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Solar Energy 84 (2010) 1809–1815

Thermal stress analysis of eccentric tube receiver usingconcentrated solar radiation

Fuqiang Wang, Yong Shuai, Yuan Yuan, Guo Yang, Heping Tan *

School of Energy Science and Engineering, Harbin Institute of Technology, 92, West Dazhi Street, Harbin 150001, PR China

Received 28 March 2010; received in revised form 24 June 2010; accepted 8 July 2010Available online 21 August 2010

Communicated by: Associate Editor Brian Norton

Abstract

In the parabolic trough concentrator with tube receiver system, the heat transfer fluid flowing through the tube receiver can inducehigh thermal stress and deflection. In this study, the eccentric tube receiver is introduced with the aim to reduce the thermal stresses oftube receiver. The ray–thermal–structural sequential coupled numerical analyses are adopted to obtain the concentrated heat flux distri-butions, temperature distributions and thermal stress fields of both the eccentric and concentric tube receivers. During the sequentialcoupled numerical analyses, the concentrated heat flux distribution on the bottom half periphery of tube receiver is obtained byMonte-Carlo ray tracing method, and the fitting function method is introduced for the calculated heat flux distribution transformationfrom the Monte-Carlo ray tracing model to the CFD analysis model. The temperature distributions and thermal stress fields are obtainedby the CFD and FEA analyses, respectively. The effects of eccentricity and oriented angle variation on the thermal stresses of eccentrictube receiver are also investigated. It is recommended to adopt the eccentric tube receiver with optimum eccentricity and 90� orientedangle as tube receiver for the parabolic trough concentrator system to reduce the thermal stresses.� 2010 Elsevier Ltd. All rights reserved.

Keywords: Eccentric tube receiver; Thermal stress; Fitting function method; Concentrated solar radiation; Ray tracing

1. Introduction

The parabolic trough concentrator with tube receiversystem is extensively employed for solar power generation.The incoming solar radiation is converged on the bottomperiphery of tube receiver by a parabolic trough concentra-tor, and then the concentrated solar radiation is convertedto heat by the heat transfer fluid flowing through the tubereceiver. The tube receiver is enclosed by a glass envelopeto reduce the heat losses to surroundings (Almanza et al.,1997). The tube receivers are designed to operate underextremely nonuniform heat flux, cyclic weather and cloudtransient cycle conditions, which in turn will produce high

0038-092X/$ - see front matter � 2010 Elsevier Ltd. All rights reserved.

doi:10.1016/j.solener.2010.07.005

* Corresponding author. Tel.: +86 451 8641 2308; fax: +86 451 86413208.

E-mail address: tanheping77@yahoo.com.cn (H. Tan).

temperature gradients and large deflection of tube receiver.The high temperature gradients will generate the large ther-mal stresses which may cause the failure of tube receiver,and the deflection of tube receiver will induce the ruptureof glass envelop which will result in the increase of heat loss(Reddy et al., 2008). Therefore, it is necessary to seek somenew approaches to reduce the thermal stresses and deflec-tion of the tube receiver.

Hitherto, mainly three methods have been proposed toreduce the thermal stresses or deflection of receiver:

� Optimizing the size of tube receivers or operationparameters, such as, employing small diameter tubes(Lata et al., 2008), or controlling the fluid flow rate(Verlotski and Flores, 1997).� Receivers with homogenous solar radiation heat flux

distribution on the surface. Generally, these kinds of

1810 F. Wang et al. / Solar Energy 84 (2010) 1809–1815

receivers are designed using ray tracing methods toobtain the isosurface of solar radiation (David et al.,2008; Shuai et al., 2008a,b). At present, the literaturesurvey indicates that the research on receivers withhomogenous solar radiation heat flux distributionremains at the theory stage, and a large amount of man-ufacturing problems wait to solve further.� Compound wall copper–steel receiver. The compound

wall receiver is composed of two parts: the internal tubestratified is made of copper to obtain an excellent heattransfer performance to reduce the temperature gradi-ents, and the external tube stratified is made of steel tostrengthen the intensity of the tube receiver. The com-pound wall copper–steel tube receivers have beenapplied to the Solar Power Plant of the National Uni-versity of Mexico (Flores and Almanza, 2004). Thoughthe compound wall copper–steel receiver can reducethe deflection of tube receiver, it will introduce the con-tact resistance if the two stratifications cannot contactwell and the efficiency of solar radiation absorption willbe affected.

