Applied Energy 87 (2010) 1317–1324

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Applied Energy

journal homepage: www.elsevier .com/locate /apenergy

Parametric optimization design for supercritical CO2 power cycle using geneticalgorithm and artificial neural network

Jiangfeng Wang a, Zhixin Sun a, Yiping Dai a,*, Shaolin Ma b

a Institute of Turbomachinery, Xi’an Jiaotong University, No.28 Xianning West Road, Xi’an 710049, PR Chinab Dongfang Steam Turbine Works, Deyang 618201, PR China

a r t i c l e i n f o

Article history:Received 16 April 2009Received in revised form 16 June 2009Accepted 28 July 2009Available online 20 August 2009

Keywords:Artificial neural networkGenetic algorithmOptimizationPower cycleSupercritical CO2

Waste heat recovery

0306-2619/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.apenergy.2009.07.017

* Corresponding author. Tel./fax: +86 029 8266870E-mail address: freego810211@gmail.com (Y. Dai)

a b s t r a c t

Supercritical CO2 power cycle shows a high potential to recover low-grade waste heat due to its bettertemperature glide matching between heat source and working fluid in the heat recovery vapor generator(HRVG). Parametric analysis and exergy analysis are conducted to examine the effects of thermodynamicparameters on the cycle performance and exergy destruction in each component. The thermodynamicparameters of the supercritical CO2 power cycle is optimized with exergy efficiency as an objective func-tion by means of genetic algorithm (GA) under the given waste heat condition. An artificial neural net-work (ANN) with the multi-layer feed-forward network type and back-propagation training is used toachieve parametric optimization design rapidly. It is shown that the key thermodynamic parameters,such as turbine inlet pressure, turbine inlet temperature and environment temperature have significanteffects on the performance of the supercritical CO2 power cycle and exergy destruction in each compo-nent. It is also shown that the optimum thermodynamic parameters of supercritical CO2 power cyclecan be predicted with good accuracy using artificial neural network under variable waste heat conditions.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

In recent years, there have been a large number of waste heatsbeing released into the environment, such as exhaust gases fromturbines and engines and waste heat from industrial plants, whichlead to serious environmental pollution. Therefore, more and moreattention has been paid to low-grade waste heat recovery for itspotential in reducing fossil fuel consumption and alleviating envi-ronmental problems.

Supercritical CO2 power cycle shows a high potential to recoverlow-grade waste heat. Because there is a better temperature glidematching between heat source and working fluid in the heat recov-ery vapor generator, and CO2 can easily reach its supercritical state(the critical pressure and temperature of CO2 are 7.38 MPa and31.1 �C, respectively). Supercritical CO2 power cycle also showsno pinch limitation in the heat recovery vapor generator. In addi-tion, CO2 is inexpensive, non-toxic, non-explosive, and abundantin nature, and the knowledge of its thermodynamic properties issufficient. Therefore, some researchers has explored the supercrit-ical CO2 power cycle.

Chen et al. [1] examined the performance of the CO2 trans-critical power cycle utilizing energy from low-grade waste heatin comparison to an Organic Rankine cycle (ORC) using R123 as

ll rights reserved.

4..

working fluid. They found that when utilizing the low grade heatsource with equal mean thermodynamic heat rejection tempera-ture, the carbon dioxide trans-critical power cycle had a slightlyhigher power output than the ORC. Zhang et al. investigated a ther-modynamic cycle powered by solar energy for both power andheat generation using supercritical carbon dioxide as a workingfluid on theoretical aspect [2–4], and they examined the effectsof various design conditions and climate conditions on the perfor-mances of this CO2-based Rankine cycle. They also set up an exper-imental system to validate the feasibility of this supercriticalcarbon dioxide cycle [5–8]. Cayer et al. [9] expressed a detailedanalysis of a carbon dioxide trans-critical power cycle using anindustrial low-grade stream of process gases as its heat source.They examined the effect of high pressure on the thermal effi-ciency, exergy efficiency, total UA values (UA is the product ofthe overall heat transfer coefficient by the area) and total heat ex-change surface with fixed temperature and mass flow rate of theheat source, fixed maximum and minimum temperatures in thecycle and a fixed sink temperature.

Up to the present, none of the published studies on the super-critical CO2 power cycle focused on the parameter optimizationto convert low-grade waste heat to useful work as much as possi-ble under the given waste heat condition. So, the aim of this studyis to conduct the parameter optimization for the supercritical CO2

power cycle to recover as much waste heat as possible from indus-trial production.

