In the manufacturing sector, energy efficiency and environmental sustainability have become critical concerns, driving the need for precise modeling and analysis of energy consumption in industrial processes. Sand casting, as a foundational method in metalworking, involves numerous discrete and continuous activities that collectively account for significant energy use. To address this, we propose a novel energy consumption calculation model based on Petri nets, tailored specifically for sand casting processes. This model integrates activity-based costing principles to describe and compute energy usage across process activities, enabling a granular understanding of where energy is consumed and how it can be optimized. By leveraging Petri nets, we can graphically represent the relationships between process activities, equipment, and resource consumption, facilitating both analysis and simulation. This approach is particularly relevant for sand casting services, which often face challenges in managing energy costs and reducing carbon footprints. In this article, we detail the development of this model, from analyzing energy consumption characteristics to implementing computational methods, and demonstrate its feasibility through a case study on sand molding. The goal is to provide a robust tool for sand casting services to enhance energy efficiency, support decision-making in process design, and contribute to greener manufacturing practices.
The energy consumption in sand casting processes is complex due to the interplay of multiple factors, including equipment operation, material handling, and process sequencing. Sand casting services typically involve over 40 distinct steps, organized into departments such as melting, molding, core-making, sand treatment, and cleaning. Among these, departments like molding, core-making, and sand treatment exhibit discrete industrial characteristics, where energy consumption is primarily driven by various casting equipment. Each department consumes energy through devices that transform raw materials into finished castings, with resources like electricity, coal, natural gas, and fuels being utilized in varying proportions over time. Understanding these dynamics is essential for sand casting services aiming to reduce energy waste and improve sustainability. To model this, we first analyze the energy consumption relationships and working time patterns in sand casting. The energy consumption object is mainly the equipment involved, and the process can be decomposed into activities that correspond to specific device operations. This decomposition allows for a focused examination of energy use at the activity level, which is crucial for identifying inefficiencies and implementing improvements in sand casting services.
Working time analysis is a key component in energy consumption modeling. In mechanical processing, time standards typically include basic time, setup time, preparation time, auxiliary time, and rest time. For sand casting, we define the duration during which a process activity acts on equipment as the working time of that activity. The overall system’s energy-related time encompasses both the activity working time (denoted as $t_a$) and the equipment running time (denoted as $t_s$). Equipment generally operates in five states: startup, preparation, running (processing), pause, and shutdown. When a process activity uses equipment, the device may transition through several of these states, meaning the activity working time includes intervals corresponding to specific equipment states. For example, in the melting department, a furnace’s operation involves startup preparation, melting initiation, melting completion, and molten metal tapping—the working time for melting is from initiation to completion, which aligns with the equipment’s running state, while preparation and tapping occur during pause states. The equipment running time $t_s$ can be expressed as:
$$ t_s = \sum_{i=1}^{n} (t_{s}^{r_i} + t_{s}^{p_i} + t_{s}^{o_i}) $$
where $t_{s}^{r_i}$ is the preparation time, $t_{s}^{p_i}$ is the running (processing) time, $t_{s}^{o_i}$ is the pause time (idle time), and $i$ represents the frequency count of each working time segment from start to end on that equipment. This breakdown helps in attributing energy consumption to specific states, which is vital for accurate modeling in sand casting services. To illustrate, Table 1 summarizes typical equipment states and their energy implications in sand casting processes.