In this study, a new type of tube receiver for the para-bolic trough concentrator system is introduced with theaim to reduce the thermal stresses. The ray–thermal–struc-tural sequential coupled numerical analyses are adopted toobtain the concentrated heat flux distributions, tempera-ture distributions and thermal stress fields of tube receiver.

2. Construction of the new type receiver

2.1. The aim of the new type receiver is to

� Reducing the thermal stresses effectively;� Without adding the mass of tube receiver;� Easy to manufacture.

Bottom half periphery

Top half periphery

rout

r

ϕx

y

Fig. 1. Schematic diagram of physical domain and coordinate system forthe eccentric tube receiver.

2.2. Construction of eccentric tube receiver

To meet the above requirements of the new type recei-ver, the eccentric tube receiver for parabolic trough concen-trator system is introduced.

Fig. 1 shows the diagram of the eccentric tube receiver.The eccentric tube receiver is proposed on the basis of con-centric tube receiver. As seen from this figure, the center ofinternal cylinder surface of concentric tube receiver ismoved upward (or other directions), which is not locatedat the same coordinate position with the center of externalcylinder surface. Therefore, the wall thickness of the bot-tom half section of tube receiver will increase without add-ing any mass to the entire tube receiver. With the sameboundary conditions for numerical analyses, the increaseof wall thickness will not only strengthen the intensity toenhance the resistance of thermal stress, but also canincrease the thermal capacity, which in turn will be benefitto alleviate the extremely nonuniform temperature distri-bution situation.

As seen from Fig. 1, the origin of coordinate system isplaced at the center of the external cylinder surface. In thisstudy, the vector eccentric radius~r (the origin of coordinatesystem points to the center of the internal cylinder surface);the vector eccentricity~e (the projection of vector~r on the y-axis); and the oriented angle / (the angle between the vec-tor~r and the x-axis) are introduced to describe the shape ofeccentric tube receiver (Manglik and Fang, 1995). The ray–thermal–structural sequential coupled numerical analysesare adopted to obtain the concentrated heat flux distribu-tions, temperature distributions and thermal stress fieldsof both the eccentric and concentric tube receivers. Theeffects of eccentricity and oriented angle variation on thethermal stresses of eccentric tube receiver are also investi-gated in this study.

3. Methodology

At the first step, the concentrated solar radiation heatflux distribution qc on the bottom half periphery of tubereceiver, which is used as the input data for the CFD anal-yses, will be calculated by the solar concentration systemprogram with the Monte-Carlo ray tracing method devel-oped in Harbin Institute of Technology (Shuai et al.,2008a,b). The Monte-Carlo ray tracing method is a power-ful tool for performing radiative equilibrium calculations,even in complex geometries. Due to the Monte-Carlomethod is a stochastic technique and does not have thetruncation error as the other numerical methods of discretedifferential–integral equation, the solutions of the Monte-Carlo method are generally used to be the reference datumof the other numerical methods (Tan et al., 2006). The ther-mal model proposed for the solar parabolic concentratorwith tube receiver system is illustrated in Fig. 2. The geo-metrical parameters of the parabolic trough concentratorand tube receiver for this study are illustrated in Table 1.As seen from this table, the transmissivity of the glass

Fluid Outlet

Sun

Fluid Inlet

x

y

z

Fluid Inlet Fluid Outlet

External Cylinder Surface

Internal Cylinder Surface

Fig. 2. Schematics diagram of the tube receiver with solar parabolictrough system.

F. Wang et al. / Solar Energy 84 (2010) 1809–1815 1811

envelop is highly close to 1 and the thickness of glassenvelop is very thin, therefore, the values and distributionof heat flux are impacted very slightly when passingthrough the glass envelop. Hence, the impact of the glassenvelop in this investigation is neglected. During the heatflux distribution calculation process, the external cylindersurface of tube receiver will be discretized to 300 nodesalong the circumference and 300 nodes along the tubelength direction. Therefore, the solar concentration systemprogram will obtain 300 � 300 heat flux values on the dis-crete nodes. No optical errors or tracking errors were con-sidered for the solar concentration system program, andthe calculation conditions are: the non-parallelism angleof sunlight is 160 and the solar radiation flow is 1000 W/m2 (Hasuike et al., 2006), the sunshape is taken to be a dis-tribution with a circumsolar-ratio of 0.05 and a limb dark-ening parameter of 0.8 (Shuai et al., 2008a,b).