Nomenclature

d expected outputE exergy, kWEr errorh enthalpy, kJ kg�1

I exergy destruction, kWIs sum of the weighted inputsm mass flow rate, kg s�1

o actual outputp pressure, MPaQ heat addition, kWs entropy, kJ kg�1 K�1

t temperature, �CT temperature, Kw weight valueW power, kW

Greek symbolsg efficiency, learning ratel momentum factor

SubscriptsCND condenserexg second law of thermodynamicsg waste heatHRVG heat recovery vapor generatorin inputloss exergy lossout outputPUMP pumpthm first law of thermodynamicsTBN turbine0 environment state1,2,3,4 state point

1318 J. Wang et al. / Applied Energy 87 (2010) 1317–1324

In the present paper, based on the examination of the effects ofthe major thermodynamic parameters on the system performanceand the exergy destruction in each component for the supercriticalCO2 power cycle, a parameter optimization is conducted to maxi-mize the exergy efficiency by means of genetic algorithm underthe given waste heat condition. Because the parametric optimiza-tion using genetic algorithm takes a long time to get the optimumvalues of thermodynamic parameters, which is not competent forconducting the online optimization operation, an artificial neuralnetwork with the multi-layer feed-forward network type andback-propagation training is also used to achieve parametric opti-mization design rapidly.

Fig. 1. Schematic diagram of the supercritical CO2 power system.

Fig. 2. T–s diagram of the supercritical CO2 power system.

2. Thermodynamic analysis of supercritical CO2 power cycle

The schematic diagram and T–s diagram of the supercritical CO2

power system are shown in Figs. 1 and 2, respectively. The cycle iscomposed of following processes.

Process 2–3: a constant-pressure heat absorption process in theheat recovery vapor generator;Process 3–4: a non-isentropic expansion process in the turbine;Process 4–1: a constant-pressure heat rejection process in thecondenser;Process 1–2: a non-isentropic compression process in thepump.

For the cycle performance simulation, the following assump-tions are also made:

(1) The system reaches a steady state, and pressure drop inpipes and heat losses to the environment in the condenser,heat recovery vapor generator, turbine and pump areneglected.

(2) The condenser outlet state is saturated liquid, and its tem-perature is assumed to be approximately 6 �C higher thanthe environment temperature.

(3) The isentropic efficiency of the turbine and pump are 75%and 70%, respectively.

In the HRVG, when the turbine inlet parameters such as pres-sure and temperature are given, the turbine inlet state is obtained.

The pump outlet temperature could be obtained by pump model.By giving the temperature difference between pump outlet andwaste heat exhaust from the HRVG, the temperature of waste heatexhaust could be obtained. The heat transfer process can be de-scribed by the energy balance.

mgðhin;g � hout;gÞ ¼ mðh3 � h2Þ ð1Þ

J. Wang et al. / Applied Energy 87 (2010) 1317–1324 1319

The absorption heat from the waste heat to the working fluids is

Q in ¼ mðh3 � h2Þ ð2Þ

In the turbine, the isentropic efficiency of the turbine can be ex-pressed as

gTBN ¼h3 � h4

h3 � h4sð3Þ

The generating power can be given as

WTBN ¼ mðh3 � h4Þ ð4Þ

In the condenser, the rejecting heat to the environment is expressedas

Q out ¼ mðh4 � h1Þ ð5Þ

In the pump, the isentropic efficiency of the pump can be expressedas

gPUMP ¼h2s � h1

h2 � h1ð6Þ

The work input by the pump is

WPUMP ¼ mðh2 � h1Þ ð7Þ

The thermal efficiency of the supercritical CO2 power cycle is de-fined on the basis of the first law of thermodynamics as the ratioof net power output to the heat addition.

gthm ¼WTBN �WPUMP

Q inð8Þ

From the viewpoint of the first law of thermodynamics and energyconservation used to determine the overall thermal efficiency, workand heat are equivalent. On the other hand, based on the second lawof thermodynamics, exergy quantifies the difference between workand heat in terms of irreversibility. Therefore, the exergy efficiencycan evaluate the cycle performance from the energy quality and it ischosen to be the criterion for the cycle performance evaluation torecover low-grade waste heat.

