Explicit expression for temperature distribution of receiver of parabolic trough concentrator considering bimetallic absorber tube

Applied Thermal Engineering 103 (2016) 323–332

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Research Paper

Explicit expression for temperature distribution of receiver of parabolic trough concentrator considering bimetallic absorber tube Sourav Khanna a,⇑,1, Vashi Sharma b,⇑,1, Suneet Singh a, Shireesh B. Kedare a 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

h i g h l i g h t s
Explicit expression is derived for temperature distribution of bimetallic tube.
Design calculations consume significantly lesser time using the expression.
Material with higher thermal-conductivity should be used as outer layer.
Best rim angle is found out for a given aperture width of trough.

a r t i c l e

i n f o

Article history: Received 25 October 2015 Accepted 21 April 2016 Available online 23 April 2016 Keywords: Parabolic trough Absorber tube Bimetallic Solar flux Temperature

a b s t r a c t The portion of the absorber tube facing the trough surface receives the concentrated sun-rays and the other side of the absorber tube receives the sun-rays directly. Consequently, the temperature of the absorber tube is non-uniform across the circumference which leads to differential expansion of the material of the tube. Thus, the tube experiences compression and tension in its different parts. This may lead to bending of the tube. In literature, the temperature of the absorber tube is computed using CFD software which take large computational time. Thus, in the previous work, an explicit analytical expression was derived for finding the distribution of absorber’s temperature and it was found that the temperature gradient across the circumference of the absorber tube can lead to significant bending. Thus, in the current work, a bimetallic tube has been studied that can reduce the temperature gradient and an explicit analytical expression is derived for finding the temperature distribution of a bimetallic absorber tube. The study of the effects of thicknesses and material selection of the inner and outer layer of bimetallic tube on temperature distribution is a must for choosing right materials and dimensions. The appropriate thicknesses and materials of inner and outer layers can be found out from the current work. The issue of whether to use high conducting material on outside or inside has also been addressed in the current work and concluded that the material with higher thermal-conductivity should be used as outer layer of the bimetallic tube to minimize the non-uniformity across the circumference. It is also concluded that for Schott-2008-PTR70-receiver, 126°, 135° and 139° respectively are the appropriate rim-angles for trough’s aperture-width = 3 m, 6 m and 9 m corresponding to minimum non-uniformity across the circumference. 72°, 100° and 112° respectively correspond to maximum solar-flux at the absorber tube. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction The electricity generation using solar parabolic trough concentrators is one of the economically feasible renewable technologies. The parabolic trough concentrates the sun-rays at its focal line, ⇑ Corresponding authors. E-mail addresses: [email protected] (S. Khanna), [email protected] (V. Sharma). 1 Both are first author. http://dx.doi.org/10.1016/j.applthermaleng.2016.04.110 1359-4311/Ó 2016 Elsevier Ltd. All rights reserved.

when tracked appropriately. A selectively coated tube with a concentric glass cover (used for reducing heat losses) is generally used as receiver which is placed such that its central axis is aligned with the focal line of the trough. The absorber tube receives the concentrated solar flux only on the portion facing the reflector. Consequently, the temperature of the absorber tube is non-uniform. Almanza et al. [1] have measured the circumferential difference in the temperature of the absorber tubes made up of steel and copper. Almanza et al. [2] have extended the previous work [1] to analyze the stratified fluid flow. Flores and Almanza [3] have analyzed

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Nomenclature dF1–2 f hf Ibn k L qA qc qL r T UL

vw

w

view factor of differential surface 1 with respect to differential surface 2 focal length of the parabolic trough collector (m) convective heat transfer coefficient on the inner surface of absorber tube (W/m2 K) beam normal radiation (W/m2) thermal conductivity of the material of absorber tube (W/m K) length of the parabolic trough and the absorber tube (m) solar flux absorbed by the absorber tube per unit outer surface area (W/m2) radiative heat loss from the inner surface of glass cover (W/m2) rate of heat loss from the outer surface of absorber tube (W/m2) radius (m) temperature (K) over all heat loss coefficient of the receiver (W/m2 K) wind velocity (m/s) width of the aperture of parabolic trough (m)

