HAL Id: hal-01344014https://hal.archives-ouvertes.fr/hal-01344014v1Preprint submitted on 11 Jul 2016 (v1), last revised 16 Nov 2017 (v3)

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A comprehensive dynamic model for opticalcharacterization of a solar power tower with a

multi-tube cavity receiverYu Qiu, Ya-Ling He, Peiwen Li, Bao-Cun Du

To cite this version:Yu Qiu, Ya-Ling He, Peiwen Li, Bao-Cun Du. A comprehensive dynamic model for optical character-ization of a solar power tower with a multi-tube cavity receiver. 2016. �hal-01344014v1�

1

A comprehensive dynamic model for optical characterization of a

solar power tower with a multi-tube cavity receiver

Yu Qiua, Ya-Ling He

a,*, Peiwen Li

b, Bao-Cun Du

a

a Key Laboratory of Thermo-Fluid Science and Engineering of Ministry of Education , School of Energy and

Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China

b Department of Aerospace and Mechanical Engineering, The University of Arizona, Tucson, AZ 85721, USA

Abstract: A comprehensive dynamic optical model for the Solar Power Tower (SPT) with a

Multi-Tube Cavity Receiver (MTCR) was developed using Monte Carlo Ray Tracing (MCRT)

method. After validation, the model was used to study the optical performance of the

DAHAN plant. The visualization results illustrate that the solar flux in the MTCR exhibits a

significant non-uniformity, showing a maximum flux of 5.141×105

W·m-2

on the tubes. A

comparison of the tracking models indicates that it is a good practice to treat the tracking

errors as the random errors of the tracking angles for considering the random effect on the

solar flux distribution. Study also indicates that multi-point aim strategy helps homogenizing

the flux and reducing the energy maldistribution among the tubes, which should be

recommended. Additionally, the cavity effect on the efficiency was revealed quantitatively,

which indicates that the reflection loss can be reduced significantly by the cavity effect,

especially when the coating absorptivity is relatively low. At the end of this study, dynamic

variations of the optical efficiencies were investigated, and the yearly efficiency for the

energy absorbed by the tubes was found to be 65.9%. The simulation results indicate that the

present model is accurate and suitable for predicting both the detailed solar flux and the

dynamic efficiency of SPT.

Keywords: Solar power tower; Multi-tube cavity receiver; Monte Carlo ray tracing; Dynamic

optical model; Solar flux distribution; Dynamic performance

1. Introduction

The global warming caused by carbon dioxide emitted through fossil fuel combustion

has become a pressing issue for years [1, 2]. Efficient utilization of solar energy is being

considered as one of the promising solutions to this challenge [3, 4]. The Concentrating Solar

Power (CSP) technology, mainly including the Solar Power Tower (SPT)[5-7], Parabolic

Dish Collector[8-11], Parabolic Trough Collector [12, 13], and linear Fresnel reflector[14,

15], has become a promising choice to utilize solar energy during the past few decades.

Relatively, the SPT is considered as an advanced and promising technology for large scale

utilization of solar energy.

A typical SPT consists of a heliostat field, a receiver mounted on a tower, and thermal

energy storage and conversion modules [16, 17]. There are four typical configurations of

receivers including Multi-Tube Cavity Receiver (MTCR), multi-tube external receiver,

volumetric receiver, and direct-absorption receiver for SPT [18]. Among these configurations,

2

the MTCR has been widely applied for the high efficiency [5, 19]. In the SPT using a MTCR,

the heliostats will track the sun and concentrate the sun rays into the MTCR firstly. Then, the

solar radiation will be absorbed by the absorber tubes and walls after multiple reflections. It is

commonly known that the absorbed solar flux on the tubes is exceedingly uneven and varies

greatly over time, which would result in extreme fluctuant non-uniform temperature and

stress, and lead to negative effects on the performances and safety of the system [20, 21].

Hence, the accurate dynamic simulation of the solar flux in MTCR is of great importance for

the performance optimization, system design, and safe operation of the SPT [22-24].

Many studies have focused on this topic, and computer codes have been developed, such

as UHC, DELSOL, HFLCAL based on convolution, MIRVAL, HFLD, and SOLTRACE based

on ray tracing [22, 25]. Vant-Hull [24] used UHC to design the aiming strategies and control

the incident flux on the cylinder receiver of solar two. Salomé and Chhel [7] used HFLCAL to

control the incident flux on the MTCR’s aperture of THEMIS plant. Rinaldi and Binotti [26]

computed the incident flux on the simplified tube panels of a MTCR in PS10 by DELSOL3.

Mecit and Miller [27] used MIRVAL to compute the incident flux on the aperture of a particle

receiver in Sandia’s NSTTF. Wei and Lu [28] developed the HFLD code and used it to

compute the incident flux on the MTCR’s aperture in DAHAN and optimize the heliostat field.

Similar work has been done for DAHAN by Yu and Wang [29], and the dynamic variation of

the incident flux on the simplified tube panels was revealed. Sanchez-Gonzalez and Santana

[30] used SOLTRACE to simulate the incident flux on a cylinder receiver, and the results are

used to validate a projection method for flux prediction. Yellowhair and Ortega [22] also used

SOLTRACE to evaluate some novel complex receivers with fins for the enhancement of the

solar radiation absorption.

