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PROCEEDINGS OF ECOS 2012 - THE 25TH INTERNATIONAL CONFERENCE ON

EEEEFFICIENCY, CCCCOST, OOOOPTIMIZATION, SSSSIMULATION AND ENVIRONMENTAL IMPACT OF ENERGY SYSTEMS

JUNE 26-29, 2012, PERUGIA, ITALY

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Exergy Analysis and Genetic Algorithms for the

Optimization of Flat-Plate Solar Collectors

Soteris A. Kalogirou

Department of Mechanical Engineering and Materials Science and Engineering, Cyprus University of

Technology, P. O. Box 50329, 3603, Limassol, Cyprus, email: Soteris.kalogirou@cut.ac.cy

Abstract:

This paper employs exergy analysis to derive a general equation for the exergy efficiency of flat platecollectors and thus optimize its design and operation. Exergy analysis of a flat-plate solar collector is a moreeffective method of finding the optimum relationship between flow rate and the collector area. The fixed

parameters in this optimization are the collector inlet temperature, available solar radiation, the collectortransmittance-absorptance product and the ambient temperature. The collector heat loss coefficient isestimated according to the collector plate temperature, wind convection loss, number of glass covers,collector inclination, ambient temperature and emittance of collector plate and glass cover. In order to findthe optimum value of this multivariable problem genetic algorithms are used which are based on theprinciples of genetics and survival of the fittest. The optimization parameter is exergy efficiency and theobjective is to maximize this parameter. Genetic algorithms proved suitable and very quick in obtaining therequired results. These results prove that the exergy efficiency of a flat-plate solar collector is maximized forsmall distances between the riser tubes and for very small diameter of these tubes. By using a morepractical distance of 10 centimetres between the tubes, and excluding this parameter from the optimizationprocedure, very small differences are observed in maximum exergy efficiency and if the cost of thematerials is accounted this is a more cost-effective solution. Other findings are that the exergy efficiencyincreases considerably at higher solar radiation and that the transmittance absorptance product affects to a

great extent the exergy efficiency.

Keywords:

Exergy analysis, Genetic algorithms, Flat-plate collectors, Optimisation, Efficiency.

1. IntroductionSolar energy collectors are special kind of heat exchangers that transform solar radiation energy to

internal energy of a transport fluid. Flat plate collectors are the most popular type of solar devises

for low temperature applications. The main use of these collectors is in solar water heating systems

operating at maximum temperatures of 80-90C. Most of these systems are operating

thermosiphonically at very small flow rate created by the small density differences between the hot

and cold water. A number of researchers have used exergy analysis to design flat plate collectors.Badescu [1] optimized the width and thickness of the fins of a flat-plate collector by minimizing the

cost per unit useful heat flux. The proposed procedure allows computation of the necessary

collection surface area. A rather involved, but still simple, flat plate solar collector model is used in

the calculations. Model implementation requires a specific geographical location with a detailed

meteorological database available. Fins of both uniform and variable thickness were considered.

The optimum fin cross section is very close to an isosceles triangle. The fin width is shorter and the

seasonal influence is weaker at lower operation temperatures. Fin width and thickness at the base

depend on the season. The optimum distance between the tubes is increased by increasing the inlet

fluid temperature, and it is larger in the cold season than in the warm season.

Badescu [2] also considered the best operation strategies for open loop flat-plate solar collectorsystems. A direct optimal control method (the TOMP algorithm) is implemented. A detailed

collector model and realistic meteorological data from both cold and warm seasons are used. The

maximum exergetic efficiency is low (usually less than 3%), in good agreement with experimental

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measurements reported in literature. The optimum mass-flow rate increases near sunrise and sunset

and by increasing the fluid inlet temperature. The optimum mass-flow rate is well correlated with

global solar irradiance during the warm season. Also, operation at a properly defined constant mass-

flow rate may be close to the optimal operation.

