A review of steady-state thermal and mechanical modelling on tubular solar receivers

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A review of steady-state thermal and mechanical modelling on tubular solar receivers Tim Conroy *, Maurice N. Collins, Ronan Grimes Stokes Laboratories, Bernal Institute, School of Engineering, University of Limerick, Ireland

A R T I C L E I N F O

A B S T R A C T

Keywords: Concentrated solar power Liquid tubular receiver Modelling Thermal Mechanical

Tower systems are forecast to become the dominant CSP technology in the future due to the potential to achieve high working fluid temperatures, thereby enhancing thermodynamic efficiency in the power block and facili­ tating dispatchable electricity through thermal energy storage. The receiver links the solar collector field and power conversion cycle in a tower plant, and is therefore a critical component that requires careful consideration. Tubular receivers represent the most prominent in commercial scale applications, with many research efforts devoted to the characterisation and modelling of such concepts. This article compiles literature engaged in steady-state thermal and mechanical modelling of tubular solar receivers. The discussion outlines contrasting approaches adopted by various authors, while also detailing some important findings from their investigations. Recent studies concerned with evaluating receiver thermal performance indicates a trend towards semi-empirical techniques, offering greater flexibility and accuracy than simplified analytical methods, without imposing a considerable computational expense that is inherent with more detailed numerical models. Such advantages allow for the screening of a large number of geometries, configurations, heat transfer media, tube materials, and operational scenarios at the receiver design stage. Mechanical reliability investigations generally consider thermal and pressure induced stresses, estimating potential damage of the component across its desired lifetime using design code guidelines or tube material data. The selection of thermal stress theory and damage evaluation method is critical to the overall mechanical life prediction, with different approaches presented.

1. Introduction Concentrated solar power (CSP) is fast emerging as a feasible tech­ nology that can alleviate fossil fuel dependence in regions with a good solar resource (e2000 kWh=m2 =year [1]), and is forecast to contribute approximately 11–12% to global electricity production by 2050 [2,3]. Today there is 5:8 GWe of installed CSP capacity worldwide, and a further 4 GWe either in construction or under development (as of late 2019) [4]. There are four main CSP systems; parabolic trough, linear Fresnel, parabolic dish, and tower, illustrated in Fig. 1. The focus of this work is on tower systems (Fig. 1c), which are expected to become the dominant CSP technology in the future. In a tower system, a large array of automated mirrors, known as heliostats, track and reflect the suns energy onto a target receiver located atop the tower, where a large solar concentration (� 103 suns) raises the temperature of a heat transfer medium. The receiver working fluid is then directly or indirectly used in the power cycle, typically of the steam Rankine type, although high-temperature Brayton cycles are

being explored [5]. The point-focus nature of tower systems presents the opportunity to generate very high working fluid temperatures in the receiver, allowing for an increase in thermodynamic efficiency as the fluid feeds into the power cycle, and excellent thermal storage potential using certain media in order to mitigate intermittency concerns. These benefits mean that tower technology is expected to play a major role in the future of CSP, despite the fact that the vast majority of CSP plants built to date have utilised trough technology [6]. Of the 5:8 GWe of installed CSP capacity worldwide, over 1:1 MWe employs tower tech­ nology. The benefits of tower systems are now being realised with an increasing number of commercial plants coming on-line in recent years, and of the 5 GWe of plants either under construction or announced as of 2017, nearly 3 GWe are based on this technology [7]. The important role tower systems are expected to play in the future of renewable energy serves as a motivation for the present work. Receiver concepts are typically categorised by the phase of state of the heat transfer medium employed; solid, liquid, or gas [8]. This paper is concerned specifically with liquid tubular based designs, however for

* Corresponding author. E-mail address: [email protected] (T. Conroy). https://doi.org/10.1016/j.rser.2019.109591 Received 30 November 2018; Received in revised form 7 October 2019; Accepted 12 November 2019 1364-0321/© 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: Tim Conroy, Renewable and Sustainable Energy Reviews, https://doi.org/10.1016/j.rser.2019.109591

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various approaches taken in modelling the thermal and mechanical response of tubular solar receivers under differing design and operating conditions, as well as offering a discussion on investigations in which these tools are applied. Brief descriptions of the various models are provided throughout the text, however it is advised that references are consulted for greater detail on individual models and investigations. The first section provides information on a number of past and present CSP tower projects (test/demonstration/pilot and commercial scale), high­ lighting the important role tubular receivers have played in the progress of the technology. The second section describes thermal modelling ac­ tivities for tubular solar receivers, focussing on model structure, key boundary conditions, and outcomes of the various studies in which the developed models are applied. The third section briefly outlines features of various heliostat aiming strategy models, as these are closely linked to the thermal model with regards to the incident heat flux boundary condition. Finally, modelling techniques used to investigate receiver mechanical limitations are then discussed, such as the selection of thermal stress theory and means of establishing component reliability, along with some interesting results of the various investigations. 2. CSP tower projects The knowledge and experience gained from the operation of a CSP tower plant is key to the development and refinement of the technology. There have been a number of small-scale projects built and operated since the surge of interest in CSP in the early 1980’s which have paved the way for valuable data collection from plant operation, and for the optimisation of receiver design and operational strategies. Some of these projects include the National Solar Thermal Test Facility (NSTTF) at Sandia National Laboratories (SNL) in New Mexico, Plataforma Solar de Almeria (PSA) in Spain, and the pioneering 10 MWe Solar One plant in California (succeeded by Solar Two). A summary of the key details of small-scale tower projects (< 10 MWe test/demonstration/pilot) is provided in Table 1. Numerous commercial-scale parabolic trough plants have been in development since the 1980’s, however it wasn’t until 2007 that the first commercial tower plant commenced operation; the 11 MWe Planta Solar 10 (PS10) operated by Abengoa Solar, near Seville, Spain (Fig. 2b) [3]. A second generation 20 MWe plant known as PS20 commenced operation in 2009 [3,32]. The 19:9 MWe Gemasolar Thermosolar plant developed by Torresol Energy outside Seville, Spain, was the first commercial CSP tower plant to employ molten salt (Fig. 2c), permitting 15 hr of thermal storage capacity [33,34]. The 377 MWe Ivanpah Solar Electric Gener­ ating System (SEGS) in California, USA, is currently the largest CSP plant in the world of any type (Fig. 2d), employing direct steam receivers [3]. At 110 MWe , the Crescent Dunes Solar Energy plant in Nevada, USA operated by SolarReserve was one of the first large-scale tower plants to employ molten salt receiver technology (Fig. 2e), providing 10 hr of thermal energy storage [7,35]. The 50 MWe Khi Solar One plant devel­ oped by Abengoa Solar opened in the Northern Cape, South Africa in 2016 [7] (Fig. 2f), and uses direct steam receivers [36]. In Morocco, the 150 MWe Noor Ouarzazate III (NOOR III) tower plant commenced operation in early 2019, complementing the NOOR I and NOOR II parabolic trough units in situ [37]. The central tower supports a cylin­ drical molten salt receiver design, and is equipped with a two-tank direct molten salt storage system with 7.5 hr capacity. In April 2019, the 121 MWe Ashalim (Plot B) from Megalim Solar Power Ltd. commenced operation in Israel’s Negev desert [6] (Fig. 2i). The 240 m tower is the tallest of any CSP project, and houses a water/steam cylindrical tubular receiver that acts as the boiler and superheater for the turbine. The tower unit does not operate with any storage capacity, however off-peak generation is facilitated by molten salt storage from a parabolic trough system.

Fig. 1. Schematic of CSP technologies: (a) Parabolic trough, (b) Linear Fresnel, (c) tower, (d) Parabolic dish © [2].

a more detailed discussion on a range of receiver designs and concepts, the authors direct the reader to comprehensive review articles by Tan & � Chen [9], Avila-Marín [10], Behar et al. [11], Ho & Iverson [8], Ho [12], and Ho [7]. Liquid tubular concepts represent the most popular design in the realm of solar receivers for CSP tower systems, and form the main focus of this paper. There has been considerable research interest in liquid tubular receivers since the 1970’s, with some of the earliest demonstration and commercial CSP systems utilising this technology [13]. At present, they are the only receiver design that has reached commercial deployment in CSP tower plants, with large scale projects all using some form of liquid tubular receiver (Section 2). The similarities in design and operation that these concepts share with conventional tubular heat exchanger/boilers has aided in the progress of the tech­ nology [14]. Significant testing and research work has been completed on receiver concepts over the past four decades, with early experimental and design endeavours completed at Sandia National Laboratories (SNL), THEMIS, and Plataforma Solar de Almeria (PSA) [8]. The rapid development and prominence of the technology has encouraged numerous authors to devise mathematical models that simulate the thermal and mechanical response of tubular receivers under a variety of operational scenarios and boundary conditions, thus aiding in the design and optimisation of components and systems. Varying approaches and methodologies are adopted, resulting in differing degrees of complexity, resolution, and accuracy. The motivation for this review article stems from the large number of thermal and mechanical modelling efforts found in literature, with an opportunity to develop a comprehensive review detailing the methodologies adopted and results gleaned there­ fore presented. This paper is concerned with steady-state heat transfer models that model receiver thermal response using techniques that can be described as semi-empirical and analytical, whilst the mechanical modelling sec­ tion reviews analytical tube stress and reliability models. Although briefly mentioned, a review of numerical techniques (FEA, CFD, etc.) is beyond the scope of the present article. In order to gauge the state-ofthe-art, both in terms of receiver design and modelling approach, there is a need to contextualise the various investigations, framing the discussion around model formulation, and key results uncovered from the accompanying analysis. This review article informs the reader on 2

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Table 1 Details of selected test, demonstration, and pilot tower plants. Country

Facility

MWe

Years active

Receiver

Working fluid

Storage

Notes, references

NSTFF

US DOE, SNL

USA

Test

–

1978 -

see Notes

see Notes

None

EURELIOS

European Commission, industrial consortium

Italy

Demonstration

1

1981–1986

2:8 MWth cavity

Water/steam

0:5 hr buffer, water and molten salt

Used to test a variety of receiver concepts and subsystem components [7], Poor performance attributed to low insolation and component design issues [15,16],

IEA-SSPS at PSA

IEA member states

Spain

Demonstration, test

0.5

1981–1986

Sodium

2 hr, two-tank direct

Early demonstration of sodium potential, large sodium spray fire hindered progress [16–18],

CESA-1 at PSA

CENSOLAR, SNL

Spain

Demonstration, test

1.2

1983-

2:8 MWth cavity & 2:5 MWth billboard 8 MWth cavity

Water/steam

3:5 hr indirect molten salt

THEMIS

AFME, EDF (France)

France

Demonstration, test

2

1983–1986

8:9 MWth cavity

Molten salt

5 hr, two-tank direct

Solar One

US DOE, SNL

USA

Pilot

10

1982–1988

43:4 MWth cavity

Water/steam

4 hr, oil/rock thermocline

Early demonstration of molten salt potential, now a test facility for subsystem components, no longer producing electricity [16,19], Field and tower now an experimental facility for the European Next-CSP project [16,20], Demonstration of a reliable CSP commercial-scale tower plant, inefficient storage [16,21,22],

Solar Two

US DOE, SNL

USA

Pilot

10

1996–1999

43 MWth external cylinder

Molten salt

3 hr, two-tank direct

Successful demonstration of commercial-scale dispatchable electricity from CSP [21,23]

SEDC

BrightSource

Israel

Demonstration

1

2008-

6 MWth external billboard

Water/steam

None

Used to test equipment, materials and procedures, construction and operating methods [24]

Jülich

DLR

Germany

Demonstration, test

1.5

2008-

10 MWth volumetric

Air

1:5 hr, ceramic bed

Sierra SunTower Solar Field (1 þ 2)

eSolar

USA

5

2009–2013

Cavity and external

Water/steam

None

CSIRO

Australia

Small-scale commercial Demonstration, test

–

2010-

Tubular

Water/steam, gaseous

None

Used for testing of higher-temperature receiver concepts [10,25] Poor performance and lack of storage preceded closure [13] Two fields, used to test supercritical and Brayton receiver cycles

ACME

ACME Group, eSolar

India

2.5

2011-

Cavity

Water/steam

None

First CSP tower plant in India [26]

Lake Cargelligo

Graphite Energy

Australia

Small-scale commercial Demonstration

3

2011–2016

5 MWth solid-based

Water/steam

4 hr, indirect molten salt

Modular layout, novel graphite receiver and storage concept [3,7]

Coalingha

Chevron, BrightSource

USA

Process steam

–

2011–2014

29 MWth billboard

Water/steam

None

Process steam generated by CSP for oil recovery, 13 MWe electrical equivalent [27]

DAHAN

Chinese Academy of Sciences

China

Demonstration, test

1

2012-

Cavity concepts

Water/steam, molten salt

1 hr, saturated steam/ oil, two-tank direct

First CSP tower plant in China, used to test different receiver concepts [28,29]

Greenway CSP

Greenway CSP

Turkey

Demonstration

1

2013-

5 MWth billboard

Water/steam

4 hr, indirect molten salt

First CSP tower plant in Turkey [3]

Sundrop CSP

Sundrop Farms, Aalborg CSP

Australia

Electricity, desalination, heating

1.5

2016-

36:6 MWth billboard

Water/steam

None

Multi-purpose plant for electricity generation (1:5 MWe ) and horticultural applications [30]

Jemalong CSP

Vast Solar

Australia

Pilot

1.1

2017-

5 � 1:2 MWth billboard

Sodium

3 hr, two-tank direct

Development underway to scale to 30 MWe , modular field and receiver layout [31]

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Sponsor/developer

3

Project

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Fig. 2. Commercial scale CSP tower plants and receivers: (a) Solar Two [23], (b) PS10 and PS20 [40], (c) Gemasolar Thermosolar [34], (d) Ivanpah SEGS [41], (e) Crescent Dunes [35], (f) Khi Solar One [42], (g) SUPCON Solar Phase II [43], (h) Shouhang Dunhuang Phase II [44], and (i) Ashalim Plot B [45].

