Study on a Moisture Ratio Curve Model for Refractance Window Drying Based on a D-Optimal Mixture Design

Abstract

Refractance window (RW) drying is a new thin-layer drying technology that can control well the heating temperature to effectively reduce the loss of heat-sensitive substances. Here, an experiment on tomato pulp drying was carried out to study the drying characteristics of RW drying based on a D-optimal mixture design. The fitting of the classical model of thin-layer drying was studied, and SAS and 1stOpt calculation software were used to analyze the test data. The result showed that the RW drying equipment could dry 8 mm of tomato pulp in 120 min, and the maximum drying speed could reach 0.40 g/(g·min). Based on an effective diffusion coefficient under different conditions, the activation energy was 27.35 kJ/mol at an air speed of 3 m/s. When comparing the fitting of the moisture ratio curve in four classic thin-layer drying models, it was found that the R-square value of the modified Page model was 0.9960, which had better fitting properties. Then, the polynomial fitting model of thin-layer drying reflects the regression relationship between the coefficient of the classic model and drying conditions including temperature, wind speed, and time. After comparison with the classic model and validation experiment, the results showed that there is no significant difference between the polynomial fitting model and the validations under a confidence level of 0.95, which could well predict the change in the water content ratio over time under different conditions.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Most traditional industrial drying equipment used hot air as the medium to achieve drying [1]. However, during the heating process, some heat-sensitive substances, such as vitamins and volatile compounds, will be destroyed due to high temperature, resulting in material loss or deactivation, which could have a critical impact on food quality [2]. Refractance window (RW) drying was first proposed by Magoon in 1986, who applied for a patent. Generally, the RW drying method spread out the material evenly on Mylar film, and the heat was transferred to the material through a heated liquid under the film to achieve drying [3]. RW drying can control the temperature well by using liquid as the main heat transfer medium to avoid product overheating and finally restain heat-sensitive substances [4]. RW drying is a new thin-layer drying technology coupled with heat conduction and radiation. In the RW drying process, the liquid will be heated by the heat source and the heat energy will be transferred through the form of convection in the whole liquid. In general, the liquid will release energy through evaporation after the absorption of energy. However, in RW drying, the heat energy will be blocked and refracted back into the liquid under the influence of the film which is placed over the liquid. When the high wet materials are over the film, the materials act as a “window” that can permit heat transmission from liquid to the materials, and the moisture in the materials will be heated and evaporated. Once the drying process is accomplished and materials become dry products, the transfer of heat energy will be blocked again by the film. During the whole drying process, radiative heat transfer contributes to less than 5% of the total heat transmission [5, 6]. As it can be seen, there is a “window” in the RW drying process which can start and finish the drying process automatically based on the moisture content in materials. Thus, it can avoid high temperature at the end of drying to achieve the purpose of saving energy [7].

In recent years, research on RW drying has focused more on drying effect comparisons between different drying methods [8–12] and less on theoretical study. Ortiz Jerez and Ochoa-Martínez [13] of Columbia University studied pumpkin pieces based on heat transfer theory. They used temperature sensors to obtain the temperature in the upper, middle, and lower parts of the pumpkin as the data source and compared the different experimental conditions of natural convection and forced convection of the air above the material and whether the Mylar film was covered with aluminum foil. Based on an analysis of variance theory, the forced convection in the top air had significant effects on the temperature distribution of pumpkin pieces. Forced convection decreased the temperature of the pumpkin pieces. Meanwhile, the aluminum foil covering had a significant effect on the bottom temperature only. This shows that the radiation between the hot water and the pumpkin had positive effects on the bottom temperature increase of the pumpkin. Franco et al. [14] using salmon beef and apple as the research material to carry out the RW drying research studied the drying in the refraction window and found no improvements for lean beef and salmon compared with the traditional drying effect. Compared with the traditional drying method, the drying speed could be faster when drying fruits, and during the RW drying process, there was often a diffusion behavior that was different from that predicted by Fick’s second law, that is, abnormal diffusion behavior. Puente-Díaz et al. [15] fitted moisture curves under different experimental conditions to take advantage of five classic models and found that the R-square value of each model was higher than 0.85, indicating a good fitting property. These studies did not establish a relationship between experimental conditions and model parameters and did not provide a reference for experiments under other experimental conditions.

