Influence of the displacement of solar receiver tubes on the performance of a parabolic-trough collector

Accepted Manuscript Influence of the displacement of solar receiver tubes on the performance of a parabolic-trough collector

Asma Aichouba, Mustapha Merzouk, Loreto Valenzuela, Eduardo Zarza, Nachida Kasbadji-Merzouk PII:

S0360-5442(18)31213-1

DOI:

10.1016/j.energy.2018.06.148

Reference:

EGY 13190

To appear in:

Energy

Received Date:

22 February 2018

Accepted Date:

21 June 2018

Please cite this article as: Asma Aichouba, Mustapha Merzouk, Loreto Valenzuela, Eduardo Zarza, Nachida Kasbadji-Merzouk, Influence of the displacement of solar receiver tubes on the performance of a parabolic-trough collector, Energy (2018), doi: 10.1016/j.energy.2018.06.148

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ACCEPTED MANUSCRIPT 1

Influence of the displacement of solar receiver tubes on the

2

performance of a parabolic-trough collector

3

Asma Aichoubaa,b, Mustapha Merzoukc, Loreto Valenzuelad,*, Eduardo Zarzad, Nachida

4

Kasbadji-Merzoukb

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a

b

Faculté de Technologie, Département de Mécanique, Université Saad Dahlab Blida 1, 09000, W. Blida, Algeria

Unité de Développement des Equipements Solaires, UDES/Centre de Développement des Energies

8 9 10 11

Renouvelables, CDER, 42004, W. Tipaza, Algeria c Faculté

d

de Technologie, Département des Energies Renouvelables, Université Saad Dahlab Blida 1, 09000, W. Blida, Algeria

Plataforma Solar de Almería, Crta. de Senes, km. 4.5, 04200 Tabernas, Almería, Spain

12 13

* Corresponding author, email address: [email protected] (L. Valenzuela)

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ABSTRACT:

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The aim of this paper is to study the influence of thermal contraction and expansion

16

phenomenon on the behaviour of the absorber tube and its alignment along the optical focal

17

line of a state-of-the-art parabolic-trough collector for commercial solar power plants. The

18

study performed reveals the influence that different ranges of operating temperatures of the

19

thermal oil used in the collectors installed in a typical commercial solar field, and other

20

parameters such the seasonal variations of sun elevation, PTC orientation, slope error, might

21

have in the amount and distribution of the solar energy intercepted by the receiver tube. The

22

study was carried out by means of a ray tracing method. A decrease of the intercepted solar

23

radiation from 2.8% to up to 38% in the range of typical operating temperatures of the solar

24

field was revealed, the decrease even exceeded the 75% when the circulating thermal oil was

25

assumed at lower temperatures (~100°C).

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Keywords

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Solar energy; parabolic-trough collector; receiver tube; intercept factor; ray-tracing; thermal

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expansion

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Acronyms

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CNRS-PROMES

Centre National de la Recherche Scientifique-Laboratoire PROcédés Matériaux et Energie Solaire

3 4

DNI

Direct Normal Irradiance

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FVM

Finite Volume Method

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HTF

Heat Transfer Fluid

7

MCRT

Monte Carlo Ray Tracing

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PSA

Plataforma Solar de Almería

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PTC

Parabolic-Trough Collector

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SEGS

Solar Electric Generating Systems

11

1.

12

Facing the great energy demand, the environmental concerns and the anticipated fossil fuel

13

shortage, moving towards a green energy policy was and still is considered as a crucial

14

alternative. Nowadays producing electricity by means of concentrated sun rays, is worldwide

15

known through the important number of power plants built all over the word. Commercial

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power plants using parabolic trough collector’s (PTC) technology were first installed in the

17

80s at the USA, experiencing a significant commercial deployment in several other countries

18

during the last decade. PTC technology is actually considered the most mature technology

19

among the current existing solar power systems.

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To produce electricity, a large number of parabolic-trough concentrators track the sun rays

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daily to concentrate them along a focal line in which is lodged a coated, glass-covered and

22

evacuated receiver tube, where the sunlight is converted into heat and transported by means of

23

a heat transfer fluid. The heat collected is then used to feed a power block. PTC technology,

24

which is currently used in most commercial power plants, works at a range of temperatures of

25

up to 400°C. Parabolic-trough concentrator and receiver tube are the key components of the

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PTC systems. Due to hard work conditions in concentrating solar power plants such as highly

27

concentrated solar irradiation, high operating temperature, large temperature difference

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between the ambient and the working temperature, the non-uniformity of the concentrated

29

solar flux and temperature distribution, receiver tubes are subjected to mechanical and thermal

30

stresses.

