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Solar Energy 83 (2009) 1232–1244 www.elsevier.com/locate/solener

Modeling, simulation and optimization of a solar collector driven water heating and absorption cooling plant J.V.C. Vargas a,*, J.C. Ordonez b, E. Dilay a, J.A.R. Parise c a

Programa de Po´s-Graduacßa˜o em Engenharia, PIPE, Centro Polite´cnico, Universidade Federal do Parana´, Caixa Postal 19011, Curitiba, PR 81531-990, Brazil b Department of Mechanical Engineering, Florida State University, Tallahassee, FL 32310-6046, USA c Departamento de Engenharia Mecaˆnica, Pontifı´cia Universidade Cato´lica, Rio de Janeiro, RJ 22453-900, Brazil Received 28 July 2008; received in revised form 8 February 2009; accepted 11 February 2009 Available online 14 March 2009 Communicated by: Associate Editor Ruzhu Wang

Abstract A cogeneration system consisting of a solar collector, a gas burner, a thermal storage reservoir, a hot water heat exchanger, and an absorption refrigerator is devised to simultaneously produce heating (hot water heat exchanger) and cooling (absorption refrigerator system). A simpliﬁed mathematical model, which combines fundamental and empirical correlations, and principles of classical thermodynamics, mass and heat transfer, is developed. The proposed model is then utilized to simulate numerically the system transient and steady state response under diﬀerent operating and design conditions. A system global optimization for maximum performance (or minimum exergy destruction) in the search for minimum pull-down and pull-up times, and maximum system second law eﬃciency is performed with low computational time. Appropriate dimensionless groups are identiﬁed and the results presented in normalized charts for general application. The numerical results show that the three way maximized system second law eﬃciency, gII; max;max;max , occurs when three system characteristic mass ﬂow rates are optimally selected in general terms as dimensionless heat capacity rates, i.e., ðwsp;s ; wwx;wx ; wH;s Þopt ﬃ ð1:43; 0:23; 0:14Þ. The minimum pull-down and pull-up times, and maximum second law eﬃciencies found with respect to the optimized operating parameters are sharp and, therefore important to be considered in actual design. As a result, the model is expected to be a useful tool for simulation, design, and optimization of solar collector based energy systems. Ó 2009 Elsevier Ltd. All rights reserved. Keywords: Systems engineering; Solar collector; Absorption refrigeration

1. Introduction Solar driven power, refrigeration and air conditioning systems are capable in many parts of the world of making a signiﬁcant contribution to reducing energy consumption since, in principle, they make very small demands on man-produced energy sources and rely on working ﬂuids that have reduced environmental impacts. However, a solar driven plant is considerably larger than a conven-

*

Corresponding author. Tel.: +55 41 33613307; fax: +55 41 33613129. E-mail address: [email protected] (J.V.C. Vargas).

0038-092X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.solener.2009.02.004

tional plant, a feature which many potential users ﬁnd unattractive. Clearly, more solar driven plants are likely to be brought into use if they can be made smaller. One of the purposes of this paper is to evolve broad quantitative criteria which will assist designers in the process of minimizing plant size, in particular the matching of the solar energy converter (concentrator dish/collector) and the hot side heat exchanger, the largest of the plant elements. Heat driven cycles can make use of waste heat and solar energy for refrigeration and air conditioning. Sorption systems, either by absorption or adsorption processes are utilized to produce cooling. However, in order to make those systems economically viable, their size must be reduced.

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Nomenclature a A B

collector heat leak fraction area, m2 total radiation leaving a surface per unit of area, W m2 c speciﬁc heat, J kg1 K1 ~c dimensionless speciﬁc heat, Eq. (6) C1,C2,C3 coeﬃcients, Eq. (27) D diameter, m e speciﬁc ﬂow exergy, J kg1 ~ E dimensionless exergy rate f friction factor H incident radiation per unit of area, W m2 I_ local solar irradiation rate per unit of area, W m2 L length, m LHV lower heating value, J kg1 m mass, kg ~ m dimensionless mass, Eq. (5) m_ mass ﬂow rate, kg s1 NTU number of heat transfer units Pr Prandtl number, ma1 T _Q heat transfer rate, W ~ Q dimensionless heat transfer rate, Eq. (5) s speciﬁc entropy, J kg1 K1 entropy generation rate, W K1 S_ gen t time, s ~t dimensionless time, Eq. (5) T temperature, K u velocity, m s1 U global heat transfer coeﬃcient, W m2 K1 _ W work rate, W ~ W dimensionless work rate, Eq. (5) x, y, z cartesian coordinates, Fig. 2c Greek symbols a absorptivity c dimensionless thermal conductance, Eq. (4) e emissivity ~e dimensionless emitted solar collector net radiation

For example, eﬀorts in that direction have been made by several authors in adsorption refrigeration: innovative adsorbent materials and heat pipes (Wang and Oliveira, 2006) improved speciﬁc cooling power (SCP) and achieved a coeﬃcient of performance (COP, deﬁned as the obtained refrigeration capacity rate divided by the heat input rate) of 0.39; numerical simulation and optimization (Khattab, 2006; Aghbalou et al., 2004; Li and Wang, 2002; Alam et al., 2001) obtained typical values for the COP of around 0.15; experimental two-stage systems (Saha et al., 2001), and various diﬀerent cycles (e.g., continuous heat recovery, mass recovery, thermal wave, cascade multi eﬀect, hybrid

g k nch q r s w

eﬃciency eﬀectiveness, Eq. (12) chemical exergy, J kg1 density, kg m3 Stefan–Boltzman constant, 5.67 108 W m2 K4 dimensionless temperature, Eq. (5) dimensionless heat capacity rate, Eq. (6)

Subscripts air air c solar collector, Fig. 2b cc combustion chamber cs cold space thermal load C reversible f fuel ﬂuid any speciﬁc ﬂuid fr friction g glass h hot water heat exchanger hx hot water heat exchanger solid material H regenerator in input I 1st Law of thermodynamics II 2nd Law of thermodynamics L refrigerated space m metal max maximum value min minimum value pd pull-down pu pull-up s thermal ﬂuid set setpoint sp serpentine (coil) t energy storage tank ts cross section v constant volume w wall wx water in the hot water heat exchanger 0 ambient

heating and cooling) (Wang, 2001) reported values for the COP close to 0.5. Tierney (2007) simulated novel chiller-trough and evacuated ﬂat plate collector combinations to investigate the potential for reducing gas-ﬁring requirements in single and double-eﬀect absorption lithium bromide chillers, ﬁnding the largest potential savings of 86% for the doubleeﬀect chiller and trough collector. Hasan and Goswami (2003) presented an exergy analysis of a solar driven power and refrigeration cycle ﬁnding that increasing the heat source temperature leads to higher ﬁrst law eﬃciencies, but associated to larger exergy destruction. Sierra et al.