Torres-Reyes et al. [3] presented a procedure to establish the optimal performance parameters for

the minimum entropy generation during the collection of solar energy. The Entropy Generation

Number, Ns, and the criterion for the optimal thermodynamic operation of a collector under non-isothermally, finite-time conditions, are reviewed. The Mass Flow Number, M, corresponding to the

optimum flow of working fluid as a function of the solar collection area, is also considered. A

general method for the preliminary solar collector design, based on Ns, M and the SunAir or

stagnation temperature, is developed. This last concept is defined as the maximum temperature that

the collector reaches at non-flow conditions for a given geographic location, geometry and

construction materials. The thermodynamic optimization procedure was used to determine the

optimal performance parameters of an experimental solar collector.

Torres-Reyes et al. [4] also presented the thermodynamic optimization of flat plate collectors based

on the first and the second law, developed to determine the optimal performance parameters and to

design a solar to thermal energy conversion system. An exergy analysis is presented to determine

the optimum outlet temperature of the working fluid and the optimum path flow length of solar

collectors with various configurations. The collectors used to heat the air flow during solar-to-

thermal energy conversion are internally arranged in different ways with respect to the absorber

plates and heat transfer elements. The exergy balance and the dimensionless exergy relationships

are derived by taking into account the irreversibilities produced by the pressure drop in the flow of

the working fluid through the collector. Design formulas for different air duct and absorber plate

arrangements are obtained.

The use of genetic algorithms (GAs) for the optimal design of solar collectors is well known [5, 6].

Genetic algorithms have been used as a design support tool by Loomans and Visser [7] for the

optimization of large hot water systems. The tool calculates the yield and the costs of solar hot

water systems based on technical and financial data of the system components. The geneticalgorithm allows for the optimization of separate variables as the collector type, the number of

collectors, the heat storage capacity and the collector heat exchanger area.

Kalogirou [8] used also genetic algorithms together with a neural network for the optimization of

the design of solar energy systems. The method is presented by means of an example referring to an

industrial process heat system. The genetic algorithm is used to determine the optimum values of

collector area and the storage tank size of the system which minimize the solar energy price.

According to the author the solution reached is more accurate than the traditional trial and error

method and the design time is reduced substantially.

Krause et al. [9] presented a study in which two solar domestic hot water systems in Germany have

been optimized by employing validated TRNSYS models in combination with genetic algorithms.Three different optimization procedures are presented. The first concerns the planning phase. The

second one concerns the operation of the systems and should be carried out after about one year of

data is collected. The third one examines the daily performance by considering predictions of

weather and hot water consumption and actual temperature level in the storage tank.

The objective of the present work is to maximize the exergy efficiency of flat-plate collectors. The

fixed parameters in this optimization are the collector inlet temperature, available solar radiation,

the collector transmittance-absorptance product and the ambient temperature. The collector heat

loss coefficient is estimated according to the collector plate temperature, wind convection loss,

number of glass covers, collector inclination, ambient temperature and emittance of collector plate

and glass cover. For this purpose an evolution strategy based on genetic algorithms is used to

determine the optimum solution.

2. Energy Analysis

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In this section various relations that are required in order to determine the useful energy collected

and the interaction of the various constructional parameters on the performance of a collector are

presented.

The useful energy collected from a collector can be obtained from the following formula [10]:

( ) ( )u c R t L f ,i a p f ,o f ,iQ A F G ( ) U T T mc T T = = & (1)where FR is the heat removal factor given by [10]:

p L cR

c L p

mc U F'AF 1 Exp

A U mc

=

&

&(2)

In (2) F is the collector efficiency factor which is calculated by considering the temperature

distribution between two pipes of the collector absorber and by assuming that the temperature

gradient in the flow direction is negligible [10]. This analysis can be performed by considering the

sheet-tube configuration shown in Fig. 1, where the distance between the tubes is W, the tube

diameter is D, and the sheet thickness is .

