Materials and Design 33 (2012) 284–291

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Materials and Design

journal homepage: www.elsevier .com/locate /matdes

Effects of material selection on the thermal stresses of tube receiver underconcentrated solar irradiation

Fuqiang Wang a,⇑, Yong Shuai a,b, Yuan Yuan a, Bin Liu a

a School of Energy Science and Engineering, Harbin Institute of Technology, 92, West Dazhi Street, Harbin 150001, PR Chinab Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0325, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 30 June 2011Accepted 20 July 2011Available online 26 July 2011

Keywords:G. Thermal analysisH. Material selection chartsH. Failure analysis

0261-3069/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.matdes.2011.07.048

⇑ Corresponding author. Tel.: +86 451 8641 2308; fE-mail addresses: Wangfq@hit.edu.cn, W_fuqiang@

Material selection of tube receiver is a critical issue to ensure the reliability during its whole service life.In this study, the thermal stress analyses of tube receiver under concentrated solar irradiation conditionusing various materials are carried out. The concentrated solar irradiation heat flux distribution isobtained by Monte-Carlo ray tracing method and used as boundary conditions of the Computational FluidDynamics (CFD) analysis. The CFD analysis will solve the temperature fields and the resulted temperaturefields defined at the nodes of CFD analysis meshes are interpolated as input data to the nodes of the ther-mal stress analysis meshes. The temperature fields, thermal stress fields of various material conditionsare obtained in the numerical analysis. The numerical results show that the temperature gradients andeffective stresses of the stainless steel and silicon carbide (SiC) conditions are significantly higher thanthat of the aluminum and copper conditions. The stress failure ratio is introduced to assess the thermalstress level of each material condition. The stainless steel condition has the highest stress failure ratio andthe copper condition has the lowest stress failure ratio. From the standpoint of thermal stress, copper isrecommended to be the material of tube receiver.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The parabolic trough concentrator with tube receiver is one ofthe most suitable systems for solar power generation. The para-bolic trough concentrator concentrates the incoming solar irradia-tion on the periphery of the tube receiver. The tube receiverconverts the concentrated solar irradiation into heat which trans-fers heat to the circulating fluid through convection. As designedto operate with concentrated heat fluxes, the tube receiver willbe subjected to the high thermal stresses which may cause the fail-ure of glass envelope and receivers [1]. For example: in the SolarPower Plant of the National University of Mexico (SPNUM), thestainless steel tube receiver with parabolic troughs have occurredfrequently deflection and glass envelope rupture during experi-mental test and application [2–4]. Greater understanding of thetemperature distributions and thermal stress fields of tube receiverusing various materials by numerical methods can give very usefulinstructions for application [5].

Numerous studies have been carried out to investigate the tem-perature distributions and thermal stress fields of tubes and receiv-ers with various material conditions. A numerical analysis hadbeen conducted by Chen and Liu [6] to study the effect of using

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porous material for the receiver on temperature distributions.Experiments were conducted by Fend [7] to research the tempera-ture distributions on the volumetric receivers used two novel por-ous materials. A finite element analysis was conducted byIslamoglu [8] to study the temperature distribution and the ther-mal stress fields on the tube heat exchanger using the SiC material.To reduce the thermal stresses, Agrafiotis et al. [9] employed por-ous monolithic multi-channeled SiC honeycombs as the materialfor an open volumetric receiver. Low cycle fatigue test of the recei-ver materials was conducted at different temperatures by Lataet al. [10], the results showed that the high nickel alloys had excel-lent thermo-mechanical properties compared to the austeniticstainless steel. Almanza and Flores [4] have proposed a bimetallicCu–Fe type receiver, and the experimental test results showed that,when operated at low pressure, the bimetallic Cu–Fe type receiverhad a lower thermal gradient and less thermal stress strain thanthe steel receiver. In the SPNUM, the stainless steel tube receiverfor parabolic trough concentrator deflected as a wave during theexperimental tests to examine the receiver behavior. When a cop-per tube receiver was used instead of a steel one, no appreciablebending of the pipe was observed during application [3].

