Analysis of the influence of heat loss factors on the overall performance of utility-scale parabolic trough solar collectors

Accepted Manuscript Analysis of the influence of heat loss factors on the overall performance of utilityscale parabolic trough solar collectors

Li Xu, Feihu Sun, Linrui Ma, Xiaolei Li, Guofeng Yuan, Dongqiang Lei, Huibin Zhu, Qiangqiang Zhang, Ershu Xu, Zhifeng Wang PII:

S0360-5442(18)31362-8

DOI:

10.1016/j.energy.2018.07.065

Reference:

EGY 13330

To appear in:

Energy

Received Date:

25 September 2017

Accepted Date:

11 July 2018

Please cite this article as: Li Xu, Feihu Sun, Linrui Ma, Xiaolei Li, Guofeng Yuan, Dongqiang Lei, Huibin Zhu, Qiangqiang Zhang, Ershu Xu, Zhifeng Wang, Analysis of the influence of heat loss factors on the overall performance of utility-scale parabolic trough solar collectors, Energy (2018), doi: 10.1016/j.energy.2018.07.065

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ACCEPTED MANUSCRIPT

1

Analysis of the influence of heat loss factors on the overall

2

performance of utility-scale parabolic trough solar collectors

3

Li Xu, Feihu Sun, Linrui Ma, Xiaolei Li, Guofeng Yuan, Dongqiang Lei, Huibin Zhu, Qiangqiang

4

Zhang*, Ershu Xu, Zhifeng Wang

5

Key Laboratory of Solar Thermal Energy and Photovoltaic System of Chinese Academy of

6

Sciences, Beijing Engineering Research Center of Solar Thermal Power, Institute of Electrical

7

Engineering, Chinese Academy of Sciences, Beijing 100190, China

8

Abstract:

9

For parabolic trough solar collectors, several factors (such as the amount of the

10

gas in the evacuated annulus, the absorber emissivity, the wind speed and temperature

11

distributions of the absorber, the glass envelope and the heat transfer fluid) are critical

12

to influence their heat losses and consequently their overall performance. Therefore,

13

this study develops a mathematical model for thermal behaviors of parabolic trough

14

solar collectors in consideration of these impact factors. Additionally, to validate this

15

model, experimental data were measured for a test facility. This facility includes a

16

utility-scale loop of parabolic trough solar collectors which can be applicable to solar

17

thermal power plants. The comparison indicates a good agreement between predicted

18

and measured temperatures of the heat transfer fluid at the outlet of the collectors. Using

19

this model, parametric studies were conducted for impact factors. These factors are the

20

pressure of the H2 or air from 0.01 to 1E5 Pa, the absorber emissivity from a measured

21

basis to its four times, the wind speed from 2 to 12 m/s, and temperature distributions

22

with and without the concentrated solar flux. Consequently, several conclusions were

23

drawn by analyzing how they influence heat losses and further overall performance

24

under specified boundary conditions.

*

Corresponding author. Tel: +86 10 82547268. E-mail address: [email protected] 1 / 39

ACCEPTED MANUSCRIPT 1

Keywords: Parabolic trough solar collector; Concentrating solar Power; Solar thermal

2

Electricity; Heat loss; Overall performance

3 4

1. Introduction

5

Concentrating Solar Power (CSP) or Solar Thermal Electricity (STE) is a prime

6

choice in developing an affordable, feasible, global energy source that is able to

7

substitute for fossil fuels in the sunbelts around the world[1]. The installed capacity of

8

CSP plants will achieve 1000 GW and share 11 % of the global electricity capacity by

9

2050, resulting in the reduction in the emissions of up to 2.1 Gt of carbon dioxide

10

annually[2]. Hence, as the most mature CSP technology, the technology of parabolic

11

trough solar collectors (PTCs) is playing a leading role in the development of the clean

12

energy. In addition to the CSP, PTCs are also widespread in other applications such as

13

the industrial processes, the desalination, the cooling and the solar chemistry[3-5]. Thus,

14

there is a significant amount of the research on the thermal characteristics of the PTCs

15

in order to improve their efficiency and reduce their cost.

16

Some of previous studies made efforts to enhance the heat transfer of the fluid in

17

the PTCs with the simulation methods for demonstrating the effects of novel structures

18

of the absorber on the overall performance. Kumar et al.[6] made a 3-D numerical

19

analysis of the porous disc line receiver for the PTCs based on renormalization-group

20

(RNG) k–ε turbulent model. Muñoz et al.[7] analyzed the effect of the use of internal

21

finned tubes for the design of the PTCs with the computational fluid dynamics tool.

22

Cheng et al.[8] presented numerical computation results on the turbulent flow and the

23

coupled heat transfer enhancement using the unilateral milt-longitudinal vortexes in a

24

novel parabolic trough solar absorber tube.

25

Additionally, some studies focused more attention on the heat losses from the

26

PTCs by investigating impact factors causing heat losses or measuring the total heat

27

loss. Gong, et al. [9] pointed out that the coating’s emittance and vacuum conditions

28

had a significant impact on parabolic trough receiver’s heat loss while the influences of 2 / 39

ACCEPTED MANUSCRIPT 1

environmental conditions were quite indirect and negligible. Liu et al.[10] measured

2

variations of composition and partial pressure of residual gases with temperature in the

3

receiver and showed that the hydrogen was the main residual gas in the annular space

4

of receiver without getter and the nitrogen was the main gas released in the receiver

5

with getter. Navarro-Hermoso et al.[11] tested three receivers for heat losses, and their

6

results indicate at the 400 °C operating temperature the losses are 225 W/m(for the 70

7

mm diameter absorber tube), 231 W/m(for the 70 mm diameter absorber tube), and 322

8

W/m (for the 90 mm diameter absorber tube). Espinosa-Rueda et al.[12] evaluated the

9

annulus gas and the thermal performance of receivers in a 50 MW plant after five years

10

of operation, and they found that 9 % of receivers showed vacuum loss to some degree

11

and the vacuum losses added up extra heat losses of 0.6 MWt in the studied plant under

12

operating conditions (8 % over the reference performance). Kumaresan et al.[13]

13

summarized several major reasons for heat loss in the parabolic trough receiver, which

14

include the improper maintenance of vacuum between the glass envelope and the

15

receiver tube, the degradation of coating at high temperature, the temperature

16

distribution in the receiver tube, and the hydrogen accumulation and permeation in the

17

receiver tube. Valenzuela et al.[14] proposed an indirect method to determine heat

18

losses from collector’s global efficiency data, which were obtained from tests

19

performed in steady state conditions at various working fluid temperatures and with the

20

zero incidence angle of solar radiation.

21

However, few research demonstrated how critical impact factors associated with

22

heat losses affected the overall thermal performance of utility-scale parabolic trough

23

solar collectors under specified boundary conditions. Therefore, this study firstly

24

established a numerical model, and then validated it with the experimental data

25

collected for a test facility including a utility-scale loop of parabolic trough solar

26

collectors. Also, it analyzed the amount of the gas in the evacuated annulus, the

27

emissivity of the absorber, the wind speed and temperature distributions of the absorber,

28

the glass envelope and the heat transfer fluid by quantifying how their variations

29

resulted in the change of heat losses and the overall thermal performance under various

30

boundary conditions. 3 / 39

ACCEPTED MANUSCRIPT 1 2

3

2. Mathematical Model 2.1 Overall model

4

Parabolic trough solar collectors concentrate the sunlight onto the receivers by

5

driving their trough-shaped reflectors in order to convert the solar energy into the useful

6

thermal energy. The Fig. 1 illustrates their structure, which consists the receiver, the

7

mirror, the frame, the support bracket, the pylon and the driver. Moreover, the

8

evacuated-type receiver with the glass envelope is usually used as the heat collection

9

element (HCE) so that the convective heat loss from the absorber is reduced and the

10

selective coating on the outer wall of the absorber is protected from the oxidation. As

11

shown in Fig. 2, the evacuated-type receiver consists of the coated steel tube, the glass

12

envelope, the vacuum annulus, the getter, the expansion bellows and the metal-to-glass

13

seal. When the heat transfer fluid (HTF) is pumped into the HCEs, the heat from the

14

inner wall of the absorber is transferred into the HTF and consequently the useful

15

energy is brought out from the PTCs.

