Effect of number of supports on the bending of absorber tube of parabolic trough concentrator

Energy 93 (2015) 1788e1803

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Energy journal homepage: www.elsevier.com/locate/energy

Effect of number of supports on the bending of absorber tube of parabolic trough concentrator Sourav Khanna a, *, 1, Vashi Sharma b, 1 a b

Department of Energy Science and Engineering, Indian Institute of Technology Bombay, Powai, Mumbai, India Centre for Energy and Environment, Malaviya National Institute of Technology, J.L.N. Marg, Jaipur, India

a r t i c l e i n f o

a b s t r a c t

Article history: Received 21 June 2015 Received in revised form 7 September 2015 Accepted 7 October 2015 Available online 19 November 2015

The circumferential non-uniformity in the temperature of absorber tube of parabolic trough leads to bending of the tube. The absorber tube is considered to be equidistantly supported at various points along the length. An analytical expression is presented in the current work for finding the bending and the results are compared with the experimental measurements. Further, the effect of number of supports and spacing between them on the shape of bent tube is analyzed using analytical equations. It is found that, keeping spacing between adjacent supports fixed, as number of supports and length of tube increase, the maximum bending does not vary beyond a certain length of tube. For the chosen system (keeping spacing between adjacent supports as 4 m), the results show that if lengths of tubes are larger than 16 m, the maximum bending remains same. The shape of bent tube is symmetric around the mid length. For one half of the tube, the bending is towards the vertex line of trough for the portion lying between 1st and 2nd support. It is away from the vertex line for the portion lying between 2nd and 3rd support and so on up to the mid length of the tube. © 2015 Elsevier Ltd. All rights reserved.

Keywords: Parabolic trough Absorber tube Deflection Bending

1. Introduction The periphery of the receiver (of parabolic trough) facing the sun receives incident rays directly and the periphery facing the reflector receives concentrated rays. It results in non-uniform distribution of solar flux on the receiver. Many studies are available that analyzed the solar flux availability on the receiver of parabolic trough which are as follows. 1.1. Solar flux distribution Evans [1] has analytically evaluated the solar flux distribution on the flat absorber for j ¼ 0 (where, j is the angle of incidence of sun rays at trough's aperture). Nicolas and Duran [2] have extended the work of Evans [1] for all values of j. Rabl [3] and Guven and Bannerot [4] have analyzed the tubular receiver and analytically studied the effect of optical errors on the optical efficiency of parabolic trough. However, the solar flux distribution on the receiver has not been evaluated. In their studies, the spread of solar * Corresponding author. E-mail address: [email protected] (S. Khanna). 1 Both are first author. http://dx.doi.org/10.1016/j.energy.2015.10.020 0360-5442/© 2015 Elsevier Ltd. All rights reserved.

flux due to optical errors is considered to be normally distributed. Jeter [5] has analytically derived the circumferential distribution of solar flux on the tubular receiver. In the study, the sun-shape has been considered whereas optical errors have not been considered. The spread of solar flux due to sun-shape has been considered to be uniformly distributed within the angular width of sun. Jeter [6] has extended the previous work [5] by considering the optical errors. The spread of solar flux due to sun-shape and optical errors has been considered to be normally distributed. However, Hegazy et al. [7] have done deterministic analysis of the effects of optical errors (slope error in reflector surface and tracking errors) on the circumferential distribution of solar flux on tubular receiver. Apart from analytical equations, ray tracing softwares are also used in some studies which are as follows. Thomas and Guven [8] have used ray tracing techniques to study the effect of optical errors on the circumferential distribution of solar flux on the tubular receiver. In the study, the spread of solar flux due to optical errors has been considered to be normally distributed. Yang et al. [9] have used MCRT (Monte Carlo Ray Tracing) method to compute the solar flux distribution on the tubular receiver. In the study, the spread of solar flux due to sun-shape has been considered to be uniformly distributed within the angular width of sun. The effects of tracking errors and rim angle on the solar flux distribution have also been

S. Khanna, V. Sharma / Energy 93 (2015) 1788e1803

Nomenclature distance between adjacent supports to absorber tube (m) dqA(q, q0 ) the fraction of qA,p(q0 ) absorbed by a differential surface located at angle q (W/m2) E modulus of elasticity of the material of absorber tube (Pa) f focal length of the trough (m) hf convective heat transfer coefficient on inner surface of absorber tube (W/m2-K) I moment of inertia of the cross section of absorber tube with respect to its centroidal axis (m4) Ibn instantaneous beam normal radiation (W/m2) k thermal conductivity of the material of absorber tube (W/m-K) L length of the parabolic trough and the continuous absorber tube (m) M bending moment (N-m) MT moment induced in the absorber tube due to circumferential temperature gradient (N-m) n number of supports to absorber tube qA solar flux absorbed on the surface of absorber tube per unit outer surface area of absorber tube (W/m2) 0 qA,p(q ) solar flux absorbed by a differential surface located at angle q0 considering the sun to be a point source and no optical errors (W/m2) r radius (m) R reaction (N) T temperature of absorber tube (K) Ta ambient temperature (K) Tf fluid temperature (K) UL over all heat loss coefficient of the receiver (W/m2-K) vw wind velocity (m/s) w width of the aperture of trough (m)

d

d

Greek symbols absorptivity of absorber tube thermal expansion coefficient of the material of absorber tube (/K)

a ath

studied. Cheng et al. [10] have developed a general MCRT method for finding the solar flux distribution on the receivers of parabolic trough, paraboloid dish and solar tower. Cheng et al. [11] have used MCRT method to compute the solar flux distribution on the absorber tube of parabolic trough. The maximum solar flux density on the circumference, average solar flux density and nonuniformity in the solar flux distribution have been found out for various values of geometrical parameters. Wang et al. [12] have analyzed the absorber tube of parabolic trough with secondary reflector. MCRT method is used to compute the solar flux distribution on the absorber tube. It has been concluded that the circumferential non-uniformity in the distribution of solar flux can be reduced significantly by using the secondary reflector. Wang et al. [13] have analyzed the elliptical glass cover for the absorber tube. It is concluded that the circumferential non-uniformity in the distribution of solar flux can be reduced significantly by using elliptical glass cover as compared to circular glass cover.

