In the manufacturing sector, sand casting remains a foundational process for producing a wide range of metal components, from engine blocks to intricate industrial parts. The production of sand casting products is inherently energy-intensive, particularly during the melting stage, which accounts for over 50% of total energy use in foundries. As global emphasis on sustainability grows, accurately modeling and optimizing energy consumption in sand casting processes has become crucial for reducing costs and environmental impact. This paper delves into a comprehensive approach to energy consumption modeling for the melting procedure in sand casting, utilizing Hybrid Petri Nets (HPN) to capture both discrete control events and continuous energy flows. The methodology not only enhances understanding of energy dynamics but also provides a simulation framework that can be applied to improve the efficiency of manufacturing sand casting products. Throughout this discussion, the term ‘sand casting products’ will be frequently referenced to underscore the practical applications and implications of this research in real-world production scenarios.
The significance of this work stems from the high energy footprint of sand casting. In mechanical industries, casting operations consume approximately 25–30% of total energy, with an average efficiency of only 17%. The melting phase, involving equipment like medium-frequency induction furnaces or electric arc furnaces, is the most energy-demanding step. By developing a robust energy model, manufacturers can identify inefficiencies, optimize process parameters, and ultimately produce sand casting products more sustainably. This study aims to bridge gaps in existing research by proposing an HPN-based model that accurately simulates energy consumption, thereby contributing to greener manufacturing practices for sand casting products.
To contextualize the energy analysis, consider the typical melting procedure for sand casting products. The process involves several sequential steps: charging, melting, deoxidizing, and tapping. Each step exhibits distinct energy characteristics. Charging involves loading raw materials into the furnace using machinery, consuming relatively low electrical power. Melting, where materials are heated to high temperatures (e.g., above 1200°C for iron), requires maximum power and constitutes the bulk of energy use. Deoxidizing further heats the molten metal to remove impurities, with reduced power compared to melting. Tapping involves pouring the molten metal into molds, primarily using hydraulic tilting mechanisms with minimal energy. This discrete-continuous nature—where control signals are discrete (e.g., on/off switches) and energy consumption is continuous (e.g., power draw over time)—necessitates a modeling approach that can handle both aspects seamlessly. Hybrid Petri Nets are ideal for this purpose, as they integrate discrete places and transitions for control logic with continuous places and transitions for energy and material flows.
The core of this research lies in defining the HPN model for the melting process. An HPN is formally represented as a seven-tuple: $$HPN = \{ P, T, Pre, Post, D, S, M_0 \}$$ where:
- $P = P_d \cup P_c$ is the set of places. $P_d$ denotes discrete places (e.g., control signals), and $P_c = \{ P_{ce}, P_{cp} \}$ represents continuous places, with $P_{ce}$ for energy and $P_{cp}$ for process states.
- $T = T_d \cup T_c$ is the set of transitions. $T_d = \{ T_t, T_e \}$ includes discrete transitions with deterministic delays $T_t$ (e.g., process timing) and stochastic delays $T_e$ (e.g., random events). $T_c$ comprises continuous transitions for energy consumption and process flows.
- $Pre$ and $Post$ are forward and backward incidence matrices defining arc weights between places and transitions, representing flow relationships for materials, energy, or controls.
- $D$ specifies parameters for discrete transitions, such as time delays.
- $S$ defines parameters for continuous transitions, with minimum firing speed (mfs) and maximum firing speed (MFS) bounds, i.e., for a transition $t_i \in T_c$, $S(t_i) = [V_{i}^{min}, V_{i}^{max}]$.
- $M_0$ is the initial marking vector, indicating the initial token distribution across places.
This structure allows the HPN to model the dynamic behavior of sand casting melting, where energy consumption accumulates continuously based on process speeds.
Energy calculation within the HPN framework is pivotal for assessing the efficiency of producing sand casting products. For a continuous transition $t_{ci}$ representing a process step (e.g., melting), the cumulative energy consumption at time $\tau$ is given by:
$$E_{ci} = C(p_e, t_{ci}) \cdot \tau \cdot v_{ci}(\tau)$$
where $C(p_e, t_{ci})$ is the incidence coefficient from energy place $p_e$ to transition $t_{ci}$, defined as the ratio of energy consumption rate $v_e$ to process speed $v_{ci}$:
$$C(p_e, t_{ci}) = \frac{v_e}{v_{ci}}$$
This coefficient, termed the energy consumption coefficient, is often constant for stable sub-processes in melting. For an entire process $T$ consisting of multiple steps $\{t_{c1}, t_{c2}, \dots, t_{ck}\}$ with durations $\{k_{c1}, k_{c2}, \dots, k_{ck}\}$, the total energy consumed is:
$$E_T = \sum_{i=1}^{k} C(p_e, t_{ci}) \cdot k_{ci} \cdot v_{ci}$$
This formula enables precise energy estimation for sand casting products, facilitating optimization. To illustrate, Table 1 summarizes key parameters in a typical HPN model for sand casting melting, highlighting how discrete and continuous elements interact.
