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Energy Conversion and Management 185 (2019) 130–142
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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Comparative analysis of an organic Rankine cycle with different turbine efficiency models based on multi-objective optimization
T
⁎
Peng Li, Zhonghe Han , Xiaoqiang Jia, Zhongkai Mei, Xu Han, Zhi Wang Key Lab of Condition Monitoring and Control for Power Plant Equipment, School of Energy, Power and Mechanical Engineering, North China Electric Power University, Baoding 071000, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Organic Rankine cycle Constant turbine efficiency Variable turbine efficiency System thermal efficiency Multi-objective optimization
The radial-inflow turbine is a key component of the organic Rankine cycle (ORC) system, and its efficiency is related to working fluid properties and working conditions. In this paper, multi-objective algorithm was employed to conduct a comparative analysis of an ORC system with different turbine efficiency models (constant and variable). The system thermal efficiency and multi-objective optimization results between the constant turbine efficiency ORC system (CTORC) and the variable turbine efficiency ORC system (VTORC) were compared, and the reasons for the difference between the CTORC and VTORC system were analyzed. Sensitivity analysis was performed to compare the difference between the CTORC system and the VTORC system at different waste flue gas inlet temperature. The results show that the system thermal efficiency of both the CTORC and the VTORC increases with the increasing evaporation temperature and decreases with the increasing condensation temperature, while the variation rate of system thermal efficiency is different between CTORC system and VTORC system. The predicted turbine efficiency is significantly different for different working fluids and different working conditions. Considering the system comprehensive performance, R236ea is the optimal working fluid for the CTORC system, while R365mfc is for the VTORC system. For different working fluids, the error caused by using constant turbine efficiency model is different. With the increasing waste flue gas inlet temperature, the error caused by using constant turbine efficiency increases for R236ea, while the error caused by using constant turbine efficiency increases first and then decreases for R365mfc.
1. Introduction With the rapid development of the economy and society, the demand for fossil fuels has been increasing daily, and environmental pollution has aroused a growing concern. The utilization of renewable energy and the recovery of waste heat are two effective solutions to alleviate the energy risk and the environment deterioration [1–3]. Organic Rankine cycle (ORC) exhibits a high potential in conversion of low-grade heat source into electricity due to its simple structure, economic feasibility and easy maintenance [4–6]. In the last few decades, many studies have been performed investigating the ORC system [7–10]. Bademlioglu et al. [11] compared different parameters’ impact weights on the ORC thermal efficiency by employing statistical methods, and found that the influence of the evaporation temperature, condensation temperature and turbine efficiency on the ORC thermal efficiency is more significant. Javanshir et al. [12] conducted a thermodynamic analysis of a simple subcritical and supercritical ORC system with different working fluids. It can be
⁎
Corresponding author. E-mail address: [email protected] (Z. Han).
https://doi.org/10.1016/j.enconman.2019.01.117 Received 6 December 2018; Accepted 31 January 2019 0196-8904/ © 2019 Elsevier Ltd. All rights reserved.
found that isentropic working fluids provide a higher thermal efficiency and a higher specific heat capacity is advantages to improve the net power output. Zhang et al. [13] performed a sustainability evaluation of an ORC system adopting an emergy analysis and life cycle method. They found that the sustainability of the ORC system is less than that of renewable source (wind, hydro and geothermal) power plants, but much greater than that of fossil fuel (oil and coal) power plants. In order to maximize the utilization of the waste heat source and simultaneously keep the specific investment cost low, Garg et al. [14] proposed a novel objective function that can reveal the tradeoff between the utilization extent of waste heat and the specific investment cost. The ORC system was optimized by using particle swarm optimization algorithm based on the novel objective function. Li et al. [15] investigated the effects of the evaporation temperature, the pinch point temperature difference in the evaporator and condenser on the ORC system economic performance. The electricity production cost was selected as an evaluation criterion to optimize the parameters for different working fluids at different heat source temperatures.
Energy Conversion and Management 185 (2019) 130–142
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ξ φ ν ψ α β ϕ ρ
Nomenclature A c Ċ D D¯ 2 Ė h L m M Q TCI u w W Ż u¯1
area, m2 average cost rate per unit exergy, $ GJ−1; absolute velocity, m s−1 cost rate, $ h−1 diameter, m hub-diameter ratio exergy rate, kW specific enthalpy, kJ kg−1 depth of rotor, m mass flow rate, kg s−1 molecular weight, kg mol−1 heat, kW total capital investment, $ peripheral velocity, m s−1 relative velocity, m s−1 power, kW capital cost rate, $ h−1 peripheral velocity ratio
loss coefficient Nozzle velocity coefficient specific volume, m3 kg−1 rotor velocity coefficient absolute velocity angle relative velocity angle maintenance factor degree of reaction
Subscripts 0, 1, 2, 3, 4, 5, 6, 7, 9, 10 state points con condenser eva evaporator k kth component pump pump s isentropic the thermal tur turbine wf working fluid
Greek letters η
efficiency
thermodynamic and economic performance indicators, respectively, and the optimization of the ORC with four different working fluids was performed using non-dominated sorting genetic algorithm. Almost all of the theoretical studies related to ORC system consider a constant turbine efficiency model. It is evident that different working fluid properties and different system operating conditions would lead to different turbine efficiency. Thus, the results from the conventional CTORC system are not necessarily accurate. Da Lio et al. [24] calculated the turbine efficiency based on a mean-line model of the radial-inflow turbine. R245fa with real gas properties was selected as the working fluid, and it was found that different working conditions lead to different turbine efficiency. Rosset et al. [25] calculated the isentropic efficiency of a radial-inflow turbine based on a preliminary design map accounting for the effect of the pressure ratio. Constrained multi-objective optimization of the ORC system was conducted with the net power output and the heat exchange area as objective functions. Rahbar et al. [26] proposed an optimized model of the VTORC system, and the
Multi-objective optimization can make a trade-off between the thermodynamic performance and economic performance to achieve a better comprehensive performance [16–19]. Yang et al. [20] established a multi-objective model of an ORC system to maximize the net power output and minimize the total capital cost. The model was solved using genetic algorithm to screen the optimal working fluid and determine the corresponding optimal operating parameters. Gimelli et al. [21] addressed a multi-objective optimization problem of an ORC system using the electricity efficiency and the overall heat exchangers area as performance indicators, with MDM as working fluid. In Pareto optimal front solutions, the range of the electricity efficiency is from 14.1% to 18.9%, and the overall heat exchangers area is between 446 m2 and 1079 m2. To find the optimum point of an ORC system with R123 as the working fluid, multi-objective optimization with respect to the exergy efficiency and the total product unit cost as performance criteria was conducted based on genetic algorithm [22]. Özahi et al. [23] employed net power output and total cost rate as the
Fig. 1. The schematic diagram of a basic ORC. 131
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into a cost stream Ċ.
DIRECT algorithm was used to maximize the ratio of the cycle net electric power output to the turbine oversize. They found that the predicted turbine efficiency varies with working fluids, and the maximum difference in the turbine efficiency between R245fa and isobutane is 6.13%. However, the influence of turbine efficiency model (constant and variable) selection on the ORC system comprehensive performance (thermodynamic and economic) based on multi-objective optimization are not extensively studied. In the present study, a variable turbine efficiency model is presented to calculate the turbine efficiency, which is unique for each set of operating conditions and working fluid properties. The system thermal efficiency of the CTORC and the VTORC were compared. Then, the influence of the turbine efficiency model selection on the ORC system comprehensive performance was investigated. Considering the thermodynamic performance and economic performance, system thermal efficiency and the cost per exergy unit of the product are selected as objective functions. Non-dominated sorting genetic algorithm-II (NSGA-II) was employed to optimize the CTORC system and the VTORC system. Taking R365mfc and R236ea as examples, the difference in the optimization results between the CTORC system and the VTORC system were analyzed in detail. Moreover, sensitivity analysis of the ORC system was conducted with respect to the heat source temperature, and the difference in the results between the CTORC and the VTORC was also compared.
Ċ = cE ̇
For each component of the ORC system, the cost balance equation is established according to the exergy balance equation:
CṖ = CḞ − CḊ + Z ̇
Ż =
CRFϕ TCI τ
CRF =
̇ ̇ (c F,total EF,total + ∑k Z ̇ k) Cp,total = ̇ ̇ Ep,total Ep,total
(3)
where ĖF and ĖP represent the exergy rate of the fuel and the product, respectively. ĖL describes the exergy rate that is lost to the ambient or leaves the system. ĖD is the exergy destruction rate. The total exergy of a working fluid stream is expressed as follows:
E ̇ = m [(h − henv ) − Tenv (s − senv )]
(9)
A constant turbine efficiency model is employed in the conventional ORC system analysis process, but, as mentioned in the introduction, this assumption is not necessarily accurate. In this study, to investigate the influence of the turbine efficiency model selection on the ORC system performance, the constant turbine efficiency model is replaced by the variable turbine efficiency model to predict the turbine efficiency that is unique for each set of operating conditions and working fluids. Both ORC analysis procedures are presented in Fig. 3. As shown in Fig. 3, the calculation model of the radial-inflow turbine efficiency and the ORC system are coupled with each other. A one-dimensional assumption is adopted in the calculation model of the radial-inflow turbine efficiency. Fig. 4 shows the schematic diagram of the radial-inflow turbine (volute and diffuser are neglected) in the meridional plane. The simplified flow process in the radial-inflow turbine is described in Fig. 5. It can be found that the incoming working fluid sequentially expands in the nozzle (0–1) and rotor (1–2), and the
Based on the second law of thermodynamics, the exergy balance equations on the ORC system components can be expressed as:
EḊ = EḞ − EṖ − EL̇
(8)
2.2. Analysis and calculation of the turbine efficiency
Conceptually, the operating principles of the organic Rankine cycle are similar to the conventional steam Rankine cycle, except that it employs low boiling point organic materials as working fluids instead of steam. A basic ORC system consists of four main components, including a pump, an evaporator, a turbine (expander) and a condenser, as presented in Fig. 1. The liquid working fluid from the condenser is pressured by the pump and enters the evaporator, where it is heated by the waste flue gas and becomes saturated vapor. The saturated working fluid vapor is directed into the turbine and expands to generate the mechanical power of the shaft, which is used to drive the generator. Then, the vapor exhausted from the turbine is condensed to liquid by the condenser, and the liquid working fluid re-enters the pump. The corresponding T-s diagram of a basic ORC system is presented in Fig. 2. In this study, each component of the ORC system is considered to be a control volume. The equations that are used to calculate the heat absorbed or rejected by the organic working fluid and the power generation or consumption are based on the mass and energy conservations and are given in Table 1. In addition to these equations, the net power output and the thermal efficiency of the ORC system are calculated as follows:
(2)
i (1 + i)n (1 + i)n − 1
In this study, the cost per exergy unit of the product is selected as an optimization criterion for the ORC system.