The literature survey shows that few research papers have beenpublished on the thermal stress field’s analysis of tube receiverunder concentrated solar irradiation condition using various mate-rials. In this study, four different materials (stainless steel, alumi-num, copper, SiC) are employed for the numerical analysis. The

Nomenclature

Ai the area of ith surface (m2)Ap the area of concentrator (m2)Ar the area of receiver (m2)Cr concentration ratiodin inner diameter of receiver (m)dout outer diameter of receiver (m)E modulus of elasticity (GPa)Esun;Dkk

sun average spectral irradiance at the spectral band Dkk

Fc failure coefficient (%)L length of tube receiver (m)Mb total spectral bandsNi total bundlesns number of bundle samplingsqb heat flux on bottom surface (W/m2)qr,j concentrated heat flux (W/m2)qs solar irradiation heat flux (W/m2)qt heat flux on top surface (W/m2)r radius (m)

ri inner radius of tube receiverro outer radius of tube receiverRr random numberRDij radiative exchange factorTf fluid temperature (K)Tin fluid inlet temperature (K)

Greek Symbolsa thermal expansion coefficienth angle (degree)l fluid velocity (m/s)lin inlet fluid velocity (m/s)m Poisson’s ratioreff effective stress (MPa)rr radial stress (MPa)rz axial stress (MPa)rh tangential stress (MPa)Dkk spectral band

Parabolic trough concentrator

Tube Receiver

xy

qs

qr,

ri

Tube Receiver Fluid inlet

ro

j

θ

Fig. 1. Schematics of the tube receiver with solar parabolic trough system.

F. Wang et al. / Materials and Design 33 (2012) 284–291 285

concentrated solar irradiation heat flux distribution is obtained byMonte-Carlo ray tracing method and is used as boundary conditionsfor the steady state heat transfer analysis. The heat transfer analysiswill solve the temperature distributions and the resulted tempera-ture distributions defined at the nodes of the CFD meshes are inter-polated as input data to the nodes in the thermal stress analysismeshes. The temperature gradients, thermal stress fields and stressfailure ratios of different material conditions are obtained and com-pared to give suggestions on material selection.

2. Methodology

A thermal model proposed for the tube receiver with solar par-abolic trough concentrator system is shown in Fig. 1. As seen fromthis figure, the incoming solar irradiation is concentrated on thebottom surface of the tube receiver by the parabolic trough con-centrator. Due to the highly concentrated solar irradiation, the tubereceiver may be subjected to considerable degree of thermal stress.

2.1. The calculation of heat flux distribution

A Monte-Carlo ray tracing computational code, which is basedon the radiative exchange factor (REF) theory, is developed to pre-dict the heat flux distribution on the bottom periphery of the tubereceiver. The REF RDij is defined as the fraction of the emissivepower absorbed by the jth element in the overall power emittedby the ith element. The jth element can absorb the emissive powerwithin the system by means of direct radiation, direct reflectionand multiple reflections. The values of the RDij are determined byboth the geometrical and radiative characteristics of the computa-tional elements.The REF within the spectral band Dkk

(k = 1, 2, ...., Mb) can be expressed as follows:

RDi;j;Dkk¼ Ni;j=Ni ð1Þ

where Ni is the total bundles emitted by the ith element, Ni,j is thebundles absorbed by the jth element, and Mb is the total spectralbands of the wavelength-dependent radiation characteristics ofthe surface. As shown in Fig. 1, the concentrated heat flux distribu-tion on the bottom surface of the tube receiver can be expressed asfollows:

qr;j ¼Ai

Aj

XMb

k¼1

RDi;j;DkkEsun;Dkk

ð2Þ

where qr,j is the heat flux of the jth surface element of the tube re-ceiver, Ai is the area of the imaginary emission surface, Aj is the areaof the jth surface element of the tube receiver, and Esun;Dkk

is the sunaverage spectral irradiance within the spectral band Dkk. Moreinformation about the Monte-Carlo ray tracing method can befound in Ref. [11].

The geometrical parameters of the parabolic trough concentra-tor and the tube receiver for this study are chosen from Ref. [12]and the parameters are illustrated in Table 1. In order to increasethe absorption of solar radiation and minimize radiation heat loss,

Table 1Geometrical parameters of the parabolic trough concentrator and tube receiver.

