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Definition and comparison of mixed expansion efficiency for cooled turbine
Accepted Manuscript Definition and Comparison of Mixed Expansion Efficiency for Cooled Turbine Wei Ba, Xiao-chen Wang, Xue-song Li, Xiao-dong Ren, Chun-wei Gu PII: DOI: Reference:
S1359-4311(18)30634-3 https://doi.org/10.1016/j.applthermaleng.2018.05.125 ATE 12264
To appear in:
Applied Thermal Engineering
Received Date: Revised Date: Accepted Date:
29 January 2018 25 May 2018 29 May 2018
Please cite this article as: W. Ba, X-c. Wang, X-s. Li, X-d. Ren, C-w. Gu, Definition and Comparison of Mixed Expansion Efficiency for Cooled Turbine, Applied Thermal Engineering (2018), doi: https://doi.org/10.1016/ j.applthermaleng.2018.05.125
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Definition and Comparison of Mixed Expansion Efficiency for Cooled Turbine
Wei Ba
Xiao-chen Wang
Xue-song Li
Xiao-dong Ren*
Chun-wei Gu
Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Energy and Power Engineering, Tsinghua University, Beijing 100084, China *Corresponding author. E-mail address: [email protected]
Abstract Over the years, there has been continual increase in the temperature inside a turbine inlet through improvements in the gas turbine efficiency and specific power, and cooling technologies to deal with such temperature increases have been under development for decades. However, the definition of “efficiency” remains ambiguous for cooled turbines, with no consensus achieved on the ideal expansion process. This paper first reviews several proposed definitions, Hartsel, MP (mainstream-pressure), FR (fully reversible), and WP (weighted-pressure) of efficiency and then presents a new one to overcome the existing shortage. The increase in entropy related to mixing is first analysed to demonstrate that FR efficiency consistently provides a significantly lower value than the other definitions, assuming the unavoidable entropy generation to be zero. The flow mixing in a smooth adiabatic channel is then analysed to show that the WP ideal process is not sufficiently “ideal.” A fully mixed (FM) definition is then proposed based on a physical mixing process, followed by isentropic expansion, the value of which does not differ significantly from most of the proposed definitions, and is able to 1
consider the influence of the inlet Mach number on the hypothetical ideal process. Finally, this paper compares several definitions in view of the thermodynamic cycle, and shows that the FM definition can determine the performance potential of an existing turbine better than the other definitions, which means that the FM efficiency can be used as a reliable tool for the performance evaluation and optimization of a cooled turbine.
Keywords: Cooled turbine; Efficiency; Mixed Expansion
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Nomenclature A Cp fc G h R T P V Xgm, Xcm
cross section area, m2 constant pressure specific heat capacity, J/kg/K coolant fraction mass flow rate, kg/s specific enthalpy, kJ/kg specific gas constant, J/kg/K temperature, K power, kW, or pressure, Pa velocity, m/s mole fractions of the mainstream and coolant after mixing
Greek symbols γ η ρ σ
specific heat ratio efficiency density, kg/m3 entropy created per unit mass of mixture, J/kg/K.
Subscripts 3 4 ac c C g Hart i id is Mix, m P t T
turbine inlet turbine outlet actual process coolant condition related to composition change gas condition Hartsel definition inlet ideal process isentropic process mixture related to pressure change total condition turbine, or related to temperature change
Acronyms FR FM MP WP
fully reversible fully mixed mainstream pressure weighted pressure
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1. Introduction Improvements in the performance of modern gas turbines have led to an increase in the turbine inlet temperature, and compound cooling technologies [1-2] have thus been widely applied to protect the hot components. Therefore, a definition of the efficiency of an uncooled turbine should be further developed to characterize the performance of a cooled turbine. However, there is still no consensus on the most suitable definition for cooled turbine efficiency, and such ambiguity has resulted in inconvenience and confusion in practical use. For turbine designers and researchers, turbine efficiency has two main functions in terms of practical design [3-7] and optimization[8], to compare the performance of different turbines [9-12] and evaluate the optimization potential [13-16], and to optimise the design of the gas turbine cycle [17-21]. Most turbine efficiency definitions can be written as Eq. (1), where Pac represents the actual power output and Pid represents the ideal power output.
T
Pac Pid
(1)
There are not many ambiguities in the determination of the actual power output. The value of Pac can be determined either experimentally or through a numerical simulation, and the only difference is that the disk windage and bearing loss can be precisely determined during the experiment. Instead of estimating the disk windage and bearing loss based on empirical correlations, in this paper Pac is determined through the aerodynamic parameters of the main flow and coolant at the inlet and outlet for simplicity. Ambiguities in a cooled turbine efficiency definition are always based on the determination of Pid, which is mainly defined in two ways in the literature. The first is 4
an unmixed efficiency definition, presented by Hartsel [22], which assumes that the mainstream gas and coolant expand separately and isentropically during an ideal process. The other type is called mixed expansion efficiency, and is referred to as MP [23], FR, or WP [24, 25]. All these definitions assume a mixing process of the mainstream and coolant prior to isentropic expansion toward the total outlet pressure. The Hartsel and MP efficiencies have already been fully discussed in the open literature [24], and it has been revealed that a shortage make them unqualified to properly characterize the performance of a cooled turbine. The FR and WP efficiencies are further investigated in this paper, and it is shown that the FR definition always has a significant lower value than the other definitions and overestimates the ideal turbine power capability by neglecting the unavoidable entropy increase associated with temperature equilibration of the mainstream and coolant. The hypothetical WP ideal process was also studied in this paper, and it was revealed that this process is not sufficiently ideal, and the WP efficiency consistently underestimates the ideal turbine power capability. The fully mixed (FM) definition is presented in this paper to overcome the shortage of the proposed efficiency definition. Although the ideal FM process also contains a mixing process prior to isentropic expansion, the FM mixing process is assumed to be applied in a smooth adiabatic channel, which can deliver a physically based ideal process for an efficiency definition, and properly consider the unavoidable increase in entropy. The study described herein on the FM efficiency shows that its value does not differ significantly from most of the proposed definitions, and is capable of considering the influence of the inlet Mach number on the ideal turbine power capability. A
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thermodynamic cycle analysis also shows that the FM efficiency can be used to evaluate the performance potential of an existing turbine better than other definitions, making it a better measurement for the performance of a cooled turbine.
2. Proposed Cooled Turbine Efficiencies 2.1 Uncooled turbine efficiency The uncooled turbine efficiency is first described because the proposed cooled turbine efficiency consistently follows a similar approach. For an uncooled turbine, Eq. (1) can be written as follows, and is usually called the total-to-total isentropic efficiency.