16 Frame

Receiver (HCE)

Mirror Support bracket

Driver

17 18

Pylon

Fig. 1 Schematic of parabolic trough solar collectors.

4 / 39

ACCEPTED MANUSCRIPT

Absorber

Expansion bellows

Vacuum annulus

Glass envelope nnulus

Metal-to-glass seal bellows

Getter

1 2

Fig. 2 Schematic of an HCE evacuated tube.

3 4

Therefore, the thermal analysis on the PTCs is carried out on the energy balance

5

of the HCEs due to their working mechanism, as showed in Fig. 3. As the solar energy

6

incident on the selective coating of the absorber is absorbed, some of the absorbed

7

energy is transferred into the HTF by the convection while some of absorbed energy is

8

transferred to the glass envelope through the vacuum annulus by the convection and

9

radiation. The energy gained by the glass envelope ultimately is lost to the surroundings

10

by the convection and radiation. In this study, the 𝑞abs is the rate of the heat transfer

11

absorbed by the absorber from the concentrated solar flux which is reflected onto the

12

absorber outer surface by the parabolic mirrors. In reality, several factors such as the

13

cosine loss, the end loss, the cleanliness of both reflector and glass envelope, the

14

reflectance of the reflector, the transmittance of the glass envelope, the absorptance of

15

the selective coating on the outer surface of the absorber, the intercept factor of the

16

PTCs and the incident angle modifier contribute to the reduction in the available direct

17

normal solar irradiance (DNI). Therefore, the 𝑞abs is equal to the product of the DNI,

18

these impact factors and the aperture area, as expressed in Eq.(1).

19

𝑞abs = 𝐺DNIcos 𝜃𝐹end𝐹cl𝜌(𝜏𝛼𝛾)𝑛𝐾𝐴

20

where 𝐺DNI is the DNI which can be measured by a pyrheliometer, 𝜃 is the incident

21

angle between the sun’s direct rays and the normal to the collector aperture plane, 𝐹end

22

is the end loss factor, 𝐹cl is the cleanliness factor, 𝜌 is the reflectance of the reflector,

(1)

a

5 / 39

ACCEPTED MANUSCRIPT 1

(𝜏𝛼𝛾)𝑛 is the effective transmittance–absorptance–intercept factor product when the

2

direct solar rays are perpendicular to the collector aperture[15], 𝐴a is the aperture area

3

of the PTCs, and the incident angle modifier 𝐾 will use the empirical correlation [16].

4

Absorber

Vacuum annulus

qHTF qabs qb,c

qg,r qg,c

qb,r

Glass envelope

5 6 7

Fig. 3 Thermal analysis on the cross-section of the HCE.

8

In addition, the Fig. 3 also shows the rate of the radiative heat loss from the glass

9

envelope to the surroundings 𝑞g,r, the rate of the convective heat loss from the glass

10

envelope to the surroundings 𝑞g,c, the rate of the radiative heat loss from the absorber

11

to the glass envelope 𝑞b,r, the rate of the convective heat loss from the absorber to the

12

glass envelope through the vacuum annulus 𝑞b,c, and the rate of the convective heat

13

transfer from the absorber to the HTF 𝑞HTF, which will be given in details in the

14

following sections respectively.

15

As shown in Fig. 4, the HCEs as well as their inner HTF are divided into control

16

volumes in the axial direction. There is a node at the center of every single control

17

volume and these nodes are evenly spaced over the entire length of the HCEs or HTF.

18

The index i is from 1 to NL which follows the HTF flow direction. For instance, i =1

19

means that the node is located at the inlet of the PTCs and i =NL means at the outlet. 6 / 39

ACCEPTED MANUSCRIPT 1

Additionally, the 𝑇HTF, 𝑇b and 𝑇g are temperatures of the HTF, the absorber and the

2

glass envelope, respectively. Furthermore, in any single control volume,

3

thermophysical properties and the boundary as well as initial conditions are assumed to

4

be homogeneous.

5

T g,1

T g,i

T g,NL

T b,1

T b,i

T b,NL

T HTF,i 6

Fig. 4 Schematic of control volumes for HCEs and HTF.

7 8 9

Hence, for every single control volume of the absorber, the governing differential

10

equation based on the transient energy balance is

11

𝑞b,c + 𝑞b,r + 𝜌b𝑐𝑝b𝑉b d𝜏 + 𝑞HTF = 𝑞abs + 𝑞cond.

12

where the 𝜌b𝑐𝑝b𝑉b d𝜏 is the rate of the energy storage inside the control volume and

13

the 𝑞cond is the rate of the conductive heat transfer through the absorber at the axial

14

direction.

15

d𝑇b

(2)

d𝑇b

Similarly, for every single control volume of the glass envelope, the unsteady-state

16

energy balance determines the temperature rise across it, which is represented as

17

𝑞g,c + 𝑞g,r + 𝜌g𝑐𝑝g𝑉g d𝜏 = 𝑞b,c + 𝑞b,r + 𝑞g,cond.

18

where the 𝜌g𝑐𝑝g𝑉g d𝜏 is the rate of the energy storage inside the control volume of the

19

glass envelope and the 𝑞g,cond is the rate of the conductive heat transfer through the

20

glass envelope at the axial direction.

d𝑇g

(3)

d𝑇g

21

Furthermore, the net thermal output ultimately brings the solar energy into use as

22

the HTF experiences through the absorber. The rate of the heat transfer from the

23

absorber to the HTF can be calculated for every single control volume using the

24

Newton’s convective heat transfer law in Eq. (4). 7 / 39

ACCEPTED MANUSCRIPT 1

𝑞HTF = ℎHTF𝐴in,b(𝑇b ‒ 𝑇HTF)

2

where the 𝐴in,b is the inner surface area of the absorber and the ℎHTF is the convection

3

heat transfer coefficient that is determined by the Nusselt number.

(4)

4

Moreover, the heat carried into the next control volume by the HTF flow is equal

5

to the heat gained by the HTF from the absorber. So for the control volumes from 2 to

6

NL, the energy balance in the Eq. (5) is established using a fully implicit step by step

7 8

method in the flow direction of the HTF. (𝑇HTF,i + 1 ‒ 𝑇HTF,i) = 𝜋𝑑ℎHTF,i(𝑇b,i ‒ 𝑇HTF,i) 𝑐𝑝HTF,i𝑚 𝐿

9

where the 𝑐𝑝HTF,i is the specific heat capacity at constant pressure of the HTF for the

10

control volume i, the ℎHTF,i is the convection heat transfer coefficient for the control

11

volume i, the 𝑚 is the mass flow rate of the HTF, 𝑑 is the diameter of the inner surface

12

of the absorber and the 𝐿 is the length between two adjacent nodes.

13

2.2 Heat losses from the absorber

(5)

14

For every single control volume, the heat losses from the outer surface of the

15

absorber to the inner surface of the glass envelope consist of the radiation and

16

convection which exist in parallel.

17

2.2.1 Radiation heat loss

18

According to the total radiation resistance analysis as the two surface resistance

19

and the space resistance based on the diffuse gray surface radiation exchange, the net

20

rate of the radiative heat loss emitted by the outer surface of the absorber to the inner

21

surface of the glass envelope can be expressed in the Eq.(6). 4

22

𝑞b,r = 𝜎1 ‒ 𝜀

4

𝑇b ‒ 𝑇 g b

𝜀 𝐴 b b

+𝐴

b

1 𝐹𝑉

1‒𝜀

+ bg

(6)

g

𝜀 𝐴 g g

23

where the 𝜎 is the Stefan-Boltzmann constant, the 𝐹𝑉b𝑔 is the view factor from the

24

convex outer surface of the absorber to the concave inner surface of the glass envelope, 8 / 39

ACCEPTED MANUSCRIPT 1

the 𝜀b is the absorber emissivity, the 𝜀g is the glass envelope emissivity, and the

2

radiative heat transfer areas 𝐴b and 𝐴g are the outer surface area of the absorber and the

3

inner surface area of the glass envelope respectively.