Fig. 1. Schematic diagram of a flat-plate sheet and tube configuration

As the sheet metal is usually made from copper or aluminum which are good conductors of heat, the

temperature gradient through the sheet is negligible, therefore the region between the centerline

separating the tubes and the tube base can be considered as a classical fin problem. By following

this analysis the equation to estimate F is given by [10]:

[ ]

+++

=

fiibL

L

hDCFDWDUW

UF

11

)(

1

1

' (3)

In (3), Cb is the bond conductance which can be estimated from knowledge of the bond thermal

conductivity, the average bond thickness, and the bond width. The bond conductance can be very

important in accurately describing the collector performance and generally it is necessary to have

good metal-to-metal contact so that the bond conductance is greater that 30 W/mK and preferably

the tube should be welded to the fin.

Factor F in (3) is the standard fin efficiency for straight fins with rectangular profile, obtained from:

[ ]

2/)(

2/)(tanh

DWn

DWnF

= (4)

where n is given by:

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k

Un L= (5)

The collector efficiency factor is essentially a constant factor for any collector design and fluid flow

rate. The ratio of UL to Cb, the ratio of UL to hfi, and the fin efficiency F are the only variables

appearing in (3) that may be functions of temperature. For most collector designs F is the most

important of these variables in determining F. The factor F is a function of UL and hfi, each of

which has some temperature dependence, but it is not a strong function of temperature.

Additionally, the collector efficiency factor decreases with increased tube center-to-center distances

and increases with increases in both material thicknesses and thermal conductivity. Increasing the

overall loss coefficient decreases F while increasing the fluid-tube heat transfer coefficientincreases F.

Therefore it is obvious from the above analysis that by increasing F more energy can be intercepted

by the collector. By keeping all other factors constant increase of F can be obtained by decreasing

W. However, decrease in W means increased number of tubes and therefore extra cost would be

required for the construction of the collector.

The collector efficiency is found by dividing Qu given in (1), by the incident radiation AcGt. Bydoing so the following Equation is obtained:

( )p f ,o f ,if ,i aR R L

t c t

mc T TT TF ( ) F U

G A G

= =

&

(6)

By plotting against /Gt a straight line is obtained with the slope equal to FRUL, called the loss

coefficient and the intercept on the y-axis equal to FR(), called optical efficiency.

Generally the efficiency of a solar thermal system increases by increasing the flow rate. This

increase is asymptotic, i.e., the increase is rapid at small flow rates and becomes almost asymptotic

to a certain maximum value at higher values of flow rate. In a similar way the collector outlet

temperature increases with the collector area and decreases with increasing flow rate and vice versa.

This creates difficulties in selecting appropriate values of flow rate and collector area.

In a real system the variation of the fluid inlet temperature depends to a great extent on the storage

tank configuration, thermal load demand and the consequent make-up water to the storage tank

which affects the fluid inlet temperature to the collector.

3.Exergy AnalysisAccording to Kalogirou [10] the temperature at any position y at a fluid inlet temperature T f,i is

given by:

Lf a f ,i a

L L p

U F ' NWyS ST T T T ExpU U mc

= + &(7)

For y=L the collector area is given by Ac = NWL. If Tf,i = Ta the fluid temperature variation at the

exit from the solar collector is:

cLf ,o a

L p

AU F'ST T T 1 Exp

U c m

= = &

(8)

The flow rate of exergy transferred from the sun to the fluid that is heated while crossing the riser

pipe is:

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( ) ( )f f f ,o f ,i a f ,o f ,iE me m h h T s s = = & & & (9)

Where hf,o - hf,i = cp(Tf,o - Tf,i) is the variation of specific enthalpy. The variation of the specific

entropy is:

f ,o

f ,o f ,i pf ,i

T

s s c ln T

= (10)

Therefore:

f ,o

f p a

f ,i

TE mc T T ln

T

=

& & (11)

And the exergy efficiency is given by dividingf

E& by the available solar radiation Qs,in which is

equal to AcGt:

f ,oa

f ,ifex p

S,in c t

TT T lnTE

mcQ A G

= =

&

&& (12)