2.1. Thermodynamic and exergoeconomic modelling
ηthe = Wnet / Qeva
(7)
where CRF is the capital recovery factor; ϕ represents the maintenance factor; and τ represents the annual plant operation hours (7000 h). The CRF is the function of the average annual interest rate, i (10%) and the plant economic life, n (20 years).
c p,total =
(1)
(6)
The cost balance equations and the corresponding auxiliary equations for each component of the ORC system are listed in Table 1. Ż represents the cost rate for each component of the ORC system, which includes the total capital investment (TCI) and the operation and maintenance (O&M) costs. The equations of TCI for different components are given in Table 1. The capital cost rate can be calculated as:
2. Modelling and analysis methods
Wnet = Wtur − Wpump
(5)
(4)
The exergoeconomic analysis method integrates the thermodynamic (exergy) aspect and economic principle. In this method, the exergy stream Ė is multiplied by an average cost per exergy unit c to convert it
Fig. 2. The T-s diagram of a basic ORC. 132
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Table 1 Thermodynamic and cost balance equations for each component of the ORC system. Components
Thermodynamic equations
Cost equations
Total capital investment
Wpump = mwf (h5 − h4 )
̇ pump + Zpump ̇ cWpump = cWtur C5̇ = C4̇ + CW
0.71 TCIpump = 422Wpump [1.41 + 1.41(
Qeva = mwf (h 0 − h5)
̇ c8 = c 7 C8̇ + C0̇ = C5̇ + C7̇ + Zeva
TCIeva = 10, 000 + 324A0.91
Wtur = mwf (h 0 − h2) = mwf (h 0 − h2s)·ηtur
̇ tur = C0̇ + Ztur ̇ C2̇ + CW c2 = c0
0.7 TCItur = 6000Wtur
Qcon = mwf (h2 − h4 )
̇ = C2̇ + C9̇ + Zcon ̇ c2 = c4 c9 = 0 C4̇ + C10
TCIcon = 10, 000 + 324A0.91
= mwf (h5s − h4 )/ ηpump
1 − 0.8 )] 1 − ηpump
Pump
Evaporator
Turbine
Condenser
enthalpy of the working fluid is converted into the mechanical power of the radial-inflow turbine. The initial state of the organic vapor (from the evaporator) at the radial-inflow turbine inlet is represented by point 0. In practical operations, due to the flow losses in the passages, the state of the organic vapor at the nozzle outlet changes from point 1s to point 1, and the state of the organic vapor at the rotor outlet changes from point 2s to point 2. Δhs is the total isentropic enthalpy drop of the organic working fluid across the entire radial-inflow turbine, and Δh is the total actual enthalpy drop. Fig. 6 shows the velocity triangles, which are used to express the distribution of absolute velocity, relative velocity and velocity angles in the nozzle and rotor. The peripheral efficiency of the radial-inflow turbine is determined from:
ηu =
(c12 − c22) + (u12 − u22) + (w22 − w12) C02
(10)
The relative velocity is defined as the ratio of each velocity to the ideal velocity, and the ideal velocity can be calculated as follows:
c0 =
2Δhs
(11)
Therefore, the peripheral efficiency of the radial-inflow turbine can be expressed as:
ηu = c¯12 − c¯22 + u¯ 12 − u¯ 22 + w¯ 12 − w¯ 22 = 2(¯c1u u¯1 − c¯2u u¯2)
Fig. 3. ORC system analysis procedure with constant and variable turbine efficiency. 133
(12)
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w¯ 12 = c¯12 + u¯ 12 − 2¯c1 u¯1 cos α1
(17)
c¯1 = φ 1 − ρ
(18)
Substitute Eqs. (17) and (18) into Eq. (16):
w¯ 2t =
ρ + φ2 (1 − ρ) + D¯ 22 u¯ 12 − 2u¯1 φ cos α1 1 − ρ
(19)
Therefore, the peripheral efficiency of the radial-inflow turbine can be calculated by a dimensionless equation:
ηu = 2u¯1 [φ cos α1 1 − ρ − D¯ 22 u¯1 + D¯ 2 ψ cos β2 ρ + φ2 (1 − ρ) + D¯ 22 u¯ 12 − 2u¯1 φ cos α1 1 − ρ ] (20) It can be determined that the peripheral efficiency of the radialinflow turbine is influenced by seven different dimensionless parameters. The peripheral velocity ratio u¯1 is the ratio of the rotor peripheral velocity to the ideal velocity, and the degree of reaction ρ is the ratio of the enthalpy drop in the rotor to the enthalpy drop in the entire radial-inflow turbine. The two parameters have a greater effect on the peripheral efficiency of the radial-inflow turbine [27]. The peripheral velocity ratio and degree of reaction are optimized using a Lagrange multiplier method with the peripheral efficiency of the radial-inflow turbine as the objective. The optimal peripheral velocity ratio and degree of reaction can be calculated as follows [28]:
Fig. 4. Schematic diagram of the radial-inflow turbine in the meridional plane.
u¯1opt =
ψ 2
cos β D¯ 22 ( 2 2 − ψ2 ) +
(1 − m2ψ4 ) φ2 cos2 α1
m
ρopt = 1 − [
(1
m2ψ2 (1 − φ2)
(21)
− mψ2) φ cos α1 u¯1 2 ] mψ2 (1 − φ2)
(22)
where m is a combined parameter that can be expressed as:
m=
1 [1 ± ψ2
D¯ 22 (1 − φ2)(1 − cos2 β2 ψ2) ] cos2 α1 φ2 + D¯ 22 ((1 − φ2)
(23)
From Eq. (16), w¯ 12 can be expressed as:
w¯ 12 = w¯ 2t2 − ρ + (u¯ 12 − u¯ 22)
(24)
Substitute Eqs. (18) and (24) into Eq. (12), another expression of the peripheral efficiency of the radial-inflow turbine can be presented as follows: Fig. 5. Flow process of the working fluid in the radial-inflow turbine.
ηu = 1 − (1 − ρ)(1 − φ2) − w¯ 2t2 (1 − ψ2) − c¯22 = 1 − ξ n − ξr − ξ e
(25)
w¯ 2t2 (1
ϕ2)(1
ψ2 )
c¯22
− − ρ ) , ξr = where ξn = (1 − and ξe = represent the nozzle loss coefficient, rotor loss coefficient and leaving velocity loss coefficient, respectively. In addition to the three types of aforementioned losses, two other types of losses (friction loss and leakage loss) need to be considered in the calculation of the turbine efficiency. The friction loss coefficient can be calculated as: ξf = f · Fig. 6. Velocity triangles of the radial-flow turbine.
D12 u1 3 1 1 ·( )· · ν1 100 1.36 m f ·Δhs
(26)
The leakage loss coefficient can be calculated as:
D u¯2 = 2 u¯1 = D¯ 2 u¯1 D1
(13)
c¯1u = c¯1 cos α1 = φ 1 − ρ cos α1
(14)
c¯2u = u¯2 − w¯ 2 cos β2 = D¯ 2 u¯1 − ψw¯ 2t cos β2 w¯ 2t =
ρ + w¯ 12 − u¯ 12 + D¯ 2 u¯ 12
δ
ξ1 =
⎧1 − 1.3· 1m , 0.015 < ⎨ 0.95 − 0.31· δ , 1m ⎩
δ 1m
δ 1m
⩽ 0.05
> 0.05
(27)
Differing from the other four loss coefficients, the leakage loss coefficient does not represent the ratio of the leakage loss to the isentropic enthalpy drop. Thus, the radial-inflow turbine efficiency can be expressed as:
(15)
ηtur = (ηu − ξf )·ξl = (1 − ξn − ξr − ξe − ξf )·ξl
(16) 134
(28)
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selection, crossover and mutation operators. The parent population and the offspring population are combined to form a new population Rt = Pt ∪ Qt, whose size is 2 N. (3) Since Rt includes all the members in the parent population and the offspring population, elitism is preserved. Non-dominated sorting is conducted for population Rt, then population Rt is classified into different non-dominated fronts F1, F2…, a higher ranking means a larger distance from the optimal solutions. (4) A new population Pt+1 with a size of N is created, and it is filled with members of different non-dominated fronts. The new population only has N slots, which is less than the population Qt size 2 N. The filling begins from the members of F1; if the number of members of F1 is less than N, the filling
An overview of the calculation model of the radial-inflow turbine efficiency is presented in Fig. 7. A real gas formulation is used in this model, and Matlab software is used to solve the calculation model. Since there are not sufficient experimental data for a radial-inflow turbine using organic working fluid, an air radial-inflow turbine is used to validate the calculation model of the radial-inflow turbine efficiency. Jones [29] performed rig test of a radial-inflow turbine with air as working fluid and drew the turbine efficiency characteristic curve (experimental curve of the turbine efficiency vs the peripheral velocity ratio), as shown in Fig. 8. The turbine efficiency characteristic curve obtained by the calculation model of the radial-inflow turbine efficiency was also presented in Fig. 8 to compare the calculation results with the experimental results. The turbine efficiency increases first and then decreases with the increasing peripheral velocity ratio, which is accord with Ref. [27]. In addition, it can be seen that the predicted turbine efficiency coincide with the experimental value well with a maximum error of 0.573%. Therefore, the radial-inflow turbine efficiency calculation model is considered accurate and can be used in this study. 2.3. Optimization model and simulation method The system thermal efficiency and the cost per exergy unit of the product are selected as objective functions to investigate the influence of the turbine efficiency model selection on the multi-objective optimization results of the ORC system. NSGA-II is used to maximize the thermal efficiency and minimize the cost per exergy unit of the product simultaneously. Multi-objective optimization model of the ORC system can be described as:
⎧ max(ηthe) = f1 (T0, T4 ) ⎨ ⎩ min(c p,total ) = f2 (T0, T4 )
(29)
The physical constraints for the multi-objective optimization problem are listed as follows: (1) The evaporation temperature should less than the heat source temperature and critical temperature of the working fluid, and the pinch point temperature difference in the evaporator should higher than 5 K.
⎧ T0 < T7 T0 < Tcr ⎨ ⎩ ΔTeva > 5
(30)
(2) The condensation temperature should less than the evaporation temperature and higher than the ambient temperature. In this study, the condensation temperature varies in the range of 313.15–333.15 K.
⎧ Tamb < T4 < T0 ⎨ ⎩313.15 < T4 < 333.15
(31)
NSGA-II is a computationally efficient multi-objective optimization algorithm, that was proposed by Deb et al. [30] on the basis of natural selection in the biological genetic process. Two special mechanisms, including non-dominated sorting and crowding distance, are used in NSGA-II. Such mechanisms are advantageous to ensure both the convergence and diversification of the population. NSGA-II has been widely used by many researchers in different research fields, including power system optimization [31,32], heat exchanger design [33], and green building optimization design [34,35]. The main procedures of the NSGA-II are summarized as follows: (1) Initial parent population P0 is created randomly with a size of N. To conveniently describe the procedures of the algorithm, the tth generation was used in the follow description. (2) The offspring population Qt is generated from the parent population Pt through binary tournament
Fig. 7. Flowchart of the turbine efficiency calculation. 135
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0.90 0.89
Turbine efficiency
0.88
increment for the CTORC system is larger than that for the VTORC system, resulting in the increasing rate of the system thermal efficiency for the CTORC system is higher than that for the VTORC system. For both the CTORC system and the VTORC system, the difference in the system thermal efficiency for different working fluids is small in the low evaporation temperature range, and the difference value gradually increases with the increasing evaporation temperature. In addition, compared with the CTORC system, the difference in the system thermal efficiency for different working fluids in the VTORC system is much higher at a high evaporation temperature. Among the six working fluids, R236ea always has the lowest thermal efficiency for the CTORC system, while for the VTORC system, the lowest thermal efficiency working fluids change from R236ea to ipentane when the evaporation temperature exceeds 376.15 K. The sequence order of some working fluids is also different between the CTORC system and the VTORC system. For the CTORC system, the sequence order is pentane, ipentane and R365mfc in terms of thermal efficiency, while the sequence order for the same three working fluids changes to R365mfc, pentane and ipentane for the VTORC system. The turbine efficiency with different working fluids at various evaporation temperatures is shown in Fig. 12. The turbine efficiency decreases with the increasing evaporation temperature. For different working fluids the predicted turbine efficiency is different, and the difference increases with the increment of the evaporation temperature. The radial-inflow turbine with R365mfc as working fluid has the largest turbine efficiency among the six working fluids, and the maximum turbine efficiency is 0.827 when the evaporation temperature is 353.15 K. The radial-inflow turbine with ipentane as working fluid has the smallest turbine efficiency, and the minimum turbine efficiency is 0.678 when the evaporation temperature is 412.15 K. The difference between the maximum turbine efficiency and the minimum turbine efficiency is 0.149, which indicates that using constant turbine efficiency for different working fluids and different operating conditions will lead to a serious error. To analyze and compare the differences between the CTORC system and the VTORC system, the system thermal efficiency difference is defined as follows:
Experimental value(Ref. [29]) Predicted turbine efficiency
0.87 0.86 0.85 0.84 0.83 0.82 0.81 0.80 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80 0.82
Velocity ratio Fig. 8. Validation of the radial-inflow turbine efficiency calculation model.
continued with the members of F2, F3…. Say that Fl is the last nondominated front that can be accommodated by population Pt+1. In general, the number of solutions in Fl is more than the remaining slots in the population Pt+1. Thus, a novel comparison operator called crowding distance was proposed to sort the solutions in Fl. In the same rank, the solutions with a larger crowding distance are better and are chosen prior to fill population Pt+1. (5) If the stopping conditions are not met, a new offspring population Qt+1 is created, and the procedure returns to step (2). The iteration ends once the stopping conditions are met. The basic procedures of NSGA-II are shown in Fig. 9. 3. Results and discussion 3.1. Simulation conditions and working fluids In this study, an ORC system utilizing the waste flue gas is investigated. The waste flue gas inlet temperature is assumed to be 433.15 K (a common temperature of the waste flue gas) [5,36]. To prevent the low-temperature corrosion, the waste flue gas outlet temperature should exceed the acid dew point, thus the waste flue gas outlet temperature is assumed to be 363.15 K [37,38]. Additionally, to compare the comprehensive performance of different working fluids more accurate, the utilized total waste heat is assumed to be constant at 1 MW [39]. The constant turbine efficiency is assumed to be 0.8 in this study [40,41]. Six organic working fluids, R114, pentane, R236ea, ipentane, R365mfc and hexane, were analyzed and compared between the CTORC and VTORC. Basic thermodynamic properties of the selected working fluids are given in Table 2.
