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Preliminary design of radial inflow turbine and working fluid selection based on particle swarm optimization
Energy Conversion and Management 199 (2019) 111933
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Preliminary design of radial inflow turbine and working fluid selection based on particle swarm optimization
T
⁎
Zhonghe Han, Xiaoqiang Jia, Peng Li
School of Energy, Power and Mechanical, North China Electric Power University, Baoding 071000, People's Republic of China
A R T I C LE I N FO
A B S T R A C T
Keywords: Organic Rankine cycle Turbine design Particle swarm optimization Working fluid selection Off-design performance
Organic Rankine cycle (ORC) is an efficient technique to recycle the low temperature heat sources. The design method of the critical component, radial inflow turbine, is a main focus of research. A preliminary design method was developed to optimize eight critical parameters based on particle swarm optimization (PSO) algorithm. The isotropic efficiency was the objective function of the algorithm. Six working fluids were selected to conduct turbine design based on the design method. The turbine with R245fa was determined to be optimal due to its small geometry size, high exergy efficiency (0.929) and high load coefficient (1.1027). The off-design performance of R245fa turbine was investigated with the variation of pressure ratio (PR), stator inlet temperature and power output. The results indicate that the turbine efficiencies increase with the reduction of PR and turbine inlet temperature and the increase of power output. The exergy efficiency and isotropic efficiency drop slightly with the increase of turbine inlet temperature. So the turbine with R245fa exhibited good off-design performance.
1. Introduction With the rapidly increasing consumption of fossil energy, the sustainable use of energy resources has received increasing attention. There are many researches on the utilization method of low temperature heat sources [1] such as ocean thermal energy (278.15 K–301.15 K) [2], geothermal energy (423.15 K) [3] and solar energy (423.15 K) [4] and low-grade heat sources like industrial waste heat (423.15 K) [5]. Many thermodynamic cycles can be applied in the recycling of low temperature heat source, in which organic Rankine cycle (ORC) is a main method. Sun et al. [6] proposed a double-pressure ORC system and conducted performance analysis and optimization on it. Astolfi et al. [7] compared CO2 cycle and ORC system based on different heat sources and presented a set of selection maps for their application. Sun et al. [8] investigated the thermodynamic influence factors of ORC system and carried out exergy efficiency analysis on the ORC-based combined cycles. There are a large number of literatures about the ORC system. As one of the main components of ORC, turbines including Tesla turbine [9], axial turbine and radial turbine [10], have a remarkable influence on the system efficiency [11]. The characteristics of small size and high efficiency make radial inflow turbines suitable for large scale heat recovery plants [12]. In recent years, many studies have focused on the
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turbine design method and ORC system optimization. And many intelligence algorithms were used in the optimization process. Zhai et al. [13] coupled optimization design of an ORC system with radial inflow turbine using the constrained genetic algorithm (GA). Li et al. [14] proposed a design method for the radial inflow turbine within the ORC thermodynamic model based on multi-objective optimization. Bahadormanesh et al. [15] conducted multi-objective optimization of radial inflow turbine in ORCs using firefly algorithm. Bekiloğlu et al. [16] optimized the system parameters based on turbine design through the multi-objective optimization method NSGA-II. Then 28 different working fluids were compared and ranked according to TOPSIS results. Particle swarm optimization (PSO) was employed for global optimization of an engine-ORC combined system in Ref. [17]. Compared with traditional algorithms, PSO has a fast calculation speed and an efficient global search ability. However, most researches considered turbine design method based on the optimization of ORC system and the system performance was selected as the optimization objective. Few of them were devoted to optimize the turbine performance in a comprehensive method including thermodynamic and aerodynamic design method based on PSO algorithm. Working fluid selection is another topic of interest in the ORC research field. A part of researches conducted working fluid selection based on physical and chemical properties. Saloux et al. [18]
Corresponding author. E-mail address: [email protected] (P. Li).
https://doi.org/10.1016/j.enconman.2019.111933 Received 8 May 2019; Received in revised form 29 July 2019; Accepted 11 August 2019 0196-8904/ © 2019 Elsevier Ltd. All rights reserved.
Energy Conversion and Management 199 (2019) 111933
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Ω ρ δ ν Φ Ψ
Nomenclature A Ar b c ca D h l m Ma n P T u w W xa Z
flow passage area, m2 ratio of needed area to actual area at rotor outlet chord length, mm absolute velocity, m·s−1 assumed expansion velocity diameter, m enthalpy, kJ kg−1 blade height, mm mass flow rate, kg·s−1 Mach number rotational speed, rpm pressure, MPa temperature, K peripheral velocity, m·s−1 relative velocity, m·s−1 power, kW velocity ratio blades number
degree of reaction density, kg·m−3 tip clearance kinematic viscosity flow coefficient load coefficient
Subscripts 0, 1, 2 b cr ex f h l m min s; i sh u
State points installation critical exergy friction loss hub leakage loss meridional minimum isentropic shroud peripheral
Greek letters Abbreviations η ζ φ ψ α β Δ
efficiency loss coefficient nozzle velocity coefficient rotor blade velocity coefficient absolute flow angle, ° relative flow angle, ° exhaust edge thickness, mm
GWP ODP ORC PR STM
introduced a selection method of the optimal working fluid based on the environmental and actual design conditions. Groniewsky et al. [19] proposed a rule of thumb to find new organic fluids according to the degree of freedom and heat capacity at constant volume. Yu et al. [20] studied the working fluid design model for a flue gas pre-compression system coupled with ORC system. Ma et al. [21] investigated the impact of physical and chemical properties of 70 fluids and compared the working fluids based on specific turbine volume (STV). Dai et al. [22] evaluated the thermal stability of the hydrocarbons experimentally and screened for the suitable working fluids of supercritical ORCs. Qiu [23] compared and optimized eight working fluids and ranked them based on the spinal point method. Also, there were some studies considered ORC system performance as selection criterion. Scaccabarozzi et al. [24] performed ORC working fluids selection from over 40 kinds of fluids to obtain the most promising choice of ORC system. In Ref. [25], a new ORC system with special expansion part was proposed and the performance of it with different organic fluids was analyzed. Guo et al. [26] compared 26 working fluids based on the net power output, outlet volume flow rate, expansion ratio and the total UA and determined that R245fa was the optimal fluid. Zhai et al. [27] investigated the influence of fluid properties on ORC system performance and obtained the most suitable working fluids for heat sources with different temperature. Yang et al. [28] analyzed the influence of working fluids composition ratios on the performance of ORC system. However, there were few researches select working fluid on the basis of geometry parameters and performance of turbine as stated in Ref. [29]. Generally, off-design performance analysis of organic turbines is important to the ORC research. But most studies were conducted based on system model. Yang et al. [30] introduced a new evaluation parameter called net power output index to assess the performance of ORC system and analyzed the influence of turbine inlet temperature on the system performance. Hu et al. [31] conducted off-design performance analysis of ORC system based on the design model of it. Zheng et al.
global warming potential ozone depletion potential organic Rankine cycle pressure ratio specific turbine mass
[32] analyzed the off-design performance of turbine on the basis of design model and CFD method. Du et al. [33] compared the off-design performances of basic ORC and dual-pressure ORC systems based on the components models. The literatures related to turbine performance mainly focused on the turbine efficiency and power output through CFD
Fig. 1. Enthalpy entropy diagram of radial inflow turbine. 2
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Table 2 Experienced value ranges of the eight key parameters. Optimized variables
Units
Range
Reference
Degree of reaction (Ω) Velocity ratio (xa) Stator velocity coefficient (φ) Rotor velocity coefficient (ψ) Ratio of wheel diameter (μ) Absolute velocity angle at rotor inlet (α1) Relatively velocity angle at rotor outlet (β2) Rotational speed (n)
– – – – – ˚ ˚ r/min
0.4–0.6 0.63–0.72 0.92–0.98 0.8–0.85 0.5–0.6 14–20 25–45 12000–25000
[38,40] [31] [38,41] [38,41] [41,42] [32] [31,32] [31,43]
Table 3 The operation parameter values of PSO [45].
Fig. 2. Model and velocity triangles of radial inflow turbine. Table 1 The constraints on aerodynamic parameters of turbine [38]. Parameters
Units
Constraints
Incidence angle (i) Mach number of rotor inlet (Ma) Relative velocity (w) Relative pitch of stator blade (t¯1) l1/b
° – m/s – –
−10°–5° < 1.35 w2 < w1 0.65–0.75 0.3–0.45
Operation parameters
Value
Population Number of generation Inertia weight (w) Population learning factor (c1) Global learning factor (c2) Constraint of velocity
80 200 0.6 1.5 1.5 [−0.1,0.1]
were compared and the best working fluid is obtained on account of the geometry size, turbine efficiency and specific fluid mass (SFM). Finally, the off-design performance of turbine was investigated based on the analysis of turbine efficiencies and flow conditions. 2. Preliminary design method The preliminary design of radial inflow turbine includes thermodynamic design and aerodynamic design. The parameters needed in aerodynamic design can be obtained from the thermodynamic design results, so the thermodynamic design is carried out at first. 2.1. Thermodynamic cycle and velocity triangles In the design process of radial inflow turbine, the boundary conditions including pressure and temperature at turbine inlet and the PR, are determined at first in consideration of the heat source temperature and fluid properties at evaporator outlet. Assuming that there is no energy loss in the radial clearance between the rotor and stator, the h-s diagram of flow process in turbine is displayed in Fig. 1. It is shown that the organic fluid flow through stator (0–1) and rotor (1–2) sequentially. The enthalpy and entropy of states 0 and 2 are calculated from the initial parameters and fluid properties obtained from REFPROP 9.0 [37]. Then the isotropic enthalpy drop Δhs can be calculated as follow: (1)
Δhs = h 0 − h2s′
To obtain the velocity triangles, ideal expansion velocity is calculated at first:
ca =
Fig. 3. Flow passage of radial inflow turbine.