Dr DTf Dz Dq, Dq0 ε

q0 qrim qshd r rta(j) stot t ta(j) j

1789

deflection in the central axis of absorber tube from the focal line of trough due to bending caused by nonuniform tube's temperature (m) interval in which the value of integral is considered to be remained constant in radial direction (m) desired rise in fluid temperature per unit length of absorber tube (averaged over the whole length of absorber tube) (K/m) length of each segment of absorber tube (m) interval in which the value of integral is considered to be remained constant in angular direction (rad) emissivity for long wavelength radiation circumferential angle (rad) rim angle of the trough (rad) angle up to which the circumference of the absorber tube does not receive concentrated rays due to the shadow cast by absorber tube on trough (rad) reflectivity of the trough surface effective reflectanceetransmittanceeabsorptance product for concentrated rays at j angle of incidence of sun rays equivalent rms angular spread caused by optical errors and sun-shape (rad) transmissivity of glass cover effective transmittanceeabsorptance product for direct incident rays at j angle of incidence of sun rays angle made by incident sun ray with the normal to aperture plane (rad)

Abbreviation MCRT Monte Carlo Ray-Trace Subscripts c glass cover ci inner surface of glass cover co outer surface of glass cover i ith segment of absorber tube and glass cover inlet inlet of absorber tube j jth support t absorber tube ti inner surface of absorber tube to outer surface of absorber tube

1.2. Temperature distribution of absorber tube The non-uniformity in the distribution of solar flux and variation in fluid's temperature lead to non-uniform distribution of receiver's temperature. Most of the available studies have not considered the circumferential non-uniformity in receiver's temperature. However, few studies have computed the temperature distribution of absorber tube by using CFD softwares which are as follows. In these studies, different types of receivers have been analyzed. Reddy and Satyanarayana [14] have analyzed the receiver having porous fins inside the absorber tube. Temperature distributions of the absorber tube have been computed for four different shapes (square, triangular, trapezoidal and circular) of fins. Cheng et al. [15] have computed the temperature distribution of the absorber tube having a concentric tube as a flow restriction device inserted inside the absorber tube. Wang et al. [16] have compared the temperature

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distributions of eccentric and concentric absorber tubes. Munoz and Abanades [17,18] have analyzed the receiver having helical fins inside the absorber tube to reduce the circumferential gradient in the temperature of absorber tube. Cheng et al. [19] have studied the effect of residual gases (in the space between absorber tube and glass cover instead of assuming vacuum) on the temperature distribution of absorber tube. Cheng et al. [20] have analyzed the receiver having longitudinal vortex generators inside the absorber tube to enhance the heat transfer between absorber tube and working fluid. It is concluded that the circumferential gradient in the temperature of absorber tube can be reduced by using longitudinal vortex generators. Islam et al. [21] have analyzed the temperature distribution of absorber tube having a concentric tube as flow restriction device inserted inside the absorber tube. Wang et al. [22] have analyzed the temperature distribution of absorber tube for various materials. Wang et al. [23] have computed the temperature distribution of absorber tube having metal foams inserted inside. Roldan et al. [24] have compared the simulated temperature distribution of absorber tube with that of experimental measurements. Yaghoubi et al. [25] have computed the temperature distribution of absorber tube in their study for vacuum and no vacuum in the space between absorber tube and glass cover. Cheng et al. [26] have studied the effects of geometrical parameters on the temperature distribution of absorber tube. Wang et al. [12] have analyzed the receiver with secondary reflector. It has been concluded that the circumferential non-uniformity in the distribution of absorber's temperature can be reduced significantly by using the secondary reflector. Song et al. [27] have analyzed the receiver having helical screw-tape inside the absorber tube to reduce the non-uniformity in temperature distribution. Wu et al. [28] have computed the temperature distribution of the absorber tube and glass cover. Chang et al. [29] have presented the correlations for the circumferential temperature distribution of absorber tube. Natarajan et al. [30] have analyzed the temperature distribution of absorber tube having triangular, inverted triangular and semi circular inserts inside the absorber tube. Patil et al. [31] have computed the temperature distribution of absorber tube considering sun to be a point source. Wang et al. [32] have analyzed the temperature distribution of absorber tube for various values of fluid's inlet temperature, fluid's velocity and solar radiation. 1.3. Bending and stresses in absorber tube The non-uniformity in the distribution of absorber's temperature leads to differential expansion of the absorber tube. Thus, the absorber tube experiences tension and compression in its different parts which leads to bending in the absorber tube. Almanza et al. [33] have reported the circumferential difference in absorber's temperature and the corresponding maximum deflection in steel and copper receivers. Almanza et al. [34] have extended the previous work [33] to analyze the steel receiver with stratified fluid flow. Flores and Almanza [35] have extended the previous work [34] to analyze the bimetallic Copper-Steel receiver. Flores and Almanza [36] have extended the previous work [35] to analyze the case when concentrated radiation falling on one side of absorber tube instead of the lower periphery. Apart from the experimental measurements, numerical studies are also available which are as follows. Wang et al. [16] have used CFD software to compute the stresses induced in the absorber tube when it is not allowed to bend and expand along the length. Yaghoubi and Akbarimoosavi [37] have used CFD software to compute the deflection in the absorber tube when it is free to bend and expand. Wang et al. [22] have extended their previous work [16] to compare different materials of the absorber tube in terms of the induced stresses. Akbarimoosavi and Yaghoubi [38] have

extended their previous work [37] by considering two supports in between the ends of absorber tube. Wang et al. [32] have used CFD software compute the stresses and deflection in absorber tube when it is free to bend and expand. 1.4. Conclusions of literature review The circumferential solar flux distribution on the absorber tube of parabolic trough has been reported in literature. The temperature distribution of absorber tube has also been reported. The deflection in absorber tube (from the focal line of trough) due to bending has been reported only for the case when tube is kept free to bend and for the case when two supports are considered in between the ends. However, in power plants, a continuous absorber tube is equidistantly supported at various points along the length (Kreith and Goswami [39]). The supports can move along the length so that the absorber tube can elongate freely on heating. At the supporting points, the supports allow the absorber tube to rotate in the plane passing through the vertex line and the focal line but do not allow to deflect from the focal line. For such supports, analytical expression is presented in the current work for finding the deflection in absorber tube from the focal line. For same type of supports, an experimental set-up has been created to verify the analytical calculations. Further, the effect of number of supports and spacing between them on the shape of bent tube has been analyzed using analytical equation. 2. Analytical derivation In the current work, a parabolic trough is considered having an absorber tube with a concentric glass cover at its focus (Fig. 1). Cartesian coordinates (x, y, z) and cylindrical coordinates (r, q, z) are chosen to define the geometry. The points O and O0 are taken as origins of the cartesian and cylindrical coordinate system respectively. The angle q is measured in anticlockwise direction from OO0 . The aperture width of trough is denoted by w, rim angle by qrim and length by L. Inner and outer radii of the absorber tube are denoted by rti and rto respectively and that of glass cover by rci and rco respectively. The assumptions made in this work are as follows. (i) The space between the glass cover and the absorber tube is evacuated (ii) The absorber tube is considered to be fully filled with fluid. Only axial variation in the fluid temperature is considered (iii) Inner and outer radii of the absorber tube are considered to remain unchanged even after heating

Fig. 1. Cross sectional view of the trough-receiver system.

S. Khanna, V. Sharma / Energy 93 (2015) 1788e1803

(iv) Bending moment induced in the absorber tube due to its weight is not considered

2.1. Solar flux distribution on the absorber tube

Zp qA ðqÞ ¼

dqA ðq; q0 Þ

cp ¼

dp ¼

The circumferential distribution of the solar flux on the absorber tube considering the optical errors and the sun-shape can be given by the following expression [40].

