Cpc Absorption

ARTICLE IN PRESS

Renewable Energy 33 (2008) 2064–2076 www.elsevier.com/locate/renene

Two-phase flow modelling of a solar concentrator applied as ammonia vapor generator in an absorption refrigerator N. Ortegaa,, O. Garcı´ a-Valladaresb, R. Bestb, V.H. Go´mezb a

Posgrado en Ingenierı´a (Energı´a), Universidad Nacional Auto´noma de Me´xico, Privada Xochicalco s/n, Temixco, Morelos 62580, Me´xico Centro de Investigacio´n en Energı´a, Universidad Nacional Auto´noma de Me´xico, Privada Xochicalco s/n, Temixco, Morelos 62580, Me´xico

b

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 35 m 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 power generation capacity problems, that tend to increase as the need for energy intensive conventional air conditioning and refrigeration also increases. Alternative cooling methods are required to decrease the power demand, and the conventional high global warming potential (GWP) and ozone depletion potential (ODP) refrigerant usage. In addition, developed countries need new refrigeration technologies as an alternative to conventional compression refrigeration to meet their air-conditioning

Corresponding author. Tel.: +52 55 56 22 97 36; fax: +52 55 56 22 97 91. E-mail address: [email protected] (N. Ortega).

0960-1481/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2007.11.016

and cooling demands without increasing greenhouse gases emissions [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 solar cooling devices for at least three decades. These studies include the combination of solar energy technologies with thermal refrigeration technologies (as absorption, adsorption, and desiccant) to produce cooling and refrigeration using 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 evaluation and improvement of solar devices, to full systems in test stage for power generation [7,8]. In addition, some solar concentrators applied as generators for intermittent absorption refrigerators have been developed [9,10].

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Nomenclature Aa Ac Ar At Atabs C COP Cp D f FR g Gbn h H I k L m _ m p P q qu q0u qwall R S t T UL V ~x V w x xg X

absorber tube heat transfer area, m2 cover heat transfer area, m2 reflector heat transfer area, m2 fluid flow cross section area, m2 absorber tube cross-section area (p(D2outD2in)/4), m2 area concentration ratio, dimensionless coefficient of performance, dimensionless specific heat, J/(kg K) diameter, m friction factor, dimensionless flow ratio, dimensionless gravitational constant, ( ¼ 9.81 m/s2) beam irradiance normal to the plane, W/m2 enthalpy, J/kg height, m solar irradiance, W/m2 thermal conductivity, W/(m K) length, m mass, kg mass flow rate, kg/s perimeter, m pressure, bar heat flow per unit area, W/m2 useful energy gain per absorber unit area, W/m2 useful energy gain per length unit, W/m heat flux per absorber unit area from fluid to wall, W/m2 thermal resistance, (m2 K)/W solar absorbed energy per unit area, W/m2 time, s temperature, K overall heat loss coefficient, W/(m2 K) volume, m3 velocity in the axial direction, m/s cover width, m axial coordinate vapor quality, dimensionless ammonia weight concentration, dimensionless

Z j m yC r s u z Dx Dt F

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efficiency, dimensionless angle of involute generation, degree viscosity, kg/(m s) acceptance half-angle, degree density, kg/m3 stefan–Boltzman constant, ( ¼ 5.6697  108 W/ (m2 K4)) wind velocity, m/s effectiveness, dimensionless spatial discretization step, m temporal discretization step, s two-phase frictional multiplier, dimensionless

Dimensionless numbers Pr Re

prandtl number, ( ¼ mCp/l) ~D=mÞ reynolds number ð¼ rV

subscripts a c co en EV ex f g GE i in inv j l o out par r ra s sk tp

absorber tube wall cover conductive environment evaporation external fluid gas generation inlet inner involute number of control volume liquid phase outlet outer parabola reflector radiative saturation sky two-phase

Greek letters Superscripts a b e eg f

heat transfer coefficient, W/(m2 K)) inclination angle of absorber tube, degree emittance, dimensionless void fraction, dimensionless generic dependent variable

Flat plate collectors have been applied to direct refrigerant evaporation in solar-assisted heat pumps, where a two-phase refrigerant flows through the collectors instead of utilizing a heat exchanger between the collector and the evaporator [11].

–  o

arithmetical average over a CV integral average over a CV value of previous instant

Since 1990, parabolic trough solar concentrators have been used to evaporate water to produce steam directly on the absorber tube [12]. The technology developed is known as direct steam generation (DSG), where the vapor produced is mainly applied for power generation. DSG

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presents many advantages compared to the heating oilbased technology, since DSG eliminates costly synthetic oil, intermediate heat transport piping, special type equipment to run the high-temperature oil, and the oil for steam heat exchanger [13]. Compound parabolic concentrators (CPCs) are a good choice for applications in direct evaporation, since these stationary collectors have a good quality rate between cost and performance at medium temperature levels [14]. Based on the main advantages of DSG, a CPC was designed in order to directly generate ammonia from an ammonia–water solution. Ammonia vapor would be utilized in an ammonia–water absorption solar refrigerator. Other applications not analyzed here could be the use of the CPC as a heat source for a direct ammonia–water solution evaporator for applications in combined power and cooling thermodynamic cycles, as proposed by Goswami and Xu [15]. In an attempt to reduce heat losses and demonstrate its feasibility, a CPC was modelled and designed in order to generate ammonia vapor inside its absorber tube. The theoretical analysis of the evaporation process inside the CPC was emphasized, through a detailed one-dimensional numerical simulation of the thermal and fluid-dynamic behavior of two-phase flow. The CPC model was coupled to a complete single-stage absorption refrigeration cycle model in order to calculate the theoretical cooling capacity and coefficient of performance (COP) under different working conditions. To our knowledge, solar concentrators have not been applied as direct ammonia vapor generators in a continuous thermal refrigeration system.

