Homogenization of solar flux distribution in a carbon aerosol entrapped cavity receiver

Energy 182 (2019) 21e36

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Homogenization of solar flux distribution in a carbon aerosol entrapped cavity receiver Yabin Jin a, Jiabin Fang b, Jinjia Wei a, b, *, Mumtaz A. Qaisrani a, Xinhe Wang a a b

State Key Laboratory of Multiphase Flow in Power Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049, China School of Chemical Engineering and Technology, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 21 November 2018 Received in revised form 31 May 2019 Accepted 2 June 2019 Available online 6 June 2019

An uneven heat flux distribution on the receiver's surface can lead to a highly non-uniform temperature distribution and high local temperature on the receiver tubes which results in fatigue failure. In the present work, a carbon aerosol entrapped cavity receiver of “DAHAN” power plant was numerically simulated to achieve homogenized heat flux distribution with improved safety of the receiver. A threedimensional Monte Carlo Ray Tracing (MCRT) and Finite Volume Method (FVM) coupled model was developed to simulate the radiation-conduction-convection heat transfer in the receiver. Firstly, the MCRT method is used to simulate the solar heat flux distribution on the surface of the receiver. Then, the thermal performance and heat losses in the receiver were investigated by the coupled model. Finally, the levelized cost of energy (LCOE) was calculated and stress analysis was performed to predict the lifespan of the receiver. Moreover, with this strategy, the peak solar heat flux on the back panel significantly dropped from 290 kW/m2 to 135 kW/m2, while the peak temperature dropped from 652K to 620K. Carbon aerosol particle slightly decreases the thermal performance of the receiver. However, it decreases stress concentration on the receiver panels. Also, the economic analysis revealed that carbon aerosol entrapped receiver is more economical. © 2019 Elsevier Ltd. All rights reserved.

Keywords: CSP cavity receiver Non-uniform temperature distribution MRCT -FVM coupled model, Coupled photothermal convection

1. Introduction With the increase in global warming, the awareness for environmental protection is also increasing. As a result, the utilization of renewable energy is gaining increasing attraction. According to the International Energy Agency prediction, by 2050 the proportion of renewable energy in the global energy mix will be 79%, of which solar energy accounts for 30% [1]. CSP being economical, efficient and having the capability for large electricity production has started gaining attention and acceptance around the world. Typical concentrated solar power (CSP) system mainly include heliostat field, heat collection system, heat storage system, power generation system, and control system [2,3]. It is estimated that the receiver costs about 20% of the total power plant cost [4,5]. The cavity receiver is a vital constituent of any solar tower power plant responsible for the photo-thermal conversion, and its heat transfer performance directly affects the performance and safety of the

* Corresponding author. State Key Laboratory of Multiphase Flow in Power Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China. E-mail address: [email protected] (J. Wei). https://doi.org/10.1016/j.energy.2019.06.005 0360-5442/© 2019 Elsevier Ltd. All rights reserved.

solar tower power plant [6]. The heliostat field focuses and reflects the sunlight, towards the cavity receiver, where the boiling tubes absorbs the sunlight energy and transmits it to the heat transfer fluid. During this operation, the low-density solar radiation focused by the heliostats is projected onto the inner surface of the cavity receiver. A single focal-point aiming strategy of the heliostat field results in a non-uniform solar flux distribution on the aperture plane and eventually inside the cavity receiver as the photons strike the other surfaces inside the cavity receiver via the aperture plane only. The highly non-uniform solar flux distribution on the receiver's surface leads to a high temperature gradient and localized high-temperature on the surface of the tubes [7,8]. Local overheating may bring adverse results such as damaging the main components, the decomposition of the HTF, structural weakness or other similar problems heading to a troublesome situation. The solar power tower starts up at dawn, stops operating at sunset and remains shut down throughout the night. Furthermore, cloudy climatic conditions during the operation also affect the heat flux distribution. This results in a non-uniform temperature distribution and low-cycle fatigue failure of the tubes, further compromising the safety of cavity receiver. Meanwhile, significantly high

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Y. Jin et al. / Energy 182 (2019) 21e36

Nomenclature

Roman C Cus D d E EFC f g h K k L n p Q RD △SP S T R u 0 u W

cost, $ convection factor distance,m particle diameter, mm annual total electricity, GWhe modulus of elasticity, Mpa coefficient of scattering phase function asymmetry factor convective heat transfer coefficient, W/m2$K
1 factor interest height, m years scattering phase function coefficient, m
1 radiative heat transfer factor total equivalent amplitude stress, Mpa alternating equivalent stress, Mpa temperature, K random number normal vector of the surface direction cosines Width, m

Greek symbols absorptivity ε emissivity h thermal efficiency of the receiver q azimuthal angle, degree

a

temperature, varying input loads, creeping effect and the fatigue phenomenon aggravates the effect of non-uniform heat flux distribution on the receiver tubes and thus, worsening the situation [9]. Therefore, the problem of uneven heat flux distribution on the surface of the receiver is a critical challenge for solar power tower systems. Compared with the parabolic dish and linear Fresnel reflector systems, solar power tower systems require a higher heat flux and temperature, and non-uniform heat flux possesses even greater challenges for solar tower power systems. In particular, receivers having molten salt as HTF are subjected to a high non-uniform heat flux above 1 MW/m2 [10]. These operating conditions demand higher safety and reliability requirements for the tower power system in CSP. Numerous approaches have been presented to deal with the problems triggered by the non-uniform solar heat flux. Not only does it optimize the solar tower concentrating system but it also enhances the heat transfer performance of the absorber tubes [11e13]. A typical heliostat field uses a single-focal point method, as it has the advantage of elevating the truncation of the cavity receiver and reducing energy losses. The single-focal point method results in very high solar heat flux on the aperture plane and internal surfaces of the cavity receiver. However, the multi-focal point method can obviously reduce the peak heat flux on the center of the aperture plane and improve the uniformity of solar heat flux distribution on the cavity receiver's inner surfaces. Besarati et al. [14] presented a novel optimization approach based on the genetic algorithm principle to obtain an optimum flux distribution on the

l r s t F u

receiver wavelength, mm reflectance mean square deviation of temperature, K transmittance circle angle, degree albedo