This study used tomato slurry as the experimental material, and an RW drying bedstand built in a laboratory was used as an apparatus. The drying properties of RW drying were studied, a polynomial model was established based on classic models, and variance analysis of the model was carried out. The purpose of this paper was to predict the change in the moisture ratio in the drying process of RW drying under different experimental conditions and to provide a reference for other modelling studies.

2. Materials and Methods

2.1. Materials

Tomatoes used for the experiment were purchased from the Yimaisheng Supermarket in Jiayuan Square in Nanjing, China, and complied with the corresponding food safety regulations. Then, tomatoes were divided into 11 groups and the water content of each group was determined under pressure of 101.3 kPa and temperature of 105°C which confirms the direct drying method standardized in the National Food Safety Standard Determination of Moisture in Food [16]. The result shows that the average value of the moisture content was 96.2%.

2.2. Instruments and Apparatus

RW drying equipment was designed and assembled in a laboratory and mainly consisted of a constant temperature heating water bath, a 0.25 mm thick Mylar film, and a fan [17]. The length, width, and height of the tank are 180 mm × 180 mm × 140 mm, and the Mylar film is placed still on the water. There is a fan installed on the tank which can provide the required wind speed. The structure is shown in Figure 1. When the equipment starts working, water is added to the water tank and heated with a heater. The heater can work intermittently according to the temperature set so that the difference between the water temperature and the set temperature did not exceed 2°C. The moisture meter used in the experiment was the MB27 model of the American Ohaus, the electronic balance was the DT5000 model of the China G and G, and the blender was the MJ-WBL2501A model of the China Midea.

[figure(s) omitted; refer to PDF]

2.3. Experiment

2.3.1. Determination of Experimental Design

To improve the fitting properties of the polynomial fitting model, this experiment adopted a D-optimal mixture design by using SAS and 1stOpt software. In response to surface experiments, values such as D-value, G-value, and A-value could be used to evaluate the accuracy of the fitting equation. Among them, the use of D-value was the most common. The D-optimal design was an experimental design plan that pursued the optimal D-value in all experimental plans. The D-optimal design was based on Wald’s determinant maximum criterion of the information matrix. A higher determinant value deduced a smaller variance of the predicted value of the regression coefficient as well as a smaller variance of the predicted value. The experimental design that reached the smallest variance was the best experimental design, thus named the D-optimal design [18]. Generally, the D-optimal design did not fit orthogonality and rotation. Therefore, based on the regression combination design, a D-optimal mixture design was proposed, which provided features of D-optimality, orthogonality, and rotation.

This experiment adopted the D-optimal mixture design scheme, assigned drying temperature, drying air speed, and drying time as variables, and took weight as the measured value for the experimental design. The R311 D-optimal experimental design table shown in Table 1 was used for the experiment. The first column of the design table was the experiment number, and the remaining tables described the coded value of different factors under each experiment number. In the table, different levels were represented by coded values, and the corresponding relationship between coded values and actual values is shown in Table 2.

Table 1

R311 experiment design table.

Number	Temperature	Air speed	Time
1	0	0	1.414
2	0	0	−1.414
3	−1	−1	0.707
4	1	−1	0.707
5	−1	1	0.707
6	1	1	0.707
7	1.414	0	−0.707
8	−1.414	0	−0.707
9	0	1.414	−0.707
10	0	−1.414	−0.707
11	0	0	0

Table 2

Comparison table of coding level and actual value.

	−1.414	−1.000	−0.707	0.000	0.707	1.000	1.417
Temperature (°C)	60	63.70		72.50		81.30	85
Air speed (m/s)	1	1.60		3		4.40	5
Time (min/min)	0		0.25τ	0.5τ	0.75τ		τ

Tontul et al. [19] held that to ensure the quality of fruits and vegetables, the drying temperature should not exceed 90°C. After pre-experimental results, when the surface temperature exceeded 85°C, the dried sample appeared obviously browning, so the highest temperature in this experiment was set to 85°C. Ochoa-Martínez et al. [20] indicated that the material temperature was approximately 10°C lower than that of a heat-transfer medium during an RW drying process. To ensure drying efficiency, the minimum temperature is selected as 60°C. To ensure that the experiment had reference in the maximum air speed range, the air speed range was set as 1–5 mm/s. As the total drying times under different temperatures and air speed are different, in order to make the time length consistent under different conditions, dimensionless time was adopted. Given that the total length of time was the unit length τ, the time range was 0 to τ. The comparison between the coding value and the actual value of each factor is shown in Table 2.