Replacing given by (8) in (12), the following relation for the exergy efficiency can be obtained:

p L c L cex a

c t L p a L p

mc U F'A U F'AS S1 Exp T ln 1 1 Exp

A G U mc T U mc

= +

&

& &(13)

As can be seen from the exergy analysis, a general equation for the exergy efficiency of flat plate

collectors is derived. This efficiency depends on the values of flow rate, collector area and collector

efficiency factor. The latter depends on the distance between consequent riser tubes, the diameter of

the riser tubes, the collector fin efficiency, the internal riser tube diameter (depends on outside tube

diameter) and the convection coefficient inside the tube - which depends on the mass flow rate and

tube inside diameter, which affect the Reynolds number and thus the Nusselt number.

4. Method DescriptionThe objective of this work is to find the parameters that maximize the exergy efficiency. In order to

do this a numbers of parameters need to be considered as constants. These are shown in Table 1.

Table 1. Constant parameters used in exergy optimization

Parameter Value

Fluid specific heat, cp

Available solar radiation, Gt

Transmittance-absorptance product, ()

Ambient temperature, Ta

Bond conductance, Cb

Absorbing plate thickness,

Thermal conductivity of fluid, k

4185 J/kg-K

500-1000 W/m2

0.60-0.90

25C

100 W/m2-K

0.5 mm

410 W/m-K

From these constant parameters a number of other parameters are evaluated. These are:

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1. Heat loss coefficient, UL, estimated from [10]:

( )( )

( )

2 2

p a p a

Lg g

g0.33

gp g pp a

p g w

T T T T1U

N 2N f 11N

0.05N 1T TC 1

T N f h

+ += +

+ +

+ +

+

(14)

Where:

( )( )2w w gf 1 0.04h 0.0005h 1 0.091N= + + (15)( )2C 365.9 1 0.00883 0.0001298= + (16)

0.6

w 0.4

8.6Vh

L= (17)

This estimation ignores the bottom and edge losses and considers the wind velocity to be 1 m/s,

collector slope, , equal to 45, one glass cover and emittance values of 0.8. The estimation is also

done at a plate temperature Tp to be 20C above the mean collector fluid temperature.

2. Factor n estimated from (5)

3. Fin efficiency F, estimated from (4)

4. The convection heat transfer coefficient inside the pipe, hf,i, estimated from the principles ofheat transfer and according to the type of flow, turbulent or laminar according to the mass flow

rate and the riser tube diameter.

5. The collector efficiency factor F, estimated from (3)

For the above parameters the inputs required are the distance between successive riser tubes, W, the

riser tube diameter, D, and the collector mass flow rate. These are varied during the optimizationprocess until the exergy efficiency is maximized using a genetic algorithm described in the

following section.

It should be noted that a proper exergy optimization should include optimization of both the

process parameters and the size of equipment by considering two key parameters, i.e., exergy

efficiency and cost, which include the capital investment (proportional to the collector area) and

operating cost due to the pressure drop. However, in this work the pressure drop is not included as

the collector is assumed to operate at very small flow rate (similar to the thermosyphonic one),

therefore, no exergy consumption for the fluid pumping is included in the analysis and this should

shift the optimum to the smaller diameters of tubes and distances between the tubes. This

assumption will be proved by the obtained results. Additionally, no other cost-connected parameter

is included in the optimization, such as exergy cost of equipment, so the optimization related to thecollector area only affects the amount of radiation collected by the system.5. Genetic AlgorithmThe genetic algorithm (GA) is a model of machine learning, which derives its behavior from a

representation of the processes of evolution in nature. This is done by the creation within a

machine/computer of a population of individuals represented by chromosomes. Essentially these are

a set of character strings that are analogous to the chromosomes that we see in the DNA of human

beings. The individuals in the population then go through a process of evolution.