31

Because of their crucial role and due to the hard work conditions mentioned above, several

32

studies had focused their interests on the receiver tubes and the effects of such hard conditions

INTRODUCTION

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on their efficiency. Ifran et al., 2009 [1] studied and analysed the thermal stresses in radiant

2

tubes due to axial, circumferential and radial temperature distributions. Wang et al., 2010, [2]

3

in the aim to reduce the thermal stresses of receiver tubes installed in parabolic troughs,

4

introduced and investigated a new configuration of an eccentric tube. Wang et al., 2012, [3]

5

carried out an investigation that treated the effects of material selection on the thermal stresses

6

of receiver tubes under concentrated solar irradiation.

7

In current commercial power plants parabolic trough collectors usually operate under a

8

temperature range from about 293°C to 393°C [4], with a metallic absorber tubes made

9

typically of stainless steel, which like a large number of materials has the capacity to elongate

10

as its temperature increases and to retract as its temperature decreases. Tube deflection may

11

happen daily during the Sun tracking operation. If extreme operating conditions are achieved

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thermal stress may act cyclically and cause creep and permanent deformation in the absorber

13

tube. When a tube deforms, it will go out of the focal line and can cause different problems

14

such as the breakage of the glass envelope and it also provokes a drop of the optical and

15

thermal efficiency of the collector [4]. The failure of the glass-to-metal seal was also a quite

16

often problem of damaging to the receiver tubes at SEGS VI–IX solar power plants after 9 to

17

11 years of operation (30%-40%). This failure was derived from the concentrated flux hitting

18

the glass-to-metal seal [5]. To understand, analyse and remedy these problems several

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theoretical, numerical and experimental studies have been carried out by the references [5-

20

12], with the aim of avoiding the breakage of the absorber tube glass envelope, decreasing

21

tube deflection, preventing optical efficiency drop, keeping high factor of safety and reducing

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daily cyclic deflection amplitude of absorber tubes caused by non-uniform solar flux and

23

temperature distribution.

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Despite the complexity of the governing "thermal and optical "phenomenon occurring in the

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parabolic trough collectors and especially in the receiver tubes, several complex studies have

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been carried out, thanks to some existing computer software and codes able to handle

27

complete and complicated optical and thermal studies. Evaluation of the concentrated solar

28

irradiation, heat flux and temperature distribution, and prediction of optical and thermal

29

performances for different optical and physical characteristics of the parabolic trough

30

collectors by taking into account an uncountable number of scenarios, can be performed now

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in an easier way than in the last two decades, using Monte-Carlo ray tracing method, finite

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volume and finite element method, Computational Fluid Dynamics software, Ansys, Matlab,

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SolidWorks and other numerical codes, which facilitate the successful development of

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studies despite its complexity, as shown in references [13-21]

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Unlikely what was done in the references cited above, the aim of the current work is to study

4

the influence of a misalignment of the receiver tube caused by the cyclic-elastic deformations,

5

more precisely the thermal contraction-expansion phenomenon of the metallic absorber tubes

6

that happens daily in parabolic trough collectors installed in commercial solar power plants

7

when the oil temperature is increased during the morning start-up.

8

In fact, during the plant daily operation, the thermal oil circulates through assemblies called

9

“collector loops”, which are composed by several collectors connected in series. In current

10

solar power plants one “loop” contains four collectors connected in series, and the thermal oil

11

used as working fluid gradually increases its temperature about 100°C from the inlet to the

12

outlet of each “collector loop”. As heat is collected gradually from one collector to another,

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each collector is working at a specific temperature (within the solar field operating

14

temperature range), when the thermal oil reaches the nominal temperature, the receiver tubes

15

at each collector tend to occupy its appropriate position along the focal line of the parabola.

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With the aim of avoiding and minimizing the deflection and bending phenomenon, flexible

17

supporting arms are designed to allow the movement of the receiver tubes when the thermal

18

expansion phenomenon occurs, so that the well aligned position of receiver tubes is reached

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only when the thermal oil is at nominal temperature. The interest of the study is focused on

20

investigating the effect of the misalignment of metallic absorber tubes (of a state-of-art

21

parabolic trough collector) when it is working at temperatures lower than that allowing its full

22

extension to occupy the perfect position along the focal line of the reflective surface, on the

23

amount and the distribution of the flux. The particularity of this work is that it is relating two

24

“existing” crucial phenomenon, the daily thermal contraction/expansion and the possible

25

misalignment of the receiver tube, which may provoke a significant reduction in the optical

26

performances of a PTC. The study was done in a practical way because all the cases studied

27

were selected taking into account the temperature profile and real operating conditions of PTC

28

power plants.

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2.

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METHODOLOGY 2.1.

