Optimizing Exergy Efficiency of Flat Plate Solar Collectors Using SQP and Genetic Algorithm

Maryam Khademi1,a, Farzad Jafarkazemi1,b, Emad Ahmadifard1,c and Saman younesnejad2,d

1Islamic Azad University, South Tehran Branch, No. 209,

North Iranshahr St., Tehran 11365-4435, Iran

2Faculty of Mechanical Engineering, Islamic Azad University, Takestan Branch, Iran

akhademi@azad.ac.ir, bf_jafarkazemi@azad.ac.ir,

cemad_ahmadifard@yahoo.com, dsaman.y.n@gmail.com

Keywords: flat plate solar collector, exergy efficiency, Genetic algorithm, SQP method, optimization process.

Abstract. An increase in exergy efficiency of flat plate solar collector leads to a considerable

improvement in collector’s performance. Different parameters influence the performance of

collector. In this paper, Sequential Quadratic Programming (SQP) and Genetic Algorithm (GA) have

been employed for optimizing exergy efficiency of the flat plate solar collector. Absorber plate area

and mass flow rate of inlet water have been considered as optimization’s variables. The results show

the possibility to reach higher exergy efficiency with lower absorber area and consequently lower

price. Also it is obvious that SQP method performs optimization process with higher convergence

speed but lower accuracy than GA.

Introduction

Solar collector is the main part of solar thermal systems which absorbs solar irradiance and

converts it to heat like a heat exchanger. The major part of the initial cost for setting up solar thermal

systems is spent on collector. Among various types of solar collectors, flat plate type is the world's

most widely used collector because of simpler technology, lower price and easier maintenance.

Performance of collector is significantly affected by variations of its design and operating parameters.

Finding the optimal values of these parameters can largely affect the increase of efficiency and reduce

costs of the collector. Obtaining these parameters based on the first law of thermodynamics is not a

complete analysis due to weaknesses of this law in the analysis of system’s energy quality, neglecting

the environmental conditions and also unspecified loss and internal irreversibility of the system [1].

Exery analysis uses the second law of thermodynamics alongside the first law to provide a low-defect

and more complete analysis than the energy analysis does [2]. In this paper exergy efficiency function

of the flat plate solar collector is used as the objective-function in the optimization process.

Complexity of heat transfer equations of flat plate collectors has resulted in many of the analysis

being performed in the past with the assumption that parameters such as heat loss coefficient and heat

dissipation coefficient of the collector were assumed constant [3-5]. Torres- Reyes et al. [6]

determined the optimum design parameters of flat plate collectors based on the minimum entropy

generation. They did not investigate design variables, simultaneously, due to lack of using

evolutionary algorithms for optimizing. In reference [7] exergy efficiency function of the flat plate

collector was fairly expanded using temperature distribution equations and then optimal values of the

surface area and fluid flow through the collector to maximize the exergy efficiency function has been

investigated using the computational program REX1. In this study also parameters such as heat loss

coefficient and heat removal factor are assumed to be constant. Said Farahat et al. [8] performed

optimization process using SQP algorithm based on the minimization of exergy losses in the

collector. SQP algorithm, like other gradient-based methods cannot be effectively used for complex

optimization problems due to dependence on starting point and differentiability condition of the

objective function, direct and non-intelligent search methods due to being time consuming and

Applied Mechanics and Materials Vols. 253-255 (2013) pp 760-765Online available since 2012/Dec/13 at www.scientific.net© (2013) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMM.253-255.760

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non-purposive. Because of these reasons many intelligent direct methods have been proposed by

researchers in recent years, most of which are modeled after natural processes such as ant colony,

particle swarm optimization, artificial immune system and evolutionary algorithms[9]. In this paper,

exergy efficiency function of flat plate solar collector is obtained based on all design parameters and

coefficients to be variable. Then considering exergy efficiency function as the objective function,

optimization process is performed by SQP and Genetic Algorithms and results are compared with

each other.

Methodology

All of the correlations have been written in steady state, steady flow condition and it is assumed the

pressure drop inside collector tubes to be negligible.

Theoretical Analysis. In the exergy analysis of thermodynamic systems it is essential to identify

the exergy source and sink (consumer), in order to calculate the system’s exergy efficiency according

to the following equation:

∑∑=

SourceExergy

SinkExergyexη . (1)

In solar collectors, exergy source is the amount of radiant energy that is radiated on the collector

surface and is derived as follows:

( )[ ]spTradiationin TTAIEx 0,

.