China is becoming an increasingly important market for commercial CSP ventures, with a national commitment to develop a number of projects1 by 2020 that will help the country meet climate goals and drive a self-sufficient domestic CSP industry [38]. As of late 2019, the total installed capacity of commercial scale CSP in China stands at more than 300 MWe , with over 250 MWe facilitated by tower technology, and many more projects under construction and in development (see Table 3). In August 2016, the 10 MWe SUPCON Solar Project (Phase I) in the Qinghai province became the first molten salt tower project in China, with 6 hr of direct storage charged by a cylindrical receiver design. In late 2018, a larger 50 MWe installation (SUPCON Solar Phase II) commenced production at the same site, with 7 hr storage [6] (Fig. 2g). In late 2016, the Shouhang Dunhuang 10 MWe Phase I plant commenced operation in the Gansu province. The plant also uses a cylindrical molten salt receiver design to charge a two-tank direct thermal storage unit, with 15 hr capacity [6]. The Shouhang Dunhuang 100 MW Phase II began production in December 2018 at the same site, also employing molten salt receiver and storage technology (Fig. 2h). In September 2019, the 50 MWe Luneng Haixi and 50 MWe Qinghai Gonghe were connected to the national grid, both using molten salt receiver tech­ nology, with 6 and 12 hr of direct storage capacity respectively [38,39]. Table 2 summarises some key details of commercial CSP tower plants currently operational (as of late 2019), with Fig. 3 providing a graphical representation. The ability to store thermal energy for the generation of dispatchable electricity is a key motivator for future tower projects. There are a number of tower plants that have been announced, are in development, or under construction, with many employing molten salt receiver tech­ nology. Key details of these future projects are summarised in Table 3.

through irradiated tubes via inlet and outlet headers, with the final fluid temperature a function of irradiance, mass flow rate, tube geometry, thermal losses, and heat transfer fluid. External designs utilise an array of tubes aligned to form a quasi-billboard shape for equator facing fields, known as the billboard receiver (Fig. 4a). A number of tube panels may be interlinked to approximate a cylinder for surrounding heliostat fields, with the fluid flowing through a series of panels in a serpentine path, known as a cylindrical receiver (Fig. 4b). The Ivanpah SEGS receiver arranges tube panels in a cardinal-like manner on a central tower, resembling a cube structure rather than a cylinder. External configura­ tions with exposed tube panels are most popular in CSP tower projects (see Tables 1 and 2). In an internal receiver (Fig. 4c), concentrated solar radiation from the heliostat field is focussed through an aperture into a box-like cavity structure which houses the tubes, known as a cavity receiver [16]. In terms of commercial deployment, cavity receiver de­ signs are currently in use at the PS10 and PS20, with the Khi Solar One tower housing three cavity receivers facing different cardinal points of the heliostat field [6]. The primary difference between external and internal designs in terms of performance is reflected by the modes of heat loss, with cavity designs yielding slightly higher thermal effi­ ciencies [8]. In a cavity receiver, low ambient view factors reduce radiative and reflective losses to the environment when compared to an external design, however limiting the view between the active heat transfer surfaces and heliostat field poses a greater risk to spillage losses, and restrictions placed on heliostat aiming results in greater heat flux non-uniformity. To cope with restricted view to the field, cavity designs will generally have a larger heat transfer area and must be placed on a taller tower structure than an equivalent external cylindrical design, leading to greater expense [16]. The selection of receiver configuration will depend on the heliostat field layout, working fluid, and cost con­ siderations. The most common fluids used in these designs are water/­ steam and molten salt, however liquid metals have also been proposed, with sodium currently being trialled at the Vast Solar facility in NSW, Australia, in a modular plant layout that employs a number of low-cost billboard receiver designs. The selection of heat transfer fluid is critical to the performance of the CSP system, dictating plant configuration, thermohydraulic and mechanical performance, thermal storage characteristics, and costs.

3. Liquid tubular receivers Liquid tubular receivers are generally classified as being internal (cavity) or external in configuration, with these designs illustrated in Fig. 4. Both internal and external configurations shuttle a working fluid

1

50

100 MWe scale. 4

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Table 2 Details of operational commercial-scale CSP tower plants as of late 2019. Project

Developer

Country

Start

MWe

GWh=yr

Field (m2 )

Receiver

Working fluid

Storage

PS10

Abengoa Solar

Spain

2007

11

23.4

75,000

Cavity

Water/steam

< 1 hr, steam accumulation

PS20

Abengoa Solar

Spain

2009

20

48

150,000

Cavity

Water/steam

< 1 hr, steam accumulation

Gemasolar

Torresol Energy

Spain

2011

19.9

80

305,000

External

Molten salt

15 hr, two-tank direct

Ivanpah SEGS Crescent Dunes

BrightSource SolarReserve

USA USA

2014 2015

377 110

1079 500

2,600,000 1,200,000

External External

Water/steam Molten salt

None 10 hr, two-tank direct

Khi Solar One

Abengoa Solar

S. Africa

2016

50

180

576,800

Cavity

Water/steam

2 hr, steam accumulation

SUPCON Phase I

SUPCON Solar

China

2016

10

–

–

External

Molten salt

6 hr, two-tank direct

Dunhuang Phase I

Beijing Shouhang IHW

China

2016

10

–

175,375

External

Molten salt

15 hr, two-tank direct

SUPCON Phase II

SUPCON Solar

China

2018

50

146

542,700

External

Molten salt

7 hr, two-tank direct

Dunhuang Phase II

Beijing Shouhang IHW

China

2018

100

390

1,380,000

External

Molten salt

11 hr, two-tank direct

NOOR III

ACWA

Morocco

2019

150

500

1,320,900

External

Molten salt

7:5 hr, two-tank direct

Ashalim Plot B Luneng Haixi

Megalim Solar Luneng Group

Israel China

2019 2019

121 50

320 160

1,052,480 607,200

External External

Water/steam Molten salt

None from the tower system 12 hr, two-tank direct

Qinghai Gonghe

SUPCON Solar

China

2019

50

150

515,900

External

Molten salt

6 hr, two-tank direct

Table 3 Details of future CSP tower projects as of late 2019 (in development/planning, under construction). Project

Status

Developer

Country

Online

MWe

Working fluid

Storage

Reference

Jemalong CSP

Planning/development

Vast Solar

Australia

> 2020

30

Sodium

10 hr, indirect molten salt

[46]

Cerro Dominador

Under construction

Abengoa, Acciona

Chile

> 2020

110

Molten salt

17:5 hr, direct

[47]

Copiap� o

Planning/development

SolarReserve

Chile

> 2021

260

Molten salt

14 hr, direct

[48]

Likana

Planning/development

SolarReserve

Chile

> 2021

390

Molten salt

13 hr, direct

[49]

Tamarugal

Planning/development

SolarReserve

Chile

> 2021

450

Molten salt

13 hr, direct

[50]

Golden Tower

Planning/development

SunCan

China

> 2021

100

Molten salt

8 hr, direct

[51]

Golmud

Under construction

Qinghai CSP

China

> 2020

200

Molten salt

15 hr, direct

[51]

Hami

Under construction

CPECC

China

> 2019

50

Molten salt

12 hr, direct

[52]

Qinghai Delingha

Under construction

BrightSource

China

> 2021

135

Water/steam

4 hr, indirect

[51]

Shangyi

Under construction

Chinese Academy of Sciences

China

> 2020

50

Water/steam

4 hr, indirect

[51]

Yumen 50 MW

Under construction

Parasol Energy, Xinchen CSP

China

> 2019

50

Molten salt

6 hr, direct

[51]

Yumen 100 MW

Planning/development

SunCan

China

> 2021

100

Molten salt

10 hr, direct

[51]

MINOS

Planning/development

NUR Energie

Greece

> 2021

52

Water/steam

5 hr, indirect

[53]

Redstone

Planning/development

ACWA, SolarReserve

South Africa

> 2021

100

Molten salt

12 hr, direct

[54]

DEWA Tower

Under construction

ACWA Power

UAE

> 2021

100

Molten salt

15 hr, direct

[55]

� Water/steam - The implementation of water/steam receiver systems in solar thermal plants dates back to the 1980’s [59]. In these de­ signs, feed water is pumped from the power block to the receiver, where it is then evaporated and superheated to produce steam that feeds the turbine. Direct steam generation (DSG) is employed in a number of commercial CSP tower projects; PS10, PS20, Ivanpah SEGS, Khi Solar One, and Ashalim Plot B (see Section 2), and also several parabolic trough plants. The advantage of this technology largely lies in its ability to dispense with costly intermediate heat exchange equipment by using the receiver as the boiler, thus simplifying the system and reducing the potential for inefficiencies. The capacity factor of water/steam systems is quite low, with extensive thermal energy storage of high temperature steam proving complicated, despite having practically no supply cost. The PS10, PS20, and Khi Solar One plants utilise short term thermal energy storage (< 2 hours), while Ivanpah SEGS and Ashalim Plot B (tower unit only) possesses no storage at all [6]. This diminishes the competitive advantage CSP has over many other renewable energy technologies, which is the ability to generate dispatchable electricity by coupling large-scale thermal storage with a synchronous power unit. The thermal conductivity of water/steam is orders of magni­ tude lower than that of molten salts and liquid metals, resulting in relatively low tubular heat transfer coefficients. This, combined with a necessity for thick tube walls to withstand high pressures, means receiver tubes carrying water/steam are susceptible to large thermal

Fig. 3. Commercial scale CSP tower plants by rating (MWe ), working fluid, storage capacity, and start year.

Very high receiver temperatures may be facilitated through gaseous media such as air, supercritical carbon dioxide (sCO2), or helium (He), however these concepts have yet to reach commercial maturity and are limited to small-scale test units, or modelling efforts [7]. A brief sum­ mary of more common liquid-based working media is presented here. 5

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Fig. 4. Tubular receivers, (a) schematic of a billboard design [56], (b) cylindrical configuration [57], and (c) cavity receiver concept [58].

and pressure induced stresses, resulting in relatively low allowable heat fluxes of � 0:5 MW=m2 [60]. Corrosion and compatibility issues with high temperature steam and tube alloys poses further concerns [59]. � Molten salts - Molten salt working fluids have been pursued since the early days of CSP, both as a receiver heat transfer fluid and sensible storage media. In a molten salt receiver, the temperature of the fluid is raised by concentrated sunlight, it is then stored and released through a steam generator to feed the turbine when required. The pioneering Solar One tower project identified the ne­ cessity for a high temperature receiver working fluid that could provide dispatchable energy for the grid, leading to the development of the first large scale molten salt tower system with the the Solar Two project in the 1990’s [21]. Molten salts have a high volumetric heat capacity, allowing for large quantities of sensible heat storage (� 0:75 kWh=m3 K) at temperatures up to � 600∘ C [61]. Commercial scale molten salt plants such as Gemasolar and Crescent Dunes have built upon the experience gained from Solar Two, utilising 15 and 10 hr of sensible storage respectively, while more recently the Shouhang Dunhuang, SUPCON Solar, Luneng Haixi, and Qinghai Gonghe plants in China, and NOOR III in Morocco, have also incor­ porated molten salt working fluids. The majority of large-scale CSP tower projects under construction and/or development will incor­ porate molten salt as a receiver working fluid and storage medium (see Section 2). A mixture of sodium and potassium nitrates known as Solar Salt is the most common molten salt composition, which is stable between temperatures of � 200 600∘ C, and has a cost of 0:5 $=kg [62]. HITEC and HITEC XL offer lower melting points and higher heat capacities than Solar Salt, however these suffer from a lower thermal stability limit � 500∘ C. Other molten salt composi­ tions have been developed for higher temperature applications, such as Li–Na–K carbonates and fluorides, Halotechnics, and Na–K–Zn chlorides. For a comprehensive discussion on molten salt mixtures and other liquid heat transfer media, the reader is directed to the paper from Vignarooban et al. [59]. High thermal conductivities of molten salts (� 10 1 100 W=mK) enable large tubular heat trans­ fer coefficients (in comparison to water/steam), thus enabling inci­ dent heat fluxes of � 1 MW=m2 . As heat transfer to molten salts in the turbulent regime is dominated by eddy conductivity, these fluids are receptive to heat transfer enhancement devices, as demonstrated experimentally by Yang et al. [63]. Despite the low-cost, high availability, and excellent sensible storage potential, concerns arise in the form of containment material compatibility. There is consid­ erable interest in characterising the effects of molten salt induced corrosion on metallic alloys, with Vignarooban et al. [59] offering a discussion on research efforts conducted on a number of different

mixtures. Current commercial molten salt mixtures have a relatively narrow operational temperature range, therefore one of the DOE SunShot initiatives concerning molten salt receivers is to generate fluid outlet temperatures of > 720∘ C with thermal efficiencies > 90% [64], thus facilitating greater power cycle efficiency and ther­ mal storage potential. Halide (fluorides and chlorides) and carbonate salts currently under development can yield higher fluid outlet temperatures than traditional solar salt (> 600∘ C), however practical concerns with extreme corrosion on containment metals and alloys have been raised [7]. � Liquid metals - Liquid metals have long been identified as poten­ tially excellent receiver working fluids. This is largely due to high operating temperature ranges in the liquid phase and efficient heat transfer performance in tubes, enabling the removal of large quan­ tities of heat from limited exchanger surfaces. Receiver concepts employing a liquid metal working fluid operate on much the same principal as molten salt based designs, with the heated working fluid used to generate steam for the turbine indirectly. Liquid metals such as sodium (Na) and lead-bismuth eutectic (Pb Bi) have been pro­ posed as candidate fluids that could deliver higher fluid outlet tem­ peratures than molten salts (> 600∘ C), also allowing for more compact receiver designs that alleviates cost and enhances thermal efficiency due to excellent heat flux capabilities ð> 2:5 MW =m2 Þ [61]. Liquid metals are impaired by relatively low specific heat ca­ pacities however, affecting their ability to operate as cost-effective thermal storage media, with both sodium and lead-bismuth pos­ sessing approximately half the volumetric heat capacity of molten salt ðkWh =m3 KÞ, and a per-unit cost of 2 $=kg and 13 $=kg; respec­ tively [62]. Corrosion with containment materials is an area of considerable concern with heavy liquid metals such as lead-bismuth, however sodium exhibits greater compatibility with stainless steels and other metallic alloys [59]. Currently there are no commercial CSP plants that operate with liquid metals, however the nuclear in­ dustry has found application for these fluids as early as the 1940’s as coolants in LMFBR cores, and the solar industry stands to benefit from this experience. A number of sodium solar receiver tests were conducted in the 1980’s from a joint effort by Rockwell International and the US DOE [65], and also at the IEA-SSPS program at PSA in Spain [17,18], however the only recent activity in terms of plant operation has been that undertaken by Vast Solar. Research groups such as the Solar Thermal Group at the Australian National Univer­ sity (ANU) in ACT, Australia, the Institute for Solar Research at DLR in Stuttgart, Germany, and the Institute for Nuclear and Energy Technologies at the Karlsruhe Institute of Technology (KIT), are currently engaged in liquid metals research for solar thermal applications. 6

