Evolution and Modelling of the Moisture Diffusion in Walnuts during the Combination of Hot Air and Microwave–Vacuum Drying

Abstract

To understand the moisture transfer mechanism of walnuts during the combination of hot air (HA) and microwave–vacuum (MV) drying (HA-MVD) process, the drying characteristics and moisture diffusion characteristics of walnut during HA-MVD were investigated. The results indicated that the HA-MVD of walnuts occurred mainly in the falling-rate stage. The value of effective moisture diffusivity (D_eff) dropped continuously with the decrease in moisture content (MC) during the HA drying, while switching to MV drying could truncate the decrease in D_eff and still maintain a high value until the end of drying. The HA temperature, MC of the transition point, microwave power, and MV thermostatic temperature have significant effects on the moisture diffusion characteristics of walnuts. The values of D_eff for walnuts ranged from 2.33 × 10⁻⁹ m²/s to 6.89 × 10⁻⁸ m²/s. The third-order polynomial prediction model of D_eff related to the sample MC and drying conditions was established to describe the dynamic change in the D_eff of walnuts during the HA-MVD process. The application of MVD in the final stage of drying could rapidly increase the internal vapor pressure of the walnuts, accelerate the diffusion speed of the internal moisture, and re-enhance the drying rate. The findings have practical value for the development of efficient and energy-saving drying methods in the walnut industry.

1. Introduction

Walnuts are important economic crops and have attracted increasing attention for their multiple health benefits and extensive applications in food, chemistry, and medicine [1,2,3]. The walnut kernel has a large number of nutrients, mainly including proteins, fats, amino acids, minerals, and vitamins, with a high unsaturated fatty acid content of more than 50% and a protein content of 20% [4,5]. Freshly harvested walnuts typically have a high initial moisture content (IMC) and physiological activity [6,7]. Such characteristics make fresh walnuts susceptible to contamination by bacteria during storage [8]. As a result, drying is an essential step in the primary processing of fresh walnuts to remove water, avoid microbial spoilage, and maintain the quality [9,10]. Meanwhile, it can also reduce transportation weight and storage costs, and increase consumption diversity. Currently, hot air drying (HAD) is still predominantly used in industrial walnut production due to processing ability and running costs [11]. However, HAD of whole walnuts is time consuming because of their hard and thick shell, which prevents heat and mass transfer [12]. Thus, there is an urgent need for new drying approaches to specifically improve the drying process of walnut.

Microwave drying (MD), as a volumetric heating method, is characterized by high heating efficiency, high processing capacity and easy controllability [13,14]. Therefore, using MD instead of HAD can reduce the drying time and improve the product properties [15,16,17]. Microwave–vacuum drying (MVD) overcomes the shortcomings of MD alone, e.g., uneven heating of food materials with a high MC or difficulties in controlling the temperature of the materials [18,19]. Previous studies have shown that the application of MVD at the final drying stage could rapidly increase the internal vapor pressure of the food, accelerate the migration of the internal moisture, and re-enhance the drying rate, which would give the advantages of shortening the drying time and ensuring product quality [20,21]. The combination of hot air (HA) and microwave–vacuum (MV) drying (HA-MVD) is an emerging method of drying and it is suitable for drying different food and agriculture materials due to its high drying efficiency, low energy consumption, and good product quality [21,22,23]. However, to the best of our knowledge, no reports have been published studying the HA-MVD of walnut or other similar tree nuts.

Many researchers have studied the HA-MVD of food materials. Zhou et al. investigated the changes in MC and quality properties of cranberries under HA-MVD [21]; Wu et al. discussed the effects of IMC, microwave intensity, microwave irradiation time, and vacuum degree on the product quality of puffed duck breast and determined the optimal combination of HA-MVD process parameters [24]; Figiel researched the drying kinetics and quality properties of beetroots under HA-MVD [22]. However, most of these studies have focused on drying characteristics, kinetics, quality, and process, and few systematic reports have been made on the internal moisture diffusion characteristics during HA-MVD, which is very important for understanding and accurately grasping the HA-MVD process for foods. The moisture diffusion of food materials is usually greatly affected by drying conditions and sample moisture content (MC), but previous studies on the moisture diffusion in food materials were performed based on the assumption that effective moisture diffusivity (D_eff) remains constant throughout the drying process, which usually does not yield relevant information about the internal moisture dynamic diffusion of materials during the HA-MVD process [25,26,27,28]. Bal et al. found a significant correlation between the D_eff and MCs of bamboo shoot slices during the MD process [29]. Dadalı et al. determined the D_eff of okra under MD and established it as third-order polynomial function of MC [30]. Nevertheless, until now, limited documented information is available on the evolution of the moisture diffusion characteristics of walnuts during HA-MVD, and in particular, the relationship between D_eff and sample MC and the drying conditions has not been clarified.

Therefore, this paper takes walnuts as the research object, studies the change pattern of moisture and D_eff during the HA-MVD process of walnuts, explores the influence of HA temperature, the MC of the transition point, microwave power, and MV thermostatic temperature on the drying characteristics and internal moisture diffusion characteristics of walnuts, and constructs a mathematical model of D_eff related to the MC and drying conditions. This study provides (i) an improved understanding of moisture transfer mechanisms during the HA-MVD drying of walnuts and (ii) a theoretical reference for applying HA-MVD technology to drying walnuts and other similar nuts.