It should be noted that evolution as occurring in nature or elsewhere is not a purposive or directed

process, i.e., there is no evidence to support the assertion that the goal of evolution is to produceMankind. Indeed, the processes of nature seem to end to different individuals competing for

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resources in the environment. Some are better than others are, those that are better are more likely

to survive and propagate their genetic material.

In nature, the encoding for the genetic information is done in a way that admits asexual

reproduction, which typically results in offspring that are genetically identical to the parent. Sexual

reproduction allows the creation of genetically radically different offspring that are still of the same

general species.

In an over simplified consideration, at the molecular level what happens is that a pair ofchromosomes bump into one another, exchange chunks of genetic information and drift apart. This

is the recombination operation, which in GAs is generally referred to as crossover because of the

way that genetic material crosses over from one chromosome to another.

The crossover operation happens in an environment where the selection of who gets to mate is a

function of the fitness of the individual, i.e., how good the individual is at competing in its

environment. Some GAs use a simple function of the fitness measure to select individuals

(probabilistically) to undergo genetic operations such as crossover or asexual reproduction, i.e., the

propagation of genetic material remains unaltered. This is fitness - proportionate selection. Other

implementations use a model in which certain randomly selected individuals in a subgroup compete

and the fittest is selected. This is called tournament selection. The two processes that mostcontribute to evolution are crossover and fitness based selection/reproduction. Mutation also plays a

role in this process.

GAs are used for a number of different application areas. An example of this would be

multidimensional optimization problems in which the character string of the chromosome can be

used to encode the values for the different parameters being optimized.

In practice, this genetic model of computation can be implemented by having arrays of bits or

characters to represent the chromosomes. Simple bit manipulation operations allow the

implementation of crossover, mutation and other operations.

When the GA is executed, it is usually done in a manner that involves the following cycle [11]:

Evaluate the fitness of all of the individuals in the population. Create a new population by performing operations such as crossover, fitness-proportionate

reproduction and mutation on the individuals whose fitness has just been measured.

Discard the old population and iterate using the new population.

One iteration of this loop is referred to as a generation. More details on genetic algorithms can be

found in Goldberg [12].

The first generation of this process operates on a population of randomly generated individuals.

From there on, the genetic operations, in concert with the fitness measure, operate to improve the

population. Genetic algorithms (GA) are suitable for finding the optimum solution in problems

where a fitness function is present. Genetic algorithms use a fitness measure to determine which

of the individuals in the population survive and reproduce. Thus, survival of the fittest causes goodsolutions to progress. A genetic algorithm works by selective breeding of a population of

individuals, each of which could be a potential solution to the problem. The genetic algorithm is

seeking to breed an individual, which either maximizes, minimizes or it is focused on a particular

solution of a problem. In this case, the genetic algorithm is seeking to breed an individual that

maximizes the exergy efficiency of the solar collector.

The larger the breeding pool size, the greater the potential of it producing a better individual.

However, the fitness value produced by every individual must be compared with all other fitness

values of all the other individuals on every reproductive cycle, so larger breeding pools take longer

time. After testing all of the individuals in the pool, a new generation of individuals is produced

for testing.

A genetic algorithm is not gradient based, and uses an implicitly parallel sampling of the solutions

space. The population approach and multiple sampling means that it is less subject to becoming

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trapped to local minima than traditional direct approaches, and can navigate a large solution space

with a highly efficient number of samples. Although not guaranteed to provide a globally optimum

solution, GAs have been shown to be highly efficient at reaching a very near optimum solution in a

computationally efficient manner.

During the setting up of the GA the user has to specify the adjustable chromosomes, i.e. the

parameters that would be modified during evolution to obtain the maximum or minimum values of

the fitness functions. In this work, the fitness function is exergy efficiency given by (13).Additionally the user has to specify the range of the input parameters called constraints.

The genetic algorithm parameters used in the present work are:

Population size=50Population size is the size of the genetic breeding pool, i.e., the number of individuals contained in

the pool. If this parameter is set to a low value, there would not be enough different kinds of

individuals to solve the problem satisfactorily. On the other hand, if there are too many in the

population, a good solution will take longer to be found because the fitness function must be

calculated for every individual in every generation.

