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Energy 48 (2012) 298e306

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Energy

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A detailed thermal model of a parabolic trough collector receiver

Soteris A. Kalogirou*

Department of Mechanical Engineering and Materials Science and Engineering, Cyprus University of Technology, P. O. Box 50329, 3603 Limassol, Cyprus

a r t i c l e i n f o

Article history:Received 26 October 2011Received in revised form24 March 2012Accepted 8 June 2012Available online 11 July 2012

Keywords:Parabolic troughHeat lossConductionConvection radiationReceiver thermal performance

* Tel.: þ357 2500 2621; fax: þ357 2500 2637.E-mail address: Soteris.kalogirou@cut.ac.cy.

0360-5442/$ e see front matter � 2012 Elsevier Ltd.http://dx.doi.org/10.1016/j.energy.2012.06.023

a b s t r a c t

Parabolic trough collectors are made by bending a sheet of reflective material into a parabolic shape. Ametal black pipe, covered with a glass tube to reduce heat losses, is placed along the focal line of thecollector. The concentrated radiation reaching the receiver tube heats the fluid that circulates through it,thus transforming the solar radiation into useful heat. It is sufficient to use a single axis tracking of thesun and thus long collector modules are produced. In this paper a detailed thermal model of a parabolictrough collector is presented. The thermal analysis of the collector receiver takes into consideration allmodes of heat transfer; convection into the receiver pipe, in the annulus between the receiver and theglass cover, and from the glass cover to ambient air; conduction through the metal receiver pipe andglass cover walls; and radiation from the metal receiver pipe and glass cover surfaces to the glass coverand the sky respectively. The model is written in the Engineering Equation Solver (EES) and is validatedwith known performance of existing collectors and subsequently is used to perform an analysis of thecollector we are going to install at Archimedes Solar Energy Laboratory at the Cyprus University ofTechnology.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

As shown in Fig. 1, a parabolic trough collector (PTC) is made bybending a sheet of reflective material into a parabolic shape. Ametal black pipe, covered with a glass tube to reduce heat losses, isplaced along the focal line of the collector. When the parabola ispointed towards the sun, the parallel rays incident on the reflectorare reflected and focused onto the receiver tube. The concentratedradiation reaching the receiver tube heats the fluid that circulatesthrough it, thus transforming the solar radiation into useful heat. Itis sufficient to use a single axis tracking of the sun and thus longcollector modules are produced [1,2].

The collector can be orientated in an east-west direction,tracking the sun from north to south, or orientated in a north-southdirection, tracking the sun from east to west. The advantages of theformer tracking mode is that very little collector adjustment isrequired during the day and the full aperture always faces the sunat noon time but the collector performance during the early andlate hours of the day is greatly reduced due to large incidenceangles (cosine loss). North-south orientated troughs have theirhighest cosine loss at noon and the lowest in the mornings andevenings when the sun is due east or due west. Over the period ofone year, a horizontal north-south trough field usually collectsslightly more energy than a horizontal east-west one. However the

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north-south field collects a lot of energy in summer and much lessin winter. The east-west field collects more energy in winter thana north-south field and less in summer, providing a more constantannual output. Therefore, the choice of orientation usually dependson the application and whether more energy is needed duringsummer or during winter [1,2].

Parabolic trough technology is the most advanced of the solarthermal technologies because of considerable experience with thesystems and the development of a small commercial industry toproduce and market these systems. Parabolic trough collectors arebuilt in modules that are supported from the ground by simplepedestals at either end. Photographs of PTC collectors are shownin Fig. 2.

Parabolic trough collectors are themostmature solar technologyto generate heat at temperatures up to 400 �C for solar thermalelectricity generation or process heat applications. The biggestapplication of this type of system is the Southern California powerplants, known as Solar Electric Generating Systems (SEGS), whichhave a total installed capacity of 354 MWe [3]. SEGS I is 14 MWe,SEGS IIeVII are 30MWe each and SEGSVIII and IX are 80MWe each.

The receiver of a parabolic trough is linear. Usually a tube isplaced along the focal line to form an external surface receiver (seeFig. 1). The size of the tube, and therefore the concentration ratio, isdetermined by the size of the reflected sun image and themanufacturing tolerances of the trough. The surface of the receiveris typically platedwith selective coating that has a high absorptancefor solar irradiation but a low emittance for thermal radiation.

Fig. 1. Schematic of a parabolic trough collector.

S.A. Kalogirou / Energy 48 (2012) 298e306 299

A glass cover tube is usually placed around the receiver tube toreduce the convective heat loss from the receiver, thereby furtherreducing the heat loss coefficient. A disadvantage, resulting fromthe use of the glass cover tube, is that the reflected light from theconcentrator must pass through the glass to reach the receiver,adding a transmittance loss of about 0.9, when the glass is clean.The glass envelope usually has an anti-reflective coating to improvetransmissivity. One way to further reduce convective heat loss fromthe receiver tube and thereby increase the performance of thecollector, particularly for high temperature applications, is toevacuate the space between the glass cover tube and the receiver.The total receiver tube length of PTCs is usually from 25m to 150m.