Δηthe = ηthe, CTORC − ηthe, VTORC
(32)
The system thermal efficiency difference between the CTORC and the VTORC is presented in Fig. 13. For pentane and ipentane, the system thermal efficiency difference becomes higher with the increasing evaporation temperature, and the variation rate gradually increases. The reason for this outcome is that only in a low evaporation temperature range the turbine efficiency for pentane and ipentane are slightly higher than the constant turbine efficiency 0.8, and the remaining turbine efficiency for pentane and ipentane are lower than the constant turbine efficiency 0.8. Additionally, the higher the evaporation temperature is, the lower the turbine efficiency is, and the larger the deviation between the predicted turbine efficiency and the constant turbine efficiency is. Therefore, a higher evaporation temperature
3.2. Comparison of the CTORC system and the VTORC system results To investigate the influence of the turbine efficiency model selection on the ORC system, a thermodynamic analysis of both the CTORC system and the VTORC system with different working fluids was conducted at various operation conditions. The variation of the system thermal efficiency with the increasing evaporation temperature for the CTORC system and the VTORC system are presented in Figs. 10 and 11, respectively. The system thermal efficiency increases with the increment of the evaporation temperature for both the CTORC and the VTORC, and the increasing rate for both systems is significant different. For the CTORC system, the turbine efficiency is constant, and the enthalpy drop in the radial-inflow turbine increases with the increasing evaporation temperature. For the VTORC system, the turbine efficiency decreases with the increment of the evaporation temperature (just as presented in Fig. 12), reducing the increment of the enthalpy drop in the radial-inflow turbine. Therefore, the amount of net power output
Fig. 9. Basic procedures of NSGA-II. 136
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0.84
Table 2 Basic thermal properties of the selected working fluids.
0.82
M/g·mol−1
Tbolling/K
Pcritical/MPa
Tcritical/K
R114 Pentane R236ea Ipentane R365mfc Hexane
170.92 72.15 152.04 72.15 148.07 86.18
276.741 309.21 279.34 300.98 313.3 341.86
3.25 3.37 3.50 3.34 3.27 3.03
145.68 469.70 412.44 460.35 460.00 507.82
0.80
Turbine efficiency
Working fluids
13
0.76
R114 pentane R236ea ipentane R365mfc hexane
0.74 0.72 0.70 0.68
12
System thermal efficiency/%
0.78
0.66 350
360
370
11
390
400
410
420
Fig. 12. Turbine efficiency at different evaporation temperatures.
10
R114 pentane R236ea ipentane R365mfc hexane
8
360
370
380
390
400
410
2.5
System thermal efficiency difference/%
9
7 350
420
Evaporation temperature/K Fig. 10. Variation of system thermal efficiency with the evaporation temperature for the CTORC. 13
11
1.5
1.0
0.5
0.0
360
370
380
390
400
Evaporation temperature/K
410
420
Fig. 13. Variation of the system thermal efficiency difference with the evaporation temperature.
10
R114 pentane R236ea ipentane R365mfc hexane
8
360
370
380
390
400
410
11.0 10.5
System thermal efficiency/%
9
7 350
R114 pentane R236ea ipentane R365mfc hexane
2.0
-0.5 350
12
System thermal efficiency/%
380
Evaporation temperature/K
420
Evaporation temperature/K Fig. 11. Variation of system thermal efficiency with the evaporation temperature for the VTORC.
would lead to a larger system thermal efficiency difference between the CTORC and the VTORC. For R365mfc, R114, hexane and R236ea, when the evaporation temperature is low, the turbine efficiency is higher than the constant turbine efficiency 0.8, thus the CTORC system thermal efficiency is lower than the VTORC system thermal efficiency. As the evaporation temperature increases gradually, the system thermal efficiency difference decreases due to the decreasing turbine efficiency, as shown in Fig. 13. When the evaporation temperature reaches a value that makes the predicted turbine efficiency equal 0.8, the system thermal efficiency difference equals 0. With a further increasing evaporation temperature, the turbine efficiency continues to decrease, thus the CTORC system thermal efficiency is higher than the VTORC system thermal efficiency, and the system thermal efficiency difference begins
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to increase. Additionally, the higher the evaporation temperature is, the larger the system thermal efficiency difference is. Figs. 14 and 15 present the variation of the system thermal efficiency with the increasing condensation temperature for the CTORC and the VTORC, respectively. The system thermal efficiency decreases 137
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Fig. 17. Variation of system thermal efficiency difference with the condensation temperature.
VTORC decreased 2.58%. With the increasing condensation temperature, the enthalpy drop in the radial-inflow turbine decreases, while the turbine efficiency increases, as shown in Fig. 16, reducing the decrement of the enthalpy drop in the radial-inflow turbine. Therefore, the decrement of the net power output for the VTORC system is less than that for the CTORC, leading to the decreasing rate of the system thermal
linearly with the increasing condensation temperature for the CTORC and the VTORC, where the decreasing rate for the VTORC is lower than that for the CTORC. Taking ipentane as an example, in the investigated condensation temperature range the system thermal efficiency for the CTORC decreased 3.20%, while the system thermal efficiency for the 138
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efficiency for the VTORC being lower than that for the CTORC. With the increasing condensation temperature, the difference in the system thermal efficiency for different working fluids decreases, and the variation of the difference in system thermal efficiency for different working fluids in the VTORC system is slightly more violent in the investigated condensation temperature range. When the condensation temperature increases from 313.15 K to 333.15 K, the difference in the system thermal efficiency between hexane and R236ea changes from 0.74 to 0.51 for the CTORC system, while the difference in the system thermal efficiency between hexane and R236ea changes from 1.00 to
0.57 for the VTORC system. The sequence order of some working fluids in terms of the system thermal efficiency for the CTORC system is pentane > ipentane > R365mfc, and the sequence order of the same working fluids changes to R365mfc > pentane > ipentane for the VTORC system. Fig. 16 shows the turbine efficiency with different working fluids at various condensation temperatures. As the condensation temperature 139
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are different, as shown in Fig. 21. The reason for this outcome is that the predicted turbine efficiency in the VTORC system is lower than the constant turbine efficiency 0.8, and the maximum deviation reaches 0.077, as shown in Fig. 21; in addition, the maximum relative deviation reaches 9.62%. The low predicted turbine efficiency leads to the difference in the Pareto optimal front between the CTORC system and the VTORC system, and the difference increases with the decreasing predicted turbine efficiency, as shown in Fig. 21. For the CTORC system, the turbine efficiency is the same for R365mfc and R236ea, and R236ea is better than R365mfc in terms of comprehensive performance. However, for the VTORC system, the predicted turbine efficiency of R365mfc is higher than that of R236ea, making R365mfc better than R236ea in terms of comprehensive performance. Additionally, it can also be found that the influence of the turbine efficiency model selection on the multi-objective optimization results of the ORC system is related to the working fluid properties; for different working fluids, the influences are different.
increases, the turbine efficiency increases linearly, and the difference of turbine efficiency among different working fluids decreases. Among the six working fluids, R365mfc has the largest turbine efficiency, and ipentane has the smallest turbine efficiency. A maximum turbine efficiency of 0.821 is obtained by R365mfc when the condensation temperature is 333.15 K, and a minimum turbine efficiency of 0.745 is obtained by ipentane when the condensation temperature is 313.15 K. The difference between the maximum turbine efficiency and the minimum turbine efficiency is 0.076. Fig. 17 presents the system thermal efficiency difference between the CTORC and the VTORC. For pentane and ipentane, with the increasing condensation temperature, the system thermal efficiency difference decreases, and the decreasing rate also gradually decreases. It is due to the fact that the turbine efficiency for pentane and ipentane is lower than the constant turbine efficiency 0.8, and as the condensation temperature increases, the turbine efficiency increases; in addition, the deviation between the predicted turbine efficiency and the constant turbine efficiency decreases. For hexane, R114 and R365mfc, with the increasing condensation temperature, the system thermal efficiency difference between the CTORC and the VTORC increases, while the increasing rate gradually decreases. The reason is that the turbine efficiency for hexane, R114 and R365mfc is higher than the constant turbine efficiency 0.8, and with the increasing condensation temperature, the turbine efficiency increases, and the deviation between the predicted turbine efficiency and the constant turbine efficiency increases. For R236ea, the turbine efficiency is lower than the constant turbine efficiency 0.8 in the low condensation temperature range, and with the increasing condensation temperature, the turbine efficiency gradually increases to over 0.8. Therefore, the system thermal efficiency difference between the CTORC and the VTORC decreases first and then increases. NSGA-II was employed to conduct multi-objective optimization of the CTORC system and the VTORC system based on the system thermal efficiency and the cost per exergy unit of the product. The Pareto optimal fronts for the CTORC system and the VTORC system are presented in Figs. 18 and 19, respectively. It is evident that there is a significant difference in the distribution of the Pareto optimal front between the CTORC system and the VTORC system. For all working fluids except for R365mfc, the distribution ranges of the system thermal efficiency of the CTORC are not only higher than that for the VTORC, but also wider than that for the VTORC. Taking R236ea as an example, the distribution range of the system thermal efficiency for the CTORC is from 9.11% to 11.77%, while the distribution range of the system thermal efficiency of the VTORC is from 8.66% to 10.53%. Based on the comprehensive performance, the sequence order of some working fluids for the CTORC system is different from that for the VTORC system. For the CTORC system, R236ea is better than R365mfc, and ipentane is better than hexane, while for the VTORC system, the working fluids sequence order is the opposite; that is, R365mfc is better than R236ea, and hexane is better than ipentane. The optimal working fluids for the CTORC system and the VTORC system are also different. For the CTORC system, R236ea is the optimal working fluid among the six working fluids. However, for the VTORC system, R365mfc is the optimal working fluid. To analyze the reason that lead to the optimal working fluids being different between the CTORC system and the VTORC system in detail, Figs. 20 and 21 show a comparison of the Pareto optimal fronts between the CTORC system and the VTORC system for R365mfc and R236ea, respectively. Additionally, the predicted turbine efficiency corresponding to the Pareto optimal fronts of the VTORC system are also presented in Figs. 20 and 21. As shown in Fig. 20, for R365mfc, the Pareto optimal front of the CTORC system and the VTORC system are essentially in coincidence. That is due to the fact that there is a small deviation between the predicted turbine efficiency in the VTORC system and the constant turbine efficiency 0.8. The maximum deviation between the predicted turbine efficiency and the constant turbine efficiency 0.8 is only 0.014, as shown in Fig. 20. However, for R236ea, the Pareto optimal fronts of the CTORC system and the VTORC system
