Effect of Drying Conditions on Kinetics, Modeling, and Thermodynamic Behavior of Marjoram Leaves in an IoT-Controlled Vacuum Dryer

Abstract

The current study aimed to investigate mathematical modeling, drying kinetics, and thermodynamic properties for cost-effectively drying marjoram leaves under different operating pressures (OPs) and drying temperatures (DTs). Three DTs of 40, 50, and 60 °C and three OPs of (atm) atmospheric, −5 kPa, and −10 kPa were used in this study. All drying processes were conducted using the developed vacuum dryer (DVD) and a constant layer thickness of 1 cm and initial moisture content of 817.43 on a dry basis (d.b.). The results obtained showed that increasing the DT from 40 to 60 °C at OPs of atm, −5 kPa, and −10 kPa led to a decrease in the drying time by about 55.6%, 36.4%, and 42.9%, respectively. On the other hand, decreasing the OP from atm to −10 kPa resulted in a decrease in drying time of about 58.8%, 45.5%, and 50% at DTs of 40, 50, and 60 °C, respectively. The moisture diffusivity (D_eff) ranged between 1.13 and 5.51 × 10⁻⁹ m²/s, with the highest value of D_eff observed at an OP of −10 kPa and a DT of 60 °C. Under these conditions, the activation energy (AE) was minimal, at approximately 2.68 kJ/mol. Mathematical modeling revealed that the Modified Midilli (I) model was the most suitable for describing the drying kinetics of marjoram leaves under experimental conditions. Among the thermodynamic parameters of marjoram leaves, it was observed that enthalpy values decrease with increasing DT and decreasing OP. Additionally, all tests showed negative entropy, suggesting that the chemical adsorption and/or structural modifications of the adsorbent are responsible for these results. The economic analysis revealed that drying marjoram leaves at an OP of 10 kPa and a DT of 60 °C resulted in yearly cost savings of up to USD 2054.19 and reduced the investment payback period to approximately 0.139 years (about 2 months).

1. Introduction

Marjoram (Origanum majorana) is a perennial herb belonging to the mint family (Lamiaceae), cultivated for culinary purposes. The fresh or dried leaves and blooming tops are utilized to season many cuisines, providing a warm, aromatic, mildly pungent, and slightly bitter flavor. Marjoram is especially valued for the flavor it imparts to sausages, meats, poultry, stuffing, fish, stews, eggs, vegetables, and salads. Indigenous to the Mediterranean region and western Asia, marjoram is frequently grown as an annual in northern latitudes where winter temperatures are lethal to the plant [1,2]. Medicinally, marjoram is recognized for its anti-inflammatory, antioxidant, and antimicrobial properties, which aid digestion, relieve stress, and boost immunity. It has been used in traditional medicine to treat ailments like colds, headaches, and muscle pain. Rich in vitamins A and C, as well as minerals like calcium and magnesium, marjoram supports overall health. Its soothing scent also makes it popular in aromatherapy, promoting relaxation and mental well-being [2,3].

Drying is a fundamental post-harvest process crucial for extending the shelf life and preserving the sensory, nutritional, and therapeutic qualities of marjoram (Origanum majorana L.). However, the method employed for drying significantly influences the final product’s quality and economic value. Traditional drying techniques, particularly open sun drying, are still commonly practiced due to their low operational costs and ease of implementation. Yet, these methods often fail to preserve the delicate biochemical and structural properties of marjoram. Exposure to direct solar radiation and fluctuating ambient conditions often results in uneven moisture removal and mechanical stress on plant tissues, leading to the loss of aroma, color fading, and degradation of essential phytochemicals [4,5,6,7,8]. Industrial drying systems, such as hot-air dryers, offer higher throughput and better process control. However, they generally operate at elevated temperatures, which pose a risk to heat-sensitive constituents, such as essential oils, flavonoids, and phenolic acids. The excessive heat accelerates the volatilization and oxidation of these compounds, significantly reducing marjoram’s aromatic strength and medicinal efficacy [9,10,11,12,13]. Additionally, industrial drying systems are energy-intensive, which not only increases operational costs but also contributes to higher carbon emissions and environmental degradation [7,14,15]. Open sun drying, despite its simplicity and affordability, presents several limitations. It is highly dependent on weather conditions—such as temperature, humidity, wind speed, and solar radiation—which introduces variability and unpredictability in the drying process [12,16,17,18,19]. Moreover, direct exposure to environmental contaminants, including dust, insects, and airborne microbes, compromises the safety, hygiene, and overall quality of the dried herb [20,21,22,23]. Extended exposure to ultraviolet (UV) radiation and inconsistent temperatures can also degrade volatile constituents and pigments, leading to a loss of aroma, discoloration, and a decline in consumer appeal [24,25].

To overcome these limitations, vacuum drying has gained attention as a highly effective alternative. This advanced drying technique operates under reduced pressure and low temperature in an oxygen-deprived environment. These conditions significantly suppress oxidative reactions and thermal degradation, thus preserving the structural integrity, vibrant green color, and volatile profile of marjoram [26,27,28]. Vacuum drying not only retains the herb’s therapeutic potency and aromatic richness but also improves its visual quality—factors that are essential for high-end culinary and pharmaceutical applications. Furthermore, vacuum drying offers several technological and environmental advantages. It consumes less energy compared to conventional thermal drying methods due to the lower temperature requirements and shorter drying times. Its adaptability enables precise control over drying parameters, thereby enhancing process efficiency and product uniformity. From a sustainability standpoint, vacuum drying contributes to reduced energy consumption and a lower environmental footprint, making it a viable option for eco-conscious herb processing industries [29,30]. Overall, the superior preservation of marjoram’s functional, sensory, and medicinal attributes through vacuum drying highlights its growing importance in the post-harvest processing sector. By ensuring high-quality output with extended shelf life and minimal nutrient loss, vacuum drying significantly enhances the herb’s commercial and therapeutic value, reinforcing its relevance in modern agribusiness and value-added herb production [31,32].

Refs. [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]. Several previous studies have been conducted on drying marjoram leaves. However, all of them were focused on studying the effect of drying methods on the antioxidant capacity and bioactive and phenolic constituents [33], using dried marjoram leaves as a green inhibitor to prevent steel from rusting in an acidic solution [34], studying the influence of biological preparations and drying methods on the content of essential oils in basil and marjoram [35], studying the effect of different levels of DTs and layer thickness on drying parameters, power consumption, and costs [36], studying the effect of drying methods on the flavor quality [37], and studying the effect of drying methods on the aroma [38].

To our knowledge, no publications in the literature describe the effects of combining DTs and OPs on mathematical modeling, D_eff, AE, drying parameters, and thermodynamic properties of marjoram leaves. The current study aimed to utilize a developed vacuum dryer (DVD) equipped with control units for temperature and humidity, as well as an electrical heater for drying marjoram leaves. To provide essential data to find the most effective drying model for improving the commercial drying process of marjoram leaves. This study’s findings will be particularly relevant to post-harvest technology specialists seeking to improve the effectiveness of their drying techniques.

2. Materials and Methods

2.1. Experimental Setup

The marjoram leaves belonged to the local Libyan variety, harvested at full maturity during the peak growing season. The raw material was sourced from farms located in Omar Al-Mukhtar University, Libya. After harvesting, the leaves were stored under controlled conditions—at a temperature of approximately 4 °C and relative humidity of 60%—to maintain their freshness and prevent degradation prior to drying experiments. Marjoram leaves were small and delicate, typically measuring 1.0 to 2.5 cm in length and 0.5 to 1.5 cm in width. Their shape ranges from oval to oblong, featuring a slightly rounded tip that tapers gently. The leaves are thin and soft-textured, with a velvety surface due to the presence of fine, downy hairs (trichomes), giving them a subtle, fuzzy appearance. The drying process was conducted at Omar Al-Mukhtar University, Libya, in the summer of 2021. The average initial moisture content of marjoram leaves was 825.93% on a dry basis (d.b.) (89.2% on a wet basis). After that, the marjoram leaves were distributed above three drying trays with a 1 cm layer thickness. Then, the drying experiments were conducted using the developed vacuum dryer (DVD) shown in Figure 1, at three drying temperatures (DTs) of T1 = 40, T2 = 50, and T3 = 60 °C, and three operating pressures (OPs) of P1 = normal atmospheric pressure (atm), P2 = −5 kPa, and P3 = −10 kPa. Each experiment was conducted in triplicate to ensure the reliability and reproducibility of the results. The average moisture content from the three replicates was used in all subsequent calculations, graphical representations, and mathematical modeling to minimize visual clutter in the figures caused by the large number of variables under investigation. This approach allowed for a clearer interpretation and presentation of the data trends. During the drying process, relative humidity, temperature, and sample weight were measured automatically using the developed control and measuring electronic circuit (Figure 2) every 15 min.

