Drying Kinetics and Mass Transfer Characteristics of Walnut under Hot Air Drying

Abstract

This study was conducted to investigate the drying kinetics and internal and external mass transfer characteristics of walnuts for an understanding of the drying mechanism. The drying characteristics, mass transfer characteristics, and color of walnut during hot air drying (HAD) were investigated under different initial moisture content (IMC) (0.35, 0.39, and 0.43 g water/g wet mass) and drying temperatures (50, 60, 70, and 80 °C). The results indicated that the IMC and drying temperature both have significant effects on the drying process of walnut, showing the higher the IMC, the longer the preheating time, the smaller the effective moisture diffusivity (D_eff) and mass transfer coefficient (h_m), and the longer the drying time, but reverse results for drying temperature. The values of D_eff and h_m for walnut ranged from 4.94 × 10⁻¹⁰ to 1.44 × 10⁻⁹ m²/s and 1.24 × 10⁻⁷ to 3.90 × 10⁻⁷ m/s, respectively. The values of activation energy for moisture diffusion and mass transfer ranged from 21.56 to 23.35 kJ/mol and 28.92 to 33.43 kJ/mol, respectively. Multivariate linear prediction models were also established for estimating the D_eff and h_m as a function of the HAD process parameters. The drying temperature has a greater effect on the walnut kernel lightness than the IMC. The Verma et al model could be used to describe the HAD process of the walnut. The findings contribute to the understanding of moisture transfer mechanisms in walnuts and have practical value for the evaluation and improvement of drying systems.

1. Introduction

Walnut has an essential economic and medicinal value due to its high content of unsaturated fatty acids, proteins and amino acids and health benefits [1,2]. Due to the high initial moisture content (IMC) in fresh walnuts, drying is important for the primary processing of walnuts to remove water, avoid microbial spoilage, maintain quality, reduce transportation costs, and increase consumption diversity [3,4]. Over recent decades, drying technology has been increasingly used in the walnut industry. Although different drying technologies such as infrared drying [5], microwave drying [6], vacuum drying [7], radio-frequency drying [7,8,9], intermittent oven drying [10,11], and combined drying [7] have been investigated in a laboratory for efficient drying, hot air drying (HAD) is still predominantly used in industrial production due to limitations in processing ability and running cost [12]. However, HAD usually has low drying efficiency, high energy consumption, and poor quality, which seriously restricts the development of the walnut industry [13,14]. Hence, new walnut HAD methods are demanded for efficient drying, which requires a good understanding of the walnut drying process.

Many studies have been conducted on the HAD of walnuts. Hassan-Beygi et al. investigated the hot air (HA) drying characteristics and effective moisture diffusivity [15]; Zhu et al. studied the effects of HA temperature, loading capacity, and air velocity on the HA drying characteristics of walnuts and determined the optimal combination of process parameters [16]; Chen et al. investigated the effect of walnut structure on the hydration and HA drying characteristics [12]. Most of these studies have successfully explored the drying process of walnuts under the assumption that the IMC of each batch of walnuts is uniform and consistent [7,12,15,16]. However, studies have shown that freshly harvested food and agricultural materials, even under the same environmental conditions, have a significant moisture content (MC) bias [13,17]. The presence of the IMC difference was suspected to affect the change pattern of moisture and the mass transfer characteristics during the walnut drying processes and thus may influence the drying time [13,18,19,20]. Thus, it is necessary to study the influence of the IMC on the change pattern of moisture and the transfer characteristics of the walnuts systematically. In addition, most previous studies have focused on drying characteristics, kinetics, energy consumption, quality and process, and few systematic reports have been made on the internal and external mass transfer characteristics during walnut drying, which is very important for understanding and accurately grasping the drying process of walnuts. The internal and external mass transfer characteristics of food materials are usually significantly affected by their intrinsic properties and drying conditions [5,21,22,23]. Teymori-Omran et al. found a significant correlation between the effective moisture diffusivity (D_eff) and mass transfer coefficient (h_m) of apple slices and drying temperature and radiation intensity during the combined HA–infrared drying process [24]. Chayjan et al. determined the D_eff of squash seeds in a semi fluidized and fluidized bed drying and established two-order polynomial models of air velocities and D_eff [25]. Nevertheless, until now, limited documented information is available on the sample IMC—and drying temperature—dependency of D_eff and h_m in the walnuts under HAD.

Therefore, this paper takes walnuts as the research object, studies the change pattern of moisture and temperature during the HAD process of walnuts, explores the influence of IMC and drying temperature on the drying characteristics, internal and external mass transfer characteristics and color of walnuts, constructs the mathematical model of the D_eff and h_m related to the IMC and drying temperature, and determines a kinetic model suitable for describing the moisture change pattern of walnuts during the drying process. This study provides (i) an improved understanding of moisture transfer mechanisms during the HAD of walnuts and (ii) a theoretical reference for the improvement of drying technology and the optimization of the operating conditions.

2. Materials and Methods

2.1. Material Preparation

Fresh walnut samples of the Wen-185 variety were provided by the Woodland, Wensu County, Aksu, Xinjiang in the 2023 harvest season. All walnut samples were vacuum-packed in 30 plastic bags, each of which weighed 4.50 ± 0.01 kg, and stored in an air-conditioned room at a relative humidity of 60% and a temperature of 21 °C to prevent water loss [26]. The IMC of walnut samples was measured using an HA dryer at 105 ± 1 °C for 24 h, according to Equation (1):

$$IMC=\frac{W_0-W_e}{W_0}$$

(1)

where, W₀ was the initial mass of the sample (g); W_e was the dry mass of the sample (g); IMC was the initial MC of the walnut (g water/g wet mass).

