Predicting the dielectric behavior of orange and other citrus fruit juices at 915 and 2450 MHz

ABSTRACT

Dielectric properties (DP: relative permittivity and loss factor) and electrical conductivity are important parameters for evaluating the microwave penetration depth and designing applicators for pasteurization processes. In this context, generalized equations that can predict the DP of a group of food products as a function of their characteristics are useful. The dielectric behaviours of eight citrus fruit juices (Pera, Lima, and Navel oranges, Ponkan tangerine, Murcott tangor, Mediterranean mandarin, Sicilian lemon, and Persian lime) and three Pera/Lima blends (25, 50, 75%) for temperatures between 0 and 90 °C and frequency between 500 and 3000 MHz were studied. The results were correlated with temperature at frequencies of 915 and 2450 MHz. It was possible to quantify the ionic and dipolar contributions. DPs were modelled based on the behaviour of pure water in a semiempirical approach using correction factors, which were correlated with temperature and juice physicochemical characteristics (pH, Brix, conductivity). Polynomial regressions can successfully predict relative permittivity with mean absolute errors of 0.49±0.35% and 0.58±0.48% and the loss factor with errors of 4.1±3.9% and 2.8±2.3% at 915 and 2450 MHz, respectively, for juices with Brix/acid ratio between 1.26 and 29.9.

KEYWORDS:

• Dielectric constant
• Dielectric loss
• Dielectric properties
• Electrical conductivity
• Pasteurization
• Microwave processing
• Fruit juices
• Orange juice

Introduction

Ready-to-drink fruit juices are usually thermally processed for inactivating undesired or harmful enzymes and microorganisms. Although exposure to high temperature can ensure food safety and extend shelf life, it also causes loss of nutritional and sensory qualities. Alternative processing technologies such as microwave heating have been studied to reduce unnecessary exposure to high temperatures and improve product quality.^[1] When operated in a continuous flow mode, microwave heating can provide a fast and relatively more homogeneous heating. Focused microwave heaters have been studied for pasteurization applications. However, key information needed for the design and simulation of such devices is the dielectric behaviour of the food product.^[2–4]

Microwaves are a form of electromagnetic radiation with frequency between 300 MHz and 300 GHz. Important microwave frequencies allocated by the Federal Communications Commission (FCC) for industrial, scientific, and medical applications are 869, 915, 2375, and 2450 MHz.^[5] Microwave interaction with food products can promote volumetric heating. The conversion of electromagnetic energy to thermal energy at microwave frequencies is achieved by dipole rotation (rapid oscillation of polar molecules, such as water) and ionic conduction (movement of ions subject to oscillating electric field). This interaction can be characterized by the relative complex permittivity of the material:

(1)

where is the relative electrical permittivity, is the dielectric loss factor, and .^[5] The relative permittivity represents the electric polarization of the material subject to an external electric field while the loss factor represents the energy dissipation as heat.^[6] The two mechanisms that contribute to dielectric heating in microwave frequencies can be expressed as follows:

(2)

where is the loss factor due to dipolar rotation and is the loss factor due to ionic conduction, which can be calculated as

(3)

where is the DC electrical conductivity (S/m), = 8.854 × 10^–12 F/m is the electrical permittivity of the free space, and is the angular frequency of the field (rad/s).^[7]

The dielectric properties (DPs) of a material (, ) depend on its composition, temperature, structure, and frequency of the alternating electric field.^[8] These properties have been studied in different frequency and temperature ranges for various liquid foods of interest such as fruit juices^[9–12], green coconut water^[13,14], honey^[15,16], milk^[17-22], wine^[23], soy beverages,^[19] and liquid eggs^[24].

DPs of citrus fruits and juices

Orange juice is the main product of interest of this work due to its economic importance in Brazil, which is a leading country in the production and exportation of orange juice.^[18] The Pera variety was considered since it is an important mid-season orange cultivar in the State of São Paulo. The other seven citrus varieties commercialized in the same region and used for juices, blends, or nectars were included in this study to investigate the range of DPs of citrus fruit juices and if they can be correlated with juice characteristics, such as soluble solids content (SS).

There are some published studies about the DPs of tissues of citrus fruits and few about the DPs of citrus juices regarding microwave pasteurization. Seaman and Seals^[25], for instance, determined the DPs of the skin and pulp of some fruits, including tangelo and orange at room temperature. Nelson et al.^[26,27] determined the DPs of the core of four citrus fruits (pink grapefruit, lemon, lime, and navel orange) at 23°C for frequencies between 200 MHz and 20 GHz. The relative permittivity of various fresh fruits and vegetables was plotted against moisture content (between 71% and 97%) for the limiting frequencies and a positive linear correlation was observed but with low coefficients of determination (R² = 0.477 and 0.825, respectively) because of the differences in the tissue structure and nature of water binding. Kuang and Nelson^[28] adjusted the Debye model to the data and tried to correlate the static permittivity and the relaxation frequency with the moisture content and total soluble solids, respectively. The obtained coefficients of determination were also low (R² = 0.637 and 0.696, respectively), but a trend was noticeable. According to the authors, better correlations might be expected for individual kinds of fruits and vegetables with wider ranges of moisture content.

Nelson^[29,30] studied the DPs of nine fresh fruit and vegetables, including the navel orange. In this study, frequencies were between 10 MHz and 1.8 GHz and the temperature of the sample tissue varied between 5°C and 95°C. Since radio frequencies ( < 300 MHz) were also tested, very high values of loss factors were obtained ( near 10³ at 95°C and 10 MHz), which masked interesting behaviours that occur at microwave frequencies. Moreover, high values of relative permittivity were also observed ( near 370 at 95°C and 10 MHz) that could be explained by the ionic polarization of the tissue cells (Maxwell–Wagner effect).

The DPs of fruit pulps were studied by Wang et al.^[31], which included navel orange and grapefruit. The frequency range was from 1 to 1800 MHz, including radio and microwave bands, using a coaxial probe and temperatures between 20°C and 60°C. Results of and were reported at discrete temperatures and frequencies.

For providing DPs data of fruit juices for pasteurization applications, Zhu et al.^[11] determined these properties for juices of apple, pear, orange, grape, and pineapple from 20 to 4500 MHz and temperatures between 15°C and 95°C. Results were correlated with temperature at frequencies of 27.12, 40.68, 915, and 2450 MHz. Power penetration depths were calculated and it was verified that, at lower frequencies, it significantly decreases with temperature. At 2450 MHz, the penetration depth showed less variation and increased with temperature. It was generally verified that ionic conduction was the dominant loss mechanism in the juices at lower frequencies. Moreover, for frequencies below 1.0 GHz, increased with increasing temperature, whereas above 3.0 GHz, it decreased as the temperature decreased. Consequently, different behaviours were observed at commercial frequencies of 915 and 2450 MHz.