2. Materials and Methods

2.1. Material Preparation

Fresh walnut samples of the Wen-185 variety were bought in 2023. All walnut samples were vacuum-packed in 30 plastic bags, each of which weighed 4.50 ± 0.01 kg, and were stored in an air-conditioned room at a relative humidity of 60% and a temperature of 21 °C to prevent water loss [31]. Walnut samples with similar dimensions and weights were selected for testing to reduce the variation in IMC and improve the representativeness of the sample selection [32]. The IMC of walnut samples was measured using an HA dryer at 105 ± 1 °C for 24 h according to Equation (1) [14]:

$$IMC=\frac{W_0-W_e}{W_0}\times 100\%$$

(1)

where W_e and W₀ are the dry and initial masses of the sample, respectively (g). The IMC of the walnuts is 41.43 ± 1.25% on a wet basis (w.b.).

2.2. Drying Equipment

The experimental equipment primarily consists of an HA dryer (BoXun, GZX-9140MBE, Shanghai, China) and an MV dryer (Huaiteng, HTWB-01, Nanjing, China) (Figure 1). The MV dryer was equipped with an infrared temperature sensor (Opus, RT-350 °C, Stamford, Germany) for real-time monitoring of the surface temperature of the samples. To ensure product quality and reduce energy consumption, a drying strategy is used in which the microwave can only operate below the maximum temperature set by the system, and when the temperature exceeds the maximum temperature, the microwave is automatically turned off [21,33]. The vacuum was set at −0.08 MPa. In order to improve the uniformity of microwave drying, the samples were rotated on the turntable at a speed of 6 rpm during the MVD process [14,34,35].

2.3. Drying Experiments

A fix three variables to change one variable design was used in the experimental scheme. The detailed plans of the HA-MVD test for walnuts are summarised in

Table 1. Details of operating parameters for the HA-MVD experimental.

Test No.	Hot Air Temperature (°C)	Moisture Content at the Transition Point (%)	Microwave Power (W)	Microwave–Vacuum Thermostatic Temperature (°C)
1	45	30	600	50
2	60	30	600	50
3	75	30	600	50
4	90	30	600	50
5	105	30	600	50
6	60	36	600	50
7	60	30	600	50
8	60	24	600	50
9	60	18	600	50
10	60	14	600	50
11	60	30	200	50
12	60	30	400	50
13	60	30	600	50
14	60	30	800	50
15	60	30	1000	50
16	60	30	600	30
17	60	30	600	40
18	60	30	600	50
19	60	30	600	60
20	60	30	600	70

.

The HAD oven was turned on to preheat, and after the system reached thermal equilibrium, the walnuts were rapidly placed on a HA dryer for drying. The HAD stage was stopped when the MC of the walnut reached the MC at the transition point (36, 30, 24, 18 and 14%). The walnuts were then rapidly transferred to the MVD stage [21]. Note that the sample experimental steps should be completed within 5 s to improve the accuracy of the test results [20]. To track the MC variation in the walnuts during the HA-MVD, 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 they were then put back until the end of drying. For all walnut drying tests, three replicates were performed.

2.4. Drying Characteristic Parameters

The moisture ratio of the walnuts was calculated using Equation (2) [25]:

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

(2)

where M_∞ and M_t are the MC at the drying end (%) and the MC at time t (%), respectively.

The drying rate of the walnuts was calculated using Equation (3) [9]:

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

(3)

where DR is the drying rate (%/min); M_t+Δt is the MC of the walnuts at time t + ∆t (%); ∆t is the time difference between t + ∆t and t.

2.5. Effective Moisture Diffusivity

The D_eff is the most important key parameter required for the analysis of intrinsic moisture diffusion characteristics during the biological or food material drying process [36,37]. 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 analyse the process of moisture diffusion inside the material [32,38]. Generally, the walnut has been viewed as a sphere in the theoretical research due to its relatively high sphericity and axially symmetry [9,39]. Therefore, the walnuts were treated as spheres in this study.

The D_eff of walnut at corresponding MCs under different drying conditions was calculated based on the Fick’s diffusion equation by assuming a spherical shape with negligible volumetric shrinkage [9,32]. Similar to those made in previous studies [9,29,32,40], the following important assumptions were made:

(1) In the beginning, the moisture was evenly distributed throughout the mass of the sphere;

(2) In relation to the centre of the sphere, the mass transfer was symmetrical;

(3) Compared to the internal resistance of the walnut sample, the surface mass transfer resistance was negligible;

(4) D_eff was constant only within the short time interval between the two adjacent measuring points and was piecewise continuous throughout the whole drying process.

Note that assumption (4) was proposed in order to compensate for the inaccuracy due to the constant diffusivity [32].

The general form of the Fick’s second law is expressed as [32]:

$$\frac{\partial MC}{\partial t}=\frac{\partial }{\partial r}\left(D_{eff}\frac{\partial MC}{\partial r}\right)$$

(4)

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 [9,41]:

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

(5)

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

For long drying times n = 1, then Equation (5) can be simplified to the following form [29,40]:

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

(6)

The r can be calculated as [42]:

$$r=\frac{\sqrt[3]{L\cdot W\cdot T}}{2}$$

(7)

where, L, W and T are length (m), width (m), and thickness (m), respectively.