2.3.2. Experimental Process

An appropriate amount of water was added to the RW drying equipment to make the water surface and the Mylar film just contact, and then the equipment was preheated to the given temperature required by the experiment. The fan was turned on, and the fan speed was adjusted to meet the experimental requirements by monitoring the air speed in the middle of the Mylar film with an anemometer. After the temperature reached the experimental temperature, the tomatoes were washed, cut into pieces of approximately 20 mm in cubes, and placed in a small tray, and then, all the tomato pieces and juice were poured into a blender, which was run for 5 minutes to mix the tomatoes into a fine pulp and uniform slurry without visible particles. The Mylar film size was 180 mm × 180 mm and initially weighed before the experiment. Then, 200 g of tomato slurry was weighed and placed on the film and a steel sheet was used to scrape the tomato slurry evenly. A height vernier caliper was used to measure the thickness at 8 different points after leveling and the average thickness was approximately 8 mm. Then, every 15 minutes, the Mylar film and the tomato slurry were weighed as a whole and water on the bottom of the film was cleaned before weighing. The sample weight which is the difference between measured weight and the mass of the film was recorded in the experimental data until the weight no longer changed. Measuring weight can visually observe the water loss of the sample; furthermore, it can be used for calculating the drying speed of the sample. Eleven experiments were carried out according to the R311 design table.

2.4. Data Calculation Method

The moisture ratio (MR) refers to the residual moisture content of materials under certain drying conditions, which could indirectly reflect the drying speed under such conditions. The calculation method is shown in the following equations:$$\begin{array}{}\text{(1)}& \text{MR}=\frac{M_t-M_e}{M_0-M_e},\text{(2)}& M_t=\frac{m_t-m}{m},\end{array}$$where $\text{MR}$ is the moisture ratio; $M_0,M_t$ is the dry basis moisture content of tomato slurry at the initial time and T time, respectively, $g/g$; $M_e$ is the moisture content of the dry basis of the tomato slurry at the final moment of drying, $g/g$; $m_t$ is the weight of tomato slurry at time t, $g$; m is the absolute drying quantity of the tomato slurry, $g$. Generally, $M_e$ is relatively small to $M_t{,M}_0$. Therefore, equation (2) could be simplified as the following equation [21]:$$\begin{array}{}\text{(3)}& \text{MR}=\frac{M_t}{M_0}.\end{array}$$

The drying rate (DR), $g/g·\min$, which reflected the dehydration rate of the tomato slurry, could be represented by the following equation [22]:$$\begin{array}{}\text{(4)}& \text{DR}=-\frac{M_{t+∆t}-M_t}{∆t},\end{array}$$where $M_{t+∆t},M_t$ referred to the moisture content ratio on a dry basis at the times of $t+∆t$ and t, respectively, $g/g$.

The RW drying of the tomato slurry was a process of water transfer from inside to outside under the action of hot water. According to the analytical solution of Fick’s second law and the experimental data, the effective diffusion coefficient ($D_{\text{eff}}$) of moisture in the drying process of the tomato slurry could be calculated as the following equation [23, 24]:$$\begin{array}{}\text{(5)}& \text{lnMR}=\ln \frac{8}{{\pi }^2}-\frac{{\pi }^2{D}_{\text{eff}}t}{l^2},\end{array}$$where $D_{\text{eff}}$ is the effective diffusion coefficient, $m^2/s$; $t$ is the drying time, s; and $l$ is the half thickness of the tomato slurry, m.

According to equation (5), there was a linear relationship between $\text{LnMR}$ and time. $\text{lnMR}$ and $t$ were fitted with SAS software to obtain the value of the slope k, and then $D_{\text{eff}}$ was calculated. The relationship between the activation energy and $D_{\text{eff}}$ could be set up according to the Arrhenius equation using the following equation [25, 26]:$$\begin{array}{}\text{(6)}& D_{\text{eff}}=D_0\exp -\frac{E_a}{RT},\end{array}$$where $D_0$ is the number of prefactors of the Arrhenius equation and it is a constant, $m^2/s$; $E_a$ is the activation energy, $\text{KJ}/\text{mol}$; $R$ is the gas molecular constant, 8.314 $J/\text{mol}\cdot k$; and $T$ is the drying temperature, $K$. To simplify the calculation, the natural logarithm on both sides of the equal sign of equation (7) could be used to obtain the linear relationship between ${\ln D}_{\text{eff}}$ and $1/T$.$$\begin{array}{}\text{(7)}& {\ln D}_{\text{eff}}={\ln D}_0-\frac{E_a}{\text{RT}}.\end{array}$$