Prediction of DPs of groups of foods

Often the DPs of a food product are measured and then correlated with temperature for a given frequency and several examples are available in the literature^[32,33]. Correlation with moisture content is also usual for microwave-assisted drying process, such as the study of the influence of moisture and salt content on the DPs or pistachio kernels by Ling et al.^[34]. There are studies where the DPs are correlated with a characteristic or property of a given food product, for instance the fat and protein content in cow milk^[22,35]. However, predictive correlations in the literature are restricted to specific foods. The prediction of the dielectric behaviour of a group of food products with different characteristics is challenging.

In search of composition-based correlations, the contribution of individual components has been studied for some model systems such as simulated frozen lean tuna with tylose water paste and NaCl^[36], simulated coconut water^[13], gellan gel with sucrose and NaCl^[37], solution of NaCl, D-sorbitol, and sucrose,^[38] and solutions of different carbohydrates^[39–42].

Sun et al.^[43] gathered DPs data from various foods and explored whether it was possible to correlate and with the composition. They verified that the inclusion of all types of foods (meats, fruits, and vegetables) inhibited any useful correlation with composition, but better results were obtained for particular types of foods. For instance, the permittivity and loss factor of meats (2450 ± 50 MHz) were correlated with temperature, water content, and ash content. Protein, carbohydrate, and fat contents did not improve the correlation significantly. The ash content is a good indicator of the total salts in a food material; however, ashes include both free and bound mineral ions that have different effects on the DPs.

Funebo and Ohlsson^[44] linearly correlated the permittivity and loss factor (2800 MHz, temperatures between 20°C and 60°C) from a group of fruits and vegetables (mushroom, apple, chervil, parsley, and strawberry) with temperature, water content, and bulk density, but large prediction errors were obtained (20% and 30%, respectively). Sun et al.^[43] and Funebo and Ohlsson^[44] verified that is harder to correlate with food composition than . In this work, it will be investigated whether it is possible to correlate the DPs of a group of juices from different citric fruits with their physicochemical characteristics in order to provide a generalized predictive model.

Objective

The objectives of this research were: (1) study the DPs and electrical conductivity of different citrus fruit juices that are relevant for microwave pasteurization at frequencies of 915 and 2450 MHz and temperatures between 0 and 90 °C, regarding properties ranges, temperature dependence, prevalent loss mechanism, and radiation penetration depth; (2) propose a generalized correlation for the prediction of the dielectric behaviour of citrus fruit juices at 915 and 2450 MHz for a wide range of temperature and physicochemical characteristics, such as the Brix/acid ratio. This model is based on the modification of the dielectric behaviour of water by the presence of soluble solids and ions.

Materials and methods

Sample preparation

The citrus fruits studied are shown in Table 1. Around 9 kg of each citrus fruit was purchased in a local market in the city of São Paulo (SP, Brazil) between July and September 2014. Fresh mature fruits were selected and washed and their juices were squeezed using a Fresh’N Squeeze Multi-Fruit juicer (FMC, Araraquara, Brazil) on the same day of purchase. This juicer can process 20 fruits per minute and each fruit is squeezed between two cups while the juice is collected and filtered from the inside avoiding contact with the peel and, consequently, the peel oil. Test tubes with 10 mL samples were stored at −30°C in a plasma freezer 349 FV (Fanem, São Paulo, Brazil) prior to the physicochemical characterization and determination of DPs and electrical conductivity.

Table 1. List of citrus fruits studied in this work and abbreviations used in text.

Common name	Scientific name	Know in Brazil as	Abbreviation
Pera sweet orange	(Citrus × sinensis (L.) Osbeck)	Laranja Pera	Pera
Lima acid-less sweet orange	(Citrus × sinensis (L.) Osbeck)	Laranja Lima	Lima
Navel or Bahia orange	(Citrus × sinensis var. brasiliensis Tanaka)	Laranja Baía	Navel
Ponkan mandarin or tangerine	(Citrus reticulata Blanco)	Tangerina Ponkan	Ponkan
Murcott tangor	(Citrus × tangor Tanaka)	Tangor Murcote	Murcott
Mediterranean or Willowleaf mandarin	(Citrus deliciosa Tenore)	Mexerica do Rio	Mediter
Sicilian lemon	(Citrus × limon ‘Bearss’)	Limão Siciliano	Sicilian
Persian or Tahiti lime	(Citrus × latifolia Tanaka)	Limão Tahiti	Tahiti

Blends between orange juices of different varieties can be made at the industry to reach the desired levels of acidity and sweetness. For instance, Lima orange juice can be used to reduce the acidity of Pera orange juice due to its higher .^[18,45] Therefore, samples of these two juices were mixed in proportions of 25%, 50%, and 75% of Pera juice and their DPs and electrical conductivity were determined. The objective is to verify whether there are any interaction effects on the DPs or whether the blend behaves as an ideal mixture.

Physicochemical characterization

The juices were characterized by their (°Brix), total solids content (, g/100g), potential of hydrogen (, –), titratable acidity (, %citric acid), and electrical conductivity at 20°C (, mS/cm). All the measurements were conducted in triplicate. Soluble solids were determined with a portable refractometer 711849 (Carl Zeiss, Jena, Germany) at 20°C. Total solids were determined according to AOAC^[46] using a vacuum oven MA030 (Marconi, Piracicaba, Brazil) at 100 mmHg and 70°C for 6 h. The pH at 20°C was measured using a pH meter Tec-3MP (Tecnal, Piracicaba, Brazil). The TA was determined according to AOAC^[46] using a pH-Stat PHM-290 (Radiometer, Copenhagen, Denmark) and expressed as citric acid percentage. Electrical conductivity at 20°C was determined using a YSI3200 conductivity meter with a YSI3552 cell (YSI, Yellow Springs, USA).

Electrical conductivity measurements

The electrical conductivity (, mS/cm) of the juices were measured at temperatures between 0°C and 90°C with increments of 10°C. The temperature was maintained using a TC500 thermostatic oil bath (Brookfield, OH, USA) and verified with a digital thermometer (Mileto, São Paulo, Brazil). Conductivity measurements were made in triplicates as described in the previous section.

DPs measurements

The relative electrical permittivity and the dielectric loss factor (, ) of the juices were determined using the open-ended coaxial line probe method. An 85070E Dielectric Probe Kit (performance configuration) connected to an E5061B Network Analyzer (Agilent Technologies, Bayan Lepas, Malaysia) was employed. Based on the reflection coefficient of the sample, the software 85070E calculated the DPs. An 85093C Electronic Calibration Module was used to minimize interferences. After warming up, calibration of the equipment was performed with three standards: open circuit (air), short circuit, and deionized water at a known room temperature. After calibration, the probe and cable were not moved.