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

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

(8)

Then, the D_eff can be calculated using Equation (9) [29,40,43]:

$$D_{eff}=\frac{-0.101\cdot \ln (MR)-0.0504}{(t/r^2)}$$

(9)

The D_eff was calculated at each corresponding MC in this study. The average D_eff (D_eff,ave) was determined as an arithmetic average of all D_eff values at different MC [32,43]:

$$D_{\mathrm{e}ff,av\mathrm{e}}=\frac{1}{n}\left({\displaystyle \sum _1^n{D}_{eff}}\right)$$

(10)

The mathematical relationship between D_eff and sample MC and the drying conditions was developed with polynomial equations as Equation (11) [29,32,40].

$$D_{\mathrm{e}ff}=A\cdot M{C}^3+B\cdot M{C}^2+C\cdot MC+D$$

(11)

where, A, B, C, and D are the coefficients of the third-order polynomial relationship.

2.6. 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 residual sum of squares (RSS) reflects the degree of variation between the actual values and the expected values, so that these two criteria can be used to determine the optimal D_eff models and to judge the models’ merits. A higher quality of fit was associated with a lower RSS and higher R² [25,29]. Note that the nonlinear regression kit in the Origin software (version 2022) was used to fit regression parameters in the D_eff model.

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

(12)

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

(13)

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

3. Results and Discussion

3.1. Drying Characteristics

3.1.1. Effect of Hot Air Temperature on Drying Characteristics

In order to investigate the effects of HA temperatures of 45, 60, 75, 90, and 105 °C on the drying characteristics at an MC at the transition point of 30%, a microwave power of 600 W, and an MV thermostatic temperature of 50 °C, the drying curves and drying rate curves of walnut at different HA temperatures are shown in Figure 2A,B. As shown in Figure 2A, the MCs of walnuts decayed exponentially in two stages during the drying time: the HAD and MVD stages, which were commonly observed for combined drying of food materials [20,21,22,23,24]. The MVD stage was much shorter than the HAD stage. The HAD stage took from 50 to 195 min, while the MVD stage only took from 25 to 30 min (

Table 2. Moisture content (MC_E), drying times (t_HAD, t_MVD, t_T), and drying rates (DR_HAD, DR_MVD, DR_AVG) of walnuts subjected to HA-MWD.

Test No.	HAD Stage		MVD Stage		MCE	tT	DRAVG
	tHAD	DRHAD	tMVD	DRMVD
1	195 ± 4	0.064 ± 0.001	30 ± 1	0.976 ± 0.006	7.99 ± 0.5	225 ± 5	0.516 ± 0.006
2	135 ± 4	0.085 ± 0.001	30 ± 1	0.971 ± 0.006	7.95 ± 0.5	165 ± 4	0.579 ± 0.006
3	75 ± 3	0.152 ± 0.001	28 ± 1	1.030 ± 0.008	7.80 ± 0.5	103 ± 4	0.669 ± 0.006
4	60 ± 3	0.191 ± 0.001	25 ± 1	1.138 ± 0.008	7.88 ± 0.5	85 ± 3	0.759 ± 0.006
5	50 ± 3	0.225 ± 0.001	27 ± 1	1.120 ± 0.008	7.58 ± 0.5	77 ± 3	0.758 ± 0.006
6	45 ± 3	0.125 ± 0.001	37 ± 1	1.032 ± 0.008	7.90 ± 0.5	82 ± 3	0.862 ± 0.006
7	135 ± 4	0.085 ± 0.001	30 ± 1	0.971 ± 0.006	7.95 ± 0.5	165 ± 4	0.528 ± 0.006
8	270 ± 5	0.068 ± 0.001	26 ± 1	0.842 ± 0.006	7.95 ± 0.5	296 ± 5	0.304 ± 0.005
9	470 ± 6	0.061 ± 0.001	22 ± 1	0.576 ± 0.006	7.98 ± 0.5	492 ± 6	0.164 ± 0.004
10	670 ± 8	0.054 ± 0.001	15 ± 0.5	0.472 ± 0.005	7.89 ± 0.5	685 ± 8	0.100 ± 0.004
11	135 ± 4	0.085 ± 0.001	125 ± 3	0.268 ± 0.004	7.68 ± 0.5	260 ± 5	0.181 ± 0.004
12	135 ± 4	0.085 ± 0.001	55 ± 2	0.741 ± 0.006	7.59 ± 0.5	190 ± 4	0.430 ± 0.006
13	135 ± 4	0.085 ± 0.001	30 ± 1	0.971 ± 0.006	7.95 ± 0.5	165 ± 4	0.528 ± 0.006
14	135 ± 4	0.085 ± 0.001	20 ± 1	1.746 ± 0.01	7.66 ± 0.5	155 ± 4	0.915 ± 0.006
15	135 ± 4	0.085 ± 0.001	13 ± 0.5	2.193 ± 0.01	7.99 ± 0.5	148 ± 4	1.077 ± 0.008
16	135 ± 4	0.085 ± 0.001	270 ± 4	0.114 ± 0.004	7.96 ± 0.5	405 ± 6	0.102 ± 0.008
17	135 ± 4	0.085 ± 0.001	90 ± 2	0.450 ± 0.005	7.89 ± 0.5	225 ± 5	0.286 ± 0.005
18	135 ± 4	0.085 ± 0.001	30 ± 1	0.971 ± 0.006	7.95 ± 0.5	165 ± 4	0.528 ± 0.006
19	135 ± 4	0.085 ± 0.001	19 ± 0.5	0.196 ± 0.004	7.99 ± 0.5	154 ± 4	0.967 ± 0.006
20	135 ± 4	0.085 ± 0.001	15 ± 0.5	2.365 ± 0.01	8.01 ± 0.5	150 ± 4	1.082 ± 0.008

The test numbers are the same as in Table 1. Symbols: MC—moisture content, %; t—drying time, min; DR—drying rate, %/min. Subscripts: HAD—hot air drying; MVD—microwave–vacuum drying; AVG—average; E—equilibrium; T—total.