absorbed. The study included reflector conduction, highwave radiative interchange, and heat removal in the tubular absorber. Tchinda et al. [19] analyzed the heat exchange in a CPC collector, where axial heat transfer in the tubular absorber was included. They developed an explicit expression in order to calculate the fluid temperature as a function of the coordinate space in the flux direction and the timedependent solar intensity. In this paper, a simple method was carried out in order to establish the energy balances in a CPC, where the absorber tube operates as an ammonia–water mixture direct vapor generator in a solar absorption refrigerator. The evaporation process was studied in order to fulfil the thermal and fluid-dynamic characterization inside the CPC absorber tube. The system under investigation consisted of a troughtype CPC with a steel tubular absorber without an evacuated glass shell. Thermodynamic equilibrium between the liquid and vapor phases was supposed. A onedimensional numerical simulation of the thermal and fluid-dynamic behavior of two-phase flow was developed. The governing equations (continuity, momentum, and energy) inside the tube, together with the energy equation in the tube wall and the thermal analysis in the solar concentrator, were solved iteratively in a segregated manner. The discretized governing equations in fluid flow were coupled using an implicit step-by-step method in the flow direction. By means of the model results, a CPC module was designed and theoretically evaluated as ammonia generator in an ammonia–water absorption solar refrigerator.

1.1. A brief description of the CPC models

1.2. CPC module

Initially, the models developed to describe CPC optical and thermal performance were restricted to the flat absorber type [16]. In these models, convective heat transfer was usually represented by flat plate film coefficients. The simplest models for CPC with tubular absorber have not considered absorption of high wavelength over reflective surfaces [16]. Hsieh developed the mathematical formulation for the thermal processes in a tubular CPC, where heat exchange between components was predicted [17]. Chew et al. [18] developed a finite-element model for a CPC with tubular absorber; they considered that the absorber tube and the cover were isothermic, while the reflectors were considered as adiabatic boundaries. Eames and Norton [16] developed and validated a two-dimensional model in steady state in order to simulate the optical and thermal behavior of a through-type CPC. Ray tracing and finite-element analysis of convection heat transfer were applied. Solar beam and diffuse radiation were considered in the optical analysis, irradiance and absorption were assumed homogeneous, and that the energy reaching the absorber tube was completely

In a previous work [20], a mathematical model was developed in order to evaluate the temperature distribution of a CPC array proposed to be used as a vapor generator in an absorption ammonia–water refrigeration system. It was established that the mixture temperature increases and wall absorber tube temperature decreases when the ammonia–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 evaporation process at the solar concentrator absorber tube was numerically simulated. This analysis was made with a control volume (CV) method on the absorber tube, and the discretized equations were coupled using a fully implicit step-by-step method in the flow direction. The conduction in the internal tube wall was solved using the TDMA algorithm. A separated flow model was applied and two different two-phase flow convective heat transfer coefficients were used. A CPC prototype of 2 m length, 0.66 m width and 0.84 m height was designed, with a solar concentration of 3.5  and an aperture angle of 151.

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In this paper a final CPC model was developed and used for the design analysis. An auxiliary heater was added to the complete refrigeration system in order to maintain a constant refrigeration load. The study consists of the thermal analysis of the CPC performance during a typical operating day. A new subroutine was developed to simulate the final CPC model coupled with the absorption ammonia–water refrigerator. 2. Ammonia–water absorption refrigerator Fig. 1 shows the single-stage ammonia–water absorption solar refrigerator. The proposed solar refrigerator includes the following components: a generator (CPC), a rectifier, a condenser, an evaporator, an absorber, a flash tank, an economizer, a pre-cooler, a pump, and two expansion valves. A model for steady-state single-stage ammonia– water absorption refrigeration system was developed to simulate the results obtained by the CPC model in order to evaluate the performance of the complete cycle. 2.1. Operative description Following the schematic diagram in Fig. 1, ammonia vapor (99.5 wt%) leaves the rectifier as overheated vapor at state 4, at the high pressure of the system. The refrigerant vapor is cooled and liquefied in the condenser as saturated liquid, 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

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two-phase mixture at state 7. Liquid ammonia enters the evaporator, where on extracting heat from the cooling water, it is converted into vapor, producing the refrigerating effect, and then exits as saturated vapor in state 8. It is then superheated in the pre-cooler (state 9). The relatively cold ammonia vapor then enters the absorber, where it is condensed and absorbed by the weak ammonia–water solution. The absorption of ammonia is exothermic, so a heat exchange equipment in the absorber is needed in order to cool the hot solution and improve its absorption capacity. The strong ammonia solution leaves the absorber at state 10 and enters the pump, leaving at high pressure at state 11. It is then introduced in an economizer, where it receives heat and leaves state 12. It then enters the CPC generator, where it receives solar generated heat, reaches the saturation point, and vaporizes, leaving state 13 as a vapor–liquid mixture. If additional heat is necessary it is added to the mixture by the auxiliary heater in order to reach the operating conditions at state 1. The two-phase high-pressure mixture, enters the flash tank where liquid and vapor are separated, the liquid phase is mixed with the condensed vapor originating from in the rectifier (state 3). This weak ammonia solution enters the economizer at state 14, where heat is extracted, and leaves state 15. It then passes through an expansion valve, where the pressure is reduced (state 16) in order to enter the absorber. The vapor coming from the flash tank separator enters the rectifier, in which through heat removal and partial condensation, water, leaving state 4 is removed. In this way the operation of the cycle is completed.