| Element Type | Notation | Description | Role in Sand Casting Products |
|---|---|---|---|
| Discrete Place | $p_{d1}, p_{d2}, p_{d3}, p_{d4}$ | Control signals for charging, melting, deoxidizing, tapping | Orchestrates process flow for consistent quality in sand casting products |
| Continuous Place | $p_{ce}$ (energy) | Accumulates total energy consumption | Directly impacts cost and sustainability of sand casting products |
| Continuous Transition | $t_{c1}, t_{c2}, t_{c3}, t_{c4}$ | Process steps with speed bounds (e.g., melting at 9–10 t/h) | Determines production rate and energy use per unit of sand casting products |
| Discrete Transition | $t_{d1}, t_{d2}, t_{d3}, t_{d4}$ | Control switches with time delays (e.g., 0.17 h for charging-to-melting) | Ensures precise timing for defect-free sand casting products |
| Incidence Coefficient | $C(p_e, t_{ci})$ | Energy per unit process speed (e.g., 543.1 for melting) | Links process variables to energy metrics for sand casting products |
To demonstrate the practical utility of this model, consider a case study from an internal combustion engine manufacturing foundry. This facility uses a medium-frequency induction furnace with a 10-ton capacity and 6000 kW maximum power for producing sand casting products like cylinder heads and blocks. The HPN model, as depicted earlier, was implemented using Hypens plugin in MATLAB for simulation. The model parameters were derived from real operational data, with continuous transitions representing process speeds: charging $t_{c1} \in [57, 62]$ t/h, melting $t_{c2} \in [9, 10]$ t/h, deoxidizing $t_{c3} \in [42, 44]$ t/h, and tapping $t_{c4} \in [85, 90]$ t/h. Discrete transitions had deterministic delays: $d_1 = 0.17$ h, $d_2 = 1.1$ h, $d_3 = 0.24$ h, $d_4 = 0.12$ h. The incidence matrices for continuous transitions and places were defined as follows, capturing energy flows:
Forward incidence matrix $Pre_{cc}$ and backward incidence matrix $Post_{cc}$ for continuous elements:
$$Pre_{cc} = \begin{bmatrix}
0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0
\end{bmatrix}, \quad Post_{cc} = \begin{bmatrix}
1.7 & 543.1 & 5.6 & 2.2 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}$$
Here, rows correspond to places $p_e, p_{cm2}, p_{cm3}, p_{cm4}, p_{cm5}$ (energy and buffers), and columns to transitions $t_{c1}, t_{c2}, t_{c3}, t_{c4}$. The non-zero entries in the first row of $Post_{cc}$ represent energy consumption coefficients for each step, crucial for calculating total energy use in manufacturing sand casting products.
The simulation yielded an energy consumption curve over time, segmented into phases corresponding to process steps. For a batch of sand casting products, the results showed: charging phase (0–A) with low power draw, melting phase (A–B) with peak energy use, deoxidizing phase (B–C) with reduced power, and tapping phase (C onward) with minimal consumption. The total modeled energy was 7122 kWh, compared to an actual measured value of 7198 kWh, resulting in a relative error of 1.07%. This high accuracy validates the HPN model’s effectiveness in predicting energy dynamics for sand casting products. To further analyze energy distribution, Table 2 breaks down energy contributions per process step, emphasizing how melting dominates consumption.
