Multi-objective optimization of supercritical carbon dioxide recompression Brayton cycle considering printed circuit recuperator design

Energy Conversion and Management 201 (2019) 112094

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Multi-objective optimization of supercritical carbon dioxide recompression Brayton cycle considering printed circuit recuperator design Zhenghua Rao1, Tianchen Xue1, Kaixin Huang, Shengming Liao

T

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School of Energy Science and Engineering, Central South University, Changsha 410083, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Supercritical carbon dioxide Recompression Brayton cycle Printed circuit recuperator Multi-objective optimization

Supercritical carbon dioxide recompression Brayton cycle is well suited to a broad range of applications including nuclear and concentrated solar energy. As printed circuit recuperators are employed to optimize the thermal performance of the cycle, the recuperator optimal design is required with the objective of maximizing the cycle thermal efficiency and minimizing the total cycle cost. In this paper, a thermo-economic model of recompression Brayton cycle with the S-shaped fin printed circuit recuperator is developed to perform multiobjective optimization considering the recuperator design parameters (i.e. mass fluxes and enthalpy efficiencies of recuperators and recompression fraction). Nondominated sorting genetic algorithm is used to obtain Pareto frontier. The results show that compared to the mass fluxes, the enthalpy efficiencies of recuperators and recompression fraction play more important roles in the optimization. From the Pareto frontier, the optimum range of the cycle thermal efficiency is 0.4303–0.5380 and that of the total cycle cost is 7.468 M$–12.31 M$. As high cycle thermal efficiency is preferred, the high recompression fraction, high mass flux and high enthalpy efficiency of low temperature recuperator, low mass flux and high enthalpy efficiency of high temperature recuperator are required. In contrast, as low cycle cost is preferred, the opposite selections of design parameters are required.

1. Introduction Supercritical carbon dioxide (sCO2) Brayton cycle has been considered as one of the most prospective power cycles due to distinct advantages, including the high thermal efficiency, simple cycle layout, compactness of component and potential capital cost savings [1]. These advantages are attributed to the high density and less compressibility of CO2 near its critical point (7.38 MPa, 30.98 °C), which significantly reduces the compression work and makes it possible to use compact turbomachines and heat exchangers [2]. In addition, sCO2 Brayton cycle can be efficiently operated with dry-cooling system which is especially suitable for application in arid regions to reduce water consumption [3]. Therefore, sCO2 Brayton cycle can be a competitive alternative to steam Rankine cycle for the future power generation with diverse energy sources, such as fossil fuel, nuclear and solar thermal power plants [4]. Up to date, the experimental loops of sCO2 Brayton cycles have been built up and tested in the laboratory scale by several research institutions such as Sandia National Laboratory [5], Bettis Atomic Power Laboratory [6], Tokyo Institute of Technology [7] and Korean Atomic

Energy Research Institute [1]. Researchers has been paying much attention to thermodynamic performances of different sCO2 cycle layouts such as simple recuperation cycle, recompression cycle, intercooling cycle and some more complex configurations. Dostal et al. [2] compared the performances of several cycle layouts and found that the recompression cycle was the most efficient one under the pressure of 20 MPa and temperature of 550 °C at turbine inlet. Al-Sulaiman and Atif [8] compared five sCO2 cycles integrated in a concentrating solar power (CSP) system, and found that the recompression cycle reached the highest thermal efficiency and the highest net power output. Turchi et al. [9] studied the effects of adding re-heat on four different sCO2 cycles, and found the reheating significantly improved the performances of the partial cooling cycle and main-compression with intercooling cycle. Dyreby et al. [10] studied the effects of compressor inlet temperature and pressure, turbine inlet temperature, recuperator conductance and recompression fraction on the thermal efficiency of recompression cycle, and pointed out the optimization of these parameters were interrelated. They also studied the part-load performances of sCO2 cycle to optimize thermal efficiency under off-design condition [11]. Floyd et al. [12] investigated the effects of compressor speed on

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Corresponding author. E-mail address: [email protected] (S. Liao). 1 The authors contributed equally to this work. https://doi.org/10.1016/j.enconman.2019.112094 Received 27 July 2019; Received in revised form 10 September 2019; Accepted 21 September 2019 0196-8904/ © 2019 Elsevier Ltd. All rights reserved.

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Nomenclature Aflow Ah C cp dch dhy f G H h k l m NTU Nu nch P pf Pr Q Re Rw s T tp UA u V W w wch wf

Greek symbols θ ΦRC λ ε ηth ρ

cross-sectional area of a channel (m2) hot side heat transfer area per unit length (m2/m) cost ($) or heat capacity (kJ) specific heat (J/g K) channel depth (mm) hydraulic diameter (mm) friction factor mass flux (kg/m2 s) enthalpy (kJ) convective heat transfer coefficient (kW/m2 K) or specific enthalpy (kJ/kg) overall heat transfer coefficient (kW/m2 K) channel length (m) mass flow rate (kg/s) dimensionless number of transfer units Nusselt number channel number pressure (MPa) fin pitch (mm) Prandtl number heat transfer rate (kW) Reynold number conductive thermal resistance of channel wall (K m2/kW) specific entropy (kJ/kg K) temperature (°C) plate thickness (mm) conductance (kW/K) velocity (m/s) volume (m3) work (kJ) specific work (kJ/kg) channel width (mm) fin width (mm)

fin angle (°) recompression fraction thermal conductivity (W/m K) heat exchanger enthalpy efficiency cycle thermal efficiency density (kg/m3)