Crossover rate=90%Crossover rate determines the probability that the crossover operator will be applied to a particular

chromosome during a generation.

Mutation rate=1%Mutation rate determines the probability that the mutation operator will be applied to a particular

chromosome during a generation.

Generation gap=96%Generation gap determines the fraction of those individuals that do not go into the next generation.

It is sometimes desirable that individuals in the population be allowed to go into next generation.

This is especially important if individuals selected are the most fit ones in the population.

Chromosome type=continuousPopulations are composed of individuals, and individuals are composed of chromosomes, which are

equivalent to variables. Chromosomes are composed of smaller units called genes. There are two

types of chromosomes, continuous and enumerated. Continuous are implemented in the computer as

binary bits. The two distinct values of a gene, 0 and 1, are called alleles. Multiple chromosomes

make up the individual. Each partition is one chromosome, each binary bit is a gene, and the value

of each bit (1, 0, 0, 1, 1, 0) is an allele. Enumerated chromosomes consist of genes, which can have

more allele values than just 0 and 1.

The genetic algorithm is usually stopped after best fitness remained unchanged for a number of

generations or when the optimum solution is reached. In this work the genetic algorithm was

stopped after best fitness remained unchanged for 75 generations.

6. ResultsThe input parameters (adjustable chromosomes) were used in a genetic algorithm program to find

the values that maximize collector exergy efficiency. The whole model was set up in a

spreadsheet program in which the various parameters are entered into different cells. The optimum

values of the various parameters were used in (13) to estimate the collector exergy efficiency which

is the fitness function that needs to be maximized. The various input parameters are constrained to

vary between certain values. The ones used in this work are shown in Table 2.

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Table 2. Constrains of the input parameters

Parameter Symbol Range

Mass flow rate

Collector area

Distance between riser tubes

Riser tube diameter

m& Ac

W

D

0.001-0.08 kg/s

1-10 m2

0.03-0.15 m

0.004-0.022 m

The optimum results can be presented graphically or in tables. Here the table presentation is

preferred so as to show the exact values obtained. The results are shown in Tables 3-5. It should be

noted that for each run of the program the optimum solution was reached in less than 5 seconds on a

Pentium 3.2 GHz machine, which is very fast.

Table 3 presents the results of the optimization process with respect to the effect of inlet

temperature and radiation.

Table 3. Results of the optimization process-effect of inlet temperature and radiation

Gt (W/m2) () Tin (C) W (m) D (m) ex (%) Tout (C) Ac (m

2) m& (kg/s)

500 0.82527

30

35

0.030.03

0.03

0.03

0.0040.004

0.004

0.013

3.074.85

7.90

13.77

66.0259.86

52.24

43.27

7.499.98

9.61

8.76

0.01060.0196

0.0304

0.0797

According to the results shown in Table 3 by increasing the collector inlet temperature the collector

outlet temperature reduces, the optimum flow rate increases and the exergy efficiency increases. As

can be seen the distance between the riser tubes remains to the minimum value, same as the riser

tube diameter except the last value which differs from the minimum value possible, determined by

the constrains. It is apparent that the optimum collector area reaches a maximum value at the inlet

temperature of 27C and then drops for bigger values. The effect of solar radiation is shown in

Table 4.

Table 4. Results of the optimization process-effect of solar radiation

Gt (W/m2) () Tin (C) W (m) D (m) ex (%) Tout (C) Ac (m

2) m& (kg/s)

500

500

1000

1000

0.8

25

27

25

27

0.03

0.03

0.03

0.03

0.004

0.004

0.004

0.004

3.07

4.85

5.19

6.21

66.02

59.86

95.88

92.22

7.49

9.98

8.24

9.93

0.0106

0.0196

0.0140

0.0190

As can be seen from Table 4 by increasing the solar radiation and keeping the other parameters

constant, the collector outlet temperature increases, as expected, and the same applies to the

optimum collector area and the optimum mass flow rate. Both the riser tube spacing, W, anddiameter, D, remain at their minimum values. The effect of solar radiation and the value of the

transmittance-absorptance product are presented in Table 5.