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4. Thermal modelling Attempts have been made by several authors to model the thermal performance of liquid tubular receivers, dating back to the early stages of CSP development in the 1980’s. Heat transfer models are generally classified as analytical or numerical in nature, and are used to model receiver performance under steady-state and/or transient conditions. Numerical models generally rely on commercial CFD codes that require several thousand iterations to solve conditions over the internal flow field and tube surface, offering solutions of a high accuracy and reso­ lution. However, coupling heat transfer and fluid motion inside a tube with material conduction effects and environmental thermal energy losses renders a significant computational requirement with numerical methods, meaning that resolving the full thermal profile across every tube on a liquid tubular concept is often impractical, particularly at the R & D and preliminary design stage where multiple receiver configu­ rations, heat transfer fluids, tube materials, and operational conditions are to be probed. Purely analytical models are much more straightfor­ ward, offering an almost immediate estimate of maximum tube tem­ peratures, which may then be used to infer mechanical damage and heat flux limits (described further in Section 6). Despite their simplicity and usefulness in mechanical reliability studies, analytical models are un­ able to provide an accurate thermal profile over the full heat transfer surface, required in order to estimate receiver performance. Semiempirical models have found favour in recent literature, and may be viewed as a bridge between highly-accurate but computationally expensive numerical models, and simple but low resolution analytical models. Semi-empirical methods typically employ a finite number of discrete nodal elements over the receiver surface, from which tube temperatures and heat losses may be calculated through heat and mass balances, with the accuracy dependent on the adopted resolution and modelling approach. The semi-empirical method is similar in ways to numerical models in that the heat transfer surface is discretised into a mesh of localised elements, and generally also require a number of it­ erations to solve. However, this method differs from numerical ap­ proaches by employing empirical correlations and analytical theory to resolve thermal conditions over the receiver surface, rather than solving governing equations for mass, momentum, and energy over a large so­ lution domain. Fewer computational nodes and iterations-toconvergence (� 101 vs � 103 ) are required by semi-empirical models when compared to numerical codes, and as a result, can generate suf­ ficiently accurate results for receiver design and analysis purposes without suffering a large computational expense. Utilising these more simplified thermal models allows for a relatively fast calculation of receiver temperatures and efficiency, thus proving a very useful tool at the design stage for the receiver component in a CSP plant. This review article is largely concerned with thermal models that can be considered as semi-empirical in design.

Fig. 5. Schematic of receiver tube discretisation into axial and circumferential elements (zel ; θel ) for heat transfer modelling, detailing inside and outside temperatures (Tsi; zel ; θel , Tso; zel ; θel ) [61].

Fig. 5 [61]), with a full three-dimensional model defined as one which can calculate thermal conditions along the tube axis, around the circumference, and through the wall thickness. Models adopting a lower resolution approach may elect to model the average temperatures on a tube, or at a single axial location (two-dimensional over the tube cross-section) for use in mechanical reliability studies for example. Models that give consideration to temperature and heat flux varia­ tion around the circumference and along the axis of the exposed tube surface represent some of the most detailed found in literature, key to­ wards providing an accurate description of thermal and mechanical response. It is common in literature for the incident heat flux to be assumed as parallel to the receiver surface normal (collimated), with the variation in heat flux due to tube curvature then calculated using a cosine/sinusoidal approximation of the angular position away from the crown. Fig. 6a illustrates the circumferentially non-uniform heat flux profile generated using a cosine approximation [66]. Jianfeng et al. [67] presents a mathematical model used to investigate the thermal perfor­ mance of an isolated solar receiver tube carrying HITEC molten salt, subjected to an axially uniform incident heat flux, variable over one-half the tube circumference. The authors consider local tube temperatures and efficiency over the exposed tube surface, however pipe wall resis­ tance is omitted from the model. Boerema et al. [56] has developed heat transfer models to compare the performance of a number of liquid so­ dium receiver designs. Each receiver tube is discretised into elements in the axial direction, thus allowing the thermal model to solve flow rates and tube surface temperatures based on desired outlet temperature, fluid properties, ambient conditions, and receiver geometry. Local sur­ face temperatures are calculated using internal heat transfer coefficients in conjunction with the bulk fluid temperature and circumferentially �nchez et al. [68] describes an important varying heat flux. Rodríguez-Sa heat transfer model developed for cylindrical molten salt receivers, representing thermal conditions on the component by simulating a single tube on each panel, accounting for local temperatures and losses by discretising the tube into finite elements in the axial and circumfer­ �nchez et al. [69] adopts this model for ential directions. Rodríguez-Sa investigations into novel ‘bayonet’ receiver concepts, described in greater detail below. Two similar models presented in Higher-detail models (HTM and HHFM, described in Section 4.2.3) from

4.1. Modelling approach and boundary conditions The discussion of literature in this section is framed around the different approaches taken to model heat transfer and temperatures at a localised tube level (Section 4.1.1), the treatment of incident heat flux at a receiver level (Section 4.1.2), the calculation of environmental heat losses (Section 4.1.3), and the means in which the thermal profile is resolved (Section 4.1.4). A summary of heliostat aiming strategy designs is provided in Section 5, as these tools are used to model the heat flux distribution on the receiver, thus forming an important boundary con­ dition for thermal modelling. 4.1.1. Thermal modelling at the localised tube level In many of the modelling studies discussed in this section, thermal conditions on a receiver tube are resolved by performing local calcula­ tions of temperature development and heat losses. Localised modelling often involves discretising tubes into a number of finite elements (see 7

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tubes are discretised into axial elements, with temperatures assumed constant across the wall and around the circumference for each node. Pye et al. [83] investigates the energetic and exergetic performance of a number of different heat transfer fluids in a cylindrical receiver model by calculating the temperature development using discrete axial ele­ ments. A simplified model presented in Rodríguez-S� anchez et al. [57] involves calculating molten salt temperature development along the axis for a cylindrical receiver design, while ignoring circumferential varia­ tions (standard model, SM). The ASTRID code (Advanced Solar Tubular Receiver Design) developed by DLR is described by Frantz et al. [78]. A low detail modelling stage investigates a large receiver design space (variable heat flux levels, tube diameter, heat transfer area, number of tubes etc.) using a modelling procedure that discretises the tubes into a small number of axial elements. The low detail stage evaluates thermal, hydraulic, and economic performance based on desired temperature, pressure drop, and power targets in order to quickly identify and reject poorly performing designs for a high detail study, which involves more localised modelling of tube temperatures and heat flux distribution using an FEM model. Flesch et al. [84] incorporates a simplified thermal model into a heliostat aiming strategy procedure in order to provide a fast calculation of thermal performance, discretising tubes into elements in the axial direction only, assuming a constant circumferential heat flux on the front side of the receiver tubes. Fig. 6b illustrates the tube dis­ cretisation process from Flesch et al. [84], which assumes a constant heat flux on the front side. Such a simplification results in a convenient and relatively fast models that may provide general estimates of receiver thermal efficiency, however Ref. [57] shows that such models are un­ suited for the design optimisation and mechanical reliability studies as they do not consider the circumferential temperature gradient. There are examples in literature of receiver thermal models that effectively assume a mean fluid temperature along the length of the tube. This approach greatly simplifies modelling complexity, thus allowing for an efficient means of screening receiver designs, however it offers a poor representation of local wall temperatures. In the investi­ gation by Singer et al. [79], tube temperatures are modelled for a single tube on each panel of a cylindrical receiver design, with convection, radiation, and reflection losses considered. The panel model uses the average of the fluid inlet and outlet temperatures to calculate wall temperatures using internal heat transfer coefficients and one-dimensional wall conduction. Li et al. [58] adopts a similar low-resolution approach to that of Ref. [79], where the average inlet-outlet temperature is used to calculate wall surface temperatures from mean internal heat transfer coefficients and one-dimensional cross-wall conduction. Xu et al. [85] describes a similar model to that of Ref. [58], with application to a first and second law analysis of a molten salt cavity receiver. Thermal models presented by Kistler [86], Grossman et al. [87], Pacheco et al. [88], Kolb [88], Liao et al. [89], and Luo et al. [90] are used to investigate temperatures over a cross-section (two-dimensional, radial and circumferential profiles only) of an non-axisymmetrically heated tube for a specified fluid temperature, flow rate/convection co­ efficient, and incident heat flux. The estimated temperatures act as an input for mechanical investigations; this is described in greater detail in Section 6. The models employed in these studies may be considered as simplified analytical, as they require no energy/mass balancing pro­ cedures and only a few closed-form analytical expressions to solve. One-dimensional radial conduction is assumed across the wall, with heat flux variable around the circumference in all cases (cosine), although only inner and outer surface temperatures are extracted for mechanical modelling. Interesting results are presented in using these models with regards to mechanical reliability and allowable heat flux limits, however they are of no use for calculation of temperatures across the heat transfer surface, thus unable to provide an estimate of receiver thermal efficiency and power output. The models by Logie et al. [91,92] similarly calcu­ lates tube temperatures over a cross-section for specified boundary conditions, however environmental heat losses are considered, and the

Fig. 6. (a) Description of circumferentially non-uniform heat flux profile over a tube cross-section [66], and (b) Illustration of a discrete tube axial element incorporating an assumption of uniform circumferential heat flux [84].

�nchez et al. [57] discretise the tube into axial and Rodríguez-Sa circumferential elements for local wall temperature calculations. S� anchez-Gonz� alez et al. [70] employs the thermal model described by Ref. [68] to calculate allowable heat flux limits on a molten salt cylin­ drical receiver for use in an aiming strategy model. The analysis by �nchez et al. [71,72] models the heat transfer characteris­ Rodríguez-Sa tics of molten salt cylindrical receiver tubes subjected to a non-uniform heat flux distribution, again employing the model from Ref. [68]. Con­ roy et al. [61,66,73] discretises receiver tubes into elements in the axial and circumferential direction, establishing inner and outer surface temperatures for a variety of heat flux boundary conditions, fluids, and design configurations; recent modelling studies by Conroy et al. [74,75] also employs this model. Kim et al. [76] considers the axial and circumferential heat flux and temperature variation in an investigation into a variety of working fluids. A similar approach to tube discretisation can be applied to numerical models, such as in Refs. [77,78]; the present discussion is limited to semi-empirical and analytical models. For simplicity, certain models ignore the effects of circumferential temperature variation, instead providing general assessment of heat transfer conditions over the receiver surface. Singer et al. [79] employs this simplification in order to investigate a large number of receiver designs and heat transfer fluids, as does Li et al. [58] in a design opti­ misation study of a molten salt cavity receiver concept. Wagner et al. [80] presents a semi-empirical model for the Solar Two molten salt receiver that assumes a homogeneous flux over the tube circumference. Yu et al. [81] investigates the performance of a water/steam cavity receiver using a heat transfer model that approximates the thermal conditions of each panel by simulating a single tube, implementing a uniform circumferential heat flux. Boerema et al. [14] also assumes a constant heat flux over the circumference, however the fluid and wall temperature development along the tube axis is accounted for in the analysis which compares liquid sodium and molten salt receiver designs by discretising the tube into a number of axial elements. Chang et al. [82] conducted a parametric design analysis on a molten salt cavity receiver configuration using a numerical heat transfer model coupled with a Monte Carlo ray tracing (MCRT) technique. The cavity receiver 8

