Multi-objective optimization and improved analysis of an organic Rankine cycle coupled with the dynamic turbine efficiency model

Applied Thermal Engineering 150 (2019) 912–922

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Research Paper

Multi-objective optimization and improved analysis of an organic Rankine cycle coupled with the dynamic turbine efficiency model

T

Peng Li, Zhongkai Mei, Zhonghe Han , Xiaoqiang Jia, Lu Zhu, Shan Wang ⁎

Key Lab of Condition Monitoring and Control for Power Plant Equipment, School of Energy, Power and Mechanical, North China Electric Power University, Baoding 071000, People's Republic of China

HIGHLIGHTS

dynamic turbine efficiency model is combined with the multi-objective model. • The efficiency significantly affects the variation of net power output. • Turbine optimal working fluid is different for two types of optimization. • The quantitative analysis of turbine efficiency is employed. • Specific • Sensitivity analysis of the heat source temperature is conducted. ARTICLE INFO

ABSTRACT

Keywords: Organic Rankine cycle Multi-objective optimization Variable turbine efficiency Thermodynamic performance Economic factor

Considering the effect of variable turbine efficiency on organic Rankine cycle (ORC) system working fluid selection and parameters determination, a multi-objective optimization model which considered the thermodynamic performance and economic factors simultaneously was coupled with a dynamic turbine efficiency model. The Pareto optimal solutions of optimization with constant turbine efficiency and with variable turbine efficiency were compared, specific quantitative analysis of turbine efficiency on an ORC system is presented and sensitivity analysis of the heat source temperature was conducted. The results reveal that the variation of net power output between constant turbine efficiency and variable turbine efficiency is different, and the distribution of Pareto optimal solutions with constant turbine efficiency and with variable turbine efficiency is also different. With constant turbine efficiency, R245ca is the optimal working fluid. However, with variable turbine efficiency, R365mfc is the optimal working fluid. In specific quantitative analysis, it can be found that the difference in the optimization results is related to the working fluid properties. For R245fa, the variation of turbine efficiency significantly affects the distribution of the Pareto optimal solutions. However, for R365mfc, the variation tendency of turbine efficiency is gentle, which only leads to a small deviation in the optimization results.

1. Introduction In recent years, the increasing energy demand leads to the depletion of fossil fuels and to environmental pollution [1–4]. Considering the existing energy dilemma, the utilization of low-grade renewable energy such as solar energy [5,6], geothermal energy [7,8] and waste heat [9–11] is a feasible solution to achieve sustainable development. Owing to easy maintenance and high efficiency, the organic Rankine cycle (ORC) has received considerable attention. As for the problems about ORC optimization, working fluids selection is a hot topic. Wang et al. [12] built a thermodynamic model and

⁎

analysed nine kinds of pure working fluids for engine waste heat recovery. The results showed that in terms of the environment-friendly aspect, R245fa and R245ca are the optimal working fluids in a given heat source. Lai et al. [13] conducted thermodynamic analysis on alkane, aromatic and linear siloxane based equations of state BACKONE and PC-SAFT. They concluded that for the given working fluid candidates, the thermal efficiencies can achieve approximately seventy percent of the Carnot efficiency with internal heat recovery. Fernández et al. [14] analyzed regenerative ORC at high temperature and found that MM and MDM are the optimal candidates. Apart from the thermodynamic analysis above, researchers also considered

Corresponding author. E-mail address: [email protected] (Z. Han).

https://doi.org/10.1016/j.applthermaleng.2019.01.058 Received 9 May 2018; Received in revised form 2 January 2019; Accepted 20 January 2019 Available online 21 January 2019 1359-4311/ © 2019 Elsevier Ltd. All rights reserved.

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Nomenclature A Bo C c cp D D¯ 2 e f F G h hin hout iattacking l l¯1 m Nu P Pr Q q Re rfg s T U u v W w x xa

ΔT δ η ξ ρ ρl ρv τ ϕ φ ψ

heat transfer area, m2 boiling number capital cost, $ absolute velocity, m s−1 specific heat, J kg−1 K−1 diameter, m ratio of wheel diameter exergy, kJ s−1 friction loss coefficient factor mass velocity, kg m−2 s−1 enthalpy, kJ kg−1 heat transfer coefficient for the inside, W m−2 K−1 heat transfer coefficient for the outside, W m−2 K−1 attacking angle rotor blade height, m ratio of the rotor inlet blade height to rotor inlet diameter mass flow rate, kg s−1 Nusselt number pressure, kPa Prandtl number heat load, kW average imposed wall heat flus, W m−2 Reynolds number enthalpy of vaporization, J kg−1 entropy, kJ kg−1 temperature, K overall heat transfer coefficient, W m−2 K−1 peripheral velocity, m s−1 specific volume, m3 kg−1 power, kW relative velocity, m s−1 vapor quality velocity ratio

temperature difference, K tip clearance, m efficiency loss coefficient degree of reaction liquid density, kg m−3 vapor density, kg m−3 blockage factor relative velocity ratio nozzle velocity coefficient rotor velocity coefficient

Subscripts 0, 1, 2, 2s 3, 4, 5, 5s, 6, g1, g2, pp1, pp2 state points within the cycle amb ambient blade blade angle c condenser critical critical point e evaporator eq equivalent exg exergy f fluid g gas gs geometric structure h hub i ith parts in inlet m log mean max maximum min minimum net net p pump s isentropic T turbine u peripheral