2.2. Description of the DVD

Figure 1 illustrates the DVD, which utilizes materials sourced from local markets in Libya. The main body of the DVD was cylindrical, with an average capacity of 200 L, and was manufactured from 2 mm thick sheet metal. The dryer chamber is fully insulated on all sides with a 5 cm thick layer of high-performance glass wool insulation, specifically selected for its excellent thermal resistance and low thermal conductivity. This insulation significantly minimizes heat loss to the external environment, thereby enhancing the overall energy efficiency of the drying system. By maintaining a stable internal temperature and reducing thermal fluctuations, the insulation also helps ensure more consistent drying conditions, which is critical for preserving the quality and uniformity of heat-sensitive plant materials such as marjoram leaves. It was installed on a movable frame with three wheels to ease the transportation process. The drying system was equipped with a 0.5 horsepower (hp) vacuum pump (Model: VP245, China) to create and maintain the desired sub-atmospheric pressure levels within the drying chamber. This pump facilitated the precise control of operating pressures, which is essential for conducting vacuum-assisted drying processes. In addition, a vacuum gauge was integrated into the system to continuously monitor and regulate the internal pressure, ensuring accurate adherence to the set operating pressure values (e.g., atmospheric, −5 kPa, −10 kPa). This configuration enabled greater control over the drying environment, promoting efficient moisture removal at lower temperatures and protecting the thermosensitive compounds in the marjoram leaves. Additionally, the system was equipped with a 600-watt electric heater to raise the internal air temperature of the DVD, enabling temperature adjustments to meet the specific thermal requirements of the drying process. To ensure accurate monitoring and control of the drying parameters—particularly the real-time measurement of sample weight loss and air temperature within the drying chamber—a custom-designed electronic measurement and control circuit was integrated into the DVD. This circuit was developed to provide both automatic regulation and precise data acquisition throughout the drying process. As illustrated in Figure 2, the circuit comprises several key components: a load cell sensor for continuously tracking the weight loss of the drying samples, a DHT-22 sensor for monitoring both air temperature and relative humidity, an Arduino Uno microcontroller board for data processing, and control logic execution, a relay for managing the operation of the electric heater, and a USB cable for data communication and power supply. An LCD screen was included in the setup to visually display real-time temperature and sample weight readings, enabling immediate user feedback and system status checks. During laboratory trials, however, the collected sensor data were transmitted directly to a connected laptop via a USB cable. The data were logged and saved in an Excel spreadsheet, facilitating detailed post-processing and analysis of the drying behavior under various operating conditions. This integrated electronic system ensured precise control over drying variables, thereby enhancing the repeatability and reliability of the drying experiments while also supporting the optimization of drying parameters for improved product quality and energy efficiency.

2.3. Evaluation Processes of the DVD

2.3.1. Drying Kinetics

Moisture Content (μ)

Estimating the initial moisture content (μ, %) of a material is crucial for the drying process for several reasons, as the initial $\mu$ helps in calculating the total amount of water that needs to be removed. This directly influences the drying time required to achieve the desired final $\mu$ [39,40]. Also, knowing the initial $\mu$ allows for the optimization of energy use. Over-drying consumes unnecessary energy, while under-drying may not meet product specifications. Accurate estimation helps in setting the right parameters for energy-efficient drying [7]. During the current study, the initial $\mu$ was estimated, according to AOAC [41], by drying the marjoram samples at 105 °C until reaching balance weight. Then, the $\mu$ was calculated according to Equation (1).

$$\mu (\%)={\displaystyle \frac{W_i-W_f}{W_f}}\times 100$$

(1)

where $W_i$ and $W_f$ are the initial and final weights of marjoram samples, g.

The weight loss of marjoram samples was calculated using Equation (2) [17,42].

$$Weight loss \left(g\right)=W_t-W_{t+15 min}$$

(2)

where $W_t$ is the weight of the dried sample at any time, g, and $W_{t+15 min}$ is the subsequent weight after 15 min, g.

Drying Rate (DR)

The DR was calculated using Equation (3) [43].

$$\mathrm{D}\mathrm{R}(g_{water}/g_{dry matter}.h)={\displaystyle \frac{Weight loss \left(g\right)}{∆t \left(h\right)}}$$

(3)

Moisture Ratio (MR)

MR calculation is an essential statistic in drying processes, representing the ratio of residual $\mu$ ($M_t$) to the initial $\mu$ ($M_0$) moisture content of a material. Assessing drying efficiency, determining drying times, and ensuring product quality are essential. By analyzing MRs, industries such as food, agriculture, and pharmaceuticals may optimize drying conditions, save energy consumption, and prevent both over-drying and under-drying. Accurate calculations contribute to maintaining consistency, prolonging shelf life, and enhancing overall process efficiency [19,22,44,45,46]. The MR of the desiccated marjoram leaf samples was ascertained utilizing Equation (4) [47].

$$MR={\displaystyle \frac{{\mu }_t-{\mu }_e}{{\mu }_0-{\mu }_e}}$$

(4)

Therefore, the MR can be represented as shown in Equation (5) [48], after neglecting the equilibrium $\mu$ (${\mu }_e$) form Equation (4).

$$MR={\displaystyle \frac{{\mu }_t}{{\mu }_0}}$$

(5)

2.3.2. Drying Constant (k)

The mathematical modeling of thin-layer drying generates the drying constant (k) by combining numerous drying variables. The drying constant was obtained using Equation (6).

$$MR=\mathrm{A\; exp}(-k\times t)$$

(6)

where A is the initial dry basis $\mu$ (

Table 1. Effect of DTs and OPs on “A” values of marjoram leaves dried using the DVD.

Coefficient	atm			−5 kPa			−10 kPa
	40 °C	50 °C	60 °C	40 °C	50 °C	60 °C	40 °C	50 °C	60 °C
A	0.765	0.765	0.793	0.772	0.828	0.802	0.797	0.779	0.804

), g_water/g_{dry matter}, and $t$ is the drying time, min.

2.3.3. Moisture Diffusivity (D_eff)

Fick’s Second Law of Diffusion is crucial for understanding the movement of moisture during drying. It describes how moisture concentration changes over time and space in a material. This law helps predict how moisture gradients evolve during drying, enabling the optimization of drying time, temperature, and airflow [49]. The general equation for mass transfer in a slab shape is shown in Equation (7) [50,51]:

$${\displaystyle \frac{\partial \mu }{\partial t}}=D_{eff}\left({\displaystyle \frac{{\partial }^2\mu }{\partial r^2}}+{\displaystyle \frac{2}{r}}{\displaystyle \frac{\partial \mu }{\partial r}}\right)$$

(7)

With the appropriate initial and boundary conditions:

$${\mu \left.\left(r,t\right)\right|}_{t=0}={\mu }_0$$

(8)

$${\left.{\displaystyle \frac{\partial \mu \left(r,t\right)}{\partial \mu }}\right|}_{t=0}=0$$

(9)

$${\mu \left.\left(R,t\right)\right|}_{t>0}={\mu }_e$$

(10)

where t is the drying time, s.

Equation (8) assumes that moisture is evenly distributed throughout the marjoram leaf sample at the beginning of the drying process. Equation (9) reflects that mass transfer occurs symmetrically with respect to the center of the leaf sample. Meanwhile, Equation (10) establishes that the surface $\mu$ of the marjoram leaf quickly reaches μ_e in response to the surrounding air conditions. Notably, the μ_e is relatively low.

Equation (11) can be condensed based on the above assumptions.

$${\mu \left.\left(\mathrm{R},\mathrm{t}\right)\right|}_{\mathrm{t}>0}=0$$

(11)

Following the numerical procedure, assume a solution of the following form to separate the variables:

$$\mu \left(r,t\right)=\mathrm{F}\left(r\right)\ast \mathrm{G}\left(t\right)$$

(12)

where (F) is a function of (r) only, and (G) is a function of (t) only.

$${\displaystyle \frac{{\mu }_0-\mu }{{\mu }_0}}=1-{\displaystyle \frac{8}{{\pi }^2}}\sum _{n=0}^{\mathrm{\infty }}\left({\displaystyle \frac{1}{n^2}}exp\left({\displaystyle \frac{{-\mathsf{\pi }}^2\times {\mathrm{D}}_{\mathrm{e}\mathrm{f}\mathrm{f}}\times \mathrm{t}}{4{\mathrm{L}}^2}}\right)\right)$$

(13)

Equation (14) was generated by simplifying Equation (13).

$${\displaystyle \frac{\mu }{{\mu }_0}}={\displaystyle \frac{8}{{\pi }^2}}\sum _{n=0}^{\mathrm{\infty }}\left({\displaystyle \frac{1}{n^2}}exp\left({\displaystyle \frac{{-\mathsf{\pi }}^2\times {\mathrm{D}}_{\mathrm{e}\mathrm{f}\mathrm{f}}\times \mathrm{t}}{4{\mathrm{L}}^2}}\right)\right)$$

(14)

In accordance with the MR previous equation and form third condition, Equation (15) was simplified:

$$\mathrm{M}\mathrm{R}={\displaystyle \frac{8}{{\mathsf{\pi }}^2}}\times \sum _{\mathrm{n}=1}^{\mathrm{\infty }}{\displaystyle \frac{1}{{\mathrm{n}}^2}}\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{{-\mathsf{\pi }}^2\times {\mathrm{D}}_{\mathrm{e}\mathrm{f}\mathrm{f}}\times \mathrm{t}}{4{\mathrm{L}}^2}}\right)$$

(15)

The diffusion coefficients are typically determined by plotting experimental drying data in terms of ln (MR) versus drying time (t) because the plot gives a straight line with a slope as $\left[{\pi }^2{\displaystyle \frac{D_{eff}}{R^2}}\right]$ [49,52,53,54,55]:

$$\mathrm{M}\mathrm{R}={\displaystyle \frac{8}{{\mathsf{\pi }}^2}}\times \mathrm{A} \mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{-{\mathsf{\pi }}^2\times {\mathrm{D}}_{\mathrm{e}\mathrm{f}\mathrm{f}}\times \mathrm{t}}{4{\mathrm{L}}^2}}\right)$$

(16)

Also, Equation (17) has been obtained mathematically from Equation (16),

$$\ln \left(\mathrm{M}\mathrm{R}\right)=\ln \left({\displaystyle \frac{8}{{\mathsf{\pi }}^2}}\right)-\left({\displaystyle \frac{{\mathsf{\pi }}^2\times {\mathrm{D}}_{\mathrm{e}\mathrm{f}\mathrm{f}}\times \mathrm{t}}{4{\mathrm{L}}^2}}\right)$$

(17)

Finding the diffusion coefficient involves plotting experimental drying data in terms of ln(MR) versus time (s).

