A Modified Formula for Calculating Dielectric Properties of Granular Agricultural Products in the Microwave Band

Dielectric properties are important characteristic parameters in the microwave-assisted application of hygroscopic agricultural media. To break through the limitation of traditional general effective medium (GEM) formula and develop the new calculation formula suitable for agricultural products, a modified formula, that is, modified general agricultural products effective medium (abbreviated as MGAPEM) formula, was presented in this study for the dielectric property calculation of granular agricultural products in the microwave band. The correctness, validity, and accuracy of MGAPEM formula were verified by using literature and experimental measurement data. Results show that the maximum errors of the dielectric constant and dielectric loss factor are 0.42% and 0.98%, respectively, in the conditions of microwave frequency (2.0–12.2 Ghz) and moisture content (2.0%-19.7%, wet basis). The accuracy of MGAPEM is higher than some traditional theoretical formulas. The MGAPEM formula provides a theoretical formula for the dielectric property analysis of granular agricultural products in the microwave band at room temperature (24℃).


Introduction
Dielectric properties are important parameters in microwave-assisted applications of hygroscopic agricultural media. In the past few decades, scholars have conducted numerous measures about the dielectric properties of agricultural products by different techniques, the agricultural materials are equated to a mixture of conductive particles surrounded by continuous inclusions (air) [1] , and some models and mixed formulas have proposed to calculate and predict the dielectric properties of granular agricultural products [2][3][4] . The early models and theoretical studies of the dielectric properties of granular agricultural products mainly focused on the analysis of the linear relationship between the dielectric properties and density and the dielectric mixing equation of the two-component gassolid (air and particle) mixture [5] . Kent first proposed that the dielectric constant ( ) H c and loss factor ( ) H cc of fish meal are linearly related to density [6] . Klein proposed that the square root of dielectric properties is linear with density [7] . Nelson adopted the cubic root linear function [called 1/3 ( ) vs H U c )] to predict the dielectric constants of pulverized coal, whole wheat flour, and whole wheat, he proposed that the prediction accuracy of cubic root linear function is better than square root linear function [3] . In addition, Nelson used several different dielectric mixing equations and linear extrapolation functions to analyse the dielectric properties of granular agricultural products, and the results showed that the Landau-Lifshitz-Looyenga(LLL) equation is considered suitable for the analysis of the dielectric properties of grain-like particulate matter. However, the values of some parameters corresponding to different particulate matters are different in these formulas. These parameters must be determined through a large number of experimental measurements even for a specific particulate matter. In recent years, scholars have successively established polynomial expressions to determine the relationship between the dielectric properties and the frequency, moisture content, temperature, and density of various agricultural particles, such as grain, wheat, cotton seeds, and coffee, through experimental measurements [4] . However, the obtained expression is not general, and this expression is almost a formula for a class of particulate matter, and it cannot be used to predict the dielectric properties of other types of granular agricultural products; the prediction error of some expressions is relatively large (3.0%-10.0%) [5] . To date, a large error between the predicted and the measured values can still be observed even if the famous Maxwell-Garnett, Bruggeman, Lichtenecker, and other mixed equations are used to predict the dielectric properties of agricultural products [5] .
During agricultural production and processing, reliance on experimental measurement methods alone to obtain the dielectric properties of agricultural products is unrealistic. Research on the prediction formula for the dielectric properties of granular agricultural products under different parameters has attracted the attention of scholars . In the present work, a modified general agricultural product effective medium (MGAPEM) mixing formula is used to analyse the dielectric properties of granular agricultural products in the microwave band. The effective dielectric properties of millet-air with different frequencies (2-6 Ghz), different moisture contents (2.0%-19.7% wet basis), and room temperature condition (24 °C) are measured. The correctness, accuracy, and validity of the MGAPEM formula are verified by the results of several mixed formulas (functions).