At the second step, the concentrated heat flux distribu-tion calculated by the Monte-Carlo ray tracing method will

Table 1Geometrical parameters of the parabolic trough concentrator and tubereceiver.

Parabolic trough concentrator and tube receiver Value

Focal length of parabolic trough concentrator 2000 mmLength of parabolic trough concentrator 2000 mmOpening radius of parabolic trough concentrator 500 mmHeight of parabolic trough concentrator 1500 mmOuter diameter of tube receiver (rout) 70 mmInner diameter of tube receiver (rin) 60 mmGlass cover diameter 100 mmLength of tube receiver 2000 mmReflectivity of parabolic trough collector 0.95Absorptivity of tube receiver 0.9Transmissivity of glass over 0.965

be employed as input data for the CFD analyses by meansof using the boundary condition function in Ansys soft-ware. In Steven’s study (Steven and Macosko, 1999), thereceiver is divided into 16 sections, and the average solarradiation heat flux of each section is calculated. The aver-age heat flux is used as boundary condition for each corre-sponding section in the thermal analysis model. Thismethod is fairly straightforward and simple, but the devia-tions generated during the heat flux transformation processare enormous. In this study, the fitting function method isintroduced for the calculated heat flux distribution trans-formation from the Monte-Carlo ray tracing model tothe CFD analysis model. The solar radiation heat flux dis-tribution calculated by the Monte-Carlo ray tracingmethod along the bottom half periphery of tube receiverwill be divided into several sections, and the heat flux dis-tribution of each section will be fitted by a polynomialregression function with highly fitted precision. The calcu-lated heat flux distribution on the bottom half periphery oftube receiver is shown in Fig. 3. Six polynomial regressionfunctions are employed as the fitted functions and illus-trated as follows:

q ¼ 12 x 2 ½�35;�17:82�q ¼ 13740:23þ 770556:99� x x 2 ½�17:82;�16:54�q ¼ 43418:96þ 2:57� x x 2 ½�16:54; 0�q ¼ 43418:96� 2:57� x x 2 ½0; 16:54�q ¼ 13740:23� 770556:99� x x 2 ½16:54; 17:82�q ¼ 12 x 2 ½17:82; 35�

8>>>>>>>><>>>>>>>>:

ð1Þ

The six fitted function curves are also drawn in Fig. 3. Asseen from this figure, the fitted function curves can matchthe calculated heat flux distribution well with highprecision.

At the third step, the CFD analyses will obtain the tem-perature distributions. Thermal oil (Syltherm 800) andstainless steel are used as the heat transfer fluid and the

-30 -20 -10 0 10 20 30

0

10000

20000

30000

40000

Hea

t F

lux

W/m

2

X mm

Fitted Curves Calculated

Fig. 3. Calculated heat flux distribution on the bottom half periphery oftube receiver and the fitted function curves.

Table 2Thermal-physical properties of heat transfer fluid and tube receiver.

Property Fluid Tube receiverThermal oil Stainless steel

Density (kg m�3) 881.68 7900Specific heat (J kg�1 K�1) 1711 500Viscosity (10�6 Pa s) 3.86 –Thermal conductivity (W m�1 K�1) 0.1237 25Poisson ratio – 0.25Young’s modulus (GPa) – 220Thermal expansion coefficient (10�6 K�1) – 17.2

0 60 120 180 240 300 360320

340

360

380

400

420

Tem

pera

ture

K

θ o

Concentric Eccentric

θ

Fig. 4. Temperature profiles along the internal circumference at the outletsection for both the concentric and eccentric tube receivers.

1812 F. Wang et al. / Solar Energy 84 (2010) 1809–1815

material of tube receiver, respectively (Kumar and Reddy,2009). The thermal-physical properties of the thermal oiland stainless steel are presented in Table 2. As the keypoint of this paper is to put forward a new type of tubereceiver, the comparisons between the new type receiverand concentric tube receiver are carried out under the sameboundary condition with the hypothesis that there are noair bubbles in the flow. The boundary conditions appliedon both the eccentric and concentric tube receivers areillustrated as follows:

� The flow has a uniform velocity u = 0.15 m/s at theatmosphere temperature at the tube receiver inlet;� The top half periphery of tube receiver is subjected to a

uniform heat flux distribution which is the sun averageradiation in the air (the value is 1000 W/m2);� The bottom half periphery of tube receiver is subjected

to the concentrated heat flux distribution calculated bythe Monte-Carlo ray tracing method which is fitted bysix polynomial regression functions;� Zero pressure gradient condition is employed across the

fluid outlet boundary.