2.3.4. Activation Energy (AE)

AE is vital in drying as it represents the minimum energy needed to remove moisture. It helps optimize drying conditions, ensuring efficiency and preventing damage to heat-sensitive materials. Understanding drying kinetics enables the scaling of processes and ensures product quality is maintained. Additionally, it reduces energy consumption and operational costs. The AE was also found using the Arrhenius law, which is carried out in the same manner as the diffusion coefficient [56].

$${\mathrm{D}}_{\mathrm{e}\mathrm{f}\mathrm{f}}={\mathrm{D}}_0\exp \left(-{\displaystyle \frac{{\mathrm{E}}_{\mathrm{a}}}{\mathrm{R}\mathrm{T}}}\right)$$

(18)

2.3.5. Mathematical Modeling (MM)

The experimental data collected from drying marjoram leaves under varying DTs and OPs were analyzed using ten established thin-layer drying models, as outlined in Table 2. To determine the best-fitting models, non-linear regression analysis was conducted in Microsoft Excel (version 2016), allowing for the estimation of model-specific parameters along with key statistical indicators, including the root mean square error (RMSE), coefficient of determination (R²), and adjusted coefficient of determination ($R_{adj.}^2$), as defined in Equations (19)–(21). The selection of the most suitable model was based on the combined criteria of minimizing RMSE while maximizing both R² and $R_{adj.}^2$, in accordance with methodologies reported in previous studies.

Table 2. List of mathematical models used for marjoram leaves at different levels of OPs and DTs [57,58,59].

No.	Model Name	Model Equation *	Refs.
1	Aghbashlo	$MR=\exp \left(-{\displaystyle \frac{k_{1}t}{1+k_{2}t}}\right)$	[60,61]
2	Logarithmic (Asymptotic)	$\mathrm{M}\mathrm{R}=\mathrm{a}\ast \mathrm{e}\mathrm{x}\mathrm{p}\left(-\mathrm{k}\mathrm{t}\right)+c$	[62,63,64]
3	Midilli	$\mathrm{M}\mathrm{R}=\mathrm{a}\ast \mathrm{e}\mathrm{x}\mathrm{p}\left(-\mathrm{k}{\mathrm{t}}^n\right)+bt$
4	Modified Midilli I	$\mathrm{M}\mathrm{R}=\mathrm{e}\mathrm{x}\mathrm{p}\left(-\mathrm{k}{\mathrm{t}}^n\right)+bt$	[65,66]
5	Modified Midilli II	$\mathrm{M}\mathrm{R}=\mathrm{a}\ast \mathrm{e}\mathrm{x}\mathrm{p}\left(-\mathrm{k}{\mathrm{t}}^n\right)+b$	[65]
6	Modified Page	$\mathrm{M}\mathrm{R}=\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\left(\mathrm{k}\mathrm{t}\right)}^{\mathrm{n}}\right)$	[62,63,64]
7	Page	$\mathrm{M}\mathrm{R}=\mathrm{e}\mathrm{x}\mathrm{p}\left(-\mathrm{k}{\mathrm{t}}^{\mathrm{n}}\right)$
8	Wang-Sigh	$MR=1+bt+a{t}^2$
9	Weibullian	$\mathrm{M}\mathrm{R}=\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\left({\displaystyle \frac{t}{\alpha }}\right)}^{\beta }\right)$	[65,66]
10	Weibullian I	$\mathrm{M}\mathrm{R}={10}^{-{\left({\displaystyle \frac{t}{\delta }}\right)}^n}$

* t: drying time, min; k₁, k₂ and k: drying constants, min⁻¹; a, b, c, n, α, β and δ: models constants. All statistical analysis was conducted at p ≤ 0.05.

Table 2. List of mathematical models used for marjoram leaves at different levels of OPs and DTs [57,58,59].

No.	Model Name	Model Equation *	Refs.
1	Aghbashlo	$MR=\exp \left(-{\displaystyle \frac{k_{1}t}{1+k_{2}t}}\right)$	[60,61]
2	Logarithmic (Asymptotic)	$\mathrm{M}\mathrm{R}=\mathrm{a}\ast \mathrm{e}\mathrm{x}\mathrm{p}\left(-\mathrm{k}\mathrm{t}\right)+c$	[62,63,64]
3	Midilli	$\mathrm{M}\mathrm{R}=\mathrm{a}\ast \mathrm{e}\mathrm{x}\mathrm{p}\left(-\mathrm{k}{\mathrm{t}}^n\right)+bt$
4	Modified Midilli I	$\mathrm{M}\mathrm{R}=\mathrm{e}\mathrm{x}\mathrm{p}\left(-\mathrm{k}{\mathrm{t}}^n\right)+bt$	[65,66]
5	Modified Midilli II	$\mathrm{M}\mathrm{R}=\mathrm{a}\ast \mathrm{e}\mathrm{x}\mathrm{p}\left(-\mathrm{k}{\mathrm{t}}^n\right)+b$	[65]
6	Modified Page	$\mathrm{M}\mathrm{R}=\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\left(\mathrm{k}\mathrm{t}\right)}^{\mathrm{n}}\right)$	[62,63,64]
7	Page	$\mathrm{M}\mathrm{R}=\mathrm{e}\mathrm{x}\mathrm{p}\left(-\mathrm{k}{\mathrm{t}}^{\mathrm{n}}\right)$
8	Wang-Sigh	$MR=1+bt+a{t}^2$
9	Weibullian	$\mathrm{M}\mathrm{R}=\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\left({\displaystyle \frac{t}{\alpha }}\right)}^{\beta }\right)$	[65,66]
10	Weibullian I	$\mathrm{M}\mathrm{R}={10}^{-{\left({\displaystyle \frac{t}{\delta }}\right)}^n}$

* t: drying time, min; k₁, k₂ and k: drying constants, min⁻¹; a, b, c, n, α, β and δ: models constants. All statistical analysis was conducted at p ≤ 0.05.

$$R^2=1-{\displaystyle \frac{\sum _{i=1}^N{{(MR}_{pre,i}-{MR}_{obs,i})}^2}{\sum _{i=1}^N{{(\overline{M}R}_{pre}-{MR}_{obs,i})}^2}}$$

(19)

$$R_{adj.}^2=1-\left(1-R^2\right)\ast {\displaystyle \frac{N-1}{N-n}}$$

(20)

$$RMSE=\sqrt{{\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{{(MR}_{pre, i}-{MR}_{obs, i})}^2}$$

(21)

where ${MR}_{obs, i}$ and ${MR}_{pre, i}$ are the ith experimental and predicted values; ${\overline{M}R}_{pre}$ is the average predicted values; N is the number of observations; n is the number of constants in a model [67].

2.3.6. Thermodynamic Parameters

Thermodynamic parameters, including enthalpy (ΔH in J/K.mol), entropy (ΔS in J/K.mol), and Gibbs free energy ($∆G$ in kJ/mol), are essential for comprehending the energy transformations and stability of dehydrated products. Enthalpy represents thermal energy, indicating the energy required or released during the drying process. Entropy quantifies disorder, affecting the microstructure and quality of the product. Gibbs free energy dictates the spontaneity of moisture extraction and the stability of the result. These parameters facilitate the optimization of drying processes, enhance energy efficiency, and preserve the intended physical, chemical, and sensory characteristics of the desiccated product. The thermodynamic parameters, ΔH and ΔS, were calculated using Equation (22) [68,69].

$$\ln \left({\displaystyle \frac{\mathrm{k}}{T}}\right)=\left[\left({\displaystyle \frac{\mathrm{l}\mathrm{n}{\mathrm{k}}_{\mathrm{B}}}{\mathrm{h}}}\right)+\left({\displaystyle \frac{∆\mathrm{S}}{\mathrm{R}}}\right)-\left(\left({\displaystyle \frac{∆\mathrm{H}}{\mathrm{R}}}\right)\times \left({\displaystyle \frac{1}{\mathrm{T}}}\right)\right)\right]$$

(22)

where k_B is the Boltzmann’s constant (1.38065 × 10⁻²³ J/K), and h is the Planck constant (6.62608 × 10⁻³⁴ J.s).