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temperature profile over the full circumference is input into a detailed thermal stress model in Ref. [92]. It is worth noting that the model described by Ref. [88] is one of few receiver thermal models in literature that makes reference to the local Nusselt number method presented by €rtner et al. [93], along with Refs. [66,73-75], which account for a Ga circumferential heat flux variation. The majority of authors tend to employ uniformly heated tube Nusselt number correlations, despite the presence of highly non-uniform thermal boundary conditions. The application of the local Nusselt number method as opposed to employing a uniformly heated tube Nusselt number can significantly affect the wall temperature profile, which largely underpins the mechanical integrity of the tube. However, this is considered more critical for liquid metals (Pr≪1) as opposed to more conventional fluids like water or molten salts (Pr � 101 ), where turbulent eddy transport plays a larger role in the heat transfer mechanism, and the local Nusselt number approaches the asymptotic value [93]. Receiver heat transfer models typically rely on an assumption of radial-dominant conduction across the tube wall (one-dimensional) thus dispensing with computationally expensive models that account for circumferential and axial wall conduction. An important investigation conducted by Marug� an-Cruz et al. [94] highlights the role that the relationship between tube material conductivity and internal heat transfer coefficients plays in this assumption. A conjugate heat transfer model is used to investigate heat conduction through a circumferentially non-uniformly heated tube, and internal forced convection to a heat transfer fluid. The investigation is concerned with evaluating the rela­ tive magnitudes of heat conduction in the circumferential and radial directions based on internal convection conditions and wall thermal conductivity, characterised by the Biot number ðBi ¼ hx =kÞ. The objective of the analysis from a thermal modelling standpoint is to identify a Bi threshold that indicates where conduction dominates in the radial direction over the circumferential coordinate. Bi > 0:3 implies radial-dominant conduction due to relatively large internal heat transfer coefficients, thus allowing for wall heat conduction modelling to be completed in the radial direction only (one-dimensional). For Bi < 0:3, the heat transfer fluid fails to remove heat sufficiently from the wall in comparison to heat travel over the tube material, meaning multi-dimensional wall conduction modelling must be completed if an accurate solution is to be obtained (> 5% error between analytical and numerical model from Ref. [94]), greatly increasing modelling complexity and computational expense for semi-empirical approaches. For turbulent molten salt and liquid metal flows encountered in receiver models presented in literature, Bi > 0:3 is generally encountered due to large heat transfer coefficients.

simulate the heat flux profile from a heliostat field over a number of novel billboard receiver concepts. A Gaussian heat flux profile is also implemented by Chang et al. [82], for a cavity receiver design. Both low and high detail models presented by Frantz et al. [78] account for a realistic heat flux distribution over all receiver tubes on a cylindrical design, relying on a ray-tracing developed profile. The model from Flesch et al. [84] is used to calculate tube temperatures on cylindrical receiver panels using the incident heat flux profile generated by a he­ liostat aiming strategy. Conroy et al. [73] demonstrates sodium receiver performance when subjected to non-uniform incident heat flux profiles generated by various aiming strategy configurations. The model in­ terpolates the heat flux map formed by the cumulative distribution of heliostat images distributed over the target in order to deliver tube temperatures for thermal and mechanical assessment. Conroy et al. [74] employs a similar aiming strategy as that from Ref. [73]. The analysis simulates receiver thermal response under cumulative heat flux profiles generated from different spillage settings assigned to the aiming strategy model, with the performance of sodium receiver designs and operating temperatures then compared based on input power from the field (spillage reciprocal), and allowable flux density on the receiver. Simi­ larly, Conroy et al. [75] utilises this aiming strategy for diurnal simu­ lations of sodium receiver and heliostat field interaction. Rodríguez-S� anchez et al. [71] considers two different modelling ap­ proaches: the coarse grid model (CGM), and fine grid model (FGM). The CGM has been utilised in studies by Refs. [57,68–70] whereby a single tube is modelled using the average heat flux on the panel (variable in the axial direction), and the calculated thermal profile assumed represen­ tative of all tubes on the panel. The FGM considers the incident heat flux on every tube on the receiver panel (using the aiming strategy described by S� anchez-Gonz� alez & Santana [96]), and can therefore offer a solution of significantly higher detail, but is three orders of magnitude more computationally expensive. The CGM proves a useful tool in estimating receiver performance, providing thermal efficiency predictions within 2:5% of the more detailed FGM. The authors do however acknowledge that the CGM is not appropriate for the investigation of receiver limi­ tations and design optimisation, as it fails to capture higher temperature excursions experienced by tubes exposed to higher-than-average heat fluxes on the panel, thus under-predicting thermal stresses required for mechanical reliability estimates and over-predicting the allowable heat flux. Variations in heat flux and surface temperature resolution using the CGM and FGM models is illustrated in Fig. 7. A brief review of aiming strategies is provided in Section 4, as these models can be used to generate the instantaneous heat flux profile on the receiver surface. 4.1.3. Calculation of heat losses Models that assume an absorbed heat flux do not require a calcula­ tion of thermal energy losses, however consideration of different loss modes plays a vital role in the calculation procedure of more detailed investigations. Heat losses considered by various authors include con­ vection, radiation, reflection, and conduction to a lesser extent. Con­ duction heat losses occur when thermal energy is conducted from high temperature receiver surfaces to lower temperature locations, particu­ larly through insulation material and support structures. Conduction typically accounts for a very small fraction of the total heat losses from a receiver [97], as the contact area between support structures and high temperature components is small and modern thermal insulation pos­ sesses very low conductivity values. It is therefore common for heat transfer models in literature to treat conduction losses as negligible with an adiabatic surface assumed at the back of the tube. Nevertheless, certain studies do incorporate this loss mode for completeness. The models presented by Refs. [58,82,85] all adopt a similar methodology, where the temperature of the insulation surface opposite the rear tube wall is first calculated, with a convective heat transfer coefficient then used to estimate heat losses to the environment. Radiative heat losses are generally a function of the assumed emis­ sivity of the tube coating, view factor, and surface temperature. The

4.1.2. Thermal modelling at the receiver level Few thermal modelling analyses exist in literature where tubes on the receiver are modelled using a realistic incident heat flux profile over the full heat transfer surface, accounting for tube-to-tube variation. Models concerned with simulating a single receiver tube, such as those from Refs. [14,58,61,66,67,76,79,83] simply assume an average heat flux over the tube length. For cylindrical receiver designs in particular, it is common practice to simulate a single receiver tube on each panel using an averaged heat flux, with the calculated thermal profile assumed similar for all tubes on the panel. This simplified approach is adopted by Refs. [57,68–70] for cylindrical receivers, with the influence of reflec­ tion and re-radiation between adjacent tubes considered. Wagner et al. [80] assumes all tubes on a cylindrical receiver panel share the same energy balance due to the resolution of the available flux map, with a single tube modelled using the temperature and heat flux variations over discrete elements along the tube axis. Yu et al. [81] and Yu et al. [95] model a single tube on panels of a cavity receiver. Of those models that do consider a non-uniform heat flux profile, the computational expense is increased as each tube on the receiver is modelled. Boerema et al. [56] uses a Gaussian-like distribution to 9

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Fig. 7. (a) Heat flux map and (b) corresponding tube surface temperatures over a cylindrical receiver design for CGM (one tube per-panel simulated) and FGM (all tubes per-panel simulated) [71].

reflective loss is influenced by the solar absorptivity of the tube surface and view factor, is independent of temperature, and often represents the largest heat loss mode on a receiver (� 5 10% efficiency reduction) [58]. Pyromark high temperature black paint is a selective solar absorber, used to coat receiver tubes in order to maximise absorptivity and minimise emissivity in the solar spectrum [98]. The coating is used on many conventional tubular designs, including the pioneering Solar One and Solar Two receivers in the 1980’s and 1990’s [23]. Several authors have elected to model receiver performance assuming the properties of this paint. Ho et al. [98] and Coventry & Burge [99] pro­ vide correlations and data for absorptivity and emissivity for variations in incidence angle, temperature, substrate, wavelength (specular and solar-weighted), and coating thickness of Pyromark 2500, however constant values are often assumed. View factor between receiver sur­ faces and environment also has a significant influence on the thermal profile calculation, however many authors simply assume full reflection and emission to the environment (Fview ¼ 1). Ignoring tube curvature effects and considering receiver surfaces as a whole, the receiver-environment view factor is effectively unity for flat billboard and convex cylindrical receiver designs, however with cavity receivers the concave box-like structure is designed to ‘trap’ incident sunlight by minimising the view factor between the heat transfer surface and envi­ ronment. Li et al. [58] and Xu et al. [85] use the ratio of aperture opening to heat transfer surface to estimate the receiver-environment view factor, while Refs. [81,95] also include the view angle between concave panels. The billboard receiver model by Boerema et al. [14] assumes an averaged value over the half tube circumference exposed to sunlight, Boerema et al. [56] uses a cosine function to vary the view factor with circumference, while Conroy et al. [61,66] applies a two-dimensional crossed-strings method. Chang et al. [82] describes in detail the view factor interaction between heat transfer surface ele­ ments, and environmental radiative and reflective heat losses using the Monte Carlo method. Refs. [57,68–72] use the two-dimensional cross­ ed-strings method to determine local view factors between tube sur­ faces, refractory backing, and environment for cylindrical designs, with tube-to-tube interaction incorporated in the model. The two-dimensional method establishes the view factor between circum­ ferential elements at complementary axial locations on neighbouring tubes, with Ref. [71] stating an almost negligible error committed by neglecting axial variations (three-dimensional view factor). Conroy et al. [73] describes a detailed three-dimensional view factor calculation for a model that considers non-uniform heat fluxes over the surface, used to compute reflection and re-radiation between discrete tube elements. For many thermal models that consider view factor, calculations of tube-to-tube reflection and re-radiation is simplified by calculating thermal losses to the environment directly using the view factor, rather than explicitly calculating the radiative and reflective energy exchange between tubular surfaces. This approach is valid for receiver models that

assume the same temperature and heat flux profile for all tubes on a surface, however such an assumption does not account for variations in local temperatures of neighbouring tubes due to light trapping effects. More detailed models may consider tube temperature variation due to view factor using a radiation balance between discrete elements on �nchez et al. [71] and Conroy neighbouring tubes, such as in Rodríguez-Sa et al. [73]. It is expected that net radiation exchanges between neigh­ bouring tubes will be negligible due to near identical thermal loading conditions, thus qualifying the use of a simplified view factor model in the energy balance [80]. Convection generally contends with radiation as the dominant heat loss mode after reflection [8]. Cavity receivers are expected to experi­ ence higher convection losses than radiation due to a larger heat transfer surface requirement, while the opposite is true for billboard and cylin­ drical designs [16]. Convective heat losses are a function of receiver temperatures and the convective heat transfer coefficient between the heated surface and environment. Nusselt number correlations control the heat transfer coefficient, and it may be forced, free, or mixed in nature, and has a dependency on ambient conditions and component geometry. Receiver thermal models assume either a fixed heat transfer coefficient for the calculation of convection losses, or use empirical models that vary the convective heat transfer coefficient based on ambient conditions (wind, film temperature), and geometry. Fixed heat transfer coefficients, typically in the range of 10 30 W=m2 K (free, mixed, forced), are assumed in models by Refs. [14,56,67,83,91,92]. In other studies, Nusselt number correlations based on environmental conditions and receiver geometry are used. Correlations developed through experimental and analytical work by Siebers et al. [100] and Siebers & Kraabel [101] are used by many receiver thermal models in literature. Experiments by Ref. [100] for horizontal flow over vertical flat plates allow for the development and identification of useful cor­ relations for billboard receiver panels (and by extension cavity designs), while Ref. [101] provides a theoretical framework specific to cylindrical and cavity configurations, with convective heat loss measurements taken by Stoddard [97] on the Solar One receiver in good agreement with theory developed for cylindrical designs. Theoretical models pre­ sented in Refs. [78,84] use average and local heat transfer coefficients devised from a recent CFD study by Uhlig et al. [102] for cylindrical designs. Refs. [81,95] use models developed by Clausing [103,104] for averaged convective heat transfer coefficients from cavity receivers. Potter et al. [105] passes surface temperature calculations from a receiver heat transfer model to a CFD solver to return accurate con­ vection coefficients for a cavity geometry. 4.1.4. Solution structure For a receiver subjected to an incident heat flux, thermal models can be set to iterate around the absorbed heat using an energy balancing procedure in conjunction with environmental heat losses. A 10

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convergence criterion used to define the point where an energy balance is reached; this is may be a very small value of the incident heat [66]. Internal flow conditions affect tube wall temperatures, which in turn influence heat losses, and finally calculations of receiver thermal power output. Therefore, an iterative energy balancing procedure allows the model to solve for receiver thermal performance based on a prescribed set of boundary conditions, such as receiver geometry, heat transfer fluid, inlet-outlet temperatures, and environmental conditions. Mass balancing techniques also play a key role in formulating a solution if the mass flow rate is not fixed, with receiver/tube flow rate iterated until the desired outlet temperature is obtained based on the absorbed power identified through the energy balance. Various energy balancing ap­ proaches are described for models developed by Jianfeng et al. [67], Li et al. [58], Wagner et al. [80], Xu et al. [85], Boerema et al. [14,56], Chang et al. [82], Rodríguez-S� anchez et al. [68] (and related models), Kim et al. [76] and Conroy et al. [66] (and related models). Closed-form analytical models are primarily concerned with establishing maximum tube temperatures for mechanical investigations (such as those from Refs. [60,86–90]), these therefore do not rely on energy or mass bal­ ances, rather operate on assumptions of absorbed heat flux and internal flow rate.