4

Furthermore, for every single control volume, if the view factor from the outer

5

surface of the absorber to the inner surface of the glass envelope is assumed to be 1

6

because the outer surface of the absorber and the inner surface of the glass envelope

7

can be viewed nearly as the enclosure, then the Eq.(6) can be reduced to be 4

4

𝑇b ‒ 𝑇 g

.

8

𝑞b,r = 𝜎

9

2.2.2 Convective heat loss

(7)

1‒𝜀 1 g +𝜀𝐴 𝜀 𝐴 b b g g

10

In reality, the pressure of the annulus space of HCEs is generally required to be

11

less than 0.013332 Pa when HCEs are manufactured. However, the H2 generation or

12

the air penetration in the vacuum annulus might occur, resulting in the increase in heat

13

losses. Additionally, the calculation of the convective heat loss in the vacuum annulus

14

is determined by the geometry, temperature, pressure and type of the gas according to

15

the theory of conduction heat transfer and kinetic theory. Furthermore, heat transfer

16

mechanisms in the vacuum annulus are mainly the molecular conduction and the natural

17

convection. The heat transfer occurs as the free molecular conduction when collisions

18

between molecules are relatively rare due to the low pressure of the gas, while the

19

natural convection eventually takes place with the pressure of the gas increasing. Then,

20

once the natural convection or the free molecular conduction is determined, the

21

convective heat transfer in the vacuum annulus can be calculated by the Eq.(8). 2𝜋𝑘eff𝐿(𝑇b ‒ 𝑇g) 𝑞b,c = 𝐷

22

ln

() g

𝐷

(8)

b

23

where the L is the length of the control volume, the 𝐷g is the diameter of the inner

24

surface of the glass envelope, the 𝐷b is the diameter of the outer surface of the absorber,

25

and the 𝑘eff is the effective thermal conductivity which can be expressed in Eq.(9) for 9 / 39

ACCEPTED MANUSCRIPT 1

the convective heat transfer in a concentric tube annulus.

2

𝑘eff = ℎeff 2 ln

3

where the ℎeff is the effective heat transfer coefficient.

𝐷b

( ). 𝐷g

(9)

𝐷b

4

(1) Free molecular conduction regime

5

The effective heat transfer coefficient can be estimated by the coefficient

6

correlation[17, 18] as expressed in Eq.(10), which covers the range from pure

7

conduction to free molecular heat transfer.

[

𝐷b

() 𝐷g

)]

‒ 5)𝜆mo 𝐷 (9𝑐𝑝 𝑐𝑣 b +1 + 𝑐𝑝 𝐷g 2(𝑐𝑣 + 1)

(

‒1

𝑘std.

(10)

8

ℎeff =

9

where the 𝜆mo is the mean-free-path between collisions of gas molecules, the 𝑘std is the

10

thermal conductivity at the standard temperature and pressure, the 𝑐𝑝 is the specific

11

heat capacity at constant pressure and the 𝑐𝑣 is the specific heat capacity at constant

12

volume.

2

ln

𝐷b

Further, according to the kinetic theory of gases[19], the mean-free-path can be

13 14

calculated by

15

𝜆mo =

16

where the Bc is Boltzmann’s constant (1.381E−23 J/K), the 𝑇bg is the arithmetic

17

average temperature of the absorber and the glass envelope temperatures, the 𝑃 is the

18

annulus gas pressure, and the 𝐷mo is the molecular diameter of the annulus gas.

(11)

2 2𝜋𝑃𝐷mo

In addition, the parameters of the H2 and air molecules used in Eqs.(10) and (11)

19 20

Bc𝑇bg

are listed in Table 1.

21

Table 1 Molecular parameters [17, 19]

22 Gas type

H2 Air

𝑘std

𝐷mo

(W/(m K))

(10-10m)

cv (J/(kg K))

cp/cv (-)

0.1769 0.02551

2.74 3.72

4.659+710-4T 4.924+1.710-4T+3.110-7T2

1.408 1.4034

10 / 39

ACCEPTED MANUSCRIPT 1 2

(2) Natural convection regime

3

The annulus of the HCE may be viewed as the annular space between long,

4

horizontal, concentric cylinders. As the temperature of the absorber is rising due to the

5

incident concentrated solar flux, the gas in the annulus of the HCE will circulate by the

6

natural convection. Consequently, the gas in the annular space will ascend along the

7

hot outer surface of the absorber while the gas will descend along the inner surface of

8

the glass envelope. In this study, a correlation is used to calculate the gas effective

9

thermal conductivity for the natural convection, as expressed by Eq.(12)[20].

10

Thermophysical properties in this correlation are evaluated at the arithmetic average

11

temperature of the absorber temperature and the glass envelope temperature.

12

𝑘eff = 0.386

13

where its valid range is 0.7
14

characteristic length 𝐿ch for Ra is determined by

15

(0.861PrRa+ Pr)

𝐿ch =

[ln (𝐷g/𝐷b)]4/3

(𝐷 ‒ b0.6 + 𝐷 ‒ g0.6)

0.25

(12)

𝑘std

.

(13)

5/3

16

Moreover, it should note that the 𝑘eff must be replaced by 𝑘std if its value

17

calculated by Eq.(12) is less than 𝑘std, because the natural convection is suppressed

18

eventually as the pressure decreases from the standard pressure and the minimum heat

19

transfer rate in the annular space cannot fall below the conduction limit.

20

2.3 Heat losses from the glass envelope

21

Similar to the outer surface of the absorber, the outer surface of the glass envelope

22

loses its heat to the surroundings by the thermal radiation and the convection in parallel.

23

2.3.1 Radiation heat loss

24

For every single control volume, if we assume that the view factor between the 11 / 39

ACCEPTED MANUSCRIPT 1

outer surface of the glass envelope and the surroundings is 1, and the surroundings are

2

viewed as the effectively black body, then the net rate of radiation transfer from the

3

outer surface of the glass envelope to the surroundings will be expressed as

4

𝑞g,r = 𝜀g𝐴s𝜎 𝑇g ‒ 𝑇sur .

5

where the 𝐴s is the outer surface area of the glass envelope and the 𝑇sur is the

6

temperature of the surroundings.

7

2.3.2 Convective heat loss

(

4

4

)

(14)

8

Like the rate of the convective heat loss from the absorber, the rate of the

9

convective heat loss from the glass envelope to the surroundings is also estimated using

10

the Newton’s convective heat transfer law, as expressed in Eq. (15).

11

𝑞g,c = ℎg,c𝐴s(𝑇g ‒ 𝑇sur)

12

where the convection heat loss coefficient ℎg,c is determined by the wind speed.

(15)

13

In practice, there are two types of the convective heat loss from the outer surface

14

of the glass envelope to the surroundings according to the wind speed. One type is the

15

external forced convection induced by the wind flow towards the glass envelope. The

16

other type is the natural convection induced by the density differences in the air

17

surrounding the heated glass envelope without the wind influence. The details of the

18

calculation of these two types of the convective heat loss are provided in the following.

19

(1) Forced convection

20

The Nusselt number for the forced convection can be estimated using the well-

21

accepted correlation in Eq.(16) [21, 22], if the wind flow towards the glass envelope is

22

viewed as the external flow across a cylinder. In addition, thermophysical properties in

23

this correlation are evaluated at the arithmetic average temperature of the glass

24

envelope temperature and the air temperature in the surroundings.

25

Nud,s = CRed,sPr

26

where the Prs is the Prandtl number evaluated at the air temperature in the surroundings,

N

1/3 s

(16)

12 / 39

ACCEPTED MANUSCRIPT 1

the characteristic length for both the Nusselt number Nud,s and the Reynolds number

2

Red,s is the outer diameter of the glass envelope, and values of the constants C and N

3

are tabulated according to the variation in the Reynolds number in Table 2.