2.2. Drying Experiment

Figure 1 presents a diagram of the HAD apparatus. The experimental apparatus mainly consists of a HA dryer (BoXun, GZX-9140MBE, Shanghai, China) and a temperature recorder (A-BF, BCL3008P, Guangdong, China). K-type thermocouples (Omega, GG-K-36-SLE, Stamford, NY, USA) equipped with the temperature recorder were used to monitor the real-time temperature of the walnut sample. Since walnuts are spherical, solid materials, the temperature at one location cannot represent the overall temperature. Therefore, the temperatures at the four test points were selected for monitoring and recording, and the average of the four test point temperatures was used to characterize the walnut temperature. Specifically, one K-type thermocouple tip (1) was fixed to the surface of the walnut with heat-conducting tape, and the other three K-type thermocouple tips (2, 3, and 4) were inserted into the different positions inside the walnut. The four temperature test points of walnut samples were: surface temperature, upper-middle temperature, middle temperature, and lower-middle temperature (Figure 1).

During the harvest season, the IMC of the walnuts varied. In this study, the overall range of the walnut IMC_S was between 0.35 and 0.43 g water/g wet mass, and this range was divided evenly into 3 groups with 0.04 g water/g wet mass interval to define the impact of walnut IMC on moisture diffusion characteristics. For simplicity, the specific HAD test program for walnuts is given in

Table 1. Program for the HAD tests of the walnut.

Test No.	IMC (g Water/g Wet Mass)	Drying Temperature (°C)
1	0.35	50
2	0.35	60
3	0.35	70
4	0.35	80
5	0.39	50
6	0.39	60
7	0.39	70
8	0.39	80
9	0.43	50
10	0.43	60
11	0.43	70
12	0.43	80

.

The HAD oven was turned on to preheat, and after the system reached thermal equilibrium, samples were quickly placed in the pallets to start the drying test. To track the MC variation of the walnuts during the drying process, we swiftly removed the 30 marked walnuts from the dryer and the weight of each sample was measured on a digital balance (Anting, FA1104, Shanghai, China) and then put back until the end of drying. Note that the sample experimental steps should be completed within 5 s to improve the accuracy of the test results [27]. For all walnut drying tests, three replicates were performed.

2.3. Drying Characteristic Parameters

The moisture ratio of walnut was calculated using Equation (2):

$$MR=\frac{M_t-M_{\infty }}{IMC-M_{\infty }}$$

(2)

where, M_∞ and M_t were the MC at the drying end (g water/g wet mass) and the MC at time t (g water/g wet mass), respectively.

The drying rate of walnut was calculated using Equation (3):

$$DR=\frac{M_{t+\Delta t}-M_t}{\Delta t}$$

(3)

where, DR was the drying rate (g water/g wet mass/min); M_t+Δt was the MC of the walnuts at time t + Δt (g water/g wet mass); Δt is the time difference between t + Δt and the t.

2.4. Effective Moisture Diffusivity

The D_eff is the most important key parameter required for the analysis of intrinsic moisture diffusion characteristics during the food material drying process [23,28]. When the drying of rectangular, spherical and other shapes of materials is mainly controlled by the descending drying stage, Fick’s second law can be adopted to analyze the process of moisture diffusion inside the material [29]. Generally, the walnut was viewed as a sphere in the theoretical research due to its relatively high sphericity [30]. Therefore, as per Hassan-Beygi et al., the sphere geometric radius (r_g) was derived by measuring the dimension parameters for two hundred walnuts [15].

The D_eff of walnut under different drying conditions was calculated based on Fick’s diffusion equation by assuming spherical shape with negligible volumetric shrinkage [5,15]. In order to improve the inaccuracy, as those made in previous studies [5,15,31], the following important assumptions were made:

(1)

The moisture diffuses radially from inside to the surface of the particle and evaporates at the surface.

(2)

In the beginning, the moisture was evenly distributed throughout the mass of a sphere;

(3)

In relation to the center of the sphere, the mass transfer was symmetrical;

(4)

Compared to the internal resistance of the walnut sample, the surface mass transfer resistance was negligible.

(5)

The D_eff was constant.

The general form of Fick’s second law for spherical particles with purely radial diffusion is expressed as:

$$\frac{\partial M}{\partial t}=\frac{\partial }{\partial r}(D_{eff}\frac{\partial M}{\partial r})$$

(4)

The initial and the boundary conditions at the surface (r = r_g) and at the center of a grain of walnut (r = 0) are:

$$\begin{gathered} M(t=0,0\le r<r_{\mathrm{g}})=M_0 \\ M(t\ge 0,r_{\mathrm{g}})=M_{\mathrm{e}} \\ {\left(\frac{\partial M}{\partial r}\right)}_{r=0}=0 \end{gathered}$$

(5)

Crank proposed the analytical solution of this partial differential equation for the one-dimensional moisture transfer in spherical geometry in the way that infinite series [5,32]:

$$MR=\frac{M_{\mathrm{t}}-M_{\mathrm{e}}}{M_0-M_e}=\frac{6}{{\pi }^2}{\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^2}}\exp \left(-\frac{n^2{\pi }^2{D}_{eff}t}{{r_{\mathrm{g}}}^2}\right)$$

(6)

where, n was the number of terms; t was the drying time (s); r_g was the geometric radius (m); D_eff was the effective moisture diffusivity of the samples (m²·s⁻¹).