While the $∆G$ was determined for each temperature using Equation (23) [69].

$$∆G=∆H-T∆S$$

(23)

2.3.7. Economic Analysis

The economic assessment of solar dryers is crucial for determining their feasibility and sustainability as an environmentally friendly alternative to traditional drying methods. Evaluating expenses, savings, and payback times helps determine the financial viability of solar dryers compared to traditional drying techniques. This analysis examines key elements, including initial investment, operational costs, energy savings, and maintenance charges. It also highlights the enduring benefits, including reduced fuel reliance and lower greenhouse gas emissions. Ultimately, economic research confirms that solar dryers are both environmentally advantageous and cost-efficient, rendering them accessible to small-scale farmers, companies, and communities pursuing sustainable drying solutions [70,71,72,73]. The economic evaluation of the DVD was performed using Equations (24)–(33), which encompass the key financial indicators relevant to system viability.

Table 3. Calculation assumptions of economic analysis of the DVD based on economic aspects in Libya.

Parameter	Nomenclature	Unit	Value
Interest rate	$d$	%	3%
Maintenance cost	$C_m$	USD/year	3% of the annual capital cost
Salvage value	$V_a$	%	8% of the annual capital cost
Operating life	τ	year	20 years
Inflation rate	$i$	%	2.5%
Drying days per year	$D_d$	day	350
Cost of fresh marjoram leaves	$C_{fd}$	USD/kg	1.0
Selling price of dried marjoram leaves	${SP}_c$	USD/kg	5.0

summarizes the assumptions applied in these calculations, carefully tailored to reflect the prevailing economic conditions and cost structures specific to Libya.

The annual investment cost ($C_a$ in USD/year) of the DVD was calculated using Equation (24).

$$C_a=C_{ac}+C_m-V_a$$

(24)

where $C_{ac}$ is the annual capital cost, is the maintenance cost, and is the salvage cost.

The annual capital cost ($C_{ac}$) of the DVD was estimated using Equations (25) and (26).

$$C_{ac}=C_{cc}\times F_c$$

(25)

$$F_c={\displaystyle \frac{{d(1+d)}^{\tau }}{{(1+d)}^{\tau }-1}}$$

(26)

where $C_{cc}$ is the capital cost and $F_c$ is the recovery factor.

The drying cost per kg of marjoram leaves ($C_s$) dried using the DVD ($C_s$), was calculated using Equation (27) [74].

$$C_s={\displaystyle \frac{C_a}{M_y }}$$

(27)

The amount of dried marjoram leaves using the DVD per year ($M_y$) is calculated using Equation (28),

$$M_y={\displaystyle \frac{M_d\times D}{D_d}}$$

(28)

The cost of one kilogram of dried marjoram leaves ($C_{ds}$) was calculated using Equation (29) [74,75].

$$C_{ds}=C_{dp}+C_s$$

(29)

where $C_{dp}$ is the cost of fresh marjoram leaves per kg of dried product, which is calculated using Equation (30),

$$C_{dp}=C_{fd}\times {\displaystyle \frac{M_f}{M_d}}$$

(30)

where $M_f$ is the quantity of fresh marjoram leaves loaded inside the DVD.

Also, the savings obtained per kg of dried marjoram leaves ($S_{kg}$) are given by Equation (31).

$$S_{kg}={SP}_c-C_{ds}$$

(31)

The savings obtained from the DVD after “j” number of years is given by Equation (32).

$$S_j={\displaystyle \frac{S_{kg}\times M_d}{D}}\times D\times {\left(1+j\right)}^{j-1}$$

(32)

The payback time (Ŧ) for the DVD is calculated using Equation (33) [74,76].

$$\mathrm{Ŧ}={\displaystyle \frac{ln\left[1-\frac{C_{cc}}{S_1}(d-i)\right]}{\ln \left(\frac{1+i}{1+d}\right)}}$$

(33)

where $S_1$ is the savings obtained from the DVD after the first year.

2.4. Uncertainty Analysis

The measurement uncertainties for critical drying parameters were quantified using Equation (34), yielding values of 0.32% for temperature, 0.28% for relative humidity, and 0.19% for operating pressure. Propagating these individual uncertainties through the system efficiency calculations resulted in a combined uncertainty of ±1% for the overall dryer performance evaluation.

$${\mathcal{W}}_r={\left[{\left({\displaystyle \frac{\partial R}{\partial x_1}}{\mathcal{W}}_1\right)}^2+{\left({\displaystyle \frac{\partial R}{\partial x_2}}{\mathcal{W}}_2\right)}^2+\dots +{\left({\displaystyle \frac{\partial R}{\partial x_3}}{\mathcal{W}}_3\right)}^2\right]}^{1/2}$$

(34)

3. Results and Discussion

3.1. Accumulated Weight Loss and Moisture Content of Marjoram Leaves at Different Levels of OPs and DTs

Figure 3 and Figure 4 show the accumulated weight losses and $\mu$ of marjoram leaves at different levels of OPs (atm, −5 kPa, and −10 kPa) and DTs (40, 50, and 60 °C). The accumulated weight loss ranged between 42.92 g and 47.03 g. The average ${\mu }_0$ of all marjoram leaf samples was 825.93% (d.b.). All drying samples were dried until reaching the ${\mu }_e$. Where the final $\mu$ of different marjoram samples were 16.14%, 16.01%, and 21.66%. The combination of DT and OP has a significant impact on drying time. Higher ATs increase evaporation rates, reducing drying time. Additionally, lower OPs decrease the boiling point of water, thereby accelerating drying, particularly for heat-sensitive materials. As shown in Figure 3 and Figure 4, increasing the DT from 40 to 60 °C resulted in a decrease in drying time from 540 to 240 min, 330 to 210 min, and 210 to 120 min at OPs of atm, −5 kPa, and −10 kPa, respectively. This means that the dying time decreased by about 55.6%, 36.4%, and 42.9% when the DT increased from 40 to 60 °C, at OPs of atm, −5 kPa, and −10 kPa, respectively. On the other hand, the illustrated data in the same figures showed that decreasing the OP from atm to −10 kPa resulted in a decrease in drying time from 540 to 210 min, 330 to 180 min, and 240 to 120 min at DTs of 40, 50, and 60 °C, respectively. This means that the dying time decreased by about 58.8%, 45.5%, and 50% when the OP decreased from atm to −10 kPa at DTs of 40, 50, and 60 °C, respectively.

For medicinal plants (such as marjoram), lower drying temperatures are typically preferred to preserve their active compounds, which can degrade at higher temperatures. Moderate to low pressures may also be used to enhance drying efficiency without compromising the plant’s medicinal properties. Thus, a combination of controlled, lower DTs and optimized OP is ideal for drying medical plants [27,77,78].

Table 4. Drying constant (k) and coefficient determination (R²) at different levels of Ops and DTs.

Coefficient	atm			−5 kPa			−10 kPa
	40 °C	50 °C	60 °C	40 °C	50 °C	60 °C	40 °C	50 °C	60 °C
k, min−1	0.007	0.011	0.016	0.011	0.014	0.018	0.016	0.021	0.033
R2	0.990	0.992	0.993	0.992	0.996	0.994	0.99	0.993	0.989

displays the drying coefficient (k) and determination coefficient (R²) for marjoram leaves under different OPs and DTs. The information in the same table showed a direct link between DT and the drying constant (k). The (k) values increased from 0.007 to 0.016, 0.011 to 0.018, and 0.016 to 0.033 as DT increased from 40 to 50 °C at 1, 5, and 10 kPa. This trend aligns with previous studies [17,79,80,81]. The tabulated values of (R²) showed that the maximum value is 0.996, and it was observed at DT of 50 °C and OP of −5 kPa. While the lowest value of (R²) is 0.989, it was observed at DT of 60 °C and OP of −10 kPa.

3.2. MR of Marjoram Leaves at Different Levels of OPs and DTs

Figure 5 presents the fluctuation of MR for different marjoram leaf samples at different levels of OPs and DTs versus drying time. The MR of marjoram leaves was elevated throughout all operational periods, OPs, and DTs. The elevated free water content in marjoram leaf samples, coupled with the intensified thermal energy associated with shorter drying durations, accelerates the depletion of moisture retention. Interestingly, as the DT decreases, moisture evaporation from the leaves becomes more sustained rather than abrupt, indicating a complex interplay between thermal input and internal moisture migration. The dehydration process of high-moisture-content agricultural products, such as marjoram leaves, generally unfolds in two distinct phases. In the initial phase, rapid moisture removal occurs primarily due to the evaporation of surface water, which is readily accessible and requires minimal energy to be removed. A portion of the supplied thermal energy is used to vaporize the surface-bound moisture while the remaining energy diffuses into the internal structure of the leaf tissue, thereby increasing its temperature. During this period, capillary forces within the leaf matrix actively drive the movement of internal water toward the surface through capillary action. Concurrently, a mass transfer process takes place at the leaf-air interface, where the accumulated moisture is released into the surrounding environment. This two-stage drying mechanism ensures a systematic reduction in both surface and internal moisture content. However, the overall rate and efficiency of dehydration are significantly influenced by multiple interrelated factors, including the uniformity of heat distribution, the moisture concentration gradient, and the anatomical characteristics of the leaf tissue. These parameters collectively govern the drying kinetics and determine the effectiveness of the drying process. The decrease in OPs leads to a swifter achievement of ${\mu }_e$ and a shorter drying period. The drying process was accelerated by the minimal MR, which enhanced moisture removal and reduced the drying duration [82,83,84]. Furthermore, the observed MR curves exhibited a pronounced rapid decline in MR during the initial phase of the drying process, corroborating the findings of Ambawat et al. [85].