An Erlenmeyer flask with 240 mL of the juice was placed in a TC-550 thermostatic oil bath (Brookfield, OH, USA) to reach the desired temperature, which was verified with a digital thermometer (Mileto, São Paulo, Brazil). The flask with the sample was then taken to the probe stand and raised so that the probe tip was submerged with at least 1 cm of juice around the tip without the presence of attached air bubbles. The measurement lasted 35 s. After that, the flask was returned to the bath and the temperature was adjusted to the next level. This procedure was repeated in triplicate with different samples of the same batch. The frequency sweep was from 500 MHz (lower limit of the probe) to 3000 MHz (upper limit of the network analyzer) at 101 discrete frequencies on a logarithmic scale. After each measurement, the probe was cleaned with distilled water and dried.

Data analysis

Physicochemical determinations, electrical conductivity measurements, and DP measurements were made in triplicates (different samples from the same batch). Mean values and standard deviations were calculated. Tukey’s test was used to analyze statistical differences among means of the physicochemical parameters ( < 0.05). All statistical analyses were made with software Statgraphics Centurion XV (StatPoint, VA, USA).

The penetration depth (, m) of the incident microwaves over a semi-infinite body is defined as the depth in which its power is reduced to 36.8% (inverse of the Euler number) and it was calculated for each temperature and frequency as

(4)

where is the speed of light in free space (2.9979 × 10⁸ m/s) and is the frequency of the electromagnetic wave (Hz).^[47] Relative permittivity and loss factor were plotted against frequency for all temperatures. DPs at the frequencies of 915 and 2450 MHz, which are commonly used for microwave heating, were calculated by linear interpolation for each run. The main results (, , ) at these two frequencies and the electrical conductivity () were correlated with temperature (0 ≤ ≤ 90°C) using a polynomial regression tool. The polynomial order was selected based on the coefficient of determination (), on the statistical significance of the parameters ( < 0.01), and on the residuals distribution.

The relative contributions of the ionic conduction and dipolar rotation mechanisms to the loss factor were calculated as follows:

(5a)

(5b)

where the loss due to ionic conduction () was calculated from Eq. (3) using the measured electrical conductivity and the loss due to dipolar rotation () was obtained by difference from Eq. (2).^[12] From Eqs. (2), (5a), and (5b), it can be seen that + = 100%. The results are plotted against temperature for the frequencies of 915 and 2450 MHz. Relative permittivity, loss factor, and electrical conductivity of the blends of Pera and Lima juices were measured to verify whether it is possible to estimate these properties from the linear combination of the values for the juices (at 915 and 2450 MHz):

(6a)

(6b)

(6c)

where is the volumetric fraction of Pera juice in the blend.

Lastly, DPs (, ) of the various citrus juices were grouped and correlated with physicochemical parameters (, , , , ) and temperature () at the frequencies of 915 and 2450 MHz. Since the temperature range was broader than the range of the physicochemical parameters, it was necessary to isolate the temperature effect, at least partially, before correlating data with the physicochemical parameters. To perform this, the DPs of the juices were modelled based on the behaviour of pure water using a semiempirical approach.

The main contribution to the relative permittivity of the juice comes from the polarization of the free water molecules. The presence of dissolved substances can disrupt the alignment of these molecules and the permittivity of the juice should be lower than that of pure water at the same temperature and frequency.^[13] Moreover, the presence of soluble and insoluble material has a low volumetric contribution to polarization compared with pure water.^[43] Consequently, the relative permittivity of the juice was expressed as

(7)

where is the water permittivity correction factor (0 < < 1) and is the relative permittivity of pure water at the same temperature and frequency. In the case of the loss factor, the presence of dissolved substances shifts the relaxation frequency of water to lower frequencies, thus increasing its dipolar loss for frequencies below the relaxation peak.^[37] Consequently, the loss factor of the juice was expressed, based on Eqs. (2) and (3), as follows:

(8)

where is the water dipolar loss correction factor ( > 1) and is the loss factor of pure water at the same temperature and frequency. In this work, the permittivity correction factor was calculated from Eq. (7) using the experimental values of permittivity of the juices and the permittivity of water from Franco et al.^[13] (0 < < 90°C):

(9a)

(9b)

This dipolar loss correction factor was calculated from Eq. (8) using the experimental values of loss factor and electrical conductivity of the juices at the same temperature (replicates were paired to propagate the uncertainty to ). The loss factor of water was estimated from Franco et al.^[13] (0 < < 90 °C):

(10a)

(10b)

First, the values obtained for and at 915 and 2450 MHz were correlated with temperature for each juice through polynomial regression to verify the model consistency and to see how the temperature dependency was like. Then, the correction factors from all juices were correlated with , , , and . and TA were not used as factors because it was verified that they correlated with and , respectively. Squared factors , , , and were tested to check whether there was a significant improvement in the model fitting. The multiple regression tool of Statgraphics Centurion XV (StatPoint, VA, USA) was used and the correlations obtained were evaluated based on the statistical significance of the factors ( < 0.01), on the adjusted coefficient of determination (, on the number of model parameters, and on the residuals distribution.

Results and discussion

Physicochemical properties

The physicochemical parameters determined for the juices are presented in Table 2. The pH ranged from 2.3 (Sicilian and Tahiti) to 5.9 (Lima) and the SS varied between 7 °Brix (Sicilian) and 15 °Brix (Ponkan). The Brix/acid ratio was calculated as and three main levels were observed: around 1.5 (Sicilian and Tahiti), between 17 and 30 (Pera, Navel, Ponkan, Murcott and Mediter), and 175 (Lima).

Table 2. Physicochemical characterization of the citrus fruit juices.¹

Fruit^2		(% citric acid)	(°Brix)	(g/100g)	Brix/Acid Ratio	(mS/cm)
Pera	3.82 ± 0.07^bc	0.727 ± 0.01^c	12.2 ± 0.2^d	13.78 ± 0.06^e	16.7 ± 0.5^b	3.23 ± 0.06^c
Lima	5.85 ± 0.06^e	0.070 ± 0.006^a	12.3 ± 0.1^d	13.76 ± 0.03^e	175 ± 12^c	2.75 ± 0.07^ab
Navel	3.97 ± 0.10^cd	0.614 ± 0.007^bc	12.2 ± 0.2^d	13.22 ± 0.06^d	19.8 ± 0.6^b	3.83 ± 0.09^d
Ponkan	4.02 ± 0.01^d	0.486 ± 0.004^b	14.5 ± 0.1^f	16.63 ± 0.06^g	29.9 ± 0.4^b	2.88 ± 0.04^b
Murcott	3.74 ± 0.02^b	0.709 ± 0.010^c	13.9 ± 0.1^e	15.75 ± 0.12^f	19.7 ± 0.3^b	2.52 ± 0.11^a
Mediter.	3.88 ± 0.03^bcd	0.590 ± 0.003^bc	10.2 ± 0.2^c	11.81 ± 0.05^c	17.2 ± 0.3^b	2.86 ± 0.18^b
Sicilian	2.29 ± 0.03^a	5.52 ± 0.06^d	7.0 ± 0.1^a	8.77 ± 0.09^a	1.26 ± 0.02^a	4.25 ± 0.09^e
Tahiti	2.26 ± 0.09^a	5.57 ± 0.20^d	7.7 ± 0.1^b	9.15 ± 0.04^b	1.38 ± 0.05^a	3.73 ± 0.13^d

¹Results are means of repetitions ± standard derivation. In columns, different letters denote significant differences (p < 0.05) according to Tukey’s test.

