Thermodynamic optimization of flat plate solar collectors

Benhamad Maha1, Snoussi Ali, 2Ben Brahim Ammar 3

Department of chemical Engineering and process National School of Engineers of Gabès (ENIG)

Gabès,Tunisia mahabenhamad@gmail.com

Ali_snoussi@yahoo.fr

Abstract— In this paper a thermodynamic optimization of flat plate solar collector is developed to determine the optimal performance. In fact the energetic and exergetic analyzes are achieved by the determination of tow efficiencies. The influence of some geometrical parameters on the performance is also studied. It is noted that the exergetic efficiency of the collector has an optimum and that the energetic efficiency cannot be increased above an upper limit which is less than 1 and the exergy content of extracted energy tends to 0.The effect of some parameters shows that those limit values depends mainly on the collector and spacing between tubes. A simulation program is used for this calculation.

Keywords—component; flat plate solar collector; energy; exergy; efficiency.

I. INTRODUCTION Solar energy is one of the most significant renewable energy sources that the world needs. The major applications of solar energy can be classified into two categories: photovoltaic (PV) system, which converts solar energy to electrical energy, and solar thermal energy system, which converts solar energy to thermal energy. The use of solar heating in domestic hot water system has increased and developed, the flat plate solar collector is the main component of this system. Hence, the optimal performance of the solar collector is highly important. On the other hand, the thermodynamic optimization based on the first and the second law is developed to determine the optimal performance parameters .The second law is more informative .Much research has been carried in this category. Bejan [1] studied the solar collector using concepts such as exergy output, exergy efficiency, and entropy generation. Luminsou and Farahat [2] developed the optimal operation of flat plate solar collector by means of exergy analysis using numerical simulation. Therefore, the consideration of this article will be detailed energy and exergy analysis of flat plate solar collectors for evaluating the thermal and exergetic performance. The effect of design parameters on the energy efficiency, exergy efficiency, the efficiency factor and the performance factor fin of flat plate collector has been examined.

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II. ENERGY ANALYSIS OF FPC The useful energy (Qu) gain by the working fluid is calculated from the following equation:

( ), ,Q (1)u f p f o u t f inm C T T= − Where Tf,in, Tf,out , Cp and �f are fluid inlet temperature ,fluid

outlet temperature, heat capacity and mass flow rate of the working fluid, respectively. An energy balance on the absorber plate yields the following equation for a steady state, considering the heat losses from the solar collector to the atmosphere:

( )0Q ( 2 )u c l p aA G U T Tη= − − The terms of Equation (2) Tp ,Ta and Ac are the average temperature of the absorber plate ,the ambient temperature and area of the absorber plate , respectively. G is the incident solar energy per unit area of the absorber plate .The term 0( )η is the optical efficiency is equal with the effective product transmittance-absorptance ( )τα .Where Ul is the overall loss coefficient, which during the previous studies assumed as a constant factor or a variable with little effect [3]. The collector efficiency factor F’, represents the ratio of the actual useful energy gain to the useful gain that would result if the collector absorbing surface had been at the local fluid temperature, can be calculated as follows [4] :

( )

'

1 ( 3 )

1 1 1l

b i il

UF

WC D hU D D W F π

=

+ ++ −

Where:

F is the Fin efficiency factor: ( )t a n h2

( )2

m W D

Fm W D

−

=−

lU

mλδ

=

Considering the correlations of the temperature distribution in the collector, the following correlation will be obtained [5]:

The fifth International Renewable Energy Congress IREC 2014 March 25 27, Hammamet, TUNISIA

,

,

'e x p ( 4 )f o u t a

l l

f pf in a

l

GT TU F U W

LG m CT TU

− −= −

− −

The maximum temperature of a flat-plate collector Tmax, stagnation temperature, is the temperature of the absorbing plate when the flow rate is equal to zero, is given by [6]:

0m a x ( 5 )a

l

GT T

Uη= +

Using the formula of the maximum temperature, use (4) the following form is:

, m ax m ax ,'( ) ex p ( ) (6 )l

f o u t f inf p

F U WT T T T L

m C= − − −

The thermal efficiency of the collector can be obtained: ( 7 )uQ

G W Lη =

III. EXERGY ANALYSIS In the order to conduct a qualitative analysis on the collector and to determine the system thermal capabilities in terms of works, performing an analysis based on the second law of thermodynamics seems to be essentials. The exergy takes the following form:

( 8 )EeW Lψ

Δ=

Under steady-state conditions, the exergy rate per unit length of tube is calculated from following equation:

(9 )f ad E d H d Sm Td x d x d x

= −

Where H and S are respectively the specific enthalpy and the specific entropy of the fluid given by :

( 1 0 )