In this experiment, four models commonly used in thin-layer drying were selected to study the drying process of tomato pulp by RW drying. The four models are, respectively, logarithmic, Page, modified Page, and Wang and Singh, and the expressions are, respectively, the following equations:$$\begin{array}{}\text{(8)}& \text{MR}={\text{ae}}^{-k\tau +b},\text{(9)}& \text{MR}={\text{ae}}^{-{\tau }^n},\text{(10)}& \text{MR}={\text{ae}}^{-{k\tau }^n},\text{(11)}& \text{MR}=1-a\tau -b{\tau }^2,\end{array}$$where MR was the water ratio; τ was the dimensionless time; a, b, k, and n are the parameters to be fitted.

3. Results and Discussion

3.1. Experimental Data

According to the code value given in the R311 experimental design table, the experimental conditions were determined after calculation. We randomly assign values to eleven experimental groups and carry out eleven experiments in order of the size of the assignment. According to the guidance of the R311 experimental design table, repeated in the sample center, which helped to improve the accuracy of the fitting equation. The experimental data are recorded in Table 3. Equation (4) was used to calculate the drying rate, and Figure 2 shows the relationship between MR and DR under 1–11 different experimental conditions of temperature and wind speed. In Figure 2, the DR of each line increases first and then decreases with the increase of the MR, which was similar to Yuda’s results when exploring potatoes RW drying [12]. Most of these lines had a maximum value when the MR was 0.3. Studying these data, we found that the RW drying equipment could quickly dehydrate an approximately 8 mm tomato slurry within 120 minutes, with the moisture rate decreasing from 96.20% to 5.0%, and the fastest drying speed could reach 0.4035 $g/g·\min$. Using equation (5), we calculated $D_{\text{eff}}$ under different test conditions and recorded the results in Table 4.

Table 3

Experiment datasheet.

Time (min)	60°C 3 m/s		63.7°C 1.6 m/s				72.5°C 1 m/s		72.5°C 3 m/s		72.5°C 3 m/s		72.5°C 3 m/s
	Weight (g)	MR (%)	Weight (g)			MR (%)	Weight (g)	MR (%)	Weight (g)	MR (%)	Weight (g)	MR (%)	Weight (g)		MR (%)
0	200	100.00	200			100.00	200	100.00	200	100.00	200	100.00	200		100.00
15	188.0	93.76	184.0			91.68	186.0	92.72	174.0	86.49	176.0	87.53	176.0		87.53
30	176.0	87.53	170.0			84.41	170.0	84.41	154.0	76.09	150.0	74.01	150.0		74.01
45	156.0	77.13	156.0			77.13	154.0	76.09	128.0	62.58	124.0	60.50	124.0		60.50
60	146.0	71.93	142.0			69.85	136.0	66.74	96.0	45.95	96.0	45.95	96.0		45.95
75	130.0	63.62	118.0			57.38	116.0	56.34	70.0	32.43	70.0	32.43	70.0		32.43
90	112.0	54.26	104.0			50.10	98.0	46.99	48.0	21.00	48.0	21.00	48.0		21.00
105	94.0	44.91	86.0			40.75	76.0	35.55	30.0	11.64	32.0	12.68	32.0		12.68
120	76.0	35.55	66.0			30.35	56.0	25.16	18.0	5.41	18.0	5.41	18.0		5.41
135	58.0	26.20	48.0			21.00	40.0	16.84	12.0	2.29	12.0	2.29	12.0		2.29
150	42.0	17.88	32.0			12.68	28.0	10.60	8.0	0.21	8.0	0.21	8.0		0.21
165	28.0	10.60	18.0			5.41	22.0	7.48
180	18.0	5.41	14.0			3.33	18.0	5.41
195	12.0	2.29	12.0			2.29	12.0	2.29
210	10.0	1.25	10.0			1.25	8.0	0.21
225	8.0	0.21	8.0			0.21
Time (min)	72.5°C 5 m/s			63.7°C 4.4 m/s			81.3°C 1.6 m/s		85°C 3 m/s		81.3°C 4.4 m/s
	Weight (g)	MR (%)		Weight (g)	MR (%)		Weight (g)	MR (%)	Weight (g)	MR (%)	Weight (g)	MR (%)
0	20	100.00		200	100.00		200	100.00	200	100.00	200	100.00
15	176.0	87.53		186.0	92.72		176.0	87.53	178.0	88.57	170.0	84.41
30	152.0	75.05		158.0	78.17		148.0	72.97	150.0	74.01	124.0	60.50
45	126.0	61.54		136.0	66.74		118.0	57.38	118.0	57.38	96.0	45.95
60	96.0	45.95		110.0	53.22		84.0	39.71	86.0	40.75	64.0	29.31
75	64.0	29.31		84.0	39.71		48.0	21.00	48.0	21.00	34.0	13.72
90	36.0	14.76		62.0	28.27		22.0	7.48	22.0	7.48	28.0	10.60
105	20.0	6.44		44.0	18.92		12.0	2.29	12.0	2.29	14.0	3.33
120	12.0	2.29		26.0	9.56		8.0	0.21	8.0	0.21	8.0	0.21
135	10.0	1.25		16.0	4.37
150	8.0	0.21		10.0	1.25
165				8.0	0.21