²Abbreviations listed in Table 1.

Pera is one of the main varieties used for orange juice production in Brazil and the Ministry of Agriculture, Livestock and Food Supply regulates the limits of ≥ 10.5 °Brix and Ratio ≥ 7.0.^[18,48] According to the United States Department of Agriculture, the requirements for Grade A unsweetened pasteurized orange juice are ≥ 11.0 °Brix and 11.5 ≤ Ratio ≤ 18.0.^[49] The lower acidity of the Lima variety makes it suitable to correct an orange juice with low ratio.^[45]

Electrical conductivity

The electrical conductivity of the juices is related to the mobility of solvated ions and it is an important parameter for processing foods using pulsed electric fields, ohmic heating, radio frequency heating, or microwave heating. For all the juices tested, a linear correlation with temperature was observed (Fig. 1), as is common with this type of food product.^{[12,13,50–52]} The coefficients of the linear regression are presented in Table 3 along with the coefficient of determination and the standard error of estimate (). The conductivity can be calculated as

(11)

Figure 1. Electrical conductivity of citrus juices as a function of temperature including linear regressions from Eq. (11) and Table 3.

Table 3. Linear regression parameters determined for the temperature dependence on the electrical conductivity of the citrus fruit juices (between 0 and 90°C), along with standard errors of estimate and coefficients of determination.

Citrus fruit juice^1	(mS.cm^−1.°C^−1)	(mS.cm^−1)	(mS.cm^−1)	R^2
Pera	0.0840 ± 0.0007	1.634 ± 0.035	0.10	0.9983
Lima	0.0711 ± 0.0008	1.381 ± 0.043	0.13	0.9965
Navel	0.1003 ± 0.0010	1.868 ± 0.056	0.16	0.9970
Ponkan	0.0801 ± 0.0010	1.299 ± 0.053	0.16	0.9957
Murcott	0.0764 ± 0.0012	1.116 ± 0.065	0.19	0.9929
Mediter	0.0800 ± 0.0010	1.268 ± 0.055	0.16	0.9953
Sicilian	0.1044 ± 0.0008	2.292 ± 0.045	0.13	0.9982
Tahiti	0.0879 ± 0.0008	2.050 ± 0.041	0.12	0.9979

¹Abbreviations listed in Table 1.

Table 4. Polynomial regression parameters determined for the temperature dependence on the relative electric permittivity (ε’) of the citrus fruit juices (between 0 and 90°C) for the frequencies of 915 and 2450 MHz, along with standard errors of estimate and coefficients of determination.

Citrus fruit juice^1	(MHz)	(°C^−2)	(°C^−1)	(–)	(–)	R^2
Pera	915	–	-(2.27 ± 0.02)×10^−1	+(8.06 ± 0.01)×10^1	0.4	0.9971
	2450	-(7.46 ± 0.97)×10^−4	-(1.00 ± 0.09)×10^−1	+(7.40 ± 0.02)×10^1	0.4	0.9945
Lima	915	–	-(2.35 ± 0.03)×10^−1	+(8.10 ± 0.02)×10^1	0.4	0.9961
	2450	-(8.47 ± 1.20)×10^−4	-(9.44 ± 1.15)×10^−2	+(7.43 ± 0.02)×10^1	0.5	0.9923
Navel	915	–	-(2.31 ± 0.02)×10^−1	+(8.11 ± 0.01)×10^1	0.3	0.9980
	2450	-(8.56 ± 0.78)×10^−4	-(8.56 ± 0.78)×10^−2	+(7.42 ± 0.02)×10^1	0.3	0.9959
Ponkan	915	–	-(2.24 ± 0.02)×10^−1	+(7.94 ± 0.01)×10^1	0.3	0.9974
	2450	-(1.08 ± 0.11)×10^−3	-(6.25 ± 1.06)×10^−2	+(7.20 ± 0.02)×10^1	0.5	0.9916
Murcott	915	–	-(2.27 ± 0.03)×10^−1	+(7.99 ± 0.02)×10^1	0.5	0.9949
	2450	-(1.13 ± 0.12)×10^−3	-(6.39 ± 1.10)×10^−2	+(7.25 ± 0.02)×10^1	0.5	0.9916
Mediter	915	–	-(2.40 ± 0.02)×10^−1	+(8.16 ± 0.01)×10^1	0.3	0.9981
	2450	-(8.93 ± 1.24)×10^−4	-(9.77 ± 1.16)×10^−2	+(7.50 ± 0.02)×10^1	0.5	0.9918
Sicilian	915	–	-(2.39 ± 0.02)×10^−1	+(8.11 ± 0.01)×10^1	0.4	0.9973
	2450	-(4.43 ± 0.68)×10^−4	-(1.55 ± 0.06)×10^−1	+(7.62 ± 0.01)×10^1	0.3	0.9980
Tahiti	915	–	-(2.41 ± 0.02)×10^−1	+(8.07 ± 0.01)×10^1	0.3	0.9979
	2450	-(4.47 ± 0.80)×10^−4	-(1.57 ± 0.08)×10^−1	+(7.60 ± 0.01)×10^1	0.3	0.9971

¹Abbreviations listed in Table 1.

DPs

Fig. 2 shows the results (average of triplicate) of the DPs determination for the Pera orange juice for temperatures between 0°C and 90°C and in the frequency range of 500–3000 MHz. The top plot shows the relative electrical permittivity () and frequency and temperature increases can be seen to have negative effects on this parameter. As the temperature increases, the molecular motion intensifies and, consequently, the polarization of the media is less effective. As for the frequency effect, as it increases, the polarization of the water molecules also become less effective, thus promoting dipolar loss.^[8,13]

Figure 2. Dielectric properties of Pera orange juice (average of triplicate) for temperatures between 0°C and 90°C and in the frequency range of 500–3000 MHz. Top: relative electrical permittivity (). Middle: dielectric loss factor (). Bottom: Cole–Cole plot of the complex permittivity ().