3.3. DR of Marjoram Leaves at Different Levels of OPs and DTs

Figure 6 illustrates the DR curves of marjoram leaves at different levels of OPs (atm, −5 kPa, and −10 kPa) and DTs (40, 50, and 60 °C). Also, Figure 7 illustrates the correlation between the $\mu$ and the DR of marjoram leaves at different levels of OPs and DTs. According to the plotted data in Figure 6 and Figure 7, it can be observed that the maximum DR was 0.034 g_water/g_{dry matter}.min, and it was observed at OP of −10 kPa and DT of 60 °C. Additionally, increasing the DT from 40 to 60 °C led to increasing the DR from 0.012 to 0.024 g_water/g_{dry matter}.min, 0.015 to 0.025 g_water/g_{dry matter}.min, and 0.024 to 0.024 g_water/g_{dry matter}.min, at OPs of atm, −5 kPa, and −10 kPa, respectively. On the other hand, the same figures showed that lowering the OP from atm to −10 kPa caused the DR to rise from 0.012 to 0.024 g_water/g_{dry matter}.min at 40 °C, 0.015 to 0.029 g_water/g_{dry matter}.min at 50 °C, and 0.024 to 0.034 g_water/g_{dry matter}.min at 60 °C. This means that increasing the OP and DT had a significant impact on maximizing the DR during the drying process of marjoram leaf samples.

The findings of the current study are consistent with those reported in several previous investigations, including the work of Elshehawy and Mosad [86], who highlighted that increasing the amount of AT significantly accelerates moisture migration from the interior of the drying material to its surface. This enhanced internal moisture diffusion facilitates more rapid evaporation at the product’s surface, ultimately resulting in a higher overall drying rate. Similarly, Darvishi et al. [87] observed that the DR tends to decline over time or as the $\mu$ of the product decreases. In the initial phase of drying, the rate is at its peak due to the efficient removal of unbound (free) moisture, particularly from the surface and outer layers of the henna leaves. However, as drying progresses, the remaining moisture—primarily bound water—requires more energy and time to be removed, leading to a gradual reduction in the drying rate. This transition reflects a shift from the constant rate period to the falling rate period, which dominated the drying behavior across all tested layer thicknesses. The constant rate period, if present, was brief and occurred at the very beginning of the process, indicating limited surface moisture availability. The majority of the drying process occurred during the falling rate period, where internal diffusion became the controlling mechanism. Therefore, internal mass transfer resistance emerged as the key limiting factor governing the drying duration, especially during the latter stages of drying. Moreover, as supported by Elshehawy and Mosad [86], the rate of MR reduction was relatively slower at the beginning of the drying process and accelerated towards the end. This pattern highlights the dynamic nature of internal moisture gradients and the evolving resistance to moisture movement as drying proceeds [86,87].

3.4. Moisture Diffusivity (D_eff) of Marjoram Leaves at Different Levels of OPs and DTs

Figure 8 presents the relation between ln (MR) and drying time, where the relationship between the natural logarithm of the MR (ln (MR)) and drying time is crucial for understanding and modeling the drying kinetics of materials. This relation yielded a linear relationship. Where the drying occurred during the falling rate period of the drying process. The slope of the ln(MR) vs. time plot provides the DR constant (k), a key parameter for characterizing the drying process. A higher (k) indicates faster drying. Also, the linear relationship allows for the prediction of the drying time required to achieve a specific $\mu$. The D_eff for basil leaves was determined by applying Fick’s second law of diffusion (Equations (7)–(17)). Then, the data obtained from the D_eff were plotted against time (Figure 9). As shown in Figure 9, the D_eff increased by increasing the DT from 40 to 60 °C, led to increasing the D_eff from 2.13 to 2.63 × 10⁻⁹ m²/s, 1.87 to 3.06 × 10⁻⁹ m²/s, and 2.62 to 5.51 × 10⁻⁹ m²/s, at OPs of atm, −5 kPa, and −10 kPa, respectively. This finding aligns with previous studies, which have shown that increasing DT generally enhances D_eff, as higher DT increases the kinetic energy of water molecules, thereby facilitating their movement through the material. This accelerates DRs but must be balanced to prevent damage to heat-sensitive products [88,89,90]. On the other hand, the illustrated data in the same figures showed that decreasing the OP from atm to −10 kPa led to increasing the D_eff from 1.13 to 2.62 × 10⁻⁹ m²/s, 1.87 to 2.53 × 10⁻⁹ m²/s, and 2.63 to 5.51 × 10⁻⁹ m²/s, at DTs of 40, 50, and 60 °C, respectively. This phenomenon is because reducing OP under atmospheric pressure increases D_eff by lowering the boiling point of water and reducing resistance to moisture movement. This enhances DRs, especially for heat-sensitive materials [91,92].

Table 5. Comparison between the obtained D_eff with previous drying studies for some medical and leafy products.

Ref.	Drying System	Product	Deff, m2/s
[50]	Convection oven	sage leaves	1.1 to 3.7 × 10−12
[93]	Microwave	sage leaves	1.73 × 10−9
[94]	Solar cabinet drier	sage leaves	1.62 to 5.73 × 10−9
[95]	Hot air convective drying	Marjoram leaves	1.7 to 8.75 × 10−10
[95]	Microwave drying	Marjoram leaves	2.915 to 2.964 × 10−9
[96]	Microwave	Purple basil	0.162 to 7.09 × 10−8
Current study	DVD	Marjoram leaves	1.13 to 5.51 × 10−9

illustrates a comparison between the obtained D_eff with previous drying studies for some medical and leafy products.

3.5. Activating Energy (AE) of Marjoram Leaves at Different Levels of OPs and DTs

The AE can be computed using an approach analogous to that employed for determining the D_eff. Utilized the previously obtained D_eff value to construct linear correlations between them and the inverse of the absolute drying temperature data, as illustrated in Figure 10. Consequently, the AE is determined by calculating the slope of the linear correlation, as shown in Figure 11. Figure 11 illustrates that the AE values for Ops at atm, −5 kPa, and −10 kPa were 21.33 kJ/mol, 32.21 kJ/mol, and 2.68 kJ/mol, respectively. The findings of this investigation are deemed plausible, supported by several pertinent literature sources that indicate the AE of vegetables and other plant-derived materials ranges from 12.7 to 110 mol/K [7,39,90].

Table 6. A comparative analysis between the AE obtained in this study and those reported in prior research on the drying behavior of medicinal herbs and leafy products.

Ref.	Drying System	Product	Drying Conditions	AE, kJ/mol
[97]	Sun and vacuum dryers	Basil leaves	45, 55, and 65 °C	38.54–20.32
[91]	Oven drier	Basil leaves	30–70 °C	32.34
[23]	Hybrid solar dryer	Basil leaves	50, 55, and 60 °C	1.945–15.37
[95]	Hot air convective drying	Marjoram leaves	50, 60, 70 and 80 °C	1.055
[95]	Microwave drying	Marjoram leaves		4.85
[98]	Hot air convective drying	Rue leaves	40, 50, 60 and 70 °C	60.58
Current study	DVD	Marjoram leaves	OP of −10kPa and 60 °C	2.68

compares the reported AE with prior drying investigations of several medicinal and leafy goods.

3.6. Mathematical Modeling of Marjoram Leaves at Different Levels of OPs and DTs

Table 7. Model constants and statistical goodness-of-fit parameters for marjoram leaves dried under varying operating pressures and drying temperatures [57,58,59].