[figure(s) omitted; refer to PDF]

Table 4

$D_{\text{eff}}$ under different conditions.

Conditions	Slope	$D_{\text{eff}}$	Conditions	Slope	$D_{\text{eff}}$
72.5°C 3 m/s	−0.0006967	$4.5178\times {10}^{-9}$	85°C 3 m/s	−0.00091642	$5.9426\times {10}^{-9}$
72.5°C 3 m/s	−0.00069334	$4.4960\times {10}^{-9}$	60°C 3 m/s	−0.0004607	$2.9874\times {10}^{-9}$
63.7°C 1.6 m/s	−0.00048007	$3.1131\times {10}^{-9}$	72.5°C 5 m/s	−0.0007756	$5.0294\times {10}^{-9}$
81.3°C 1.6 m/s	−0.00091502	$5.9335\times {10}^{-9}$	72.5°C 1 m/s	−0.00046196	$2.9956\times {10}^{-9}$
63.7°C 4.4 m/s	−0.0006528	$4.2331\times {10}^{-9}$	72.5°C 3 m/s	−0.00069843	$4.5290\times {10}^{-9}$
81.3°C 4.4 m/s	−0.00086818	$5.6298\times {10}^{-9}$

In Table 4, the $D_{\text{eff}}$ increased with increasing temperature when the air speed remained the same. When the temperature was lower than 81.3°C, $D_{\text{eff}}$ increased with increasing air speed. However, the relationship between $D_{\text{eff}}$ and air speed still needed to be explored when the temperature was too high. In Table 4, the $D_{\text{eff}}$ at different temperatures when the air speed was 3 m/s used SAS to fit ${\ln D}_{\text{eff}}$ and $1/T$. The experiment was repeated three times at 72.5°C. These three experiments existed independently. The repetition at the center of the sample helped to improve the accuracy of the variance equation. In Figure 3, the ${\ln D}_{\text{eff}}$ and $1/T$ had a linear relationship when the air speed was 3 m/s, and the linear equation could be expressed as ${\ln D}_{\text{eff}}=-9.73-3289.191/T$. According to equation (7), $E_a$ was 27.346 $\text{kJ}/\text{mol}$.

[figure(s) omitted; refer to PDF]

3.2. Prediction of Total Drying Time

Excel software is used to visualize the change trend of average drying time under different drying conditions of wind speed and temperature as shown in Figures 4 and 5. It is shown that the drying time decreases with the increase in wind speed and drying temperature.

[figure(s) omitted; refer to PDF]

In order to better solve the relationship between temperature and air speed and total drying time, a three-dimensional scatter plot with total drying time as a function value and temperature and air speed as independent variables was drawn by Origin, and these points were connected as planes to form Figure 6.

