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Renewable Energy 33 (2008) 2064–2076

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Two-phase flow modelling of a solar concentrator applied as ammoniavapor generator in an absorption refrigerator

N. Ortegaa,�, O. Garcıa-Valladaresb, R. Bestb, V.H. Gomezb

aPosgrado en Ingenierıa (Energıa), Universidad Nacional Autonoma de Mexico, Privada Xochicalco s/n, Temixco, Morelos 62580, MexicobCentro de Investigacion en Energıa, Universidad Nacional Autonoma de Mexico, Privada Xochicalco s/n, Temixco, Morelos 62580, Mexico

Received 9 January 2007; accepted 30 November 2007

Available online 28 January 2008

Abstract

A detailed one-dimensional numerical model describing the heat and fluid-dynamic behavior inside a compound parabolic

concentrator (CPC) used as an ammonia vapor generator has been developed. The governing equations (continuity, momentum, and

energy) inside the CPC absorber tube, together with the energy equation in the tube wall and the thermal analysis in the solar

concentrator were solved.

The computational method developed is useful for the solar vapor generator design applied to absorption cooling systems. The effect

on the outlet temperature and vapor quality of a range of CPC design parameters was analyzed. These parameters were the acceptance

half-angle and CPC length, the diameter and coating of the absorber tube, and the manufacture materials of the cover, the reflector, and

the absorber tube. It was found that the most important design parameters in order to obtain a higher ammonia–water vapor production

are, in order of priority: the reflector material, the absorber tube diameter, the selective surface, and the acceptance half-angle.

The direct ammonia–water vapor generation resulting from a 35m long CPC was coupled to an absorption refrigeration system model

in order to determine the solar fraction, cooling capacity, coefficient of performance, and overall efficiency during a typical day of

operation. The results show that approximately 3.8 kW of cooling at �10 1C could be produced with solar and overall efficiencies up to

46.3% and 21.2%, respectively.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Compound parabolic concentrator; CPC; Ammonia–water mixture; Direct vapor generation; Absorption refrigeration; Mathematical model

1. Introduction

The majority of the developing countries have powergeneration capacity problems, that tend to increase as theneed for energy intensive conventional air conditioningand refrigeration also increases. Alternative coolingmethods are required to decrease the power demand,and the conventional high global warming potential(GWP) and ozone depletion potential (ODP) refrigerantusage. In addition, developed countries need new refrigera-tion technologies as an alternative to conventionalcompression refrigeration to meet their air-conditioning

e front matter r 2007 Elsevier Ltd. All rights reserved.

nene.2007.11.016

ing author. Tel.: +5255 56 22 97 36;

2 97 91.

ess: noa@cie.unam.mx (N. Ortega).

and cooling demands without increasing greenhouse gasesemissions [1].Solar energy has the evident advantage that cooling is

generally required when solar radiation is available [2].This is the main reason for sustained research into solarcooling devices for at least three decades. These studiesinclude the combination of solar energy technologies withthermal refrigeration technologies (as absorption, adsorp-tion, and desiccant) to produce cooling and refrigerationusing medium to high-temperature solar technologies(from 80 to 250 1C) [2–6].Solar concentrator designs applied to steam generation

are found in diverse development stages, from evaluationand improvement of solar devices, to full systems in teststage for power generation [7,8]. In addition, some solarconcentrators applied as generators for intermittentabsorption refrigerators have been developed [9,10].

ARTICLE IN PRESS

Nomenclature

Aa absorber tube heat transfer area, m2

Ac cover heat transfer area, m2

Ar reflector heat transfer area, m2

At fluid flow cross section area, m2

At�abs absorber tube cross-section area (p(D2out�D2

in)/4),m2

C area concentration ratio, dimensionlessCOP coefficient of performance, dimensionlessCp specific heat, J/(kgK)D diameter, mf friction factor, dimensionlessFR flow ratio, dimensionlessg gravitational constant, ( ¼ 9.81m/s2)Gbn beam irradiance normal to the plane, W/m2

h enthalpy, J/kgH height, mI solar irradiance, W/m2

k thermal conductivity, W/(mK)L length, mm mass, kg_m mass flow rate, kg/s

p perimeter, mP pressure, barq heat flow per unit area, W/m2

qu useful energy gain per absorber unit area, W/m2

q0u useful energy gain per length unit, W/mqwall heat flux per absorber unit area from fluid to

wall, W/m2

R thermal resistance, (m2K)/WS solar absorbed energy per unit area, W/m2

t time, sT temperature, KUL overall heat loss coefficient, W/(m2K)V volume, m3

~Vx velocity in the axial direction, m/sw cover width, mx axial coordinatexg vapor quality, dimensionlessX ammonia weight concentration, dimensionless

Greek letters

a heat transfer coefficient, W/(m2K))b inclination angle of absorber tube, degreee emittance, dimensionlesseg void fraction, dimensionlessf generic dependent variable

Z efficiency, dimensionlessj angle of involute generation, degreem viscosity, kg/(m s)yC acceptance half-angle, degreer density, kg/m3

s stefan–Boltzman constant, ( ¼ 5.6697� 10�8W/(m2K4))

u wind velocity, m/sz effectiveness, dimensionlessDx spatial discretization step, mDt temporal discretization step, sF two-phase frictional multiplier, dimensionless

Dimensionless numbers

Pr prandtl number, ( ¼ mCp/l)Re reynolds number ð¼ r~VD=mÞ

subscripts

a absorber tube wallc coverco conductiveen environmentEV evaporationex externalf fluidg gasGE generationi inletin innerinv involutej number of control volumel liquid phaseo outletout outerpar parabolar reflectorra radiatives saturationsk skytp two-phase

Superscripts

– arithmetical average over a CV� integral average over a CVo value of previous instant

N. Ortega et al. / Renewable Energy 33 (2008) 2064–2076 2065

Flat plate collectors have been applied to directrefrigerant evaporation in solar-assisted heat pumps, wherea two-phase refrigerant flows through the collectors insteadof utilizing a heat exchanger between the collector and theevaporator [11].

