The manufacturing industry faces increasing pressure to enhance energy efficiency and reduce its environmental footprint. Within this sector, foundry operations, particularly those producing sand casting parts, are recognized as significant consumers of energy and sources of emissions. To effectively manage and mitigate this impact, a precise and dynamic method for quantifying energy consumption at the granular level of process activities is essential. Traditional assessment methods often lack the formal structure to capture the complex, discrete-event nature of the interactions between equipment, resources, and process steps in a casting line. This work presents a formal modeling framework that integrates the concept of process activities with Petri net theory to describe, analyze, and calculate the energy consumption of sand casting parts production. By providing a graphical and mathematical representation, this model allows for a detailed breakdown of energy flows, identifying key consumption points and enabling targeted strategies for energy savings in the manufacture of sand casting parts.
Energy Consumption Characteristics in Sand Casting
The production of sand casting parts is a multi-stage process involving several major departments: melting, molding, core making, sand preparation, and finishing/cleaning. Among these, departments like molding, core making, and sand handling exhibit characteristics typical of discrete manufacturing, where operations are event-driven and asynchronous. The primary energy-consuming entities in this system are the various pieces of equipment (e.g., molding machines, furnaces, conveyors). Each department executes a sequence of processes, which are themselves composed of finer-grained activities. These activities, when performed by equipment, consume energy resources such as electricity, natural gas, and compressed air. The total energy consumption for sand casting parts is thus an aggregate of the energy consumed by all equipment across all activities over time.
A critical aspect of modeling energy use is understanding equipment operating states and their corresponding time durations. Generally, equipment can be in one of several states: Startup, Preparation (or Ready), In-Operation (Processing), Idle (Pause), and Shutdown. The relevant time periods for energy calculation are the Process Activity Work Time ($t_a$) and the broader Equipment Run Time ($t_s$).
Definition 1: Process Activity Work Time. The duration for which a sand casting process activity acts upon a piece of equipment is defined as its work time.
An activity’s work time may correspond to only a subset of the equipment’s state durations. For instance, a melting activity’s work time might align with the furnace’s “In-Operation” state, while loading and tapping are associated with “Idle” or “Preparation” states. The total run time of a piece of equipment can be expressed as the sum of its state durations over all cycles:
$$ t_s = \sum_{i=1}^{n} (t_{s_{ri}} + t_{s_{pi}} + t_{s_{oi}}) $$
where $t_{s_{ri}}$ is the preparation time, $t_{s_{pi}}$ is the in-operation (processing) time, $t_{s_{oi}}$ is the idle (pause) time for the $i$-th cycle, and $n$ is the total number of cycles from start to finish.
Process Activity Modeling for Sand Casting
To enable precise energy accounting, the sand casting process is decomposed from an energy perspective into fundamental units called Process Activities.
Definition 2: Sand Casting Process Activity. A process activity is defined as the basic, energy-relevant action performed during a sand casting process step.
The set of all activities in a process can be denoted as $S(A) = \{A_1, A_2, …, A_n\}$. Each activity $A_i$ is uniquely associated with parameters such as its work time and the specific equipment it uses, forming a tuple $A_i = \{t_{a_i}, De_i\}$.
Defining these activities requires analyzing the correlations between traditional process steps based on key factors. The most influential factors are determined, and their weights $(U_W)$ can be established using methods like the Analytic Hierarchy Process (AHP). This results in an ordered vector of influencing factors:
$$ U_{factor} = ((U_{N1}, U_{W1}), (U_{N2}, U_{W2}), …, (U_{Nj}, U_{Wj})) $$
where $U_{W1} \geq U_{W2} \geq … \geq U_{Wj}$.
For sand casting, analyses typically show that “Equipment Used” and “Process Sequence/Functional Dependency” are the highest-weighted factors, guiding the logical grouping of steps into activities.
