Minh Ha Nguyen
| Luan Nguyen Thanh*
© 2025 The authors. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
OPEN ACCESS
This study aims to determine the drying kinetics of Arctium lappa roots in a combined infrared–vacuum dryer. The effects of 45-65℃ drying temperatures and 6-24 kPa system pressures on the drying kinetics were considered. The results indicated that the sample dried faster as the drying temperature increased and the system pressure decreased. At a 6 kPa system pressure, the drying time was reduced by up to 23.8% and 46.7% compared to drying at 15 kPa and 24 kPa system pressures, respectively. In a drying regime with 45-65℃ drying temperatures and 6 kPa system pressure, the mathematical model in the form of a third-order polynomial yielded the best prediction of the moisture content reduction. The effective moisture diffusivity coefficient was 1.80 ´ 10-10-3.56 ´ 10-10 m2/s. The results provide valuable data regarding the theoretical and practical aspects of drying Arctium lappa roots in a combined infrared–vacuum dryer.
infrared–vacuum dryer, Arctium lappa, drying kinetics
Infrared radiation has the advantages of fast heating, low energy consumption, and bactericidal ability [1, 2]. Therefore, combining infrared radiation with other drying technologies has been considered. Studies on infrared-assisted convective drying [3-5], infrared-assisted heat pump drying [6, 7], combined infrared–vacuum drying [8-10], and infrared-assisted freeze drying [11-14] indicated that combining infrared radiation offered significant benefits regarding the efficiency and quality of dried products. Combined infrared–vacuum drying has the potential to yield high-quality dried products. Under the influence of infrared radiation and a vacuum environment, a low drying temperature ensures an appropriate drying time, and the vacuum environment limits material oxidation in the drying process. Therefore, combined infrared–vacuum is a potential drying method for temperature-sensitive materials. Recently, researchers have conducted drying kinetics studies for many different materials, such as mushroom [8], pumpkin [9], kiwifruit [10], lemon [15], grapefruit [16], banana bract powder [17], Cistanche [18], and potato [19]. The reports provide valuable data related to drying in a combined infrared–vacuum dryer.
Arctium lappa (ARL) is a herb of the Asteraceae family commonly grown in Asian countries. It can be used to treat colds, edema, and anti-inflammatory. ARL is often harvested seasonally; so, drying is a simple way to prolong its shelf life. The simplest drying method is sun-drying; however, it is dependent on the weather and can accumulate dust. Thus, artificial drying has been preferentially applied. Nguyen et al. [20] studied the drying kinetics of ARL roots in a hot air dryer. Xia et al. [21] examined the color and flavor substances of ARL roots using different drying methods. The flavor composition of dried ARL roots was also analyzed in the report of Lu et al. [22]. Our review of the literature indicated that while the drying characteristics of ARL have been considered in a few studies, to our knowledge, data on the mathematical model and effective moisture diffusivity coefficient have not been provided. These data are important in predicting drying times for design and operation. Therefore, this research gap is addressed in the present work. The results provide a useful reference for drying ARL roots in a combined infrared–vacuum dryer.
2.1 Materials
Fresh samples after harvest were selected and stored in vacuum-sealed bags at 5℃. The samples were sliced into elliptical shapes with dimensions of (45 ± 5 mm) ´ (20 ± 3 mm), and each slice was 4 ± 0.2 mm thick. The initial moisture content of ARL was determined by the oven drying method [23]. It was approximately 0.78 w.b, with an uncertainty of less than 0.2%.
2.2 Experimental description
Figure 1 shows the combined infrared–vacuum dryer used in this study. In the dryer, an infrared radiation lamp was placed 20 cm from the surface of the drying tray; the tray was 38 cm × 38 cm in size and had a digital scale beneath with an accuracy of 0.01 g to obtain the parameters during testing. A temperature sensor was used to monitor the temperature in the drying chamber (resolution of 0.1℃), and a digital vacuum pressure gauge was used to measure the system pressure (resolution of 0.1 kPa). The material arrangement density was approximately 2.43 kg/m2. Each experiment was replicated three times, and the values were averaged. Data were collected every ten minutes. The process was conducted from the initial moisture content to a final moisture of 10% w.b.
