significant.

3. Results and discussion

The inactivation of E. coli and S. Typhi are revealed in Fig. 2 subjected to 6 min of sonication treatment. The examinations were conducted in three thermal levels of 50, 60 and 70 °C. In all of the cases, the initial populations of microorganisms were set as 8.1 log (CFU/mL). The removal of both microorganisms was increased as a result of applied sonication duration and temperature level. For the temperature levels of 60 and 70 °C, it was observed that 6 min was sufficient to nearly remove all of the bacterial count; however, for the lowest examined temperature (i.e. 50° C) the remained microbial count of E. coli and S. Typhi was observed to be 1 and 0.6 log (CFU/mL), respectively. Moreover, it was evident that for E. coli the procedure duration of 5 min was almost sufficient for a total bacterial inactivation and beyond that instance no significant inactivation was observed.

Fig. 2.

Effects of sonication duration and temperature on inactivation of a) E. coli and b) S. Typhi.

It should be noted that the removal of the microbial count was mainly a result of the physical disruption of their cell walls caused by shockwaves generated by cavitation, or the detrimental harm caused by changes in permeability due to the increased volume of cellular content during the formation of pores [28].

Fig. 3 shows the velocity contours at a frequency of 37 kHz above the ultrasonic transducer at different time instances. The transducer's oscillations on the tip surface resulted in the generation of multiple pressure waves that pass through the fluid, creating an upward flow stream. This flow stream transports energy to the beakers placed above. Focusing on the snapshots of the flow field, the creation of the upward stream, its propagation within the tank, and its impact on the upper beaker from 0.008 to 9.387 s can be observed. The flow jet created by the vibrating transducer induced considerable circulation throughout the domain. Furthermore, it is important to acknowledge that the maximum velocity value differs at different time intervals. This observation was a result of the high-frequency velocity oscillations, making it highly unlikely to attain a constant or consistently increasing velocity variation [29].

Fig. 3.

The velocity contours at a frequency of 37 kHz above the ultrasonic transducer at different time instances.

It should be mentioned that the selection of an effective combination of time and velocity for microbial inactivation is based on the variability in the maximum velocity over different time intervals, as observed in Fig. 3. Therefore, when selecting an effective combination of time and velocity for microbial inactivation, one must consider the variability in the maximum velocity over different time intervals. It is necessary to understand the specific time instances and corresponding velocities that result in optimal microbial inactivation. This variability should be taken into account to ensure that sufficient energy is delivered to the microbial population for effective inactivation.

Fig. 4 presents the results of the acoustic simulation, displaying the total power dissipation density and surface intensity magnitude for frequency of 37 kHz. Acoustic total power dissipation density was a measure of the rate at which acoustic energy was dissipated or lost in a medium. In simpler terms, it referred to the amount of acoustic power that was converted into other forms of energy, such as heat, as sound propagated through a medium. The acoustic total power dissipation density was influenced by various factors, including the properties of the medium (such as its viscosity and thermal conductivity) and the frequency and intensity of the sound wave [30].

Fig. 4.

The results of the acoustic simulation: a) Total power dissipation density and b) surface intensity magnitude.

The total dissipation of power directly impacted the release of energy in the flow field. The findings indicated that the distribution of power dissipation density was not uniform across the entire domain, including the beaker area, with a maximum value of 27.8 × 10³ W/m³ (Fig. 4-a).

The magnitude of acoustic intensity referred to the measurement of sound energy transmitted through a medium per unit of time and area. As shown in Fig. 4-b, the acoustic intensity magnitude was the highest in close proximity to the sound source, specifically above the transducer's surface, and significantly decreased as it moved further away [31]. The maximum magnitude of acoustic intensity recorded was 1.81 × 103 W/m².

The mathematical modeling of the inactivation processes using the kinetic models provided useful information regarding the physical trend of microbial removal. The parameters and validation indices of the applied mathematical models are listed in Table 1. The main parameter for the first-order kinetics is $D_T$, which determines the resistance of the bacteria to the imposed harmful agent. Its magnitude specifies the required time to remove 90 % of the initial population and is inversely related to the reaction rate (k). According to the results, at the lower temperature the resistance to sonication was 9.4 % higher for E. coli, while as the temperature increased S. Typhi had a higher durability by 5.4 % compared to E. coli. Increasing of thermal level by 20 °C from 50 to 70 °C reduced the $D_T$ parameter by 15.8 and 27.0 % for S. Typhi and E. coli, respectively.

Table 1.

Mathematic models assessment: parameters and validation indices.

