Introduction

The manufacturing industry accounts for a significant part of the world’s energy consumption and waste and is responsible for nearly one third of global energy consumption and 29.2% of global electricity-related CO2 emissions (Dunham 2015; Sieminski 2016). Energy efficiency has become one of the key factors of the manufacturing industry (Anderberg et al. 2010). Die-casting processes are widely known for being highly energy demanding. Approximately 25% of the total cost for die-casting products is associated to energy consumption (Brevick et al. 2004). In many countries, legislations have become stricter and stricter over the years regarding the energy consumption and emissions of die-casting processes (Salonitis et al. 2016a). Production managers face great challenges when trying to implement energy efficiency initiatives under productivity concerns (Mohr et al. 2012; Trianni et al. 2013). It is also noted that increasing resource utilization during production could reduce the amount of energy consumption in die-casting processes by 20 to 30% (Shao 2017; Trianni and Cagno 2012).

Researchers suggest that the efforts to reduce energy consumption should start with evaluating the energy consumption of manufacturing systems more accurately (Cao et al. 2012; He et al. 2017; Zhong et al. 2016). However, traditional methods of studying energy consumption of die-casting processes heavily depend on experiments. Parameter calculation for resulting models are widely considered weak due to the lack of the needed data or model validation. For high-energy consumption processes like die casting, modeling their energy consumption is complex, time-consuming, and challenging. Specifically, aluminum casting has experienced continuous growth (Das and Yin 2007; Heinemann 2016) and dominates the nonferrous sector in general, comprising 78% of total nonferrous shipments (Rosen and Lee 2009). Some researchers also pointed out that the energy consumption of the aluminum casting process is of the order of 6–17 GJ per ton in using electricity and natural gas, which means the order of 36–100 billion GJ for the global industry in 2017. Salonitis et al. (2017) stated that there are huge opportunities for the metal casting industry to adopt the best energy practices based on energy modeling. Therefore, energy modeling and efficiency analysis of aluminum die-casting processes are crucial for the energy efficiency of the manufacturing industry.

Using real data of multiple machines and products at aluminum die-casting workshops through an energy data-acquiring system, this research built the mathematical relationship between specific energy consumption (SEC), including both gas and electricity, and production rate for aluminum die-casting processes. The high energy efficiency zone was defined after various nonlinear regression methods were investigated. The performance of different methods was compared to obtain the optimum one, which was further validated by experiments. The modeling results can be used to analyze the energy efficiency of aluminum die-casting workshops and further to support production scheduling with consideration on energy usage. The study showed that the modeling results can lead to 10 to 15% of energy savings without sacrificing profits. All the data-acquiring and case studies were carried out in die-casting workshops of a furniture manufacturer in Zhejiang Province, China.

Literature review

Energy consumption of die-casting processes has been long studied in literature. Much of the research focused on energy consumption using measurement and statistical methods. For example, Lazzarin and Noro (2015) presented energy audits performed in five Italian cast iron foundries and identified energy utilization in various processes, from iron melting to the end of casting. They further discussed energy efficiency opportunities in service plants of cast iron foundries in Italy (Lazzarin and Noro 2016). A report of the Office of Scientific and Technical Information Technical (Schwam 2012) addressed multiple aspects of the aluminum smelting and handling in die operations for energy efficiency improvement based on data from North American Die Casting Association (NADCA). Eglins and Röders (2005) presented a model about the overall heat transfer within an aluminum die-casting cell to find measures for an energy demand reduction. DETR (Department of Environment 1997) analyzed the energy ratio of the key energy-consuming equipment and phases in the casting process. However, energy consumption data are scattered, inconsistent, and not helpful to guide energy-saving measures.

Simulation and visualization techniques have been used by several researchers to calculate and compare the energy and material flows without physical implementation. For example, Salonitis et al. (2017) presented a case study of selecting energy- and resource-efficient casting processes based on simulation. Mishra and Sharma (2017) investigated melting of the bulk Al-7039 alloy during in situ microwave casting through experimental trials in a multi-mode applicator cavity and numerical simulation. Pagone et al. (2016) developed a simulation tool to undertake a systematic analysis of energy and material flows in the casting process. Henninger et al. (2016) conducted simulations based on studies of energy-saving measures in the aluminum tooling and die-casting industry. Krause et al. (2012) provided a discrete event simulation (DES)-based model of an aluminum die-casting process to represent energy-oriented material flows. Singh et al. (2012) proposed a new computer-aided system named Sustainability Analyzer for die-casting processes using three sustainability indicators including energy use, solid water, and carbon emissions. However, the energy data for simulations are mostly based on earlier data and statistical analysis.