Subscripts a alt abs d ext e

distance travelled prior to absorption event alternating absorption debt extinction distance a photon travels before interacting with carbon aerosol in a unit f fatigue strength reduction h heliostat i unit i invest investment n,x n,y n,z x,y,z direction O&M operation and maintenance costs qw quartz window sac scattering solar-i solar flux on unit i sun-i unit i on aperture plane T temperature 1 inner tuber 2 surface of quartz window Abbreviations DNI Direct Normal Irradiance HTF Heat Transfer Fluid LCOE Levelized Cost of Energy

surface of the receiver. The result shows that the heat flux is decreased by a factor of 10, reaching around 500 kW/m2. Moreover, they also examined the relationship between the number of focusing points and the size of the targeted surface. Yu et al. [15] established a multi-focal model to achieve uniform heat flux distribution for “DAHAN” solar tower power facility. The model established on the TABU metaheuristic method was introduced to optimize the solar flux distribution on the surface of the receiver. The ensuing outcomes indicated that the multi-focal method increased solar flux uniformity among the inner surface of the receiver and the method also improved the safety of the system. Tu et al. [16,17] presented a simulation model for the evaluation of the thermal performance of a saturated water/steam cavity receiver. They came up with an optimal distribution for the absorptivity in the solar wavelength band. Their results revealed that surface heat flux uniformity could be significantly improved with a decrease in maximum mean square deviation in the temperature. Tu et al. [18] extended the depth of the cavity receiver and found an optimum depth based on the peak thermal efficiency of the receiver and the uniformity of flux distribution on the receiver's surface. Buck et al. [19] introduced a novel dual receiver concept to elevate the adaption of steam cycles of the central receivers. Incorporating the tubular section to the open volumetric receiver the dual receiver concept benefited from these two straightforward concepts and evaded the problems. The ensuing results revealed several advantages of the new concept, particularly the increase in thermal efficiency along with a decrease in the receiver's thermal and parasitic losses. An increase of 27% in annual output was reported

Y. Jin et al. / Energy 182 (2019) 21e36

because of these improvements as compared to the solar air heating system. Zheng et al. [20] employed a porous medium in central receiver tubes to improve the convection heat transfer due to turbulent flow. Examining the tube with single-side absorption, a number of porous media filling techniques are presented and the relation between the non-uniform heat flux and heat transfer capacity of heat transfer fluid is studied. The simulation results showed that the partially porous medium filled ERT has a higher thermal efficiency than the entirely filled ERT. Also, the ERT with the porous insert can be used to avoid hot spots on the surface of the tube. Montes et al. [21] optimized the flow pattern based on non-uniform heat flux distribution in the receiver by adjusting the number of pipes, the pipe diameter and the flow direction. Their results depict that the coating technology improves uniformity of heat flux distribution by changing the spectral absorption rate of the surface of the tube. In the current study, a method of inserting carbon aerosol in a cavity receiver is proposed to homogenize the uniformity of heat flux distribution on the surface of the tubes. At present, there are only a few research carried out on the particles entrapped cavity receivers. Steinfeld et al. [22] studied the Finite Volume Method (FVM) and the Monte Carlo Ray Tracing (MCRT) method coupled heat transfer radiation-convection-conduction in the receiver. They calculated the average temperature of the mixture and numerically evaluated transition mixture compositions at the exit of the reactor. The results were validated by carrying out an experiment using a 5 kW simulator. Miler et al. [23] presented a six-flux model and numerically simulated the 3D radiant flux as well as the temperature distribution in a cavity receiver with carbon particles media. The results indicate that receiver work condition under extremely high heat flux, and the relationship between the thermal performance of the cavity receiver and particle density was established. Most of the studies carried out on this topic focused on enhancing the thermal efficiency or investigated chemical reactions in the receiver instead of uniforming the heat flux distribution. Our previous study carried out to homogenize the heat flux on the twodimensional square shaped cavity receiver's surface is further extended in this research to a three-dimension model. 1MWe “DAHAN” is China's first independent research and development of solar tower power plant and it serves as the research object for the present study. In the current work, the authors developed a three-dimensional Monte Carlo Ray Tracing (MCRT) and Finite Volume Method (FVM) coupled model for the numerical investigations on the temperature distribution characteristics and the thermal efficiency based on a carbon aerosol entrapped cavity receiver of DAHAN. By using the MCRT-FVM coupled model, the entire complex steady photothermal conversion process in the receiver can be examined and the steady-state performance and the mass flow rate of the steam produced in the tubes can be precisely calculated. Also, the MCTRFVM coupled model is used to find an optimized density of carbon aerosol medium and water-steam circulation mode to enhance the surface heat flux uniformity. Ultimately, the authors established an economic evaluation model of the carbon aerosol entrapped receiver. The thermal stress and fatigue analyses were also carried out to further examine the fatigue failure of the tubes.

23

Fig. 1. Heliostat field for 1MWe DAHAN in Beijing, China.

Fig. 2 represents the non-uniform solar flux distribution as obtained on the aperture plane. The highest heat flux is obtained at the center of the receiver, decreasing gradually towards the receiver boundaries. The total incident sunlight energy on the receiver's aperture surface is 6343 kW, with a peak value of 1400 kW/m2 at the center of the receiver. A saturated water/steam solar cavity receiver is shown in Fig. 3 (a). The cavity receiver has a symmetrical geometry, with an aperture of 4 m  4 m. The aperture is enclosed by a quartz window of 10 mm having high resistance against the temperature. The parameters of the receiver are listed in Table 1. A serpentine configuration is adopted for pipe layout as shown in Fig. 3 (b) and 3(c). The boiling tubes are made up of 1Cr18Ni9Ti with a diameter of F ¼ 50  4 mm at the back-panel and F ¼ 40  4 mm at the remaining other panels. Tubes' inlet is located at the receiver's bottom whereas the fluid leaves the receiver from the top in all panels. There are a total of 7 turns between the inlet and the outlet for the back panel and 6 turns for the left-back panel and right-back panels. In order to absorb more energy, the adjacent tubes are connected through a membrane wall. The outer wall of the cavity receiver is shielded with an insulation layer to attenuate the heat losses.

2. Computational model description The heliostat field of DAHAN plant, situated at the base of the Great Wall in the city of Beijing, is shown in Fig. 1. It comprises 100 heliostats, each having dimensions of 10 m by 10 m [24]. During the operation, the sun rays are reflected and focused into the aperture of the cavity receiver. The cavity receiver is mounted at the top of the 78 m high tower with a downward inclination of 21.8 .

Fig. 2. Solar flux distribution on the aperture plane.

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Y. Jin et al. / Energy 182 (2019) 21e36

Fig. 3. The cavity geometry and tube layouts,(a)cavity receiver,(b)tube layout a, (c)tube layout b,(d)schematic diagram of thermal fatigue evaluation model.