Greek letters

Abbreviations

α β βch Δh

PER CEPCI ORC

absolute flow angle relative flow angle chevron angle enthalpy drop, kJ kg−1

thermoeconomic criteria. By calculating the capital cost of the ORC system, they evaluated the system performance of different working fluids. Mosaffa et al. [15] compared four ORC systems driven by liquefied natural gas (LNG) and geothermal water and found that a dualfluid system can achieve the highest net power output. Wang et al. [16] investigated an ORC system combined with a cement production line and evaluated working fluids according to the economic criteria. The results showed that R601 is the optimal working fluid in terms of economic performance. In addition to thermoeconomic criteria, exergoeconomic analysis has been carried out for an ORC system. ElEmam et al. [17] applied exergoeconomic criteria for a regenerative ORC system driven by geothermal water and concluded that the evaporator and the condenser account for most of the exergy destruction rates. The introduction of multi-objective optimization algorithm can improve the analysis of an ORC system and coordinate different evaluation criteria. Barbazza et al. [18] selected evaporation pressure, equipment geometry and minimum allowable temperature differences as decision variables and employed genetic algorithm to conduct multiobjective optimization. The result revealed that R1234yf and R1234ze

system total cost per unit net power output, $/kW chemical engineering plant cost index organic Rankine cycle

are the optimal working fluids. Pierobon et al. [19] considered thermal efficiency, total volume of the system and net present value as objective functions and designed shell and tube heat exchangers in detail. The result showed that cyclopentane is the optimal working fluid in terms of thermal efficiency and net present value. Imran et al. [20] built a multiobjective optimization model based on thermal efficiency and specific investment cost, and the Non-dominated Sorting Genetic Algorithm-II (NSGA-II) was adopted to solve the model. They found that the increase in superheat at the turbine inlet leads to a slight increase in thermal efficiency, while specific investment cost rises significantly. Ganjehkaviri et al. [21] chose system exergy efficiency and total cost rate of the system as objective functions, and an equation was derived to decide the optimized point. The results showed that the increase in turbine inlet pressure leads to the increase in total cost rate of the system, while the increase in turbine inlet pressure results in the increase in system exergy efficiency. Apart from the traditional ORC system, some research emphasis focused on the novel combined ORC system to achieve higher power output. Akrami et al. [22] proposed a multi-generation energy system to produce heating, cooling, electricity and hydrogen and conducted 913

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exergetic, energetic and exergoeconomic analysis to design the system. Ahmadi et al. [23] developed a novel system that can be applied to produce electricity, cooling, hydrogen hot water and fresh water. They assessed the system performance from both thermodynamic and environmental aspect and derived the equation about total cost rate and exergy efficiency. Wang et al. [24] presented a combined cooling, heating and power system, and built the multi-objective optimization model based on average useful output and total heat transfer area. They analysed the system performance under three different modes and obtained the optimal operating parameters. Zhang et al. [25] investigated a three-stage cycle system driven by LNG and low-temperature waste heat to achieve power output. They found that n-pentane is the optimal working fluid for the multi-stage system, and multi-objective optimization is more appropriate than single objective optimization in terms of engineering practice. In the conventional performance analysis of the ORC system, the turbine efficiency is specified as a constant. However, it is evident that the turbine efficiency is directly related to the working fluid properties and the system operating conditions. Considering that constant turbine efficiency is not necessarily accurate, Pan et al. [26] employed an internal efficiency analysis method to compute turbine efficiency under different evaporation temperatures. They found that it is irrational to assume a constant turbine isentropic efficiency for different working fluids and conditions. Song et al. [27] built a one-dimensional aerodynamic analysis model for the radial-inflow turbine and calculated the efficiency of the heat exchanger for different working fluids. The results showed that as the variation of turbine efficiency is considered, R123 is the optimal working fluid. Through empirical correlation, these papers could estimate turbine efficiency and evaluate system performance. Apparently, the optimization result with variable turbine efficiency is more precise than the optimization result with constant turbine efficiency. In addition, some researchers have conducted optimization design of a radial-inflow turbine based on a certain algorithm. Through the design of the radial-inflow turbine, they derived the optimal geometry parameters under different working conditions. Zhai et al. [28] optimized radial-inflow turbine efficiency by designing an iteration model that combined the variation of nozzle and rotor energy loss coefficients. They compared four kinds of working fluids and obtained the optimal parameters based on the genetic algorithm. The results showed that the increase in the heat source outlet temperature leads to the increase in thermal efficiency monotonously. Rahbar et al. [29] proposed an optimized modelling approach to carry out the design of the main component of a radial-inflow turbine. The results revealed that with the dynamic efficiency approach, the turbine efficiency between various working fluids is divergent. Bahadormanesh et al. [30] defined thermal efficiency and size parameter as objective functions and solved the optimization problem using the firefly algorithm. They found that tension and vibration constraints significantly affected decision-making. As introduced above, different working fluids and operating conditions lead to variation of turbine efficiency. In the current study, little research considers the influence of variable turbine efficiency on working fluid selection and parameters determination from both thermodynamic performance and economic factors. Exergy efficiency and system total cost per unit net power output were chosen as objective functions to investigate the influence of variable turbine efficiency on multi-objective optimization results. The dynamic turbine efficiency model was applied to replace the constant turbine efficiency, and the velocity ratio and degree of reaction were optimized in this model. R245fa, R114, R245ca, R236ea and R365mfc were specified as working fluid candidates. The multi-optimization results with variable turbine efficiency and with constant turbine efficiency were compared and analysed. Two working fluids R245fa and R365mfc were selected to analyse the difference in the optimization results between ORC system with constant turbine efficiency and with variable turbine efficiency.

0 G Generator Turbine 2

Waste flue gas Working fluid Cooling water Evaporator Condenser 5

Pump 4

Fig. 1. Schematic diagram of a basic ORC.