Review of the literature indicates that the first five codes mentioned above are limited to

standard receiver geometries [25] such as flat plate, cylinder, and simplified cavity receiver

without considering the tubes and cavity effect coming from the multiple reflections and

absorptions on the tubes and cavity walls, although they can predict the dynamic performances

at different time and locations. It can also be found that there is almost no limit on geometries

of the receiver in SOLTRACE; however, SOLTRACE has no function to predict the dynamic

performances, because the sun position and heliostat tracking angles cannot be updated

automatically in the code [22]. The current status is that no studies have developed a model to

manage both the complex geometry with complex optical processes in the MTCR and the

dynamic performance prediction for the SPT.

To provide better studies to the optical system of SPT, present work focuses on

developing a comprehensive dynamic optical model for the SPT with a MTCR by Monte

Carlo Ray Tracing (MCRT) [31]. Based on the model, the optical performance of the DAHAN

plant [32] with a redesigned molten salt MTCR is studied, and the results are discussed.

2. Physical model

The DAHAN plant located at 40.4°N, 115.9°E in Beijing is taken into consideration for

the current physical model [33]. The heliostat field and a new designed molten salt MTCR

including 30 panels and 620 tubes are shown in Fig. 1 and Fig. 2, respectively. The detailed

parameters are given in Table 1. Due to the lack of published data, the optical errors of the

3

heliostat are assumed to be the same as those of PS10[26].

0

-50

-100

-150

-200

-250

-300

-350

-150 -100 -50 0 50 100 150 200

Heliostat

Uninstalled Tower

Xg /

m

S-N

Yg / m W-E

B

C

D

E

O

Aperture

O

Xr

Yr

Zr

Fig. 1. Radial staggered heliostat field in DAHAN plant. Fig. 2. Sketch of the MTCR in DAHAN plant.

Table 1 Parameters and assumptions of DAHAN plant [33-35].

Parameters Dim. Parameters Dim.

Heliostat number nh 100 Tube distance in a panel 1 mm

Heliostat shape Spherical Distance between panels 1 mm

Heliostat width Wh 10 m Aperture height 5m

Heliostat height Lh 10 m Aperture width 5m

Heliostat center height 6.6 m Heliostat reflectivity ρh,1 0.9

Tower height 118 m Heliostat cleanliness ρh,2 0.97

Tower radius 10 m Altitude tracking error σte,1=σte 0.46 mrad

Receiver Height HO 78 m Azimuth tracking error σte,2=σte 0.46 mrad

Receiver altitude αr 5π/36 Heliostat slope error σse 1.3 mrad

Panel number 31 Coating absorptivity αt 0.9

Tubes in a rear panel 25 Coating diffuse reflectance ρt,d 0.1

Tubes in a side panel 20 Cavity wall absorptivity αw 0.6

Tube radius 19 Wall diffuse reflectance ρw,d 0.4

3. Mathematical model

The radiation transfer from the sun to the MTCR can be divided in two parts. One is the

process in the heliostat field, and the other is the process within the MTCR. A comprehensive

dynamic MCRT model and corresponding code named after SPTOPTIC were developed to

simulate the two processes, with the flow chart of the simulation shown in Fig. 3.

To describe the model, several Cartesian right-handed coordinate systems are established

(Fig. 4). The ground system is defined as XgYgZg, where the tower base G is the origin, and Xg,

Yg, and Zg points to the south, east, and zenith, respectively. The heliostat system is defined as

XhYhZh, where the center of each heliostat H is the origin. Xh is horizontal, and Yh is normal to

the tangent plane at H and points upwards. Zh is perpendicular to XhYh plane. The

incident-normal system is defined as XiYiZi, where the point which is hit by the ray on the

heliostat is the origin, and Zi points towards the sun. Xi is horizontal and normal to Zi, and Yi

is perpendicular to XiZi plane and points upwards. The receiver system is defined as XrYrZr,

4

where the aperture center is the origin. Xr points to the east, and Yr points upwards. Zr is

perpendicular to XrYr plane. The tube system is defined as XtYtZt and the tube center T is the

origin. Xt is parallel to XrYr, and Yt is coincident with the tube centerline and points upwards.

Zt is normal to XtYt plane. The wall system is defined as XwYwZw in the similar way as that of

XtYtZt (Fig. 4). The local system on tube is defined as XlYlZl, and the relation between XtYtZt

and it is illustrated in Fig. 4. The transformation matrixes including M1 ~ M14 among these

systems are summarized in the Appendix.

Define the date, time and photon number.

Define the geometric parameters.

Define the optical parameters.

Start

Initialize photon distribution in the heliostat field

Compute the solar density and position

Reflected by heliostat ?

Blocked by heliostats ?

N

Y

Compute the specular reflection

Hit the aperture ？

N YN

Hit the tube or cavity wall？

N

Y

Y

N YNY

Reflection type？

Compute

specular

reflection

Compute

diffuse

reflection

Specular Diffuse

Last photon ?N

Calculate the position on aperture, tube or wall

Abandoned

Count the photon distributions

Count the solar flux distributions

on the tubes and walls

End

Calculate the optical efficiency

(1)

(2)

Shadowed by tower or heliostats ?

Absorbed by tube or wall？

Y

Y

Fig. 3. The flow diagram of the SPTOPTIC code.