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(1993) tested an absorption refrigeration system driven by a solar pond, which was simulated by a lower than 80 °C heat source, reaching generation temperatures up to 73 °C and evaporation temperatures as low as 2 °C, and a COP in the range of 0.24–0.28. Ejector refrigeration systems either alone or combined with absorption systems have also been the subject of research by several authors with solar energy heat input, either simulated by electrical sources in the laboratory (Sankarlal and Mani, 2006) or in actual systems (Kalogirou, 2004), as well as by potential implementation assessment through meteorological data (Sozen and Ozalp, 2005) and exergy analysis (Pridasawas and Lundqvist, 2004). Due to the growing interest in absorption refrigeration cycles driven by low temperature heat sources, Medrano et al. (2001) conducted numerical simulations to test the alternative organic ﬂuid mixtures triﬂuoroethanol (TFE)/tetraethylenglycol dimethylether (TEGDME or E181) and methanol/TEGDME in series ﬂow and vapor exchange double-lift absorption cycles, which showed higher performances than the ammonia/water mixture (e.g., 15% higher with the mixture TFE/TEGDME). In sum, heat driven refrigeration systems have long been proven feasible and practical, but low COP and large size are hurdles to be overcome. Thermodynamic optimization is one possible way of achieving better performance and size reduction in order to make these systems commercially competitive. The method of entropy generation minimization has emerged during the last three decades as a distinct subﬁeld in heat transfer (Bejan, 1982, 1996, 2002; Bejan and Mamut, 1999). The method consists of the simultaneous application of heat transfer and thermodynamic principles in the pursuit of realistic models of heat transfer processes, devices, and installations, i.e., models that account for the inherent irreversibility of heat, mass, and ﬂuid ﬂow pro-

cesses. In engineering, the entropy generation minimization method is known also as thermodynamic optimization and thermodynamic design. The method has been gaining more importance in the context of shape and structure in engineering and nature, as one of the grounds of constructal theory (Bejan, 2000). In the refrigeration area, several authors optimized systems based on the method of entropy generation minimization that had power input and heat rejection to the ambient (Bejan, 1989; Bejan et al., 1995; Vargas et al., 1996), as in the case of the vapor compression cycle (Klein, 1992; Radcenco et al., 1994; Buzelin et al., 2005). They were optimized by maximizing the refrigeration load (rate of heat extraction from the cold space), which corresponds to minimizing the rate of entropy generation of the cold space of the refrigeration plant. However, no transient mathematical model was found in the literature of solar collector and gas-ﬁred driven water heating and refrigeration plants that addresses in general terms, i.e., non-dimensionally, the variation of design and operating parameters, and their eﬀect on performance, and that has been utilized for general system thermodynamic optimization (system pull-up and pulldown times, and second law eﬃciency). The objectives of this paper are: (i) to introduce a general (dimensionless) transient mathematical model for a solar collector and gas-ﬁred driven water heating and refrigeration plant and (ii) to minimize system pull-up and pull-down times and extract maximum exergy input rate from the solar and/or fuel sources. The system is shown schematically in Fig. 1, where it should be noted that the refrigerator could be either an absorption or ejector system. The approach is crossdisciplinary and pursues simultaneously: (i) the local optimization of components and processes with (ii) the optimal global integration and conﬁguration of the system. The model is represented by a system of ordinary diﬀerential equations with respect to

thermal-fluid level . I

Coil

Absorption Refrigerator

.

Energy Storage . m sp, Tsp Tank

m H , TH

vent

T0 .

Q0

Twx

T0 C

Hot Water Heat Exchanger Solar Collector

Tt

Combustion Chamber

. QH

Twx

TH

pump 3

Reversible Compartment (C) THC

. m wx, Twx,in

TLC .

m sp , T t

pump 2

pump 1

. m H, Tt

Regenerator Heat Exchanger

.

mf, h f

Fig. 1. Schematic diagram of the proposed solar and gas-ﬁred driven water heating and refrigeration plant.

. QL

TL

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1235

time, the solution of which consists of the temperatures of each control volume, from which, the resulting water heating and refrigeration heat transfer rates, exergy rates and second law combined eﬃciency are computed as functions of time, to be able to analyze the integrated system transient operation. The model is simple enough to ensure small computational time requirements, so that it is possible to simulate the system steady state and transient response in a large number of competing geometric and operating conﬁgurations.

In order to present general results for the system conﬁguration proposed in Fig. 1, dimensionless variables are needed. First, since the thermal conductances ðUAÞi present in all subsystems i shown in Fig. 1 are commodities in short supply, it makes sense to recognize the thermal conductance inventory (hardware) as a constraint. For example, by selecting the absorption refrigerator system, one may deﬁne the total external conductance inventory, UA (hardware), as a design constraint:

2. Solar system mathematical model

As a result, for the whole system, dimensionless thermal conductances are deﬁned by:

A complete diagram of the proposed system is shown in Fig. 1. In the context of systems engineering, the overall system is divided in three integrated subsystems: (I) solar collector; (II) energy storage tank and hot water heat exchanger, and (III) regenerator heat exchanger and absorption refrigerator. Therefore, the mathematical model is developed according to that division. 2.1. Solar collector The thermal radiation interactions are calculated for the surfaces involved in the system. A balance of energy states that the net radiation heat transfer rate for the collector surface, as shown in Fig. 2a, is calculated by (Sparrow and Cess, 1978): Q_ c ¼ ðH c Bc ÞAc

ð1Þ

where H c is the incident radiation, i.e., the total radiation heat transfer rate per unit of area striking the collector surface and coming from all directions; Bc is the radiosity, i.e., the total radiation heat transfer rate per unit of area leaving the surface, and Ac is the exposed collector surface area. The radiation net heat transfer rate at the solar collector gray surface area is given by: Q_ c ¼ fac I_ ec rðT 4c T 40 ÞgAc

a

ð2Þ

Bi

Hi

Solar Collector

b

.

m sp, T t

T0

. m sp, Tsp Fig. 2. (a) Thermal radiation energy interactions on the collector surface and (b) solar collector details.