T
G3 ht 3 ht 4 h h t3 t 4 G3 ht 3 ht 4is ht 3 ht 4is
(2)
where ht3 is the total enthalpy of a turbine inlet, ht4 is the actual total enthalpy of a turbine outlet, and ht4is is the ideal total enthalpy of a turbine outlet. As we can see, for an uncooled turbine, Pac is the difference between the actual total enthalpy at the turbine inlet and outlet, and Pid is the difference between the actual total enthalpy at the inlet and the ideal total enthalpy at the outlet, which is determined through an isentropic expansion.
2.2 Actual Power Output of Cooled Turbine Determination of the actual power output is essential to every efficiency definition, and the difference between the actual shaft power output and actual turbine power generation should be mentioned. The actual power output of the shaft is always determined experimentally, considering the negative effect caused by the disk windage and bearing loss, and should be used as the actual power output when an experiment is 6
conducted to test the shaft power. The actual turbine power generation is always determined through a numerical simulation, and there is no way to estimate the disk windage and bearing loss precisely. In this paper, the actual turbine power generation is regarded as the actual power output for simplicity.
Fig. 1 Schematic of gas turbine cycle and control volume selection
Next, the main argument is whether the rotor coolant flow has the ability to do work. To avoid this argument, the actual power output is calculated based on the control volume of the turbine, as shown in Fig. 1, and the details of the flow in a turbine are neglected by applying an energy conservation equation to the control volume. With an adiabatic assumption of the turbine, the actual power output can be written as follows.
Pac G3ht 3 Gc htc G4 ht 4
(3)
2.3 Hartsel and MP ideal processes The Hartsel and MP efficiencies have been used by turbine designers, and are prototypes of unmixed and mixed expansion efficiency definitions, respectively. Both
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definitions have been well documented in the literature, and therefore only a brief introduction is provided here. Hartsel assumes that the mainstream and the coolant expand separately and isentropically during the ideal process, and the ideal power output can be calculated as follows:
Pid , Hart G3 ht 3 ht 4is , g Gc htc ht 4is ,c
(4)
where ht4is,g and ht4is,c are separately determined by each isentropic expansion toward the total outlet pressure. The Hartsel process assumes an adiabatic and fully reversible expansion process; however, it has been criticized because its value may be greater than unity when considering the real properties of a gas and coolant. MP efficiency is referred to as the mixed efficiency of the mainstream pressure, which assumes that the mainstream and the coolant mix at a constant pressure (mainstream pressure) prior to the isentropic expansion during the ideal process. For all mixed expansion definitions, the ideal power output can be written as follows:
Pid ,Mix G3 Gc htm,3 htm,is
(5)
where htm,3 is the total enthalpy of the mixture at the turbine inlet, and htm,is is the total enthalpy of the ideal mixture expanded isentropically from the total pressure of the inlet mixture to the total outlet pressure. For the MP definition, the total pressure of the inlet mixture is simply equal to the total pressure of the mainstream inlet. Ptm,3 Pt 3
8
(6)
Compared to the Hartsel definition, the MP definition is capable of considering the mixing procedure of the mainstream and the coolant, but has been criticized for neglecting the influence of inlet pressure of the coolant on the ideal power output.
2.4 FR ideal process Considering the shortages of the Hartsel and MP definitions, the FR ideal process was presented as a fully reversible mixed expansion process. The ambiguity of a mixed expansion efficiency definition is how to determine the state parameter of the mixture prior to expansion. The total temperature can be easily determined through the energy conservation equation, but the total pressure has to be determined based on a different assumption, such as being equal to the total pressure of the mainstream in MP ideal process. To obtain a fully reversible adiabatic mixing process, the total pressure of the mixture is determined to make the total entropy generation zero during the FR ideal process. In the mixing process, the entropy generation [23] per unit mass of the mixture can be written as follows:
T P C
(7)
Where
T 1 f c C pg ln
P 1 f c Rg ln
Ttm,3 Tt 3
f cC pc ln
Ttm,3 Ttc
Pt 3 P f c Rc ln tc Ptm,3 Ptm,3
C 1 fc Rg ln X gm fc Rc ln X cm
9
(8)
(9)
(10)
The term σc represents the diffusional entropy creation owing to a compositional change, which cannot be avoided. Therefore, only (σT+σP) = 0 is assumed in the FR definition. The total pressure of the mixture can then be determined as follows: Ptm,3 Pt13 fc Ptc fc exp T R
(11)
The FR efficiency definition is seemed to be reasonable for the cooled turbine efficiency, and for a fully reversible adiabatic mixing process combined with the following isentropic expansion process. However, the value of the FR efficiency is significant lower than the other definitions, even if the Hartsel efficiency also assumes an adiabatic and fully reversible expansion process. It will then be discussed how this came to be through the neglecting of the unavoidable entropy increase associated with the temperature equilibration of the mainstream and coolant.
Fig. 2 Relative values of entropy creation terms σT and σP
In the FR ideal process, the σP term is required to be the negative value of σT. Therefore, the relative values of these two terms in a normal mixing process are worth comparing. σP1, σP2 and σP3 are the entropy production related to pressure change in the
10
mixing process, where the mixture pressure is assumed to be the mainstream pressure, the weighted average pressure, and the coolant pressure separately. As shown in Fig. 2, the absolute value of σT is much bigger than that of σP in each case, which means that a zero total entropy generation assumption may lead to a much higher mixture pressure than the pressure of each component. The FR ideal mixture pressures at different coolant fractions are then compared with the mainstream and coolant pressures, as shown in Fig. 3, where the mass flow rate of coolant is 10%, 20% and 30% of the mainstream mass flow rate. The results show that the total pressure of the FR ideal mixture is higher than the pressure of the mainstream and coolant, and will further increase with the coolant fraction, thereby resulting in zero total entropy generation.
Fig. 3 Relative value of FR mixture and component pressures
With a higher mixture pressure and a constant turbine outlet pressure, the FR ideal power output will increase as the expansion ratio increases. This explains why the value of the FR efficiency is significantly lower than in the other definitions.
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A zero total entropy generation assumption leads to an unreasonable FR ideal mixture pressure, which deserves further discussion with regard to the reasonability of the assumption. In most modern turbines, coolant is introduced to protect the hot components from the high temperature of the mainstream, which unavoidably increases the entropy associated with the temperature equilibration of the mainstream and coolant. Therefore, similar to the σc term, σT is also unable to be avoided in the mixing process, and instead produces a useful operation. In other words, the FR definition has an unreasonably large ideal power output when assuming the unavoidable entropy increase to be zero.