(1)

1791



qp k p rtip1 þ hf;i rtip



ep þ f p   p1 p qp k p rti  hf;i rti ep þ f p

(6)

(7a)

   ep ¼ k p rtip1  hf;i rtip UL;i rtop  k p rtop1

(7b)

   p1 p p1 p þ hf ; i rti f p ¼ k p rto þ UL;i rto k p rti

(7c)

q0 ¼p

where, q0 is the circumferential angle which can be further explained as follows. In Eq. (1), dqA(q, q0 ) is the fraction of qA,p(q0 ) absorbed by a differential surface of tube located at angle q. qA,p(q0 ) is the solar flux absorbed by a differential surface located at angle q0 considering the sun to be a point source and no optical errors [41]. The expression of dqA(q, q0 ) can be given by

8 > > 0 > > > > > > > > 2f rtaðjÞIbn cosj > " # > > 0 > > b2 ðq; q0 Þ g < rto f1 þ cosq p ffiffiffiffiffiffi ffi exp db dqA ðq; q0 Þ ¼ 2 s2tot ðjÞ stot ðjÞ 2p > > > > > > þ > > ½taðjÞIbn cosjðcosq0 Þ > > > > > > > : 0

if jq0 j2ðqshd ; qrim ; jq  q0 j  fðq0 Þ

 cp rp þ dp rp cospq

otherwise

(3)

parts which leads to bending in the absorber tube. Thus, the central axis of absorber tube deflects from the focal line. Since the absorber tube is considered to be fully filled with fluid and only axial variation in the fluid temperature is considered, the absorber's temperature distribution is symmetric about the plane passing through focal line and vertex line of trough. Due to the temperature symmetry and the non-consideration of bending moment due to weight, the deflection (d) in absorber tube (due to bending caused by non-uniformity in temperature distribution) occurs in the plane passing through the focal line and vertex line of trough. The moment induced in the tube due to non-uniform temperature distribution can be given by Ref. [40].

p¼1

Zp Zrto Tðr; qÞ r2 cosq dr dq

MT ¼ E ath

where,

(8)

p rti

       q þ UL;i Ta hf; i lnðrti Þ  rkti  hf ; i Tf;i UL;i lnðrto Þ þ rkto     a0 ¼ UL;i hf; i lnðrti Þ  rkti  hf; i UL;i lnðrto Þ þ rkto (4)     hf ; i q þ UL;i Ta  UL;i hf; i Tf;i     b0 ¼ hf; i UL;i lnðrto Þ þ rkto  UL;i hf ; i lnðrti Þ  rkti

(2)

if jq0 j2ðqrim ; p; q ¼ q0

The non-uniformity in the distribution of solar flux and variation in fluid's temperature lead to non-uniform distribution of receiver's temperature. Considering radial and circumferential variations but assuming constant temperature along the length within a segment of length Dz, the temperature distribution of ith segment of the absorber tube can be given by the following expression of the previous work (Khanna et al. [40]). 10  X

The non-uniformity in the distribution of absorber's temperature leads to differential expansion of the absorber tube. Thus, the absorber tube experiences tension and compression in its different

if jq0 j2½0; qshd 

2.2. Temperature distribution on the absorber tube

Ti ðr; qÞ ¼ a0 þ b0 lnr þ

2.3. Bending in absorber tube

(5)

Supports to absorber tube highly affect the bending. The absorber tube is equidistantly supported at various points as shown in Fig. 2. The supports can move along the length so that the absorber tube can elongate freely on heating. At the supporting points, the supports allow the absorber tube to rotate in the plane passing through the vertex line and the focal line but do not allow to deflect from the focal line. Thus,

dðzÞ ¼ 0

for z ¼ 0; d; 2d; …; ðn  1Þd

(9)

The freeebody diagram is drawn in Fig. 3. Rj (1  j  n) is the reaction at jth support. MT (Eq. (8)) is the moment induced due to

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L Absorber Tube

d

y

x

1st Support

d

2nd Support

(n-2)th Support

3rd Support

nth Support

(n-1)th Support

z Fig. 2. View of supports to absorber tube as seen along x axis.

non-uniform temperature distribution. d(z) follows the following equation (Beer et al. [42])

d2 fdðzÞg dz2

if 2d  z  3d (11c)

MðzÞ ¼ EI

(10)

where, E is the modulus of elasticity of absorber's material, I is moment of inertia of the cross section of absorber tube with respect to its centroidal axis and M(z) is the bending moment at any point z along the length of the absorber tube. Taking moment equilibrium, the expression for M(z) can be written as (Fig. 4)

MðzÞ ¼ MT þ R1 z

MðzÞ ¼ MT þ R1 z þ R2 ½z  d þ R3 ½z  2d

if 0  z  d

MðzÞ ¼ MT þ R1 z þ R2 ½z  d

Thus, the general expression of M(z) is

MðzÞ ¼ MT þ R1 z þ R2 ½z  d þ … þ Rj ½z  ðj  1Þd if ðj  1Þd  z  jd By solving the differential equation (Eq. (10)),

(11a)

if d  z  2d

(11b)

dðzÞ ¼

  1 MT z2 R1 z3 þ þ C1 z þ D 1 EI 2 6

Absorber Tube

MT

MT

R2

R3

Rn-2

y d x

if 0  z  d

(12a)

where, C1 and D1 can be found out by solving d(z ¼ 0) ¼ 0 and d(z ¼ d) ¼ 0.

Lt

R1

(11d)

d

z

Fig. 3. Free-body diagram of absorber tube.

Rn-1

Rn

S. Khanna, V. Sharma / Energy 93 (2015) 1788e1803

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Fig. 4. Free-body diagram of various portions of absorber tube.

dðzÞ ¼

" # 1 MT z2 R1 z3 R2 ½z  d3 þ þ þ C2 z þ D2 if d  z  2d EI 2 6 6

" # Rj ½z  ðj  1Þd3 1 MT z2 R1 z3 R2 ½z  d3 þ þ þ…þ dðzÞ ¼ EI 2 6 6 6 þ Cj z þ D j

if ðj  1Þd  z  jd

(12b)

(12d)

where, C2 and D2 can be found out by solving d(z ¼ d) ¼ 0 and d(z ¼ 2d) ¼ 0.

where, Cj and Dj can be found out by solving d[z¼(j1)d] ¼ 0 and d(z ¼ jd) ¼ 0. 3. Experimental set-up

"

dðzÞ ¼

z2

1 MT EI 2

þ

þ C3 z þ D3

z3

R1 6

3

þ

R2 ½z  d R ½z  2d þ 3 6 6

3

# (12c)

In order to verify the analytical equations, an experimental set up is created. In this section, (i) description of the set-up and (ii) measurements of the distribution of tube's temperature and the corresponding bending in the tube have been presented.

if 2d  z  3d 3.1. Description of the set-up

where, C3 and D3 can be found out by solving d(z ¼ 2d) ¼ 0 and d(z ¼ 3d) ¼ 0. Thus, the general expression of d(z) is

A tube having inner and outer diameter as 38.9 mm and 42.2 mm respectively and length as 5.86 m is used which is made

