Environ. Sci. Technol. 2011, 45, 751–754
Power-Law Relationships for Estimating Mass, Fuel Consumption and Costs of Energy Conversion Equipments M A R L O E S C A D U F F , * ,†,‡ MARK A. J. HUIJBREGTS,§ HANS-JOERG ALTHAUS,† AND A. JAN HENDRIKS§ Empa, Swiss Federal Laboratories for Materials Testing and Research, Technology and Society Laboratory, Ueberlandstrasse 129, CH-8600 Duebendorf, Switzerland, ETH Zurich, Institute of Environmental Engineering, CH-8093 Zurich, Switzerland, and Department of Environmental Science, Institute for Wetland and Water Research, Faculty of Science, Radboud University Nijmegen, P.O. Box 9010, NL-6500 GL Nijmegen, The Netherlands
Received September 10, 2010. Accepted November 22, 2010.
To perform life-cycle assessment studies, data on the production and use of the products is required. However, often only few data or measurements are available. Estimation of properties can be performed by applying scaling relationships. In many disciplines, they are used to either predict data or to search for underlying patterns, but they have not been considered in the context of product assessments hitherto. The goal of this study was to explore size scaling for commonly used energy conversion equipment, that is, boilers, engines, and generators. The variables mass M, fuel consumption Q, and costs C were related to power P. The established power-law relationships were M ) 100.73.. 1.89 P0.64.. 1.23 (R2 g 0.94), Q ) 100.06.. 0.68P0.82.. 1.02 (R2 g 0.98) and C ) 102.46.. 2.86P0.83.. 0.85 (R2 g 0.83). Mass versus power and costs versus power showed that none of the equipment types scaled isometrically, that is, with a slope of 1. Fuel consumption versus power scaled approximately isometrically for steam boilers, the other equipments scaled significantly lower than 1. This nonlinear scaling behavior induces a significant size effect. The power laws we established can be applied to scale the mass, fuel consumption and costs of energy conversion equipments up or down. Our findings suggest that empirical scaling laws can be used to estimate properties, particularly relevant in studies focusing on early product development for which generally only little information is available.
Introduction In many disciplines, scaling relationships are developed to allow such an estimation of properties when no measurements are available and to facilitate the understanding of the underlying mechanisms. Scaling relationships classically * Corresponding author phone: +41 44 823 48 12; fax: +41 44 823 40 42; e-mail:
[email protected]. † Empa. ‡ ETH Zurich. § Radboud University Nijmegen. 10.1021/es103095k
2011 American Chemical Society
Published on Web 12/06/2010
relate two variables to each other in the form of a power law, y ) a xb, with a being a normalization constant and b the scaling factor. These laws are classified as allometric, geometric or isometric according to their slope b, being multiples of 1/4, and 1, respectively. Scaling is found in a wide range of fields, such as in physics (1-3), socioeconomics (4-6), biology (7-9), and economics (10-12). This predictive feature of scaling is also utilized in technical systems, where cost scaling and cost estimation techniques have a long tradition. In cost engineering practices, the “sixtenths rule” with a scaling factor b of 0.6 is used as a rule of thumb to scale production costs (y) with the capacity of an equipment (x) (10). This relationship is still commonly used to estimate capital costs of equipments (13-15). Life cycle assessment (LCA) is an instrument used for evaluating the potential environmental impacts of products throughout its life cycle, from raw material extraction to endof-life (16). To assess the potential impacts, life cycle inventories (LCI) of the products in question have to be compiled and the inputs and outputs for a given product system quantified. Gathering LCI information may in many cases be difficult, costly and time-consuming (17). Various efforts have been taken to overcome such a limitation by presenting tools to estimate missing inventory data, building better databases and more integrated LCA approaches (e.g., refs 18-20). However a nonlinear approach such as power laws are rarely used to account for scale effects in the field of both life cycle assessment (LCA) and life cycle costing (LCC). Power laws which link power to mass, fuel consumption and costs might aid in estimating LCI properties as power is commonly known and mass, fuel consumption and costs are input parameters in LCA or LCC studies. The main objective of this paper is to present scaling relationships for mass, energy consumption and costs of commonly used energy conversion equipment. Energy conversion equipments were chosen since they are widely used in all kinds of technical set-ups and commonly found in background systems of life-cycle assessments. A second reason is that such equipment exists over a wide range of sizes and shapes. For instance, small engines combust gasoline and have one or two cylinders, whereas large marine engines have as many as 18 cylinders. This study includes the following equipments: mobile gasoline engines, industrial diesel engines, marine engines, generators and steam boilers. We also compare our results with other scaling studies to explore the possibility of common scaling rules across different scientific disciplines. Regression analysis was used to relate these properties to power output P of the equipment, as power is chosen as the functional unit of engines and generators in this study.