2.5 WP ideal process Considering the extremely low FR efficiency, the WP definition assumes σP to be zero, and the WP ideal mixture pressure can then be determined as follows:
Ptm,3 Pt13 fc Ptc fc
(12)
However, differing from σT, which is always positive and unavoidable, the lowest value of σP may be negative, which will make the WP definition underestimate the ideal power output of a turbine. Considering the mixing process of two streams in a smooth adiabatic channel and applying conservation equations to the whole control volume to neglect the flow details, the entropy generation related to the mixing process can be determined. More details on the mixing calculation are provided in the following chapter. The contours of the σP value during the mixing process at different temperatures and pressures of the components are shown in Fig. 4. To determine all component inlet
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conditions, assumptions of equal inlet static pressure, equal inlet static temperature, and equal inlet velocity are made separately to carry out the calculation. As shown in Fig. 4, the value of σP, and the differences in the total pressure and temperature of the components, are all non-dimensionalised. The two inlet stream are distinguished by subscript “1” and “2”, and “bar” means the mean value of pressure, temperature or velocity.
P
P ,coef
PCoef
Pt1 Pt 2 i
P P t
TCoef
(13)
1 2 Vi 2
i
Tt1 Tt 2 i
T T t
(14)
(15)
i
The value of σP is always positive when the mixing components have the same inlet static pressure, but is not the same when assumptions of the same inlet static temperature or same inlet velocity are made. As shown in Fig. 4 (b), with a constant static inlet temperature, a negative σP occurs when one component has a higher total temperature and the other has higher total pressure. For a constant inlet velocity, shown in Fig. 4 (c), a negative σP occurs when one component has both a higher total temperature and a higher total pressure.
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(a) Equal inlet static pressure
(b) Equal inlet static temperature
(c) Equal inlet velocity
Fig. 4 Value of σP with different total temperatures and pressures of the components
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The calculation above shows that the lowest value of σP may not be zero when the mixing process occurs in a smooth adiabatic channel, and therefore the simple assumption of σP = 0 in the WP definition is insufficiently accurate, which will lead to an underestimation of the ideal turbine power output.
3. FM Cooled Turbine Efficiency Studies on FR and WP definitions have shown that the critical issue of a mixed expansion efficiency definition is about how to determine the ideal mixing process of the mainstream and coolant. Previous ideal mixing processes were determined by simply assuming the entropy generation, or part of it, to be zero; however, such an assumption neglects too much physical matter in the turbine design and operation, and results in a different shortage. A reasonable ideal mixing process should be developed as a physically based ideal process, which will accept the unavoidable entropy generation during a turbine operation, such as the creation of diffusional entropy owing to a change in composition, and entropy creation associated with a temperature equilibration. In addition, the physically based ideal mixing process should also minimize the part of the entropy change that can be optimized. The ideal mixed expansion process for a cooled turbine can then be determined and the ideal power output can be specified. The physically based efficiency definition may not only become a reliable tool for characterizing the performance of a cooled turbine, it may also be capable of evaluating the theoretical optimization potential by comparing the efficiency value with unity.
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3.1 FM ideal mixing process Different from the ideal mixing process determined by restricting the entropy generation directly, the FM definition presented in this paper assumes an ideal mixing process based on a physically based mixing process.
Fig. 5 Schematic of mixing process in smooth adiabatic channel
As shown in Fig. 5, the FM definition assumes an ideal mixing process in a smooth adiabatic channel, and the mainstream and coolant are further simplified as two flows in the same direction with different flow conditions. Once the outlet flow is assumed to be uniform, the flow condition at the channel outlet can be easily determined based on the governing equations of the entire control volume, without too much concern regarding the details of the flow mixing. The governing equations can be written as follows: The continuity equation is
A1 A 1V1 2V2 2
A1 1 =mVm A2
(16)
The momentum equation is
A1 2 2 A P1 1V1 P2 2V2 2 The energy equation is
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A1 2 1 =Pm mVm A2
(17)
1 fc Cp1Tt1 fcCp2Tt 2 TtmCpm
(18)
Based on the provided inlet flow conditions, the mass flow ratio, and the state equation, the flow condition of the mixture at the outlet can be determined.
R a a 4 RmTtm 1 m b b 2Cpm Vm R 2 m Cpm 2
Ttm
Ptm
1 fc Cp1Tt1 fcCp2Tt 2 Cpm a bVm
(19)
(20)
(21)
m
m 1 Vm 2 T Ttm tm 2Cpm
where a and b are two constants, as follows:
A a 1 P1 1V12 P2 2V22 A2 A b 1 1V1 2V2 A2
A1 1 A2
A1 1 A2
(22)
(23)
As we can see in Eq. (19), there may be two values for the same set of inlet flow conditions, corresponding to the subsonic and supersonic outlet flow conditions. Because the mainstream at the turbine inlet and the coolant at the cooling holes are always subsonic, only a subsonic mixture flow condition is considered in this paper.
3.2 Comparison of efficiency values The total pressure of FM ideal mixture before expansion is determined through Eq. (21), and the FM ideal power output can then be determined from Eq. (5). Using the given actual turbine power output, the FM efficiency can then be obtained. 17
A typical high-pressure turbine stage is considered here to compare the value of the different efficiency definitions for different coolant fractions. Because the actual stage performances with different coolant fractions are not provided, the calculations are carried out by assuming that the stage expansion ratio and MP efficiency are fixed, and the MP efficiency is also taken as the reference efficiency to provide another efficiency value.
Fig. 6 Comparison of different efficiencies for varying coolant flow fractions
As shown in Fig. 6, the relative efficiency values of the cooling stage under different definitions of the varying coolant flow fractions are compared. When the coolant fraction equals zero, all relative efficiencies are unity, which means that all the proposed definitions will have the same value when there is no coolant in the turbine. This is an expected behaviour because all cooled turbine efficiency definitions will degenerate into the total-to-total isentropic efficiency mentioned in the second chapter, when no coolant is applied. The FR efficiency has a significantly lower value than the other definitions, and the difference increases notably with the coolant fraction. This phenomenon again demonstrates that the FR efficiency assumes a much larger ideal power output than the
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other definitions, particularly for a large coolant fraction. This large difference will not only increase the difficulty to translate experiences from the past to a new design goal, it will also cause an overestimation of the theoretical optimization potential in the design. The Hartsel, WP, and FM definitions seem to have similar values for varying coolant fractions, all of which are smaller than the MP efficiency. This phenomenon can be expected when the total pressure of the coolant is higher than that of the mainstream, which is a usual situation found in cooled turbines. A more detailed observation reveals that the FM efficiency consistently has a lower value than the WP definition in this calculation, which means that the FM ideal process has a larger ideal power output than that of the WP definition.