3. Mathematical model 3.1. MCRT-FVM coupled model To calculate the thermal performance of the carbon aerosol entrapped cavity receiver, an MCRT-FVM coupling calculation model was established. The key processes of the calculation model are listed below. I. Photons transmittance inside the cavity receiver via quartz glass; II photo-thermal conversion inside the cavity receiver having carbon aerosol. III. Radiation-convection-conduction heat transferred between carbon aerosol and the inner surfaces of the cavity receiver. IV. Water/steam phase transformation process occurring inside the tubes; V. Radiation-convection-conduction heat transfer process between the quartz glass window and the surrounding environment. The following assumptions are made in this model:⑴the outer walls of the receiver are adiabatic as these are by an insulation layer and the heat losses through them are negligible; ⑵density of carbon aerosol distribution remains same throughout the cavity receiver.⑶the scattering process of carbon aerosol is purely elastic and it only diverts photon's direction without losing any energy and satisfies the independent scattering condition. The detailed flowchart of MCRT-FVM model is shown in Fig. 4. It can be observed that the coupled model consists of four main steps. Firstly, the Mie theory is used to compute the radiation characteristics of carbon aerosol. Then, the behavior of the photons is simulated by the MCRT method to attain a non-uniform solar heat flux distribution on the cavity receiver's surface. Afterward, using the apposite heat transfer correlations and identification criteria, the outer surface temperature of the boiling tubes and the heat transfer coefficient of water/steam mixture are calculated. In the end, the carbon aerosol domain inside the receiver can be simulated by using the ANSYS FLUENT 15. The above mentioned four parts are coupled to obtain the thermal performance and heat loss from the cavity receiver, and the mechanism for heat flux homogenization on the receiver's surface in the cavity receiver is achieved.

Fig. 4. Flowchart of MCRT coupled with FVM to calculate the thermal efficiency of the receiver.

3.2. Modeling of photon transfer in the MCRT MCRT method simulates radiative transfer by tracking the path of photons. The modus operandi of MCRT for radiation is that the radiative transfer process is divided into a series of sub-processes, including emission, scattering, reflection, absorption and escaping from the system [25]. The MCRT process-flowchart of MCRT is presented in Fig. 5. The cavity is divided into many surface and volume units. A certain number of photons enter the cavity receiver through the aperture. The trajectory of each photon is tracked to determine, either it is absorbed or reflected by the inner surface of the cavity, or if it is absorbed or scattered by the carbon particles or if it's reflected or transmitted outside the calculation domain by the quartz glass window. Finally, the total number of photons absorbed by each unit is counted and the radiative heat transfer factor is obtained. In the present work, the photons carry no energy. The carbon aerosol is considered as a parametric medium, having a scattering effect on the photons. 3.2.1. Photon launching model Fig. 1 shows the heliostat field layout. The field is “fan-symmetrically” arranged and distributed along the normal aperture plane of the cavity receiver. Due to the fan-shaped arrangement of the heliostat field, the reflected photon irradiates within a certain angle range, i.e., 70 , from left-to-right-direction, while each side

Table 1 Parameters of the DAHAN cavity receiver and plant. Parameters

value

Parameters

value

Heliostat width Wh[m] Heliostat height Lh[m] Heliostat center height[m] Tower height/m Inclination angle of the receiver Particle diameter d[mm] Thickness of quartz [mm] absorber tubes for sunlight Quartz transmittance tl
0:1
5:0 Quartz reflectance Rl
0:1
5:0 Quartz transmittance tl
5:0
20:0 Quartz reflectance Rl
5:0
20:0

10 10 6.6 118 21.8 30 20.0 0.80 0.94 0.06 0.0 0.1

Distance between panels[mm] Panels number Tubes in the back panel Tubes in the east/west panel Back panel tube radius[mm] Left and right panel tube radius[mm] Cavity wall absorptivity for sunlight Quartz absorptivity al
0:1
5:0 Thickness of quartz glass [mm] Back panel tubes thickness[mm] Left and right panel tubes thickness[mm] Quartz absorptivity al
5:0
20:0

7 60 20 20 25 20 0.85 0.0 20.0 4 4 0.9

Y. Jin et al. / Energy 182 (2019) 21e36

25

Fig. 5. Flowchart of the MCRT process.

having a range of 35 . Whereas, its range is 40 from top-to-bottom direction, with 10 downward and 30 upward. The range of solar irradiation angles is depicted in Fig. 6.

3.2.2. Photon reflection inside the cavity receiver (1). Photon transmission through the quartz window Refraction is ignored through the quartz window, as quartz is a semi-transparent medium. The photon interaction with the quartz window is similar to the interaction of a photon between two

mirrors. The probability model comprises the following equations. R  aqw Absorption. aqw  R  aqw þrqw Reflection.

R > aqw þ rqw Refraction. The vector normal to the surface of quartz glass and the receiver, where the photon strikes, is evaluated by employing the relation obtained through Eq (1), given below.

8 > > vFn ðx; y; zÞ > > > un;x ¼ > > vx > > > < vFn ðx; y; zÞ un;y ¼ > vy > > > > > vF ðx; y; zÞ > > > un;z ¼ n > : vz

(1)

The photon strikes the quartz window after being scattered by the carbon aerosol. As specular reflection occurs, the photons traveling position is replaced by the new position.

8 0 > > < u0n;x ¼ ux þ k0 un;x un;y ¼ uy þ k0 un;y > > : u0 ¼ uz þ k0 un;z n;z

Fig. 6. The range of solar irradiation angles.

(2)

where the coefficient, (k0 ¼ 2  jUjU ¼ ðux ;uy ;uz Þ,ðun;x ;un;y ;un;z Þ, is the dot product of the photon propagation direction vector and the normal vector of the quartz window surface. WhileU > 0,ðun;x ; un;y ; un;z Þ is taken ð
un;x ;
un;y ;
un;z Þto get the correct solution.

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Y. Jin et al. / Energy 182 (2019) 21e36

4. Photon reflections in the receiver

expressions:

Once the photon reaches the wall or the tubes, it is judged whether the photon is absorbed or reflected. The zenith angle q and the azimuth angle 4 are sampled statistically to define the new direction of the reflected photon. The probability distribution for the zenith angle q is defined by Lambert function, and the azimuth angle 4is uniformly distributed between 0 and2p. Thus, the zenith angle q and the azimuth angle 4 are given as:

4 ¼ 2pR

pffiffiffi q ¼ arcsin R

(3)

4 ¼ 2pR

(4)

After evaluating the photon's zenith angle q and azimuth angle4, the new direction after scattering can be evaluated as:

2

un;i ¼ Tsc usc;i

Once the photon's zenith angle q and azimuth angle 4are determined, the new direction of the reflected photon can be calculated by the following expressions:

8 > > < ux ¼ sin q cos 4 uy ¼ sin q sin 4 > > : uz ¼
cos 4

(5)

When a photon travels through the carbon aerosol, its direction will change with every collision. De represents the distance a photon travels before interacting (absorbed or scattered) with carbon aerosol in a volume unit. It is calculated by the probability distribution sampling technique for the photon's free path. The De can be expressed as:

(6)

P0P1 represents the distance a photon travels in a volume unit. If the distance De is greater than P0P1, it suggests photon doesn't collide with the particle and its direction remains unchanged in the volume unit. While if De is lesser than P0P1 the photon collides with the carbon particle, the albedo is employed to judge whether the photon is absorbed or scattered. In the case of scattering, there is a change in photon's direction. The probability distribution for the zenith angle q is based on Delta-Eddington function, while the azimuthal angle 4is evenly distributed in the range 0-2p [26].