In comparison with the electrical permittivity of water reported by Franco et al.^[13] for the same frequency and temperature ranges, the permittivity of the juice is slightly smaller (between 0% and 7%). This can be explained by the reduction in water activity and by the volumetric contribution of soluble and insoluble solids that have a lower permittivity.^[12,43]

The middle plot in Fig. 2 shows the dielectric loss factor (). As shown in Eq. (2), two mechanisms contribute to the dipolar loss in this frequency range: ionic and dipolar. Ionic loss is favoured by higher temperatures (improvement of ionic motion) and lower frequencies (increase of the length of ionic motion). This behaviour can be better observed at the lowest frequency (500 MHz). On the other hand, dipolar loss is related to the dipolar relaxation of the water molecule. The relaxation peak of water can be found at frequencies higher than those studied in this work (17.0 GHz for pure water at 20°C^[53]), but the loss increase with frequency can be clearly seen for the lower temperatures (0°C, 10°C, and 20°C). As a temperature raise shifts the relaxation peak to higher frequencies^[37,53], the dipolar dielectric loss decreases for a given frequency. This trend can be better seen at the highest frequency (3000 MHz) in Fig. 2. The behaviour observed in the loss factor plot from 500 MHz to 3000 MHz can be interpreted as the transition from an ionic-governed loss to a dipolar-governed loss.^[7,53]

The results for the other juices are not shown here since the behaviour is quite similar to the one shown in Fig. 2 for the Pera orange juice, as was observed by Siguemoto and Gut^[12] for the juices of different apple varieties. The main differences observed among varieties are related to ionic conduction. Juices with higher conductivity have higher loss factors at lower frequencies, for instance, Sicilian lemon juice can reach = 46 at 90°C and 500 MHz. At higher frequencies, the dipolar loss is mostly linked to water content, which is similar among juices (Table 2); thus, the variation was smaller. Differences among juices will be better discussed ahead for the commercial frequencies of 915 and 2450 MHz.

The bottom plot in Fig. 2 is the Cole–Cole plot that represents the permittivity in the complex plane. For some temperatures, an inflection point can be clearly noted in the convex curve. This point indicates the shift of the dominant loss mechanism between ionic and dipolar as frequency changes. Taking the 20°C curve, for instance, as the frequency increases from 500 MHz to 3000 MHz, the dominant loss mechanism changes from ionic to dipolar with a minimum loss close to 1.4 GHz.

The DPs and electrical conductivity of the orange juices obtained in this study follow the same trends observed in other liquid foods for similar ranges of frequency and temperature, such as fruit juices^[9,11,51], fruit pulps^[25], apple juices^[12], green coconut water,^[13,14] and milk^[21,54].

Dielectric behaviour at commercial frequencies

Results for relative permittivity, loss factor, and penetration depth ( calculated with Eq. (4)) are presented in Fig. 3 for the commercial frequencies of 915 and 2450 MHz and for temperatures between 0°C and 90°C. Permittivity of the juices showed a linear decrease with temperature at 915 MHz and a quadratic decrease at 2450 MHz. Polynomial regressions for are presented in Table 4, along with the standard errors of the estimate, which were less than 0.5, and the coefficients of determination. The relative permittivity can be calculated as

(12)

Figure 3. Relative electrical permittivity (), dielectric loss factor (), and microwave penetration depth (, mm) of the citrus fruit juices for temperatures between 0°C and 90°C and at frequencies of 915 MHz and 2450 MHz including the polynomial regressions from Eqs. (12), (13), and (14) and Tables 4, 5, and 7.

Relative permittivity did not show a strong variation among juices at these frequencies, as can be seen in Fig. 3, which could be explained by the similar water content (Table 2) as the water molecule is responsible for media polarization. On the other hand, the loss factor presented more distinct results among juices at 915 MHz, but not at 2450 MHz. This can be related to the ionic loss, which is more relevant at lower frequencies and depends on the electrical conductivity of the juice. Fig. 1 shows that Sicilian, Navel, and Tahiti are juices with high conductivities. This observation is consistent with the loss factor plot in Fig. 4 at 915 MHz as these juices also show high values of . The polynomial regressions for the loss factor data in Fig. 3 are presented in Table 5, including the standard errors of the estimate, which were under 0.6, and the coefficients of determination. The loss factor of the juices can thus be calculated as

(13)

Table 5. Polynomial regression parameters determined for the temperature dependence on the dielectric loss factor (ε”) of the citrus fruit juices (between 0 and 90°C) for the frequencies of 915 and 2450 MHz, along with standard errors of estimate and coefficients of determination.

Citrus fruit juice^1	(MHz)	(°C^−3)	(°C^−2)	(°C^−1)	(–)	(–)	R^2
Pera	915	-(1.19 ± 0.23)×10^−5	+(3.15 ± 0.31)×10^−3	-(1.07 ± 0.11)×10^−1	+(1.48 ± 0.01)×10^1	0.2	0.9944
	2450	-(2.13 ± 0.29)×10^−5	+(5.29 ± 0.39)×10^−3	-(4.26 ± 0.15)×10^−1	+(2.29 ± 0.01)×10^1	0.3	0.9954
Lima	915	-(1.49 ± 0.16)×10^−5	+(3.64 ± 0.22)×10^−3	-(1.57 ± 0.09)×10^−1	+(1.43 ± 0.01)×10^1	0.1	0.9958
	2450	-(2.03 ± 0.22)×10^−5	+(5.24 ± 0.30)×10^−3	-(4.47 ± 0.12)×10^−1	+(2.31 ± 0.01)×10^1	0.2	0.9977
Navel	915	-(1.71 ± 0.23)×10^−5	+(3.88 ± 0.32)×10^−3	-(1.11 ± 0.12)×10^−1	+(1.57 ± 0.01)×10^1	0.2	0.9962
	2450	-(2.28 ± 0.22)×10^−5	+(5.44 ± 0.30)×10^−3	-(4.20 ± 0.11)×10^−1	+(2.32 ± 0.01)×10^1	0.2	0.9967
Ponkan	915	-(1.33 ± 0.46)×10^−5	+(3.43 ± 0.62)×10^−3	-(1.32 ± 0.23)×10^−1	+(1.48 ± 0.02)×10^1	0.4	0.9741
	2450	-(1.58 ± 0.60)×10^−5	+(4.35 ± 0.82)×10^−3	-(3.88 ± 0.31)×10^−1	+(2.26 ± 0.03)×10^1	0.6	0.9793
Murcott	915	-(1.50 ± 0.40)×10^−5	+(3.60 ± 0.54)×10^−3	-(1.32 ± 0.20)×10^−1	+(1.49 ± 0.02)×10^1	0.4	0.9810
	2450	-(2.24 ± 0.66)×10^−5	+(5.40 ± 0.90)×10^−3	-(4.33 ± 0.34)×10^−1	+(2.31 ± 0.03)×10^1	0.6	0.9764
Mediter	915	-(1.61 ± 0.13)×10^−5	+(3.74 ± 0.18)×10^−3	-(1.23 ± 0.07)×10^−1	+(1.49 ± 0.01)×10^1	0.1	0.9983
	2450	-(2.54 ± 0.15)×10^−5	+(5.91 ± 0.21)×10^−3	-(4.50 ± 0.08)×10^−1	+(2.28 ± 0.01)×10^1	0.1	0.9986
Sicilian	915	-(1.16 ± 0.18)×10^−5	+(2.82 ± 0.25)×10^−3	-(3.10 ± 0.90)×10^−2	+(1.48 ± 0.01)×10^1	0.2	0.9986
	2450	-(2.33 ± 0.16)×10^−5	+(5.17 ± 0.22)×10^−3	-(3.65 ± 0.08)×10^−1	+(2.09 ± 0.01)×10^1	0.1	0.9974
Tahiti	915	-(1.25 ± 0.13)×10^−5	+(2.76 ± 0.18)×10^−3	-(5.18 ± 0.66)×10^−2	+(1.39 ± 0.01)×10^1	0.2	0.9986
	2450	-(2.37 ± 0.25)×10^−5	+(5.26 ± 0.34)×10^−3	-(3.86 ± 0.13)×10^−1	+(2.06 ± 0.01)×10^1	0.2	0.9946