Table 5. Results of the optimization process-effect of radiation and transmittance absorptance

product

Gt (W/m2) () Tin (C) W (m) D (m) ex (%) Tout (C) Ac (m

2) m& (kg/s)

500

500

1000

1000

0.6

0.9

0.6

0.9

27

0.03

0.03

0.03

0.03

0.004

0.004

0.004

0.004

3.57

5.60

4.10

7.39

49.56

64.56

77.32

99.03

10.0

9.96

9.68

9.26

0.0224

0.0190

0.0180

0.0180

The results presented in Table 5 reveal that the increase of the transmittance-absorptance product

causes increase in the collector outlet temperature, and a slight decrease in the optimum collector

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area and the optimum mass flow rate. The exergy efficiency however almost doubles. It is thus

required to achieve a high value of () as possible in flat-plate collectors. Again both, the riser tube

spacing, W, and diameter, D, remain at their minimum values.

As seen from the above tables the distance between the riser tubes, W, is very small. In fact it could

be zero but was stuck to the minimum value specified as a constrain. For this reason a more

practical distance of 10 centimetres between the tubes is used and this parameter is excluded from

the optimization procedure. By doing so the results shown in Table 6 are obtained.

Table 6. Optimization results for a fixed distance between riser tubes of 10cm

Gt (W/m2) () Tin (C) D (m) ex (%) Tout (C) Ac (m

2) m& (kg/s)

500 0.8 27 0.004 4.72 59.6 9.98 0.0193

1000 0.8 27 0.004 6.02 92.1 8.38 0.0156

As can be seen for this more practical case, and by comparing the values shown with the values

presented in previous tables, the exergy efficiency is not affected too much. This case however is

much more cost effective because the minimum the distance between the riser tubes the maximum

would be the number of tubes and the collector will cost more.

Another practical case that needs to be investigated is the effect of riser pipe diameter on the exergy

efficiency. For this exercise the distance between the riser pipes is kept to 10 cm and the other

parameters, like optimum collector area and mass flow rate, as the ones presented in Table 6. The

results of this exercise are shown in Table 7.

Table 7. Effect of pipe diameter for a fixed distance between riser tubes of 10 cm

Gt (W/m2) () Tin (C) Ac (m

2) m& (kg/s) D (m) ex (%)

500 0.8 27 9.98 0.0193 0.004

0.008

0.012

0.015

4.72

4.71

4.70

4.691000 0.8 27 8.38 0.0156 0.004

0.008

0.012

0.015

6.02

6.01

5.99

5.99

As can be seen the pipe diameter can be increased from the optimum small diameter without any

problem as the exergy efficiency is marginally affected for the bigger sizes. So this needs to be

decided solely on cost, which increases for bigger diameter pipes and the friction factor imposed by

the smaller diameter pipe, which needs to be kept as small as possible in order not to block the

thermosiphonic flow.

It should be noted that in all results presented in Tables 3-7 a very small flow rate is given as theoptimum and the actual value approaches the thermosiphonically created one which is the most

frequently used operation mode for flat-plate collectors. So the assumption made earlier about the

small flow rate and the exclusion of fluid pumping from the analysis is proved.

7. ConclusionsIt is proved in this paper that exergy analysis and genetic algorithms are very suitable for obtaining

the required results quickly. These results prove that the exergy efficiency of a flat-plate solar

collector is maximized for small distances between the riser tubes and for very small diameter of

these tubes. By using a more practical distance of 10 centimeters between the tubes, and excluding

this parameter from the optimization procedure, very small differences are observed in maximumexergy efficiency and if the cost of the materials is accounted this is a more cost-effective solution.

