# A General Mathematical Framework for Calculating Systems-Scale Efficiency of Energy Extraction and Conversion: Energy Return on Investment (EROI) and Other Energy Return Ratios

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## Abstract

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## 1. Introduction

## 2. Energy Ratios and Their Uses

#### 2.1. Development of Energy Return Ratios

#### 2.2. The Usefulness of Energy Return Ratios

#### 2.3. The Limitations of Energy Return Ratios

#### 2.4. Existing NEA Methods

## 3. Developing a General Bottom-Up Model of ERRs

#### 3.1. Types of Processing Stages

- Harvesting, capturing, or gathering of the primary energy carrier (the initial form of the principal energy flow) from the natural environment. This might include the lifting of crude oil from the subsurface or the conversion of energy contained in photons to electrical energy in a PV panel.
- Shifting energy availability in space. This includes the transport of a principal energy flow from one location in space to another, as in the movement of crude oil from an oil field to a refinery via pipeline or the transport of solar PV electricity from generators to consumers.
- Shifting energy availability in time. This is the shifting of a principal energy flow from one time period to another later time period (storage). Examples include the storage of solar-PV-generated electricity in batteries for use at night, or the storage of crude oil in oil tanks before processing.
- Upgrading or improvement in the quality of a principal energy flow. This is a chemical or physical conversion that does not change the fundamental character of the energy type, but improves its usefulness or quality. An example might be an oil refinery hydrotreating unit that removes sulfur from distillate fuels (e.g., creating ultra-low-sulfur diesel fuel from conventional diesel fuel). Another example might be the transformation of DC solar PV electricity to AC power in an inverter.
- Conversion from one fundamental energy type to another. This is the use of a technology to convert the principal energy flow from one physical form to another. This is typically a more profound and fundamental change than upgrading the principal energy flow. For example, this might involve the conversion from chemical potential energy in gasoline (which exists due to hydrocarbons being out of equilibrium with an oxygen-rich atmosphere) to rotational kinetic energy using a heat engine.

#### 3.1.1. Example of a Processing Stage

**Figure 2.**Observed flows in US refining sector, 2008 [45], neglecting indirect (embodied) consumption. Deviation from our simple model occurs due to the inclusion of some energy content from external natural gas inputs into the principal energy stream in the form of H${}_{2}$ (called flow $dE$ in this figure). This flow exists in reality but is not included in our simple model of a processing stage. Flows are rounded independently and may not sum exactly to 100. Flow ${X}_{11}$ is of uncertain magnitude (refined products use within refinery process boundary, e.g., diesel fuel use on refinery site).

#### 3.1.2. Processing Stage Efficiency

#### 3.2. Energy Pathway Analysis

- The flow of principal energy type ${F}_{s}$ leaving a given processing stage s;
- The total self consumption inputs to a stage s from all other stages within the pathway ($\sum _{f}}{X}_{fs$);
- The total external inputs to a stage s from all external energy types (adjusted for indirect consumption, $\sum _{p}}{E}_{ps$);
- The total embodied energy induced in all economic sectors c due to external energy and materials consumed in stage s ($\sum _{c}}{I}_{cs$).

**Table 1.**NER and GER formulations based on flows of energy (flow-based) and process stage efficiencies (efficiency-based). The EROI as generally practiced would fall under the heading of a life-cycle based GER.

ERR Type | Net Energy Return (NER) | Gross Energy Return (GER) |
---|---|---|

Consumptive (Flow) | $NE{R}_{n}^{\beta}=\frac{{F}_{n}^{*}}{{X}_{01}+{\displaystyle \sum _{f}}{\displaystyle \sum _{t}}{X}_{ft}}$ | $GE{R}_{n}^{\beta}=\frac{{F}_{n}}{{X}_{01}+{\displaystyle \sum _{f}}{\displaystyle \sum _{t}}{X}_{ft}}$ |

Consumptive (Efficiency) | $NE{R}_{n}^{\beta}=\frac{{\displaystyle \prod _{s=1}^{n}}{\varphi}_{s}{\eta}_{s}^{\alpha}}{{\displaystyle \prod _{s=1}^{1}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}\left(\frac{1}{{\eta}_{1}^{\beta}}-1\right)+\cdots +{\displaystyle \prod _{s=1}^{n}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}\left(\frac{1}{{\eta}_{n}^{\beta}}-1\right)}$ | $GE{R}_{n}^{\beta}=\frac{{\displaystyle \prod _{s=1}^{n}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}}{{\displaystyle \prod _{s=1}^{1}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}\left(\frac{1}{{\eta}_{1}^{\beta}}-1\right)+\cdots +{\displaystyle \prod _{s=1}^{n}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}\left(\frac{1}{{\eta}_{n}^{\beta}}-1\right)}$ |