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S. Khanna, V. Sharma / Energy 93 (2015) 1788e1803

up of stainless steel 316. Four pairs of clamps (equidistantly spaced along the length of tube) are used to grip the tube (Figs. 5a and 6). Unthreaded bolts (Figs. 5a and 6) are welded on the clamps. Along the length, the tube is supported at 4 equidistant points. Each support (Figs. 5a and 6) is made up of two plates. Height, width and thickness of plate are 10 cm, 10 cm and 1 cm respectively. C-clamps are used to join both plates as shown in Fig. 5a. In order to plug the unthreaded bolts (welded on clamps) inside the plates, a piece is cut out from each plate (Figs. 5b and 6). With this arrangement, the tube is allowed to elongate freely on heating and to rotate freely in the vertical plane. In order to heat the tube electrically, a heater (made up of nichrome wire and ceramic beads) is placed inside the tube (Figs. 5a and 7). The length of heater is same as that of tube. The bending of the tube will not be restrained due to the fact that the heater is flexible. To achieve a larger circumferential nonuniformity in tube's temperature, a mica sheet (having length, width and thickness as 6 m, around 38.9 mm and 2 mm respectively) is inserted inside the tube so that the upper half cannot be heated convectively and radiatively (Figs. 5a and 7). Due to bending caused by circumferential non-uniformity in tube's temperature, forces will act upwards at 2nd and 3rd supports and downwards at 1st and 4th supports. In order to restrict the upward movement of tube at its supporting point, weights (around 18 kg) are put on the supports (Fig. 5a). For the measurement of tube's temperature, 56 T type thermocouples are used. Temperatures of 7 different cross sections (lying at z ¼ 0.2 m, 1.1 m, 2.1 m, 3.1 m, 4.1 m, 4.7 m and 5.7 m) are measured. At each cross section, temperature is measured at 8 different points on the circumference (at q ¼ 180 , 135 , 90 , 45 , 0 , 45 , 90 and 135 ) as shown in Fig. 6. Data acquisition modules and a computer are used to store the readings of thermocouples which are in mV. Calibration is carried out for each thermocouple to convert the mV readings to  C. The following equation is obtained for each thermocouple

Fig. 6. Photos showing the support to tube and thermocouples.

Weight

Mild Steel Plate Mica Sheet

Mild Steel Plate

Tube

Heater

Unthreaded Bolt Clamp

Bolt C-Clamp

(a) Cross sectional view

(b) Side view Fig. 5. Support to tube.

S. Khanna, V. Sharma / Energy 93 (2015) 1788e1803

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Table 1 Values of a and b for all thermocouples with the corresponding R2 and the minimum and maximum error occurred in converting the mV values to  C. Thermocouple location

Fig. 7. Photo showing the heater and mica sheet.

Temperature in  C ¼ a ðTemperature in mVÞ þ b

(13)

where ‘a’ and ‘b’ are computed using linear regression and presented in Table 1 for all the thermocouples along with their R2 values. The table shows that the error occurred in the conversion of mV values to  C lies within 0.8  C to þ1.2  C. Depth gauge (Fig. 8) having least count of 0.01 mm is used to measure the deflection in the tube (in the vertical plane) due to bending caused by non-uniform tube's temperature. Readings of depth gauge are taken at every 24.3 cm along the length between the supports. Input power to heater is fixed as 1000 W using Voltage Regulator and measured by Wattmeter.

3.2. Measurements of the distribution of tube's temperature and bending in tube The procedure followed for the measurements has been described in this section. Before switching on the heater, the position of tube with respect to the floor was measured at every 24.3 cm along the length using depth gauge. Marking was also done on the floor at every 24.3 cm. In a trial experiment, it was observed that the tube elongated by 11 mm on heating. Thus, the error (due to elongation) in the measurements of the position of tube has to be noted. Thus, the position of tube was again measured with respect to the marked points on the floor after sliding the tube manually by 11 mm. Taking the average of the 2 sets of readings, the average position of tube before heating has been presented in the 2nd column of Table 2. The 3rd and 4th columns show that the error (occurred due to sliding of tube by 11 mm) in the measurement of the position of tube lies within 0.09 mm to þ0.08 mm. After that, the heater had been switched on. Beyond 80 min, it was observed that the tube's temperature did not vary much with time. Thus, the steady state had been achieved. From 80 min to 120 min, 8 sets of readings of thermocouples were taken with 5 min interval. Taking the average of all sets of readings, the average temperature of each thermocouple during steady state has been presented in the 2nd column of Table 3. Accounting the error due to variation in tube's

z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z

¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1 4.1 4.1 4.1 4.1 4.1 4.1 4.1 4.1 4.7 4.7 4.7 4.7 4.7 4.7 4.7 4.7 5.7 5.7 5.7 5.7 5.7 5.7 5.7 5.7

m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m

and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and and

q ¼ 180 q ¼ 135 q ¼ 90 q ¼ 45 q ¼ 0 q ¼ 45 q ¼ 90 q ¼ 135 q ¼ 180 q ¼ 135 q ¼ 90 q ¼ 45 q ¼ 0 q ¼ 45 q ¼ 90 q ¼ 135 q ¼ 180 q ¼ 135 q ¼ 90 q ¼ 45 q ¼ 0 q ¼ 45 q ¼ 90 q ¼ 135 q ¼ 180 q ¼ 135 q ¼ 90 q ¼ 45 q ¼ 0 q ¼ 45 q ¼ 90 q ¼ 135 q ¼ 180 q ¼ 135 q ¼ 90 q ¼ 45 q ¼ 0 q ¼ 45 q ¼ 90 q ¼ 135 q ¼ 180 q ¼ 135 q ¼ 90 q ¼ 45 q ¼ 0 q ¼ 45 q ¼ 90 q ¼ 135 q ¼ 180 q ¼ 135 q ¼ 90 q ¼ 45 q ¼ 0 q ¼ 45 q ¼ 90 q ¼ 135

a ( C/mV)

b ( C)

R2

Error ( C)

20.60 21.06 21.44 21.05 20.58 21.66 22.18 21.62 20.42 21.07 20.78 20.51 21.37 20.49 20.97 21.39 21.39 20.77 20.44 20.79 21.06 21.39 20.53 21.04 20.31 21.37 20.58 20.57 21.00 20.41 21.41 21.06 21.33 20.50 20.30 20.74 20.52 20.74 20.00 20.29 20.42 20.25 20.37 20.97 22.09 20.49 20.92 21.35 20.82 21.35 20.41 20.99 22.08 21.36 21.33 21.38

8.87 8.93 8.92 8.95 8.88 9.16 8.72 8.54 8.89 9.11 8.93 9.08 9.01 8.97 9.27 8.47 8.89 8.83 8.85 8.76 9.20 8.98 8.95 8.61 9.01 8.76 8.97 8.99 9.09 8.94 9.07 8.41 8.98 8.92 9.08 8.92 8.94 8.95 8.98 8.67 8.79 8.95 8.83 9.00 8.68 8.73 8.98 8.75 8.50 8.51 8.30 8.66 8.29 8.43 8.08 8.33