Fig. 1. Ammonia–water absorption solar refrigerator with a CPC as vapor generator.

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2.2. Methodology for the complete cycle energy analysis

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

The purpose of the calculation sequence presented here is to obtain the operation conditions of the ammonia– water absorption solar refrigerator system shown in Fig. 1. An overall energy balance has been applied to all the components of the system (with the exception of the CPC model, where a detailed numerical simulation has been developed). The following assumptions have been made:

      

The high and low pressures of the system are 11 and 2.8 bar, respectively. Pressure drop through elements is neglected (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 (99.5 wt%) 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 system is coupled with the CPC model previously developed in order to evaluate the performance of the complete ammonia–water absorption refrigeration systems through the calculation of the COP, the flow ratio, the solar fraction, and the solar and overall efficiencies. The COP for cooling is defined as the ratio between the cooling capacity (evaporation heat extracted inside the evaporator, QEV) and the generation heat (QGE): Q COP ¼ EV . QGE

A numerical analysis was carried out for the designed CPC illustrated in Fig. 2, which has the geometrical and optical characteristics established from the parametric analysis shown below. A 35 m long row is considered as a CPC module in this calculation, which could be scaled up. The calculations were made for a non-tracking CPC, installed in Temixco, Morelos, Me´xico (18150.360 N, 99114.070 W). The CPC analyzed had an inlet temperature according to the inlet generation temperature obtained with the absorption cycle simulation, and a generator pressure 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. Subcooled ammonia–water mixture enters the tube at position 0 with _ and an inner temperature Tf,i. The absorber a mass flow m, tube receives a useful energy gain qu. Ammonia–water mixture starts to evaporate at a certain length Ls, where saturation temperature Tf,s is reached. Finally, the twophase mixture is out at position L with an outside quality xgf,o and temperature Tf,o. Ac

(1)

O

The flow ratio, FR, is the ratio between the solution flow in the circuit constituted by the generator and the absorber _ 1 ), and the refrigerant flow in the main circuit that joint (m _ 4 ). This ratio indicates the condenser and the evaporator (m the strong ammonia solution mass flow needed to produce a unit of refrigerant vapor, in this case, ammonia vapor: _1 m FR ¼ . _4 m

(2)

Solar fraction was defined as the percentage of the total energy required to generate the ammonia that was achieved by the CPC. Solar efficiency is the ratio between the useful energy gain obtained by the absorber tube area (Arqu) and the solar irradiance that reaches the aperture area (AaI): Zsolar ¼

QGE Ar qu ¼ . Aa I Aa I

(3)

Overall efficiency is the product of COP and solar efficiency: Zoverall ¼ Zsolar COP:

(4)

L

W

θC H D M

N

Ar

Aa

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.

quΔx

r

m

x Tf,i

xgf,o,Tf,o 0

x

x+Δx

Ls

Tf,s

L

Fig. 3. Absorber tube cross-section showing one-phase and two-phase zones of ammonia–water mixture.

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The study was divided in three subroutines: fluid flow inside the absorber tube, heat transfer in the wall tube, and solar thermal analysis.

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4.2. Solar thermal analysis

4.1. Fluid flow inside the absorber tube

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

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

Ac S Aa U L  ðT a  T en Þ. (8) L L The useful energy gain can be obtained from the last expression as

  

One-dimensional flow: P(x, t), h(x, t), T(x, t), etc. Non-participant radiation medium and negligible radiant heat exchange between surfaces. Axial heat conduction inside the fluid was neglected.

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

q0u ¼

Qu ¼ Ac S  Aa U L ðT a  T en Þ.

Then, the useful energy gain per unit of absorber area qu is obtained as qu ¼

Ac S  U L ðT a  T en Þ ¼ CS  U L ðT a  T en Þ. Aa

Continuity: _ jj1 þ ½m



qm ¼ 0. qt

(5)

Momentum: ~_ qm qt ¼ ½Pjj1 At  t~ p Dx  mg sin b.

_ g ng jj1 þ ½m _ l nl jj1 þ Dx ½m



ð6Þ

Energy: ~_ l j þ ½m _ g ðeg  el Þjj1 þ ð~eg  e~l Þ m½e j1

qmg qt

q~eg q~el qP~ þ ml  At Dx qt qt qt qm ~ ¼ q_ u p Dx, þ ð~el  e¯ l Þ ð7Þ qt ~ represents the integral volume average of a where f ¯ its arithmetic generic variable f over the CV and f average between the inlet and outlet of the CV. The subscript and superscript in the brackets indicate ½X jj1 ¼ X j  X j1 , i.e., the difference between the quantity X at the outlet section and the inlet section. þ mg

In the governing equations, the evaluation of the shear stress is performed by means of a friction factor f. This factor is defined from the expression: t ¼ F(f/4)(G2/2r), where F is the two-phase factor multiplier. The onedimensional model also requires the knowledge of the twophase flow structure, which is evaluated by means of the void fraction eg. Finally, heat transfer through the absorber tube wall and fluid temperature are related by the convective heat transfer coefficient a, which is defined as a ¼ q_ wall =ðT wall  T fluid Þ.