External (Flow) | $NE{R}_{n}^{\gamma}=\frac{{F}_{n}^{*}}{{X}_{01}+{\displaystyle \sum _{f}}{\displaystyle \sum _{t}}{X}_{ft}+{\displaystyle \sum _{p}}{\displaystyle \sum _{s}}{E}_{ps}}$ | $GE{R}_{n}^{\gamma}=\frac{{F}_{n}}{{X}_{01}+{\displaystyle \sum _{f}}{\displaystyle \sum _{t}}{X}_{ft}+{\displaystyle \sum _{p}}{\displaystyle \sum _{s}}{E}_{ps}}$ |

External (Efficiency) | $NE{R}_{n}^{\gamma}=\frac{{\displaystyle \prod _{s=1}^{n}}{\varphi}_{s}{\eta}_{s}^{\alpha}}{{\displaystyle \prod _{s=1}^{1}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}\left(\frac{1}{{\eta}_{1}^{\gamma}}-1\right)+\cdots +{\displaystyle \prod _{s=1}^{n}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}\left(\frac{1}{{\eta}_{n}^{\gamma}}-1\right)}$ | $GE{R}_{n}^{\gamma}=\frac{{\displaystyle \prod _{s=1}^{n}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}}{{\displaystyle \prod _{s=1}^{1}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}\left(\frac{1}{{\eta}_{1}^{\gamma}}-1\right)+\cdots +{\displaystyle \prod _{s=1}^{n}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}\left(\frac{1}{{\eta}_{n}^{\gamma}}-1\right)}$ |

Life cycle (Flow) [EROI] | $NE{R}_{n}^{\delta}=\frac{{F}_{n}^{*}-{r}_{i}{\displaystyle \sum _{c}}{\displaystyle \sum _{s}}{I}_{cs}}{{X}_{01}+{\displaystyle \sum _{f}}{\displaystyle \sum _{t}}{X}_{ft}+{\displaystyle \sum _{p}}{\displaystyle \sum _{s}}{E}_{ps}+{\displaystyle \sum _{c}}{\displaystyle \sum _{s}}{I}_{cs}}$ | $GE{R}_{n}^{\delta}=\frac{{F}_{n}}{{X}_{01}+{\displaystyle \sum _{f}}{\displaystyle \sum _{t}}{X}_{ft}+{\displaystyle \sum _{p}}{\displaystyle \sum _{s}}{E}_{ps}+{\displaystyle \sum _{c}}{\displaystyle \sum _{s}}{I}_{cs}}$ |

Life cycle (Efficiency) | $NE{R}_{n}^{\delta}=\frac{{\displaystyle \prod _{s=1}^{n}}{\varphi}_{s}{\eta}_{s}^{\alpha}-{r}_{i}{\displaystyle \sum _{j=1}^{n}}\left[{\displaystyle \prod _{s=1}^{j}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}\left(\frac{1}{{\eta}_{j}^{\delta}}-\frac{1}{{\eta}_{j}^{\gamma}}\right)\right]}{{\displaystyle \prod _{s=1}^{1}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}\left(\frac{1}{{\eta}_{1}^{\delta}}-1\right)+\cdots +{\displaystyle \prod _{s=1}^{n}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}\left(\frac{1}{{\eta}_{n}^{\delta}}-1\right)}$ | $GE{R}_{n}^{\delta}=\frac{{\displaystyle \prod _{s=1}^{n}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}}{{\displaystyle \prod _{s=1}^{1}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}\left(\frac{1}{{\eta}_{1}^{\delta}}-1\right)+\cdots +{\displaystyle \prod _{s=1}^{n}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}\left(\frac{1}{{\eta}_{n}^{\delta}}-1\right)}$ |

**Table 2.**NEER and GEER formulations based on flows of energy (flow-based) and process stage efficiencies (efficiency-based).

