Sensitivity analysis and thermoeconomic comparison of ORCs (organic Rankine cycles) for low temperature waste heat recovery

Sensitivity analysis and thermoeconomic comparison of ORCs (organic Rankine cycles) for low temperature waste heat recovery

Energy xxx (2015) 1e14 Contents lists available at ScienceDirect Energy journal homepage: www.elsevier.com/locate/energy Sensitivity analysis and t...

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Energy xxx (2015) 1e14

Contents lists available at ScienceDirect

Energy journal homepage: www.elsevier.com/locate/energy

Sensitivity analysis and thermoeconomic comparison of ORCs (organic Rankine cycles) for low temperature waste heat recovery Yongqiang Feng a, Yaning Zhang a, Bingxi Li a, *, Jinfu Yang b, Yang Shi a a b

School of Energy Science and Engineering, Harbin Institute of Technology, Harbin, China Institute of Engineering Thermo-physics Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 12 October 2014 Received in revised form 13 January 2015 Accepted 24 January 2015 Available online xxx

The sensitivity analysis for low temperature ORCs (organic Rankine cycles), as well as the thermoeconomic comparison between the basic ORC and regenerative ORC using Non-dominated sorting genetic algorithm-II (NSGA-II), are conducted in this paper. The derivatives of five system parameters on system performance are used to evaluate the parametric sensitiveness. The exergy efficiency and the APR (heat exchanger area per unit net power output) are selected as the objective functions for multi-objective optimization using R123 under the low temperature heat source of 423 K. The Pareto frontier solution with bi-objective for maximizing exergy efficiency and minimizing APR is obtained and compared with the corresponding single-objective solutions. The results indicate that the prior consideration of improving thermal efficiency and exergy efficiency is to increase the evaporator outlet temperature. A fitting curve can be yielded from the Pareto frontier between the thermodynamic performance and economic factor. The optimum exergy efficiency and APR of the regenerative ORC obtained from the Pareto-optimal solution are 59.93% and 3.07 m2/kW, which are 8.10% higher and 15.89% lower than that of the basic ORC, respectively. The Pareto optimization compromises the thermodynamic performance and economic factor, therefore being more suitable for decision making. © 2015 Elsevier Ltd. All rights reserved.

Keywords: Organic Rankine cycles (ORCs) Thermoeconomic comparison Heat exchanger area per unit net power output (APR) Non-dominated sorting genetic algorithm-II (NSGA-II)

1. Introduction Nowadays, with the increasing consumption of fossil fuels and the environmental problems (e.g. global warming, air pollution, acid rain, etc.), various thermodynamic cycles such as the ORC (organic Rankine cycle), Kalina cycle [1,2], Goswami cycle [3] and trilateral flash cycle [4] have been proposed for the conversion of low-grade heat sources into power [5]. Compared with the Kalina cycle's complex system structure and trilateral flash cycle's difficult two-phase expansion, ORC has the characteristics of simple structure, high reliability and easy maintenance. The ORC applies the principle of the steam Rankine cycle, but uses organic working fluids with low boiling points to recover heat from low-grade waste heat and renewable energy, such as solar energy [6,7], biomass [8,9], geothermal heat [10,11] and heat recovery from mechanical units [12]. Therefore, researches on how to improve the cycle

* Corresponding author. Tel./fax: þ86 451 86412078 E-mail address: [email protected] (B. Li).

performance, increase the net power output and select of working fluids matching to heat source are of great significance. As a mature energy conversion technology, extensive research on the ORC has been carried out from the view of different aspects [13,14], such as working fluids selection [16e37], application of ORC on different heat source [38e50], and optimization with different evaluation criterion [13e51]. The selection of working fluids exhibits a significant effect on the efficiency of system, the sizes of the system components, the design of expansion machine, and the economic performance [15]. Various types of fluids such as hydrocarbons [16e21], hydroflurocarbon [22e27], ammonia [28e31], carbon dioxide [32,33], and mixtures [34e37] have been investigated. The detail review of different working fluids selection can be found in Chen et al. [5] and Bao et al. [15], and the selection of working fluid is a tedious process, influenced by heat source types, temperature level and the performance indexes. The ORC system has a widely application in recovering different heat sources, such as industrial waste heat [16,18], solar energy [38], geothermal energy [39], biomass energy [40,41], ocean energy [42], internal combustion engine [43e45], micro turbine [46] and gas turbine [47,48]. The comprehensive review of the ORC application can be

http://dx.doi.org/10.1016/j.energy.2015.01.075 0360-5442/© 2015 Elsevier Ltd. All rights reserved.

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Nomenclature A APR

a B0 C0 d E Fr h m

Nu Pr Q

Re s T U W

heat transfer area, m2 ratio of the heat transfer area to net power output, m2/ kW forced convection heat transfer coefficient, W/m2 K boiling number convection number diameter, m exergy, kW Froude number specific enthalpy, kJ/kg mass flow rate, kg/s; derivative Nusselt number Prandtl number energy, kW Reynolds number specific entropy, kJ/kg K temperature, K overall heat transfer coefficient, W/m2 K power, kW