Focal length of parabolic concentrator (m) 2Length of parabolic concentrator (m) 2Aperture of parabolic concentrator (m) 0.6Outer diameter of tube receiver (m) 0.07Inner diameter of tube receiver (m) 0.06Length of tube receiver (m) 2Reflectivity of parabolic trough concentrator 0.95Absorptivity of tube receiver 0.95Emissivity of tube receiver 0.13Diameter of glass envelop 0.1Transmissivity of glass envelop 0.965

286 F. Wang et al. / Materials and Design 33 (2012) 284–291

the tube receiver is coated with selective absorptive layer which isblack chromium. The specula absorptivity of tube receiver over thesolar spectrum (0.3–3 lm, 97% energy of solar energy is in thiswavelength range) is 0.95 and the emissivity (wavelength >3 lm)of tube receiver at the working temperature is 0.13 [3]. Besides,the glass envelop are also coated with selective layer to increasethe transmittance of solar radiation and minimize radiation heatloss. Due to the low emissivity of tube receiver at working temper-ature and the selective layer of glass envelop, the radiation heatloss at working temperature (wavelength >3 lm) is very smalland not considered in this research. As seen in Table 1, the trans-missivity of the glass envelop is highly close to 1 and the thicknessof glass envelop is very thin, therefore, the values and distributionof heat flux are impacted very slightly when passing through theglass envelop. Hence, the impact of glass envelop on concentratedheat flux distribution is neglected in this investigation.

As the key point of this research is to stress the impact of usingdifferent materials on thermal stress under the same boundarycondition and different solar power plants have different trackingerrors, the tube receiver with no tracking errors is adopted to sim-plify the numerical loads and highlight the thermal stress distribu-tion with different materials. For more details about the effects oftracking errors, sun shape, surface slope error, rim angle and posi-tion of focal plane on heat flux distribution, please refer to the pre-vious researches studied by the author [11].

The heat flux distribution on the bottom surface of tube receivercalculated by Monte-Carlo ray tracing method is presented inFig. 2. As seen from this figure, the heat flux distribution alongthe bottom half periphery is highly uneven. Using the boundarycondition function in the Ansys software, the heat flux distributionshown in Fig. 2 will be adopted as boundary conditions for the CFDanalysis.

0

1

2

3

4

5

0

30

60

90

120

150

180

210

240

270

300

330

0

1

2

3

4

5

Hea

t F

lux

(104

W/m

2 )

Fig. 2. Heat flux distribution on the bottom surface of tube receiver.

2.2. Boundary conditions

For the numerical analysis, the flow is considered as hydro-dynamically developed and thermally developing [12]. The proper-ties of the heat transfer fluid (thermal oil) and the four differentmaterials of tube receiver used for the analysis are assumed tobe constant. Tables 2 and 3 show the thermal–physical and struc-tural properties of thermal oil and the four different materialsrespectively. The steady state heat transfer analysis performed byAnsys 11.0 software is conducted to calculate the temperature dis-tributions. Two phenomena have been considered in the heattransfer analysis: the conduction inside the tube receiver walland the convection from the inner surface of the tube receiver tothe fluid flowing inside it. The analysis is based on steady state,three dimensional continuity, momentum equation and energyequation. The boundary conditions applied for the numerical anal-ysis are expressed as follows:

Fluid inlet boundary condition:The flow has the uniform velocity at the fluid inlet boundary

l ¼ lin; Tf ¼ Tin ¼ 300 K at L ¼ 0; 0 6 r 6 ri;

0� 6 h 6 360�

Wall boundary condition:No-slip conditions exist at the inside surface of the tube wall

l ¼ 0; at r ¼ ri; 0� 6 h 6 360�; 0 6 L 6 2

The top half periphery of the receiver is subjected to the uni-form heat flux

qt ¼ qs; at r ¼ ro; 0� 6 h 6 180�; 0 6 L 6 2

The bottom half periphery of the receiver is subjected to theconcentrated solar irradiation

qb ¼ qr;j; at r ¼ ro; 180� 6 h 6 360�; 0 6 L 6 2

Zero pressure gradient condition is employed across the fluidoutlet boundary.