ERR Type | Net External Energy Return (NEER) | Gross External Energy Return (GEER) |
---|---|---|

Consumptive (Flow) | $NEE{R}_{n}^{\beta}=\frac{{F}_{n}^{*}}{0}=\infty$ | $GEE{R}_{n}^{\beta}=\frac{{F}_{n}}{0}=\infty$ |

Consumptive (Efficiency) | $NEE{R}_{n}^{\beta}=\frac{{\displaystyle \prod _{s=1}^{n}}{\varphi}_{s}{\eta}_{s}^{\alpha}}{0}$ | $GEE{R}_{n}^{\beta}=\frac{{\displaystyle \prod _{s=1}^{n}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}}{0}$ |

External (Flow) | $NEE{R}_{n}^{\gamma}=\frac{{F}_{n}^{*}}{{\displaystyle \sum _{p}}{\displaystyle \sum _{s}}{E}_{ps}}$ | $GEE{R}_{n}^{\gamma}=\frac{{F}_{n}}{{\displaystyle \sum _{p}}{\displaystyle \sum _{s}}{E}_{ps}}$ |

External (Efficiency) | $NEE{R}_{n}^{\beta}=\frac{{\displaystyle \prod _{s=1}^{n}}{\varphi}_{s}{\eta}_{s}^{\alpha}}{{\displaystyle \prod _{s=1}^{1}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}\left(\frac{1}{{\eta}_{1}^{\gamma}}-\frac{1}{{\eta}_{1}^{\beta}}\right)+\cdots +{\displaystyle \prod _{s=1}^{n}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}\left(\frac{1}{{\eta}_{n}^{\gamma}}-\frac{1}{{\eta}_{n}^{\beta}}\right)}$ | $GEE{R}_{n}^{\beta}=\frac{{\displaystyle \prod _{s=1}^{n}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}}{{\displaystyle \prod _{s=1}^{1}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}\left(\frac{1}{{\eta}_{1}^{\gamma}}-\frac{1}{{\eta}_{1}^{\beta}}\right)+\cdots +{\displaystyle \prod _{s=1}^{n}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}\left(\frac{1}{{\eta}_{n}^{\gamma}}-\frac{1}{{\eta}_{n}^{\beta}}\right)}$ |

Life cycle (Flow) [EROI] | $NEE{R}_{n}^{\delta}=\frac{{F}_{n}^{*}-{r}_{i}{\displaystyle \sum _{c}}{\displaystyle \sum _{s}}{I}_{cs}}{{\displaystyle \sum _{p}}{\displaystyle \sum _{s}}{E}_{ps}+{\displaystyle \sum _{c}}{\displaystyle \sum _{s}}{I}_{cs}}$ | $GEE{R}_{n}^{\delta}=\frac{{F}_{n}}{{\displaystyle \sum _{p}}{\displaystyle \sum _{s}}{E}_{ps}+{\displaystyle \sum _{c}}{\displaystyle \sum _{s}}{I}_{cs}}$ |

Life cycle (Efficiency) | $NEE{R}_{n}^{\beta}=\frac{{\displaystyle \prod _{s=1}^{n}}{\varphi}_{s}{\eta}_{s}^{\alpha}-{r}_{i}{\displaystyle \sum _{j=1}^{n}}\left[{\displaystyle \prod _{s=1}^{j}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}\left(\frac{1}{{\eta}_{j}^{\delta}}-\frac{1}{{\eta}_{j}^{\gamma}}\right)\right]}{{\displaystyle \prod _{s=1}^{1}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}\left(\frac{1}{{\eta}_{1}^{\delta}}-\frac{1}{{\eta}_{1}^{\beta}}\right)+\cdots +{\displaystyle \prod _{s=1}^{n}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}\left(\frac{1}{{\eta}_{n}^{\delta}}-\frac{1}{{\eta}_{n}^{\beta}}\right)}$ | $GEE{R}_{n}^{\beta}=\frac{{\displaystyle \prod _{s=1}^{n}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}}{{\displaystyle \prod _{s=1}^{1}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}\left(\frac{1}{{\eta}_{1}^{\delta}}-\frac{1}{{\eta}_{1}^{\beta}}\right)+\cdots +{\displaystyle \prod _{s=1}^{n}}{\varphi}_{s-1}{\eta}_{s}^{\alpha}\left(\frac{1}{{\eta}_{n}^{\delta}}-\frac{1}{{\eta}_{n}^{\beta}}\right)}$ |

Stages and Flows | ||||
---|---|---|---|---|

Pathways | ${P}_{p}$ | ${P}_{1},{P}_{2},\dots ,{P}_{m}$ | m other energy extraction and conversion pathways, with counter index p | Energy extraction and processing pathways |

Stages | ${S}_{s}$ | ${S}_{1},{S}_{2},\dots ,{S}_{n}$ | n stages counted with counter index s (or $s1,s2$) | Energy extraction and processing stages |

Distribution points | ${D}_{s}$ | ${D}_{1},{D}_{2},\dots ,{D}_{n}$ | n distribution points counted with counter index s | Energy distribution points for principal energy flows |

Flows | ${F}_{s}$ | ${F}_{1},{F}_{2},\dots ,{F}_{n}$ | n flows leaving energy extraction and processing stages (counter index s) | Flows of principal energy (MJ of principal energy type s) |

