# Development of a Semi-Empirical Model for Estimating the Efficiency of Thermodynamic Power Cycles

## Abstract

**:**

## 1. Introduction

#### 1.1. Power Plants Efficiency

_{2}emissions on a yearly basis [2]. The population increase and new lifestyle trends are basic reasons for the present and future energy situation. Power plants produce the majority of the required electricity; therefore, great interest is given to them, with the aim of increasing their performance [3]. Moreover, the incorporation of renewable energies (e.g., solar energy [4]) is vital for reducing the associated CO

_{2}emissions. The optimization of power plants [5] is also an important weapon for achieving sustainability and leading to suitable units that are ideal for reducing the cost of electricity and simultaneously increasing CO

_{2}avoidance. Another important option is the optimization of power cycles in order to increase their thermodynamic efficiency, which would reduce fuel consumption and consequently lead to sustainable designs.

#### 1.2. Brief Literature Review

_{low}) and the high cycle temperature (T

_{high}), as shown below [6]:

_{m}) [15]:

#### 1.3. The Scope of the Present Work

## 2. Material and Methods

#### 2.1. Basic Mathematical Background

_{th}) of a heat engine. More specifically, the following expression is used [13]:

_{low}) and the high cycle temperature (T

_{high}) are used in the previous formula, while parameter (a) determines the cycle’s performance. Specifically, the different values of the parameter (a) can lead to different cases of the usual thermodynamic cycles.

_{th}) and the respective Carnot cycle efficiency (η

_{carnot}), is given below:

#### 2.2. Followed Methodology

_{low}), the high cycle temperature (T

_{high}) and the cycle efficiency (η

_{th}) are extracted, aiming to estimate superscript (a), which describes the cycle behavior according to Equation (10). Linear regression methods are applied to estimate the suitable values of parameter (a) in every case. In this work, data for various cycles have been used, and more specifically, for the organic Rankine cycle (ORC), water-steam Rankine cycle (WS-RC), Stirling engine, combined cycle (CC), air gas turbine (Air-GT) and supercritical carbon dioxide gas turbine (SCO

_{2}-GT). The maximum examined temperature is 1800 K for the air gas turbine, while the minimum is 353 K for ORC. It is useful to state that the data for the ORC were separated into two categories: one for low-temperature ORC (LT-ORC) and one for high-temperature ORC (HT-ORC). The low-temperature ORC presents a maximum cycle temperature of up to 110 °C, while the cases with higher temperatures are included in the high-temperature ORC.

## 3. Results and Discussion

_{th}) according to Equation (10), and the results are reported in Table 1. Moreover, the respective results for the Carnot and the nonreversible cycle are reported. It is clear that the Carnot efficiency is always greater than the reported efficiency, while the endoreversible efficiency is close to the reported efficiency data. Moreover, it is useful to add that the endoreversible efficiency is a bit higher than the respective Carnot efficiency, as was mentioned in the introduction part regarding Equation (3).

^{2}) is also given for every case. It is very important to highlight that parameter (a) was calculated with a linear regression of the logarithmic factors, according to Equation (10). It is obvious that the reported (R

^{2}) values are high; therefore, the regressions are assumed as reliable. Specifically, the values of (R

^{2}) ranged from 96.47% up to 99.91%—a fact that verifies the validity of the regression procedures.

- ➢
- For the LT-ORC, the total (a) is found at 0.3481 while the reported results indicate a variation from 0.2996 up to 0.3753.
- ➢
- For the HT-ORC, the total (a) is found at 0.5813 while the reported results indicate a variation from 0.4770 up to 0.6399.
- ➢
- For the WS-RC, the total (a) is found at 0.5295 while the reported results indicate a variation from 0.4750 up to 0.5793.
- ➢
- For the Stirling cycle, the total (a) is found at 0.4267 while the reported results indicate a variation from 0.3432 up to 0.5653.
- ➢
- For the SCO
_{2}-GT, the total (a) is found at 0.5189 while the reported results indicate a variation from 0.5051 up to 0.5470. - ➢
- For the Air-GT, the total (a) is found at 0.4808 while the reported results indicate a variation from 0.4617 up to 0.4929.
- ➢
- For the CC, the total (a) is found at 0.4220 while the reported results indicate a variation from 0.3667 up to 0.4440.