Subscripts CO2 c ch comp cycle f HTR h in isen LTR mc max min out PC PCHE PHX p RC rc recup turb w

the off-design performances of the sCO2 recompression cycle coupled with the sodium cooled reactor. Pham et al. [13] and Padilla et al. [14] performed the exergy analysis of sCO2 cycles to identify the location and magnitude of the irreversibility, and found that the heat exchangers had the second largest contributions to the exergy losses following the nuclear reactor in the nuclear power system and the solar receiver in the CSP system. The above studies showed that the recompression cycle was considered as a promising sCO2 configuration with a desirable high efficiency. The main reason is that the CO2 hot stream was split at the outlet of low temperature recuperator (LTR) to minimize the specific heat difference between hot stream and cold stream. It was reported that the recupertor performance had great effects on the recompression cycle performance. Clementoni et al. [15] found that the efficiency of CSP plant could be almost doubled by incorporating the recuperator to the sCO2 Brayton cycle. Neises and Turchi [16] found that the thermal efficiency of the recompression cycle was more sensitive to the recuperator conductance. Dyreby et al. [10] concluded that the thermal efficiency of recompression cycle was significantly improved by increasing the total recuperator conductance. Wang et al. [17] found the recuperator effectiveness had more significant impact on exergy efficiency as compared to the compressor isentropic efficiency. Sarkar and Bhattacharyya [18] found that the thermal efficiency of recompression cycle was more sensitive to the high temperature recuperator (HTR) effectiveness than the low temperature recuperator (LTR) effectiveness. The requirement for high-effectiveness recuperators in sCO2 cycles motivates researches on the compact heat exchanger. Recently, printed circuit heat exchanger (PCHE) has been of great interest to the sCO2

carbon dioxide cold stream channel compressor sCO2 Brayton cycle fin high temperature recuperator hot stream inlet isentropic process low temperature recuperator main compressor maximum minimum outlet precooler printed circuit heat exchanger primary heat exchanger pitch recompression recompressor recuperator turbine wall

Brayton cycle, because it has great compactness and capability to withstand the high temperature and pressure [19]. Due to the coupling of the sharply varied thermophysical properties of the fluid and the special geometrical characteristics of different channels, the fluid flow and heat transfer in PCHE become very complex [20]. Many researchers studied the effects of different channel shapes on the thermal and hydraulic performances of PCHE. Kim et al. [21] studied the thermalhydraulic performance of zigzag PCHE and observed a significant pressure drop near the bending point of a zigzag channel. Kim and No [22] and Khan et al. [23] proposed the correlations of heat transfer coefficient and friction factor for the zigzag PCHE considering the inclination angle of channel. The channel with S-shaped fin was proposed and showed better thermal-hydraulic performances. It was reported that as compared to the zigzag PCHE, the S-shaped fin PCHE had the comparable heat transfer ability but only 1/5 of the pressure drop [24] and increase the sCO2 cycle efficiency potentially by 0.85% [25]. In general, the PCHE design is important to satisfy different requirements of the cycle. Recently, multi-objective optimization algorithms were widely used in cycle system design taking various objectives into account, such as cycle thermal efficiency, exergy efficiency and product unit cost. These algorithms were used to find tradeoffs between different objectives and obtain a set of optimal solutions, known as Pareto optimal solutions [26]. Many researchers performed multi-objective optimizations to obtained the optimum design of different cycle systems such as combined cycles [27] and cogeneration power cycles [28]. In terms of sCO2 Brayton cycle system, Mohagheghi et al. [29] carried out a multi-objective optimization of the single-recuperated and recompression cycles by maximizing cycle efficiency and 2

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2. Methods

specific power. Padilla et al. [30] compared four sCO2 Brayton cycle configurations by using multi-objective optimization with four different sets of objectives. Wang et al. [31] carried out a multi-objective optimization of five different sCO2 cycles applied in solar power tower system with system efficiency and specific work as objective functions. Zhao et al. [32] conducted a multi-objective optimization of the recompression cycle to maximize the cycle exergy efficiency and minimize the cycle cost. However, although the existing studies involved variables such as turbine inlet temperature, compressor inlet pressure and pressure ratio, the recuperator designs have not been fully taken into consideration in the optimization. The recuperator design parameters significantly affect the cycle thermo-economic performances. For example, the high PCHE effectiveness for high cycle efficiency can lead to a large portion of the cost of the power block [33]. The mass flux of channel directly affects the pressure drop and heat transfer coefficient of the recuperator, which brings a conflict between pumping power and recuperator cost [34]. For recompression cycle, the recompression fraction related to the LTR heat recovery should also be optimized considering the less expensive cycle cost and high thermal efficiency. In this study, we perform a multi-objective optimization of the sCO2 recompression Brayton cycle in order to determine the optimal design parameters of printed circuit recuperators by achieving the tradeoff between the high cycle thermal efficiency and low total cycle cost. First, a specific thermo-economic model of the sCO2 recompression Brayton cycle is developed and validated. Then, the parametric studies on the mass flux and enthalpy efficiency of recuperators and the recompression fraction are conducted to examine the conflict between the thermal efficiency and cycle cost. The multi-objective optimization is subsequently performed between efficiency and cost objectives by using nondominated sorting genetic (NSGA-II) algorithm to obtain the Pareto frontier. The optimum solution is gained by TOPSIS method. Finally, the sensitivity analysis is conducted to investigate the effects of the variation of PCHE unit cost on the optimization results.

A typical sCO2 recompression Brayton cycle layout includes ten main processes, as shown in Fig. 1. The sCO2 is heated in the primary heat exchanger (5 → 6), and then enters the turbine to generate power output (6 → 7). The hot stream from the turbine passes through the high temperature recuperator (HTR) (7 → 8) and low temperature recuperator (LTR) (8 → 9) for heat recovery. Thus, the cold streams from the compressors are preheated in the LTR (2 → 3) and HTR (4 → 5). At the outlet of LTR hot side (state 9), the flow is split into two streams. One stream rejects heat in the precooler (9 → 1) and is then compressed by a main compressor (1 → 2), and another stream is only compressed by a recompressor (9 → 10) without rejecting heat. These two streams then merge into one stream at the outlet of LTR (10 + 3 → 4). In this section, a specific mathematical model of sCO2 recompression Brayton cycle is encoded in MATLAB [35]. Based on the cycle model, the multiobjective optimization is then conducted to determine the optimal solution. 2.1. Thermo-economic model The mathematical model of the sCO2 recompression cycle consists of models of the main components. The thermodynamic model is based on energy balance in each component, which applies the following assumptions:

• Expansion and compression processes are adiabatic but non-isentropic with constant isentropic efficiencies. • Heat exchanger pressure drops due to the entrance loss, exit loss and • •

acceleration effect are neglected, and only the frictional loss along the channel length is considered. Kinetic energy and potential energy in each component are neglected. The heat loss and pressure drop due to pipe lines and junctions are neglected.

