Optimal process of solar to thermal energy conversion and design of irreversible flat-plate solar collectors

Energy 28 (2003) 99–113 www.elsevier.com/locate/energy

Optimal process of solar to thermal energy conversion and design of irreversible flat-plate solar collectors E. Torres-Reyes a,∗, J.J. Navarrete-Gonza´lez a, A. Zaleta-Aguilar b, J.G. Cervantes-de Gortari c a

b

Instituto de Investigaciones Cientı´ficas, Universidad de Guanajuato, L. de Retana No. 5, Guanajuato, Gto. 36000, Mexico Facultad de Ingenierı´a Meca´nica Ele´ctrica y Electro´nica, Universidad de Guanajuato, Prolongacio´n Tampico S/N, Salamanca, Gto. 36730, Mexico c Departamento de Termoenergı´a y Mejoramiento Ambiental, Facultad de Ingenierı´a, Universidad Nacional Auto´noma de Me´xico, Mexico, D.F. 04510 Mexico Received 1 December 2001

Abstract Thermodynamic optimization based on the first and the second law is 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 solarto-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.  2002 Elsevier Science Ltd. All rights reserved.

1. Introduction Thermal performance of solar air heaters is limited by the heat transfer efficiency between the absorber and the fluid. Since heat transfer augmentation usually increases friction, optimum parameters, such as mass flow and collector geometry, have to be determined. The second law analysis is required to establish the best collector for a specific application. An optimization analysis was performed by Altfeld et al. [1,2] on the net flow of exergy in flatplate collectors for air heating. They found that the collection surface characteristics, especially ∗

Corresponding author. Tel.: +52-473-73-27555; fax: +52-473-73-26468. E-mail address: [email protected] (E. Torres-Reyes).

0360-5442/03/$ - see front matter  2002 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0360-5442(02)00095-6

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Nomenclature AC Af AP CP Cp Cpa Cpv Dh EVC E˙ x E˙ xQ Exvc F⬘ f G G⬙ h∗pf hpf h H K K⬘ k L M m ˙ m ˙a NEx,Q NEx,f Ns NTUf P ˙0 Q ˙S Q ˙U Q R⬙ R RT RB Rfc

absorber area of the collector, m2 cross flow area, m2 heating surface area, m2 product of mass by heat capacity, kW/K heat capacity of the fluid, kJ/(kg K) heat capacity of the air, kJ/(kg K) heat capacity of the steam water, kJ/(kg K) hydraulic diameter, m energy of the control volume, kJ fluid exergy flow, kW exergy flow due to the heat flow, kW exergy of the control volume, kJ efficiency factor friction factor mass velocity, kg/(s m2) incident solar energy per absorber area unit, kW/m2 apparent heat transfer coefficient between absorber plate and fluid, kW/(m2 K) heat transfer coefficient between absorber plate and fluid, kW/(m2 K) convection heat transfer coefficient, kW/(m2 K) fluid enthalpy, kW pressure constant constant isentropic constant collector length, m mass flow number fluid mass flow, kg/s dry air mass flow, kg/s dimensionless thermal exergy dimensionless fluid exergy flow entropy generation number fluid number of transfer units fluid pressure, Pa heat transfer rate to the environment, kW incident energy in the collection area, kW useful heat, kW energy absorbed per unit of absorber area, kW/m2 universal gas constant, kJ/(kg K) resistance from the top cover to surroundings, m2 K/kW resistance from the back collector to surroundings, m2 K/kW resistance between fluid and cover

E. Torres-Reyes et al. / Energy 28 (2003) 99–113

Rfi Rfp Rpc Rpi S˙ gen St SVC T0 Tf Tp TS t UL

101

resistance between fluid and insulation surface, m2 K/kW resistance between fluid and absorber plate, m2 K/kW resistance between plate and cover, m2 K/kW resistance between plate and insulation, m2 K/kW entropy generated in the system, kW/K Stanton number entropy generation of the control volume, kJ/K ambient temperature, K fluid temperature, K plate temperature, K apparent sun temperature, K time, s overall heat transfer coefficient, kW/(m2 K)