2.3. Numerical procedure

In the CFD analysis, the steady state heat transfer analysis withconcentrated solar irradiation had been conducted to investigatethe temperature distributions. The Ansys 11.0 software was usedto solve the governing equations. The standard turbulent Re-Nor-malization Group (RNG) k–e model was used for the simulationsof forced convection inside the receiver. The standard RNG k–emodel was expected to yield improved prediction of near wallflows, separated flows and flows in curved geometries such astubular and intricate surfaces [13]. To solve the temperature equa-tions, the Preconditioned Generalized Minimum Residual (PGMR)method was selected as it was the recommended solver for ill-con-ditioned heat transfer problems. The enhanced Semi-ImplicitMethod for Pressure Linked Equations (SIMPLEN) scheme was usedto handle the coupling between the pressure and momentumequations. The SIMPLEN algorithm was a fast and robust variant

Table 2Thermal–physical properties of working fluid.

Property FluidThermal oil

Density (kg m�3) 938Specific heat (J kg�1 K�1) 1970Viscosity (10�6 Pa s) 15.3Thermal conductivity (W m�1 K�1) 0.118

Table 3Thermal–physical and structural properties of the four different materials.

Property Materials of tube receiver

Stainless steel Aluminum Copper SiC

Density (kg m�3) 7900 2698 8930 3210Specific heat (J kg�1 K�1) 500 879 386 2540Thermal conductivity (W m�1 K�1) 48 247 384 42Young’s modulus (Gpa) 220 70 128 427Poisson ratio 0.25 0.32 0.31 0.17Thermal expansion coefficient (10�6 �C�1) 17.2 23.6 17.1 4.8Tensile strength (MPa) 450 130 270 400

F. Wang et al. / Materials and Design 33 (2012) 284–291 287

algorithm for finite-element simulations of incompressible flows,which can improve the rate of convergence significantly if pres-sure–velocity coupling dominated the convergence [14]. A conver-gence criterion of 10�3 was imposed on the residuals of thecontinuity equation. Similarly, for the momentum equation, a con-vergence criterion of 10�3 was set for the energy equation and aconvergence criterion of 10�6 was set for the energy equation.

In the thermal stress analysis, the resulted temperature distri-butions obtained in the CFD analysis were interpolated to thenodes of thermal-stress analysis meshes; this method was definedas submodeling method. Because no recirculation of temperaturewas necessary, the submodeling method was a fairly straightfor-ward method for thermal stress analysis. In this study, the receiveris unrestrained and freely-expanded. Therefore, the thermal stressis caused by temperature gradients and internal (geometric) con-straints, like bends in the tube ends which are very prominent dur-ing tube receiver application.

The governing thermal stress equations for hollow cylinders[15] are expressed as follows:

rz ¼E � a

ð1� mÞ � r2 �2

r20 � r2

i

�Z ro

ri

TðrÞ � r � dr � TðrÞ" #

ð3Þ

rr ¼E � a

ð1� mÞ � r2 �r2 � r2

i

r20 � r2

i

�Z ro

ri

TðrÞ � r � dr �Z r

ri

TðrÞ � r � dr

" #ð4Þ

rh ¼E � a

ð1� mÞ � r2

� r2 þ r2i

r20 � r2

i

�Z ro

ri

TðrÞ � r � dr þZ r

ri

TðrÞ � r � dr � TðrÞ � r2

" #ð5Þ

According to the Von-Mises theory, the effective stress equation[15] is expressed as follows:

reff ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

z þ r2r þ r2

h � ðrzrr þ rrrh þ rhrzÞq

ð6Þ

The details of the computational meshes are presented in Fig. 3.All of the meshes were generated with O-grid method by AnsysWorkbench. The grid independent studies were carried out beforethe actual CFD and thermal stress analysis. After the studies, it wasfound that 24 000 mesh elements in solid part and 62 000 meshelements in fluid part were sufficient with minimum deviationfor the CFD analysis. For the thermal stress analysis, finer solidparts with 123 280 mesh elements were used to produce a reason-ably accurate degree of freedom solution.

3. Model validation

The literature survey shows that little numerical investigationshave been performed on the thermal stress analysis of tube recei-ver using Monte-Carlo ray tracing and FEM (Ansys software)

combined method, therefore, the sequential validation steps areadopted in this research.