¹Abbreviations listed in Table 1.

Figure 4. Contribution of ionic conduction to the loss factor () of the citrus fruit juices at frequencies of 915 MHz and 2450 MHz as calculated from Eqs. (2) and (6a) using the correlations in Eq. (11) and Table 3 for electrical conductivity and Eq. (13) and Table 5 for loss factor. Dotted lines indicate the region of contribution between 45% and 55%.

Fig. 3 shows that, for the frequency of 915 MHz, temperatures above 30°C provide increasing losses; however, at 2450 MHz, the loss factor drops with temperature from 0°C to 60°C and reaches approximately stable values. The different effects of temperature at these two frequencies indicate differences in the dominant loss mechanism. To better evaluate the ionic and dipolar contributions to the loss factor of the juices, Eqs. (5a)–(5b) were used to calculate the relative contributions using the adjusted polynomials in Table 4. The results of are displayed in Fig. 4. The dotted lines delimit the region for between 45% and 55%, where the ionic and dipolar contributions to the dielectric loss are closely balanced. This balance happens around 25°C at 915 MHz and around 75°C at 2450 MHz, depending on the juice.

When the loss mechanisms are balanced, temperature variations have a smaller effect on the loss factor and, consequently, the penetration depth of the microwaves remains constant, as can be noted in Figure 3. When the main loss mechanism is ionic, has a positive effect on ; therefore, the penetration depth decreases as the hot surface starts to absorb more power. This phenomenon is commonly referred to as “thermal runaway”: the high-temperature parts of the product have a higher heating rate and, consequently, temperature distribution becomes increasingly nonuniform. On the other hand, when the main loss mechanism is dipolar, has a negative effect on , and provides a “temperature levelling effect”, in which the high-temperature parts of the product have a smaller heating rate, thus providing a more uniform temperature distribution.^[5]

Regarding temperature uniformity, heating citrus juices at 2450 MHz would prevent thermal runaway, but the penetration depth is smaller than at 915 MHz. Nevertheless, the heating pattern depends strongly on the shape and dimensions of the sample and the geometry of the microwave cavity. Computational simulation of the microwave heating of a food product, using correlations for the DPs, is important to analyze the heating pattern and optimize the process.^[3,4,55]

Zhu et al.^[11] determined the DPs of the juice of navel oranges (Gannan variety) purchased in Xi´an China for temperatures between 15°C and 95°C and presented regression polynomials for predicting and at 915 and 2450 MHz. The juice had a TS of 10.7% ± 0.3%, = 4.72 ± 0.13, and = 3.05 ± 0.19 mS/cm at 25°C. In comparison, the Navel juice in this work had = 12.2 ± 0.2 °Brix, = 3.97 ± 0.10, and = 4.38 ± 0.16 mS/cm at 25°C (from Eq. (11) and Table 3). Comparing the experimental data in this work with the predictions from Zhu et al.^[11] for temperatures between 20°C and 90°C, there is a mean absolute difference of 2.0 ± 1.4 at 915 MHz and of 1.9 ± 1.2 at 2450 MHz. The curves show different slopes and cross near 45°C. For the loss factor, the curves show the same trend, but the results from this work are higher, with a mean absolute difference of 4.5 ± 1.5 at 915 MHz and 2.9 ± 1.7 at 2450 MHz. This difference can be explained by the lower electrical conductivity of the Gannan navel orange juice.

Dielectric behaviour of juice blends

Since blending juices from different citrus varieties is practiced by the industry, the DPs and electrical conductivity of the blend can be predicted by the linear combination of the values of the unmixed juices at the frequencies of 915 and 2450 MHz. Samples of Pera and Lima juices were mixed in proportions of 25%, 50%, and 75% of Pera juice and the properties predicted by Eqs. (6a), (6b), and (8) were compared to the experimental values. Linear behaviour (ideal mixture) was confirmed, with no mixture or interactions effects. Table 6 shows that the mean absolute errors for and are quite similar and mostly under 1%. Thakur et al.^[56] observed interesting interaction effects in the DPs of a mixture of condensed milk and distilled water at room temperature and frequencies between 0.6 and 3.0 GHz. As water was added to the condensed milk, the electrical conductivity increased because of the release of bounded ions; consequently, the dielectric loss also increased. This trend continued until the addition of water started only to dilute the free ions, then the dielectric loss started to decrease. As for the relative permittivity, the value of of the mixture was slightly higher than that expected by linear combination (non-interactive and distributive system). Since the major contribution to comes from the presence of water in the mixture, the difference can be explained by changes in the form of water bonding to the soluble matter.^[39,43]

Table 6. Absolute errors for the prediction of dielectric properties (, ) and electrical conductivity () of orange juice blends made with Pera and Lima fruits (means and standard deviation for temperatures between 0 and 90°C)