QðqÞ ¼ 2f sð1
cosðqÞ þ ð1
f Þð1 þ 3g cosðqÞÞ

(7)

4 ¼ 2pR

(8)

Integrating Delta-Eddington phase function in 0-p range and the probability density function, the phase function is gained.

ðu RðuÞ ¼


1 ð1
1

6 ux 6 6 6 6 uy 6 ¼6 6 6 6 6 uz 4

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1:0
u2x ux uy qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1:0
u2x ux uz qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1:0
u2x

3 7 7 7 uz 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 27
1:0
ux 7 7 7 7 uy qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 5 2
1:0
ux 0

2

3 cos q 6 7 7 6 4 sin q cos 4 5 sin q sin 4

(12)

Ifjux j > 0:999, the new direction of a scattered photon can be evaluated by the expressions given below

8 > > > ux > > cos q un;x ¼ > > jux j <

4.1. Photon transmission in the carbon aerosol

lnð1
RÞ De ¼
Qext

(11)

ð1
f Þð1 þ 3guÞdu (9)

un;y ¼ cos 4 sin q > > > > > > un;z ¼ sin q sin 4 > :

(13)

4.2. Energy distribution statistics While performing the numerical simulation for photon tracing, if the photon is absorbed it is added to the absorption element j. The radiation transfer factor i.e.,RDi;j is the ratio of the radiation energy absorbed by the element j to the total energy radiated by the element i after an incidence or one or more reflections or scattering. It can be expressed as:

RDi;j ¼

Ni;j Ni

(14)

The transmission, absorption, reflection and scattering of photons are continuous processes. The data records the energy absorbed by each surface and volume unit. However, when a photon was absorbed by the wall of the receiver or carbon aerosol or it escaped from the receiver through the quartz window, its trajectory record was deleted. During the tracing of the photons, photons are absorbed recorded for each unit. The total solar flux of each grid element can be calculated as:

qw
j ¼

n X

 RDi;j qsolar
i Smesh

(15)

i¼1

ð1
f Þð1 þ 3guÞdu

u ¼ cosðqÞ, the q is obtained with

0 B
1 þ B
1 B q ¼ cos B @

4.3. Selection of heat transfer correlations for tubes

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 1
6g 1
3g 2
2R C C C C 3g A

(10)

The azimuthal angle4can be represented by the following

In the CSP system, the subcooled water delivered by the circulating pump into the tubes is heated into saturated water and steam. The heat flux on the surface of the tubes is extremely high and the phase change occurs as the water flows through the tubes. In the present work, the phase change process is divided into three separate regions; single-phase flow region, subcooled boiling flow

Y. Jin et al. / Energy 182 (2019) 21e36

region, and saturated boiling flow region. Based on the heat transfer mechanism, the subcooled boiling flow region is further subdivided into the partial boiling region, fully developed boiling region and significant void region. The detailed heat transfer correlations and the identification criteria for each flow region are listed in Table 2 below.

27

v v ðrui kÞ ¼ vxi vxi



mþ

mt st

mþ

mt st



 vk þ Gk
rε vxi

(20)

 vε ε þ ðc1 Gk
c2 rεÞ vxi k

(21)

ε equation

v v ðrui εÞ ¼ vxi vxi





wheremt is the turbulent viscosity and given by. 4.4. Governing equations for FVM The flow considered in this study is three-dimensional, steady and turbulent. For evaluating the natural convection inside the cavity receiver, by simultaneously solving the equations of conservation of mass, momentum and energy of the system, the flow and heat transfer simulations are carried out. Continuity equation [34]:

v ðrui Þ ¼ 0 vxi

(16)

Momentum equation:

" !#   v rui uj vðrÞ v vui vuj 2 vul ðut þ uÞ ¼
þ þ
sij vxi vxi vxj vxi 3 vxl vxi  þ brgi T
Tref

(17)

(18)

where Sh in Eq. (17) is the inner source term. A user-defined function is employed to load solar energy Sh on the internal surface of the receiver, quartz glass and the carbon aerosol air-carbon particle mixture. It can be expressed as follows:

Sh ¼ qw
j ðx; yÞ

(19)

k equation

Cm rk2 ε

Gk ¼ mt

vui vxj

(22) vui vuj þ vxj vxi

! (23)

The following standard constants are implemented in the current model: Cu ¼ 0.09, c1 ¼1.44, c2 ¼ 1.92, sk ¼ 1.0, sε ¼ 1.0 and sT ¼ 0.85. Discrete Ordinates (DO) model is coupled with the turbulence natural convection model for the numerical simulations. The radiation heat transfer equation for the gray body is given below [35],

! !  dIl ð r ; s Þ ! ! ! ! ! ¼
bl Il ð r ; s Þ þ Qabs ð r Þ þ Qgas ð r Þ Il ð r Þ ds ! 4ðp ! Qsca ð r Þ 0 ! ! ! þ Il s ; s Pð r ; s Þ 4p

! ! ! The relationship betweenQgas ð r Þ, Qabs ð r Þand Qsca ð r Þ of the carbon aerosol, independent of position vector r, and bl can be expressed by:

bl ¼ Qgas þ Qabs þ Qsca

(25)

4.5. Calculating the thermal performance of the receiver When calculating the heat loss, surface temperature and surface heat flux of the receiver, the convective heat transfer coefficient h1

Table 2 Heat transfer correlation and identification criteria for each flow regain. Flow region

Identification criteria

Single-phase

e

Heat transfer correlation Gnielinski(1976) [27] ðRelo
1000ÞPrl ðf =8Þ f ¼ ½1:82lgðReÞ
1:64
2 pffiffiffiffiffiffiffiffi 2=3 1 þ 12:7 f =8ðPrl
1Þ Kandlikar (1998) [29] qFDB
qONB a ¼ qONB
bðDTSat:ONB Þm q ¼ a þ bðDTsat Þm b ¼ Þ
ðDTSat:ONB Þm

Nulo ¼ hlo D=ll ¼ Partial boiling regain Hsu(1962) [28]

" 4dTsat vlg hl 1þ ll ilg sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# lj ilg DTsub ¼ ½lj ilg =8dTsat vlg ðDTsat:ONB Þ2 1þ q 2dTSat nlg hl ONB

DTsat:ONB ¼

Fully developed boiling region Significant void regain

Saturated boiling region

(24)

0

Energy equation:

    v  v m mt vT þ cp þ Sh rui cp T ¼ vxi vxi Pr Prt vxi

mt ¼

Thom et al.(1965) and Bowing(1962) [30] qFDB ¼ 1:4qD

Thom et al.(1965) [30]

x0

Gunger and Winterton [33]