New developments in the field of parabolic trough collectorsaim at cost reduction and improvements of the technology. In onesystem the collector can be washed automatically thus reducingdrastically the maintenance cost, which is the mostly used processrequired [1].

A comprehensive review on parabolic trough collectors andtheir applications is presented by Fernandez-Garcia et al. [4]. Thereview covers a historical survey, types of collectors and theircharacteristics and applications by region/country and by processcoupled to the collector such as concentrating solar power (CSP),industrial process heat (IPH), hot water and space heating, airconditioning and refrigeration, pumping irrigation water, desali-nation and solar chemistry.

The purpose of this paper is to present a detail thermal model ofthe receiver of the collector. Many researchers presented studies ofenergy models of parabolic trough collectors. The most importantones are the study of Karimi et al. [5], Forristall [6] and Gong et al.[7]. Karimi et al. [5] applied a piecewise two-dimensional model ofthe receiver by considering the circumferential variation of solarflux, performed by dividing the receiver into longitudinal andisothermal nodal sections and applying the principle of energybalance to the glazing and receiver nodes. Forristall [6] build andanalysed both a 1-D and a 2-D heat transfer model of a PTC receiver

Fig. 2. Photos of parabolic trough collectors (left picture is Eurot

implemented in EES. A recent study presented by Gong et al. [7]presented an optimised model and tested China’s first hightemperature parabolic trough receiver.

Various other groups publishedmodels of PTCs. Premjit et al. [8]presented a numerical investigation of parabolic trough receiverperformance with outer vacuum shell whereas Munoz et al. [9]presented the thermal regimes in solar thermal linear collectors.The type of collector considered however in this last paper is linearFresnel type which is different than the PTC. Tao and He [10] pre-sented also a numerical study on coupled fluid flow and heattransfer process in a PTC tube. In this work a unified two-dimensional model is developed and the temperature distribu-tions in the receiver are presented. Huang et al. [11] presented theperformance simulation of a PTC. The paper deals with the opticalperformance of a PTC and a new analytical model is given. Theeffects of the optical error, tracking error, position error, opticalproperties of reflector, transmittance and absorptivity of vacuumtube receiver and efficiencies of the trough systems are simulatedand analysed.

Some other important designs of solar collectors published arethe works presented by Issa et al. [12] and concern v-troughconcentrators, and Al-Nimr et al. [13,14] and concern size optimi-sation and a tubeless solar collector respectively.

The model presented in this paper takes into consideration allmodes of heat transfer; convection into the receiver pipe, in theannulus between the receiver and the glass cover, and from theglass cover to ambient air; conduction through the metal receiverpipe and glass cover walls; and radiation from the metal receiverpipe to glass cover and from glass cover to the sky.

2. The energy model

Although for low-temperature applications bare tube receiverscanbeused theusual case is tohave aglazed receiver, soonly this caseis considered in this paper. For the annulus between the receiver and

rough, right picture is industrial solar technology collector).

S.A. Kalogirou / Energy 48 (2012) 298e306300

the glass cover two conditions are considered the vacuumand the aircase. The former is usually used in high temperature applications.

The model is written in Engineering Equation Solver (EES) [15].This is done for two reasons; the EES includes routines to estimatethe properties of various substances by specifying any two prop-erties, such as temperature and pressure and EES can be called fromTRNSYS which allows the development of a model which can usethe capabilities of both programs. The model is validated withknown performance of existing collectors and subsequently wasused to perform an analysis of the collector we are going to installat Archimedes Solar Energy Laboratory at the Cyprus University ofTechnology.

The collector performance model uses an energy balancebetween the fluid flowing through the receiver, usually a heattransfer fluid (HTF), and the atmosphere. It includes all equationsnecessary to predict the various expressions of the energy balance,which depend on the ambient conditions and the collector receiveroptical properties and condition.

A cross-section of the collector receiver and the subscript defi-nitions are shown in Fig. 3a whereas Fig. 3b shows the steady-statethermal resistance model obtained from an energy balance of thereceiver. The model assumes that all temperatures, heat fluxes, andthermodynamic properties are uniform around the circumferenceof the receiver. This is not very true as the radiation profile is notuniform and the bottom part receives much higher solar flux thanthe top part because of the radiation reflected by the parabolicmirror. For small solar collectors however, this simplification doesnot introduce severe inaccuracies. In the resistance model theincoming solar energy and optical losses have been omitted forclarity. The optical losses are due to imperfections in the collectormirrors, tracking errors, shading and cleanliness of the mirror andreceiver glazing. The incoming solar energy, which effectively isequal to the solar energy input minus optical losses, is absorbed bythe glass envelope (qgo,SolAbs) and receiver pipe (qpo,SolAbs). Most ofthe energy that is absorbed by the receiver is conducted throughthe receiver pipe material (qpi-po,cond) and eventually transferred tothe HTF by convection (qf-pi,conv). The remaining energy is trans-mitted back to the glass envelope by convection (qpo-gi,conv) andradiation (qpo-gi,rad). The energy reaching the glass cover fromradiation and convection then passes through the glass envelope