For long drying times, the above series converges very rapidly and the first term can be used to approximate the above series with high accuracy, while higher-order terms (n > 1) approach zero as t increases. Hence, ignoring higher-order terms, n = 1, Equation (6) can be simplified to the following form [5,15,31]:

$$MR=\frac{6}{{\pi }^2}\exp \left(-\frac{{\pi }^2{D}_{eff}t}{{r_{\mathrm{g}}}^2}\right)$$

(7)

The conversion of Equation (7) can be obtained as follows:

$$\ln (MR)=\ln \frac{6}{{\pi }^2}-\frac{{\pi }^2{D}_{eff}t}{{r_{\mathrm{g}}}^2}$$

(8)

Equation (8) can be further expressed in the form of slope:

$$k=\frac{\ln MR-\ln \frac{6}{{\pi }^2}}{t}=-\frac{{\pi }^2{D}_{eff}}{{r_{\mathrm{g}}}^2}$$

(9)

2.5. Mass Transfer Coefficient

Mass transfer coefficient (h_m) is used to compute the mass transfer rate, and it can be computed with the Biot number (Bi) [5,24]:

$$Bi=\frac{h_{m}L}{D_{eff}}$$

(10)

where, h_m is the mass transfer coefficient (m·s⁻¹); L is the characteristic length (m). L is calculated as the walnut volume divided by the walnut area.

Bi can be calculated with Dincer number (Di) [33]:

$$Bi=\frac{24.85}{D{i}^{0.375}}$$

(11)

Di can be calculated as [34]:

$$Di=\frac{V}{k_{c}L}$$

(12)

$$-k_c(M_t-M_e)=DR$$

(13)

where, V is the flow velocity (m/s); k_c is the drying constant (1/s); M_e is the equilibrium moisture content (g water/g wet mass).

2.6. Activation Energy

Most researchers in the field of food drying have linked their D_eff and h_m to the reciprocal of the absolute drying temperature, which usually remains constant during kinetic studies [24,25,35]. This approach can be acceptable when dealing with low MC foods, where the sample temperature quickly attains thermal equilibrium with the drying temperature. Nonetheless, in high MC foods, the sample temperature stays well below the drying temperature for a considerable fraction of the drying time. Therefore, in order to accurately obtain the activation energy of moisture diffusion and mass transfer for foods with high MC, the relationship between the D_eff and h_m, as well as temperature, must be established with a representative sample temperature value instead of with the drying temperature [35]. In this study, the walnut average temperature was determined from the steady temperature portion of the measured temperature curves. The activation energy for moisture diffusion and mass transfer of walnut were determined from the Arrhenius equation given in Equations (14) and (15) [24,36].

$$D_{eff}=D_0\cdot \exp \cdot \left(-\frac{E_D}{RT}\right)$$

(14)

$$h_m=h_0\cdot \exp \cdot \left(-\frac{E_H}{RT}\right)$$

(15)

where, D₀ and h₀ were the pre-exponential factors; R was the universal gas constant (J·mo⁻¹·K⁻¹); T was the sample temperature (K); E_D and E_H were the activation energies for moisture diffusion and mass transfer, respectively (KJ·mol⁻¹). The equation can be transformed as follows:

$$\ln D_{eff}=\ln D_0-\frac{E_D}{R}\frac{1}{T}$$

(16)

$$\ln h_m=\ln h_0-\frac{E_H}{R}\frac{1}{T}$$

(17)

2.7. Lightness

The lightness value of walnut kernels was evaluated using a colorimeter (Sanenshi, SC-10, Shenzhen, China) as an indicator of browning.

2.8. Drying Kinetics

Accurately describing and predicting the moisture dissipation of walnuts during HAD is of great significance for the study of the drying characteristics of walnuts [19]. Ten typical mathematical models were selected from the classical drying kinetic models (as shown in

Table 2. Mathematical models considered for describing drying curves.

No.	Model Name	Model Formula	Parameter
1	Newton	$MR=\exp (-kt)$	k
2	Page	$MR=\exp (-k{t}^n)$	k, n
3	Henderson and Pabis	$MR=a\exp (-kt)$	a, k
4	Two term	$MR=a\exp (k_{0}t)+b\exp (k_{1}t)$	a, b, k0, k1
5	Two-term exponential	$MR=a\exp (-kt)+(1-a)\exp (-kat)$	a, k
6	Verma et al	$MR=a\exp (-kt)+(1-a)\exp (-gt)$	a, g, k
7	Logarithmic	$MR=a\exp (-kt)+c$	a, c, k
8	Wang and Singh	$MR=1+at+b{t}^2$	a, b
9	Midilli et al	$MR=a\exp (-k{t}^n)+bt$	a, n, k
10	Weibull distribution	$MR=a-b\exp (-k{t}^n)$	a, b, k, n

) to fit the walnut drying curves, and the most suitable mathematical model for describing and predicting the change rule of moisture in walnut hot air drying was selected according to the judgment criterion [37]. Note that the OriginPro software 2021 was used to fit parameters in the mathematical model.