[figure(s) omitted; refer to PDF]

In Figure 6, time was roughly distributed in a plane that was formed by temperature and air speed. In order to find the relationship between them, we used the least square method to calculate the estimated values of the following equation parameters:$$\begin{array}{}\text{(12)}& \theta ={\displaystyle \sum }_{i=1}^n{y_i-{\widehat{y}}_i}^2,\end{array}$$where $y_i$ was the actual value, ${\widehat{y}}_i$ was the estimated value corresponding to $y_i$, and $\theta$ was their difference. In order to ensure the good fit of the formula, it was necessary to constantly adjust the variance parameter to ensure that $\theta$ was the minimum value to obtain the parameters of the equation. This process was complicated, so we used SAS software to calculate. The experimental data referred to Table 3, and the variance analysis referred to Table 5. The analysis results showed that the $P$ value of the model was 0.0014, the $P$ value of the intercept was less than 0.001, the $P$ value of the temperature regression coefficient was 0.0008, and the $P$ value of the air speed regression coefficient was 0.0236. Given that the confidence coefficient was 95%, we could conclude that there was a linear relationship between the total drying time, the drying temperature, and the drying air speed. The R-square value was 0.8878, the RMSE (root mean square error) was 17.02, the mean value of the dependent variable was 166.67, and the coefficient of variation was 10.21%. It was generally considered that a coefficient of variation below 15% is acceptable, and the chi-square value was 5.01. In Figure 7, the drying time was similarly evenly distributed on both sides of the fitting curve. Combining the above values, it could be considered that the model had a good fit. The expression is shown in the following equation:$$\begin{array}{}\text{(13)}& t=511.39-4.22T-12.84v,\end{array}$$where $t$ was the total drying time, min; $T$ was the drying temperature, °C; and $v$ was the drying air speed, m/s.

Table 5

Analysis of variance for prediction of total drying time.

Source	Analysis of variance
	DF	Sum of squares	Mean square	F value	Pr > F	RMSE	R-square value
Model	2	13759	6879.42878	23.33	0.0005	17.02	0.8878
Error	8	2359.32425	294.91553	—	—	—	—
Corrected total	10	16118	—	—	—	—	—

[figure(s) omitted; refer to PDF]

3.3. Establishment of Classic Thin-Layer Drying Model

The thin-layer drying model was widely used in agricultural product drying [27]. Four commonly used models in thin-layer drying were selected for this experiment [28–30]. At the same time, based on the experimental data, the parameters in the model were fitted with the software 1stOpt using the Marquardt method and the global optimization method [31]. The model selection and calculation results are shown in Table 6.

Table 6

Classic model fitting result table.

Temp (°C)	Air speed (m/s)	Logarithmic$\text{MR}=a{e}^{-k\tau +b}$				Page$\text{MR}=a{e}^{-{\tau }^n}$
		$a_1$	$k_1$	$b_1$	$R^2$	$a_2$	$n_2$	$R^2$
72.5	3	0.031	2.51	3.58	0.9779	0.73	1	0.9828
63.7	1.6	2.35	2.53	17.69	0.942	0.73	1	0.9839
81.3	1.6	0.0058	2.32	5.25	0.9349	0.78	1	0.9829
63.7	4.4	51.51	2.509	−3.83	0.956	0.74	1	0.9909
81.3	4.4	2.99	2.65	10.49	0.9707	0.71	1	0.9855
85	3	0.0019	2.31	6.38	0.9321	0.78	1	0.9814
60	3	62.99	2.35	−4.02	0.9373	0.77	1	0.986
72.5	5	0.33	2.69	3.51	0.9472	0.71	1	0.9789
72.5	1	3.37	2.44	10.41	0.949	0.75	1	0.9934
Temp (°C)	Air speed (m/s)	Modified page$\text{MR}=a{e}^{-{k\tau }^n}$				Wang and Singh$\text{MR}=1-a\tau -b{\tau }^2$
		$a_3$	$k_3$	$n_3$	$R^2$	$a_4$	$b_4$	$R^2$
72.5	3	0.96	3.9	1.79	0.9969	1.6	−0.6	0.9921
63.7	1.6	0.94	4.36	2.03	0.9945	1.52	−0.45	0.9855
81.3	1.6	0.96	4	2.05	0.9949	1.37	−0.31	0.9926
63.7	4.4	0.98	3.81	1.76	0.9974	1.54	−0.49	0.9924
81.3	4.4	0.99	3.68	1.52	0.9966	1.77	−0.76	0.9944
85	3	0.97	4.04	2.08	0.9955	1.34	−0.28	0.9839
60	3	0.95	3.88	2.06	0.9948	1.32	−0.25	0.9881
72.5	5	0.96	4.85	1.96	0.9958	1.71	−0.66	0.9853
72.5	1	0.96	3.92	1.92	0.9969	1.48	−0.43	0.9881

In the classic model, only time was an independent variable, which could only be fitted under known drying conditions and had no guiding role for other drying conditions without experimentation. For example, in Meric’s study, although 11 drying model parameters were obtained, none of them could predict models under other experimental conditions [32]. To give the classic model a better applicability under other experimental conditions, a regression relationship between the coefficients obtained in the model and the experimental conditions was established. Based on the data in Table 6, a multiple linear regression model was established by SAS software with experimental conditions as variables and parameters as response values. If the linear model was not applicable, a nonlinear model was established. A relationship between the experimental parameters and the coefficients in the classic thin-layer drying model was obtained.