observation was that calculated receiver thermal efficiencies were considerably smaller (< 78%) than those reported previously for molten salt designs (80 90%); this is attributed to the inclusion of circum­ ferential temperature variation in the model. The STRAL thermal model described by Flesch et al. [84] is integrated into the aiming strategy procedure, with the calculated thermal power output used to modify the aiming point distribution while respecting pre-determined heat flux �nchez-Gonza �lez et al. [70] incorporates the model from limits. Sa Ref. [68] into an aiming strategy procedure, using the approximate thermal power output as a cost function. The model is also used to define allowable heat flux limits for tubes on a cylindrical molten salt receiver, based on corrosion and thermal stress limits at different fluid tempera­ tures for Alloy 800H tubes. The investigations conducted by Refs. [57, 71] are concerned with model resolution rather than uncovering receiver performance characteristics. 4.2.2. Novel receiver concepts A number of novel receiver concepts have been proposed in recent literature, with thermal models used to gauge the relative performance of the novel design against more conventional counterparts. Boerema et al. [56] compares a number of liquid sodium billboard receiver de­ signs to a conventional single pass (Fig. 4a). The concepts include a multi-pass configuration based on the billboard design investigated in the IEA/SSPS project [18], a configuration that employs tubes with variable diameters, and a concept using non-uniform mass flow rates in the tubes (shown in Fig. 8a–c). These novel designs have been devised in order to handle non-uniform heat flux distributions over the heat transfer surface more effectively, delivering more homogeneous outlet temperatures from parallel tubes. The new concepts outperform the conventional single diameter design in terms of maximising efficiency and lowering surface temperatures. The multi-pass configuration is the most promising as it minimises the surface area under high temperatures and augments tube velocity, with careful control of the flux distribution deemed a concern for operation of the variable diameter receiver, and the practicality of implementing a non-uniform mass flow rate into the standard billboard design requiring further investigation. For a given heat transfer area, the multi-pass design increases the flow path for the fluid, resulting in an increased tube flow rate/velocity relative to the single pass design, augmenting efficiency. Conroy et al. [74] compares the performance of a number of multi-pass geometries using sodium. A 1-pass, 3-pass, and 5-pass design are compared, with designs also varying in terms of tube diameter and material. Increasing the panel number and reducing the tube diameter raises thermal efficiency and allowable flux density, which facilitates a greater power output, how­ ever pressure drop also increases. The direction of flow through the panels also influences the temperature profile and power output. A similar mechanism of flow path increase for a given receiver area can also be achieved by increasing the geometrical aspect ratio, as demon­ strated by Conroy et al. [61]. A new concept is presented by Rodríguez-S� anchez et al. [69], with concentric tubes used to heat molten salt in a cylindrical external configuration. In the scheme of the ‘bayonet’ system, cold fluid enters the inner tube and then returns through the outer annulus, while the external tube wall is irradiated. The bayonet design exhibits promising results when compared to conventional molten salt receiver tubes [68], with thermal efficiency improvements of 2% and maximum wall tem­ perature reductions of > 100∘ C reported. A variable velocity (VVR) �n­ molten salt cylindrical receiver concept is proposed by Rodríguez-Sa chez et al. [72], with the thermal model presented by Ref. [68] used to investigate thermohydraulic performance. In a conventional cylindrical receiver, tubes on a panel are connected to a single inlet and outlet header, with the panels then connected in series to form a serpentine flow path. A high number of panels allows for a reduction in surface temperatures due to an increased flow velocity, resulting in better thermal and mechanical performance, but a large pressure drop due to an increased flow path - while the opposite is true for a low number of

4.2. Thermal modelling studies This section describes the nature of investigations undertaken by various authors using thermal models described in Section 4.1, detailing interesting results and findings. Receiver thermal models presented in literature have found application in simulation of a broad range of liquid tubular receiver designs and concepts; these are mostly conventional molten salt, however some analyses have investigated more novel and inventive configurations. Models that simulate a single tube are largely independent of boundary conditions specific to the receiver design as a whole, providing a more general assessment of thermal performance for a prescribed set of operating conditions. 4.2.1. Cylindrical receiver designs Conventional cylindrical receiver designs are investigated by Refs. [78,79,83] for a variety of heat transfer fluids, while Refs. [57,68,70,71, 84] consider molten salt working fluids. The designs considered in these studies are quite large, as cylindrical receivers must be sized appropri­ ately for surrounding heliostat fields, and molten salt heat flux limits must be respected. For example, the analysis by Rodríguez-S� anchez et al. [68] identified an optimum molten salt receiver design with 576 tubes, 10:5 m in length, and 0:0422 m in diameter, based on satisfaction of thermohydraulic and mechanical performance criteria. Such large heat transfer surfaces require a significant computational effort if thermal modelling is to be completed using a non-uniform heat flux in the axial and circumferential directions, as is the case for the majority of models described above. For this reason, studies by Refs. [57,68,70,79] have elected to model a single tube on each panel and assume the results for �nchez [71] highlights all tubes on the panel, however Rodríguez-Sa limitations with this low-resolution approach. Singer et al. [79] in­ vestigates the potential of a number of candidate working fluids in a cylindrical receiver for a 50 MWe power plant operating on the ultra-supercritical steam cycle, using LCOE as the objective function. A significant LCOE reduction (� 15%) can be obtained using liquid metals at high temperature as the receiver HTF when compared to molten salts. A recommendation for delivering dispatchable electricity from the high temperature ultra-supercritical steam cycle is to use a liquid metal as the receiver working fluid in conjunction with a separate thermal storage media. Variations in tube diameter, number of tubes per panel, and number of panels on a 360∘ cylindrical receiver design are explored by Rodríguez-S� anchez et al. [68]. Similar to other design optimisation studies, the authors recommend the use of small diameter tubes to maximise thermal and mechanical performance, however the resultant pressure drop penalty must also be considered. An important 11

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Fig. 8. Novel billboard receiver concepts; (a) non-uniform mass flow rate, (b) variable diameter, (c) multi-pass, from Ref. [56].

panels [68]. The VVR concept aims to emulate the thermal and hy­ draulic advantages of high and low panel numbers respectively, by accelerating localised flow velocity in high heat flux regions to reduce wall temperatures, and extending operating hours by maintaining tur­ bulent flow under low incident power levels. This is achieved using a complex system of valves and split headers that facilitate better localised flow velocity control in order to match variable incident heat flux dis­ tributions more appropriately and maintain temperatures within the safe operating envelope, without suffering a large pumping parasitic (see Fig. 9). Conroy et al. [61] investigates the heat transfer performance of a molten salt receiver with heat transfer enhancement techniques using empirical Nusselt number and friction factor correlations. Surface roughness mechanisms (ribbed surface, wire coil) are more effective at enhancing heat transfer in the turbulent regime than twisted tape structures, enhancing the net power output (absorbed heat minus pumping power) over a smooth tube design. The performance benefits of liquid metals over more conventional fluids in solar receivers are highlighted by numerous authors [59,62, 106,107]. However few works cited thus far have investigated the per­ formance of liquid metals in solar receivers using thermal models, and as there has been very little activity in this field since the 1980’s, in­ vestigations of this type are considered relatively novel. Nevertheless, the small number of studies that do exist do offer some useful insights.

The analysis by Singer et al. [79] demonstrates some of the critical ad­ vantages of liquid metals over molten salts in terms of thermal efficiency and cost reduction, namely large heat transfer coefficients which lower surface temperatures, and high temperature capabilities that can supply advanced power cycles. Boerema et al. [14] compared the performance of sodium and HITEC molten salt in their receiver model. Based on a fixed 1 MWth input and maximum pressure drop allowance of 0:5 bar, the optimum receiver size required to deliver a fixed outlet temperature is adjusted in an iterative procedure. The investigation found that a sodium design could be sized � 57% smaller than a molten salt unit for an equivalent thermal rating, due to significantly larger heat flux allowance. Pye et al. [83] demonstrates the superior performance of liquid sodium over more conventional fluids in a cylindrical receiver concept, both in terms of heat transfer performance and high tempera­ ture capabilities. The authors recommend its use as an effective receiver heat transfer fluid provided practical limitations can be overcome. The analysis conducted by Boerema et al. [56] studied the performance of liquid sodium in a number of novel receiver concepts. Logie et al. [91] compares the performance of liquid sodium to molten salt in a single tube. The large conductivity of sodium means that it is much less sen­ sitive to changes in flow rate, diameter, and wall thickness than molten salt in terms of efficiency (consistently > 90%). Kim et al. [76] in­ vestigates the optimum heat flux conditions over receiver designs

Fig. 9. Molten salt flow path for (d) a conventional cylindrical receiver, and (e) the variable velocity configuration Ref. [72]. 12

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carrying different working fluids. Allowable heat fluxes ranging from 1:1 0:4 MW=m2 are identified for a large sodium inlet-outlet temper­ ature of 550 750∘ C. Potter et al. [105] integrates optical and thermal models to simultaneously optimise the geometry of a small 1 MWth so­ dium cavity receiver operating from 500 700∘ C and an equator-facing heliostat field layout, objectively maximising thermal power output. The authors report a design point thermal efficiency of 91:6%, with a peak heat flux of 1:1 MW=m2 . Conroy et al. [66] investigates the influence of heat transfer area, tube diameter, and tube material, on the performance of sodium bill­ board receiver designs. The thermal, hydraulic, and mechanical per­ formance is investigated in the analysis, however some desirable characteristics are potentially conflicting. Thermal performance is enhanced considerably with a decrease in diameter (thermal efficiency > 90%, see Fig. 10), resulting in lower heat losses and encouraging larger incident heat fluxes (� 2 MW=m2 ), however this comes at the penalty of an increased pressure drop. Reducing the heat transfer area (increasing the solar concentration) minimises convective and radiative losses, however this increases temperatures and susceptibility to me­ chanical damage. LCOE valuation is used as the decision making tool in the multi-objective problem for the identification of optimum designs. Conroy et al. [61] compares the thermohydraulic performance of so­ dium, lead-bismuth, and molten salt in a billboard receiver concept. Both liquid metals permit considerably larger heat fluxes than molten salt (> 2 MW=m2 vs 1 MW=m2 ), sodium in particular, however the analysis also demonstrates potential performance gains through the application of heat transfer enhancement techniques to the molten salt fluid. An LCOE study demonstrates the excellent potential of sodium as a working fluid and storage medium for short term capacities (< 3 hour), with lead-bismuth, somewhat predictably, performing very poorly as a cost-effective thermal storage medium. Conroy et al. [73] models the performance of a sodium-cooled billboard receiver under heat flux profiles generated by an aiming strategy, highlighting the significance of optimising heliostat targeting to affect receiver efficiencies beyond > 90%. Logie et al. [92] demonstrates the superior performance of liquid sodium in comparison to molten salt with regards to lowering peak thermal stresses, referencing fluid conductivity as key to lowering tube temperature gradients. Conroy et al. [74] simulates the elevated tem­ perature performance of sodium-cooled receiver concepts when inte­ grated with next-generation power cycles (550 750∘ C - gaseous Brayton, SC/USC/A-USC steam Rankine). An optical-thermal-mechanical model compares receiver geometry, flow configuration, tube diameter, material, and outlet temperature. Based on solar-to-electric efficiency, receiver designs constructed of Ni-based superalloys and operating to 650 700∘ C yield a 4% power output improvement over the 550∘ C baseline case (conventional power block temperatures).

Water/steam and molten salts have been studied extensively, as these fluids are representative of those employed in almost all com­ mercial and test/demonstration facility. 4.2.3. Thermal model validation Numerous authors have sought to qualify the integrity of receiver thermal models through validation against experimental data or benchmarked models when possible. In many instances however, unique receiver designs/concepts are investigated using developed mathemat­ ical models, meaning pertinent experimental data is often unavailable for validation purposes. Pacheco et al. [23] has published experimental data from the pio­ neering Solar Two project, which has been intercepted by a number of authors for validation of molten salt receiver models. Jianfeng et al. [67] simulates the thermal efficiency of the Solar Two receiver at design point conditions using their model, reporting good agreement with measured values (� 85 89%). Liao et al. [60] computes the allowable heat flux for the Solar Two design. The calculated limit of 0:88 MW=m2 is higher than the reported design value of 0:8 MW=m2 , with the authors citing possible conservatism employed by designers as reasoning for the discrepancy. Frantz et al. [78] simulates the geometry and operational parameters of the Solar Two molten salt receiver using their ASTRID model, with calculations comparing well to reported efficiency values given in Ref. [23]. The thermal model from Rodríguez-S�anchez et al. [68], or variations thereof, has appeared in a number of significant publications that investigate the performance of molten salt cylindrical designs ([57,69–72]). When compared to published thermal efficiency values from the Solar Two project [23], the consideration of localised wall temperature variation in the model from Rodríguez-S� anchez et al. [68] results in a greater estimate of radiative losses, resulting in a considerably lower thermal efficiency calculation. Comparable thermal efficiency values are identified when the [68] model is simplified to effectively neglect the circumferential temperature variation however, resulting in an averaged wall temperature condition that affects lower �nchez et al. [108] subse­ radiative losses according to T 4 . Rodríguez-Sa quently revisited the means of thermal efficiency calculation for the Solar Two receiver described by Pacheco et al. [23] (Power-On Method). The authors highlight a key limitation with this method, which is the removal of the incident power from the calculation of thermal energy losses, with an assumption that receiver tube temperatures are equal to the arithmetic mean of the inlet and outlet temperature. Also, thermal resistance in the tube wall is diminished with respect to internal con­ vection, resulting in under-predicted circumferential temperature vari­ ations. A revised Power-On Method calculation from Ref. [108] yields thermal efficiency estimates that agree more favourably with the model from Ref. [68], showing that previously reported results from Pacheco et al. [23] are a reasonable first approximation (as acknowledged by Ref. [23]), but not wholly accurate as tube temperature and thermal loss estimates invoke the interpretation of incident power localised heat flux conditions. Li et al. [58] use their cavity receiver tube model to simulate the efficiency of a 5 MWth molten salt design from the SNL molten salt electric experiment (MSEE) [109], and compare to experimental results. The simulated values are close to reported thermal efficiencies of � 87:5%. Xu et al. [85] also compares a receiver model to experimental results of the MSEE [109], with predicted values close to the measure­ ments and results from Li et al. [58]. Yu et al. [81] cites a lack of experimental data from the DAHAN receiver at the time of writing, with model simulation results for steam parameters instead compared to design values. The simulation results compare well to the design point, however the authors cite the need for experimental data for a more robust validation. Boerema et al. [56] compares their sodium receiver model to the HOTREC model from Schiel & Geyer [18], which compares well to experimental data from the IEA/SSPS high flux sodium receiver tests. Simulations from the Ref. [56] model compares well to the