The relationship between an activity and equipment is central to energy modeling. Let $U_{device} = \{De_1, De_2, …, De_j\}$ be the set of all equipment. The association between an activity $A_j$ and equipment $De_j$ is defined as a relation $R_{ij} = \{(De_j, A_j) | De_j \in U_{device}, A_j \in S(A), A_j \rightarrow De_j \}$, where $A_j \rightarrow De_j$ signifies that activity $A_j$ requires equipment $De_j$. The “strength” of this relationship is quantified by the activity’s work time on that equipment. Two primary scenarios exist:
- One-to-One: A single activity uses one specific piece of equipment. The relationship strength is simply the activity’s work time.
- Many-to-One: Multiple activities utilize the same piece of equipment (e.g., a conveyor used between different stations). In this case, the equipment’s energy consumption must be allocated to each activity. A common method is pro-rata allocation based on the work time each activity claims on the shared equipment.
Petri Net-Based Energy Consumption Model (Model G)
To graphically and formally describe the discrete sand casting process with its resource flows, an extended Petri net model, termed Model G, is constructed. This model integrates the concepts of process activities, resources, and energy consumption.
$$ G = (P, T, I, O, K, W, M_0, D) $$
Where:
- $P$ is a finite set of places, categorized into: Process Activity places ($P_a$), Equipment places ($P_d$), Energy Resource places ($P_e$), Material places ($P_m$), Pollutant places ($P_p$), and Other places ($P_o$).
- $T$ is a finite set of transitions representing events or state changes (e.g., start/end of an activity).
- $I$ and $O$ are input and output functions.
- $M_0$ is the initial marking.
- $K$ is the capacity function of places (often set to infinite).
- $D$ is a set of time delays associated with places and transitions.
- $W$ is the arc weight function, crucial for quantifying resource consumption and emissions.
Specific definitions for the arc weight function $W$ in the context of energy modeling are established:
- Definition 5-6: General arc weight between a place $p_i$ and a transition $t_j$ (or vice versa).
- Definition 7: $W(p_d, t)$: The idle time of equipment ($t_{s_o}$) required to enable the firing of transition $t$.
- Definition 8: $W(t, p_d)$: The preparation time ($t_{s_r}$) of the equipment after a transition fires.
- Definition 9: $W(p_m, t)$: The quantity of material consumed when transition $t$ fires.
- Definition 10: $W(p_e, t)$: The quantity of energy resource (e.g., kWh of electricity) consumed when transition $t$ fires.
- Definition 11: $W(t, p_p)$: The quantity of pollutant emitted when transition $t$ fires.
This structure allows for a clear representation of how a sequence of activities (a process) consumes energy and materials while using equipment and generating waste.
Definition 13: A sequence of process activity places is called a Sand Casting Process Activity Sequence, $S_{pr} = \{p_{a1}, p_{a2}, …, p_{ar}\}$.
Correspondingly, the firing sequence of transitions from an initial marking $M_0$ to a final marking $M_r$ is $S_r = \{t_1, t_2, …, t_r\}$.
Definition 14: The relationship between a transition $t$ and the various resource places is encapsulated in a relation $R_{tp} = ((t), (p_a, p_d, p_e, p_m, p_p))$, which can be represented using matrices for analytical purposes.
Energy Consumption Calculation Based on Model G
Model G provides the foundation for calculating the energy consumption of producing sand casting parts. The calculation is derived from the relationship between activities, equipment operating times, and equipment power ratings.
Let $A_i$ be the $i$-th type of process activity ($i = 1, 2, …, m$). Let $Q_j$ be the energy consumption rate (e.g., average power in kW) of the $j$-th piece of equipment ($j = 1, 2, …, n$). If an activity $A_i$ uses equipment in a one-to-one manner, its energy consumption $ES_{A_i}$ is:
$$ ES_{A_i} = t_{s_{i1}}Q_1 + t_{s_{i2}}Q_2 + … + t_{s_{in}}Q_n = \sum_{j=1}^{n} t_{s_{ij}} Q_j $$
where $t_{s_{ij}}$ is the work time of activity $A_i$ on equipment $j$.