Glass cover

pi f

(sky) s

po

gi go

(air) a

Heat transfer fluid

Receiver pipe

s

ago gi f pi po

Nomenclature

Rf-pi,conv Rpi-po,condRpo-gi,rad

Rpo-gi,conv

Rgi-go,condRgo-s,rad

Rgo-a,conv

Thermal resistance model

HTF PIPE ANNULUS GLASS

SKY

AIR

qgo,SolAbs

qpo,SolAbs

a

b

Fig. 3. Collector receiver model a) nomenclature, b) Thermal resistance network forthe cross-section of the receiver.

wall by conduction (qgi-go,cond) and along with the energy absorbedby the glass envelope wall (qgo,SolAbs) is lost to the environment byconvection to ambient air (qgo-a,conv) and radiation towards the sky(qgo-s,rad).

The energy balance equations are determined by consideringthat the energy is conserved at each surface of the receiver cross-section, shown in Fig. 3. Therefore:

qf�pi;conv ¼ qpi�po;cond (1)

qpo;SolAbs ¼ qpo�gi;conv þ qpo�gi;rad þ qpi�po;cond (2)

qpo�gi;conv þ qpo�gi;rad ¼ qgi�go;cond (3)

qgi�go;cond þ qgo;SolAbs ¼ qgo�a;conv þ qgo�s;rad (4)

qHeatLoss ¼ qgo�a;conv þ qgo�s;rad (5)

It should be noted that the solar absorption at the outside pipe,qpo,SolAbs and outside glass, qgo,SolAbs surfaces are treated as heatflux expressions, which simplifies the solar absorption expressionsas it considers the heat conduction through the receiver pipe andglass envelope wall to be linear. In reality, the solar absorption inthe glass envelope wall (semitransparent material) and receiverpipe (opaquemetal material) are volumetric phenomena. However,it is well known from heat transfer textbooks [16] that most of theabsorption in the metallic surfaces (receiver pipe) occurs very closeto the surface (within a few mm) and although solar absorptionoccurs throughout the thickness of the glass envelope wall, theabsorptance is very small (a¼ 0.02). Thus, the error in treating solarabsorption as a surface phenomenon is very small.

The various heat transfer interactions are analysed in differentsections below, starting from the heat transfer fluid inside towardsthe air and sky outside the receiver assembly.

2.1. Convection heat transfer between the HTF and the receiver pipe

Newton’s law of cooling states that the convection heat transferfrom the inside surface of the receiver pipe to the HTF is given byhA(Ts � TN). Therefore, in the case of the PTC model and using thenomenclature adopted in Fig. 3:

qf�pi;conv ¼ hfpDpi

�Tpi � Tf

�(6)

and the convection heat transfer coefficient at the inside pipediameter, hf is given by:

hf ¼ NuDpi

kfDpi

(7)

where: NuDpi¼ Nusselt number based on Dpi (�)

In Eq. (6), both Tf and Tpi are independent of angular andlongitudinal directions of the receiver. The same applies for alltemperatures and properties in the energy model.

The Nusselt number depends on the type of flow through thereceiver pipe. The model includes conditional statements todetermine the type of flow, although usually, at typical operatingconditions, the flow in the receiver pipe is well within the turbulentflow region. Additionally it is assumed that the flow in the pipe isthermally and hydrodynamically fully developed, which is fullycorrect for the turbulent flow regime (except for the initial lengthcorresponding to the first 10 pipe diameters). It is however not fullycorrect for the laminar flow regime but as these collectors areusually very long and rarely operate in the laminar flow regime (to

S.A. Kalogirou / Energy 48 (2012) 298e306 301

have an increased heat transfer coefficient) the possible errorintroduced is not important and can be considered as a safetymargin, as the Nuselt number and thus the convection coefficient,in the developing region is bigger than the one in the fully devel-oped region. Additionally, in the laminar flow the entrance region isnot that big as the circulating fluid is water with a relatively lowPrandtl number.

When the Reynolds number is lower than 2300, laminar flowexists in the receiver pipe and the Nusselt number is constant. Forpipe flow, the constant value, assuming constant heat flux, as in thecase of a PTC, is equal to 4.36 [16].