4

Table 2

5

Constants of Equation for the cylinder in cross flow[21]

Red,s

C

N

0.4 – 4

0.989

0.330

4 – 40

0.911

0.385

40 – 4 000

0.683

0.466

4 000 – 40 000

0.193

0.618

40 000 – 400 000

0.0266

0.805

6 7

(2) Natural convection

8

If no wind flows over the glass envelope, then the natural convection determines

9

the heat loss from the glass envelope to the surroundings. Hence, using an weighted

10

average method for a horizontal cylinder in consideration of the laminar and turbulent

11

parts in Eq.(17) [23], the Nusselt number for the natural convection can be obtained. In

12

addition, this correlation is valid for the interval 10

13

properties of the air used in the calculation of the Ra are evaluated at the arithmetic

14

average temperature of the air and the glass envelope temperatures.

15

Nunc = Nulam + Nutur

16

where the laminar Nusselt number Nulam and the turbulent Nusselt number Nutur are

17

given by Eqs.(18) and (19) respectively.

18

Nulam =

(

10 0.1

)

10

(

𝐶 cy 0.386𝐶 Ra lam

19

and

20

Nutur = 0.1Ra

0.25

7

≤Ra≤10 . Thermophysical

(17)

2𝐶cy ln 1 +

‒ 10

(18)

)

1/3

13 / 39

ACCEPTED MANUSCRIPT 1

(19)

2

where the laminar coefficient 𝐶lam and the cylinder coefficient 𝐶cy are given by Eqs.

3

(20) and (21) respectively.

4

𝐶lam = 0.671(1 + 0.671Pr

5

(20)

6

and

7 8 9

𝐶cy =

{

‒ 0.5625 ‒ 4/9

(

1 ‒ 0.13 0.772𝐶lamRa

)

0.25 ‒ 0.16

)

0.8

𝑓𝑜𝑟 10

𝑓𝑜𝑟 10

‒4

‒ 10

≤ Ra ≤ 10 7

‒4

.

< Ra ≤ 10

(21) 7

Moreover, if the Rayleigh number Ra is greater than 10 , the Nusselt number for

10

the natural convection will be given by[23]

11

Nunc = 0.6 +

12

2.4 Conduction through the absorber/glass envelope

[

0.387Ra

1/6

(1 + 0.721Pr ‒ 0.5625)

]

8/27

2

(22)

.

13

For the conduction heat transfer through the absorber or the glass envelope along

14

the axial direction of the HCEs, two kinds of control volumes are treated separately due

15

to their different sizes. One is the internal control volume in the range from 2 to NL-1

16

which experiences the heat conduction with two adjacent control volumes through both

17

edge surfaces. The other is the control volume 1 or NL, whose heat conduction is

18

transferred with only one adjacent control volume at its one edge surface while the other

19

edge surface is assumed as the adiabatic surface. Therefore, the rate of the conductive

20

heat transfer through the absorber or the glass envelope is given by the Eq. (23).

21

𝑞cond or 𝑞g,cond =

14 / 39

ACCEPTED MANUSCRIPT

1

{

𝜆i ‒ 1𝐴

an

(𝑇i ‒ 1 ‒ 𝑇i) 𝐿 𝜆1𝐴

𝜆i𝐴

an

+ ( 𝑇2 ‒ 𝑇1) an

(𝑇i + 1 ‒ 𝑇i) 𝐿

𝐿 𝜆NL ‒ 1𝐴 (𝑇NL ‒ 1 ‒ 𝑇NL) an

𝐿

for nodes from 2 to NL ‒ 1 for node 1

(23)

for node NL

2

where the 𝜆i is the thermal conductivity between nodes i and i+1 which is evaluated at

3

the arithmetic average temperature between the 𝑇i and the 𝑇i + 1, the 𝐴an is the annulus

4

zone area for the cross section of the absorber or the glass envelope, and the 𝐿 is the

5

discrete spatial step which is the distance between the two adjacent nodes.

6 7

3. Results and discussion

8

3.1 Validation

9

In order to make sure the reliability and accuracy of the model for further analyses,

10

a set of experimental data was obtained by measuring a utility-scale loop of parabolic

11

trough solar collectors and then was compared with the simulated results from the

12

model. Additionally, the experimental facility including this loop of parabolic trough

13

solar collectors is located at Yanqing, Beijing, China as shown in Fig. 5(1). The

14

parabolic trough solar collectors used in this study track the sun lights along an east-

15

west horizontal axis. Moreover, the total length of these used parabolic trough solar

16

collectors is nearly 600 m and their aperture area is 3317.8 m2. The detailed structure

17

parameters of these parabolic trough solar collectors were listed in Table 3. Furthermore,

18

in this experimental setup as shown in Fig. 5(2), the measuring instruments include the

19

inlet and outlet HTF temperature sensors which are the platinum resistance

20

thermometers with an accuracy of less than ±2.1 °C, an ambient temperature sensor

21

which is the platinum resistance thermometer with an accuracy of less than ±0.15 °C, a

22

first-class pyrheliometer to measure the DNI with an uncertainty of less than ±0.5 %,

23

and a flow meter to measure the HTF volumetric flow meter with an uncertainty of less 15 / 39

ACCEPTED MANUSCRIPT 1

than ±1 %.

2

Test loop Inlet

3

Outlet (1)

4

F

T

Flow meter Temperature sensor(inlet)

PTCs

T

PTCs

Temperature sensor(outlet)

Control valve

Heat exchanger Pump

Tank

Pyrheliometer Temperature sensor(ambient)

5 6

(2)

7 8

Fig. 5 The experimental set-up of parabolic trough solar collectors: (1) photograph and (2) schematic diagram.

9

Table 3 Structure parameters of parabolic trough solar collectors

10

Parameter

value

units

Aperture width

5.76

m

Focal length

1.71

m

HCE absorber outer diameter

0.07

m

HCE absorber thickness

0.003

m

HCE envelope outer diameter

0.125

m

HCE envelope thickness

0.003

m

Single HCE length

4.06

m

11 16 / 39

ACCEPTED MANUSCRIPT 1

Then, to verify the model predictions, a process of the HTF temperature rise was

2

carried out to make parabolic trough solar collectors experience the range of the

3

operation temperatures as sufficiently as possible. Thus, the experimental data were

4

collected in this operation process under normal weather conditions as shown in Fig. 6.

5

The Fig. 6 (1) illustrates the measured DNI in the experimental process, which indicates

6

that relatively steady solar resources of approximately 900 W/m2 except that partially

7

cloudy time appeared nearly at the end of the experiment. The flow rate of the HTF, as

8

shown in Fig. 6 (2), underwent a small step change from nearly 31.8 m3/h to 35.3 m3/h

9

at about 12:00 in the experimental process. Furthermore, the slight variation in the

10

ambient temperature, as shown in Fig. 6 (3), existed almost between 27 °C and 30 °C.

11

In reality, the measured HTF inlet and outlet temperatures as shown in Fig. 6 (4)

12

indicate that the whole experiment was a startup process. This process tried to let

13

parabolic trough solar collectors concentrate the sun lights to heat the HTF for the

14

requirement of the whole system in order to run parabolic trough solar collectors at the

15

required HTF inlet temperature (for instance, 290°C). In addition to showing the

16

measured data, the Fig. 6 (4) also illustrates the HTF outlet temperature simulated by

17

the model. Then, the comparison between the model prediction and the measurement

18

shows that a reasonable agreement in consideration of the thermal capacity and

19

accuracy of the thermal resistance. Therefore, it proves that it is reasonable and reliable

20

to apply the model to further analyses of the influence of heat loss factors on the overall

21

performance of parabolic trough solar collectors in the following sections.

22

17 / 39

ACCEPTED MANUSCRIPT 1000

36

900 35

800

Flowrate (m 3/h)

DNI (W/m 2)

700 600 500 400 300 200

34

33

32

31

100 0 10:30

11:00

11:30

12:00

12:30

13:00

Time ( HH:MM )

1

30 10:30

13:30

(1)

11:30

12:00

12:30

13:00

Time ( HH:MM )

30

13:30

(2)

400

29.5

350

29

Temperature (  C)

Ambient temperature (  C)

11:00

28.5

28

27.5

300

250

200

150

27

26.5 10:30

11:00

11:30

12:00

12:30

13:00

100 10:30

13:30

Outlet simulation Outlet experiment Inlet 11:00

11:30

12:00

12:30

13:00

13:30

3

Time ( HH:MM ) (3) (4) Fig. 6. Measured and simulated data: (1) measured DNI, (2) measured flow rate, and

4

(3) measured ambient temperature, and (4) measured inlet and outlet HTF

5

temperatures and simulated outlet HTF temperature.