1−= . (2)

where Ap, T0, IT are the area of the absorber plate, the ambient temperature and the incident solar

energy per unit area of the absorber plate, respectively. Ts is apparent sun temperature as exergy

source which is approximately 75% of the blackbody temperature of the sun, and is assumed to be

4500 K [10].

Increase in the exergy of the collector’s working fluid is considered as exergy consumer, and can

be obtained from the following equation:

( )

−−=−

inf

outf

infoutfpinfoutfT

TTTTCmExEx

,

,

0,,

.

,

.

,

.

ln . (3)

where finxE ,

� , foutxE ,

� , Tf,in, Tf,out, Cp and m� are the exergy of inlet fluid, the exergy of outlet fluid, the

fluid inlet temperature, fluid outlet temperature, heat capacity and mass flow rate of the working

fluid, respectively.

According to Eq. 1 the exergy efficiency of the collector is derived by dividing Eq. 3, which

illustrates the changes in sink exergy, by Eq. 2, which illustrates the changes in source exergy.

( )

−

−−

=

)(1

ln,

,,,

.

sr

aTp

inf

outfainfoutfp

ex

T

TIA

T

TTTTCm

η . (4)

Considering the correlations of temperature distribution in the collector, the following correlation

will be obtained [11]:

−=−−

−−

p

pL

L

Toinf

L

Tooutf

Cm

FAU

U

ITT

U

ITT

.

'

0,

0,

expη

η

. (5)

ηo and UL are the optical efficiency and the overall heat loss coefficient of collector, respectively.

Applied Mechanics and Materials Vols. 253-255 761

Using the above-mentioned correlations, the working fluid outlet temperature component can be

omitted from Eq. 4 and therefore, the correlation of exergy efficiency of the collector can be

rephrased into the following form:

−

+

−−

−

−

−

−

−

−−

=

)(1

1

1exp

ln1exp ,,

.

'

.

.

'

,

.

s

aTp

Lainf

inf

p

pL

ap

p

pL

Lainfp

ex

T

TIA

U

STT

T

Cm

FAU

TCm

Cm

FAU

U

STTCm

η . (6)

As it is clear from Eq. 6, exergy efficiency of the collector can be considered as a function of the

following parameters:

),,,,,,,( , pLToainfpex AUITTCmf ηη �= .

(7)

The effect of parameters Tf,in and Cp is quite clear. Increase in Tf,in causes an increase in the fluid’s

thermal value and thus increases the exergy efficiency of the collector. Cp is related to the type of

working fluid and is mainly selected among water, ethylene glycol and a mixture of water and

propylene glycol. Among these fluids water has the best performance and the highest exergy

efficiency [12]. Also it could be said that an increase in the optical efficiency of the collector will

certainly have a direct impact on its exergy efficiency.

T0 and IT are also components of environmental conditions which are out of our control. m� and Ap

are independent parameters and can be considered as variables in the optimization process. UL

depends on other design and performance parameters of the collector. For calculating UL and uQ�

(The useful energy gain by the working fluid) we use the method which mentioned in previous

research [12].

It should also be mentioned that the energy efficiency of the collector, considering the amount of

useful heat transferred to the fluid, is derived from the following equation:

Tp

uen

IA

Q�=η .

(8)

Algorithms

SQP algorithm. Sequential Quadratic Programming (SQP) is an iterative efficient method for

solving nonlinear optimization problems and is used on problems for which the objective function

and the constraints are twice continuously. SQP methods solve a sequence of optimization sub

problems, optimizes a quadratic model of the objective function subject to a linearization of the

constraints.

Genetic algorithm. Genetic algorithm (GA) is a random search process based on the laws of

biology such as natural selection and Darwin‘s survival of the fittest that is introduced by John

Holland [13].The superiority of GA is its suitability in solving nonlinear and complex problems. In

recent years, revealing ever more capabilities, flexibility and speed of GA, the application of this

method in optimization problems is increasing. In the beginning, GA needs an initial population. This

algorithm evolves the initial population by changing the genetic of individuals so that gradually finds

the answer to the optimization problem by creating new generations. The algorithm uses probabilistic

operators such as selection, cross over and mutation for evolving this population.