Turbulent and transitional cases occur at Reynoldsnumber > 2300. Therefore, the following Nusselt number correla-tion developed by Gnielinski [17] is used for the convective heattransfer from the receiver pipe to the HTF:

NuDpi¼

fpi

8�ReDpi

� 1000�Prf

1þ 12:7ffiffiffiffiffiffiffiffiffiffiffifpi=8

q �Pr2=3f � 1

� PrfPrpi

!0:11

For 0:5 < Prf < 2000 and 2300 < ReDpi < 5� 106

(8)

with

fpi ¼h1:82 log

�ReDpi

�� 1:64

i�2(9)

where

Prf ¼ Prandtl number evaluated at the HTF temperature, Tf (�)Prpi ¼ Prandtl number evaluated at the receiver pipe insidesurface temperature, Tpi (�)

Except for Prpi, all fluid properties are evaluated at themean HTFtemperature, Tf. The correlation assumes uniform heat flux andtemperature, and assumes that the receiver pipe has a smoothinside surface.

The above equations are valid for both turbulent pipe flow andthe transitional flow which occur for Reynolds numbers between2300 and 4000 [16]. Furthermore, the above correlations areadjusted for fluid property variations between the receiver pipewall temperature and the bulk fluid temperature. If the correlationis used out of the range of validity, shown in Eq. (8), the programwill display a warning message.

2.2. Conduction heat transfer through the receiver pipe wall

Conduction heat transfer through the receiver pipe wall isdetermined by the Fourier’s law of conduction through a hollowcylinder given by [16]:

qpi�po;cond ¼ 2pkpipe�Tpi � Tpo

�ln

Dpo

Dpi

! (10)

where

kpipe ¼ receiver pipe thermal conductivity at the averagereceiver pipe temperature (Tpi þ Tpo)/2 (W/m-�C)

In this equation the thermal conductivity is considered asconstant, and evaluated at the average temperature between theinside and outside receiver pipe surfaces.

The thermal conductivity depends on the receiver pipe materialtype. The receiver performance model includes three types ofstainless steels (304L, 316L, and 321H) and one copper, which canbe chosen by the user at the beginning. If copper is chosen, thethermal conductivity is constant equal to 385 W/m-�C. If stainlesssteel 304L or 316L is chosen, the thermal conductivity is calculatedwith the following equation:

kpipe ¼ ð0:013ÞTpi�po þ 15:2 (11)

and if stainless steel 321H is chosen:

kpipe ¼ ð0:0153ÞTpi�po þ 14:775 (12)

Both equations were determined by linearly fitting data fromDavis [18].

2.3. Heat transfer from the receiver pipe to the glass envelope

As was mentioned before, between the receiver pipe and theglass envelope heat transfer occur by convection and radiation.Convection heat transfer depends on the annulus pressure [19]. Atlow pressures (<0.013 Pa), heat transfer is by molecular conduc-tion, whereas at higher pressures is by free convection. Becausethere is a difference in temperature between the outsider receiverpipe surface and the inside glass envelope surface, radiation heattransfer also occurs. The radiation heat transfer calculation issimplified by assuming that the glass envelope wall is opaque toinfrared radiation and gray surfaces, for which (r ¼ a). All these areexamined separately in the following sections.

2.3.1. Convection heat transferAs mentioned above, two heat transfer mechanisms are

considered in the determination of the convection heat transferbetween the receiver pipe and glass envelope wall (qpo-gi,conv).These are the free-molecular and natural convection [19]. The casesof vacuum and pressure in the annulus are examined separately.

a) Vacuum in annulus: When the annulus is under vacuum(pressure <0.013 Pa), the convection heat transfer between thereceiver pipe and glass envelope occurs by free-molecularconvection and is given by [20]:

qpo�gi;conv ¼ pDpohpo�gi�Tpo � Tgi

�(13)

where

hpo�gi ¼ kstdDpo

2 ln�Dgi

Dpo

�þ bl

Dpo

Dgiþ 1

!

For : RaDgi <�Dgi=

�Dgi � Dpo

��4(14)

and

b ¼ ð2� aÞð9g� 5Þ2aðgþ 1Þ (15)

l ¼ 2:331� 10�20�Tpo�gi þ 273�

�Pad

2� (16)

where kstd ¼ thermal conductivity of the annulus gas at standardtemperature and pressure (W/m-�C)

This correlation slightly overestimates the heat transfer for verysmall pressures (<0.013 Pa). The molecular diameters of air, d, isobtained from Marshal [21] and is equal to 3.55 � 10�8 cm, the

S.A. Kalogirou / Energy 48 (2012) 298e306302

thermal conductivity of air is 0.02551 W/m-�C, the interactioncoefficient is 1.571, the mean-free-path between collisions ofamolecule is 88.67 cm, and the ratio of specific heats for the annulusair is 1.39. These are for average fluid temperature of 300 �C andpressure equal to 0.013 Pa. Using these values, the convection heattransfer coefficients (hpo-gi) is equal to 0.0001115 W/m2-�C.