	0.00	0.25	0.50	0.75	1.00
	(0.40 ± 0.17) %	(0.41 ± 0.41) %	(0.49 ± 0.24) %	(0.26 ± 0.21) %	(0.32 ± 0.21) %
	(0.46 ± 0.26) %	(0.46 ± 0.33) %	(0.54 ± 0.36) %	(0.27 ± 0.36) %	(0.35 ± 0.19) %
	(0.35 ± 0.27) %	(1.43 ± 0.88) %	(0.83 ± 0.59) %	(0.78 ± 0.42) %	(0.59 ± 0.35) %
	(0.36 ± 0.24) %	(0.84 ± 0.54) %	(0.79 ± 0.57) %	(0.68 ± 0.61) %	(0.56 ± 0.29) %
	(2.6 ± 3.6) %	(1.7 ± 2.3) %	(3.4 ± 2.2) %	(2.5 ± 2.6) %	(1.7 ± 2.5) %

Strong interaction effects in mixtures were also observed by Smith Jr. et al.^[57] when studying the relative permittivity and relaxation frequency of mixtures of water and alcohols (methanol, ethanol, and propanol). The excess permittivity of the mixtures was negative as opposed to the results from Thakur et al.^[56] for condensed milk and water mixtures. The changes in the dielectric relaxation times of the water and alcohols mixtures are linked with the structures of the hydrogen bounding and clustering in the mixture. Álvarez et al.^[58] also detected nonideal mixing of grape pomace, ethanol, and water. In the case of the mixture of the two orange juices in this work, no interaction effects could be detected since they share the same solvent and similar composition. Therefore, the DPs of juice blends can be calculated by linear combination using the volumetric fraction.

Prediction of the dielectric behaviour based on physicochemical parameters

As observed in Fig. 3, there is a variation among the DPs of the various citrus fruit juices. The temperature dependence was empirically modelled through polynomial regression and the results for and are presented in Tables 4 and 5, respectively. The objective of this part of the study was to correlate the DPs of the juices at the commercial frequencies with the physicochemical parameters in order to obtain a general correlation for citrus juices.

The proposed model in Eqs. (7) and (8) is based on the DPs of water, on the calculated ionic loss (Eq. (3)), and on two parameters: the permittivity correction factor () and the dipolar loss correction factor (). First, these two parameters were calculated for the frequencies of 915 and 2450 MHz using Eqs. (7) and (8), the polynomial correlations for the DPs of pure water in Eqs. (9a), (9b), (10a), and (10b), the linear correlations for electrical conductivity of the juices in Table 3 and Eq. (11), and the polynomial correlations for the DPs of the juices in Tables 4 and 5 and Eqs. (12) and (13). The results are shown in Fig. 5.

Figure 5. Correction factors for the relative electrical permittivity of water () and for the dipolar loss factor of water () calculated for the citrus fruit juices at frequencies of 915 MHz and 2450 MHz using Eqs. (7)–(10b) and the correlations in Eqs. (12) and (13) Tables 3, 4, and 5.

It was verified that < 1 and > 1, implying that the presence of the soluble and insoluble substances of the juices reduces the permittivity of water and increases the dipolar loss. The permittivity reduction was more prominent at 2450 MHZ than at 915 MHz, whereas the dipolar loss increase was more intense at 915 MHz than at 2450 MHz. Even though these correction factors were defined based on the behaviour of pure water, noticeable temperature dependence (no-linear) was still observed. In order to better evaluate the significance of this temperature dependence, the experimental uncertainty was introduced. Factors and were once more calculated with Eqs. (7) and (8), but now using the experimental values of DPs and electrical conductivity (paired triplicates). Results were correlated with temperature for each juice and it was verified that can be correlated with temperature by a second-order polynomial and that correlates linearly with temperature ( < 0.01).

The next step was to correlate and with the physicochemical parameters as well. The parameters available were: , , , and electrical conductivity at 20°C (). Since it was verified that and were correlated, likely as and , the physicochemical factors selected were only , , and . Significance of the squared factors , , and was also tested to determine whether they bring any improvement to the model fitting.

The data of and at 915 and 2450 MHz was correlated with the selected factors using a multiple regression tool. The uncertainties on , , and were introduced by means of their standard deviations and paired with the DPs and conductivity data. The total number of observations were 8 juices × 10 temperatures × 3 measurements = 240. Regression results were analysed based on the statistical significance of the factors ( < 0.01), on the adjusted coefficient of determination (), on the number of model parameters, and on the residuals distribution. Table 7 brings the parameters of the best correlations obtained, which are valid for citrus juices with 2.26 ≤ ≤ 5.85, 7.0 ≤ ≤ 14.5, and 2.52 ≤ ≤ 4.25 mS/cm at temperatures between 0°C and 90°C. Squared temperature and squared showed important contribution as factors for some correlations. The correction factors and can thus be calculated as

(14)

Table 7. Multiple regression parameters determined for the correlation of the corrections factors ( and at 915 and 2450 MHz) of the citrus fruit juices with temperature and physicochemical parameters, along with standard errors of estimate and adjusted coefficients of determination

	(–)	(°C^−1)	(°C^−2)	(°Brix^−1)	(–)	(–)	(mS^−1.cm)	(–)
	+(8.84±0.05)×10^−1	+(1.26±0.05)×10^−3	-(7.46±0.52)×10^−6	-(4.88±0.31)×10^−3	+(4.48±0.34)×10^−2	-(4.51±0.38)×10^−3	–	0.006	0.9119
	+(8.67±0.06)×10^−1	+(2.06±0.06)×10^−3	-(1.30±0.06)×10^−5	-(6.33±0.37)×10^−3	+(3.71±0.41)×10^−2	-(3.57±0.46)×10^−3	–	0.007	0.9438
	+(1.85±0.39)×10^0	+(1.55±0.07)×10^−2	–	–	+(7.16±1.30)×10^−1	-(1.04±0.15)×10^−1	-(4.43±0.52)×10^−1	0.3	0.7268
	+(5.81±1.15)×10^−1	+(7.74±0.22)×10^−3	–	–	+(3.83±0.38)×10^−1	-(4.51±0.45)×10^−2	-(6.93±1.54)×10^−2	0.1	0.8643

Factor correlated with and , as was verified individually for each juice (Figure 5). SS and pH were important factors to distinguish among juices. Since the permittivity of the juice comes majorly from the water molecules, the presence of soluble solids reduces this permittivity by bounding water molecules and by a poor volumetric contribution to polarization. Consequently, a negative contribution from was determined ( in Table 7). This was also observed by Kuang and Nelson^[28] when studying the permittivity of various fresh fruits and vegetables and by Zhang et al.^[37], who studied the DPs of a gellan gel as affected by the addition of sucrose and NaCl.

The linear and quadratic contributions from seem to indirectly introduce particular characteristics of the juices in the model, since there is no straight connection between permittivity and pH. The adjusted coefficient of determination and the standard errors of estimate were good (Table 7) and the parity charts for the observed versus predicted values of (not shown) presented a fairly good distribution of residuals with preferably positive or negative residuals for each juice.