Fig. 10. Demonstration of tube diameter and heat transfer area influence on thermal efficiency [66]. 13

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HOTREC code in terms of sodium and tube temperatures, and thermal efficiency. Sodium receiver models developed by Conroy et al. [61,66, 73-75] are validated using experimental measurements from the IEA/SSPS high flux sodium receiver tests [17,18], and also temperature profiles from HOTREC model [110]. As well as comparison to Solar Two data, Frantz et al. [78] also compare their model to measured data from the SOLUGAS gas receiver, with tube temperatures in good agreement. Flesch et al. [83] compares calculations from the STRAL thermal model, which is used for aiming strategy optimisation, to both Solar Two thermal efficiency data and simulations from the more detailed ASTRID model [78], with a deviation of less than 1%. Few models described in literature are compared to more detailed numerical solutions, however such comparisons can provide useful insight into the merits of semi-empirical techniques where computa­ tional expense is traded against accuracy. An analysis by RodríguezS� anchez et al. [57] compares three simplified heat transfer models of a molten salt cylindrical receiver to more detailed, but computationally expensive, CFD simulations in ANSYS Fluent. The most straightforward model (standard model, SM) assumes an axial temperature variation only, with uniform circumferential temperatures. The remaining models offer a more detailed calculation of circumferentially varying wall temperature (described in Ref. [68]); one assumes a homogeneous temperature boundary condition (HTM), while the other assumes a homogeneous heat flux boundary condition (HHFM). Both HTM and HHFM rely on the same solution structure and tube discretisation pro­ cess, however their differences lie in the initial assumption of wall surface condition. The analysis compares the simplified models to CFD simulations for a single tube on a cylindrical receiver panel, investi­ gating variations in mass flow rate and environmental conditions for sensitivity purposes. The three simplified models compare well in terms of molten salt temperature evolution, however SM fails to capture the circumferential temperature variation which is necessary for

mechanical design. The HTM and HHFM models offer practically iden­ tical results across all cases considered, and are quite accurate when compared to the CFD simulation. The primary advantage of the simplified semi-empirical models is a reduced computational expendi­ ture than CFD; the simplified models utilise � 1000� less calculation elements, and takes 15 s to solve compared to 3 � 104 s. Such perfor­ mance benefits means a full receiver may be simulated in a relatively quick manner, thus allowing for greater flexibility when it comes to receiver design and optimisation studies. 5. Heliostat aiming strategies The interaction between the receiver component and heliostat field is critical for efficient and safe plant operation, as reflecting heliostats control the incident heat flux boundary condition. The cumulative effect of focussing heliostats on the receiver surface is an inhomogeneous heat flux profile that could lead to inefficient performance and significant thermal stresses at the location of peak heat fluxes and tube tempera­ tures. It has therefore been the pursuit of numerous authors to devise heliostat aiming strategies that will homogenise the heat flux over the receiver surface in order to maximise thermal performance and mitigate mechanical reliability issues. Almost every aiming strategy design in literature follows a procedure of focussing each heliostat in the field to a discrete number of aiming points on the surface in order to flatten the profile and lower peak heat fluxes. The means in which this is achieved varies between strategies, with a brief description of these works pro­ vided in Table 4. An aiming strategy is referred to as ’closed-loop’ if the model explicitly relies on the instantaneous receiver thermal and/or mechanical response for regulation (via measurement or thermo­ mechanical simulations). Some authors do not reference the use of such tools in their aiming strategy models, however it is important to note that these may be configured to acknowledge such limits (such as

Table 4 Key details of various heliostat aiming strategies. Reference

Field design

Receiver

Optical model

Optimisation approach

Notes

García-Martín et al. [114]

CESA-1, equatorfacing

CESA-1 volumetric air, flat

HELIOS

Dynamic control

Salom�e et al. [115]

THEMIS, equatorfacing Equator-facing

Flat

HFLCAL

TABU metaheuristic

Flat, CPV receiver

Equator-facing

Flat

STRAL raytracer HFLCAL

DAHAN, equatorfacing DAHAN, equatorfacing Surround

DAHAN, cavity

Ant colony optimisation metaheuristic Genetic algorithm metaheuristic TABU metaheuristic

Astolfi et al. [122]

Gemasolar, surround

Gemasolar, cylindrical

Gaussian curve fit to DELSOL

Multiple optimisation tools

S� anchez-Gonz� alez et al. [70] Flesch et al. [84]

Gemasolar, surround Redstone, surround Vast Solar, equator-facing CESA-1, equatorfacing Shouhang Dunhuang, surround NOOR III (like), surround NOOR III (like), surround

Cylindrical

Model from Ref. [96] STRAL raytracer HFLCAL

Aiming factor

Closed-loop, heliostat positions adjusted based on a feedback of receiver temperature measurements and predefined limits Minimises deviation between maximum and minimum heat flux while respecting spillage limits Closed-loop simulation, simplified CPV thermal model used to modulate aiming strategy Can successfully minimise peak heat flux, spillage limited by centralising peak flux in self-modifying algorithm Heliostats are grouped in order to minimise computational expense, employs a fixed spillage limit Multi-objective optimistation tools used to homogenise flux while minimising spillage Novel heat flux model presented, aiming factor method distributes images over the receiver based on an assumed beam radius fraction [121] Grouping heliostats according to field position and breaking into sub problems decreases computational expense Closed-loop simulation, allowable flux density data and thermal model used in optimisation procedure Closed-loop simulation, allowable flux density data and thermal model used in optimisation procedure Thermal and mechanical model used to model receiver response under yielded heat flux profiles Computational expense is minimised by grouping heliostats and solving multiple smaller problems Heliostat images alligned with receiver edge using assumed beam radius according to aiming factor

Belhomme et al. [116] Besarati et al. [117] Yu et al. [118] Wang et al. [119] S� anchez-Gonz� alez and Santana [96].

Conroy et al. [73] Gallego et al. [123] S� anchez-Gonz� alez et al. [124] Collado and Guallar [125] Collado and Guallar [126]

DAHAN, cavity Cylindrical

Cylindrical Vast Solar, billboard (flat) Flat Shouhang Dunhuang, cylindrical Cylindrical Cylindrical

Novel MCRT method Novel MCRT method [120] Novel analytical model

HFLCAL, UNIZAR Model from Ref. [96]

Non-dominated sorting genetic algorithm Aiming factor

Ant colony optimisation metaheuristic, Ref. [115] Simulated annealing metaheuristic Multiple optimisation tools Aiming factor

HFLCAL

Model from Ref. [96]

HFLCAL

Aiming factor, two parameter

14

Heliostat images aligned with receiver edge using assumed beam radius according to aiming factor New parameter increases homogeneity relative to method from Refs. [96,121,126]

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allowable flux density). A number of different optical modelling tech­ niques are employed in order to approximate radiative flux distributions from heliostats for use in aiming strategies. Further detail on these op­ tical models (numerical ray tracing, analytical etc.) and theory can be found in review articles by Garcia et al. [111], Bode & Gauch� e [112], and Li et al. [113]. 6. Mechanical modelling Securing structural integrity across the desired life of the plant is a major concern with solar receiver design. Solar receivers are designed with an objective of maximising thermal power output, however an additional thermal resistance between the concentrated sunlight and heat transfer fluid, in the form of a tube wall, limits possible design options and operational capabilities. Mechanical models are used to investigate the durability of receiver designs through a period of service, or to check whether stress/strain or heat flux limits are satisfied. Receiver mechanical models typically rely on a thermal model to calculate temperatures and wall heat flux conditions; there is therefore some crossover between literature described in the present section and in Section 4. Mechanical modelling literature is framed in this discussion by approaches adopted (Section 6.1), and also key findings uncovered by the various studies (Section 6.2).

Fig. 11. Schematic of a tube cross-section detailing pressure and temperature components influencing mechanical stresses.

�nchez et al. [68], Potter et al. [105], Nithya­ et al. [137], Rodríguez-Sa �nchez-Gonza �lez et al. nandam & Pitchumani [138], Kim et al. [76], Sa �nchez et al. [72]. Application of these theoretical [70], and Rodríguez-Sa models is deemed suitable if the magnitude of the radial temperature gradient (temperature difference across a heated wall) is significantly larger than that of the circumferential profile [68]. However, for instances where receiver tubes are subjected to large circumferentially inhomogeneous sunward-side heat fluxes, this may not be the case as the rear of the tube is at a much lower temperature than the crown, and more suitable theory need sought. Babcock & Wilcox [139] develops simplified one-dimensional thermal stress theory that assumes a cosine flux distribution over the front half of the tube, reliant on the maximum temperature difference across the tube thick­ ness and depth. Similar theory is attributed to Young & Budynas [130], and is applied in the thermomechanical investigations of Kistler [86], Grossman et al. [87], Kolb [89], Liao et al. [60], and Luo et al. [90]. In these studies, simplified thermal models (described in Section 4) are used to establish inner and outer surface temperatures at the tube crown from an assumed bulk fluid temperature, from which thermomechanical stress and strain components may be established over the tube cross-section and used to check for failure. A similar method of calcu­ lating thermal stresses for non-uniform circumferential temperatures is presented in Faupel & Fisher [129], however it is limited for the particular case of a thin-walled tube with a cosinusoidally varying temperature around the wall circumference. Kim et al. [76] employs equations from Hetnarski & Eslami [140] for a nonaxisymmetric tem­ perature distribution, accounting for the radial and circumferential thermal profile (Tðr;θÞ). The case of thermal stresses on a tube subjected to a combination of radial and circumferential temperature gradients is presented by Goodier [141], and is applied in the analysis by Conroy et al. [66]. The analysis requires the circumferential temperature vari­ ation to be represented by the average temperature and the first har­ monic of a Fourier series expression. The theoretical equations provided are used to calculate thermal stresses caused by the radial temperature difference across the tube thickness, and temperature profile around the circumference separately, with results superimposed to solve for the full thermal stress state over the wall cross-section (σ∝Tðr; θÞ). Although not explicitly concerned with establishing the mechanical reliability of receiver components, some important studies in literature have investigated the relative merits of different thermal stress models. Irfan & Chapman [127] compares analytical thermal stress equations from Timoshenko & Goodier [128], Goodier [141], and Young & Budynas [130], to finite element models for radiantly heated tubes

6.1. Modelling approach and boundary conditions The discussion of literature in this section is framed around the different models used to calculate thermal and pressure induced stresses on individual tubes (Section 6.1.1), and the methods in which me­ chanical reliability of the receiver through a period of service is un­ derstood (Section 6.1.2). 6.1.1. Thermal and pressure induced stress calculation Due to the non-uniform thermal conditions persistent on solar receiver tubes, it is common across mechanical investigations in litera­ ture to consider thermal stresses, while some analyses also include components of pressure induced stress over a tube cross-section. At a particular axial location, internal fluid pressures exert an equal hydro­ static force in all directions, meaning that pressure induced stresses vary in the radial direction in order to satisfy the laws of equilibrium for internal and external forces and moments. Theory for pressure induced stresses over a tube wall is generally well agreed upon in literature with regards to receiver mechanical modelling. The same level of agreement is not apparent for thermal stress theory however, largely due to differing assumptions of tube boundary conditions. Fig. 11 details the normal stress components (σr ; σθ ; σ z ) acting on a receiver tube wall in the presence of temperature ( Tsi ; Tso ) and pressure gradients (Pi ; Po ) over the inner and outer diameters (Di ; Do ). When non-uniform heating conditions persist, the presence of a temperature differential will result in differing thermal expansion con­ ditions that cannot be freely accommodated on a continuous body, inducing thermal stresses. This is therefore a critical consideration in receiver design as a non-uniform temperature profile prevails in the axial, radial and circumferential directions due to the unique heat transfer process across the tubes. Consideration of the temperature gradient in the radial direction is the most critical, with the analysis by Irfan & Chapman [127] demonstrating significantly larger stresses for a radial temperature drop than for an equivalent temperature change over the tube circumference. It is commonplace in recent literature to represent the thermal stress state on irradiated tubes using analytical thermoelastic stress equations presented by Timoshenko & Goodier [128], Faupel & Fisher [129], and Young & Budynas [130] for example, which only require the radial temperature gradient to compute stresses at a particular wall location (σ ∝TðrÞ). These equations are referenced in studies by Alzaharnah et al. [131], Yapici & Albayrak [132], Wang et al. [133], Fork et al. [134], Wang et al. [135], Neises et al. [136], Flores 15

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under differing boundary conditions, some comparable to those on receiver tubes. The analytical models compare well to numerical simu­ lations for their respective boundary conditions. From a solar receiver viewpoint, the analysis shows that thermal stresses are negligible for a non-uniform axial temperature distribution, and both radial and circumferential thermal gradients should be studied, as the combination could potentially be a significant source of stress. Marug� an-Cruz et al. [94] studies thermal stresses through a circumferentially non-uniformly heated thin-walled tube, using a conjugate convection-conduction heat transfer model that establishes tube temperatures under different Biot number conditions. Thermal stress equations from Faupel & Fisher [129] for a radial-only temperature drop (σ ∝TðrÞ) are compared to so­ lutions from Gatewood [142], which accounts for combined radial and circumferential temperature gradients (σ ∝Tðr;θÞ). Results of the analysis indicates that the equations from Faupel & Fisher [129] are insufficient when a non-uniform circumferential temperature profile persists, and that thermoelastic stress equations involving multi-dimensional thermal gradients should be employed if Bi < 10. More recently, Logie et al. [92] conducted a comparative analysis between simplified one-dimensional thermal stress calculations of Babcock & Wilcox [139] to a synthesised method compiling theory from different authors for thermal stresses where the temperature varies around the wall and through the thick­ ness, subsequently employed by Conroy et al. [73,74]. Contributions from Refs. [128,141,143,144]) [140] are unified in the development new analytical thermoelastic stress equations specific to the thermal boundary conditions encountered on receiver tubes, with thermal stress components at a particular wall position a function of the radial and circumferential temperature profile (σ∝Tðr; θÞ). The simplified one-dimensional method was deemed unsatisfactory when compared to the more detailed synthesised approach, largely due to the omission of the axial stress component. The authors place particular emphasis on tube axial stress, a component which is often ignored in similar thermal stress investigations.