Modeling of exergy optimization. The exergy efficiency function which obtained from Eq. 6

considered as a function of the optimization algorithms: SQP and GA. The algorithms developed in

MATLAB software are based on minimizing the objective function. Therefore in this program, it is

762 Sustainable Development of Urban Infrastructure

necessary to make changes in the objective function for maximazing exergy efficiency of the

collector. So the following function has been introduced as the objective function of optimization

algorithms:

Objective function = 1- exη . (9)

The model of optimization problem is posed:

min objective function=Eq.(9)

skgm

mAp

/1.0001.0

51 2

≤≤

≤≤

�

m� and Ap are design variables of optimization process. Water is considered as the operating fluid.

Other variables and specifications of the collector are considered according to Table 1.

Table 1: Design parameters for the flat plate solar collector

Value Parameter

76% Optical Efficiency, oη

5% Emissivity of the glass cover, gε

92% Emissivity of the absorber plate, pε

50 [mm] Thickness of the back insulation, bδ

0.75 [mm] Thickness of absorber plate, pδ

237 [W/m.K] Thermal conductivity of the absorber plat, Kp

0.04 [W/m.K] Thermal conductivity of the insulation, Ki

13 [mm] Inner diameter of pipes, Di

72 [mm] Inner tubes centre to centre distance, W

408 Collector tilt angle , β

Negligible Adhesive resistance, 1/Cb

300 [K] Water inlet temperature, Tf,in

300 [K] Ambient temperature, T0

Results and Discussion

The optimization procedure’s results.

SQP Method. Optimization using SQP algorithm ended after 9 iterations and the following

parameters are the results of the optimization:

%1728.6,5,/004365.0 2 === exp mAskgm η�

Based on these results and parameters mentioned in Table 1, other design and performance

parameters of the collector are derived as:

KmWUmWQKT Lenuoutf ./7978.3%,4519.46,/9.1621,997.388 22.

, ==== η

Genetic Algorithm (GA). Optimization using GA considering population=500 and

Generation=150 was finished after 51 generations and the following results were obtained:

%0002.7,12.3,/002178.0 2 === exp mAskgm η�

Based on these results and parameters mentioned in Table 1 other design and performance parameters

of the collector are derived as:

KmWUmWQKT Lenuoutf ./2922.3,%9486.44,/8.1493,684.40722

.

, ==== η

Applied Mechanics and Materials Vols. 253-255 763

Comparison between optimization results. Fig. 1 illustrates the trend in the two optimization

processes.

Fig. 1: Optimization processes

As indicated in this diagram the speed of operations in the SQP method is much higher than the

GA. But the higher accuracy of GA in calculating the maximum exergy efficiency is clearly seen.

SQP method is very dependent on the starting point which is provided by the user while the GA

with respect to the optimal random selection method would not need to a starting point. The results

indicate that smaller collector can have similar and even better performance than the collector with

larger surface area. Observing this point can be very effective in reducing the production cost of the

collector. Also it is clear from the results that the highest exergy efficiency is obtained in low mass

flow rate.

Conclusions

Exergy efficiency is an accurate criterion to evaluate the performance of flat plate solar collector.

Increase in the exergy efficiency is a result of decrease in the internal irreversibility and also more

utilization of the potential of solar energy. In this paper, at first the exergy efficiency has been

obtained without assuming the design parameters as constant, so that the function is suitable for any

collector with determined specifications, and then the optimum parameters of inlet water flow rate

and absorber surface area of collector were obtained by using SQP and GA algorithms.

SQP algorithm convergence rate is very favorable. But high dependence on the starting point and

first and second order derivatives of the objective function and also stopping in local optimum points

are the weaknesses of this algorithm. The results of GA represent more accuracy of the algorithm.

SQP method reaches the optimal solution after 9 iterations and GA algorithm after 51 generations.

This indicates the higher convergence rate of SQP.

Optimal exergy efficiency obtained by SQP method and GA are equal to 6.1728% in surface area

of 5 m2 and mass flow rate of 0.0044 kg/s and 7.0002% in surface area of 3.12 m

2 and the mass flow

of 0.0022 kg/s, respectively. Comparison of efficiencies indicates that the GA computed the optimal

point in a lower surface area and higher exergy efficiency. This reduces construction costs in addition

to increasing the efficiency of the collector.

Also it is obvious that with specified surface area, the exergy efficiency increases by decreasing

mass flow rate and increasing fluid inlet temperature.

764 Sustainable Development of Urban Infrastructure

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