The Monte-Carlo ray tracing method codes are validated againsttwo cases that have been investigated by independent researchers.The first validation case is the comparison of dimensionless heatflux distribution between numerical results calculated by theMonte-Carlo ray tracing method codes and experimental resultsperformed by Johnston [16]. The second validation case is the com-parison of heat flux distributions on the focal plane for ideal parab-oloidal dish concentrator between the Monte-Carlo ray tracingmethod codes and the analytical results investigated by Jeter[17]. The comparison results of the two cases are shown in Figs.4 and 5 respectively. As seen from these two figures, the resultscalculated by the Monte-Carlo ray tracing method codes are ingood agreement with both the experimental results conductedby Johnston [16] and the analytical results performed by Jeter [17].

In order to validate the thermal stress analysis method, thenumerical results is compared with that calculated by Islamoglu[8] under the same conditions. The validation case is the thermalstress analysis of ceramic tube heat exchanger under both axialuniform and non-uniform convective heat transfer coefficientboundary conditions. The inner surface of ceramic tube receiveris under convective heat transfer boundary conditions and the out-er surface of ceramic tube receiver is under a uniform heat flux dis-tribution. The schematic model and the coordinate system for thisvalidation case are illustrated in Fig. 6. The steady state radialstress variation on the inner surface of tube receiver along thelength of computational region is shown in Fig. 7. As seen from thisfigure, the numerical results obtained in this work are in goodagreements with those obtained by Islamoglu under both the axialuniform and non-uniform convective heat transfer coefficientsboundary conditions.

It should be noted that, the method for thermal stress analysisperformed in Ref. [8] is coupled-field element method of Ansyssoftware and the method adopted for validation is submodelingmethod. The advantage of coupled-field element method is thatthis method can solve the energy and structural equations simulta-neously and no need to create different meshing models for ther-mal analysis and structural analysis respectively. However, thecoupled-field element is not suitable for thermal stress analysiswith heat transfer (for example: fluid flow in tube) as the fluidhas no stiffness for structural analysis. Therefore, the submodelingmethod is adopted in this research for thermal stress analysis. Thevalidation of the submodeling method have been conducted byseveral investigators and the comparisons between the simulationresults and the experimental results reveal high level of compli-ance [18–20].

4. Results and discussion

Fig. 8 shows the temperature profiles across the circumferenceon the tube inner surface at the tube outlet section. For all the fourdifferent material conditions, the temperature distributions varied

(a) Meshes for CFD analysis (b) Finer meshes for thermal stress analysis

Fig. 3. Computational meshes for the CFD and thermal stress analysis.

-6 -4 -2 0 2 4 6

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Dim

ensi

onle

ss H

eat

Flu

x

Angular Distance (mrad)

Calculated in this paper Measured in Ref. [16]

Fig. 4. Comparison of dimensionless heat flux distribution between Monte-Carloray tracing method and experimental data.

-0.02 -0.01 0.00 0.01 0.02

0

1x104

2x104

3x104

4x104

Hea

t Flu

x (k

W/m

2 )

X dimension (m)

Ref. [17] This paper

Fig. 5. Comparison of heat flux distributions on the focal plane for ideal solar dishconcentrator with Ref. [17] results.

Heat flux

Non-uniform convective heat

transfer coefficient profile

FEM

y

o x

Fig. 6. Schematic model and the coordinate system for validation of thermal stressanalysis method.

0 1 2 3 4

-200

0

200

400

600

y (mm)

Non-uniform, Ref. [8]Non-uniform, This PaperUniform, Ref. [8]Uniform, This paper

Rad

ial S

tres

s (k

Pa)

Fig. 7. Comparison of radial stress variation on the inner surface of tube receiveralong the length of computational region with Ref. [8] results.

288 F. Wang et al. / Materials and Design 33 (2012) 284–291

with angle h exhibit as egg-shaped profiles, and the peak temper-atures of all the four profiles are found at h = 270�. The lowerconductivity of the material is; the larger aspect ratio of the egg-shaped profile and temperature gradient is, which can cause higherthermal stress. Among the four different material conditions, theSiC condition has the highest maximum temperature.