Xr

Yr

O

ZrA

Xg

Zg (zenith)

H

Zh

αh

Ah

Asαs θh

+

+

-

-

I

R

Sun

Sun rays

G

δI

R

Zi

Heliostat Ⅰ

Heliostat Ⅱ

HO

λh

αr

Xr

Zr

O

Xt

Zt

Xl

Zl

θt+

Optical processes in MTCR

Reflection

Reflection loss

θi

-

Zi

Xi

Yi

Xi

Yi

Ray

Ph

Zw

Xw

Yw

Zr

Xr

Yrαw,r

+-

Zt

Xt

YtZr

Xr

Yrαt,r

+-

The tube, XtYtZt and XrYrZr

The wall, XwYwZw and XrYrZr

T

W

South

Fig. 4. Sketch of the SPT with a MTCR showing the solar ray transfer and coordinate systems.

3.1 Modeling of the solar ray transfer in the heliostat field

3.1.1 Tracking equations of the heliostat

5

The altitude (αh) and azimuth (Ah) of the heliostat’s center normal are defined in Eq.(1),

where the quadrant ambiguity of Ah should be recognized when the sun rays come from the

north[36]. The tracking errors are treated as the angles’ errors (Model A) [37]. This treatment

is different from another model (Model B) which treats the tracking errors as equivalent slope

error and calculates the total slope error including the effects of tracking and slope errors by 2 2 2

se te,1 te,2 [36].

1 s hh te,1

i

1 h h s sh te,2

h h s s

sin cos=sin

2cos

sin sin sin costan

cos sin cos cos

R

AA R

A

(1)

where h is the azimuth of the heliostat in the field, which is calculated using Eq.(2);

Computed by Eq.(3) is h , which is the angle between the line HA and local vertical; Given

in Eq.(4) are H and A which are the heliostat’s center and the aim point in XgYgZg,

respectively; i is the incident angle of the principle ray at the heliostat center; αs and Αs are

the solar altitude and azimuth given in Eq.(6) and (7) [38], respectively; 2

te,1 te,10 ),~ (R N and

2

te,2 te,20 ),~ (R N are the tracking errors of αh and Ah, respectively.

1 2 2

h ,g ,g ,g ,gcos / , 0x x y y H H H H

(2)

1

h ,g ,g ,cos /z z D A H H A

(3)

T T

,g ,g ,g ,g ,g ,gx y z x y z H H H A A AH A, (4)

1

i s h s h h s

2cos sin cos cos sin cos( ) 1

2A

(5)

1

s sin sin sin cos cos cos (6)

1

s

sin sin sincos , 0

cos cosA

(7)

In the above equations, ,D

H Ais the distance between H and A; φ, δ, and ω are the latitude,

declination, and hour angle, respectively; the heliostat azimuth in the field should be 2π-θh

when ,g 0y

H; the solar azimuth should be –As when ω>0.

3.1.2 Solar model and photon initialization

The shape effect of the sun is considered, and the photons initialized on a point of the

6

heliostat is treated as a cone with a half apex angle of δ=4.65 mrad (Fig. 4). So, the unit

vector (I) of an incident photon in XiYiZi can be written in Eq.(8). A solar radiation model

given in Eq.(10) is applied to predict the Direct Normal Irradiance (DNI) at any time in a year

[39]. The energy carried by each photon on the heliostats (ep) is calculated by Eq.(11).

T

2

i s s s s scos sin 1

I (8)

1 2

s 1 s 2=sin sin , =2 (9)

day s

s

2 sin1367 1 0.033cos

365 sin 0.33

NDNI

(10)

h

p h h cos p1( ) /

n

ie DNI L W i N

(11)

where each ξ is a uniform random number between 0 and 1, i.e. ξ ~U[0,1]; Nday is the day

number in a year; ηcos(i) is the cosine efficiency of the ith heliostat; Np is the total number of

the photons traced in the field; Lh and Wh are the height and width of the heliostat,

respectively.

The solar radiation is assumed to be uniform, so the photons are initialized uniformly on

the heliostat, and the intersection of the photon and the heliostat is initialized by Eq.(12).

,h h 3

h ,h h 4

2 2 2,h

, , ,h ,h

( 0.5)

= ( 0.5)

2 4

x L

y W

z D D x y

P

P

PH O H O P P

P (12)

where DH,O is the distance between H and O in Fig. 4; and the heliostat radius equals to twice

of DH,O.

3.1.3 Specular reflection on the heliostat

When the photon hits the heliostat, the reflection computation will be conducted. Firstly,

a random number (ξ5) is generated to determine the optical process by Eq.(13). Then, if the

photon is reflected, the incident vector Ii will be transformed from XiYiZi to XhYhZh by Eq.(14).

Finally, the reflected vector Rh at Ph in XhYhZh will be calculated by Eq. (15). The slope error is

assumed to follow the Gaussian distribution [40], and the real normal vector (Nh) at Ph is

expressed in Eq.(16).

5 h,1 h,2

h,1 h,2 5

0 , specular reflection

1 , abandone

 

d

(13)

T

h hi hi hi 4 3 2 1 icos cos cos = I M M M M I (14)

h h h h h2 R = I N N I (15)

7

2 T

h 6 5 h h h h h

2

h se 6 h 7

cos sin 1

2 ln(1 ), 2

N = M M (16)

where M1 and M2 are the transformation matrixes from XiYiZi to XgYgZg; M3 and M4 are the

transformation matrixes from XgYgZg to XhYhZh; M5 and M6 are the transformation matrixes to

introduce slope error; ρh and φh are the radial and tangential angles of Nh caused by slope

error [37].