Greek symbols a absorptivity of the absorber tube DTf rise in fluid temperature per unit length of receiver (averaged over the whole length) (K/m) Dz length of each segment of the receiver (m)
emissivity for long wavelength radiation hrim rim angle of the parabolic trough (rad) angle up to which the circumference of the absorber hshd tube does not receive concentrated rays due to the shadow cast by absorber tube on trough (rad) q reflectivity of the surface of parabolic trough qsa(w) effective product of reflectivity, transmissivity and absorptivity for concentrated rays at w angle of incidence

the bimetallic receiver made up of steel and copper. Flores and Almanza [4] have analyzed the case when the concentrated solar radiation falling on one side of the absorber tube instead of the lower periphery. Apart from the experimental measurements, numerical studies are also available in which the distribution of solar flux is computed using Monte Carlo Ray Tracing (MCRT) software and the distribution of absorber’s temperature is computed using Computational Fluid Dynamics (CFD) software. In these studies, different types of receivers have been analyzed which are as follows. Reddy and Satyanarayana [5] have considered the porous fins (square, triangular, trapezoidal and circular) inside the absorber tube. Cheng et al. [6] have considered a flow restriction device (a concentric tube) inside the absorber tube. Wang et al. [7] have analyzed an eccentric absorber tube. Munoz and Abanades [8,9] have carried out the calculations considering the helical fins inside the absorber tube. Cheng et al. [10] have considered the residual gases in the space between the absorber tube and the glass cover. Cheng et al. [11] have studied the effect of the longitudinal vortex generators inside the absorber tube. Wang et al. [12] have studied the selection of the material of the absorber tube. Wang et al. [13] have considered the metal foams inside the absorber tube. Roldan et al. [14] have compared the numerical calculations of the absorber’s temperature with the experimental measurements. Yaghoubi et al. [15] have analyzed the receiver by considering vacuum in the space between the absorber tube and the glass cover and

r Stefan–Boltzmann constant (W/m2 K4) rsun,w=0° equivalent rms angular width of sun, at w = 0°, in line roptical rtot

s sa(w) w

focus geometry (rad) equivalent rms angular spread caused by all optical errors (rad) equivalent rms angular spread caused by sun-shape and optical errors (rad) transmissivity of the glass cover effective product of transmissivity and absorptivity for direct incident rays at w angle of incidence angle of incidence of sun-rays at trough’s aperture, i.e. the angle between the incident sun-ray and the normal to aperture plane (rad)

Abbreviation CFD Computational Fluid Dynamics MCRT Monte Carlo Ray-Trace Subscripts a ambient c glass cover ci inner surface of glass cover co outer surface of glass cover f fluid fr flow restriction device i inner layer of bimetallic absorber tube inlet inlet of receiver int interface of inner and outer layers of bimetallic absorber tube j jth segment of absorber tube and glass cover o outer layer of bimetallic absorber tube sky sky t absorber tube ti inner surface of absorber tube to outer surface of absorber tube

compared it with the case in which vacuum is not considered. Cheng et al. [16] have analyzed the effects of various parameters on the distribution of absorber’s temperature. Wang et al. [17] have investigated the receiver with a secondary reflector. Song et al. [18] have considered a helical screw-tape inside the absorber tube. Wu et al. [19] have presented the distributions of the temperature of absorber tube and glass cover. Natarajan et al. [20] have carried out the calculations considering the inserts (triangular, inverted triangular and semi circular) inside the absorber tube. Patil et al. [21] have computed the distribution of absorber’s temperature considering the sun to be a point source. Wang et al. [22] have analyzed the effect of the fluid’s inlet temperature, the fluid’s velocity and the solar radiation on the distribution of absorber’s temperature. Guiqiang et al. [23] have analyzed the distribution of the solar flux on the flat receiver of compound parabolic concentrator and found that the distribution for concentrator having lens-walls is more uniform in comparison to the one having mirror-walls. Guiqiang et al. [24] have extended the previous work [23] and found that the optical efficiency of the concentrator can be increased by using air gap between the lens-walls and the reflector. Guiqiang et al. [25] have extended the previous work [24] to analyze the concentrated photovoltaic/thermal system. Thus, to summarize, the distribution of the temperature of absorber tube of parabolic trough has been reported using CFD software. However, using explicit expressions, the design