b) Pressure in annulus: If the receiver annulus vacuum is lost orthe receiver is filled or partially filled with ambient air(pressure > 0.013 Pa), the convection heat transfer mechanismbetween the receiver pipe and glass envelope occurs bynatural convection. For this purpose the Raithby andHolland’s correlation for natural convection in an annular space(enclosure) between horizontal concentric cylinders is used, givenby [16]:

qpo�gi;conv ¼ 2pkeff

ln�Dgi

Dpo

��Tgi � Tpo�

For : 0:7 � Prpo�gi

� 6000 and 102 � FcylRapo�gi � 107

(17)

keffkag

¼ 0:386

Prpo�gi

0:861þ Prpo�gi

!14�FcylRaDpo

�14 (18)

Fcyl ¼

ln�Dgi

Dpo

�4

L3c�D�3=5gi � D�3=5

po

�5 (19)

in these equations the critical length is given by: Lc ¼ ðDgi � DpoÞ2

.where

kag ¼ thermal conductivity of annulus gas at Tpo-gi (W/m-�C)Prpo-gi ¼ Prandtl number for gas properties evaluated at Tpo-gi(�)RaDpo ¼ Rayleigh number evaluated at Dpo (�)

This correlation assumes long, horizontal, concentric cylindersat uniform temperatures. All physical properties are evaluated atthe average temperature (Tpo þ Tgi)/2.

2.3.2. Radiation Heat TransferIn deriving an equation for the radiation heat transfer, several

assumptions were made as follows:

� Non-participating gas in the annulus,� The surfaces are gray,� Diffuse reflections and irradiation� Long concentric isothermal cylinders, and� The glass envelope is opaque to infrared radiation.

Not all these assumptions are completely accurate. For instance,the glass envelope wall and the selective coatings are not gray, andthe glass envelope wall is not completely opaque for the entirethermal radiation spectrum [22]. However, any errors associatedwith the assumptions are relatively small.

The radiation heat transfer between the receiver pipe and glassenvelope (qpo-gi,rad) is estimated with the following equation [16]:

qpo�gi;rad ¼spDpo

�T4po � T4gi

�

13po

þ �

1� 3gi�Dpo

3giDgi

!! (20)

2.4. Conduction heat transfer through the glass envelope

The anti-reflective treatment on the inside and outside surfacesof the glass envelope is assumed not to introduce any thermalresistance or to have any effect on the glass emissivity. This isreasonably accurate since the treatment is usually a chemicaletching which does not add any additional elements to the glasssurface [6]. The conduction heat transfer through the glass enve-lope uses the same equation as the conduction through the receiverpipe wall described in Section 2.2. As in the receiver case, thetemperature distribution is assumed to be linear. Furthermore, thethermal conductivity of the glass (kglass) is assumed constant e asexplained in Section 2.1 e with a value of 1.04, which correspondsto Pyrex glass [22].

2.5. Heat transfer from the glass envelope to the atmosphere

The heat transfer from the glass envelope to the atmosphereoccurs by convection and radiation. Depending onwhether there iswind the convectionwill either be forced or natural. Radiation heatloss occurs due to the temperature difference between the glassenvelope and sky. All these are examined separately below.

2.5.1. Convection heat transferThe convection heat transfer is determined by knowing the

Nusselt number, which depends on whether the convection heattransfer is natural (no wind) or forced (wind case). When there iswind, the convection heat transfer from the glass envelope to theatmosphere presents a much bigger heat loss. This is estimatedfrom Newton’s law of cooling:

qgo�a;conv ¼ hgo�apDgo�Tgo � Ta

�(21)

hgo�a ¼ kairDgo

NuDgo(22)

where

hgo-a ¼ convection heat transfer coefficient for air at (Tgo � Ta)/2(W/m2-�C)kair ¼ thermal conductivity of air at (Tgo � Ta)/2 (W/m-�C)NuDgo ¼ average Nusselt number based on the glass envelopeoutside diameter Dgo (�)

a) Nowind:When there is no wind, the convection heat transferfrom the glass envelope to the environment occurs by naturalconvection and the correlation developed by Churchill and Chu isused to estimate the Nusselt number [16]:

NuDgo¼240:60þ

0387R1=6Dgo�an1þ �0:559=Prgo�a

� 916

o 827

352

105 < RaDgo < 1012

(23)

RaDgo¼ gb

�Tgo � Ta

�D3go

v2go�aPrgo�a (24)

b ¼ 1Τgo�a

(25)

Prgo�a ¼ vgo�a

ago�a(26)

S.A. Kalogirou / Energy 48 (2012) 298e306 303

where

RaDgo ¼ Rayleigh number for air based on the glass envelopeoutside diameter, Dgo (�)ago-a ¼ thermal diffusivity for air at Tgo-a (m2/s)Prgo-a ¼ Prandtl number for air at Tgo-a (�)ngo-a ¼ kinematic viscosity for air at Tgo-a (m2/s)

This correlation assumes a long isothermal horizontal cylinder.Also, all the fluid properties are determined at the mean filmtemperature, (Tgo þ Ta)/2.

b) Wind: When there is wind, the convection heat transfer fromthe glass envelope to the environment occurs by forced convection.The Nusselt number in this case is estimated with Zhukauskas’correlation for external forced convection flow normal to anisothermal cylinder [23]:

NuDgo¼C RemDgo

Prna

�PraPrgo

�14

0:7 < Pra < 500;

and 1 < ReDgo < 106(27)

the constants C andm are given in Table 1, obtained from Incroperaet al. [23] whereas the constant n is equal to 0.37 for Pr � 10 and isequal to 0.36 for Pr > 10.