UA ¼ ðUAÞH þ ðUAÞL þ ðUAÞ0

ci ¼

ðUAÞi UA

ð3Þ

ð4Þ

Dimensionless heat transfer and work rates, temperatures, time, and mass are deﬁned by: _ _ t ~ i; W ~ i Þ ¼ ðQi ; W i Þ ; si ¼ T i ; ~t ¼ ; ðQ T0 UAT 0 ðmH;s cs Þ=ðUAÞ mi ~i ¼ m mH;s

ð5Þ

Dimensionless heat capacity rates, speciﬁc heats, solar collector absorbed, and emitted net radiation are deﬁned by: wi;j ¼

ci ac Ac I_ ec Ac rT 30 m_ i cj ; ~ci ¼ ; ~I ¼ ; ~ec ¼ UA cs UA UAT 0

ð6Þ

In Eqs. (4)–(6), subscripts i and j refer to a particular location or subsystem in the system shown in Fig. 1. In a dimensionless model, all variables are directly proportional to the actual dimensional ones, as Eqs. (4)–(6) demonstrate. Therefore, this allows for scaling up or down any system with similar characteristics to the system analyzed by the model. Another important aspect is that any dimensionless variable value used in the simulations could represent an entire and numerous set of dimensional values by varying appropriately the parameters in the dimensionless variables deﬁnition, which by itself stresses the generality of the dimensionless model. Physically, the set of results of a dimensionless model actually represent the expected system response to numerous combinations of system parameters (geometry, architecture) and operating conditions (e.g., ambient conditions, mass ﬂow rates), without having to simulate each of them individually, as a dimensional model would require. Based on that reasoning, it is convenient to search for an alternative formulation that eliminates the physical dimensions of the problem, aiming to achieve two main objectives: 1. Numerical stability: the nondimensionalization of the variables is done based on appropriate physical scales for the original variables, i.e., by using scale analysis (Bejan, 1993). In Eq. (4), UA is comparable in magnitude to all other thermal conductances in a realistic sys-

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Table 1 Properties and constant values used in the simulations. Ac ¼ 40 m2 Dsp ; Dh ; DH ¼ 0:01 m ~ cc ¼ 0:0396 E f ¼ 0:01 I_ ¼ 1400Wm2 Lsp ¼ 100 m Lh ; LH ¼ 10 m mH;s cs ¼ 2:391 JK1 m ~ sp;s ¼ 4 ~ c ~cc ¼ 4:53 m ~ h ~ch ¼ 4:26 m ~ H ~cH ¼ 1:77 m m ~ L ~cL ¼ 22:58 ~ t ~ct ¼ 407:7 m

Dpi ¼ 2f ~ cc ¼ 0:0403 Q ~ cs ¼ 0:00034 Q T 0 ¼ 298:15K UA ¼ 500 WK-1 cc ¼ 0:1 cH ¼ 0:25 cH;w ¼ 0:06 cL ¼ 0:25 cL;w ¼ 0:1 csp ¼ 1 ct;w ¼ 0:1 ct;wx ¼ 0:5 ec ¼ 0:9 swx;in ¼ 1

tem of the type proposed in Fig. 1, therefore, ci ¼ Oð1Þ, as it is shown in Table 1. The same reasoning applies to Eqs. (5) and (6), except to time, which is allowed to progress as long as demanded by the transient simulation, and ~t t. In this way, the calculated dimensionless variables are as close as possible to the unity, therefore avoiding the divergence of the algorithm, which is possible to occur with the original dimensional variables, and 2. Generalization of the results: with the nondimensionalization of the variables, the results are normalized, i.e., the graphs and numerical tables are valid to any geometric conﬁguration (or architecture) with functional and physical characteristics similar to the system analyzed by the model. Two more equations are needed to evaluate the solar collector and serpentine (coil) outlet temperatures as functions of time. For that, two control volumes are deﬁned based on Fig. 2b, one for the solar collector and coil pipe and another only for the ﬂuid inside the coil. The ﬁrst law of thermodynamics applied to those two control volumes, respectively, states that: dsc ~ c cc ðsc 1Þ Q ~ sp g 1 ¼ fQ ~ ~ c~cc m dt dssp ~ sp W ~ fr;sp g 1 ¼ fwsp;s ðst ssp Þ þ Q ~ sp;s d~t m

ð7Þ ð8Þ

~ c ¼ ~I ~ec s4 1 ; Q ~ sp ¼ csp sc ssp , and m ~ c~cc ¼ where Q c ðmsp csp þ mc;air cv;air þ mc;g cc;g þ mc;m cc;m Þ=ðmH;s cs Þ. It is assumed that the selected thermal ﬂuid does not undergo a change of phase in the coil. In Eq. (8), the work the ﬂuid has to do to overcome fric~ fr;i , is calculated as a result of the pressure drop in tion, W the coil pipe as follows (Bejan, 1993): _ ~ fr;i ¼ W fr;i ¼ m_ i Dpi =qfluid W UAT 0 UAT 0 where

ð9Þ

Li q u2 Di fluid i

ð10Þ

with ui ¼ qfluidm_ iAi;ts , and subscript i refers to a particular location or subsystem in the system shown in Fig. 1. 2.2. Energy storage tank and hot water heat exchanger The collected solar energy is stored in a tank containing a mass of thermal ﬂuid (e.g., ethylene-glycol). According to the Fig. 1, balances of energy applied to two control volumes in the tank (thermal ﬂuid/tank walls/combustion chamber and hot water heat exchanger) are written as follows: dst ~ wx þ Q ~ cc ct;w ðst 1Þ þ wsp;s ðssp st Þ Q ~ Hs g 1 ¼ fQ ~ t~ct m d~t ð11Þ ~ wx ¼ct;wx ðswx st Þ; Q ~ Hs ¼wH;s ðst sH Þ; m ~ t~ct ¼ðmt;s cs þ where Q mt;w ct;w þmcc ccc Þ=ðmH;s cs Þ, and ~ cc ¼ gI;cc LHVf m_ f Q UAT 0 dswx ~ wx W ~ fr;wx g 1 ¼ fwwx;wx ðswx;in swx Þ Q ~ h~ch m d~t

ð12Þ ð13Þ

~ h~ch ¼ ðmwx cwx þ mhx chx Þ=ðmH;s cs Þ and the friction where m ~ fr;wx is calculated work in the hot water heat exchanger, W according to the Eq. (9). Eq. (13) is written based on the fact that the water does not undergo a change of phase in the heat exchanger and on the incompressible substance model. 2.3. Regenerator heat exchanger and absorption refrigerator The absorption refrigerator system shown in Fig. 1 has negligible work input. The cycle is driven by the heat transfer rate Q_ H (generator) received from the regenerator hot side stream at an average temperature T H . The refrigeration load Q_ L (evaporator) is removed from the refrigerated space at T L , and the heat transfer rate Q_ 0 (condenser and absorber) is rejected to the ambient, T 0 . The refrigerator operates irreversibly due to the entropy generation mechanisms that are present (e.g., heat transfer, mixing, throttling). The heat input for the absorption refrigeration system is provided by a regenerator heat exchanger. The eﬀectiveness of the regenerator is deﬁned as: k¼