( 1 1 )

d Pd H T d S

d Td S C pT

ρ= +

=

The integration between the inlet and outlet permits to resort the total exergy extracted from the fluid across the tube:

( ) ,, ,

,

2 ( 1 2 )

f o u tf f o u t f i n a

f i n

f

TE m C p T T T L n

T

m d p Ld xρ

Δ = − −

+

Dividing EΔ by the term W Lψ , use (6) is obtained the exergy efficiency (e) of the collector is calculated by dividing the increase in working:

( )'

max ,

'

max max ,

,

1 exp( )

(13)( ) exp( )

Lf in

f p

f p fL

f inf p

af in

U F WLT Tm C

m C m dpU F WLe T T TWL W dxm CT Ln

T

ψ ψ ρ

− − −

= +− − −−

IV. DIMENSIONLESS ALNALYSIS

The use of dimensionless parameters in analysis is particularly powerful, since it can be used to generlaize some results.In this paper, we build on this existing work to develop a new dimensions design approach, are defined as :

TABLE I. DIMENSIONLESS NUMBERS

Fluid temperature 1f

fa

TT

θ = −

Solar flux *G *3/2 3/2

a

GGCp Tρ

=

Geometric parameters

The spacing between tubes W* *

1 / 2( )WW

W L=

Tube length L* *1/2( )

LLWL

=

Inside diameter of tube *iD

*1 / 2( )

ii

dD

W L=

The plate thickness of absorbent *

a b sδ *1/2( )

absabs WL

δδ =

The dimensionless thickness of the tube i

Di

D DeD−=

Thermal parameters

Viscosity of the heat transfer fluid *fμ

*1/ 2( )

f af

CpTG W Lμ

μ =

Thermal conductivity of the heat transfer fluid *fk *

1/ 2( )f a

f

k Tk

G W L=

Thermal conductivity of metal absorber *k *1/ 2( )

akTkG W L

=

In addition, we will consider the following dimensionless numbers in heat transfer [7], given by:

• The Reynolds number ( ReiD):

4Re (14)

i

fD

i f

mDπ μ

=

• The Prandtl number ( P r ): Pr (15)f

f

Cpk

μυα

= =

• The Nusselt number (iDNu ): (16)i i

Dif

h DNuk

=

With dimensionless parameters as defined above, the previous

expressions become :

• The maximum temperature:

( )

'0

max maxmax

4exp (17)Pr Re

i

fs fea D f i

G F WLT k D

ηθ θ θ θπθ

= − − −

• The performance factor fin: * *

0* *

m a x

* *0

* *m a x

(1 )t a n h2

( 1 8 )(1 )

2

i D

i D

W D ek

FW D e

k

ηθ δη

θ δ

− +

=− +

• The efficiency factor of the flat plate collector :

( )

1**

' 0** * *

m ax

(19)(1 ) (1 ) iD fi D i D

WWFN u kW D e F D e

ηπθ

−

= +− + + +

• The energy efficiency factor:

( )* *

m a x* *

' * *0

* *m a x

P r R e4

41 e x p ( 2 0 )P r R e

i

i

D f ie

D f i

k DW L

F W Lk D

πη θ θ

ηπ θ

= −

− −

• The exergy efficiency factor:

( )

( ) 2

* * ' * *0

m ax* * * *m ax

' * *0

max * * 2 * 3 *m ax

*2 *

Re Pr 41 exp4 Pr R e

41 expPr R e 8 R e

11

i

i

i i

D f ie

D f i

eD f i D f

e i

k D F W Le

W L k D

F W Lk D G

LnD W

π ηθ θα πθ

ηθ θπθ π μ

θ α

= − − −

− − −

− + −+

(21)

V. RESULT AND DISCUSSION The thermal and exergetic models presented in the previous sections have been transposed into a MATLAB computational program. In this program, most of the geometric parameters and operating conditions can be varied.

TABLE II. ENVIRONMENTAL AND DESIGN CONDITIONS FOR THE SOLAR COLLECTOR

Collector parameters Value

Glazing Single

Agent fluid in flow ducts water

Fluid inlet and ambient temperature, Tin=Ta

310 K

Inside diameter of the tube 0.0065m Outside diameter of the tube 0.0075m Spacing between tubes 0.1 m Tube length 1 .2 m

Thickness of the absorber 0.0015m

Conductivity of the absorber (Aluminium)

211 W/mk

Absorptance of the absorber plate, a

0.95

Transmittance of the cover, c 0.93 Incident solar energy per unit area of the absorber plate, G 857.014 Wm-2

Incident solar exergy per unit area of the absorber plate, � 778.938 Wm-2

Fig. 1, shows the behavior of the exergy efficiency as a function of the energy efficiency and the Reynolds number, in laminar flow. As seen,the increase in the Reynolds numbers leads to an increase in the energy efficiency without presenting a maximum points. However, the exergy efficiency takes an extermum point, the exergy analysis is more informative in the optimization of solar collector.