2

6

Time ( HH:MM )

3.2 Gas in the evacuated annulus

7

According to the calculation method in the section 2.2.2 Convective heat loss and

8

the geometry of HCE described in Table 3, the Fig. 7 illustrates the effective thermal

9

conductivity of the H2 and the air, respectively, at the temperature 230 °C as a function

10

of the pressure. Four distinct regimes of the H2 or air pressure appear obvious, as shown

11

in Fig. 7. To take the H2 for an instance, the effective gas thermal conductivity is less

12

than 1.55E-3 W/(m K) when the pressure of the H2 is below 0.1 Pa, which indicates the

13

convective heat loss in the annulus is very small because the very large mean-free path

14

between molecules makes intermolecular collisions rarely happen. As the pressure of

15

the H2 increases to about 14 Pa, the specific volume of the H2 begins to decrease while

16

the mean-free path decreases and consequently, the effective gas thermal conductivity

17

increases to 2.27E-2 W/(m K). With the further increase in the pressure of the H2 from 18 / 39

ACCEPTED MANUSCRIPT 1

about 14 Pa to 6850 Pa, the effective thermal conductivity of the H2 in this pressure

2

range is almost independent of the pressure and becomes nearly constant at about

3

2.54E-2 W/(m K). When the pressure of the H2 continues increasing from 6850 Pa, the

4

natural convection starts to take place, resulting in the considerable increase in the

5

effective thermal conductivity of the H2. In addition, a comparison between the Fig. 7

6

(1) and (2) shows that the effective thermal conductivity of the air is much smaller than

7

that of the H2 at the same conditions. For example, at the pressure 100 Pa, the effective

8

thermal conductivity is 0.025 W/(m K) if the air is the gas in the evacuated annulus

9

while it is 0.172 W/(m K) if the H2 is the gas. So it means that the H2 generation must

10

be noted as parabolic trough solar power plants run.

11 0.3

0.1 0.09 0.08 0.07

0.2

keff(W/(m K))

keff(W/(m K))

0.25

0.15

0.1

0.06 0.05 0.04 0.03 0.02

0.05

0.01 0 10 -2

12 13

10 0

10 2

Pressure (Pa)

10 4

0 10 -2

10 6

(1)

10 0

10 2

Pressure (Pa)

10 4

10 6

(2)

Fig. 7. Effective gas thermal conductivity at 𝑡bg = 230 °C for (1) H2 and (2) air.

14 15

In this study, the primary interest of the model is to demonstrate how the essential

16

impact factors associated with heat losses affect the overall thermal performance of

17

parabolic trough solar collectors under specified normally boundary conditions.

18

Therefore, to clearly compare among the results from the variations in different impact

19

factors, the transient processes are not considered in view of their complexity of the

20

coupled heat transfer behaviors although the model can simulate these unsteady

21

characteristics. Then, it means that the energy analyses in this study are independent of

22

the initial conditions. Additionally, a set of specified conditions was selected as a basis

23

and its details of boundary conditions are as follows: A loop of PTCs with structure

24

parameters described in the section 3.1 Validation is exposed to sun lights with the DNI 19 / 39

ACCEPTED MANUSCRIPT 1

= 900 W/m2. The surrounding air is at 𝑡sur = 25 °C and the wind blows across the PTCs

2

with the velocity u = 4 m/s. The HTF at 290 °C enters the inlet of the PTCs with the

3

flow rate 𝑉 = 40.35 m3/h, which can let the HTF outlet temperature be the required

4

value 394 °C.

5

Furthermore, a concept of the heat gain efficiency is proposed in this study. The

6

definition of this efficiency is the ratio of the energy removed by the HTF over a

7

specified time period to the absorbed solar energy by the HCE for the same period in

8

the steady-state process, which is given by the Eq. (24). Additionally, it should note

9

that different from the definitions of the thermal efficiency of solar collectors in the

10

most references[24-26], the definition of the heat gain efficiency does not include

11

optical losses in order to show how variations in parameters associated with the heat

12

influence the thermal performance of the PTCs.

13

𝜂HTF =

∫𝑞HTFd𝜏 ∫𝑞absd𝜏

.

(24)

14

Thus, four values of the annulus pressure of the H2 (0.01 Pa, 10 Pa, 1E3 Pa and

15

1E5 Pa) were selected to calculate the thermal performance of parabolic trough solar

16

collectors respectively, and resulting temperatures of the absorber, the glass envelope

17

and the HTF are shown as a function of the length position in the Fig. 8. Because of the

18

same HTF inlet temperature, the difference among the absorber temperatures at the inlet

19

of the PTCs for these four values of the annulus pressure is no more than 0.6 °C, as

20

shown in Fig. 8 (1). While at the outlet, the absorber temperature difference between

21

the cases at 0.01 Pa and 10 Pa is 11.5 °C and the absorber temperature difference

22

between the cases at 0.01 Pa and 1E5 Pa is about 20 °C. Moreover, the glass envelope

23

temperatures resulted from these four values of the annulus pressure are obviously

24

different as shown in Fig. 8 (2). For instance, an average temperature difference of the

25

glass envelope between the cases at 0.01 Pa and 10 Pa is about 35 °C, and the

26

temperature of the glass envelope at the outlet achieves 118.6 °C when the annulus

27

pressure of the H2 is 1E5 Pa. Further, as shown in Fig. 8 (3), for 10 Pa and 1E3 Pa, the

28

temperature distributions of the HTF are very similar and even the temperature

29

maximum difference which occurs at the outlet of collectors is no more than 2.4 °C. In 20 / 39

ACCEPTED MANUSCRIPT 1

addition, Table 4 shows the detailed analysis that reveals convection and radiation heat

2

losses from the absorber and the glass envelope as well as the heat gain efficiency.

3

Resulted from the increase in the annulus pressure of the H2, the rate of the convective

4

heat loss from the absorber 𝑞b,c continues increasing. The 𝑞b,c at 1E5 Pa is almost 300

5

times larger than that at 0.01 Pa. Moreover, the slight reduction in the rate of the

6

radiation heat loss from the absorber 𝑞b,r is caused by the decrease in the temperature

7

difference between the absorber and the glass envelope as shown in Fig. 8 (1) and (2).

8

Furthermore, Table 4 shows that because of the increase in the glass envelope

9

temperature, both rates of the convective and radiation heat losses from the glass

10

envelope (𝑞g,c and 𝑞g,r) increase as the annulus pressure of the H2 increases.

11 400

120 0.01 Pa 10 Pa 1000 Pa 100000 Pa

100

Temperature (°C)

Temperature (°C)

380

0.01 Pa 10 Pa 1000 Pa 100000 Pa

110

360

340

320

90 80 70 60 50

300 40 280

12

0

100

200

300

400

500

Length position (m)

30

600

(1)

0

100

200

300

400

Length position (m)

500

600

(2)

400 0.01 Pa 10 Pa 1000 Pa 100000 Pa

Temperature (°C)

380

360

340

320

300

280

0

100

200

300

400

500

600

14

(3) Fig. 8. Temperature as a function of the length position for various values of the

15

annulus pressure of the H2 for (1) the absorber, (2) glass envelope and (3) HTF.

13

Length position (m)

16 17

Table 4 Energy analysis for various values of the annulus pressure of the H2 21 / 39

ACCEPTED MANUSCRIPT

Pressure(Pa)

𝑞g,c(W)

𝑞g,r(W)

𝑞b,c(W)

𝑞b,r(W)

𝜂HTF(%)

0.01

1.01E+05

2.70E+04

1.36E+03

1.27E+05

94.5

10

2.73E+05

8.75E+04

2.45E+05

1.15E+05

84.6

1E3

3.09E+05

1.03E+05

3.00E+05

1.12E+05

82.3

1E5

3.79E+05

1.37E+05

4.09E+05

1.07E+05

77.9

1 2

Similarly, the air is taken as the gas which varies in the annulus pressure, and then

3

the resulting temperatures of the absorber, glass envelope and HTF are shown as a

4

function of the length position in the Fig. 9. Since the absorber temperature difference

5

between the cases at 0.01 Pa and 1000 Pa at the outlet is only 2.4 °C as shown in Fig.