Since 1990, parabolic trough solar concentrators havebeen used to evaporate water to produce steam directly onthe absorber tube [12]. The technology developed is knownas direct steam generation (DSG), where the vaporproduced is mainly applied for power generation. DSG

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presents many advantages compared to the heating oil-based technology, since DSG eliminates costly syntheticoil, intermediate heat transport piping, special typeequipment to run the high-temperature oil, and the oilfor steam heat exchanger [13].

Compound parabolic concentrators (CPCs) are a goodchoice for applications in direct evaporation, since thesestationary collectors have a good quality rate between costand performance at medium temperature levels [14].

Based on the main advantages of DSG, a CPC wasdesigned in order to directly generate ammonia from anammonia–water solution. Ammonia vapor would beutilized in an ammonia–water absorption solar refrigera-tor. Other applications not analyzed here could be the useof the CPC as a heat source for a direct ammonia–watersolution evaporator for applications in combined powerand cooling thermodynamic cycles, as proposed byGoswami and Xu [15].

In an attempt to reduce heat losses and demonstrate itsfeasibility, a CPC was modelled and designed in order togenerate ammonia vapor inside its absorber tube. Thetheoretical analysis of the evaporation process inside theCPC was emphasized, through a detailed one-dimensionalnumerical simulation of the thermal and fluid-dynamicbehavior of two-phase flow. The CPC model was coupledto a complete single-stage absorption refrigeration cyclemodel in order to calculate the theoretical cooling capacityand coefficient of performance (COP) under differentworking conditions.

To our knowledge, solar concentrators have not beenapplied as direct ammonia vapor generators in a contin-uous thermal refrigeration system.

1.1. A brief description of the CPC models

Initially, the models developed to describe CPC opticaland thermal performance were restricted to the flatabsorber type [16]. In these models, convective heattransfer was usually represented by flat plate filmcoefficients. The simplest models for CPC with tubularabsorber have not considered absorption of high wave-length over reflective surfaces [16].

Hsieh developed the mathematical formulation for thethermal processes in a tubular CPC, where heat exchangebetween components was predicted [17]. Chew et al. [18]developed a finite-element model for a CPC with tubularabsorber; they considered that the absorber tube and thecover were isothermic, while the reflectors were consideredas adiabatic boundaries.

Eames and Norton [16] developed and validated atwo-dimensional model in steady state in order to simulatethe optical and thermal behavior of a through-typeCPC. Ray tracing and finite-element analysis of convectionheat transfer were applied. Solar beam and diffuseradiation were considered in the optical analysis, irradianceand absorption were assumed homogeneous, and thatthe energy reaching the absorber tube was completely

absorbed. The study included reflector conduction, high-wave radiative interchange, and heat removal in thetubular absorber.Tchinda et al. [19] analyzed the heat exchange in a CPC

collector, where axial heat transfer in the tubular absorberwas included. They developed an explicit expression inorder to calculate the fluid temperature as a function of thecoordinate space in the flux direction and the time-dependent solar intensity.In this paper, a simple method was carried out in order

to establish the energy balances in a CPC, where theabsorber tube operates as an ammonia–water mixturedirect vapor generator in a solar absorption refrigerator.The evaporation process was studied in order to fulfil thethermal and fluid-dynamic characterization inside the CPCabsorber tube.The system under investigation consisted of a trough-

type CPC with a steel tubular absorber without anevacuated glass shell. Thermodynamic equilibrium betweenthe liquid and vapor phases was supposed. A one-dimensional numerical simulation of the thermal andfluid-dynamic behavior of two-phase flow was developed.The governing equations (continuity, momentum, andenergy) inside the tube, together with the energy equationin the tube wall and the thermal analysis in the solarconcentrator, were solved iteratively in a segregatedmanner. The discretized governing equations in fluid flowwere coupled using an implicit step-by-step method in theflow direction.By means of the model results, a CPC module was

designed and theoretically evaluated as ammonia generatorin an ammonia–water absorption solar refrigerator.

1.2. CPC module

In a previous work [20], a mathematical model wasdeveloped in order to evaluate the temperature distributionof a CPC array proposed to be used as a vapor generator inan absorption ammonia–water refrigeration system.It was established that the mixture temperature increasesand wall absorber tube temperature decreases when theammonia–water mixture reaches saturation conditions,which improves the heat transfer process.Ortega et al. [21,22] developed a more accurate model,

where the thermal and fluid-dynamic behavior of evapora-tion process at the solar concentrator absorber tubewas numerically simulated. This analysis was madewith a control volume (CV) method on the absorber tube,and the discretized equations were coupled using a fullyimplicit step-by-step method in the flow direction.The conduction in the internal tube wall was solvedusing the TDMA algorithm. A separated flow model wasapplied and two different two-phase flow convectiveheat transfer coefficients were used. A CPC prototype of2m length, 0.66m width and 0.84m height was designed,with a solar concentration of 3.5� and an apertureangle of 151.

ARTICLE IN PRESSN. Ortega et al. / Renewable Energy 33 (2008) 2064–2076 2067

In this paper a final CPC model was developed and usedfor the design analysis. An auxiliary heater was added tothe complete refrigeration system in order to maintain aconstant refrigeration load. The study consists of thethermal analysis of the CPC performance during a typicaloperating day. A new subroutine was developed tosimulate the final CPC model coupled with the absorptionammonia–water refrigerator.

2. Ammonia–water absorption refrigerator

Fig. 1 shows the single-stage ammonia–water absorptionsolar refrigerator. The proposed solar refrigerator includesthe following components: a generator (CPC), a rectifier, acondenser, an evaporator, an absorber, a flash tank, aneconomizer, a pre-cooler, a pump, and two expansionvalves. A model for steady-state single-stage ammonia–water absorption refrigeration system was developed tosimulate the results obtained by the CPC model in order toevaluate the performance of the complete cycle.