2Δhs
(2)
The velocities of rotor inlet and outlet are calculated according to Eqs. (3)–(10). The velocity triangle is obtained as shown in Fig. 2.
method as described in Refs. [34–36]. However, the flow conditions parameters such as Mach number and mass flow rate in turbine flow passage also have great impact on turbine performance. Therefore, it is significant to analyze flow conditions parameters and turbine efficiencies based on mathematic method. In this paper, a preliminary design method of radial inflow turbine including thermodynamic and aerodynamic design is presented in which PSO is employed to optimize eight key parameters. The isotropic efficiency of turbine was selected as fitness function. Considering the impact of organic fluids properties, six fluids were selected as working fluid candidates to carry out turbine design. Then the designed turbines
c1 = φ 2Δh1s ,
φ=
c1 , c1t
w1 =
c1 sin α1 sin β1
u2 = μu1 3
(3) (4)
u1 = x a ca β1 = arctan
Δh1s = (1 − Ω)Δhs
c1 sin α1 c1 cos α1 − u1
(5) (6) (7)
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Fig. 4. The flowchart of PSO. Table 4 Comparison of design model and Ref. [46]. Variables
Design model
Ref. [46]
Error (%)
mf (kg/s) n (rpm) D1 (mm) Z2 ηi (%)
1.51 19,418 166 12 86
1.58 20,000 173 12 84
−4.43 −2.91 −4.05 0 2.38
c2m c cos(α2 − 90) = 2 u1 u1
(11)
Load coefficient: Ψ =
uc −uc Δh = 1 1u 2 2 2u u12 u1
(12)
where, c2m is the absolute meridional velocity at rotor outlet; c1u and c2u are periphery components of absolute velocity at rotor inlet and outlet. The establishment of loss model is important to determine the turbine efficiency. The peripheral losses of turbine are defined as follows:
Table 5 Different working fluids with heat sources. Heat source temperature
Flow coefficient: Φ =
Nozzle loss coefficient: ξ1 = (1 − Ω)(1 − φ2) Working fluid
Turbine inlet temperature
2
w 1 Rotor loss coefficient: ξ2 = ⎛ 2 ⎞ ⎜⎛ 2 − 1⎞⎟ ⎝ ca ⎠ ⎝ ψ ⎠ ⎜
373.15 K–393.15 K 353.15 K–373.15 K 373.15 K 403.15 k 393.15 K 433.15 K
– – 367.15 K – 333 K 351.91 K
w2 = ψ 2ΩΔhs + w12 − (u12 − u22) ,
α2 = 90 + arctan
u2 − w2 cos β2 w2 sin β2
(13)
Reference
R124 R245fa R600 R134a R152a R236fa
ψ=
w2 w2t
[25] [26] [30] [27] [29] [28]
⎟
(14)
2
c Exit loss coefficient: ξc 2 = ⎛ 2 ⎞ ⎝ ca ⎠ ⎜
⎟
(15)
The peripheral efficiency is expressed in Eq. (16).
ηu = 1 − ξ1 − ξ2 − ξc 2 (8)
(16)
In addition, the mass flow rate is calculated in Eq. (17):
mf = (9)
W ηi Δhs
(17) −1
c2 =
(u2 − w2 cos β2 )2 + (w2 sin β2 )2
The specific fluid mass, SFM (kg·kJ ), can be defined as the initial mass consumption per unit of turbine power output:
(10)
On the basis of velocity triangles, the flow coefficient (Φ) and load coefficient (Ψ) defined in Eqs. (11) and (12) are obtained:
SFM = 4
mf W
(18)
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Table 6 Properties of working fluid candidates. Working fluid
Molecular weight (kg/kmol)
Tcr (°C)
Pcr (kPa)
ODP
GWP
Safety data*
Type
Price ($/kg)
R124 R245fa R600 R134a R152a R236fa
137 134 58 102 66 152
122 153 152 101 113 125
3613 3610 3800 4060 4520 3200
0.026 0 0 0 0 0
620 950 20 1430 2.8 9810
A1 B1 A3 A1 A2 A1
dry dry dry Isentropic dry dry
19.30 11.87 20.79 4.80 6.68 13.07
* ASHRAE Standard 34-2016 (designation and safety classification of refrigerants)-A: Low toxicity/B: High Toxicity/1: No flammability/2: Low flammability/3: High flammability. Table 7 Initial parameters for the design method. Initial condition
Value
P0 (kPa) T0 (K) PR W (kW)
725 350 4.11 100
2.2. Aerodynamic design 2.2.1. Stator The TC-4P blade model is selected for the stator blade since it has better aerodynamic performance and more reliable processing technology [38]. The relative blade pitch t¯ has a significant influence on blade profile loss. The range of it varies with the blade model of stator. When the relative blade pitch is oversized, the suction surface of tailing edge will appear flow separation which results in flow loss. The optimal relative blade pitch range of TC-4P blade model is 0.65–0.75 when the Mach number of stator outlet is greater than 1 [38]. The flow capacity of stator is measured by the blockage factor τ which is defined in Eq. (19):
τ1 = 1 −
Z1 Δ1 πD1 sin α1 πD1 , bt¯1
(19) 60u1 . 1000nπ
Δ1 = 0.1, D1 = where, Z1 = The blade height of rotor l1 is calculated in Eq. (20): l1 =
mf ρ1 c1 πD1 τ1 sin α1
(20)
These aerodynamic parameters of stator are coupled with others, so iteration method is employed in the design process to find the optimal solution.
Fig. 5. Size of main geometry parameters of stator and rotor.
2.2.2. Rotor The number of rotor blades should be within a suitable range. Unreasonable number of blades will lead to additional losses and decrease turbine efficiency. The number of rotor blades is determined by Glassman's empirical formula [39].
Z2 =
π (20 + α1) tan(90 − α1) 30
(21)
Limited by the flow angle of stator outlet, there is a minimum
Table 8 Results of the design method with six working fluids. Classification Independent parameters
Objective
Ω xa φ ψ μ α1 β2 n ηi
R124
R245fa
R600
R134a
R152a
R236fa
0.4002 0.6598 0.9799 0.8494 0.5002 14.7856 25.3143 22,792 0.8960
0.4159 0.6943 0.9800 0.8314 0.5004 15.8597 25.0240 22,529 0.8923
0.4006 0.6524 0.9798 0.8475 0.5081 15.6633 26.0738 18,900 0.8893
0.4012 0.6761 0.9798 0.8499 0.5004 14.6323 25.0618 17,973 0.8967
0.4020 0.6776 0.9798 0.8245 0.5001 15.4008 25.0065 23,872 0.8904
0.4119 0.6678 0.9792 0.8446 0.5006 18.2375 25.2041 23,148 0.8931
5
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Fig. 6. Mach number and absolute velocity at rotor inlet.
number of rotor blades.
Z2 min =
2π sin(β2b − α1) sin α1 sin β2b
Fig. 8. Loss coefficient values of turbine with different working fluids.
(22)
When Z2 > Z2min, the number of rotor blades is rounded to Z2, otherwise, the number of rotor blades is rounded to Z2min. The shroud and hub diameters of rotor outlet are calculated as follows:
A2 =
mf w2 sin β2 ρ2 τ2
D2sh =
D22 +
2 A2 π
Fig. 7. Loss contribution of different working fluids. 6
(23)
(24)
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Fig. 12. The variation of loss and efficiencies with PR.
Fig. 9. Turbine efficiencies with different fluids.
Fig. 13. The variation of flow conditions with PR.
Fig. 10. Flow coefficient and load coefficient of turbines with different working fluids.
Fig. 14. The variation of loss and efficiencies with temperature.
D2h =
D22 −
2 A2 π
(25)
Thus, the blade height of rotor outlet l2 is obtained.
l2 =
Fig. 11. Fluid consumption of turbines with different working fluids.
D2sh − D2h 2
(26)
As shown in Table 1, some optimization constraints are included in the design model to guarantee the reliable results of the designed turbine. 7
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velocity angle at rotor outlet (β2) and rotational speed (n). The ranges of them are given in Table 2. An original MATLAB [44] code was developed to perform the entire design process of radial-inflow turbine. Particle swarm optimization (PSO) was applied in the optimization of the eight key parameters. Compared with traditional algorithms, PSO has a fast calculation speed and an efficient global search ability [17]. In addition, PSO is not very sensitive to population size, so it is suitable for solving extremum problems of continuous function. Since the ranges of the eight parameters are different, this paper normalizes all parameters in the optimization algorithm at first, and then decodes them into original parameters in the calculation of population fitness as shown in Eq. (27).