3.1.4 Shading and blocking

The shading is the part of heliostat shadowed by the adjacent heliostats or the tower, and

the blocking is the part of reflected rays blocked by nearby heliostats (Fig. 4). The blocking

here is taken as an example to illustrate the modeling of the two processes. First, the

initialized location (PI) on heliostat I and the reflection vector (RI) at PI are transformed from

XhYhZh(I) to XhYhZh(II) and expressed as PI,II (Eq.(17)) and RI,II (Eq. (18)), respectively. Then,

the equation of the reflected ray in system II can be derived using PI,II and RI,II. Finally, the

intersection of the ray and heliostat II surface is calculated, and if it is within heliostat II, the

ray is blocked.

4 3 8 7, ⅠⅡ Ⅰ Ⅰ ⅡⅡ Ⅰ

P M M M M P + H H (17)

4 3 8 7, = ⅠⅡ ⅠⅡ ⅠR M M M M R (18)

where M7 and M8 are the transformation matrixes from XhYhZh to XgYgZg.

3.2 Modeling of the solar ray transfer in the MTCR

3.2.1 Intersection with the surfaces in MTCR

When a ray is reflected and arrives at the focal plane of the field, i.e., the MTCR’s

aperture (Fig. 4), the intersection Pa,r in XrYrZr is calculated by transforming Ph and Rh to

XrYrZr, which are expressed as Ph,r and Rr in Eq.(19) and Eq.(20), respectively. When the ray

gets through the aperture and hits the tube or wall, the intersection will be calculated. The

intersection with the tube is taken as an example to illustrate this process. Firstly, the Pa,r and

Rr are transformed from XrYrZr to XtYtZt by Eq.(21) and Eq.(22) and expressed as Pa,t and It,

respectively. Then, the intersection (T

,t ,t tt,t ,x y z P P P

P ) in XtYtZt is computed by solving the

ray and the tube equations. The intersection of the ray and the wall can be calculated in the

similar way.

h,r 9 8 7 h g g P M M M P + H O (19)

r 9 8 7 h= R M M M R (20)

a,t 11 10 a r r= ,

P M M P T (21)

t 11 10 r= I M M R (22)

where gO is the origin of XrYrZr in XgYgZg; M9 is the transformation matrix from XgYgZg to

XrYrZr; M10 and M11 are the transformation matrixes from XrYrZr to XtYtZt; rT is the origin of

8

XtYtZt in XrYrZr.

3.2.2 Multiple reflections among the tubes and walls

When the photon hits the cavity walls or the tubes (Fig. 4), a random number (ξ8) is

generated to determine the optical process by Eq.(23). If the photon is reflected diffusely, the

reflected vector (Rl) in XlYlZl will be computed by Eq. (24) based on the Lambert law [15, 41].

If the photon is reflected specularly, Rl will be calculated by Fresnel’s Law in the similar way

as that on the heliostat [42].

t,d

t,d t

t,

8

8

8d t,s

0 , diffuse reflection

1 , specular reflection

1, absorptio

 

n

(23)

T

l d d d d d

1

d 9 d 10

= sin cos sin sin cos

=cos , =2

R

(24)

After the reflection, firstly, Rl will be transformed from XlYlZl to XtYtZt and expressed

as Rt in Eq.(25). Then, Rt and Pt,t are transformed from XtYtZt to XrYrZr and expressed as Rr

and Pt,r in Eq.(25) and Eq.(26). Then we should go back to section 3.2.1 and begin to

calculate the next intersection between the ray and other surfaces using the new Rr and Pt,r.

These processes will continue until the ray is absorbed or lost.

t 12 l r 14 13 t, = R M R R M M R (25)

t ,r 14 13 t,t r P M M P + T (26)

where T

,t ,t tt,t ,x y z P P P

P is the intersection on the tube in XtYtZt; M12 is the transformation

matrix from XlYlZl to XtYtZt; M13 and M14 are the transformation matrixes from XtYtZt to XrYrZr.

3.2.3 Statistics of the photon and flux

The quadrilateral meshes are generated on the tubes and walls, and when a photon is

absorbed by these surfaces, the statistics of the photon would be conducted in the following

way. First, the photons absorbed in each mesh (np,mesh) would be counted. Then, the local

solar flux in each mesh (ql) would be computed after the tracing of the last photon by Eq.(27).

The mesh of 20 (circumferential) ×200 (length) for each tube is considered for present

MTCR.

l p p,mesh mesh/q e n S (27)

where Smesh is the area of the mesh.

3.3 Parameter definitions

Some performance indexes are defined below to characterize the optical performance.