0:5

DTSat ¼ 22:65ðq=106 Þ ep=8:410 Kandlikar(1991) [32] Saha and Zuber(1974) [31] 8 ðh
hNVG Þðx
xNVG Þ > > hTP ¼ hNVG
x¼0 > > xNVG < xNVG ¼
0:0022 qD ¼
0:0022BoReto PrReto Pr < 70000l rl ilg ll > > > > xNVG ¼
153:8BoReto Pr > 70000 : 6


0:55

hTP ¼ Ehlo þ ShB hb ¼ 55p0:12 ð
0:4343 lnPr Þ r 1:37X
0:86 S ¼ ð1 þ 1:15  n


1 10
6 E2 Re0:17 Þ l

M
0:5 q0:67 E ¼ 1 þ 24000Bo1:16 þ

28

Y. Jin et al. / Energy 182 (2019) 21e36

and the temperature T1 on the quartz glass are needed. The convective heat transfer correlations are used to calculate the water temperature T2 and the heat transfer coefficient h2 of the inner surface of the tubes. However, it is necessary to know the water conditions at the inlet of the tubes and the heat flux on the surface of the tubes to employ appropriate heat transfer correlations. And for the calculation of the airflow field around the receiver, it is necessary to know the temperature T1 of the quartz glass. The above three calculations are needed to be coupled together in three iterative loops to calculate the thermal performance of the receiver, as shown in Fig. 7. The whole iterative scheme is mentioned in detail below: 5. Code and simulation verification Due to the lack of available data for direct validation of MCRTFVM model of the carbon aerosol system, the model is indirectly verified in two steps. Firstly, the MCRT program calculation results are compared with the model presented by Farmer et al. [36]. Secondly, the coupling model is used to calculate the solar radiation transfer in the receiver of the existing Spanish CESA-1 power plant,

and the simulated solar flux distribution is compared with the published data of Spanish CESA-1 power plant. 5.1. MCRT model validation A carbon aerosol entrapped cavity receiver model was developed based on the MCRT method and validated with a similar method presented by Farmer et al. and Hsu. A cuboid geometry with dimensions L ¼ 2 m, H ¼ 3 m, and W ¼ 5 m was investigated by Farmer et al. and Hsu and the medium used was a mixture of carbon dioxide (the volume fraction of carbon dioxide was 0.21 and N ¼ 2.0  108 particle/m3) and nitrogen. The mixture was specified at 1atm and the temperature of the mixture was assumed 1000K. Fig. 8(a) shows the divergence of radiative heat flux against position along the Y-axis, and Fig. 9(b) represents the surface of radiative heat flux along the center of the wall in the X direction at Y ¼ 0 and Z ¼ 1.5 m. It can be observed from the results depicted in Fig. 8(aeb) that the MCRT model presented in this study is in agreement with the models in the literature [36,37] which indicates that the MCRT model proposed in this study is reliable and accurate. 5.2. Validation of the simulation method To validate the coupled simulation model presented in this paper, the method is implemented to simulate the solar radiation transfer for the heliostat field and the receiver of an existing solar power tower facility CESA-1 power plant located in Spain. The average absorptivity ratio of the tubes is 0.87. Further details about the heliostat field layout and the cavity receiver are mentioned in Ref. [38]. Fig. 9 presents a comparison of the solar flux distribution on the surface of the tubes. Fig. 9 shows that the solar-flux distribution as simulated in this study is similar in profile in the published literature. Furthermore, the solar heat flux profile on the surface of the receiver is computed and compared with that of CESA-1. The parity plot between the current model and the CESA-1 case is shown in Fig. 10. It can be seen that the maximum deviation of the peak fluxes is less than 5%. 5.3. Gird independence test

Fig. 7. Flowchart for calculating the thermal performance of the cavity receiver. (1) The Mie theory is implemented to compute the radiative properties of the carbon aerosol. The MCRT simulation is employed to compute the radiative heat transfer factor RDi,j prior to iterations. The solar heat flux at the aperture's plane can be calculated from the heliostat field. 0 (2) Assume a circulation ratio r for water/steam flow inside the tubes, heat 0 0 transfer coefficient h2 on the surface of the quartz glass and the heat flux qi on the tubes' outer surface. (3) With the help of suitable heat transfer correlations and identification criteria, the temperature on the outer surface of the tubes is calculated based 0 0 on the supposed circulation ratio r and heat flux qi .Then, the carbon aerosol flow field in the receiver is simulated by using FLUENT to acquire a new heat flux qi and the temperature T1 on the quartz glass. (4) Once the wall temperature of the quartz glass is gained, the air flow domain surrounding the receiver is simulated by using FLUENT to obtain the new heat transfer coefficient h2 on the quartz glass. (5) When the two inner loops converge, the circulation ratio r is calculated 0 based on the outlet quality of the steam. Then, r is compared withr , if the difference between the two is higher than the acceptable errorε1, the cir0 culation ratio r is reassumed and the process is repeated to recalculate the inlet temperature of tubes. The whole iterative process is repeated until convergence is achieved for the heat transfer coefficients for quartz glassh2 , heat flux qi on the tubes surface and the circulation ratio rat tubes.

The grid independence test was conducted with several gird systems, having the total solar energy 6343 kW on the aperture of the receiver, N ¼ 1.0  108 particle/m3, and the water was used as the heat transfer fluid. The results are shown in Table 3, where the average temperature of the back-panel and left-back panel is examined for each grid number. It is found that the grid of 2.0  106 cells is adequate. The solar heat flux distribution in the receiver is calculated by the MCRT code, and it is treated as an inner heat source term Si in Eq (18) for FVM model in the FLUENT. The self-developed user-defined function (UDF) code is then employed to input the flux distribution on the receiver's surface. 6. Results and discussion 6.1. Ray tracing results of the receiver Using the MCRT method, the photon's propagation in the receiver is simulated. The solar heat flux distribution obtained on the receiver's surface and carbon aerosol at the equinox noon is shown in Fig. 11(aeb) It can be observed that the back panel and its adjacent side panels have a non-uniform solar heat flux distribution. Whereas the highest heat flux is observed at the central region of the panels, particularly the back panel, with a maximum value of 290 kW/m2. With the increment in carbon aerosol particles density

Y. Jin et al. / Energy 182 (2019) 21e36

29

Fig. 8. Comparison of solutions in non-gray medium:(a) divergence of radiative heat flux (b)radiative heat flux.

distribution on the carbon aerosol in different sections. An increase in carbon aerosol density (N ¼ 0-2  108 particle/m3) noticeably increases the solar energy absorbed by the receiver. Solar heat flux distribution is non-uniform but symmetrically distributed. The peak solar heat flux is observed in the central region behind the aperture. The region is marked by a in Fig. 11 (b).