Greek symbols efficiency DTm logarithmic mean temperature difference l thermal conductivity, W/m2 K m dynamic viscosity of liquid, Ns/m2 x average dryness fraction r density, kg/m3 g latent heat, J/kg b chevron angle of the plates ε heat exchanger effectiveness

h

found in Tchanche et al. [49] and Fredy et al. [50], and the ORC system has an enormous potential to produce heat or electrical energy from renewable energy. With respect to the optimization with different evaluation criterion, numerous researchers devoted main efforts on singleobjective optimization with either thermodynamic performance (thermal efficiency, net power output, exergy efficiency [18,32e55]) or economic factors (levelized energy cost [51,53], heat exchanger area per unit power output [31,52], specific invest cost [17]) as the single-objective function. Meanwhile, a few studies focused on the bi-objective optimization [13e56] considering both thermodynamic performance and economic factor simultaneously. For one thing, from the view of the single-objective optimization [17,18,31,51e55], the thermodynamic performance or the economic factors is selected as the objective function. Zhang et al. [51] presented the parameter optimization of the subcritical and supercritical ORCs to minimize the LEC (levelized energy cost) and APR (heat exchanger area per unit power output). Hettiarachchi et al. [31] and Delgado-Torres et al. [52] addressed the ratio of total heat exchange area to net power output as the objective function to find the optimum design for the ORC system. Li et al. [53] presented a parametric optimization on subcritical ORC using EPC (electricity production cost) as the evaluation criterion. Wang et al. [32] compared the cycle performances of 13 working fluids and conducted a parametric optimization by simulated annealing algorithm. Roy et al. [18] examined the parametric optimization of nonregenerative ORC using R12, R123, R134a and R-717. Rashidi et al. [54] employed a parametric optimization for the regenerative ORCs with three different objective functions: thermal efficiency, exergy

Subscripts 0 base state 1e4,i state points cs cold side d destruction ex exergy efficiency g generator; gas h hydraulic diameter hs hot side in inlet l liquid max maximum min minimum net net out outlet p constant pressure pp pinch point s isentropic sup degree of superheat tot total th thermal efficiency wf working fluid Acronyms C condenser E evaporator Exp expander GWP global warming potential H heat source IHE inter heat exchanger ODP ozone depletion potential P pump

efficiency, as well as specific work. The similar study for maximizing the system net power generation and the system thermal efficiency was implemented by Sun et al. [55] for the ORC system using the ROSENB optimization algorithm. For the other thing, from the view of the bi-objective optimization [13e56], the combination of thermodynamic performance and economic factors is selected as the objective function. Ahmadi et al. [56,57] investigated thermo-economic optimization of biomass-based integrated energy system using the total cost rate of the system and the exergy efficiency as objective functions. Wang et al. [58] conducted a multi-objective optimization for the basic ORC in combination of maximizing exergy efficiency and minimizing the overall capital cost. Taking the environmental impact of the plant and the specific total cost as the objective function, the multi-objective optimization is applied by Salcedo et al. [59] to the solar ORC coupled with reverse osmosis desalination. The thermoeconomic optimization for a small-scale ORC was applied by Quoilin et al. [17] to maximize the thermal efficiency and minimize the specific investment cost. Similar study for MW-size ORC system was employed by Pierobon et al. [60] to find an optimum operation parameters from the view of the thermodynamic performance and the economic factor. Furthermore, Muhammad et al. [13] applied the same objective function to the basic ORC, single stage regenerative and double stage regenerative ORC for waste heat recovery applications. In addition, some scholars put a particular emphasis on the regenerative ORC [54,61e64] to improve the performance of ORC system. Bees colony and artificial neural network was applied by Rashidi et al. [54] to optimize the regenerative ORC system. Mago

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et al. [61] compared the thermal efficiency and exergy efficiency between the basic ORC and the regenerative ORC using dry working fluids. Wang et al. [62] evaluated the performance of five different types of ORC, showing that the exergy destruction rate of the regenerative ORC was the lowest. Xi et al. [63] compared the performance of the basic, single stage regenerative and double stage regenerative ORC from the view of exergy efficiency. Roy et al. [64] optimizated of the regenerative ORC using thermal efficiency and exergy efficiency as evaluated criteria. As mentioned above, many studies concerning on parametric effects on system performance only considered that which parameters exhibited a higher influence, but not the sensitive degree. Meanwhile, limited studies concerned the thermoeconomic comparison between the regenerative ORC and basic ORC with the Pareto-optimal solution in combination of thermodynamic performance and economic factors. Additionally, the performance comparison between the bi-objective optimization using NSGA-II and the other two single-objective optimizations using GA (Genetic algorithm) (GA), i.e. maximum exergy efficiency and minimum APR, is not well studied or reported. Therefore, the purpose of this present study is (a) to determine the sensitive degree of different parameters on system performance; (b) to compare the Pareto-optimal solution of the regenerative ORC and basic ORC in combination of thermodynamic performance and economic factor; and (c) to investigate the comparative analysis between biobjective optimization and the other two single-objective optimizations, i.e. maximum exergy efficiency and minimum APR.

2. Modeling 2.1. System description The schematic of a simple ORC system is shown in Fig. 1. It includes four parts: evaporator, expander, condenser and pump. The pump supplies the working fluid (state 5) to the evaporator (state 1) where the working fluid is heated and vaporized by the heat source. The high pressure vapor flows into the expander (state 2) and its enthalpy is converted into work. The low pressure vapor exits the expander and is led to the condenser (state 4) where it is liquefied with water. The liquid is available at the condenser outlet, and then it is pumped back to the evaporator and a new cycle begins. A temperature of 423 K and a mass flow of 0.33 kg/s are used in the simulations for the pressurized (0.5 MPa) air. It should be

Fig. 1. Schematic diagram of the ORC system for low temperature waste heat.