Consider p0 and T0 to be the environment pressure and temper-ature as the specified dead reference state. The following assump-tions are made to calculate the exergy of each state point:

(a) It is assumed that only physical exergies are used for flue gasand steam flows.

(b) Chemical exergies of the substances are neglected.(c) Kinetic and potential exergies of materials are ignored.

The exergy of the state point can be considered as

Ei ¼ m½ðhi � h0Þ � T0ðsi � s0Þ� ð9Þ

The exergy balance for an open thermodynamic system can be ex-pressed as [10]X

Ein �X

Eout ¼ I ð10Þ

The exergy destruction in the HRVG can be given as

IHRVG ¼ Ein þ E2 � Eout � E3 ð11Þ

The exergy destruction in the turbine can be given as

ITBN ¼ E3 �WTBN � E4 ð12Þ

The exergy destruction in the condenser can be given as

ICND ¼ E4 þ E5 � E1 � E6 ð13Þ

The exergy destruction in the pump can be given as

IPUMP ¼WPUMP þ E1 � E2 ð14Þ

The exergy efficiency of the supercritical CO2 power cycle can begiven as

gexg ¼Ein �

PI � Eloss

Einð15Þ

where Ein is the exergy of waste heat fluid which is input to the cy-cle and Eloss is the exergy which is carried by the exhaust and cool-ing water.

3. Methodology for parametric optimization

In order to recover as much waste heat as possible, it is neces-sary to optimize the performance of the supercritical CO2 powercycle. The exergy efficiency is selected as objective function forparameter optimization.

3.1. Genetic algorithm

The GA, which is presented firstly by professor Holland [11], is astochastic global search method that simulates the natural biolog-ical evolution. Based on the Darwinian survival-of-fittest principle,the genetic algorithm operates on a population of potential solu-tions to produce better and better approximations to the optimalsolution. The GA differs from more traditional optimization tech-niques because it involves a search from a population of solutionsand not from a single point and it can prevent the convergence tosuboptimal solutions.

The GA encodes a potential solution to a specific domain prob-lem on a simple chromosome-like data structure (which consti-tutes an individual), where genes are parameters of the problemto be solved. In the present study, the float-point coding is usedin parameter optimization for the supercritical CO2 power cycle.Each chromosome vector is coded as a vector of floating pointnumbers of the same length as the dimension of the search space.Chromosome is defined as a real number vector, X = (x1, x2, . . ., xn),xi 2 R, i = 1,2, . . ., n, where i is ith parameter for the supercriticalCO2 power cycle, n is the number of optimizing parameters.

The GA uses fitness function to evaluate adaptability of individ-ual without external information in the evolution search. Theadaptability is expressed by the fitness value. A bigger fitness valuemeans a better adaptability subjected to constraints and a betterviability of the individual. Fitness function which is not con-strained by definition domain, continuity and differentiability, re-quires that the objective function is defined as a form of non-negative maximum. In this optimization, the exergy efficiency isselected as the fitness function.

The GA operators include selection operator, crossover operatorand mutation operator. Selection operator is responsible for select-ing the parents to create the next generation of solutions. The par-ent is chosen with a probability based on its fitness. The higher thefitness is, the higher the probability of selection is. The rank-basedmodel is selected for this optimization in the supercritical CO2

power cycle.Crossover operator is the basic operator for producing new

chromosomes. It produces new individuals that have some partsof both parent’s genetic material. The simple arithmetic crossoveris applied to this optimization problem due to very simple opera-tion, which is presented as follows:

c1 ¼ af1 þ ð1� aÞf2

c2 ¼ af2 þ ð1� aÞf1

�ð16Þ

where a is a random number between 0 and 1, f1 and f2 are parentsindividuals which are selected to crossover each other, c1 and c2 arechildren individuals which are produced by crossover.

1320 J. Wang et al. / Applied Energy 87 (2010) 1317–1324

Mutation is needed because even if selection and crossoversearch new solutions together, they tend to cause rapid conver-gence and there is a danger of losing potentially useful geneticmaterial. The role of mutation in GA is to restore lost or unexploredgenetic material into the population to prevent the premature con-vergence of the GA to suboptimal solutions. Random mutation isadopted to optimize the parameters for the supercritical CO2

power cycle. It is achieved by selecting individuals from the rangeof the parameter according to mutation probability.

Fig. 4. Schematic diagram of a multi-layer feed forward neural network.