2.9. Statistical Analysis

The determination coefficient (R²) is the most important indicator for evaluating the fit of the model and is used to indicate the close relationship between the variables. The root mean square error (RMSE) and sum of squared deviations (χ²) reflect the degree of variation between the actual values and the expected values, so that these three criteria can be used to determine the optimal drying kinetic models and judge the models merits [37]. A higher quality of fit was associated with lower RMSE and χ², and higher R².

$$R^2=\frac{{\displaystyle \sum _{i=1}^N(M{R}_i-M{R}_{\mathrm{pre},i})(M{R}_i-M{R}_{\exp ,i})}}{\sqrt{\left[{\displaystyle \sum _{i=1}^N{(M{R}_i-M{R}_{\mathrm{pre},i})}^2}\right]\left[{\displaystyle \sum _{i=1}^N{(M{R}_i-M{R}_{\exp ,i})}^2}\right]}}$$

(18)

$$RMSE={\left[\frac{1}{N}{\displaystyle \sum _{i=1}^N\left(M{R}_{\mathrm{pre},i}-M{R}_{\exp ,i}\right)}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$$

(19)

$${\chi }^2=\frac{{\sum }_{i=1}^N{(M{R}_{\mathrm{pre},i}-M{R}_{\exp ,i})}^2}{N-n}$$

(20)

where, MR_exp,i, MR_pre,i and MR_i were experimental MR, predicted MR and the average of the experimental MR, respectively; N and n are the total data and the number of constants, respectively.

3. Results and Discussion

3.1. Drying Characteristics

The drying curves of the walnut at different IMCs and drying temperatures are shown in Figure 2. As shown in Figure 2, the MCs of walnuts decreased gradually with the drying time, which was commonly observed for food and agricultural materials [5,21,28,38,39]. It was found that as the IMC was larger, the drying time was substantially longer, and the drying curves were flatter. When the IMC was 0.35, 0.39, and 0.43 g water/g wet mass, the drying time at 50 °C was 1300, 1600, and 1950 min, respectively; the drying time at 60 °C was 1000, 1200, and 1500 min, respectively; the drying time at 70 °C was 800, 950, and 1250 min, respectively; and the drying time at 80 °C was 660, 800, and 1000 min, respectively (

Table 3. Moisture contents (IMC, MC_E), temperatures (T_d, T_S), drying time (t), and drying rates (DR_MAX, DR_AVG) of walnuts subjected to HAD.

No.	IMC	Td	MCE	t	DRMAX	DRAVG	TS
1	0.351 ± 0.017	50	0.064 ± 0.003	1300 ± 8	0.00053 ± 0.00002	0.00027 ± 0.00001	47.57 ± 0.8
2	0.356 ± 0.015	60	0.063 ± 0.003	1000 ± 5	0.00063 ± 0.00002	0.00031 ± 0.00001	57.56 ± 1.3
3	0.351 ± 0.012	70	0.062 ± 0.003	800 ± 5	0.00073 ± 0.00002	0.00043 ± 0.00001	67.84 ± 0.6
4	0.348 ± 0.016	80	0.061 ± 0.003	660 ± 5	0.00084 ± 0.00002	0.00050 ± 0.00001	77.32 ± 0.9
5	0.386 ± 0.012	50	0.063 ± 0.003	1600 ± 8	0.00061 ± 0.00002	0.00028 ± 0.00001	47.34 ± 1.1
6	0.389 ± 0.013	60	0.064 ± 0.003	1200 ± 6	0.00072 ± 0.00002	0.00034 ± 0.00001	57.29 ± 1.2
7	0.394 ± 0.016	70	0.065 ± 0.003	950 ± 5	0.00091 ± 0.00002	0.00044 ± 0.00001	67.43 ± 0.8
8	0.395 ± 0.014	80	0.063 ± 0.003	800 ± 5	0.00116 ± 0.00003	0.00053 ± 0.00001	77.08 ± 1.0
9	0.433 ± 0.018	50	0.068 ± 0.003	1950 ± 8	0.00075 ± 0.00002	0.00030 ± 0.00001	47.28 ± 1.1
10	0.427 ± 0.013	60	0.063 ± 0.003	1500 ± 8	0.00085 ± 0.00002	0.00034 ± 0.00001	57.33 ±1.3
11	0.432 ± 0.017	70	0.065 ± 0.003	1250 ± 6	0.00107 ± 0.00003	0.00043 ± 0.00001	67.43 ± 0.7
12	0.430 ± 0.020	80	0.067 ± 0.003	1000 ± 5	0.00124 ± 0.00003	0.00053 ± 0.00001	77.22 ± 0.9

Mean values and standard deviations are presented in the table. Symbols: MC—moisture content, g water/g wet mass; t—drying time, min; DR—drying rate, g water/g wet mass/min; T—temperature, °C. Subscripts: d—drying; E—equilibrium; MAX—maximum; AVG—average; s—sample.

). This is attributed to the different water removal required for walnuts with different IMC [19].