3.3. Sensitivity analysis of the CTORC system and the VTORC system The waste flue gas inlet temperature varies with the factory and environmental conditions. In this section, sensitivity analysis of the CTORC system and the VTORC system were conducted to investigate the influence of the turbine efficiency model selection on the ORC system at different waste flue gas inlet temperatures. R365mfc and R236ea were selected as working fluids. Multi-objective optimization of the CTORC and the VTORC were conducted at each waste flue gas inlet temperature, and the aid of an ideal point method [40,42] was used to determine the final optimal solution. The process of decision-making for the aid of an ideal point method is shown in Fig. 18. A hypothetical point (ideal point) that maximizes the system thermal efficiency and simultaneously minimizes the cost per exergy of the product is assumed, and the point of the Pareto optimal front that is nearest to the ideal point is considered as the final optimal solution. Figs. 22 and 23 show the optimal evaporation temperature and condensation temperature at different waste flue gas inlet temperatures for R236ea and R365mfc, respectively. As shown in Fig. 22, for R236ea, the optimal evaporation temperature and the optimal condensation temperature for the CTORC system and the VTORC system increase with the increasing waste flue gas inlet temperature, and the increasing rate gradually decreases. It is apparent that the optimal evaporation temperature of the CTORC is always higher than that of the VTORC. Moreover, as the waste flue gas inlet temperature increases, the difference in the optimal evaporation temperature between the CTORC system and the VTORC system increases, indicating that the error caused by using constant turbine efficiency becomes larger with the increasing waste flue gas inlet temperature. The optimal condensation temperature of the CTORC system varies in the range of 320.0–321.5 K, while the optimal condensation temperature of the VTORC system varies in a slightly higher range of 321.0–323.5 K. The difference in the optimal condensation temperatures between the CTORC system and the VTORC system also increases with the increment of the waste flue gas inlet temperature. As shown in Fig. 24, for R236ea, with the increasing waste flue gas inlet temperature, the predicted turbine efficiency decreases and the deviation between the predicted turbine efficiency and the constant turbine efficiency 0.8 increases. Therefore, the error from using the constant turbine efficiency in an ORC system with R236ea as working fluid increases with the increment of the waste flue gas inlet temperature. As shown in Fig. 23, for R365mfc, as the waste flue gas inlet temperature increases, the optimal evaporation temperature and the optimal condensation temperature for the CTORC system and the VTORC system increase, and the increasing rate gradually increases. The optimal evaporation temperature of the CTORC system is lower than that of the VTORC system, while the optimal condensation temperature of the CTORC system is higher than that of the VTORC system in the low 140
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waste flue gas inlet temperature range. With the increasing waste flue gas inlet temperature, the difference in the optimal evaporation temperature and the difference in the optimal condensation temperature between the CTORC system and the VTORC system decreases. When the waste flue gas inlet temperature is higher than 435.9 K, the optimal evaporation temperature of the CTORC system is higher than that of the VTORC system, while the optimal condensation temperature of the CTORC system is lower than that of the VTORC system. With the increasing waste flue gas inlet temperature, both the difference in the optimal evaporation temperature and the difference in the optimal condensation temperature between the CTORC system and the VTORC system increase. The reason for the aforementioned results is that the predicted turbine efficiency of R365mfc is higher than the constant turbine efficiency 0.8 when the waste flue gas inlet temperature is low, as shown in Fig. 24. With the increasing waste flue gas inlet temperature, the predicted turbine efficiency decreases, and the deviation between the constant turbine efficiency 0.8 and the predicted turbine efficiency also decreases. When the waste flue gas inlet temperature is higher than 435.9 K, the predicted turbine efficiency is lower than the constant turbine efficiency 0.8. With the increasing waste flue gas inlet temperature, the deviation between the constant turbine efficiency 0.8 and the predicted turbine efficiency begins to increase. Therefore, the error from using constant turbine efficiency in an ORC system with R365mfc as working fluid increases first and then decreases with the increasing waster flue gas inlet temperature. Fig. 24 shows the predicted turbine efficiency of R236ea and R365mfc at different waste flue gas inlet temperatures. It is evident that the predicted turbine efficiency of R365mfc is higher that of R236ea, and the difference between the two predicted turbine efficiency increases in the investigated waste flue gas inlet temperature range. The predicted turbine efficiency of R236ea and R365mfc decreases with the increasing waste flue gas inlet temperature. The variation rate of the predicted turbine efficiency of R236ea gradually decreases, while the variation rate of the predicted turbine efficiency of R365mfc gradually increases. As shown in Figs. 22 and 23, the variation range of the optimal condensation temperature is obviously smaller than the variation range of the optimal evaporation temperature; thus, the optimal evaporation temperature is the main factor that influences the predicted turbine efficiency. For R236ea, the increasing rate of the optimal evaporation temperature decreases in the high waste flue gas inlet temperature range, thus the decreasing rate of the predicted turbine efficiency also decreases accordingly. However, for R365mfc, the increasing rate of the optimal evaporation temperature increases in the high waste flue gas inlet temperature range, so the decreasing rate of the predicted turbine efficiency also increases accordingly. It can also be found that the deviation between the constant turbine efficiency 0.8 and the predicted turbine efficiency of R365mfc decreases first and then increases with the increasing waste flue gas inlet temperature, and the maximum deviation value is 0.014. The deviation between the predicted turbine efficiency of R236ea and the constant turbine efficiency 0.8 increases with the increasing waste flue gas inlet temperature, and the maximum deviation value is 0.076. Therefore, it can be concluded that the error caused by using constant turbine efficiency is related to the working fluid properties, and the error for R365mfc is smaller than that for R236ea.
turbine efficiency model selection on the ORC system at different waste flue gas inlet temperatures. (1) When the constant turbine efficiency is replaced with the variable turbine efficiency, the variation rate of the system thermal efficiency with evaporation temperature or condensation temperature decreases, and the sequence order of some working fluids also changes. For different working fluids and different working conditions, the predicted turbine efficiency is quite different. (2) Different turbine efficiency models lead to different multi-objective optimal results for ORC system. The distribution of Pareto optimal fronts between the CTORC system and the VTORC is significantly different. R236ea is the optimal working fluid for CTORC system, while R365mfc is the optimal working fluid for VTORC system. (3) For different working fluids, the influence of turbine efficiency model selection on the ORC system is different. As the waste flue gas inlet temperature increases, the error caused by using constant turbine efficiency increases for R236ea, while the error caused by using constant turbine efficiency increases first and then decreases for R365mfc. The predicted turbine efficiency of both R236ea and R365mfc decreases with the increasing waste flue gas inlet temperature. Declaration of interests The authors declared that there is no conflict of interest. Acknowledgements 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] Nawi ZM, Kamarudin SK, Abdullah SS, Lam SS. The potential of exhaust waste heat recovery (WHR) from marine diesel engines via organic rankine cycle. Energy 2019;166:17–31. [2] Landelle A, Tauveron N, Haberschill P, Revellin R, Colasson S. Organic Rankine cycle design and performance comparison based on experimental database. Appl Energy 2017;204:1172–87. [3] Zhang Z, Li H, Chang H, Pan Z, Luo X. Machine learning predictive framework for CO2 thermodynamic properties in solution. J CO2 Util 2018;26:152–9. [4] Li P, Han Z, Jia X, Mei Z, Han X, Wang Z. Analysis and comparison on thermodynamic and economic performances of an organic Rankine cycle with constant and one-dimensional dynamic turbine efficiency. Energy Convers Manage 2019;180:665–79. [5] Mahmoudi A, Fazli M, Morad MR. A recent review of waste heat recovery by Organic Rankine Cycle. Appl Therm Eng 2018;143:660–75. [6] Mahmoudzadeh Andwari A, Pesiridis A, Karvountzis-Kontakiotis A, Esfahanian V. Hybrid electric vehicle performance with organic rankine cycle waste heat recovery system. Appl Sci 2017;7:437. [7] Xu G, Song G, Zhu X, Gao W, Li H, Quan Y. Performance evaluation of a direct vapor generation supercritical ORC system driven by linear Fresnel reflector solar concentrator. Appl Therm Eng 2015;80:196–204. [8] Quoilin S, Van Den Broek M, Declaye S, Dewallef P, Lemort V. Techno-economic survey of Organic Rankine Cycle (ORC) systems. Renew Sustain Energy Rev 2013;22:168–86. [9] Pethurajan V, Sivan S, Joy GC. Issues, comparisons, turbine selections and applications – an overview in organic Rankine cycle. Energy Convers Manage 2018;166:474–88. [10] Kazemi N, Samadi F. Thermodynamic, economic and thermo-economic optimization of a new proposed organic Rankine cycle for energy production from geothermal resources. Energy Convers Manage 2016;121:391–401. [11] Bademlioglu AH, Canbolat AS, Yamankaradeniz N, Kaynakli O. Investigation of parameters affecting Organic Rankine Cycle efficiency by using Taguchi and ANOVA methods. Appl Therm Eng 2018;145:221–8. [12] Javanshir A, Sarunac N. Thermodynamic analysis of a simple Organic Rankine Cycle. Energy 2017;118:85–96. [13] Zhang H, Guan X, Ding Y, Liu C. Emergy analysis of Organic Rankine Cycle (ORC) for waste heat power generation. J Clean Prod 2018;183:1207–15. [14] Garg P, Orosz MS. Economic optimization of Organic Rankine cycle with pure fluids and mixtures for waste heat and solar applications using particle swarm optimization method. Energy Convers Manage 2018;165:649–68. [15] Li YR, Du MT, Wu CM, Wu SY, Liu C, Xu JL. Economical evaluation and
4. Conclusions To investigate the influence of the turbine efficiency model selection on the organic Rankine cycle system, a comparative analysis on the system thermal efficiency and the multi-objective optimization results between the CTORC system and the VTORC system was conducted. Both the reasons for the difference of the system thermal efficiency and for the difference of the multi-objective optimization results were analyzed in detail. Sensitivity analysis of the CTORC system and the VTORC system were conducted to investigate the influence of the 141
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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Comparative analysis of an organic Rankine cycle with different turbine efficiency models based on multi-objective optimization
T
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Peng Li, Zhonghe Han , Xiaoqiang Jia, Zhongkai Mei, Xu Han, Zhi Wang Key Lab of Condition Monitoring and Control for Power Plant Equipment, School of Energy, Power and Mechanical Engineering, North China Electric Power University, Baoding 071000, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Organic Rankine cycle Constant turbine efficiency Variable turbine efficiency System thermal efficiency Multi-objective optimization
The radial-inflow turbine is a key component of the organic Rankine cycle (ORC) system, and its efficiency is related to working fluid properties and working conditions. In this paper, multi-objective algorithm was employed to conduct a comparative analysis of an ORC system with different turbine efficiency models (constant and variable). The system thermal efficiency and multi-objective optimization results between the constant turbine efficiency ORC system (CTORC) and the variable turbine efficiency ORC system (VTORC) were compared, and the reasons for the difference between the CTORC and VTORC system were analyzed. Sensitivity analysis was performed to compare the difference between the CTORC system and the VTORC system at different waste flue gas inlet temperature. The results show that the system thermal efficiency of both the CTORC and the VTORC increases with the increasing evaporation temperature and decreases with the increasing condensation temperature, while the variation rate of system thermal efficiency is different between CTORC system and VTORC system. The predicted turbine efficiency is significantly different for different working fluids and different working conditions. Considering the system comprehensive performance, R236ea is the optimal working fluid for the CTORC system, while R365mfc is for the VTORC system. For different working fluids, the error caused by using constant turbine efficiency model is different. With the increasing waste flue gas inlet temperature, the error caused by using constant turbine efficiency increases for R236ea, while the error caused by using constant turbine efficiency increases first and then decreases for R365mfc.