Meanwhile, the overall life cycles of products made by die-casting processes are also common research topics. For example, Yilmaz et al. (2015) used life cycle assessment (LCA) as a decision support tool to evaluate best available techniques (BATs) for cleaner production of iron casting. Mitterpach et al. (2017) conducted environmental evaluation of gray cast iron via LCA. LCA is mostly used under certain conditions, such as neglecting complicated real-time situations and variations of the parameters, and does not provide accurate energy data. There is also much research using modeling methods to look into the relation between energy consumption and other process parameters. From the process viewpoint, Selvaraj et al. (2017) presented mathematical modeling of raw material preheating in metal casting processes. Børset et al. (2016) explored the potential for waste heat recovery during metal casting with thermoelectric generators by using on-site experiments and mathematical modeling. Salonitis et al. (2016b) pointed out that constrained rapid induction melting single shot up-casting (CRIMSON) has advantages compared to conventional sand casting processes regarding energy savings. Watkins et al. (2013) used the sustainability characterization methodology to evaluate the sustainability of die-casting manufacturing processes. Thiriez and Gutowski (2006) studied the energy consumption of an injection modeling process and created a model to represent the SEC trends exhibited by each machine. However, the applications of process improvement for energy savings are often restricted because of the difficulty of its implementation and expansion. Respectively, from the operational viewpoint, Sharma et al. (2017) conducted the sustainability modeling of die-casting processes based on the process plan information. Bettoni and Zanoni (2011) presented a power model focusing on the relationship between SEC and the production rate in order to improve energy efficiency in die-casting processes. The model was based on previous work by Gutowski et al. (2006). However, the data provided by them are insufficient and there is no clear evidence to support that the power form used by them is the optimum one to characterize the regressive relation between SEC and production rate.

In summary, energy modeling and efficiency analysis of die-casting processes and the relationship with production rate deserve further study.

Energy data acquisition and analysis system of aluminum die-casting processes

This research collected energy consumption data for die-casting processes at a typical die-casting workshop with two casting machines. Each machine was designed to produce two product types with corresponding types of dies equipped. They are automatic, middle-sized, horizontal cold chamber die-casting machines using electricity as their power supply. Each was also equipped with a continuous furnace using natural gas. During data collection, the workshop was under full operations with three shifts per day and 6 days per week. In order to maintain their temperature, the furnaces were continuously operated except cases of emergency and necessary maintenance. We developed an energy data acquisition and analysis system with four main modules. The first module consisted of digital meters for measuring energy consumption, including voltage and current sensors for electricity and flow sensors for gas. In addition, infrared sensors were used to measure the production rate, and digital platform scales were used to obtain weight of used materials. Once the real-time energy and other data were collected by the first module, they were transferred to the second module by the standard serial communication protocol of RS232/RS485. The second module was an integrated control panel with the installed data acquisition software, which received the original data, transformed them to the html form, and then transferred them to the system server. The data acquisition software was developed with the Microsoft Foundation Class (MFC) framework on the Visual Studio 2015 platform. The database was developed using SQL Server 2010. The relevant analysis tools and interfaces were developed using Java Spring MVC framework. The third module was used to receive and store the data and to respond to query requests for analysis. The last module was the analysis kit to conduct the energy efficiency analysis and then assist users in making decisions for improving production schedules. The four modules worked together to monitor the real-time state of energy efficiency so that managers were able to control the production rate or adjust the production plan. Figure 1 depicts the structure of the energy data acquisition and analysis system developed in this research. Table 1 shows example data of energy consumption and production rates for machines producing different products, including gas consumption in m3, electricity consumption in KW∙h, used material in kg, and pieces of products produced. A sample data for gas consumption is shown in Fig. 2.