2.1. Operative description

Following the schematic diagram in Fig. 1, ammoniavapor (99.5 wt%) leaves the rectifier as overheated vapor atstate 4, at the high pressure of the system. The refrigerantvapor is cooled and liquefied in the condenser as saturatedliquid, at state 5; it is then subcooled in the pre-cooler(state 6) and thereafter passes through an expansion valve,where the pressure is reduced, giving as a result a cooled

Fig. 1. Ammonia–water absorption solar refr

two-phase mixture at state 7. Liquid ammonia enters theevaporator, where on extracting heat from the coolingwater, it is converted into vapor, producing the refrigerat-ing effect, and then exits as saturated vapor in state 8. It isthen superheated in the pre-cooler (state 9). The relativelycold ammonia vapor then enters the absorber, where it iscondensed and absorbed by the weak ammonia–watersolution. The absorption of ammonia is exothermic, so aheat exchange equipment in the absorber is needed in orderto cool the hot solution and improve its absorptioncapacity. The strong ammonia solution leaves the absorberat state 10 and enters the pump, leaving at high pressure atstate 11. It is then introduced in an economizer, where itreceives heat and leaves state 12. It then enters the CPCgenerator, where it receives solar generated heat, reachesthe saturation point, and vaporizes, leaving state 13 as avapor–liquid mixture. If additional heat is necessary it isadded to the mixture by the auxiliary heater in order toreach the operating conditions at state 1. The two-phasehigh-pressure mixture, enters the flash tank where liquidand vapor are separated, the liquid phase is mixed with thecondensed vapor originating from in the rectifier (state 3).This weak ammonia solution enters the economizer at state14, where heat is extracted, and leaves state 15. It thenpasses through an expansion valve, where the pressure isreduced (state 16) in order to enter the absorber. The vaporcoming from the flash tank separator enters the rectifier, inwhich through heat removal and partial condensation,water, leaving state 4 is removed. In this way the operationof the cycle is completed.

igerator with a CPC as vapor generator.

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2.2. Methodology for the complete cycle energy analysis

The purpose of the calculation sequence presented hereis to obtain the operation conditions of the ammonia–water absorption solar refrigerator system shown inFig. 1. An overall energy balance has been applied toall the components of the system (with the exceptionof the CPC model, where a detailed numerical simulationhas been developed). The following assumptions havebeen made:

�

The high and low pressures of the system are 11 and2.8 bar, respectively. Pressure drop through elements isneglected (with the exception of the CPC model). � Fluid leaves the condenser as saturated liquid (state 5). � A saturated vapor (state 8) exits from the evaporator. � Ammonia vapor (99.5wt%) leaves the rectifier (state 4). � Expansion valves are considered isenthalpic. � The pre-cooler has an effectiveness of 0.5 and an

economizer of 0.86.

� The ammonia–water solution pump driving power is

negligible.

The energy balance analysis over each component of the

O

M

W

DAr

Ac

Aa

θC

N

H

L

Fig. 2. CPC section showing acceptance half-angle yC, aperture area Ac,

tubular absorber area Aa, reflecting area Ar, reflector segments MN and

NO, absorber tube diameter D, concentrator height H, width W, and

length L.

0

r

L

x

x x+ΔxTf,i Ls Tf,s

m

xgf,o,Tf,o

quΔx

Fig. 3. Absorber tube cross-section showing one-phase and two-phase

zones of ammonia–water mixture.

system is coupled with the CPC model previously devel-oped in order to evaluate the performance of the completeammonia–water absorption refrigeration systems throughthe calculation of the COP, the flow ratio, the solarfraction, and the solar and overall efficiencies.

The COP for cooling is defined as the ratio between thecooling capacity (evaporation heat extracted inside theevaporator, QEV) and the generation heat (QGE):

COP ¼QEV

QGE

. (1)

The flow ratio, FR, is the ratio between the solution flow inthe circuit constituted by the generator and the absorber( _m1), and the refrigerant flow in the main circuit that jointthe condenser and the evaporator ( _m4). This ratio indicatesthe strong ammonia solution mass flow needed to producea unit of refrigerant vapor, in this case, ammonia vapor:

FR ¼_m1

_m4. (2)

Solar fraction was defined as the percentage of the totalenergy required to generate the ammonia that was achievedby the CPC.

Solar efficiency is the ratio between the useful energygain obtained by the absorber tube area (Arqu) and thesolar irradiance that reaches the aperture area (AaI):

Zsolar ¼QGE

AaI¼

Arqu

AaI. (3)

Overall efficiency is the product of COP and solarefficiency:

Zoverall ¼ Zsolar COP: (4)

3. Coupling between the CPC model and a single-stage

absorption system model

A numerical analysis was carried out for the designedCPC illustrated in Fig. 2, which has the geometrical andoptical characteristics established from the parametricanalysis shown below. A 35m long row is considered as aCPC module in this calculation, which could be scaled up.The calculations were made for a non-tracking CPC,

installed in Temixco, Morelos, Mexico (18150.360N,99114.070W). The CPC analyzed had an inlet temperatureaccording to the inlet generation temperature obtainedwith the absorption cycle simulation, and a generatorpressure of 11 bar.The study consists of the thermal analysis of the CPC

performance during a typical operation day (March 15th,1996).

4. Mathematical formulation

Fig. 3 shows the absorber tube cross-section. Subcooledammonia–water mixture enters the tube at position 0 witha mass flow _m, and an inner temperature Tf,i. The absorbertube receives a useful energy gain qu. Ammonia–watermixture starts to evaporate at a certain length Ls, wheresaturation temperature Tf,s is reached. Finally, the two-phase mixture is out at position L with an outside qualityxgf,o and temperature Tf,o.

ARTICLE IN PRESSN. Ortega et al. / Renewable Energy 33 (2008) 2064–2076 2069

The study was divided in three subroutines: fluid flowinside the absorber tube, heat transfer in the wall tube, andsolar thermal analysis.

4.1. Fluid flow inside the absorber tube

Taking into account the characteristic geometry ofthe absorber tube (diameter, length, roughness, and angle),the governing equations have been integrated assuming thefollowing assumptions:

�

One-dimensional flow: P(x, t), h(x, t), T(x, t), etc. � Non-participant radiation medium and negligible radi-

ant heat exchange between surfaces.