The energy maldistribution index (σE) among the tubes is defined in Eq.(28). The

instantaneous optical efficiency of the SPT (ηi,T) is defined as the ratio of the energy absorbed

by the tubes (Qij,T) and the maximum solar energy that can be accepted by the heliostats (Qij,H)

in Eq.(29). The daily and yearly optical efficiencies are defined as ηd,T and ηy,T in the similar

9

way in Eq.(30) and Eq.(31), where the SPT is assumed to operate when the solar altitude is

larger than 10o . The instantaneous optical efficiency (ηi,A) of the energy entering the aperture

(Qij,A) is defined as the ratio of Qij,A and Qij,H in Eq.(33), and the daily and yearly efficiencies

of the energy entering the aperture are defined in the similar way. The instantaneous

efficiency of the receiver (ηi,R) and the reflection loss (Qi,loss) from the MTCR are defined in

in Eq.(34).

t

t

2

1 t t t

E t t

t t1

/ -1 1= ,

i

i

n

nE i E

E E in

nE

(28)

i,T ,T ,H ,H m m m/ ,ij ij ij ijQ Q Q DNI L W n (29)

2 2

1 1d,T m m m/ ( )

s s

s s

t t

ij iji t i t

Q DNI L W n

(30)

2

1

2

1

365

1

y,T 365

m m m1( )

s

s

s

s

t

ijj i t

t

ijj i t

Q

DNI L W n

(31)

o

1 2( ) ( ) 10s st t (32)

i,A ,A ,H/ij ijQ Q (33)

i,R ,T ,A i,loss ,A ,T/ ,ij ij ij ijQ Q Q Q Q (34)

where nt is the number of the tubes; Et(i) is the power absorbed by ith tube; ( )st is the

solar altitude at the solar time of ts, DNIij is the DNI at i o'clock in jth day in a year,

respectively.

4. Uncertainty analysis and model validation

The uncertainty introduced by the MCRT is an aleatoric uncertainty which mainly

depends on the random photon number and thus can decrease with the increase of photon

number [43, 44]. Figure 5 shows the maximum flux (ql,max) on the tubes and ηi,T with different

photon sizes at summer solstice noon, where Model B is applied. It is seen that there is no

obvious change in ql,max and ηi,T when the photon number is larger than 5×108 and 2×10

7,

respectively. The uncertainty analysis was also conducted under all other conditions.

To validate the model, the computed incident flux and power on the MTCR’s tube panels

(simplified as flat plates) of PS10 is compared with those in literature [26] as shown in Fig. 6.

It is seen that the two patterns of the fluxes in Fig. 6(a) and 6(b) agree well with each other.

The deviations of the peak fluxes and the total powers are less than 0.1% and 0.4%,

respectively. Furthermore, the flux curves on a MTCR’s tubes in a linear Fresnel reflector [15]

were compared with those of SOLTRACE at normal incidence. It is seen in Fig. 7 that the

present curves agree with those of SOLTRACE quite well. The good agreement indicates that

the present model is accurate for modeling both the heliostat field and the MTCR.

10

105

106

107

108

109

1010

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Ma

xim

um

flu

x o

n t

ub

e q

l,m

ax /

×10

6 W

·m-2

Photon number Np

Maximum solar flux on tube ql,max

70.24

70.26

70.28

70.30

70.32

70.34

70.36

70.38

70.40

Inst. optical efficiency i,T

Inst

. o

pti

cal

effi

cien

cy

i,T/

%

Fig. 5. Uncertainty analysis of the MCRT model

(a) Data from Ref.[26]. Total Power =54.8 MW, Peak

flux =714.0 kW·m-2

(b) Present result. Total Power =55.0 MW, Peak flux

=714.9 kW·m-2

Fig. 6. Comparison of the incident flux distributions between published data and present result

(Equinox noon, DNI=970 W·m-2

).

0 60 120 180 240 300 3600

10

20

30

40

50

Circumferential angle variable on the tube θ / °

Lo

cal

sola

r fl

ux

ql /

kW

·m-2 Tube 1: SolTrace , MCRT

Tube 4: SolTrace , MCRT

DNI=1000 W·m-2

Fig. 7. Comparison of the flux curves on the tubes between MCRT and SOLTRACE.

5. Results and discussion

5.1 Visual presentation of typical solar flux distribution

11

Figure 8 shows the visual presentation of typical solar flux distributions in the MTCR at

12:00, spring equinox, where all the heliostats aim at O in Fig. 2. It is seen from Fig. 8(a) that

local solar flux (ql) on the aperture decreases from the center to the margin, and the maximum

flux (ql,max) of 2.622×106 W·m

-2 appears at the center. From Fig. 8(b) and (c), it is observed

that two high flux regions appear on the side panels, and the ql,max of 5.141×105 W·m

-2 occurs

on tube 443 in Fig. 2. It is also seen that most energy is concentrated on the middle part of

each tube, and other parts along the length are barely utilized. From Fig. 8(d), it is seen that a

hot spot appears on the rear wall, because the direct radiation from the field shines on the wall

from the tube gaps. However, there is no spot on the side walls for the reason that the direct

radiation is blocked by the tubes installed along the side walls which are very steep in the

depth direction of the MTCR. Also, the hot spots appear on the upper and lower walls due to

the diffuse reflections among the surfaces in the MTCR. Figure 8 (e) illustrates the whole

flux distribution in the MTCR combining the tubes and cavity walls, and this detailed

distribution could be applied in heat transfer analysis of the MTCR and performance

evaluation of the SPT in the future.

(a) Aperture

ql,max=2.622×106

W·m-2

, ηi,A=80.7%

(b) Tubes

ql,max=5.141×105

W·m-2

, ηi,T=75.7%, σE =65.2%

(c) Detailed solar fluxes on tubes 441~445, and the ql,max

locates on tube 443.

