This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 148.204.18.220

This content was downloaded on 07/03/2014 at 20:29

Please note that terms and conditions apply.

A simple relationship between the sunlight concentration factor and the thermal conductance

in a class of photothermal engines

View the table of contents for this issue, or go to the journal homepage for more

1998 J. Phys. D: Appl. Phys. 31 1742

(http://iopscience.iop.org/0022-3727/31/14/019)

Home Search Collections Journals About Contact us My IOPscience

J. Phys. D: Appl. Phys. 31 (1998) 1742–1744. Printed in the UK PII: S0022-3727(98)88912-2

A simple relationship between thesunlight concentration factor and thethermal conductance in a class ofphotothermal engines

J A Rocha-Mart ınez†, T D Navarrete-Gonz alez† andF Angulo-Brown ‡

† Area de Fısica, Departamento de Ciencias Basicas,Universidad Autonoma Metropolitana Azcapotzalco, Avenida San Pablo 180,02200 Mexico DF, Mexico‡ Departamento de Fısica, Escuela Superior de Fısica y Matematicas,Instituto Politecnico Nacional, Edificio 9 UP Zacatenco, 07738 Mexico DF, Mexico

Received 3 November 1997, in final form 26 February 1998

Abstract. In this brief paper we present an addendum to a recently publishedanalysis of a photothermal engine model. Here, we numerically demonstrate thatthe design parameters, the sunlight concentration factor and the thermalconductances of materials employed as thermal conductors are linked by a simplerelationship, if one wishes to obtain the maximization of the power output of thephotothermal engine.

In a recent paper [1], we have analysed a photothermalconversion engine model based on a combination of theMuser and Curzon–Ahlborn engines as depicted in figure 1.This model was proposed by De Vos [2] as a suitable arrayfor reproducing some solar conversion devices such as theEurelios farm in Adrano, Italy. We called this array ofengines the Muser–Curzon–Ahlborn (MCA) engine. Inour previous study of that model we found that the poweroutput W without concentrators (equation (9), in [1]) isa convex function of the temperatureT1 of the absorbingbody (see figure 1 of [1]). We also found that the so-calledecological functionE [3] (equation (12), in [1]) and thesolar conversion efficiencyω [2, 4] (equation (16) in [1])only depend onT1 by means of convex functions. Inthe second part of that paper we study the MCA enginewith concentrators, that is, with optical arrays (lenses andmirrors) for artificially increasing the radiation incident overthe solar collector atT1, which is the managing temperatureof W , E and ω. The impact of the concentrators ismeasured in terms of the concentration factorC, whichis in the interval 1≤ C ≤ 1/f , wheref is the dilutionfactor [2] (f = R2

s /r2, with RS the radius of the sun

and r the mean radius of the earth’s orbit around thesun) with a valuef = 2.16× 10−5. ThusC is betweenC = 1 without concentrators andCmax = 46 300 in thecase of fully concentrated sunlight, that is,Cmaxf σT 4

S =σT 4

S , with TS the temperature of the sun’s surface (TS =5762 K) andσ the Stefan–Boltzmann constant (σ = 5.67×10−8 W m−2 K−4). In the case of a MCA engine with

C > 1, we obtained that bothW andω (ω ≡ W/(f σT 4S ),

equation (22) in [1]) depend onT1 and C by means ofconvex functions, that is, there exist values ofT1 andC atwhich the power output and the solar conversion efficiencyreach maximum points. In this brief paper we present anaddendum to our previous paper [1], which consists of anumerical demonstration that the design parametersg (thethermal conductance) andC (the concentration factor) arenot independent. These parameters are linked by a simplerelationship, which suggests that, if one wishes to obtain themaximization of the power output of a MCA engine, oneought to choose the array of lenses and/or mirrors linkedto the materials of the steam generator and the condenser(see figure 5.11 of [2]).