3

noted that the evaporator is a heat exchanger characterized by a double-pipe construction which is composed of an outer pipe and an outward convex corrugated inner tube, while the plate heat exchanger is used as the condenser. 2.2. Mathematical model The following general assumptions are used in this study: each component is considered as a steady-state flow system, the kinetic and potential energies as well as the heat and friction losses are neglected, both the isentropic efficiencies of the pump and turbine are 0.8, heat losses in the pipes and components are also neglected [65]. The thermodynamic process is illustrated on a temperatureentropy (Tes) diagram as shown in Fig. 2. 2.2.1. Process 4e5 (pump) _ P ) can be expressed as: The pump power (W

_ P ¼ m_ wf ðh5s  h4 ÞhP ¼ m_ ðh5  h Þh W 4 P wf hs;P

(1)

where hs;P and hP are the pump isentropic efficiency and the generator efficiency, respectively, h5s and h5 are the specific enthalpies of the working fluid at the outlet of the pump under ideal and actual conditions, respectively, h4 is the specific enthalpy of the working fluid at the outlet of the condenser, and m_ wf is the mass flow rate of the working fluid. 2.2.2. Process 5e1 (evaporator) This is an isobaric heat absorption process. The heat transfer rate (Q_ E ) from the evaporator into the working fluid is given by:

Q_ E ¼ m_ wf ðh1  h5 Þ

(2)

where h1 and h5 are the specific enthalpies of the working fluid at the outlet and inlet of the evaporator, respectively. 2.2.3. Process 1e2 (expander) The superheated vapor working fluid passes through the expander to generate the mechanical power. For the ideal case, it is _ t ) is given by: an isentropic process. The expander power (W

_ t ¼ m_ ðh  h Þh h ¼ m_ ðh  h Þh W 2s s;t g 2 g wf 1 wf 1

(3)

Fig. 2. Typical T-s diagram for the ORC system.

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where hs;t and hg are the expander isentropic efficiency and the generator efficiency, respectively, h2 is the specific enthalpy of the working fluid at the outlet of the evaporator, and h2s is the ideal case of h2. 2.2.4. Process 2e4 (condenser) The exhaust vapor exits the expander and is led to the condenser where it is condensed by the cooling water. This is an isobaric condensation process. The condenser heat rate (Q_ C ) can be expressed as:

Q_ C ¼ m_ wf ðh2  h4 Þ

(5)

2.2.6. Thermal efficiency The total heat recovery heat from the heat source (Q_ H ) is expressed as:

Q_ H ¼ m_ H ðh11  h12 Þ

(6)

where m_ H is the mass flow rate of the heat source. The thermal efficiency (h_ th ) of the ORC is the ratio of the net power output to the heat input. It can be expressed as:

h_ th ¼

_ net W Q_

(7)

E

The mass flow of ORC system can be expressed as:

m_ wf

m_ ðh  h12 Þ ¼ H 11 h1  h5

(8)

(13)

DTpp;C ¼ T3  T10

(14)

The degree of superheat (DTsup ) is the temperature difference between the evaporator outlet temperature (T1 ) and saturation temperature (T7 ) corresponding to evaporator outlet pressure, which is given by:

DTsup ¼ T1  T7

(9)

(15)

2.3. Calculation of the heat exchanger's area Based on the variation of heat transfer coefficient caused by different phase state, the evaporator is divided into three sections, while the condenser is divided into two sections, as shown in Fig. 3. Note that the logarithmic mean temperature difference method is used in this present study.

DTm ¼

DTmax  DTmin max ln DT DTmin

(16)

whereDTm is the logarithmic mean temperature difference, DTmax and DTmin are the maximal and minimal temperature differences at the ends of the heat exchangers, respectively. The heat transfer (Qi ) in each section can be calculated by:

Qi ¼ Ui Ai DTm

2.2.7. Exergy efficiency The exergy of the state point can be considered as:

i h E_ ¼ m_ ðhi  h0 Þ  T0 ðsi  s0 Þ

DTpp;E ¼ T9  T6

(4)

2.2.5. Net power output _ net ) produced by expander can be The net power output (W expressed as:

_ tW _P _ net ¼ W W

In addition, the pinch point temperature difference and degree of superheat are two important parameters in the ORC system. The pinch point temperature difference [66] can be defined as the smallest heat transfer temperature difference between the heat source and the evaporator. The pinch point temperature difference in the evaporator (DTpp;E ) and condenser (DTpp;C ) can be expressed as:

(17)

where, Ui and Ai are the overall heat transfer coefficient and the heat transfer area of each section, respectively.

where hi and si are the specific enthalpy and specific entropy of the working fluid at each point, respectively; h0 and s0 are the specific enthalpy and specific entropy of the base state, respectively. The equation of the exergy destruction rate (E_ d ) can be expressed as

E_ d ¼

X

E_ in 

X

_ E_ out  W

(10)

P _ P _ Ein and Eout are the total exergy at the inlet and outlet where _ is the power output of the of the system, respectively, and W system. The exergy destruction rate of ORC system (E_ d ) can be expressed:

E_ d ¼ E_ d;E þ E_ d;EXP þ E_ d;C þ E_ d;P

(11)

The exergy efficiency of ORC can be expressed as:

h_ ex ¼

_ net W _ net E_ d þ W

(12)

Fig. 3. The schematic diagram of evaporator and condenser model, including heat source and cooling water.

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1 1 d 1 ¼ þ þ Ui ai;hs l ai;cs

(18)

where ai;hs and ai;cs are the heat transfer coefficient for the hot side and cold side of the heat exchanger, respectively, d and l are the thickness and the thermal conductivity of the material, respectively. 2.3.1. Evaporator The evaporator is divided into three parts: heating section (5e6), evaporating section (6e7) and superheating section (7e1), as shown in Fig. 3. In the heating section (5e6), the heat transfer rate (Q56 ) and the heat transfer area (A56 ) are calculated as follows:

Q56 ¼ mwf h6  h5 DTm;56 ¼

A56 ¼

ðT12  T5 Þ  ðT9  T6 Þ T5 ln TT129 T 6

Q56 U56 DTm;56

(19) (20)

(21)

Using the similar method, the heat transfer area of the evaporating section (6e7) and superheating section (7e1) can be expressed as:

5

  mð1  xÞd 0:8 Pr0:4 l ll al ¼ 0:023 ml d  C0 ¼

1x x

0:8  0:5 rg rl

(29)

(30)