Fig. 9 shows the effective stress profiles across the circumfer-ence on the tube inner surface at the tube outlet section. The effec-tive stress profiles varied with angle h exhibit as crisscross shapefor all the four different material conditions. For the effective stressprofiles of the stainless steel and SiC conditions, the values of bot-tom sections of the effective stress profiles are much higher thanthe values of top sections, and the values of left section and rightsection are symmetrical. However, for the effective stress profiles

320

340

360

380

400

0

30

60

90

120

150

180

210

240

270

300

330

320

340

360

380

400

Aluminum

SiC Copper

Stainless Steel

Tem

pera

ture

(K

)

Fig. 8. Temperature profiles across the circumference on the tube inner surface atthe tube outlet section.

0

20

40

60

0

30

60

90

120

150

180

210

240

270

300

330

0

20

40

60

Aluminum

SiC Copper

Stainless Steel

Eff

ecti

ve S

tres

s (M

Pa)

Fig. 9. Effective stress profiles across the circumference on the tube inner surface atthe tube outlet section.

0.0 0.5 1.0 1.5 2.0

0

1

2

3

Rad

ial S

tres

s (M

Pa)

Z (m)

Stainless steelAluminumCopperSiC

Fig. 10. Radial stress profiles on the tube inner surface along the length direction ath = 270�.

F. Wang et al. / Materials and Design 33 (2012) 284–291 289

of the copper and aluminum conditions, the values of the bottomsections are close to the values of the top sections. These phenom-enon can be explained by Ifran and Chapmans’ research resultsthat temperature differences of a radiant tube are determinativein the generation of thermal stress [21]: for the stainless steeland SiC conditions, the temperature profile fluctuations at the bot-tom section (shown in Fig. 8) are much more drastic than that atthe top section, and the big temperature fluctuation difference be-tween the bottom section and top section cause the big thermalstress difference; however, the temperature profiles for the copperand aluminum conditions tend to degenerating from egg-shapedprofiles to rotund profiles due to the high conductivity, and the

small temperature fluctuation differences between the bottom sec-tion and top section induce the small thermal stress differences.Among the four different material conditions, the stainless steelcondition has the highest maximum effective stress which is67.5 MPa and the copper condition has the lowest maximum effec-tive stress which is only 5.0 MPa.

The numerical result shows that the maximum effective stres-ses are found at the circumference on the tube inner surface atthe tube outlet section at h = 270�. For all the four different mate-rial conditions. Fig. 10 shows the radial stress profiles for the fourdifferent material conditions on the tube inner surface along thelength direction at h = 270�. For all the four different material con-ditions, the radial stresses behave as tensile stress at the tube inletends, and then the radial stress values change sharply to compres-sive stress. From z = 0.1 m to z = 1.9 m, the radial stress values keepalmost constant values which are near to zero. The radial stressvalues change to tensile stress at the tube outlet ends. Amongthe four different material conditions, the stainless steel conditionhas the highest maximum radial stress which is only 3.06 MPa. Themaximum radial stress for SiC condition is 1.46 MPa and the max-imum radial stress for aluminum and copper conditions are verysmall which are about 0.5 MPa. It is worth noting that, comparedwith the effective stresses of each material condition, the radialstresses are very small and contributes little to the effectivestresses.

Fig. 11 shows the axial stress profiles for the four differentmaterial conditions on the tube inner surface along the lengthdirection at h = 270�. As seen from the figure, the axial compressivestress values for stainless steel condition are much higher than theaxial stresses values for the other three material conditions. At thetube inlet ends, the axial stress values are close to zero for all thefour different material conditions and then the axial stresses in-crease sharply to a higher compressive stress values which areclosed to the peak compressive axial stress values of each profile.From z = 0.1 m to z = 1.9 m, the axial stress values are almostinvariant. At the tube outlet ends, the axial stress values decreasesharply to about zero. These phenomena can be explained by thetemperature profiles along the axial direction at the tube inner sur-face shown in Fig. 12. As can be seen from this figure, the temper-ature profiles at the two tube receiver ends increase almostlinearly along the axial direction which cannot create any stresses[21]; therefore, the axial stress at the two ends are close to zero. Inthe temperature profiles for the four different material conditions,the inflexions at z = 0.1 m induce the sudden rises of axial stressvalues in the axial stress profiles.