6.2. Effect of N on dT Fig. 9. Comparison between simulations results (left) and published data of Spain CESA-1[kW/m2].

The mean square deviation sT is used to evaluate the nonuniformity of temperature on the outer surface of the boiling tubes. It is expressed as

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N  u1 X 2 sT ¼ t Ti
Ti N 1

Fig. 10. Comparison of the solar heat flux profiles on the surface of present receiver and CESA-1.

Table 3 Gird independence test. Grid number/105

T Back panel

5T Lift back panel

17 20 25 30

598.34 599.15 599.30 599.45

619.87 621.05 621.15 621.26

(N ¼ 0-1.5  108 particle/m3) the solar flux distribution across the back panel and the adjacent side panels drops significantly from 290 kW/m2 to 135 kW/m2, while at the other surfaces of the receiver it increases slightly. Fig. 11 (b) shows solar heat flux

(26)

In the present study, two different water vapor circulation modes are designed and examined, as shown in Fig. 12(a-b). In Fig. 12 (a), tubes are attached on the cavity's left-back, right-back and back wall only. While the tubes layout on the remaining walls is shown in Fig. 12 (b). Inlet pressure in the tubes is 7.0Mpa; inlet temperature is 378K while the circulation rate is 140 t/h at the inlet. These boundary conditions are kept same for both the circulation modes, i.e., a and b. Fig. 13(aeb) depicts the temperature distribution on the surface and different cross sections of the receiver under varying carbon aerosol densities under the tube layout a. It can also be observed that the temperature at the top, bottom, and right-front panels is higher as compared to the left-back panel, back panel, and rightback panel, with the top panel having the peak temperature. With the increase of carbon aerosol density N (0-1.5  108 particle/ m3), a significant temperature drop is observed at the boiling tubes of the left-back panel, back panel and right-back panel, precisely, the highest temperature drop occurs at the back-panel pipes, i.e., 652Ke620K. Not only the temperature gradient at the back panel in regions A and B is reduced from 15K/m to 3K/m, but also the uniformity of temperature distribution on the back panel is significantly improved. The average temperature at the left-front, top, and bottom walls was increased from 827K, 806K, and 800Ke972K, 930K, and 880K, respectively. The reason is that the solar heat flux density absorbed in the region right below the top surface (i.e., region A) increased with the increase in the carbon aerosol density N. The left-front, top, bottom, and right-front walls get more radiation energy emitted from carbon aerosol, and therefore, the temperature at left-front, top, bottom, and right-front walls increase significantly.

30

Y. Jin et al. / Energy 182 (2019) 21e36

(a). Solar heat flux on the surface of the receiver. (b). Detail solar heat flux in the receiver on different sections Fig. 11. Effects of N on solar flux distribution of cavity receiver.

(a)

(b) Fig. 12. Two different water-steam circulation modes for the tube layout.

(a). The temperature profiles at the surface of the receiver. (b). detail temperature profiles of different cross sections inside the receiver. Fig. 13. Temperature distributions on the receiver at different N under circulation mode a.

(a). The temperature profiles at the surface of the receiver. (b). detail temperature profiles of different cross sections inside the receiver. Fig. 14 (a) and (b) depicts the temperature distribution on the surface and different cross sections of the receiver for varying carbon aerosol density under tube layouts b. It can be observed that

the temperature at the left-back panel, back panel, and right-back panel is higher as compared to the other panels, with the center of the back panel having the peak temperature. With the increase of carbon aerosol density N (0-1.5  108 particle/m3), a significant temperature drop is observed at the boiling tubes of the left-back panel, back panel and right-back panel, particularly the tubes located at the center of the back panel pipe have the highest

Y. Jin et al. / Energy 182 (2019) 21e36

31

(a). The temperature profiles at the surface of the receiver. (b). detailed temperature profiles of different cross sections inside the receiver. Fig. 14. Temperature distributions on the receiver at different N under b circulation mode.

temperature drop i.e., 651Ke617K. The temperature gradient at the back panel in regions A and B is reduced from 14K/m to 2.5K/m. The average temperature at the left-front, top, and bottom walls increased from 564K, 566K, and 564Ke579K, 584K, and 574K, respectively. By comparing the tube layouts a and b, it can be observed that the temperature profile of the tube layout b is more uniform. Fig. 15(aeb) depict the mean square deviation of the temperature on the receiver's surface under varying carbon aerosol density, for tube layouts a and b, respectively. With the increase in carbon aerosol density N (0-1.5  108 particle/m3), the mean square deviation of the temperature at the left-back panel, back panel and right-back panel is decreased from 23K, 34K, and 23Ke14K, 17K, and 14K under tube layout a and circulation mode a, respectively. However, the mean square deviation of the temperature at the top, bottom, left-front, and right-front is increased. It can be seen from Fig. 15(b) that with the increase of carbon aerosol density N (01.5  108 particle/m3), the mean square deviation of the temperature at the left-back panel, back panel and right-back panel also decreased from 23K, 34K, and 23Ke14K, 17K, and 14K. In comparison with Fig. 15 (a) that the mean square deviation of the temperatures at the left-back panel, back panel and right-back panel is slightly increased.

(a)

6.3. Effect of N on heat loss and thermal performance of the receiver The velocity contours of the receiver with varying flow directions in range 0 e160 are shown in Fig. 16. Six different directions have been investigated. With the increase in wind angle towards the aperture the wind velocity increases. The maximum wind velocity is observed at a ¼ 90 . Fig. 17 shows heat transfer coefficient h2 on the outer surface of the quartz glass with different flow direction ranges of 0 e160 and wind velocities. It is found that convection heat loss gradually increased with the increase and different wind velocity and the highest convection heat losses are also observed at a ¼ 90 . The effects of the carbon aerosol density on the heat loss of the receiver under the different tube layouts and circulation modes are shown in Fig. 18(a) and Fig. 18(b). The total heat loss of the receiver mainly consists of the reflection loss Qlost,ref,q, the radiation loss,Qlost,rad,q, and the convection loss Qlost,con,q through the quartz window and the reflection loss of the receiver Qlost,ref,r. From Fig. 18(a) and Fig. 18(b) it may be noted that Qlost,rad,q、Qlost,con,q and Qlost,ref,r increases with increase in carbon aerosol density N (0-1.5  108 particle/m3). While radiation and convective losses through the quartz window slightly decreased. Also the energy drop in the leftback panel, back panel and right-back panel caused a decrease in energy absorbed by the water in the tubes on these panels. This is due to the fact that the increase in carbon aerosol density also

(b)

Fig. 15. distributions on the receiver at different N:(a) tube layout a and circulation mode a (b) tube layout b and circulation mode b.

