Design of Enhanced Virtual Reality Training Environments for Industrial Rotary Dryers Using Mathematical Modeling

Abstract

Rotary dryers are widely used in industry for their ease of operation in processing large volumes of material continuously despite persistent challenges in energy efficiency, cost-effectiveness, and safety. Addressing the need for effective operator training, the purpose of this study is to develop virtual reality (VR) environments for industrial rotary dryers. Visual and behavioral aspects were considered in the methodology for developing the environments for two application cases—ammonium nitrate and low-rank coal drying. Visual aspects considered include the industrial-scale geometry and detailed components of the rotary dryer, while behavioral aspects were governed by mathematical modeling of heat and mass transfer phenomena. The case studies of ammonium nitrate and low-rank coal were selected due to their industrial relevance and contrasting drying characteristics, ensuring the versatility and applicability of the developed VR environments. The main contribution of this work is the embedding of validated mathematical models—expressed as ordinary differential equations—into these environments. The numerical integration of these models provides key process variables, such as solid temperature and moisture content along the rotary dryer, thereby enhancing the behavioral realism of the developed VR environments.

1. Introduction

Drying is a unit operation that is widely used in the chemical, food, agricultural, wastewater treatment, and pharmaceutical industries to remove water from solids through heat and mass exchanges.

One of the most used equipment to perform drying is the rotary dryer because of its adaptability and flexibility in processing materials that are not heat-sensitive or fragile [1]. A rotary dryer is composed of a slightly inclined cylindrical shell or drum (length: 5–90 m, diameter: 0.3–5 m) with internal lifting flights. The cylinder rotates around a shaft in such a manner that the solid to be dried is fed in at the upper part of the cylinder and descends through the flights towards the discharging end. The drying medium (usually hot air) can be directly or indirectly in contact with the wet solid, and it can be operated in concurrent or countercurrent flow mode [2].

There exist diverse evaluation criteria for the performance of a rotary dryer and some of these are as follows: drying efficiency which is computed considering initial, final, and equilibrium moisture content [3] and energy efficiency that represents the ratio between the amount of energy used for the evaporation of moisture and the total energy consumption [4]. Although rotary dryers are characterized by their relative high drying efficiency for diverse wet solids (e.g., around 85% in the case of sewage sludge [5], 95% in the case of wood particles [6] and up to 98% in the case of lateritic ore [7]), they are also distinguished by their relative low energy efficiency (between 2% and 17% for paddy drying [8]). According to Kaveh et al. (2021), this fact could lead to an increase in emissions of greenhouse gases (GHGs) with negative environmental effects [9]. Furthermore, Perazzini et al. (2021) indicate that 10 to 20% of the total energy in industrialized countries is used for drying, making it a highly demanding power operation [10]. Another drawback of rotary dryers is related to their economic performance. In the case of biological sludge dewatering, the internal rate of return for a thermal rotary dryer can be around 27%, which is higher than that usually expected. In addition, the ratio between the total operating cost and the total capital cost can reach more than 0.5 [11]. Moreover, drying has been reported as a hazardous industrial operation for a number of reported incidents, with serious results for personnel and equipment [12].

Despite these limitations, rotary dryers are still widely used in industry due to their ability to continuously dry large quantities of material in a relatively short period and their ease in adjusting operating conditions such as residence time and rotation speed [3]; therefore, the training of students or professionals in rotary dryers is required. An innovative tool that has been recently considered for training is Virtual Reality (VR) [13]. Basically, VR is a digital or computer-generated representation (in 3D) of a real (or invented) space that can be explored from diverse angles and in which interaction is possible through head-mounted displays and other accessories. In addition, VR can serve as an immersive interface to interact with digital twins—a digital replica of a physical system that continuously receives real-time data from its physical counterpart—and with virtual commissioning, an approach that uses digital twins to test industrial automation prior to physical implementation. Recently, VR has been applied in diverse fields, demonstrating instructional impacts. In science and engineering, VR has shown a moderate and significant impact on practical skills (e.g., equipment handling, technical assembly or disassembly) [14] while in healthcare education, VR has been particularly effective in enhancing knowledge acquisition, improving skill scores and satisfaction [15]. Furthermore, VR has also been employed in industry for training sessions aimed at transferring knowledge related to assembly and maintenance processes [16] and to enhance chemical safety in plant operators [17]. Applying VR-based training for rotary dryers could yield similar instructional impact on practical skills while significantly reducing efficiency, environmental, economic, and industrial safety issues associated with the use and training in real rotary dryers. Therefore, VR-based training for rotary dryers may be regarded as a sustainable educational strategy and it can be considered as an immersive and effective learning experience if the VR environment accurately replicates the real system. To achieve this, the VR simulation should integrate both visual and behavioral aspects of the real-world equipment, system, or process [18]. A clear research gap has been identified as follows: while most efforts to engineer digital versions of equipment in such environments focus on enhancing their visual aspects, comparatively little has been given to improving their behavioral realism with the notable exception proposed by Hassan et al., 2024 [19], which is discussed in Section 5.3.