2. Analysis methods 2.1. Thermodynamic model As shown in Fig. 1, the basic ORC system contains four main components, including evaporator, turbine, condenser and pump. In the beginning, the working fluid absorbs the waste heat and converts from liquid to vapor in the evaporator. This paper focusses on basic ORC without superheat, which means that the working fluid is in a saturation state at the turbine inlet. The organic vapor expands in the turbine to generate mechanical work. Then, the exhaust vapor from the turbine is cooled by the cooling water in the condenser. Finally, the working fluid is pumped to return to the evaporator, and the basic ORC is completed. The T-s diagram is shown in Fig. 2. The heat transferred from the exhaust gas to the working fluid (5–0) can be calculated as

Qe = m f (h 0

(1)

h5)

In the turbine (0–2), the power produced can be expressed as

WT = mf (h 0

h2s)

T

= mf (h 0

(2)

h2)

In the condenser (2–4), the condensation heat can be computed as

T

Tg1

Tpp1 Tg2

Te

0

6

2

5

5s

4 Tw1

3

Tc Tpp2

2s Tw2

Fig. 2. T-s diagram of an basic ORC.

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Qc = m f (h2

(3)

h 4)

components (evaporator, condenser, turbine and pump). Through calculating the net power output, the pumping power and heat transfer area, the capital cost for each component can be expressed as

The power consumption of the pump (4–5) is

Wp = mf (h5s

h 4)/

p

= mf (h5

h 4)

(4)

CBM = Cb Fbm

(5)

lg Cb = K1 + K2 lg Z + K3 (lg Z)2

The net power output is expressed as

Wnet = WT

Wp

The basic cost for each component is given by

The system exergy efficiency for the ORC system is exg

=

Tamb (sin

sin = samb + cp,g (ln Tin

The coefficient Fp is given by

(7)

samb )

The coefficients in the above equations can be found in Table 1 in detail, and the capital cost for the ORC system can be estimated as

2.2. Economic model According to Refs. [31,32], the system total cost per unit net power output (PER) is selected as an economic criterion. Since variable turbine efficiency affects the system net power output, PER is more rationally than the system capital cost as an economic criterion. The heat exchanger area is calculated by the logarithmic mean temperature difference (LMTD) method. Plate heat exchangers are chosen as the evaporator and the condenser, and the heat exchange process can be expressed as:

6

(11)

Considering the dramatic variation of physical properties in the two-phase region, the working fluid enthalpy rise from the saturated fluid to the saturated vapor is divided evenly into N parts. For each section, the Nusselt number during the evaporation process can be calculated as

Nui = 1.926Re0.5Pr 0.33 Bo0.3 eq [1

x + x ( l / v )0.5]

(12)

The Boiling numbers can be expressed as

Boeq,i =

u

q (13)

Geq,i rfg

Geq,i = G [1

(21)

In the condenser, the Nusselt number for the single phase can be estimated as (15)

Nu i = 4.118Re0.4Pr1/3

(22)

=

hu hs

(23)

Considering the fact that velocity ratio xa and the degree of reaction ρ have a significant influence on the peripheral efficiency, the two parameters are optimized before the working fluid selection. In terms of optimization for peripheral efficiency, the limiting effect of attacking angle and relative velocity ratio are considered in this paper. The attacking angle iattacking is defined as the difference between the blade angle βblade and the relative velocity angle β1 at the rotor inlet, which can be expressed as

(14)

x + x ( l / v )0.5]

C2014 Wnet

According to Ref. [35], the radial-inflow turbine is appropriate for the ORC system with low efficiency levels owing to good performance and affordable price. To simplify analysis, the one-dimensional assumption and mean-line method were employed to conduct the preliminary design for the radial-inflow turbine. As shown in Fig. 3, the flow procedure for the working fluid in a radial-inflow turbine consists of two parts (in the nozzle and in the rotor), while expansion in the volute and in the diffuser are neglected. In the nozzle, the organic working fluid vapor expands from state point 0 to state point 1, with enthalpy decreasing and velocity increasing. Then, in the rotor, organic working fluid vapor continues to expand from state point 1 to state point 2, achieving power output. Δhs is the isentropic enthalpy drop of working fluid in the entire radial-inflow turbine, andΔhu is the actual enthalpy drop in which the energy of the leaving velocity is not included. Like the axial flow turbine, absolute velocity, relative velocity and velocity angles at the rotor inlet and outlet are computed according to the velocity triangle of the radial-inflow turbine, as shown in Fig. 4. Peripheral efficiency is the ratio of peripheral power (actual enthalpy drop Δhu, as shown in Fig. 3) to the isentropic power (isentropic enthalpy drop Δhs, as shown in Fig. 3).

(10)

ch 0.646 ) Re0.583Pr 0.33

C2014 = C1996 × CEPCI2014/ CEPCI1996

2.3. The dynamic turbine efficiency model

In the single phase of heat exchangers, the Nusselt number is calculated to evaluate the forced convection heat transfer coefficients, which can be expressed as

Nu = 0.724(

(20)

PER =

The overall heat transfer coefficient U consists of two parts (in the hot side and in the cold side), while the heat transfer resistance through the wall is neglected.