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calculations consume significantly lesser time and lesser computational effort due to the absence of iterations and the problems related to the convergence of solution. Thus, in the previous work [26], an explicit analytical expression was derived for finding the distribution of absorber’s temperature and it was found that the temperature gradient across the circumference of the absorber tube can lead to significant bending [26–30]. Thus, in the current work, a bimetallic tube has been studied that can reduce the temperature gradient and an explicit analytical expression is derived for finding the temperature distribution of a bimetallic absorber tube. The appropriate pair of thicknesses and materials of inner and outer layers can also be found out from the current work. The issue of whether to use high conducting material on outside or inside has also been addressed in the current work. The best rim angle for a given aperture-width of the trough is also found out corresponding to the minimum non-uniformity in the absorber’s temperature across its circumference. 2. Methodology The system under consideration consists of a parabolic trough. The aperture width, rim angle and length of the trough are denoted by w, hrim and L respectively. A bimetallic absorber tube with a concentric glass cover is aligned to the focal line of trough. The absorber tube is made up of two concentric layers of different materials (Fig. 1). The geometries of both absorber tube and glass cover are defined by cylindrical coordinates (r, h, z). The angle h is named as circumferential angle of the absorber tube and is measured from OA in anti-clockwise direction (Fig. 1). The inner and outer radii of the glass cover are rci and rco respectively and those of the bimetallic absorber tube are rti and rto respectively. The radius of the interface of the layers of the absorber tube is named as rint. It means that the thicknesses of inner and outer layers of the absorber tube are (rint  rti) and (rto  rint) respectively. The thermal conductivities of the materials of the inner and the outer layers are ki and ko respectively. The temperature of the absorber tube can be given by




Tðr; h; zÞ ¼

T i ðr; h; zÞ

if rti 6 r 6 rint

T o ðr; h; zÞ if rint 6 r 6 r to

ð1Þ

The methodology described in this work is based on the following assumptions (i) The solidity factor of the trough is considered to be unity. (ii) The variations in the temperature of the glass cover are considered only in circumferential and axial directions and ignored in radial direction. (iii) The space between the absorber tube and the glass cover is perfectly evacuated and the convective heat transfer is negligible between the two.

(iv) Since the absorber tube is fully filled with the fluid, the fluid is in contact with the whole inner surface of the absorber tube. (v) The variation in the fluid temperature is considered only along the length. 2.1. Solar flux distribution on the absorber tube Based on the availability of the solar flux, the circumference of the absorber tube can be divided into different portions which are as follows: 0° 6 h 6 hshd, hshd < h 6 hrim, hrim < h 6 90° and 90° < h 6 180° (where hshd is the angle up to which the circumference of the absorber tube does not receive the solar flux due to the shadow cast by the absorber on the trough before considering the sun-shape and the optical errors) [31,32]. For hrim > 90°, the division will be as follows: 0° 6 h 6 hshd, hshd < h 6 90°, 90° < h 6 hrim and hrim < h 6 180° [31,32]. The expression for the distribution of the solar flux absorbed by the absorber tube (incorporating the effect of Gaussian sun shape and optical errors) can be written as follows [32]

qA ðhÞ ¼

h0 ¼p

dqA ðh; h0 Þ

ð2Þ

where

8 0 > > > > 2f qsaðwÞIbn cos w h 2 0i > < rto f1þcospffiffiffiffi ðh;h Þ h0 g exp b db 2r2tot ðwÞ rtot ðwÞ 2p dqA ðh; h0 Þ ¼ > þ > > ½saðwÞIbn cos wð cos h0 Þ > > : 0

if jh0 j 2 ½0;hshd  if jh0 j 2 ðhshd ;hrim ;jh  h0 j 6 /ðh0 Þ if jh0 j 2 ðhrim ; p;h ¼ h0 otherwise ð3Þ

2.2. Temperature distribution on the absorber tube Some fraction of the absorbed solar flux will be lost from the outer surface of the absorber tube to the glass cover and then glass cover to surroundings. The circumferential non-uniformity in the distribution of solar flux leads to non-uniform distribution of heat losses across the circumference. It is observed that there exists a distance Dz such that the radiative loss by any differential surface of the absorber tube located at (rto, h, z) to the differential surface of the glass cover located at (rci, h + c, z + l) is negligible if |l| > Dz/2. Dividing the absorber tube and the glass cover into small sections of length Dz and assuming uniform temperatures along the length within the section, the distribution of radiative loss (qL,j) from the outer surface of the jth section of the absorber tube to the inner surface of the glass cover can be written (using radiosityirradiation method [26,33]) as follows:

qL;j ðhÞ


t

qc;j ðhÞ


c

Fig. 1. Cross sectional view of the system.

Z p

¼

¼

qt;j ðhÞ


t

¼

  1 qt;j ðh þ cÞdF to—to ðcÞ
t c¼p    1 þ  1 qc;j ðh þ cÞdF to—ci ðcÞ
c Z p h rfT 4t;j ðhÞ  T 4t;j ðh þ cÞgdF to—to ðcÞ þ c¼p i þ rfT 4t;j ðhÞ  T 4c;j ðh þ cÞgdF to—ci ðcÞ



1

  1 qt;j ðh þ cÞdF ci—to ðcÞ
t c¼p    1 þ  1 qc;j ðh þ cÞdF ci—ci ðcÞ
c Z p ½rfT 4c;j ðhÞ  T 4t;j ðh þ cÞgdF ci—to ðcÞ þ c¼p n o þ r T 4c;j ðhÞ  T 4c;j ðh þ cÞ dF ci—ci ðcÞ Z p