7

Main Compressor

Generator

Turbine

2 6

ĭRC

1

Recom pressor

Motor

Precooler

10

Primary heat exchanger

5 3

9

4

8

Low temperature recuperator

High temperature recuperator

Fig. 1. Supercritical CO2 recompression Brayton cycle layout. 3

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a hot side channel, as shown in Fig. 3. The channel unit is then discretized into n sub-heat exchangers along the channel length, as shown in Fig. 4. For each recuperator model, the input parameters include the inlet temperatures of hot stream (Th,in) and cold stream (Tc,in), the inlet pressure of cold stream (Pc,in), the outlet pressure of hot stream (Ph,out), the enthalpy efficiency of recuperator (ε), the mass flux of channel (G) and the geometric parameters of recuperator, while the output parameters include the outlet temperatures of cold stream (Tc,out) and hot stream (Th,out), the outlet pressure of cold stream (Pc,out), the inlet pressure of hot stream (Ph,in) and the required recuperator volume (Vrecup). For each element, the bulk average temperature is assumed to determine the temperature-dependent thermophysical properties of sCO2 in the calculation. The usual heat exchanger effectiveness ε is defined as

2.1.1. Compressor and turbine models Given the constant isentropic efficiencies ηcomp, the outlet enthalpies of main compressor and recompressor are calculated by

wcomp, isen = hcomp, out , isen − hcomp, in

wcomp =

(1)

wcomp, isen ηcomp

(2)

hcomp, out = hcomp, in + wcomp

(3)

where hcomp,out,isen is the specific enthalpy at the outlet of compressor if the fluid is isentropically processed to the outlet pressure, hcomp,in is the specific enthalpy at the inlet of compressor, ηcomp is the compressor isentropic efficiency, wcomp,isen is the isentropic specific work of compressor, wcomp is the specific work of compressor. The total power input to the main compressor and recompressor is given by

Wcomp = mCO2 ·(1 − ΦRC )·wmc + mCO2 ·ΦRC ·wrc

ε=

(4)

wturb, isen = hturb, in − hturb, out , isen

(5) (6)

hturb, out = hturb, in − wturb

(7)

where hturb,out,isen is the specific enthalpy at the outlet of turbine if the fluid is isentropically processed to the outlet pressure, hturb,in is the specific enthalpy at the inlet of turbine, ηturb is the turbine isentropic efficiency, wturb,isen is the isentropic specific work of turbine, wturb is the specific work of turbine. The power generated by the turbine is given by

Wturb = mCO2 ·wturb

(9)

where Ch is the hot stream capacitance rate, Cmin is the minimum capacitance rate of cold and hot streams, Th,in and Tc,in are the inlet temperature of hot stream and cold stream of the sub-heat exchanger respectively, Th,out is the outlet temperature of hot stream. As the mass flow rates of hot stream and cold stream are different, the minimum heat capacity in LTR is uncertain. In this study, instead of traditional effectiveness, the enthalpy efficiency considering the maximum enthalpy drop in the recuperator is used to characterize the recuperator performance. The total heat transfer rate of a channel unit Qch is calculated by Eq. (10) based on the enthalpy efficiency ε and the maximum enthalpy drop in the recuperator Δhmax.

where mCO2 is the total CO2 mass flow rate in the loop, ΦRC is the recompression fraction, wmc is the specific work of main compressor, wrc is the specific work of recompressor. The outlet enthalpy of the turbine is calculated by

wturb = wturb . isen·ηturb

Ch·(Th, in − Th, out ) Cmin·(Th, in − Tc, in )

Qch = ε ·Δhmax

Δhmax =

(8)

(10)

⎧ min {mch, h ·(1 − ΦRC )(hc*, out − hc, in ), ⎪ m ·(h − hh*, out )}(For LTR) ⎨ ch, h h, in ⎪ min {mch, h ·(hc*, out − hc, in ), mch, h ·(hh, in − hh*, out )}(For HTR) ⎩ (11)

2.1.2. Recuperator model Fig. 2 demonstrates the schematic geometry of S-shaped fin PCHE for LTR and HTR. The plates are chemically etched to form the S-shaped fin, and then are diffusion-bonded into a single block. The inlets of LTR for hot stream and cold stream, respectively, correspond to states 8 and 2, as shown in Fig. 1; and the inlets of HTR for hot stream and cold stream, respectively, correspond to states 7 and 4, as shown in Fig. 1. In this study, a detailed S-shaped fin PCHE model is employed with constant channel geometry parameters in order to determine the channel length, outlet temperature and pressure drop of LTR and HTR. By taking the periodic boundary conditions, the recuperator model is simplified as a single channel unit consisting of a cold side channel and

where h*c,out is the outlet specific enthalpy of cold stream evaluated at Th,in, hc,in is the inlet specific enthalpy of cold stream, hh,in is the inlet specific enthalpy of hot stream, h*h,out is the outlet specific enthalpy of hot stream evaluated at Tc,in. mch,h is the mass flow rate of hot stream of a channel. The mass flux of cold stream in LTR can be determined with the mass flux of hot stream multiplying the recompression fraction, while the mass flux of cold stream in HTR is equal to that of hot stream. The mass flux is defined as the rate of mass flow per unit area. With the given mass flux of hot stream Gh, the corresponding mass flow rate mch,h can be calculated by

mch, h = Gh·Aflow

Cold Side Cold stream inlet

Hot Side Hot stream outlet

Cold Side

Fig. 2. Schematic geometry of an S-shaped fin PCHE. 4

(12)

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Hot stream inlet

Periodic

dch tp

Periodic

Periodic

Periodic

wch

ș Pf

Cold stream inlet

Fig. 3. Schematic geometry of single channel unit of an S-shaped fin PCHE.