B0 ¼

q mg

(31)

Frl ¼

m2 9:8r2l d

(32)

where al is the forced convection heat transfer coefficient for liquid single-phase, C0 and B0 are the convection number and the boiling number, respectively, Frl is the Froude number of liquid, x is the average dryness fraction, g denotes latent heat, and r is density. The value of C1, C2, C3, C4 and C5 are set to 1.136, 0.9, 667.2, 0.7 and 0.3, respectively. 2.3.2. Condenser The condenser is divided into two parts: cooling section (2e3) and condensation section (3e4). The heat transfer area of those two sections can be expressed as follows:

A23 ¼

Q23 U23 DTm;23

(33)

Q34 U34 DTm;34

(34)

A67 ¼

Q67 U67 DTm;67

(22)

A34 ¼

A71 ¼

Q71 U71 DTm;71

(23)

The total heat transfer area (AC,tot) in the condenser can be obtained as:

AC;tot ¼ A23 þ A34 The total heat transfer area in the evaporator (AE;tot ) can be obtained as:

AE;tot ¼ A56 þ A67 þ A71

(24)

For the cooling process (2e3), the Chisholm and Wanniarachchi correlation [69] is employed and the heat transfer coefficient is calculated as follows:

 For the heating process (5e6) and superheating process (7e1), the DittuseBoelte correlation [67] is applied to calculate the heat transfer coefficient for the single-phase working fluids.

ai;cs

l ¼ 0:023 Re0:8 Pr n d

mdh Re ¼ m Pr ¼

cp m l

(25)

(26)

(27)

where Re and Pr are the Reynolds number and Prandtl number, respectively, cp is the specific heat, m is the dynamic viscosity, and dh is the hydraulic diameter of flow channel. Note that n can be set to 0.4 for the heating process and 0.3 for the condensing process. For the evaporating process (6e7), the Kandlikar correlation [68] is used for the two-phase region and the heat transfer coefficient is calculated as follows:

ai;cs ¼ C1 ðC0 ÞC2 ð25Frl ÞC5 þ C3 ðB0 ÞC4 Frl al

(28)

(35)

Nu ¼ 0:724

ai;hs ¼

 6b 0:646 0:583 1=3 Re Pr p

(36)

l,Nu dh

where Nu and b are Nusselt number and the chevron angle of the plates, respectively. For the condensation process (3e4), the heat transfer coefficient [70] for the two-phase is calculated by:

Nu ¼

hdh 1=3 ¼ 4:118Re0:4 Prl l

(37)

2.4. Economical modeling From the Section 2.3, the total heat transfer area (Atot) of the basic ORC can be described as:

Atot ¼ AE;tot þ AC;tot

(38)

In this present study, the economic performance of ORC system is evaluated using the APR [31,51,52], which is the ratio of the total heat transfer area of total net power output.

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APR ¼

Atot _ net W

(39)

2.5. Accuracy and validation To validate the numerical solution, the energy balance equations are solved with the same operating conditions as in Ref. [20] using R123 as the working fluid. The comparison between the simulation result and the result reported by Liu et al. [20] is shown in Fig. 4. It can be observed that the simulation result agrees well with that in the reference. The maximum derivation from the date in reference is 4.7% for thermal efficiency, which indicates that the numerical solution is accurate enough for most applications. 3. Sensitivity analysis The effects of different system operation parameters on the system performance have been conducted for many times, and however the sensitive degree of operation parameters on the system performance is not well studied. Therefore, the derivatives of system performance on operation parameters are proposed as the representative of parametric sensitiveness. A parametric analysis is conducted to investigate the effect of five different operating parameters on the system performances, including the net power _ net , thermal efficiency h_ , exergy efficiency h_ and APR. output W ex th The ORC simulation is performed in combination of Matlab and Refprop. The main assumptions for the ORC system are listed in Table 1. R123 belongs to hydrochlorofluorocarbon family of refrigerants [64] with the lower environmental impacts [71] of small ODP (0.020), small GWP (76) and short half-life (1.3 years). Meanwhile, R123 gives the best characteristics to recover low temperature waste heat [14,51,72]. Therefore, R123 is chosen as the working fluid in this study due to its excellent comprehensive performance with lower environment impact and better thermodynamic performance.

Table 1 Main assumptions for the ORC system. Item

Unit

Value

Heat sources temperature Expander isentropic efficiency Generator efficiency Pump isentropic efficiency Pump efficiency Heat exchanger effectiveness Cooling water temperature Degree of superheat Temperature difference in evaporator Temperature difference in condenser Condenser temperature Environmental temperature Mass flow of heat sources

K [%] [%] [%] [%] [%] K K K K K K Kg/s

423 80 90 80 90 90 293 7 10 5 303 293 0.33

than the environmental temperature. Fig. 5 presents the variations of the system performances with the evaporator outlet temperature (T1). _ net increases from When the T1 varies from 320 K to 400 K, W 1.94 kW to the maximum value of 2.68 kW, and then declines to 1.21 kW. With the increase of T1, the enthalpy difference (h1-h2) in the expander increases and the m_ wf decreases. At the same time, the enthalpy difference increases faster than the decrease in m_ wf , _ net increases at first. However, as the effect of the and thus the W dwindling m_ wf gradually outweighs that of the increasing enthalpy _ net decreases later. Still, it can difference through the expander, W be found that better performances of h_ th and h_ ex are accompanied

3.1. Effect of evaporator outlet temperature The degree of superheat is assumed to be constant at 10 K to ensure a superheated state at the expander exit while the condensation temperature is set to be 303 K which is 10 K higher

Fig. 4. Validation of the numerical model with the previously published data [20] for ORC system using R123.

Fig. 5. Variations of the system performances with evaporator outlet temperature.