Materials and Methods Data Selection. Two independent data sets were applied. The first data set was used to relate mass [kg] and fuel consumption [kWh/h] to power output [kW]. Sample sizes varied between 35 and 112 samples per equipment type. Power output was included as the power for continuous use, the mass of the equipment was defined as the shipping weight, without liquids or supporting structures. Fuel consumption was reported according to ISO standards. In order to guarantee high data quality, boiler producers were selected upon their membership to American Society of Mechanical Engineers (ASME), securing that all boilers comply with the respective norms. The type of power output varies per equipment. The output power of an engine is mechanical power or shaft power, whereas the output power VOL. 45, NO. 2, 2011 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 1. Power range [kW] of the used process equipments. n indicates the sample size. Two independent data sets are displayed using the colors black and gray, respectively. Black symbolizes the data set which relates mass M and fuel consumption Q to power P. Gray indicates the data set used to relate costs C to power P.
FIGURE 2. a. Mass M (kg) versus power P (kW) b. fuel use Q (kW) versus power P (kW) and c. price C (US$) versus power P (kW). The regression lines were calculated using ordinary least-squares linear regression (OLS). (O) gasoline engine, (0) diesel engine, (4) marine engine, (*) diesel generator, and (×) steam boiler. of a generator is electric power. A steam boiler on the other hand exports energy as thermal energy. All properties were retrieved from freely accessible technical specification sheets. The second data set was used to relate costs [$] to power output [kW]. The costs were included as current U.S. market prices, including production costs as well as profit margins. The cost data was available in 2010 US dollar and retrieved online from four suppliers. Cost data from industrial and marine engines as well as generators were found, resulting in a total of 117 data points for industrial engines, 19 for marine engines, and 651 for generators. Technologies available on the market at the present time were included only. Data and corresponding literature references are provided in the Supporting Information (SI). Regression and Statistics. All data were log10-transformed as commonly done in scaling. Regressions on log-log plots of the data were performed with the ordinary least-squares (OLS) linear regression. To test whether the results are steady, 752
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the reduced major axes (RMA) regression was used as an additional regression method (21). The regression analysis was performed with the Statistical Package for the Social Sciences (SPSS) statistical software for Windows, version 16.0 (SPSS, Chicago, IL). The intercept is presented as a [95% CI] and the scale factors are presented as b [95% CI]. We also report the standard error and the explained variance (R2) of the regression.