First-order kinetics
	S. Typhi			E. coli
	50 °C	60 °C	70 °C	50 °C	60 °C	70 °C
$D_T$	0.8096 a A	0.7165 bA	0.6816 cA	0.8861a B	0.7303b B	0.6468cB
R2	0.9812	0.9883	0.9784	0.9656	0.983	0.9483
Adj-R2	0.9812	0.9883	0.9784	0.9656	0.983	0.9483
RMSE	0.3974	0.3312	0.4639	0.5125	0.4111	0.7333
Weibull model
	S. Typhi			E. coli
	50 °C	60 °C	70 °C	50 °C	60 °C	70 °C
b	0.9146c A	1.44b A	1.678a A	0.69c B	1.138b B	2.104aB
n	1.195a A	0.9794b A	0.912c A	1.318a B	1.12b B	0.798cB
R2	0.9921	0.9884	0.9816	0.9908	0.9875	0.9681
Adj-R2	0.9905	0.9861	0.9779	0.9889	0.985	0.9617
RMSE	0.2833	0.3604	0.4691	0.2911	0.3858	0.6309
Log-logistic model
	S. Typhi			E. coli
	50 °C	60 °C	70 °C	50 °C	60 °C	70 °C
A	−6.223 bB	−6.135 bA	−5.674a B	−5.6b A	−6.095c A	−5.083a A
$\sigma$	−3.035bA	−2.654a A	−2.565a B	3.069b A	−3.053bB	−2.44a A
$\tau$	1.576 a A	1.426bB	1.256c A	1.552aB	1.444b	1.011c B
R2	0.9994	0.9976	0.9959	0.9991	0.998	0.9922
Adj-R2	0.9991	0.9964	0.9939	0.9986	0.9971	0.9882
RMSE	0.08778	0.1839	0.2471	0.1036	0.1711	0.3495

All values are means of three determinations with coefficient of variations (CV = SD/mean × 100) < 5 %. Different lowercase letters for the same in each parameter at different temperatures show significantly different (p < 0.05). Different capital letters for each microbe at the same temperature and parameter indicate a significant difference (p < 0.05).

The Weibull model is considered as a simplified but strong alternative to the linear kinetic model for analyzing microbial inactivation. It introduces the concept of treating microbial survival as a distribution of the remaining population, taking into account the variations in treatment intensity or the differing resistance levels of microorganisms [32]. The results for the n-value of the Weibull model were interesting and provided useful information regarding the effect of temperature level on the progressive resistance of the remaining microbial count. For S. Typhi an downward concavity (n > 1) was reported for 50 °C, while by increasing the temperature the trend altered to upward concavity (n < 1) [33]. In other words, at 50 °C a decreasing resistance behavior was present which changes to increasing trend for 60 and 70 °C. The same trend was observed for E. coli with the difference that the decreasing resistance trend was observable for both of 50 and 60 °C.

The log-logistic model operates under the assumption that microorganisms consist of subgroups resistant to treatment, subgroups capable of self-repair and sensitive subgroups formed due to the absence or loss of repair mechanisms. This model suggests a treatment duration parameter that is specific to the bacterial strain and relies on the treatment time, enabling the prediction of the most effective inactivation time.

The magnitude of $\sigma$ as the maximum inactivation rate was obtained to decrease by increasing of temperature for both microorganisms. The mentioned reduction was obtained to be 15.5 and 20.5 % for S. Typhi and E. coli, respectively. On the other hand, the $\tau$ parameter which represents the log time to achieve the maximum inactivation rate conversely relates to the lethality of the imposed agent. The mentioned parameter was reduced by 20.3 and 34.9 % by increasing of the thermal level by 20 °C for S. Typhi and E. coli, respectively which emphasize the higher effect of sonication at elevated temperatures. Lee et al. (2009) reported the combination of heat and/or sonication with or without pasteurization significantly reduced the process time to reduce E. coli K12 by 5 logs, and the biphasic model had the best fit to the inactivation data compared to the non-linear models [34]. The use of microwaves to inactivate Salmonella enterica serovar Enteritidis was investigated and the inactivation data were analyzed using the Weibull model. The results showed that a temperature of 70 °C or more reduced the Salmonella population (i.e. 300 W-80 s; 450 W-60 s or 600 W/800 W-40 s) by at least 4-log [35]. It was observed that the effect of temperature is more than the amplitude of pulsed ultrasound for the inactivation of Listeria monocytogenes, Shigella sonnei and Saccharomyces cerevisiae, while the effect of these two parameters was the same for Byssochlamys fulva. It was also reported that pulse thermosonication of tangerine juice for the inactivation of the mentioned microorganisms was well described by models such as polynomial, log-logistic, Weibull, biphasic linear and modified Gompertz models [36].

The magnitude of microbial inactivation Gibbs, heat, and entropy plays an important role in understanding and interpreting the process. These parameters are listed in Table 2 for both of the microorganism at three different temperature levels.

Table 2.

The thermodynamic properties for the inactivation process.

E. coli				S. Typhi
Ea (J/mol)		A		Ea (J/mol)		A
1.45 × 104		255.4432		1.08 × 104		64.97483
T (°C)	ΔH (J/mol)	ΔG (J/mol)	ΔS (J/mol.K)	T (°C)	ΔH (J/mol)	ΔG (J/mol)	ΔS (J/mol.K)
50	11853.33	79004.02	−207.80	50	8063.33	78892.07	−219.18
60	11770.19	81083.30	−208.05	60	7980.19	81085.16	−219.43
70	11687.05	83165.07	−208.30	70	7897.05	83280.76	−219.68

All values are means of three determinations with coefficient of variations (CV = SD/mean × 100) < 5 %.