MMs	Pressure	T, °C	Parameters	Models’ Constants Values			Goodness of Fit Indices
				Values	S.E.	p-Value	RMSE	R2	R2adj.
Aghbashlo	atm	40	k1	0.01252	0.00028	1.14 × 10−32 *	0.01408	0.99656	0.99647
			k2	0.00205	0.00015	2.74 × 10−15 *
		50	k1	0.02011	0.00053	8.82 × 10−21 *	0.01331	0.99738	0.99726
			k2	0.00304	0.00029	1.09 × 10−9 *
		60	k1	0.01086	0.00276	0.00134 *	0.10870	0.84058	0.82996
			k2	−0.00417	0.00197	0.05133
	−5 kPa	40	k1	0.01979	0.00055	1.98 × 10−20 *	0.01389	0.99717	0.99704
			k2	0.00294	0.00030	3.10 × 10−9 *
		50	k1	0.02269	0.00074	2.28 × 10−16 *	0.01500	0.99692	0.99674
			k2	0.00320	0.00041	5.53 × 10−7 *
		60	k1	0.03054	0.00097	1.18 × 10−13 *	0.01282	0.99798	0.99783
			k2	0.00424	0.00054	2.52 × 10−6 *
	−10 kPa	40	k1	0.02789	0.00104	9.45 × 10−13 *	0.01531	0.99703	0.99680
			k2	0.00466	0.00060	3.21 × 10−6 *
		50	k1	0.01028	0.00434	0.03712 *	0.16765	0.67538	0.64586
			k2	−0.00556	0.00323	0.11372
		60	k1	0.05117	0.00139	2.83 × 10−9 *	0.00811	0.99944	0.99936
			k2	0.00552	0.00076	0.00017 *
Logarithmic (Asymptotic)	atm	40	k	0.01039	0.00040	3.89 × 10−24 *	0.02369	0.99056	0.99000
			a	0.90502	0.01629	5.84 × 10−35 *
			c	0.04675	0.00674	5.45 × 10−8 *
		50	k	0.01740	0.00078	1.30 × 10−15 *	0.02341	0.99229	0.99152
			a	0.92787	0.01920	3.45 × 10−22 *
			c	0.04477	0.00822	2.48 × 10−5 *
		60	k	0.02317	0.00141	1.54 × 10−10 *	0.02773	0.99032	0.98893
			a	0.91425	0.02482	2.44 × 10−15 *
			c	0.04259	0.01151	0.00238 *
	−5 kPa	40	k	0.01723	0.00078	1.43 × 10−15 *	0.02356	0.99225	0.99148
			a	0.93012	0.01928	3.55 × 10−22 *
			c	0.04447	0.00833	3.19 × 10−5 *
		50	k	0.01914	0.00096	1.05 × 10−12 *	0.02380	0.99269	0.99178
			a	0.92107	0.02028	2.43 × 10−18 *
			c	0.03734	0.00995	0.00174 *
		60	k	0.02632	0.00138	2.40 × 10−10 *	0.02316	0.99392	0.99291
			a	0.93307	0.02147	1.43 × 10−14 *
			c	0.03776	0.01024	0.00311 *
	−10 kPa	40	k	0.02383	0.00144	1.30 × 10−9 *	0.02567	0.99229	0.99101
			a	0.91080	0.02345	5.48 × 10−14 *
			c	0.05359	0.01232	0.00094 *
		50	k	0.03247	0.00217	3.54 × 10−8 *	0.02804	0.99174	0.99009
			a	0.93049	0.02723	1.09 × 10−11 *
			c	0.04240	0.01274	0.00763 *
		60	k	0.04682	0.00211	5.54 × 10−7 *	0.01688	0.99793	0.99724
			a	0.96304	0.01789	2.76 × 10−9 *
			c	0.03076	0.00939	0.01687 *
Midilli	atm	40	k	0.03067	0.00345	2.82 × 10−10 *	0.01398	0.99681	0.99652
			a	1.01610	0.01344	1.44 × 10−38 *
			b	6.36 × 10−6	1.43 × 10−5	0.66019
			n	0.76226	0.02257	3.57 × 10−27 *
		50	k	0.03936	0.00521	3.84 × 10−7 *	0.01528	0.99688	0.99639
			a	1.01138	0.01505	4.64 × 10−24 *
			b	2.33 × 10−5	3.04 × 10−5	0.45260
			n	0.79242	0.03008	2.02 × 10−16 *
		60	k	0.06541	0.00196	5.42 × 10−14 *	0.00333	0.99987	0.99984
			a	1.00064	0.00331	2.14 × 10−26 *
			b	−6.56 × 10−5	1.25 × 10−5	0.00015 *
			n	0.72015	0.00752	6.58 × 10−20 *
	−5 kPa	40	k	0.03818	0.00537	9.14 × 10−7 *	0.01615	0.99654	0.99599
			a	1.01224	0.01589	1.29 × 10−23 *
			b	2.41 × 10−5	3.22 × 10−5	0.46377
			n	0.79770	0.03191	5.33 × 10−16 *
		50	k	0.04882	0.00220	6.94 × 10−13 *	0.00497	0.99970	0.99964
			a	1.00227	0.00492	3.11 × 10−27 *
			b	−5.04 × 10−5	1.54 × 10−5	0.00506 *
			n	0.75988	0.01081	2.59 × 10−20 *
		60	k	0.05976	0.00279	2.54 × 10−10 *	0.00492	0.99975	0.99968
			a	1.00092	0.00490	4.81 × 10−21 *
			b	−3.47 × 10−5	2.06 × 10−5	0.11987
			n	0.76883	0.01209	1.80 × 10−15 *
	−10 kPa	40	k	0.06102	0.00260	9.67 × 10−11 *	0.00440	0.99979	0.99974
			a	1.00056	0.00438	1.42 × 10−21 *
			b	−5.22 × 10−5	2.24 × 10−5	0.03947 *
			n	0.73534	0.01102	1.06 × 10−15 *
		50	k	0.08172	0.00472	3.25 × 10−8 *	0.00591	0.99967	0.99956
			a	1.00050	0.00590	4.38 × 10−17 *
			b	−6.72 × 10−5	3.27 × 10−5	0.07026
			n	0.72398	0.01586	5.80 × 10−12 *
		60	k	0.07523	0.00922	0.00045 *	0.01009	0.99938	0.99901
			a	1.00065	0.01009	1.97 × 10−9 *
			b	3.88 × 10−5	8.42 × 10−5	0.66440
			n	0.83701	0.03722	3.23 × 10−6 *
Modified Midilli I	atm	40	k	0.02810	0.00241	1.98 × 10−13 *	0.01409	0.99666	0.99646
			b	9.65 × 10−6	1.40 × 10−5	0.49439
			n	0.77757	0.01868	9.33 × 10−31 *
		50	k	0.03728	0.00407	1.37 × 10−8 *	0.01513	0.99678	0.99646
			b	2.63 × 10−5	2.96 × 10−5	0.38614
			n	0.80265	0.02646	3.37 × 10−18 *
		60	k	0.06523	0.00167	1.07 × 10−15 *	0.00321	0.99987	0.99985
			b	−6.53 × 10−5	1.19 × 10−5	8.33 × 10−5 *
			n	0.72069	0.00676	8.90 × 10−22 *
	−5 kPa	40	k	0.03600	0.00418	3.60 × 10−8 *	0.01600	0.99643	0.99607
			b	2.72 × 10−5	3.13 × 10−5	0.39543
			n	0.80879	0.02805	9.11 × 10−18 *
		50	k	0.04830	0.00181	1.09 × 10−14 *	0.00484	0.99970	0.99966
			b	−4.95 × 10−5	1.48 × 10−5	0.00418 *
			n	0.76197	0.00954	3.04 × 10−22 *
		60	k	0.05953	0.00239	1.07 × 10−11 *	0.00471	0.99975	0.99971
			b	−3.43 × 10−5	1.96 × 10−5	0.10595
			n	0.76963	0.01088	4.22 × 10−17 *
	−10 kPa	40	k	0.06086	0.00221	3.28 × 10−12 *	0.00421	0.99979	0.99976
			b	−5.19 × 10−5	2.13 × 10−5	0.03123 *
			n	0.73585	0.00986	2.23 × 10−17 *
		50	k	0.08157	0.00416	2.59 × 10−9 *	0.00561	0.99967	0.99960
			b	−6.69 × 10−5	3.10 × 10−5	0.05589
			n	0.72437	0.01446	2.43 × 10−13 *
		60	k	0.07507	0.00808	8.80 × 10−5 *	0.00922	0.99938	0.99918
			b	3.91 × 10−5	7.68 × 10−5	0.62945
			n	0.83750	0.03330	2.60 × 10−7 *
Modified Midilli II	atm	40	k	0.02947	0.00343	6.36 × 10−10 *	0.01384	0.99687	0.99659
			a	1.00589	0.01752	1.19 × 10−34 *
			b	0.00870	0.00857	0.31740
			n	0.77400	0.02544	1.01 × 10−25 *
		50	k	0.03800	0.00517	5.75 × 10−7 *	0.01502	0.99699	0.99651
			a	0.99769	0.01955	8.47 × 10−22 *
			b	0.01286	0.01056	0.23803
			n	0.80595	0.03377	1.26 × 10−15 *
		60	k	0.06525	0.00211	1.47 × 10−13 *	0.00363	0.99985	0.99981
			a	1.02064	0.00618	5.50 × 10−23 *
			b	−0.01995	0.00459	0.00079 *
			n	0.71368	0.00944	1.41 × 10−18 *
	−5 kPa	40	k	0.03681	0.00533	1.38 × 10−6 *	0.01589	0.99665	0.99612
			a	0.99812	0.02067	2.41 × 10−21 *
			b	0.01328	0.01114	0.24803
			n	0.81166	0.03583	3.27 × 10−15 *
		50	k	0.04863	0.00230	1.40 × 10−12 *	0.00521	0.99967	0.99961
			a	1.01847	0.00837	6.98 × 10−24 *
			b	−0.01622	0.00583	0.01396 *
			n	0.75558	0.01287	3.84 × 10−19 *
		60	k	0.05955	0.00289	3.89 × 10−10 *	0.00505	0.99973	0.99966
			a	1.00884	0.00792	8.68 × 10−19 *
			b	−0.00796	0.00558	0.18177
			n	0.76697	0.01422	1.10 × 10−14 *
	−10 kPa	40	k	0.06059	0.00266	1.30 × 10−10 *	0.00460	0.99977	0.99971
			a	1.01437	0.00890	2.96 × 10−18 *
			b	−0.01383	0.00708	0.07678
			n	0.73255	0.01331	8.76 × 10−15 *
		50	k	0.08165	0.00480	3.80 × 10−8 *	0.00603	0.99966	0.99954
			a	1.01564	0.01058	7.32 × 10−15 *
			b	−0.01513	0.00822	0.09880
			n	0.71824	0.01879	2.86 × 10−11 *
		60	k	0.07411	0.00944	0.0005 *	0.00997	0.99940	0.99904
			a	0.99376	0.01523	1.60 × 10−8 *
			b	0.00682	0.01107	0.56466
			n	0.84506	0.04271	6.09 × 10−6 *
Modified Page	atm	40	k	0.01006	0.00014	8.90 × 10−40 *	0.01398	0.99661	0.99652
			n	0.76840	0.01267	4.82 × 10−37 *
		50	k	0.01645	0.00031	6.78 × 10−24 *	0.01503	0.99666	0.99650
			n	0.78635	0.01816	5.05 × 10−22 *
		60	k	0.02301	0.00020	1.72 × 10−23 *	0.00562	0.99957	0.99955
			n	0.74939	0.00783	2.54 × 10−22 *
	−5 kPa	40	k	0.01625	0.00032	1.68 × 10−23 *	0.01589	0.99630	0.99612
			n	0.79189	0.01924	1.45 × 10−21 *
		50	k	0.01902	0.00016	2.56 × 10−26 *	0.00619	0.99947	0.99944
			n	0.78615	0.00821	1.12 × 10−24 *
		60	k	0.02576	0.00021	2.57 × 10−21 *	0.00509	0.99968	0.99966
			n	0.78384	0.00797	4.63 × 10−20 *
	−10 kPa	40	k	0.02257	0.00017	1.08 × 10−21 *	0.00499	0.99968	0.99966
			n	0.75523	0.00719	1.97 × 10−20 *
		50	k	0.03175	0.00039	1.21 × 10−16 *	0.00654	0.99951	0.99946
			n	0.74856	0.01109	9.38 × 10−16 *
		60	k	0.04529	0.00087	2.48 × 10−10 *	0.00871	0.99936	0.99927
			n	0.82524	0.02145	2.08 × 10−9 *
Page	atm	40	k	0.02918	0.00187	2.43 × 10−17 *	0.01398	0.99661	0.99652
			n	0.76840	0.01267	4.82 × 10−37 *
		50	k	0.03957	0.00327	6.38 × 10−11 *	0.01503	0.99666	0.99650
			n	0.78635	0.01816	5.05 × 10−22 *
		60	k	0.05921	0.00198	8.70 × 10−15 *	0.00562	0.99957	0.99955
			n	0.74940	0.00783	2.54 × 10−22 *
	−5 kPa	40	k	0.03829	0.00336	1.89 × 10−10 *	0.01589	0.99630	0.99612
			n	0.79189	0.01924	1.45 × 10−21 *
		50	k	0.04438	0.00160	1.42 × 10−15 *	0.00619	0.99947	0.99944
			n	0.78615	0.00821	1.12 × 10−24 *
		60	k	0.05682	0.00187	1.83 × 10−13 *	0.00509	0.99968	0.99966
			n	0.78384	0.00797	4.63 × 10−20 *
	−10 kPa	40	k	0.05709	0.00174	7.14 × 10−14 *	0.00499	0.99968	0.99966
			n	0.75523	0.00719	1.97 × 10−20 *
		50	k	0.07559	0.00334	1.39 × 10−10 *	0.00654	0.99951	0.99946
			n	0.74856	0.01109	9.38 × 10−16 *
		60	k	0.07778	*	3.60 × 10−6 *	0.00871	0.99936	0.99927
			n	0.82524	0.02145	2.08 × 10−9 *
Wang-Sigh	atm	40	b	−0.00542	0.00023	3.29 × 10−23 *	0.10854	0.79599	0.79017
			a	7.17 × 10−6	5.36 × 10−7	2.56 × 10−15 *
		50	b	−0.00887	0.00047	1.16 × 10−14 *	0.10825	0.82697	0.81874
			a	1.91 × 10−5	1.80 × 10−6	6.31 × 10−10 *
		60	b	−0.01206	0.00077	1.00 × 10−10 *	0.11102	0.83370	0.82262
			a	3.54 × 10−5	4.01 × 10−6	2.49 × 10−7 *
	−5 kPa	40	b	−0.00884	0.00046	9.80 × 10−15 *	0.10697	0.83232	0.82433
			a	1.90 × 10−5	1.77 × 10−6	5.62 × 10−10 *
		50	b	−0.01056	0.00059	1.79 × 10−12 *	0.10133	0.85925	0.85098
			a	2.73 × 10−5	2.74 × 10−6	1.66 × 10−8 *
		60	b	−0.01379	0.00090	1.02 × 10−9 *	0.10686	0.85975	0.84896
			a	4.61 × 10−5	5.33 × 10−6	9.41 × 10−7 *
	−10 kPa	40	b	−0.01307	0.00083	7.89 × 10−10 *	0.09918	0.87531	0.86572
			a	4.28 × 10−5	4.95 × 10−6	9.51 × 10−7 *
		50	b	−0.01619	0.00121	3.66 × 10−8 *	0.11488	0.84757	0.83371
			a	6.33 × 10−5	8.32 × 10−6	1.05 × 10−5 *
		60	b	−0.02413	0.00203	6.69 × 10−6 *	0.10756	0.90214	0.88816
			a	0.00014	2.06 × 10−5	0.00026 *
Weibullian	atm	40	β	0.76840	0.01267	4.82 × 10−37 *	0.01398	0.99661	0.99652
			α	99.432	1.36805	8.90 × 10−40 *
		50	β	0.78635	0.01816	5.05 × 10−22 *	0.01503	0.99666	0.99650
			α	60.78	1.14099	6.78 × 10−24 *
		60	β	0.74940	0.00783	2.54 × 10−22 *	0.00562	0.99957	0.99955
			α	43.465	0.37919	1.72 × 10−23 *
	−5 kPa	40	β	0.79189	0.01924	1.45 × 10−21 *	0.01589	0.99630	0.99612
			α	61.554	1.20705	1.68 × 10−23 *
		50	β	0.78615	0.00821	1.12 × 10−24 *	0.00619	0.99947	0.99944
			α	52.576	0.43929	2.56 × 10−26 *
		60	β	0.78384	0.00797	4.63 × 10−20 *	0.00509	0.99968	0.99966
			α	38.816	0.31574	2.57 × 10−21 *
	−10 kPa	40	β	0.75523	0.00719	1.97 × 10−20 *	0.00499	0.99968	0.99966
			α	44.304	0.33727	1.08 × 10−21 *
		50	β	0.74856	0.01109	9.38 × 10−16 *	0.00654	0.99951	0.99946
			α	31.494	0.38715	1.21 × 10−16 *
		60	β	0.82524	0.02145	2.08 × 10−9 *	0.00871	0.99936	0.99927
			α	22.08	0.42294	2.48 × 10−10 *
Weibullian I	atm	40	n	0.76840	0.01267	4.82 × 10−37 *	0.01398	0.99661	0.99652
			δ	294.385	4.85876	4.94 × 10−37 *
		50	n	0.78635	0.01816	5.05 × 10−22 *	0.01503	0.99666	0.99650
			δ	175.54	3.95367	3.01 × 10−22 *
		60	n	0.74940	0.00783	2.54 × 10−22 *	0.00562	0.99957	0.99955
			δ	132.278	1.349	1.77 × 10−22 *
	−5 kPa	40	n	0.79189	0.01924	1.45 × 10−21 *	0.01589	0.99630	0.99612
			δ	176.467	4.17165	8.22 × 10−22 *
		50	n	0.78615	0.00821	1.12 × 10−24 *	0.00619	0.99947	0.99944
			δ	151.891	1.5382	6.70 × 10−25 *
		60	n	0.78384	0.00797	4.63 × 10−20 *	0.00509	0.99968	0.99966
			δ	112.489	1.083	2.29 × 10−20 *
	−10 kPa	40	n	0.7552	0.0072	1.97 × 10−20 *	0.00499	0.99968	0.99966
			δ	133.675	1.2503	1.57 × 10−20 *
		50	n	0.74856	0.01109	9.38 × 10−16 *	0.00654	0.99951	0.99946
			δ	95.967	1.33706	4.77 × 10−16 *
		60	n	0.82524	0.02145	2.08 × 10−9 *	0.00871	0.99936	0.99927
			δ	60.661	1.323	6.14 × 10−10 *