1 1 1 = + U h in hout

C1996 = CBM,e + CBM,c + CBM,p + CBM,T

The system total cost per unit net power output is given by

(9)

Q = UA Tm

(19)

lg Fp = C1 + C2 lg P + C3 (lg P)2

(8)

ln Tamb )

(18)

Fbm = B1 + B2 Fm Fp

(6)

hamb

(17)

The coefficient Fbm can be estimated as

Wnet mg ein

ein = h in

(16)

The capital cost of the basic ORC system contains four main Table 1 Coefficients required for the cost evaluation of each component [33,34]. Components

K1

K2

K3

C1

C2

C3

B1

B2

Fm

Fbm

Z

Heat exchanger Pump Turbine

3.2138 3.3892 3.5140

0.2688 0.0536 0.5890

0.0796 0.1538 0

0.0649 0.3935 0

0.0502 0.3957 0

0.01474 −0.0022 0

1.80 1.89 0

1.50 1.35 0

1.25 1.5 0

− − 3.50

A Wp WT

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h

0.80

P0

xamin,β

0.75

0

1

hs

1s

3

h2s

2

velocity ratio

P1

h1s

hu P2 h2

B

0.60

0.50 0.42

1 1

2

w1

c2

0.48

w2

min, gs

=

(D¯ 22

2l¯1

D¯ 22h )

·

x amax, =

(24)

According to Ref. [36], the increasing positive iattacking leads to the increasing energy loss and the increasing possibility of flow separation. Considering this point, the iattacking is restricted from −20° to 10°, while βblade is specified as 90°. Considering the influence of the attacking angle, the velocity ratio is constrained in the range as follows [37]

x a max, =

cos

1

1

(1

tan 1 ) tan 1,max

(25)

x a min, =

cos

1

1

(1

tan 1 ) tan 1,min

(26)

(27)

On one side, the minimum of the relative velocity ratio is restricted in terms of the flow condition at the hub of rotor and the restriction of geometric structure for the rotor. According to Ref. [37], if relative velocity at the hub of the rotor outlet (w2,h) is smaller than w1, the flow condition will be accompanied by increasing pressure and decreasing velocity, which should be avoided in the previous turbine design. Considering this point, the restriction of ϕ can be expressed as [37]

=

1+

2 (D ¯ 22

sin2

1

/sin2 (

c

0.63

0.66

1 2

·

sin

1

sin

2

·

v2 v1

(29)

2

1

D¯ 22 +

sin2 1 ( max 2 sin2 ( 1 1)

1

D¯ 22 +

sin2 1 ( min 2 sin2 ( 1 1)

2

1)

(30)

1)

(31)

= (1

2 )(1

)

(32)

The rotor loss coefficient is given by [26]

D¯ 22h) 1

0.60

As the restriction conditions above are considered, the relationship curve of velocity ratio and degree of reaction for the radial-inflow turbine using R245fa as the working fluid is presented in Fig. 5. The velocity ratio and degree of reaction can be selected from the shadow area. According to Ref. [38], the selection of the B point is better than the selection of A point in terms of peripheral efficiency. For the conventional Rankine cycle with radial-inflow turbine, the Mach at the rotor inlet should be less than a unit to avoid supersonic flow. However, for the ORC, the supersonic flow at the rotor inlet is permitted due to the special physical properties of organic working fluids, which means that the small degree of reaction can be selected. Therefore, the optimal point should be B point rather than A point. In addition, the optimization problem with the curved boundary can be solved by the genetic algorithm embedded in Matlab. As the turbine efficiency is selected as objective function, the result revels that the optimal point is extremely closet to B point. Therefore, the B point is selected as the final optimal result. Considering variable control, the velocity ratio and degree of reaction are selected uniformly for different working fluids to conduct working fluid selection, and the optimization flowchart of the turbine is shown in Fig. 6. The initial parameter for the radial inflow turbine is shown in Table 2. Three kinds of flow loss are considered in the calculation of peripheral efficiency. The nozzle loss coefficient is calculated as [26]

Apart from the limiting effect of the attacking angle, the relative velocity ratio is also considered. The relative velocity ratio ϕ is defined as the ratio of the relative velocity at the rotor outlet (w2) to the relative velocity at the rotor inlet (w1), which can be given by

min, h

0.57

The increase in the relative velocity ratio leads to the increase in energy loss in the turbine. Therefore, an appropriate upper boundary should be given. In terms of the boundary for the relative velocity ratio, the restriction condition for velocity ratio can be expressed as [37]

Fig. 4. Velocity triangle of a radial-flow turbine.

w2 w1

0.54

rational design, the restriction condition can be given by [37]

2

x amin, =

1

0.51

Fig. 5. The relationship curve of velocity ratio and degree of reaction for radialinflow turbine with R245fa.

u2

u1

0.45

degree of reaction

Fig. 3. Flow process of the working fluid in a radial-flow turbine.

c1

A 0.65

0.55

S

=

xamin,Φ

2s

2s

blade

xamax,Φh

0.70

h1

iattacking =

xamax,Φgs

xamax,β

1)

(28)

b

On the other side, the minimum of the relative velocity ratio is affected by the geometric structure. Considering the feasibility and

=

w22 1 ( 2 hs 2

1)

The leaving velocity loss coefficient can be calculated as [26] 916

(33)

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Start

Star

Set initial parameters and constrain conditions

set operating condition and logical bounds

Select xa, according to point B

Initialize population

Compute c1,u1 and the velocity triangle at section 1

Evaluate fitness fuction

Compute c2,u2 and the velocity triangle at section 2

Selection process

Compute ηu,ξf,ξ l

Rank and sort population

Cross over and mutation

Compute ηT

Create next generation

End

Calculate fitness function

Fig. 6. The dynamic turbine efficiency model for the radial-inflow turbine.