Self consumption | ${X}_{ft}$ | ${X}_{11},{X}_{12},\dots ,{X}_{nn}$ | $n\times n$ flows leaving energy extraction and processing stage f and going into stage t ($f=1,\dots ,n$, $t=1,\dots ,n$) | Flows of principal energy (MJ of principal energy type f) |

External consumption | ${E}_{ps}$ | ${E}_{11},{E}_{12},\dots ,{E}_{mn}$ | $m\times n$ flows leaving energy extraction and conversion pathway m into stage s | Flows of principal energy of type m (MJ of principal energy type m) |

Indirect consumption | ${I}_{cs}$ | ${I}_{11},{I}_{12},\dots ,{I}_{qn}$ | $q\times n$ flows induced in economic sector c due to direct consumption of materials and energy in stage s | Indirect consumption of all energy types in sector c (MJ of all energy types) |

Dependent Variables and Efficiencies | ||||

Distribution fraction | ${\varphi}_{s}$ | ${\varphi}_{1}$, ${\varphi}_{2}$, …, ${\varphi}_{n}$ | Ratio of outflow ${F}_{s}^{*}$ from distribution point s to inflow ${F}_{s}$ | Dimensionless ratio |

Flow efficiency | ${\eta}_{s}^{\alpha}$ | ${\eta}_{1}^{\alpha}$, ${\eta}_{2}^{\alpha}$, …, ${\eta}_{n}^{\alpha}$ | Ratio of outflow ${F}_{s}$ from stage s to inflow ${F}_{s-1}^{*}$ | Dimensionless ratio |

Consumptive efficiency | ${\eta}_{s}^{\beta}$ | ${\eta}_{1}^{\beta}$, ${\eta}_{2}^{\beta}$, …, ${\eta}_{n}^{\beta}$ | Ratio of outflow ${F}_{{s}_{2}}$ from stage ${s}_{2}$ to sum of inflow ${F}_{{s}_{2}-1}^{*}$ and energy consumed from within pathway (${\sum}_{s1}{X}_{{s}_{1}{s}_{2}}$) | Dimensionless ratio |

External efficiency | ${\eta}_{s}^{\gamma}$ | ${\eta}_{1}^{\gamma}$, ${\eta}_{2}^{\gamma}$, …, ${\eta}_{n}^{\gamma}$ | Ratio of outflow ${F}_{{s}_{2}}$ from stage ${s}_{2}$ to sum of inflow ${F}_{{s}_{2}-1}^{*}$ and energy consumed from within pathway (${\sum}_{s1}{X}_{{s}_{1}{s}_{2}}$) and from outside the pathway (${\sum}_{p}{E}_{p{s}_{2}}$) | Dimensionless ratio |

Life cycle efficiency | ${\eta}_{s}^{\delta}$ | ${\eta}_{1}^{\delta}$, ${\eta}_{2}^{\delta}$, …, ${\eta}_{n}^{\delta}$ | Ratio of outflow ${F}_{{s}_{2}}$ from stage ${s}_{2}$ to sum of inflow ${F}_{{s}_{2}-1}^{*}$ and energy consumed from within pathway (${\sum}_{s1}{X}_{{s}_{1}{s}_{2}}$) and from outside the pathway (${\sum}_{p}{E}_{p{s}_{2}}$) as well as indirect consumption in other economic sectors (${\sum}_{c}{I}_{c{s}_{2}}$) | Dimensionless ratio |

Indirect self-use ratio | ${r}_{i}$ | – | The fraction of indirect energy consumed that originally arose from the pathway in question. Can be determined using society-wide energy mix or sector-specific energy mixes. | Dimensionless ratio |

## 4. Results and Discussion

#### 4.1. A Simple Applied Example: Solar Photovoltaic Energy Capture and Processing

**Figure 4.**Electricity production from a PV panel with data from [54,55,56] modeled as an energy extraction and conversion pathway. Collection losses are neglected in this case (see text for discussion). In reality a proportion of the finished product would be fed back into the production process. Unfortunately no data are available on this.

**Table 4.**Efficiencies and energy return ratios for the example PV production process. As discussed in the text, collection losses are omitted from the analysis.