Z p

ð4aÞ

1

ð4bÞ

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where
t and
c are the emissivities for long wavelength radiation of the absorber tube and the glass cover respectively. qt,j is the radiative heat loss by a differential surface of the jth section of the absorber tube and qc,j is the radiative heat loss by a differential surface of the inner surface of the glass cover. dF1–2 is the view factor of the differential surface 1 with respect to differential surface 2 [26]. Since the heat losses from the absorber tube to the glass cover is same as that of the glass cover to the surroundings, qL,j(h) can be written as

h

n oi r co qL;j ðhÞ ¼ hw fT c;j ðr co ; hÞ  T a g þ r2c T 4c;j ðr co ; hÞ  T 4sky  r to

ð4cÞ

ð4dÞ

The difference of the absorbed solar flux and the heat losses from the absorber tube is the remaining heat flux that enters the jth section of the outer layer of the absorber tube through conduction and can be written as

  @T o;j ðr; hÞ ko ¼ qA ðhÞ  qL;j ðhÞ @r r¼r to

ð5Þ

Following the below conduction equation for the jth section of the absorber tube, the heat flux reaches the interface of the outer and inner layers of the absorber tube. 2

2

@ T o;j ðr; hÞ 1 @T o;j ðr; hÞ 1 @ T o;j ðr; hÞ þ 2 þ ¼ 0; @r 2 r @r r @h2

8r 2 ðr int ; r to Þ

ð6Þ

Equating the heat flux at the interface of the outer and inner layers of the absorber tube (at r = rint), the following energy balance can be written

    @T o;j ðr; hÞ @T i;j ðr; hÞ ko ¼ ki @r @r r¼r int r¼r int

T i;j ðr; hÞ ¼ T i;j ðr; hÞ

ð13Þ

T o;j ðr; hÞ ¼ T o;j ðr; hÞ

ð14Þ

Eqs. (13) and (14) give 0

en ¼ f n ¼ e0n ¼ f n ¼ 0

ð7Þ

ð8Þ

The heat flux at the interface (at r = rint), enters the inner layer of the absorber tube and follows the below conduction equation

@ 2 T i;j ðr; hÞ 1 @T i;j ðr; hÞ 1 @ 2 T i;j ðr; hÞ þ 2 þ ¼ 0; @r 2 r @r r @h2

8r 2 ðr ti ; r int Þ

  @T i;j ðr; hÞ ki ¼ hf ;j ½T i;j ðr ti ; hÞ  T f ;j  @r r¼rti

T i;j ðr; hÞ ¼ a þ b ln r þ

1 X n¼1

þ

1 X ðen r n þ f n r n Þ sin nh

ð11Þ

are

ð16Þ

b¼h

hf ;j T f ;j  hf ;j a hf ;j lnðr ti Þ  rktii

i

ð17Þ

n1 00  n 00 

 00  n1 00 ko r b r c þ rnint  ko rn r int c  rn1 int b int 



 cn ¼ n1 00 int n1  int n n n 00 n1 00 00 ki rint a  ki r int rn rn1 int c þ r int  ko r int þ ko r int a int c  r int ð18Þ

dn ¼ a00 cn a0 ¼ a þ

ð19Þ

   ki lnðr int Þ b 1 ko

ki b ko

0

b ¼

ð20Þ

ð21Þ



c0n

 00  n1 00 ko r n1 þ ki r n1 int b int  ki r int a cn

n1  ¼ 00 ko rint  ko r n1 int c

0

00

dn ¼ b þ c00 c0n where

q¼

1 2p

qn ¼

1

p

Z p p

Z p p

ð22Þ ð23Þ

qA ðhÞdh

ð24Þ

qA ðhÞ cos nhdh

ð25Þ

ki nr n1  hf ;j rnti ti n1 ki nr ti þ hf ;j rn ti

ð26Þ

b ¼

qn U L;j r n  ko nr n1 to to

ð27Þ

c00 ¼

n ko nr n1 to þ U L;j r to n1 ko nr to  U L;j rn to

ð28Þ

a00 ¼ 00

ðcn rn þ dn r n Þ cos nh

and

0 dn

ti

ð10Þ

By applying the method of separation of variables [35], the Eqs. (6) and (9) are solved and the solutions are as follows:

c0n

h i h  n oi ðqþU L;j T a Þ hf ;j lnðrti Þ rktii ðhf ;j T f ;j Þ U L;j 1 kkoi lnðrint Þþ kkoi rktoo þU L;j lnðrto Þ h i h  n oi a¼ U L;j hf ;j lnðr ti Þ rki hf ;j U L;j 1 kkoi lnðrint Þþ kkoi rktoo þU L;j lnðrto Þ

ð9Þ

The heat flux that reaches the inner surface of the inner layer of the absorber tube will go to the fluid of jth section and can be found out using following energy balance

0

The following expressions for a, b, cn, dn, a , b , derived by solving Eqs. (A.1)–(A.8) of Appendix A.