). When the HA temperature was 45, 60, 75, 90, and 105 °C, the total drying time for walnuts was 225, 165, 103, 85, and 77 min, respectively (Table 2). The results showed that, when the HA temperature increased from 45 °C to 105 °C, the total drying time for walnuts decreased by 65.78%.

It can be seen from Figure 2B that the HAD stage had significant preheating and falling-rate drying periods, but the average drying rate was very small, about 0.516 to 0.758%/min (Table 2). When the HAD reached the MC of the transition point, the drying process converted to MVD. The drying rate of the walnut increased significantly, which may be attributed to the fact that the use of electromagnetic radiation under reduced pressure could significantly accelerate water transfer [16,19,44]. In addition, it was found that the maximum drying rate in the HAD stage increased significantly when the HA temperature was increased, with the first peak of the corresponding curve increasing from 0.133%/min to 0.303%/min. This is because the IMC of walnuts is high, and increasing the drying temperature decreases the relative humidity around the walnuts, and the increased humidity difference between walnuts and their surroundings, which leads to a faster migration of water [45,46].

3.1.2. Effect of Moisture Content at the Transition Point on Drying Characteristics

In order to investigate the effects of MCs at the transition point of 36, 30, 24, 18 and 14% on the drying characteristics at a HA temperature of 60 °C, a microwave power of 600 W and an MV thermostatic temperature of 50 °C, the drying curves and drying rate curves of walnuts at different MCs at the transition point are shown in Figure 3A,B. As shown in Figure 3A, the MC at the transition point was higher and the HAD curves were shorter the earlier it entered the MVD stage, and the total drying time was shorter. When the MC at the transition point increased from 14% to 36%, the total drying time decreased from 685 min to 82 min for walnuts, and the total drying time decreased by 88.03% (Table 2). The results indicate that the use of MV in walnut drying was found to be very useful, especially in the final stage of drying.

It can be seen from Figure 3B that an increase in the MC at the transition point could effectively improve the drying rate of the MVD stage, especially at the beginning of the MVD stage. The highest drying rates in the MVD stage increased from 0.767%/min to 2.260%/min for walnuts, and the average drying rates in the HA-MVD stage increased from 0.100%/min to 0.862%/min (Table 2). This phenomenon is related to the interaction between the absorbing medium and the microwave. Since water is the main radiation-absorbing component, the higher the MC of walnuts, the more microwave energy is absorbed, which increases the kinetic energy in the water molecules and thus accelerates the migration of water [20]. The results show that the sooner the process enters the MVD stage, the larger the total drying rate is, and the shorter the total drying time is.

3.1.3. Effect of Microwave Power on Drying Characteristics

In order to investigate the effects of microwave powers of 200, 400, 600, 800 and 1000 w on the drying characteristics at a HA temperature of 60 °C, an MC at the transition point of 30% and an MV thermostatic temperature of 50 °C, the drying curves and drying rate curves of walnuts at different microwave power are shown in Figure 4A,B. As shown in Figure 4B, similar to the HAD process, when the walnut underwent MVD, the drying rate curves of walnuts under different microwave powers successively passed through preheating and falling-rate drying periods. The preheating period of the MVD stage corresponds to the stage of internal energy accumulation of walnuts under microwave radiation when the electromagnetic energy was converted into internal thermal energy in the walnut [20]. Although a constant-rate drying period can be expected for a high-moisture food like walnuts, this was not observed in this study, probably because of the single-layer arrangement and because the microwave heating under a vacuum was too fast to provide immediate drying [14,21,47]. With the increase in microwave power, the microwave radiation intensity that the per unit mass of walnuts receives increased. Under this effect, the average drying rates of the walnut for all five power levels (200 W, 400 W, 600 W, 800 W, and 1000 W) significantly increased as follows: 0.181, 0.430, 0.528, 0.915, and 1.077%/min, and microwave drying time decreased (260 min, 190 min, 165 min, 155 min, and 148 min, respectively) (Figure 4A and Table 2). Similar results were found in the drying of hyacinth, where the drying rate increased and the drying time was shortened as the microwave power was increased [25].

It can also be seen from Figure 4B that the effect of microwave power on the drying rate was significantly greater when the MC was higher. As the MC decreases, the difference in drying rate between different powers gradually decreases, which indicates that the internal mass transfer resistance of the material at low MC is significant. Walnuts with a high MC and a high dielectric loss factor that is in contact with microwave radiation increases the dipolar rotation in the water molecules, which promotes the absorption of microwave energy and thus accelerates water migration [19,48]. As drying proceeds, the loss of moisture in the walnut reduces the absorption of microwave energy and leads to a decrease in drying rate [19,49]. In addition, it was shown that MVD led to a significant increase in the drying rate (almost 3 to 25 times) compared to HAD (Table 2). Xong et al. also reported that the drying rate for the MVD of cranberries was up to 17 times higher than the drying rate for HAD [21].