With a confidence coefficient of 0.95, the parameters in the logarithmic model could not establish a quadratic polynomial regression relationship with the experimental indices. Parameter $a_2$ in the Page model could establish a nonlinear relationship as equation (14) with the experimental conditions at a confidence coefficient of 0.95.$$\begin{array}{}\text{(14)}& a_2=1.68-0.031T+0.12v+0.00025T^2-0.0016v\times T-0.0014v^2.\end{array}$$

The parameters of the modified Page model could not establish a polynomial regression relationship with the experimental conditions. Parameter $a_4$ in the Wang and Singh model could establish a nonlinear relationship with the experimental conditions as equation (15) at a confidence coefficient of 0.9.$$\begin{array}{}\text{(15)}& a_4=-4.60+0.19T-0.55v-0.0015T^2+0.0077v\times T+0.0088v^2.\end{array}$$

A nonlinear relationship between parameter $b_4$ and the experimental conditions could be established at a confidence coefficient like equation (16) at a confidence coefficient of 0.9.$$\begin{array}{}\text{(16)}& b_4=7.75-0.25T+0.52v+0.0019T^2-0.0083v\times T.\end{array}$$

3.4. Establishment of the Polynomial Regression Model

Among the four classic models used in this paper, only the parameters of the Wang and Singh model could establish a nonlinear relationship with the drying conditions, but the confidence coefficient was not good. At the same time, the establishment of a nonlinear relationship between each parameter and the drying conditions required the determination of the model parameters under different drying conditions, and the model could not be established directly and quickly. We found that the existing thin-layer drying models were all third-order derivative functions. For such functions, we could write them in polynomial form by using Taylor expansion, so the thin-layer drying models were unified in form. Since they could be written in the form of polynomials, it was better to establish a polynomial model of thin-layer drying based on the original drying data. Time was used as the variable for regression analysis and a quadratic polynomial model was established in which moisture ratio as function and drying temperature, wind speed, and time as three independent variables. To make the prediction effect of the model more accurate, the D-optimal mixture design scheme was adopted in the actual experimental design, which is shown in Table 3. The total time of each drying was not the same due to different drying temperatures and air speeds. Measuring the time in minutes resulted in large errors and reduced the accuracy of the multiform model prediction. Therefore, dimensional time was used to establish a polynomial model applicable to different temperatures and air speeds. Then, according to the prediction of temperature and air speed for the total drying time, τ could be calculated.

The data selection of the regression model was mainly based on the guidance of the R311 experimental design table. However, the total drying time τ was not known before the experiment. If the drying sample was weighed frequently, the drying effect could be greatly affected. Therefore, the sample was weighed every 15 minutes during the experiment. However, the sampling points needed to process the data may not be the same as the actual sampling points. In order to calculate the weight of sampling points required for data analysis (required sampling points for short), the drying process is regarded as the uniform drying stage. When the required sampling point falls between two actual sampling points, the weight of the required sampling point is calculated according to the weight measured at the two actual sampling points before and after the required sampling point. The original weights of all drying samples were 200 g, and the dried weights of samples were 8 g, see Table 7 for detailed data.

Table 7

Polynomial model data table.

