In the mechanical industry, sand casting is a foundational process for producing complex metal parts, but it is also highly energy-intensive, accounting for 25% to 30% of total energy consumption in the sector. Among its various procedures, melting is the most energy-consuming step, often exceeding 50% of the total energy use. This high energy demand, coupled with low efficiency rates averaging around 17%, underscores the critical need for effective energy management. My research focuses on developing a robust energy consumption model for the melting procedure in sand casting, leveraging Hybrid Petri Nets (HPN) to capture both discrete control actions and continuous energy flows. This approach aims to provide insights for optimizing energy usage and reducing costs in manufacturing sand casting parts.
The melting process in sand casting involves several key stages: charging, melting, deoxidizing, and tapping. Each stage exhibits distinct energy characteristics. Charging involves loading raw materials like iron scraps into a furnace, typically using equipment such as chargers, and consumes minimal electrical energy. Melting, where materials are heated to high temperatures (e.g., above 1200°C for cast iron), requires maximum power and dominates energy consumption. 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, and also has low energy usage. Understanding these stages is essential for accurate modeling, as the intermittent control signals (discrete) and continuous energy consumption patterns must be integrated. This dual nature makes HPN an ideal tool for representing such hybrid systems, enabling precise simulation of energy dynamics in producing sand casting parts.
Hybrid Petri Nets extend traditional Petri nets by incorporating both discrete and continuous elements, making them suitable for systems like sand casting melting. An HPN model is defined as a seven-tuple: HPN = {P, T, Pre, Post, D, S, M0}. Here, P = Pd ∪ Pc, where Pd is the set of discrete places representing control signals, and Pc is the set of continuous places, further divided into energy places (Pce) and process places (Pcp). T = Td ∪ Tc, with Td as discrete transitions (including timed transitions Tt and stochastic transitions Te) for control delays, and Tc as continuous transitions for energy consumption and process flows. The matrices Pre and Post define arc weights and directions, D specifies delays for discrete transitions, S sets speed bounds for continuous transitions, and M0 is the initial marking. This formalism allows me to model the melting procedure’s start-stop controls and real-time energy usage, crucial for analyzing the production of sand casting parts.
To calculate energy consumption in the HPN model, I derive formulas based on transition firing speeds. For a continuous transition tci representing a sub-process like melting, the cumulative energy consumption at time τ is given by:
$$ E_{ci} = C(p_e, t_{ci}) \int_0^\tau v_{ci}(t) \, dt $$
where C(p_e, t_{ci}) is the energy coefficient, defined as the ratio of energy consumption speed v_e to process speed v_{ci}:
$$ C(p_e, t_{ci}) = \frac{v_e}{v_{ci}} $$
In sand casting melting, since power input and heating rates are relatively stable, C(p_e, t_{ci}) can be treated as constant for each sub-process. For an entire process T consisting of k sub-processes T = {t_{c1}, t_{c2}, …, t_{ck}}, each with duration k_{ci} and average speed v_{ci}, the total energy consumption is:
$$ E_T = \sum_{i=1}^k C(p_e, t_{ci}) \cdot k_{ci} \cdot v_{ci} $$
This formula enables quantitative analysis of energy usage across different stages, helping identify inefficiencies in manufacturing sand casting parts. For instance, in melting, high energy coefficients highlight areas for potential optimization.
I applied this HPN model to a real-world sand casting melting process at an industrial facility. The setup uses a medium-frequency induction furnace with a capacity of 10 tons and a maximum power of 6,000 kW, typical for producing large sand casting parts. The HPN model for this process includes discrete places for control signals, continuous places for energy and buffers, and transitions for each sub-process. Key parameters are summarized in the table below:
| Place/Transition | Meaning |
|---|---|
| pd1, pd2, pd3, pd4 | Control signals for charging, melting, deoxidizing, tapping |
| td1, td2, td3, td4 | Discrete transitions for process switching with delays |
| pdm1 | Raw material input (e.g., iron scraps for sand casting parts) |
| pdm2 | Deoxidizing agent input |
| pcm2, pcm3, pcm4 | Buffer places between sub-processes |
| pcm5 | Output place for finished molten metal |
| pe | Total energy consumption place |
| tc1, tc2, tc3, tc4 | Continuous transitions for charging, melting, deoxidizing, tapping |
The continuous transitions have firing speed ranges: tc1 ∈ [57, 62] t/h for charging, tc2 ∈ [9, 10] t/h for melting, tc3 ∈ [42, 44] t/h for deoxidizing, and tc4 ∈ [85, 90] t/h for tapping. Discrete transitions have delays: d1 = 0.17 h for charging-to-melting, d2 = 1.1 h for melting-to-deoxidizing, d3 = 0.24 h for deoxidizing-to-tapping, and d4 = 0.12 h for tapping-to-charging. The association matrices define the flow relationships. For continuous places and transitions, the forward and backward matrices are:
$$ \text{Pre}_{cc} = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{bmatrix}, \quad \text{Post}_{cc} = \begin{bmatrix} 1.7 & 543.1 & 5.6 & 2.2 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
Here, the first row corresponds to the energy place pe, with coefficients representing energy consumption rates per ton for each sub-process. For example, the coefficient 543.1 for tc2 (melting) indicates high energy use per ton, critical for sand casting parts production. Other rows represent buffer places with unitary weights to maintain material balance. Discrete associations are defined similarly, ensuring control logic integrity.