0.99981 0.99980 0.99980 0.99979 0.99981 0.99973 0.99981 0.99979 0.99978 0.99977 0.99979 0.99977 0.99978 0.99978 0.99977 0.99977 0.99970 0.99972 0.99971 0.99973 0.99970 0.99971 0.99972 0.99969 0.99973 0.99976 0.99973 0.99973 0.99972 0.99973 0.99973 0.99971 0.99976 0.99978 0.99977 0.99978 0.99977 0.99978 0.99978 0.99977 0.99980 0.99979 0.99979 0.99978 0.99978 0.99978 0.99977 0.99976 0.99975 0.99976 0.99976 0.99975 0.99978 0.99976 0.99980 0.99976

0.6 0.6 0.6 0.6 0.6 0.7 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.7 0.8 0.7 0.7 0.8 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.7 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.7 0.6 0.6 0.6 0.6 0.6 0.6 0.6

1.0 1.0 1.0 1.0 1.0 1.1 1.0 1.0 1.0 1.1 1.0 1.0 1.0 1.0 1.0 1.0 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.2 1.1 1.1 1.1 1.1 1.2 1.1 1.1 1.2 1.1 1.1 1.1 1.0 1.1 1.0 1.0 1.1 1.0 1.0 1.0 1.0 1.1 1.0 1.1 1.1 1.1 1.1 1.1 1.1 1.0 1.1 1.0 1.1

temperature with time during steady state and the one due to the conversion of mV values to  C, the total error is tabulated in the last two columns of Table 3. The results show that the error occurred in the measurements of the tube's temperature during steady state lies within 2.7  C to þ2.7  C. The circumferential distributions of tube's temperature at various cross sections (during steady state) are plotted in Fig. 9. From 80 min to 120 min, 4 sets of readings of depth gauge (with 10 min interval) were also taken. Taking the average of all the 4 sets of readings, the average position of tube after heating during steady state is presented in the 5th column of Table 2 and the corresponding errors in 6th and 7th columns. The difference between the position of tube after heating (during

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measuring the deflection which lies within the range 0.11 mm to þ0.10 mm. The deflection is plotted in Fig. 10. After that, the heater had been switched off. In order to check the repeatability of the results, the experiment was done thrice as shown in Figs. 9 and 10. 4. Comparison of analytical calculation with experimental measurements

Fig. 8. Photo showing the depth gauge.

steady state) and before switching on the heater gives the deflection in the tube (in the vertical plane) due to bending caused by non-uniform circumferential tube's temperature and is tabulated in 8th column of Table 2. The last two columns give the error in

The solar flux distribution is calculated using Eq. (1). The integration (for q0 ¼ 180 to 180 ) involved in the equation is solved numerically considering the integral to be remained constant within Dq0 . It is found that if Dq0 is decreased beyond 0.5 , the solar flux distribution does not change much. Thus, the calculations are carried out using Dq0 ¼ 0.5 . The moment MT is calculated using Eq. (8). The integration involved in the equation is solved numerically considering the integral to be remained constant within radius Dr and an angle Dq. It is found that if Dr is decreased beyond 1 mm and Dq beyond 0.5 , the moment MT does not change much. Thus, the calculations are carried out using Dr ¼ 1 mm and Dq ¼ 0.5 . Thus, the tube's temperature [T(r,q)] is calculated using Eq. (3) for r ¼ rti to rto with 1 mm interval and for q ¼ 180 to þ180 with 0.5 interval. The deflection [d(z)] in the absorber tube due to bending is calculated using Eq. (12) for z ¼ 0 m to L with 0.25 m interval. The comparison of the analytical calculations with the experimental measurements has been done as follows. Putting the measured distribution of tube's temperature (Fig. 9) in the analytical equations, the corresponding bending is calculated and plotted in Fig. 10 along with the experimental measurements. The analytical calculations differ from the measurements within the range 15.4% to 16.9% (i.e. 0.26 mm to þ0.50 mm). Most of the measured values of bending are lesser than the results of equations. The reason can be explained as follows. The measured distribution of tube's temperature is not perfectly symmetric around the vertical plane passing through the central axis of tube. However, in the equations, it is considered to be symmetric and, thus, the moment due to temperature gradient is considered to be induced in the aforementioned vertical plane. Since, in case

Table 2 Reading of Depth Gauge while measuring the position of tube (before and after heating) and the corresponding minimum and maximum error. z (cm)

Reading of Depth Gauge before heating (mm)

Reading of Depth Gauge after heating (mm)

Deflection (mm)

Average

Error

Error

Average

Error

Error

Average

Error

Error

25.56 49.88 74.19 98.50 122.81 147.13 171.44 220.06 244.38 268.69 293.00 317.31 341.63 365.94 414.56 438.88 463.19 487.50 511.81 536.13 560.44

19.01 19.11 18.74 20.27 21.45 21.06 19.21 21.01 19.90 19.87 21.94 21.22 21.60 21.49 19.89 18.57 18.52 17.61 15.49 15.74 16.81

0.02 0.03 0.01 0.02 0.05 0.03 0.02 0.02 0.03 0.02 0.05 0.09 0.03 0.02 0.05 0.05 0.01 0.04 0.01 0.06 0.03

0.01 0.04 0.01 0.02 0.06 0.02 0.02 0.02 0.02 0.02 0.05 0.08 0.03 0.02 0.04 0.04 0.02 0.04 0.02 0.06 0.04

17.22 16.26 15.51 17.23 19.00 19.32 18.44 21.64 20.82 21.04 23.19 22.40 22.54 22.05 19.22 17.21 16.46 15.16 12.91 13.45 15.40

0.02 0.02 0.02 0.03 0.02 0.02 0.01 0.02 0.02 0.03 0.03 0.02 0.01 0.01 0.03 0.04 0.03 0.02 0.03 0.03 0.02

0.01 0.02 0.02 0.02 0.02 0.03 0.01 0.02 0.03 0.03 0.02 0.02 0.01 0.02 0.02 0.04 0.02 0.02 0.02 0.03 0.02

1.79 2.85 3.23 3.04 2.45 1.73 0.77 0.62 0.92 1.17 1.25 1.18 0.94 0.56 0.67 1.36 2.06 2.45 2.58 2.29 1.41

0.03 0.06 0.04 0.04 0.07 0.05 0.03 0.04 0.05 0.05 0.08 0.11 0.04 0.03 0.08 0.08 0.04 0.06 0.04 0.09 0.05

0.03 0.06 0.02 0.04 0.08 0.05 0.03 0.03 0.05 0.05 0.07 0.10 0.04 0.04 0.06 0.08 0.04 0.06 0.04 0.10 0.06

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Table 3 Temperature of tube during steady state and the corresponding minimum and maximum error. Thermocouple location z z z z z z z z z z z z z z z z z z z z z z z z z z z z

¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 3.1 3.1 3.1 3.1

m m m m m m m m m m m m m m m m m m m m m m m m m m m m

and and and and and and and and and and and and and and and and and and and and and and and and and and and and

q ¼ 180 q ¼ 135 q ¼ 90 q ¼ 45 q ¼ 0 q ¼ 45 q ¼ 90 q ¼ 135 q ¼ 180 q ¼ 135 q ¼ 90 q ¼ 45 q ¼ 0 q ¼ 45 q ¼ 90 q ¼ 135 q ¼ 180 q ¼ 135 q ¼ 90 q ¼ 45 q ¼ 0 q ¼ 45 q ¼ 90 q ¼ 135 q ¼ 180 q ¼ 135 q ¼ 90 q ¼ 45