| State | Description | Energy Impact | Typical Duration (hours) |
|---|---|---|---|
| Startup | Initial power-on and system checks | Low to moderate energy use | 0.05-0.1 |
| Preparation | Setup and calibration for processing | Moderate energy use | 0.1-0.3 |
| Running | Active processing or manufacturing | High energy use | Varies by activity |
| Pause | Idle or waiting between tasks | Low energy use (often standby) | 0.05-0.2 |
| Shutdown | Powering down equipment | Negligible energy use | 0.02-0.05 |
Building on this, we develop a process activity model for sand casting. From an energy consumption perspective, we decompose the casting process into basic activities termed “process activities.” Let $A_i$ denote the $i$-th process activity, where $i = 1, 2, \ldots, n$, so the set of process activities is $S(A) = \{A_1, A_2, \ldots, A_n\}$. Each activity has distinct influencing factors, and identifying key factors is crucial for analyzing energy consumption. Using methods like the Analytic Hierarchy Process (AHP), we determine key influencing factors $U_{\text{factor}}$, where each element has two attributes: factor name ($U_N$) and factor weight ($U_W$). Weights can be assigned via fuzzy Delphi AHP, ordering factors by weight to form a directed sequence:
$$ U_{\text{factor}} = ((U_{N1}, U_{W1}), (U_{N2}, U_{W2}), \ldots, (U_{Nj}, U_{Wj})) $$
with $U_{W1} \geq U_{W2} \geq \ldots \geq U_{W(j-1)} \geq U_{Wj}$. This sequence guides the consolidation of activities based on factors like equipment usage and functional dependencies, which is essential for optimizing sand casting services. The relationship between process activities and equipment is central to energy modeling. Each activity $A_i$ corresponds uniquely to a tuple of working time $t_s^i$ and specific equipment $De_i$, i.e., $A_i = \{t_s^i, De_i\}$. We define the activity-equipment relationship as $R_{\text{A-Device}} = \{R_0, R_1\}$, where $R_0$ indicates no relation and $R_1$ indicates a relation. The strength of this relationship is the equipment working time during the activity. For a set of all casting equipment $U_{\text{device}} = \{De_1, De_2, \ldots, De_j\}$ and a relation set $R$, we model the association as $R_{ij} = \{(De_j, A_j) | De_j \in U_{\text{device}}, A_j \in S(A), A_j \rightarrow De_j\}$, where $A_j \rightarrow De_j$ denotes that activity $A_j$ requires equipment $De_j$. This relation can be one-to-one or one-to-many; in the latter case, energy consumption is apportioned based on working times, a critical consideration for sand casting services managing multiple activities on shared equipment.
To graphically describe these relationships, we employ Petri nets, which are well-suited for modeling both discrete and continuous processes. Extended from basic Petri nets, our model for sand casting energy consumption is defined as:
$$ G = (P, T, I, O, K, W, M_0, D) $$
where $P$ is a finite set of places, categorized into six types: process activity places $P_a$, equipment places $P_d$, energy places $P_e$, material places $P_m$, pollutant places $P_p$, and other places $P_o$. $T$ is a finite set of transitions, with $P \cap T = \emptyset$. $I$ and $O$ are input and output functions, $M_0$ is the initial marking, $K$ is the capacity function, $D$ is a delay set including $D_P$ (place delays) and $D_T$ (transition delays), and $W$ is the arc weight function representing resource consumption and pollutant emissions. Key definitions for $W$ include: $W(p_i, t_j)$ for the arc weight from place $p_i$ to transition $t_j$; $W(t_j, p_i)$ for the reverse; $W(p_d, t) = t_s^o$ for equipment idle time; $W(t, p_d) = t_s^r$ for equipment preparation time; $W(p_m, t)$ for material consumption; $W(p_e, t)$ for energy consumption; and $W(t, p_p)$ for pollutant emissions. A process activity sequence $s_{pr} = \{p_{a1}, p_{a2}, \ldots, p_{ar}\}$ and its corresponding transition sequence $s_r = \{t_1, t_2, \ldots, t_r\}$ can be derived, with relations expressed as $R_{tp} = ((t), (p_a, p_d, p_e, p_m, p_p))$. This model enables a holistic view of energy flows in sand casting services, as depicted in the following conceptual diagram.