MMs: mathematical models; k₁, k_2, and k: drying constants, min⁻¹; a, b, c, n, α, β and δ: mathematical models constants; S.E.: Standard error; * significant at p ≤ 0.05.

presents the results of fitting experimental drying data for marjoram leaves, obtained under various DTs and OPs, to ten different thin-layer drying models. The effectiveness of each model was checked using common statistical measures, such as the RMSE, the R², and the $R_{adj.}^2$ Among all the models assessed, the Modified Midilli (I) model consistently demonstrated superior performance across all experimental conditions. It exhibited the lowest RMSE values, indicating minimal deviation between the predicted and observed MR values. Additionally, it achieved the highest R² and R²_adj values, reflecting a strong correlation and a better model fit compared to the other models. Based on these statistical indicators—namely, the minimal RMSE and the maximum values of R² and $R_{adj.}^2$—the Modified Midilli (I) model was identified as the most accurate and reliable for describing the drying behavior of marjoram leaves under the tested conditions. To further validate this finding, a comparison between the predicted and experimental MR values over time was graphically illustrated in Figure 12, demonstrating a strong agreement and reinforcing the suitability of the model. The selection of the best-fit model was guided by well-established model selection criteria, emphasizing high goodness-of-fit indices and low estimation error, as recommended in previous studies [57,58,59]. These findings suggest that the Modified Midilli (I) model is well-suited for predicting moisture reduction kinetics in marjoram drying processes and can be effectively applied for process optimization and design in similar herb-drying systems.