(d) Cavity walls. ql,max=12710 W·m-2

.

12

(e) Both the tubes and cavity walls.

Fig. 8. Visual presentation of typical solar flux distributions in the MTCR at 12:00, spring equinox

(Model A , σte =0.46 mrad, DNI=961 W·m-2

).

5.2 Comparison of the tracking models

Figure 8(a) and (b), Fig. 9, and Fig. 10 show the effects of the tracking error models on

solar flux distribution, maximum flux (ql,max), maldistribution index (σE), and instantaneous

efficiencies (ηi,A, ηi,T), where all the heliostats aim at O in Fig. 2.

(a) Model B, aperture

ql,max=2.478×106 W·m

-2, ηi,A=80.5%

(b) Model B, tubes

ql,max=5.016×105 W·m

-2, ηi,T=75.4%, σE =63.5%

Fig. 9. Solar fluxes on the aperture and tubes at 12:00, spring equinox

(Model B , σte =0.46 mrad, DNI=961 W·m-2

).

It is seen in Fig. 8(a), (b) and Fig. 9 that the variation of the flux distribution is

insignificant when σte=0.46 mrad; the ql,max on the aperture and the tube for Model A are only

about 5.8 % and 2.5 % respectively larger than that for Model B. However, it is seen in Fig.

10 that the random effect on the flux distributions becomes significant for Model A when

σte=1.0 mrad; the ql,max on the aperture and the tube for Model A are 24.9 % and 11.2%

respectively larger than that for Model B. Under this condition, the σE for Model A is 8.2%

larger than that for Model B. As a result, a deviation in ηi,T of 1.5 percent is also observed.

These results indicate that the random effects of the tracking errors are smoothed by the

widely-used Model B, which however is revealed by Model A more clearly. Since the

13

accurate prediction of ql,max etc. is important for the safe operation of the plant, the random

effect should be considered, and therefore Model A is recommended from the current study,

especially, when σte is relatively large.

(a) Model A, aperture

ql,max=2.254×106

W·m-2

, ηi,A=80.1%

(b) Model A, tubes

ql,max=4.787×105

W·m-2

, ηi,T=75.0%, σE =64.2%

(c) Model B, aperture

ql,max=1.804×106

W·m-2

, ηi,A=78.6%

(d) Model B, tubes

ql,max=4.304×105

W·m-2, ηi,T=73.5%, σE =59.5%

Fig. 10. Solar fluxes on the aperture and tubes at 12:00, spring equinox

(Model A and Model B, σte =1.00 mrad, DNI=961 W·m-2).

5.3 Effects of multi-point aim strategy

Figure 8(b) and Fig. 11 show the heat flux obtained using the 5 point aim strategy with

d=0.7 m, as indicated in Fig. 1 and Fig. 2, and the traditional 1 point strategy where all

heliostats aim at O. It is seen that at time of 12:00 ql,max on the tube and σE drop 10.5 % and

31.6 %, respectively; and the values are 12.0 % and 33.7 % for the time of 15:00 when the 5

point strategy is applied. It is also seen that longer tubes can be utilized when the multi-point

aim strategy is applied to. These results indicate that the flux can be greatly homogenized by

the multi-point aim strategy with just a little drop in efficiency. Therefore, this method should

be recommended to study SPT and is used in the following sections.

14

(a) 12:00, 5 points: ql,max=4.599×10

5 W·m

-2, ηi,A=80.0 %, ηi,T=74.7%, σE =44.6%

(b) 15:00, 1 point:

ql,max=4.320×105 W·m

-2, ηi,T=69.5 %, σE =63.5 %

(c) 15:00, 5 points:

ql,max=3.799×105 W·m

-2, ηi,T=68.3 %, σE =42.1 %

Fig. 11. Effects of aim strategies on solar flux distribution, ql,max, σE, and ηi,opt at spring equinox.

(DNI=855 W·m-2

at 15:00).

5.4 Impact of cavity effect and tube absorptivity

Figure 12 shows the instantaneous efficiency (ηi,T) for the power absorbed by the tubes

and the reflection loss (Qi,loss), against the coating absorptivity (αt). It is seen that the ηi,T with

cavity effect is larger than that of without cavity effect at the same αt, due to the fact that the

Qi,loss is reduced by the cavity effect that renders the multiple reflections and absorptions

among the tubes and walls. For example, Qi,loss decreases from 2658 kW to 1617 kW after

considering the cavity effect at αt =0.65, and the corresponding increases of the absorbed

power (Qij,T) and ηi,T are 1041 kW and 10.8 percent, respectively. It is also seen that the

decrease of Qi,loss due to cavity effect becomes less when αt is higher. For instance, at αt =0.90

the cavity effect makes the increase of Qij,T and ηi,T being 340 kW and 3.5 percent,,

respectively. Therefore, it is clear that the impact of cavity effect is more significant at low αt

than that at high αt. These results quantitatively reveal the impact of cavity effect on the

MTCR’s performance, which show that the reflection loss can be reduced greatly due to

cavity effect, especially when αt is relatively low.

15

60 65 70 75 80 85 90 95 10050

55

60

65

70

75

80

85

10.8 percent

340 kW

3.5 percent1041 kW

Coating absorptivity t / %

Ref

lect

ion

lo

ss Q

i,lo

ss /

kW

Inst

. o

pti

cal

effi

cien

cy

i,T /

%

i,R

0

500

1000

1500

2000

2500

3000

3500

Qi,loss

Cavity effect No Yes

Fig. 12. Variations of ηi,T and Qi,loss with αt at 12:00, spring equinox (DNI=961 W·m-2

).