Greeks r h0 hI hII q p ta t w

density, kg/m3 surface overall efficiency first law efficiency second law efficiency dimensionless temperature dimensionless fluid pressure effective transmittance–absorptance product dimensionless temperature ratio air mass humidity

Subscripts c f i in max out p

cover fluid insulation inlet flow maximum outlet flow plate

those of the extended surface, diminished the air flow, thus increasing the net flow of exergy and the thermal efficiency of the device. Optimized geometries have been derived for a solar ventilation air preheater [3]. Offset stripe fins did not show an improved performance compared to optimally spaced continuous fins, due to larger electrical power for this geometry. However, offset stripe fins yielded high net energy gains for large fin spacing.

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Kolb et al. [4] described the development and testing of an efficient single-glazed solar matrix air collector. This is an improved collector design that avoids certain technical problems of the matrix in conventional air collectors. The investigations have covered a wide range of geometrical parameters of wire mesh screen matrix (wire diameter, pitch and number of layers) [5]. Generalized correlations have been developed for heat transfer coefficient and friction factor for air flowing through the bed of wire mesh screen matrix packed in the solar air heater duct in a cross flow arrangement. A method to select an optimal number of metal vanes, as well as an appropriate depth for the flowing air duct, has been developed [6]. The results showed that a high efficiency can be achieved with the use of the metal vanes, particularly at smaller depths of the air duct. In a previous work [7], Torres-Reyes et al. presented a thermoeconomic model based on second law analysis. It was developed to determine the annualized total cost for air heating, in a specific application of solar air heaters. The optimum performance parameters and the solar collector surface of the analyzed device [8] were previously determined based on Entropy Generation Minimization Method [9]. In this paper, a generalized methodology is established to determine the optimum path flow length of the working fluid by means of a thermohydraulic model developed from the first and the second law points of view. Relationships are derived for different air duct and absorber plate arrangements, which can be used to calculate the optimal thermal performance for a specific application and a given solar collector geometry. The thermodynamic procedure incorporates the irreversibilities produced by the pressure drop in the flow of the working fluid along the collector during the process of solar-to-thermal energy conversion. 2. Exergy analysis The thermal balance of the solar collector (Fig. 1), if the effects due to the kinetic and potential energy changes are neglected, is given by ˙ S⫺Q ˙0 ⫽ Q

dEVC dE ˙ U ⫽ VC ⫹ H ˙ out⫺H ˙ in, ⫹Q dt dt

Fig. 1.

Energy, entropy and exergy flows, in a solar collector.

(1)

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103

˙ S is the solar energy absorbed by the solar collection surface and it is evaluated by where Q the expression ˙ S ⫽ G⬙(ta)AC, Q

(2)

where ta is the effective product transmittance–absorptance of the solar collector, Ac the solar ˙ 0 is the energy transferred from the absorber plate to the ambient, given by collection area, and Q ˙ 0 ⫽ ACUL(TP⫺T0). Q

(3)

˙ S⫺H ˙ e represents the In Eq. (1), dEVC/dt is the energy accumulated in the control volume and H ˙ U. enthalpy change of the fluid through the control volume, usually called the useful heat, Q The enthalpy change of the moisture air in the collector is given by ˙U ⫽ H ˙ S⫺H ˙e ⫽ m Q ˙ a(Cpa⫺wCpv)(Tf,out⫺Tf,in),

(4)

where Cpa is the air heat capacity, Cpv the steam heat capacity, w the air humidity and m ˙ a is the dry air mass flow. The next two expressions take care of the heat capacity of the wet air mass flow, and of the dimensionless isentropic constant of the ideal gases CP ⫽ m ˙ Cp ⫽ m ˙ a(Cpa⫺wCpv),

(5)

m ˙ R k⫺1 ⫽ . CP k

(6)

The second law in terms of the entropy generation for this system, can be written as

冉 冊 冉 冊 冊

dEx T0S˙ gen ⫽ ⫺ dt

VC

冉

T0 Tf,out ⫹ 1⫺ ACG⬙(ta)⫺CP(Tf,out⫺Tf,in) ⫹ T0CP ln TS Tf,in

(7)

k⫺1 Pout ln . ⫺ k Pin

The second law efficiency is expressed by T0S˙ gen hII ⫽ 1⫺ ˙ S. [1⫺(T0 / TS)]Q

(8)