θ = θ0 ·(θmax − θmin ) + θmin
(27)
θ0 ∈ [0, 1]
PSO describes the motion state of a particle with two parameters: position and velocity. Then the historical optimal position of particle (pBest) and global population (gBest) are recorded and combined with the inertia of the particle to update the position and velocity of the particle. The updated formula of PSO is as follows:
Fig. 15. The variation of flow conditions with temperature.
Vt +1 = w·Vt + c1 r1·(pBest − Xt ) + c2 r2·(gBest − Xt )
(28)
Xt +1 = Xt + Vt +1
(29)
where, t is number of generation, X is the position, V is the velocity, w is the inertia weight, c is the learning factor, r is random number. The objective function of PSO is the turbine isotropic efficiency ηi which is defined in Eqs. (30)–(32). The operation parameter values and the flowchart of PSO are shown in Table 3 and Fig. 4, respectively.
Friction loss coefficient: ξf =
f D2 u13 · 1· 1.36 × 106 ν1 m f ·Δh s
δ D Leakage loss coefficient: ξl = 0.47 ⎛1 + 2 ⎞ l2 ⎝ l2 ⎠ ⎜
⎟
δ = 0.01~0.20 l2
Turbine isotropic efficiency: ηi = 1 − ξ1 − ξ2 − ξc 2 − ξf − ξl Fig. 16. The variation of loss and efficiency with power output.
(30)
(31) (32)
2.4. Validation In order to validate the accuracy of this design method, the results of the design model are compared with Ref. [46] as shown in Table 4. The initial conditions are the same as those in Ref. [46]. It can be seen that the results of the design model are in good agreement with Ref. [46]. The turbine isotropic efficiency of design model is higher than that of Ref. [46], the reason is that a part of loss is ignored in the design model. The mass flow rate of design model is lower. It is noted that the error of diameter of nozzle outlet D1 and mass flow rate mf are more than that of ratio of rotational speed n, but they are limited in 5%. In a word, the design model is precise and reliable. 3. Working fluid selection An appropriate working fluid selection is important to the performance of ORC system and radial inflow turbine. Many studies have investigated the working fluid selection based on heat sources in different temperature. When the fluid at turbine inlet is superheated, the thermal efficiency can be improved but the system cost is increased. Therefore, the working fluid is set to saturated states to reduce the cost and irreversible loss. As shown in Table 5, six working fluids are recommended in these literatures for recycling the energy of heat sources with temperature of 353.15–433.15 K. The main focuses of working fluid selection are the critical pressure and temperature, safety, and thermodynamic type. Additionally, the environmental impact is considered in working fluid selection. The properties of the six fluids are shown in Table 6. The molecular weight mainly influences the geometry size of turbine. The thermodynamic state point is limited by critical temperature (Tcr) and pressure (Pcr).
Fig. 17. The variation of flow conditions with power output.
On the basis of the calculation process above, the geometry parameters are obtained as shown in Fig. 3. 2.3. PSO optimization In the process of thermodynamic and aerodynamic design, there are eight key parameters including degree of reaction (Ω), velocity ratio (xa), stator velocity coefficient (φ), rotor velocity coefficient (ψ), wheel diameter ratio (μ), absolute velocity angle at rotor inlet (α1), relatively 8
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The environmental impact of organic fluid is indicated by ODP and GWP. From Table 6, it is noted that the environmental impact and safety of these organic fluids are acceptable. The small molecular weight leads to a minor geometry size of the turbine. The temperature of the heat source studied in this paper ranges from 350 to 400 K, so the thermodynamic types guarantee that these organic fluids can be kept in superheated state during the whole working process. As recommended in Refs. [25–30], the thermodynamic performance of these candidates are satisfied. Therefore, they are selected as the working fluid candidates to recycle the heat source. It is indicated from Ref. [31] that the total to total efficiency of turbine increases with the pressure ratio when the inlet temperature is low. As recommended in Ref. [46], the inlet pressure, pressure ratio and inlet temperature are set to 725 kPa, 4.11 and 350 K, respectively. It is known from Ref. [47] that, the general power of single-stage radial inflow turbine is 100 kW in actual ORC plant. Considering the properties of the six working fluids and heat source, the initial operating conditions are given as shown in Table 7.
Figs. 7 and 8. It is noted that the proportion and value of friction losses of turbines with R124, R245fa and R236fa are smaller than the others and can be ignored, as a result of small geometry size and low periphery velocity. As shown in Fig. 8, due to similar Ω and φ, the nozzle losses of turbines with all fluids are approximately the same, and the proportions of them also have slight variation. The rotor loss is the maximum loss among turbines with all fluids, the rotor loss of turbine with R600 reaches 0.0374 and accounts for 32% of the total loss. The reason is that the friction loss of it reaches the largest of 0.005, the irreversible loss makes the expansion process in turbine deviating from the isotropic expansion process. The leakage losses of turbines with R245fa and R236fa are less than the others due to their small rotor size and reasonable rotational speed. The turbine efficiencies with different fluids are shown in Fig. 9. According to Fig. 9, the variation trend of exergy efficiency is similar to the isotropic efficiency. Although the periphery efficiencies of turbines with R124 is slightly higher than R245fa turbine, the isotropic efficiency of R124 turbine is lower than R245fa turbine on accounts of the large leakage loss as shown in Fig. 8. For the reason that the shroud curve and passage turning are reasonable, the irreversible losses of R245fa and R236fa turbine like friction loss and leakage loss are less than the other turbines. Therefore, the exergy efficiencies of turbines with R245fa and R236fa are higher than the other turbines in which the exergy efficiency of R236fa turbine reach the highest of 0.9313. The flow coefficient (Φ) and load coefficient (Ψ) defined in Eqs. (11) and (12) are important evaluation indices for turbine performance. They can efficiently assess the dimensionless volume flow rate and total enthalpy drop of the whole stage [48]. Rahbar K et al. [29] claimed that the increase of Ψ can lead to the increase of turbine efficiency, while the impact of Φ is opposite. Fig. 10 shows the flow coefficient and load coefficient of turbines with different working fluids. It is indicated that the variation trend of Ψ is more pronounced than that of the Φ. The Ψ of turbine with R245fa reaches the highest of 1.1027, so the conversion efficiency of mechanical work and working capacity of R245fa turbine are higher than the others. The Φ of turbines with R245fa and R600 are higher than the others, so the volume flow rate of them are higher and they need larger flow passage area. SFM characterizes the mass of working fluid consumed in the initial stage of system construction. Specific fluid cost is applied in assessing the system initial cost on working fluid when SFM multiplied with the prices of working fluids. As shown in Fig. 11, due to small molecular weight and large critical pressure, the SFM of turbines with R600, R134a and R152a are low, but the price of R600 is high. Therefore, the lower cost ones are turbines with R134a and R152a which are 0.15 $·kW−1 and 0.14 $·kW−1. From the above discussion, the turbine with R245fa has small geometry size, rational flow passage, low irreversible loss and high efficiency. Therefore, R245fa is selected to recycle the heat from heat source at the temperature of 350 K–400 K.
4. Results and analysis 4.1. Characteristic comparison It is rational to consider the working fluid properties during the turbine design process. Based on the six working fluids in the last section, six different radial inflow turbines are designed using the design method. The designed parameters of them are shown in Table 8. It can be seen that the degree of reaction and velocity ratio of turbine with R245fa reach the highest of 0.4159 and 0.6943, respectively. Therefore, the working capacity and accelerate ability of turbine with R245fa are better than the others. The wheel diameter ratios are almost the same. The rotational speed of turbines with R600 and R134a are slower than the other turbines due to the large diameters and blade heights of rotors. In addition, the isotropic efficiencies of these turbines are high, so the six fluids are suitable to work at the design condition in Table 7. It also can be seen from Table 8 that, for all working fluids, the optimal angles at rotor outlet is almost the same while the variation of the angles at rotor inlet is larger. The α1 affect the passage turning and nozzle loss of stator. The α1 of turbine with R236fa reaches the largest of 18.2375°. Therefore, the change of passage turning and the nozzle loss of it are relatively small. Fig. 5 shows the main geometry parameters of stator and rotor. The b and l1 express the size of stator while D2 and l2 express the size of rotor. The D2sh/D1 indicate the shroud curvature and blade loss of rotor. As shown in Fig. 5(a), the variations of b and l1 are similar while the variation of D1 is opposite to them. So in order to increase the D1, the b and l1 should be reduced. In addition, it can be noted from Fig. 5(b) that variations of D2 and l2 are similar while the variation of D2sh/D1 is opposite to them. Although the b and l1 of turbines with R124 and R245fa are larger than others, the D1, D2 and l2 of them is smaller. So the geometry sizes of turbines with R124 and R245fa are smaller than the other turbines. The D2sh/D1 of turbine with R245fa is the largest so that the blade loss of rotor is smaller than the others [29]. Except for the geometric characteristics, turbine performance has also been the main focus of recent researches. Fig. 6 shows the Mach number (Ma) and absolute velocity (c1) at rotor inlet with the six working fluids. It is indicated that the variation of c1 is more pronounced than that of Ma. The absolute velocity of turbines with R124 and R236fa is low, but the Mach number of them are high which are 1.2074 and 1.2029, respectively. The reason is that the enthalpy drop of the whole stages are low and degree of reaction are almost the same, but the accelerate ability of them are better than others. The absolute velocity of the R152a turbine is high, but the Mach number of it reaches the lowest of 1.1794. The losses contribution of turbines with the fluids are given in
4.2. Off-design performance analysis In order to deal with the changes in generation demand, the radial inflow turbine operates at off-design conditions in some cases. R245fa turbine was selected to analyze the off-design performance of the turbine. An important off-design adjustment mode of turbine is controlling back pressure. When the inlet pressure is constant, the variation of back pressure is equal to the pressure ratio (PR) [49]. When the geometry parameters and the eight variables are fixed, the impact of PR on the turbine efficiency is shown in Fig. 12. It is indicated that the periphery efficiency is constant due to the constant geometry parameters. The enthalpy drop and flow velocities increase with increasing PR, so the friction loss also increases, but the increase is relatively small. Therefore, the isotropic efficiency slowly decreases with the increase of PR. It 9
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and rational flow passage. In addition, it has a high efficiency whose exergy efficiency reaches 0.929 and loading coefficient reaches 1.1027. Therefore, R245fa is more suitable to recycle the heat source as the working fluid of radial inflow turbine. The turbine efficiency increases with the reduction of PR and turbine inlet temperature and the increase of power output. The flow conditions will improve with the increase of PR and inlet temperature, but the increasing power output can lead to worse flow conditions. Therefore, the back pressure and power output should be kept in a rational range to guarantee the flow condition and efficiency of turbine. On the basis of performance analysis, the turbine with R245fa can operate in a stable and efficient status at off-design conditions in terms of temperature.