_{2}-GT, while the next cycles that follow are the Air-GT, Stirling cycle, combined cycle and LT-ORC, respectively. At this point, it is critical to state that value (a) can be variable when a thermodynamic cycle is optimized, but the present results give a general overview of typical cycle cases.

^{2}) is 98.06% for this case, and thus this regression is acceptable. Figure 2 depicts the reported data and the approximated data with the calculated approximation model. Specifically, Figure 2a shows the thermodynamic efficiency results for different values of the high cycle temperature, while in Figure 2b, they are shown as a function of the temperature ratio (low temperature to high temperature). The global approximation model for all the cycles is described by the next equation:

^{2}is found to be 96.47%, which is the smallest reported value among the cycles; however, it is an acceptable value for the present analysis.

## 4. Conclusions

^{2}of 98.06%, which indicates high accuracy. Also, different equations have been separately developed for each power cycle with high-accuracy indexes. More specifically, the following points describe the examined cycles:

- ➢
- In the LT-ORC case, the mean (a) is found at 0.3481 while the reported results indicate a variation from 0.2996 up to 0.3753.
- ➢
- In the HT-ORC case, the mean (a) is found at 0.5813 while the reported results indicate a variation from 0.4770 up to 0.6399.
- ➢
- In the WS-RC case, the mean (a) is found at 0.5295 while the reported results indicate a variation from 0.4750 up to 0.5793.
- ➢
- In the Stirling cycle case, the mean (a) is found at 0.4267 while the reported results indicate a variation from 0.3432 up to 0.5653.
- ➢
- In the SCO
_{2}-GT case, the mean (a) is found at 0.5189 while the reported results indicate a variation from 0.5051 up to 0.5470. - ➢
- In the Air-GT case, the mean (a) is found at 0.4808 while the reported results indicate a variation from 0.4617 up to 0.4929.
- ➢
- In the CC case, the mean (a) is found at 0.4220 while the reported results indicate a variation from 0.3667 up to 0.4440.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

a | Superscript of the temperature ratio |

i | Irreversibility factor |

R^{2} | Regression coefficient |

T_{high} | High cycle temperature, K |

T_{low} | Low cycle temperature, K |

T_{m} | Medium temperature, K |

Greek Symbols | |

η_{carnot} | Carnot efficiency |

η_{endor} | Endoreversible efficiency |

η_{th} | Thermodynamic efficiency |

η_{th,appr} | Approximated thermodynamic efficiency |

Subscripts | |

max | Maximum reported value for the specific cycle type |

min | Minimum reported value for the specific cycle type |

total | Total value for the specific cycle type |

Abbreviations | |

Air-GT | Air Gas Turbine |

CC | Combined Cycle |

EXPER | Experimental work |

HT-ORC | High-Temperature Organic Rankine Cycle |

LT-ORC | Low-Temperature Organic Rankine Cycle |

ORC | Organic Rankine cycle |

SCO_{2}-GT | Supercritical Carbon Dioxide Gas Turbine |

THEOR | Theoretical work |

WS-RC | Water-Steam Rankine Cycle |

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**Figure 1.**Thermodynamic cycle efficiency for different cycles (

**a**) as a function of the high cycle temperature (T

_{high}); (

**b**) as a function of the temperature ratio (T

_{low}/T

_{high}).

**Figure 2.**Thermodynamic cycle efficiency for all the reported data (

**a**) as a function of the high cycle temperature (T

_{high}); (

**b**) as a function of the temperature ratio (T

_{low}/T

_{high}).

**Figure 3.**Thermodynamic cycle efficiency for all the reported data, Carnot efficiency and endoreversible efficiency (

**a**) as a function of the high cycle temperature (T

_{high}); (

**b**) as a function of the temperature ratio (T

_{low}/T

_{high}).