5.5 Dynamic performance of the system

Figures 13 and 14 illustrate the variations of instantaneous efficiencies (ηi,A, ηi,T, ηi,R) on

three typical days. It is seen in Fig. 13 that the work time increases from the winter solstice to

the summer solstice due to the variation of the sunshine duration. It is also observed that the

instantaneous efficiencies increases in the morning and decreases in the afternoon in every

day, and the maximum ηi,A of 80.0 % and the maximum ηi,T of 74.7 % can be achieved at the

noon of spring equinox, which is the design point of the plant. It is seen in Fig. 14 that ηi,R

varies at around 93.5 % for winter solstice and spring equinox, and for summer solstice it

varies at around 93.0 %. The cavity effect physically improves the optical performance and

thus causes an increase of the solar power absorption, which is not affected by the date/time

in a year.

4 6 8 10 12 14 16 18 205

15

25

35

45

55

65

75

85

Aperture

Aperture Aperture

Tubes

ηi,

T o

f p

ow

er a

bso

rbed

by

tu

bes

/ %

ηi,

A o

f p

ow

er e

nte

rs a

per

ture

/ %

The local solar time ts / h

Aperture Tubes

Summer solstice

Spring equinox

Winter solstice 20

30

40

50

60

70

80

90

100

Fig. 13. Variations of ηi,A and ηi,T on three typical days.

16

4 6 8 10 12 14 16 18 2089

90

91

92

93

94

Coating absorptivity t=0.90

The local solar time ts / h

Inst

. ef

fici

ency

of

MT

CR

ηi,

R /

%

Summer solstice

Spring equinox

Winter solstice

Fig. 14. Variation of ηi,R on three typical days.

Figure 15 illustrates the variations of the daily efficiencies (ηd,A, ηd,T) versus the days in

a year (Nday). There are two peaks and one valley on each curve, and the valley is at around

the summer solstice. It is found that the ηy,A of 70.5% and ηy,T of 65.9% can be achieved in

the plant.

0 60 120 180 240 300 36060

65

70

75

80

Day number in a year Nday / day

Da

ily

op

tica

l ef

fici

enci

es (

ηd

,A,

ηd

,T)

/ %

Enters the aperture ηd,A

Absorbed on the tubes ηd,T

Fig. 15. The variations of daily efficiencies (ηd,A, ηd,T) in a year.

6. Conclusions

This work focuses on developing a comprehensive dynamic model for optical

characterization of a solar power tower (SPT) with a multi-tube cavity receiver (MTCR).

Based on the model, the dynamic optical performance of the DAHAN plant was studied. The

following conclusions are derived.

(1) Validation study showed that the comprehensive dynamic model and corresponding

code were valid and reliable. The non-uniform characteristics of the real-time solar fluxes on

all the surfaces in the MTCR were calculated and illustrated. The maximum flux (ql,max) of

5.141×105 W·m

-2 was found to appear on the tubes under typical condition.

(2) The tracking-error model treats the tracking errors as the errors of the tracking angles,

rather than the equivalent slope error as usually used. The tracking-error model is able to

consider the random effects of the errors on the flux uniformity and efficiency, and thus is

recommended for the accurate prediction of the SPT performance.

(3) The concentrated solar flux can be greatly homogenized by multi-point aim strategy

17

(compared to single-point strategy) with just a minor drop in efficiency. This method is also

recommended to SPT; for example, ql,max and the non-uniform index on the tubes drops 10.5%

and 31.6%, respectively, at spring equinox noontime with a sacrifice of only 1 percent in the

efficiency.

(4) The cavity effect on the MTCR’s performance was revealed quantitatively, which

indicates that the reflection loss can be reduced significantly by the cavity effect, especially,

when the coating absorptivity (αt) is relatively low. Furthermore, variations of the optical

efficiencies were also studied, and the yearly efficiency for the power absorbed by the tubes

was found to be 65.9%.

The simulation results indicate that the present model is suitable for dealing with the

complex geometry and optical processes in SPT with a MTCR, which predicts both the

detailed solar flux and the dynamic efficiency accurately. The results of this study can be

applied in the heat transfer analysis of the MTCR and the performance evaluation of the

plant.

Acknowledgements

The study is supported by the funding for Key Project of National Natural Science

Foundation of China (No.51436007) and the Major Program of the National Natural Science

Foundation of China (No. 51590902).

Appendix

The transformation matrixes among the seven Cartesian right-handed coordinate systems

are summarized as follows:

(1) M1 and M2 are the transformation matrixes from XiYiZi to XgYgZg:

1 s s

s s

s s

2 s s

1 0 0

0 cos( / 2 ) sin( / 2 )

0 sin( / 2 ) cos( / 2 )

cos( / 2) -sin( / 2) 0

sin( / 2) cos( / 2) 0

0 0 1

A A

A A

M

M

(1)

where αs and Αs are the solar altitude and azimuth.