Average ( C)

Error ( C)

121.5 121.3 128.6 137.3 149.5 133.6 121.5 119.9 115.5 120.6 120.4 138.4 141.4 124.4 114.9 114.7 106.1 116.6 124.5 125.6 124.8 107.3 105.7 102.7 106.0 113.6 116.0 135.2

1.0 1.9 2.2 1.6 1.5 2.1 2.7 2.3 1.0 1.2 1.9 1.3 1.1 1.2 1.5 1.3 0.9 1.2 1.2 1.4 1.3 1.0 1.0 1.1 1.1 1.1 1.6 1.4

Thermocouple location 1.3 1.9 2.0 1.6 2.2 2.0 2.7 2.4 1.6 1.6 2.6 2.1 1.5 1.7 2.0 2.0 1.5 1.4 1.6 1.9 1.7 1.2 1.4 1.6 1.5 1.6 2.4 1.7

of experiments, the distribution is slightly asymmetric, the moment is divided in more than one plane and the bending occurs in the resultant plane which is slightly different from the aforementioned vertical plane. The resultant bending will be lesser than the case when the moments lie in same plane. Thus, the measured values of bending are lesser than the results of equations. 5. Results and discussion For carrying out the calculations, the dimensions (Table 4) of LS3 parabolic trough collector with Schott 2008 PTR70 receiver and Therminol VP1 heat transfer fluid are chosen. To achieve the desired rise in the fluid temperature (DTf), flow rate of the fluid is appropriately chosen. DTf is the rise in fluid temperature per unit length (averaged over the whole length of the absorber tube). 5.1. Effect of number of supports keeping length of tube fixed The deflection in the tube from the focal line (due to bending caused by non-uniformity in tube temperature) has been plotted in Fig. 11 for different values of DTf and number of supports keeping length of the tube fixed as 12 m. The results show that the bending increases as DTf increases. It is due to the fact that lesser mass flow rate is required to achieve higher DTf. And lesser mass flow rate leads to lesser heat transfer coefficient and thus higher tube's temperature and larger circumferential non-uniformity in tube's temperature. Thus, higher DTf leads to larger bending. The results also show that the maximum deflection decreases as number of supports increases which is as expected. The shape of bent tube changes as number of supports changes. For n ¼ 2, the bending of the tube is towards the vertex line of trough. The reason can be explained as follows. Since the reflector facing periphery of the tube is hotter than the sun facing periphery, the reflector facing portion wants to expand more. Thus, the tube, if kept free, would

z z z z z z z z z z z z z z z z z z z z z z z z z z z z

¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼

3.1 3.1 3.1 3.1 4.1 4.1 4.1 4.1 4.1 4.1 4.1 4.1 4.7 4.7 4.7 4.7 4.7 4.7 4.7 4.7 5.7 5.7 5.7 5.7 5.7 5.7 5.7 5.7

m m m m m m m m m m m m m m m m m m m m m m m m m m m m

and and and and and and and and and and and and and and and and and and and and and and and and and and and and

q ¼ 0 q ¼ 45 q ¼ 90 q ¼ 135 q ¼ 180 q ¼ 135 q ¼ 90 q ¼ 45 q ¼ 0 q ¼ 45 q ¼ 90 q ¼ 135 q ¼ 180 q ¼ 135 q ¼ 90 q ¼ 45 q ¼ 0 q ¼ 45 q ¼ 90 q ¼ 135 q ¼ 180 q ¼ 135 q ¼ 90 q ¼ 45 q ¼ 0 q ¼ 45 q ¼ 90 q ¼ 135

Average ( C)

Error ( C)

135.0 119.7 108.4 102.7 108.7 112.9 118.7 125.8 130.6 117.8 116.6 107.1 113.6 116.2 127.7 133.2 147.7 124.1 119.7 110.9 105.7 107.0 111.3 120.9 132.8 121.4 110.3 103.4

1.1 1.0 1.1 1.6 1.1 1.1 1.1 1.1 1.0 1.6 1.2 1.3 1.4 1.6 1.5 1.5 1.0 1.0 1.2 1.3 1.0 1.2 1.2 1.2 1.1 1.0 1.2 1.3

1.6 1.5 1.7 1.8 2.5 1.5 1.5 1.7 1.6 2.0 1.8 2.1 1.9 1.8 1.7 1.7 1.7 1.4 1.5 1.6 1.5 1.9 2.1 1.9 1.4 1.4 1.4 1.5

bend towards the vertex line. For n ¼ 2, the tube is free to bend. Thus, for this case, the bending is towards the vertex line. For n ¼ 3, the shape of the bent tube is as expected. The results show that, for n ¼ 4, the bending between 1st and 2nd support is towards the vertex line. Between 2nd and 3rd support, it is away from the vertex line. It is again towards the vertex line for the portion of tube lying between 3rd and 4th support. The reason can be explained as follows. It is already explained that the tube, if kept free, would bend towards the vertex line. However due to 2nd and 3rd support, the reactions are acting away from the vertex line at these supports and thus the bending is away from the vertex line for the portion lying between 2nd and 3rd support. Similarly the shape of the bent tube for n ¼ 5 can be explained by inserting an additional support at z ¼ 6 m to the bent tube of the case n ¼ 4. Thus, in general, the shape of bent tube is symmetric around the mid length. For one half of the tube, the bending is towards the vertex line of trough for the portion lying between 1st and 2nd support. It is away from the vertex line for the portion lying between 2nd and 3rd support and so on up to the mid length of the tube. The results also show that the ends of bent absorber tube have maximum slope (Fig. 11) and the slope at the supporting points reduces as one moves along the length from one end towards the middle. It is due to the fact that the ends of absorber tube are free to rotate and the rotation at other supporting points is not easy due to continuation of absorber tube around these points. 5.2. Effect of number of supports (n) and length of tube (L) keeping spacing between adjacent supports (d) fixed The deflection in the tube from the focal line (due to bending caused by non-uniformity in tube temperature) has been plotted in Fig. 12 for different values of DTf, number of supports (n) and length of tube (L) keeping spacing between adjacent supports (d) as 4 m. The results show that as one shifts from n ¼ 2 (L ¼ 4 m) to n ¼ 3

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Fig. 9. Temperature of tube after heating during steady state.

S. Khanna, V. Sharma / Energy 93 (2015) 1788e1803

1799

Fig. 10. Deflection in the tube (in the vertical plane passing through the central axis of the tube) due to bending caused by circumferential non-uniformity in the tube's temperature.

