# Analysis of an Air Powered Engine System Using a Multi-Stage Radial Turbine

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## Abstract

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## 1. Introduction

## 2. Description of the System

- Mass flow: m = 1 kg/s.
- Ambient temperature: T
_{0}= 293 K - External heat source temperature: T
_{6}= 293 K to 793 K - Ambient pressure: P
_{0}= 1 bar - Inlet pressure: P
_{0'}= 70 bar

## 3. Analysis Methodology

#### 3.1. Thermodynamic Analyses

_{n}of the compressed air (State n) is given as:

_{n}is a function of pressure P

_{n}and temperature T

_{n}. The exergy of the compressed air (State n) is given by:

_{n-n'}, defined as the ratio of the actual work (W

_{ac,n−n'}) to the ideal isentropic work (W

_{id,n−n'}):

_{ac,n-n'}of the expansion process (Process n−n') is therefore given by:

_{id,n'}is a function of entropy s

_{n}and pressure P

_{n'}, and the pressure P

_{n'}is given as:

_{n}is the expansion ratio of the nth stage radial turbine and satisfies the equation:

#### 3.2. Evaluation Criteria

_{0'}, Ex

_{0'}are the enthalpy and exergy at the entry of a single stage radial turbine with heating system, Q

_{s}and Ex

_{s}are the heat and exergy absorbed during the heating process, W is the work output of the radial turbine, the thermal efficiency (η

_{th}) and exergy efficiency (η

_{ex}) of the single radial turbine can be given as:

_{0'}, Q

_{s}, Ex

_{0'}and Ex

_{s}are constant at a given working condition. As a result, thermal efficiency, exergy efficiency and power output are equivalent for single stage radial turbine, that is to say, maximum thermal efficiency means maximum exergy efficiency and maximum power output for single stage radial turbine, so any of them can be considered as the criterion for evaluating a single stage radial turbine’s performance. In other words, given the working conditions, a good design of a single stage radial turbine with or without a heating system will have a high thermal efficiency, a high exergy efficiency and a large power output, but in multistage radial turbine, the heat and exergy absorbed during the heating process are not constant at a given working condition and they are influenced by expansion ratio of each stage, so theoretical thermodynamic analysis is conducted as follows to answer if this is also applicable for evaluating multistage radial turbines with inter-heating.

_{0’}and the heat absorbed during four heating processes (Q

_{0'−1}, Q

_{1'−2}, Q

_{2'−3}and Q

_{3'−4}). The energy output of the system is the work output of four radial turbines (W

_{ac,1−1'}, W

_{ac,2−2'}, W

_{ac,3−3'}and W

_{ac,4−4'}) and the enthalpy of the compressed air at the outlet H

_{4'}. Therefore the power output and thermal efficiency of the four-stage radial turbine can be expressed as:

_{n}at State n after heating is greater than that of exergy Ex

_{(n−1)'}at State n−1' before heating, the compressed air absorbs the heat during the heating processes and the heating processes increases the exergy of the compressed air. The exergy input is the sum of the exergy of the compressed air at the entry and the exergy absorbed during the heating processes, so the exergy efficiency of the four-stage radial turbine is defined as:

_{(n−1)'}before heating is greater than that of exergy Ex

_{n}after heating as in Figure 4b. In this case, the exergy of the compressed air decreases though it absorbs the heat during the heating processes. In other words, the compressed air gets the energy but losses the ability to generate available work. Figure 4b shows the case in which the temperature before heating (T

_{0'}, T

_{1’}, T

_{2’}and T

_{3’}) is lower than ambient temperature. In this case, Ex

_{0'-1},Ex

_{1’-2}, Ex

_{2’-3}and Ex

_{3’-4}are negative which implies that Ex

_{0'-1}, Ex

_{1’-2}, Ex

_{2’-3}and Ex

_{3’-4}are exergy output of the system. Its exergy input is only the exergy of the compressed air at the entry (Ex

_{0’}). On the other hand, the effective output does not change, i.e., (W

_{ac,1−1’}+ W

_{ac,2−2’}+ W

_{ac,3−3’}+ W

_{ac,4−4’}). Therefore, the definition of exergy efficiency can be expressed as:

#### 3.3. Numerical Methodology

## 4. Results and Discussion

#### 4.1. Maximum Thermal Efficiency

_{4’}), hence increasing the total thermal efficiency. This is why the expansion ratios of the fourth stage at efficiencies investigated (70%, 80% and 90%) all reach the expansion ratio limit. From Equation (19) it can be derived that there are two ways to increase the thermal efficiency of the four-stage radial turbine, one is decreasing the energy leaving the system (H