also leads to the decrease of exergy efficiency for the increase of irreversible loss. As a result, the back pressure cannot be set to a small value. The impact of PR on the flow condition is shown in Fig. 13. It is obvious that the Mach number increases with the PR for which the enthalpy drop in stator becomes larger with the increasing pressure. When PR is lower than 3, the fluid at nozzle outlet is in subsonic state, when PR is larger than 3, the fluid will expand in the chamfered portion and be accelerated into supersonic state, the aerodynamic performance and expansion ability of TC-4P blade type is fully utilized. The variable Ar is the ratio of needed rotor outlet area to the actual rotor outlet area, as defined in Eq. (33). It is known that the increasing PR leads to the reduction of Ar, for the reason that the rotor outlet pressure and mass flow rate reduces with the increasing PR. The flow capacity of rotor is not fully utilized which leads to large flow loss as shown in Fig. 12. So the back pressure cannot be set to a large value.
Ar =
A2 − need A2 − actual
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
(33)
In addition, the temperature of turbine inlet is another important influence factor. When the other conditions are fixed, the impact of temperature on the turbine efficiency is shown in Fig. 14. It is noted that the exergy efficiency and isotropic efficiency drop slightly with the increase of turbine inlet temperature. In addition, the enthalpy drop and flow velocities increase with the increase of inlet temperature, so the friction loss rises slightly. Consequently, the turbine can operate at a stable and efficient state with the variation of turbine inlet temperature, the off-design performance of the turbine is reliable. The impact of temperature on the flow condition is shown in Fig. 15. It can be seen that the Mach number increases with the increase of temperature due to the increase of enthalpy drop in stator, but the Ar and mass flow rate reduce with the increase of temperature. The internal energy of working fluid will rise with the increase of temperature. When the power output is constant, the increasing enthalpy drop will result in the reduction of mass flow rate and the needed rotor outlet area. So the inlet temperature should be kept in a rational range. Furthermore, the power output of turbine also varies with the change of heat source. When the other conditions are fixed, the influence of power output on the turbine efficiency is shown in Fig. 16. According to the figure, the isotropic efficiency increases slightly with the power output as a consequence of the reduction of friction loss. In addition, it also leads to the increase of exergy efficiency. Therefore, the power output of turbine cannot be set to a small value, when it is low, the turbine is inefficient. The influence of power output on the flow conditions is shown in Fig. 17. It is indicated that the Mach number remains stable with the variation of power output for the fact that the boundary conditions of the turbine is fixed. It is apparent that the Ar and mass flow rate rise with the power output for the increasing demand of working fluid. So the actual geometry size of the turbine cannot meet the demand of the increasing power output. Therefore, the power output of turbine cannot be set to a large value, when it is large, the flow loss of turbine will increase.
Acknowledgments This work was supported by the National Natural Science Foundation of China (NO. 51306059) and the Fundamental Research Funds for the Central Universities in China (NO. 2017XS120). References [1] Zhai H, An Q, Shi L, Lemort V, Quoilin S. Categorization and analysis of heat sources for organic Rankine cycle systems. Renew Sustain Energy Rev 2016;64:790–805. [2] Min-Hsiung Y, Rong-Hua Y. Analysis of optimization in an OTEC plant using organic Rankine cycle. Renew Energy 2014;68:25–34. [3] Liu C, Gao T. Off-design performance analysis of basic ORC, ORC using zeotropic mixtures and composition-adjustable ORC under optimal control strategy. Energy 2019;171:95–108. [4] Delgado-Torres Agustín M, García-Rodríguez Lourdes. Analysis and optimization of the low-temperature solar organic Rankine cycle (ORC). Energy Convers Manage 2010;51:2846–56. [5] White MT, Oyewunmi OA, Haslam AJ, Markides CN. Industrial waste-heat recovery through integrated computer-aided working-fluid and ORC system optimisation using SAFT-γ Mie. Energy Convers Manage 2017;150:851–69. [6] Sun Q, Wang Y, Cheng Z, Wang J, Zhao P, Dai Y. Thermodynamic and economic optimization of a double-pressure organic Rankine cycle driven by low-temperature heat source. Renew Energy 2018. [7] Astolfi M, Alfani D, Lasala S, Macchi E. Comparison between ORC and CO2 power systems for the exploitation of low-medium temperature heat sources. Energy 2018;161:1250–61. [8] Sun W, Yue X, Wang Y. Exergy efficiency analysis of ORC (Organic Rankine Cycle) and ORC-based combined cycles driven by low-temperature waste heat. Energy Convers Manage 2017;135:63–73. [9] Talluri L, Fiaschi D, Neri G, Ciappi L. Design and optimization of a Tesla turbine for ORC applications. Appl Energy 2018;226:300–19. [10] Al Jubori A, Daabo A, Al-Dadah RK, Mahmoud S, Ennil AB. Development of microscale axial and radial turbines for low-temperature heat source driven organic Rankine cycle. Energy Convers Manage 2016;130:141–55. [11] Kim D, Kim Y. Preliminary design and performance analysis of a radial inflow turbine for ocean thermal energy conversion. Renew Energy 2017;106:255–63. [12] Bao J, Zhao L. A review of working fluid and expander selections for organic Rankine cycle. Renew Sustain Energy Rev 2013;24:325–42. [13] Zhai L, Xu G, Wen J, Quan Y, Fu J, Wu H, et al. An improved modeling for lowgrade organic Rankine cycle coupled with optimization design of radial-inflow turbine. Energy Convers Manage 2017;153:60–70. [14] Li P, Han Z, Jia X, Mei Z, Han X, Wang Z. Comparative analysis of an organic Rankine cycle with different turbine efficiency models based on multi-objective optimization. Energy Convers Manage 2019;185:130–42. [15] Bahadormanesh N, Rahat S, Yarali M. Constrained multi-objective optimization of radial expanders in organic Rankine cycles by firefly algorithm. Energy Convers Manage 2017;148:1179–93. [16] Bekiloğlu HE, Bedir H, Anlaş G. Multi-objective optimization of ORC parameters and selection of working fluid using preliminary radial inflow turbine design. Energy Convers Manage 2019;183:833–47. [17] Zhao R, Zhang H, Song S, Yang F, Hou X, Yang Y. Global optimization of the diesel engine–organic Rankine cycle (ORC) combined system based on particle swarm optimizer (PSO). Energy Convers Manage 2018;174:248–59. [18] Saloux E, Sorin M, Nesreddine H, Teyssedou A. Reconstruction procedure of the thermodynamic cycle of organic Rankine cycles (ORC) and selection of the most appropriate working fluid. Appl Therm Eng 2018;129:628–35.