Description of the system

31

For most collectors designs installed in commercial solar power plants using synthetic oil as

32

heat transfer fluid, solar fields are composed by parallel rows and loops of collectors. 4

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Typically each loop contains four collectors placed in series, each collector is supposed to

2

deliver a temperature increase of about 25°C to the circulating heat transfer fluid, which

3

enters the loop usually at 293°C and leaves it at 393°C (see Figure 1). At the highest

4

temperature within the previously mentioned operating range temperature, the receiver tube

5

(last collector in the loop from figure 1), is supposed to be well aligned along the focal line of

6

the parabolic trough collector, while at lower temperature the absorber tube tends to retract so

7

it goes down in the vertical line (optical axis), [4], (see Figures 2 and 3). Before achieving the

8

highest operating temperature, the receiver tube is not correctly positioned along the focal

9

line, and it is closer to the parabolic trough reflecting surface. Gradually and till achieving the

10

maximum operating temperature range, the absorber tube will take its place along the focal

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line, so due to thermal expansion, the receiver tube tends to elongate. During the engineering

12

phase of a solar field of parabolic-trough collectors it is possible to define different design

13

lengths for the receiver supports to minimize the effect of not having the receiver tube placed

14

in the focal line during nominal operation, but in practice it may happen that all collectors are

15

built with the same configuration and then the effect explained could appear.

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Figure 1 displays the location of each collector in the loop and its corresponding range of

17

temperature in which it is working.

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Figure 1. Sketch of parabolic-trough collector’s loop composed of four collectors connected in series. Exemplary average HTF (thermal oil) temperatures considered in the study are presented.

22

Figure 2 shows a sketch of a complete parabolic trough collector composed of two sections

23

(segments) of receiver tube “joined” and fixed at one side (at the middle of the PTC where the

24

drive pylon is located) and supported by flexible supports in its other two sides (ends), taking 5

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into consideration the thermal expansion of the receiver tube. The both ends of the receiver

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tube are allowed to move with a displacement ∆y and ∆z, respectively, in the Y and Z axis

3

directions (see Fig. 2 and 3).

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Figure 2. Simplified schematic diagram of a large-size parabolic-trough collector designed for commercial solar power plants showing the longitudinal and transversal displacement of the absorber tube: (top) under cold conditions; and (bottom) under hot conditions.

8 9

Figure 3, shows a cross section of a parabolic-trough collector demonstrating the displacement that might have receiver tubes from the focal line of the parabola.

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Figure 3. Schematic diagram of the cross section of a parabolic-trough collector demonstrating the displacement of the receiver tube from the focal line of the parabola when the declination occurs (at lower temperature).

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For the purpose of the study, and referring to the current operating parabolic-trough solar

2

collector power plants, five working HTF temperatures were selected. The five cases analyzed

3

were considered as follows:

4

-

First and reference case corresponds to a well aligned receiver tube along the focal

5

line of the parabolic trough aperture, it refers to the fourth and last collector of the

6

loop which is assumed to be working under an average HTF temperature T1 = 380.5°C

7

(refer to Figure 1). In this position it is assumed that the receiver supporting arms are

8

perpendicular to the aperture of the parabola.

9

-

Second case is assumed to be at the range of temperature of about T2 = 355.5°C. T2is the mean temperature in which the third collector of the loop is working.

10 11

-

Third case corresponds to the second collector of the loop which is working under the mean working temperature T3 = 330.5°C.

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-

The fourth case is assumed to be at T4 = 305.5°corresponding to the average working temperature in which the first collector of the loop is working.

14 15

-

And finally, the fifth and last case is assumed to be at 100°C.This temperature is

16

considered as the minimum temperature in which the circulating heat transfer fluid is

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maintained at the start-up or shutdown of the plant to prevent and avoid the

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solidification of the HTF.

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For the purpose of the study, the geometrical dimensions of a Eurotrough-150 collector type

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and the diameter of the absorber tube of the receiver model PTR-70 by Schott were

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considered (see Table 1) [4,22].

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Table 1. Simulation parameters: Solar collector characteristics. Parameter Focal length of the collector Width of the reflective aperture Effective length of the tube receiver Outer diameter of the absorber tube Reflectivity of the mirrors

Value 1.71 5.76 148 0.07 0.93

Unit m m m m -

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2.2.

Simulation method.