This contribution seeks to enhance the behavioral realism of a digital rotary dryer in a VR environment by incorporating and numerically integrating validated mathematical models given by ordinary differential equations. The model depicts the drying process taking place within the digital rotary dryer with the following variables: water content and temperature of both the solid to be dried and the drying medium (i.e., hot air) along and within the rotary dryer. Two case studies were selected to demonstrate the applicability of the VR environment. The first case focuses on the drying of ammonium nitrate, chosen for its critical role in the fertilizer industry where precise moisture control is essential for product quality and safety. The second case involves the drying of low-rank coal (LRC), or sub-bituminous coal, a promising energy resource whose value depends heavily on maintaining low moisture content to ensure efficient combustion. These cases were selected not only for their industrial significance but also because they present distinct drying characteristics, making them ideal for comprehensive training scenarios.

The paper is organized as follows. The Methodology section describes visual aspects of a proposed digital industrial rotary dryer. Behavioral aspects of the digital dryer are also addressed as a part of the Methodology section. This is conducted through the description of the mathematical model that governs the digital dryer and their corresponding drying kinetics for ammonium nitrate and low-rank coal. Next, the Results section addresses the following topics: description of the VR environment, interaction in the VR environment, expansion of such an environment, and simulation results of the mathematical model for both case studies. These simulation results were embedded in two different VR environments, one for each case study, that are identical in what concerns visual aspects but with differences in their behavioral aspects since they are governed by different drying kinetics. In the Discussion section, the behavior of the state variables considered in the mathematical models is analyzed and the enhancement of the VR environment is demonstrated. A comparison with related work is also included in the Discussion section as well as the workflow diagram. Finally, the Conclusions section summarizes the main learnings and outlines future work.

2. Materials and Methods

The main challenge addressed in this work—and the key contribution—was the successful embedding of mathematical models into VR environments. This challenge was overcome by achieving the following objectives: defining the visual aspects of the digital rotary dryer within the virtual environment and establishing its behavioral characteristics through the definition of a general mathematical model and the customization of this model to specific case studies.

2.1. Visual Aspects of the Digital Rotary Dryer in the VR Environment

The rotary dryer in the VR environment was designed to work in concurrent flow mode with the following components: cylindrical drum, internal lifting flights, roller rings supported by carrier rollers distributed along the drum, a primary gear, or gear ring connected to a motor gear by a drive chain, a feed hopper, a feeding cover, a burner, an induced draft fan, and a couple of control panels (see Figure 1). The individual components of the rotary dryer were first designed separately using Blender 3.4, a 3D computer graphics software. The modeling process strictly adhered to industrial-scale dimensions to ensure realism and applicability. These dimensions were sourced from Abbasfard et al. (2013) and are consistent with the values referenced earlier in the introduction of this paper [20]. Specifically, the rotary dryer was designed with a length of 18 m, an internal diameter of 3.324 m, and a drum slope of 0.025 m per meter. By basing the virtual models on these characteristic measurements, the resulting virtual environment accurately reflects real-world equipment, thereby improving the fidelity of the simulation.