6

9 (1), the tiny amount of the air does not matter to the heat loss. Moreover, as shown in

7

Fig. 9 (2), an average temperature difference of the glass envelope between the cases at

8

1E3 Pa and 1E5 Pa is about 20 °C, which gives the main change range in the heat loss

9

for the air. Furthermore, as shown in Fig. 9 (3), even when the annulus pressure of the

10

air increases from 0.01 Pa to 1E5 Pa, the HTF outlet temperature is reduced by less than

11

9 °C. Particularly for 10 Pa and 1000 Pa, it seems that the absorber or HTF temperatures

12

remain almost the same. This is because the difference in their effective thermal

13

conductivities is very small, which are similar to 0.0221 W/(m K) and 0.0255 W/(m

14

K), respectively in the Fig. 7 (2). Additionally, Table 5 indicates that when the annulus

15

pressure of the air is below 1E3 Pa, the variation in the amount of the air affects the

16

heat gain efficiency by no more than 2%. Therefore, for the air in the vacuum annulus,

17

the significant influence on the overall thermal performance of parabolic trough solar

18

collectors only occurs in the natural convection regime.

22 / 39

ACCEPTED MANUSCRIPT

400

90 0.01 Pa 10 Pa 1000 Pa 100000 Pa

360

340

320

300

280

1

0.01 Pa 10 Pa 1000 Pa 100000 Pa

80

Temperature (°C)

Temperature (°C)

380

70

60

50

40

0

100

200

300

400

500

Length position (m)

30

600

(1)

0

100

200

300

400

500

Length position (m)

600

(2)

400 0.01 Pa 10 Pa 1000 Pa 100000 Pa

Temperature (°C)

380

360

340

320

300

280

0

100

200

300

400

500

600

3

(3) Fig. 9. Temperature as a function of the length position for various values of the

4

annulus pressure of the air for (1) the absorber, (2) glass envelope and (3) HTF.

2

Length position (m)

5 6

Table 5 Energy analysis for various values of the annulus pressure of the air Pressure(Pa)

𝑞g,c(W)

𝑞g,r(W)

𝑞b,c(W)

𝑞b,r(W)

𝜂HTF(%)

0.01

1.01E+05

2.68E+04

3.62E+02

1.27E+05

94.5

10

1.33E+05

3.66E+04

4.45E+04

1.25E+05

92.8

1E3

1.37E+05

3.80E+04

5.07E+04

1.25E+05

92.5

1E5

2.34E+05

7.21E+04

1.88E+05

1.18E+05

86.9

7

8

3.3 Emissivity of the absorber

9

Selective coatings on the outer surface of the absorbers are widely adopted, which

10

provide the high absorptance for the solar radiation and the low emissivity for the long-

11

wave radiation. Usually, the polynomial curve fit equation of the emissivity for the

12

HCE as a function of the temperature is determined by experimental data in the 23 / 39

ACCEPTED MANUSCRIPT 1

laboratory. This equation plays an essential role in calculating the radiation heat loss

2

from the absorber. In the environment of the vacuum, the selective coatings are

3

supposed to work well. However, the emissivity will increase due to the degradation of

4

coatings especially at high temperature once the amount of the gas such as the air

5

increases. Hence, to analyze how the variation in the emissivity of the absorber affects

6

the thermal performance of the PTCs, this study defines the emissivity as a function of

7

the absorber temperature and the emissivity multiplier 𝐹emi, as expressed by Eq.(25),

8

based on the emissivity fit equation in the reference[27]. The emissivity fit equation is

9

made as the basis when the 𝐹emi is equal to 1 in this study.

[

10

𝜀b = (6.282𝐸 - 2) + (1.208𝐸 ‒ 4)(𝑇b - 273.15) + (1.907𝐸 ‒ 7)(𝑇b - 273.15)

11

𝐹emi

]

2

(25)

12

Then, four values of the emissivity multiplier (0.5, 1, 2 and 4) were selected to

13

calculate the emissivity of the absorber and the resulting curves are plotted as the

14

function of the absorber temperature in the Fig. 10. Thus, temperatures of the absorber,

15

the glass envelope and the HTF were calculated with these four emissivity curves

16

respectively, which are shown as a function of the length position in the Fig. 11. With

17

the same HTF inlet temperature, the maximum difference among the absorber

18

temperatures at the inlet using these four different emissivity curves is no more than 0.4

19

°C, as shown in Fig. 11 (1). As the HTF passes through the PTCs, the absorber

20

temperature difference between the cases for 𝐹emi = 4 and 𝐹emi = 1 achieves 15.6 °C

21

at the outlet, and the consequent emissivities at these two outlet temperatures of the

22

absorber are 0.47 and 0.12 respectively. Furthermore, the difference among the glass

23

envelope temperatures resulted from these four different emissivity curves is obvious

24

as shown in Fig. 11 (2). Particularly at the outlet, the glass envelope temperature for

25

𝐹emi = 4 is higher than that of the basis by more than 61 °C, and the glass envelope

26

temperature for 𝐹emi = 0.5 is lower than that of the basis by less than 15 °C. Moreover,

27

as shown in Fig. 11 (3), compared with that in the basis, the outlet HTF temperature for 24 / 39

ACCEPTED MANUSCRIPT 1

𝐹emi = 4 is lower by 14.5 °C due to the increase in the radiation heat loss of the absorber.

2

The outlet HTF temperature for 𝐹emi = 0.5 is increased by only 3 °C although the

3

emissivity is halved from the basis. Further, Table 6 shows the detailed analysis that

4

reveals convection and radiation heat losses from the absorber and the glass envelope

5

as well as the heat gain efficiency. The larger emissivity multiplier will increase the

6

rate of the radiation heat loss from the absorber, leading to the slightly lower absorber

7

temperature and the higher glass envelope temperature. Consequently, both rates of

8

convection and radiation heat losses from the glass envelope will be increased. Then,

9

the heat gain efficiency will be reduced by 13 % if the emissivity multiplier is four

10

times larger than the basis.

11 0.5 0.45 0.4

0.5 1 2 4

Emissivity (-)

0.35 0.3 0.25 0.2 0.15 0.1 0.05

12

0 100

150

200

250

300

350

400

Temperature (°C)

13

Fig. 10. Emissivity of the absorber as a function of the absorber temperature for

14

various values of the emissivity multiplier.

25 / 39

ACCEPTED MANUSCRIPT

420

120 0.5 1 2 4

0.5 1 2 4

110 100

380

Temperature (°C)

Temperature (°C)

400

360 340 320

90 80 70 60 50

300 280

1

40

0

100

200

300

400

500

30

600

0

100

(1)

Length position (m)

200

300

400

500

Length position (m)

600

(2)

400 0.5 1 2 4

Temperature (°C)

380

360

340

320

300

280

2

0

100

200

300

400

Length position (m)

500

600

(3)

3

Fig. 11. Temperature as a function of the length position for various values of the

4

emissivity multiplier for (1) the absorber, (2) glass envelope and (3) HTF.

5 6

Table 6 Energy analysis for various values of the emissivity multiplier 𝐹emi

𝑞g,c(W)

𝑞g,r(W)

𝑞b,c(W)

𝑞b,r(W)

𝜂HTF(%)

0.5

5.33E+04

1.34E+04

1.42E+03

6.53E+04

97.1

1

1.01E+05

2.70E+04

1.36E+03

1.27E+05

94.5

2

1.86E+05

5.49E+04

1.25E+03

2.40E+05

89.7

4

3.22E+05

1.11E+05

1.08E+03

4.31E+05

81.5

7

8

3.4 Wind speed

9

Because the convective heat loss from the glass envelope to the surroundings can

10

contribute to the overall heat loss from the PTCs, the effects of the wind speed on the

11

performance evaluation should be taken into consideration. The following study 26 / 39

ACCEPTED MANUSCRIPT 1

analyzes the thermal behaviors of the PTCs by giving the relative magnitude of heat

2

losses with respect to different wind speeds.