For the more general case where $n$ activities share a single piece of equipment $j$, the energy consumption must be allocated. Based on the pro-rata principle, the energy allocated to the $k$-th activity ($A_k$) from shared equipment $j$ is:
$$ ED_{A_k} = \frac{t_{s_{k}}}{\sum_{i=1}^{n} t_{s_{i}}} \cdot Q_j $$
Therefore, the total energy consumption for a sand casting process, which is the sum of all dedicated and shared equipment energy usage across all activities, is given by:
$$ E_{pa} = \sum_{i=1}^{n} ES_{A_i} + \sum_{k=1}^{n} ED_{A_k} = \sum_{i=1}^{n} (t_{s_{i}} Q_i) + \sum_{k=1}^{n} \left( \frac{t_{s_{k}}}{\sum_{i=1}^{n} t_{s_{i}}} Q_k \right) $$
Refining this with detailed equipment state power ratings leads to a more precise formulation. Let $Pr_{pd}$ be the average power during equipment Preparation, $Pp_{pd}$ be the average power during In-Operation (Processing), and $Po_{pd}$ be the average power during Idle (Off-load) states. The total process energy calculation becomes:
$$
\begin{aligned}
E_{pa} = & \sum_{i=1}^{n} \left( Pr_{pd_i} t_{s_{ri}} + Pp_{pd_i} t_{s_{pi}} + Po_{pd_i} t_{s_{oi}} \right) + \\
& \sum_{k=1}^{n} \left( Pr_{pd_k} \frac{t_{s_{rk}}}{\sum_{i=1}^{n} t_{s_{ri}}} + Pp_{pd_k} \frac{t_{s_{pk}}}{\sum_{i=1}^{n} t_{s_{pi}}} + Po_{pd_k} \frac{t_{s_{ok}}}{\sum_{i=1}^{n} t_{s_{oi}}} \right)
\end{aligned}
$$
This formula allows for a comprehensive and accurate assessment of energy consumption, accounting for different operational modes of equipment involved in making sand casting parts.
Application Case: Energy Analysis of a Molding Line
To validate the proposed framework, it is applied to a sand molding department, a core area for producing sand casting parts. The primary process steps include filling, compaction, stripping, core setting, mold closing, and various auxiliary transfer steps.
Using the AHP method with factors like Work Time, Function, Equipment, and Process Sequence, the weight vector was determined as $U_{factor} = ((Equipment, 0.521), (Function, 0.22), (Sequence, 0.21), (Work Time, 0.049))$. Guided by the high-weight factors (Equipment and Sequence), the molding process is aggregated into six key activities:
- Molding Activity ($pa_1$)
- Drying Activity ($pa_2$)
- Stripping & Turning Activity ($pa_3$)
- Mold Closing Activity ($pa_4$)
- Auxiliary/Transfer Activity ($pa_5$)
- Core Setting Activity ($pa_6$)
Following the definitions of Model G, the corresponding Petri net places are established for equipment, energy, materials, and pollutants. The resulting Petri net model visually maps the flow of activities and resources. From the model, the identified process activity sequence is $S_{pr} = \{pa_1, pa_5, pa_2, pa_5, pa_3, pa_6, pa_4, pa_5\}$. The auxiliary activity ($pa_5$) is invoked multiple times between other main activities.
For energy calculation, equipment data from a representative foundry producing sand casting parts is used. The primary equipment and their parameters are summarized below:
| Equipment Place | Equipment Description | Avg. Operating Power, $Pp_{pd}$ (kW) | Avg. Process Time, $t_{s_p}$ (h) |
|---|---|---|---|
| $pd_1$ | Molding Machine | 21.0 | 0.260 |
| $pd_2$ | Drying Machine | 9.0 | 0.100 |
| $pd_3$ | Stripping Machine | 6.0 | 0.015 |
| $pd_4$ | Turning Machine | 4.0 | 0.015 |
| $pd_5$ | Conveyor System | 6.0 | — |
| $pd_6$ | Mold Closing Machine | 4.0 | 0.015 |
| $pd_7$ | Pushing Machine | 5.0 | 0.015 |
| $pd_8$ | Dust Collector | 12.0 | 0.350 |
| $pd_9$ | Core Setter | 2.0 | 0.020 |
Assuming integrated machinery minimizes idle states between connected steps, and neglecting short start-up times, the energy for dedicated activities ($ES_{pa}$) simplifies to their processing power consumption. The shared conveyor ($pd_5$) has its total energy allocated across its three invocations ($pa_1 \rightarrow pa_2$, $pa_2 \rightarrow pa_3$, $pa_3 \rightarrow pa_4$).