1. Introduction With the rapid development of the economy and society, the demand for fossil fuels has been increasing daily, and environmental pollution has aroused a growing concern. The utilization of renewable energy and the recovery of waste heat are two effective solutions to alleviate the energy risk and the environment deterioration [1–3]. Organic Rankine cycle (ORC) exhibits a high potential in conversion of low-grade heat source into electricity due to its simple structure, economic feasibility and easy maintenance [4–6]. In the last few decades, many studies have been performed investigating the ORC system [7–10]. Bademlioglu et al. [11] compared different parameters’ impact weights on the ORC thermal efficiency by employing statistical methods, and found that the influence of the evaporation temperature, condensation temperature and turbine efficiency on the ORC thermal efficiency is more significant. Javanshir et al. [12] conducted a thermodynamic analysis of a simple subcritical and supercritical ORC system with different working fluids. It can be
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Corresponding author. E-mail address: [email protected] (Z. Han).
https://doi.org/10.1016/j.enconman.2019.01.117 Received 6 December 2018; Accepted 31 January 2019 0196-8904/ © 2019 Elsevier Ltd. All rights reserved.
found that isentropic working fluids provide a higher thermal efficiency and a higher specific heat capacity is advantages to improve the net power output. Zhang et al. [13] performed a sustainability evaluation of an ORC system adopting an emergy analysis and life cycle method. They found that the sustainability of the ORC system is less than that of renewable source (wind, hydro and geothermal) power plants, but much greater than that of fossil fuel (oil and coal) power plants. In order to maximize the utilization of the waste heat source and simultaneously keep the specific investment cost low, Garg et al. [14] proposed a novel objective function that can reveal the tradeoff between the utilization extent of waste heat and the specific investment cost. The ORC system was optimized by using particle swarm optimization algorithm based on the novel objective function. Li et al. [15] investigated the effects of the evaporation temperature, the pinch point temperature difference in the evaporator and condenser on the ORC system economic performance. The electricity production cost was selected as an evaluation criterion to optimize the parameters for different working fluids at different heat source temperatures.
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ξ φ ν ψ α β ϕ ρ
Nomenclature A c Ċ D D¯ 2 Ė h L m M Q TCI u w W Ż u¯1
area, m2 average cost rate per unit exergy, $ GJ−1; absolute velocity, m s−1 cost rate, $ h−1 diameter, m hub-diameter ratio exergy rate, kW specific enthalpy, kJ kg−1 depth of rotor, m mass flow rate, kg s−1 molecular weight, kg mol−1 heat, kW total capital investment, $ peripheral velocity, m s−1 relative velocity, m s−1 power, kW capital cost rate, $ h−1 peripheral velocity ratio
loss coefficient Nozzle velocity coefficient specific volume, m3 kg−1 rotor velocity coefficient absolute velocity angle relative velocity angle maintenance factor degree of reaction
Subscripts 0, 1, 2, 3, 4, 5, 6, 7, 9, 10 state points con condenser eva evaporator k kth component pump pump s isentropic the thermal tur turbine wf working fluid
Greek letters η
efficiency
thermodynamic and economic performance indicators, respectively, and the optimization of the ORC with four different working fluids was performed using non-dominated sorting genetic algorithm. Almost all of the theoretical studies related to ORC system consider a constant turbine efficiency model. It is evident that different working fluid properties and different system operating conditions would lead to different turbine efficiency. Thus, the results from the conventional CTORC system are not necessarily accurate. Da Lio et al. [24] calculated the turbine efficiency based on a mean-line model of the radial-inflow turbine. R245fa with real gas properties was selected as the working fluid, and it was found that different working conditions lead to different turbine efficiency. Rosset et al. [25] calculated the isentropic efficiency of a radial-inflow turbine based on a preliminary design map accounting for the effect of the pressure ratio. Constrained multi-objective optimization of the ORC system was conducted with the net power output and the heat exchange area as objective functions. Rahbar et al. [26] proposed an optimized model of the VTORC system, and the
Multi-objective optimization can make a trade-off between the thermodynamic performance and economic performance to achieve a better comprehensive performance [16–19]. Yang et al. [20] established a multi-objective model of an ORC system to maximize the net power output and minimize the total capital cost. The model was solved using genetic algorithm to screen the optimal working fluid and determine the corresponding optimal operating parameters. Gimelli et al. [21] addressed a multi-objective optimization problem of an ORC system using the electricity efficiency and the overall heat exchangers area as performance indicators, with MDM as working fluid. In Pareto optimal front solutions, the range of the electricity efficiency is from 14.1% to 18.9%, and the overall heat exchangers area is between 446 m2 and 1079 m2. To find the optimum point of an ORC system with R123 as the working fluid, multi-objective optimization with respect to the exergy efficiency and the total product unit cost as performance criteria was conducted based on genetic algorithm [22]. Özahi et al. [23] employed net power output and total cost rate as the
Fig. 1. The schematic diagram of a basic ORC. 131
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into a cost stream Ċ.
DIRECT algorithm was used to maximize the ratio of the cycle net electric power output to the turbine oversize. They found that the predicted turbine efficiency varies with working fluids, and the maximum difference in the turbine efficiency between R245fa and isobutane is 6.13%. However, the influence of turbine efficiency model (constant and variable) selection on the ORC system comprehensive performance (thermodynamic and economic) based on multi-objective optimization are not extensively studied. In the present study, a variable turbine efficiency model is presented to calculate the turbine efficiency, which is unique for each set of operating conditions and working fluid properties. The system thermal efficiency of the CTORC and the VTORC were compared. Then, the influence of the turbine efficiency model selection on the ORC system comprehensive performance was investigated. Considering the thermodynamic performance and economic performance, system thermal efficiency and the cost per exergy unit of the product are selected as objective functions. Non-dominated sorting genetic algorithm-II (NSGA-II) was employed to optimize the CTORC system and the VTORC system. Taking R365mfc and R236ea as examples, the difference in the optimization results between the CTORC system and the VTORC system were analyzed in detail. Moreover, sensitivity analysis of the ORC system was conducted with respect to the heat source temperature, and the difference in the results between the CTORC and the VTORC was also compared.
Ċ = cE ̇
For each component of the ORC system, the cost balance equation is established according to the exergy balance equation:
CṖ = CḞ − CḊ + Z ̇
Ż =
CRFϕ TCI τ
CRF =
̇ ̇ (c F,total EF,total + ∑k Z ̇ k) Cp,total = ̇ ̇ Ep,total Ep,total
(3)
where ĖF and ĖP represent the exergy rate of the fuel and the product, respectively. ĖL describes the exergy rate that is lost to the ambient or leaves the system. ĖD is the exergy destruction rate. The total exergy of a working fluid stream is expressed as follows:
E ̇ = m [(h − henv ) − Tenv (s − senv )]
(9)
A constant turbine efficiency model is employed in the conventional ORC system analysis process, but, as mentioned in the introduction, this assumption is not necessarily accurate. In this study, to investigate the influence of the turbine efficiency model selection on the ORC system performance, the constant turbine efficiency model is replaced by the variable turbine efficiency model to predict the turbine efficiency that is unique for each set of operating conditions and working fluids. Both ORC analysis procedures are presented in Fig. 3. As shown in Fig. 3, the calculation model of the radial-inflow turbine efficiency and the ORC system are coupled with each other. A one-dimensional assumption is adopted in the calculation model of the radial-inflow turbine efficiency. Fig. 4 shows the schematic diagram of the radial-inflow turbine (volute and diffuser are neglected) in the meridional plane. The simplified flow process in the radial-inflow turbine is described in Fig. 5. It can be found that the incoming working fluid sequentially expands in the nozzle (0–1) and rotor (1–2), and the
Based on the second law of thermodynamics, the exergy balance equations on the ORC system components can be expressed as:
EḊ = EḞ − EṖ − EL̇
(8)
2.2. Analysis and calculation of the turbine efficiency
Conceptually, the operating principles of the organic Rankine cycle are similar to the conventional steam Rankine cycle, except that it employs low boiling point organic materials as working fluids instead of steam. A basic ORC system consists of four main components, including a pump, an evaporator, a turbine (expander) and a condenser, as presented in Fig. 1. The liquid working fluid from the condenser is pressured by the pump and enters the evaporator, where it is heated by the waste flue gas and becomes saturated vapor. The saturated working fluid vapor is directed into the turbine and expands to generate the mechanical power of the shaft, which is used to drive the generator. Then, the vapor exhausted from the turbine is condensed to liquid by the condenser, and the liquid working fluid re-enters the pump. The corresponding T-s diagram of a basic ORC system is presented in Fig. 2. In this study, each component of the ORC system is considered to be a control volume. The equations that are used to calculate the heat absorbed or rejected by the organic working fluid and the power generation or consumption are based on the mass and energy conservations and are given in Table 1. In addition to these equations, the net power output and the thermal efficiency of the ORC system are calculated as follows:
(2)
i (1 + i)n (1 + i)n − 1
In this study, the cost per exergy unit of the product is selected as an optimization criterion for the ORC system.
2.1. Thermodynamic and exergoeconomic modelling
ηthe = Wnet / Qeva
(7)
where CRF is the capital recovery factor; ϕ represents the maintenance factor; and τ represents the annual plant operation hours (7000 h). The CRF is the function of the average annual interest rate, i (10%) and the plant economic life, n (20 years).
c p,total =
(1)
(6)
The cost balance equations and the corresponding auxiliary equations for each component of the ORC system are listed in Table 1. Ż represents the cost rate for each component of the ORC system, which includes the total capital investment (TCI) and the operation and maintenance (O&M) costs. The equations of TCI for different components are given in Table 1. The capital cost rate can be calculated as:
2. Modelling and analysis methods
Wnet = Wtur − Wpump
(5)
(4)
The exergoeconomic analysis method integrates the thermodynamic (exergy) aspect and economic principle. In this method, the exergy stream Ė is multiplied by an average cost per exergy unit c to convert it
Fig. 2. The T-s diagram of a basic ORC. 132
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Table 1 Thermodynamic and cost balance equations for each component of the ORC system. Components
Thermodynamic equations
Cost equations
Total capital investment
Wpump = mwf (h5 − h4 )
̇ pump + Zpump ̇ cWpump = cWtur C5̇ = C4̇ + CW
0.71 TCIpump = 422Wpump [1.41 + 1.41(
Qeva = mwf (h 0 − h5)
̇ c8 = c 7 C8̇ + C0̇ = C5̇ + C7̇ + Zeva
TCIeva = 10, 000 + 324A0.91
Wtur = mwf (h 0 − h2) = mwf (h 0 − h2s)·ηtur
̇ tur = C0̇ + Ztur ̇ C2̇ + CW c2 = c0
0.7 TCItur = 6000Wtur
Qcon = mwf (h2 − h4 )
̇ = C2̇ + C9̇ + Zcon ̇ c2 = c4 c9 = 0 C4̇ + C10
TCIcon = 10, 000 + 324A0.91
= mwf (h5s − h4 )/ ηpump
1 − 0.8 )] 1 − ηpump
Pump
Evaporator
Turbine
Condenser
enthalpy of the working fluid is converted into the mechanical power of the radial-inflow turbine. The initial state of the organic vapor (from the evaporator) at the radial-inflow turbine inlet is represented by point 0. In practical operations, due to the flow losses in the passages, the state of the organic vapor at the nozzle outlet changes from point 1s to point 1, and the state of the organic vapor at the rotor outlet changes from point 2s to point 2. Δhs is the total isentropic enthalpy drop of the organic working fluid across the entire radial-inflow turbine, and Δh is the total actual enthalpy drop. Fig. 6 shows the velocity triangles, which are used to express the distribution of absolute velocity, relative velocity and velocity angles in the nozzle and rotor. The peripheral efficiency of the radial-inflow turbine is determined from:
ηu =
(c12 − c22) + (u12 − u22) + (w22 − w12) C02
(10)
The relative velocity is defined as the ratio of each velocity to the ideal velocity, and the ideal velocity can be calculated as follows:
c0 =
2Δhs
(11)
Therefore, the peripheral efficiency of the radial-inflow turbine can be expressed as:
ηu = c¯12 − c¯22 + u¯ 12 − u¯ 22 + w¯ 12 − w¯ 22 = 2(¯c1u u¯1 − c¯2u u¯2)
Fig. 3. ORC system analysis procedure with constant and variable turbine efficiency. 133
(12)
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w¯ 12 = c¯12 + u¯ 12 − 2¯c1 u¯1 cos α1
(17)
c¯1 = φ 1 − ρ
(18)
Substitute Eqs. (17) and (18) into Eq. (16):
w¯ 2t =
ρ + φ2 (1 − ρ) + D¯ 22 u¯ 12 − 2u¯1 φ cos α1 1 − ρ
(19)
Therefore, the peripheral efficiency of the radial-inflow turbine can be calculated by a dimensionless equation:
ηu = 2u¯1 [φ cos α1 1 − ρ − D¯ 22 u¯1 + D¯ 2 ψ cos β2 ρ + φ2 (1 − ρ) + D¯ 22 u¯ 12 − 2u¯1 φ cos α1 1 − ρ ] (20) It can be determined that the peripheral efficiency of the radialinflow turbine is influenced by seven different dimensionless parameters. The peripheral velocity ratio u¯1 is the ratio of the rotor peripheral velocity to the ideal velocity, and the degree of reaction ρ is the ratio of the enthalpy drop in the rotor to the enthalpy drop in the entire radial-inflow turbine. The two parameters have a greater effect on the peripheral efficiency of the radial-inflow turbine [27]. The peripheral velocity ratio and degree of reaction are optimized using a Lagrange multiplier method with the peripheral efficiency of the radial-inflow turbine as the objective. The optimal peripheral velocity ratio and degree of reaction can be calculated as follows [28]:
Fig. 4. Schematic diagram of the radial-inflow turbine in the meridional plane.