5. Conclusion This paper presents a preliminary radial inflow turbine design method based on PSO. A complete process of thermodynamic design and aerodynamic design are conducted to optimize the turbine isotropic efficiency. Considering the critical parameters and environment impact of working fluids, six organic fluids were selected to recycle the thermal energy of low temperature heat source at 350 K–400 K. Six turbines were designed based on the working fluids and heat source conditions using the design method. Comparing the geometry parameters and performance characteristics of the turbines, the turbine with R245fa has small geometry size 10
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radial inflow turbines. Appl Therm Eng 2018;138:18–34. [34] Kim D, Kim Y. Preliminary design and performance analysis of a radial inflow turbine for organic Rankine cycles. Appl Therm Eng 2017;120:549–59. [35] Sauret E, Gu Y. Three-dimensional off-design numerical analysis of an organic Rankine cycle radial-inflow turbine. Appl Energy 2014;135:202–11. [36] Nithesh KG, Chatterjee D. Numerical prediction of the performance of radial inflow turbine designed for ocean thermal energy conversion system. Appl Energy 2016;167:1–16. [37] Lemmon EW, Huber ML, McLinden MO. NIST standard reference database 23: reference fluid thermodynamic and transport properties-REFPROP, version 9.0, standard reference data program. Gaithersburg (MD): National Institute of Standards and Technology; 2010. [38] Li YS, Lu GL. Radial-inflow turbine and centrifugal compressor. 1984. Beijing. [39] Glassman AJ. Computer program for design analysis of radial-inflow turbines. NASA Tech Pap; 1976. [40] Al Jubori AM, Al-Dadah RK, Mahmoud S, Daabo A. Modelling and parametric analysis of small-scale axial and radial-outflow turbines for Organic Rankine Cycle applications. Appl Energy 2017;190:981–96. [41] Xia J, Wang J, Wang H, Dai Y. Three-dimensional performance analysis of a radialinflow turbine for an organic Rankine cycle driven by low grade heat source. Energy Convers Manage 2018;169:22–33. [42] Razaaly N, Persico G, Congedo PM. Impact of geometric, operational, and model uncertainties on the non-ideal flow through a supersonic ORC turbine cascade. Energy 2019;169:213–27. [43] Cao Y, Gao Y, Zheng Y, Dai Y. Optimum design and thermodynamic analysis of a gas turbine and ORC combined cycle with recuperators. Energy Convers Manage 2016;116:32–41. [44] MathWork website (n.d.) https://it.mathworks.com/products/matlab. [45] Das S, Abraham A, Konar A. Particle swarm optimization and differential evolution algorithms: technical analysis, applications and hybridization perspectives. Stud Comput Intell 2008;116:1–38. [46] Kang SH. Design and experimental study of ORC (organic Rankine cycle) and radial turbine using R245fa working fluid. Energy 2012;41:514–24. [47] Larjola J. Electricity from industrial waste heat using high-speed organic Rankine cycle (ORC). Int J Prod Econ 1995;41:227–35. [48] Wu H, Pan K. Optimum design and simulation of a radial-inflow turbine for geothermal power generation. Appl Therm Eng 2018;130:1299–309. [49] Mohtaram S, Chen W, Zargar T, Lin J. Energy-exergy analysis of compressor pressure ratio effects on thermodynamic performance of ammonia water combined cycle. Energy Convers Manage 2017;134:77–87.
[19] Groniewsky A, Györke G, Imre AR. Description of wet-to-dry transition in model ORC working fluids. Appl Therm Eng 2017;125:963–71. [20] Yu H, Eason J, Biegler LT, Feng X, Gundersen T. Process optimization and working fluid mixture design for organic Rankine cycles (ORCs) recovering compression heat in oxy-combustion power plants. Energy Convers Manage 2018;175:132–41. [21] Ma W, Liu T, Min R, Li M. Effects of physical and chemical properties of working fluids on thermodynamic performances of medium-low temperature organic Rankine cycles (ORCs). Energy Convers Manage 2018;171:742–9. [22] Dai X, Shi L, An Q, Qian W. Screening of hydrocarbons as supercritical ORCs working fluids by thermal stability. Energy Convers Manage 2016;126:632–7. [23] Qiu G. Selection of working fluids for micro-CHP systems with ORC. Renew Energy 2012;48:565–70. [24] Scaccabarozzi R, Tavano M, Invernizzi CM, Martelli E. Comparison of working fluids and cycle optimization for heat recovery ORCs from large internal combustion engines. Energy 2018;158:396–416. [25] Kajurek J, Rusowicz A, Grzebielec A, Bujalski W, Futyma K, Rudowicz Z. Selection of refrigerants for a modified organic Rankine cycle. Energy 2019;168:1–8. [26] Guo T, Wang H, Zhang S. Fluid selection for a low-temperature geothermal organic Rankine cycle by energy and exergy. In: Asia-Pacific power energy eng conf APPEEC; 2010. [27] Zhai H, Shi L, An Q. Influence of working fluid properties on system performance and screen evaluation indicators for geothermal ORC (organic Rankine cycle) system. Energy 2014;74:2–11. [28] Yang M, Yeh R. The effects of composition ratios and pressure drops of R245fa/ R236fa mixtures on the performance of an organic Rankine cycle system for waste heat recovery. Energy Convers Manage 2018;175:313–26. [29] Rahbar K, Mahmoud S, Al-Dadah RK, Moazami N. Parametric analysis and optimization of a small-scale radial turbine for Organic Rankine Cycle. Energy 2015;83:696–711. [30] Yang M, Yeh R. Economic performances optimization of an organic Rankine cycle system with lower global warming potential working fluids in geothermal application. Renew Energy 2016;85:1201–13. [31] Hu D, Li S, Zheng Y, Wang J, Dai Y. Preliminary design and off-design performance analysis of an Organic Rankine Cycle for geothermal sources. Energy Convers Manage 2015;96:175–87. [32] Zheng Y, Hu D, Cao Y, Dai Y. Preliminary design and off-design performance analysis of an Organic Rankine Cycle radial-inflow turbine based on mathematic method and CFD method. Appl Therm Eng 2017;112:25–37. [33] Du Y, Yang Y, Hu D, Hao M, Wang J, Dai Y. Off-design performance comparative analysis between basic and parallel dual-pressure organic Rankine cycles using
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Contents lists available at ScienceDirect
Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Preliminary design of radial inflow turbine and working fluid selection based on particle swarm optimization
T
⁎
Zhonghe Han, Xiaoqiang Jia, Peng Li
School of Energy, Power and Mechanical, North China Electric Power University, Baoding 071000, People's Republic of China
A R T I C LE I N FO
A B S T R A C T
Keywords: Organic Rankine cycle Turbine design Particle swarm optimization Working fluid selection Off-design performance
Organic Rankine cycle (ORC) is an efficient technique to recycle the low temperature heat sources. The design method of the critical component, radial inflow turbine, is a main focus of research. A preliminary design method was developed to optimize eight critical parameters based on particle swarm optimization (PSO) algorithm. The isotropic efficiency was the objective function of the algorithm. Six working fluids were selected to conduct turbine design based on the design method. The turbine with R245fa was determined to be optimal due to its small geometry size, high exergy efficiency (0.929) and high load coefficient (1.1027). The off-design performance of R245fa turbine was investigated with the variation of pressure ratio (PR), stator inlet temperature and power output. The results indicate that the turbine efficiencies increase with the reduction of PR and turbine inlet temperature and the increase of power output. The exergy efficiency and isotropic efficiency drop slightly with the increase of turbine inlet temperature. So the turbine with R245fa exhibited good off-design performance.
1. Introduction With the rapidly increasing consumption of fossil energy, the sustainable use of energy resources has received increasing attention. There are many researches on the utilization method of low temperature heat sources [1] such as ocean thermal energy (278.15 K–301.15 K) [2], geothermal energy (423.15 K) [3] and solar energy (423.15 K) [4] and low-grade heat sources like industrial waste heat (423.15 K) [5]. Many thermodynamic cycles can be applied in the recycling of low temperature heat source, in which organic Rankine cycle (ORC) is a main method. Sun et al. [6] proposed a double-pressure ORC system and conducted performance analysis and optimization on it. Astolfi et al. [7] compared CO2 cycle and ORC system based on different heat sources and presented a set of selection maps for their application. Sun et al. [8] investigated the thermodynamic influence factors of ORC system and carried out exergy efficiency analysis on the ORC-based combined cycles. There are a large number of literatures about the ORC system. As one of the main components of ORC, turbines including Tesla turbine [9], axial turbine and radial turbine [10], have a remarkable influence on the system efficiency [11]. The characteristics of small size and high efficiency make radial inflow turbines suitable for large scale heat recovery plants [12]. In recent years, many studies have focused on the
⁎
turbine design method and ORC system optimization. And many intelligence algorithms were used in the optimization process. Zhai et al. [13] coupled optimization design of an ORC system with radial inflow turbine using the constrained genetic algorithm (GA). Li et al. [14] proposed a design method for the radial inflow turbine within the ORC thermodynamic model based on multi-objective optimization. Bahadormanesh et al. [15] conducted multi-objective optimization of radial inflow turbine in ORCs using firefly algorithm. Bekiloğlu et al. [16] optimized the system parameters based on turbine design through the multi-objective optimization method NSGA-II. Then 28 different working fluids were compared and ranked according to TOPSIS results. Particle swarm optimization (PSO) was employed for global optimization of an engine-ORC combined system in Ref. [17]. Compared with traditional algorithms, PSO has a fast calculation speed and an efficient global search ability. However, most researches considered turbine design method based on the optimization of ORC system and the system performance was selected as the optimization objective. Few of them were devoted to optimize the turbine performance in a comprehensive method including thermodynamic and aerodynamic design method based on PSO algorithm. Working fluid selection is another topic of interest in the ORC research field. A part of researches conducted working fluid selection based on physical and chemical properties. Saloux et al. [18]
Corresponding author. E-mail address: [email protected] (P. Li).
https://doi.org/10.1016/j.enconman.2019.111933 Received 8 May 2019; Received in revised form 29 July 2019; Accepted 11 August 2019 0196-8904/ © 2019 Elsevier Ltd. All rights reserved.