0

1000

2000

3000

0

0.5

1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

nombre de Reynolds(Re)efficacité thermique(n)

effic

acité

exe

rgét

ique

(e)

Fig. 1. The evolution of energy and exergy efficiency for different Reynolds

number.

As seen Fig. 2, the coordinate of the maximum point are ( 0, 4984, 0 .07317 )opt op teη = = corresponding with the Reynolds number equal to 40. Note also that when the energy efficiency tends to 0, the upper limit of the thermal efficiency is about 0.81.This figure illustrates the operating limits of all flat plate collectors.

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

efficacité thermique(n)

X: 40Y: 0.4984

Z: 0.07317

effic

acité

exe

rgét

ique

(e)

Fig. 2. The varaitions of the exergy efficiency versus the energy efficiency.

A. The effects of Geometric Parameters • The effect of the spacing between the tubes W * :

Fig.3, shows the spacing between the tubes on the exergy and energy efficiencies. The optimal point decreases from ( 0 .5 , 0 .0 7 5 )o p t o p teη = = to ( 0 .45, 0 .068)o p t op teη = = if

the value of spacing pass from 0.2 to 0.5. As it seen in Fig.4 increasing the spacing leads to a considerable decrease in the variation of efficiency factor (F’) and the performance factor (F).

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

efficacité thermique(n)

effic

acité

exe

rgét

ique

(e)

0.1

0.2

0.30.4

0.5

Fig. 3. Exergy and energy efficiency versus the spacing between the tubes.

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.86

0.88

0.9

0.92

0.94

0.96

0.98

1

espacement entre les tubes(W*)

le r

ende

men

t d’

aile

tte

(F)

et e

ffic

acité

de

plaq

ue (

F’)

F

F'

Fig. 4. The variations of efficiency factor (F’) and the performance factor fin

(F) versus the spacing .

The variation in the spacing between the tubes is important. It shows a loss surface relative to the quantity of recoverable energy by conduction. For best performance, it is necessary to reduce this spacing.

• The effect of tube length *L and thickness of the

absorber *absδ :

As shown in Fig.5, the variations of these geometric parameters has no considerable effect on the energy and exergy efficiencies .

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

efficacité thermique(n)

effic

acité

exe

rgét

ique

(e)

L*=1

L*=2L*=3

L*=4L*=5

L*=6L*=7L*=8L*=9

(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

efficacité thermique(n)

effic

acité

exe

rgét

ique

(e)

d*=0.029d*=0.024

d*=0.019

d*=0.014

d*=0.009d*=0.004

(b)

Fig. 5. (a) The effect of tube length L *. (b) The effect of the absorber thickness.

B. The effect of the Working Fluid

Fig.6, represents the effect of three different working fluids including water and a mixture of water and glycol with two concentrations of 25% and 44% [8].As it can be seen, water is the best working fluid. However, in some cases, using water as the working fluid is not possible due to environmental conditions such as freezing probability in winter. In these cases a mixture of water and glycol can be a good choice .

0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

efficacité thermique(n)

effic

acité

exe

rgét

ique

(e)

eauPglycol-eau(25%)Pglycol-eau(44%)

Fig. 6. The varaitions of exergy versus energy efficiency for three different

working fluids.

C. The effects of Absorber Material

The material of absorber should be taken into consideration. Fig.7, represents the comparison between the effects of the thermal conductivity for three material: steel, aluminum and copper. As it can be seen that the energy and exergy efficiencies increase of 0.78 to 0.82 and 0.07 to 0.074 respectively, when an absorber plate steel is replaced by an aluminum plate and practically no change in the values between copper and aluminum plate. Fig.8, shows the efficiency factor (F’) increases from 0.897 à 0.93 and the performance factor fin (F) increases from 0.0956 to 0.993, if a steel plate is replaced with a copper plate, it can be noted that the two factors increase with increasing conductivity material. Additionally, other factors should be taken into consideration, such as : absorber should be as light as possible, the corrosion resistance, the plate geometry and economic study material.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

efficacité thermique(n)

effic

acité

exe

rgét

ique

(e)

Acier

AluminumCuivre

Fig. 7. The varaitions of exergy versus energy efficiency for three different

material absorber palte.

0 50 100 150 200 250 300 350 400 4500.88

0.9

0.92

0.94

0.96

0.98

1

Acier Aluminium Cuvire 52W/mK 211 W/mK 385 W/mK

le ren

dem

ent d’

aile

tte

(F) et

effic

acité

de

plaq

ue (F’)

Fig. 8. The variations of efficiency factor (F’) and the performance factor fin

(F) versus the three type of material palte .