3.1.4. Effect of Microwave–Vacuum Thermostatic Temperature on Drying Characteristics

In order to investigate the effects of MV thermostatic temperatures of 30, 40, 50, 60, and 70 °C on the drying characteristics at a HA temperature of 60 °C, a MC at the transition point of 30% and a microwave power of 600 W, the drying curves and drying rate curves of walnuts at different MV thermostatic temperatures are shown in Figure 5A,B. As can be seen from Figure 5, since the HAD stage of walnut drying had the same conditions, the drying characteristics curves were highly similar to those under the five drying conditions, a characteristic that can also be observed in Figure 4. As shown in Figure 5A, when the MV thermostatic temperature is 30 °C, the drying time in the MVD stage is the longest, and the total drying time is the longest, which reaches 405 min, while when the MV thermostatic temperature rises to 40 °C, the drying time in the MVD stage is significantly reduced, and the total drying time is reduced to 225 min (Table 2). These very long drying times indicate that even if HA-MVD could be used, a long drying time would be required to complete drying if the MV thermostatic temperature is very low. In contrast, when the MV thermostatic temperature was 50 °C, 60 °C, and 70 °C, the corresponding drying time was reduced to 150~165 min (Table 2). This indicated that, when HAD was converted into higher temperature MVD after reaching the MC of the transition point, the drying rate improved significantly. Similar results were found for the drying of amaranth seeds [50].

As shown in Figure 5B, the drying rate increased significantly with increasing MV thermostatic temperature in the early period of the MVD stage. The average drying rates in the HA-MVD stage increased from 0.114%/min to 2.365%/min (Table 2). However, as the amount of water inside the walnut continued to decrease (such as the 12% MC), the drying rates under different MV thermostatic temperature conditions become smaller and more similar. The moisture migration is affected by the material’s own moisture, structure, and other aspects [46]. The moisture migration becomes more difficult due to the moisture protection mechanism of walnut shells [46]. In addition, Figure 2B, Figure 3B, Figure 4B, and Figure 5B show that HA-MVD of walnuts occurred mainly in the falling-rate stage, which suggesting that internal moisture diffusion is the main mechanism of moisture transfer of the walnuts [46,51,52,53]. Hence, Equation (9) was applicable for obtaining the D_eff of the walnut under the drying conditions tested in this study.

3.2. Effective Moisture Diffusivity

The mechanism of moisture movement within a walnut during the falling-rate period could be represented by the phenomenon of effective moisture diffusion according to a diffusion model [26,29,30]. The D_effs were calculated from Equation (9) at different MCs according to the drying curves. Figure 6 shows the changes of D_effs of a walnut with an MC under the HA-MVD process at the different drying conditions. The results show that the trends in D_eff changes are similar for different HA temperatures, different MCs at the transition point, different microwave powers, and different MV thermostatic temperatures. It was found that the value of D_eff dropped sharply with the decrease in the MC during the HAD stage (corresponding to the high moisture ranges), and when the drying entered the MVD stage (corresponding to the low moisture ranges), the value of D_eff changed slightly with the decrease in MC. Such patterns were considered to be a typical phenomenon of liquid water diffusion based on the drying principle [26,32,54]. These results suggest that the D_eff cannot be assumed to be constant, especially in the HAD stage. These conclusions are consistent with the results reported in Thuwapanichayanan et al. for banana slices [27]. In addition, it was also found that the trends in changes in D_eff were consistent with the drying rate in the HAD stage, while the trends in changes were not consistent at the MVD stage.

As shown in Figure 6A, D_eff increases with increases in HA temperature. This can be attributed to the fact that a high temperature increases the kinetic energy in the water molecules, thus accelerating moisture diffusion [46]. It was also found that the higher the HA temperature, the faster the water diffusion in the subsequent MVD process. As thermal energy is moved from the outside to the inside by heat conduction in terms of the HAD, the outside is heated faster and has a higher temperature compared to the inside [12]. Therefore, the drying rate on the outside is obviously larger than on the inside when the HA temperature increases, resulting in walnuts entering the MVD stage with more internal moisture, which may increase its internal temperature and produce more water vapor, and hence provide an additional driving force for the water diffusion [20,32].

As shown in Figure 6B, D_eff is very high in the early period of the HAD stage, but very low in the later period of the HAD stage. It was found that D_eff decreases continuously with decreasing MC during the HAD process, while switching to MVD at any MCs at the transition point could truncate the decrease in D_eff and still maintain a high value until the end of drying. The results showed that, despite the fact that walnut samples were dried with HAD up to a certain extent, the ensuing MVD had a strong positive effect on their water diffusion. This is due to the high penetration power and volumetric heating of MD. Specifically, microwaves rapidly penetrate walnut shells, providing the energy required for internal water diffusion. Volumetric heating of intracellular water creates a high pressure within the cell walls, which accelerates water diffusion [14,15,18,21]. The results showed that the application of MVD in the final stage of drying could rapidly increase the internal vapor pressure of the walnut, accelerate the diffusion speed of the moisture, and re-enhance the drying rate (Figure 3 and Figure 6B).

It can be seen from Figure 6C that the D_eff increases with an increase in microwave power. The increase is attributed to the significant effect of microwave power in generating molecular motion, increasing the water vapor pressure inside the material, and decreasing the equilibrium MC at the material’s surfaces [21,25]. It was found that the D_eff decreases with decreasing MC at a microwave power of 200 W, while it increases slightly with decreasing MC at microwave powers of 400, 600, 800 and 1000 W. It can be seen from Figure 6D that a higher MV thermostatic temperature may cause the moisture molecules in the walnut to evaporate faster, thus resulting in a faster decrease in MC and a correspondingly higher value of D_eff. It was also found that the D_eff decreases with decreasing MC at MV thermostatic temperatures of 30 and 40 °C, while it increases slightly with decreasing MC at MV thermostatic temperatures of 50, 60, and 70 °C. This phenomenon is related to the drying time in the MVD stage [20,25]. Specifically, when the microwave power was 200 W and the MV thermostatic temperature was 30 and 40 °C, the drying time was very long (225–405 min) and D_eff decreased with decreasing MC, while when the microwave power was 400, 600, 800, and 1000 W and the MV thermostatic temperature was 50, 60, and 70 °C, the drying time was shorter (148–190 min) and D_eff increased with decreasing MC (Table 2). In addition, comparing Figure 6C,D, the moisture diffusion law under different MV thermostatic temperatures is similar to that under different microwave powers, and the effect of MV thermostatic temperature on the moisture diffusion characteristics was greater than that of microwave power. The values of D_eff found in this study were in the range of 2.33 × 10⁻⁹ m²/s to 6.89 × 10⁻⁸ m²/s, which is a typical range for drying food and agricultural materials [25,26,27,28,29,30,32,36,37,38,40,42,43,52,55,56].