(10)

Cover and absorber tube area were defined as Ac ¼ wL;



(9)

Aa ¼ pDout L.

(11)

The useful energy gain depends on the absorbed solar radiation S that is equal to the cover incident solar energy reduced by optical losses in the concentrator [23]; thermal losses in the cover, the reflector, and the absorber tube are represented 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 conditions that depend on solar time (solar radiation, solar position, and ambient temperature). Absorber solar radiation was calculated with the method presented by Duffie and Beckman [23]. The overall heat loss coefficient UL depends on the temperatures of the CPC components through the individual heat loss coefficients:  Rcen Rren UL ¼ Rcen þ Rren þ Rrc  1 )1 1 1 þ , ð12Þ þ Rac þ Rcenr Rar þ Rrenc where Rren Rrc , Rcen þ Rren þ Rrc Rcen Rrc ¼ , Rcen þ Rren þ Rrc

Rrenc ¼ Rcenr

ð13Þ

and finally Rcen ¼ ðaco;cen þ ara;csk Þ1 , Rac ¼ ðaco;ac þ ara;ac Þ1 , Rren ¼ ðaco;ren þ ara;rsk Þ1 , Rrc ¼ ðaco;rc þ ara;rc Þ1 , Rar ¼ ðaco;ar þ ara;ar Þ1 .

ð14Þ

The convective heat transfer coefficient between the reflector and the cover aco,rc was fixed at a constant value

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of 5 W/(m2 K), as it has been previously evaluated by Prapas et al. [24] and Hsieh [17]. The convective heat transfer coefficients between the cover and the ambient, and between the reflector and the ambient are, respectively, [23] aco;cen ¼ ð5:7 þ 3:8uÞ

Ac , Aa

(15)

aco;ren ¼ ð5:7 þ 3:8uÞ

Ar , Aa

(16)

where the reflector area Ar was calculated by "

j2 1 Ar ¼ Dout Dx inv þ pffiffiffi 4 2

Z

fpar finv

ðp=2Þ þ yC þ j  cosðj  yC Þ ½1 þ sinðj  yC Þ3=2

# dj .

two well-defined sections: subcooled liquid region and equilibrium liquid–vapor region, a slope change is expected, due to the use of different empirical heat transfer correlations and their magnitudes for both regions. Thus, after comparing different empirical correlations presented in the technical literature, the following ones have been selected: 4.3.1. Subcooled liquid region The Gnielinski [25] correlation was used to calculate the heat transfer coefficient assuming constant heat flux in the case of laminar flow: af;l ¼ maxðaf;l ; 4:364Þ, where

(17) The convective heat transfer coefficients between the absorber tube and the reflector and between the absorber tube and the cover were expressed, respectively, by [17] aco;ar

ðT a  T r Þ ¼ 3:25 þ 0:0085 , 2Dout

(18)

ðT a  T c Þ , 2Dout

(19)

aco;ac ¼ 3:25 þ 0:0085

ara;rsk ¼ r sðT 2r þ T 2sk ÞðT r þ T sk Þ

Ar , Aa

(20)

ara;csk ¼ c sðT 2c þ T 2sk ÞðT c þ T sk Þ

Ac , Aa

(21)

ara;rc ¼

sðT 2c þ T 2r ÞðT c þ T r Þ Ar , ð1  c Þ=c þ ðð1  r Þ=r ÞðAc =Ar Þ Aa

(22)

ara;ac ¼

sðT 2a þ T 2c ÞðT a þ T c Þ , ð1=c Þ þ ðAc =Aa Þðð1=a Þ  1Þ

(23)

ara;ar ¼

sðT 2a þ T 2r ÞðT a þ T r Þ . ð1  r Þ=r þ ðð1  a Þ=a ÞðAr =Aa Þ

(24)

The temperatures of cover and reflector a necessary in order to solve Eq. (10). Both were determined by means of the energy balances in each CPC component: Tc ¼

ðara;ac þ aco;ac ÞT a þ ara;csk T sk þ aco;cen T en þ ðaco;rc  ara;rc ÞT r , ara;ac þ aco;ar þ ara;csk þ aco;cen þ aco;rc  ara;rc

(25) Tr ¼

(27)

ðara;ar þ aco;ar ÞT a þ aco;ren T en þ ara;rsk T sk þ ðaco;rc  ara;rc ÞT c . ara;ar þ aco;ar þ aco;ren þ ara;rsk þ aco;rc  ara;rc

(26)

4.3. Evaluation of empirical coefficients The mathematical model of fluid flow inside the absorber tube requires some additional local information obtained from empirical correlations. Since the fluid flow presents

af;l ¼

ðf =8ÞðRe  1000ÞPr k pffiffiffiffiffiffiffiffiffiffiffi ,  ðf =8Þ Pr2=3  1 Din

1 þ 12:7

f ¼ ð1:82log10 Re  1:64Þ2 .