3. Dimensionless thermodynamic analysis The dimensionless temperatures are given as qp ⫽ Tp / T0⫺1,qf,out ⫽ Tf,out / T0⫺1,qf,in ⫽ Tf,in / T0⫺1,qS ⫽ TS / T0⫺1,qmax ⫽ Tmax / T0

(9)

⫺1, where qp, qf,out, qf,in, qS, qmax, account for the absorber surface, outlet fluid, inlet fluid, solar ˙ S /T0 results in temperature and maximum collector temperature, respectively. Dividing Eq. (1) by Q

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ACUL 1⫺ ˙ q ⫽ M(qf,out⫺qf,in). QS / T0 P

(10)

The dimensionless number M known as the mass flow number, was determined for a nonisothermal solar collector as a function of the dimensionless temperatures and the NTU [7]; it is expressed here by CP M⫽ ˙ . QS / T0

(11)

Eq. (1) can be reduced by substituting the dimensionless maximum temperature of the solar collector, qmax [10] ACUL 1⫺ ˙ q ⫽ 0. QS / T0 max

(12)

Eq. (12) can be expressed by ˙ S / T0 Q . qmax ⫽ ACUL

(13)

Substituting Eq. (13) in Eq. (10), the following relationship is obtained: qP . M(qf,out⫺qf,in) ⫽ 1⫺ qmax

(14)

Pressure drop expression for square duct is given by Bejan [9] 4L G2 ⌬P ⫽f , P D 2rP

(15)

dimensionless pressure drop can be written as follows: kK 2 Pin⫺Pout M, ⫽ Pin k⫺1 were K⫽

冉

(16)

冊

k⫺1 4L 1 G⬙(ta)eAC 2 f . k D 2rP CpT0 Af

(17)

Substituting Eqs. (11), (13), (16) and (17) in Eq. (7) and considering that ⌬P/Pin is considerably smaller than unity, in this limit the entropy generation number can be linearized to read

冉

冊

qP qf,out ⫹ 1 1 ⫹ KM2 ⫺ ⫹ . NS ⫽ M ln qf,in ⫹ 1 qS ⫹ 1 qmax The exergy losses result as NS ⫽

冉

冊

冉

(18)

冊

qf,out ⫹ 1 qS ⫺M(qf,out⫺qf,in) ⫹ M ln ⫹ KM2 . qS ⫹ 1 qf,in ⫹ 1

(19)

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105

From Eq. (19) various dimensionless exergy components can be distinguished. The dimensionless irreversibility is given by the entropy generation number NS. The dimensionless inlet exergy flow is also represented by the expression NEx,Q ⫽

冉

冊

qS . qS ⫹ 1

(20)

The exergy change of the air flow through the solar collector can be expressed by

冉

NEx,f ⫽ M qf,out⫺qf,in⫺ln

冊

qf,out ⫹ 1 ⫺KM2 . qf,in ⫹ 1

(21)

The dimensionless exergy balance can be summarized by the following general expression: NS ⫽ NEx,Q⫺NEx,f

(22)

or, as function of the dimensionless temperatures NEx,f ⫽

qS⫺NS(qS ⫹ 1) . qS ⫹ 1

(23)

When the dimensionless balance is obtained, it is easy to determine the second law efficiency under the above described concepts hII ⫽

NEx,f NS ⫽ 1⫺ , NEx,Q NEx,Q

(24)

substituting Eqs. (20) and (21) into Eq. (24), the second law efficiency can be predicted by hII ⫽

M[qf,out⫺qf,in⫺ln(qf,out ⫹ 1) / (qf,in ⫹ 1)⫺KM2] . [qS / (qS ⫹ 1)]

(25)

4. Thermo-hydraulic analysis Four geometries have been considered with one pass of the working fluid flowing through the collector. The different geometries are distinguished by the position of the absorber plate with respect to the air flow. Fig. 2 shows the arrangements that can be made between the different absorber finned plates and the air flow. The fluid temperature profile developed in relation to the duct wall and the thermal resistance arrangements is also shown in Fig. 2(see also Fig. 3). Energy balance in geometry III G⬙(ta) ⫹ h∗pf(Tf⫺Tp) ⫹ (UT ⫹ UB)(T0⫺Tp) ⫽ 0.