Fig. 12. Sample of DNI data, exhibiting the intermittent nature of the solar resource [146].

loading condition, which incorporates a stress range at a certain tem­ perature. For each fatigue cycle type, the number of actual cycles the component will experience is assigned, and the number of allowable cycles is ascertained based on the equivalent strain magnitude, tem­ perature, and material; creep follows a similar structure. The fractional damage for each band is then calculated for each fatigue cycle type and creep loading condition by dividing the actual cycles/loads by the allowable, with the linear damage rule then summing together the contributions of all bands to arrive at a damage estimate [145]. Nar­ ayanan et al. [147] recommends using an interim solar receiver design code from Berman et al. [148] to calculate creep-fatigue of a molten salt receiver design, which is effectively a modified version of ASME B&PV Code Case N-47. The interim standard from Ref. [148] aims reduce the level of conservatism of Code Case N-47, which is ordinarily used to design nuclear components to a large factor of safety [136]. Kistler [86] and Grossman et al. [87] use rules of Code Case N-47 to investigate mechanical damage on receiver tubes. In both of these studies, discrete fatigue cycle types are used to represent the erratic thermal cycling conditions experienced by the component over time, however the effects of creep are ignored. Kistler [86] attributes this assumption to small structural and pressure loads, while Grossman et al. [87] highlights the short time periods spent at elevated temperature as reasoning to neglect a creep damage evaluation. It is worth noting that the mechanical models of Refs. [60,68,70,89,90] do not state the inclusion creep; however in some instances allowable stresses are[89 based on material limits corrected for long-term operation. An assumption of negligible may be justified for low internal fluid pressures of molten salt receivers (100 MPa), and also moderate fluid outlet temperatures (< 600∘ C) coupled with low allowable heat fluxes (< 1 MW=m2 ) that curtail ma­ terial temperatures and stresses. Direct water/steam receivers operates at considerably higher pressures (101 MPa), with larger wall thicknesses required [89]; creep may therefore be worthy of investigation in such cases. More recent analyses by Fork et al. [134], Neises et al. [136], Nithyanandam & Pitchumani [138], and Kim et al. [76] consider the mechanical reliability of next-generation receivers using high temper­ ature and pressure working fluids, such as air or sCO2 , which are pro­ posed to deliver higher outlet temperatures for Brayton power cycles (> 650∘ C), often operating at very large internal pressures (� 25 MPa). The consideration of creep damage as well as fatigue is therefore considered necessary due to excursions to very high temperatures and pressures. A slight modification to the rules of ASME B&PV Code: Sec­ tion III - Subsection NH are applied in the investigation by Fork et al. [134], with the code also employed by Conroy et al. [66,73]. The US

6.1.2. Mechanical reliability estimation As well as the thermal stress theory employed, literature concerned with thermomechanical modelling of solar receiver tubes also varies in terms of the approach used to evaluate receiver damage. Models pre­ sented by Pacheco et al. [88] and Flores et al. [137] investigate thermal stress/strain under specified operational conditions, rather than explore damage. Pacheco et al. [88] manipulates thermoelastic strain equations to demonstrate the influence of heat flux, fluid temperature, and heat transfer coefficient on strain at the crown. A simplified approach �nchez et al. [68], Sa �nchez-Gonza �lez et al. [70], adopted by Rodríguez-Sa �nchez et al. [72] relates the effective thermal stress to and Rodríguez-Sa material ultimate tensile strength in order to determine resistance against fatigue. A more detailed analysis of receiver reliability involves the use of the linear damage rule found in the ASME Boiler & Pressure Vessel Code: Section III - Subsection NH, earlier known as Code Case N-47 [145], which accounts for the accumulative nature of fatigue cy­ cles and creep loads over the operational lifetime of the receiver. The nature of CSP plant operation means that the receiver is exposed to large fluctuations in tube temperature due to cloud passages and diurnal cy­ cles; the cumulative effect of such cyclic thermal events has the potential to induce significant fatigue damage. Continuous operation of the receiver over a plant lifetime means that tubes experience a prolonged exposure to material stresses at high temperature, potentially facili­ tating mechanisms of creep deformation. A schematic of the variable DNI resource is shown in Fig. 12.. Due to the erratic nature of thermal cycling and stress loads on high temperature components, in particular solar receivers due to the vari­ ability in resource, the linear damage rule conveniently defines incon­ sistent fatigue cycles and creep loading conditions using more discrete bands. A band for fatigue damage is described as a fatigue cycle type, typically defined as a range of equivalent strain over a particular tem­ perature cycle, while a band for creep damage is described as a creep 16

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The selection of thermal stress theory and approach adopted for damage evaluation are crucial factors in estimating the structural integrity of a tubular solar receiver design. The maximum thermal stresses on receiver tubes are typically of the order 101 102 MPa, with values varying according to the thermal loading conditions, operating temperature, tube geometry and material, chosen stress model, bound­ ary conditions, and heat transfer fluid. With regards to thermal stresses, models from Babcock & Wilcox [139] and Young & Budynas [130] are one-dimensional in nature and do not consider the large axial stress component, shown to be inaccurate by Logie et al. [92]. Deployment of models from Timoshenko & Goodier [128] and Faupel & Fisher [129] that account for radial-only temperature gradients are most likely

DOE SunShot initiative sets a safety requirement of 10; 000 design point fatigue cycles, and 100; 000 hours for operation under design point creep loads for next-generation receivers [138,149], with a number of recent works following these criteria. Nithyanandam & Pitchumani [138] evaluates damage based on measured creep-fatigue data for design point creep loads and fatigue cycle equivalent strains. Neises et al. [136] references the use of the method devised by Berman et al. [148], again evaluating damage based on design point creep loads and fatigue cycle strains. Conroy et al. [74,75] employs 10,000 design point fatigue cycles and 100,000 creep hours as limiting criteria in the development of an allowable flux density model for high temperature sodium receivers. Table 5 Details of various receiver mechanical investigations. Reference

Receiver

Material

Thermal stress model

Analysis type

Notes

Narayanen et al. [147]

Cavity, molten salt

304 SS, 316 SS, Alloy 800H

Multi-dimensional, σ∝Tðr; θÞ, Ref. [151]

Creep-fatigue damage estimate

Kistler [86]

Cylindrical, sodium

316 SS

One-dimensional, Refs. [139,151]

Fatigue damage estimate

Modified version of ASME B&PV Code Case N47 employed, all materials satisfy creep and fatigue limits for specified operational conditions Negligible creep assumed, employs ASME B&PV Code Case N47, failure resisted across a full service life, 2� design point cycles recommended

Grossman et al. [87]

Cavity, molten salt

Alloy 800H

One-dimensional,

Fatigue damage estimate

Pacheco et al. [88]

Solar Two, molten salt Radiantly heated tube

316SS

One-dimensional, Ref. [130] see Notes

Tube strain calculation Thermal stress investigation

Kolb [89]

Solar Two, molten salt

One-dimensional, Ref. [130]

Fork et al. [134]

Air, high temperature

Alloy 800H, Inconel alloy 625, Haynes 230 Inconel alloy 617

Allowable flux density and material strain Creep-fatigue damage estimate

Flores et al. [137]

Molten salt tube

316L

Liao et al. [60]

Solar Two, multiple fluids considered

Neises et al. [136]

Irfan & Chapman [127]

–

σ∝TðrÞ, Ref. [139]

Multi-dimensional,

σ∝TðrÞ, Ref. [129]

Negligible creep assumed, employs ASME B&PV Code Case N47, relatively large damage reported for short term operation Model presented for tube strain calculation at the crown Analysis demonstrates the influence of the temperature gradient components on thermal stresses on radiantly heated tubes Employs a similar method as Ref. [86] to establish material flux and strain limits, Haynes 230 most suitable for higher temperature excursions Damage investigated using ASME B&PV Section III: Subsection NH and measured creep-fatigue data, creep more critical than fatigue Temperature and thermal stress of different receiver tube configurations are investigated

σ∝TðrÞ, Ref. [129]

Multi-dimensional,

Thermal stress investigation

Multiple material

One-dimensional, Ref. [130]

Allowable flux density

sCO2 , high temperature and pressure

Haynes 230

Multi-dimensional, σ∝TðrÞ, Ref. [128]

Creep-fatigue damage estimate

Rodríguez-S� anchez et al. [68]

Cylindrical, molten salt

Alloy 800H

Multi-dimensional,

Fatigue damage estimate

Luo et al. [90]

Direct steam receiver tubes Cavity, sodium

304H, Alloy 800H

One-dimensional, Ref. [130] Multi-dimensional σ∝TðrÞ[128],

Allowable flux density Design optimisation

Marug� an-Cruz et al. [94]

Radiantly heated tube, various Pr fluids

–

Multi-dimensional, Refs. [129,142]

Thermal stress investigation

Thermal stress theory from Ref. [129] (σ∝TðrÞ) deemed appropriate for high Bi numbers only, distribution over full cross-section otherwise recommended (σ∝Tðr; θÞ)

Nithyanandam & Pitchumani [138]

sCO2 , high temperature and pressure

Haynes 230

Multi-dimensional, σ∝TðrÞ, Ref. [128]

Creep-fatigue damage estimate

SunShot goals for creep and fatigue employed, interim code from Ref. [148], parametric analysis optimises thermal and mechanical performance

Kim et al. [76]

Molten salt, sodium, sCO2 tubes

Haynes 230

Multi-dimensional, σ∝Tðr; θÞ Refs. [140]

Design optimisation

Receiver geometry iterated in order to satisfy thermal and mechanical constraints

Conroy et al. [66]

Billboard, sodium

304 SS, 316 SS, Alloy 800H

Multi-dimensional,

Creep-fatigue damage estimate

Conroy et al. [73]

Vast Solar, sodium billboard

304 SS

Multi-dimensional,

Creep-fatigue damage estimate

Logie et al. [92]

Molten salt, sodium

316 SS

Multi-dimensional,

Thermal stress investigation

Conroy et al. [74]

Billboard, sodium

304 SS, 316 SS, Alloy 800H, Inconel 617, Haynes 230

Multi-dimensional,

Allowable flux density

Rules of ASME B&PV Section III: Subsection NH employed, parametric study optimises receiver design to achieve lifelong mechanical reliability Rules of ASME B&PV Section III: Subsection NH employed, mechanical investigation demonstrates importance of aiming strategy in affecting mechanical durability Investigates various thermal stress models in literature, composes a synthesised approach that considers the axial component, lower thermal stresses with sodium tubes than molten salt Allowable flux density model used to regulate incident power on a variety of receiver designs, Ni-based superalloys necessary for high temperature excursions

Potter et al. [105]

Haynes 230

σ∝TðrÞ, Ref. [127]

σ∝Tðr; θÞ, Ref. [141] σ∝Tðr; θÞ, Ref. [92] σ∝Tðr; θÞ, see Notes

σ∝Tðr; θÞ, Ref. [92]

17

Model presented for allowable flux density calculation, molten salt permits a considerably greater heat flux than water/steam, Alloy 800H highest performing material SunShot goals for creep and fatigue employed, interim code from Ref. [148], negligible fatigue damage, considerable creep contribution Fatigue failure is confirmed if the effective stress as a fraction of the material ultimate tensile strength exceeds 50% Follows on from the work from Refs. [60,89], allowable flux based on fatigue, creep assumed negligible Receiver and field geometry iterated, ASME material stresses used as the mechanical constraint

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Renewable and Sustainable Energy Reviews xxx (xxxx) xxx

unsuitable for highly non-uniform circumferential temperature profiles encountered by receiver tubes subjected to large heat fluxes [94]. Studies by Irfan & Chapman [127], Marug� an-Cruz et al. [94] and Logie et al. [92] provide some much needed clarification on thermal stress models specific to solar receiver tube boundary conditions going for­ ward, and should be consulted in the development of a mechanical analysis. There appears to be no agreed standard or method for evalu­ ating reliability upon investigation of the different mechanical models found in literature, which can largely be attributed to the relative nov­ elty of the technology. Methodologies used to estimate damage vary in conservatism depending on the design code applied, with some analyses neglecting creep altogether, while more simplified models use a pass/­ fail criterion based on material stress/strain limits. Recent literature indicates a trend towards use of the ASME B&PV Code [145] (or a modified version thereof, according to Ref. [148]) in order to provide a comprehensive mechanical reliability prediction for high temperature designs. Perhaps given the lack of agreement on the most appropriate method to evaluate the mechanical reliability of a solar receiver, sub­ sequent analyses should involve thermal stress theory best suited to the boundary conditions of a solar receiver tube [92]. Also, guidelines of a well established industry standard such as the ASME B&PV Code should be followed when possible (material qualified etc.), albeit considered somewhat conservative for solar applications. Key details of various mechanical modelling studies are summarised in Table 5.