25

In order to estimate and analyse the amount of the incident solar radiation for each of the five

26

cases and the distribution of the concentrated solar radiation on the metallic absorber tubes of

27

the solar collectors considered, numerical simulations were carried out by using a ray tracing

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code, Tonatiuh, which is an open source software, based on a Monte Carlo ray tracing

29

method, developed and maintained by the national Renewable Energy Center of Spain, and 7

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experimentally validated with different solar concentrators from test facilities located at the

2

Plataforma Solar de Almería (Spain) and the CNRS-PROMES (France) [23, 24, 25, 26]. The

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software includes an extend and complete catalogue of surfaces and shapes in its graphical

4

interface, in which optical and geometrical properties can be defined and fulfilled depending

5

on the desired solar optical device to design, which makes the handling of the software

6

relatively easy. A total of 108 simulations were carried out by considering the parameters

7

detailed in Table 1 and Table 2 (below) and the five cases corresponding to different

8

temperature of the thermal oil in the collectors composing each loop in a typical solar field of

9

a power plant.

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Table 2. Simulation parameters: Solar radiation data, sun coordinates and some optical specifications. Parameter Direct solar normal irradiance Azimuth angle Sun elevation angle Slope errors of the reflective surfaces, standard deviation Sun distribution

Value 1000 0 and 90 30, 60 and 90 1.7; 3.0 Buie

Unit W/m² degree degree mrad

12 13

The modelling of the first case, when the receiver tube is perfectly aligned along the focal line

14

of the parabolic trough collector’s aperture, was performed using the graphical interface of

15

Tonatiuh to model a complete receiver tube, which is put straight along the focal line of the

16

PTC.

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For the modelling of the other four cases, we considered a complete receiver tube divided into

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two equal segments, the cut sides were fixed (situated at the middle of the parabolic trough

19

where the drive pylon is supposed to be located) to allow the cyclic dilatation (vertical and

20

longitudinal displacement) of the metallic absorber tube due to its thermal expansion. The

21

displacement will then act only in the direction of the flexible supports located at the southern

22

and northern extremes of the receiver tube. The results of the simulations ran for the southern

23

and northern parts of the receiver tubes are then summed for each case to form a complete

24

receiver tube of the PTC. This explains why in section 3 some results are presented under the

25

nomination southern/ northern receiver tubes.

26

The simulations were performed for: a) different sun elevations, to simulate representative

27

changes of position of the sun in a day and also along the year, and b) for a “Normal”

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distribution of the sun disk and specific standard deviations for the slope error, and c) for 8

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different PTC orientations, with azimuth angles of 90° and 0° to compare the North-South-

2

and East-West orientations. Results and outputs of the simulations are displayed and discussed

3

in section 3 they reveal the impact of the misalignment of the receiver tube and the mentioned

4

criteria on the amount and distribution of the concentrated solar radiation, intercepted by the

5

metallic receiver tube.

6

2.3.

Governing equations

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2.3.1. Optical efficiency of a parabolic-trough collector

8

The optical efficiency of a parabolic-trough collector at normal incidence depends on the solar

9

reflectance of the mirrors 𝜌, the transmittance of the glass cover of the receiver 𝜏, the

10

absorptance of the absorber tube of the receiver 𝛼, and the intercept factor of the whole

11

collector 𝛾 (see Equation (1)) [4].

12

𝜂𝑜𝑝𝑡,0° = 𝜌 ∙ 𝜏 ∙ 𝛼 ∙ 𝛾

( 1)

13

For the three first parameters (𝜌, 𝜏, 𝛼), data provided by the manufacturers of the mirrors and

14

the receiver tubes or by accredited laboratories are used. For the intercept factor, which

15

accounts for the ratio between solar energy intercepted by the receiver and the solar energy

16

reflected by the parabolic-trough concentrator and includes and considers the effects of

17

optical errors [27], different techniques (e.g. photogrammetry [28]) can be applied to get a

18

representative value. Some factors that may decrease the intercept factor, and therefore the

19

energy collected in the receiver, may be a misalignment of the single facets composing the

20

optical concentrator, deformation of the supporting structure and deviation of the receiver

21

from the focal line.

22

2.3.2. Linear thermal expansion coefficient

23

As the aim of this work is the study of the influence of the thermal expansion of metallic

24

absorber tubes on the performance of a parabolic-trough collector working at five different

25

temperature ranges, Equation (2) is used to calculate the corresponding lengths L1, L2, L3, L4,

26

L5 and retractions ∆L of the receiver tubes from the reference one, which corresponds to the

27

length of the receiver at the maximum working temperature.

28

The expression of the linear thermal expansion is given by, Equation (2) [29]:

29

𝐿𝑡 = 𝐿0[ 1 +  (𝑇 ‒ 𝑇0)]

( 2)

30

Where Lt is the absorber tube length at mean temperature T (in Celsius), L0 is the absorber

31

tube length at ambient temperature (in m),  is the average coefficient of linear thermal 9

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expansion, which takes a value of 17.3·10-6 °C-1 according to the literature [30], T is the

2

working mean temperature (in Celsius), and T0 is the ambient temperature, referring to the

3

literature [30], it was assumed equal to 23°C.