The D_eff,ave values of walnuts under different drying conditions were calculated using Equation (10) and are summarised in

Table 3. Regression coefficients for moisture diffusivity model at different drying conditions.

Test No.	Deff,ave	A	B	C	D	R2	RSS
1	1. 386 × 10−8	7.83 × 10−12	−4.69 × 10−10	8.40 × 10−9	−3.62 × 10−8	0.944	9.120 × 10−17
2	1.591 × 10−8	7.69 × 10−12	−4.52 × 10−10	7.90 × 10−9	−2.98 × 10−8	0.957	8.530 × 10−17
3	2.239 × 10−8	9.12 × 10−12	−5.19 × 10−10	8.79 × 10−9	−2.63 × 10−8	0.951	1.239 × 10−16
4	2.636 × 10−8	1.00 × 10−11	−5.67 × 10−10	9.57 × 10−9	−2.59 × 10−8	0.960	1.427 × 10−16
5	2.922 × 10−8	9.79 × 10−12	−5.43 × 10−10	9.01 × 10−9	−1.92 × 10−8	0.951	1.977 × 10−16
6	2.679 × 10−8	6.59 × 10−12	−4.14 × 10−10	7.86 × 10−9	−2.08 × 10−8	0.820	3.399 × 10−16
7	1.591 × 10−8	7.69 × 10−12	−4.52 × 10−10	7.90 × 10−9	−2.98 × 10−8	0.957	1.427 × 10−16
8	1.210 × 10−8	6.87 × 10−12	−3.97 × 10−10	7.01 × 10−9	−3.10 × 10−8	0.963	1.515 × 10−16
9	1.083 × 10−8	6.42 × 10−12	−3.67 × 10−10	6.53 × 10−9	−3.12 × 10−8	0.960	1.767 × 10−16
10	1.005 × 10−8	6.17 × 10−12	−3.51 × 10−10	6.31 × 10−9	−3.17 × 10−8	0.958	1.919 × 10−16
11	1. 463 × 10−8	7.86 × 10−12	−4.71 × 10−10	8.58 × 10−9	−3.78 × 10−8	0.958	1.512 × 10−16
12	1.538 × 10−8	7.74 × 10−12	−4.58 × 10−10	8.12 × 10−9	−3.22 × 10−8	0.956	1.507 × 10−16
13	1.591 × 10−8	7.69 × 10−12	−4.52 × 10−10	7.90 × 10−9	−2.98 × 10−8	0.957	1.427 × 10−16
14	1.611 × 10−8	7.80 × 10−12	−4.58 × 10−10	7.94 × 10−9	−2.85 × 10−8	0.956	1.450 × 10−16
15	1.655 × 10−8	7.88 × 10−12	−4.63 × 10−10	8.02 × 10−9	−2.88 × 10−8	0.958	1.353 × 10−16
16	1.297 × 10−8	7.38 × 10−12	−4.40 × 10−10	8.15 × 10−9	−3.96 × 10−8	0.954	1.791 × 10−16
17	1.465 × 10−8	7.50 × 10−12	−4.37 × 10−10	7.84 × 10−9	−3.51 × 10−8	0.958	1.493 × 10−16
18	1.591 × 10−8	7.69 × 10−12	−4.52 × 10−10	7.90 × 10−9	−2.98 × 10−8	0.957	1.427 × 10−16
19	1.638 × 10−8	7.96 × 10−12	−4.72 × 10−10	8.29 × 10−9	−3.12 × 10−8	0.958	1.380 × 10−16
20	1.687 × 10−8	8.02 × 10−12	−7.74 × 10−10	8.25 × 10−9	−3.03 × 10−8	0.960	1.290 × 10−16

. Figure 7 shows the effect of different drying conditions on the D_eff,ave of walnuts. It was found that, as the HA temperature increased from 45 to 105 °C, the value of D_eff,ave for walnut increased linearly from 1.386 × 10⁻⁸ to 2.922 × 10⁻⁸ m²/s (Figure 7A); as the MC at the transition point increased from 14 to 36%, the value of D_eff,ave for the walnut increased slowly and then rapidly, ranging from 1.005 × 10⁻⁸ to 2.679 × 10⁻⁸ m²/s (Figure 7B); as the microwave power increased from 200 to 1000 W, the value of D_eff,ave for the walnut increased linearly from 1.463 × 10⁻⁸ to 1.655 × 10⁻⁸ m²/s (Figure 7C); as the MV thermostatic temperature increased from 30 to 70 °C, the value of D_eff,ave for the walnut increased rapidly and then slowly, ranging from 1.297 × 10⁻⁸ to 1.686 × 10⁻⁸ m²/s (Figure 7D). Comparing Figure 7A–D, HA temperature and the MC at the transition point have a greater effect on the D_eff,ave than the microwave power and MV thermostatic temperature, which suggests that the D_eff,ave is more sensitive to increases in the HA temperature and the MC at the transition point than to increases in microwave power and MV thermostatic temperature during the HA-MVD process of walnut.