3.3 Sensitivity study of turbine inlet Mach number Differing from other proposed efficiency definitions, the FM efficiency is capable of considering the influence of the inlet Mach number on the ideal turbine power output. Therefore, a sensitivity study of the inlet Mach number is essential for a new definition. Because the WP ideal process is similar to that of the FM definition, the WP efficiency under the same operating conditions was chosen for a comparison. It should be recalled that the WP efficiency has nothing to do with the inlet Mach number, as long as the inlet stagnation parameters are specified. All stage parameters applied in the following calculation are the same as those in chapter 3.2.
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Fig. 7 Sensitivity of inlet Mach number on FM efficiency
As shown in Fig. 7, the FM efficiency has a different value under different inlet Mach number conditions, and all the FM efficiencies decrease with the coolant fraction. We can also see that the FM efficiency decreases with the inlet Mach number at a specified coolant fraction. This is because the total pressure of the mixture increases with the mainstream inlet Mach number under the same total inlet conditions during the FM ideal mixing process, which increases the expansion ratio of the ideal expansion process and leads to a larger ideal power output. As shown in the comparison with the WP efficiency, we can see that most of the FM efficiency has a lower value than the WP efficiency, which means that the FM efficiency defines a more ideal mixed expansion process than that of the WP under such conditions. However, there are still certain conditions in which the inlet Mach number is relatively small, and thus the FM efficiency has a higher value. We can see that the value of the σP term is positive under these FM ideal mixing processes, leading to a smaller ideal power output than the WP definition.
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4. Thermodynamic Cycle Analysis Turbine efficiency has always been a critical parameter in a thermodynamic cycle analysis, and a further investigation into all four mixed expansion cooled turbine efficiencies was carried out from the perspective of the thermodynamic cycle. All conditions in which the value of the different definitions equals unity are marked on the T-S chart shown in Fig. 8. For all cycles, the total pressure loss in the burner is considered, and the air and gas are assumed to be perfect gases with different specific heat ratios.
Fig. 8 T-s chart of different perfect gas turbine cycle
The four cycles, in which the values of MP, FR, WP and FM efficiencies are unity, have the same coolant fraction. Therefore, the total temperatures of the four ideal mixtures prior to expansion are the same, which can be calculated based on the energy conservation equation. However, the total pressures of the four ideal mixtures are not
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the same, and are determined separately to meet different assumptions during the four different ideal mixing processes. Perfect cooled and uncooled cycles are both studied here, not only to compare the ideal power output, but also to further investigate the ideal process and develop a proper standard to evaluate the theoretical optimization potential of a designed turbine. Cycle 1-2’-3-4’ is the perfect gas turbine cycle without turbine cooling technology, and is called an uncooled ideal cycle for short. Although the total pressure loss in the burner may make this cycle not as ideal, it is essential to consider the pressure difference between the compressor outlet and turbine inlet so as to distinguish between the ideal MP cycle and ideal WP cycle. Cycle 1-2’-3-3a-4a is the perfect MP cycle, where the coolant fraction that bypasses the burner mixes with the mainstream under a constant pressure. Therefore, the total pressure of the mixture is the same as the total pressure of the turbine inlet, as shown at point 3a. The mixture is then expanded to the turbine outlet total pressure isentropically. Cycle 1-2’-3-3b-4b is the perfect FR cycle, where there is no entropy increase during the ideal mixing process. Therefore, the mixture property is located at the intersection of line 2’-3 and the isotherm line of the mixture. It can be easily seen that the power output in the perfect FR cycle is larger than that of the perfect MP cycle, and is thus also proven to be the largest power output among the four ideal definitions. Cycle 1-2’-3-3c-4c is the perfect WP cycle, where the total pressure of the mixture is determined by assuming the σP term to be zero. Under a usual situation, where the coolant total pressure is larger than that of the mainstream, the total pressure of the mixture is larger than that in the perfect MP cycle. Therefore, the power output in the
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perfect WP cycle is larger than that in the perfect MP cycle, leading to higher cycle efficiency. Cycle 1-2’-3-3d-4d is the perfect FM cycle, which assumes a different ideal mixing process from the other definitions. The FM ideal mixture process is influenced by the Mach number of the turbine inlet, and under usual turbine operating conditions will have a slightly higher total pressure of the mixture than that of the perfect WP cycle, as shown at point 4d.
Fig. 9 Cycle efficiency of different perfect cycles for varying coolant fractions
The cycle efficiencies of the perfect uncooled cycle and four perfect cooling cycles for varying coolant fractions are shown in Fig. 9. The FR definition has the highest cycle efficiency (with the exception of the perfect uncooled cycle) and seems to have the most ideal mixed expansion process; however, it has been proven that an ideal FR mixed expansion process does not exist or is even desirable in a real turbine design. The FM definition has a higher value than the other two definitions, which means the perfect FM cycle provides the most ideal process among the achievable mixing expansion
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process, and can therefore better evaluate the theoretical optimization potential of a designed turbine by comparing the efficiency value with unity.
5. Conclusions This paper presents the definition of a new efficiency for a cooled turbine, which is based on an ideal physically based mixed expansion process. Compared with the previously proposed cooled turbine efficiency definitions, the new definition (FM efficiency) can consider the influence of the inlet mainstream Mach number on the ideal turbine power output, and can provide a more reliable method to evaluate the theoretical optimization potential of a designed turbine. Some currently used cooled turbine efficiency definitions were reviewed in this paper, and both the increase in mixing entropy and the flow mixing in a smooth adiabatic channel were studied to analyse and test the proposed definitions. The FR efficiency was proved to provide a significantly lower value when assuming the unavoidable entropy generation to be zero, and the ideal mixed expansion process with regard to the WP efficiency was found to be insufficiently ideal. Therefore, FM efficiency was presented based on an ideal physically based mixed expansion process. The values of all definitions for varying coolant fractions were compared, the results of which show that the FM efficiency does not differ significantly from most of the other proposed definitions. Further comparison in view of the thermodynamic cycle also shows that the FM definition can better evaluate the performance potential of existing turbines. Therefore, the newly presented FM efficiency can be a reliable tool for evaluating and optimising the performance of a cooled turbine.
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Acknowledgement The authors wish to thank Mr. Zhang of Tsinghua University for the advice and discussion provided on an ideal physically based mixing process, and would like to express their gratitude for the work of the reviewers and editors in evaluating this manuscript.
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1.
FM definition assumes the unavoidable entropy generation to be zero
2.
WP definition assumes a non-ideal process to determine the ideal work
3.