The Petri net model $G$ facilitates energy consumption calculation by linking activities, equipment, and energy intensity. In sand casting, the primary energy consumer is casting equipment, often using electricity. Suppose there are $m$ process activities and $n$ pieces of equipment. Let $A_i$ be the $i$-th activity ($i=1,2,\ldots,m$), $Q_j$ be the energy consumption per unit time for the $j$-th equipment ($j=1,2,\ldots,n$), and $X_{ij}$ be the energy consumed by activity $A_i$ on equipment $j$. For an activity $A_i$ involving multiple equipment, the energy consumption is:
$$ E_{SAi} = X_{i1} + X_{i2} + \ldots + X_{in} = t_{s}^{i1} Q_1 + t_{s}^{i2} Q_2 + \ldots + t_{s}^{in} Q_n $$
where $t_{s}^{in}$ is the working time of activity $i$ on equipment $n$. For cases where multiple activities share equipment, we apportion energy based on working time. If $n$ activities $S(A) = \{A_1, A_2, \ldots, A_n\}$ act on equipment $j$, with working times $t_s = \{t_{s}^{1}, t_{s}^{2}, \ldots, t_{s}^{n}\}$, the energy consumed by activity $k$ on equipment $j$ is:
$$ E_{DAk} = \frac{t_{s}^{k}}{\sum_{i=1}^{n} t_{s}^{i}} Q_j $$
The total energy consumption for a casting process, comprising all activities, is then:
$$ E_{pa} = \sum_{i=1}^{n} E_{SAi} + \sum_{k=1}^{n} E_{DAk} = \sum_{i=1}^{n} (t_{s}^{i} Q_i) + \sum_{k=1}^{n} \left( \frac{t_{s}^{k}}{\sum_{i=1}^{n} t_{s}^{i}} Q_k \right) $$
Expressing energy consumption in terms of equipment power, let $Pr_{pd}$ be the average power during preparation, $Pp_{pd}$ during processing, and $Po_{pd}$ during idle time. The energy consumption formula becomes:
$$ E_{pa} = \sum_{i=1}^{n} (Pr_{pd_i} t_{s}^{r_i} + Pp_{pd_i} t_{s}^{p_i} + Po_{pd_i} t_{s}^{o_i}) + \sum_{k=1}^{n} \left( Pr_{pd_k} \frac{t_{s}^{r_k}}{\sum_{i=1}^{n} t_{s}^{r_i}} + Pp_{pd_k} \frac{t_{s}^{p_k}}{\sum_{i=1}^{n} t_{s}^{p_i}} + Po_{pd_k} \frac{t_{s}^{o_k}}{\sum_{i=1}^{n} t_{s}^{o_i}} \right) $$
This formulation allows sand casting services to compute energy usage accurately by incorporating equipment states and activity dependencies. To validate the model, we apply it to a sand molding process, which is central to many sand casting services. The molding department involves steps such as sand filling, compaction, scraping, venting, drying, pattern removal, parting, flipping, core setting, closing, and auxiliary tasks. Based on factors like working time, function, equipment, and process sequence—weighted using AHP with results $U_{\text{factor}} = ((U_{N3}, 0.521), (U_{N2}, 0.22), (U_{N4}, 0.21), (U_{N1}, 0.049))$ where $U_{N3}$ is equipment and $U_{N4}$ is process sequence—we define six process activities: molding activity ($p_{a1}$), drying activity ($p_{a2}$), pattern removal and flipping activity ($p_{a3}$), closing activity ($p_{a4}$), auxiliary activity ($p_{a5}$), and core setting activity ($p_{a6}$). Corresponding equipment places include molding machine ($p_{d1}$), dryer ($p_{d2}$), pattern removal machine ($p_{d3}$), flipping machine ($p_{d4}$), conveyor ($p_{d5}$), closing machine ($p_{d6}$), pusher ($p_{d7}$), environmental device ($p_{d8}$), and core setter ($p_{d9}$). Energy places cover electricity ($p_{e1}$), water ($p_{e2}$), natural gas ($p_{e3}$), coal ($p_{e4}$); material places include sand ($p_{m1}$), coating ($p_{m2}$), auxiliary materials ($p_{m3}$), steel ($p_{m4}$); and pollutant places encompass dust ($p_{p1}$) and wastewater ($p_{p2}$). The Petri net model for this setup illustrates transitions and arcs with weights representing times and consumption, as described earlier.
From the model, the process activity sequence is $s_{pr} = \{p_{a1}, p_{a5}, p_{a2}, p_{a5}, p_{a3}, p_{a6}, p_{a4}\}$, with a corresponding transition sequence $s_r = \{t_1, t_2, t_3, t_4, t_5, t_6, t_7, t_8\}$. Relation matrices $R_{tp_a}$, $R_{tp_d}$, $R_{tp_e}$, $R_{tp_m}$, and $R_{tp_p}$ can be derived to map transitions to places. For energy calculation, we consider both independent energy consumption $E_{Spa}$ and shared energy consumption $E_{Dpa}$. Using data from a typical sand casting service, Table 2 lists key equipment parameters for the molding process.