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_ net and Q_ E . The with higher T1. From Eq. (7), h_ th is affected by W _ net and Q_ E . However, increase of T1 leads to an increase in the W _ net increases faster than that of Q_ E , which leads to an increase in W h_ th . From Fig. 5, h_ ex increases with T1, which is attributed to the _ net . Furthermore, the APR decreases at first and then increase of W increases with T1. The increase of T1 enables the mean heat transfer temperature difference to decrease, which results in an increase in _ net increases heat transfer area of evaporator. At the same time, W faster than the increase in the heat transfer area with T1. And thus, the APR decreases at first. However, with the further increase of T1, _ net becomes gradually obvious, and thus the APR the decrease of W increases afterwards.

3.2. Effect of evaporator outlet pressure The variations of system performances with the evaporator outlet pressure (P1) are shown in Fig. 6. It can be found that with the _ net increases at first and then decreases, while increase of P1, the W the APR presents the opposite trend. Moreover, the h_ th and h_ ex keep rising with P1. Similar with the influence of T1 , the increase of P1 yields an increase in enthalpy difference (h1-h2) in expander and a decrease _ net increases at first which is attributed to in m_ wf , and thus the W the former. With the further increase of P1, the decrease of the m_ wf _ net decreases. becomes gradually obvious, and therefore the W Furthermore, the P1 exhibits positive influence on the h_ th and h_ ex . _ net from the Eq. (22). The The APR is affected by Ref. Atot and W

Fig. 6. Variations of the system performances with evaporator outlet pressure.

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_ net . And thus, the increase of P1 results in higher AE,tot and more W _ net . After reaching the first decrease of APR is caused by the more W minimum APR of 3.03 m2/kW, the effect of Atot overweighs that of _ net , resulting in the increase in APR. W 3.3. Effect of condenser temperature The pinch point temperature difference in condenser is set to a constant of 5 K and the condenser temperature varies from 303 K to 312 K. Fig. 7 depicts the variations of the system performances with _ net , h_ and h_ keep condenser temperature (T4). Obviously, the W ex th decreasing with the T4, whereas the APR presents the opposite trend. According to the definition of Carnot efficiency, higher h_ th is obtained by increasing evaporator outlet temperature or _ net , decreasing condenser temperature. As indicated in Fig. 7, the W h_ th and h_ ex decrease with T4. Reason for this is that the increment of the T4 leads to a decrease of enthalpy difference in evaporator and a _ net . And thus, the h_ reduction in m_ wf , resulting in a decline in W th and h_ ex decrease accordingly. From the view of the APR, the increase of T4 leads to an increase in DTm;c in condenser, resulting in a _ net exhibits higher effect decrease inAC,tot. At the same time, the W on the APR than that ofAC,tot. Therefore, the APR increases with T4. 3.4. Effect of degree of superheat The evaporator outlet temperature is set constant at 403 K and the condenser temperature is 303 K. Fig. 8 displays the variations of

Fig. 7. Variations of the system performances with condenser temperature.

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the system performances with degree of superheat (DTsup ). It can _ net decreases, while be found that with the increase of DTsup , the W the h_ th and APR increase. However, the h_ ex exhibits a slight variation with DTsup . Hung et al. [16] stated that with an increment in the degree of superheat, the thermal efficiency of dry working fluids decreases while that of wet fluids increases. The increase of DTsup yields an increase in enthalpy difference of expander and a reduction in m_ wf . The m_ wf decreases faster than the increase of enthalpy difference in _ net decreases. The h_ increases accordexpander, and thus the W th ingly and the h_ ex remains almost constant. Regarding of the APR, the increase of DTsup enables Atot to increase. Moreover, the Atot _ net , and thus the APR increases faster than the increase of W increases. 3.5. Effect of pinch point temperature difference Fig. 9 demonstrates the variations of the system performances with pinch point temperature difference (DTpp;E ). The evaporator outlet temperature is set constant at 403 K and the condenser temperature is 303 K. It shows that with the increase of DTpp;E , the _ net and h_ decrease, while the APR increases. However, the h_ W ex th presents a slight variation. When the DTpp;E increases, the m_ wf and the enthalpy difference _ net decreases. Furthermore, in expander decrease, and thus the W the h_ th nearly remains constant with theDTpp;E . The decline of h_ ex is _ net with DTpp;E . At the same time, the caused by less m_ wf and less W increase of DTpp;E leads to the Atot decreases. TheAtot decreases _ net , and thus the APR increases. faster than the decrease of W

Fig. 9. Variation of the system performances with pinch point temperature difference.

3.6. Sensitivities of system parameters In order to verify the sensitivity of the system performances on the operating parameters, the derivatives of different operation parameters on system performance are used to compare the sensitive degree of operation parameters. The derivative (m) indicates the ratio of two infinitesimal quantities and can be expressed by,



Fig. 8. Variation of the system performances with degree of superheated.

dy dx

(40)

where dx is the infinitesimal change in x, and dy is the increment in y corresponding to the dx. The average derivatives of system performance on different parameters for R123 are depicted in Table 2. In the case of the evaporator outlet temperature, the average derivatives of net power output is 0.0283. It indicates that when the evaporator outlet temperature increases 1 K, 0.0283 kW more net power output can be obtained. It can be seen that the evaporator outlet pressure owns the highest sensitive degree. Regarding of the net power output, the highest average derivatives of 1.23 is yielded for evaporator outlet pressure, which is approximately 42 times higher than that of evaporator outlet temperature, and 4e6 times higher than that of condenser temperature, degree of superheat and pinch point temperature difference. In case of the APR, the highest average derivatives of 2.83 is obtained for evaporator outlet pressure, followed by pinch point temperature difference (0.238), condenser

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Table 2 Sensitivity analysis of the net power output, thermal efficiency, exergy efficiency and APR. Parameters