FM definition is based on an ideal physically based expansion process
4.
FM definition can consider the inlet mainstream Mach number influence
5.
FM definition can evaluate the theoretical optimization potential of turbine
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S1359-4311(18)30634-3 https://doi.org/10.1016/j.applthermaleng.2018.05.125 ATE 12264
To appear in:
Applied Thermal Engineering
Received Date: Revised Date: Accepted Date:
29 January 2018 25 May 2018 29 May 2018
Please cite this article as: W. Ba, X-c. Wang, X-s. Li, X-d. Ren, C-w. Gu, Definition and Comparison of Mixed Expansion Efficiency for Cooled Turbine, Applied Thermal Engineering (2018), doi: https://doi.org/10.1016/ j.applthermaleng.2018.05.125
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Definition and Comparison of Mixed Expansion Efficiency for Cooled Turbine
Wei Ba
Xiao-chen Wang
Xue-song Li
Xiao-dong Ren*
Chun-wei Gu
Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Energy and Power Engineering, Tsinghua University, Beijing 100084, China *Corresponding author. E-mail address: [email protected]
Abstract Over the years, there has been continual increase in the temperature inside a turbine inlet through improvements in the gas turbine efficiency and specific power, and cooling technologies to deal with such temperature increases have been under development for decades. However, the definition of “efficiency” remains ambiguous for cooled turbines, with no consensus achieved on the ideal expansion process. This paper first reviews several proposed definitions, Hartsel, MP (mainstream-pressure), FR (fully reversible), and WP (weighted-pressure) of efficiency and then presents a new one to overcome the existing shortage. The increase in entropy related to mixing is first analysed to demonstrate that FR efficiency consistently provides a significantly lower value than the other definitions, assuming the unavoidable entropy generation to be zero. The flow mixing in a smooth adiabatic channel is then analysed to show that the WP ideal process is not sufficiently “ideal.” A fully mixed (FM) definition is then proposed based on a physical mixing process, followed by isentropic expansion, the value of which does not differ significantly from most of the proposed definitions, and is able to 1
consider the influence of the inlet Mach number on the hypothetical ideal process. Finally, this paper compares several definitions in view of the thermodynamic cycle, and shows that the FM definition can determine the performance potential of an existing turbine better than the other definitions, which means that the FM efficiency can be used as a reliable tool for the performance evaluation and optimization of a cooled turbine.
Keywords: Cooled turbine; Efficiency; Mixed Expansion
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Nomenclature A Cp fc G h R T P V Xgm, Xcm
cross section area, m2 constant pressure specific heat capacity, J/kg/K coolant fraction mass flow rate, kg/s specific enthalpy, kJ/kg specific gas constant, J/kg/K temperature, K power, kW, or pressure, Pa velocity, m/s mole fractions of the mainstream and coolant after mixing
Greek symbols γ η ρ σ
specific heat ratio efficiency density, kg/m3 entropy created per unit mass of mixture, J/kg/K.
Subscripts 3 4 ac c C g Hart i id is Mix, m P t T
turbine inlet turbine outlet actual process coolant condition related to composition change gas condition Hartsel definition inlet ideal process isentropic process mixture related to pressure change total condition turbine, or related to temperature change
Acronyms FR FM MP WP
fully reversible fully mixed mainstream pressure weighted pressure
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1. Introduction Improvements in the performance of modern gas turbines have led to an increase in the turbine inlet temperature, and compound cooling technologies [1-2] have thus been widely applied to protect the hot components. Therefore, a definition of the efficiency of an uncooled turbine should be further developed to characterize the performance of a cooled turbine. However, there is still no consensus on the most suitable definition for cooled turbine efficiency, and such ambiguity has resulted in inconvenience and confusion in practical use. For turbine designers and researchers, turbine efficiency has two main functions in terms of practical design [3-7] and optimization[8], to compare the performance of different turbines [9-12] and evaluate the optimization potential [13-16], and to optimise the design of the gas turbine cycle [17-21]. Most turbine efficiency definitions can be written as Eq. (1), where Pac represents the actual power output and Pid represents the ideal power output.
T
Pac Pid
(1)
There are not many ambiguities in the determination of the actual power output. The value of Pac can be determined either experimentally or through a numerical simulation, and the only difference is that the disk windage and bearing loss can be precisely determined during the experiment. Instead of estimating the disk windage and bearing loss based on empirical correlations, in this paper Pac is determined through the aerodynamic parameters of the main flow and coolant at the inlet and outlet for simplicity. Ambiguities in a cooled turbine efficiency definition are always based on the determination of Pid, which is mainly defined in two ways in the literature. The first is 4
an unmixed efficiency definition, presented by Hartsel [22], which assumes that the mainstream gas and coolant expand separately and isentropically during an ideal process. The other type is called mixed expansion efficiency, and is referred to as MP [23], FR, or WP [24, 25]. All these definitions assume a mixing process of the mainstream and coolant prior to isentropic expansion toward the total outlet pressure. The Hartsel and MP efficiencies have already been fully discussed in the open literature [24], and it has been revealed that a shortage make them unqualified to properly characterize the performance of a cooled turbine. The FR and WP efficiencies are further investigated in this paper, and it is shown that the FR definition always has a significant lower value than the other definitions and overestimates the ideal turbine power capability by neglecting the unavoidable entropy increase associated with temperature equilibration of the mainstream and coolant. The hypothetical WP ideal process was also studied in this paper, and it was revealed that this process is not sufficiently ideal, and the WP efficiency consistently underestimates the ideal turbine power capability. The fully mixed (FM) definition is presented in this paper to overcome the shortage of the proposed efficiency definition. Although the ideal FM process also contains a mixing process prior to isentropic expansion, the FM mixing process is assumed to be applied in a smooth adiabatic channel, which can deliver a physically based ideal process for an efficiency definition, and properly consider the unavoidable increase in entropy. The study described herein on the FM efficiency shows that its value does not differ significantly from most of the proposed definitions, and is capable of considering the influence of the inlet Mach number on the ideal turbine power capability. A
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thermodynamic cycle analysis also shows that the FM efficiency can be used to evaluate the performance potential of an existing turbine better than other definitions, making it a better measurement for the performance of a cooled turbine.
2. Proposed Cooled Turbine Efficiencies 2.1 Uncooled turbine efficiency The uncooled turbine efficiency is first described because the proposed cooled turbine efficiency consistently follows a similar approach. For an uncooled turbine, Eq. (1) can be written as follows, and is usually called the total-to-total isentropic efficiency.