0.0 0.5 1.0 1.5 2.0-60

-40

-20

0

20

Axi

al S

tres

s (M

Pa)

Z (m)

Stainless steelAluminumCopperSiC

Fig. 11. Axial stress profiles on the tube inner surface along the length direction ath = 270�.

0.0 0.5 1.0 1.5 2.0

300

320

340

360

380

Z (m)

Stainless steelAluminumCopperSiC

Tem

pera

ture

(K

)

Fig. 12. Temperature profiles along the axial direction at the tube inner surface.

0.0 0.5 1.0 1.5 2.0-20

0

20

40

60

80

Tan

gent

ial S

tres

s (M

Pa)

Z (m)

Stainless steelAluminumCopperSiC

Fig. 13. Tangential stress profiles on the tube inner surface along the lengthdirection at h = 270�.

0

4

8

12

16

0

30

60

90

120

150

180

210

240

270

300

330

0

4

8

12

16

Aluminum

SiC Copper

Stainless Steel

Stre

ss F

ailu

re R

atio

(%

)

Fig. 14. Stress failure ratio profiles across the circumference on the tube innersurface at the tube outlet section.

290 F. Wang et al. / Materials and Design 33 (2012) 284–291

Fig. 13 shows the tangential stress profiles for the four differentmaterial conditions on the tube inner surface along the length direc-tion at h = 270�. The tangential stress absolute values at the two tubefree ends are much higher than the tangential stress absolute valuesat the other positions. These phenomenon are caused by the bendingmovement of the tube receiver at the two free ends due to the out-ward defection, and these phenomenon are also observed duringIfran and Walts’ research on thermal stresses of a radiant tube[21]. At the tube inlet ends, the tangential stresses behave as tensilestress and then the tensile tangential stress values change sharply tocompressive stress. From z = 0.1 m to z = 1.9 m, the tangential stressvalues for the four different material conditions keep almost con-stant. At the tube outlet ends, the compressive tangential stresseschange sharply to tensile stress. Compared with the axial stressand radial stress, the tangential stress makes a major contributionto the maximum effective stress. The maximum tangential stressesfor the stainless steel and SiC conditions are 69 MPa and 41 MParespectively. Compared with the stainless steel and SiC conditions,the maximum tangential stresses for copper condition are verysmall which only 4.8 MPa is.

In this study, the stress failure ratio Fc (Fc = deff/db � 100) is intro-duced to assess the thermal stress level of each material condition.Fig. 14 presents the stress failure ratio profiles across the circumfer-ence on the tube inner surface at the tube outlet section for the fourdifferent material conditions. As seen from this figure, the coppercondition has the lowest stress failure ratio and the stainless steel

condition has the highest stress failure ratio which is 7.47 times ofthe copper condition. Therefore, from the standpoint of thermalstress, copper is recommended as the material of tube receiver[22]. This is the reason why that when a copper receiver was usedto take place of a steel receiver, no appreciable bending of the tubereceiver were observed during application in SPNUM [2].

5. Conclusions

The effects of material selection on the thermal stresses of tubereceiver under concentrated solar irradiation are investigated bynumerical analyses. Four different materials of tube receiver areemployed for the numerical analyses. The following conclusionsare drawn:

(1) Among the four different material conditions, the tube recei-ver made of SiC condition has the highest maximumtemperature.

F. Wang et al. / Materials and Design 33 (2012) 284–291 291

(2) The temperature gradients and effective stresses of the tubereceiver made of stainless steel and SiC conditions are signif-icantly higher than those of the tube receiver made of alumi-num and copper conditions.

(3) The tube receiver made of stainless steel has the higheststress failure ratio, and the tube receiver made of copperhas the lowest stress failure ratio.

(4) From the standpoint of thermal stress, copper is recom-mended to be the material of tube receiver.

Acknowledgments

This work was supported by the National Key Basic ResearchSpecial Foundation of China (No. 2009CB220006), the key programof the National Natural Science Foundation of China (Grant No.50930007) and the National Natural Science Foundation of China(Grant No. 50806017). A very special acknowledgment is madeto the editors and referees whose constructive criticism has im-proved this paper.

References

[1] Lei DQ, Wang ZF, Li J. The analysis of residual stress in glass-to-metal seals forsolar receiver tube. Mater Des 2010;31:1813–20.