4

The temperature ranges considered and the corresponding lengths and retractions of the

5

receiver tubes for the five cases are defined in Table 3. The declinations (angular deviations

6

from the focal line as it is defined below) of each receiver tube are also defined in the last

7

column.

8 9 10

Table 3. Description of the five cases studied: the ranges of temperature considered, corresponding, lengths, retractions from the reference length at the maximum temperature, and declinations of the receiver tubes. Cases Case #1 Case #2 Case #3 Case #4 Case #5

Temperature ranges (°C) T1 = 380.5 T2 = 355.5 T3 = 330.5 T4 = 305.5 T5 = 100.0

Receiver tubes lengths (m) L1 = 147.820 L2 = 147.756 L3 = 147.693 L4 = 147.630 L5 = 147.110

Retractions |∆L| (m) ∆L1 = 0 ∆L2 = 0.064 ∆L3 = 0.127 ∆L4 = 0.19 ∆L5 = 0.71

Declinations (mrad) Dec1 = 0 Dec2 = 0.433 Dec3 = 0.859 Dec4 = 1.285 Dec5 = 4.803

11

Values given in the fourth column of Table 3 are the differences (in absolute value) in length

12

with case #1. The declinations Dec1 to Dec5 are the vertical deviations from the focal line

13

when retraction phenomenon occurs to receiver tubes. This parameter is calculated by using

14

the simplified equation 3, which is the ratio of the elongation of the receiver tube ∆L by the

15

effective length of the receiver tube corresponding to the ideal conditions, L1=147.82 m.

16

𝐷𝑒𝑐𝑖 =

∆𝐿𝑖 𝐿1

( 3)

17

3.

RESULTS AND DISCUSSION

18

To get a comprehensive understanding of the influence of the receiver tube defocusing in a

19

state-of-the-art parabolic-trough collector installed in a commercial solar field, a total of 108

20

simulations were run varying parameters according to what is defined in Tables 1, 2 and 3.

21

In a first approach the amount of the total power and average flux were studied and compared.

22

Below we are representing some outputs provided by Tonatiuh. Figures 4 to 5 give an

23

overview on the amount of the total power and average flux intercepted by the receiver tubes

24

for each case study.

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In a second approach we studied and compared the flux profile and distribution for each case.

2

Here below, Figures from 09 to 15 represent outputs provided by Tonatiuh, and show the flux

3

distribution for most representative simulation case studies.

4

3.1.

Amount of total power and average flux:

5

Referring to the outputs displayed for each of the 108 simulations ran we could draw the

6

histograms presented below in Figures 4 and 5. Each figure represents the amount of total

7

solar power and average flux intercepted by the external surfaces of metallic absorber tubes

8

for the five cases and for North-South and East-West orientations. The total power and

9

average flux are expressed, respectively in kW and kW/m². It is worth mentioning that the

10

concentrator (parabola) has been modelled as a continuous reflective surface and optical

11

losses in the glass envelope of the receivers have been neglected.

12 13

Figure 4. Amount of total solar power received on metallic receiver tubes for:

14

a. North-South oriented PTC, b. East-West oriented PTC.

15 11

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Figure 5. Average flux received on the external area of metallic receiver tubes for:

2

a. North-South oriented PTC, b. East-West oriented PTC.

3

The histograms from Figures 4 to 5 reveal a significant difference in the amount of total solar

4

power and average flux intercepted by the receiver tubes in each of the 5 cases studied

5

depending on thermal expansion, and therefore defocusing (angular deviation from the

6

horizontal line (focal line)) of the receiver tubes. The histograms reveal also the influence that

7

may cause each of the parameters assumed in this study such sun elevation, PTC orientation

8

and optical error considered for the reflective surface.

9

Comparative histograms showing the difference in kW and in percentage are shown in

10

Figures 6 and 7 respectively to concretize more the analysis done. Total power received by

11

the absorber tube from case #1 and the other 4 cases, for both orientations (North-South/East-

12

West) are presented in these figures. Figure 6 shows a comparison between the total power (in

13

kW) received on well aligned receiver tube and the other 4 cases considered, where the

14

receiver tubes are defocused from the focal line of: in the left a.) PTC North-South oriented;

15

and, in the right b.) PTC oriented in East-West direction. Figure 7 shows the same comparison

16

in percentage (%).

17 18

Figure 6. Comparative histograms showing the differences in kW between total power

19

received by the receiver tube from case #1 and the other 4 cases, for:

20

a. North-South PTC orientation, b. East-West PTC orientation.

12

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1 2

Figure 7. Comparative histograms showing the differences in % between total power received

3

by the receiver tube from case #1 and the other 4 cases, for:

4

a. North-South PTC orientation, b. East-West PTC orientation.