At the forth step, the finite element analysis (FEA) willobtain the Von-Mises thermal stress fields, which is a syn-thesis stress of radial stress, axial stress and circumferentialstress. The governing thermal stress equations for hollowcylinders (Fauple and Fisher, 1981) are expressed asfollows:

rz ¼E � a

ð1� mÞ � r2� 2

r20 � r2

i�Z ro

ri

T ðrÞ � r � dr � T ðrÞ� �

ð2Þ

rr ¼E � a

ð1� mÞ � r2

� r2 � r2i

r20 � r2

i�Z ro

ri

T ðrÞ � r � dr �Z r

ri

T ðrÞ � r � dr� �

ð3Þ

rh ¼E � a

ð1� mÞ � r2

� r2 þ r2i

r20 � r2

i�Z ro

ri

T ðrÞ � r � dr þZ r

ri

T ðrÞ � r � dr � T ðrÞ � r2

� �

ð4Þ

rvon ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

z þ r2r þ r2

h � ðrzrr þ rrrh þ rhrzÞq

ð5Þ

where rr, rz, rh and rvon are the radial stress, axial stress,circumferential stress and Von-Mises stress, respectively.The resulted temperature fields defined at the nodes ofCFD analysis meshes are interpolated as input data tothe nodes of the thermal stress analysis meshes. This sim-ulation approach is fairly straightforward and has beenadopted by many investigators (Steven and Macosko,1999; Qin et al., 2004; Jae and Allan, 2006; Friedrichet al., 2008; Knaus et al., 2005; Wetzel et al., 2007).The validation of this simulation approach have been de-scribed in Friedrich et al. (2008), Knaus et al. (2005),Wetzel et al. (2007), and the comparisons between thesimulation results and the experimentations reveals a highlevel of compliance. Compared to meshes of the CFDanalysis, a much finer solid part meshes are used for theFEA analysis to produce a reasonably accurate degreesof freedom solution.

4. Results and discussion

4.1. Comparison between the concentric and eccentric tube

receiver

The eccentric tube receiver with the center of internalcylinder surface 3 mm moved upward along the y-axis(the magnitude of vector eccentricity~r is 3 mm, and the ori-ented angle / is 90�) is chosen for the comparison research.The temperature distributions and thermal stress fields ofeccentric tube receiver are compared with those of concen-tric tube receiver under the same boundary conditions andmaterial physical properties.

Fig. 4 shows the temperature distributions along theinternal circumference at the outlet section for both theconcentric and eccentric tube receivers. As seen from thisfigure, the concentric tube receiver has a higher value ofpeak temperature which is about 11 �C higher than thatof eccentric tube receiver. Along the bottom half internal

-3 -2 -1 0 1 2 3

80

120

160

200

240

280

320

The

rmal

Str

ess

(M

Pa)

Eccentricity ε

Concentric Eccentric

ε

Fig. 6. Relationship between eccentricity variation and peak thermalstress values.

F. Wang et al. / Solar Energy 84 (2010) 1809–1815 1813

circumference (the h is between 180� and 360�) where thepeak temperatures of both the concentric and eccentrictube receivers are found, the temperature gradients of con-centric tube receiver are higher than those of eccentric tubereceiver which can lead to the higher thermal stresses Ifranand Chapman (2009, 2010), the cause of this phenomenonshould be attributed to the thermal capacity increase on thebottom section of tube receiver due to the wall thicknessincrease on this section.

The Von-Mises thermal stress fields along the internalcircumference at the outlet section for both the concentricand eccentric tube receivers are presented in Fig. 5. Thepeak thermal stress values of the two profiles are bothfound at h = 270� where the peak temperature values arealso located at. Attributed to the lower temperature gradi-ents and intensity strengthen on the bottom half section oftube receiver, the peak thermal stress value of the eccentrictube receiver which is only 80.7 MPa is much lower com-pared to that of the concentric tube receiver which is137.01 MPa. Therefore, adopting eccentric tube receiveras the tube receiver for the parabolic trough concentratorsystem can reduce the Von-Mises thermal stresses effec-tively up to 41.1%, which means the eccentric tube receivercan meet the requirements of the new type receiver.

4.2. Effect of the eccentricity variation on the thermal

stresses

The way how the variation of eccentricity affecting thethermal stresses of eccentric tube receiver is investigatedwith the objective to give instructions to the designer ofeccentric tube receiver.