Nu = 0.174·Re 0.593Pr 0.43

(18)

The hydraulic diameter dhy is calculated by

dhy =

2·wch·dch wch + dch

(19)

where wch is the channel width, dch is the channel depth. The thermal resistance of the channel wall Rw is calculated by

Rw =

Fig. 4. Schematic sketch of a counter-flow PCHE model.

Aflow = wch·dch

where Aflow is the cross-sectional area of the channel, wch is the channel width, dch is the channel depth. The total heat transfer rate of the channel unit δQch is equally allocated to each element as

Ah = 2.18·(wch + dch)

(14)

n

ΔP =

+ Rw +

1 hc

(22)

(23)

The volume of recuperator Vrecup is calculated by

Vrecup = Vch nch (16)

(24)

2mCO2 mch, h

(25)

Vch = (wch + wf )·tp·l

(26)

nch =

where Rw is the wall thermal resistance, h is the convective heat transfer coefficient determined by Eq. (17).

λ CO2 ·Nu h= dhy

hy

ρui2 ·δli 2

f = 0.4545·Re−0.34

1 1 hh

·

where ρ is the fluid density, u is the fluid velocity, f is the friction factor given by Eq. (23) [36].

(15)

where ΔTi is the arithmetic mean temperature difference between the hot stream and cold stream of the ith element, k is the local overall heat transfer coefficient given by

k=

f

∑ di i=1

δQch ki·ΔTi ·Ah

(21)

where the factor value of 2.18 other than 2 is taken because the heat transfer area of S-shaped PCHE is slightly larger than the zigzag PCHE with the same volume [36]. Once the length of each element δli is determined, the pressure drop along the channel can be calculated by

Thus, the temperature distribution along the channel length can be obtained. Based on the calculated temperature and heat transfer rate in each element, the length of the ith element can be calculated as

δli =

(20)

λw

where tp is the channel thickness, λw is the thermal conductivity of PCHE material. For the channel, the heat transfer area per unit length at the hot side is calculated by

(13)

Q δQch = ch n

tp − dch

n

(17)

l=

∑ δli i=1

where λ CO2 is the thermal conductivity of CO2, dhy is the hydraulic diameter. Nu is determined by Eq. (18) [36], in which Re and Pr are determined based on the bulk average temperature of each element.

(27)

where Vch is the volume of a single channel, nch is the total number of channels in the recuperator, wf is the fin width, tp is the plate thickness, 5

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l is the total length of the channel.

Start

2.1.3. Primary heat exchanger and precooler models The primary heat exchanger and precooler are discretized into a set of sub-heat exchangers. Given the fixed mass flux and geometric parameters, the pressure drops in the primary heat exchanger and precooler are assumed to be constant. In this study, the heat input into the primary heat exchanger is fixed so that the total CO2 mass flow rate in the loop mCO2 can be calculated by

mCO2 =

QPHX hPHX , out − hPHX , in

Input design parameters of cycle and geometric parameters of recuperator

Initialization P1=Pcomp,in P6=Pcomp,in-0.1MPa P2=P3=P4=P5=P10=Pcomp,out P7=P8=P9=Pcomp,in +0.1MPa

(28)

where QPHX is the heat transfer rate in primary heat exchanger, hPHX,in and hPHX,out are the specific enthalpies of CO2 at the inlet and outlet of the primary heat exchanger, respectively. The heat transfer rate in the precooler is calculated by

QPC = mCO2 ·(1 − ΦRC )·(hPC , in − hPC , out )

Calculate T7 with Turbine model

Calculate T2 with Compressor model

(29)

where hPC,in and hPC,out are the specific enthalpies of CO2 at inlet and outlet of the precooler, respectively. The conductance model is used in the primary heat exchanger and precooler. The conductance of each sub-heat exchanger UAi is defined as

(

1 − ε·C

CR =

Calculate T3, T9, P3, P8 with Recuperator model

(30)

UAi = NTU ·Cmin R log ⎧ 1−ε ⎪ 1 − CR NTU = ⎨ ε ⎪ 1−ε ⎩

Initialization of T8

)

Calculate T10 with Compressor model Calculate T4 with T3&T10

if CR ≠ 1 otherwise

Cmin Cmax

(31)

Calculate T8*, T5, P7*, P5* with Recuperator model

(32)

where NTU is dimensionless number of transfer units for each sub-heat exchanger, C is the average capacitance rate, Cmin and Cmax are the minimum and maximum capacitance rate of the cold and hot streams respectively. The total conductance is the sum of each sub-heat exchanger conductance.

If |T8-T8*|<0.2

No

Adjust T8

Yes If |P5-P5*|<0.01MPa If |P7-P7*|<0.01MPa

No

P5=P5* P7=P7*

n

UA =

∑ UAi i=1

Yes

(33) Calculate Șth, Ccycle

The thermal efficiency of the cycle is defined as

ηth =

QPHX − QPC QPHX

Return Șth, Ccycle

(34)

Fig. 5. The flowchart of iteration process for the thermo-economic model of sCO2 recompression Brayton cycle.

2.1.4. Economic model Appropriate economic indicator is needed to assess the economic performance of a power generation system [37]. In this study, the total cycle cost is taken as the economic indicator based on the calculation of each component cost. The compressor and turbine cost models are only functions of their corresponding working power according to the previous study [33]. 0.7865 Ccomp = 6898·Wcomp

(35)

0.6842 Cturb = 7790·Wturb

(36)

where UAPHX and UAPC are the primary heat exchanger conductance and precooler conductance in unit of kW/K, respectively. In order to investigate the effect of the mass flux of recuperator on its cost, the recuperators cost Crecup is calculated by Eq. (39) based on the PCHE cost per unit volume which is taken as 1.027 × 103 $/m3 [34].

Crecup = 1.027 × 103·V where V is the total volume of recuperators in unit of m3. Therefore, the total cycle cost can be evaluated as

where Wcomp and Wturb are the total compressor inlet power and turbine outlet power in unit of kW, respectively. The PHX and PC cost models are functions of their corresponding conductance [33].