Fig. 1
figure 1

Structure of the data acquisition and analysis system

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Table 1 Examples of data of energy consumption and production rate
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Fig. 2
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Sample data for gas consumption

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Energy modeling of aluminum die-casting processes

This study built the regressive relation model between SEC and production rate P in [piece/h] or [kg/h] following Gutowski et al. [28]. The SEC is defined as the amount of energy required for processing a certain amount of one kind of product. For casting processes, it can be expressed as SECgas in [m3/kg] for gas used in the furnace for melting raw materials, and SECelec in [KW∙h/piece] for electricity used for machining finished products. The production rate P is defined as the production rate of quality products rather than the overall one and varies over shifts. Managers are able to control the energy consumption state and effective production rate P of each shift by taking measures such as rescheduling the production plan to accelerate or decelerate the production rate, reducing the machine idle time, optimizing machine parameters, and enhancing workers’ skills to improve the energy efficiency and quality rate. Since there are two machines producing two kinds of products, capital letters are used to denote the machine and lowercase letters are used to denote the product type. For example, SECgasAb means the SEC of furnace A melting raw material for product b using gas, and SECelecBb is the SEC of machine B producing product b using electric energy. PgasAb in [kg/h] and PelecAb in [piece/h] are production rates for furnace A and machine A for producing product b respectively. For a fixed cycle time Tc of process c, the production rate P is related to stand-by idling time tidle and production time tp. P increases when tidle decreases and tp increases. Moreover, the SEC decreases when the production rate P increases (Gutowski et al. 2006; Bettoni and Zanoni 2011). Figure 3 illustrates the general relationship between SECgasAb and production rate PgasAb based on data acquired by the developed system. The area circled out in red is considered the high-efficiency zone, which is defined as the aggregation of data points near the Pareto frontier of productivity and energy efficiency. The energy model is used to describe the relation between SEC and P within the zone, which is marked by green. Once the mathematical model is established, the real-time state of the production should be within the high-efficiency zone as much as possible to improve the energy efficiency and productivity of the overall production.

Fig. 3
figure 3

Relationship between SECgasAb and PgasAb

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To obtain the specific optimum mathematical form for the nonlinear regressive relation between SEC and P, the most used modeling method in literature is nonlinear regression. The general mathematical form for the relation can be represented by fSEC-P in Eq. (1), where θ1, θ2, …, θn represent the unknown coefficients, and n represents the number of coefficients. Apparently, the meaning and number of θ1, θ2, …, θn are determined by the specific mathematical form of fSEC-P. ε is the error between the calculation result by fSEC-P and the real value of SEC:

$$ SEC={f}_{SEC-P}\left(P,{\theta}_1,{\theta}_2,\dots, {\theta}_n\right)+\varepsilon $$
(1)

Following the least square method, the optimum fSEC-P under a certain mathematical form can be found by determining the optimum group of θ1, θ2, …, θn leading to the least aggregation of ε, as Eq. (2). Here, θ represents the vector of (θ1, θ2, …, θn)T, F(θ) is the function to get the least relative error, N represents the sample group number of observed SEC and P used for the modeling, SECi represents one of the observed value of SEC, and Pi represents the relevant observed value of P.

$$ \underset{\theta }{\min }F\left(\theta \right)=\underset{\theta }{\min}\sum \limits_{i=1}^N{\left({SEC}_i-{f}_{SEC-P}\left({P}_i,{\theta}_1,{\theta}_2,\dots, {\theta}_n\right)\right)}^2. $$
(2)

The minimum value of F(θ) can be obtained by solving a differential equation group (3), formed by n equations of fSEC − P’s partial deviation to θi (i = 1, 2, …, n). The optimum values of θ1, θ2, …, θn are represented by a vector of \( \widehat{\theta}={\left(\widehat{\theta_1},\widehat{\theta_2},\dots, \widehat{\theta_n}\right)}^T \). This research used the Newton iterative method to obtain the solution of \( \widehat{\theta} \):

$$ {\left.\frac{\partial F}{\partial {\theta}_i}\right|}_{\theta_i=\widehat{\theta_i}}=-2\sum \limits_{j=1}^N\left({SEC}_j-{f}_{SEC-P}\left({P}_j,\widehat{\theta}\right)\right){\left.\frac{\partial {f}_{SEC-P}}{\partial {\theta}_i}\right|}_{\theta_i=\widehat{\theta_i}}=0 $$
(3)

Based on the data collected over 5 months, various non-linear regression curves, such as reciprocal, power, polynomial cubic, exponential, and logarithmic functions, were analyzed to obtain the optimum one for each pair of machine and product. The expression and coefficients of each fSEC − P is listed in Table 2, in which AP, BP, CP, and DP are the coefficients of polynomial cubic, and aR, bR, aP, bP, aE, bE, cE, aL, bL, and cL are the coefficients of reciprocal, power, exponential, and logarithmic respectively.