Fig. 6 presents the relationship between the eccentric-ity variation and the peak Von-Mises thermal stress val-ues of tube receiver, while keeps the center of internalcylinder surface locating at the y-axis. As seen from thisfigure, not all the eccentric tube receivers can reduce the

0 60 120 180 240 300 3600

40

80

120

160

Eff

ecti

ve S

tres

sM

Pa

θ o

Concentric Eccentric

θ

Fig. 5. Thermal stress profiles along the internal circumference at theoutlet section for both the concentric and eccentric tube receivers.

thermal stresses of tube receiver. The peak thermal stressvalues of tube receivers are increased significantly whenthe direction of vector eccentricity switches from positiveto negative (the center of internal cylinder surface ismoved downward). The thermal stress reduction of tubereceiver only occurs at the positive direction of vectoreccentricity ~r when adopting eccentric tube receiver asthe tube receiver for parabolic trough concentrator sys-tem, and the larger magnitude of vector eccentricity is,the greater reduction of thermal stresses is. However,the manufacturing cost will increase with the magnitudeof vector eccentricity increasing. The eccentric tube recei-ver should employ an optimum magnitude of eccentricityto minimize the thermal stresses and prevent the failureof tube receiver while not incurring excessive manufac-turing cost.

-90 -60 -30 0 30 60 90

80

120

160

200

240

280

320

The

rmal

Str

ess

(M

Pa)

Oriented Angle ϕ

Concentric Eccentric

ϕ

Fig. 7. Relationship between oriented angle variation and peak thermalstress values.

1814 F. Wang et al. / Solar Energy 84 (2010) 1809–1815

4.3. Effect of the oriented angle variation on the thermal

stresses

The effect of the oriented angle variation on the Von-Mises thermal stresses of eccentric tube receiver is also per-formed with a constant eccentric radius value (r = 3 mm).The eccentric tube receiver has symmetric property withthe variation of oriented angle, therefore, instead of dealingwith the whole oriented angle range, analyses of half rangeof the oriented angle can yield complete results. As seenfrom Fig. 7, the variation of oriented angle has a bigimpact on the thermal stresses of eccentric tube receiver.The peak thermal stress value of eccentric tube receiver isless than that of the concentric tube receiver only whenthe oriented angle varies between 0� and 90�. The eccentrictube receiver with 90� oriented angle has the lowest peakthermal stress value.

Therefore, it is recommended to employ eccentric tubewith 90� oriented angle and optimum eccentricity as thetube receiver for parabolic trough concentrator system toreduce the thermal stresses.

5. Conclusion

Aiming at reducing the thermal stresses of tube recei-ver, the eccentric tube receiver is introduced in this inves-tigation. The ray–thermal–structural sequential couplednumerical analyses are adopted to obtain the concen-trated heat flux distributions, temperature distributionsand thermal stress fields of both the eccentric and con-centric tube receivers. The fitting function method isintroduced for the calculated heat flux distribution trans-formation from the Monte-Carlo ray tracing model tothe CFD analysis model, and the fitted function curvescan match the calculated heat flux distribution well withhigh precision.

The sequential coupled numerical analysis results indi-cate that adopting eccentric tube as the tube receiver forparabolic trough concentrator system can reduce theVon-Mises thermal stress effectively up to 41.1%. Theway how the variation of eccentricity and oriented angleaffect the thermal stresses of eccentric tube receiver arealso investigated with the objective to give instructionsto the designer of eccentric tube receiver. When adoptingeccentric tube receiver, the thermal stress reduction oftube receiver only occurs at the positive direction of vec-tor eccentricity, and the larger magnitude of vector eccen-tricity is, the greater reduction of thermal stresses is. Theoriented angle has a big impact on the thermal stresses ofeccentric tube receiver. The thermal stress reduction oftube receiver only occurs when the oriented angle isbetween 0� and 90�.

Therefore, employing eccentric tube receiver with opti-mum eccentricity and oriented angle for parabolic troughconcentrator system can reduce the thermal stress andenhance the reliability of tube receiver effectively.

Acknowledgements

This work was supported by the National Key Basic Re-search Special Foundation of China (No. 2009CB220006),the key program of the National Natural Science Founda-tion of China (Grant No. 50930007) and the National Nat-ural Science Foundation of China (Grant No. 50806017).A very special acknowledgement is made to the editorsand referees whose constructive criticism has improved thispaper.

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