32

Y. Jin et al. / Energy 182 (2019) 21e36

Fig. 16. Top views of air velocity field surrounding the equal environmental wind velocity of v ¼ 9 m/s.(a)flow direction a ¼ 0 (b)flow direction a ¼ 30 (c)flow direction a ¼ 60 (d) flow direction a ¼ 90 (e)flow direction a ¼ 120 (f)flow direction a ¼ 160 .

increased the probability of scattering between the carbon aerosol particles and the photons. Fig. 18 (a-b) show the comparison of heat losses between the tube layouts a and b at the same carbon aerosol density. It can be noticed that tube layout b has lesser Qlost,rad,q,Qlost,con,q and Qlost,ref,r as compared to tube layout a. The reason being tube layout b has lesser temperature on top, bottom, leftfront, right-front walls and the quartz glass at the aperture. Energy balance is expressed as below.

Q lost;rad;q þ Q lost;con;q þ Q lost;ref;r þ Q lost;ref;q ¼ ð1
hr ÞQsolar (27) 8

As the carbon aerosol density increases N (0-1.5  10 particle/ m3), there is a decrease in the steam flow rate as well as the efficiency of the receiver as shown in Fig. 19. The drop in efficiency for tube layout a is about 9.3%, i.e., from 89.5% to 80.2%, and that for tube layout b is about 8.7% i.e., from 91.0% to 82.3%. The thermal efficiency and steam flow rate of tube layout b are higher than that of a at the same carbon aerosol density N. Fig. 17. Average heat transfer coefficient h2 around the quartz glass.

6.4. Thermal stress and LCOE analysis of the receiver The performance of a CSP plant heavily relies on weather and

(a)

(b)

Fig. 18. Energy distributions on the receiver at different N:(a) tube layout a and circulation mode a (b)tube layout b and circulation mode b.

Y. Jin et al. / Energy 182 (2019) 21e36

33

stress with the criterion set by Von Mises.

seq

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 2 2 2 ðs1
s2 Þ þ ðs2
s3 Þ þ ðs1
s3 Þ ¼ 2

(29)

The elastic stress analysis and the thermal analysis were performed using the commercial software package ANSYS workbench 15.0. Whereas, the fatigue assessment of the panel structure was carried out using the methodology described in (ASME Boiler, 2013, VIII) [39]. The fatigue curves can be expressed as below.

N ¼ 10X

(30)

where Fig. 19. Thermal efficiency and steam flow rate as a function of N under different circulation modes.

climatic conditions. During the rainy/cloudy days and at night the receiver shuts down. Such conditions result in high thermal stresses on the pipes and the structure of the membrane walls between the tubes and thus high thermal fatigue damages the welds and the pipes. As shown in Fig. 11 the maximum heat flux density in the receiver lies at the center of the back wall. Hence, for this reason, a tube from this section is considered for the thermal stress analysis. The temperature results in thermal stress in the tubes in three directions i.e. the radial, axial and circumferential direction. The longitudinal stress (z-direction) in the tubes is mainly composed of two parts. I. The stress due to the temperature gradientDTz =h. However, in the longitudinal direction, it is much smaller than in the cross-sectional directionDTr =th. So, the thermal stress is only prominent in the cross-sectional plane. II. The internal pressure causes stress in the longitudinal direction. It can be calculated using Eq (28).

sz ¼

PD 4s

(28)

The stresses at the longitudinal direction are much lesser than in the cross-sectional direction, and thus can be neglected. The total stress at the pipe can be calculated by evaluating the stress at the cross-sectional plane of the pipe. During the modeling, the left and right membrane walls were fixed, and the influence of local constraints on the weld was eliminated according to the Saint-Venant principle. The boundary conditions i.e., heat flux; surface temperature and fluid temperature of the tubes were calculated using the method described in section 5.1-5.2 of this manuscript for thermal stress analysis. The modulus of elasticity E and thermal expansion coefficient a are found as a function of temperature T. The following expressions were used to evaluate the equivalent

X¼

C1 þ C3 Y þ C5 Y þ C7 Y þ C9 Y þ C11 Y 1 þ C2 Y þ C4 Y þ C6 Y þ C8 Y þ C10 Y 



Y¼

Salt Cus

EFC ET

Salt ¼

Ke Kf DSP 2

(31)

 (32)

The economic evaluation has been investigated based on the data summarized in Tables 2 and 3. The levelized cost of energy (LCOE) is calculated by Eq (33)

!n n

LCOE ¼

n

kd ð1þkd Þ n ð1þkd Þ
1

þ kinsurance E

Cinvest þ CO&M (33)

In order to evaluate the economics of a carbon aerosol entrapped cavity receiver, an evaluation function model was established. It is expressed as below.

f ðn; hÞ ¼

LCOEN0 LCOEN

(34)

The temperature distribution on the cross section of the tube under different carbon aerosol density for tube layout a is illustrated in Fig. 20. It can be observed that the temperature distribution on the cross section of the tube is highly non-uniform, with a peak temperature of about 650K in the regions I and II. The temperature difference between the outer and the inner surface of the tube is 54K. As the carbon aerosol density increases N (01.5  108 particle/m3), the temperature in the regions I and II decreases as expected, and the temperature distribution on the cross section of the tube becomes more uniform. Carbon aerosol addition

Fig. 20. Temperature distribution on the tube's cross-section under tube layout a.

34

Y. Jin et al. / Energy 182 (2019) 21e36

into the receiver prevents local overheating at the surface of the tube which in turns extends the lifespan of the receiver. The equivalent thermal stress distribution at the cross section of the tube under different carbon aerosol density for tube layout a is illustrated in Fig. 21. Due to highly non-uniform temperature distribution across the surface of the pipes, the equivalent thermal stress distribution on the cross section of the tube is highly nonuniform as well, with a peak value of about 478Mpa in the region III. As expected, an increase in the carbon aerosol density N (01.5  108 particle/m3) decreases the equivalent thermal stress in region III. The daily cold start-up of the receiver causes the low cycle fatigue damage on the tubes. Table 4 summarizes the fatigue life and economic evaluation function of the back panel with the tube layouts a and b, when the aperture receives total solar energy Qtotal ¼ 6434 kW. It can also be observed that the difference in weld quality levels (i.e., level 1e7) have a significant effect on the life of the back panel. With both the tube layouts a and b the service life of the receiver is predicted to be less than 1.5 years with weld quality level 7 in region III and the structure of the heat pipe can be damaged. However, with the increase of carbon particle density N (0-1.5  108 particle/m3), the stress in region III. decreases and this also makes it more economical. Although the weld quality levels (1, 3 and 5) reduce the stresses in the region III they tend to have a slight adverse economic effect. Fig. 22 illustrates the relationship between the effective equivalent alternating stress and the number of life cycles of the receiver. The point a shows the predicted lifespan of the receiver (i.e., 20 years) and the equivalent amplitude stress Salt is less than 490 MPa. The literature survey shows the solar flux on the aperture at noon can reach up to 7800 kW. And the heat flux at the center of the

Fig. 21. The equivalent thermal stress distribution on the tube's cross-section under the tube layout a.