[2] Almanza RF, Lentz A, Jimenez G. Receiver behavior in direct steam generationwith parabolic toughs. Sol Energy 1997;67:275–8.

[3] Almanza RF, Flores VC, Lentz A, Valdés A. Compound wall receiver for DSG inparabolic troughs. In: Proceedings of the 10th international symposium ofsolar thermal: solar PACES, Sydney, Australia; 2002. p. 131–5.

[4] Flores VC, Almanza RF. Behavior of compound wall copper–steel receiver withstratified two-phase flow regimen in transient states when solar irradiance isarriving on one side of receiver. Sol Energy 2004;76:195–8.

[5] Yapici HS, Albayrak B. Numerical solutions of conjugate heat transfer andthermal stresses in a circular pipe externally heated with concentrated solarirradiation heat flux. Energy Convers Manage 2004;45:927–37.

[6] Chen W, Liu W. Numerical analysis of heat transfer in a composite wall solar-collector system with a porous absorber. Appl Energy 2004;78:137–49.

[7] Fend T, Paal RP, Reutter O, Bauer J, Hoffschmidt B. Two novel high-porositymaterials as volumetric receivers for concentrated solar radiation. Sol EnergyMater Sol Cells 2004;84:291–304.

[8] Islamoglu YS. Finite element model for thermal analysis of ceramic heatexchanger tube under axial concentrated solar irradiation convective heattransfer coefficient. Mater Des 2004;25:479–82.

[9] Agrafiotis CC, Mavroidis I, Konstandopoulos AG, Hoffschmidt B, Stobbe P,Romero M, et al. Evaluation of porous silicon carbide monolithic honeycombsas volumetric receivers/collectors of concentrated solar radiation. Sol EnergyMater Sol Cells 2007;91:474–88.

[10] Lata JM, Rodriguez MA, Lara MA. High flux central receivers of molten salts forthe new generation of commercial stand-alone solar power plants. ASME J SolEnergy Eng 2008;130:0211002/1–02/5.

[11] Shuai Y, Xia XL, Tan HP. Radiation performance of dish solar concentrator/cavity receiver systems. Sol Energy 2008;82:13–21.

[12] Kumar NS, Reddy KS. Thermal analysis of solar parabolic collector with porousdisc receiver. Appl Energy 2009;86:1804–12.

[13] Biswas G, Eswaran V. The k–e model, the RNG k–e model and the phaseaveraged model. Turbulent flows fundamentals, experiments and modeling.New Delphi, India; 2002. p. 360–2.

[14] Wang G. A fast and robust variant of the SIMPLE algorithm for finite-element simulations of incompressible flows. Comput Fluid Solid Mech2001;2:1014–6.

[15] Fauple JH, Fisher FE. Engineering design – a synthesis of stress analysis andmaterial engineering. New York: Wiley; 1981.

[16] Johnston G. Focal region measurements of the 20 m2 tiled dish at theAustralian national university. Sol Energy 1998;63:117–24.

[17] Jeter SM. The distribution of concentrated solar radiation in paraboloidalcollectors. J Sol Energy Eng 1986;108:219–25.

[18] Friedrich B, Wolfram K, Yang C. Virtual temperature cycle testing ofautomotive heat exchangers by coupled fluid structure simulation. SAETechnical Paper. No. 2008-01-1210.

[19] Knaus H, Weise S, Kühnel W, Krüger U. Overall approach to the validation ofcharge air coolers. SAE Technical Paper. No.2005-01-2064.

[20] Wetzel T, Brotz F, Löffler B, Boermsa A. Coupled aerothermodynamics andstructural analysis of transient charge air cooler operation modes. VTMS8:07VTMS-38; 2007.

[21] Ifran MA, Chapman WC. Thermal stresses in radiant tubes due to axial,circumferential and radial temperature distributions. Appl Therm Eng2009;29:1913–20.

[22] Shuai Y, Wang FQ, Xia XL, Tan HP. Ray–thermal–structural coupled analysis ofparabolic trough solar collector system. In: Reccab MO, editor. Solarcollectors and panels, theory and applications. Rijeka: Sciyo-Publishing Inc.;2010. p. 341–56.