Compare fitness between current generation and previous generation

Table 2 Initial parameters for the radial-inflow turbine. Parameter

Symbol

Value

Nozzle velocity coefficient Rotor blade velocity coefficient Ratio of wheel diameter Absolute velocity angle at the rotor inlet Relative velocity angle at the rotor outlet

ϕ ψ Dr α1 β2

0.95 0.85 0.5 15 30

No

Is it the last generation? Yes Optimal solution

Fig. 7. Flow chart of multi-objective genetic algorithm.

c2 = 2 2 hs

B

2.5. Optimization algorithm

(34)

Therefore, the peripheral efficiency can be expressed as u

=1

c

b

The multi-objective genetic algorithm (MOGA) embedded in Matlab is selected to solve the multi-objective optimization model. The MOGA is a random search algorithm based on evolutionary process in a natural system, which is convenient to solve non-linear problems. The flow chart of MOGA is shown in Fig. 7.

(35)

B

The friction loss coefficient and the leakage loss coefficient can be given by [26,36]

= f·

f

l

D12 1

·(

u1 3 1 1 )· · 100 1.36 m f · h s

D = 0.47 (1 + 2 ) l2 l2

l2

= 0.01

3. Working fluids and conditions

(36)

0.20

The heat source inlet temperature is set to 423.15 K, and the outlet temperature is 363.15 K. The recovery heat of the exhaust gas is 1.2 MW, which means that the mass flow rate of the heat source can be calculated. The water at ambient temperature (293.15 K) is used as cooling source. The isentropic efficiency for the cycle pump is 0.7. For the basic ORC system without superheat, dry or isentropic working fluids are adopted. Considering the environmental friendly and safety aspect, R245fa, R114, R245ca, R236ea and R365mfc were selected as working fluid candidates, and the relative physical properties are shown in Table 3.

(37)

The turbine efficiency can be expressed as T

=

u

f

l

(38)

2.4. Multi-optimization model In this paper, exergy efficiency and system total cost per unit net power output were selected as objective functions to evaluate working fluids, and multi-objective genetic algorithm was adopted to maximize exergy efficiency and minimize system total cost per unit net power output. Evaporation temperature and condensation temperature were chosen as decision variables, and the pinch point temperature in the evaporator and condenser is higher than 5 K. The specific constrain conditions can be expressed as

T0 < Tcritical, T0 < Tg1, Te > 5, Tc > 5 303.15 < T4 < 323.15

Table 3 Physical parameters of the working fluid candidates.

(39) 917

Working fluid

Critical pressure/ MPa

Critical temperature/K

Molecular weight/g mol−1

Dry or isentropic

R245fa R114 R245ca R236ea R365mfc

3.65 3.25 3.92 3.50 3.27

427 418 447 412 460

134 170 134 152 148

isentropic Dry Dry Dry Dry

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0.20

Friction loss coefficient

Turbine efficiency

0.87 0.86 0.85 0.84 0.83

0.18

0.12

0.008

0.08

0.004

365

370

375

380

385

0.000 395

390

Evaporation temperature/ K

temperature is, the bigger the error for optimization with constant turbine efficiency is. As the evaporation temperature varies from 363.15 K to 393.15 K, the variation of turbine efficiency is in the range of 6–11%, which will significantly influence the performance of the ORC system. The variation value for turbine efficiency is influenced by the type of working fluid, which also means that working fluid selection with constant turbine efficiency would lead to certain error. When the turbine efficiency is derived from empirical correlation, optimization with variable turbine efficiency is more precise than with constant turbine efficiency. In order to further analyse the loss coefficients of turbine efficiency, Fig. 10 shows the effect of evaporation temperature on the friction loss coefficient and the leakage loss coefficient. It can be found that the friction loss is much larger than the leakage loss, which is accord with the results of Ref. [40]. R245fa has the largest friction loss coefficient and the leakage loss coefficient, thus its turbine efficiency is the lowest. As the evaporation temperature increases, the increase in the friction loss is more violent than the increase in the leakage loss. For various working fluids, the variation of friction loss coefficient is from 5% to 10%, while the variation of the leakage loss coefficient is from 0.5% to 1%. Figs. 11 and 12 show the difference in the net power output between constant turbine efficiency and variable turbine efficiency. As the turbine efficiency is constant, the increase in evaporation temperature leads to the linear increase in net power output. However, it is not maintained as the variation turbine efficiency is considered, the increasing trend gradually decreases with the increasing evaporation temperature. The net power output is related to the varying tendency of

4.1. Validation of the dynamic turbine efficiency model There are rarely experimental data about the radial-inflow turbine using organic working fluid in the current literatures. Therefore, air was selected as working fluid for a radial-inflow turbine to validate the dynamic turbine efficiency model. Fig. 8 shows the comparison of the simulation results of dynamic turbine efficiency model and the reference experimental data (in literature [39]). As shown Fig. 8, the simulation results of dynamic turbine efficiency model are basically in line with the reference experimental data, and the maximum error is less than 1%. Therefore, the dynamic turbine efficiency model can be used to calculate the turbine efficiency that is used to replace the constant turbine efficiency in the ORC system. 4.2. Analysis of the variable turbine efficiency With the condensation temperature at 313.15 K, the increase in the evaporation temperature leads to the decrease in turbine efficiency, as shown in Fig. 9. According to Ref. [32], R245fa is a suitable working fluid for the basic ORC system. However, as the variable turbine efficiency is considered, the turbine efficiency for R245fa is the lowest among the six working fluids, while the turbine efficiency for R365mfc is the highest. As the evaporation temperature increases, the downtrend of the turbine efficiency gradually accelerates, which means that with a certain heat source temperature, the higher the evaporation

150

0.82 0.80

140

Net power output/ kW

0.78

Turbine efficiency

0.012

0.10

Fig. 10. The variations in friction loss coefficient and leakage loss coefficient with increasing evaporation temperature.

4. Results and discussion

0.76 0.74

0.66 360

0.016

0.14

0.02 360

Fig. 8. Comparison of the simulation results of turbine efficiency and the reference experimental data.