3.2. Artificial neural network

ANN is a computational system that simulates the microstruc-ture of a biological neurons system. It mimics the learning processof a natural brain, organizes itself to gain knowledge from givenexamples and applies the knowledge to solve new problems. Itoperates like a ‘‘black box”, only cares about the inputs and out-puts, and requires no detailed information about the system.ANN is able to handle noisy and incomplete data, and deal withnon-linear problems. Once it is trained it can perform predictionat high speed for highly complex and ill-defined problems, justas humans usually decide on an intuitional basis. Because of itsadvantages, ANN has been widely used in diverse applications suchas classification, forecasting, control systems, optimization anddecision making [12].

The ANN consists of many units that represent neurons. An arti-ficial neuron architecture is shown in Fig. 3. Each unit is a basicunit of the information process. Units are interconnected via linksthat contain weight values. Weight values help the neural-networkto express knowledge. Each input is multiplied by a connectionweight and summed, then passes through a transfer function togenerate a result, and finally the output is obtained. In ANN, thedata used as inputs is transmitted through the network, layer bylayer, and a set of outputs is obtained.

The multi-layer feed forward (MLFF) neural network is the mostwidely used network. A MLFF neural network typically employsthree or more layers for the architecture: an input layer, an outputlayer, and at least one hidden layer, shown as in Fig. 4. In MLFF net-works, neurons are arranged in layers with connectivity betweenthe neurons of different layers. The layer that receives inputs iscalled the input layer, and the layer that gives the output is calledthe output layer. Other layers, as they do not receive any direct in-put or contribute to output directly, are called hidden layers. How-ever, it is not necessary to have more than one hidden layer, sinceit has been proved that one hidden layer is enough to approximateany continuous function as long as it has a sufficient number ofneurons [13].

w1

w2

wn

yxiwi

x2

xn

f

weights

inputs

output

transferfunction

∑

x1

Fig. 3. An artificial neuron model.

The transfer function for neurons in a MLFF network can be lin-ear or non-linear. A sigmoid function is a widely used non-linearactivation function whose output lies between 0 and 1 and is de-fined as:

f ðxÞ ¼ 11þ e�x

ð17Þ

Once a neural network is configured, a training process must bedone to perform a particular function. Training is normally accom-plished through an adaptive procedure or algorithm that adjustsweights of the connections between the neurons, either from theinformation outside the network or by the neurons themselves inresponse to the input. The ANN reads the input and output valuesin the training data set and changes the values of the weights tominimize the differences between the actual outputs and expectedoutputs using a learning algorithm (e.g., back-propagation algo-rithm). Initially, due to the random weights assigned randomly tothe connections, the differences between the actual outputs and ex-pected outputs are usually large, but through many training cycles,the differences are reduced. If a satisfactory level of accuracy isreached, the training stops, and the network uses the weights tomake decisions in test data.

There are different kinds of learning algorithms. One popularalgorithm is the back-propagation algorithm. Back-propagationtraining is a gradient descent algorithm. It tries to improve the per-formance of the neural network by reducing the total error bychanging the weights along its gradient.

The learning of ANN is a procedure of modifying the weights.Fig. 5 shows the schematic diagram of error back propagation.The error for each neuron is the difference squared between the ex-pected output and the actual output, defined as:

Erkþ1m ¼ 1

2okþ1

m � dm� �2 ð18Þ

where okþ1m is the actual output for each neuron and dm is the ex-

pected output. Each weight w(N + 1) is updated from its previousstate value w(N) according to

1, −kkjiw kk

mjw ,1+

1+kmEk

jE

Fig. 5. Schematic diagram of error back propagation.

Fig. 6. The effect of turbine inlet pressure on the exergy efficiency for different heatsource temperature.

J. Wang et al. / Applied Energy 87 (2010) 1317–1324 1321

wk;k�1ji ðN þ 1Þ ¼ wk;k�1

ji ðNÞ þ Dwk;k�1ji ðNÞ ð19Þ

where Dw(N) is the incremental change in the weight, which can bewritten as

Dwk;k�1ji ðNÞ ¼ �gdk

j ok�1i ð20Þ

where g is the learning rate, o is the corresponding output, and dkj is

defined as@Erk

j

@okj

f 0ðIskj Þ, Is is the sum of the weighted inputs.

The weight change Dw(N) in the output and hidden layer neu-rons can be calculated using Eqs. (18) and (19), respectively.