(26)

Useful heat gained by the working fluid ACh∗pf(Tp⫺Tf) ⫽ QU. Where the heat transfer coefficient is given by

(27)

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Fig. 2. Various absorber finned plates (fluid flow profiles depend on the duct wall and the thermal resistance arrangements).

h∗pf ⫽

Ap hh . AC 0 pf

(28)

Efficiency factor F⬘ of the solar collector is defined as follows: h∗pf ˙ U, A [G⬙(ta)⫺UL(Tf⫺T0)] ⫽ Q h∗pf ⫹ UL C

(29)

or F⬘ ⫽

h∗pf . h∗pf ⫹ UL

(30)

Overall heat losses coefficient is given by the sum of the top losses coefficient and the bottom losses coefficient UL ⫽ UT ⫹ UB .

(31)

Four different terms have been introduced in the expressions of Fig. 4, and are not described in the text of the paper. These terms are referred to dimensionless temperature and pressure relations, given by p⫽

P ⫺1,tc ⫽ qc⫺qf,tp ⫽ qp⫺qf,ti ⫽ qi⫺qf. P0

The number of transfer units of the fluid is given in the next expression

(32)

E. Torres-Reyes et al. / Energy 28 (2003) 99–113

Fig. 3.

NTUf ⫽

107

Thermal balance for the four geometries presented in Fig. 2.

Aph0h . CP

(33)

As can be observed in the energy and exergy balances of Fig. 4, in order to factorize NTU the same heat transfer coefficient was considered in all duct walls, as it is usually done [11].

5. Optimization procedure This optimization procedure requires the temperature and the pressure drop profiles developed by the fluid flow through the solar device. A differential analysis of the solar–thermal energy conversion is done in order to obtain numerically the optimum parameters to the entropy generation minimization in the process dNEx,f dNS ⫽⫺ ⫽ 0. dM dM

(34)

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Fig. 4. Exergy balance and dimensionless exergy balance for the four collector geometries presented in Fig. 2.

The temperature profile of the solar collector as a function of the mass flow number M, can be obtained by solving Eq. (14)

冉

冊

F⬘ qf,out⫺qf,in ⫽ 1⫺exp ⫺ . qmax⫺qf,in Mqmax From Eq. (21) the next expression can be written as

(35)

E. Torres-Reyes et al. / Energy 28 (2003) 99–113

冉

NEx,f ⫽ M qf,out⫺qf,in⫺ln

冊

qf,out ⫹ 1 ⫺KM3. qf,in ⫹ 1

109

(36)

Optimal outlet fluid temperature of the solar collector is determined by maximizing the exergy change of the air stream with respect to mass flow number, taking into account the irreversibilities due to the pressure drop, as follows:

冋冉

d dNEx,f qf,out ⫹ 1 ⫽ M qf,out⫺qf,in⫺ln dM dM qf,in ⫹ 1

冊册

⫺3KM2 ⫽ 0.

(37)

The maximum exergy flow number can now be determined. From the solution of Eq. (37) results an expression that is a function of mass flow number and the outlet fluid temperature M

冉

冊 冉

冊

dqf,s qf,out qf,out ⫹ 1 ⫹ qf,out⫺qf,in⫺ln ⫺3KM2 ⫽ 0. dM qf,in ⫹ 1 qf,in ⫹ 1

(38)

From Eq. (35) the relation of outlet fluid temperature with the mass flow number is expressed by

冉

冊

F⬘(qmax⫺qf,in) F⬘ dqout ⫽⫺ exp ⫺ . dM M2qmax Mqmax

(39)

The relation between the maximum exergy flow number as a function of the mass flow number and the maximum temperature of the collector gives the general expression to predict the optimum temperature dimensionless ln(qf,out,opt ⫹ 1) ⫽ 0.7064ln(qmax ⫹ 1) ⫹ 0.0014.