2.2. Behavioral Aspects of the Digital Rotary Dryer in the VR Environment

In order to enhance behavioral realism of the rotary dryer in the VR environment, a steady state one-dimensional mathematical model is proposed to be included in such an environment. This model was originally proposed by Abbasfard et al. (2013) [20] and describes the water content and temperature of both the solid being dried and the air along and within the rotary dryer drum. The model was developed under the following assumptions: the heat exchange from the drum to the surroundings is neglected, the solids to be dried have a spherical shape and their dimensions and heat capacities remain constant, the drying process is conducted below critical moisture content (i.e., in the falling-rate period of the drying process), the flows of solids and air follow a plug flow model, diffusion and dispersion through the axial direction are not considered, and back-mixing, potential, and kinetic energy are neglected. The model is given by the following set of ordinary differential equations:

$${\displaystyle \frac{dX}{dZ}}=-{\displaystyle \frac{RM}{S}}$$

(1)

$${\displaystyle \frac{dY}{dZ}}={\displaystyle \frac{RM}{G}}$$

(2)

$${\displaystyle \frac{d{T}_s}{dZ}}=-{\displaystyle \frac{-Q-\lambda RM}{S\left({C_P}_S+X{C_P}_{lw}\right)}}$$

(3)

$${\displaystyle \frac{d{T}_g}{dZ}}={\displaystyle \frac{Q+{C_P}_{W}RM\left(T_s-T_g\right)}{G\left({C_P}_A+Y{C_P}_W\right)}}$$

(4)

where $Z$ is a dimensionless normalized length of the rotary dryer drum (i.e., $0\le Z\le 1$), defined as $Z=x/L,$ where $L$ is the length of the drum, whereas $x$ is the coordinate indicating the direction of both solid and air flows. In addition, $X$, $T_s$ denote the solid moisture (kg water/kg dry solid) and temperature in the solid whereas $Y$ and $T_g$ represent the air absolute humidity (kg water/kg dry air) and temperature of the air. Boundary conditions of the model are denoted by $X_O$, ${T_s}_O$, $Y_O$, and ${T_g}_O$. In addition, $R$ is the drying rate or kinetics, $M$ is the dryer total load which is computed as $M=\tau S$ ($\tau$ is the average residence time and $S$ is the dry solid mass flow rate), $G$ is the dry air mass flow rate, $Q$ is the internal heat transfer rate, and $\lambda$ is the latent heat of vaporization. According to Arruda (2006) [21], an expression for $Q$, which is widely accepted for industrial rotary dryers and will therefore be used in this work, is given by the following equation: $Q=U_{CGV}V(T_s-T_g)$ where $V$ is the dryer volume (i.e., $V=AL$) and $U_{CGV}$ is the global volumetric heat transfer coefficient that can be computed with $U_{CGV}=0.394{\left({\displaystyle \frac{G}{A}}\right)}^{0.289}{\left({\displaystyle \frac{S}{A}}\right)}^{0.541}$ where $A$ is the dryer cross-sectional area. In addition, $\lambda$ can be computed with the following model proposed by Watson (1943) $\lambda ={\lambda }_1{\left[{\displaystyle \frac{T_C-T_s}{T_C-T_1}}\right]}^{0.38}$ where ${\lambda }_1$ is the latent heat of vaporization at temperature $T_1$ whereas $T_C$ is the critical temperature [22]. Furthermore, ${C_P}_S$, ${C_P}_{lw}$, ${C_P}_A$, and ${C_P}_W$ are the specific heats of the dry solid, of liquid water, of dry air and of water vapor, respectively. For the case of ${C_P}_A$ and ${C_P}_W$, the following equation proposed by NASA is considered (1971) $C_{P_i}={\displaystyle \frac{\underset{\_}{R}}{M{M}_I}}\left({\alpha }_i+{\beta }_i{T}_g+{\gamma }_i{T}_g^2+{\delta }_i{T}_g^3+{\epsilon }_i{T}_g^4\right)$ where $i$ = $A$ or $W$ (Air or water), $M{M}_I$ is the molar mass of the $i$-th species, $T_g$ must be provided in K. The numerical values for ${\alpha }_i$, ${\beta }_i$, ${\gamma }_i$, ${\delta }_i$, and ${\epsilon }_i$ are given in

Table 1. Parameters for the heat capacities of air and water.