Dwkþ1;kmj ðNÞ ¼ �gdkþ1

m okj ¼ �gðokþ1

m � dmÞf 0ðIskþ1m Þok

j ð21Þ

Dwk;k�1ji ðNÞ ¼ �gdk

j ok�1i ¼ �g

Xm

dkþ1m wkþ1;k

mj

!f 0ðIsk

j Þok�1i ð22Þ

Since back-propagation employs a form of gradient descent, it isvery easy for the training process to get trapped in a local minimum.However, it can be avoided by adding a momentum term to theweight change as follows:

DwðN þ 1Þ ¼ �gdoþ lDwðNÞ ð23Þ

where l is the momentum factor. Thus, the new value of weight be-comes equal to the previous value of the weight plus the weightchange, which includes the momentum term. This training methodis known as back-propagation with momentum.

A higher learning rate has the advantage of faster learning, butit may cause the weights to bounce around error minimum, thusfailing to learn properly. In contrast, a small learning rate is moresafe, but it drives the learning slowly. A momentum term is widelyapplied to accelerate the learning and reduce an oscillation. Appro-priate values of these parameters aid the network learning.

The configuration of the multilayer ANN has to be determinedby experience since there are no definitive rules to select the num-ber of hidden layers and the number of neurons in each hiddenlayer. In the present study, the input of the system is selected asheat source temperature, and the outputs are selected as turbineinlet pressure, turbine inlet temperature and exergy efficiency.So, the input layer in ANN has one neuron for the heat source tem-perature, the hidden layer has six neurons, and the output layerhas three neurons for the turbine inlet pressure, the turbine inlettemperature and the exergy efficiency. Neurons in input layer haveno transfer function. Logistic sigmoid (logsig) transfer function hasbeen used in hidden layer and pureline transfer function in outputlayer.

4. Results and discussion

The supercritical CO2 power cycle can be heated by the wasteheat from industrial production, solar energy, geothermal heat orany other heat sources. In the present study, the waste heat, whichis composed of 70% steam and 30% air, is used as heat source tosimulate the supercritical CO2 power cycle. The initial parametersused to simulate the supercritical CO2 power cycle are shown inTable 1.

Table 1Initial parameters for simulating the supercritical CO2 power cycle.

Environment temperature (�C) 15.00Environment pressure (MPa) 0.10135Heat source mass flow rate (kg/s) 210Turbine isentropic efficiency (%) 75.00Pump isentropic efficiency (%) 70.00Cooling water initial temperature (�C) 15.00Cooling water mass flow rate (kg/s) 2000

The simulation of the supercritical CO2 power cycle is carriedout using a simulation program written in Fortran by authors.The thermodynamics properties of CO2 were calculated by REF-PROP 6.01 [14] developed by the National Institute of Standardsand Technology of the United States.

The parametric analysis is performed to evaluate the effects ofeach key parameter on the supercritical CO2 power cycle, such asturbine inlet pressure, turbine inlet temperature and so on.

Fig. 6 shows the effect of turbine inlet pressure on the exergyefficiency for different heat source temperature while the otherparameters are kept constant as those in Table 1. It can be seen thatthe exergy efficiency increases at first to maximum and then de-creases as the turbine inlet pressure increases. It is known thatthe enthalpy drop across the turbine increases as the pressure ratioincreases, thus the turbine power output increases. By subtractingpump input from the turbine power output, the net power outputincreases. This is why the exergy efficiency increases at first. Butthe enthalpy gains from an increased pressure ratio do not makeup for the decrease in vapor flow rate generated by HRVG, thusthe net power output decreases afterwards, resulting in a decreasein exergy efficiency. In addition, it is found that the exergy

Fig. 7. The effect of turbine inlet temperature on the exergy efficiency for differentheat source temperature.

Fig. 10. The exergy destruction in each component vs. turbine inlet temperature.

Fig. 9. The exergy destruction in each component vs. turbine inlet pressure.

1322 J. Wang et al. / Applied Energy 87 (2010) 1317–1324

efficiency increases with an increasing in heat source temperature.It is obvious that higher heat source temperature conforms to high-er exergy efficiency.

Fig. 7 shows the effect of turbine inlet temperature on the exer-gy efficiency for different heat source temperature while the otherparameters are kept constant as those in Table 1. It can be seen thatthe exergy efficiency increases as the turbine inlet temperature in-creases. In addition, it is obvious that the exergy efficiency in-creases with an increase in heat source temperature.