� Axial heat conduction inside the fluid was neglected.

The semi-integrated governing equations over a finiteCV have the following form:

�

Continuity:

½ _m�jj�1 þqm

qt¼ 0. (5)

�

Momentum:

½ _mgng�jj�1 þ ½ _mlnl�

jj�1 þ Dx

q ~_mqt

¼ �½P�jj�1At � ~tpDx�mg sin b. ð6Þ

�

Energy:

~_m½el�jj�1 þ ½ _mgðeg � elÞ�

jj�1 þ ð~eg � ~elÞ

qmg

qt

þmgq~egqtþml

q~elqt� At Dx

q ~Pqt

þ ð~el � elÞqm

qt¼ ~_qupDx, ð7Þ

where ~f represents the integral volume average of ageneric variable f over the CV and f its arithmeticaverage between the inlet and outlet of the CV. Thesubscript and superscript in the brackets indicate½X �

jj�1 ¼ X j � X j�1, i.e., the difference between the

quantity X at the outlet section and the inlet section.

In the governing equations, the evaluation of the shearstress is performed by means of a friction factor f. Thisfactor is defined from the expression: t ¼ F(f/4)(G2/2r),where F is the two-phase factor multiplier. The one-dimensional model also requires the knowledge of the two-phase flow structure, which is evaluated by means of thevoid fraction eg. Finally, heat transfer through the absorbertube wall and fluid temperature are related by theconvective heat transfer coefficient a, which is defined asa ¼ _qwall=ðTwall � T fluidÞ.

4.2. Solar thermal analysis

The useful energy gain per CPC length unit q0u, expressedin terms of the local absorber temperature Ta and theabsorber solar radiation per aperture unit S, is [23]

q0u ¼AcS

L�

AaUL

LðTa � TenÞ. (8)

The useful energy gain can be obtained from the lastexpression as

Qu ¼ AcS � AaULðTa � T enÞ. (9)

Then, the useful energy gain per unit of absorber area qu isobtained as

qu ¼Ac

AaS �ULðTa � T enÞ ¼ CS�ULðTa � T enÞ. (10)

Cover and absorber tube area were defined as

Ac ¼ wL; Aa ¼ pDoutL. (11)

The useful energy gain depends on the absorbed solarradiation S that is equal to the cover incident solar energyreduced by optical losses in the concentrator [23]; thermallosses in the cover, the reflector, and the absorber tube arerepresented as the overall heat loss coefficient UL.Absorbed solar radiation S is a function of the radiative

properties of the CPC components (reflectance, emittance,absorptance, and transmittance) and environmental con-ditions that depend on solar time (solar radiation, solarposition, and ambient temperature). Absorber solar radia-tion was calculated with the method presented by Duffieand Beckman [23].The overall heat loss coefficient UL depends on the

temperatures of the CPC components through the indivi-dual heat loss coefficients:

UL ¼Rc�enRr�en

Rc�en þ Rr�en þ Rr�c

�

þ1

Ra�c þ Rc�en�rþ

1

Ra�r þ Rr�en�c

� ��1)�1, ð12Þ

where

Rr�en�c ¼Rr�enRr�c

Rc�en þ Rr�en þ Rr�c,

Rc�en�r ¼Rc�enRr�c

Rc�en þ Rr�en þ Rr�c, ð13Þ

and finally

Rc�en ¼ ðaco;c�en þ ara;c�skÞ�1,

Ra�c ¼ ðaco;a�c þ ara;a�cÞ�1,

Rr�en ¼ ðaco;r�en þ ara;r�skÞ�1,

Rr�c ¼ ðaco;r�c þ ara;r�cÞ�1,

Ra�r ¼ ðaco;a�r þ ara;a�rÞ�1. ð14Þ

The convective heat transfer coefficient between thereflector and the cover aco,r�c was fixed at a constant value

ARTICLE IN PRESSN. Ortega et al. / Renewable Energy 33 (2008) 2064–20762070

of 5W/(m2K), as it has been previously evaluated byPrapas et al. [24] and Hsieh [17].

The convective heat transfer coefficients between thecover and the ambient, and between the reflector and theambient are, respectively, [23]

aco;c�en ¼ ð5:7þ 3:8uÞAc

Aa, (15)

aco;r�en ¼ ð5:7þ 3:8uÞAr

Aa, (16)

where the reflector area Ar was calculated by

Ar ¼ Dout Dxj2inv

4þ

1ffiffiffi2p

Z fpar

finv

ðp=2Þ þ yC þ j� cosðj� yCÞ

1þ sinðj� yCÞ½ �3=2dj

" #.

(17)

The convective heat transfer coefficients between theabsorber tube and the reflector and between the absorbertube and the cover were expressed, respectively, by [17]

aco;a�r ¼ 3:25þ 0:0085ðTa � T rÞ

2Dout, (18)

aco;a�c ¼ 3:25þ 0:0085ðTa � T cÞ

2Dout, (19)

ara;r�sk ¼ �rsðT2r þ T2

skÞðT r þ T skÞAr

Aa, (20)

ara;c�sk ¼ �csðT2c þ T2

skÞðT c þ T skÞAc

Aa, (21)

ara;r�c ¼sðT2

c þ T2r ÞðTc þ T rÞ

ð1� �cÞ=�c þ ðð1� �rÞ=�rÞðAc=ArÞ

Ar

Aa, (22)

ara;a�c ¼sðT2

a þ T2cÞðTa þ T cÞ

ð1=�cÞ þ ðAc=AaÞðð1=�aÞ � 1Þ, (23)

ara;a�r ¼sðT2

a þ T2r ÞðTa þ T rÞ

ð1� �rÞ=�r þ ðð1� �aÞ=�aÞðAr=AaÞ. (24)

The temperatures of cover and reflector a necessary inorder to solve Eq. (10). Both were determined by means ofthe energy balances in each CPC component:

Tc ¼ðara;a�c þ aco;a�cÞTa þ ara;c�skT sk þ aco;c�enTen þ ðaco;r�c � ara;r�cÞT r

ara;a�c þ aco;a�r þ ara;c�sk þ aco;c�en þ aco;r�c � ara;r�c,

(25)

T r ¼ðara;a�r þ aco;a�rÞTa þ aco;r�enT en þ ara;r�skT sk þ ðaco;r�c � ara;r�cÞTc

ara;a�r þ aco;a�r þ aco;r�en þ ara;r�sk þ aco;r�c � ara;r�c.