All fluid properties are evaluated at atmospheric temperature,Ta, except Prgo, which is evaluated at the glass envelopewall outsidesurface temperature.

2.5.2. Radiation heat transferIn this model, the useful solar irradiance is considered in the

solar absorption expressions. Thus, the radiation transfer betweenthe glass envelope wall and sky is caused by the temperaturedifference between the glass cover and the sky. This is done byassuming that the cover is a small convex gray object in a largeblackbody cavity, the sky. In this case, net radiation transferbetween the glass envelope and sky is given by [16]:

qgo�s;rad ¼ s 3gopDgo

�T4go � T4

s

�(28)

It should be noted that the sky, especially during non-clearconditions, does not act as a blackbody; however, it is commonpractice to model it as such and to use an effective sky temperatureto compensate for the difference [1]. To simplify the model, theeffective sky temperature is approximated as Ta-8 �C, despite thefact that several relations have been proposed to relate the effectivesky temperature for clear skies to measured meteorological data.

2.6. Solar irradiance absorption

In this model the optical efficiency terms are estimated andcombined to form an effective optical efficiency, which is subse-quently used to determine the optical loss and solar absorptionexpressions. The optical properties used in the collector perfor-mance model were obtained from a combination of sources.

The parameters used to estimate effective optical efficiencies aregenerated from the National Renewable Energy Laboratory (NREL)

Table 1Constants for Equation (27).

ReD C m

1e40 0.75 0.440e1000 0.51 0.51000e200000 0.26 0.6200000e1000000 0.076 0.7

report [24],whichwasbasedonfield tests conductedbyDudleyet al.[25], and software performance modelling. These are as follows:

esh ¼ Receiver shadowing (bellows, shielding, supports), 0.974(�)etr ¼ Tracking error, 0.994 (�)ege ¼ Geometry error (mirror alignment), 0.98 (�)rcl ¼ Clean mirror reflectance, 0.935 (�)edm ¼ Dirt on mirrors (reflectivity/rcl) [reflectivity is an inputparameter, usual value: 0.88e0.93]eda ¼ Dirt on receiver, (1 þ edm)/2 (�)eun ¼ Unaccounted, 0.96 (�)

The terms, esh, etr, ege, and eun, are estimates. The clean mirrorreflectance rcl is a known value, and the two dirt effects edm and edaare obtained from recommendations by Duffie and Beckman [26]. Itshould be noted that these parameters are valid only for normalsolar incidence irradiance. To account for incident angle losses, theincident angle modifier is used, which accounts for end shading ofthe trough, reflection and refraction loses, and selective coatingincident angle effects.

The above list of parameters account for collector geometriceffects (shadowing, tracking, alignment), mirror and glass envelopetransmittance effects (mirror reflectance and dirt), and a parameterfor unexplained differences between field test data and modelleddata. All these values can be altered by the user if better and moreaccurate values become available.

Generally, the incident anglemodifier is used to account for caseswhen the solar irradiance is not normal to thecollectoraperture [1,2].It is a function of the solar incidence angle (q) to the normal of thecollector aperture. The equation determined from a collector testingcarried out at Sandia National Laboratory (SNL) is given by [25]:

Kq ¼ cosðqÞ þ 0:000884q� 0:00005369q2 (29)

Other optical properties required include the selective coatingabsorptance and emittance, and the glass envelope transmittance,absorptance and emittance. The glass envelope absorptance andemissittance are constant (independent of temperature) andindependent of selective coating type. The values used in themodelare a ¼ 0.02 and 3¼ 0.86 and can be changed by the user if there isa need. The glass envelope transmittance and the selective coatingabsorptance and emittance depend on the type of selective coating.Both the envelope transmittance and the coating absorptance areconstants; whereas the coating emittance is a function of temper-ature. The properties of the Luz cerment selective coating type usedin the model are as follow [6]:

� Envelope transmittance ¼ 0.935 (�)� Coating absorptance ¼ 0.92 (�)� Coating emittance ¼ 0.06 at 100 �C and 0.15 at 400 �C (�)

The emittance equation used for the selective coating consid-ered, which coincide with the emittance values given above is [6]:

Coating Emittance; 3po ¼0:000327ðT þ 273:15Þ� 0:065971

(30)

The emittance values between the two reference points, of100 �C and 400 �C, are nearly linear. It should be noted that thetemperature in Eq. (30) is in degrees Celsius.