Fig. 8 shows the effect of environment temperature on the exer-gy efficiency for different heat source temperature while the otherparameters are kept constant as those in Table 1. It is obvious thatthe exergy efficiency decreases with an increase in environmenttemperature. The reason for this is that an increase in environmenttemperature results in an increase in condensing pressure, whichreduces the turbine power.

Exergy analysis has been performed to evaluate the exergydestructions in the system. The exergy destruction of each compo-nent is calculated based on the assumptions in Table 1. Figs. 9–11show the effects of thermodynamic parameters on the exergydestruction in each component. It can be seen that the biggestexergy destruction basically occurs in the HRVG followed by theturbine, the condenser or the pump. The major exergy destructionin the HRVG or the condenser is due to heat transfer over a finitetemperature difference, and the exergy destruction in the turbineor pump is due to the friction losses of the flow through the turbineor the pump, the non-ideal adiabatic expansion or compression inthe turbine or the pump, and the corresponding irreversibilities.

The turbine inlet pressure has great effect on the exergydestruction in each component, shown in Fig. 9. As the turbine in-let pressure increases, the exergy destruction in the HRVG de-creases. With the increasing turbine inlet pressure, the exergydestruction in the turbine or pump increases. This is because an in-crease in the turbine inlet pressure results in an increase in pres-sure difference through the turbine or pump, which leads to anincrease in entropy through the turbine or pump. In addition, whenthe turbine inlet pressure increases, the turbine outlet temperaturedecreases. This could result in a decrease in heat transfer temper-ature difference for the condenser, thus, the exergy destruction inthe condenser decreases.

As shown in Fig. 10, it can be seen that as the turbine inlet tem-perature increases, the exergy destruction in the HRVG increases,and the exergy destruction in the condenser increases. This is be-cause an increase in the turbine inlet temperature can result in adecrease in the heat transfer temperature difference for the HRVG.In addition, due to an increase in the turbine inlet temperature, the

Fig. 11. The exergy destruction in each component vs. environment temperature.

Fig. 8. The effect of environment temperature on the exergy efficiency for differentheat source temperature.

turbine outlet temperature increases, thus, the heat transfer tem-perature difference in the condenser increases, resulting in an in-crease in the exergy destruction. It also can be seen that as theturbine inlet temperature increases, the exergy destruction in thepump and the turbine decreases.

As shown in Fig. 11, it can be seen that the exergy destruction inthe turbine and pump decrease as the environment temperature

Table 3The predicted optimum parameters using ANN and the comparison of optimum parameters between ANN and GA.

ANN GA Error (%) ANN GA Error (%) ANN GA Error (%)

Heat source initial temperature (�C) 85 75 65Turbine inlet pressure (MPa) 10.348 10.35 0.0193 9.5505 9.552 0.0157 8.7616 8.764 0.0274Turbine inlet temperature (�C) 74.928 74.97 0.0560 64.927 64.95 0.0354 54.913 54.94 0.0491Exergy efficiency (%) 29.555 29.58 0.0845 27.446 27.46 0.0510 24.687 24.7 0.0526

Table 2The optimization results of the supercritical CO2 power cycle under given waste heat conditions using GA.

Conditions Population size 50 50 50 50Crossover probability 0.95 0.95 0.95 0.95Mutation probability 0.05 0.05 0.05 0.05Stop generation 200 200 200 200Heat source initial temperature (�C) 90 80 70 60

Optimization results Turbine inlet pressure (MPa) 10.751 9.948 9.155 8.376Turbine inlet temperature (�C) 79.93 69.94 59.91 49.96Exergy efficiency (%) 30.44 28.57 26.16 23.00