Q_ Hs _QH; max

ð14Þ

where Q_ Hs ¼ m_ H cs ðT t T H Þ is the actual heat transfer rate delivered by the thermal ﬂuid, and Q_ H; max ¼ m_ H cs ðT t T HC Þ is the maximum possible heat transfer rate, recognizing that maximum heat transfer would occur if the regenerator hot side outlet temperature matched the regenerator

J.V.C. Vargas et al. / Solar Energy 83 (2009) 1232–1244

cold side irreversible free temperature, T HC . Therefore, it is possible to write: sH ð1 kÞst ð15Þ k where, for a counterﬂow heat exchanger in which one of the streams experiences a change of phase at nearly constant pressure (Bejan, 1993), the eﬀectiveness is calculated by: ð16aÞ k ¼ 1 expðNTUH Þ UA cH ¼ ð16bÞ NTUH ¼ cH m_ H cs wH;s sHC ¼

The ﬁrst law of thermodynamics applied to the regenerator hot side states that: dsH ~ Hs Q ~ H cH;w ðsH 1Þ W ~ fr;H g 1 ¼ fQ ð17Þ ~ H~cH m d~t ~ fr;H is calcuwhere the friction work in the regenerator, W ~ H~cH ¼ ðmH;m cH;m þ lated according to the Eq. (9), m mH;s cs Þ=ðmH;s cs Þ, and with sHC calculated from Eq. (15), it follows that ~ H ¼ cH ðsH sHC Þ Q

ð18Þ

The thermal inertia of the absorption system evaporator, condenser and absorber are neglected in presence of larger inertias present in the system (solar concentrator, energy storage tank, regenerator heat exchanger, and cold (evaporator) and dsd0C space). Therefore dsdLC ~t ~t (condenser/ absorber) are neglected and the absorption refrigeration system equations are developed for steady state conditions, as follows: ~ 0 ¼ c0 ðs0C 1Þ Q ð19Þ ~ L ¼ cL ðsL sLC Þ Q ~L ¼ Q ~0 ~H þ Q Q ~H Q ~L ~0 Q Q þ ¼ sHC sLC s0C

ð20Þ ð21Þ ð22Þ

In the present model, the refrigerator is considered to be internally reversible (endoreversible), i.e., internal irreversibilities are assumed negligible in presence of the heat exchangers irreversibilities due to ﬁnite temperature diﬀerences (Bejan et al., 1995; Vargas et al., 1996, 2000). The label (C) in Fig. 1 is an allusion to Carnot’s name to suggest a system that operates reversibly. So, the model considers the refrigerator to operate irreversibly with all irreversibilities happening in the heat exchangers. Eq. (22) is the second law of thermodynamics applied to the reversible compartment of the absorption refrigerator shown in Fig. 1, i.e., in that compartment S_ gen ¼ 0. The unknowns ~ L ; sLC ; s0C and sL . The equation to compute sL ~ 0; Q are Q follows from a balance of energy in the cold space: dsL ~ cs Q ~ L cL;w ðsL 1Þg 1 ¼ fQ ~ ~ L~cL m dt

ð23Þ

~ cs is a known thermal load inside the cold space. where Q

1237

From the design constraint stated in Eq. (3), the absorption refrigerator dimensionless thermal conductances are related as follows: c0 ¼ 1 cL c H :

ð24Þ

3. Thermodynamic optimization The optimization problem focuses initially on determining the mass ﬂow rate m_ sp , an important operating parameter in the system. The products of the proposed system are the hot water (heating rate production) and the refrigeration rates (cooling rate production). The existence of a maximum heating and cooling rate production with respect to m_ sp is expected based on the analysis of two extremes: (i) when m_ sp ! 0, the heat transfer rate collected by the thermal ﬂuid approaches zero, and the ﬂow exergy rate picked up by the thermal ﬂuid stream approaches zero as well and (ii) when m_ sp ! 1, collector heat loss to the ambient and the work the ﬂuid has to do to overcome friction increase, and the serpentine outlet temperature of the thermal ﬂuid stream is almost the same as the inlet temperature, since the heat exchanger length is ﬁnite, therefore as ﬂow speed increases, the temperature variation along the coil goes to zero and heat transfer as well, and the ﬂow exergy received by the m_ sp stream is also small. As a result, in both extremes, the system heating and cooling rate production are negligible. This asymptotic behavior at zero and very high values of m_ sp implies the existence of an intermediate (and optimal) ﬂow rate m_ sp that maximizes the system heating and cooling rate production. The dimensionless total exergy input rate (Bejan, 1997) to the system is given by: ~c 1 1 ; E ~ cc ¼ gII;cc nch;f m_ f ; E ~ in ¼ E ~c þ E ~ cc ð25Þ ~c ¼ Q E UAT 0 sc Assuming negligible pressure drop in the heat exchanger and the incompressible substance model, the heat rate production is evaluated by the dimensionless ﬂow exergy rate picked up by the water stream being heated in the hot water heat exchanger shown in Fig. 1, as follows: ~ wx ¼ m_ wx ðewx;out ewx;in Þ ¼ wwx;wx swx swx;in ln swx E swx;in UAT 0 ð26Þ where e represents the speciﬁc ﬂow exergy, i.e., e ¼ ðh h0 Þ T 0 ðs s0 Þ. The second product is the absorption refrigerator cooling rate production. It is evaluated by the exergy content of the heat transfer ðQ_ L ; T L ; T 0 Þ that must be deposited in the T L – cold space (Bejan, 1997), as follows: 1 ~ ~ 1 ð27Þ EL ¼ QL sL

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The proper dimensionless measure of the thermodynamic optimum is the maximum reached by the combined second law eﬃciency of the entire system, which is calculated by: gII ¼

~L ~ wx þ E E : ~ in E

ð28Þ

~ H , so that system (17)–(20) is solved for (18) evaluates Q ~ ~ Q0 ; QL ; s0C , and sLC at each time step of the numerical integration of Eqs. 7, 8, 11, 13, 17, and 23. System (19)–(22) is ~L solved by analytical substitution, delivering ﬁrst Q through the following quadratic equation: ~ 2 þ C2Q ~ L C3 ¼ 0 C1Q L