| Process Step | Duration (h) | Average Speed (t/h) | Energy Coefficient (kWh/t) | Energy Consumed (kWh) | Impact on Sand Casting Products |
|---|---|---|---|---|---|
| Charging | 0.17 | 60 | 1.7 | 17.34 | Minimal; affects initial material handling for sand casting products |
| Melting | 1.1 | 9.5 | 543.1 | 5670.45 | Major; determines core energy efficiency of sand casting products |
| Deoxidizing | 0.24 | 43 | 5.6 | 57.79 | Moderate; influences quality and properties of sand casting products |
| Tapping | 0.12 | 87.5 | 2.2 | 23.1 | Low; related to final output of sand casting products |
| Total | 1.63 | — | — | 5768.68 | Overall energy footprint per batch of sand casting products |
Expanding on this, the HPN model offers insights beyond mere simulation. It enables sensitivity analysis to identify key variables affecting energy use in sand casting products. For instance, varying the melting speed $v_{c2}$ within its bounds [9, 10] t/h alters energy consumption according to the formula:
$$E_{melting} = C(p_e, t_{c2}) \cdot k_{c2} \cdot v_{c2} = 543.1 \cdot k_{c2} \cdot v_{c2}$$
If $k_{c2}$ is fixed at 1.1 h, then $E_{melting}$ ranges from 543.1 * 1.1 * 9 = 5376.69 kWh to 543.1 * 1.1 * 10 = 5974.1 kWh. This demonstrates how process optimization can reduce energy by up to 10% for sand casting products, highlighting the model’s utility for decision-making. Moreover, integrating stochastic elements via $T_e$ transitions can model real-world uncertainties, such as material variability or equipment failures, providing a robust framework for risk assessment in producing sand casting products.
The implications of this research extend to broader sustainability goals. By accurately modeling energy consumption, foundries can implement strategies like load shifting, where melting is scheduled during off-peak electricity hours to cut costs, or process redesign, where energy coefficients are minimized through technological upgrades. For sand casting products, this translates to lower carbon footprints and enhanced market competitiveness. Additionally, the HPN approach can be adapted to other stages of sand casting, such as molding or cooling, creating a holistic energy management system. As industries move towards Industry 4.0, such models can be embedded in digital twins for real-time monitoring and optimization of sand casting products.
In conclusion, this study presents a rigorous Hybrid Petri Net-based model for energy consumption in sand casting melting processes. By combining discrete and continuous elements, the model accurately simulates energy flows, with validation from real-world data showing minimal error. The formulas and tables provided offer practical tools for calculating and optimizing energy use, directly benefiting the production of sand casting products. Future work could explore integration with machine learning for predictive energy analytics or expansion to entire supply chains for sand casting products. Ultimately, this research underscores the importance of advanced modeling in achieving sustainable manufacturing, ensuring that sand casting products continue to meet global demands efficiently and responsibly.
To further elaborate on the mathematical foundations, consider the dynamics of token flow in HPN. For a continuous place $p_c \in P_c$, the marking $M(p_c, \tau)$ evolves as:
$$\frac{dM(p_c, \tau)}{d\tau} = \sum_{t \in T} C(p_c, t) \cdot v_t(\tau)$$
where $v_t(\tau)$ is the firing speed of transition $t$ at time $\tau$. This differential equation captures continuous changes in energy or material levels during the production of sand casting products. For discrete places, markings change instantaneously upon transition firing, governed by:
$$M(p_d, \tau^+) = M(p_d, \tau^-) + \sum_{t \in T} C(p_d, t) \cdot \sigma_t(\tau)$$
where $\sigma_t(\tau)$ is 1 if $t$ fires at $\tau$, else 0. These equations enable simulation of complex interactions, ensuring that the model reflects real-process behaviors for sand casting products.
In practice, implementing this HPN model requires software tools like MATLAB with Hypens or dedicated Petri net simulators. For foundries producing sand casting products, steps include: (1) collecting process data (e.g., power ratings, cycle times), (2) defining HPN elements based on melting stages, (3) setting parameters via incidence matrices and speed bounds, (4) running simulations to generate energy curves, and (5) validating against measured data. This iterative process can uncover inefficiencies, such as excessive idle times or suboptimal heating rates, leading to targeted improvements. As sand casting products vary in size and complexity—from small gears to large machinery components—the model can be scaled by adjusting parameters like furnace capacity or material types, making it versatile for diverse applications.
Moreover, the energy consumption model has direct economic implications. Reducing energy use by 1% in melting can save thousands of dollars annually for a medium-sized foundry, directly lowering the production cost of sand casting products. By incorporating energy pricing models, the HPN framework can evaluate cost trade-offs, such as balancing faster production speeds against higher power tariffs. This aligns with circular economy principles, where resource efficiency is paramount for sustainable sand casting products. Additionally, regulatory pressures on carbon emissions make such models invaluable for compliance and reporting, as they provide quantifiable data on energy intensity per unit of sand casting products.
In summary, the Hybrid Petri Net approach offers a powerful methodology for energy modeling in sand casting. Through detailed formulas, tables, and case studies, this paper demonstrates its accuracy and applicability. By repeatedly emphasizing sand casting products, the research highlights the end-goal: enabling manufacturers to produce high-quality components with minimal energy waste. As technology advances, integrating this model with IoT sensors and AI could lead to autonomous energy optimization, further revolutionizing the sand casting industry and its products.