0.68

0.020

0.16

0.04

Velocity ratio

0.70

0.024

0.06

Reference experimental data (Ref. [39]) Simulation results of turbine efficiency

0.82 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

0.72

0.028

R245fa R114 R245ca R236ea R365mfc

0.22

Leakage loss coefficient

0.88

R245fa R114 R245ca R236ea R365mfc

R245fa R114 R245ca R236ea R365mfc

130

120

110

100 365

370

375

380

385

Evaporation temperature/ K

390

360

395

Fig. 9. The variations in turbine efficiency with increasing evaporation temperature.

365

370

375

380

385

Evaporation temperature/ K

390

395

Fig. 11. The effect of evaporation temperature on net power output with constant turbine efficiency at 0.8. 918

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120

110

0.075

0.014

R245ca R236ea R365mfc

0.012 0.010 0.008

0.060

370

375

380

385

Evaporation temperature/ K

390

0.006

R245fa R114 R245ca

0.004 304

395

306

308

310

0.002 314

312

Condensation temperature/ K

Fig. 14. The variations in friction loss coefficient and leakage loss coefficient with increasing condensation temperature.

The variation in the turbine efficiency is less than 3% in the investigated condensation temperature. 4.3. Comparison of multi-optimization results To analyse the impact of variable turbine efficiency on multi-objective optimization results, Figs. 15 and 16 show the comparison between optimization results with constant turbine efficiency and with variable turbine efficiency. According to Refs. [41,42], the constant turbine efficiency is selected to be 0.8 as a common choice. In the decision-making process, the point with minimum PER and maximum exergy efficiency is considered the ideal point. For each working fluid, the point that is the closest to the ideal point is selected as the optimal point, taking R245ca as an example, the decision-making process is presented in Fig. 16. For the same working fluid, the distribution of Pareto optimal solutions with constant turbine efficiency and with variable turbine efficiency is different. When the constant turbine efficiency is replaced with variable turbine efficiency, the PER in the Pareto optimal solutions becomes higher and the exergy efficiency becomes lower. According to Fig. 15, R245ca has the lowest PER and the highest exergy efficiency among the five working fluids, therefore it is the optimal working fluid. However, as the variable turbine efficiency is considered, the optimal working fluid becomes R365mfc. The reason for the difference between the two optimization results is that the turbine efficiency for R365mfc is higher than the turbine efficiency for R245ca. Therefore, taking the constant turbine efficiency may lead to deviation about working fluid selection. 6600

R236ea R365mfc

R245fa R114 R245ca R236ea R365mfc

6400 6200

0.78

6000

PER $/kW

Turbine efficiency

R245fa R114

0.090

0.030 302

365

turbine efficiency, which relates to the physical properties of working fluids. For different working fluids, the difference in the variation of net power output between constant turbine efficiency and variable turbine efficiency is different. With constant turbine efficiency, the net power output for R245fa is higher than the net power output for R114. While with variable turbine efficiency, when the evaporation temperature is lower than 375 K, the net power output for R114 is close to the net power out put for R245fa. Owing the turbine efficiency for R114 is higher than the turbine efficiency for R245fa, the net power output for R114 increases faster than the net power output for R245fa, and the difference in net power output between R114 and R245fa gradually increases. Because of the low turbine efficiency, as the evaporation temperature increases, the net power output for R245fa basically remains stable after 385 K. Because the turbine efficiency for R114 is higher than the turbine efficiency for R245ca, the two curves intersect in the high evaporation temperature range. Therefore, as the variation of turbine efficiency is considered, the net power output for different working fluids varies significantly, which affects working fluid selection and system parameter optimization. With the evaporation temperature at 373.15 K, the increase in the condensation temperature leads to the increase in turbine efficiency, as shown in Fig. 13. The turbine outlet pressure increases with the increase in the condensation temperature, which causes the reduction of the friction loss coefficient, as shown in Fig. 14. The leakage loss coefficient basically remains stable with increasing condensation temperature. Compared to the evaporation temperature, the influence of the condensation temperature for the turbine efficiency is rather small.

0.79

0.016

0.105

0.045

Fig. 12. The effect of evaporation temperature on net power output with variable turbine efficiency.

0.80

0.018

0.120

130

100 360

0.020

0.135

Friction loss coefficient

Net power output/ kW

140

0.022

0.150

R245fa R114 R245ca R236ea R365mfc

Leakage loss coefficient

150

0.77 0.76

5800 5600

0.75

5400

0.74

5200

0.73 302

5000 0.44

304

306

308

310

Condensation temperature/ K

312

314

Fig. 13. The variations in turbine efficiency with increasing condensation temperature.

Optimal point

Ideal point 0.45

0.46

0.47

0.48

0.49

0.50

Exergy efficiency

0.51

0.52

0.53

0.54

Fig. 15. The Pareto optimal solutions of working fluids with constant turbine efficiency. 919