2.6.1. Solar irradiance absorption in the glass envelopeAs stated in Section 2.1, to simplify the model and although

physically this is not true, the solar absorption into the glass

Fig. 5. Comparison of measured against predicted heat lossereceiver annulus vacuum.

S.A. Kalogirou / Energy 48 (2012) 298e306304

envelopewall is treated as a heat flux. In fact, the solar absorption inthe glass envelope wall is a heat generation phenomenon and assuch is a function of the glass wall thickness. However, thisassumption introduces an insignificant error since the glass enve-lope wall is relatively thin and the solar absorptance coefficient forglass is very small, 0.02 [22]. Additionally, the optical efficiency isused to calculate the solar absorption in the glass envelope given by:

qgo;SolAbs ¼ qsolhenvaenv (31)

with

henv ¼ eshetregeedmedaeunrclKq (32)

All parameters in Equation (32), except the incidence anglemodifier (Kq), are taken from the above list. Furthermore, the solarirradiance term (qsol) in Equation (31) is determined bymultiplyingthe direct normal solar irradiance (DNI) by the projected normalreflective surface area of the collector, i.e., aperture area, anddividing by the receiver length. In both equations, all terms areassumed to be independent of temperature.

2.6.2. Solar irradiance absorption in the receiver pipeAs stated before, the solar energy absorbed by the receiver pipe

occurs essentially at the surface; therefore, it is treated as a heatflux (see Section 2.1). Therefore, the equation for the solarabsorption in the receiver pipe is given by:

qpo;SolAbs ¼ qsolhabsaabs (33)

With habs ¼ henvsenv (34)

In Equation (33), the effective optical efficiency of the glassenvelope, henv is obtained by Equation (32) and as before, all termsare assumed to be independent of temperature.

2.7. Heat removal factor

The heat removal factor represents the ratio of the actual usefulenergy gain that would result if the collector-absorbing surface hadbeenat the localfluid temperature. InEquation formthis is givenby [1]:

FR ¼ _mcpArUL

�1� Exp

� ULF 0Ar

_mcp

�(35)

where F΄ is the collector efficiency factor, given by [1]:

Fig. 4. Comparison of measured against predicted thermal efficiencyereceiver annulusvacuum.

F 0 ¼1UL

1UL

þ Dpo

hfDpiþ Dpo

2kfln

Dpo

Dpi

! (36)

the factor UL represents the collector heat loss coefficient which isthe summation of the coefficients for conduction through the glasscover, convection from the outside of the receiver pipe to theannulus space and ambient air, and radiation from the outside ofthe receiver pipe to the sky, given in Sections 2.3e2.5. So if thisparameter needs to be estimated, the above equations can beemployed to calculate it quickly.

3. Code testing

The code developed is tested using known performancemeasurements from test carried out at SNL and presented inDudley et al. [25]. The information required to input to EES code isthe following:

1. Direct Normal Irradiance (DNI) (W/m2)2. Inlet temperature (�C)3. Wind speed (m/s)4. Ambient temperature (�C)5. Solar incidence angle (�)6. Coating absorptance (�)7. Coating emittance at 100 �C (�)8. Coating emittance at 400 �C (�)9. Mirror reflectivity (�)

Fig. 6. Comparisonofmeasuredagainstpredictedthermal efficiencyereceiverannulus air.

Fig. 7. Comparison of measured against predicted heat lossereceiver annulus air.

500525550575600625650675700725750

50 75 100 125 150 175 200HTF temperature (°C)

He

at (W

/m

)

5052.55557.56062.56567.57072.575

Effic

ie

nc

y

Heat gainEfficiency

Fig. 8. Performance of the collector that will be installed at Archimedes solar energylaboratory.

S.A. Kalogirou / Energy 48 (2012) 298e306 305

10. Glass envelope transmittance (�)11. Annulus pressure or vacuum (�)12. Annulus absolute pressure (kPa)13. HTF flow rate (m3/s)14. Type of heat transfer fluid (label)15. Receiver inside diameter (m)16. Receiver outside diameter (m)17. Glass envelope inside diameter (m)18. Glass envelope outside diameter (m)19. Collector aperture area (m2)20. Shadowing (�)21. Tracking error (�)22. Dirt factor on glass envelope (�)23. Dirt factor on mirror (�)

A comparison of the performance of the code developed and thetests conducted at SNL is shown in the following figures. Figs. 4 and5 show a comparison of the actual efficiency and heat loss of thecollector with the values determined from the EES code developed,when vacuum exists in the receiver annulus.

Similar results for air in the receiver annulus are presented inFigs. 6 and 7. In all cases the agreement between the experimentalresults and those obtained by the EES code is very acceptable. Theagreement is better for the air case whereas in both cases thedifference increases with increasing operating temperature. Thepossible reason for the deviation presented in the case of heat lossis the dependence of the optical properties on the temperature,which was ignored.