Dedicated Activity Energy ($ES_{pa}$):
$ES_{pa_1} = Pp_{pd_1} \cdot t_{s_{p1}} + Pp_{pd_8} \cdot t_{s_{p8}} = (21 \times 0.26) + (12 \times 0.35) = 9.66 \text{ kW}$
$ES_{pa_2} = 9 \times 0.10 = 0.90 \text{ kW}$
$ES_{pa_3} = 6 \times 0.015 = 0.09 \text{ kW}$
$ES_{pa_4} = 4 \times 0.015 = 0.06 \text{ kW}$
$ES_{pa_6} = (4 \times 0.015) + (2 \times 0.02) = 0.10 \text{ kW}$
$ES_{pa} = 9.66 + 0.90 + 0.09 + 0.06 + 0.10 = 10.81 \text{ kW}$
Shared Activity (Conveyor) Energy Allocation ($ED_{pa}$):
The conveyor has an idle power $Po_{pd_5} = 3.5$ kW and operating power $Pp_{pd_5} = 6.0$ kW. Measured idle and operating times for its three segments are:
| Segment | Idle Time, $t_{s_o}$ (h) | Operating Time, $t_{s_p}$ (h) |
|---|---|---|
| $pa_1 \rightarrow pa_2$ | 0.035 | 0.100 |
| $pa_2 \rightarrow pa_3$ | 0.025 | 0.150 |
| $pa_3 \rightarrow pa_4$ | 0.030 | 0.080 |
| Total | $ \Sigma t_{s_o} = 0.09$ | $ \Sigma t_{s_p} = 0.33$ |
Applying the allocation formula $ED_{A_k} = Po_{pd} \frac{t_{s_{ok}}}{\Sigma t_{s_o}} + Pp_{pd} \frac{t_{s_{pk}}}{\Sigma t_{s_p}}$:
$ED_{(pa1 \rightarrow pa2)} = 3.5 \times \frac{0.035}{0.09} + 6.0 \times \frac{0.100}{0.33} \approx 3.180 \text{ kW}$
$ED_{(pa2 \rightarrow pa3)} = 3.5 \times \frac{0.025}{0.09} + 6.0 \times \frac{0.150}{0.33} \approx 3.700 \text{ kW}$
$ED_{(pa3 \rightarrow pa4)} = 3.5 \times \frac{0.030}{0.09} + 6.0 \times \frac{0.080}{0.33} \approx 2.620 \text{ kW}$
$ED_{pa} = ED_{pd_5} = 3.180 + 3.700 + 2.620 = 9.50 \text{ kW}$
Total Process Energy Consumption:
$$ E_{total} = ES_{pa} + ED_{pa} = 10.81 \text{ kW} + 9.50 \text{ kW} = 20.31 \text{ kW} $$
This calculation reveals the energy profile for the molding line segment producing sand casting parts. The analysis shows that the Molding Activity ($pa_1$) and the Auxiliary Transfer Activity ($pa_5$) together account for a dominant share of the total energy consumption (approximately 94% in this case), clearly highlighting the key areas for potential energy-saving interventions. This level of detail is invaluable for process engineers aiming to optimize the energy efficiency of manufacturing sand casting parts.
Conclusion
The modeling framework presented in this work addresses the need for a structured, analytical approach to energy consumption in sand casting, a vital process for producing metal components. By decomposing the process into fundamental energy-related activities and employing an extended Petri net (Model G), it becomes possible to create a dynamic and graphical representation of the entire system. This model formally captures the relationships between process sequences, equipment states, resource consumption, and emissions. The derived energy calculation formulas, which account for both dedicated and shared equipment usage across different operational states, provide a practical and accurate method for quantifying the energy footprint of producing sand casting parts. The application to a molding line demonstrates the framework’s feasibility and utility in identifying major energy-consuming activities, thereby offering a powerful tool for foundries to diagnose, analyze, and ultimately reduce energy consumption and environmental impact in their pursuit of sustainable manufacturing.