E. O. T. E. O. P. C. T. D. S. P. P. T. D.

_ net W

h_ th

h_ ex

APR

Range

Average

Range

Average

Range

Average

Range

Average

0.04~0.06 2.4~1.983 0.38~0.36 0.25~0.2 0.37~0.37

0.028 1.230 0.370 0.225 0.370

0.0016~0.0008 0.16~0.04 0.0156~0.0156 0.001~0.002 0.000~0.000

0.0122 0.0948 0.0156 0.0010 0.0000

0.0622~0.0007 0.885~0.014 0.0012~0.0013 0.0028~0.0025 0.00668~0.00114

0.03145 0.44950 0.00125 0.00265 0.00391

0.0933~0.0574 10.93~0.731 0.211~0.428 0.0178~0.1238 0.0068~0.470

0.075 5.830 0.220 0.071 0.238

E. O. T.¼Evaporator outlet temperature, E. O. P.¼Evaporator outlet pressure, C. T.¼Condenser temperature, D. S.¼Degree of superheated, P. P. T. D.¼Pinch point temperature difference.

temperature (0.220); and the degree of superheat (0.071) and evaporator outlet temperature (0.0754) exhibits relatively lower average derivatives. Therefore, without respect to the economic factor (APR), besides the evaporator outlet pressure, the prior consideration of improving net power output is to decrease condenser temperature or increasing pinch point temperature difference and degree of superheat. Taking the economic factor into account, increasing the degree of superheat is a prior consideration to improve the net power output because of its relatively higher average derivatives on net power output of 0.225 and lowest average derivatives on APR of 0.071. With respect to the thermal efficiency and exergy efficiency, the average derivatives of degree of superheat and pinch point temperature difference are relatively lower. The average derivatives of degree of superheat on the thermal efficiency is relatively low (0.001), implying that the degree of superheat presents a slight influence on the thermal efficiency. The average derivatives of pinch point temperature difference on thermal efficiency is kept at zero, indicating that the pinch point temperature difference exhibits no influence on the thermal efficiency. Therefore, the improvement of the thermal efficiency and exergy efficiency should preferentially consider to increase the evaporator outlet temperature for its relatively higher average derivatives of 0.012 on thermal efficiency and 0.031 on exergy efficiency and relatively lower average derivatives on APR of 0.075. Nevertheless, higher evaporator outlet pressure enables better net power output, thermal efficiency and exergy efficiency with the higher average derivatives on APR of 5.83. It means that when the evaporator outlet pressure increases 1 MPa, the increment is 1.23 for net power output, 0.0948 for thermal efficiency and 0.449 for exergy efficiency. However, the APR increases 5.83 simultaneously. Improving the thermodynamic performance enables the economic factor to deteriorate. Therefore, the multi-objective optimization considering the thermodynamic performance and economic factor is should be conducted to find the optimum system parameters.

4.1. Objective functions and decision variables Exergy analysis can help to develop strategies and guidelines for more efficient and effective use of energy. Therefore, two objective functions including exergy efficiency and APR, are considered in this present study for bi-objective optimization. Meanwhile, the five system parameters in section 3 are selected as the decision variables. 4.2. Genetic algorithm NSGA-II (Non-dominated Sorting Genetic Algorithm II), which is improved version of NSGA (K. Deb, 1994), is applied to conduct the multi-objective optimization of the ORC system. NSGA-II is one of the contemporary multi-objective evolutionary algorithms with a fast sorting procedure and an elitism preservation mechanism, which is based on the concept of Pareto dominance and optimality. The concept of Pareto dominance and optimality can be expressed as follows for a multi-objective minimization problem:

h minFðXÞ ¼ f1 ðXÞ; f2 ðXÞ; f3 ðXÞ:::; fn ðXÞT

(41)

subject to

gi ðXÞ  0; i ¼ 1; :::; m hi ðXÞ  0; j ¼ 1; :::; n xk;min  xk  xk;max

4. Multi-objective optimization From the view of the thermodynamic performance and economic factor, a multi-objective optimization based on evolutionary algorithm is applied to determine the best design parameters for the ORC system.

Table 3 Decision variables and their lower and upper boundaries. Decision variables

Lower bound

Upper bound

Evaporator outlet temperature (K) Evaporator outlet pressure (MPa) Condenser temperature (K) Degree of superheat (K) Pinch point temperature difference (K)

330 0.27 303 3 3

410 1.66 308 10 10

Fig. 10. Pareto frontier of the basic ORC for APR with exergy efficiency.

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Y. Feng et al. / Energy xxx (2015) 1e14

Table 4 Optimum values of objective and decision variables for points A-C on the Pareto optimal front of the basic ORC. Design parameter

A

B

C

APR (m2/kW) Exergy efficiency (%) Evaporator outlet temperature (K) Evaporator outlet pressure (MPa) Condenser temperature (K) Degree of superheat (K) Pinch point temperature difference (K)

3.02 52.25 378.05 0.88 303.00 3.58 10.00

3.65 55.44 391.63 1.16 303.17 5.63 4.40

4.93 56.77 399.82 1.37 303.01 5.79 3.00

where X denotes the vector of decision variables to be optimized, F(X) is the vector of objective function, gi ðXÞ is the inequality constraints while hi(X) is the equality constraints, respectively, xk,min and xk,max are the bottom and top boundary of the decision variables, respectively. NSGA-II has been widely applied in various disciplines for its potential of reducing the overall running time and improving the running efficiency. Details about the procedure adopted by NSGA-II can be found in Refs. [73,74].

Fig. 12. Scattered distribution of evaporator outlet pressure with population in Pareto frontier.