Results The mass M of the engines and generators were strongly related to power P (Figure 2a and Table 1). The scaling factors b ranged from 0.64 [0.61-0.68] for industrial diesel engines up to 1.23 [1.14-1.33] for marine engines. All equipment scaled with an exponent significantly smaller than 1.0, with exception of the marine engines, which had a slope larger than 1.0. Figure 2b and Table 1 show the relationship between
TABLE 1. Exponent b and Intercept a for the Parameters Mass M (kg), Fuel Consumption Q (kWh/h) and Costs C (US$) versus Power P (kW) using OLS, Ordinary Least Squares. n: Number of Observations; R 2: Coefficient of Determination; SE: Standard Error; 95% CI: 95% Confidence Interval product M ) a · Pb gasoline engine diesel engine marine engine generator steam boiler Q ) a · Pb diesel engine marine engine generator steam boiler C ) a · Pb diesel engine marine engine generator
OLS
n
R2
SE
b (95% CI)
a (95% CI)
0.77 (0.71-0.83) 0.64 (0.61-0.68) 1.23 (1.14-1.33) 0.68 (0.63-0.72) 0.87 (0.84-0.90)
0.73 (0.66-0.79) 43 1.36 (1.29-1.43) 89 0.19 (-0.19-0.57) 35 1.89 (1.82-1.96) 60 0.95 (0.85-1.04) 112
0.94 0.95 0.95 0.94 0.97
0.08 0.05 0.10 0.10 0.10
0.93 (0.92-0.94) 0.96 (0.95-0.97) 0.82 (0.79-0.85) 1.02 (1.01-1.03)
0.55 (0.53-0.58) 0.48 (0.45-0.51) 0.68 (0.63-0.73) 0.06 (0.04-0.08)
1.00 1.00 0.98 1.00
0.02 0.01 0.07 0.02
0.85 (0.79-0.91) 2.46 (2.34-2.58) 0.83 (0.67-1.02) 2.57 (2.09-3.06) 0.83 (0.81-0.85) 2.86 (2.81-2.91)
75 35 59 61
117 0.85 0.21 19 0.83 0.24 651 0.90 0.21
fuel use Q and power P. The diesel and marine engines scaled with exponents between 0.93 and 0.96, while the values for the generators were 0.82. The diesel and marine engines consumed more fuel than the generators with the same power. The steam boilers scaled almost isometrically with 1.02. The costs related to power within a 0.8-0.9 range. The three derived scaling exponents for the market prices of diesel and marine engines and generators versus power did not significantly differ from each other. The intercept of the generators was in all three relationships higher than the other equipment, specifically for the mass versus power relationship. The intercepts resulting from fuel consumption versus power were all larger than zero. A measure of the strength of the relationships between the mass or fuel consumption and power is explained by the value of the coefficient of determination (R 2). In this study R 2 for both mass versus power and fuel consumption versus power was 0.94 and higher, which means that in minimum 94% of the variation of the dependent variables was explained by the relationship with power. The results from the RMA regression did not differ significantly from the OLS regression (for RMA details see SI).
Discussion Mass and Fuel Consumption versus Power. All equipment scaled with mass M to power P with exponents between 0.64 and 0.87 with the exception of the marine engines which scaled with 1.23. The apparent increase of the M:P ratio in marine engines may be attributed to the increased requirements for cooling, as indicated by McMahon and Bonner (22) for 39 internal combustion engines with a power range of 0.34 kW to 20 730 kW. The slope of the mass M versus power P relationships for the generators was within the range of the exponents noted for the diesel and gasoline engines. The diesel generators, however, consistently generated less power P than equally heavy diesel and gasoline engines. This is explained by the difference in construction and type of energy. A generator converts shaft power from the engine to electric power by the means of an electric generator. In the presented data this conversion is done with an efficiency of 25-68% and thus reduces the power output accordingly. Furthermore, the diesel generator consists of a diesel engine plus generator, implying that a generator is evidently heavier than an engine with equal power. The Q-P relationship approaches linearity, specifically for the steam boilers which almost scaled isometrically for the Q:P ratio, implying that no significant efficiency increase over size can be observed. The intercepts from the Q:P ratio were all close to or
larger than zero, which is in itself logic since one or more units of fuel input is needed to produce one unit of power. The intercept of the steam boiler is 0.06 which corresponds well to an average efficiency of 80% (1/100.06). The intercepts of the engines and generators correspond to average efficiencies between 21% and 33%, whereas the lowest efficiency plausibly matches with the generators. As mentioned in the Materials and Methods section, technological effects were minimized by only including products that are being offered to the market. Nevertheless, technological differences are still present. For instance, electronically controlled fuel injection, turbo technology or lean combustion technology are likely to influence the weight, fuel consumption, and costs of an engine. These effects might lead to a larger spread among the data. Cost versus Power. The market prices scaled between 0.83 and 0.85 to power with no significant difference between the engine types and generator. In cost estimation literature a generic scaling factor of 0.6 is commonly advised, if no other information is available (10). The empirical values found in this work are higher and hence more conservative compared to the generic scaling factor. Note, however, that