CPHX = 3500·UAPHX

(37)

CPC = 2300·UAPC

(38)

(39)

Ccycle = Ccomp + Cturb + CPHX + CPC + Crecup

(40)

2.1.5. Numerical approach to the model The above thermodynamic and economic models are coupled and 6

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value of xij, M is the number of variables. In this study, the objectives are cycle thermal efficiency and total cycle cost; the variables are the LTR mass flux, HTR mass flux, LTR enthalpy efficiency, HTR enthalpy efficiency and recompression fraction.

solved in an iterative method to determine the state points 1 to 10 of the cycle and calculate the cycle thermal efficiency ηth and total cycle cost Ccycle, as shown in Fig. 5. The thermodynamic and transport properties of CO2 are extracted from NIST’s REFPROP [38]. The inputs of the model contain the key design parameters of recompression cycle and geometric parameters of the LTR and HTR, and both the geometric parameters of each recuperator are set as the same. Firstly, the temperature and pressure of each the state point are initialized. Secondly, T2 and T7 are, respectively, calculated by the compressor model and turbine model with the known parameters. Given an initial guess value of T8, the recuperator model is used for LTR to calculate T3, T9, P3, P8 and channel length. Then, T10 is determined by the compressor model, and T4 is determined by mixing the stream 3 and stream 10 based on the energy balance. The recuperator model is used for HTR to calculate T8*, T5, P7*, P5* and the channel length. Repeat the above iterative solution until the convergence criterion |T8 − T8*| < 0.2 °C is satisfied. During the above process, pressure drops through the two recuperators are also calculated. P5 and P7 are updated using the newly calculated values of P5* and P7* and return the second step until |P5 − P5*| < 0.1 MPa and |P7 − P7*| < 0.1 MPa. Finally, the converged solution is obtained. Then the objective functions, cycle thermal efficiency ηth and total cycle cost Ccycle, are calculated.

(43)

2

∑ (Sj+ − wj yij )2 ,

i = 1, 2, ..., M , j = 1, 2 (44)

j=1 2

Di − =

∑ (wj yij − Si −)2 ,

i = 1, 2, ..., M , j = 1, 2 (45)

j=1

The relative closeness Ci for each Pareto solution is defined according to the following equation:

Ci =

Di − , i = 1, 2, …, M Di + + Di −

(46)

The final optimal solution is the one with the maximum Ci. 2.3. Validation of modeling approach In order to validate the model of sCO2 recompression Brayton cycle presented in this study, the cycle thermal efficiencies under the different turbine inlet temperature are calculated and compared with the simulation results from Turchi et al. [9]. The input parameters used for the validation are given in Table 1, and the recompression fraction is optimized for the different turbine inlet temperature. As showed in Fig. 6, it can be seen that the modeling results are consistent with the existing results with the maximum deviation in efficiency less than 3.9%. The slight deviations could be caused by the differences in the thermodynamic properties database and the optimization approaches used for calculation. Thus, the model allows for the further parametric analysis and multi-objective optimization.

2.2.1. Non-dominated sorting genetic optimization method In this work, the LTR mass flux, HTR mass flux, LTR enthalpy efficiency, HTR enthalpy efficiency and recompression fraction (i.e. GLTR, GHTR, εLTR, εHTR and ΦRC) are selected as variables. The objectives are the cycle thermal efficiency and total cycle cost composing two-dimensional array [ηth, Ccycle] as the chromosome. The algorithm is started with a randomly generated initial population [ηth, Ccycle]n. Then the non-dominated sorting is performed based on front or crowding distance of each individuals. After the non-dominated sorting, the parent is selected from population by using selection operator, and the first offspring is generated from parent by using crossover and mutation operators. Starting from the second offspring, the parent and child are merged to create a new larger population. Then the algorithm performs the non-dominated sorting, selection, crossover and mutation again to generate new population. The basic operation steps are repeated until the maximum generation number is met. When the algorithm terminates, the non-dominated solution of the final population is the Pareto frontier.

3. Thermo-economic analysis In this section, the comparison of different printed circuit recuperator are first conducted. Then the parametric studies are carried out to investigate the effects of recuperator design parameters on the cycle thermal efficiency and total cycle cost. The input parameters including the design conditions of recompression cycle [10] and the design geometric parameters of S-shaped fin PCHE [34,36] are

2.2.2. TOPSIS decision making method The first step of TOPSIS method is to obtain the ideal solution, Sj+, and non-ideal solution, Sj−, which are both hypothetical. The ideal solution is identified with the point in which each objective function obtains the best (i.e. the maximum cycle thermal efficiency and the minimum total cycle cost). Similarly, the non-ideal solution is identified with the point in which each objective function obtains the worst (i.e. the minimum cycle thermal efficiency and the maximum total cycle cost).

Table 1 Parameters for the validation of cycle modeling approach.

x ij M

Sj − = {(Min (wj yij )|j = 1), (Max (wj yij )|j = 2), i = 1, 2, ...,M }

Di + =

For a multi-objective optimization problem, there is no absolute single optimal solution as the best because several sub-objectives have to be optimized simultaneously. In this study, the NSGA-II algorithm proposed by Deb et al. [39] is used to find a set of optimal solutions called Pareto frontier. As each solution in Pareto frontier is optimal, the decision making method TOPSIS [40] is used to find the final optimal solution with a compromise between the objectives.

∑i = 1 x ij2

(42)

where wj is the weight value of jth objective. In this work, j = 1 corresponds to the benefit attribute which is the cycle thermal efficiency ηth in this work, and j = 2 corresponds to the cost attribute which is the total cycle cost Ccycle. The TOPSIS method consider the Euclidian distance from each solution to the ideal solution and to the non-ideal solution, denoted as Di+ and Di- separately.