To provide a good relation between exergy efficiency and APR, a curve is fitted to the optimized points and the expression is: 4.3. Optimization results The genetic algorithm parameters are specified according to the following values: population size 40, generation size 100, crossover fraction 0.8 and migration fraction 0.2. The decision variables and their ranges are listed in Table 3. Fig. 10 shows the Pareto frontier solution for the ORC system with two objectives regarding of the exergy efficiency and APR. It should be noted that every point of this Pareto frontier represents a potentially optimum solution for maximum exergy efficiency and minimum APR. When the exergy efficiency increases from 52% to 53%, the APR presents a moderately increasing trend. The APR rises steeply when the exergy efficiency is in the range of 53e57%. At the same time, there is a clear trade-off between exergy efficiency and APR. The highest thermodynamic performance, which exists at point C, is 56.77% in exergy efficiency with the worst economic performance of 4.93 m2/kW in APR. Meanwhile, the best economic factor, which appears in point A, is 3.02 m2/kW in APR with the lowest thermodynamic performance of 52.25% in exergy efficiency.

Fig. 11. Scattered distribution of evaporator outlet temperature with population in Pareto frontier.

APR ¼

35:69h3ex  3:27h2ex þ 2:44hex þ 7:85 37:77h4ex  7:58h3ex þ 18:74h2ex  4:62hex þ 2:24 (42)

The optimal point does not exist to maximize exergy efficiency and minimize APR simultaneously. What should be emphasized is that the optimum thermodynamic performance (maximum exergy efficiency) can be yielded at design point C using exergy efficiency as the single-objective function, while the optimum economic factor (minimum APR) can be owned at design point A using APR as the single-objective function. Therefore, a process of decision-making is conducted based on engineering experiences. The process of decision-making is usually performed with the aid of a hypothetical point [57e61] (ideal point). It should be noted that the ideal point (as shown in Fig. 10) owns the highest exergy efficiency and lowest APR but not exist actually. So the point which is the nearest point of the Pareto

Fig. 13. Scattered distribution of condenser temperature with population in Pareto frontier.

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the condenser temperature is in the range of 303.00e303.54 K, meaning that lower condenser temperature ensures higher optimal results. 5. Pareto optimization applied to thermoeconomic comparison between the basic ORC and regenerative ORC

Fig. 14. Scattered distribution of degree of superheat with population in Pareto frontier.

frontier to the ideal point, is chosen as a desired final optimal solution and defined as Pareto-optimal solution. Based on the definition of Pareto-optimal solution, the point B in Fig. 10 is the optimal solution for the Pareto optimization in combination of maximum thermodynamic performance and minimum economic factor, and owns the optimum exergy efficiency of 55.44% and APR of 3.65 m2/kW. The detail parameters of point A-C for the basic ORC are shown in Table 4. The distributions of five decision variables for the Pareto frontier in their respective ranges are displayed in Figs. 11e15. It can be found that the evaporator outlet temperature, evaporator outlet pressure, condenser temperature, degree of superheat and pinch point temperature difference exhibit great influence on the exergy efficiency and APR. The optimum evaporator outlet temperature is in the range of 377.81e400.96 K and the evaporator outlet pressure is in the range of 0.88e1.37 Mpa, indicating that better optimization results can be obtained with higher evaporator outlet temperature or evaporator outlet pressure. Furthermore, the optimum degree of superheat and pinch point temperature difference are varied from the minimum to maximum values in their design ranges. However,

Fig. 15. Scattered distribution of pinch point temperature difference with population in Pareto frontier.

As mentioned in section 4.3, the optimal point can be obtained to ensure better thermodynamic performance and economic factor using NSGA-II. At the same time, the performance comparison between the basic ORC and regenerative ORC is carried out with respect to the single-objective of thermodynamic performance or economic factor. However, limited study considered the thermoeconomic comparison using Pareto-optimal solution between the basic ORC and regenerative ORC. Therefore, the Pareto optimization of the regenerative ORC is applied using similar process in Section 4.3 and compared with the basic ORC. Further investigation on the comparison between the Pareto-optimal solution and the other two single-objective optimizations, i.e. maximum exergy efficiency and minimum APR is employed in Section 5.2. 5.1. Thermoeconomic comparison of the basic ORC and regenerative ORC For the regenerative ORC, the IHE (internal heat exchanger) is used to heat the working fluid at the pump outlet and cool the working fluid at the expander outlet. The schematic of regenerative ORC and T-s plot are shown in Figs. 16 and 17, respectively. The IHE effectiveness [75] can be expressed as 0

εIHE ¼

qactual h2  h2 ¼ qmax h2  h5

(43)

For the internal heat exchanger, the Eq. (25) and Eq. (37) are used to calculate the heat transfer coefficient of the hot side vapor and cold side liquid working fluids, respectively. The heat transfer (QIHE ) and the heat transfer area (AIHE ) can be expressed as follows:

Fig. 16. Schematic diagram of the regenerative ORC system for low temperature waste heat.

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Y. Feng et al. / Energy xxx (2015) 1e14 Table 5 Optimum values of objective and decision variables for points A-C on the Pareto optimal front of the regenerative ORC.

Fig. 17. Typical T-s diagram for the regenerative ORC system.