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Ω ρ δ ν Φ Ψ
Nomenclature A Ar b c ca D h l m Ma n P T u w W xa Z
flow passage area, m2 ratio of needed area to actual area at rotor outlet chord length, mm absolute velocity, m·s−1 assumed expansion velocity diameter, m enthalpy, kJ kg−1 blade height, mm mass flow rate, kg·s−1 Mach number rotational speed, rpm pressure, MPa temperature, K peripheral velocity, m·s−1 relative velocity, m·s−1 power, kW velocity ratio blades number
degree of reaction density, kg·m−3 tip clearance kinematic viscosity flow coefficient load coefficient
Subscripts 0, 1, 2 b cr ex f h l m min s; i sh u
State points installation critical exergy friction loss hub leakage loss meridional minimum isentropic shroud peripheral
Greek letters Abbreviations η ζ φ ψ α β Δ
efficiency loss coefficient nozzle velocity coefficient rotor blade velocity coefficient absolute flow angle, ° relative flow angle, ° exhaust edge thickness, mm
GWP ODP ORC PR STM
introduced a selection method of the optimal working fluid based on the environmental and actual design conditions. Groniewsky et al. [19] proposed a rule of thumb to find new organic fluids according to the degree of freedom and heat capacity at constant volume. Yu et al. [20] studied the working fluid design model for a flue gas pre-compression system coupled with ORC system. Ma et al. [21] investigated the impact of physical and chemical properties of 70 fluids and compared the working fluids based on specific turbine volume (STV). Dai et al. [22] evaluated the thermal stability of the hydrocarbons experimentally and screened for the suitable working fluids of supercritical ORCs. Qiu [23] compared and optimized eight working fluids and ranked them based on the spinal point method. Also, there were some studies considered ORC system performance as selection criterion. Scaccabarozzi et al. [24] performed ORC working fluids selection from over 40 kinds of fluids to obtain the most promising choice of ORC system. In Ref. [25], a new ORC system with special expansion part was proposed and the performance of it with different organic fluids was analyzed. Guo et al. [26] compared 26 working fluids based on the net power output, outlet volume flow rate, expansion ratio and the total UA and determined that R245fa was the optimal fluid. Zhai et al. [27] investigated the influence of fluid properties on ORC system performance and obtained the most suitable working fluids for heat sources with different temperature. Yang et al. [28] analyzed the influence of working fluids composition ratios on the performance of ORC system. However, there were few researches select working fluid on the basis of geometry parameters and performance of turbine as stated in Ref. [29]. Generally, off-design performance analysis of organic turbines is important to the ORC research. But most studies were conducted based on system model. Yang et al. [30] introduced a new evaluation parameter called net power output index to assess the performance of ORC system and analyzed the influence of turbine inlet temperature on the system performance. Hu et al. [31] conducted off-design performance analysis of ORC system based on the design model of it. Zheng et al.
global warming potential ozone depletion potential organic Rankine cycle pressure ratio specific turbine mass
[32] analyzed the off-design performance of turbine on the basis of design model and CFD method. Du et al. [33] compared the off-design performances of basic ORC and dual-pressure ORC systems based on the components models. The literatures related to turbine performance mainly focused on the turbine efficiency and power output through CFD
Fig. 1. Enthalpy entropy diagram of radial inflow turbine. 2
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Table 2 Experienced value ranges of the eight key parameters. Optimized variables
Units
Range
Reference
Degree of reaction (Ω) Velocity ratio (xa) Stator velocity coefficient (φ) Rotor velocity coefficient (ψ) Ratio of wheel diameter (μ) Absolute velocity angle at rotor inlet (α1) Relatively velocity angle at rotor outlet (β2) Rotational speed (n)
– – – – – ˚ ˚ r/min
0.4–0.6 0.63–0.72 0.92–0.98 0.8–0.85 0.5–0.6 14–20 25–45 12000–25000
[38,40] [31] [38,41] [38,41] [41,42] [32] [31,32] [31,43]
Table 3 The operation parameter values of PSO [45].
Fig. 2. Model and velocity triangles of radial inflow turbine. Table 1 The constraints on aerodynamic parameters of turbine [38]. Parameters
Units
Constraints
Incidence angle (i) Mach number of rotor inlet (Ma) Relative velocity (w) Relative pitch of stator blade (t¯1) l1/b
° – m/s – –
−10°–5° < 1.35 w2 < w1 0.65–0.75 0.3–0.45
Operation parameters
Value
Population Number of generation Inertia weight (w) Population learning factor (c1) Global learning factor (c2) Constraint of velocity
80 200 0.6 1.5 1.5 [−0.1,0.1]
were compared and the best working fluid is obtained on account of the geometry size, turbine efficiency and specific fluid mass (SFM). Finally, the off-design performance of turbine was investigated based on the analysis of turbine efficiencies and flow conditions. 2. Preliminary design method The preliminary design of radial inflow turbine includes thermodynamic design and aerodynamic design. The parameters needed in aerodynamic design can be obtained from the thermodynamic design results, so the thermodynamic design is carried out at first. 2.1. Thermodynamic cycle and velocity triangles In the design process of radial inflow turbine, the boundary conditions including pressure and temperature at turbine inlet and the PR, are determined at first in consideration of the heat source temperature and fluid properties at evaporator outlet. Assuming that there is no energy loss in the radial clearance between the rotor and stator, the h-s diagram of flow process in turbine is displayed in Fig. 1. It is shown that the organic fluid flow through stator (0–1) and rotor (1–2) sequentially. The enthalpy and entropy of states 0 and 2 are calculated from the initial parameters and fluid properties obtained from REFPROP 9.0 [37]. Then the isotropic enthalpy drop Δhs can be calculated as follow: (1)
Δhs = h 0 − h2s′
To obtain the velocity triangles, ideal expansion velocity is calculated at first:
ca =
Fig. 3. Flow passage of radial inflow turbine.
2Δhs
(2)
The velocities of rotor inlet and outlet are calculated according to Eqs. (3)–(10). The velocity triangle is obtained as shown in Fig. 2.
method as described in Refs. [34–36]. However, the flow conditions parameters such as Mach number and mass flow rate in turbine flow passage also have great impact on turbine performance. Therefore, it is significant to analyze flow conditions parameters and turbine efficiencies based on mathematic method. In this paper, a preliminary design method of radial inflow turbine including thermodynamic and aerodynamic design is presented in which PSO is employed to optimize eight key parameters. The isotropic efficiency of turbine was selected as fitness function. Considering the impact of organic fluids properties, six fluids were selected as working fluid candidates to carry out turbine design. Then the designed turbines
c1 = φ 2Δh1s ,
φ=
c1 , c1t
w1 =
c1 sin α1 sin β1
u2 = μu1 3
(3) (4)
u1 = x a ca β1 = arctan
Δh1s = (1 − Ω)Δhs
c1 sin α1 c1 cos α1 − u1
(5) (6) (7)
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Fig. 4. The flowchart of PSO. Table 4 Comparison of design model and Ref. [46]. Variables
Design model
Ref. [46]
Error (%)
mf (kg/s) n (rpm) D1 (mm) Z2 ηi (%)
1.51 19,418 166 12 86
1.58 20,000 173 12 84
−4.43 −2.91 −4.05 0 2.38
c2m c cos(α2 − 90) = 2 u1 u1
(11)
Load coefficient: Ψ =
uc −uc Δh = 1 1u 2 2 2u u12 u1
(12)
where, c2m is the absolute meridional velocity at rotor outlet; c1u and c2u are periphery components of absolute velocity at rotor inlet and outlet. The establishment of loss model is important to determine the turbine efficiency. The peripheral losses of turbine are defined as follows:
Table 5 Different working fluids with heat sources. Heat source temperature
Flow coefficient: Φ =
Nozzle loss coefficient: ξ1 = (1 − Ω)(1 − φ2) Working fluid
Turbine inlet temperature
2
w 1 Rotor loss coefficient: ξ2 = ⎛ 2 ⎞ ⎜⎛ 2 − 1⎞⎟ ⎝ ca ⎠ ⎝ ψ ⎠ ⎜
373.15 K–393.15 K 353.15 K–373.15 K 373.15 K 403.15 k 393.15 K 433.15 K
– – 367.15 K – 333 K 351.91 K
w2 = ψ 2ΩΔhs + w12 − (u12 − u22) ,
α2 = 90 + arctan
u2 − w2 cos β2 w2 sin β2
(13)
Reference
R124 R245fa R600 R134a R152a R236fa
ψ=
w2 w2t
[25] [26] [30] [27] [29] [28]
⎟
(14)
2
c Exit loss coefficient: ξc 2 = ⎛ 2 ⎞ ⎝ ca ⎠ ⎜
⎟
(15)
The peripheral efficiency is expressed in Eq. (16).