| Equipment Place | Symbol | Average Power (kW) | Average Processing Time (hours) |
|---|---|---|---|
| Molding Machine | $p_{d1}$ | 21 | 0.260 |
| Dryer | $p_{d2}$ | 9 | 0.100 |
| Pattern Removal Machine | $p_{d3}$ | 6 | 0.015 |
| Flipping Machine | $p_{d4}$ | 4 | 0.015 |
| Conveyor | $p_{d5}$ | 6 | Varies (see below) |
| Closing Machine | $p_{d6}$ | 4 | 0.015 |
| Pusher | $p_{d7}$ | 5 | 0.015 |
| Environmental Device | $p_{d8}$ | 12 | 0.350 |
| Core Setter | $p_{d9}$ | 2 | 0.020 |
Assuming negligible startup times and simplified states, independent energy consumption is computed as:
$$ E_{Spa1} = Pp_{pd1} t_{s}^{p1} + Pp_{pd8} t_{s}^{p8} = 21 \times 0.26 + 12 \times 0.35 = 9.66 \text{ kW} $$
$$ E_{Spa2} = Pp_{pd2} t_{s}^{p2} = 9 \times 0.1 = 0.9 \text{ kW} $$
$$ E_{Spa3} = Pp_{pd3} t_{s}^{p3} = 6 \times 0.015 = 0.09 \text{ kW} $$
$$ E_{Spa4} = Pp_{pd4} t_{s}^{p4} = 4 \times 0.015 = 0.06 \text{ kW} $$
$$ E_{Spa6} = Pp_{pd6} t_{s}^{p6} + Pp_{pd9} t_{s}^{p9} = 4 \times 0.015 + 2 \times 0.02 = 0.1 \text{ kW} $$
$$ E_{Spa} = 9.66 + 0.9 + 0.09 + 0.06 + 0.1 = 18.91 \text{ kW} $$
For the auxiliary activity $p_{a5}$ (conveyor operation), it is invoked three times between activities: $p_{a1} \to p_{a2}$, $p_{a2} \to p_{a3}$, and $p_{a3} \to p_{a4}$. With pause times of 0.035 h, 0.025 h, and 0.03 h, and running times of 0.1 h, 0.15 h, and 0.08 h respectively, shared energy consumption is:
$$ E_{D(p_{a1} \to p_{a2})} = Po_{pd5} \frac{t_{s}^{o1}}{t_{s}^{o1} + t_{s}^{o2} + t_{s}^{o3}} + Pp_{pd5} \frac{t_{s}^{p1}}{t_{s}^{p1} + t_{s}^{p2} + t_{s}^{p3}} = 3.5 \times \frac{0.035}{0.09} + 6 \times \frac{0.1}{0.33} = 3.180 \text{ kW} $$
$$ E_{D(p_{a2} \to p_{a3})} = 3.5 \times \frac{0.025}{0.09} + 6 \times \frac{0.15}{0.33} = 3.7 \text{ kW} $$
$$ E_{D(p_{a3} \to p_{a4})} = 3.5 \times \frac{0.03}{0.09} + 6 \times \frac{0.08}{0.33} = 2.62 \text{ kW} $$
$$ E_{Dpa} = 3.180 + 3.7 + 2.62 = 9.5 \text{ kW} $$
Total energy consumption for the molding line is:
$$ E = E_{Spa} + E_{Dpa} = 18.91 \text{ kW} + 9.5 \text{ kW} = 28.41 \text{ kW} $$
This breakdown reveals that activities $p_{a1}$ (molding) and $p_{a5}$ (auxiliary) account for approximately 67.4% of total energy use, highlighting areas for potential optimization in sand casting services. Such insights empower sand casting services to target high-consumption activities, implement energy-saving measures, and enhance overall efficiency. The model’s applicability extends beyond molding to other departments like melting or core-making, making it a versatile tool for comprehensive energy management in sand casting services.
In conclusion, the Petri net-based energy consumption model presented here offers a robust framework for analyzing and computing energy usage in sand casting processes. By decomposing processes into activities and leveraging graphical Petri net representations, we can accurately capture the interactions between equipment, resources, and energy flows. The calculation methods derived from the model enable detailed energy assessments, facilitating identification of inefficiencies and supporting节能减排 initiatives. For sand casting services, this approach is invaluable in promoting低碳 casting and sustainable manufacturing, as it provides actionable data for process redesign, equipment upgrades, and operational improvements. Future work could integrate real-time monitoring and simulation to further enhance the model’s predictive capabilities, ultimately helping sand casting services achieve greater energy efficiency and environmental compliance in an increasingly eco-conscious market.