(2) M3 and M4 are the transformation matrixes from XgYgZg to XhYhZh:

h h

3 h h

4 h h

h h

cos( / 2) sin( / 2) 0

= sin( / 2) cos( / 2) 0

0 0 1

1 0 0

= 0 cos ( / 2 ) sin ( / 2 )

0 sin ( / 2 ) cos ( / 2 )

A A

A A

M

M

(2)

where αh and Αh are the altitude and azimuth of the heliostat’s center normal.

(3) M5 and M6 are the transformation matrixes to introduce slope error:

18

1 1

5 2 2 6 1 1

2 2

1 0 0 cos( / 2) -sin( / 2) 0

0 cos -sin , sin( / 2) cos( / 2) 0

0 sin cos 0 0 1

M M (3)

1 2 2

h,ideal h,ideal h,ideal h,ideal

11 2 2

h,ideal h,ideal h,ideal h,ideal

2 h,ideal

cos cos / cos cos ,cos 0

2 cos cos / cos cos ,cos 0

(4)

22 2

,h ,h ,h ,h ,

h,ideal2

2 2

h,ideal h,ideal ,h ,h ,h ,h ,

2h,ideal 2 2

,h , ,h ,h ,h ,

/ 2cos

cos / 2

cos2 / 2

x x y z D

y x y z D

z D x y z D

P P P P H O

P P P P H O

P H O P P P H O

N (5)

where θ1 and θ2 are angle variables; Nh,ideal is the ideal normal vector at

T

h ,h ,h ,hx y z P P PP .

(4) M7 and M8 are the transformation matrixes from XhYhZh to XgYgZg:

7 h h

h h

h h

8 h h

1 0 0

= 0 cos( / 2 ) sin( / 2 )

0 sin( / 2 ) cos( / 2 )

cos( / 2) sin( / 2) 0

= sin( / 2) cos( / 2) 0

0 0 1

A A

A A

M

M

(6)

(5) M9 is the transformation matrix from XgYgZg to XrYrZr:

9 r r

r r

1 0 00 1 0

= 0 cos sin 1 0 02 2

0 0 1

0 -sin cos2 2

M (7)

where r is the altitude of the MTCR.

(6) M10 and M11 are the transformation matrixes from XrYrZr to XtYtZt:

t,r t,r

10 t,r t,r

cos( / 2) sin( / 2) 0

= sin( / 2) cos( / 2) 0

0 0 1

A A

A A

M (8)

19

11 t,r t,r

t,r t,r

1 0 0

= 0 cos ( / 2 ) sin ( / 2 )

0 sin ( / 2 ) cos ( / 2 )

M (9)

where αt,r and Αt,r are the altitude and azimuth of the tube in XrYrZr, respectively, as shown in

Fig. 4. For present MTCR, αt,r=90° and Αt,r =-90° for all the tubes.

(7) M12 is the transformation matrix from XlYlZl to XtYtZt:

t t,t ,t t

12 t

,t ,tt t

cos 0 sinarcos / , 0

= 0 1 0 ,arcos / , 0

-sin 0 cos

z xr

z xr

P P

P P

M (10)

where T

,t ,t tt,t ,x y z P P P

P the intersection a ray and a tube in XtYtZt; t is the angle shown

in (Fig. 4), tr is the tube radius.

(8) M13 and M14 are the transformation matrixes from XtYtZt to XrYrZr:

13 t,r t,r

t,r t,r

t,r t,r

14 t,r t,r

1 0 0

= 0 cos( / 2 ) sin( / 2 )

0 sin( / 2 ) cos( / 2 )

cos( / 2) sin( / 2) 0

= sin( / 2) cos( / 2) 0

0 0 1

A A

A A

M

M

(11)

Nomenclature

A, B, C, D, E aim points of the heliostats

As solar azimuth (rad, o)

Αh azimuth of heliostat’s center normal

DNI Direct Normal Irradiance (W·m-2

)

d aim point coordinate value (m)

ep power carried by each photon (W)

Et(i) power absorbed by ith tube

G tower base

H center of each heliostat

Ho height of aperture center (m)

I, N, R incident / normal / reflection vector

M1~M14 matrix

Lh length of the heliostat (m)

nt, nh number of absorber tube / heliostat

Np total number of the photon traced in the field

Nday the number of the day in a year

O aperture center

P point

20

Q solar power (W)

q heat flux (W·m-2

)

Rte tracking error (rad)

S area (m2)

ts solar time (h)

Wh width of the heliostat (m)

X, Y, Z Cartesian coordinates (m)

Greek symbols

αs solar altitude (rad, o)

αh altitude of heliostat’s center normal

αr altitude of the MTCR

αt, αw absorptivity of coating / cavity wall

αi incident angle between I and Yg (rad)

η efficiency (%)

θi incident angle on surface (rad, o)

θh heliostat azimuth in the field (rad, o)

ξ random number

ρt,s, ρt,d specular / diffuse reflectance of coating

ρh1, ρh2 reflectance / cleanliness of heliostat

ρw,s, ρw,d specular / diffuse reflectance of the wall

σE energy maldistribution index among the tubes (%)

σte , σse standard deviation of tracking / slope error (mrad)

φ local latitude (rad, o)

ω hour angle (rad, o)

Subscripts

g, h, r, t, w ground / heliostat / receiver / tube / wall parameter

i instantaneous or incident parameter

l local parameter

d,y daily/yearly parameter

T,H,R tube / heliostat field/ receiver symbol for efficiency

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