In addition, it was found that the drying time was significantly reduced with the increase in drying temperature. When the drying temperature increased from 50 °C to 80 °C, the drying time decreased from 1300 min to 660 min for walnuts with an IMC of 0.35 g water/g wet mass, and the drying time decreased by 49.23%; the drying time decreased from 1600 min to 800 min for walnuts with an IMC of 0.39 g water/g wet mass, and the drying time decreased by 50%; and the drying time decreased from 1950 min to 1000 min for walnuts with an IMC of 0.43 g water/g wet mass, and the drying time decreased by 48.72% (Table 3). The results indicated that the drying time was reduced by about 50% when the drying temperature was increased from 50 °C to 80 °C, regardless of the IMC. This may be due to the increased kinetic energy and activity of water molecules in walnuts at high temperatures and the increased rate of water migration [40].

The average temperature curves of the walnut at different IMC and drying temperatures are shown in Figure 3. As shown in Figure 3, the average temperature of walnuts rises continuously with the drying time and maintains a stable value when it rises to a certain value. A hysteresis in the temperature curves was found in Figure 3A–D, as it took a longer time for the preheating of walnuts with high IMC than walnuts with low IMC. This may be due to the fact that the specific heat of water is much higher than the dry mass of walnut, and the higher IMC required more heat energy to evaporate, resulting in a slow increase in the temperature of the walnuts [12]. It was found that the preheating time for walnuts decreased with increasing drying temperatures. This was attributed to the fact that higher temperatures provided more heat energy, thus reducing the preheating time [29]. On the other hand, since the heat energy of the drying medium is transferred from the surface to the interior, the higher temperature resulted in large temperature gradients, thus accelerating the heat transfer [18]. In addition, it was also found that the lag time for temperature changes became shorter as the temperature increased. Therefore, we considered that it should be applied with high temperature treatment at the early stage of drying to improve the heat transfer effect and the drying performance and then enhance the drying rate of walnuts.

As can be seen from the temperature curves, the walnut temperature was always lower than the drying temperature during the HAD process. When walnut samples are placed in hot air, theoretically, the walnut temperature and drying temperature should be the same, but due to material moisture evaporation during HAD, the surface temperature is lower than the air temperature. At the same time, moisture evaporation by heat after the HA is properly cooled down during HAD, allowing the HA to reach the walnut surface on the way to heat diffusion. When the HA reaches the walnut center position, the heat diffusion is the most serious, resulting in the center temperature always being the lowest. Similar behavior was observed for the HAD of the almond, where the material temperature was lower than the drying temperature [41].

The rate curves of the walnut at different IMCs and drying temperatures are shown in Figure 4. As shown in Figure 4, the drying of walnuts occurred mainly in the falling-rate periods, which indicated that the walnut drying process is controlled by the internal mass transfer rate and the transfer mechanism is diffusion [35,40]. This drying rate curve is a typical drying behavior for food materials with porous structures or cellular structures, i.e., garlic [42], brown rice [43], apple [44], wolfberry [45], and cistanche [46]. As can be seen in Figure 3, the drying rate curve shows two falling-rate drying periods: the first and second falling-rate drying periods. In the first falling-rate drying period, the surface and internal free moisture of walnuts evaporated rapidly, resulting in a high drying rate. As the drying progressed, the drying evaporation zone gradually moved inward from the outside surface to the internal. This results in elongated inner moisture transfer paths, and therefore the drying rate dropped rapidly, indicating that the drying entered the second falling-rate drying period [12,40].

It can also be seen from Figure 4 that the drying rates at the early drying stages increase with the increase in IMC. During the drying process of the material, the dissipation of surface moisture and the diffusion of internal moisture occur simultaneously, but the drying rates at the early drying stages are mainly influenced by the rate of surface moisture dissipation [13,40]. Therefore, the larger the IMC of walnut, the greater the surface moisture, and thus the greater the drying rates at the early drying stages. It was found that an increase in drying temperature could significantly increase the highest drying rate and average drying rate of walnuts (Table 3). This is due to the high temperatures improving heat transfer and increasing the thermal energy absorbed by the walnuts, which in turn increases the drying rate [29]. The highest drying rates increased from 0.00053 to 0.00084 g water/g wet mass/min for walnuts with an IMC of 0.35 g water/g wet mass, from 0.00061 to 0.00116 g water/g wet mass/min for walnuts with an IMC of 0.39 g water/g wet mass, and from 0.00075 to 0.00124 g water/g wet mass/min for walnuts with an IMC of 0.43 g water/g wet mass, respectively (Table 3). The combination of high IMC and drying temperature leads to a significant increase in highest drying rates, reaching a peak of approximately 0.00124 g water/g wet mass/min.

3.2. Effective Moisture Diffusivity

Generally, the D_eff is employed and represents an internal moisture diffusion characteristics in the material since the limited information on the moisture movement mechanism during drying [22]. It can be seen from Equation (9) that the natural logarithm of the moisture ratio (lnMR) of walnuts during HAD is linearly related to the drying time. Figure 5 shows the changes of lnMR of walnut with drying time under the drying condition. The D_eff were determined by linear regression, as shown in

Table 4. Effective moisture diffusivity of walnuts at different drying conditions.