2.2. Multi-objective optimization model

yij =

Sj + = {(Max (wj yij )|j = 1), (Min (wj yij )|j = 2), i = 1, 2, ...,M }

(41)

where xij is the jth objective value of the ith variable, yij is the normalized 7

Parameter

Value

Unit

Turbine inlet temperature, Tturb,in Main compressor inlet temperature, Tmc,in Main compressor inlet pressure, Pmc,in Main compressor outlet pressure, Pmc,out Turbine isentropic efficiency, ηturb Main compressor isentropic efficiency, ηmc Recompressor isentropic efficiency, ηrc LTR enthalpy efficiency, εLTR HTR enthalpy efficiency, εHTR Recompression fraction, ΦRC

500–850 32 7.69 20 0.93 0.89 0.89 0.95 0.95 Optimized

°C °C MPa MPa

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there exists an optimal value of the GLTR and GHTR to obtain the tradeoff between the high cycle thermal efficiency and the low total cycle cost, which is demonstrated in Section 4. 3.3. Effect of recuperator enthalpy efficiency Fig. 8(a) shows the effects of the LTR enthalpy efficiency (εLTR) on the cycle thermal efficiency and total cycle cost. As the εLTR increases, both the cycle thermal efficiency and total cycle cost increase significantly. The cycle thermal efficiency increases almost linearly due to the more heat recovery. While the increase of cycle cost is quite small at low εLTR and becomes significant at high εLTR. The increase of εLTR requires a larger recuperator area, leading to the increase of recuperator cost as shown in Fig. 8(c). In addition, the costs of other components such as precooler, and turbine also change apparently with the enthalpy efficiency, due to the variation on the inlet condition of these components. Nevertheless, the increase of cycle cost is dominated by the rise of recuperator cost. Fig. 8(b) and (d) show the effects of the HTR enthalpy efficiency (εHTR) on the cycle thermal efficiency, total cycle cost and component cost. The similar phenomena are observed for HTR. It should be noted that in the same range, the εHTR results in a larger variation of cycle thermal efficiency and total cycle cost as compared to the εLTR, Therefore, the result shows these two objectives are more sensitive to the εHTR due to higher temperature difference in HTR. This result also demonstrates the possibility to find optimal values of εLTR and εHTR with the high cycle thermal efficiency and low total cycle cost as objectives, as discussed in Section 4.

Fig. 6. Comparison of cycle thermal efficiency with the results from Turchi et al. [9].

summarized in Table 2. According to some projections of mature and commercial size turbomachinery, the efficiency of compressor is predicted to be around 89%, and that of turbine is expected to reach 93% [16]. 3.1. Comparison of different channel shapes The cycle performances with recuperators in different channel shapes are compared based on the same recuperator design parameters listed in Table 3. The results of comparison are given in Table 4. It can be seen that the S-shape fin increases cycle efficiency by 1.1% and 4.3% higher than that of straight and zigzag type recuperators, respectively. In terms of the pressure drop and overall heat transfer coefficient of recuperators, the S-shaped recuperator is obviously superior to the straight channel recuperator. While compared with the zigzag type recuperator, the overall heat transfer coefficient of S-shaped recuperator in the same enthalpy efficiency is lower by around 28%, which leads to a slightly larger recuperator volume. Nevertheless, the pressure drop of S-shaped recuperator is 1/5–1/4 lower than that of zigzag recuperator, which is consistent with the results from Ngo et al. [41]. The lower pressure drop in S-shaped recuperator helps reduce the compression power, leading to a higher cycle thermal efficiency. As a result, the S-shaped recuperator is selected for the following analysis.

3.4. Effect of recompression fraction Fig. 9(a) shows the effect of recompression fraction (ΦRC) on the cycle thermal efficiency, total cycle cost and heat capacity ratio of cold stream and hot stream in the LTR (i.e. cp,c·(1 − ΦRC)/cp,h). As shown in Fig. 9(a), the value of ΦRC corresponding to the maximum cycle efficiency is 0.28, where the heat capacity ratio between cold stream and hot stream is 1.05 in LTR. This is consistent with the result from Sarkar et al. [18]. The heat capacity ratio in the vicinity of 1 leads to an extremely proximity between the outlet state of recompressor and that of cold stream in LTR, which can bring about the highest cycle efficiency. A local peak point for the total cycle cost is also obtained at the optimal ΦRC (0.28) for the maximum cycle efficiency. However, the total cycle cost increases again when the value of ΦRC exceeds 0.38 and becomes even higher than the local peak at the optimal ΦRC (0.28). Fig. 9(b) shows the effect of the ΦRC on costs of components. It can be seen the maximum total recuperator cost also occurs at the ΦRC of 0.28. As the ΦRC increases, the total compressor cost increases substantially because more CO2 flows across the recompressor with an increase in the recompression work. In addition, A large ΦRC also means less cold stream

3.2. Effect of recuperator mass flux Fig. 7(a) shows the effects of the LTR mass flux (GLTR) on the cycle thermal efficiency and total cycle cost. As the GLTR increases, both the cycle thermal efficiency and total cycle cost decrease, which indicates the GLTR makes the conflict between the cycle efficiency and investment. The increase of GLTR leads to a rising pressure drop in LTR, which reduces the turbine inlet pressure but increases the turbine outlet pressure due to the fixed inlet and outlet pressures of main compressor. Thus, the turbine work decreases, leading to the drop of cycle efficiency. Fig. 7(c) shows the effects of the GLTR on component costs. It can be noticed that the reduction of cycle cost is attributed to the decrease of total recuperator cost as other component costs remain almost constant. Fig. 7(b) and (d) show the effects of the HTR mass flux (GHTR) on the cycle thermal efficiency, total cycle cost and component costs. The similar phenomena are observed for HTR. While compared to GLTR, the same range of GHTR generates the larger variation of cycle thermal efficiency and total cycle cost, which indicates the cycle thermodynamic and economic objectives are more sensitive to the GHTR. This can be attributed to the higher specific volume of CO2 in HTR. The results indicate that the low mass fluxes of recuperators could produce the desirable high thermal efficiency but also increase the cycle cost. Thus,

Table 2 Fixed parameters of the calculation.