~H Q =cL ; sHC ð1 cH cLÞ 1 cH cL ~H ~H Q Q sL 1 C2 ¼ 1 þ 1 cH cL sHC ð1 cH cL Þ ~ ~ 1 QH QH 1þ ; and þ 1 1 cH c L cL sHC ~H Q 1 ~ H sL : C3 ¼ 1 1þ Q 1 cH c L sHC

where C 1 ¼ 4. Numerical method, results, and discussions The problem consisted of integrating Eqs. 7, 8, 11, 13, 17, and 23 in time and solving the nonlinear system (19)– (22) at each time step. The objectives are to minimize the pull-down and pull-up times ~tpd and ~tpu to reach speciﬁed setpoints, i.e., a refrigerated space temperature sL;set , and a hot water temperature swx;set , respectively, in transient operation. In steady state operation, such objectives are translated into maximizing the system combined second law eﬃciency, deﬁned by Eq. (28). Properties and constant values are selected, according to the Table 1, and initial conditions ðsc ; ssp ; st ; swx ; sH ; sL Þ0 are established to complete the initial value problem formulation. It is important to point out that an optimal absorption refrigerator thermal conductance allocation has been found in previous studies (Bejan et al., 1995; Vargas et al., 1996) for achieving maximum refrigeration rate, i.e., ðc0 ; cH ; cL Þopt ¼ ð0:5; 0:25; 0:25Þ, which is also roughly insensitive to the external temperature levels ðsH ; sL Þ. Therefore, such architecture is the one utilized in this work. To test the model and for conducting the analysis presented in this section, the other parameters listed in Table 1 were selected according to the dimensionless parameters deﬁned by the model, assuming a small absorption refrigeration unit with a low total thermal conductance (UA = 500 W K1), such that all other components’ thermal conductances are referenced to that by assigning appropriate values to ci , where subscript ‘‘i” indicates a particular system component. Ethylene-glycol was assumed as the thermal ﬂuid, hydrogen as the fuel, a 50% eﬃciency in the combustion chamber, and a thermal load of 50 W in the cold space, in order to produce the other necessary dimensionless parameters for the simulations. The numerical method calculates the transient behavior of the system, starting from a set of initial conditions, then the solution is marched in time (and checked for accuracy) until a desired condition is achieved (temperature setpoints or steady state). The equations are integrated in time explicitly using an adaptive time step, 4th–5th order Runge–Kutta method (Kincaid and Cheney, 1991). The time step is adjusted automatically according to the local truncation error, which is kept below a speciﬁed tolerance of 10–6. Steady state conditions are veriﬁed when i j 6 106 , where subscript i stands for each one of the j @s @~t calculated temperatures in the simulation. During the integration of the ordinary diﬀerential equations, using the current value of sH , Eqs. (15) and (16b) deliver sHC , and Eq.

ð29Þ !

1

þ

1

In Eq. (27), only the positive root is selected through: 2 1=2 C 2 þ C 2 4C 1 C 3 ~L ¼ Q ð30Þ 2C 1 ~ 0 ; s0C and sLC are calculated with sys~ L is known, Q Once Q tem (19)–(22) and the current value of sL for a particular time step. The following analysis is conducted for the system deﬁned in Figs. 1 and 2, with the parameters listed in Table 1. The transient evolution of the dimensionless heat transfer rates that drive the system and computed in the simulation is shown in Fig. 3. At ~t ¼ 0, all heat transfer rates are equal to zero, since there are no driving temperature diﬀer~ cc , since fuel is assumed to burn from ~t ¼ 0. ences, except Q The system response to heat input ﬂuctuations is analyzed by varying both the solar irradiation and the combustion chamber heat input, which directly aﬀects the computed heat transfer rates, as shown in Fig. 3. In Figs. 3 and 4 ~ cc ¼ 0:04 was the following conditions were considered: Q ‘‘on” for the entire simulation time, except when turned ‘‘oﬀ” by the control action, and the maximum solar _ ¼ 1400 Wm2 0 6 ~t 103 < 3 or irradiation was set as I ~t 103 P 12 , being the well known universal solar constant (Kreith and Bohn, 2001), I_ ¼ 0 Wm2 ð3 6 ~t 103 < 5Þ, and I_ ¼ 700 Wm2 ð5 6 ~t 103 < 12Þ. It is noted that such conditions are not the same as real conditions observed in a day light cycle, since solar irradiation actually increases slowly in time up to its highest value. However, the idea of using step functions is to show how the model responds to sudden changes in solar irradiation due to shades caused by clouds for example. For the parameters listed in Table 1, the total dimensionless simulation time, ~t 103 ¼ 18 corresponds to approximately 12 h. A control strategy was devised for the combined system and analyzed by the simulation, and consisted of: (i) the thermal ﬂuid mass ﬂow rate through the collector, m_ sp , was ‘‘on” only when the collector coil output temperature was greater than the tank temperature; (ii) the cold chamber temperature control was based on two setpoints,

J.V.C. Vargas et al. / Solar Energy 83 (2009) 1232–1244

Q c , Q sp , Q cc

a

0.15

ψ sp , s , ψH, s , ψwx,wx

0.1

0.3 35, 0.239,0.5

~ Qc

~ Qc

~ Qsp

~ Q sp

0.05 ~ Q cc

0 0

2

4

6

8

10

~ t 10

QL , QH , Q0

b

0.02

ψsp, s , ψH, s , ψwx,wx

12

14

16

18

3

0.3 35, 0.239,0.5

1239

size constraint, which could be investigated by the present model, but it is not within the scope of this study. The heat transfer rate actually collected by the thermal ﬂuid through ~ sp , is shown to be roughly 50% of the collector the coil, Q ~ c , showing that at high net radiation heat transfer rate, Q collector temperatures, thermodynamic losses can be high, and need to be accounted for in collector design. Fig. 3b shows the transient behavior of the heat transfer rates associated to subsystem (III) regenerator heat exchanger and absorption refrigerator. Energy conservation dic~H ~ 0 has to be always equal to the sum of Q tates that Q ~ and QL , which is indeed observed in the simulation results. The regenerator mass ﬂow rate was switched on and oﬀ according to the established setpoints for the cold chamber temperature level, as it is reﬂected in all subsystem III heat ~ H and Q ~ L . Using the deﬁnition of coef~ 0, Q transfer rates, Q ﬁcient of performance for an absorption refrigerator, i.e., ~ H , by direct inspection of Fig. 3b, one evalu~ L =Q COP ¼ Q ates for the refrigerator considered in the present system that COP 0.5, which is a typical value in absorption refrigeration.

Q0 0.01

a

QH

0 0

2

4

6

8

10

t 10

12

14

16

18

3

τ t , τ c , τ sp

QL

Fig. 3. The transient behavior of: (a) subsystem (I) and combustion chamber, and (b) subsystem (III).