ηu = 1 − ξ1 − ξ2 − ξc 2 (8)
(16)
In addition, the mass flow rate is calculated in Eq. (17):
mf = (9)
W ηi Δhs
(17) −1
c2 =
(u2 − w2 cos β2 )2 + (w2 sin β2 )2
The specific fluid mass, SFM (kg·kJ ), can be defined as the initial mass consumption per unit of turbine power output:
(10)
On the basis of velocity triangles, the flow coefficient (Φ) and load coefficient (Ψ) defined in Eqs. (11) and (12) are obtained:
SFM = 4
mf W
(18)
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Table 6 Properties of working fluid candidates. Working fluid
Molecular weight (kg/kmol)
Tcr (°C)
Pcr (kPa)
ODP
GWP
Safety data*
Type
Price ($/kg)
R124 R245fa R600 R134a R152a R236fa
137 134 58 102 66 152
122 153 152 101 113 125
3613 3610 3800 4060 4520 3200
0.026 0 0 0 0 0
620 950 20 1430 2.8 9810
A1 B1 A3 A1 A2 A1
dry dry dry Isentropic dry dry
19.30 11.87 20.79 4.80 6.68 13.07
* ASHRAE Standard 34-2016 (designation and safety classification of refrigerants)-A: Low toxicity/B: High Toxicity/1: No flammability/2: Low flammability/3: High flammability. Table 7 Initial parameters for the design method. Initial condition
Value
P0 (kPa) T0 (K) PR W (kW)
725 350 4.11 100
2.2. Aerodynamic design 2.2.1. Stator The TC-4P blade model is selected for the stator blade since it has better aerodynamic performance and more reliable processing technology [38]. The relative blade pitch t¯ has a significant influence on blade profile loss. The range of it varies with the blade model of stator. When the relative blade pitch is oversized, the suction surface of tailing edge will appear flow separation which results in flow loss. The optimal relative blade pitch range of TC-4P blade model is 0.65–0.75 when the Mach number of stator outlet is greater than 1 [38]. The flow capacity of stator is measured by the blockage factor τ which is defined in Eq. (19):
τ1 = 1 −
Z1 Δ1 πD1 sin α1 πD1 , bt¯1
(19) 60u1 . 1000nπ
Δ1 = 0.1, D1 = where, Z1 = The blade height of rotor l1 is calculated in Eq. (20): l1 =
mf ρ1 c1 πD1 τ1 sin α1
(20)
These aerodynamic parameters of stator are coupled with others, so iteration method is employed in the design process to find the optimal solution.
Fig. 5. Size of main geometry parameters of stator and rotor.
2.2.2. Rotor The number of rotor blades should be within a suitable range. Unreasonable number of blades will lead to additional losses and decrease turbine efficiency. The number of rotor blades is determined by Glassman's empirical formula [39].
Z2 =
π (20 + α1) tan(90 − α1) 30
(21)
Limited by the flow angle of stator outlet, there is a minimum
Table 8 Results of the design method with six working fluids. Classification Independent parameters
Objective
Ω xa φ ψ μ α1 β2 n ηi
R124
R245fa
R600
R134a
R152a
R236fa
0.4002 0.6598 0.9799 0.8494 0.5002 14.7856 25.3143 22,792 0.8960
0.4159 0.6943 0.9800 0.8314 0.5004 15.8597 25.0240 22,529 0.8923
0.4006 0.6524 0.9798 0.8475 0.5081 15.6633 26.0738 18,900 0.8893
0.4012 0.6761 0.9798 0.8499 0.5004 14.6323 25.0618 17,973 0.8967
0.4020 0.6776 0.9798 0.8245 0.5001 15.4008 25.0065 23,872 0.8904
0.4119 0.6678 0.9792 0.8446 0.5006 18.2375 25.2041 23,148 0.8931
5
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Fig. 6. Mach number and absolute velocity at rotor inlet.
number of rotor blades.
Z2 min =
2π sin(β2b − α1) sin α1 sin β2b
Fig. 8. Loss coefficient values of turbine with different working fluids.
(22)
When Z2 > Z2min, the number of rotor blades is rounded to Z2, otherwise, the number of rotor blades is rounded to Z2min. The shroud and hub diameters of rotor outlet are calculated as follows:
A2 =
mf w2 sin β2 ρ2 τ2
D2sh =
D22 +
2 A2 π
Fig. 7. Loss contribution of different working fluids. 6
(23)
(24)
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Fig. 12. The variation of loss and efficiencies with PR.
Fig. 9. Turbine efficiencies with different fluids.
Fig. 13. The variation of flow conditions with PR.
Fig. 10. Flow coefficient and load coefficient of turbines with different working fluids.
Fig. 14. The variation of loss and efficiencies with temperature.
D2h =
D22 −
2 A2 π
(25)
Thus, the blade height of rotor outlet l2 is obtained.
l2 =
Fig. 11. Fluid consumption of turbines with different working fluids.
D2sh − D2h 2
(26)
As shown in Table 1, some optimization constraints are included in the design model to guarantee the reliable results of the designed turbine. 7
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velocity angle at rotor outlet (β2) and rotational speed (n). The ranges of them are given in Table 2. An original MATLAB [44] code was developed to perform the entire design process of radial-inflow turbine. Particle swarm optimization (PSO) was applied in the optimization of the eight key parameters. Compared with traditional algorithms, PSO has a fast calculation speed and an efficient global search ability [17]. In addition, PSO is not very sensitive to population size, so it is suitable for solving extremum problems of continuous function. Since the ranges of the eight parameters are different, this paper normalizes all parameters in the optimization algorithm at first, and then decodes them into original parameters in the calculation of population fitness as shown in Eq. (27).
θ = θ0 ·(θmax − θmin ) + θmin
(27)
θ0 ∈ [0, 1]
PSO describes the motion state of a particle with two parameters: position and velocity. Then the historical optimal position of particle (pBest) and global population (gBest) are recorded and combined with the inertia of the particle to update the position and velocity of the particle. The updated formula of PSO is as follows:
Fig. 15. The variation of flow conditions with temperature.
Vt +1 = w·Vt + c1 r1·(pBest − Xt ) + c2 r2·(gBest − Xt )
(28)
Xt +1 = Xt + Vt +1
(29)
where, t is number of generation, X is the position, V is the velocity, w is the inertia weight, c is the learning factor, r is random number. The objective function of PSO is the turbine isotropic efficiency ηi which is defined in Eqs. (30)–(32). The operation parameter values and the flowchart of PSO are shown in Table 3 and Fig. 4, respectively.
Friction loss coefficient: ξf =
f D2 u13 · 1· 1.36 × 106 ν1 m f ·Δh s
δ D Leakage loss coefficient: ξl = 0.47 ⎛1 + 2 ⎞ l2 ⎝ l2 ⎠ ⎜
⎟
δ = 0.01~0.20 l2
Turbine isotropic efficiency: ηi = 1 − ξ1 − ξ2 − ξc 2 − ξf − ξl Fig. 16. The variation of loss and efficiency with power output.
(30)
(31) (32)
2.4. Validation In order to validate the accuracy of this design method, the results of the design model are compared with Ref. [46] as shown in Table 4. The initial conditions are the same as those in Ref. [46]. It can be seen that the results of the design model are in good agreement with Ref. [46]. The turbine isotropic efficiency of design model is higher than that of Ref. [46], the reason is that a part of loss is ignored in the design model. The mass flow rate of design model is lower. It is noted that the error of diameter of nozzle outlet D1 and mass flow rate mf are more than that of ratio of rotational speed n, but they are limited in 5%. In a word, the design model is precise and reliable. 3. Working fluid selection An appropriate working fluid selection is important to the performance of ORC system and radial inflow turbine. Many studies have investigated the working fluid selection based on heat sources in different temperature. When the fluid at turbine inlet is superheated, the thermal efficiency can be improved but the system cost is increased. Therefore, the working fluid is set to saturated states to reduce the cost and irreversible loss. As shown in Table 5, six working fluids are recommended in these literatures for recycling the energy of heat sources with temperature of 353.15–433.15 K. The main focuses of working fluid selection are the critical pressure and temperature, safety, and thermodynamic type. Additionally, the environmental impact is considered in working fluid selection. The properties of the six fluids are shown in Table 6. The molecular weight mainly influences the geometry size of turbine. The thermodynamic state point is limited by critical temperature (Tcr) and pressure (Pcr).
Fig. 17. The variation of flow conditions with power output.
On the basis of the calculation process above, the geometry parameters are obtained as shown in Fig. 3. 2.3. PSO optimization In the process of thermodynamic and aerodynamic design, there are eight key parameters including degree of reaction (Ω), velocity ratio (xa), stator velocity coefficient (φ), rotor velocity coefficient (ψ), wheel diameter ratio (μ), absolute velocity angle at rotor inlet (α1), relatively 8
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The environmental impact of organic fluid is indicated by ODP and GWP. From Table 6, it is noted that the environmental impact and safety of these organic fluids are acceptable. The small molecular weight leads to a minor geometry size of the turbine. The temperature of the heat source studied in this paper ranges from 350 to 400 K, so the thermodynamic types guarantee that these organic fluids can be kept in superheated state during the whole working process. As recommended in Refs. [25–30], the thermodynamic performance of these candidates are satisfied. Therefore, they are selected as the working fluid candidates to recycle the heat source. It is indicated from Ref. [31] that the total to total efficiency of turbine increases with the pressure ratio when the inlet temperature is low. As recommended in Ref. [46], the inlet pressure, pressure ratio and inlet temperature are set to 725 kPa, 4.11 and 350 K, respectively. It is known from Ref. [47] that, the general power of single-stage radial inflow turbine is 100 kW in actual ORC plant. Considering the properties of the six working fluids and heat source, the initial operating conditions are given as shown in Table 7.