ultimate tensile strength, however none of the designs investigated violate this limit when using thermal stress theory from Young & Budynas [130]. S� anchez-Gonz� alez et al. [70] again uses thermal stress theory from Ref. [130] to investigate the allowable flux limits of Alloy 800H receiver tubes at different temperatures. The allowable flux limit from a thermal stress standpoint is based on the ultimate tensile strength of the material and temperature, with mechanical results combined with those of a molten salt corrosion study to generate a database for consultation of an aiming strategy. Maximum allowable heat fluxes remain relatively stable between 0:8 1:0 MW=m2 across the receivers operational temperature range of 290 565∘ C for thermal stresses, however a strong dependence on corrosion with temperature and mass flow rate significantly lowers the heat flux limit to 0:3 0:4 MW=m2 with salt temperature evolution. Liao et al. [60] evaluates the perfor­ mance differences between molten salt and water/steam working fluids for a number of tube materials in terms of allowable heat flux density at design point conditions (290 565∘ C). The method in determining the maximum allowable heat flux is similar to that of Kolb [89], relating calculated strain at the crown to material fatigue data. The results of the analysis compared well with peak heat flux data for Solar Two, with a 316 Stainless Steel tube carrying molten salt allowing 0:88 MW=m2 , Alloy 800H permits 1:0 MW=m2 , while the allowable heat flux for water/steam working fluids are less than 0:4 MW=m2 regardless of material. Small diameter, thin walled tubes of a high strength material are recommended for receiver construction to maximise the allowable heat flux. Luo et al. [90] employs a similar model to that of [60] to investigate the allowable flux density of direct steam receiver con­ structed of 304 Stainless Steel (superheater) and Alloy 800H (boiler), demonstrating the role of steam pressure, velocity, and tube diameter. Potter et al. [105] employs allowable stress limits for Haynes 230 from ASME B&PV Code: Section II, Part D, in a study that optimises the ge­ ometry of a sodium cavity receiver and heliostat field. Kim et al. [76] employs allowable stress limits for Haynes 230 from ASME B&PV Code: Section II, Part D in a mechanical model that considers heat flux con­ ditions over tubular receivers carrying molten salt, liquid sodium, and sC02 . The analysis demonstrates the variation in allowable heat flux with bulk temperature development in the receiver, with different inlet-outlet temperature combinations considered. Conroy et al. [74,75] integrates the allowable flux density tool with an aiming strategy model to regulate incident heat flux and power levels on a variety of sodium receiver designs. Ni-based superalloys Inconel 617 and Haynes 230 demonstrate far greater allowable heat fluxes across the temperature range of interest (inlet-outlet) than more conventional alloys, namely; 304 Stainless Steel, 316 Stainless Steel, and Alloy 800H. The Ni-based superalloys therefore facilitate much higher incident heat fluxes and power levels when operating at high temperatures, allowing for superior power output.

6.2. Mechanical modelling studies The following section details some important results and findings of the mechanical models introduced in Section 6.1. The differing ap­ proaches adopted by the various authors means that the method by which mechanical performance results are represented also differ, with authors electing to demonstrate mechanical performance in a manner using allowable stress, strain, or heat flux limits, or through a pass/fail criterion determined using creep-fatigue damage. 6.2.1. Allowable stress/heat flux limits Explorations of maximum allowable heat fluxes and stress/strain limits on receiver tube materials are relatively common in literature, with violation of these limits deemed unacceptable for safe operation of the component across the desired plant lifetime. Kolb [89] applied the cumulative fatigue damage rule [145] to investigate the allowable strain range for Alloy 800H, Inconel 625, and Haynes 230 materials at high temperatures, using material data from different sources. The allowable strain range is defined as the maximum tolerated by the material in order to resist failure for a combination of cycles, with strain magnitudes and frequency devised by Kistler [86] used in the investigation. The maximum allowable heat flux for a molten salt receiver design is then derived based on calculated tube strains and the allowable strain range for each material. The model relies on heat flux and fluid temperature inputs, from which maximum wall temperatures and strain may be calculated using simplified thermoelastic strain equations from Ref. [130]. The model is also used to evaluate appropriate tube di­ ameters and wall thickness based on fluid pressures and salt induced corrosion. Inconel 625 exhibits the lowest material strains of the three candidates, however is only permitted for operation by the design code to 565∘ C [145]. Haynes 230 has an acceptable strain range and is capable of operation to 650∘ C. Alloy 800H has the lowest allowable strain range of the candidates, marginally delivering on a 30 year desired life for a maximum temperature of 600∘ C due to a larger wall thickness required to cope with internal pressures and corrosion. Vari­ ations in tube diameter and panel number were explored for a cylin­ �nchez et al. [68]. drical molten salt receiver design by Rodríguez-Sa Alloy 800H is chosen for construction of the receiver tubes, as it has a higher film temperature limit with molten salt than other candidate alloys (650∘ C) [150]. Receiver fatigue failure is confirmed if the maximum effective thermal stress value exceeds 40% of the material

6.2.2. Creep-fatigue damage Different investigations uncover the mechanical reliability of solar receivers using design codes, such as ASME B&PV Code Section III: Subsection NH [145], modifications to Code Case N-47, and the interim design code from Berman et al. [148]. Narayanan et al. [147] in­ vestigates the creep-fatigue resistance of molten salt cavity receiver tubes, constructed of 304 and 316 Stainless Steels, and Alloy 800H materials on different panels. The analysis assumes 11; 000 diurnal and 19; 000 cloud cycles of the same severity over a 30 year plant lifetime, and 100; 000 hours total operational time at design point. The analysis predicts that all panels on the receiver will survive a lifetime of service, however the linear damage rule indicates that the 304 Stainless Steel panels are the most susceptible to damage, followed by 316 Stainless Steel, while Alloy 800H has the greatest resistance to creep-fatigue. Kistler [86] reduces real weather data into thermal cycle definitions in order to assess the fatigue life of a cylindrical liquid sodium receiver undergoing cloud and diurnal events, using rules of Code Case N-47 18

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Renewable and Sustainable Energy Reviews xxx (xxxx) xxx

(minus creep), and theory from Ref. [139] (inlelastic strain calculated relating analytical model to Ref. [151]). Results show that only 1= 3rd of the allowable fatigue damage is used up through the life of the receiver, when using a 316 Stainless Steel construction and maximum tempera­ tures of 590∘ C. Based on these findings, the authors recommend designing the receiver component by multiplying the maximum number of design point fatigue cycles by two. A similar methodology is employed by Grossman et al. [87] to investigate damage of a scaled-down molten salt receiver constructed of Alloy 800H tubes, operated over 264 hr at SNL to maximum temperatures of 566∘ C. The authors describe the use of the rain-flow counting algorithm used to reduce solar insolation data into thermal cycle definitions for use in the mechanical damage estimation, applying rules of Code Case N-47, again ignoring the effects of creep. Using the linear damage rule, a cumulative fatigue damage fraction that amounts to 37:5% of the allowable thermal fatigue cycles (of all types) before failure was estimated. This level of damage is significant given the relatively short operational time of the component, and is largely attributed to poor heat flux homogeneity on the receiver. The creep-fatigue life of a pressurised-air solar receiver with Alloy 617 tubes is modelled by Fork et al. [134] through two methods: (1) through rules of the ASME B&PV Code Section III: Subsection NH [145], and (2) using measured creep and fatigue data. The authors model the mechanical response of a receiver tube under conditions described as steady, which is similar to fossil fuel powered operation, sunny, which accounts for diurnal events only, and cloudy, which models both diurnal and cloud cycles. Fatigue damage is not a significant factor for steady operation, which was found to be less damaging than all sunny and cloudy simulations. In almost all sunny and cloudy cases simulated, fatigue damage was also found negligible in comparison to creep. A significant finding from the analysis is that large variations exist be­ tween both prediction methods, particularly for creep damage evalua­ tion, with the ASME B&PV method deemed more conservative. An investigation into the creep-fatigue damage of high pressure ð25 MPaÞ, high temperature sCO2 receiver constructed of Haynes 230 alloy is completed by Neises et al. [136]. Tube temperatures are established using a commercial thermodynamic solver, operating to maximum fluid temperatures of 650∘ C. The mechanical analysis is conducted using simplified design rules of Code Case N-47 from Ref. [148], with 10; 000 design point fatigue cycles, and 100; 000 design point creep hours considered. Thermal stresses are calculated using theory for a radial gradient only from Ref. [128]; this is deemed appropriate given the analysis assumes a uniform circumferential heat flux. The thermo­ mechanical model is also used to identify the maximum allowable heat flux along the tube axis based on fluid temperature, wall thickness, and internal pressure, with heat fluxes of � 0:6 MW=m2 permitted for lower fluid temperatures of 470∘ C, decreasing to � 0:1 MW=m2 for 650∘ C. Results show that the high temperatures and pressures drive significant creep damage, but negligible fatigue, for the failure criteria considered. The authors acknowledge that the assumption of uniform circumferen­ tial wall temperatures could lead to an under-estimation of damage. A creep-fatigue analysis of asC02 receiver tube constructed of Haynes 230 is conducted by Nithyanandam & Pitchumani [138] for high pres­ sure ð� 25 MPaÞ and high temperature ð> 650∘ CÞ operation. A numer­ ical model is used to establish the coupled flow-thermal conditions inside the tube, with the established temperature profile and internal fluid pressure used in the mechanical investigation. Creep and fatigue damage is established using empirical material data, with 10; 000 design point fatigue cycles, and 100; 000 operational hours at design point temperatures for creep set as a criteria for mechanical reliability. A parametric analysis is conducted by varying tube diameter, wall thick­ ness, mass flow rate, and irradiance, revealing optimised design and operational conditions that will maximise thermal efficiency, while retaining mechanical integrity. In an investigation into creep-fatigue damage on sodium-cooled receiver designs, Conroy et al. [66] reduces

real weather data into cloud/diurnal cycle types and frequency, and also time periods spent at a particular DNI level using the rain-flow counting technique. Alloy 800H exhibits greater resistance to creep-fatigue damage than both 304 and 316 Stainless Steel grades, with the anal­ ysis demonstrating a strong dependency on tube diameter selection and heat flux concentration on reliability. The large CAPEX of Alloy 800H is detrimental to its economic competitiveness when compared to the more conventional stainless steels, with both 304 and 316 grades per­ forming reasonably well given their moderate cost. Similar to the in­ vestigations above, creep is found to be more critical than fatigue due to the large temperatures and heat fluxes (driving stress) encountered on receiver cooled by liquid metals. Conroy et al. [73] adopts a similar approach, investigating the creep-fatigue reliability of a single billboard receiver design constructed of 304 Stainless Steel when subjected to a variety of non-uniform heat flux distributions from an aiming strategy. The analysis demonstrates the significance of parameters such as spillage constraint and aiming point distribution in ensuring life-long durability. 7. Conclusions The prominence of liquid tubular receiver designs in CSP tower systems is clear, despite the emergence of more novel gaseous and particle based concepts in recent years. This is evidenced by their overwhelming deployment in small and commercial-scale tower pro­ jects, and the significant number of modelling efforts devoted to uncovering the thermal performance and mechanical durability of these systems. Thermal and mechanical models have proven to be useful tools at the design stage of the receiver, where multiple designs, working media, materials, and operational scenarios are considered. This review article offers a discussion on semi-empirical and analytical techniques used for thermal and mechanical modelling of tubular solar receivers. Numerical methods are largely beyond the scope of the present work, however a comprehensive review of these techniques would also be of benefit to the CSP community. 7.1. Thermal modelling Simplified two-dimensional heat transfer models are suitable for obtaining a thermal profile of the tube cross-section given specified operational conditions. Such analytical models can be useful for general mechanical reliability studies, however these are of little use in estab­ lishing an accurate thermal profile for the receiver component, with three-dimensional semi-empirical models (radial, circumferential, axial) instead required to solve energy and mass balances based on the incident heat flux profile. These models can resolve temperatures and heat losses across the full component in conjunction with a realistic heat flux map, thus accounting for complex interactions between the receiver and he­ liostat field, and providing more accurate flow rate and thermal effi­ ciency calculations. The semi-empirical model can be simplified by modelling heat transfer on a single tube using an idealised heat flux profile, significantly diminishing the computational requirement at the expense of full component accuracy. The flexibility and relatively inexpensive computational requirements facilitated through the semiempirical method has elevated these models to the methods-of-choice in recent literature, with an increasing number of authors offering different approaches and investigations. It is expected that these methods will continue to grow in popularity in line with the increasing deployment of tubular concepts, and will play an important role in the design and analysis of future designs. 7.2. Mechanical modelling Mechanical reliability studies usually incorporate analytical theory for thermal and pressure induced stresses, in conjunction with a design standard or material data. Literature suggests that the selection of 19

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Renewable and Sustainable Energy Reviews xxx (xxxx) xxx

thermal stress theory is somewhat contentious, with model preference dependent on assumptions related to the tube thermal profile. Thermal stress models presented for solar receiver tubes often account for stresses due to temperature drops across the wall only (radial gradient), with others also including the contribution of the non-uniform circumferen­ tial profile. Studies explicitly concerned with the selection of thermal stress theory have signalled limitations with various approaches, and should be consulted in the development of future mechanical models. There are also numerous means of establishing component reliability, such as investigating creep-fatigue through anticipated periods of ser­ vice, or by identifying maximum allowable heat flux and stress/strain limits. Recently, investigations into creep-fatigue damage indicate a significant contribution from creep, in particular for tubes subjected to large temperatures and pressures, however authors concerned with more conventional receiver designs and working fluids generally elect to model fatigue damage alone. Large levels of conservatism associated with design codes warrants the development of a code more specific to solar receiver boundary conditions in order to avoid overly-safe designs.

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