Eq. (1) gives the following equality

T o;j ðr int ; hÞ ¼ T i;j ðrint ; hÞ

ð15Þ 0

where hw is the convective heat transfer coefficient on the outer surface of the glass cover, i.e. the wind heat transfer coefficient. The heat losses from the absorber tube can also be written in terms of overall heat loss coefficient (UL) as [34]

qL;j ðhÞ ¼ U L;j fT o;j ðrto ; hÞ  T a g

where n is an integer. Since, the fluid temperature is assumed to vary only in axial direction and the distribution of solar flux is symmetric about h = 0° (Fig. 1), the distribution of tube temperature follows the below equations

There exists a finite integer ‘m’ which satisfies the following equations

n¼1

0

T o;j ðr; hÞ ¼ a0 þ b ln r þ

1 X 0 ðc0n r n þ dn r n Þ cos nh

T i;j ðr; hÞ ¼ a þ b ln r þ

n¼1

n¼1 1 X 0 ðe0n r n þ f n r n Þ sin nh þ n¼1

1 X ðcn r n þ dn r n Þ cos nh

ð12Þ

m X  a þ b ln r þ ðcn r n þ dn r n Þ cos nh n¼1

ð29Þ

S. Khanna et al. / Applied Thermal Engineering 103 (2016) 323–332 0

T o;j ðr; hÞ ¼ a0 þ b ln r þ

327

1 X 0 ðc0n r n þ dn rn Þ cos nh n¼1

m X 0  a þ b ln r þ ðc0n r n þ dn rn Þ cos nh 0

0

ð30Þ

n¼1

3. Verification Sukhatme [36] has presented a method to solve the conduction equation numerically by following an iterative process. Thus, in order to compare the results obtained from the proposed explicit expression (Eqs. (29), (30)), Eqs. (5)–(10) are solved numerically in this section. For this, all the parameters are assumed to be uniform within the span of Dr and Dh in radial and circumferential directions respectively. Thus, in order to solve Eqs. (5)–(10) numerically, they are approximated by Eqs. (31)–(36) respectively as follows:

3 2

ko

T o;j ðr to ; hÞ  2T o;j ðr to  Dr; hÞ þ 12 T o;j ðr to  2Dr; hÞ Dr



¼ qA ðhÞ  qL;j ðhÞ 8h 2 ½p þ Dh; p

ð31Þ

T o;j ðr  Dr; hÞ  2T o;j ðr; hÞ þ T o;j ðr þ Dr; hÞ ðDrÞ2 1 T o;j ðr þ Dr; hÞ  T o;j ðr  Dr; hÞ þ r 2 Dr 1 T o;j ðr; h  DhÞ  2T o;j ðr; hÞ þ T o;j ðr; h þ DhÞ þ 2 ¼0 r ðDhÞ2

8r 2 ½rint þ Dr; r to  Dr and h 2 ½p þ Dh; p

ð32Þ

 ko

  32 T o;j ðr int ; hÞ þ 2T o;j ðrint þ Dr; hÞ  12 T o;j ðrint þ 2Dr; hÞ Dr 3  T ðr ; hÞ  2T ðr  Dr; hÞ þ 12 T i;j ðr int  2Dr; hÞ i;j int i;j int ¼ ki 2 Dr

8h 2 ½p þ Dh; p

ð33Þ

T o;j ðr int ; hÞ ¼ T i;j ðr int ; hÞ 8h 2 ½p þ Dh; p

ð34Þ

T i;j ðr  Dr; hÞ  2T i;j ðr; hÞ þ T i;j ðr þ Dr; hÞ ðDrÞ2 1 T i;j ðr þ Dr; hÞ  T i;j ðr  Dr; hÞ þ r 2 Dr 1 T i;j ðr; h  DhÞ  2T i;j ðr; hÞ þ T i;j ðr; h þ DhÞ ¼0 þ 2 r ðDhÞ2

8r 2 ½rti þ Dr; rint  Dr and h 2 ½p þ Dh; p

Fig. 2. Circumferential distribution of temperature of bimetallic absorber tube.