1.5

(ψ

s p, s

, ψ H, s , ψ wx,wx ) = ( 0.3 35, 0.239,0.5)

1.4

τc

1.3

τ sp

τc τ sp

τc

1.2

τ sp

τt

τt

τt

1.1 1 0.9 0

2

4

6

8

10

12

14

16

18

14

16

18

~ t × 10 −3

b

1.3

(ψ

sp , s

)

, ψ H, s , ψ wx,wx = (0 .3 35, 0.239,0.5)

1.2

τ t , τ wx τH , τL

sL;set;low ¼ 0:965 and sL;set;high ¼ 0:975, so that when sL < 0:965, m_ H was turned oﬀ, and turned on again when sL > 0:975, and so on, and iii) the hot water output temperature control was also based on two setpoints, swx;set;low ¼ 1:07 and swx;set;high ¼ 1:09, so that when swx > 1:09; m_ f was turned oﬀ, and turned on again when swx < 1:07, and so on. Fig. 3a shows the simultaneous eﬀect of the system control action and of the perturbations in the sun irradiation during the day. The combustion chamber heat transfer rate was switched on and oﬀ according to the established setpoints for the hot water temperature output level, as the ~ cc curve shows. The perturbations in the sun irradiation Q ~ sp curves. It is noted ~ c and Q are shown by the computed Q ~ sp ~ c and Q that when the sun irradiation dropped to zero, Q dropped to zero as well leaving the system to be driven solely by the tank thermal inertia and combustion chamber heat transfer rate, returning to work when the sun irradiation came back at half of its maximum value. A design issue would be the optimization of the receiver and coil ~ sp , under a ﬁxed total internal architecture to maximize Q

τt

τH

1.1

τ wx

τL

1

0.9 0

2

4

6

8

10

12

~ t × 10 −3 Fig. 4. The transient behavior of: (a) temperatures of subsystem (I) and the tank temperature and (b) temperatures of subsystems (II) and (III).

J.V.C. Vargas et al. / Solar Energy 83 (2009) 1232–1244

Fig. 4a shows the transient behavior of the dimensionless temperatures of subsystem I) solar concentrator and receiver for a given set of dimensionless thermal ﬂuid and water heat capacity ﬂow rates, until steady-periodic state is achieved. All initial temperatures are set equal to ambient temperature. The collector temperature, sc , is higher than all others, being the system component responsible to collect as much as possible of the available solar energy to be stored in the energy storage tank. The collector and the serpentine (coil) outlet temperature levels, sc and ssp , depart from each other determining the temperature diﬀerence responsible for the heat input rate collected by the ~ sp . Note that when the sun irradiation drops to zero, coil, Q the thermal ﬂuid mass ﬂow rate through the collector is switched oﬀ since the coil outlet temperature drops to lower values than the tank temperature, and switched on again when the sun irradiation returned in a later simulation time. The control action is observed accordingly, as the system reached the desired temperature setpoints. A similar behavior is observed in the dimensionless temperatures of subsystems (II) energy storage tank and hot water heat exchanger, and (III) regenerator heat exchanger and absorption refrigerator in Fig. 4b. All initial temperatures were also set equal to ambient temperature. It is observed that the refrigerator temperature levels depart from each other determining temperature diﬀerences in the heat exchangers that are responsible for the necessary heat transfer rates for appropriate hot water heat exchanger and refrigerator operation. The storage tank temperature, st , is higher than all others, being the system component responsible to deliver the necessary exergy input for the hot water heat exchanger and the refrigerator to operate. The water outlet temperature, swx , increases in time due to thermal contact with the storage tank thermal ﬂuid. Due to the energy storage tank, the system is shown to be capable of sustaining an adequate response even with the suppression ð3 6 ~t 103 < 5Þ and reduction of solar irradiation ð5 6 ~t 103 < 12Þ, for a period of time. After ~t 103 ¼ 12, the system reached a steady-periodic regime, i.e., an ‘‘on-oﬀ” pattern repeated itself since sun irradiation was no longer perturbed, with both fuel and regenerator mass ﬂow rates being switched on and oﬀ according to the system control strategy. All curves in Figs. 3 and 4 capture the expected physical trends for the proposed system, and therefore this gives reliability to the model to evaluate the transient behavior of the entire system until steady state is achieved and to perform the system thermodynamic optimization. A transient model is also mandatory if one is to evaluate system performance operating under ﬂuctuations in solar and combustion chamber heat supply. From this point, the analysis proceeds with the ﬁxed parameters listed in Table 1, i.e., assuming ﬁxed solar irradiation and no combustion chamber heat input. The search for system thermodynamic optimization opportunities started by monitoring the behavior of sL and swx in time, for three dimensionless serpentine (coil) thermal ﬂuid

capacity rates, while holding the others constant, i.e., wH;s and wwx;wx . Fig. 5 shows that there is an intermediate value of the serpentine (coil) thermal ﬂuid capacity rate, between 0.48 and 4.3, such that the temporal temperature gradient is maximum, minimizing the pull-down and pull-up times to achieve prescribed setpoint temperatures (sL;set ¼ 0:97 and swx;set ¼ 1:08). Since there are three heat capacity rates that characterize the proposed system (wsp;s ; wH;s and wwx;wx ), three levels of optimization were carried out in this study for maximum system performance. The ranges of variation of each parameter to be optimized were: 0:2 6 wsp;s 6 4:3; 0:096 6 wH;s 6 0:335, and 0:083 6 wwx;wx 6 0:584. The chosen discretization in all three ranges for the performance maximization was the coarsest set for which the optimal value of each parameter did not change as the sets became ﬁner, while the relative error was kept below 1% in all cases. The cogeneration system three-way optimization procedure utilized to obtain the results shown in Figs. 6–10 is summarized by the following algorithms: (a) Pull-down time minimization: 1. Fix the hot water heat capacity rate, wwx;wx , at any desired value. 2. Vary the regenerator thermal ﬂuid heat capacity rate: 0:096 6 wH;s 6 0:335. 3. Vary the serpentine (coil) thermal ﬂuid heat capacity rate: 0:2 6 wsp;s 6 4:3. 4. For each heat capacity rate set (wsp;s ; wH;s and wwx;wx ) use the model to determine the pull-down time, ~tpd , to reach the refrigerated space temperature setpoint, sL;set . 5. From all tested sets, for a speciﬁc hot water heat capacity rate, wwx;wx , select ðwsp;s ; wH;s Þopt that leads to the two-way minimized pull-down time, ~tpd; min;min .