5 6

The histograms show a significant decrease in the amount of total power and average flux

7

between case #1 and the other four cases, a decrease of more than a half (three quarters) was

8

revealed in case #5 in comparison with case #1 (ideal case), between 73.6% and 75.9% of

9

difference was revealed dependently on the sigma error slope and the orientation considered.

10

It is also noticed a non-negligible decrease of more than 38% in both total power and average

11

flux in case #4 and more than 22% in case #3. The differences between cases #1 and #2 were

12

not so relevant with decrease of about 2.8% to 5.7%. The decrease was more important when

13

sun elevation was low and also for slope error equals 3mrad.

14

It is worth mentioning again that the design of the receiver tubes supports considered through

15

the whole study is considering that the support is perfectly perpendicular to the collector

16

aperture when the mean temperature of the fluid inside the whole collector is at 380.5°C. If

17

the position of the supports is optimized to be deviated from the perpendicular when the fluid

18

is at the higher working temperature, the decrease in total power within the working

19

temperature range through the whole collector loop would be minimized.

20

By comparing the amount of total power and average flux gained by the receiver tubes in the

21

different five cases considered, when the standard deviations assumed were 𝜎𝑠𝑙𝑜𝑝𝑒 = 1.7 mrad

22

and 𝜎𝑠𝑙𝑜𝑝𝑒 = 3 mrad, we noticed:

23

a. For North-South PTC orientation, a difference varying between 0.6% to 6%. The

24

trends in decrease vary from case #2 to cases #3 to #5. And the higher differences in

25

the results are more relevant for the lower sun elevation considered (30°). 13

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b. For East-West PTC orientation, a decrease varying between 0.8% and 4%. The

2

decrease was constant (without exceptions) for all sun elevations assumed. The

3

highest difference in total power and average flux was noticed in cases #2 and #3,

4

while the difference was less important in case #4 and almost not apparent in case #5.

5

The choose of the orientation is one of the crucial elements in the design of a PTC field in

6

solar power plants [31]. Figure 8 represents comparative histograms of total power gained for

7

both orientations (North-South/East-West), in order to compare the thermal energy collected

8

for the five cases studied (we present the comparison, in the left for: a). standard deviation

9

𝜎𝑠𝑙𝑜𝑝𝑒 = 1.7 mrad, and in the right for: b). 𝜎𝑠𝑙𝑜𝑝𝑒 = 3 mrad).

10 11 12

Figure 8. Comparison between total power collected by the receiver tubes with PTC oriented at two different orientations, the East-West / North-South orientations, for:

13

a). slope error 1.7 mrad, b). slope error 3 mrad.

14

Referring to the outputs we notice significant differences in the amount of total power and

15

average flux gained by the PTC for each orientation, the figures above show clearly how the

16

amount of total power is affected by the orientation chosen. It reveals a decrease varying

17

from:

18 19 20 21

a. 50% to 54% (more than the half), when sun elevation is low (30°), the decrease is especially accentuated for 𝜎𝑠𝑙𝑜𝑝𝑒 = 3 mrad. b. For higher sun elevations (60°) the comparison revealed a decrease varying between 13.6% to 14.7%. The differences were more accentuated in cases #2 and #3.

22

c. In contrary to what was noticed previously, the analyse revealed a slight increase of

23

the amount of total power and average flux when sun elevation assumed was 90°, the

24

maximum increase noticed, did not exceed 0.1% . 14

ACCEPTED MANUSCRIPT 1

The seasonal variations in total power and average flux intercepted by a North-South PTC

2

orientation are obvious, the maximum amount of total power and average flux is noticeably

3

(obviously) high when sun elevation is high (i.e., 90°), in comparison with low sun elevation

4

(i.e., 30°). An important decrease was revealed, varying between 46% and 53%, (the higher

5

percentages were noticed for standard deviation, 𝜎𝑠𝑙𝑜𝑝𝑒 = 1.7 mrad). The comparison between

6

sun elevation 90° and 60° revealed a decrease in total power and average flux that exceeded

7

14% in almost all the cases. On the contrary to what was noticed for the North-South PTC

8

orientation, the solar energy delivery in East-West PTC orientation was quite constant

9

independently on the sun elevations considered.

10

As a summary of what has been explained above, concentrated total power and solar flux

11

change sensitively from each case to another and their magnitude drops gradually with the

12

increase of the misalignment of the receiver tube due to lower working temperature ranges

13

(defocusing effect). Concentrated solar flux and power also decrease depending on sun

14

elevation (time of a day or season of the year), optical errors of PTC mirrors, and the azimuth

15

(which could be assimilated also to the orientation of the PTC axis and therefore tracking

16

orientation chosen).