(28) (29)

The friction factor was evaluated from the expression proposed by Churchill [26]. In the subcooled boiling region (if it exists) the heat transfer coefficient was estimated according to Kandlikar [27]. 4.3.2. Equilibrium two-phase region In the two-phase flow region the void fraction was estimated from the equation of Rouhani and Axelsson [28]. For the convective heat transfer coefficient the flow boiling model proposed by Zu¨rcher et al. [29] was applied. The friction factor was calculated from the same equation as in the case of subcooled liquid flow using a 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 thermophysical properties were calculated using matrix functions of the pressure and enthalpy obtained using the REFPROP version7.0 [31], i.e. f ¼ fðP; hÞ where

f ¼ T; xg ; r; . . .

(30)

Transport properties (viscosity, thermal conductivity, and surface tension) were calculated with the correlations proposed by Conde [32]. 5. Numerical resolution Numerical analysis was carried out by means of a CV method. The discretized equations were coupled using a fully implicit step-by-step method in the flow direction. From the known values at the inlet section and guessed values of the wall boundary conditions, the variable values at the outlet of each CV were iteratively obtained from the discretized governing equations. This solution

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(outlet values) was the inlet values for the next CV. The procedure was carried out until the end of the absorber tube was reached. The governing equations discretized for each CV are presented for the fluid flow, the absorber tube, and the solar analysis. 5.1. Fluid flow analysis For each CV, a set of algebraic equations is obtained by a discretization of the governing Eqs. (5)–(7). The transient terms of the governing equations are discretized using the following approximation: qf/qtffi(ff1)/Dt, where f represents 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 estimated by the arithmetic mean between their values at the inlet and ~ ffi f¯  ðf þ f =2Þ. outlet sections, that is: f j j j jþ1 Based on the numerical approaches indicated above, the governing Equations. (5)–(7) can be discretized to obtain the 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 discretized continuity equation _j ¼ m _ j1  m

At Dx ðr¯ tp  r¯ otp Þ, Dt

(31)

where the two-phase density is obtained from the relation rtp=egrg+(1eg)rl. In terms of the mass flow rate, gas and liquid velocities are calculated as " #   _ _  xg Þ mx mð1 g ~ ~ Vg ¼ , (32) ; Vl ¼ rl ð1  g ÞAt rg g At The discretized momentum equation is solved for the outlet pressure: ( ¯_ 2 f¯ m Dx pDin F Pj ¼ Pj1  4 2r¯ tp A2t At  j _ m ~ ~ ðxg V g þ ð1  xg ÞV l Þ þ Dx j1  ¯_  m ¯_ o m þrtp At g sin b þ . ð33Þ Dt From the energy Equation (3) and the continuity Equation (1), the following equation is obtained for the outlet enthalpy: hj ¼

_ j þ bm _ j1 þ cAt Dx=Dt ð2pDin DxÞqwall  am , _ j1 þ r¯ otp At Dx=Dt _j þm m

where qwall ¼ af ðT a;j  T¯ f;j Þ, ~g þ ð1  xg ÞV ~l 2 þ g sin bDx  hj1 , a ¼ ½xg V j

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~g þ ð1  xg ÞV ~l 2  g sin bDx þ hj1 , b ¼ ½xg V j1 ¯ j1  P ¯ oj1 Þ  r¯ otp ðhj1  2h¯ oj1 Þ c ¼ 2ðP ~2  r¯ o V ~o2 Þ.  ðr¯ V j1 j1

ð35Þ

The above-mentioned conservation equations of mass, momentum, and energy are applicable to transient twophase flow. Situations of steady flow and/or single-phase flow (liquid or gas) are particular cases of this formulation. Moreover, the mathematical formulation in terms of enthalpy gives generality of the analysis (only one equation is needed for all the regions) and allows dealing in easy form with cases of ammonia–water mixtures. In this study the model was solved considering steady state. 5.2. Absorber tube wall The conduction equation has been written assuming one-dimensional transient temperature distribution. A characteristic CV is shown in Fig. 4, where P represents the central node, E and W indicate its neighbors. The CV faces are indicated by e, w, n, and s. Integrating the conduction equation over this CV, the following equation was obtained: qh~ , (36) qt where q~ wall was evaluated using the convective heat transfer coefficient and temperature in the fluid flow (q¯_ wall ¼ aðT wall  T fluid Þ), and the conductive heat fluxes were evaluated using the Fourier law:



qT a qT a q~_ e ¼ ke ; q~_ w ¼ kw . (37) qx e qx w

ðq~ wall ps  q~ u pn ÞDx þ ðq~_ w  q~_ e ÞAtabs ¼ m

The following equation was obtained for each node of the grid: aT a;j ¼ bT a;jþ1 þ cT a;j1 þ d,

(38)

where the coefficients were kw Atabs , Dx Atabs Dx r Cp; a ¼ b þ c þ af;j ps Dx þ Dt Atabs Dx r CpT ow;j . d ¼ ðaf;j ps T¯ f;j þ qu;j pn ÞDx þ ð39Þ Dt The coefficients mentioned above are applicable for 2pjpN1; for j ¼ 1 and j ¼ N adequate coefficients were b¼

ke Atabs ; Dx

c¼

W

(34)

j-1

w

P

n

e

j

E j+1

s x Fig. 4. Discretized absorber tube wall.