Table 2 Forms and coefficients of none-linear curves used
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For each form of regression model fSEC − P, adjusted coefficient of determination adjR2 and residual standard deviation s are used to evaluate its performance. Their calculation formula are as follows:

$$ adj{R}^2=1-\left(1-{R}^2\right)\frac{\left(N-1\right)}{\left(N-k\right)}, $$
(4)
$$ {R}^2=1-\frac{SSE}{SST}, $$
(5)
$$ \mathrm{SSE}=\sum \limits_{i=1}^N{\left({y}_i-\widehat{y_i}\right)}^2, $$
(6)

and

$$ SST=\sum \limits_{i=1}^N{\left({y}_i-\overline{y}\right)}^2 $$
(7)

Equation (8) is used for calculating residual standard deviation s.

$$ s=\sqrt{SSE/\left(N-n\right)} $$
(8)

Here, SSE is the sum of residual squares, SST is the sum of the total square, yi is the true value of the sample of SECi, \( \widehat{y_i} \) is the value of the model output, and n is the number of factors in the model. Apparently, adjR2 ≤ 1. A greater adjR2 implies that the fitted values are closer to the observations. Similarly, s can be considered the estimation of the mean square variance. The smaller its value is, the better the model. The final regression results for gas are listed in Table 3, in which the optimum ones are highlighted. Table 4 is for electricity consumption.

Table 3 Regression results for gas consumption for each pair of machine and product
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Table 4 Regression results for electricity consumption for each pair of machine and product
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Figure 4 shows the regression results of gas usage for each pair of machine and product, and so does Fig. 5 for electricity consumption. The data regression processes were carried out by the software of MATLAB 2016, Origin 8 and SPSS 2016.

Fig. 4
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Regression results for gas consumption

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Fig. 5
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Regression results for electricity consumption

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From Tables 3 and 4, the best regression model for each group of machine and product can be easily selected using the performance indices (i.e., adjR2 and s). Specifically, Eqs. (9) and (10) can be used to calculate and evaluate the gas and electricity consumption for machine A producing product a respectively.

$$ SE{C}_{gasAa}=0.513-0.0794 In\left({P}_{gasAa}-31.91\right)\left({P}_{gasAa}>31.91\right), $$
(9)

and

$$ SE{C}_{elecAa}=0.589-0.103 In\left({P}_{elecAa}-11.82\left)\right({P}_{elecAa}>11.82\right). $$
(10)

Similarly, for the case of machine A producing product b, the energy consumption models are

$$ SE{C}_{gasAb}=0.22-4.393\times 1{0}^{-4}{P}_{gasAb}-5.34\times 1{0}^{-6}{P_{gasAb}}^2+2.246\times 1{0}^{-8}{P_{gasAb}}^3, $$
(11)
$$ SE{C}_{elecAb}=0.379-0.00372{P}_{elecAb}-1.737\times 1{0}^{-5}{P_{elecAb}}^2+4.509\times 1{0}^{-7}{P_{elecAb}}^3. $$
(12)

For the case of machine B producing product c, the regression models are

$$ SE{C}_{gasBc}=1.152{P}_{gasBc}^{-0.35}, $$
(13)

and

$$ SE{C}_{elecBc}=0.387+0.526\times 0.897{P}_{elecBc}. $$
(14)

For the case of machine B producing product d, the regression models are

$$ SE{C}_{gasBd}=0.1+0.391\times 0.983{P}_{gasBd}, $$
(15)

and

$$ SE{C}_{elecBd}=0.569-0.0826\mathit{\ln}\left({P}_{elecBd}-11.46\right)\kern-12em \mathrm{when}\ {P}_{elecBd}>11.46. $$
(16)

In order to validate the resulting models (9–16), validation experiments for the high-energy-efficiency zones were carried out. The energy consumptions of the overall processes in real practice, Greal for gas and Ereal for electricity, were measured and compared to the calculation results using the corresponding regression models. For each model, the validation experiments were conducted five times using different production rates. The relative errors between the calculation results (i.e., Gc and Ec) and the real energy consumption during experiments (i.e., Greal and Ereal) were calculated by using Eqs. (17) and (18), where δgas and δelec represent the relative errors of the prediction results of gas and electricity consumptions:

$$ {\delta}_{gas}=\frac{\mid {G}_{real}-{G}_c\mid }{G_{real}}\times 100\% $$
(17)
$$ {\delta}_{elec}=\frac{\mid {E}_{real}-{E}_c\mid }{E_{real}}\times 100\% $$
(18)

The resulting δgas and δelec are listed in Table 5. All values of δgas and δelec are lower than 5%, meaning that the proposed models are rather accurate in predicting energy consumption of the overall shifts.