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6800 6600

0.82 0.80

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Exergy efficiency

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PER $/kW

0.56

R245fa R114 R245ca R236ea R365mfc

6200 6000 5800 5600

0.52

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CTE VTE TE

0.76 0.74 0.72

0.50

0.70 0.48

0.68 0.66

0.46

5400

0.64

5200 0.40

0.42

0.44

0.46

Exergy efficiency

0.48

0.50

0.44 362

0.52

In the basic ORC system, the evaporation temperature is a key parameter to affect system performance. Comparing Fig. 9 with Fig. 13, it is evaporation temperature rather than condensation temperature that dominates the variation in turbine efficiency. Therefore, the condensation temperature is assumed to be 303.15 K to investigate the impact of the evaporation temperature on the system performance with variable turbine efficiency. In addition, the optimization result with constant turbine efficiency is also presented to compare the difference, as shown in Figs. 17–19. For various working fluids, the difference in the physical properties leads to the difference in the influence of variable turbine efficiency on the ORC system. To investigate the influence of variable turbine efficiency intuitively, the working fluids with the most violent (R245fa) and the most gentle (R365mfc) changing degree in turbine efficiency were selected to analyse the effect of variable turbine efficiency on system parameters. As shown in Fig. 17, the constant turbine efficiency is assumed to be 0.8, which is represented by a red solid line. Because only a single parameter (evaporation temperature) varies, the variation of exergy efficiency and turbine efficiency is nearly monotonic. CTE represents the optimization results with constant turbine efficiency, while VTE stands for the optimization results with variable turbine efficiency. TE1 is the turbine efficiency corresponding to the evaporation temperature in the Pareto optimal solutions, while TE2 is calculated individually to supplement the variation tendency after the evaporation temperature is over 385 K for vividly comparison to the constant turbine efficiency. 0.82 0.80

Exergy efficiency

VTE TE1 TE2

0.48

0.78 0.76 0.74 0.72

0.45

0.70 0.68

0.42

Turbine efficiency

CTE

0.51

0.66 0.64 365

370

375

380

385

Evaporation temperature/K

390

366

368

370

372

374

Evaporation temperature/ K

376

378

0.62 380

For R245fa, the variation in the turbine efficiency affects the distribution of the Pareto optimal solutions. There is a common range for the distribution of optimal evaporation temperature between constant turbine efficiency and variable turbine efficiency, which is approximately from 366 K to 384 K. The exergy efficiency corresponding to the common range is highlighted with a blue rectangle. For CTE, the increase in the evaporation temperature leads to a linear increase in the exergy efficiency. With respect to VTE, the increasing tendency of exergy efficiency gradually slows down, especially when the evaporation temperature is higher than 375 K. The distribution of the optimal evaporation temperature with VTE is located in a relatively lower range compared to the distribution of the optimal evaporation temperature with CTE. In the Pareto optimal solutions, the biggest difference between constant turbine efficiency and the variable turbine efficiency is approximately 11%, which has been highlighted with a red arrow, as shown in Fig. 17. As shown in TE2, if the evaporation temperature reaches 395 K, the difference can increase to 16%, which is extremely detrimental to system performance. Therefore, in the optimization process, the consideration of variable turbine efficiency can modify the error and increase the reliability. Fig. 18 shows the optimal results for R365mfc in the Pareto optimal solutions. Different from R245fa, the biggest difference between constant turbine efficiency and the variable turbine efficiency is only 3%. Therefore, the Pareto optimal solutions with variable turbine efficiency and with constant turbine efficiency almost share the same distribution range of the optimal evaporation temperature. The difference between the two optimal results is very small. It can be concluded that for certain working fluid, the gentle variation tendency of turbine efficiency leads to a small deviation in the system parameters. Fig. 19 shows the optimal results of PER for R245fa and R365mfc. Because PER is related to net power output, the variation turbine efficiency directly influences the distribution of PER. For R365mfc, as the evaporation temperature is lower than 368 K, PER for CTE and VTE is similar. While the evaporation temperature is higher than 368 K, the deviation between the two PER becomes slightly larger. For R245fa, the significant deviation between CTE and VTE leads to a big difference between PER for CTE and VTE. When the working fluids with the most violent (R245fa) and the most gentle (R365mfc) changing degree in turbine efficiency are compared, it can be concluded that the influence of turbine efficiency on an ORC system performance is related to the working fluid properties.

4.4. Specific quantitative analysis of turbine efficiency in optimization progress

0.54

364

Fig. 18. The optimal results for R365mfc in the Pareto optimal solutions.

Fig. 16. The Pareto optimal solutions of working fluids with variable turbine efficiency.

0.39 360

Turbine efficiency

7000

4.5. Sensitivity analysis of the heat source temperature

0.62 395

The heat source temperature is related to factory and environmental conditions. Two representative working fluids (R245fa and R365mfc) were selected to investigate the influence of variable turbine efficiency

Fig. 17. The optimal results for R245fa in the Pareto optimal solutions. 920

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PER $/kW

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380

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R365mfc with CTE R365mfc with VTE R245fa with CTE R245fa with VTE

PER $/kW

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R365mfc R245fa

376 374 372 370 368

5200 4800 362 364 366 368 370 372 374 376 378 380 382 384 386 388

366 400

Evaporation temperature/ K

Fig. 19. The optimal results of PER in the Pareto optimal solutions for R245fa and R365mfc.

405

410

415

420

425

Heat source temperature/ K

430

435

Fig. 21. Variation of evaporation temperature with increasing heat source temperature for R245fa and R365mfc.

0.80

0.52

0.79

0.50

0.77 0.76

R365mfc R245fa

Exergy efficiency

Turbine efficiency

0.78

0.75 0.74

R365mfc R245fa

0.48

0.46

0.44

0.73 0.72 400

405

410

415

420

425

Heat source temperature/ K

430

0.42 400

435

Fig. 20. Variation of turbine efficiency with increasing heat source temperature for R245fa and R365mfc.