_{4'}), the other is increasing the energy entering the system (H

_{0'}, Q

_{0'-1},Q

_{1'-2}, Q

_{2'-3}and Q

_{3'-4}). The energy entering the system H

_{0'}and the heat absorbed during the first heating process Q

_{0'-1}are given by the working condition of the turbine, the others (Q

_{1'-2}, Q

_{2'-3}and Q

_{3'-4}) depend on the work output of the first three stages. The more the work output of the first three stages, the more the energy entering the system, that is to say, to maximize the energy input is to maximize the work output. If the working fluid is perfect gas, the expansion ratios of the first three stages should be identical to maximize the work output. The expansion ratio increases from the first stage to the third stage because of the properties of real air, i.e., the enthalpy drop is influenced by working pressure.

_{4'}) increases with working temperature, then but the energy input also increases, their growth rates are nearly same, so the thermal efficiency is almost identical at the temperature investigated (293 K–793 K). It can also be seen, in Figure 7, the maximum thermal efficiency increases with increasing turbine efficiency.

#### 4.2. Maximum Exergy Efficiency

#### 4.3. Maximum Power Output

## 5. Conclusions

- (1)
- It is found that the maximum thermal efficiency, maximum exergy efficiency and maximum work output of the four stage radial turbine with inter-heating are 62.6%, 91.9%, and 763.2 kJ/s, respectively. However, the thermal efficiency, exergy efficiency and work output are not equivalent.
- (2)
- At low working temperatures (below 500 K) both maximum exergy efficiency and maximum work output can be used as the design objective of the multi-stage radial turbine, however, only maximum work output can be used as the design objective of the turbine over the whole working temperature range in this work.
- (3)
- The maximum thermal efficiency can't be used as the design objective for the turbine.

## Acknowledgments

## Nomenclature:

Ex_{(n-1)'-n} | The variation of exergy during the heating process (n−1)'–n, kJ/s |

Ex_{s} | The variation of exergy during the heating process of a single stage radial turbine, kJ/s |

Ex_{tot} | Total exergy input of the system, kJ/s |

h | Specific enthalpy, kJ/kg |

h_{id} | Ideal specific enthalpy, kJ/kg |

H | Enthalpy, kJ/s |

m | Mass flow, kg/s |

p | Pressure, bar |

Q_{(n-1)'-}_{n} | Heat absorbed during the heating process (n−1)'–n, kJ/s |

Q_{s} | Heat absorbed during the heating process of a single stage radial turbine, kJ/s |

s | Entropy, kJ/(kg) (K) |

T | Temperature, K |

T_{6} | External heat source temperature, K |

w | Specific work, kJ/kg |

W | Total work output of the turbine, kJ/s |

W_{ac,n-}_{n'} | Actual Work output during the expansion process n–n', kJ/s |

W_{id,n-}_{n'} | Ideal Work output during the expansion process n–n', kJ/s |

η_{n-}_{n'} | Isentropic efficiency during the expansion process n–n' |

η_{ex} | Exergy efficiency |

η_{th} | Thermal efficiency |

π_{n} | Expansion ratio of the nth stage |

## Subscripts

0 | Ambient state |

0' | The state of air entering the system |

1 | The state of air entering the first stage radial turbine |

1' | The state of air leaving the first stage radial turbine |

2 | The state of air entering the second stage radial turbine |

2' | The state of air leaving the second stage radial turbine |

3 | The state of air entering the third stage radial turbine |

3' | The state of air leaving the third stage radial turbine |

4 | The state of air entering the fourth stage radial turbine |

4' | The state of air leaving the fourth stage radial turbine |

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**MDPI and ACS Style**

Zhang, X.; Chen, H.; Yan, X.; Zhang, X.; Tan, C.
Analysis of an Air Powered Engine System Using a Multi-Stage Radial Turbine. *Entropy* **2013**, *15*, 1186-1201.
https://doi.org/10.3390/e15041186

**AMA Style**

Zhang X, Chen H, Yan X, Zhang X, Tan C.
Analysis of an Air Powered Engine System Using a Multi-Stage Radial Turbine. *Entropy*. 2013; 15(4):1186-1201.
https://doi.org/10.3390/e15041186

**Chicago/Turabian Style**

Zhang, Xuehui, Haisheng Chen, Xiaohui Yan, Xinjing Zhang, and Chunqing Tan.
2013. "Analysis of an Air Powered Engine System Using a Multi-Stage Radial Turbine" *Entropy* 15, no. 4: 1186-1201.
https://doi.org/10.3390/e15041186