Just like in our previous paper, for simplicity, we takeg1 = g2 = g (see figure 1) and the heat transfer throughthe heat conductors governed by Newton’s law of coolinggiven by

Q1 = g(T1− θ1)

Q2 = g(θ2− T2) (1)

whereQ1, Q2, T1, θ1, T2 and θ2 are depicted in figure 1.In [1] it was shown that the solar conversion efficiencyω for the MCA engine with concentrators is given by(equation (22) in [1])

ω(T1, C) =(

1− T2

T1− 2(σ ′/g)(T ′40 − T 41 )

)×(

4C + 1

4C− T 4

1

4CT 42

)(2)

0022-3727/98/141742+03$19.50 c© 1998 IOP Publishing Ltd

Sunlight concentration and thermal conductance

Figure 1. The Muser–Curzon–Ahlborn engine. Both theeffective sky temperature and the planet’s temperature, T0and T2, respectively, are regarded as fixed [2]. In thisscheme we change the nomenclature of temperatures usedfor the Muser engine in [2]. We use T0 instead of T1, T1instead of T3 and T2 = Tp . For simplicity we takeg1 = g2 = g .

Figure 2. A family of ω against C curves for several values of g . The maxima of ω are linked with Cm by means ofω(Cm) = 0.226C 0.149

m .

where T ′0 ≈ (4C + 1)1/4T2 (with T2 = Tp, the planet’stemperature in [2], see figure 1). In equation (2),σ ′ =σA (with A = 1 m2), in such a way that the term(σ ′/g)(T ′40 − T 4

1 ) is in kelvins. This remark is necessarybecause, in the steady state described in [1], a Stefan–Boltzmann heat flux per unit area is transmitted by meansof a linear heat conductor of conductanceg. This way ofputting σ ′ = σA was not clarified in our previous study[1] and, in equation (16) in [1], we had a misprint in thesecond factor, where (

5

4− T

41

T 42

)must be read as (

5

4− T 4

1

4T 42

).

This mistake was not carried through the rest of ourcalculations and all the graphs in [1] are correct. If wemaximizeω(T1, C) given by equation (2), for a givenC,we find T ∗1 by means of(

∂ω

∂T1

)C

= 0.

Thus, we have a plot ofω(T ∗1 , C) versusC as depictedin figure 7 of [1]. Figure 7 in that paper was obtainedfor g = 100 W K−1. We wish to remark that, if we takeC = 389 for the Eurelios solar plant [2], then we obtain theexperimental efficiencyωE = 0.13 for g = 268 W K−1.If we plot the function given by equation (2) for a setof g values, we obtain the family of curves depicted infigure 2. If we fit the set of maxima for those curves, weobtain that

Cmax = (0.0457)g1.112 (3)

which can be viewed on a log–log graph in figure 3. Ifone wishes to maximize the power output of the MCA

1743

J A Rocha-Martınez et al

Figure 3. A log–log plot of the concentration factor against the conductance.

engine then the concentration factorCmax and the thermalconductanceg must be linked by such a relation asequation (3). That is, both design parametersC andg areintimately linked by means of a compromise relationshipunder the maximum power regime for photothermal enginesof the MCA type. In fact, this result is not unexpectedbecauseC has to do with the amount of energy sent tothe collector atT1, whereasg has to do with the energytransmitted through the walls of the steam generator andthe condenser; thus, in the steady state, there must exista compromise between these fluxes if one maximizes thepower output of the engine. An illustrative analogy of thiscompromise can be found in the context of a hydrostaticexample. If the collector is a big water tank suppliedby a pipe of diameterd1 (analogous toC), the waterlevel of the tank will depend both ond1 and also on thediameterd2 of a second, exit, pipe (analogous tog) used formoving a set of paddles. Thus, the maximum power in thepaddles’ mechanism will depend on a compromise between

the fluxes through these two pipes. Finally, we wish toremark that, even though our model is highly idealized, itcan serve as a guide for real photothermal engines.

Acknowledgments

This work was partially supported by the COFAA-IPN andthe Instituto Mexicano del Petroleo.

References

[1] Navarrete-Gonzalez T D, Rocha-Martınez J A andAngulo-Brown F 1997J. Phys. D: Appl. Phys.30 2490

[2] De Vos A 1992Endoreversible Thermodynamics of SolarEnergy Conversion(Oxford: Oxford University Press)

[3] Angulo-Brown F 1991J. Appl. Phys.69 7465Arias-Hernandez L A and Angulo-Brown F 1997J. Appl.

Phys.81 2973[4] De Vos A, Landsberg P T, Baruch P and Parrott J E 1993

J. Appl. Phys. 74 3631

1744