Gas	${\mathit{\alpha }}_{\mathit{i}}$	${\mathit{\beta }}_{\mathit{i}}\times 10^3$ (K−1)	${\mathit{\gamma }}_{\mathit{i}}\times 10^6$ (K−2)	${\mathit{\delta }}_{\mathit{i}}\times 10^9$ (K−3)	${\mathit{\epsilon }}_{\mathit{i}}\times 10^{12}$ (K−4)
$A$ = Air	3.653	−1.337	3.294	−1.913	0.2763
$W$ = Water	4.070	−1.108	4.152	−2.964	0.8070

[23].

3. Case Studies

3.1. Case Study: Ammonium Nitrate Plant

The first case addresses the drying of ammonium nitrate, chosen due to its widespread use in fertilizers. In this context, achieving accurate control of moisture levels is crucial, as both the outlet moisture content and the air temperature have a decisive impact on the commercial quality of the final product [20].

Abbasfard et al. (2013) [20] proposed a mathematical model with a similar structure described by Equations (1)–(4) to describe the drying process of 650 metric tons/day of ammonium nitrate in a rotary dryer. For this process, the model was validated with the following drying kinetics:

$$R=k\left(X-X_{eq}\right)$$

(5)

where $k$ (with units of min⁻¹) is the drying constant and $X_{eq}$ is the equilibrium moisture content, calculated as follows:

$$k=0.0349e^{{\displaystyle \frac{-7.95}{T_g}}}$$

(6)

$$X_{eq}=\mathsf{\Phi }\left(a{\Phi }^2+b\Phi +c\right)$$

(7)

where

$$a=2.39{\times 10}^{-6}\left({0.987}^{T_g}\right)\left(T_g^{-0.832}\right)$$

(8)

$$b=-5.76{\times 10}^{-5}+1.306\times {10}^{-6}ln\left(T_g\right)$$

(9)

$$c=0.9715\left({1.024}^{T_g}\right)\left(T_g^{-2.31}\right)$$

(10)

where $\Phi$ is air relative humidity. In this paper, $\Phi$ was computed as $\Phi ={\displaystyle \frac{P}{P_{sat}}}{\displaystyle \frac{1}{\frac{0.622}{Y}+1}}$ where $P$ is the operating pressure (assumed as 101,325 Pa, the atmospheric pressure) and $P_{sat}$ is the saturation pressure for water obtained from the following equation $P_{sat}={\displaystyle \frac{{exp}^{\left[34.494-\frac{4924.99}{T_g+237.1}\right]}}{{\left(T_g+105\right)}^{1.57}}}$ that was proposed by Huang [24].

3.2. Case Study: Low-Rank Coal (LRC)

The second case addresses the drying of low-rank coal (LRC), or sub-bituminous coal, a resource of considerable energy potential. However, its elevated moisture content (30–40%) significantly limits its utilization, increasing transportation costs, lowering energy efficiency, and contributing to higher CO₂ emissions. For these reasons, drying is a critical step to improve its combustion performance, facilitate handling and storage, and ultimately optimize its energy potential [25].

The drying of LRC was addressed using the same mathematical model defined by Equations (1)–(4), except that in this case, the parameters $k$ and $X_{eq}$ of the drying rate $R$ were obtained as follows. First, the drying rate can be related to the decrease in the moisture content over time with the following expression:

$$-{\displaystyle \frac{dX}{dt}}=R$$

(11)

Considering Equation (5), the previous equation takes the following form

$$-{\displaystyle \frac{dX}{dt}}=k\left(X-X_{eq}\right)$$

(12)