17

3.2.

Flux profile and distribution:

18

In this section only a few sketches (14 from180) are presented, to give an overview on the

19

changing solar flux distributions, Figure 9, below corresponds to results of simulations run for

20

case #1. This figure shows the profile and distribution of the solar flux onto a perfectly

21

aligned receiver tube along the focal line of the parabola.

22 23

Figures 10 to 13 correspond to results of the simulations ran for case study #3. They show

24

some profiles and distributions of the solar flux received on southern and northern sides

25

(sections) of a defocused (misaligned) receiver tube, according to the defocusing concept

26

explained in section 2, for two different sun elevations 90°and 30°.

27

Figures 14 to 15 correspond to results of the simulations run for case study #5. They show

28

some profiles and distributions of the solar flux received on southern and northern sides

29

(sections) of a defocused (misaligned) receiver tube, according to the defocusing concept

30

explained in section 2, for only one sun elevation angle 90°.

31

15

ACCEPTED MANUSCRIPT

1 2

(a) North-South orientation

3 4

(b) East-West orientation

5

Figure 9. Flux distribution (in W/m2) for case #1.Views of the solar flux intercepted by the

6

complete and correctly aligned receiver tube for two different solar azimuth angles.

7

Additional simulation parameters: Dec = 0 mrad; sun elevation = 90°; mirror slope error = 1.7

8

mrad; DNI = 1000 W/m2.

9 10

(a) North-South orientation

11

16

ACCEPTED MANUSCRIPT

1 2

(a) East-West orientation

3 4

Figure 10. Flux distribution (in W/m2) for case #3.Views of the solar flux intercepted by the

5

southern side of the receiver tube for two different solar azimuth angles. Additional

6

simulation parameters: Dec = 4.803 mrad; sun elevation = 90°; mirror slope error = 1.7 mrad;

7

DNI = 1000 W/m2.

8 9

(a) North-South orientation

10 11

(a) East-West orientation

12

Figure 11. Flux distribution (in W/m2) for case #3.Views of the solar flux intercepted by the

13

northern side of the receiver tube for two different solar azimuth values. Additional 17

ACCEPTED MANUSCRIPT 1

simulation parameters: Dec = 4.803 mrad; sun elevation = 90°; mirror slope error = 1.7 mrad;

2

DNI = 1000 W/m2.

3 4

(a) North-South orientation

5

6 7 8

(a) East-West orientation

9

Figure 12. Flux distribution (in W/m2) for case #3.Views of the solar flux intercepted by the

10

southern side of the receiver tube for two different solar azimuth angles. Additional

11

simulation parameters: Dec = 4.803 mrad; sun elevation = 30°; mirror slope error = 1.7 mrad;

12

DNI = 1000 W/m2.

13 18

ACCEPTED MANUSCRIPT 1

(a) North-South orientation

2

3 4

(b) East-West orientation

5

Figure 13. Flux distribution (in W/m2) for case #3.Views of the solar flux intercepted by the

6

northern side of the receiver tube for two different solar azimuth values. Additional

7

simulation parameters: Dec = 4.803 mrad; sun elevation = 30°; mirror slope error = 1.7 mrad;

8

DNI = 1000 W/m2.

9

10 11

(a) North-South orientation

12

13 14

(a) East-West orientation 19

ACCEPTED MANUSCRIPT 1

Figure 14. Flux distribution (in W/m2) for case #5.Views of the solar flux intercepted by the

2

southern side of the receiver tube for two different solar azimuth angles. Additional

3

simulation parameters: Dec = 4.803 mrad; sun elevation = 90°; mirror slope error = 1.7 mrad;

4

DNI = 1000 W/m2.

5 6

(a) North-South orientation

7

8 9

(a) East-West orientation

10

Figure 15. Flux distribution (in W/m2) for case #5.Views of the solar flux intercepted by the

11

northern side of the receiver tube for two different solar azimuth values. Additional

12

simulation parameters: Dec = 4.803 mrad; sun elevation = 90°; mirror slope error = 1.7 mrad;

13

DNI = 1000 W/m2.

14

As demonstrated in Figures 09 to 15, (representing cuts in a transversal section of the

15

absorber tube, the bleu dark zone at the left of the figures represents the upper part (the side

16

facing the sky) of the receiver tube, while the colourful zone represents the bottom-part (the

17

side facing the reflective mirrors where the sun rays are concentrated). It is clearly shown that

18

solar flux distribution on the absorber tube surface changes from each case to another and

19

takes different shapes and configurations.

20

ACCEPTED MANUSCRIPT 1

Higher appearance of the concentrated solar flux distribution is noticed when the receiver tube

2

is well aligned, and it changes its shape gradually and decreases with the increase of the

3

defocusing.