Temp (°C)	Air speed (m/s)	τ	Time (min)	MR	Temp (°C)	Air speed (m/s)	τ	Time (min)	MR
72.5	3.0	0.0	0.0	1.0000	81.3	4.4	0.0	0.0	1.0000
63.7	1.6	0.8	168.8	0.0541	85.0	3.0	0.0	0.0	1.0000
81.3	1.6	0.8	90.0	0.0748	60.0	3.0	0.0	0.0	1.0000
63.7	4.4	0.8	123.8	0.0852	72.5	5.0	0.0	0.0	1.0000
81.3	4.4	0.8	90.0	0.1060	72.5	1.0	0.0	0.0	1.0000
85.0	3.0	0.3	30.0	0.7401	63.7	1.6	1.0	225.0	0.0021
60.0	3.0	0.3	56.3	0.6881	81.3	1.6	1.0	120.0	0.0021
72.5	5.0	0.3	37.5	0.6830	63.7	4.4	1.0	165.0	0.0021
72.5	1.0	0.3	52.5	0.7401	81.3	4.4	1.0	120.0	0.0021
72.5	3.0	1.0	150.0	0.0021	85.0	3.0	1.0	120.0	0.0021
72.5	3.0	0.5	75.0	0.3243	60.0	3.0	1.0	225.0	0.0021
63.7	1.6	0.0	0.0	1.0000	72.5	5.0	1.0	150.0	0.0021
81.3	1.6	0.0	0.0	1.0000	72.5	1.0	1.0	210.0	0.0021
63.7	4.4	0.0	0.0	1.0000

The data were imported into SAS software, and the model was established by regression analysis. According to Table 8, the $P$ value of the model was less than 0.001, which indicates that the model had extreme prominence. The R-square value was 0.9932, which had good fitting properties.

Table 8

Significance table of the regression model.

Regression	DF	R-square	F value	Pr > F
Linear	3	0.9782	814.52	<0.0001
Quadratic	3	0.0149	12.4	0.0002
Cross product	3	0.0001	0.05	0.9863
Total model	9	0.9932	275.66	<0.0001

At the same time, according to the statistical results, we could obtain the relationship between the MR and the drying temperature and drying air speed and drying time, as shown in the following equation:$$\begin{array}{c}\text{(17)}& \text{MR}=1.41-0.011T-0.020v-1.61\tau +0.000079T^2+0.0029v^2-0.0020\tau \ast T+0.0056\tau \ast v+0.58{\tau }^2.\end{array}$$

3.5. Comparative Study of Models

The R-square average of the logarithmic model was 0.9449; the Page model was 0.9851; the modified Page model was 0.9960; the Wang and Singh model was 0.9892; and the multiple models were 0.9932. The R-square value of the polynomial model was second to that of the modified Page model, which had a better fitting property.

To better verify the fitting degree of each model, a validation experiment was carried out under a drying temperature of 70°C and at air speed of 4 m/s. At the same time, the model parameters were calculated by taking advantage of the relationship between the experimental conditions and the coefficients of the classic model. The Page model was $\text{MR}=0.7446e^{-\tau }$. The Wang and Singh model was $\text{MR}=1-1.4468\tau +0.6424{\tau }^2$. Drawing Figure 8 to show the curve of the water ratio change with time, the polynomial model was basically equivalent to the measured value. The Page model transformation was smoother, the Wang and Singh model changed more dramatically, and a negative value will appear at an early stage. SAS software was used to calculate the variance of the three models and the measured values. The Duncan method was used to compare the averages, which are shown in Figure 9. The comparison shows that there was no significant difference between the polynomial model and the measured value at a confidence coefficient of 0.95, but there was a significant difference between the Page model and Wang and Singh model.

[figure(s) omitted; refer to PDF]

4. Conclusions

This experiment investigated the law of the moisture ratio change of tomato slurry during the RW drying process. With the guidance of a D-optimal mixture experimental design, this experiment adopted the R311 experimental design table to conduct RW drying experiments. Based on the experimental data and analysis, this study shows that the drying speed of RW drying could reach 0.40 $g/g·\min$, which could finish drying apace. The total drying time could establish a linear relationship with the drying conditions, which could be used to predict the drying time. Among the four classic models selected in this paper, the modified Page model had a good fit, but the equation parameters could not connect with the experiment conditions. At the same time, the parameters of the Page model and the Wang and Singh model could connect with the test conditions. The variance analysis of the verification experiment shows that the polynomial model could better predict the change of MR over time under other test conditions than the classic model.

Acknowledgments

This research was supported by the Natural Science Foundation of Jiangsu Province (Grant no. BK2022204), National Natural Science Foundation of China (32301718), Key Research and Development Program of Jiangsu Province of China (BE2022319), and Agricultural Science and Technology Innovation Program of the Chinese Academy of production and Equipment of Fruits and Vegetables and Fundamental Research Funds for Central Nonprofit Scientific Institution (S202006-03).

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