Fig. 22. Relation between alternating equivalent stress and life cycles of the weld at region III.

back panel can reach 390 kW/m2, i.e., there exists a higher temperature gradient between the inner and the outer surface of the pipe causing the stress in region III to be even higher. Table 5 summarizes the fatigue life and economic evaluation function of the back panel with the tube layouts a and b when the aperture receives total solar energy Qtotal ¼ 7800 kW. With the increase in carbon aerosol density, the economic evaluation function values obtained are all less than 1.0 at weld quality levels (5e7). The total investment in DAHAN power station is 28.04¥ million and the receiver accounts for 14.3% of the total investment cost (1¥ ¼ 0.15 USD). The DAHAN power plant is designed for a period of 20 years with annual power generation of 2.7  106 kWh. If the cavity receiver is filled with carbon aerosol of 1  108 particles/m3, the thermal efficiency of the receiver declines by 5% as compared to a cavity receiver without any carbon aerosol. The current price stipulated by the government of China for solar thermal power generation is 1.15 ¥/kWh. The annual power generation after carbon aerosol filling drops to 1.35  105 kWh, in terms of finances, it is equivalent to a decrease of 155,250¥ every year. As shown in Table 5, if the weld quality level of the receiver is 3, the receiver needs replacement after every 5 years. The total cost of the replacement for every 5 years is 2,800,000¥ and in the 20 years it sums up to 8,400,000¥. However, if the receiver is filled with the 1  108 particle/m3 carbon particles, its life span can be increased to 18.8 years and only one replacement would be required for the 20 years designed time period. The total cost including the one-time replacement cost of the receiver and the decline in electricity production cost during the 20 years sums up to 5,905,000¥. It can be concluded that filling the receiver with the carbon

Table 4 Fatigue life of the central boiling pipe for case-a and case-b under Qtotal ¼ 6434 kW. case

[Particle/m3]

DS P

a

b

0 5.0  107 1.0  108 1.5  108 0 5.0  107 1.0  108 1.5  108

478 442 386 345 453 421 374 327

Fatigue life(cycles)

Ke

[Mpa] 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

Kf ¼ 1.0 28251 38997 67769 93547 36394 46885 83374 *

fðn; hÞ

Kf ¼ 1.7

fðN; hÞ

Kf ¼ 2.5

fðN; hÞ

Kf ¼ 3.0

fðN; hÞ

Kf ¼ 4.0

fðN; hÞ

1.000 1.035 1.085 1.110 1.000 1.031 1.082 *

3991 5383 8532 13214 4966 6452 9773 16887

1.000 1.034 1.085 1.105 1.000 1.033 1.081 1.100

1205 1483 2296 3244 1416 1783 2637 3991

1.000 1.044 1.080 1.110 1.000 1.031 1.050 1.089

743 886 1345 1832 783 956 1387 2064

1.000 1.024 1.059 1.086 1.000 1.022 1.045 1.078

364 564 615 834 423 501 662 968

1.000 0.154 0.119 0.093 1.00 0.153 0.188 0.174

Y. Jin et al. / Energy 182 (2019) 21e36

35

Table 5 Fatigue life of the central boiling pipe for case-b under Qtotal ¼ 7800 kW. 249. case

269. b

250. N 251. [particle/m3]

252. DSP 253. [Mpa]

254. Ke

270. 282. 294. 306.

271. 283. 295. 307.

272. 284. 296. 308.

0 5.0  107 1.0  108 1.5  108

610 504 416 380

1.0 1.0 1.0 1.0

255. Fatigue life(cycles) 260. Kf¼1.0

261. fðn; hÞ

262. Kf¼1.7

263. fðN; hÞ

264. Kf¼2.5

265. fðN; hÞ

266. Kf¼3.0

267. fðN; hÞ

273. 285. 297. 309.

274. 286. 298. 310.

275. 287. 299. 311.

276. 288. 300. 312.

277. 289. 301. 313.

278. 290. 302. 314.

279. 291. 303. 315.

280. 292. 304. 316.

* * * *

* * * *

particles will save up to 30% of the total receiver cost during the 20 years period. However, losses during the time period the replacement takes place are not considered in the current calculations. The receiver with the carbon aerosol can decrease the replacement frequency and the losses during the replacement period can also be avoided. This indicates that as the carbon aerosol density increases, although the efficiency of the heat absorber decreases, the overall economic performance of the heat absorber is significantly improved.

1782 3243 6785 8835

1.000 1.020 1.061 1.089

582 1358 1867 2604

1.000 0.657 0.681 0.705

361 724 1107 1387

1.000 0.087 0.086 0.085

associative interest that represents a conflict of interest in connection with the work submitted. Acknowledgments The present work is sponsored by the National Natural Science Foundation of China (No. 51506173), Yulin Science and Technology Project (NO.2017KJJH-03), Shaanxi Key Science and Technology Innovation Team (2019TD-039) and the Fundamental Research Funds for the Central Universities (No.cxtd2017004).

7. Conclusion In this study, a method is proposed to improve the heat flux uniformity over the receiver's surface. Carbon aerosol is introduced into the receiver at varying densities. Heat transfer through conduction, convection and radiation is investigated by coupling the MCRT and FVM model. Furthermore, the stress analysis was performed to evaluate the lifespan of the receiver, and the economic analysis was carried out to financially assess the benefits of inserting the carbon aerosol into the receiver. The results are summarized as follows: Compared with our previous two-dimensional coupled MCRTFVM model, the present work also considers the influence of carbon aerosol on the thermal performance of the receiver. Also, boiling flow correlations are used to calculate the wall temperature of the tube. The three-looped iterative scheme for r, h2, and h1 accurately calculate the steady-state thermal performance of the receiver. The three-dimensional MCRT-FVM coupling model proposed in the present study can simulate the complex photothermal conversion process of carbon aerosol containing cavity receiver. With the increase in carbon aerosol density N (0-1.5  108 particle/m3), the peak solar heat flux at the center of the back-panel dropped significantly, and the uniformity of the solar heat flux and thus temperature distribution on left-back panel, the back panel and the right-back panel is significantly improved. The thermal performance decreases with the increase of carbon aerosol density in the cavity receiver. The increase in heat losses responsible for the decrease in performance is mainly caused by the escape of scattered photons from the cavity receiver after striking the carbon aerosol. The peak thermal stress is mainly located at the weld (region III). A weld quality level of at least 3 is necessary to guarantee the designed lifespan of 20 years. The peak stresses can be reduced by decreasing the temperature at the surface of the tube. This can be achieved by increasing the carbon aerosol density N (0-1.5  108 particle/m3). By investigating the two different tube layouts, it is concluded that the tube layout b has 1.2% higher efficiency and 5% lower peak stress than the tube layout a. Considering the thermal efficiency and economic assessment of the receiver a carbon aerosol density of 1.0  108 number/m3 is the best choice. Conflict of interest The authors declare that we do not have any commercial or

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