^{1}

^{*}

^{2}

^{1}

^{4}

^{3}

^{4}

The authors have declared that no competing interests exist.

Conceived and designed the experiments: AL JR. Performed the experiments: AL. Analyzed the data: AL. Contributed reagents/materials/analysis tools: AL. Wrote the paper: AL JR BB MW. Manuscript reviews, revisions, and discussion: BB MW.

A sustainable global community requires the successful integration of environment and engineering. In the public and private sectors, designing cyclical (“closed loop”) resource networks increasingly appears as a strategy employed to improve resource efficiency and reduce environmental impacts. Patterning industrial networks on ecological ones has been shown to provide significant improvements at multiple levels. Here, we apply the biological metric cyclicity to 28 familiar thermodynamic power cycles of increasing complexity. These cycles, composed of turbines and the like, are scientifically very different from natural ecosystems. Despite this difference, the application results in a positive correlation between the maximum thermal efficiency and the cyclic structure of the cycles. The immediate impact of these findings results in a simple method for comparing cycles to one another, higher cyclicity values pointing to those cycles which have the potential for a higher maximum thermal efficiency. Such a strong correlation has the promise of impacting both natural ecology and engineering thermodynamics and provides a clear motivation to look for more fundamental scientific connections between natural and engineered systems.

A sustainable global community, one that meets the needs of the current generation without sacrificing those of future generations

“To be ultimately sustainable, biological ecosystems have evolved over the long term to be almost completely cyclical in nature, with ‘resources’ and ‘waste’ being undefined, since waste to one component of the system represents resources to another.” – Jelinski, et al.

In 1969, Odum recognized that ecological systems, particularly mature ones, are associated with a high degree of internal recycling of energy and materials, such that the amount of new inputs into the system is small compared to what is transformed among the system components

In this paper we use 28 familiar thermodynamic power cycles of increasing complexity to explore trends in network structure defined by the ecological metric cyclicity

Ideal Rankine and Brayton cycles composed the 28 power cycles used. The ideal Brayton cycle is used to model the gas turbine engine and the ideal Rankine cycle is the simplest representation of the vapor power cycles utilized by the electric power generating industry. The inclusions of feedwater heaters, regeneration, reheating and intercooling are all standard ways of increasing the thermal efficiency of the Rankine and Brayton cycles

To uncover the internal cycling present in the system we must first use the network approach in thermodynamics to construct a graphical model revealing system topology, referred to here as an energy flow network

Note that the link between the condenser (Node vi) and Pump 1 (Node i) is not a physical flow of energy. Since State 1 acts as an energetic reference state for the network, working fluid returning to that reference state only closes the

A structural adjacency matrix (

The adjacency matrix in _{ij}

(a) Labeled adjacency matrix for the ideal Rankine cycle with one open feed water heater – rows represent flow

With the power cycles now in matrix form, cyclicity is found by calculating the maximum real eigenvalue (λ_{max}) for each corresponding adjacency matrix. The eigenvalues of a matrix are mathematically defined as the solutions to equation 1: the determinant of the quantity of the matrix in question minus the eigenvalues times the identity matrix of the equivalent size, all equal to zero. The result of equation 1 is a set of eigenvalues (which may be both real and imaginary); MATLAB’s “_{max} is a measure of the proliferation of pathways that connect two nodes in a network. There is a greater potential for flows to remain within the system as pathways proliferate, λ_{max} is indicative of the resulting internal cycling _{max} = 1) as seen in

Cyclicity can be either 0, 1 or greater than 1. This is illustrated in

(a) No cycling λ_{max} = 0, (b) weak cycling λ_{max} = 1, (c) and strong cycling λ_{max}>1

The proof presented by Borrett et al. (2007) for the use of eigenvalues to determine the cyclicity (what Borrett et al. call “pathway proliferation rate”) of a system combines results from graph theory and linear algebra _{1}≥λ_{i}

All thermal efficiencies (_{I}_{in} and Q_{in} respectively, and the work produced by the power cycle, W_{out}, were calculated based upon enthalpies (

Rankine Cycles - water | Brayton Cycles - air |

T_{min} = 318.9 K |
T_{min} = 288.2 K |

T_{max} = 873.2 K |
T_{max} = 1273 K |

P_{pump1, input} = 10 kPa |
P_{compresser, input} = 100 kPa |

P_{boiler, input} = 15000 kPa |
r_{p} = 10 (pressure ratio) |

Analysis of 28 variations on the ideal Brayton and Rankine cycles shows a positive correlation between cyclicity and the maximum thermal efficiency. The compiled values for cyclicity and thermal efficiency, as well as the specific modifications made to the Brayton and Rankine cycles can be found in

Cycle | ThermalEfficiency (η_{I}) |
Cyclicity(λ_{max}) |

0.430 | 0 | |

0.451 | 1 | |

0.453 | 1 | |

0.463 | 1 | |

0.472 | 1.15 | |

0.453 | 1.17 | |

0.476 | 1.21 | |

0.476 | 1.30 | |

0.479 | 1.24 | |

0.480 | 1.25 | |

0.482 | 1.26 | |

0.482 | 1.27 | |

0.483 | 1.27 | |

0.470 | 1.27 | |

0.483 | 1.33 | |

0.488 | 1.43 | |

0.491 | 1.44 | |

0.492 | 1.45 | |

0.493 | 1.45 |

FWH, feed water heater.