3

Thus, four values of the wind speed (2 m/s, 4 m/s, 6 m/s and 12 m/s) were selected

4

to calculate the thermal performance of parabolic trough solar collectors respectively.

5

All parameters except the wind speed are the same as for the basis described in the

6

previous section 3.2 Gas in the evacuated annulus. Then, the resulting temperature

7

distributions of the absorber, the glass envelope and the HTF are illustrated as the

8

function of the length position of the PTCs, as shown in Fig. 12. Although the different

9

curves of the temperature distribution of the glass envelope are shown in Fig. 12 (2),

10

the influence of the wind speed is not very significant. For example, the glass envelope

11

temperature at the outlet of the PTCs decreases by about 15 °C as the wind speed

12

increases from 4 m/s to 12 m/s, and the glass envelope temperature at the outlet

13

increases by about 11 °C as the wind speed decreases from 4 m/s to 2 m/s. In addition,

14

for the absorber or the HTF, it cannot tell the difference among the temperature

15

distribution curves for various wind speeds, as shown in Fig. 12 (1) and (3). The major

16

reasons for the almost invariable temperature distributions for different wind speeds is

17

that the perfect vacuum state in the annulus space of the HCEs and the low emissivity

18

of the absorber bring out the large thermal resistance to the heat transfer between the

19

absorber and the glass envelope. Furthermore, Table 7 shows convection and radiation

20

heat losses from the absorber and the glass envelope as well as the heat gain efficiency.

21

Resulted from the increase in the wind speed, the rate of the convection heat loss from

22

the glass envelope continues increasing, which causes the reduction in the glass

23

envelope temperature. Further, the reduced glass envelope temperature lets the rate of

24

the radiation heat loss from the glass envelope decrease. The increased amount of the

25

convection is almost equal to the reduced amount of the radiation, therefore, as the wind

26

speed changes from 2 m/s to 12 m/s, the total heat loss from the PTCs slightly varies

27

and only 0.1 % variation in the heat gain efficiency occurs.

28

27 / 39

ACCEPTED MANUSCRIPT 70

400 2 m/s 4 m/s 6 m/s 12 m/s

60

Temperature (°C)

Temperature (°C)

380

2 m/s 4 m/s 6 m/s 12 m/s

65

360

340

320

55 50 45 40

300

280

1

35

0

100

200

300

400

500

Length position (m)

30

600

(1)

0

100

200

300

400

500

Length position (m)

600

(2)

400 2 m/s 4 m/s 6 m/s 12 m/s

Temperature (°C)

380

360

340

320

300

280

2

0

100

200

300

400

500

Length position (m)

600

(3)

3

Fig. 12. Temperature as a function of the length position for various values of the

4

wind speed for (1) the absorber, (2) glass envelope and (3) HTF.

5 6

Table 7 Energy analysis for various values of the wind speed Wind speed (m/s)

𝑞g,c(W)

𝑞g,r(W)

𝑞b,c(W)

𝑞b,r(W)

𝜂HTF(%)

2

8.93E+04 3.81E+04 1.32E+03 1.26E+05

94.55

4

1.01E+05 2.70E+04 1.36E+03 1.27E+05

94.51

6

1.07E+05 2.13E+04 1.39E+03 1.27E+05

94.49

12

1.17E+05 1.27E+04 1.42E+03 1.28E+05

94.45

7

8

3.5 Temperature distribution

9

Based on the energy balance, the incident solar energy of parabolic trough solar

10

collectors is distributed into the useful energy gain, thermal losses and optical losses as

11

described in the previous section 2. Mathematical Model. For the utility-scale PTCs, 28 / 39

ACCEPTED MANUSCRIPT 1

the incident solar energy and the useful energy gain can be measured directly while

2

direct measurements of the total thermal loss or the total optical loss may not be easy

3

to obtain. Therefore, some indirect methods to determine the total thermal loss or the

4

optical performance are proposed, which make use of the energy balance mentioned

5

above. The general idea of indirect methods is that, in addition to the measurement with

6

the concentrated solar flux, an independent measurement is carried out without the

7

concentrated solar flux to eliminate the influence of optical losses. However,

8

temperature distributions of the absorber, the glass envelope and the HTF are different

9

under these two measurement conditions, which affects the total heat loss. Hence, in

10

the following, this study takes an indirect measurement method for instance to analyze

11

what the role of temperature distributions is in heat losses, the overall performance and

12

the reliability of the indirect method.

13

A measurement method may be proposed to evaluate the optical performance of

14

parabolic trough solar collectors indirectly. In this method, two processes are needed.

15

First, collectors are operated with the concentrated solar flux, and thus the thermal

16

power gained by the HTF and incident solar power are measured at the same time.

17

Second, the incident solar flux is not used, and consequently the measured change in

18

the enthalpy of the HTF in the process is considered as the total heat loss from the PTCs.

19

The idea behind this method is that the total heat loss in these two processes are

20

equivalent. Thus, a simple operation may be carried out by choosing the reasonable

21

inlet temperature in the second process which lets heat losses closely approximate those

22

in the first process. So this study analyzed how different temperature distributions in

23

these two processes affect heat losses.

24

As shown in Fig. 13, temperature distributions of the absorber, the glass envelope

25

and the HTF are illustrated as the function of the length position of the PTCs. They are

26

obtained by operating the PTCs under the boundary conditions in the basis described in

27

the section 3.2 Gas in the evacuated annulus. This typical operation state is quite normal

28

in the real solar thermal power plant, so it may be chosen in the first process for the

29

indirect measurement method of the optical performance. Then, four various values of

30

the HTF inlet temperature(290, 342, 350 and 360 °C) were selected to calculate 29 / 39

ACCEPTED MANUSCRIPT 1

temperature distributions of the absorber, the glass envelope and the HTF respectively

2

and the resulting curves are plotted as the function of the length position of the PTCs

3

in the Fig. 14. When the HTF enters the PTCs at the inlet temperature 290 °C like that

4

in the basis in the Fig. 13, the HTF temperature difference between the inlet and outlet

5

of the PTCs is 3.4 °C with the HTF passing through the PTCs, as shown in Fig. 14 (3).

6

As the HTF inlet temperature increases, the HTF temperature difference between the

7

inlet and outlet also increases due to the larger heat losses at the higher temperature.

8

For example, when the HTF inlet temperature changes from 290 °C to 360 °C, the HTF

9

temperature difference between the inlet and outlet is nearly doubled. In addition, for

10

the absorber temperature as shown in Fig. 14 (1), similar phenomenon happens and the

11

absorber temperature difference between the inlet and outlet is slightly higher than the

12

HTF temperature difference between the inlet and outlet under the same conditions.

13

Moreover, as shown in Fig. 14 (2), although the glass envelope temperature difference

14

between the inlet and outlet increases as the HTF inlet temperature increases, this

15

difference is so small that the temperature distribution of the glass envelope may be

16

assumed to be uniform along the length position of the PTCs. For instance, when the

17

HTF enters the PTCs at the inlet temperature 360 °C, the glass envelope temperature

18

difference between the inlet and outlet is only 1.2 °C.

19

Additionally, the rate of the total heat loss from the PTCs is 1.28E+05 W in the

20

basis in Fig. 13. Table 8 shows the detailed analysis that reveals convection and

21

radiation heat losses from the absorber as well as the glass envelope respectively.