N/A | T_{high} (K) | T_{low} (K) | η_{th} | η_{carnot} | η_{endor} | a | Cycle | Type | Ref. |
---|---|---|---|---|---|---|---|---|---|

1 | 389 | 298 | 0.0940 | 0.2339 | 0.1247 | 0.3704 | LT-ORC | EXPER | [17] |

2 | 381 | 298 | 0.0881 | 0.2178 | 0.1156 | 0.3753 | LT-ORC | EXPER | [17] |

3 | 370 | 298 | 0.0737 | 0.1946 | 0.1026 | 0.3538 | LT-ORC | EXPER | [17] |

4 | 363 | 298 | 0.0574 | 0.1791 | 0.0939 | 0.2996 | LT-ORC | THEOR | [17] |

5 | 353 | 298 | 0.0522 | 0.1558 | 0.0812 | 0.3165 | LT-ORC | EXPER | [18] |

6 | 381 | 298 | 0.0794 | 0.2178 | 0.1156 | 0.3367 | LT-ORC | EXPER | [18] |

7 | 542 | 333 | 0.2536 | 0.3856 | 0.2162 | 0.6005 | HT-ORC | THEOR | [19] |

8 | 507 | 333 | 0.2341 | 0.3432 | 0.1896 | 0.6345 | HT-ORC | THEOR | [19] |

9 | 514 | 333 | 0.2316 | 0.3521 | 0.1951 | 0.6069 | HT-ORC | THEOR | [19] |

10 | 533 | 333 | 0.2155 | 0.3752 | 0.2096 | 0.5160 | HT-ORC | THEOR | [19] |

11 | 500 | 333 | 0.2125 | 0.3340 | 0.1839 | 0.5877 | HT-ORC | THEOR | [19] |

12 | 468 | 333 | 0.1957 | 0.2885 | 0.1565 | 0.6399 | HT-ORC | THEOR | [19] |

13 | 472 | 333 | 0.1800 | 0.2945 | 0.1601 | 0.5689 | HT-ORC | THEOR | [19] |

14 | 463 | 333 | 0.1714 | 0.2808 | 0.1519 | 0.5705 | HT-ORC | THEOR | [19] |

15 | 450 | 333 | 0.1338 | 0.2600 | 0.1398 | 0.4770 | HT-ORC | THEOR | [19] |

16 | 623 | 298 | 0.3260 | 0.5217 | 0.3084 | 0.5350 | WS-RC | THEOR | [20] |

17 | 373 | 298 | 0.1092 | 0.2011 | 0.1062 | 0.5151 | WS-RC | THEOR | [21] |

18 | 423 | 298 | 0.1702 | 0.2955 | 0.1607 | 0.5326 | WS-RC | THEOR | [21] |

19 | 473 | 298 | 0.2139 | 0.3700 | 0.2063 | 0.5209 | WS-RC | THEOR | [21] |

20 | 523 | 298 | 0.2451 | 0.4302 | 0.2452 | 0.4999 | WS-RC | THEOR | [21] |

21 | 573 | 299 | 0.2658 | 0.4782 | 0.2776 | 0.4750 | WS-RC | THEOR | [21] |

22 | 666 | 300 | 0.3700 | 0.5495 | 0.3288 | 0.5793 | WS-RC | EXPER | [22] |

23 | 1123 | 298 | 0.3870 | 0.7346 | 0.4849 | 0.3689 | Stirling cycle | EXPER | [23] |

24 | 1172 | 298 | 0.3750 | 0.7457 | 0.4958 | 0.3432 | Stirling cycle | EXPER | [23] |

25 | 773 | 298 | 0.3440 | 0.6145 | 0.3791 | 0.4423 | Stirling cycle | THEOR | [24] |

26 | 939 | 288 | 0.4873 | 0.6933 | 0.4462 | 0.5653 | Stirling cycle | THEOR | [25] |