IMC (g Water/g Wet Mass)	Td (°C)	Linear Regression Equation	Deff (m2/s)	R2
0.35	50	$\ln MR=-2.11\times {10}^{-5}t-0.0059$	6.82 × 10−10	0.995
0.35	60	$\ln MR=-2.86\times {10}^{-5}t+0.0136$	9.29 × 10−10	0.998
0.35	70	$\ln MR=-3.63\times {10}^{-5}t+0.0062$	1.18 × 10−9	0.999
0.35	80	$\ln MR=-4.44\times {10}^{-5}t+0.0096$	1.44 × 10−9	0.999
0.39	50	$\ln MR=-1.80\times {10}^{-5}t-0.0498$	5.85 × 10−10	0.998
0.39	60	$\ln MR=-2.45\times {10}^{-5}t-0.0401$	7.96 × 10−10	0.999
0.39	70	$\ln MR=-3.06\times {10}^{-5}t-0.0654$	9.94 × 10−10	0.997
0.39	80	$\ln MR=-3.81\times {10}^{-5}t-0.1025$	1.24 × 10−9	0.989
0.43	50	$\ln MR=-1.52\times {10}^{-5}t-0.0134$	4.94 × 10−10	0.987
0.43	60	$\ln MR=-2.06\times {10}^{-5}t-0.1361$	6.69 × 10−10	0.983
0.43	70	$\ln MR=-2.50\times {10}^{-5}t-0.1588$	8.12 × 10−10	0.974
0.43	80	$\ln MR=-3.07\times {10}^{-5}t-0.1676$	9.98 × 10−10	0.961

.

In order to quantitatively analyze the effect of IMC and drying temperature on the internal moisture diffusion characteristics of the walnut, Figure 6 shows the variation of D_eff with IMC and drying temperature. It was also found that the D_eff of the lower IMC walnuts was generally higher than the higher IMC walnuts. When walnuts with different IMCs were dried to the same MC, walnuts with higher IMCs underwent a longer drying time and excluded more water, and water loss was controlled by more internal water diffusion, while internal water diffusion is much more difficult than surface water dissipation, resulting in smaller D_eff values [19]. At the same time, for materials with a porous structure in their organization, the volume of pore shrinkage during drying is almost entirely used to compensate for the loss of moisture in the pores [40]. As a result, higher IMC walnuts have a greater degree of structural shrinkage, which in turn results in a more difficult diffusion of internal moisture. The results showed that walnuts with higher IMC had a longer descending drying time, entered the internal control phase earlier, and were affected by internal diffusion control effects for a longer period of time. In addition, it was also found that as the drying temperature increased from 50 to 80 °C, the value of D_eff for walnut with an IMC of 0.35 g water/g wet mass increased linearly from 6.82 × 10⁻¹⁰~1.44 × 10⁻⁹ m²/s; the value of D_eff for walnut with an IMC of 0.39 g water/g wet mass increased linearly from 5.85 × 10⁻¹⁰~1.24 × 10⁻⁹ m²/s; and the value of D_eff for walnut with an IMC of 0.43 g water/g wet mass increased linearly from 4.94 × 10⁻¹⁰~9.98 × 10⁻¹⁰ m²/s (Figure 6B and Table 4). This is because the high temperature increases the thermal energy absorbed by the walnuts, which in turn enhances the diffusion of water molecules. Similar results were found for straw-based nutrient seedling-growing bowl trays in previous studies [37].

Comparing Figure 6A,B, drying temperature has a greater effect on the D_eff than the IMC, which suggested that the D_eff is more sensitive to the increase in the drying temperature than to increases in the IMC during the HAD process of walnut. The values of D_eff found in this study were in the range of 4.94 × 10⁻¹⁰ m²/s to 1.44 × 10⁻⁹ m²/s, which is a typical range for drying of agricultural products [5,21,22,23,24,25,28,29,31,35,36,37,46,47,48,49,50]. In order to quantify the effects of IMC and drying temperature on the D_eff of walnut, a two factor relationship model of D_eff with IMC and drying temperature was developed by multiple linear regression fitting analysis of the experimental data, as shown in Equation (21). The high value of the determination coefficient of the model (R² > 0.978) suggested the goodness of the regression fittings.

$$D_{\mathrm{e}ff}=1.03{\times 10}^{-9}-3.93{\times 10}^{-9}IMC+2.11{\times 10}^{-11}{T}_{\mathrm{d}}$$

(21)

where, T_d was the drying temperature (°C).

3.3. Mass Transfer Coefficient

In order to quantitatively analyze the effect of IMC and drying temperature on the mass transfer characteristics of the walnut, Figure 7 shows the variation of h_m with IMC and drying temperature. As shown, the value of hm decreased linearly with the increased of IMC (0.35~0.43 g water/g wet mass) and increased linearly with the increased of drying temperature (50~80 °C). Comparing Figure 7A,B, drying temperature has a greater effect on the h_m than the IMC, which suggested that the h_m is more sensitive to the increase in the drying temperature than to increases in the IMC during the HAD process of walnut. Unsurprisingly, the higher the drying temperature, the faster the water molecules in the walnuts evaporate, thus the higher the h_m accordingly [24]. As the drying temperature increased linearly from 50 to 80 °C, the value of h_m for walnut with an IMC of 0.35 g water/g wet mass increased linearly from 1.45 × 10⁻⁷~3.86 × 10⁻⁷ m/s; the value of h_m for walnut with an IMC of 0.39 g water/g wet mass increased linearly from 1.34 × 10⁻⁷~3.90 × 10⁻⁷ m/s; the value of h_m for walnut with an IMC of 0.43 g water/g wet mass increased from 1.24 × 10⁻⁷~3.17 × 10⁻⁷ m/s (Figure 7B).