GEM theoretical Formula
In 1990, McLachlan proposed the calculation of the efficiency conductance of particle-filled binary composites based on a comprehensive analysis of the relationship between the efficiency conductivity of binary composites with the conductivity, volume fraction, and spatial dimension of the components . The general effective medium (GEM) theoretical formula for the composites is:

Proposition of MGAPEM Formula
The GEM formula cannot be directly used to predict or calculate the efficient dielectric properties of particlefilled binary composites under arbitrary medium parameters (arbitrary 1 H and 2 H ). With regard to the limitation of the GEM formula, this study uses the functional expressions of particle material medium parameter 1 H and matrix material medium parameter 2 H to characterize unknown characteristic parameters A and t . Moreover, given that the granular agricultural product naturally stacked in the air is a hygroscopic mixture, the moisture content is the main influencing factor of its dielectric properties, and the inclusion is air 2 . Accordingly, the modified formula for calculating the effective media of agricultural products is as follows: (2) Where H is the efficient dielectric property of the mixture; 2 is the dielectric property of particulate matter; 1 f and 2 f are the volume fraction of the particulate matter and inclusion (air) in the mixture 1 2 ( 1 ) f f , respectively; and e is the natural constant.

Literature Data Validation of the MGAPEM
Three agricultural products reported by Nelson et al. were used for theoretical calculation and comparative analysis on the basis of the data provided in the literature: ground white rice ("Lebonnet," 12.2%, 11.0 Ghz), whole kernel hard red winter wheat ("Scout 66," 11.5%, 9.4 Ghz) and ground hard red winter wheat ("Scountland," 10.9%, 11.7 Ghz) . This undertaking was performed to verify the validity and accuracy of the MGAPEM formula in the analysis of the dielectric properties of granular agricultural products.
First,  U function at the microwave frequencies and volume fractions of three agricultural products. In comparison with the LLL equation, the maximum error of the dielectric constant and dielectric loss factor obtained using the MGAPEM formula are 0.42% and 0.97%, respectively. As shown in Table 1 The efficient dielectric properties of the millet-air mixture were experimentally measured under the different microwave frequencies (2-6 Ghz), different moisture contents (2.0%, 4.7%, 7.5%, 10.0%, 15.0%, 19.7%, wet basis), and room temperature condition (24℃) using an un-calibrated coaxial transmission reflection method . This undertaking was performed to further verify the accuracy of using the MGAPEM formula for analyzing the dielectric properties of agricultural particles under different frequencies and moisture contents. Table 2 shows the measurement results of the efficient dielectric constant of the milletair mixture, where M is the moisture content, and U is the density of mixture ( 3 / g cm ). , and Böttcher in the moisture content(2%~19.7%) and the frequency (2-6 Ghz). In comparison with LLL equation, the maximum deviation amplitudes of MGAPEM for the dielectric constant and dielectric loss factor are 0.38% nd 0.98%, respectively. Böttcher is follow, and the deviation amplitude of the calculated results of the remaining formulas is large. Generally, the MGAPEM formula can be used to analyze the dielectric properties of agricultural particles under different frequencies and moisture contents. However, the calculation results of MGAPEM show a slightly increasing trend as the moisture content increases, but the deviation range is less than 1.0%.

Conclusions
This work proposes the MGAPEM formula for calculating and predicting the dielectric properties of granular agricultural products in the microwave band. We use the functional expressions of particle material medium parameter and matrix material medium parameter to characterize unknown characteristic parameters in GEM, and an ideal function expression and an optimal dimensionless parameter were obtained. The correctness, validity, and accuracy of MGAPEM formula were verified by using literature and experimental measurement data. The comparison of the calculated results of different formulas (functions) shows that the MGAPEM and LLL results are highly consistent under the conditions of microwave frequency and volume fraction calculation of all the studied subjects. The maximum errors of the calculated results for the dielectric constant and dielectric loss factor are 0.42% and 0.98%, respectively. The MGAPEM formula provides a theoretical formula for the dielectric property analysis of granular agricultural products in the microwave frequency band at room temperature (24℃).
However, the dielectric loss in agricultural products increases with the moisture content because the internal moisture molecules of hygroscopic substances highorder multiple moment contribution and multiple scattering effect additional loss under microwave, the accuracy of using the MGAPEM formula to predict the dielectric properties of agricultural product particles with high moisture content ( 20%) M ! needs to be further verified. Measurement techniques for the dielectric properties of single particles of agricultural products must also be developed.