Using the Hypens plugin in MATLAB, I implemented and simulated this HPN model. The simulation tracked energy consumption over time, producing a curve that mirrors the actual melting process. The results show distinct segments: charging (0–A) with low slope due to minimal power, melting (A–B) with steep slope reflecting high energy use, deoxidizing (B–C) with reduced slope, and tapping (C onward) with near-flat slope. The total model energy consumption was 7,122 kWh, compared to an actual measurement of 7,198 kWh, yielding a relative error of 1.07%. This close match validates the HPN model’s accuracy in capturing energy dynamics for sand casting melting. The simulation also allows sensitivity analysis; for instance, varying melting speeds affects total energy, highlighting optimization opportunities for producing sand casting parts efficiently.
To further analyze the model, I derived energy consumption per ton for sand casting parts. Using the energy coefficients, the melting stage consumes approximately:
$$ E_{\text{melting}} = C(p_e, t_{c2}) \cdot v_{c2} \cdot k_{c2} = 543.1 \, \text{kWh/t} \times 10 \, \text{t/h} \times 1.1 \, \text{h} = 5,974.1 \, \text{kWh} $$
This accounts for over 80% of total energy, emphasizing its dominance. In contrast, charging consumes about 1.7 kWh/t, deoxidizing 5.6 kWh/t, and tapping 2.2 kWh/t. These values can be tabulated for clarity:
| Sub-process | Energy Coefficient (kWh/t) | Typical Speed (t/h) | Energy per Batch (kWh for 10 tons) |
|---|---|---|---|
| Charging | 1.7 | 60 | 102 |
| Melting | 543.1 | 10 | 5,974.1 |
| Deoxidizing | 5.6 | 43 | 240.8 |
| Tapping | 2.2 | 87.5 | 192.5 |
This breakdown aids in pinpointing high-consumption stages for sand casting parts. Moreover, the HPN model can be extended to include environmental factors, such as heat loss or equipment efficiency, by adding continuous places for temperature or power metrics. For example, the energy balance during melting can be expressed as:
$$ P_{\text{input}} = P_{\text{melting}} + P_{\text{loss}} $$
where Pinput is electrical power, Pmelting is useful heating power, and Ploss is thermal loss. Integrating this into the HPN framework enhances model realism for sand casting applications.
The flexibility of HPN models also supports scenario analysis. For instance, by adjusting transition speeds or delays, I can simulate different operational strategies, such as preheating materials to reduce melting energy. Suppose preheating increases charging energy by 20% but cuts melting energy by 15%. The modified energy consumption becomes:
$$ E_{\text{new}} = (1.2 \times E_{\text{charging}}) + (0.85 \times E_{\text{melting}}) + E_{\text{deoxidizing}} + E_{\text{tapping}} $$
Plugging in values from Table 2 for a 10-ton batch: Echarging = 102 kWh, Emelting = 5,974.1 kWh, etc., yields a total of 5,330.5 kWh, a significant reduction. This illustrates how HPN models can guide energy-saving interventions in sand casting parts manufacturing.
In conclusion, the HPN-based energy consumption model effectively captures the hybrid nature of sand casting melting procedures, combining discrete control and continuous energy flows. My implementation demonstrates high accuracy in simulating real-world processes, with errors below 2%. This model provides a foundation for optimizing energy use, reducing costs, and enhancing sustainability in producing sand casting parts. Future work could integrate machine learning for predictive control or expand the model to entire foundry lines, covering multiple sand casting parts simultaneously. By leveraging HPN’s versatility, industries can achieve finer energy management and contribute to greener manufacturing practices.