405

410

415

420

425

Heat source temperature/ K

430

435

Fig. 22. Variation of exergy efficiency with increasing heat source temperature for R245fa and R365mfc. 6600

on system off-design performance. The optimal parameters were determined by the ideal point method. Fig. 20 shows variation of turbine efficiency with the increasing heat source temperature for R245fa and R365mfc. The turbine efficiency for R365mfc is obviously greater than the turbine efficiency for R245fa. Both of the turbine efficiency for R365mfc and for R245fa decrease with the increment in heat source temperature, and the decreasing value is about 3%. Fig. 21 shows the variation of optimal evaporation temperature with the increasing heat source temperature. The increase in the heat source temperature leads to a nearly linear increase in the optimal evaporation temperature for R365mfc. However, the low turbine efficiency obstructs the selection of a higher optimal evaporation temperature for R245fa, thus the increasing tendency of evaporation temperature gradually slows down. Figs. 22 and 23 show the effect of heat source temperature on exergy efficiency and PER. With the increasing heat source temperature, both the optimal exergy efficiency and the optimal PER decrease. The decreasing tendency of exergy efficiency for R365mfc is much gentler than the decreasing tendency of exergy efficiency for R245fa. The variation in the exergy efficiency for R365mfc is within 2.5% and the variation in the exergy efficiency for R245fa is within 4%. The increase in the heat source temperature leads to a nearly linear decrease in PER for R365mfc. For R245fa, as the heat source temperature is lower than 423.15 K, the decrease in PER is linear. When the heat source temperature is higher than 423.15 K, the decreasing tendency of PER slows down.

6400

PER $/kW

6200 6000

R365mfc R245fa

5800 5600 5400 400

405

410

415

420

425

Heat source temperature/ K

430

435

Fig. 23. Variation of PER with increasing heat source temperature for R245fa and R365mfc.

5. Conclusions This paper focusses on the influence of variable turbine efficiency on working fluid selection and parameters determination from both thermodynamic performance and economic factors. The dynamic turbine efficiency model is combined with a multi-objective model to analyse the difference between optimization with variable turbine 921

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efficiency and optimization with constant turbine efficiency. As the variation turbine efficiency is considered, the differences among various working fluids are compared, and the Pareto optimal solutions are analysed. The reason that leads to the difference in the optimization results between ORC system with constant turbine efficiency and ORC system with variable turbine efficiency is analysed. And sensitivity analysis for heat source temperature is also conducted. The main conclusions are summarized below:

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(1) As the evaporation temperature varies from 363.15 K to 393.15 K, the variation in the turbine efficiency is in the range of 6–11%. With condensation temperature increasing, the variation in the turbine efficiency is less than 3%. As the variation turbine efficiency is considered, the net power output for different working fluids varies significantly, which affects working fluid selection and system parameter determination. (2) The distribution of Pareto optimal solutions with constant turbine efficiency and with variable turbine efficiency is different. With constant turbine efficiency, R245ca is the optimal working fluid. However, R365mfc is the optimal working fluid for ORC system with variable turbine efficiency. (3) The influence of variable turbine efficiency on an ORC system performance is related to the type of working fluid properties. For R245fa, the variation turbine efficiency significantly affects the distribution of the Pareto optimal solutions. While for R365mfc, the variation tendency of turbine efficiency is gentle, thus the deviation of system parameters is small. (4) In the sensitivity analysis, the increase in the heat source temperature leads to a nearly linear increase in the optimal evaporation temperature for R365mfc, while the increasing tendency of the evaporation temperature gradually slows down for R245fa. Acknowledgments This work was supported by the National Natural Science Foundation of China (NO. 51306059) and the Fundamental Research Funds for the Central Universities in China (NO. 2017XS120). References [1] K.G. Nithesh, D. Chatterjee, C. Oh, Y. Lee, Design and performance analysis of radial-inflow turboexpander for OTEC application, Renew. Energy 85 (2016) 834–843. [2] P. Li, Z. Han, X. Jia, Z. Mei, X. Han, Z. Wang, Analysis and comparison on thermodynamic and economic performances of an organic Rankine cycle with constant and one-dimensional dynamic turbine efficiency, Energy Convers. Manage. 180 (2019) 665–679. [3] T.C. Hung, T.Y. Shai, S.K. Wang, A review of organic rankine cycles for the recovery of low-grade waste heat, Energy 22 (1997) 661–667. [4] T.C. Hung, S.K. Wang, C.H. Kuo, B.S. Pei, K.F. Tsai, A study of organic working fluids on system efficiency of an ORC using low-grade energy sources, Energy 35 (2010) 1403–1411. [5] B.F. Tchanche, G. Papadakis, G. Lambrinos, A. Frangoudakis, Fluid selection for a low temperature solar organic Rankine cycle, Appl. Therm. Eng. 29 (2009) 2468–2476. [6] A.M. Delgadotorres, L. Garcíarodríguez, Analysis and optimization of the lowtemperature solar organic Rankine cycle (ORC), Energy Convers. Manage. 51 (2010) 2846–2856. [7] H.D.M. Hettiarachchi, M. Golubovic, W.M. Worek, Y. Ikegami, Optimum design criteria for an organic Rankine cycle using low-temperature geothermal heat sources, Energy 32 (2007) 1698–1706. [8] G. Cammarata, L. Cammarata, G. Petrone, Thermodynamic analysis of ORC for energy production from geothermal resources, Energy Proc. 45 (2014) 1337–1343. [9] G. Yu, G. Shu, H. Tian, Y. Huo, W. Zhu, Experimental investigations on a cascaded steam-/organic-Rankine-cycle (RC/ORC) system for waste heat recovery (WHR) from diesel engine, Energy Convers. Manage. 129 (2016) 43–51. [10] Z. Han, Z.K. Mei, P. Li, Multi-objective optimization and sensitivity analysis of an organic Rankine cycle coupled with a one-dimensional radial-inflow turbine efficiency prediction model, Energy Convers. Manage. 166 (2018) 37–47. [11] Y. Dai, J. Wang, G. Lin, Parametric optimization and comparative study of organic Rankine cycle (ORC) for low grade waste heat recovery, Energy Convers. Manage.

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