The highest and lowest D_eff,ave values for the HA-MVD of walnuts were 1.005 × 10⁻⁸ m²/s and 2.922 × 10⁻⁸ m²/s, respectively. For drying walnuts with a HA dryer, Chen et al. reported D_eff,ave values in the range of 1.51 × 10⁻⁹ to 9.28 × 10⁻⁹ m²/s and 1.13 × 10⁻⁹ to 2.85 × 10⁻⁹ m²/s for the walnut shell and kernel, respectively [32]. The results show that the HA-MVD could raise the D_eff by one order of magnitude compared to the HAD. This indicates that the replacement of HA with MV in the final stage of drying walnuts greatly improved the diffusivity of internal moisture. According to the drying theory, internal moisture diffusivity is controlled by a combination of heat and moisture diffusion. Heat diffusion is water transfer caused by a temperature gradient inside the material, while moisture diffusion is water transfer caused by a moisture gradient in the material. If the moisture diffusion and heat diffusion are in the same direction, then heat diffusion catalyses moisture diffusion and increases the internal moisture diffusivity; conversely, heat diffusion hinders moisture diffusion [20,57]. During HAD, thermal energy is translated from the outside to the inside of the particle by thermal conduction, while moisture is moved from the inside to the surface of the particle due to moisture concentration differences [12,32]. Therefore, heat diffusion of the material increases the resistance to internal moisture diffusion [20,57]. In the MVD process, heat is generated inside the particles. In this case, both heat and moisture diffusion are transported from the inside to the outside, which contributes to moisture diffusion [14,19,20]. Meanwhile, the accumulation of internal heat sources can generate vapor pressure inside the particle, which promotes moisture diffusion [15,18,21]. In addition, the walnut structure is prone to producing more open pores under the combined effect of microwave and vacuum, which increases the diffusion channels and reduces the diffusion resistance, thereby accelerating water diffusion [52].

3.3. Model Development for Effective Moisture Diffusivity

D_eff models for walnuts under HA-MVD were developed using a non-linear regression method. The values of A, B, C, and D in the model, along with the values of R² and RSS, are shown in Table 3. The results show that the dynamic evolution of D_eff during the walnut HA-MVD process could be accurately represented by a third-order regression model. To include the influence of different drying conditions in the D_eff models, regression parameters were fitted using polynomial equations. These polynomial equations are shown in Equations (14)–(29). The R² values ranged between 0.943 and 0.999, which indicated the high performance of the regression fittings.

At a MC at the transition point of 30%, a microwave power of 600 W, and an MV thermostatic temperature of 50 °C:

$$A=-{6.6\times 10}^{-17}{T_{\mathrm{HA}}}^3+{1.5\times 10}^{-14}{T_{\mathrm{HA}}}^2-{9.8\times 10}^{-13}{T}_{\mathrm{HA}}+{2.9\times 10}^{-11}\begin{array}{c}(R^2=0.992)\end{array}$$

(14)

$$B=3.9\times {10}^{-15}{T_{\mathrm{HA}}}^3{-8.6\times 10}^{-13}{T_{\mathrm{HA}}}^2+{5.9\times 10}^{-11}{T}_{\mathrm{HA}}{-1.7\times 10}^{-9}\begin{array}{c}(R^2=0.996)\end{array}$$

(15)

$$C=-{6.7\times 10}^{-14}{T_{\mathrm{HA}}}^3+{1.5\times 10}^{-11}{T_{\mathrm{HA}}}^2-1.1\times {10}^{-9}{T}_{\mathrm{HA}}+{3.2\times 10}^{-8}\begin{array}{c}(R^2=0.999)\end{array}$$

(16)

$$D=2.3\times {10}^{-13}{T_{\mathrm{HA}}}^3{-5.2\times 10}^{-11}{T_{\mathrm{HA}}}^2+{4.0\times 10}^{-9}{T}_{\mathrm{HA}}{-1.3\times 10}^{-7}\begin{array}{c}(R^2=0.991)\end{array}$$

(17)

At a HA temperature of 60 °C, a microwave power of 600 W, and an MV thermostatic temperature of 50 °C:

$$A=-9.6\times {10}^{-16}M{C_{\mathrm{C}}}^3+{6.5\times 10}^{-14}M{C_{\mathrm{C}}}^2-1.3\times {10}^{-12}M{C}_{\mathrm{C}}+{1.5\times 10}^{-11}\begin{array}{c}(R^2=0.948)\end{array}$$

(18)

$$B=4.8\times {10}^{-14}M{C_{\mathrm{C}}}^3{-3.4\times 10}^{-12}M{C_{\mathrm{C}}}^2+{6.9\times 10}^{-11}M{C}_{\mathrm{C}}{-7.9\times 10}^{-10}\begin{array}{c}(R^2=0.969)\end{array}$$

(19)

$$C=-5.4\times {10}^{-13}M{C_{\mathrm{C}}}^3+{3.9\times 10}^{-11}M{C_{\mathrm{C}}}^2-8.1\times {10}^{-10}M{C}_{\mathrm{C}}+{1.2\times 10}^{-8}\begin{array}{c}(R^2=0.988)\end{array}$$