T
G3 ht 3 ht 4 h h t3 t 4 G3 ht 3 ht 4is ht 3 ht 4is
(2)
where ht3 is the total enthalpy of a turbine inlet, ht4 is the actual total enthalpy of a turbine outlet, and ht4is is the ideal total enthalpy of a turbine outlet. As we can see, for an uncooled turbine, Pac is the difference between the actual total enthalpy at the turbine inlet and outlet, and Pid is the difference between the actual total enthalpy at the inlet and the ideal total enthalpy at the outlet, which is determined through an isentropic expansion.
2.2 Actual Power Output of Cooled Turbine Determination of the actual power output is essential to every efficiency definition, and the difference between the actual shaft power output and actual turbine power generation should be mentioned. The actual power output of the shaft is always determined experimentally, considering the negative effect caused by the disk windage and bearing loss, and should be used as the actual power output when an experiment is 6
conducted to test the shaft power. The actual turbine power generation is always determined through a numerical simulation, and there is no way to estimate the disk windage and bearing loss precisely. In this paper, the actual turbine power generation is regarded as the actual power output for simplicity.
Fig. 1 Schematic of gas turbine cycle and control volume selection
Next, the main argument is whether the rotor coolant flow has the ability to do work. To avoid this argument, the actual power output is calculated based on the control volume of the turbine, as shown in Fig. 1, and the details of the flow in a turbine are neglected by applying an energy conservation equation to the control volume. With an adiabatic assumption of the turbine, the actual power output can be written as follows.
Pac G3ht 3 Gc htc G4 ht 4
(3)
2.3 Hartsel and MP ideal processes The Hartsel and MP efficiencies have been used by turbine designers, and are prototypes of unmixed and mixed expansion efficiency definitions, respectively. Both
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definitions have been well documented in the literature, and therefore only a brief introduction is provided here. Hartsel assumes that the mainstream and the coolant expand separately and isentropically during the ideal process, and the ideal power output can be calculated as follows:
Pid , Hart G3 ht 3 ht 4is , g Gc htc ht 4is ,c
(4)
where ht4is,g and ht4is,c are separately determined by each isentropic expansion toward the total outlet pressure. The Hartsel process assumes an adiabatic and fully reversible expansion process; however, it has been criticized because its value may be greater than unity when considering the real properties of a gas and coolant. MP efficiency is referred to as the mixed efficiency of the mainstream pressure, which assumes that the mainstream and the coolant mix at a constant pressure (mainstream pressure) prior to the isentropic expansion during the ideal process. For all mixed expansion definitions, the ideal power output can be written as follows:
Pid ,Mix G3 Gc htm,3 htm,is
(5)
where htm,3 is the total enthalpy of the mixture at the turbine inlet, and htm,is is the total enthalpy of the ideal mixture expanded isentropically from the total pressure of the inlet mixture to the total outlet pressure. For the MP definition, the total pressure of the inlet mixture is simply equal to the total pressure of the mainstream inlet. Ptm,3 Pt 3
8
(6)
Compared to the Hartsel definition, the MP definition is capable of considering the mixing procedure of the mainstream and the coolant, but has been criticized for neglecting the influence of inlet pressure of the coolant on the ideal power output.
2.4 FR ideal process Considering the shortages of the Hartsel and MP definitions, the FR ideal process was presented as a fully reversible mixed expansion process. The ambiguity of a mixed expansion efficiency definition is how to determine the state parameter of the mixture prior to expansion. The total temperature can be easily determined through the energy conservation equation, but the total pressure has to be determined based on a different assumption, such as being equal to the total pressure of the mainstream in MP ideal process. To obtain a fully reversible adiabatic mixing process, the total pressure of the mixture is determined to make the total entropy generation zero during the FR ideal process. In the mixing process, the entropy generation [23] per unit mass of the mixture can be written as follows:
T P C
(7)
Where
T 1 f c C pg ln
P 1 f c Rg ln
Ttm,3 Tt 3
f cC pc ln
Ttm,3 Ttc
Pt 3 P f c Rc ln tc Ptm,3 Ptm,3
C 1 fc Rg ln X gm fc Rc ln X cm
9
(8)
(9)
(10)
The term σc represents the diffusional entropy creation owing to a compositional change, which cannot be avoided. Therefore, only (σT+σP) = 0 is assumed in the FR definition. The total pressure of the mixture can then be determined as follows: Ptm,3 Pt13 fc Ptc fc exp T R
(11)
The FR efficiency definition is seemed to be reasonable for the cooled turbine efficiency, and for a fully reversible adiabatic mixing process combined with the following isentropic expansion process. However, the value of the FR efficiency is significant lower than the other definitions, even if the Hartsel efficiency also assumes an adiabatic and fully reversible expansion process. It will then be discussed how this came to be through the neglecting of the unavoidable entropy increase associated with the temperature equilibration of the mainstream and coolant.
Fig. 2 Relative values of entropy creation terms σT and σP
In the FR ideal process, the σP term is required to be the negative value of σT. Therefore, the relative values of these two terms in a normal mixing process are worth comparing. σP1, σP2 and σP3 are the entropy production related to pressure change in the
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mixing process, where the mixture pressure is assumed to be the mainstream pressure, the weighted average pressure, and the coolant pressure separately. As shown in Fig. 2, the absolute value of σT is much bigger than that of σP in each case, which means that a zero total entropy generation assumption may lead to a much higher mixture pressure than the pressure of each component. The FR ideal mixture pressures at different coolant fractions are then compared with the mainstream and coolant pressures, as shown in Fig. 3, where the mass flow rate of coolant is 10%, 20% and 30% of the mainstream mass flow rate. The results show that the total pressure of the FR ideal mixture is higher than the pressure of the mainstream and coolant, and will further increase with the coolant fraction, thereby resulting in zero total entropy generation.
Fig. 3 Relative value of FR mixture and component pressures
With a higher mixture pressure and a constant turbine outlet pressure, the FR ideal power output will increase as the expansion ratio increases. This explains why the value of the FR efficiency is significantly lower than in the other definitions.
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A zero total entropy generation assumption leads to an unreasonable FR ideal mixture pressure, which deserves further discussion with regard to the reasonability of the assumption. In most modern turbines, coolant is introduced to protect the hot components from the high temperature of the mainstream, which unavoidably increases the entropy associated with the temperature equilibration of the mainstream and coolant. Therefore, similar to the σc term, σT is also unable to be avoided in the mixing process, and instead produces a useful operation. In other words, the FR definition has an unreasonably large ideal power output when assuming the unavoidable entropy increase to be zero.