u¯1opt =
ψ 2
cos β D¯ 22 ( 2 2 − ψ2 ) +
(1 − m2ψ4 ) φ2 cos2 α1
m
ρopt = 1 − [
(1
m2ψ2 (1 − φ2)
(21)
− mψ2) φ cos α1 u¯1 2 ] mψ2 (1 − φ2)
(22)
where m is a combined parameter that can be expressed as:
m=
1 [1 ± ψ2
D¯ 22 (1 − φ2)(1 − cos2 β2 ψ2) ] cos2 α1 φ2 + D¯ 22 ((1 − φ2)
(23)
From Eq. (16), w¯ 12 can be expressed as:
w¯ 12 = w¯ 2t2 − ρ + (u¯ 12 − u¯ 22)
(24)
Substitute Eqs. (18) and (24) into Eq. (12), another expression of the peripheral efficiency of the radial-inflow turbine can be presented as follows: Fig. 5. Flow process of the working fluid in the radial-inflow turbine.
ηu = 1 − (1 − ρ)(1 − φ2) − w¯ 2t2 (1 − ψ2) − c¯22 = 1 − ξ n − ξr − ξ e
(25)
w¯ 2t2 (1
ϕ2)(1
ψ2 )
c¯22
− − ρ ) , ξr = where ξn = (1 − and ξe = represent the nozzle loss coefficient, rotor loss coefficient and leaving velocity loss coefficient, respectively. In addition to the three types of aforementioned losses, two other types of losses (friction loss and leakage loss) need to be considered in the calculation of the turbine efficiency. The friction loss coefficient can be calculated as: ξf = f · Fig. 6. Velocity triangles of the radial-flow turbine.
D12 u1 3 1 1 ·( )· · ν1 100 1.36 m f ·Δhs
(26)
The leakage loss coefficient can be calculated as:
D u¯2 = 2 u¯1 = D¯ 2 u¯1 D1
(13)
c¯1u = c¯1 cos α1 = φ 1 − ρ cos α1
(14)
c¯2u = u¯2 − w¯ 2 cos β2 = D¯ 2 u¯1 − ψw¯ 2t cos β2 w¯ 2t =
ρ + w¯ 12 − u¯ 12 + D¯ 2 u¯ 12
δ
ξ1 =
⎧1 − 1.3· 1m , 0.015 < ⎨ 0.95 − 0.31· δ , 1m ⎩
δ 1m
δ 1m
⩽ 0.05
> 0.05
(27)
Differing from the other four loss coefficients, the leakage loss coefficient does not represent the ratio of the leakage loss to the isentropic enthalpy drop. Thus, the radial-inflow turbine efficiency can be expressed as:
(15)
ηtur = (ηu − ξf )·ξl = (1 − ξn − ξr − ξe − ξf )·ξl
(16) 134
(28)
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selection, crossover and mutation operators. The parent population and the offspring population are combined to form a new population Rt = Pt ∪ Qt, whose size is 2 N. (3) Since Rt includes all the members in the parent population and the offspring population, elitism is preserved. Non-dominated sorting is conducted for population Rt, then population Rt is classified into different non-dominated fronts F1, F2…, a higher ranking means a larger distance from the optimal solutions. (4) A new population Pt+1 with a size of N is created, and it is filled with members of different non-dominated fronts. The new population only has N slots, which is less than the population Qt size 2 N. The filling begins from the members of F1; if the number of members of F1 is less than N, the filling
An overview of the calculation model of the radial-inflow turbine efficiency is presented in Fig. 7. A real gas formulation is used in this model, and Matlab software is used to solve the calculation model. Since there are not sufficient experimental data for a radial-inflow turbine using organic working fluid, an air radial-inflow turbine is used to validate the calculation model of the radial-inflow turbine efficiency. Jones [29] performed rig test of a radial-inflow turbine with air as working fluid and drew the turbine efficiency characteristic curve (experimental curve of the turbine efficiency vs the peripheral velocity ratio), as shown in Fig. 8. The turbine efficiency characteristic curve obtained by the calculation model of the radial-inflow turbine efficiency was also presented in Fig. 8 to compare the calculation results with the experimental results. The turbine efficiency increases first and then decreases with the increasing peripheral velocity ratio, which is accord with Ref. [27]. In addition, it can be seen that the predicted turbine efficiency coincide with the experimental value well with a maximum error of 0.573%. Therefore, the radial-inflow turbine efficiency calculation model is considered accurate and can be used in this study. 2.3. Optimization model and simulation method The system thermal efficiency and the cost per exergy unit of the product are selected as objective functions to investigate the influence of the turbine efficiency model selection on the multi-objective optimization results of the ORC system. NSGA-II is used to maximize the thermal efficiency and minimize the cost per exergy unit of the product simultaneously. Multi-objective optimization model of the ORC system can be described as:
⎧ max(ηthe) = f1 (T0, T4 ) ⎨ ⎩ min(c p,total ) = f2 (T0, T4 )
(29)
The physical constraints for the multi-objective optimization problem are listed as follows: (1) The evaporation temperature should less than the heat source temperature and critical temperature of the working fluid, and the pinch point temperature difference in the evaporator should higher than 5 K.
⎧ T0 < T7 T0 < Tcr ⎨ ⎩ ΔTeva > 5
(30)
(2) The condensation temperature should less than the evaporation temperature and higher than the ambient temperature. In this study, the condensation temperature varies in the range of 313.15–333.15 K.
⎧ Tamb < T4 < T0 ⎨ ⎩313.15 < T4 < 333.15
(31)
NSGA-II is a computationally efficient multi-objective optimization algorithm, that was proposed by Deb et al. [30] on the basis of natural selection in the biological genetic process. Two special mechanisms, including non-dominated sorting and crowding distance, are used in NSGA-II. Such mechanisms are advantageous to ensure both the convergence and diversification of the population. NSGA-II has been widely used by many researchers in different research fields, including power system optimization [31,32], heat exchanger design [33], and green building optimization design [34,35]. The main procedures of the NSGA-II are summarized as follows: (1) Initial parent population P0 is created randomly with a size of N. To conveniently describe the procedures of the algorithm, the tth generation was used in the follow description. (2) The offspring population Qt is generated from the parent population Pt through binary tournament
Fig. 7. Flowchart of the turbine efficiency calculation. 135
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0.90 0.89
Turbine efficiency
0.88
increment for the CTORC system is larger than that for the VTORC system, resulting in the increasing rate of the system thermal efficiency for the CTORC system is higher than that for the VTORC system. For both the CTORC system and the VTORC system, the difference in the system thermal efficiency for different working fluids is small in the low evaporation temperature range, and the difference value gradually increases with the increasing evaporation temperature. In addition, compared with the CTORC system, the difference in the system thermal efficiency for different working fluids in the VTORC system is much higher at a high evaporation temperature. Among the six working fluids, R236ea always has the lowest thermal efficiency for the CTORC system, while for the VTORC system, the lowest thermal efficiency working fluids change from R236ea to ipentane when the evaporation temperature exceeds 376.15 K. The sequence order of some working fluids is also different between the CTORC system and the VTORC system. For the CTORC system, the sequence order is pentane, ipentane and R365mfc in terms of thermal efficiency, while the sequence order for the same three working fluids changes to R365mfc, pentane and ipentane for the VTORC system. The turbine efficiency with different working fluids at various evaporation temperatures is shown in Fig. 12. The turbine efficiency decreases with the increasing evaporation temperature. For different working fluids the predicted turbine efficiency is different, and the difference increases with the increment of the evaporation temperature. The radial-inflow turbine with R365mfc as working fluid has the largest turbine efficiency among the six working fluids, and the maximum turbine efficiency is 0.827 when the evaporation temperature is 353.15 K. The radial-inflow turbine with ipentane as working fluid has the smallest turbine efficiency, and the minimum turbine efficiency is 0.678 when the evaporation temperature is 412.15 K. The difference between the maximum turbine efficiency and the minimum turbine efficiency is 0.149, which indicates that using constant turbine efficiency for different working fluids and different operating conditions will lead to a serious error. To analyze and compare the differences between the CTORC system and the VTORC system, the system thermal efficiency difference is defined as follows:
Experimental value(Ref. [29]) Predicted turbine efficiency
0.87 0.86 0.85 0.84 0.83 0.82 0.81 0.80 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80 0.82
Velocity ratio Fig. 8. Validation of the radial-inflow turbine efficiency calculation model.
continued with the members of F2, F3…. Say that Fl is the last nondominated front that can be accommodated by population Pt+1. In general, the number of solutions in Fl is more than the remaining slots in the population Pt+1. Thus, a novel comparison operator called crowding distance was proposed to sort the solutions in Fl. In the same rank, the solutions with a larger crowding distance are better and are chosen prior to fill population Pt+1. (5) If the stopping conditions are not met, a new offspring population Qt+1 is created, and the procedure returns to step (2). The iteration ends once the stopping conditions are met. The basic procedures of NSGA-II are shown in Fig. 9. 3. Results and discussion 3.1. Simulation conditions and working fluids In this study, an ORC system utilizing the waste flue gas is investigated. The waste flue gas inlet temperature is assumed to be 433.15 K (a common temperature of the waste flue gas) [5,36]. To prevent the low-temperature corrosion, the waste flue gas outlet temperature should exceed the acid dew point, thus the waste flue gas outlet temperature is assumed to be 363.15 K [37,38]. Additionally, to compare the comprehensive performance of different working fluids more accurate, the utilized total waste heat is assumed to be constant at 1 MW [39]. The constant turbine efficiency is assumed to be 0.8 in this study [40,41]. Six organic working fluids, R114, pentane, R236ea, ipentane, R365mfc and hexane, were analyzed and compared between the CTORC and VTORC. Basic thermodynamic properties of the selected working fluids are given in Table 2.