4

By analysing the flux distributions delivered by the simulations, we could clearly notice a

5

difference in the profile of the concentrated solar flux between case #1 and the other four

6

cases. In case #1 the spot of the flux takes a perfect “rectangular” shape, the intensity of the

7

flux is well apparent, by a large zone taking a dominant dark red colour and indicating a high

8

quantity of concentrated solar energy. In the other four cases the absorber tube is misaligned

9

from Dec2=0.433 mrad to Dec5=4.803mrad, and the profile of the concentrated solar flux

10

changes totally its shape from the rectangular form noticed in Figure 9 (ideal case) to take the

11

appearance of a large “˅” (the alphabetic latter “˅”) at the southern side of the receiver

12

(please refer to figures 10, 12 and 14) while taking the shape of an inverted “˅”, “˄” at the

13

northern side of the receiver tubes (please to refer to figures 11, 13 and 15). The flux profile

14

degrades gradually from case #2 to case #5 and the degradation is increasingly flagrant from

15

each case to another.

16

After analysing carefully all the flux distributions, we had noticed that the profile of the “˅”

17

and “˄” forms, becomes gradually larger in the “X” axis direction from each case to another,

18

while it decrease towards the base of the “˅”and “˄” forms ( in the “Y” axis direction).

19

On the contrary to the first case, the blue and green colours are the dominant ones in most

20

cases from case #2 to case#5, the “˅”and “˄” forms are accentuated with apparent red zones

21

situated downward the flux profile (“˅” form) at the southern side of the receiver tubes and

22

upward the flux profile (“˄” form) at the northern side of the receiver tubes.

23

When comparing the flux distribution between North-South and East-West orientations, we do

24

not notice big differences in the shape of the spot. Sometime the profile is less dense and the

25

colours are blurry and the whole image might be more granulated when the North-South PTC

26

orientation is chosen. This phenomenon is accentuated dependently on the sun elevation and

27

the standard deviation (when the mirror slope error 𝜎𝑠𝑙𝑜𝑝𝑒 = 3 mrad).

28 29

4. CONCLUSION

30

Aiming at evaluating the influence of thermal contraction- expansion of receiver tubes on its

31

misalignment along the focal line of the reflective surface, a detailed analysis of the amount

32

and distribution of the concentrated solar radiation has been performed. It has been considered

33

an absorber tube of a state-of-the art parabolic-trough collector, with geometrical 21

ACCEPTED MANUSCRIPT 1

characteristics similar to those of parabolic troughs installed in commercial solar power plants

2

in operation worldwide. A total number of 108 simulations were carried out, by considering

3

the layout of a typical collector loop in commercial power plant, which is composed of four

4

collectors connected in series. The numerical simulations were carried out by using optical

5

modelling software based on a Monte Carlo ray tracing method, the software Tonatiuh.

6

Depending on the temperature of the heat transfer fluid circulating inside the receiver tube

7

(typically synthetic thermal oil), it may appear an angular misalignment of the receiver tube

8

from the focal line of the parabola, which is maximum at both ends of the solar collector. The

9

study was completed considering North-South and East-West orientations of the collector

10

axis, seasonal variations by assuming different sun elevations, two different slope errors of the

11

solar reflectors. Simulation outputs revealed interesting results related to the concentrated

12

solar flux intercepted by the receiver tube, showing how cyclic thermal expansion affects

13

sensitively the amount and distribution of the total solar power and average flux on the

14

absorber tube. The decrease on these quantities is significant when receiver tube is deviated

15

from the focal line of about 0.4 mrad but become more important for deviations exceeding 0.8

16

mrad. During the start-up of solar commercial field using state-of-the-art parabolic-trough

17

collectors, it may appear an angular misalignment of up to 4.8 mrad, depending on the design

18

of the receiver supporting arms, which would provoke an important reduction in the total

19

concentrated solar flux collected by the absorber tube. In this study it was considered that the

20

receiver supporting arms were perpendicular to the aperture plane of the collector only when

21

the average thermal oil temperature in the collector’s receiver was 380.5°C. The numerical

22

study performed revealed a decrease in the total solar power collected of about 38% in the

23

working temperature range (293°C-393°C) and of up to 75% when the thermal oil circulating

24

through the receiver tubes is at temperatures of around 100°C.

25

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ACCEPTED MANUSCRIPT

Highlights 1. Effect of thermal expansion of receivers used in parabolic troughs was studied 2. Angular shift of the receiver tube due to expansion changes optical performance 3. Ray-tracing was applied to evaluate the change in performance for different shifts 4. Concentrated solar energy intercepted by receiver changes of up 59%