2.5 WP ideal process Considering the extremely low FR efficiency, the WP definition assumes σP to be zero, and the WP ideal mixture pressure can then be determined as follows:
Ptm,3 Pt13 fc Ptc fc
(12)
However, differing from σT, which is always positive and unavoidable, the lowest value of σP may be negative, which will make the WP definition underestimate the ideal power output of a turbine. Considering the mixing process of two streams in a smooth adiabatic channel and applying conservation equations to the whole control volume to neglect the flow details, the entropy generation related to the mixing process can be determined. More details on the mixing calculation are provided in the following chapter. The contours of the σP value during the mixing process at different temperatures and pressures of the components are shown in Fig. 4. To determine all component inlet
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conditions, assumptions of equal inlet static pressure, equal inlet static temperature, and equal inlet velocity are made separately to carry out the calculation. As shown in Fig. 4, the value of σP, and the differences in the total pressure and temperature of the components, are all non-dimensionalised. The two inlet stream are distinguished by subscript “1” and “2”, and “bar” means the mean value of pressure, temperature or velocity.
P
P ,coef
PCoef
Pt1 Pt 2 i
P P t
TCoef
(13)
1 2 Vi 2
i
Tt1 Tt 2 i
T T t
(14)
(15)
i
The value of σP is always positive when the mixing components have the same inlet static pressure, but is not the same when assumptions of the same inlet static temperature or same inlet velocity are made. As shown in Fig. 4 (b), with a constant static inlet temperature, a negative σP occurs when one component has a higher total temperature and the other has higher total pressure. For a constant inlet velocity, shown in Fig. 4 (c), a negative σP occurs when one component has both a higher total temperature and a higher total pressure.
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(a) Equal inlet static pressure
(b) Equal inlet static temperature
(c) Equal inlet velocity
Fig. 4 Value of σP with different total temperatures and pressures of the components
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The calculation above shows that the lowest value of σP may not be zero when the mixing process occurs in a smooth adiabatic channel, and therefore the simple assumption of σP = 0 in the WP definition is insufficiently accurate, which will lead to an underestimation of the ideal turbine power output.
3. FM Cooled Turbine Efficiency Studies on FR and WP definitions have shown that the critical issue of a mixed expansion efficiency definition is about how to determine the ideal mixing process of the mainstream and coolant. Previous ideal mixing processes were determined by simply assuming the entropy generation, or part of it, to be zero; however, such an assumption neglects too much physical matter in the turbine design and operation, and results in a different shortage. A reasonable ideal mixing process should be developed as a physically based ideal process, which will accept the unavoidable entropy generation during a turbine operation, such as the creation of diffusional entropy owing to a change in composition, and entropy creation associated with a temperature equilibration. In addition, the physically based ideal mixing process should also minimize the part of the entropy change that can be optimized. The ideal mixed expansion process for a cooled turbine can then be determined and the ideal power output can be specified. The physically based efficiency definition may not only become a reliable tool for characterizing the performance of a cooled turbine, it may also be capable of evaluating the theoretical optimization potential by comparing the efficiency value with unity.
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3.1 FM ideal mixing process Different from the ideal mixing process determined by restricting the entropy generation directly, the FM definition presented in this paper assumes an ideal mixing process based on a physically based mixing process.
Fig. 5 Schematic of mixing process in smooth adiabatic channel
As shown in Fig. 5, the FM definition assumes an ideal mixing process in a smooth adiabatic channel, and the mainstream and coolant are further simplified as two flows in the same direction with different flow conditions. Once the outlet flow is assumed to be uniform, the flow condition at the channel outlet can be easily determined based on the governing equations of the entire control volume, without too much concern regarding the details of the flow mixing. The governing equations can be written as follows: The continuity equation is
A1 A 1V1 2V2 2
A1 1 =mVm A2
(16)
The momentum equation is
A1 2 2 A P1 1V1 P2 2V2 2 The energy equation is
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A1 2 1 =Pm mVm A2
(17)
1 fc Cp1Tt1 fcCp2Tt 2 TtmCpm
(18)
Based on the provided inlet flow conditions, the mass flow ratio, and the state equation, the flow condition of the mixture at the outlet can be determined.
R a a 4 RmTtm 1 m b b 2Cpm Vm R 2 m Cpm 2
Ttm
Ptm
1 fc Cp1Tt1 fcCp2Tt 2 Cpm a bVm
(19)
(20)
(21)
m
m 1 Vm 2 T Ttm tm 2Cpm
where a and b are two constants, as follows:
A a 1 P1 1V12 P2 2V22 A2 A b 1 1V1 2V2 A2
A1 1 A2
A1 1 A2
(22)
(23)
As we can see in Eq. (19), there may be two values for the same set of inlet flow conditions, corresponding to the subsonic and supersonic outlet flow conditions. Because the mainstream at the turbine inlet and the coolant at the cooling holes are always subsonic, only a subsonic mixture flow condition is considered in this paper.
3.2 Comparison of efficiency values The total pressure of FM ideal mixture before expansion is determined through Eq. (21), and the FM ideal power output can then be determined from Eq. (5). Using the given actual turbine power output, the FM efficiency can then be obtained. 17
A typical high-pressure turbine stage is considered here to compare the value of the different efficiency definitions for different coolant fractions. Because the actual stage performances with different coolant fractions are not provided, the calculations are carried out by assuming that the stage expansion ratio and MP efficiency are fixed, and the MP efficiency is also taken as the reference efficiency to provide another efficiency value.
Fig. 6 Comparison of different efficiencies for varying coolant flow fractions
As shown in Fig. 6, the relative efficiency values of the cooling stage under different definitions of the varying coolant flow fractions are compared. When the coolant fraction equals zero, all relative efficiencies are unity, which means that all the proposed definitions will have the same value when there is no coolant in the turbine. This is an expected behaviour because all cooled turbine efficiency definitions will degenerate into the total-to-total isentropic efficiency mentioned in the second chapter, when no coolant is applied. The FR efficiency has a significantly lower value than the other definitions, and the difference increases notably with the coolant fraction. This phenomenon again demonstrates that the FR efficiency assumes a much larger ideal power output than the
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other definitions, particularly for a large coolant fraction. This large difference will not only increase the difficulty to translate experiences from the past to a new design goal, it will also cause an overestimation of the theoretical optimization potential in the design. The Hartsel, WP, and FM definitions seem to have similar values for varying coolant fractions, all of which are smaller than the MP efficiency. This phenomenon can be expected when the total pressure of the coolant is higher than that of the mainstream, which is a usual situation found in cooled turbines. A more detailed observation reveals that the FM efficiency consistently has a lower value than the WP definition in this calculation, which means that the FM ideal process has a larger ideal power output than that of the WP definition.