In order to quantify the effects of IMC and drying temperature on the h_m of walnut, a two factor relationship model of h_m with IMC and drying temperature was developed by multiple linear regression fitting analysis of the experimental data, as shown in Equation (22). The high value of the determination coefficient of the model (R² > 0.973) indicated the goodness of the regression fittings.

$$h_m=-3.42{\times 10}^{-8}-5.68{\times 10}^{-7}IMC+7.65{\times 10}^{-9}{T}_{\mathrm{d}}$$

(22)

3.4. Activation Energy

The activation energy is a crucial indicator in the end analysis of the drying process since it shows the difficulty of dehydrating under certain drying conditions [51]. Table 3 displays the walnut average temperatures during the HAD. These walnut average temperatures were employed in Equations (16) and (17) to research the temperature dependence of the h_m and D_eff. Specifically, Figure 8A shows the plots of the natural log of D_eff (ln D_eff) versus the reciprocal of sample temperature (1/T) for the HAD experiments of walnut. The high correlation between 1/T and ln D_eff under different IMCs was obtained with linear models (R² > 0.99). From the line slope in Figure 8A, the E_D values of walnut were calculated to range from 21.56 to 23.35 kJ/mol, with an average value of 22.73 kJ/mol. Figure 8B shows the plots of the natural log of h_m (ln h_m) versus reciprocal of sample temperature (1/T) for the HAD experiments of walnut. The high correlation between 1/T and ln h_m under different IMCs was obtained with linear models (R² > 0.99). From the line slope in Figure 8B, the E_H values of walnut were calculated to range from 28.92 to 33.43 kJ/mol, with an average value of 31.05 kJ/mol. The results show that the range of energy required to remove 1 mol of water from walnuts during drying is 21.56~33.43 kJ.

3.5. Lightness of the Walnut Kernel

The lightness of the kernel is one of the key quality indicators of dried walnuts. The lightness values of walnut kernels under different IMC and drying temperature conditions are shown in Figure 9, with lightness values ranged from 35.87 to 52.61. As shown in Figure 9, drying temperature has a greater effect on the walnut kernel lightness than the IMC, which suggested that the walnut kernel lightness is more sensitive to changes in the drying temperature than to changes in the IMC under the HAD. It can be found that lightness slightly increases and then decreases with increasing drying temperature. The darkening of the color of food materials during the drying process may be caused by both non-enzymatic and enzymatic browning reactions [52,53]. The walnuts are dried under low-temperature drying conditions for long periods of time, which leads to increased browning [13]. The high temperature heating induced the Maillard reaction of amino acids and reducing sugar within walnut kernels and led to more non-enzymatic browning [54]. Meanwhile, it was also found that the lightness values of kernels after drying of high IMC walnuts were lower than those of kernels after drying of low IMC walnuts. It is due to the fact that the high IMC walnuts require longer drying times, resulting in a higher browning degree of the kernels [19]. Additionally, the combination of high IMC (0.43 g water/g wet water) and drying temperature (70~80 °C) results in a significant decrease in lightness. This indicates that the combination of higher IMC and drying temperature can exacerbate the browning of the kernel during HAD.

3.6. Kinetic Analyses on the Hot Air Drying Process

Moisture movement in food is a dynamic and complex phenomenon. Drying kinetics studies were necessary for assessing and optimizing the walnut drying process [37,55,56]. The MR obtained from the HAD process of walnut was investigated by the mathematical models presented in Table 2, and the model results are summarized in

Table 5. Range and average value of R², χ² and RMSE of drying kinetic models for walnut.

Model	R2		RMSE		χ2
No.	Range	Average	Range	Average	Range	Average
1	0.98639~0.99986	0.9944	0.00385~0.05005	0.0254	0.00012~0.03257	0.01019
2	0.99721~0.99994	0.9988	0.00213~0.01418	0.0086	0.00004~0.00181	0.00093
3	0.98107~0.99984	0.9924	0.00355~0.03609	0.0203	0.00010~0.01693	0.00615
4	0.98107~0.99996	0.9979	0.00169~0.03609	0.0079	0.00003~0.01693	0.00179
5	0.99696~0.99985	0.9987	0.00391~0.02062	0.0098	0.00012~0.00553	0.00143
6	0.99855~0.99996	0.9997	0.00173~0.01010	0.0044	0.00003~0.00123	0.00027
7	0.99225~0.99995	0.9979	0.00189~0.02239	0.0104	0.00003~0.00652	0.00162
8	0.84622~0.99911	0.9531	0.00944~0.15905	0.0696	0.00071~0.31629	0.08745
9	0.99840~0.99994	0.9996	0.00204~0.01003	0.0048	0.00003~0.00121	0.00034
10	0.98213~0.99997	0.9972	0.00135~0.03401	0.0086	0.00002~0.01503	0.00225

. As shown, the mean value of R² for the Verma et al model is 0.99965, which indicated that the Verma et al model could be used to describe the walnut HAD processes. Additionally, the mean values of RMSE and χ² are 0.00441 and 0.00027, respectively, which are the minimum values, which could also indicated the Verma et al model was the best model to describe the drying processes. The values of drying coefficients and constants in the Verma et al model at different conditions are summarized in

Table 6. Fitted results of Verma et al models under different drying conditions.