Then, Equation (12) can be integrated resulting in

$$ln\left[{\displaystyle \frac{X-X_{eq}}{X_O-X_{eq}}}\right]=-kt$$

(13)

where $X_O$ is the moisture content at the beginning of the drying for a certain experiment. On the other hand, Rong et al. (2016) [25] evaluated the drying of LRC with a plot of $log\left(log\left({\displaystyle \frac{X_o}{X}}\right)\right)$ versus time at 101 kPa under different air temperatures ($T_g$= 64.85 °C, $T_g$= 99.85 °C and $T_g$= 149.85 °C). Extracting data from that plot using the software WebPlotDigitalizer 5.2 is possible to build Figure 2, which shows the moisture content $X$ versus time at different $T_g$. This is required to obtain k and $X_{eq}$ as a function of $T_g$ as follows. For each $T_g$, an arbitrary value for $X_{eq}$ can be assumed to plot a graph with $ln\left[{\displaystyle \frac{X-X_{eq}}{X_O-X_{eq}}}\right]$ on the y-axis and time on the x-axis. From such a graph, the slope corresponding to $k$ (as defined in Equation (13)) and the coefficient of determination or R-squared ($R^2$) can be obtained. This procedure was optimized to maximize $R^2$, yielding the values described in

Table 2. Optimized values for $k$ (with units of min⁻¹) and $X_{eq}$ for the drying of LRC at 101 kPa.

${\mathit{T}}_{\mathit{g}}$ (°C)	$\mathit{k} \left({\mathbf{m}\mathbf{i}\mathbf{n}}^{-1}\right)$	${\mathit{X}}_{\mathit{e}\mathit{q}}\left(\frac{\mathit{k}{\mathit{g}}_{\mathit{w}\mathit{a}\mathit{t}\mathit{e}\mathit{r}}}{\mathit{k}{\mathit{g}}_{\mathit{d}\mathit{r}\mathit{y}\mathit{s}\mathit{o}\mathit{l}\mathit{i}\mathit{d}}}\right)$
64.85	0.011827	0.135
99.85	0.024813	0.119
149.85	0.057254	0.105

.

Finally, $k$ and $X_{eq}$ were fitted to an exponential expression and to a linear expression with respect to $T_g$, respectively, resulting in

$$k=3.7092{\times 10}^{-3}{e}^{1.84288{\times 10}^{-2}{T}_g}$$

(14)

$$X_{eq}=-3.47945{\times 10}^{-4}{T}_g+1.56149{\times 10}^{-1}$$

(15)

It was decided to fit $k$ to an exponential expression of $T_g$ because that is a standard practice (see Equation (6)) whereas $X_{eq}$ was fitted to a linear expression of $T_g$ in order to give a simpler alternative than those usually considered (see Equation (7)).

4. Results

4.1. Results of Visual Aspects of the Proposed VR Environment

4.1.1. Description of VR Environment

The integration of each rotary dryer component, as described in Figure 1, into a VR environment was carried out using Unity 3D 2022.3.26f1. This process was performed on an Alienware Aurora R8 Gaming Desktop Computer, specifically designed to support virtual reality applications (3.2 GHz Intel Core i7-9700, 16 GB, NVIDIA GeForce RTX 2060) (see Figure 3). A conveyor belt (not shown in Figure 1) was also designed and included as an element to transport the solid to be dried from the feeding cover towards the digital rotary dryer.

Control Panel 1 was equipped with digital displays for the temperature of the air, moisture content of the solid at the inlet of the dryer (i.e., the boundary conditions ${T_g}_O$ and $X_O$), velocity of the air ($v_0$) (which is directly related to the mass flow of the air or $G$, see

Table 3. Operational conditions.

Boundary and Operational Conditions	Numerical Values or Equations
${T_s}_O$	33 °C
${T_g}_O$1	77 °C
$Y_O$	0.002
Velocity of wet air $\mathrm{at}\mathrm{the}\mathrm{inlet}(v_0$) 1	3 m/s
Dry air mass flow rate (G) in kg/min	$G=60{\displaystyle \frac{v_{0}A\rho }{1+Y_O}}$
	$\rho ={\displaystyle \frac{M{M}_{AV}P}{R{T_g}_O}}$
	$M{M}_{Av}=M{M}_{Air}{\displaystyle \frac{1}{1+Y_O}}+M{M}_{Water}{\displaystyle \frac{Y_O}{1+Y_O}}$
$\mathrm{Wet}\mathrm{solid}\mathrm{flow}\mathrm{rate}(S_{wet}$)	5 Kg/s
$\mathrm{Dry}\mathrm{solid}\mathrm{flow}\mathrm{rate}(S$) in kg/min	$S=60{\displaystyle \frac{S_{wet}}{1+X_O}}$
$\mathrm{Total}\mathrm{load}(M$) in kg of dry solid	$\tau S$

¹ These variables could be modified in the digital displays of Control Panel 1. In addition, the boundary condition $X_O$ was 0.11 for ammonium nitrate plant and 0.32 for the LRC case studies, respectively.