other studies also found scaling factors higher than 0.6. For instance, for the production of steam boilers, factors between 0.72 and 0.80 are reported (15) and a scaling factor of 0.85 was reported for processes involving the handling of solids (23, 24). The data set used in this study described the market prices of the equipments instead of the production costs. Market prices include profit margins which might differ between producers. This effect might results in a larger standard error of the costs versus power relationship than for the other relationships. If profit margins remain constant, the intercept is expected to be higher than those reported for production costs only. The used data set, however, does not allow a separation of the production costs and profit margins. Biological Systems. While the ultimate scaling factors are yet to be established, comparisons of energy use by different types of systems allows for independent tests of allometric theories. Allometric relationships are often noted as P ) f (M), whereas in this paper M ) f (P) is used. To facilitate a comparison the empirical relationships in this paper were converted to P ) f (M), hence power P of the equipment scaled to mass M, mostly with a factor larger than 1. By contrast, in biological systems resting and field metabolism of organisms increases with a body size slope of about 3/4 (7, 25, 26). Such a slope has been attributed to the fractal networks that plants and animals need for an efficient transport of nutrients (27). Metabolism of flying birds, bats, and bees was shown to scale with mass M with a slope of 1.1-1.2 (26), indicating that working engines may be comparable to rapidly moving animals (Figure 3). Scaling behavior of piston engines and running, flying or swimming animals was observed to be similar by Marden and Allen (28) too. Practical Implementation. Especially for assessing the environmental consequences and economic costs of products and production processes the established relationships can be used to simplify the estimation of properties. Scaling relationships are rarely used for instance in the field of life cycle assessment (LCA) and life cycle costing (LCC). Although it is recognized that nonlinear scale effects should not be disregarded in LCA studies (30), a systematic approach has not been established yet. For instance, within LCA databases, the specification of one product size is often included (e.g., in ref 31), but no guidelines or scaling relationships are given to scale up or down. The current study showed that mass, fuel consumption, and costs do not scale isometrically with power, hence indicating that scaling should be included. For instance, when no data on a 10 MW marine engine is available or no time can be allocated to search for more detailed information, the established power laws can be applied VOL. 45, NO. 2, 2011 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 3. Power output (kW) versus mass (kg). The regression lines are based on the ordinary least-squares regression. Biological scaling relationships connected with mammal basal metabolic rate (29) and power of locomotion (flying) for birds and bats (26). instead of linear scaling. The differences between isometric and nonisometric scaling can be substantial. The mass of a 10 MW marine engine will be a factor of 8 higher based on higher based on the empirical scaling relationship compared to linear scaling; hence in this example linear scaling profoundly underestimates the mass. Changes in type of material, has, however, not been taken into consideration in our study. This obviously needs to be addressed as well in LCA studies. On the other hand, the fuel consumption and costs of the same engine are a factor of 1.4 and 4.8, respectively, lower based on the power law relation compared to linear scaling, therefore overestimating the costs and fuel consumption. Overestimating the fuel consumption automatically also overestimates the environmental profile of the use phase of these equipments, causing an unfavorable LCA outcome. Likewise, the overestimation of the product price can negatively influence LCC outcomes. Concluding Remarks. The majority of the equipments included scaled nonlinear, inducing a significant size effect that should not be neglected in assessments. A general rule of thumb such as the “six-tenths rule” for cost relationships could not be derived for these properties, since the equipment types scaled inherently different (11). The large size range covered in this study as well as the high correlations found show that these scaling factors can be used in life-cycle assessment and lifecycle costing studies, particularly if no other data is available or to scale background systems. Future efforts should on the one hand be focused on improving the relationship, for example, by assessing material composition in addition to mass, and by including different technological stages. On the other hand, the present relationships already allow incorporation into LCA and LCC studies to improve accuracy.
Acknowledgments We thank Stefanie Hellweg and Annette Koehler for assistance and comments.
Supporting Information Available The raw data used for establishing Table 1 and Figure 2 as well as the data sources are supplied. This material is available free of charge via the Internet at http://pubs.acs.org.
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