Δηthe = ηthe, CTORC − ηthe, VTORC
(32)
The system thermal efficiency difference between the CTORC and the VTORC is presented in Fig. 13. For pentane and ipentane, the system thermal efficiency difference becomes higher with the increasing evaporation temperature, and the variation rate gradually increases. The reason for this outcome is that only in a low evaporation temperature range the turbine efficiency for pentane and ipentane are slightly higher than the constant turbine efficiency 0.8, and the remaining turbine efficiency for pentane and ipentane are lower than the constant turbine efficiency 0.8. Additionally, the higher the evaporation temperature is, the lower the turbine efficiency is, and the larger the deviation between the predicted turbine efficiency and the constant turbine efficiency is. Therefore, a higher evaporation temperature
3.2. Comparison of the CTORC system and the VTORC system results To investigate the influence of the turbine efficiency model selection on the ORC system, a thermodynamic analysis of both the CTORC system and the VTORC system with different working fluids was conducted at various operation conditions. The variation of the system thermal efficiency with the increasing evaporation temperature for the CTORC system and the VTORC system are presented in Figs. 10 and 11, respectively. The system thermal efficiency increases with the increment of the evaporation temperature for both the CTORC and the VTORC, and the increasing rate for both systems is significant different. For the CTORC system, the turbine efficiency is constant, and the enthalpy drop in the radial-inflow turbine increases with the increasing evaporation temperature. For the VTORC system, the turbine efficiency decreases with the increment of the evaporation temperature (just as presented in Fig. 12), reducing the increment of the enthalpy drop in the radial-inflow turbine. Therefore, the amount of net power output
Fig. 9. Basic procedures of NSGA-II. 136
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0.84
Table 2 Basic thermal properties of the selected working fluids.
0.82
M/g·mol−1
Tbolling/K
Pcritical/MPa
Tcritical/K
R114 Pentane R236ea Ipentane R365mfc Hexane
170.92 72.15 152.04 72.15 148.07 86.18
276.741 309.21 279.34 300.98 313.3 341.86
3.25 3.37 3.50 3.34 3.27 3.03
145.68 469.70 412.44 460.35 460.00 507.82
0.80
Turbine efficiency
Working fluids
13
0.76
R114 pentane R236ea ipentane R365mfc hexane
0.74 0.72 0.70 0.68
12
System thermal efficiency/%
0.78
0.66 350
360
370
11
390
400
410
420
Fig. 12. Turbine efficiency at different evaporation temperatures.
10
R114 pentane R236ea ipentane R365mfc hexane
8
360
370
380
390
400
410
2.5
System thermal efficiency difference/%
9
7 350
420
Evaporation temperature/K Fig. 10. Variation of system thermal efficiency with the evaporation temperature for the CTORC. 13
11
1.5
1.0
0.5
0.0
360
370
380
390
400
Evaporation temperature/K
410
420
Fig. 13. Variation of the system thermal efficiency difference with the evaporation temperature.
10
R114 pentane R236ea ipentane R365mfc hexane
8
360
370
380
390
400
410
11.0 10.5
System thermal efficiency/%
9
7 350
R114 pentane R236ea ipentane R365mfc hexane
2.0
-0.5 350
12
System thermal efficiency/%
380
Evaporation temperature/K
420
Evaporation temperature/K Fig. 11. Variation of system thermal efficiency with the evaporation temperature for the VTORC.
would lead to a larger system thermal efficiency difference between the CTORC and the VTORC. For R365mfc, R114, hexane and R236ea, when the evaporation temperature is low, the turbine efficiency is higher than the constant turbine efficiency 0.8, thus the CTORC system thermal efficiency is lower than the VTORC system thermal efficiency. As the evaporation temperature increases gradually, the system thermal efficiency difference decreases due to the decreasing turbine efficiency, as shown in Fig. 13. When the evaporation temperature reaches a value that makes the predicted turbine efficiency equal 0.8, the system thermal efficiency difference equals 0. With a further increasing evaporation temperature, the turbine efficiency continues to decrease, thus the CTORC system thermal efficiency is higher than the VTORC system thermal efficiency, and the system thermal efficiency difference begins
10.0 9.5 9.0
R114 pentane R236ea ipentane R365mfc hexane
8.5 8.0 7.5 7.0 6.5 312
314
316
318
320
322
324
326
328
330
332
334
Condensation temperature/K Fig. 14. Variation of system thermal efficiency with the condensation temperature for the CTORC.
to increase. Additionally, the higher the evaporation temperature is, the larger the system thermal efficiency difference is. Figs. 14 and 15 present the variation of the system thermal efficiency with the increasing condensation temperature for the CTORC and the VTORC, respectively. The system thermal efficiency decreases 137
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12
11.0
9.5
10
9.0 8.5
R114 pentane R236ea ipentane R365mfc hexane
8.0 7.5 7.0 6.5 312
Ideal point
11
10.0
System thermal efficiency/%
System thermal efficiency/%
10.5
314
316
318
320
322
324
326
328
330
332
334
R114 pentane R236ea ipentane R365mfc hexane
9
8
7 15
Condensation temperature/K
16
17
18
19
20
21
22
23
24
25
26
Cost per exergy of the product/$·GJ-1
Fig. 15. Variation of system thermal efficiency with the condensation temperature for the VTORC.
Fig. 18. Pareto optimal front for the CTORC system. 12
0.83 0.82
11
System thermal efficiency/%
0.80 0.79 0.78
R114 pentane R236ea ipentane R365mfc hexane
0.77 0.76 0.75 0.74 312
314
316
318
320
322
324
326
328
330
332
10
R114 pentane R236ea ipentane R365mfc hexane
9
8
7 15
334
Condensation temperature/K Fig. 16. Turbine efficiency at different condensation temperatures.
0.5 0.4 0.3
19
20
21
22
23
24
25
26
0.820
Turbine efficiency 0.815
0.2 0.1 0.0
-0.1
10.5 0.810
10.0 9.5
0.014
0.805
9.0 0.800
8.5
0.003 8.0
-0.2 -0.3 312
18
Cost per exergy of the product/$·GJ-1
Constant turbine efficiency Dynamic turbine efficeincy
11.0
System thermal efficiency/%
System thermal efficiency difference/%
11.5
R114 pentane R236ea ipentane R365mfc hexane
0.6
17
Fig. 19. Pareto optimal front for the VTORC system.
0.8 0.7
16
Turbine efficiency
Turbine efficiency
0.81
15
314
316
318
320
322
324
326
328
330
332
16
17
18
19
20
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Cost per exergy of the product/$·GJ
334
Condensation temperature/K
22
0.795 23
-1
Fig. 20. Comparison of the Pareto optimal front between the CTORC system and the VTORC system for R365mfc.
Fig. 17. Variation of system thermal efficiency difference with the condensation temperature.
VTORC decreased 2.58%. With the increasing condensation temperature, the enthalpy drop in the radial-inflow turbine decreases, while the turbine efficiency increases, as shown in Fig. 16, reducing the decrement of the enthalpy drop in the radial-inflow turbine. Therefore, the decrement of the net power output for the VTORC system is less than that for the CTORC, leading to the decreasing rate of the system thermal
linearly with the increasing condensation temperature for the CTORC and the VTORC, where the decreasing rate for the VTORC is lower than that for the CTORC. Taking ipentane as an example, in the investigated condensation temperature range the system thermal efficiency for the CTORC decreased 3.20%, while the system thermal efficiency for the 138
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11.0
0.78
0.035 0.077
0.76
10.5 10.0
0.74
9.5
Turbine efficiency
System thermal efficiency/%
11.5
390
0.80
Constant turbine efficiency Dynamic turbine efficeincy Turbine efficiency
Optimal evaporation temperature/K
12.0
0.72 9.0 8.5
16
17
18
19
20
Cost per exergy of the product/$·GJ-1
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22
0.70
CTORC system VTORC system 385
380
375
370
Optimal condensation temperature/K
Fig. 21. Comparison of the Pareto optimal front between the CTORC system and the VTORC system for R236ea.
Optimal evaporation temperature/K
410
CTORC system VTORC system
405 400 395 390 385 380
321.0
320.5
320.0 412
416
420
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432
436
440
444
448
Waste flue gas inlet temperature/K Fig. 23. Optimal evaporation temperature and condensation temperature at different waste flue gas inlet temperatures for R365mfc.
CTORC system VTORC system
323.0
0.82
322.5
0.014
322.0
0.80
0.010
0.014
Turbine efficiency
321.5 321.0 320.5 320.0 319.5 412
CTORC system VTORC system
321.5
375 370 323.5
Optimal condensation temperature/K
322.0
416
420
424
428
432
436
440
444
0.78
R365mfc R236ea
0.76
0.076 0.74
448
Waste flue gas inlet temperature/K
0.72
Fig. 22. Optimal evaporation temperature and condensation temperature at different waste flue gas inlet temperatures for R236ea.
412
416
420
424
428
432
436
440
444
448
Waste flue gas inlet temperature/K Fig. 24. Predicted turbine efficiency at different waste flue gas inlet temperature.
efficiency for the VTORC being lower than that for the CTORC. With the increasing condensation temperature, the difference in the system thermal efficiency for different working fluids decreases, and the variation of the difference in system thermal efficiency for different working fluids in the VTORC system is slightly more violent in the investigated condensation temperature range. When the condensation temperature increases from 313.15 K to 333.15 K, the difference in the system thermal efficiency between hexane and R236ea changes from 0.74 to 0.51 for the CTORC system, while the difference in the system thermal efficiency between hexane and R236ea changes from 1.00 to
0.57 for the VTORC system. The sequence order of some working fluids in terms of the system thermal efficiency for the CTORC system is pentane > ipentane > R365mfc, and the sequence order of the same working fluids changes to R365mfc > pentane > ipentane for the VTORC system. Fig. 16 shows the turbine efficiency with different working fluids at various condensation temperatures. As the condensation temperature 139
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are different, as shown in Fig. 21. The reason for this outcome is that the predicted turbine efficiency in the VTORC system is lower than the constant turbine efficiency 0.8, and the maximum deviation reaches 0.077, as shown in Fig. 21; in addition, the maximum relative deviation reaches 9.62%. The low predicted turbine efficiency leads to the difference in the Pareto optimal front between the CTORC system and the VTORC system, and the difference increases with the decreasing predicted turbine efficiency, as shown in Fig. 21. For the CTORC system, the turbine efficiency is the same for R365mfc and R236ea, and R236ea is better than R365mfc in terms of comprehensive performance. However, for the VTORC system, the predicted turbine efficiency of R365mfc is higher than that of R236ea, making R365mfc better than R236ea in terms of comprehensive performance. Additionally, it can also be found that the influence of the turbine efficiency model selection on the multi-objective optimization results of the ORC system is related to the working fluid properties; for different working fluids, the influences are different.