Figs. 7 and 8. It is noted that the proportion and value of friction losses of turbines with R124, R245fa and R236fa are smaller than the others and can be ignored, as a result of small geometry size and low periphery velocity. As shown in Fig. 8, due to similar Ω and φ, the nozzle losses of turbines with all fluids are approximately the same, and the proportions of them also have slight variation. The rotor loss is the maximum loss among turbines with all fluids, the rotor loss of turbine with R600 reaches 0.0374 and accounts for 32% of the total loss. The reason is that the friction loss of it reaches the largest of 0.005, the irreversible loss makes the expansion process in turbine deviating from the isotropic expansion process. The leakage losses of turbines with R245fa and R236fa are less than the others due to their small rotor size and reasonable rotational speed. The turbine efficiencies with different fluids are shown in Fig. 9. According to Fig. 9, the variation trend of exergy efficiency is similar to the isotropic efficiency. Although the periphery efficiencies of turbines with R124 is slightly higher than R245fa turbine, the isotropic efficiency of R124 turbine is lower than R245fa turbine on accounts of the large leakage loss as shown in Fig. 8. For the reason that the shroud curve and passage turning are reasonable, the irreversible losses of R245fa and R236fa turbine like friction loss and leakage loss are less than the other turbines. Therefore, the exergy efficiencies of turbines with R245fa and R236fa are higher than the other turbines in which the exergy efficiency of R236fa turbine reach the highest of 0.9313. The flow coefficient (Φ) and load coefficient (Ψ) defined in Eqs. (11) and (12) are important evaluation indices for turbine performance. They can efficiently assess the dimensionless volume flow rate and total enthalpy drop of the whole stage [48]. Rahbar K et al. [29] claimed that the increase of Ψ can lead to the increase of turbine efficiency, while the impact of Φ is opposite. Fig. 10 shows the flow coefficient and load coefficient of turbines with different working fluids. It is indicated that the variation trend of Ψ is more pronounced than that of the Φ. The Ψ of turbine with R245fa reaches the highest of 1.1027, so the conversion efficiency of mechanical work and working capacity of R245fa turbine are higher than the others. The Φ of turbines with R245fa and R600 are higher than the others, so the volume flow rate of them are higher and they need larger flow passage area. SFM characterizes the mass of working fluid consumed in the initial stage of system construction. Specific fluid cost is applied in assessing the system initial cost on working fluid when SFM multiplied with the prices of working fluids. As shown in Fig. 11, due to small molecular weight and large critical pressure, the SFM of turbines with R600, R134a and R152a are low, but the price of R600 is high. Therefore, the lower cost ones are turbines with R134a and R152a which are 0.15 $·kW−1 and 0.14 $·kW−1. From the above discussion, the turbine with R245fa has small geometry size, rational flow passage, low irreversible loss and high efficiency. Therefore, R245fa is selected to recycle the heat from heat source at the temperature of 350 K–400 K.
4. Results and analysis 4.1. Characteristic comparison It is rational to consider the working fluid properties during the turbine design process. Based on the six working fluids in the last section, six different radial inflow turbines are designed using the design method. The designed parameters of them are shown in Table 8. It can be seen that the degree of reaction and velocity ratio of turbine with R245fa reach the highest of 0.4159 and 0.6943, respectively. Therefore, the working capacity and accelerate ability of turbine with R245fa are better than the others. The wheel diameter ratios are almost the same. The rotational speed of turbines with R600 and R134a are slower than the other turbines due to the large diameters and blade heights of rotors. In addition, the isotropic efficiencies of these turbines are high, so the six fluids are suitable to work at the design condition in Table 7. It also can be seen from Table 8 that, for all working fluids, the optimal angles at rotor outlet is almost the same while the variation of the angles at rotor inlet is larger. The α1 affect the passage turning and nozzle loss of stator. The α1 of turbine with R236fa reaches the largest of 18.2375°. Therefore, the change of passage turning and the nozzle loss of it are relatively small. Fig. 5 shows the main geometry parameters of stator and rotor. The b and l1 express the size of stator while D2 and l2 express the size of rotor. The D2sh/D1 indicate the shroud curvature and blade loss of rotor. As shown in Fig. 5(a), the variations of b and l1 are similar while the variation of D1 is opposite to them. So in order to increase the D1, the b and l1 should be reduced. In addition, it can be noted from Fig. 5(b) that variations of D2 and l2 are similar while the variation of D2sh/D1 is opposite to them. Although the b and l1 of turbines with R124 and R245fa are larger than others, the D1, D2 and l2 of them is smaller. So the geometry sizes of turbines with R124 and R245fa are smaller than the other turbines. The D2sh/D1 of turbine with R245fa is the largest so that the blade loss of rotor is smaller than the others [29]. Except for the geometric characteristics, turbine performance has also been the main focus of recent researches. Fig. 6 shows the Mach number (Ma) and absolute velocity (c1) at rotor inlet with the six working fluids. It is indicated that the variation of c1 is more pronounced than that of Ma. The absolute velocity of turbines with R124 and R236fa is low, but the Mach number of them are high which are 1.2074 and 1.2029, respectively. The reason is that the enthalpy drop of the whole stages are low and degree of reaction are almost the same, but the accelerate ability of them are better than others. The absolute velocity of the R152a turbine is high, but the Mach number of it reaches the lowest of 1.1794. The losses contribution of turbines with the fluids are given in
4.2. Off-design performance analysis In order to deal with the changes in generation demand, the radial inflow turbine operates at off-design conditions in some cases. R245fa turbine was selected to analyze the off-design performance of the turbine. An important off-design adjustment mode of turbine is controlling back pressure. When the inlet pressure is constant, the variation of back pressure is equal to the pressure ratio (PR) [49]. When the geometry parameters and the eight variables are fixed, the impact of PR on the turbine efficiency is shown in Fig. 12. It is indicated that the periphery efficiency is constant due to the constant geometry parameters. The enthalpy drop and flow velocities increase with increasing PR, so the friction loss also increases, but the increase is relatively small. Therefore, the isotropic efficiency slowly decreases with the increase of PR. It 9
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and rational flow passage. In addition, it has a high efficiency whose exergy efficiency reaches 0.929 and loading coefficient reaches 1.1027. Therefore, R245fa is more suitable to recycle the heat source as the working fluid of radial inflow turbine. The turbine efficiency increases with the reduction of PR and turbine inlet temperature and the increase of power output. The flow conditions will improve with the increase of PR and inlet temperature, but the increasing power output can lead to worse flow conditions. Therefore, the back pressure and power output should be kept in a rational range to guarantee the flow condition and efficiency of turbine. On the basis of performance analysis, the turbine with R245fa can operate in a stable and efficient status at off-design conditions in terms of temperature.
also leads to the decrease of exergy efficiency for the increase of irreversible loss. As a result, the back pressure cannot be set to a small value. The impact of PR on the flow condition is shown in Fig. 13. It is obvious that the Mach number increases with the PR for which the enthalpy drop in stator becomes larger with the increasing pressure. When PR is lower than 3, the fluid at nozzle outlet is in subsonic state, when PR is larger than 3, the fluid will expand in the chamfered portion and be accelerated into supersonic state, the aerodynamic performance and expansion ability of TC-4P blade type is fully utilized. The variable Ar is the ratio of needed rotor outlet area to the actual rotor outlet area, as defined in Eq. (33). It is known that the increasing PR leads to the reduction of Ar, for the reason that the rotor outlet pressure and mass flow rate reduces with the increasing PR. The flow capacity of rotor is not fully utilized which leads to large flow loss as shown in Fig. 12. So the back pressure cannot be set to a large value.
Ar =
A2 − need A2 − actual
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
(33)
In addition, the temperature of turbine inlet is another important influence factor. When the other conditions are fixed, the impact of temperature on the turbine efficiency is shown in Fig. 14. It is noted that the exergy efficiency and isotropic efficiency drop slightly with the increase of turbine inlet temperature. In addition, the enthalpy drop and flow velocities increase with the increase of inlet temperature, so the friction loss rises slightly. Consequently, the turbine can operate at a stable and efficient state with the variation of turbine inlet temperature, the off-design performance of the turbine is reliable. The impact of temperature on the flow condition is shown in Fig. 15. It can be seen that the Mach number increases with the increase of temperature due to the increase of enthalpy drop in stator, but the Ar and mass flow rate reduce with the increase of temperature. The internal energy of working fluid will rise with the increase of temperature. When the power output is constant, the increasing enthalpy drop will result in the reduction of mass flow rate and the needed rotor outlet area. So the inlet temperature should be kept in a rational range. Furthermore, the power output of turbine also varies with the change of heat source. When the other conditions are fixed, the influence of power output on the turbine efficiency is shown in Fig. 16. According to the figure, the isotropic efficiency increases slightly with the power output as a consequence of the reduction of friction loss. In addition, it also leads to the increase of exergy efficiency. Therefore, the power output of turbine cannot be set to a small value, when it is low, the turbine is inefficient. The influence of power output on the flow conditions is shown in Fig. 17. It is indicated that the Mach number remains stable with the variation of power output for the fact that the boundary conditions of the turbine is fixed. It is apparent that the Ar and mass flow rate rise with the power output for the increasing demand of working fluid. So the actual geometry size of the turbine cannot meet the demand of the increasing power output. Therefore, the power output of turbine cannot be set to a large value, when it is large, the flow loss of turbine will increase.
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