In the case of factor , there was a significant and positive temperature effect ( in Table 7). Linear and quadratic effects of the pH were also important to distinguish among juices; however, the electrical conductivity “replaced” the SS in the correlation. This was not expected because the ionic contribution was theoretically separated from the measured loss factor using Eq. (8) so that the correction factor corresponds only to the dipolar loss from water molecules. Factor can be considered as an indicator of the dissolved salts, as was the ash content in the work of Sun et al.^[43].

The adjusted coefficient of determination for and the standard errors of estimate were not so good as those obtained for (Table 7) and the residuals were not evenly distributed, showing a slightly curved and increasing pattern with increasing (not shown). This means that the proposed factors (, , , , , , , and ) and linear model were not sufficiently adequate to model the behaviour of . Fortunately, higher residuals were obtained in conditions where the ionic contribution to the dielectric loss exceeds that of the dipolar contribution; consequently, when the contributions are added using Eq. (8), the influence of the residual in the prediction of is minimal. Sun et al.^[43] and Funebo and Ohlsson^[44] also verified that is harder to correlate with food composition than .

In order to better evaluate the performance of the predictive model herein proposed, Fig. 6 presents the parity charts for the predicted DPs (calculated from Eqs. (7), (8), (9a), (9b), (10a), (10b), and (14) with parameters from Table 7) as a function of the observed properties. It is possible to note a very good prediction for the relative permittivity of the juices at 915 and 2450 MHz with = 0.9961 and = 0.9898, respectively. The mean absolute error for was only 0.49% ± 0.35% at 915 MHz and 0.58% ± 0.48% at 2450 MHz.

Figure 6. Parity charts for predicting the relative electrical permittivity () and the dielectric loss factor () of the citrus fruit juices using the proposed semiempirical model and the correlation in Eq. (15) and Table 7.

For predicting the loss factor of the juices at 915 and 2450 MHz, the mean absolute error was 4.1% ± 3.9% at 915 MHz and 2.8% ± 2.3% at 2450 MHz; furthermore, = 0.9273 at 915 MHz and = 0.9722 at 2450 MHz. For < 16 at 915 MHz, larger errors can be observed in Fig. 6 for the Lima, Navel, Sicilian, and Tahiti varieties. These points correspond to temperatures of 0°C and 10 °C, which are not of importance for pasteurization processes. Fig. 4 shows that, at lower temperatures, the loss is mainly dipolar; consequently, the error in the correction factor weighs in the prediction of . For higher temperatures, the dielectric loss is balanced or mainly ionic (Fig. 4).

Future works should explore temperatures above 90°C using a sealed measurement cell and also investigate factors that contribute to the dielectric response of juices from different types of fruits in order to develop prediction models valid for larger groups of food products.

Conclusions

The ranges of the main physicochemical properties of the eight juices were: pH between 2.26 and 5.85, soluble solids between 7.0 and 14.5 °Brix, and Brix/acid ratio between 1.26 and 29.9. Electrical conductivity of the juices showed a linear positive temperature dependence varying between 2.5 and 4.3 mS/cm at 20°C. At 915 and 2450 MHz, relative permittivity decreased with temperature and there was small variation among juices since polarization comes from free water molecules and the water content was similar. On the other hand, loss factor showed significant variation among juices, especially at 915 MHz, since the ionic contribution to loss is more important at this frequency and the electrical conductivity of the juices was quite variable. Consequently, salt content is a key component for the prediction of the dielectric behaviour at 915 MHz. At 2450 MHz, the dipolar contribution to the loss was more prominent in the studied temperature range. Generally, the loss factor increased with temperature at 915 MHz and decreased with temperature at 2450 MHz, which indicates the prevalence of different loss mechanisms at these frequencies. The calculated penetration depth indicates that heating at 2450 MHz should be more uniform (temperature levelling effect), but the penetration depth is considerably smaller than at 915 MHz. The study of the juice blends did not show interaction effects regarding the DPs and electrical conductivity, which could be linearly predicted from the volumetric fractions (ideal mixture). A model to predict the dielectric behaviour of the group of citrus juices at 915 and 2450 MHz was successfully adjusted using SS (°Brix), hydrogen potential (), electrical conductivity at 20°C, and temperature as parameters. The linear temperature dependence of the electrical conductivity must be known to evaluate the ionic loss. The model, which was based on the dielectric behaviour of pure water and the calculated ionic loss, was able to predict the relative permittivity with absolute errors around 0.5% and the loss factor with errors around 4%. Such a comprehensive model is useful to predict the dielectric behaviour of citrus fruit juices taking into account variations in physicochemical characteristics instead of relying on correlations that are specific for a given variety or cultivar.

Nomenclature

	=	polynomial coefficient of order i
	=	speed of light in free space (2.9979 × 108 m/s)
	=	relative contribution of ionic conduction to the loss factor (%)
	=	relative contribution of dipole rotation to the loss factor (%)
	=	penetration depth (m)
	=	field frequency (Hz)
	=	imaginary unit (–)
	=	statistical p-value (–)
	=	potential of hydrogen (–)
	=	coefficient of determination (–)
	=	adjusted coefficient of determination (–)
	=	soluble solids content (°Brix)
	=	temperature (°C)
	=	total solids content (g/100g)
	=	titratable acidity (%citric acid)
	=	volumetric fraction in a blend (–)

Greek letters

	=	water permittivity correction factor (–)
	=	water dipolar loss correction factor (–)
	=	complex relative permittivity (–)
	=	relative electrical permittivity of free space (8.854×10−12 F/m)
	=	relative electrical permittivity (–)
	=	relative electrical permittivity of water (–)
	=	dielectric loss factor (–)
	=	dielectric loss factor of water (–)
	=	dielectric loss factor due to dipole rotation (–)
	=	dielectric loss factor due to ionic conduction (–)
	=	pi number (3.1416)
	=	DC electrical conductivity (S/m)
	=	DC electrical conductivity at 20 °C (S/m)
	=	standard error of estimate
	=	angular frequency (rad/s)

Funding

The authors acknowledge financial support from the São Paulo Research Foundation (FAPESP) under grants 2012/04073-0 and 2013/07914-8 and also the scholarships provided by the Coordination for the Improvement of Higher Education Personnel (CAPES) and by the National Council for Scientific and Technological Development (CNPq).

Additional information

Funding

The authors acknowledge financial support from the São Paulo Research Foundation (FAPESP) under grants 2012/04073-0 and 2013/07914-8 and also the scholarships provided by the Coordination for the Improvement of Higher Education Personnel (CAPES) and by the National Council for Scientific and Technological Development (CNPq).