), and rotary dryer drum speed (in rpm). Control Panel 1 also has on–off buttons, an emergency stop or E-stop button, and a selector switch for the drum’s rotation. In addition, Control Panel 2 has a couple of digital displays to register the solid that is being dried (between ammonium nitrate or LRC) and the number of solid particles that can be generated from the feed hopper in the VR environment. It was decided to show how the drying occurs in the solid using solid particles (See Figure 4b) instead of dust-type particles, since they allow for easier manipulation and more effective interaction control in VR environments. In this sense, each solid particle has the same value for temperature and moisture content at the inlet of the dryer, and these values will change according to the mathematical model and drying kinetics along the dryer drum. Control Panel 2 also includes a button to initiate the feeding of solids. The components of Control Panels 1 and 2 are described in Figure 5a,b. The VR environment for the digital rotary dryer was enriched with laboratory workbenches on which a virtual thermometer for the solid, a virtual moisture content meter for the solid, and a virtual shovel were placed (see Figure 6a,b). The shape of the virtual thermometer and the moisture content meter were inspired by common IR thermometers and by common portable moisture meters, respectively.

4.1.2. Interaction in the VR Environment

An Oculus Quest all-in-one VR gaming system headset (128 GB, model MH-B) and a pair of Oculus Touch Controllers (MR-BR, Right and MI-BL, Left) were selected as the head-mounted display (HMD) to interact in the VR environment containing the digital rotary dryer. Through this HMD, it is possible to increase or to decrease ${T_g}_O$, $X_O$, $v_O$, and the rotary dryer drum speed by selecting the up or down arrow key in the digital displays of Control Panel 1. All these variables can also be set to their nominal (arbitrary) values by selecting the set button. It is also possible to turn the burner or the induced draft fan on or off, to activate the E-stop button or to change the drum rotation in the corresponding buttons placed in Control Panel 1. Interaction also occurs when the type and quantity of solid particles to be dried are selected using the digital displays on Control Panel 2, or when the “Start” button is pressed on the same panel.

Using the HMD, users can also interact with the proposed environment by picking up the virtual shovel to manually guide solid particles from the feed hopper to the conveyor belt. Users can also interact with single solid particles from the hopper, placing them directly into the rotary dryer (see Figure 4a,b). In addition, the temperature and moisture of the solid particles at the inlet of the rotary dryer can be monitored using the thermometer and the moisture meter placed on the laboratory workbenches by pointing them directly to the solid (see Figure 4c,d).

The transport of solid particles along the rotary dryer can also be visualized using HMD, and the particles can be extracted at any point from the dryer drum to analyze their temperature and moisture content. Variations in the color of the solid particles, depending on the point from which they have been extracted from the drum (i.e., related to their temperature and moisture content), are also included as a part of the VR experience (see Figure 7a–c and Supplementary Material).

4.1.3. Expansion of the VR Environment

Interaction with the VR environment through the HMD is limited to one person. In order to expand the VR environment to multiple people, a physical space or virtual cellar at the Chemical Engineering Laboratory of the Universidad Autónoma de Guadalajara in México was set up with four projectors (Epson Home Cinema 2500 ANSI Lumens, V11H852020, Epson America, Inc., Long Beach, CA, USA) supported in the roof and connected to the Alienware Aurora desktop computer and the HMD both remaining outside the virtual cellar (see Figure 8a–c). Therefore, the VR environment was controlled by one person outside the virtual cellar (generally a teacher) while the rest of the people (generally students) can experience the VR environment inside the virtual cellar.