The variations of the Gibbs energy can be interpreted as the driving force behind the microbial inactivation and can be utilize to determine the spontaneous or non-spontaneous nature of the process. The negative values of ΔG implies the spontaneous and favor able inactivation, while for the positive values of this parameter can be interpreted as a required input energy [37]. In other words, negative values of the Gibbs energy suggest that the energy released during the process is sufficient to overcome any energy barriers and promote microbial inactivation and consequently, inactivation process is energetically favorable and likely to occur spontaneously.

For the present study, the magnitudes of ΔG were positive for both of E. coli and S. Typhi with similar range of variations. It should be also noted that the magnitude of ΔG was increased between 5.2 and 5.5 % for both of microorganism. Hence, the examination concluded that the predicted inactivation was not spontaneous.

Based on the obtained values, it was concluded that each inactivation was endothermic with higher values for both of the microorganism [38]. However, the required energy consumption was obtained to be about 47 % higher for E. coli in comparison to S. Typhi. Moreover, increasing of the thermal level slightly reduced ΔH between 1.4 and 2 %.

Finally, focusing on the obtained values of ΔS, one could deduce that the inactivation process reduced the entropy and system randomness. When microorganisms are active and present in a system, they contribute to the overall disorder by their metabolic processes, growth, and movement. However, when microbial inactivation occurs, the system becomes less chaotic as the microorganisms are no longer actively participating in these processes. However, it should be noted that the variations of ΔS with temperature and type of microorganism is very limited. Overall, the interpretation of Gibbs, heat, and entropy variations in microbial inactivation provides insights into the energetic and thermodynamics of the process and helps understand the driving forces and changes occurring during microbial inactivation.

4. Conclusion

The effectiveness of sonication treatment in inactivation of E. coli and S. Typhi was examined. It was found that sonication at temperatures of 60 °C and 70 °C for 6 min effectively eliminated almost all pathogens. However, at a temperature of 50 °C, a small number of bacterial counts remained. Increasing the temperature from 50 to 70 °C resulted in a 15.8 % decrease in the count for S. Typhi and a 27.0 % decrease for E. coli.

Flow field examination showed that the transducer's oscillations induced significant circulation and created an upward flow stream that transported energy to the beakers. Acoustic simulation revealed that the distribution of power dissipation density was not uniform, with the highest value observed in the beaker area. The magnitude of acoustic intensity was highest near the sound source and decreased as it moved away.

The first-order kinetics model showed that E. coli had higher resistance to sonication at lower temperatures, while S. Typhi showed higher durability at higher temperatures. Increasing the thermal level reduced resistance for both microorganisms. The Weibull model demonstrated that the resistance of the remaining microbial count varied with temperature levels, with an upward concavity trend for S. Typhi and a decreasing resistance trend for E. coli at 50 °C, which changed to an increasing trend at higher temperatures. The log-logistic model suggested the most effective inactivation time based on the treatment duration parameter specific to the bacterial strain. The magnitude of the maximum inactivation rate decreased with increasing temperature, while the log time to achieve the maximum inactivation rate decreased, highlighting the higher effectiveness of sonication at elevated temperatures.

Analyzing Gibbs energy, heat, and entropy variations provided insights into the energetic and thermodynamics of the microbial inactivation process. The inactivation was found to be energetically non-spontaneous and required higher energy consumption for E. coli compared to S. Typhi. Additionally, the energy consumption for E. coli was found to be approximately 47 % greater than that of S. Typhi. Furthermore, raising the temperature slightly decreased the ΔH by 1.4 to 2 %.

Based on the findings of the study, future work could focus on optimizing the sonication duration and temperature levels to achieve complete inactivation of E. coli and S. Typhi at 50 °C, as residual microbial counts were still observed at this temperature. Additionally, further investigation could be conducted to understand the mechanisms of inactivation caused by shockwaves generated by cavitation and changes in permeability due to the formation of pores. This could involve exploring different sonication frequencies and intensities to optimize the physical disruption of cell walls and maximize the removal of microorganisms. Furthermore, the mathematical models used in the study could be more refined and validated with experimental data to enhance their accuracy in predicting microbial inactivation under different conditions.

Author contribution

Elahe Abedi and Reza Roohi wrote the manuscript, conceived and designed the research and analyzed the data. Seyed Mohammad Bagher Hashemi conceived and designed the research and analyzed the data. Elahe Abedi and Reza Roohi conducted the experiments. All the authors read and approved the manuscript.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

CRediT authorship contribution statement

Seyed Mohammad Bagher Hashemi: Writing – review & editing, Resources, Project administration, Conceptualization. Reza Roohi: Software, Methodology, Formal analysis, Data curation, Conceptualization. Elahe Abedi: Writing – review & editing, Resources, Project administration, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.