ð35Þ



  32 T i;j ðr ti ; hÞ þ 2T i;j ðr ti þ Dr; hÞ  12 T i;j ðrti þ 2Dr; hÞ ki Dr ¼ hf ;j ½T i;j ðr ti ; hÞ  T f ;j  8h 2 ½p þ Dh; p

ð36Þ

For calculations, a bimetallic tube has been considered having ki = 54 W/m K, ko = 400 W/m K, rti = 0.033 m, rint = 0.0335 m and rto = 0.035 m. The values of other parameters as taken by Cheng et al. [6] have been considered. Using the proposed explicit expression (Eqs. (29) and (30)), the distributions of absorber’s temperature at z = 2 m and z = 4 m are plotted in Fig. 2 along with the results calculated using the methodology of Sukhatme [36]. The results found out using the proposed explicit expression (Eqs. (29) and (30)) differ from those of Sukhatme [36] within the range of 0.02% to +0.05%. It must be mentioned that, for m = 10, the approximated expressions of the Eqs. (29) and (30) differ from the complete expressions by less than 0.04%.

4. Results and discussion Burkholder and Kutscher [37] have carried out an analysis of the heat losses from the Schott 2008 PTR70 receiver using the LS-3 parabolic trough collector and the Therminol VP1 heat transfer fluid. In their study [37], the circumference of the absorber tube is considered to be heated uniformly. Using the same troughreceiver system and the values of other parameters [37], tabulated in Table 1, the calculations have been carried out in the current work to evaluate the distributions of tube temperature. The Schott PTR70 receiver has a monometallic absorber tube. However, in the current work, a bimetallic absorber tube, made up of two concentric layers having materials stainless steel and copper, is considered. In this section, the circumferential distributions of the tube temperature are plotted for the cross section located at the mid length of the absorber tube. The materials of the bimetallic absorber tube can be chosen in two ways: (i) ki = 400 W/m K and ko = 19 W/m K and (ii) ki = 19 W/m K and ko = 400 W/m K. Both the cases have been com-

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Table 1 Values of the parameters used for the calculations. Parameters Ibn ki ko L rci rco rint rti rto Ta Tf,inlet

vw

Values

Parameters

Values

950 W/m [37] 19 W/m K@320 °C [37] 400 W/m K@320 °C 4 m [37] 0.057 m [37] 0.06 m [37]

w

5.76 m [38] 0.96 [37,38]

0.0335 m 0.033 m [37] 0.035 m [37] 30 °C [37] 293 °C [37] 2 m/s [37]

hrim

2

a DTf Dz


c
t

q roptical rsun,w=0° s w

2 °C/m 0.01 m 0.89 [37] Function of temperature [37] 80° [38] 0.94 [37,38] 8 mrad 4.1 mrad [39] 0.96 [37,38] 20° [37]

pared in Fig. 3 in terms of circumferential non-uniformity in the temperature of the absorber tube. In Fig. 3a, the thickness of both inner and outer layers of the absorber tube is taken as 1 mm (rti = 0.033 m, rint = 0.034 m, rto = 0.035 m). The results show that both the cases lead to almost same temperature distributions. It is due to the fact that the thickness is too small to conduct the heat throughout the circumference. Thus, Fig. 3b is drawn in which the thickness of both the layers is chosen as 5 mm (rti = 0.033 m, rint = 0.038 m, rto = 0.043 m). The results show that the case having ki = 19 W/m K and ko = 400 W/m K leads to lesser circumferential non-uniformity in the absorber’s temperature. It is due to the fact that the outer layer with higher thermal conductivity helps in distributing the absorbed solar flux over the circumference in more uniform manner than the other case. Thus, the material with higher thermal conductivity should be used as outer layer of the bimetallic absorber tube in order to minimize the circumferential non-uniformity in temperature. Thus, ki = 19 W/m K and ko = 400 W/m K have been used for all the calculations henceforth. 4.1. Effect of thicknesses of both layers of bimetallic absorber tube The circumferential distributions of the temperature of bimetallic absorber tube are plotted in Fig. 4 for different values of the thicknesses of inner (rint–rti) and outer (rto–rint) layers, where other parameters are kept fixed as tabulated in Table 1. Fixing the inner (rti) and outer (rto) radii of the bimetallic absorber tube, the thicknesses of inner and outer layers are varied by changing the radius of their interface (rint). The results show that as the thickness of outer layer decreases (or rint increases), the circumferential nonuniformity in the temperature of the absorber tube increases. It is due to the fact that the outer layer is of higher thermal conductivity than the inner layer of the absorber tube and higher thermal conductivity helps in distributing the absorbed solar flux over the circumference in more uniform manner. These temperature distributions can further be used to compute the tensile and compressive stresses induced in the absorber tube in order to choose the appropriate pair of thicknesses and materials of inner and outer layers. 4.2. Comparison of monometallic absorber tube with bimetallic tube The circumferential distributions of the temperature of monometallic and bimetallic absorber tube are plotted in Fig. 5a and b respectively for different values of DTf, where other parameters are kept fixed as tabulated in Table 1. DTf is the rise in fluid temperature per unit length of the absorber tube (averaged over the whole length of the absorber tube). The results show that as DTf increases, the circumferential non uniformity in the