Cycle | ThermalEfficiency (η_{I}) |
Cyclicity(λ_{max}) |

0.482 | 1.00 | |

0.563 | 1.22 | |

0.685 | 1.39 | |

0.718 | 1.46 | |

0.733 | 1.50 | |

0.742 | 1.52 | |

0.748 | 1.53 | |

0.751 | 1.54 |

The vapor power cycles utilized for the generation of 90% of all electric power used throughout the world are modeled by the Rankine cycle _{C}) will not be reached. The Carnot efficiency, although physically unattainable, is useful in that it gives us an upper limit to strive for. If the efficiency of a real engine is significantly lower, then additional improvements may be possible. More information on efficiencies and power cycles can be found in any thermodynamic reference book, for example

There is a clear lack of data points between the values of zero and one for cyclicity in the Rankine cycles due to the nature of cyclicity being zero, 1, or greater than 1. This constraint makes it impossible to drastically increase the R^{2} value, or coefficient of determination, by obtaining data between the cyclicity values of zero and 1. Including all cycle points (^{2} values for the linear trend lines are 0.988 and 0.768 for Brayton and Rankine cycles respectively. The R^{2} value, for the Rankine cycle increases to 0.818 if we focus on those cycles which are greater than or equal to one (the Brayton cycles all contain some amount of internal structural cycling and therefore are unaffected by this refocusing).

Note: All cycles described here are ideal and optimized for maximum thermal efficiency; changes in kinetic and potential energy from one point to another have been neglected as well as losses in connections between components, such as friction losses in pipes, turbulence, and flow separation.

We conclude from this analysis that the structural method for computing energy cyclicity accurately predicts maximum thermal efficiency for both Rankine and Brayton power cycles. The correlation between cyclicity and maximal thermal efficiency ranges from 0.88 to 0.99, suggesting an extremely strong relationship between these two measures of efficiency. This suggests that increasing the cyclicity (a biological metric) in energetic networks is associated with, or perhaps partially driven by, the maximization of thermodynamic work (an engineering ‘metric’). Alternate power cycle models should be analyzed to further validate the positive relationship between cyclicity and maximum thermal efficiency. From an immediately practical perspective, the benefit of verifying this connection is in determining the relative potential efficiencies of the power cycles. When comparing two modifications to the same cycle it is a great deal easier to calculate cyclicity than to carry out a complete thermodynamic analysis. If cycle A has a higher cyclicity than cycle B, the correlation found here would lead the investigator to believe that cycle A has the potential for a higher maximum thermal efficiency. Establishing this correlation, we can now take advantage of the ecological strategies that we know increase cyclicity, use analogous solutions in human problems, and investigate the extent to which current solutions employing such principles function more effectively.

Our analysis also suggests the two power cycles differ in the extent to which each may be improved by changing the connectivity of its components. The efficiency of the Brayton cycle is extremely sensitive to how interconnected its components are with respect to the transfer of energy. The linear trend lines and coefficients of determination in

Nature’s networks and mankind’s power cycles must both obey the Laws of Thermodynamics, but connecting the two often proves less than straightforward. Although it is well appreciated that thermodynamic constraints affect energy flow in ecological systems

Finally, our results also suggest additional structural parallels between efficient human vs. natural systems, aside from relationships between structural complexity (number of links) and efficiency. Odum, in his paper

New possibilities and questions appear in the field of industrial ecology and power systems design if the link between cyclicity and thermodynamic efficiency withstands further analysis. Maximization of system work becomes an important goal when aiming to base closed loop industrial systems on ecological ones. One may ask, what is system work in a natural ecosystem? What is the analogy between the average heat input temperature of a thermodynamic power cycle and measurable quantities in an ecosystem? Although answering these answers may or may not yield better system designs, it is doubtful that one would ask the questions were it not for an apparent maximum thermal efficiency-cyclicity correlation. Other analyses will most likely continue to show the importance of cyclical connections to the efficient use and production of energy and matter. Additional cycles, including and beyond thermodynamic ones, should be investigated to broaden the positive relationship seen here to one between any network structure and its efficiency. As the resources that current systems are based on continue to diminish, engineering can only benefit from a greater theoretical structure establishing biology and nature as a source of principles, inspiration and guidance.

(TIF)

(TIF)

(TIF)

(TIF)

(TIF)

(TIF)

We thank S.M. Ghiaasiaan, S.M. Jeter, and S.R. Borrett for discussions.