1.2

ψH,s , ψwx ,wx

τ L , τ wx

1240

0.23 9,0.5

τwx

1.1 τ wx, set 1.08

1.91

ψsp,s ψsp, s

1

0.48 4.3 4.3 0.48

τL,set 0 97

τL

1.91

0.9 0

1

2

3

4

t 10

3

5

6

7

Fig. 5. The system pull-down and pull-up times for achieving sL;set ¼ 0:97 and swx;set ¼ 1:08, respectively, with respect to wsp;s .

J.V.C. Vargas et al. / Solar Energy 83 (2009) 1232–1244 4

a

(ψH,s, ψwx ,wx ) = (0.23 9,0.5)

1241

3

ψH , s = 0.239 ; τwx, set = 1.0 8

τ L ,set = 0.97 ; τ wx , set = 1. 08 ψwx ,wx = 0.58 4

~ tpu × 10 −3

~ tpd × 10 −3 ~ tpu × 10 −3

3

2

~ tpu × 10 −3

2

0.5 1

1

~ tpd × 10 −3

0.083

0 0

0.5

1

1.5

2

2.5

3

ψsp , s

Fig. 6. The one way minimization of the system pull-down and pull-up times for achieving sL;set ¼ 0:97 and swx;set ¼ 1:08, respectively, with respect to wsp;s .

b

2

2

ψsp, s,opt

3

= 0. 5 ; τL,set = 0. 9 7

1.5

1

pu

1

(~t

(~t

~ tpd × 10 −3

ψH, s = 0.335

pu

× 10 −3

)

min

0.5

0.5

0.143

ψsp, s,opt

2

× 10 − 3

)

ψwx , wx

1.5 min

a

0

4

2 ψsp , s

1

ψ

0.239

H, s

= 0 .2 39 ; τ wx ,set = 1.0 8 0

0 0

0.4

0.6

ψwx , wx

0 0.5

1

1.5

2

2.5

3

Fig. 8. (a) The one way pull-up time minimization with respect to wsp;s , and (b) the increase of pull-up time as the dimensionless water capacity rate increases.

ψsp , s

b

0.2

2

2

ψsp, s,opt 1.5

(

~ tpd × 10 −3

(~t

)

ψsp, s,opt

1

1

pd

× 10 −3

)

min

1.5

min

0.5

0.5

ψwx , wx = 0. 5 ; τ L , set = 0.9 7 0

0 0.1

0.2

0.3

0.4

ψ , H s

Fig. 7. (a) The one way pull-down time minimization with respect to wsp;s , and (b) the two way pull-down time minimization with respect to wsp;s and wH;s .

(b) Pull-up time minimization: 1. Fix the regenerator thermal ﬂuid heat capacity rate, wH;s , at any desired value.

2. Vary the hot water heat capacity rate: 0:083 6 wwx;wx 6 0:584. 3. Vary the serpentine (coil) thermal ﬂuid heat capacity rate: 0:2 6 wsp;s 6 4:3. 4. For each heat capacity rate set (wsp;s ; wH;s and wwx;wx ) use the model to determine the pull-up time, ~tpu , to reach the hot water temperature setpoint, swx;set . 5. From all tested sets, for a speciﬁc regenerator thermal ﬂuid heat capacity rate, wH;s , select ðwsp;s ; wwx;wx Þopt that leads to the two-way minimized pull-up time, ~tpu; min;min . (c) System second law eﬃciency maximization: 1. Vary the regenerator thermal ﬂuid heat capacity rate: 0:096 6 wH;s 6 0:335. 2. Vary the hot water heat capacity rate: 0:083 6 wwx;wx 6 0:584. 3. Vary the serpentine (coil) thermal ﬂuid heat capacity rate: 0:2 6 wsp;s 6 4:3.

a

J.V.C. Vargas et al. / Solar Energy 83 (2009) 1232–1244 0.095

0.08

2

ηII ,max,max

ψH, s = 0.239

ψwx , wx = 0.5 0.584

0.09

ψsp ,s ,opt

ηII ,max,max

0.07

ηII

0.083 0.06

0.085 1 0.08 0.5

0.05

0.075

0.04 0.5

b

1.5

(ψsp, s , ψwx , wx )opt

1242

1

1.5

ψsp , s

2

2.5

0.07 0.05

3

0.15

0.2

0.25

0.3

0 0.35

1.6

Fig. 10. The three way maximization of system second law eﬃciency with respect to wsp;s , wwx;wx , and wH;s .

1.5

cooling and heating. Fig. 7a shows the minimization of the refrigerated space pull-down time for diﬀerent values of the dimensionless regenerator thermal ﬂuid capacity rate, wH;s . It is clear that there is a second minimum, this time with respect to wH;s . The minimization of the pulldown time was pursued for 0:096 6 wH;s 6 0:335, and wwx;wx ¼ 0:5, and the results are shown in Fig. 7b. The optimal dimensionless collector thermal ﬂuid capacity rate is ‘‘robust” with respect to the variation of wH;s , i.e., wsp;s;opt ﬃ 1:73. The two way minimized pull-down time happens at the optimal pair ðwsp;s ; wH;s Þopt ﬃ ð1:73; 0:25Þ, for wwx;wx ¼ 0:5. Using the same procedure for obtaining the hot water heat exchanger pull-up time, the dimensionless regenerator thermal ﬂuid capacity rate was kept ﬁxed ðwH;s ¼ 0:239Þ and the dimensionless water capacity rate, wwx;wx , was varied together with wsp;s . The minimization of the pull-up time with respect to wsp;s is shown in Fig. 8a for three values of wwx;wx , ﬁnding wsp;s;opt ﬃ 1:73 for wH;s ¼ 0:239. It is observed that the one way minimized pull-up time decreases monotonically as wwx;wx decreases. This behavior is diﬀerent from what was observed in the refrigerated space pull-down time. In the cold space, the thermal load is kept ﬁxed in the optimization process, therefore the pull-down time is minimized as the refrigerator exergy input rate is maximized. Conversely, in the hot heat exchanger, the ‘‘load” to be heated decreases as the water capacity rate decreases, therefore the less water the faster it is to heat it up, as the results of Fig. 8b show. However, although the water stream is heated up faster, less exergy is available, i.e., the potential for use of the hot water reduces. This discussion brings to light the need to evaluate the performance of the combined system (or thermal systems in general) on a more concrete basis, which is provided by exergy analysis, i.e., by evaluating the resulting second law eﬃciency of the entire system, according to the Eq. (28). It is then clear that, by maximizing the second

ηII , max

ηII,max

ψsp, s,opt

0.075

ψsp, s,opt 1.4

0.07

ψH, s = 0.239 1.3

0.065 0.2

0.1

ψH, s

0.08

0

ψwx , wx , opt

0.4

0.6

ψwx , wx

Fig. 9. (a) The one way maximization of system second law eﬃciency with respect to wsp;s , and (b) the two way system second law minimization with respect to wsp;s and wwx;wx .