Fig. 3. Comparison of the circumferential temperature distribution between the cases ki > ko and ki < ko.

absorber’s temperature increases. It is due to the fact that lesser value of fluid’s mass flow rate is required to achieve higher DTf. Thus, the convective heat transfer coefficient on the inner surface of the absorber tube reduces and results in higher temperature of the absorber tube. The maximum circumferential difference in the absorber’s temperature for various values of DTf is presented in Table 2. The results show that the increase in DTf from 0.5 °C/m to 3 °C/m leads to increase in the maximum circumferential difference in the absorber’s temperature from 23 °C to 56 °C for monometallic absorber tube and from 19 °C to 40 °C for bimetallic. The circumferential distributions of the temperature of monometallic and bimetallic absorber tube are plotted in Fig. 6a and b respectively for different values of roptical, where other parameters are kept fixed as tabulated in Table 1. The results show that the circumferential non-uniformity in the absorber’s temperature decreases with increase in the optical errors. It is due to the fact that increase in the optical errors leads to increase in the divergence of reflected rays (from the trough surface). Thus the solar flux gets distributed more uniformly on the

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Fig. 4. Circumferential temperature distribution for various values of rint (or thicknesses of inner and outer layers). Fig. 5. Circumferential temperature distributions of (a) monometallic absorber tube and (b) bimetallic tube for various values of DTf.

circumference of the absorber tube resulting in lesser nonuniformity in the temperature distribution. It must also be mentioned that as optical errors increase, the total solar flux intercepted by the absorber tube decreases [31,32]. The maximum circumferential difference in the absorber’s temperature for various values of roptical is presented in Table 3. The results show that the increase in roptical from 0mrad to 16mrad leads to decrease in the maximum circumferential difference in the absorber’s temperature from 50 °C to 40 °C for monometallic absorber tube and from 38 °C to 28 °C for bimetallic. The circumferential distributions of the temperature of monometallic and bimetallic absorber tube are plotted in Fig. 7a and b respectively for different values of hrim, where other parameters are kept fixed as tabulated in Table 1. The results show that as hrim increases initially, the circumferential non-uniformity in the absorber’s temperature decreases. However beyond hrim = 135°, further increment in the rim angle leads to increase in the non-uniformity. It is due to the fact that with increase in the rim angle, the solar flux gets distributed more uniformly on the circumference of the absorber tube. However beyond

Table 2 Maximum circumferential difference in absorber’s temperature for various values of DTf.

DTf (°C/m)

0.5 1.0 1.5 2.0 2.5 3.0

Maximum circumferential difference in absorber’s temperature (°C) Monometallic

Bimetallic

23 35 43 48 53 56

19 28 32 36 38 40

hrim = 135°, the solar flux intercepted by the sun facing absorber’s periphery becomes higher than that of the reflector facing periphery. The maximum circumferential difference in the absorber’s temperature for various values of hrim is presented in Table 4. The results show that the increase in hrim from 60° to 135° leads

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Fig. 7. Temperature distributions of (a) monometallic absorber tube and (b) bimetallic for various values of hrim. Fig. 6. Circumferential temperature distributions of (a) monometallic absorber tube and (b) bimetallic tube for various values of roptical.

Table 3 Maximum circumferential difference in absorber’s temperature for various values of roptical.

roptical (mrad)

0 4 8 12 16

Table 4 Maximum circumferential difference in absorber’s temperature for various values of hrim. hrim (°)

Maximum circumferential difference in absorber’s temperature (°C) Monometallic

Bimetallic

50 49 48 45 40

38 38 36 32 28

to decrease in the maximum circumferential difference in absorber’s temperature from 57 °C to 22 °C for monometallic absorber tube and from 40 °C to 8 °C for bimetallic. However beyond hrim = 135°, the increase in hrim from 135° to 160° leads to increase in the maximum circumferential difference in the absorber’s temperature from 22 °C to 31 °C for monometallic absorber tube and from 8 °C to 16 °C for bimetallic. Thus, 135° is the appropriate