QIHE ¼ mwf ðh2  h20 Þ AIHE ¼

AIHE UIHE DTIHE

(44) (45)

The Pareto frontier of the regenerative ORC for APR with exergy efficiency is shown in Fig. 18. It can be seen that the regenerative ORC presents the same behavior in Pareto frontier with the basic ORC. The Pareto-optimal solution (point B in Fig. 18) of the regenerative ORC can be obtained with the optimum exergy efficiency of 59.93% and APR of 3.07 m2/kW, which is 8.10% higher in exergy efficiency and 15.89% lower in APR than that of the basic ORC. Meanwhile, the maximum exergy efficiency is occurred at design C with the value of 61.34%, which is 8.34% higher than that of the basic ORC. The design point A owns the minimum APR with the value of 2.83 m2/kW, which is 6.7% lower than that of the basic ORC. The detail parameters of point A-C for the regenerative ORC are shown in Table 5. 5.2. Performance comparison between Pareto-optimal solution and the other two single-objective optimizations As mentioned in Section 4.3, the point C yields the highest thermodynamic performance with exergy efficiency as the single-

Design parameter

A

B

C

APR (m2/kW) Exergy efficiency (%) Evaporator outlet temperature (K) Evaporator outlet pressure (MPa) Condenser temperature (K) Degree of superheat (K) Pinch point temperature difference (K)

2.83 54.97 374.44 0.81 303.00 3.64 9.85

3.07 58.18 388.73 1.10 303.36 5.63 4.37

5.81 61.34 406.36 1.55 303.00 7.07 3.25

objective function, and at the same time, the point A owns the best economic performance with APR as the single-objective function. Meanwhile, the point B is the Pareto-optimal solution with bi-objective optimization. And then, the net power output and the thermal efficiency of AeC for the regenerative ORC and basic ORC can be calculated using the system parameters which are listed in Tables 4 and 5, respectively. The performance comparison of the regenerative ORC and basic ORC with different objective function is shown in Table 6. With respect to the single-objective of maximum exergy efficiency, compared with the basic ORC, the regenerative ORC owns 7.47% higher exergy efficiency, 11.54% higher thermal efficiency, 17.24% more APR whereas 35.92% less net power output. When considering the minimum APR as the single-objective function, 4.95% higher exergy efficiency, 5.87% higher thermal efficiency, 3.92% more net power output but 6.71% less APR are yielded for the regenerative ORC. However, with respect to the Pareto optimization, besides the 6.18% higher exergy efficiency and 8.40% lower in APR as discussed in section 5.1, 19.14% higher thermal efficiency and 5.76% more net power output are obtained for the regenerative ORC. It demonstrates that compared with the basic ORC, the improvement of exergy efficiency, thermal efficiency, net power output and APR for the regenerative ORC under the Pareto-optimal solution is 6.18%, 19.14%, 5.76% and 8.40%, respectively. Meanwhile, highest exergy efficiency and thermal efficiency with worst economic factor and lowest net power output are obtained with maximum exergy efficiency as the single objective, while the opposite trend is yielded when minimum APR is used as the objective function. Compared with the other two single-objective optimizations, the Pareto optimization compromises the thermodynamic performance and economic factor, therefore being more suitable for decision making. 6. Conclusion The sensitivity analysis for low temperature ORCs (organic Rankine cycles), as well as the thermoeconomic comparison between the basic ORC and regenerative ORC using NSGA-II (Nondominated sorting genetic algorithm-II), are conducted in this

Table 6 The performance of the basic ORC and regenerative ORC with three different objective functions. Objective function

h_ ex

Max h_ ex  Max h_ ex Min APR

61.34% 56.76% 59.56% 55.88% 54.97% 52.25%

Min APR Fig. 18. Pareto frontier of the regenerative ORC for APR with exergy efficiency.

(þIHE) (IHE) (þIHE) (IHE) (þIHE) (IHE)

APR (m2/kW)

Wnet (kW)

h_ th

5.80 4.80 3.58 3.88 2.83 3.02

1.42 1.93 2.43 2.29 2.55 2.45

17.07% 15.10% 15.36% 12.42% 13.80% 12.99%

(þIHE) (IHE) (þIHE) (IHE) (þIHE) (IHE)

(þIHE) (IHE) (þIHE) (IHE) (þIHE) (IHE)

(þIHE) (IHE) (þIHE) (IHE) (þIHE) (IHE)

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paper. The derivatives of five system parameters on system performance are used to evaluate the parametric sensitiveness. The exergy efficiency and the APR (heat exchanger area per unit net power output) are selected as the objective functions for multiobjective optimization using R123 under the low temperature heat source of 423 K. The Pareto frontier solution with bi-objective for maximizing exergy efficiency and minimizing APR is obtained and compared with the corresponding single-objective solutions. The main conclusions can be made as follows: (1) The improvement of the net power output should preferentially consider to increase the degree of superheat, while the prior consideration of improving thermal efficiency and exergy efficiency is to increase the evaporator outlet temperature. Moreover, the degree of superheat presents a slight influence on the thermal efficiency and the pinch point temperature exhibits no effect on the thermal efficiency. (2) A tradeoff can be obtained between the thermodynamic performance and economic factor. The Pareto-optimal solution of the regenerative ORC with optimum exergy efficiency of 59.93% and APR of 3.07 m2/kW, which is 8.10% higher in exergy efficiency and 15.89% lower in APR than the basic ORC. Meanwhile, a fitting cure for the Pareto frontier of basic ORC between exergy efficiency and APR is derived as: 35:69h3ex 3:27h2ex þ2:44hex þ7:85 . 3 2 ex 7:58hex þ18:74hex 4:62hex þ2:24

APR ¼ 37:77h4

(3) Better optimization results can be obtained with higher evaporator outlet temperature or low condenser temperature. The optimum evaporator outlet temperature is in the range of 377.81e400.96 K and the evaporator outlet pressure is in the range of 0.88e1.37 Mpa under the given heat source condition. (4) Compared with the basic ORC, the improvement of exergy efficiency, thermal efficiency, net power output and APR for the regenerative ORC under the Pareto-optimal solution is 6.18%, 19.14%, 5.76% and 8.40%, respectively. At the same time, the Pareto optimization compromises the thermodynamic performance and economic factor, therefore being more better suitable for decision making.

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Please cite this article in press as: Feng Y, et al., Sensitivity analysis and thermoeconomic comparison of ORCs (organic Rankine cycles) for low temperature waste heat recovery, Energy (2015), http://dx.doi.org/10.1016/j.energy.2015.01.075