Additional results CO2 mass flow rate (kg/s) 174.300 151.620 126.855 99.383Heat source exhaust temperature (�C) 39.71 38.40 37.06 35.71Cooling water temperature (�C) 18.89 18.24 17.58 16.92Turbine outlet pressure (MPa) 5.857 5.857 5.857 5.857Turbine outlet temperature (�C) 35.09 31.28 27.75 24.61Pump inlet temperature (�C) 21.00 21.00 21.00 21.00Pump outlet temperature (�C) 29.71 28.40 27.06 25.71Turbine work (kW) 3177.945 2279.117 1510.714 884.826Pump work (kW) 1557.877 1136.782 769.522 462.196Net power output (kW) 1620.068 1142.335 741.192 422.630HRVG heat input (kW) 34129.961 28212.232 22323.017 16455.200Condenser heat rejection (kW) 32509.892 27069.897 21581.825 16032.570Thermal efficiency (%) 4.75 4.05 3.32 2.57Heat source exergy input (kW) 5322.184 3997.787 2833.513 1837.328HRVG exergy destruction (kW) 1094.358 882.877 679.861 483.946Turbine exergy destruction (kW) 995.094 721.677 483.445 285.874Condenser exergy destruction (kW) 578.731 469.873 372.299 281.291Pump exergy destruction (kW) 445.330 326.280 221.787 133.782Exhaust exergy loss (kW) 371.374 303.902 238.902 176.741Cooling water exergy loss (kW) 217.229 150.843 96.027 53.065

J. Wang et al. / Applied Energy 87 (2010) 1317–1324 1323

increases. This is because that a condensing pressure increaseswith the increasing environment temperature, thus, the pressuredifferences through the turbine and the pump decrease, resultingin an increase in entropy through the turbine or pump. In addition,due to the increasing condensing pressure, the turbine outlet tem-perature increases, and this results in an increase in the heat trans-fer temperature difference for the condenser, thus, the exergydestruction in the condenser increases.

From the discussions above, the turbine inlet pressure and tem-perature have great impact on the exergy efficiency of the super-critical CO2 power cycle under a given waste heat condition, so itis necessary to conduct the parameter optimization to obtain themaximum exergy efficiency. It is obvious that the parameters cho-sen for optimization are turbine inlet pressure and turbine inlettemperature, and the objective function is the exergy efficiency.In addition, because the waste heat from industrial plant fluctuatesgreatly than environment condition, the condition of parametricoptimization could be assumed that the waste heat temperaturevaries from 60 �C to 90 �C and the environment temperature is keptto be constant. Table 2 shows the optimization results of the super-critical CO2 power cycle using genetic algorithm under the differ-ent waste heat temperature and the same environment.

Genetic algorithm is a good method to conduct optimizationproblem, but the process is so long that it isn’t suitable for onlineoptimization. In order to obtain the optimum values of thermody-namic parameters for the supercritical CO2 power cycle rapidly toreach the optimum performance under the variable waste heat

conditions, an artificial neural network is used to achieve paramet-ric optimization design.

The parametric optimization results by the GA shown in Table 2are selected as training samples for ANN to complete the trainingprocess. After training, the ANN can achieve the parametric optimi-zation with high speed. The number of training samples is four. Aback-propagation momentum learning method with a learningrate of 0.2 and a momentum factor of 0.95 is adopted. The errorfor back-propagation training is 1 � 10�5 and the training epochis set to 1000. When the ANN finishes the training process, severalexamples gained by GA are selected as test data to predict the opti-mum parameters for the supercritical CO2 power cycle. Table 3shows the predicted optimum parameters using ANN and the com-parison results of the optimum parameters between ANN and GA.It is indicated that the predicted parameters gained by ANN are insubstantial agreement with those gained by GA. The errors be-tween ANN’s data and GA’s data are less than 0.1% with goodaccuracy.

5. Conclusions

In the present study, the effects of parameters on the thermody-namic performance and the exergy destruction in each componentare examined for the supercritical CO2 power cycle in waste heatrecovery field. Parameter optimization is conducted with exergyefficiency as objective function by means of genetic algorithmunder different waste heat conditions. An artificial neural network

1324 J. Wang et al. / Applied Energy 87 (2010) 1317–1324

with the multi-layer feed-forward network type and back-propa-gation training is used to achieve parameter optimization designrapidly.

From the discussions above, it can be concluded that the wasteheat condition such as heat source temperature and environmenttemperature, and the major thermodynamic parameters such as tur-bine inlet pressure, turbine inlet temperature have significant effectson the exergy efficiency of the supercritical CO2 power cycle.

It is also shown that the thermodynamic parameters of thesupercritical CO2 power cycle can be predicted with good accuracyby using artificial neural network under variable waste heat condi-tions to reach the optimum performance.

Acknowledgement

The study presented in this paper is financially supported byNational High Technology Research and Development Program(863 Program) of China (Grant Nos. 2009AA05Z205, 2007AA05Z251).

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