(L ¼ 8 m), the maximum deflection decreases. However, as one shifts from n ¼ 3 (L ¼ 8 m) to n ¼ 4 (L ¼ 12 m), the maximum deflection increases but remains lesser than the case n ¼ 2 (L ¼ 4 m). It again decreases as one shifts from n ¼ 4 (L ¼ 12 m) to n ¼ 5 (L ¼ 16 m). The maximum deflection does not vary beyond n ¼ 5 (L ¼ 16 m). Thus, it can be concluded (for the chosen system) that if lengths of tubes are larger than 16 m, the maximum bending remains same. 6. Conclusions In the current work, an analytical expression is presented for finding the deflection in the absorber tube from the focal line of trough (due to bending caused by non-uniformity in tube's temperature). Since the absorber tube is considered to be fully filled with fluid and only axial variation in the fluid temperature is considered, the absorber's temperature distribution is symmetric about the plane passing through focal line and vertex line of trough. Due to the temperature symmetry and the non-consideration of bending moment due to weight, the deflection in absorber tube (due to bending caused by non-uniformity in temperature distribution) occurs in the plane passing through the focal line and vertex line of trough. In the current work, the absorber tube is supported at various equidistant points along the length. At the supporting points, the supports do not allow the tube to deflect from the focal line but allow to rotate in the plane passing through the vertex line and focal line of trough. The supports can move along the length so that the absorber tube can elongate freely on heating. For same type of supports, an experimental set-up has been created to verify the analytical calculations. The tube

(length ¼ 5.86 m, inner diameter ¼ 38.9 mm and outer diameter ¼ 42.2 mm) is supported at 4 equidistant points along the length. The results show that the analytical calculations differ from the experimental measurements within the range 15.4% to 16.9%. Further, the effect of number of supports (n) on the shape of bent tube has been analyzed using analytical equation. The conclusions are as follows. (i) The ends of the bent absorber tube have maximum slope and the slope at the supporting points reduces as one moves along the length from one end towards the middle. (ii) The shape of bent tube is symmetric around the mid length. For one half of the tube, the bending is towards the vertex line of trough for the portion lying between 1st and 2nd support. It is away from the vertex line for the portion lying between 2nd and 3rd support and so on up to the mid length of the tube. (iii) Keeping spacing between adjacent supports fixed, as number of supports and length of tube increase, the maximum bending does not vary beyond a certain length of tube. For the chosen system (keeping spacing between adjacent supports as 4 m), the results show that if lengths of tubes are larger than 16 m, the maximum bending remains same. Thus, using the analytical equations, the deflection (due to bending) in the absorber tube from the focal line of trough can be calculated for any number of supports and spacing between them. Using appropriate values of these parameters, the deflection of absorber tube from the focal line can be reduced. Thus, the solar flux intercepted by the absorber tube can be increased. Further, the

Table 4 Values of the parameters used for the calculations. Parameters

Values

Parameters

Values

E Ibn

190 GPa 950 W/m2 [43] Function of temperature [43] 0.057 m [43] 0.06 m [43] 0.033 m [43] 0.035 m [43] 30  C [43] 293  C [43] 2 m/s [43] 5.76 m [44]

a ath Dz

0.96 [43,44] 17.3  106/ C [43] 0.01 m 0.89 [43] Function of temperature [43] 80 [44] 0.94 [43,44] 8 mrad 4.1 mrad [45] 0.96 [43,44] 20 [43]

k rci rco rti rto Ta Tf, inlet vw w

εc εt

qrim r soptical ssun,j¼0 t j

1800

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Fig. 11. Deflection in the tube from the focal line (due to bending caused by non-uniformity in tube temperature) for different values of DTf and number of supports (n) keeping length of the tube fixed as 12 m.

S. Khanna, V. Sharma / Energy 93 (2015) 1788e1803

1801

Fig. 12. Deflection in the tube from the focal line (due to bending caused by non-uniformity in tube temperature) for different values of DTf and number of supports (n) keeping distance between adjacent supports fixed as 4 m.

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S. Khanna, V. Sharma / Energy 93 (2015) 1788e1803

required spacing between absorber tube and glass cover can also be found out. It must be mentioned that the experimental measurements and the analytical calculations are carried out for a fixed heat input. However, in real conditions, the solar radiation keeps on changing. As solar radiation increases, the deflection in the tube (due to bending) increases because of increase in the circumferential non-uniformity in tube's temperature. During the operating hours of the parabolic trough, the solar radiation does not change much in an interval of 10 min. Thus, the bending in the tube during that time interval can be calculated using steady state analysis. It must also be mentioned that the analytical equations presented in this work are based on some assumptions presented in Section 2. The variations in the results due to the relaxation of these assumptions can be explained as follows. (i) The space between the glass cover and the absorber tube is considered to be evacuated in the work. If it is not the case, the heat loss will increase. Thus, the absorber temperature and its circumferential non-uniformity will decrease leading to decrement in bending. (ii) The variation in the fluid temperature is considered only along the length of tube due to the well mixing of the fluid. This assumption reduces the circumferential non-uniformity in tube's temperature because the mixing of fluid transfers the heat from hotter portion of tube to cooler one. Thus, if there is some circumferential non-uniformity in the fluid temperature, the circumferential non-uniformity in the tube's temperature will increase and the bending will increase. (iii) Inner and outer radii of the absorber tube are considered to remain unchanged even after heating. However, on heating, the cross section of the tube will not remain circle which will affect the bending. (iv) Bending moment induced in the absorber tube due to its weight is not considered in the work. However, if spacing between adjacent supports is very large, the bending due to weight is also considerable. If it will be considered, the maximum bending in the tube will increase and the bending will not lie in the plane passing through focal line and vertex line of trough during non-zero tracking angle of trough. It is due to the fact that the bending due to weight lies in the vertical plane and the bending due to temperature gradient lies in the plane passing through focal line and vertex line. Thus, the net bending will lie in the resultant plane.

References [1] Evans DL. On the performance of cylindrical parabolic solar concentrators with flat absorbers. Sol Energy 1977;19:379e85. [2] Nicolas RO, Duran JC. Generalization of the two-dimensional optical analysis of cylindrical concentrators. Sol Energy 1980;25:21e31. [3] Rabl A. Active solar collectors and their applications. New York: Oxford University Press; 1985. [4] Guven HM, Bannerot RB. Determination of error tolerances for the optical design of parabolic troughs for developing countries. Sol Energy 1986;36: 535e50. [5] Jeter SM. Calculation of the concentrated flux density distribution in parabolic trough collectors by a semi finite formulation. Sol Energy 1986;37:335e45. [6] Jeter SM. Analytical determination of the optical performance of practical parabolic trough collectors from design data. Sol Energy 1987;39:11e21. [7] Hegazy AS, EL-Kassaby MM, Hassab MA. Prediction of concentration distribution in parabolic trough solar collectors. Int J Sol Energy 1994;16:121e35. [8] Thomas A, Guven HM. Effect of optical errors on flux distribution around the absorber tube of a parabolic trough concentrator. Energy Convers Manage 1994;35:575e82.