60 80 100 120 140 160

Maximum circumferential difference in absorber’s temperature (°C) Monometallic

Bimetallic

57 48 37 29 22 31

40 36 26 17 9 16

rim angle (for trough’s aperture width = 5.76 m) corresponding to minimum circumferential non-uniformity in the temperature distribution of absorber tube. 5. Conclusions In the present work, an explicit expression is derived (using the method of separation of variables) for finding the distribution of

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temperature of bimetallic absorber tube in significantly lesser time than the available software. It also incorporates the effect of optical errors of the trough-receiver system and the Gaussian sun-shape. It is verified with the already existing methodology. The results found out using the proposed explicit expression differ from those of existing methodology within the range of 0.02% to +0.05%. By changing the thicknesses and the thermal conductivities of inner and outer layers of the bimetallic absorber tube, the temperature distributions are calculated for different values of desired rise in fluid temperature, optical errors and rim angle of trough keeping solar radiation, ambient conditions, receiver’s dimensions and trough’s aperture width fixed. The best rim angle for a given aperture-width of trough is also found out corresponding to the minimum circumferential non-uniformity in the absorber’s temperature. The conclusions are summarized as follows (i) It is found that the material with higher thermal conductivity should be used as outer layer of the bimetallic absorber tube in order to minimize the circumferential nonuniformity in temperature. (ii) The increase in DTf from 0.5 °C/m to 3 °C/m leads to increase in the maximum circumferential difference in the absorber’s temperature from 23 °C to 56 °C for monometallic absorber tube and from 19 °C to 40 °C for bimetallic. (iii) The increase in roptical from 0mrad to 16mrad leads to decrease in the maximum circumferential difference in the absorber’s temperature from 50 °C to 40 °C for monometallic absorber tube and from 38 °C to 28 °C for bimetallic. (iv) The increase in hrim from 60° to 135° leads to decrease in the maximum circumferential difference in the absorber’s temperature from 57 °C to 22 °C for monometallic absorber tube and from 40 °C to 8 °C for bimetallic. However beyond hrim = 135°, the increase in hrim from 135° to 160° leads to increase in the maximum circumferential difference in the absorber’s temperature from 22 °C to 31 °C for monometallic absorber tube and from 8 °C to 16 °C for bimetallic. From the calculations, it is concluded that for Schott 2008 PTR70 receiver (absorber tube with 70 mm outer diameter), 126°, 135° and 139° respectively are the appropriate rim angles for 3 m, 6 m and 9 m aperture width of trough corresponding to the minimum circumferential non-uniformity in the absorber’s temperature. And 72°, 100° and 112° respectively are the appropriate rim angles corresponding to the maximum solar flux at absorber tube for 3 m, 6 m and 9 m aperture width of trough. However, the best rim angle must be chosen only after the cost analysis because larger rim angles require larger reflector area to obtain same aperture width, which adds to cost. Appendix A 0

A.1. Equations used for solving a, b, cn, dn, a0 , b0 , c0n and dn 0

a, b, cn, dn, a0 , b0 , c0n and dn (in Eqs. (11) and (12)) can be found out by solving the following Eqs. (A.1)–(A.8) which are obtained by using Eqs. (5), (7), (8) and (10).

# Z p Z p "
@T o;j ðr; hÞ dh ¼ ko ½qA ðhÞ  U L;j fT o;j ðr to ; hÞ  T a gdh @r p p r¼rto

ðA:1Þ # # Z p "
Z p "
@T o;j ðr; hÞ @T i;j ðr; hÞ dh ¼ dh ko ki @r @r p p r¼rint r¼r int

ðA:2Þ

Z p p

T o;j ðr int ; hÞdh ¼

Z p p

T i;j ðr int ; hÞdh

# Z p Z p "
@T i;j ðr; hÞ dh ¼ ki ½hf ;j fT i;j ðr ti ; hÞ  T f ;j gdh @r p p r¼rti Z p "
p

¼

@T o;j ðr; hÞ ko @r

Z p

p

ðA:3Þ

ðA:4Þ

#



cos nh dh r¼r to

½qA ðhÞ  U L;j fT o;j ðr to ; hÞ  T a g cos nhdh

# @T o;j ðr; hÞ cos nh dh @r p r¼r int # Z p "
@T i;j ðr; hÞ ki cos nh dh ¼ @r p r¼r int

ðA:5Þ

Z p "


ko

Z p p

T o;j ðr int ; hÞ cos nhdh ¼

Z p p

T i;j ðr int ; hÞ cos nhdh

# Z p "
@T i;j ðr; hÞ ki cos nh dh @r p r¼rti Z p ¼ ½hf ;j fT i;j ðrti ; hÞ  T f ;j g cos nhdh p

ðA:6Þ

ðA:7Þ

ðA:8Þ

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