IMC (g Water/g Wet Mass)	Td (°C)	a	g	k	R2	RMSE	χ2
0.35	50	0.0318	0.0012	0.0223	0.9986	0.0101	0.00123
0.35	60	0.0046	0.0017	2.0145	0.9993	0.0067	0.00045
0.35	70	−1.6165	0.0024	0.0026	0.9998	0.0035	0.00011
0.35	80	−0.0547	0.0027	0.0068	0.9999	0.0018	0.00003
0.39	50	0.0930	0.0010	0.0107	0.9996	0.0051	0.00031
0.39	60	0.9241	0.0121	0.0014	0.9999	0.0029	0.00009
0.39	70	0.1598	0.0017	0.0088	0.9997	0.0043	0.00019
0.39	80	0.4349	0.0016	0.0059	0.9997	0.0047	0.00019
0.43	50	0.2664	0.0008	0.0056	0.9998	0.0035	0.00016
0.43	60	0.3582	0.0009	0.0046	0.9995	0.0054	0.00035
0.43	70	0.5174	0.0047	0.0010	0.9999	0.0017	0.00003
0.43	80	0.6154	0.0009	0.0048	0.9999	0.0032	0.00009

.

In order to further characterize the effect of drying factors on the Verma et al model, the equations between the drying model constants (a, g, and k) and the drying factors were developed using regression analysis. The regression equations between the Verma et al model constants and the drying factors were as follows:

IMC was 0.35 g water/g wet mass:

$$a=0.7961{T_{\mathrm{d}}}^3-5.5735{T_{\mathrm{d}}}^2+11.12T_{\mathrm{d}}-6.3112\begin{array}{cc}(R^2=0.9989)& \end{array}$$

(23)

$$g=-0.00003{T_{\mathrm{d}}}^2+0.0007T_{\mathrm{d}}+0.0005\begin{array}{cc}(R^2=0.9793)& \end{array}$$

(24)

$$k=1.0034{T_{\mathrm{d}}}^3-8.0223{T_{\mathrm{d}}}^2+19.036T_{\mathrm{d}}-11.994\begin{array}{cc}(R^2=0.9979)& \end{array}$$

(25)

IMC was 0.39 g water/g wet mass:

$$a=0.439{T_{\mathrm{d}}}^3-3.4317{T_{\mathrm{d}}}^2+8.0532T_{\mathrm{d}}-4.9675\begin{array}{cc}(R^2=0.9993)& \end{array}$$

(26)

$$g=0.0053{T_d}^3-0.0425{T_d}^2+0.1014T_d-0.0632\begin{array}{cc}(R^2=0.9996)& \end{array}$$

(27)

$$k=-0.0045{T_{\mathrm{d}}}^3+0.0353{T_{\mathrm{d}}}^2-0.0838T_{\mathrm{d}}+0.0637\begin{array}{cc}(R^2=0.9989)& \end{array}$$

(28)

IMC was 0.43 g water/g wet mass:

$$a=0.0015{T_{\mathrm{d}}}^2+0.113T_{\mathrm{d}}+0.1454\begin{array}{cc}(R^2=0.9888)& \end{array}$$

(29)

$$g=-0.0019{T_{\mathrm{d}}}^3+0.0129{T_{\mathrm{d}}}^2-0.0256T_{\mathrm{d}}+0.0158\begin{array}{cc}(R^2=0.9999)& \end{array}$$

(30)

$$k=0.0017{T_{\mathrm{d}}}^3-0.0115{T_d}^2+0.0217T_d-0.0064\begin{array}{cc}(R^2=0.9991)& \end{array}$$

(31)

To verify whether the Verma et al model can accurately predict the walnut HAD process, Figure 10 shows the comparison of experimental MR and predicted MR by the Verma et al model. According to the literature, if the curves of the experimental and predicted values are approximately at a 45° angle, it means that the predicted values represent the experimental values well [37]. As can be seen from Figure 10, the model of Verma et al showed a good agreement between experimental MR and predicted MR during the HAD processes of walnut. These data points are banded line plots at an angle of approximately 45°, which suggests that the model of Verma et al is able to make good predictions of walnut HAD processes under different drying conditions.

4. Conclusions

The HAD of walnuts mainly occurred in the falling-rate stage. The higher the IMC, the longer the preheating time, the smaller the D_eff and h_m, and the longer the drying time. The higher the drying temperature, the shorter the preheating time, the larger the D_eff and h_m, and the shorter the drying time. The walnut kernel lightness is more sensitive to changes in the drying temperature than to changes in IMC under the HAD. The values of D_eff and h_m for walnut ranged from 4.94 × 10⁻¹⁰ to 1.44 × 10⁻⁹ m²/s and 1.24 × 10⁻⁷ to 3.90 × 10⁻⁷ m/s, respectively. The values of E_D and E_H for walnut ranged from 21.56 to 23.35 kJ/mol and 28.92 to 33.43 kJ/mol, respectively. Two multivariate linear prediction models were developed to correlate the D_eff and h_m with IMC and drying temperature, respectively, with a high correlation coefficient, and these can be embedded in future FEM simulations to model the moisture transport of walnuts during HAD, which could provide a theoretical basis for elucidating the moisture transfer mechanism of walnuts during HAD. The developed kinetic model showed good potential for the prediction of the walnut HAD process, which can be applied for the optimization of drying conditions for efficient walnut drying in the industry.