(26)

4.3. Evaluation of empirical coefficients

The mathematical model of fluid flow inside the absorbertube requires some additional local information obtainedfrom empirical correlations. Since the fluid flow presents

two well-defined sections: subcooled liquid region andequilibrium liquid–vapor region, a slope change is ex-pected, due to the use of different empirical heat transfercorrelations and their magnitudes for both regions. Thus,after comparing different empirical correlations presentedin the technical literature, the following ones have beenselected:

4.3.1. Subcooled liquid region

The Gnielinski [25] correlation was used to calculate theheat transfer coefficient assuming constant heat flux in thecase of laminar flow:

af ;l ¼ maxðaf ;l; 4:364Þ, (27)

where

af ;l ¼ðf =8ÞðRe� 1000ÞPr

1þ 12:7ffiffiffiffiffiffiffiffiffiffiffiðf =8Þ

pPr2=3 � 1� � k

Din, (28)

f ¼ ð1:82log10 Re� 1:64Þ�2. (29)

The friction factor was evaluated from the expressionproposed by Churchill [26]. In the subcooled boiling region(if it exists) the heat transfer coefficient was estimatedaccording to Kandlikar [27].

4.3.2. Equilibrium two-phase region

In the two-phase flow region the void fraction wasestimated from the equation of Rouhani and Axelsson [28].For the convective heat transfer coefficient the flow boilingmodel proposed by Zurcher et al. [29] was applied.The friction factor was calculated from the same

equation as in the case of subcooled liquid flow usinga correction factor (two-phase frictional multiplier F)according to Friedel [30].

4.4. Evaluation of ammonia–water thermodynamic and

thermophysical properties

Temperature, mass fraction, and all the thermophysicalproperties were calculated using matrix functions of thepressure and enthalpy obtained using the REFPROPversion7.0 [31], i.e.

f ¼ fðP; hÞ where f ¼ T ;xg;r; . . . (30)

Transport properties (viscosity, thermal conductivity, andsurface tension) were calculated with the correlationsproposed by Conde [32].

5. Numerical resolution

Numerical analysis was carried out by means of a CVmethod. The discretized equations were coupled using afully implicit step-by-step method in the flow direction.From the known values at the inlet section and guessedvalues of the wall boundary conditions, the variablevalues at the outlet of each CV were iteratively obtainedfrom the discretized governing equations. This solution

ARTICLE IN PRESSN. Ortega et al. / Renewable Energy 33 (2008) 2064–2076 2071

(outlet values) was the inlet values for the next CV. Theprocedure was carried out until the end of the absorbertube was reached.

The governing equations discretized for each CV arepresented for the fluid flow, the absorber tube, and thesolar analysis.

x

j j+1j-1

wW n e E

s

P

Fig. 4. Discretized absorber tube wall.

5.1. Fluid flow analysis

For each CV, a set of algebraic equations is obtained bya discretization of the governing Eqs. (5)–(7). The transientterms of the governing equations are discretized using thefollowing approximation: qf/qtffi(f�f1)/Dt, where frepresents a generic dependent variable (f=h, P, T, etc.);superscript ‘‘o’’ indicates the value of the previous instant.The averages of the different variables have been estimatedby the arithmetic mean between their values at the inlet andoutlet sections, that is: ~fj ffi fj � ðfj þ fjþ1=2Þ.

Based on the numerical approaches indicated above, thegoverning Equations. (5)–(7) can be discretized to obtainthe value of the dependent variables (mass flow rate,pressure, and enthalpy) at the outlet section of each CV.The final form of the governing equations is given below.

The mass flow rate is obtained from the discretizedcontinuity equation

_mj ¼ _mj�1 �At Dx

Dtðrtp � ro

tpÞ, (31)

where the two-phase density is obtained from the relationrtp=egrg+(1�eg)rl.

In terms of the mass flow rate, gas and liquid velocitiesare calculated as

~V g ¼_mxg

rg�gAt

" #; ~V l ¼

_mð1� xgÞ

rlð1� �gÞAt

� �, (32)

The discretized momentum equation is solved for the outletpressure:

Pj ¼ Pj�1 �Dx

AtpDinF

f

4

_m2

2rtpA2t

(

þ_m

Dxðxg

~V g þ ð1� xgÞ~V lÞ

� �j

j�1

þrtpAtg sin bþ_m� _m

o

Dt

�. ð33Þ

From the energy Equation (3) and the continuity Equation(1), the following equation is obtained for the outletenthalpy:

hj ¼ð2pDin DxÞqwall � a _mj þ b _mj�1 þ cAt Dx=Dt

_mj þ _mj�1 þ rotpAt Dx=Dt

, (34)

where

qwall ¼ af ðTa;j � T f ;jÞ,

a ¼ ½xg~V g þ ð1� xgÞ~V l�

2j þ g sin bDx� hj�1,

b ¼ ½xg~V g þ ð1� xgÞ~V l�

2j�1 � g sin bDxþ hj�1,

c ¼ 2ðPj�1 � Po

j�1Þ � rotpðhj�1 � 2h

o

j�1Þ

� ðr~V2

j�1 � ro ~Vo2

j�1Þ. ð35Þ

The above-mentioned conservation equations of mass,momentum, and energy are applicable to transient two-phase flow. Situations of steady flow and/or single-phaseflow (liquid or gas) are particular cases of this formulation.Moreover, the mathematical formulation in terms ofenthalpy gives generality of the analysis (only one equationis needed for all the regions) and allows dealing in easyform with cases of ammonia–water mixtures. In this studythe model was solved considering steady state.