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used taking into account the axial heat conduction or temperature boundary conditions. The set of heat conduction discretized equations was solved using the TDMA algorithm [33]. 5.3. Numerical solver The solution process was carried out on the basis of a global algorithm that solves in a segregated manner the fluid flow inside the absorber tube, the heat conduction in the absorber tube wall, and the heat transfer in the solar concentrator. The coupling between the three main subroutines was performed iteratively following the procedure described below:

absorber tube were considered: a commercially available selective surface, cermet, and a commercial black paint. Three different cover materials were analyzed: temperate glass, a polycarbonate, and glass with antireflective surface. The reflectors studied were: mirror quality stainless steel, highly polished aluminum, and highly polished aluminum with a protective layer. The three absorber tubes evaluated were: carbon steel, stainless steel, and aluminum, since ammonia–water mixture is corrosive to copper. 6.1. Effect of tube diameter

6. Results and discussion

Fig. 5 shows the fluid temperature and vapor quality distribution along the CPC for seven different carbon steel absorber tube diameters, with a commercial selective surface and acceptance half-angle of 151. The difference between the minimum and the maximum absorber tube diameter (21.3 and 101.6 mm) in the outlet fluid temperature was around 3.4 1C, from 90.1 to 93.5 1C, respectively. For the vapor quality, the difference was 0.0117, from 0.0730 to 0.0847. The best result for both fluid temperature and vapor quality were obtained for an absorber tube diameter of 73.0 mm (outlet temperature of 93.9 1C and vapor quality of 0.0897), which practically had the same behavior as at 60.3 mm; both were followed by 101.6 mm. The 21.3 mm tube presented a higher vapor quality than 26.7, 33.4, and 48.3 mm, resulting from a higher pressure drop that helped the evaporation process. Numerical results obtained with stainless steel and aluminum as absorber tube had the same tendencies. The fluid temperature reached is directly proportional to the tube diameter; this is not so for the exit vapor quality. Therefore, a compromise exists between the heat transfer area (that depends directly on tube diameter) and the pressure drop, which affects the fluid temperature and the vapor quality distribution, in favor of one or the other.

The CPC model developed was applied to analyze the effects of design parameters; these included: the acceptance half-angle and length of the CPC, the diameter and selective surface of the absorber tube, and the material properties of the cover, the reflector, and the absorber tube. The calculations were carried out for 57 withouttracking CPC configurations installed in Temixco, Morelos, Me´xico (18150.360 N, 99114.070 W, altitude 1219 mosl), on March 15th at solar noon, with a solar irradiance of 991 W/m2, and a solar absorbed energy per aperture unit area of 649.3 W/m2. The inlet temperature, pressure, and mass flow rate of the ammonia–water solution were considered to be 81.7 1C, 11 bar, and 0.0483 kg/s, respectively. The aperture area was maintained constant at 23.2 m2 by varying the truncation percentage and length of the CPC configurations, in order to have the same energy input in all study cases. The acceptance half-angles selected for the analysis were 151, 211, 271, 301, and 401; the absorber tube diameter was between 21.3 and 101.6 mm. Three different coatings on the

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.

(1) For fluid flow inside the absorber tube, the equations were solved considering the absorber tube wall temperature distribution as a boundary condition, and evaluating the convective heat transfer in each fluid CV. (2) In the absorber tube wall, the temperature distribution was re-calculated using the fluid flow temperature and the convective heat transfer coefficient evaluated in the preceding step, and considering the useful energy gain as boundary condition. (3) The useful energy gain was obtained by means of the thermal analysis carried out on the CPC components, and the absorber tube wall temperature distribution calculated in the previous steps. Global convergence was reached when between two consecutive loops of the three main subroutines a strict convergence criterion was verified for all the CVs in the domain.

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The slope change of the fluid temperature at approximately 2 m2 of aperture area is because at this point the fluid changes from subcooled liquid to two-phase flow. Due to this, the evaluation of the heat transfer coefficient between both regions has abrupt changes that produce this tendency. Moreover, the use of different empirical heat transfer correlations for both regions produces a discontinuity in the CV where the transition occurs. This tendency appears in all the following figures when the evaporation process takes place.

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Table 1 Geometrical characteristics of the CPC configurations with several aperture half-angles yC (deg)

L (m)

W (m)

% Truncated area

C

Creal

15 21 27 30 40

35.0 100.5 100.5 100.5 100.5

0.66 0.23 0.23 0.23 0.23

46.13 84.85 76.44 71.76 53.90

3.86 2.79 2.29 2.00 1.56

3.50 1.22 1.22 1.22 1.22

6.2. Effect of acceptance half-angle Fig. 6 shows the fluid temperature distribution and vapor quality for five CPC acceptance half-angles. The carbon steel absorber tube diameter was 60.3 mm for all the cases. It was observed that 151, the lowest acceptance halfangle that corresponds to a real concentration ratio of 3.5, offered the best results in both variables, with an outlet fluid temperature of 93.9 1C, and outlet vapor quality of 0.0890. As can be seen in Table 1, in order to maintain a constant heat input it was necessary to modify the other CPC dimensions. 6.3. Effect of absorber coating and reflector material Fig. 7 shows the vapor quality and fluid temperature distribution along the CPC for three coatings of the carbon steel absorber tube, and three manufacture reflector materials. The absorber tube and cover material properties were also analysed, but no important effect on the results were found. All the curves were analyzed with temperate glass as cover, and carbon steel as absorber tube. The combination of highly polished aluminum with protective layer as reflector, and cermet as selective surface offered the best results, with an outlet fluid temperature of 95.4 1C, and outlet 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.