4.2. Results of Behavioral Aspects of the Proposed VR Environment

The mathematical model for the industrial rotary dryer given by the set of Equations (1)–(4) described in Section 2.2 was solved in Matlab R2024a using the fourth-order Runge–Kutta method (RK4) for each one of the case studies, considering their corresponding drying kinetics. In the numerical integration, the following values were considered in the equation for the heat of vaporization $\lambda$, described in Section 2.2: ${\lambda }_1$ = 2257.06 KJ/Kg at $T_1$ = 100 °C and $T_C$ = 374.14 °C [22].

In addition, the heat capacities for liquid water and for the solids were considered constants with the following values $C_{P_{lw}}$ = 4.18 KJ/(kgK) and ${C_P}_S$ = 1.56 KJ/(kgK) for the ammonium nitrate and ${C_P}_S$ = 1.3 KJ/(kgK) for the LRC. Finally, the rotary dryer drum speed was set at 3 rpm. Given the characteristic lengths of the rotary dryer (i.e., 18 m length, 3.324 m inside diameter and 0.025 m/m slope), this leads to an average residence time of 30 min (i.e., $\tau$ = 30 min) [20]. The boundary and operational conditions depicted in Table 3 were also considered in the simulation runs, yielding Figure 9 and Figure 10 for the nitrate plant and for the LRC case studies, respectively.

5. Discussion

5.1. On the Behavior of the State Variables

For both case studies, the air temperature ($T_g$) decreases along the dryer drum (see the red line in Figure 9a and Figure 10a). This can be explained due to mass and energy interactions of the air with a solid which has a lower temperature. Regarding the temperature of the solid, an increase is observed in both cases (see blue line in Figure 9a and Figure 10a). Nevertheless, in the ammonium nitrate drying, the temperature of the solid ($T_s$) reaches a maximum around $z$= 0.2 (see the blue line in Figure 9a), while in the case of LRC drying, the temperature of the solid also reaches a maximum but this occurs after $z$= 0.2 (see the blue line in Figure 10a). This different behavior can be attributed to the different parameters in the drying kinetics of the ammonium nitrate and LRC. The influence in drying kinetics is also shown in the behavior of the solid moisture ($X$) along the dryer drum (see the blue line in Figure 9b and Figure 10b). Although in both case studies $X$ decreases as expected, the decrease is noticeably faster for ammonium nitrate than for LRC. Notice that water lost by the solids is absorbed into the air, as shown by the increase in the absolute humidity of air ($Y$) in both drying cases (see the red line in Figure 9b and Figure 10b). The use of Figure 9 and Figure 10 in the context of the proposed VR environment is explained in the following section.

5.2. Enhancing the VR Environment

The development of equipment in VR environments generally focuses on improving visual aspects leaving aside behavioral realism. In this contribution, while the visual features of the digital rotary dryer in the VR environment were addressed, equal attention was given to accurately representing the behavioral aspects of the dryer. To enhance the realism of the digital dryer, the mathematical model described by Equations (1)–(4), their corresponding drying kinetics depending on the case study, and the numerical method to solve the model (i.e., RK4) were transferred from Matlab into Unity 3D. This transfer represented a technical challenge because the programming language in Matlab (M-Code) that combines C, Fortran, and Python characteristics is different than that used by Unity 3D which is C#. The challenge was overcome by using Visual Studio Code 1.86.2 as translator of the M-Code to C#. Then, it can be said that the proposed VR environment for the industrial rotary dryer is enhanced in comparison to most available VR environments, as one of its main components, the digital dryer, is governed by a validated mathematical model. This feature of the VR environment is present in the measurements recorded by the virtual thermometer and by the virtual moisture content meter that correspond to the numerical values of Ts and X, respectively, that are produced by the integrated mathematical model. At this point, it is important to remark that two different VR environments were developed, one for each case study. The VR environments are identical in terms of visual aspects but they differ from each other in their behavioral aspects because they are governed by different drying kinetics.