27 | 623 | 298 | 0.2850 | 0.5217 | 0.3084 | 0.4549 | Stirling cycle | THEOR | [26] |

28 | 573 | 298 | 0.2690 | 0.4799 | 0.2788 | 0.4793 | Stirling cycle | THEOR | [26] |

29 | 523 | 298 | 0.2310 | 0.4302 | 0.2452 | 0.4670 | Stirling cycle | THEOR | [26] |

30 | 823 | 305 | 0.4190 | 0.6294 | 0.3912 | 0.5470 | SCO_{2}-GT | THEOR | [27] |

31 | 1023 | 305 | 0.4652 | 0.7019 | 0.4540 | 0.5172 | SCO_{2}-GT | THEOR | [27] |

32 | 823 | 323 | 0.3765 | 0.6075 | 0.3735 | 0.5051 | SCO_{2}-GT | THEOR | [27] |

33 | 1023 | 323 | 0.4440 | 0.6843 | 0.4381 | 0.5092 | SCO_{2}-GT | THEOR | [27] |

34 | 1123 | 298 | 0.4800 | 0.7346 | 0.4849 | 0.4929 | Air-GT | EXPER | [28] |

35 | 1123 | 298 | 0.4580 | 0.7346 | 0.4849 | 0.4617 | Air-GT | EXPER | [28] |

36 | 1152 | 298 | 0.4800 | 0.7413 | 0.4914 | 0.4836 | Air-GT | EXPER | [28] |

37 | 788 | 298 | 0.3780 | 0.6218 | 0.3850 | 0.4883 | Air-GT | EXPER | [28] |

38 | 1200 | 298 | 0.4000 | 0.7517 | 0.5017 | 0.3667 | CC | THEOR | [29] |

39 | 1500 | 298 | 0.5000 | 0.8013 | 0.5543 | 0.4289 | CC | THEOR | [29] |

40 | 1800 | 298 | 0.5500 | 0.8344 | 0.5931 | 0.4440 | CC | THEOR | [29] |

41 | 1244 | 293 | 0.4700 | 0.7645 | 0.5147 | 0.4391 | CC | EXPER | [30] |

42 | 1561 | 288 | 0.5047 | 0.8155 | 0.5705 | 0.4157 | CC | THEOR | [31] |

**Table 2.**Summary of the parameter (a) for the different cycles by using the logarithmic approximation of the reported results.

Cycle | a_{total} | R^{2} | a_{min} | a_{max} |
---|---|---|---|---|

Low-Temperature ORC | 0.3481 | 99.44% | 0.2996 | 0.3753 |

High-Temperature ORC | 0.5813 | 99.37% | 0.4770 | 0.6399 |

Water-Steam Rankine cycle | 0.5295 | 99.51% | 0.4750 | 0.5793 |

Stirling cycle | 0.4267 | 96.47% | 0.3432 | 0.5653 |

S-CO_{2} gas turbine | 0.5189 | 99.91% | 0.5051 | 0.5470 |

Air gas turbine | 0.4808 | 99.93% | 0.4617 | 0.4929 |

Combined cycle | 0.4220 | 99.63% | 0.3667 | 0.4440 |

TOTAL | 0.4594 | 98.06% | - | - |

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**MDPI and ACS Style**

Bellos, E.
Development of a Semi-Empirical Model for Estimating the Efficiency of Thermodynamic Power Cycles. *Sci* **2023**, *5*, 33.
https://doi.org/10.3390/sci5030033

**AMA Style**

Bellos E.
Development of a Semi-Empirical Model for Estimating the Efficiency of Thermodynamic Power Cycles. *Sci*. 2023; 5(3):33.
https://doi.org/10.3390/sci5030033

**Chicago/Turabian Style**

Bellos, Evangelos.
2023. "Development of a Semi-Empirical Model for Estimating the Efficiency of Thermodynamic Power Cycles" *Sci* 5, no. 3: 33.
https://doi.org/10.3390/sci5030033