22

According to the results, the HTF inlet temperature in the second process cannot be

23

chosen as the same value (290 °C) as in the first process because of much lower heat

24

loss in the second process. Moreover, if the arithmetic average temperature of the HTF

25

inlet and outlet temperatures in the first process (342 °C) is chosen as the HTF inlet

26

temperature in the second process, then the total heat loss in the second process will be

27

1.18E+05 W. Thus, this total heat loss is smaller than that in the first process by 8 % of

28

its value (1.28E+05 W). Therefore, the HTF inlet temperature in the second process has

29

to be increased in order to let heat losses approximate those in the first process as closely

30

as possible. From the data in the table, the HTF inlet temperature in the second process 30 / 39

ACCEPTED MANUSCRIPT 1

exists between 350 °C and 360 °C. Actually, the best way to choose the closely

2

equivalent heat losses in two processes is based on the glass envelope temperature

3

rather than the HTF temperature. Unfortunately, it is very hard to accurately measure

4

the glass envelope temperature in practice. Additionally, for thermal solar power plants,

5

the variations in the solar irradiance and the ambient temperatures can affect the

6

difference of the total heat loss in those two processes. Furthermore, heat losses from

7

the PTCs are also sensitive to the quality of the HCEs including the vacuum state in the

8

annulus space and the emissivity of the absorber. Consequently, temperature

9

distributions under detailed conditions in different processes should be taken with great

10

care in order to provide reliable indirect evaluation methods. 400

Temperature (°C)

350 Tb

300

Tg T

250

HTF

200 150 100 50 0

0

100

200

300

400

500

600

Length position (m)

11 12

Fig. 13. Temperature distributions as a function of the length position for the

13

absorber, the glass envelope and the HTF.

14 360

48

46

340 290 342 350 360

330 320

Temperature (°C)

Temperature (°C)

350

°C °C °C °C

310

44 290 342 350 360

42

°C °C °C °C

40

300 38

290 280

15

0

100

200

300

400

Length position (m)

500

600

(1)

36

0

100

200

300

400

Length position (m)

31 / 39

500

600

(2)

ACCEPTED MANUSCRIPT 360

Temperature (°C)

350 340 290 342 350 360

330

°C °C °C °C

320 310 300 290 280

1

0

100

200

300

400

500

Length position (m)

600

(3)

2

Fig. 14. Temperature as a function of the length position for various values of the inlet

3

temperature for (1) the absorber, (2) glass envelope and (3) HTF.

4 5

Table 8 Energy analysis for various values of the HTF inlet temperature 𝑇HTF,1(°C)

𝑞g,c(W)

𝑞g,r(W)

𝑞b,c(W)

𝑞b,r(W)

𝑞HTF(W)

290

5.89E+04

1.49E+04

1.23E+03

7.26E+04

-7.39E+04

342

9.32E+04

2.44E+04

1.35E+03

1.16E+05

-1.18E+05

350

9.95E+04

2.63E+04

1.37E+03

1.24E+05

-1.26E+05

360

1.08E+05

2.88E+04

1.39E+03

1.35E+05

-1.37E+05

6 7

4. Conclusion

8

In this study, a numerical model was developed to quantify how several important

9

factors affect heat losses and reveal the relationship between heat losses and the overall

10

performance of parabolic trough solar collectors under various boundary conditions. In

11

order to validate this model, experimental data including the HTF inlet and outlet

12

temperatures, the HTF flow rate, the ambient temperature and the direct normal solar

13

irradiance were measured for a test facility. This facility includes a utility-scale loop of

14

parabolic trough solar collectors designed for solar thermal power plants. The results

15

show that the overall match between predicted and measured outlet temperatures of the

16

heat transfer fluid indicates a good agreement. Then, parametric studies were carried

17

out for the critical impact factors to analyze how they influence heat losses and overall

18

thermal behaviors of parabolic trough solar collectors. Those impact factors include the 32 / 39

ACCEPTED MANUSCRIPT 1

amount of the hydrogen as well as the air in the evacuated annulus of the HCEs, the

2

emissivity of the absorber, the wind speed and temperature distributions with or without

3

the concentrated solar flux.

4

Several valuable conclusions can be drawn from these analyses. (1). The effective

5

thermal conductivity of the H2 is much larger than that of the air at the same conditions.

6

For example, at 230 °C and 100 Pa, the effective thermal conductivity is 0.172 W/(m

7

K) for the H2 while it is 0.025 W/(m K) for the air. So it means that the H2 generation

8

must be noted as solar thermal power plants run. (2). The heat gain efficiency will be

9

reduced by about 10 % if the H2 pressure in the vacuum annulus increases from 0.01

10

Pa to 10 Pa. (3). The variation in the amount of the air affects the heat gain efficiency

11

by no more than 2 % when the annulus pressure of the air is below 1E3 Pa. Therefore,

12

the significant influence of the air on the overall thermal performance of parabolic

13

trough solar collectors only occurs in the natural convection regime. (4). Larger

14

emissivity multiplier increases the rate of the radiation heat loss from the absorber,

15

leading to the slightly lower absorber temperature and the higher glass envelope

16

temperature. For example, the heat gain efficiency will be reduced by 13 % if the

17

emissivity multiplier is four times larger than the basis. (5). The wind speed does not

18

have a powerful influence on heat losses because the perfect vacuum state in the

19

annulus space of the HCEs and the low emissivity of the absorber bring out the large

20

thermal resistance which dominates the problem. For example, the total heat loss from

21

the PTCs slightly varies and only 0.1 % variation in the heat gain efficiency occurs as

22

the wind speed changes from 2 m/s to 12 m/s. (6). Temperature distributions caused by

23

different inlet temperatures of the HTF with or without the concentrated solar flux

24

should be taken with great care for providing indirect evaluation methods. For example,

25

for an indirect method to evaluate the optical performance of parabolic trough solar

26

collectors, the total heat loss can deviate the expected value by as much as 8 % if the

27

arithmetic average temperature of the HTF inlet and outlet temperatures in the process

28

with the concentrated solar flux is chosen as the HTF inlet temperature in the process

29

without the concentrated solar flux.

30 33 / 39

ACCEPTED MANUSCRIPT

1

Acknowledgment

2

This work was supported by the National Natural Science Foundation of China

3

(No. 51476165, 51476164 and 61505211), and the National Key Research and

4

Development Program of China (No. 2018YFB0905102).

5 6

Nomenclature A

area (m2)

Bc

Boltzmann’s constant (1.381E−23 J/K)

cp

specific heat capacity at constant pressure (J/kg K)

cv

specific heat capacity at constant volume (J/kg K)

D

outer diameter (m)

d

inner diameter of the absorber (m)

F

impact factor or multiplier

FV

view factor

G

solar irradiance (W/m2)

h

convection heat transfer coefficient (W/m2 K)

K

incident angle factor

k

thermal conductivity (W/m K)

L

length (m)

𝑚

mass flow rate (kg/s)

Nu

Nusselt number

P

pressure (Pa)

Pr

Prandtl number

q

rate of heat transfer (W)

Ra

Rayleigh number 34 / 39

ACCEPTED MANUSCRIPT Re

Reynolds number

T

temperature (K)

t

temperature (°C)

u

wind speed (m/s)

V

volume (m3)

𝑉

volume flow rate (m3/h)

Greek symbols α

absorptance

ε

emissivity

η

efficiency

σ

Stefan-Boltzmann constant (5.67E-8 W/m2 K4)

λ

thermal conductivity (W/kg K) or mean-free-path (m)

ρ

density (kg/m3) or reflectance of the reflector

τ

Time (s) or transmittance

θ

Incident angle (°)

γ

intercept factor

Subscripts a

aperture

abs

absorbed part by the absorber

an

annulus zone

b

absorber

c

convective heat loss

ch

characteristic

cl

cleanliness

35 / 39

ACCEPTED MANUSCRIPT cond

conductive heat transfer

cy

cylinder

emi

emissivity

end

end loss

DNI

direct normal solar irradiance

eff

effective

fc

forced convection

mo

molecular

nc

natural convection

g

glass envelope

HTF

heat transfer fluid

i

node index

in

inner surface

lam

laminar

NL

total number of the nodes

n

at normal incidence

r

radiative heat loss

s

outer surface of the glass envelope

std

at the standard temperature and pressure

sur

surroundings

tur

turbulent

1 2

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Highlights > Model was validated with experiment data for utility-scale PTCs. > Parametric studies were made to analyze critical heat loss factors. > Effects of these factors on overall performance are quantified respectively. > H2’s effective thermal conductivity is much larger than air’s in tube annulus. > Wind speed barely affects heat losses due to perfect vacuum tubes.