3.7. Thermodynamic Properties of Marjoram Leaves at Different Levels of OPs and DTs

Table 8. Thermodynamic properties of marjoram leaves at different levels of OPs and DTs.

Temperature	Pressure	ΔH, kJ/mol	ΔS, KJ/mol.K	ΔG, kJ/mol
40 °C	atm	18.73	−0.113	54.12
	−5 kPa	29.61	−0.150	76.72
	−10 kPa	0.08	−0.156	48.85
50 °C	atm	18.64	−0.113	55.18
	−5 kPa	29.53	−0.151	78.15
	−10 kPa	−0.002	−0.156	50.34
60 °C	atm	18.56	−0.113	56.24
	−5 kPa	29.44	−0.151	79.59
	−10 kPa	−0.09	−0.156	51.83

shows the thermodynamic properties of marjoram leaves at different levels of OPs and DTs. Where it includes enthalpy (ΔH), entropy (ΔS), and Gibbs free energy (ΔG). According to enthalpy (ΔH), the data presented showed that it increased with increasing DT, and the highest values were observed at a DT of 60 °C, while the lowest was observed at 40 °C. Where higher DTs increase enthalpy, providing more energy to evaporate moisture from marjoram leaves. On the other hand, based on the OP, the lowest values of enthalpy were detected at an OP of −10 kPa, and the highest value was detected at an OP of −5 kPa. Enthalpy pertains to the energy required to eliminate the water associated with dry matter during drying, thereby decreasing with elevated drying temperatures [99,100]. Low enthalpy values at reduced temperatures signify an increased energy demand for the drying of marjoram leaves; analogous behavior was noted in the drying processes of Baru fruits examined by [101], Bode pepper grains investigated by Rodovalho et al. [102], and tamarind seeds analyzed by Ferreira Junior et al. [103]. According to the entropy (ΔS) data, the presented results showed that it increased with increasing OP, with the highest values observed at an OP of −10 kPa and the lowest at atmospheric pressure. On the other hand, based on the DT, no change was observed. Positive or negative entropy values indicate changes in the structural and energetic states of the material during the drying process. The negative entropy values observed suggest structural reordering or chemical adsorption phenomena, indicating a more ordered transition state between bound and free moisture molecules. A higher (less negative) entropy at lower pressures may reflect greater molecular mobility and randomness in the system, contributing to faster internal moisture diffusion and enhanced drying performance [104]. All values of Gibbs free energy (ΔG) were positive, indicating that the drying process is non-spontaneous and requires continuous input of external energy (i.e., heat). This thermodynamic requirement emphasizes that moisture removal from marjoram leaves is an endergonic process in which energy must be supplied to drive water from the interior of the leaves into the surrounding environment. Higher ΔG values correspond to more energy-demanding conditions and lower drying rates, whereas lower ΔG values (often observed at higher temperatures and vacuum conditions) reflect more efficient energy use and accelerated drying kinetics [105]. Similar trends have been observed in studies [100,106,107].

3.8. Economic Analysis

The primary goal of the economic analysis undertaken in this study was to assess the cost-effectiveness and financial feasibility of utilizing a DVD for processing marjoram leaves under various drying temperatures (DTs) and operating pressures (OPs). This analysis aimed to determine whether the investment in DVD technology would be economically justifiable, particularly in light of its energy efficiency and impact on product quality. To achieve this, a comprehensive evaluation framework was applied using Equations (24)–(33), which integrated both the cost Savings approach and the Simple Payback Period method. The cost-savings strategy enabled the estimation of long-term economic benefits by accounting for the total cost savings over the system’s operational lifespan. Meanwhile, the SPP provided a quick estimate of how long it would take for the initial investment to be recovered through energy savings or increased product value. Several critical economic parameters and assumptions, outlined in Table 3, were incorporated into the analysis. These included the initial capital cost of the dryer and its components, annual operational and maintenance costs, energy consumption rates, product market value, and system lifespan. Furthermore, the study considered macroeconomic factors relevant to the Libyan context, such as inflation rates, energy prices, and currency fluctuations. By combining technical performance data with detailed economic modeling, this analysis provides valuable insights into the sustainability and scalability of DVD technology for marjoram drying, particularly under the current economic conditions in Libya. The results aim to support informed decision-making for stakeholders seeking energy-efficient and economically viable post-harvest solutions in the herbal processing sector. The capital cost of the DVD was approximately USD 270, and its lifespan was expected to be around 20 years. Additionally, the data from the economic analysis revealed that the salvage cost and maintenance cost were USD 1.45 and USD 0.544 per year, respectively. Additionally, the annualized capital cost was approximately USD 18.15, and the annualized investment cost was USD 17.24.

Table 9. Economic analysis of drying marjoram leaves using the DVD at different levels of OPs and DTs.

Economic Analysis	DT, °C and OP, kPa
	atm			−5 kPa			−10 kPa
	40 °C	50 °C	60 °C	40 °C	50 °C	60 °C	40 °C	50 °C	60 °C
Mass of fresh marjoram leaves per patch, kg	3	3	3	3	3	3	3	3	3
Number of drying cycles per year	388.89	636.36	875	636.36	777.78	1000	1000	1166.67	1750
Mass of dried marjoram leaves annually, kg	1166.7	1909.1	2625	1909.1	2333.3	3000	3000	3500	5250
Drying cost per kg of fresh product, USD	0.54	0.39	0.32	0.39	0.365	0.308	0.308	0.275	0.1977
Cost of marjoram leaves per kg of dried product, USD	1.63	1.48	1.41	1.48	1.46	1.40	1.42	1.37	1.29
Saving after the first year, USD	237.03	543.71	857.29	549.53	698.3	990.93	969.28	1219.88	2054.19
Payback period, year	1.167	0.520	0.332	0.514	0.406	0.287	0.294	0.234	0.139

shows the economic analysis of drying marjoram leaves using the DVD at different levels of OPs and DTs. The findings show that when using the DVD for drying marjoram leaves at different DTs and OPs, the DVD has potential yearly savings of USD 2054.19 when used for drying marjoram leaves at an OP of −10 kPa and a DR of 60 °C. Furthermore, the obtained results of the current study also showed that the payback period was about 0.139 years (about 2 months) under the same operating conditions.

4. Conclusions

During the current study, a vacuum dryer was developed and integrated with a measuring and control electronic unit based on the IoT concept. Then, the developed vacuum dryer (DVD) was used for drying marjoram leaves at three drying temperatures (DTs) of 40, 50, and 60 °C, and three operating pressures (OPs) of (atm) atmospheric, −5 kPa, and −10 kPa. The results obtained were used to select the appropriate mathematical model for describing the drying kinetics of the marjoram leaves. Also, the results were used to estimate the drying kinetics and thermodynamic properties for the cost-effective drying of marjoram leaves under the different drying conditions mentioned above. The results obtained can be concluded as follows:

➢

Drying marjoram leaves at a drying temperature (DT) of 60 °C, and an operating pressure (OP) of −10 kPa resulted in a decrease in drying time by approximately 77.78% and increased the drying rate to 0.034 g_water/g_{dry matter}.min.

➢

The moisture diffusivity (D_eff) ranged between 1.13 × 10⁻⁹ and 5.51 × 10⁻⁹ m²/s, with the highest value observed at an OP of –10 kPa and a temperature difference (DT) of 60 °C.

➢

The activation energy (AE) values were 21.33, 32.21, and 2.68 mol/K at OPs of atm, −5 kPa, and −10 kPa, respectively.

➢

The modified Midilli (I) model was ideal, and it is the fittest mathematical model to describe the drying process of marjoram leaves using the DVD.

➢

Among the thermodynamic parameters of marjoram leaves, it was observed that enthalpy values decrease with increasing DT and decreasing OP. Additionally, all tests showed negative entropy.

➢

The economic analysis revealed that drying marjoram leaves at 10 kPa and 60 °C resulted in yearly cost savings of up to USD 2054.19 and reduced the investment payback period to approximately 0.139 years (about 2 months).

The present study has several limitations that should be acknowledged. Primarily, it focused on evaluating the effect of selected levels of operating pressures and temperatures on the drying performance of marjoram without exploring a broader range of processing conditions or their interactions. Additionally, the scope of the study was limited to the technological and economic evaluation of the drying process itself. It did not encompass a comprehensive assessment of the dried marjoram product, particularly in terms of changes in essential oil yield, composition, or other quality attributes, such as color, aroma, and retention of bioactive compounds. These factors are critical for determining the commercial and functional value of the final product.

Future work will focus on investigating the influence of operating pressures and drying temperatures on both the yield and quality characteristics of essential oils extracted from marjoram leaves. This includes assessing changes in chemical composition, aroma profile, and retention of bioactive compounds under varying thermal and pressure conditions. The findings will contribute to optimizing drying protocols that preserve the functional and commercial value of the essential oil.