Table 5 Relative errors between validation experiments and regression models
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The modeling results also show that there is no clear evidence to imply that a certain form of model can be used as the optimum one to characterize the regressive relation between SEC and production rate P in the high-energy-efficiency zone. However, for each pair of machine and product and for either gas or electricity, a specific optimum model can be established. Managers can use the models to predict energy consumption under a certain production rate and further use the relational models to support production scheduling with the consideration on the trade-off between energy efficiency and productivity. In general, the production should be kept within the high-efficiency zone as much as possible.

Energy efficiency analysis

The case studies carried out in this research were targeted to meet orders with the certain amount of D units and a delivery time T. Two production strategies were further studied. Under the first strategy, the machines were operated causally without considering energy consumption, which was the current practice. The second strategy was to keep the operations within the high-energy-efficiency zone as much as possible based on the proposed models. The energy consumptions of both strategies were measured using the data acquisition system and were compared to analyze the energy-saving potentials of the workshops.

The detailed operating information and results of the three cases are listed in Table 6, where Tca, Gca, and Eca are defined as the operational time, gas consumption, and electric energy consumption to meet demand D under the first strategy. Th, Gh, and Eh are those under the second strategy. Relative energy efficiency evaluation indices Rgas and Relec were calculated using Eqs. (19) and (20) to compare the energy usage of the two strategies. Figure 6 shows the energy consumption data details, and Fig. 7 shows the comparison of energy use under the two production strategies.

$$ {R}_{gas}=\frac{G_{ca}-{G}_h}{G_{ca}}\times 100\% $$
(19)
$$ {R}_{elec}=\frac{E_{ca}-{E}_h}{E_{ca}}\times 100\% $$
(20)
Table 6 Detailed information and measuring results of the three objective cases
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Fig. 6
figure 6

The energy consumption details of the three cases under both strategies

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Fig. 7
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Comparison of energy use under the two production strategies

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As indicted by Table 6 and Figs. 6 and 7, if the production was well-scheduled within the high-energy-efficiency zone, 10 to 15% of the currently used energy can be saved without sacrificing profits or production efficiency. There was even a potential to improve both energy efficiency and productivity.

Conclusions and future work

An improved method was proposed in this paper for modeling gas and electric energy consumption of aluminum die-casting processes. The modeling objective is to represent the mathematical relation between SEC and production rate. The modeling process can be well-adjusted according to complex conditions of multi-combinations of machines and products so that it can be easily extended to other die-casting situations. Environmental concerns can be well addressed together with the productivity objective. The detailed process was based on the development of an intelligent data-acquiring system (DAQS) aiming at collecting energy-related data, transferring and storing data, and conducting analysis. The metering devices and communication protocols are all standard ones in the industry and support both discrete and continuous information. Comparison among a family of nonlinear regression methods was conducted, and performances of different regression methods were analyzed using certain regression evaluation indices to obtain the best one for each engineering situation. Furthermore, the modeling results are validated by a comparison against data for validation experiments. The models were further used to support production scheduling considering both energy use and productivity. Final reports of case studies showed that if the modeling results were reasonably used and production was accordingly well-scheduled, the production rate would be kept being within the high-energy-efficiency zone as much as possible, and 10 to 15% of energy saving can be realized without losing profits. The possibly broad application of the proposed modeling method has the potential of saving energy and improving profitability of the die-casting industry. It is estimated that the overall operation costs can be reduced by 5 to 10%.

In the future, we will investigate other die-casting processes and situations to further verify the proposed procedure. Rather than monitoring production rate, we will also explore other optimization techniques, such as optimization of process parameters and engineering conditions, to obtain balanced solutions for equilibrium of energy saving and economic objectives based on the proposed modeling procedure. The data acquisition, related information collection, and model validation may also be applied to other complex engineering situations beyond casting processes. Since scheduling is just one aspect of the overall die-casting process optimization, we will study how to incorporate the proposed energy modeling method to improve the overall die-casting processes, such as job sequencing, choice of die tools, and others.