(40)

0.7 [10]. Finally, the optimum outlet fluid It can be represented by the small expression qout ⫽ qmax temperature in a flat-plate solar collector is obtained from Eq. (40) substituting the dimensionless variable definitions as follows:

冉

Tf,out,opt G⬙(ta)e ⫽ ⫹1 T0 ULT0

冊

0.7

.

(41)

Fig. 5 shows the exergy change of the air stream, named here as Exergy Flow Number as a

Fig. 5. Exergy flow number as a function of mass flow number determined for the geometry III of the solar air collector.

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Fig. 6. Optimum outlet temperature of the fluid in a solar air collector (geometry III) as a function of maximum collector temperature, without irreversibilities due to pressure drop.

function of the mass flow number, without taking into account the irreversibilities due to the pressure drop. Fig. 6 shows the optimal outlet fluid temperature of a solar collector given by Eq. (40), for values of qmax between 0.0 and 1.0. Exergy flow number as a function of mass flow number and optimal outlet fluid temperature and mass flow number as a function of the maximum temperature of a solar collector are shown in Figs. 7 and 8, respectively. In both cases, the friction losses are taken into account. The pressure drop considered in this case study is given by a value of K ⫽ 0.001. This value represents the average values of environmental conditions of the chosen geographic location for typical flatplate collectors as a function of the friction losses given by the geometry of the absorbing area. The predicted optimal parameters depend strongly on the choice of K. Finally, the optimum outlet fluid temperature in a flat-plate solar collector taking into account the friction losses, can be represented by the next expression, substituting the dimensionless variable definitions as follows:

冉

Tf,out,opt G⬙(ta)e ⫽ ⫹1 T0 ULT0

冊 再 冋 0.62

exp 0.28 f

冉

冊册 冎

4L 1 G⬙(ta)eAC D 2rPin CpT0 Af

2 0.2

.

(42)

Fig. 7. Exergy flow number as a function of mass flow number determined for geometry III of the solar air collector, taking into account the pressure drop, K ⫽ 0.001,0.0 ⬍ qmax ⬍ 1.0.

E. Torres-Reyes et al. / Energy 28 (2003) 99–113

111

Fig. 8. Optimum outlet temperature of the fluid in a solar air collector (geometry III) as a function of maximum collector temperature, taking into account the pressure drop, K ⫽ 0.001.

5.1. Optimal path flow length in a flat-plate solar collector Optimal path flow length of a solar collector is obtained identifying as independent variable to the relation between length and hydraulic diameter of the flow duct. Eq. (34) can be rewritten as follows: dNEx,f dNS ⫽⫺ ⫽ 0. d(4L / Dh) d(4L / Dh) Rearranging Eq. (21) as follows:

冉

(43)

冊

NEx,f qf,out ⫹ 1 4L ⫺K⬘ , ⫽ qf,out⫺qf,in⫺ln M qf,in ⫹ 1 Dh

(44)

were the constant K⬘ is a function of mass flow given by K⬘ ⫽

k⫺1 G2 , f k 2rPe

(45)

were G is the mass velocity term given by Kays and London [12] for pressure drop in a circular duct. This term is represented by the constant K⬘ derived here. The exergy flow number is maximized with respect to the length and hydraulic diameter of the flow duct, from Eqs. (43) and (44), this can be represented by

冉

冊

1 dNEx,f dqf,out qf,out ⫽ ⫺K⬘ ⫽ 0. Md(4L / Dh) d(4L / Dh) qf,out ⫹ 1

(46)

On the other hand, the derivate fluid dimensionless temperature as a function of the length and hydraulic diameter can be obtained from Eq. (35)

冉

冊

F⬘ dF⬘ qmax⫺qf,in dqout exp ⫺ ⫽ . d(4L / Dh) Mqmax Mqmax d(4L / Dh)

(47)

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E. Torres-Reyes et al. / Energy 28 (2003) 99–113

Efficiency factor is defined for the compact geometry III, showed in Fig. 2, as F⬘ ⫽

(4L / Dh)Sth0 . (4L / Dh)Sth0 ⫹ (1 / Mqmax)

(48)