5.2. Absorber tube wall

The conduction equation has been written assumingone-dimensional transient temperature distribution.A characteristic CV is shown in Fig. 4, where P representsthe central node, E and W indicate its neighbors. The CVfaces are indicated by e, w, n, and s. Integrating theconduction equation over this CV, the following equationwas obtained:

ð ~qwallps � ~qupnÞDxþ ð~_qw �~_qeÞAt�abs ¼ m

q ~hqt

, (36)

where ~qwall was evaluated using the convective heat transfercoefficient and temperature in the fluid flow ( _qwall ¼

aðTwall � T fluidÞ), and the conductive heat fluxes wereevaluated using the Fourier law:

~_qe ¼ �keqTa

qx

e

; ~_qw ¼ �kwqTa

qx

w

. (37)

The following equation was obtained for each node of thegrid:

aTa;j ¼ bTa;jþ1 þ cTa;j�1 þ d, (38)

where the coefficients were

b ¼keAt�abs

Dx; c ¼

kwAt�abs

Dx,

a ¼ bþ cþ af ;jps DxþAt�absDx

DtrCp;

d ¼ ðaf ;jpsT f ;j þ qu;jpnÞDxþAt�absDx

DtrCpTo

w;j. ð39Þ

The coefficients mentioned above are applicable for2pjpN�1; for j ¼ 1 and j ¼ N adequate coefficients were

ARTICLE IN PRESSN. Ortega et al. / Renewable Energy 33 (2008) 2064–20762072

used taking into account the axial heat conduction ortemperature boundary conditions. The set of heat conduc-tion discretized equations was solved using the TDMAalgorithm [33].

5.3. Numerical solver

The solution process was carried out on the basis of aglobal algorithm that solves in a segregated manner thefluid flow inside the absorber tube, the heat conduction inthe absorber tube wall, and the heat transfer in the solarconcentrator. The coupling between the three mainsubroutines was performed iteratively following the proce-dure described below:

(1)

For fluid flow inside the absorber tube, the equationswere solved considering the absorber tube walltemperature distribution as a boundary condition,and evaluating the convective heat transfer in eachfluid CV.

(2)

In the absorber tube wall, the temperature distributionwas re-calculated using the fluid flow temperature andthe convective heat transfer coefficient evaluated in thepreceding step, and considering the useful energy gainas boundary condition.

(3)

The useful energy gain was obtained by means of thethermal analysis carried out on the CPC components,and the absorber tube wall temperature distributioncalculated in the previous steps.

Fig. 5. Vapor quality and fluid temperature distribution along the CPC

for seven different diameters of the carbon steel absorber tube. One label

for each y-axis that is ordered from the top curve to the bottom curve is

shown.

Global convergence was reached when between twoconsecutive loops of the three main subroutines a strictconvergence criterion was verified for all the CVs in thedomain.

6. Results and discussion

The CPC model developed was applied to analyze theeffects of design parameters; these included: the acceptancehalf-angle and length of the CPC, the diameter andselective surface of the absorber tube, and the materialproperties of the cover, the reflector, and the absorber tube.

The calculations were carried out for 57 without-tracking CPC configurations installed in Temixco,Morelos, Mexico (18150.360N, 99114.070W, altitude1219mosl), on March 15th at solar noon, with a solarirradiance of 991W/m2, and a solar absorbed energy peraperture unit area of 649.3W/m2.

The inlet temperature, pressure, and mass flow rate ofthe ammonia–water solution were considered to be 81.7 1C,11 bar, and 0.0483 kg/s, respectively. The aperture area wasmaintained constant at 23.2m2 by varying the truncationpercentage and length of the CPC configurations, in orderto have the same energy input in all study cases.

The acceptance half-angles selected for the analysis were151, 211, 271, 301, and 401; the absorber tube diameter wasbetween 21.3 and 101.6mm. Three different coatings on the

absorber tube were considered: a commercially availableselective surface, cermet, and a commercial black paint.Three different cover materials were analyzed: temperateglass, a polycarbonate, and glass with antireflective surface.The reflectors studied were: mirror quality stainless steel,highly polished aluminum, and highly polished aluminumwith a protective layer. The three absorber tubes evaluatedwere: carbon steel, stainless steel, and aluminum, sinceammonia–water mixture is corrosive to copper.

6.1. Effect of tube diameter

Fig. 5 shows the fluid temperature and vapor qualitydistribution along the CPC for seven different carbon steelabsorber tube diameters, with a commercial selectivesurface and acceptance half-angle of 151. The differencebetween the minimum and the maximum absorber tubediameter (21.3 and 101.6mm) in the outlet fluid tempera-ture was around 3.4 1C, from 90.1 to 93.5 1C, respectively.For the vapor quality, the difference was 0.0117, from0.0730 to 0.0847. The best result for both fluid temperatureand vapor quality were obtained for an absorber tubediameter of 73.0mm (outlet temperature of 93.9 1C andvapor quality of 0.0897), which practically had the samebehavior as at 60.3mm; both were followed by 101.6mm.The 21.3mm tube presented a higher vapor quality than26.7, 33.4, and 48.3mm, resulting from a higher pressuredrop that helped the evaporation process. Numericalresults obtained with stainless steel and aluminum asabsorber tube had the same tendencies. The fluid tempera-ture reached is directly proportional to the tube diameter;this is not so for the exit vapor quality. Therefore, acompromise exists between the heat transfer area (thatdepends directly on tube diameter) and the pressure drop,which affects the fluid temperature and the vapor qualitydistribution, in favor of one or the other.

ARTICLE IN PRESS

Table 1

Geometrical characteristics of the CPC configurations with several

aperture half-angles

yC (deg) L (m) W (m) % Truncated area C Creal

15 35.0 0.66 46.13 3.86 3.50

21 100.5 0.23 84.85 2.79 1.22

27 100.5 0.23 76.44 2.29 1.22

30 100.5 0.23 71.76 2.00 1.22

40 100.5 0.23 53.90 1.56 1.22

Fig. 7. Vapor quality and fluid temperature distribution along the CPC

for three coatings of the carbon steel absorber tube, and three different

reflector materials. The close captions are distributed as reflector/coating,

where HPA-PL means highly polished aluminium with a protective layer,

HPA means highly polished aluminium, MQSS means mirror quality

stainless steel, CSS means commercial selective surface, and CBP means

commercial black paint.