3.3 Sensitivity study of turbine inlet Mach number Differing from other proposed efficiency definitions, the FM efficiency is capable of considering the influence of the inlet Mach number on the ideal turbine power output. Therefore, a sensitivity study of the inlet Mach number is essential for a new definition. Because the WP ideal process is similar to that of the FM definition, the WP efficiency under the same operating conditions was chosen for a comparison. It should be recalled that the WP efficiency has nothing to do with the inlet Mach number, as long as the inlet stagnation parameters are specified. All stage parameters applied in the following calculation are the same as those in chapter 3.2.
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Fig. 7 Sensitivity of inlet Mach number on FM efficiency
As shown in Fig. 7, the FM efficiency has a different value under different inlet Mach number conditions, and all the FM efficiencies decrease with the coolant fraction. We can also see that the FM efficiency decreases with the inlet Mach number at a specified coolant fraction. This is because the total pressure of the mixture increases with the mainstream inlet Mach number under the same total inlet conditions during the FM ideal mixing process, which increases the expansion ratio of the ideal expansion process and leads to a larger ideal power output. As shown in the comparison with the WP efficiency, we can see that most of the FM efficiency has a lower value than the WP efficiency, which means that the FM efficiency defines a more ideal mixed expansion process than that of the WP under such conditions. However, there are still certain conditions in which the inlet Mach number is relatively small, and thus the FM efficiency has a higher value. We can see that the value of the σP term is positive under these FM ideal mixing processes, leading to a smaller ideal power output than the WP definition.
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4. Thermodynamic Cycle Analysis Turbine efficiency has always been a critical parameter in a thermodynamic cycle analysis, and a further investigation into all four mixed expansion cooled turbine efficiencies was carried out from the perspective of the thermodynamic cycle. All conditions in which the value of the different definitions equals unity are marked on the T-S chart shown in Fig. 8. For all cycles, the total pressure loss in the burner is considered, and the air and gas are assumed to be perfect gases with different specific heat ratios.
Fig. 8 T-s chart of different perfect gas turbine cycle
The four cycles, in which the values of MP, FR, WP and FM efficiencies are unity, have the same coolant fraction. Therefore, the total temperatures of the four ideal mixtures prior to expansion are the same, which can be calculated based on the energy conservation equation. However, the total pressures of the four ideal mixtures are not
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the same, and are determined separately to meet different assumptions during the four different ideal mixing processes. Perfect cooled and uncooled cycles are both studied here, not only to compare the ideal power output, but also to further investigate the ideal process and develop a proper standard to evaluate the theoretical optimization potential of a designed turbine. Cycle 1-2’-3-4’ is the perfect gas turbine cycle without turbine cooling technology, and is called an uncooled ideal cycle for short. Although the total pressure loss in the burner may make this cycle not as ideal, it is essential to consider the pressure difference between the compressor outlet and turbine inlet so as to distinguish between the ideal MP cycle and ideal WP cycle. Cycle 1-2’-3-3a-4a is the perfect MP cycle, where the coolant fraction that bypasses the burner mixes with the mainstream under a constant pressure. Therefore, the total pressure of the mixture is the same as the total pressure of the turbine inlet, as shown at point 3a. The mixture is then expanded to the turbine outlet total pressure isentropically. Cycle 1-2’-3-3b-4b is the perfect FR cycle, where there is no entropy increase during the ideal mixing process. Therefore, the mixture property is located at the intersection of line 2’-3 and the isotherm line of the mixture. It can be easily seen that the power output in the perfect FR cycle is larger than that of the perfect MP cycle, and is thus also proven to be the largest power output among the four ideal definitions. Cycle 1-2’-3-3c-4c is the perfect WP cycle, where the total pressure of the mixture is determined by assuming the σP term to be zero. Under a usual situation, where the coolant total pressure is larger than that of the mainstream, the total pressure of the mixture is larger than that in the perfect MP cycle. Therefore, the power output in the
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perfect WP cycle is larger than that in the perfect MP cycle, leading to higher cycle efficiency. Cycle 1-2’-3-3d-4d is the perfect FM cycle, which assumes a different ideal mixing process from the other definitions. The FM ideal mixture process is influenced by the Mach number of the turbine inlet, and under usual turbine operating conditions will have a slightly higher total pressure of the mixture than that of the perfect WP cycle, as shown at point 4d.
Fig. 9 Cycle efficiency of different perfect cycles for varying coolant fractions
The cycle efficiencies of the perfect uncooled cycle and four perfect cooling cycles for varying coolant fractions are shown in Fig. 9. The FR definition has the highest cycle efficiency (with the exception of the perfect uncooled cycle) and seems to have the most ideal mixed expansion process; however, it has been proven that an ideal FR mixed expansion process does not exist or is even desirable in a real turbine design. The FM definition has a higher value than the other two definitions, which means the perfect FM cycle provides the most ideal process among the achievable mixing expansion
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process, and can therefore better evaluate the theoretical optimization potential of a designed turbine by comparing the efficiency value with unity.
5. Conclusions This paper presents the definition of a new efficiency for a cooled turbine, which is based on an ideal physically based mixed expansion process. Compared with the previously proposed cooled turbine efficiency definitions, the new definition (FM efficiency) can consider the influence of the inlet mainstream Mach number on the ideal turbine power output, and can provide a more reliable method to evaluate the theoretical optimization potential of a designed turbine. Some currently used cooled turbine efficiency definitions were reviewed in this paper, and both the increase in mixing entropy and the flow mixing in a smooth adiabatic channel were studied to analyse and test the proposed definitions. The FR efficiency was proved to provide a significantly lower value when assuming the unavoidable entropy generation to be zero, and the ideal mixed expansion process with regard to the WP efficiency was found to be insufficiently ideal. Therefore, FM efficiency was presented based on an ideal physically based mixed expansion process. The values of all definitions for varying coolant fractions were compared, the results of which show that the FM efficiency does not differ significantly from most of the other proposed definitions. Further comparison in view of the thermodynamic cycle also shows that the FM definition can better evaluate the performance potential of existing turbines. Therefore, the newly presented FM efficiency can be a reliable tool for evaluating and optimising the performance of a cooled turbine.
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Acknowledgement The authors wish to thank Mr. Zhang of Tsinghua University for the advice and discussion provided on an ideal physically based mixing process, and would like to express their gratitude for the work of the reviewers and editors in evaluating this manuscript.
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1.
FM definition assumes the unavoidable entropy generation to be zero
2.
WP definition assumes a non-ideal process to determine the ideal work
3.
FM definition is based on an ideal physically based expansion process
4.
FM definition can consider the inlet mainstream Mach number influence
5.
FM definition can evaluate the theoretical optimization potential of turbine
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