(20)

$$D=3.7\times {10}^{-12}M{C_{\mathrm{C}}}^3{-2.4\times 10}^{-10}M{C_{\mathrm{C}}}^2+{4.9\times 10}^{-9}M{C}_{\mathrm{C}}-6.5\times {10}^{-8}\begin{array}{c}(R^2=0.997)\end{array}$$

(21)

At a HA temperature of 60 °C, an MC at the transition point of 30%, and an MV thermostatic temperature of 50 °C:

$$A=-1.0\times {10}^{-21}{P_{\mathrm{W}}}^3+{2.9\times 10}^{-18}{P_{\mathrm{W}}}^2-2.1\times {10}^{-15}{P}_{\mathrm{W}}+{8.2\times 10}^{-12}\begin{array}{c}(R^2=0.956)\end{array}$$

(22)

$$B=8.3\times {10}^{-20}{P_{\mathrm{W}}}^3{-2.4\times 10}^{-16}{P_{\mathrm{W}}}^2+{1.9\times 10}^{-13}{P}_{\mathrm{W}}{-5.0\times 10}^{-10}\begin{array}{c}(R^2=0.977)\end{array}$$

(23)

$$C=-2.1\times {10}^{-18}{P_{\mathrm{W}}}^3+{6.1\times 10}^{-15}{P_{\mathrm{W}}}^2-5.5\times {10}^{-12}{P}_{\mathrm{W}}+{9.5\times 10}^{-9}\begin{array}{c}(R^2=0.997)\end{array}$$

(24)

$$D=1.7\times {10}^{-17}{P_{\mathrm{W}}}^3{-5.3\times 10}^{-14}{P_{\mathrm{W}}}^2+{5.4\times 10}^{-11}{P}_{\mathrm{W}}-4.7\times {10}^{-8}\begin{array}{c}(R^2=0.998)\end{array}$$

(25)

At a HA temperature of 60 °C, an MC at the transition point of 30%, and a microwave power of 600 W:

$$A=-2.3\times {10}^{-17}{T_{\mathrm{MV}}}^3+{3.5\times 10}^{-15}{T_{\mathrm{MV}}}^2-1.5\times {10}^{-13}{T}_{\mathrm{MV}}+{9.3\times 10}^{-12}\begin{array}{c}(R^2=0.996)\end{array}$$

(26)

$$B=2.9\times {10}^{-15}{T_{\mathrm{MV}}}^3{-4.5\times 10}^{-13}{T_{\mathrm{MV}}}^2+{2.1\times 10}^{-11}{T}_{\mathrm{MV}}{-7.4\times 10}^{-10}\begin{array}{c}(R^2=0.998)\end{array}$$

(27)

$$C=-6.7\times {10}^{-14}{T_{\mathrm{MV}}}^3+{1.1\times 10}^{-11}{T_{\mathrm{MV}}}^2-5.4\times {10}^{-10}{T}_{\mathrm{MV}}+{1.6\times 10}^{-8}\begin{array}{c}(R^2=0.957)\end{array}$$

(28)

$$D=1.3\times {10}^{-13}{T_{\mathrm{MV}}}^3{-2.9\times 10}^{-11}{T_{\mathrm{MV}}}^2+{2.1\times 10}^{-9}{T}_{\mathrm{MV}}-8.1\times {10}^{-8}\begin{array}{c}(R^2=0.943)\end{array}$$

(29)

where, T_HA is the hot air temperature (°C); MC_C is the moisture content at the transition point (%); P_W is the microwave power (W); and T_MV is the microwave–vacuum thermostatic temperature (°C).

To verify whether the D_eff model can accurately predict the evolution of moisture diffusion characteristics of walnuts during the HA-MVD process, Figure 8 shows a comparison of experimental D_eff and the D_eff predicted by the D_eff 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 [52]. As can be seen from Figure 8, the D_eff model provided a good conformity between experimental and predicted D_eff during the HA-MVD processes of walnuts. These data points are banded line plots at an angle of approximately 45°, which suggests that the D_eff model is able to make good predictions of the dynamic change of D_eff during the walnut HA-MVD processes under different drying conditions. Meanwhile, the 45° banded line graphs also show the accuracy of the proposed third-order polynomial functions, and these can be embedded in future FEM simulations to model the water loss of walnuts during HA-MVD, which could provide a theoretical basis for elucidating the water loss mechanism of walnuts during HA-MVD.

4. Conclusions

A HA-MAD system was established to investigate the drying characteristics and moisture diffusion properties of walnuts and to develop a D_eff model. Based on the experimental findings, several key conclusions were drawn:

(1) The HA-MVD of walnut occurred mainly in the falling-rate stage. The average drying rate for MVD was up to 3 to 25 times higher than the average drying rate for HAD. The total drying time shorted and the average drying rate increased with increasing HA temperature, MC at the transition point, microwave power, and MV thermostatic temperature.

(2) The value of D_eff dropped continuously with the decrease in MC in the HAD stage, while switching to the MVD stage could truncate the decrease in D_eff and still maintain at a high value until the end of drying.

(3) The values of D_eff for walnut ranged from 2.33 × 10⁻⁹ m²/s to 6.89 × 10⁻⁸ m²/s. The HA temperature and MC at the transition point have a greater effect on the D_eff than the microwave power and MV thermostatic temperature. The replacement of HAD with MVD in the final stage of the drying of walnuts could raise the D_eff by one order of magnitude.

(4) The third-order polynomial prediction model of D_eff related to the sample MC and drying conditions could accurately describe the dynamic change of D_eff of walnut during the HA-MVD process.