8

Parameter

Value

Unit

Thermal input, Q Turbine inlet temperature, Tturb,in Main compressor inlet temperature, Tmc,in Main compressor inlet pressure, Pmc,in Main compressor outlet pressure, Pmc,out Turbine isentropic efficiency, ηturb Main compressor isentropic efficiency, ηmc Recompressor isentropic efficiency, ηrc Recuperator channel width, wch Recuperator channel depth, dch Plate thickness, tp Fin width, wf Fin angle, θ Fin pitch, pf

10 700 32 7.69 20 0.93 0.89 0.89 2 1 1.5 0.4 52 11.54

MW °C °C MPa MPa

mm mm mm mm ° mm

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optimal value of ΦRC to achieve a tradeoff between the cycle thermal efficiency and total cycle cost.

Table 3 Initial design parameters for comparison of different PCHE channel shapes. Parameter

Value

Unit

700 700 0.35 0.95 0.95

kg/m2 s kg/m2 s

4. Multi-objective optimization LTR mass flux, GLTR (kg/m s) HTR mass flux, GHTR (kg/m2 s) Recompression fraction, ΦRC LTR enthalpy efficiency, εLTR HTR enthalpy efficiency, εHTR 2

The above results indicate that the cycle thermal efficiency and total cycle cost are influenced by several design parameters of printed circuit recuperator and conflict with each other. Therefore, the multi-objective optimization is required to search the optimal values of PCHE design parameters by maximizing the cycle thermal efficiency (ηth) and minimizing the total cycle cost (Ccycle). The specific constrain conditions of decision variables are shown in Table 5 based on the previous studies [9,41,42], while the other input parameters remain the same as those in Table 2.

Table 4 Cycle performances with different channel shapes. Performance

Straight

Zigzag

S-shape

LTR cold stream pressure drop (kPa) LTR hot stream pressure drop (kPa) HTR cold stream pressure drop (kPa) HTR hot stream pressure drop (kPa) LTR mean overall heat transfer coefficient (W/ m2 K) HTR mean overall heat transfer coefficient (W/ m2 K) LTR pinch point (°C) HTR pinch point (°C) LTR volume (m3) HTR volume (m3) Cycle thermal efficiency

10.78 68.75 62.10 165.0 159.8

24.67 169.8 139.7 363.0 1448

6.305 37.18 32.87 87.77 1034

181.7

1768

1246

9.458 17.57 2.707 4.092 0.4948

9.484 17.83 0.3092 0.4350 0.4795

9.437 17.48 0.3791 0.5410 0.5003

4.1. Optimization results Due to the optimization involves five decision variables in this study, the effect of one variable on the optimization objectives can be compensated by others. Therefore, the same optimization result [ηth, Ccycle] can be generated from different sets of variables [GLTR, GHTR, ΦRC, εLTR, εHTR] through each optimization process. Fig. 10 shows Pareto frontier curve between the cycle thermal efficiency and total cycle cost. The solution convergence reaches after 300 generations evolving with 100 individuals for population. The Pareto frontier creates a set of variables for designers to choose a desirable combination of the recuperator parameters. In Fig. 10, five typical points are highlighted according to different decision-making methods. The optimum values of objectives and decision variables corresponding to these points are listed in Table 6. It can be seen that the optimal solutions for two objectives are 0.4303 ≤ ηth ≤ 0.5380 and 7.468 M$ ≤ Ccycle ≤ 12.31 M$ respectively. The maximum cycle thermal efficiency (0.5380) occurs at point E

flows across LTR to recover heat from hot stream, which increase the heat recovery by LTR. This result leads to a smaller precooler area and reduce the precooler cost significantly. Therefore, the variation of total cycle cost with increasing ΦRC is dominated by the total compressor and precooler cost. The results indicate that the high cycle efficiency requires more cost as ΦRC < 0.38, which means it is possible to find the

Fig. 7. (a) Effects of LTR mass flux on cycle thermal efficiency and total cycle cost, (b) effects of HTR mass flux on cycle thermal efficiency and total cycle cost, (c) effects of LTR mass flux on component costs, (d) effects of HTR mass flux on component costs. 9

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Fig. 8. (a) Effects of LTR enthalpy efficiency on cycle thermal efficiency and total cycle cost, (b) effects of HTR enthalpy efficiency on cycle thermal efficiency and total cycle cost, (c) effects of LTR enthalpy efficiency on component costs, (d) effects of HTR enthalpy efficiency on component costs.

steel is estimated to be 50 $/kg for non-nuclear applications, and be around 30 $/kg for nuclear application [43]. Considering the effects of different applications on the price of PCHE, we assume the price change of ± 20% in the sensitive analysis. Fig. 11 shows Pareto frontier curves at different PCHE unit costs. As compared to the referenced case, increasing the PCHE unit cost would lead to an increase in total cycle cost at the same cycle thermal efficiency on Pareto frontier. However, a 20% increase of PCHE cost only causes the differences of the largest total cycle costs within 7%. This is because the total recuperators cost only accounts for about 18% in the total cycle cost in this study. Table 7 lists the corresponding values of decision variables and objectives. The decision variables linked to each optimal point (point A, point C and point E) have changed with the variation of PCHE cost, especially the mass fluxes of recuperators. Nevertheless, the relationship between the optimal combination of decision variables and preferred objective is unchanged by the variation of PCHE unit cost.

where the maximum total cycle cost (12.31 M$) is obtained, while the minimum total cycle cost (7.468 M$) appears at point A with the minimum cycle thermal efficiency (0.4303). Point B, C and D are solutions in the compromise between the maximum cycle thermal efficiency and the minimum total cycle cost. Note point C (ηth = 0.5100, Ccycle = 9.539 M$) is the solution selected by TOPSIS method, which is the sensible optimum solution. From Table 6, it can be noted that the optimized ranges of the LTR mass flux (GLTR), the HTR mass flux (GHTR) and the recompression faction (ΦRC) on the Pareto frontier are obviously narrowed. The value of GLTR on the Pareto frontier is located between 471.1 kg/m2 s and 584.3 kg/m2 s. While the alternative value of GHTR is generally larger than GLTR, which is located between 529.3 kg/m2 s and 619.4 kg/m2 s. The optimized value of ΦRC is located between 0.2013 and 0.3132, which is mainly related to the optimal enthalpy efficiencies of two recuperators (εLTR and εHTR). The alternative ranges of εLTR and εHTR are between 0.8 and 0.9. It can be noted that as the high cycle thermal efficiency is preferred, the optimal combination of the decision variables is the high recompression fraction (ΦRC), high LTR enthalpy efficiency (εLTR), high LTR mass flux (GLTR), high HTR enthalpy efficiency (εHTR) and low HTR mass flux (GHTR). In contrast, as low cycle cost is preferred, the low recompression fraction (ΦRC), low LTR enthalpy efficiency (εLTR), low LTR mass flux (GLTR), low HTR enthalpy efficiency (εHTR) and high HTR mass flux (GHTR) are required.