Finally, two examples are given to illustrate enhancements of the VR environments. First, the VR environment for ammonium nitrate is launched. In this case, solid samples are extracted along the dryer drum at the following positions z = 0, 0.25, 0.5, 0.75, and 1 and are placed over a virtual workbench. Note that these samples differ in color, as expected in the real world, thereby enhancing the realism of the virtual experience. When the virtual thermometer is pointed at the sample located at z = 0.75, a temperature of 54.91 °C is recorded (see Figure 11a). This value clearly corresponds to the one shown in Figure 9a for Ts at z = 0.75. Next, the VR environment for LRC is launched. Similarly, solid samples are also extracted at the same z positions and placed on a virtual workbench. In this case, the moisture content meter is taken and pointed at the sample located at z = 0.75, yielding a moisture content of 0.27 (see Figure 11b). This value corresponds to that for X at z = 0.75 in Figure 10b. Taken together, these examples demonstrate that the values generated for Ts and X in the VR environments are consistent with those provided by a validated mathematical model and shown in Figure 9 and Figure 10, depending on the case study. The main limitation of this model is that it represents the temperature and moisture content of the solid and air only under steady-state conditions, thus neglecting temporal variations in these variables. Incorporating a mathematical model based on partial differential equations that includes time as an independent variable could further enhance both the realism and the educational value of the VR experience.

5.3. Comparison with Related Work

Hassan et al. (2024) developed a virtual reality-based bioreactor system as a digital twin to train operators in both set up and operational tasks for penicillin production that was modeled by a set of ordinary differential equations (ODEs) and validated with real batch data [19]. This digital twin could be later leveraged as a virtual commissioning to test the control systems of the bioreactor before they are physically installed or operated.

These equations are simulated within the .NET framework and integrated into the Unity3D game engine. Similarly, the present work employs Unity3D as the game engine for the visual components of the proposed VR environment although it was complemented by a set of ODEs using Visual Studio Code. In addition, this study differs from that of Hassan et al. in that the ODEs describe a continuous process rather than a batch process, and the VR environment has been expanded to support multiple users through the construction and implementation of a virtual cellar.

5.4. Workflow Diagram

A workflow diagram is included to clarify the steps required to provide behavioral realism to the proposed VR environments (see Figure 12). Once the rotary dryer components are designed in Blender, two parallel development paths can be followed independently. The first path involves selecting the model, identifying case studies, determining the corresponding drying kinetics, and running simulations in Matlab through numerical integration. The second path begins with integrating the rotary dryer components into Unity 3D using appropriate hardware. This is followed by interacting with the VR environment through an HMD, and then expanding the environment to multiple users via a virtual cellar. Both paths converge when the mathematical model is embedded into the VR environment. This integration is achieved by using Visual Studio Code to translate the M-code from Matlab’s numerical integration into C#, which Unity 3D requires to perform equivalent computations. This process enhances the behavioral realism of the developed VR environments.

6. Conclusions

A couple of VR environments for the drying of ammonium nitrate and LRC in industrial rotary dryers were developed in Unity 3D. These environments provide, through digital devices, numerical values for temperature and moisture content of the solid to be dried. These values were obtained from the numerical integration of mathematical models with validated drying kinetics for the corresponding case studies. As a result, the proposed VR environments could serve not only as visualization platforms but also as impactful training systems for people dealing with industrial rotary dryers, as they are able to replicate real-world behavior using mathematical modeling techniques. It is actually quite straightforward to apply the VR environments developed here to materials beyond those analyzed in this work. Essentially, it would be a matter of adjusting the various parameters considered by the mathematical model for the solid, such as the drying kinetics, as well as the specific heat capacity of the solid to be dried and the flow rates of the wet solid and dry solid. The VR environments have been officially registered in Mexico (through INDAUTOR) under the following number: 03-2023-050410553100-01. The methodology proposed in this contribution could be expanded for the design of other virtual equipment, particularly those whose real versions can also be depicted by validated mathematical models. In this sense, future work will focus on incorporating mathematical models using time as a state variable in order to provide real-time responses in the VR environment. A longitudinal study with students from the Chemical Engineering undergraduate program at Universidad Autónoma de Guadalajara is also planned to analyze not only the assessment of learning outcomes regarding the use and operation of an industrial rotary dryer but also to have feedback from the students to improve the VR environment.