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.

worst results of the cases shown in Fig. 7 were obtained for the case with mirror quality stainless steel, and commercial selective surface, whose outlet fluid temperature and vapor quality were 91.8 1C, and 0.0617, respectively. This curve reveals that the most important influence in the quantity of vapor obtained by a CPC is the reflector material, followed by the coating of the absorber tube. From the analysis of the last three figures, a design CPC module with the geometrical and optical characteristics presented in Tables 2 and 3 was chosen to be coupled with a single-stage absorption system model. A 60.3 mm absorber tube diameter was selected since, together with the 73.0 mm one, if provides the best results in vapor quality and temperature rise, but with the advantage of a more compact CPC device and better wetting inside the diameter tube, since the fluid flow is relatively low. The simulation was carried out considering a carbon steel absorber tube, highly polished aluminum reflector, commercial selective surface, and temperate glass cover. The use of highly polished aluminum with protective layer reflector was not contemplated as this material must be imported and the total manufacture cost increases.

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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 0.91 selective surface Temperate glass cover 0.03 Highly polished aluminum 0.11 reflector

0.38

0.09

0.94 0.05

0.05 0.87

Fig. 8. Vapor quality and temperature distribution along the designed CPC (March 15th at solar noon).

The commercial selective surface was selected over cermet because 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 the reflector, the cover, the absorber tube wall, and the ammonia–water mixture, as well as the vapor quality along the design CPC module for May 15th at solar noon. The ammonia–water mixture enters the CPC with a subcooling degree of 5.4 1C. The outlet vapor quality obtained was 0.0891, which represents an ammonia vapor production of 0.0043 kg/s. The abrupt change in the absorber tube temperature when the ammonia–water mixture begins to evaporate is because, as explained before, the convective heat transfer coefficient from the subcooling liquid region and the two-phase flow presents a discontinuity, due to the use of different empirical heat transfer correlations for both regions. Fig. 9 illustrates the fluid temperature and vapor quality distribution for a typical day during seven operation hours. As expected, the outlet vapor quality increases with an increase in solar radiation, reaching a maximum of 0.0891 (ammonia vapor production of 0.0043 kg/s for the solar noon). Also a minimum quality of 0.0316 (ammonia vapor production of 0.0015 kg/s) is observed for 10:00 h. At 9:00 and 15:00 h there was no vapor production, but a fluid temperature rise of 1.2, and 2.9 1C was obtained, respectively. Due to the lower vapor production estimated at 9:00, 10:00, 14:00, and 15:00 h, an auxiliary heater is used in series after the CPC in order to reach the required outlet temperature and the vapor production obtained around 11:00 h, and therefore obtaining a reasonable cooling capacity for the complete system. The objective is to improve the cooling capacity and efficiency of the ammonia–water absorption solar refrigerator, according to the simulation results by coupling the CPC model and a single-stage absorption system model simulator.

Fig. 9. Vapor quality and fluid temperature distribution along the CPC for a typical day of operation.

Some refrigeration variables were analysed, such as cooling capacity, cooling COP, refrigeration efficiency, and flow ratio. Table 4 shows the values obtained by coupling the CPC module and the absorption refrigeration system for the seven cases analyzed during a typical operation day. It can be seen that for the CPC module, the outlet temperature varies, from 92.6 1C for the baseline case at 11:00 h to lower values 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 outlet temperature increases to 94.1 and 92.7 1C, respectively. The CPC module efficiency is 43.4% for the baseline case at 11:00 h, and lower at 9:00, 10:00, 14:00, and 15:00 h. The solar fraction, fixed as 100% at 11:00 h, is only 3.2% at 9:00 h, 51.7% at 10:00 h, 55.5% at 14:00 h, and 7.6% at 15:00 h. The cooling capacity is 3.81 kW and is higher at 12:00 and 13:00 h, being 4.79 and 3.90 kW, respectively.

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Table 4 Results comparison of a refrigeration system operated during a typical day coupled with the designed CPC Time (h) 9:00 Ammonia concentration (kg NH3/kg sol) Refrigerant (4) 0.995 Strong solution (10) 0.387 Weak solution (13) 0.363 Pressure (bar) Condensation Evaporation Mass flow rate (kg/s) Refrigerant (4) Strong solution (10) Weak solution (13) Main temperatures (1C) Inlet evaporator (7) Inlet condenser Inlet generator (12) Outlet CPC (13) Outlet generator (1) Outlet absorber (10) CPC Solar energy (kW) Useful energy gain (kW) Solar efficiency (%) Energetic behaviour Flow ratio (FR) Auxiliary energy (kW) Solar fraction (%) Cooling capacity (kW) Cooling COP Overall efficiency (%)

11.0 2.8 0.0033 0.0483 0.0448 10.80 55.00 81.65 82.87 92.60 39.19

10:00

0.995 0.387 0.363 11.0 2.8 0.0033 0.0483 0.0448 10.80 55.00 81.65 89.46 92.60 39.19

11:00

12:00

0.995 0.387 0.363

0.995 0.387 0.357

11.0 2.8

11.0 2.8

0.0033 0.0483 0.0448 10.80 55.00 81.83 92.60 92.60 39.19

0.0041 0.0483 0.0438 10.80 55.00 83.30 94.12 94.12 39.19

13:00

0.995 0.387 0.362 11.0 2.8 0.0033 0.0483 0.0447 10.80 55.00 81.94 92.74 92.74 39.19