increases, the turbine efficiency increases linearly, and the difference of turbine efficiency among different working fluids decreases. Among the six working fluids, R365mfc has the largest turbine efficiency, and ipentane has the smallest turbine efficiency. A maximum turbine efficiency of 0.821 is obtained by R365mfc when the condensation temperature is 333.15 K, and a minimum turbine efficiency of 0.745 is obtained by ipentane when the condensation temperature is 313.15 K. The difference between the maximum turbine efficiency and the minimum turbine efficiency is 0.076. Fig. 17 presents the system thermal efficiency difference between the CTORC and the VTORC. For pentane and ipentane, with the increasing condensation temperature, the system thermal efficiency difference decreases, and the decreasing rate also gradually decreases. It is due to the fact that the turbine efficiency for pentane and ipentane is lower than the constant turbine efficiency 0.8, and as the condensation temperature increases, the turbine efficiency increases; in addition, the deviation between the predicted turbine efficiency and the constant turbine efficiency decreases. For hexane, R114 and R365mfc, with the increasing condensation temperature, the system thermal efficiency difference between the CTORC and the VTORC increases, while the increasing rate gradually decreases. The reason is that the turbine efficiency for hexane, R114 and R365mfc is higher than the constant turbine efficiency 0.8, and with the increasing condensation temperature, the turbine efficiency increases, and the deviation between the predicted turbine efficiency and the constant turbine efficiency increases. For R236ea, the turbine efficiency is lower than the constant turbine efficiency 0.8 in the low condensation temperature range, and with the increasing condensation temperature, the turbine efficiency gradually increases to over 0.8. Therefore, the system thermal efficiency difference between the CTORC and the VTORC decreases first and then increases. NSGA-II was employed to conduct multi-objective optimization of the CTORC system and the VTORC system based on the system thermal efficiency and the cost per exergy unit of the product. The Pareto optimal fronts for the CTORC system and the VTORC system are presented in Figs. 18 and 19, respectively. It is evident that there is a significant difference in the distribution of the Pareto optimal front between the CTORC system and the VTORC system. For all working fluids except for R365mfc, the distribution ranges of the system thermal efficiency of the CTORC are not only higher than that for the VTORC, but also wider than that for the VTORC. Taking R236ea as an example, the distribution range of the system thermal efficiency for the CTORC is from 9.11% to 11.77%, while the distribution range of the system thermal efficiency of the VTORC is from 8.66% to 10.53%. Based on the comprehensive performance, the sequence order of some working fluids for the CTORC system is different from that for the VTORC system. For the CTORC system, R236ea is better than R365mfc, and ipentane is better than hexane, while for the VTORC system, the working fluids sequence order is the opposite; that is, R365mfc is better than R236ea, and hexane is better than ipentane. The optimal working fluids for the CTORC system and the VTORC system are also different. For the CTORC system, R236ea is the optimal working fluid among the six working fluids. However, for the VTORC system, R365mfc is the optimal working fluid. To analyze the reason that lead to the optimal working fluids being different between the CTORC system and the VTORC system in detail, Figs. 20 and 21 show a comparison of the Pareto optimal fronts between the CTORC system and the VTORC system for R365mfc and R236ea, respectively. Additionally, the predicted turbine efficiency corresponding to the Pareto optimal fronts of the VTORC system are also presented in Figs. 20 and 21. As shown in Fig. 20, for R365mfc, the Pareto optimal front of the CTORC system and the VTORC system are essentially in coincidence. That is due to the fact that there is a small deviation between the predicted turbine efficiency in the VTORC system and the constant turbine efficiency 0.8. The maximum deviation between the predicted turbine efficiency and the constant turbine efficiency 0.8 is only 0.014, as shown in Fig. 20. However, for R236ea, the Pareto optimal fronts of the CTORC system and the VTORC system
3.3. Sensitivity analysis of the CTORC system and the VTORC system The waste flue gas inlet temperature varies with the factory and environmental conditions. In this section, sensitivity analysis of the CTORC system and the VTORC system were conducted to investigate the influence of the turbine efficiency model selection on the ORC system at different waste flue gas inlet temperatures. R365mfc and R236ea were selected as working fluids. Multi-objective optimization of the CTORC and the VTORC were conducted at each waste flue gas inlet temperature, and the aid of an ideal point method [40,42] was used to determine the final optimal solution. The process of decision-making for the aid of an ideal point method is shown in Fig. 18. A hypothetical point (ideal point) that maximizes the system thermal efficiency and simultaneously minimizes the cost per exergy of the product is assumed, and the point of the Pareto optimal front that is nearest to the ideal point is considered as the final optimal solution. Figs. 22 and 23 show the optimal evaporation temperature and condensation temperature at different waste flue gas inlet temperatures for R236ea and R365mfc, respectively. As shown in Fig. 22, for R236ea, the optimal evaporation temperature and the optimal condensation temperature for the CTORC system and the VTORC system increase with the increasing waste flue gas inlet temperature, and the increasing rate gradually decreases. It is apparent that the optimal evaporation temperature of the CTORC is always higher than that of the VTORC. Moreover, as the waste flue gas inlet temperature increases, the difference in the optimal evaporation temperature between the CTORC system and the VTORC system increases, indicating that the error caused by using constant turbine efficiency becomes larger with the increasing waste flue gas inlet temperature. The optimal condensation temperature of the CTORC system varies in the range of 320.0–321.5 K, while the optimal condensation temperature of the VTORC system varies in a slightly higher range of 321.0–323.5 K. The difference in the optimal condensation temperatures between the CTORC system and the VTORC system also increases with the increment of the waste flue gas inlet temperature. As shown in Fig. 24, for R236ea, with the increasing waste flue gas inlet temperature, the predicted turbine efficiency decreases and the deviation between the predicted turbine efficiency and the constant turbine efficiency 0.8 increases. Therefore, the error from using the constant turbine efficiency in an ORC system with R236ea as working fluid increases with the increment of the waste flue gas inlet temperature. As shown in Fig. 23, for R365mfc, as the waste flue gas inlet temperature increases, the optimal evaporation temperature and the optimal condensation temperature for the CTORC system and the VTORC system increase, and the increasing rate gradually increases. The optimal evaporation temperature of the CTORC system is lower than that of the VTORC system, while the optimal condensation temperature of the CTORC system is higher than that of the VTORC system in the low 140
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waste flue gas inlet temperature range. With the increasing waste flue gas inlet temperature, the difference in the optimal evaporation temperature and the difference in the optimal condensation temperature between the CTORC system and the VTORC system decreases. When the waste flue gas inlet temperature is higher than 435.9 K, the optimal evaporation temperature of the CTORC system is higher than that of the VTORC system, while the optimal condensation temperature of the CTORC system is lower than that of the VTORC system. With the increasing waste flue gas inlet temperature, both the difference in the optimal evaporation temperature and the difference in the optimal condensation temperature between the CTORC system and the VTORC system increase. The reason for the aforementioned results is that the predicted turbine efficiency of R365mfc is higher than the constant turbine efficiency 0.8 when the waste flue gas inlet temperature is low, as shown in Fig. 24. With the increasing waste flue gas inlet temperature, the predicted turbine efficiency decreases, and the deviation between the constant turbine efficiency 0.8 and the predicted turbine efficiency also decreases. When the waste flue gas inlet temperature is higher than 435.9 K, the predicted turbine efficiency is lower than the constant turbine efficiency 0.8. With the increasing waste flue gas inlet temperature, the deviation between the constant turbine efficiency 0.8 and the predicted turbine efficiency begins to increase. Therefore, the error from using constant turbine efficiency in an ORC system with R365mfc as working fluid increases first and then decreases with the increasing waster flue gas inlet temperature. Fig. 24 shows the predicted turbine efficiency of R236ea and R365mfc at different waste flue gas inlet temperatures. It is evident that the predicted turbine efficiency of R365mfc is higher that of R236ea, and the difference between the two predicted turbine efficiency increases in the investigated waste flue gas inlet temperature range. The predicted turbine efficiency of R236ea and R365mfc decreases with the increasing waste flue gas inlet temperature. The variation rate of the predicted turbine efficiency of R236ea gradually decreases, while the variation rate of the predicted turbine efficiency of R365mfc gradually increases. As shown in Figs. 22 and 23, the variation range of the optimal condensation temperature is obviously smaller than the variation range of the optimal evaporation temperature; thus, the optimal evaporation temperature is the main factor that influences the predicted turbine efficiency. For R236ea, the increasing rate of the optimal evaporation temperature decreases in the high waste flue gas inlet temperature range, thus the decreasing rate of the predicted turbine efficiency also decreases accordingly. However, for R365mfc, the increasing rate of the optimal evaporation temperature increases in the high waste flue gas inlet temperature range, so the decreasing rate of the predicted turbine efficiency also increases accordingly. It can also be found that the deviation between the constant turbine efficiency 0.8 and the predicted turbine efficiency of R365mfc decreases first and then increases with the increasing waste flue gas inlet temperature, and the maximum deviation value is 0.014. The deviation between the predicted turbine efficiency of R236ea and the constant turbine efficiency 0.8 increases with the increasing waste flue gas inlet temperature, and the maximum deviation value is 0.076. Therefore, it can be concluded that the error caused by using constant turbine efficiency is related to the working fluid properties, and the error for R365mfc is smaller than that for R236ea.
turbine efficiency model selection on the ORC system at different waste flue gas inlet temperatures. (1) When the constant turbine efficiency is replaced with the variable turbine efficiency, the variation rate of the system thermal efficiency with evaporation temperature or condensation temperature decreases, and the sequence order of some working fluids also changes. For different working fluids and different working conditions, the predicted turbine efficiency is quite different. (2) Different turbine efficiency models lead to different multi-objective optimal results for ORC system. The distribution of Pareto optimal fronts between the CTORC system and the VTORC is significantly different. R236ea is the optimal working fluid for CTORC system, while R365mfc is the optimal working fluid for VTORC system. (3) For different working fluids, the influence of turbine efficiency model selection on the ORC system is different. As the waste flue gas inlet temperature increases, the error caused by using constant turbine efficiency increases for R236ea, while the error caused by using constant turbine efficiency increases first and then decreases for R365mfc. The predicted turbine efficiency of both R236ea and R365mfc decreases with the increasing waste flue gas inlet temperature. Declaration of interests The authors declared that there is no conflict of interest. Acknowledgements 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] Nawi ZM, Kamarudin SK, Abdullah SS, Lam SS. The potential of exhaust waste heat recovery (WHR) from marine diesel engines via organic rankine cycle. Energy 2019;166:17–31. [2] Landelle A, Tauveron N, Haberschill P, Revellin R, Colasson S. Organic Rankine cycle design and performance comparison based on experimental database. Appl Energy 2017;204:1172–87. [3] Zhang Z, Li H, Chang H, Pan Z, Luo X. Machine learning predictive framework for CO2 thermodynamic properties in solution. J CO2 Util 2018;26:152–9. [4] Li P, Han Z, Jia X, Mei Z, Han X, Wang Z. Analysis and comparison on thermodynamic and economic performances of an organic Rankine cycle with constant and one-dimensional dynamic turbine efficiency. Energy Convers Manage 2019;180:665–79. [5] Mahmoudi A, Fazli M, Morad MR. A recent review of waste heat recovery by Organic Rankine Cycle. Appl Therm Eng 2018;143:660–75. [6] Mahmoudzadeh Andwari A, Pesiridis A, Karvountzis-Kontakiotis A, Esfahanian V. Hybrid electric vehicle performance with organic rankine cycle waste heat recovery system. Appl Sci 2017;7:437. [7] Xu G, Song G, Zhu X, Gao W, Li H, Quan Y. Performance evaluation of a direct vapor generation supercritical ORC system driven by linear Fresnel reflector solar concentrator. Appl Therm Eng 2015;80:196–204. [8] Quoilin S, Van Den Broek M, Declaye S, Dewallef P, Lemort V. Techno-economic survey of Organic Rankine Cycle (ORC) systems. Renew Sustain Energy Rev 2013;22:168–86. [9] Pethurajan V, Sivan S, Joy GC. Issues, comparisons, turbine selections and applications – an overview in organic Rankine cycle. Energy Convers Manage 2018;166:474–88. [10] Kazemi N, Samadi F. Thermodynamic, economic and thermo-economic optimization of a new proposed organic Rankine cycle for energy production from geothermal resources. Energy Convers Manage 2016;121:391–401. [11] Bademlioglu AH, Canbolat AS, Yamankaradeniz N, Kaynakli O. Investigation of parameters affecting Organic Rankine Cycle efficiency by using Taguchi and ANOVA methods. Appl Therm Eng 2018;145:221–8. [12] Javanshir A, Sarunac N. Thermodynamic analysis of a simple Organic Rankine Cycle. Energy 2017;118:85–96. [13] Zhang H, Guan X, Ding Y, Liu C. Emergy analysis of Organic Rankine Cycle (ORC) for waste heat power generation. J Clean Prod 2018;183:1207–15. [14] Garg P, Orosz MS. Economic optimization of Organic Rankine cycle with pure fluids and mixtures for waste heat and solar applications using particle swarm optimization method. Energy Convers Manage 2018;165:649–68. [15] Li YR, Du MT, Wu CM, Wu SY, Liu C, Xu JL. Economical evaluation and
4. Conclusions To investigate the influence of the turbine efficiency model selection on the organic Rankine cycle system, a comparative analysis on the system thermal efficiency and the multi-objective optimization results between the CTORC system and the VTORC system was conducted. Both the reasons for the difference of the system thermal efficiency and for the difference of the multi-objective optimization results were analyzed in detail. Sensitivity analysis of the CTORC system and the VTORC system were conducted to investigate the influence of the 141
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