4. For each heat capacity rate set (wsp;s ; wH;s and wwx;wx ) use the model to determine the steady state system second law eﬃciency, gII , Eq. (28). 5. From all tested sets, select ðwH;s ; wwx;wx ; wsp;s Þopt that leads to the three-way maximized system second law eﬃciency, gII; max;max;max .

The optimization with respect to the collector thermal ﬂuid capacity rate is pursued in Fig. 6 both for minimum pull-down and pull-up times, for ðwH;s ; wwx;wx Þ ¼ ð0:239; 0:5Þ. It is observed that the optimal value of the collector thermal ﬂuid capacity rate is the same for both objectives, i.e., cooling and heating. This is explained by the fact that both the hot water heat exchanger and the refrigerator are driven by the same heat source. It is also clear that the same optmization opportunity exists for other pairs ðwH;s ; wwx;wx Þ based on the explanation given in the ﬁrst paragraph of Section 3. Next, for better interpretation of the results, the optimization is pursued separately for each desired product, i.e.,

J.V.C. Vargas et al. / Solar Energy 83 (2009) 1232–1244

law eﬃciency of the entire system, the potential for use of the two products, cooling and heating (at sL;set ¼ 0:97 and swx;set ¼ 1:08) will be maximized. Proceeding in that direction, the maximization of the steady state combined second law eﬃciency of the entire system, gII , is conducted with respect to wsp;s , for three diﬀerent values of wwx;wx , while keeping the dimensionless regenerator thermal ﬂuid capacity rate ﬁxed, wH;s ¼ 0:239. The results are seen in Fig. 9a, and it is observed that there is an intermediate dimensionless water capacity rate in which a second maximum for gII is found. Fig. 9b shows the two way maximization of the second law eﬃciency for 0:083 6 wwx;wx 6 0:584, and wH;s ¼ 0:239. Again, the optimal dimensionless collector thermal ﬂuid capacity rate is ‘‘robust” with respect to the variation of wwx;wx , i.e., wsp;s;opt ﬃ 1:43. The two way maximized second law eﬃciency is found at the optimal pair ðwsp;s ; wwx;wx Þopt ﬃ ð1:43; 0:34Þ, for wH;s ¼ 0:239. Note that, now the system exergy input rate (total system available power for heating and cooling) is maximized. This does not necessarily mean minimum pull-down and pull-up times. This is why wsp;s;opt occurs at a diﬀerent location for maximum system exergy input rate for wH;s ¼ 0:239 than for minimum pull-down and pull-up times. Finally, the three way maximization of the steady state second law eﬃciency is pursued in Fig. 10. The two way maximization procedure was repeated for 0:096 6 wH;s 6 0:335, and an optimal set of the three dimensionless heat capacity rates was found where the calculated second law eﬃciency was maximum. The results show that the two way maximized gII; max : max shows a third maximum at ðwsp;s ; wwx;wx ; wH;s Þopt ﬃ ð1:43; 0:23; 0:14Þ. The maximum is sharp, with a 25% variation of gII;max : max in the range 0:096 6 wH;s 6 0:335, therefore important to be accounted for in actual solar system design. Since all results obtained in this study are dimensionless, in the search for optimal operation, it is expected that the optima found could be utilized to set the operating mass ﬂow rates of systems of any size, similar to Fig. 1, even when the dimensionless design parameters are not close to the ones listed in Table 1. Therefore, the results of this study could be eﬀectively used in actual system design and operation, for maximum thermodynamic performance. Further investigation is needed, by using the model to evaluate how robust the optimal parameters are with respect to the variation of the design parameters listed in Table 1, and also to optimize inventory allocation subject to ﬁxed total system size. The results are still valuable even without thermoeconomic optimization as a preliminary design tool, although cost has not been addressed in the analysis, due to the fact that maximum thermodynamic performance usually implies at least in cost reduction. 5. Conclusions The basic thermodynamics problem of how to extract maximum exergy input rate from a solar collector and

1243

gas-ﬁred driven water heating and refrigeration plant has been considered. A transient solar system mathematical model was developed to obtain the system response in time and to calculate the second law eﬃciency of the entire system, as functions of operating and design parameters. Appropriate dimensionless groups were identiﬁed and the generalized results reported in dimensionless charts. Based on the results of Figs. 3–10, the main conclusions of this study are summarized as follows: 1. There exists a fundamental optimal set of three heat capacity rates that characterize the system such that maximum exergy input rate is obtained in the solar collector and gas-ﬁred driven water heating and refrigeration plant, and therefore maximum water heating and refrigeration rate, no matter how complicated the actual design may be; 2. Since this optimization principle is general, the model could be used as a preliminary project tool, to locate ðwsp;s ; wwx;wx ; wH;s Þopt for any set of system design parameters that are diﬀerent from Table 1, for maximum second law eﬃciency, and therefore optimal operation; 3. It was shown that the optimal dimensionless collector thermal ﬂuid capacity rate, wsp;s;opt , is robust with respect to the variation of the other two heat capacity rates, wH;s and wwx;wx . This is an important ﬁnding for the purpose of system scalability; 4. The minimum pull-down and pull-up times, and maximum second law eﬃciencies found with respect to the optimized operating parameters are sharp, stressing their importance for practical design, and therefore must be identiﬁed accurately in the quest for achieving solar systems higher net eﬃciencies and size reduction in order to make these systems commercially competitive.

Acknowledgements The authors acknowledge with gratitude the ﬁnancial support given by the Oﬃce of Naval Research (ONR) grant N00014-08-1-0080, CAPS and IESES at the Florida State University, the Brazilian National Council for Scientiﬁc and Technological Development (CNPq), and the Brazilian Studies and Projects Financer (FINEP) under the Grant No. 01.04.0944.00 (proj. 2254/04). References Aghbalou, F., Mimet, A., Badia, F., Illa, J., El Bouardi, A., Bougard, J., 2004. Heat and mass transfer during adsorption of ammonia in a cylindrical adsorbent bed: thermal performance study of a combined parabolic solar collector, water heat pipe and adsorber generator assembly. Applied Thermal Engineering 24 (17–18), 2537–2555. Alam, K.C.A., Saha, B.B., Akisawa, A., Kashiwagi, T., 2001. Optimization of a solar driven adsorption refrigeration system. Energy Conversion and Management 42 (6), 741–753.

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