System Boundary | ||||
---|---|---|---|---|

Ratios | α | β | γ | δ |

${\eta}_{1}$ | $\frac{18.8}{18.8}=1.00$ | $\frac{18.8}{18.8}=1.00$ | $\frac{18.8}{18.8+0.6}=0.97$ | $\frac{18.8}{18.8+0.6+3.6}=0.82$ |

${\eta}_{2}$ | $\frac{17.9}{18.8}=0.95$ | $\frac{17.9}{18.8}=0.95$ | $\frac{17.9}{18.8}=0.95$ | $\frac{17.9}{18.8+0.002}=0.95$ |

${\eta}_{3}$ | $\frac{15.9}{17.9}=0.89$ | $\frac{15.9}{17.9}=0.89$ | $\frac{15.9}{17.9}=0.89$ | $\frac{15.9}{17.9+0.06}=0.89$ |

$GE{R}_{sys}$ | - | $\frac{15.9}{1.9+2.0}=4.08$ | $\frac{15.9}{1.9+2.0+0.6}=3.53$ | $\frac{15.9}{1.9+2.0+0.6+3.6+0.002+0.06}=1.95$ |

$NE{R}_{sys}$ | - | $4.08$ | $3.53$ | $1.95$ |

$GEE{R}_{sys}$ | - | $\frac{15.9}{0}=\infty $ | $\frac{15.9}{0.6}=26.5$ | $\frac{15.9}{0.6+3.6+0.002+0.06}=3.73$ |

$NEE{R}_{sys}$ | - | ∞ | $26.5$ | $3.73$ |

#### 4.2. Application of Mathematical form to a Generic Energy Processing Chain

**Figure 5.**Comparison of NER${}^{\beta}$, NER${}^{\gamma}$ and ${F}_{s}$ as a function of processing stage s.

**Figure 6.**Comparison of final energy output given changes in location of large processing energy input. Note that a processing intensive technology further in the processing chain results in less final energy output.

**Figure 7.**Comparison between EER${}^{\gamma}$ and NER${}^{\gamma}$ with changes in the makeup of consumption between internal (X) and external (E) consumption. In all cases indirect consumption is equal to 0.

## 5. Discussion of Formulation Limitations and Next Steps

## 6. Appendix: Derivation of Multi-Stage, Single-Pathway Models

#### 6.1. Two Stage Pathways with Only Internal Consumption

#### 6.1.1. Energy Balance

#### 6.1.2. Efficiencies

#### 6.1.3. Net Energy Ratio and Other Energy Return Ratios

#### 6.2. Two Stage Pathways with only Within-Pathway Consumption

#### 6.2.1. Energy Balance

#### 6.2.2. Efficiencies

#### 6.2.3. Net Energy Ratio and Other Energy Return Ratios

#### 6.3. One Pathway, n Stages

#### 6.3.1. Energy Balance

#### 6.3.2. Efficiencies and Normalized Quantities

#### 6.3.3. Energy Return Ratios

#### 6.4. One Pathway, n Stages, with External Energy Inputs

**Figure 11.**Energy production process with one pathway and n stages and external energy inputs from outside the production pathway.

#### 6.4.1. Energy Balance

#### 6.4.2. Efficiencies

#### 6.4.3. Net Energy Ratio

#### 6.4.4. Pathways with Indirect Consumption of Energy

- Indirect consumption of all energy types due to external energy consumption;
- Indirect consumption of all energy types due to materials consumption in the energy system.

- Indirect consumption of all energy types must be added to the denominator of the NER, GER and EER equivalents;
- Indirect consumption of the principal energy flow must be subtracted from that energy available due to the production process (e.g., must be removed from the numerator of the ratios).

**Figure 12.**Energy production process with one pathway and n stages, including external (E) and indirect (I) energy inputs.

#### 6.4.5. Energy Balance

#### 6.4.6. Efficiencies

#### 6.4.7. Net Energy Ratio and Gross Energy Ratio

#### 6.4.8. External Energy Ratio

## Acknowledgements

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**MDPI and ACS Style**

Brandt, A.R.; Dale, M.
A General Mathematical Framework for Calculating Systems-Scale Efficiency of Energy Extraction and Conversion: Energy Return on Investment (EROI) and Other Energy Return Ratios. *Energies* **2011**, *4*, 1211-1245.
https://doi.org/10.3390/en4081211

**AMA Style**

Brandt AR, Dale M.
A General Mathematical Framework for Calculating Systems-Scale Efficiency of Energy Extraction and Conversion: Energy Return on Investment (EROI) and Other Energy Return Ratios. *Energies*. 2011; 4(8):1211-1245.
https://doi.org/10.3390/en4081211

**Chicago/Turabian Style**

Brandt, Adam R., and Michael Dale.
2011. "A General Mathematical Framework for Calculating Systems-Scale Efficiency of Energy Extraction and Conversion: Energy Return on Investment (EROI) and Other Energy Return Ratios" *Energies* 4, no. 8: 1211-1245.
https://doi.org/10.3390/en4081211