From Eq. (48) the derivate efficiency factor can be obtained as a function of the length and hydraulic diameter of the flow duct as follows: MqmaxSth0 dF⬘ . ⫽ d(4L / Dh) [(4L / Dh)MqmaxSth0 ⫹ 1]2

(49)

The simultaneous solution of the equation system formed by Eqs. (35), (46)–(49) gives as result the optimum path flow length of a solar collector. When this parameter is obtained, the thermal effectiveness and the number of transfer units of the solar collector can be known. The number of exergy flow is related to the dimensions of the solar collector in Fig. 9. Optimum path flow length is determined taking into account the irreversibilities due to the friction losses. From the parametric analysis done on Fig. 9 the next generalized relationships of the optimum path flow length such that the number of exergy flow is a maximum can be obtained. 4L 0.39[G⬙(ta) / ULT0]0.82 ⫽ . Dh [(k⫺1) / k](fSt / 2rPe)0.5G

(50)

At fixed values of mass velocity and Reynolds number the Stanton number and the friction factor are known, therefore, optimal path flow length in a flat-plate solar collector can be determined, using average values of G⬙ and T0 for the chosen geographic location. 6. Final remarks Solar to thermal energy conversion was studied in order to optimize the process in a flat-plate solar collector. A method was established to determine the optimum temperature of performance and the optimum path flow length of a solar collector. These optimum parameters are related to finite conditions of operation for finite size systems, including the irreversibilities due to pressure drop of the working fluid in solar devices.

Fig. 9. Exergy flow number in a solar air collector (geometry III) as a function of path flow length, K⬘ ⫽ 0.001 and qmax ⫽ 1.

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Design formulas for different air duct and absorber plate arrangements using dimensionless terms were derived. They take into account the efficiency factor of the solar energy collection given by the geometry and materials of the absorber plate. When the optimum parameters are determined, the thermal effectiveness and the number of transfer units can be known, both representing a simplified way to thermal design of solar exchangers. References [1] Altfeld K, Leiner W, Fiebig M. Second law optimization of flat-plate solar air heaters. Solar Energy 1988;41:127–32. [2] Altfeld K, Leiner W, Fiebig M. Second law optimization of flat-plate solar air heaters. Part 2: results of optimization and analysis of sensibility to variations of operating conditions. Solar Energy 1988;41:309–17. [3] Pottler K, Sippel CM, Beck A, Fricke J. Optimized finned absorber geometries for solar air heating collectors. Solar Energy 1999;67(1–3):35–52. [4] Kolb AE, Winter RF, Viskanta R. Experimental studies on a solar air collector with metal matrix absorber. Solar Energy 1999;65(2):91–8. [5] Varshney L, Saini JS. Heat transfer and friction factor correlations for rectangular solar air heater duct packed with wire mesh screen matrices. Solar Energy 1998;62(4):255–62. [6] Matrawy KK. Theoretical analysis for an air heater with a box-type absorber. Solar Energy 1998;63(3):191–8. [7] Torres-Reyes E, Ibarra-Salazar BA, Cervantes-de Gortari JG, Picon-Nun˜ ez M. Optimal design of nonisothermal flat-plate solar collectors based on minimum entropy generation method. In: ECOS 2000 From thermo-economics to sustainability. Part 1: Proceedings of ECOS 2000. Nederland: Universiteit Twente; 2000. p. 213–23. [8] Torres-Reyes E. Cylindrical cavities a solar collector design. In: Proceedings of the Ninth Congress, National Academy of Engineering of Mexico 1983, Leo´ n, Gto., Me´ xico. 1983. p. 1–5 [in Spanish]. [9] Bejan A. Entropy generation minimization. The method of thermodynamic optimization of finite-size systems and finite-time processes. New York: CRC Press, 1996. [10] Torres-Reyes E, Cervantes-de Gortari JG, Ibarra-Salazar BA, Picon-Nun˜ ez MA. Design method of flat-plate solar collectors based on minimum entropy generation. Exergy—Int J 2001;1(1):46–52. [11] Duffie JA, Beckman WA. Solar engineering of thermal processes. New York: Wiley, 1991. [12] Kays WM, London AL. Compact heat exchangers. New York: McGraw-Hill, 1984.