N. Ortega et al. / Renewable Energy 33 (2008) 2064–2076 2073

The slope change of the fluid temperature at approxi-mately 2m2 of aperture area is because at this point thefluid changes from subcooled liquid to two-phase flow.Due to this, the evaluation of the heat transfer coefficientbetween both regions has abrupt changes that produce thistendency. Moreover, the use of different empirical heattransfer correlations for both regions produces a disconti-nuity in the CV where the transition occurs. This tendencyappears in all the following figures when the evaporationprocess takes place.

6.2. Effect of acceptance half-angle

Fig. 6 shows the fluid temperature distribution andvapor quality for five CPC acceptance half-angles. Thecarbon steel absorber tube diameter was 60.3mm for all thecases. It was observed that 151, the lowest acceptance half-angle that corresponds to a real concentration ratio of 3.5,offered the best results in both variables, with an outletfluid temperature of 93.9 1C, and outlet vapor quality of0.0890. As can be seen in Table 1, in order to maintain aconstant heat input it was necessary to modify the otherCPC dimensions.

6.3. Effect of absorber coating and reflector material

Fig. 7 shows the vapor quality and fluid temperaturedistribution along the CPC for three coatings of the carbonsteel absorber tube, and three manufacture reflectormaterials. The absorber tube and cover material propertieswere also analysed, but no important effect on the resultswere found. All the curves were analyzed with temperateglass as cover, and carbon steel as absorber tube. Thecombination of highly polished aluminum with protectivelayer as reflector, and cermet as selective surface offered thebest results, with an outlet fluid temperature of 95.4 1C, andoutlet vapor quality of 0.1083. On the other hand, the

Fig. 6. Vapor quality and fluid temperature distribution along the CPC

for five different acceptance half-angles.

worst results of the cases shown in Fig. 7 were obtained forthe case with mirror quality stainless steel, and commercialselective surface, whose outlet fluid temperature and vaporquality were 91.8 1C, and 0.0617, respectively. This curvereveals that the most important influence in the quantity ofvapor obtained by a CPC is the reflector material, followedby the coating of the absorber tube.From the analysis of the last three figures, a design CPC

module with the geometrical and optical characteristicspresented in Tables 2 and 3 was chosen to be coupled witha single-stage absorption system model. A 60.3mmabsorber tube diameter was selected since, together withthe 73.0mm one, if provides the best results in vaporquality and temperature rise, but with the advantage of amore compact CPC device and better wetting inside thediameter tube, since the fluid flow is relatively low. Thesimulation was carried out considering a carbon steelabsorber tube, highly polished aluminum reflector, com-mercial selective surface, and temperate glass cover. Theuse of highly polished aluminum with protective layerreflector was not contemplated as this material mustbe imported and the total manufacture cost increases.

ARTICLE IN PRESS

Table 2

Geometric characteristics of the CPC collector

yC (deg) Creal Dout (mm) Din (mm) H (m) W (m) L (m)

151 3.5 60.3 52.5 0.76 0.66 35

Table 3

Radiative properties of the CPC components

Components Absorptance Emittance Reflectance

Carbon steel absorber/commercial

selective surface

0.91 0.38 0.09

Temperate glass cover 0.03 0.94 0.05

Highly polished aluminum

reflector

0.11 0.05 0.87

Fig. 8. Vapor quality and temperature distribution along the designed

CPC (March 15th at solar noon).

Fig. 9. Vapor quality and fluid temperature distribution along the CPC

for a typical day of operation.

N. Ortega et al. / Renewable Energy 33 (2008) 2064–20762074

The commercial selective surface was selected over cermetbecause of its lower cost and easier application.

6.4. Temperature distribution and vapor quality inside the

CPC module

Fig. 8 shows the distribution of the temperatures of thereflector, the cover, the absorber tube wall, and theammonia–water mixture, as well as the vapor qualityalong the design CPC module for May 15th at solar noon.The ammonia–water mixture enters the CPC with asubcooling degree of 5.4 1C. The outlet vapor qualityobtained was 0.0891, which represents an ammonia vaporproduction of 0.0043 kg/s. The abrupt change in theabsorber tube temperature when the ammonia–watermixture begins to evaporate is because, as explainedbefore, the convective heat transfer coefficient from thesubcooling liquid region and the two-phase flow presents adiscontinuity, due to the use of different empirical heattransfer correlations for both regions.

Fig. 9 illustrates the fluid temperature and vapor qualitydistribution for a typical day during seven operation hours.As expected, the outlet vapor quality increases with anincrease in solar radiation, reaching a maximum of 0.0891(ammonia vapor production of 0.0043 kg/s for the solarnoon). Also a minimum quality of 0.0316 (ammonia vaporproduction of 0.0015 kg/s) is observed for 10:00 h. At 9:00and 15:00 h there was no vapor production, but a fluidtemperature rise of 1.2, and 2.9 1C was obtained, respec-tively. Due to the lower vapor production estimated at9:00, 10:00, 14:00, and 15:00 h, an auxiliary heater is usedin series after the CPC in order to reach the required outlettemperature and the vapor production obtained around11:00 h, and therefore obtaining a reasonable coolingcapacity for the complete system. The objective is toimprove the cooling capacity and efficiency of theammonia–water absorption solar refrigerator, accordingto the simulation results by coupling the CPC model and asingle-stage absorption system model simulator.