Figure 1. The combined infrared–vacuum dryer
2.3 Analysis of the drying kinetics
The drying kinetics were interpreted based on the computed results of the dry basis moisture content and moisture ratio. The dry basis moisture content was calculated according to the following formula:
$M_i=\frac{G_i-G_S}{G_S}$ (1)
Ignoring the equilibrium moisture content, the moisture ratio was calculated using the following formula [24]:
$Y=\frac{M_i}{M_o}$ (2)
In this work, six mathematical models were applied for kinetics modeling of ARL roots drying in a combined infrared–vacuum dryer [23, 25]:
1/ Henderson and Pabis: Y = a.exp(-kt);
2/ Page: Y = exp(-ktn);
3/ Logarithmic: Y = a.exp(-kt) + b;
4/ Newton: Y = exp(-kt);
5/ Wang and Singh: Y = 1 + at + bt2;
6/ Third-order polynomial: Y = 1 + at + bt2 + ct3.
The present work used statistical parameters (correlation coefficient: R2, root mean square error: RMSE, and reduced chi-squared: Cr2) to select the best-fitting model. The best-fitting model had the highest R2, the lowest RMSE, and the lowest Cr2. The statistical parameters were determined using the following formulas [23, 26, 27]:
$R^2=1-\frac{\sum_{i=1}^N\left(Y_{e, i}-Y_{p, i}\right)^2}{\sum_{i=1}^N\left(Y_{e, m}-Y_{e, i}\right)^2}$ (3)
$R M S E=\sqrt{\frac{1}{N} \sum_{i=1}^N\left(Y_{p, i}-Y_{e, i}\right)^2}$ (4)
$C_r^2=\frac{1}{N-z} \sum_{i=1}^N\left(Y_{p, i}-Y_{e, i}\right)^2$ (5)
A model in the form of a third-order polynomial is proposed, based on the Wang and Singh model [28]. Drying data of several materials were used to evaluate the potential use of the third-order model. The results show that the third-order polynomial model fits the data points well (see Figure 2). Therefore, this model can be applied in drying kinetics studies.
Figure 2. The potential of using the third-order polynomial model: (a) milky mushroom [29]; (b) sweet potato [30]; (c) onion [31]; (d) paddy [27]
The effective moisture diffusivity coefficient (Deff) is an important parameter in the modeling of the drying process. According to Fick’s second law of diffusion, the differential equation for moisture diffusion is as follows:
$\frac{\partial M}{\partial t}=D_{e f f} \nabla^2 M$ (6)
We assumed the initial moisture was uniformly distributed, with constant diffusion, dismissible shrinkage, and dismissible external resistance, and the material was a slab geometry. The approximate analytical solution for Eq. (6) is [16, 32]:
$\begin{aligned} & Y=\frac{M_i-M_e}{M_o-M_e} \\ & =\frac{8}{\pi^2} \sum_{n=0}^{\infty} \frac{1}{(2 n+1)^2} \exp \left(-\frac{(2 n+1)^2 \pi^2 D_{e f f} t}{4 L^2}\right)\end{aligned}$ (7)
Assuming a long drying time and ignoring the equilibrium moisture content, the approximation of Eq. (7) has the following form [10, 16]:
$Y=\frac{M_i}{M_o}=\frac{8}{\pi^2} \exp \left(-\frac{\pi^2 D_{e f f} t}{4 L^2}\right)$ (8)
Eq. (8) can further be simplified in logarithmic form:
$\ln Y=\ln \left(\frac{8}{\pi^2}\right)-\frac{\pi^2 D_{e f f} t}{4 L^2}$ (9)
The slope of the function ln(Y) with the variable (t) is the basis for determining the Deff.
The standard deviation of the computed results was determined using the following formula [33]:
$s=\sqrt{\frac{1}{(n-1)} \sum_{j=1}^n\left(x_j-x_m\right)^2}$ (10)
The uncertainty of the calculated results was determined using the following formula [34]:
$U_F=\left(\sum_{j=1}^n\left[\left(\frac{\partial F}{\partial X_j}\right)^2\left(U_{X_j}\right)^2\right]\right)^{0.5}$ (11)
Figure 3 shows the effect of the drying temperature and system pressure on the drying time. Under constant system pressure conditions, the drying time at 65℃ was reduced by 25.0-35.4% and 45.8-65.6% compared to 55℃ and 45℃, respectively. When observed under constant temperature conditions, the drying time at 6 kPa system pressure was reduced by 12.5-23.8% and 16.1-46.7% compared to drying at 15 kPa and 24 kPa system pressures, respectively. The water molecules receive more energy as the drying temperature increases, increasing moisture diffusion from the center to the shell. Furthermore, there is a high difference in vapor pressure between the material surface and the environment in the drying chamber, leading to increased water escape from the material surface. Thus, the drying time decreased as the drying temperature increased and the system pressure decreased. Drying at a low system pressure was suitable for a reasonable drying time. Therefore, in the next part of the study, the drying kinetics were examined in detail at 6 kPa system pressure and 45-65℃ drying temperatures.