D. The effects of Optical Efficiency Fig.9, shows the optical efficiency 0η effect for some materials used in the glazing. Table III shows their factors of transmittance ( ) [9]. Should be considered that , the absorber is in aluminum. The optical efficiency increases with the transmission factor. The optical efficiency has a great effect on the exergy efficiency, for example the optimal exergy point pass from 0.055opte = ,for polyamide ,to 0.072opte = ,for glass. Other factors can influence in the optical efficiency: The use of double glazing, the wind speed and the ambient temperature, and the angle of inclination.

TABLE III. TRANSMITTANCE FACTORS OF SOME MATERIALS

Materials Transmittance ( )

M1 Polyamide (Kapton) 0.80

M2 Polyethylene-Terephthalate

(polyester)

0.84

M3 Glass 0.93

M4 Fluorinated Ethylene Propylene 0.96

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

efficacité thermique(n)

effic

acité

exe

rgét

ique

(e)

no(M1)

no(M2)no(M3)

no(M4)

Fig. 9. The effect of glazing materials on efficiencies.

E. The effects of Maximum Temperature The maximum temperature

maxθ has a great effect. Fig.10, shows that, an increase of

m a xθ from 0.4 to 0.8 leads to an increase in the exergy efficiency from 0.06 to 0.12. Fig. 11, shows, the efficiency factor (F’) and the performance factor fin (F) increase if the maximum temperature increase.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.02

0.04

0.06

0.08

0.1

0.12

efficacité thermique(n)

effic

acité

exe

rgét

ique

(e)

omax=0.4

omax=0.5

omax=0.6omax=0.7

omax=0.8

Fig. 10. The effect of the maximum temperature on the energy and exergy

efficiencies.

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.80.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

température maximale (omax)

le ren

dem

ent d’

aile

tte

(F) et

effic

acité

de

plaq

ue (F’)

varF

varF'

Fig. 11. The variations of efficiency factor (F’) and the performance factor fin

(F) versus the maximum temperature.

VI. CONCLUSION In this paper, the effect of some parameters on the energy efficiency, exergy efficiency, the efficiency factor and the performance factor fin of flat plate collector has been examined. The exergy analysis is more informative in optimization. In this paper we use dimensionless numbers to solve the equations obtained .The results show on the effects of geometric parameters, the spacing between tubes is very important than the others parameters (inner diameter, tube length and thickness of the absorber). Furthermore, Water is the best working fluid. In cold climates where water may freeze during winters, it is more appropriate to use the solution of water and glycol. However, the material of absorber should be taken into consideration. Additionally, the optical efficiency has a great effect on the exergy. The optical efficiency increases with the transmission factor of some materials used in glazing. The maximum temperature has a great effect, an increase in this parameter leads to an increase in the exergy efficiency but have a little effect on the energy efficiency.

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SUBSCRIPT

REFERENCES

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[3] S. Farahat, F. Sarhaddi, and H.Ajam, “Exergetic optimization of flat plate collectors,” Renewable Energy, vol. 34, pp.1169-1174, 2009.

[4] A.Duffie. Johan and A.Beckman. William, “Solar Engineering of Thermal Processes,” John Wiley and Sons ,New York, 1991.

[5] S.A. Kalogirou, F. Sarhaddi, and H.Ajam, “Solar thermal collectors and applications,” Energy and Combustion Science, vol. 30, pp.231-295, 2004.

[6] A. Bejan, DW. Kearney, and F. Kreith “Second law analysis and synthesis of solar collector systems,” Journal of Solar Energy Engineering (130), vol. 34, pp.23-28, 1981.

[7] K. Frank, M.Mangalik. Raja, and S.Bohn.Mark, “Principles of Heat Transfer ,” Seventh Edition,United States of America, 2011.

[8] R.Herrero Martin and all, “Simulation of an enhanced flat-plate solar liquid collector with wire-coil insert devices,” Solar Energy, vol. 85, pp. 455-469, 2011.

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Cp Specific heat of fluid (J/kg.K) G global solar radiation (W/m2) � Mass fluid Flow rate (kg/s) Q Heat flux (W) T Temperature (K) U Collector loss coefficient (W /m2K)

k, λ conductivity (W/mK) Cp Heat capacity of the fluid (J/kg.K) Ac Absorber plate’s area (m2) D diameter (m) L tube length (m) W spacing between tubes (m) F Fin efficiency factor F’ Collector efficiency factor

Absorptance Thermal efficiency

e Exergy efficiency emissivity

� overall exergy absorbed (W m-2) fluid density (Kg/m3) transmittance

μ viscosity (Kg/ms) δ thickness (m)

a Absorber plate f fluid in inlet out outlet u useful a ambient i inner p avarage o Optical l overall