14:00

0.995 0.387 0.363 11.0 2.8 0.0033 0.0483 0.0448 10.80 55.00 81.65 89.70 92.60 39.19

15:00

0.995 0.387 0.363 11.0 2.8 0.0033 0.0483 0.0448 10.80 55.00 81.65 84.54 92.60 39.19

7.154 0.278 3.889

13.968 4.514 32.317

20.120 8.731 43.398

22.601 10.455 46.257

20.120 8.896 44.213

13.968 4.844 34.683

7.154 0.660 9.224

14.85 8.45 3.18 3.81 0.437 –

14.85 4.22 51.70 3.81 0.437 –

14.85 0.00 100.00 3.81 0.437 18.94

11.82 0.00 100.00 4.79 0.458 21.19

14.51 0.00 100.00 3.90 0.439 19.40

14.85 3.89 55.48 3.81 0.437 –

14.85 8.07 7.56 3.81 0.437 –

The COP of the refrigeration cycle is between 0.437 and 0.458 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 the thermal and fluid-dynamic behavior of two-phase flow inside a CPC used as an ammonia–water vapor generator has been developed. The numerical analysis was made with a CV method on the absorber tube, and the discretized equations were coupled using a fully implicit step-by-step method in the flow direction. The numerical algorithm solves, in a segregated manner, three subroutines: fluid flow inside the absorber tube, heat conduction in the absorber tube wall, and the useful energy gain in the solar concentrator. Coupling between the three main subroutines was performed iteratively until convergence was reached. This numerical model can be used to simulate the generation process of any refrigerant–absorbent mixture at

the CPC whenever the mixture thermodynamic properties are known. The effect on the results of a range of design parameters was analyzed. These parameters were the acceptance halfangle, the diameter and coating of the absorber tube, and the manufacture material of the cover, the reflector, and the absorber tube. It was found that the most important design parameter is the reflector material selection, followed in order of priority by the absorber tube diameter and coating, and the acceptance half-angle. The material of the absorber tube and cover are not significant in the production of ammonia vapor, although corrosion could 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 refrigeration application in order to obtain certain ammonia vapor production. The system analysis (CPC model coupled to an absorption refrigeration system) was carried out for a typical operation day for 7 h (boundary conditions) in order to

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predict the solar fraction, cooling capacity, COP, and overall efficiency. It is theoretically possible to directly produce ammonia– water vapor in a 35 m long CPC module coupled to a single-stage ammonia–water refrigeration system. The system analyzed can produce more than 3.8 kW of cooling at 10 1C. For larger cooling needs a number of CPC modules can be connected in parallel to produce the ammonia–water vapor required. A CPC ammonia–vapor outlet temperature of around 93 1C can be theoretically achieved with a CPC efficiency of over 43%. The calculated overall efficiency of the system at solar noon can reach 21.2%. Acknowledgments This work had been financed by DGAPA-UNAM through the PAPIIT project IN105602-3, and by CONACyT project U44764-Y. The authors thank CONACyT, Me´xico, for the support provided for the student scholarship 118090. References [1] Mendes LF, Collares-Pereira M, Ziegler F. Supply of cooling and heating with solar assisted absorption heat pumps: an energetic approach. Int J Refrig 1998;21:116–25. [2] Syed A, Maidment GG, Missenden JF, Tozer RM. An efficiency comparison of solar cooling schemes. In: Proceedings of ASHRAE transactions, part I, vol. 1, 2002, p. 877–86. [3] Best R, Ortega N. Solar refrigeration and cooling. Renew Energy 1999;16:685–90. [4] Schweiger H, Mendes J, Benz N, Hennecke K, Prieto G, Cusı´ M., et al. The potential of solar heat in industrial processes, a state of the art review for Spain and Portugal. In: Proceedings of ISES-Europe conference, EuroSun, Copenhagen, Denmark, 2000. [5] Henning HM, Erpenbeck T, Hindenburg C, Santamaria IS. The potential of solar energy use in desiccant cooling cycle. Int J Refrig 2001;24:220–9. [6] Ziegler F. State of the art in sorption heat pumping and cooling technologies. Int J Refrig 2002;25:450–9. [7] Eck M, Zarza E, Eickhoff M, Rheinla¨nder J, Valenzuela L. Applied research concerning the direct steam generation in parabolic troughs. Sol Energy 2003;74:341–51. [8] Flores V, Almanza R. Behaviour of the compound wall copper–steel receiver with stratified two-phase flow regimen in transient states when solar irradiance is arriving on one side of receiver. Sol Energy 2004;76:195–8. [9] Erickson DC. Intermittent solar ammonia absorption cycle refrigerator. Report no US 4744224, Patent and Trademark Office, Box 9, Washington, DC, 1988. [10] Rivera CO, Rivera W. Modelling of an intermittent solar absorption refrigeration system operating with ammonia–lithium nitrate mixture. Sol Energy Mater Sol C 2003;76:417–27. [11] Aziz W, Chaturvedi SK, Kheireddine A. Thermodynamic analysis of two-component, two-phases flow in solar collectors with application to a direct-expansion solar-assisted heat pump. Energy 1999;24:247–59.

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