Some refrigeration variables were analysed, such ascooling capacity, cooling COP, refrigeration efficiency, andflow ratio.Table 4 shows the values obtained by coupling the CPC

module and the absorption refrigeration system for theseven cases analyzed during a typical operation day. It canbe seen that for the CPC module, the outlet temperaturevaries, from 92.6 1C for the baseline case at 11:00 h to lowervalues at 9:00, 10:00, 14:00, and 15:00 h, of 82.9, 89.5, 89.7,and 84.5 1C, respectively. At 12:00 and 13:00 h the outlettemperature increases to 94.1 and 92.7 1C, respectively. TheCPC module efficiency is 43.4% for the baseline case at11:00 h, and lower at 9:00, 10:00, 14:00, and 15:00 h. Thesolar fraction, fixed as 100% at 11:00 h, is only 3.2% at9:00 h, 51.7% at 10:00 h, 55.5% at 14:00 h, and 7.6% at15:00 h. The cooling capacity is 3.81 kW and is higherat 12:00 and 13:00 h, being 4.79 and 3.90 kW, respectively.

ARTICLE IN PRESS

Table 4

Results comparison of a refrigeration system operated during a typical day coupled with the designed CPC

Time (h)

9:00 10:00 11:00 12:00 13:00 14:00 15:00

Ammonia concentration (kgNH3/kg sol)

Refrigerant (4) 0.995 0.995 0.995 0.995 0.995 0.995 0.995

Strong solution (10) 0.387 0.387 0.387 0.387 0.387 0.387 0.387

Weak solution (13) 0.363 0.363 0.363 0.357 0.362 0.363 0.363

Pressure (bar)

Condensation 11.0 11.0 11.0 11.0 11.0 11.0 11.0

Evaporation 2.8 2.8 2.8 2.8 2.8 2.8 2.8

Mass flow rate (kg/s)

Refrigerant (4) 0.0033 0.0033 0.0033 0.0041 0.0033 0.0033 0.0033

Strong solution (10) 0.0483 0.0483 0.0483 0.0483 0.0483 0.0483 0.0483

Weak solution (13) 0.0448 0.0448 0.0448 0.0438 0.0447 0.0448 0.0448

Main temperatures (1C)

Inlet evaporator (7) �10.80 �10.80 �10.80 �10.80 �10.80 �10.80 �10.80

Inlet condenser 55.00 55.00 55.00 55.00 55.00 55.00 55.00

Inlet generator (12) 81.65 81.65 81.83 83.30 81.94 81.65 81.65

Outlet CPC (13) 82.87 89.46 92.60 94.12 92.74 89.70 84.54

Outlet generator (1) 92.60 92.60 92.60 94.12 92.74 92.60 92.60

Outlet absorber (10) 39.19 39.19 39.19 39.19 39.19 39.19 39.19

CPC

Solar energy (kW) 7.154 13.968 20.120 22.601 20.120 13.968 7.154

Useful energy gain (kW) 0.278 4.514 8.731 10.455 8.896 4.844 0.660

Solar efficiency (%) 3.889 32.317 43.398 46.257 44.213 34.683 9.224

Energetic behaviour

Flow ratio (FR) 14.85 14.85 14.85 11.82 14.51 14.85 14.85

Auxiliary energy (kW) 8.45 4.22 0.00 0.00 0.00 3.89 8.07

Solar fraction (%) 3.18 51.70 100.00 100.00 100.00 55.48 7.56

Cooling capacity (kW) 3.81 3.81 3.81 4.79 3.90 3.81 3.81

Cooling COP 0.437 0.437 0.437 0.458 0.439 0.437 0.437

Overall efficiency (%) – – 18.94 21.19 19.40 – –

N. Ortega et al. / Renewable Energy 33 (2008) 2064–2076 2075

The COP of the refrigeration cycle is between 0.437 and0.458 at �10 1C. The overall efficiency (Eq. (4)) is 18.9%for the base case at 11:00 h, with a maximum of 21.2%at 12:00 h.

7. Conclusions

A detailed one-dimensional numerical simulation of thethermal and fluid-dynamic behavior of two-phase flowinside a CPC used as an ammonia–water vapor generatorhas been developed. The numerical analysis was made witha CV method on the absorber tube, and the discretizedequations were coupled using a fully implicit step-by-stepmethod in the flow direction.

The numerical algorithm solves, in a segregated manner,three subroutines: fluid flow inside the absorber tube, heatconduction in the absorber tube wall, and the useful energygain in the solar concentrator. Coupling between the threemain subroutines was performed iteratively until conver-gence was reached.

This numerical model can be used to simulate thegeneration process of any refrigerant–absorbent mixture at

the CPC whenever the mixture thermodynamic propertiesare known.The effect on the results of a range of design parameters

was analyzed. These parameters were the acceptance half-angle, the diameter and coating of the absorber tube, andthe manufacture material of the cover, the reflector, andthe absorber tube.It was found that the most important design para-

meter is the reflector material selection, followed inorder of priority by the absorber tube diameter andcoating, and the acceptance half-angle. The materialof the absorber tube and cover are not significant in theproduction of ammonia vapor, although corrosioncould represent a problem inside the absorber tube,therefore, material selection must be done carefully.Once the previous design parameters are established,CPC length must be selected for a specific refrigerationapplication in order to obtain certain ammonia vaporproduction.The system analysis (CPC model coupled to an absorp-

tion refrigeration system) was carried out for a typicaloperation day for 7 h (boundary conditions) in order to

ARTICLE IN PRESSN. Ortega et al. / Renewable Energy 33 (2008) 2064–20762076

predict the solar fraction, cooling capacity, COP, andoverall efficiency.

It is theoretically possible to directly produce ammonia–water vapor in a 35m long CPC module coupled to asingle-stage ammonia–water refrigeration system. Thesystem analyzed can produce more than 3.8 kW of coolingat �10 1C. For larger cooling needs a number of CPCmodules can be connected in parallel to produce theammonia–water vapor required.

A CPC ammonia–vapor outlet temperature of around93 1C can be theoretically achieved with a CPC efficiency ofover 43%. The calculated overall efficiency of the system atsolar noon can reach 21.2%.

Acknowledgments

This work had been financed by DGAPA-UNAMthrough the PAPIIT project IN105602-3, and by CON-ACyT project U44764-Y. The authors thank CONACyT,Mexico, for the support provided for the student scholar-ship 118090.

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