Figure 3. Effect of drying regimes on the drying time
Figure 4(a) shows the data points of the three drying regimes. The six mathematical models were applied to determine the best-fitting mathematical model for the data points. The results in Table 1 indicate that the sixth mathematical model best described the drying characteristics of ARL roots due to the highest R2, the lowest RMSE, and the lowest Cr2. The coefficients in the third-order polynomial model are detailed in Table 2. Figure 4(b) confirms a good fit between the mathematical model in third-order polynomial form and the data points, with a high accuracy in prediction. Furthermore, this model is simple in form and advantageous for the practical application of predicting the moisture content variation with drying time. Therefore, this model should be considered for drying kinetics studies.
Figure 4. Experimental data and the regression: (a) experimental data; (b) predicted curve
Table 1. The statistical parameters of the mathematical models
|
Drying Mode |
Model |
Statistical Parameters |
||
|
R2 |
RMSE |
Cr2 |
||
|
T=65℃, P=6 kPa |
1 |
0.982766 |
0.042255 |
0.001928 |
|
2 |
0.996437 |
0.018839 |
0.000383 |
|
|
3 |
0.997930 |
0.013688 |
0.000211 |
|
|
4 |
0.987456 |
0.049250 |
0.002519 |
|
|
5 |
0.999492 |
0.006972 |
0.000052 |
|
|
6 |
0.999730 |
0.005005 |
0.000028 |
|
|
T=55℃, P=6 kPa |
1 |
0.981571 |
0.043610 |
0.002014 |
|
2 |
0.995427 |
0.021106 |
0.000472 |
|
|
3 |
0.998353 |
0.012186 |
0.000162 |
|
|
4 |
0.987009 |
0.051833 |
0.002763 |
|
|
5 |
0.999443 |
0.007431 |
0.000058 |
|
|
6 |
0.999605 |
0.005970 |
0.000039 |
|
|
T=45℃, P=6 kPa |
1 |
0.969571 |
0.054514 |
0.003098 |
|
2 |
0.993582 |
0.025164 |
0.000660 |
|
|
3 |
0.999016 |
0.009314 |
0.000092 |
|
|
4 |
0.977323 |
0.063870 |
0.004164 |
|
|
5 |
0.999391 |
0.007629 |
0.000061 |
|
|
6 |
0.999881 |
0.003415 |
0.000012 |
|
Table 2. The coefficients in the third-order polynomial model
|
T (℃) |
P (kPa) |
Coefficients |
||
|
a |
b |
c |
||
|
65 |
6 |
- 0.384026 |
0.0272412 |
0.00227276 |
|
55 |
6 |
- 0.281214 |
0.0145357 |
0.000855446 |
|
45 |
6 |
- 0.161599 |
0.000773081 |
0.00051556 |
The effective moisture diffusivity coefficient was calculated according to Eq (9). The result shows that the Deff obtained was from 1.80 × 10-10 to 3.56 × 10-10 m2/s (see Table 3). This result was consistent with the data from some materials provided in the previous studies [15, 16].
Table 3. The Deff of ARL roots
|
T (℃) |
P (kPa) |
Deff (m2/s) |
|
65 |
6 |
3.56 ´ 10-10 |
|
55 |
6 |
2.72 ´ 10-10 |
|
45 |
6 |
1.80 ´ 10-10 |
The present work focused on determining the drying kinetics of ARL roots in a combined infrared–vacuum dryer. The drying regime was established with 45–65℃ drying temperatures and 6–24 kPa system pressures. The main findings are as follows:
|
Gi |
weight of the sample at time i, g |
|
Gs |
weight of dry solids, g |
|
M |
dry basis moisture content |
|
Y |
moisture ratio |
|
t |
drying time, s |
|
L |
half thickness of slab, m |
|
Deff |
effective moisture diffusivity coefficient, m2.s-1 |
|
R2 |
correlation coefficient |
|
RMSE |
root mean square error |
|
Cr2 |
reduced chi-squared |
|
s |
standard deviation |
|
k |
drying constant, h-1 |
|
T |
drying temperature, ℃ |
|
P |
system pressure, kPa |
|
z |
number of constants |
|
N |
number of observations |
|
a, b, c |
coefficients |
|
U |
uncertainty |
|
$\Delta$ |
difference |
|
Subscripts |
|
|
o |
initial |
|
i |
at time i |
|
m |
mean |
|
e |
equilibrium |
|
F |
calculated results |
|
Xj |
measured parameter |
|
e,i |
experimental value |
|
p,i |
predicted value |
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