The exergy efficiency it is also insensitive to the size of the riser pipe diameter. Other findings

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prove that exergy efficiency increases considerably at higher solar radiation and that the

transmittance absorptance product affects to a great extent the exergy efficiency.

NomenclatureAc collector area, m

2

B bond width , m

C factor given by (16)

Cb bond conductance, W/(mK)

cp specific heat capacity, J/(kgK)

D riser tube outside diameter, m

Di riser tube inside diameter, m

fE& exergy flow rate, W

e specific exergy, J/kg

F collector efficiency factor

F fin efficiencyf factor given by (15)

FR heat removal factor

Gt solar radiation, W/m2

h specific enthalpy, J/kg

hfi heat transfer coefficient inside absorber tube, W/(m2K)

hw wind loss coefficient, W/(m2K)

I solar radiation, W/m2

k absorber plate thermal conductivity, W/(mK)

kb bond thermal conductivity, W/(mK)

L collector length, m

m& mass flow rate, kg/s

n factor given by (5)

N number of riser tubes

Ng number of glass covers

Qs,in available solar radiation, W/m2

Qu rate of useful energy collected, W

s specific entropy, J/(kgK)

S power absorbed per unit area of collector, W/m2

Ta ambient temperature, K

Tf fluid temperature, K

Tf,i collector inlet temperature, K

Tf,o collector outlet temperature, K

Tp plate temperature, K

UL overall heat loss coefficient, W/(m2K)

V wind speed, m/s

W distance between riser tubes, m

Greeksymbols

collector slope, degrees

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absorber (fin) thickness, m

p plate emittance

g glass cover emittance

T temperature difference, K

transmittance-absorptance product

References[1] Badescu V., Optimum fin geometry in flat plate solar collector systems, Energy Conversion and

Management 2006; 47(15-16): 2397-413.

[2] Badescu V., Optimal control of flow in solar collectors for maximum exergy extraction,International Journal of Heat and Mass Transfer 2007; 50: 4311-22.

[3] Torres-Reyes E., Cervantes-de Gortari J.G., Ibarra-Salazar B.A., Picon-Nuez M., A designmethod of flat-plate solar collectors based on minimum entropy generation, Exergy the

International Journal 2001; 1(1): 46-52.

[4] Torres-Reyes E., Navarrete-Gonzalez J.J., Zaleta-Aguilar A., Cervantes-de Gortari J.G.,

Optimal process of solar to thermal energy conversion and design of irreversible flat-plate solarcollectors, Energy 2003; 28(2): 99-113.

[5] Kalogirou S.A., Use of genetic algorithms for the optimal design of flat plate solar collectors,Proceedings of the ISES 2003 Solar World Congress on Solar energy for a sustainable future,

2003 June 14-19; Goteborg, Sweden, on CD ROM.

[6] Kalogirou S.A., Artificial neural networks and genetic algorithms for the optimisation of solarthermal systems, In: Sayigh, A.S., Editor, WREC IX: Proceedings of the IX World Renewable

Energy Congress on CD-ROM, 2006; Florence, Italy.

[7] Loomans M., Visser H., Application of the genetic algorithm for the optimisation of large solarhot water systems, Solar Energy 2002; 72: 427-39.

[8] Kalogirou, S.A., Use of artificial intelligence for the optimisation of solar systems, InternationalJournal of Renewable Energy Engineering 2002; 4: 499-505.

[9] Krause M., Valen K., Wiese F., Ackermann H., Investigations on optimizing large solar thermalsystems, Solar Energy 2002; 73: 217-25.

[10] Kalogirou S.A., Solar Energy Engineering: Processes and Systems, New York: AcademicPress, Elsevier Science; 2009.

[11] Zalzala A., Fleming P., Genetic Algorithms in Engineering Systems, London, UK: TheInstitution of Electrical Engineers; 1997.

[12] Goldberg D.E., Genetic Algorithms in Search Optimisation and Machine Learning, Reading,MA: Addison-Wesley; 1989.