5. Conclusion In this work, a multi-objective optimization of sCO2 recompression Brayton cycle is conducted considering printed circuit recuperator designs. The parametric study of five relative design parameters of Sshaped fin PCHE recuperator, including the mass fluxes, enthalpy efficiencies of recuperators and recompression fraction, are conducted to examine their effects on the thermodynamic and economic performances of the sCO2 recompression cycle. The multi-objective optimization is conducted by using NSGA-II algorithm to obtain the Pareto frontier taking the cycle thermal efficiency and total cycle cost as objective functions. The parametric study shows the enthalpy efficiencies of recuperators and recompression fraction have more important effects on the cycle efficiency and cost compared to the recuperator mass fluxes.

4.2. Sensitive analysis The sensitive analysis considering the PCHE cost is carried out by examining the effect of PCHE cost per unit volume on the optimized result based on the PCHE cost used in the above optimization (1.027 $/ m3) [34]. This price can be different in various applications. According to the British PCHE producer Heatric, the price of PCHE in stainless 10

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Table 6 The optimum results of typical points on Pareto frontier. Point

A B C D E

Unit

LTR mass flux, GLTR HTR mass flux, GHTR Recompression fraction, ΦRC LTR enthalpy efficiency, εLTR HTR enthalpy efficiency, εHTR

400–900 400–900 0.2–0.4 0.8–0.99 0.8–0.99

kg/m2 s kg/m2 s

GLTR (kg/ m2 s)

GHTR (kg/ m2 s)

ΦRC

εLTR

εHTR

ηth

Ccycle (M $)

471.1 515.8 538.6 552.8 560.2

619.4 589.3 585.4 588.5 529.3

0.2013 0.2081 0.2286 0.2865 0.3132

0.8008 0.8290 0.9519 0.9696 0.9897

0.8013 0.9106 0.9541 0.9736 0.9900

0.4303 0.4706 0.5100 0.5274 0.5380

7.468 8.305 9.539 10.62 12.31

In terms of the cycle component cost, the five design parameters all have impact on the total recuperator cost. While the increase of the enthalpy efficiencies of the LTR and HTR also lead to rises of turbine and precooler costs. In addition, the increase of the recompression fraction can significantly lead to an increase of the total compressor cost and a decrease of the precooler cost. The Pareto frontier offers a set of optimal solutions for cycle designers. The optimal solutions for two objectives are 0.4303 ≤ ηth ≤ 0.5380 and 7.468 M$ ≤ Ccycle ≤ 12.31 M$, respectively. By using TOPSIS method, the advisable optimum solution is the cycle thermal efficiency of 0.51 and the total cycle cost of 9.539 M$. A correlation between the optimal PCHE design parameters and the preferred objective is identified. As the high cycle thermal efficiency is preferred, the optimal combination of the decision variables are the high recompression fraction (ΦRC), high LTR enthalpy efficiency (εLTR), high LTR mass flux (GLTR), high HTR enthalpy efficiency (εHTR) and low HTR mass flux (GHTR). In contrast, as low cycle cost is preferred, the low recompression fraction (ΦRC), low LTR enthalpy efficiency (εLTR), low LTR mass flux (GLTR), low HTR enthalpy efficiency (εHTR) and high HTR mass flux (GHTR) are required. The results of multi-objective optimization offer a better insight into the printed circuit recuperator design in the sCO2 recompression cycle.

Table 5 Ranges of decision variables. Range

Objectives

Fig. 11. Comparison of Pareto frontier between the cycle thermal efficiency and total cycle cost in different PCHE unit cost cases.

Fig. 9. (a) Effects of recompression fraction on heat capacity ratio, cycle thermal efficiency and total cycle cost, (b) effects of recompression fraction on component costs.

Parameter

Decision variables

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Fig. 10. Pareto frontier between cycle thermal efficiency and total cycle cost.

Acknowledgements This work is supported by Hunan Provincial Natural Science 11

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Table 7 The optimum results of typical points on Pareto frontier in different PCHE unit cost cases. Point

Decision variables

Objectives

GLTR (kg/m s)

GHTR (kg/m s)

ΦRC

εLTR

εHTR

ηth

Ccycle (M$)

2

2

A

−20% PCHE unit cost Basic PCHE unit cost +20% PCHE unit cost

524.8 471.1 807.9

715.2 619.4 604.9

0.2014 0.2013 0.2001

0.8002 0.8008 0.8022

0.8013 0.8013 0.8028

0.4296 0.4303 0.4302

7.384 7.468 7.489

C

−20% PCHE unit cost Basic PCHE unit cost +20% PCHE unit cost

596.6 538.6 783.9

618.0 585.4 595.2

0.2251 0.2286 0.2213

0.9656 0.9519 0.9491

0.9574 0.9541 0.9485

0.5122 0.5100 0.5060

9.407 9.539 9.503

E

−20% PCHE unit cost Basic PCHE unit cost +20% PCHE unit cost

607.1 560.2 769.4

606.9 529.3 590.7

0.3120 0.3132 0.3137

0.9895 0.9897 0.9882

0.9897 0.9900 0.9900

0.5360 0.5380 0.5340

11.54 12.31 12.45

Foundation of China (No. 2016JJ2144) and National Science Foundation of China (51606225).

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