Finally, the code developed is usedwith the characteristics of thecollector we will erect at the premises of the Cyprus University ofTechnology and in particular at the Archimedes Solar Energy Labo-ratory (ASEL). The collector is supplied from the Australian companyNEP-SOLAR and has the characteristics presented in Table 2.

Table 2Characteristics of the collector wewill install at Archimedes solarenergy laboratory.

Parameter Value

Number of collector modules 6Collector Length 1993 mmCollector width 1208 mmParabola focal distance 647 mmMirror reflectivity 93.5%Receiver material Stainless steel 304 LReceiver external diameter 28 mmReceiver internal diameter 25 mmGlass tube transmittance 0.89Selective coating absorptance 0.93Selective coating emittance 0.18

The collector to be installed at the roof of the laboratory willhave a length of 12.2 m and consists of galvanised steel mounts,lightweight, stiff and precise parabolic reflector panels manufac-tured from reinforced polymeric material, a structurally efficientgalvanised steel torque tube, a tubular receiver and an accuratesolar tracking system.

As the collector is able to operate up to about 200 �C theselective coating properties are assumed to be constant to allpossible temperature range. The results of the program are showngraphically in Fig. 8 and give a thermal efficiency of about 58% at200 �C, which is very satisfactory. These results were obtained ata solar radiation of 900 W/m2, wind speed of 0.45 m/s, flow rate of8.8 kg/s and ambient temperature of 25 �C and ambient air atatmospheric pressure in receiver annulus. These findings are inagreement with the value of efficiency given by the manufacturerand will be validated in the near future when the collector instal-lation is finalised.

4. Conclusions

In this paper a detailed thermal model, which can be used forthe analysis of a parabolic trough collector receiver is presented.The model takes into consideration all modes of heat transfer;convection into the receiver pipe, in the annulus between thereceiver and the glass cover, and from glass cover to ambient air;conduction through the metal receiver pipe and glass cover walls;and radiation from the metal receiver pipe to the glass cover andfrom glass cover to the sky. The model is written in the EngineeringEquation Solver (EES). The validation of the model is done using theknown performance of existing collectors tested at Sandia NationalLaboratories, and its performance is very satisfactory. Finally, themodel is used to perform an analysis of the collector we are going toinstall at the Archimedes Solar Energy Laboratory of the CyprusUniversity of Technology.

Nomenclature

a accommodation coefficient (�)Ar receiver area (m2)b interaction coefficient (�)cp specific heat capacity (J/kg-�C)Dpi inside diameter of the receiver pipe (m)Dpo outside receiver pipe diameter (m)Dgi inside glass envelope diameter (m)Dgo outside glass envelope diameter (m)F0 collector efficiency factor (�)fpi friction factor for the inside surface of the receiver pipe,

Dpi (�)h convection heat transfer coefficient (W/m2-�C)k thermal conductivity (W/m-�C)

S.A. Kalogirou / Energy 48 (2012) 298e306306

kf thermal conductivity of the HTF at Tf (W/m-�C)Κq incident angle modifier (�)hf HTF convection heat transfer coefficient at Tf (W/m2-�C)g gravitational constant (¼9.81 m/s2)_m mass flow rate (kg/s)Nu Nusselt number (�)Pa annulus gas pressure (mmHg)qsol solar irradiance per receiver length (W/m)Pr Prandtl number (�)Ra Rayleigh number (�)Ta ambient air temperature (�C)Tf mean (bulk) temperature of the HTF (�C)Tpi receiver pipe inside surface temperature (�C)Tpo receiver pipe outside surface temperature (�C)Tgi inside glass envelope surface temperature (�C)Tgo glass envelope outside surface temperature (�C)Tgo-a film temperature (Tgo þ Ta)/2 (K)Tpo-gi average temperature (Tpo þ Tgi)/2 (�C)Ts effective sky temperature (K)UL collector heat loss coefficient (W/m2-�C)

Greek:aabs absorptance of receiver pipe (�)aair thermal diffusivity for air at Tgo-a (m2/s)aenv absorptance of the glass envelope (�)b volumetric thermal expansion coefficient (ideal gas) (1/K)g ratio of specific heats for the annulus gas (�)d molecular diameter of annulus gas (cm)3po receiver pipe selective coating emissivity (�)3gi glass envelope emissivity (�)3go emissivity of the glass envelope outside surface (�)habs effective optical efficiency at receiver pipe (�)henv effective optical efficiency of the glass envelope (�)q solar incidence angle (�)n kinematic viscosity (m2/s)l mean-free-path between collisions of a molecule (cm)s StefaneBoltzmann constant (¼5.67 � 10�8 W/m2-K4)senv transmittance of the glass envelope (�)

AbbreviationsASEL Archimedes Solar Energy LaboratoryCSP Concentrating Solar PowerDNI Direct Normal IrradianceEES Engineering Equation SolverHTF Heat Transfer FluidNREL National Renewable Energy LaboratoryPTC Parabolic Trough CollectorSEGS Solar Energy Generating SystemsSNL Sandia National Laboratories

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