[9] Yang B, Zhao J, Xu T, Zhu Q. Calculation of the concentrated flux density distribution in parabolic trough solar concentrators by Monte Carlo ray-trace method. In: Symposium on Photonics and Optoelectronic (SOPO). June 2010 at Chengdu, China; 2010. [10] Cheng ZD, He YL, Cui FQ. A new modelling method and unified code with MCRT for concentrating solar collectors and its applications. Appl Energy 2013;101:686e98. [11] Cheng ZD, He YL, Cui FQ, Du BC, Zheng ZJ, Xu Y. Comparative and sensitive analysis for parabolic trough solar collectors with a detailed Monte Carlo raytracing optical model. Appl Energy 2014a;115:559e72. [12] Wang K, He YL, Cheng ZD. A design method and numerical study for a new type parabolic trough solar collector with uniform solar flux distribution. Sci China Technol Sci 2014;57:531e40. [13] Wang F, Tan J, Ma L, Wang C. Effects of glass cover on heat flux distribution for tube receiver with parabolic trough collector system. Energy Convers Manage 2015;90:47e52. [14] Reddy KS, Satyanarayana GV. Numerical study of porous finned receiver for solar parabolic trough concentrator. Eng Appl Comput Fluid Mech 2008;2: 172e84. [15] Cheng ZD, He YL, Xiao J, Tao YB, Xu RJ. Three-dimensional numerical study of heat transfer characteristics in the receiver tube of parabolic trough solar collector. Int Commun Heat Mass Transf 2010;37:782e7. [16] Wang F, Shuai Y, Yuan Y, Yang G, Tan H. Thermal stress analysis of eccentric tube receiver using concentrated solar radiation. Sol Energy 2010;84: 1809e15. [17] Munoz J, Abanades A. Analysis of internal helically finned tubes for parabolic trough design by CFD tools. Appl Energy 2011a;88:4139e49. [18] Munoz J, Abanades A. A technical note on application of internally finned tubes in solar parabolic trough absorber pipes. Sol Energy 2011b;85:609e12. [19] Cheng ZD, He YL, Cui FQ, Xu RJ, Tao YB. Numerical simulation of a parabolic trough solar collector with nonuniform solar flux conditions by coupling FVM and MCRT method. Sol Energy 2012a;86:1770e84. [20] Cheng ZD, He YL, Cui FQ. Numerical study of heat transfer enhancement by unilateral longitudinal vortex generators inside parabolic trough solar receivers. Int J Heat Mass Transf 2012b;55:5631e41. [21] Islam M, Karim A, Saha SC, Miller S, Yarlagadda PK. Three dimensional simulation of a parabolic trough concentrator thermal collector. In: Proceedings of the 50th Annual Conference, Australian Solar Energy Society (Australian Solar Council), Melbourne, Australia. Dec. 2012; 2012. p. 1e11. [22] Wang F, Shuai Y, Yuan Y, Liu B. Effects of material selection on the thermal stresses of tube receiver under concentrated solar irradiation. Mater Des 2012;33:284e91. [23] Wang P, Liu DY, Xu C. Numerical study of heat transfer enhancement in the receiver tube of direct steam generation with parabolic trough by inserting metal foams. Appl Energy 2013;102:449e60. [24] Roldan MI, Valenzuela L, Zarza E. Thermal analysis of solar receiver pipes with superheated steam. Appl Energy 2013;103:73e84. [25] Yaghoubi M, Ahmadi F, Bandehee M. Analysis of heat losses of absorber tubes of parabolic through collector of Shiraz (Iran) solar power plant. J Clean Energy Technol 2013;1:33e7. [26] Cheng ZD, He YL, Wang K, Du BC, Cui FQ. A detailed parameter study on the comprehensive characteristics and performance of a parabolic trough solar collector system. Appl Therm Eng 2014b;63:278e89. [27] Song X, Dong G, Gao F, Diao F, Zheng L, Zhou F. A numerical study of parabolic trough receiver with nonuniform heat flux and helical screw-tape inserts. Energy 2014;77:771e82. [28] Wu Z, Li S, Yuan G, Lei D, Wang Z. Three-dimensional numerical study of heat transfer characteristics of parabolic trough receiver. Appl Energy 2014;113: 902e11. [29] Chang C, Li X, Zhang QQ. Experimental and numerical study of the heat transfer characteristics in solar thermal absorber tubes with circumferentially non-uniform heat flux. Energy Procedia 2014;49:305e13. [30] Natarajan M, Thundil karuppa Raj R, Raja Sekhar Y, Srinivas T, Gupta P. Numerical simulation of heat transfer characteristics in the absorber tube of parabolic trough collector with internal flow obstructions. ARPN J Eng Appl Sci 2014;9:674e81. [31] Patil RG, Panse SV, Joshi JB. Optimization of non-evacuated receiver of solar collector having non-uniform temperature distribution for minimum heat loss. Energy Convers Manage 2014;85:70e84. [32] Wang Y, Liu Q, Lei J, Jin H. Performance analysis of a parabolic trough solar collector with non-uniform solar flux conditions. Int J Heat Mass Transf 2015;82:236e49. [33] Almanza R, Lentz A, Jimenez G. Receiver behavior in direct steam generation with parabolic troughs. Sol Energy 1997;61:275e8. [34] Almanza R, Jimenez G, Lentz A, Valdes A, Soria A. DSG under two-phase and stratified flow in a steel receiver of a parabolic trough collector. J Sol Energy Eng 2002;124:140e4. [35] Flores V, Almanza R. Direct steam generation in parabolic trough concentrators with bimetallic receivers. Energy 2004a;29:645e51. [36] Flores V, Almanza R. Behavior of the compound wall copperesteel receiver with stratified two-phase flow regimen in transient states when solar irradiance is arriving on one side of receiver. Sol Energy 2004b;76:195e8. [37] Yaghoubi M, Akbarimoosavi M. Three dimensional thermal expansion analysis of an absorber tube in a parabolic trough collector. In: SolarPACES. Sept. 2011 at Spain; 2011.

S. Khanna, V. Sharma / Energy 93 (2015) 1788e1803 [38] Akbarimoosavia SM, Yaghoubi M. 3D thermal-structural analysis of an absorber tube of a parabolic trough collector and the effect of tube deflection on optical efficiency. Energy Procedia 2014;49:2433e43. [39] Kreith F, Goswami DY. Handbook of energy efficiency and renewable energy. CRC Press, Taylor and Francis Group; 2007. [40] Khanna S, Singh S, Kedare SB. Explicit expressions for temperature distribution and deflection in absorber tube of solar parabolic trough concentrator. Sol Energy 2015;114:289e302. [41] Khanna S, Kedare SB, Singh S. Analytical expression for circumferential and axial distribution of absorbed flux on a bent absorber tube of solar parabolic trough concentrator. Sol Energy 2013;92:26e40.

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[42] Beer FP, Johnston ER, Dewolf JT, Mazurek DF. Mechanics of materials. 5th ed. New Delhi: The Tata McGraw Hill Education Private Limited; 2010. [43] Burkholder F, Kutscher C. Heat loss testing of Schott's 2008 PTR70 parabolic trough receiver. Technical Report NREL/TP-550e45633. United States: National Renewable Energy Laboratory; 2009. [44] Garcia FA, Zarza E, Valenzuela L, Perez M. Parabolic-trough solar collectors and their applications. Renew Sustain Energy Rev 2010;14:1695e721. [45] Rabl A, Bendt P, Gaul HW. Optimization of parabolic trough solar collectors. Sol Energy 1982;29:407e17.