Wavy-tape insert designed for managing highly concentrated solar energy on absorber tube of parabolic trough receiver

Accepted Manuscript Wavy-tape insert designed for managing highly concentrated solar energy on absorber tube of parabolic trough receiver

Xiaowei Zhu, Lei Zhu, Jingquan Zhao PII:

S0360-5442(17)31679-1

DOI:

10.1016/j.energy.2017.10.010

Reference:

EGY 11652

To appear in:

Energy

Received Date:

15 December 2016

Revised Date:

30 September 2017

Accepted Date:

03 October 2017

Please cite this article as: Xiaowei Zhu, Lei Zhu, Jingquan Zhao, Wavy-tape insert designed for managing highly concentrated solar energy on absorber tube of parabolic trough receiver, Energy (2017), doi: 10.1016/j.energy.2017.10.010

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Highlights 1. A parabolic trough receiver equipped with a wavy-tape insert is investigated numerically. 2. Wavy-tape insert leads to highly localized heat transfer enhancement effects. 3. Highly concentrated solar energy can be carried away more efficiently by using wavy-tape insert. 4. Wavy-tape insert benefits to the decreases of entropy generation rate, heat loss and structure stress. 5. Wavy-tape insert is more effective when operating at relatively lower working fluid flow rate.

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Title: Wavy-tape insert designed for managing highly concentrated solar energy on absorber tube of parabolic

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trough receiver

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Authors: Xiaowei Zhu, Lei Zhu, Jingquan Zhao*.

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Affiliation: School of Aeronautic Science and Engineering, Beihang University, Beijing, China, 100191.

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First author: Xiaowei Zhu (E-mail: [email protected])

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Corresponding author: Jingquan Zhao

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E-mail: [email protected]; [email protected]

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Address: XueYuan Road No.37, Haidian District, Beijing, China, 100191

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Tel: +8610 82338188

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Tax: +8610 82338600

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Wavy-tape insert designed for managing highly concentrated solar

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energy on absorber tube of parabolic trough receiver

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Xiaowei Zhu, Lei Zhu and Jingquan Zhao*

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School of Aeronautic Science and Engineering, Beihang University, Beijing, China, 100191.

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Abstract

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In this paper, a swirl flow generator, namely wavy-tape insert, is tailored for the parabolic trough receiver (PTR) to

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improve its performances by mean of enhancing the heat transfer inside the absorber tube. A comprehensive

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computational fluid dynamics model is established to study the flow and heat transfer inside the full-size PTR

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equipped with wavy-tape insert. It is found that wavy-tape provokes highly-localized heat transfer enhancement

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effects, which exactly aim at the highly-concentrated solar heat load. Consequently, both the tube temperature and

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the heat loss are reduced effectively. A static mechanical analysis is performed to evaluate the thermal stress and

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deformation of absorber tube, both of which decrease in the presence of wavy-tape. However, the wavy-tape

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improves the PTR thermal-mechanical performances at the expense of increased pressure loss penalty. The overall

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effects of wavy-tape insert on PTR are evaluated based on different criterions. For a same flow rate, the total entropy

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generation rate can be reduced significantly by using wavy-tape insert. For an identical pumping power

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consumption, the insert not only reduces the thermal stress and the heat loss, but also gives raises to the specific

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enthalpy gain of the working fluid.

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Keywords: parabolic trough receiver; wavy-tape insert; computational fluid dynamics; heat transfer enhancement;

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thermal stress; heat loss. Nomenclature

𝐴𝑡

amplitude of wavy-tape, mm

𝛼

thermal diffusivity, m2 s-1

𝑐𝑝

specific heat, J kg-1 K-1

𝛼𝑇

turbulent thermal diffusivity, m2 s-1

𝐷

diameter, mm

𝛿

thickness, mm

𝑓

friction factor, = ∆𝑃 (0.5𝜌𝑢2)

𝜀

turbulent dissipation rate, m2 s-3

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𝑘

thermal conductivity, W m-1 K-1

δ

total structure deformation, m

𝐿

total length of PTR, mm

𝜃

angle of circumference, °

Nusselt number, = 𝑞𝐷(𝑇𝑤 ‒ 𝑇𝑗) ‒ 1

𝜇

viscosity, Pa s

pitch of waviness, mm

ξ

reduce-rate

pumping power consumption, = 𝑉r

𝜌

density, kg m-3

σ

Von-Mises stress, Pa

𝑁𝑢

𝑘‒1 𝑃𝑡 𝑃𝑤

∆𝑃, W 𝑃𝐸𝐶

heat flux, W m-2

𝛥𝐻

gain in specific enthalpy, J/kg

heat loss, W

𝛥𝑃

total pressure drop, Pa

𝑅𝑒

Reynolds number, = 𝜌𝑢𝑖𝑛𝐷 𝜇

𝛺

control volume, m3

𝑆𝑔𝑒𝑛

entropy generation rate, W K-1

Subscripts

𝑡𝑔𝑙𝑎𝑠𝑠

thickness of glass envelop, m

avg

𝑞 𝑄𝑙𝑜𝑠𝑠

𝑇 𝑢,𝑣,𝑤

temperature, K

circumferentially average

cir

velocity vectors, m s-1

area-weighted average

glass

glass envelop

𝑉

total volume of the tube, m3

𝑉𝑟

volumetric flow rate, L s-1

max

maximum

𝑊𝑡

width of wavy-tape, mm

tube

tube

𝑥,𝑦,𝑧 30

performance evaluation criteria factor

coordinate axis

H

W

case with hollow absorber tube

case with wavy-tape

1. Introduction

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Parabolic trough power plants are currently one of the most prevalent techniques for solar energy harvesting. In

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such systems, solar radiation is concentrated on an absorber tube by the optical focus of parabolic trough reflector

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plate, such that high-density solar energy can be absorbed by the heat transfer fluid (HTF) inside the tube, and

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thereby efficiently converted into usable power. Notably, the parabolic trough receiver (PTR) plays an important

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role in the whole system since its performance directly determines the energy collecting efficiency. PTR is an

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elaborate component which requires considerable large costs. The survey of Mokheimer et al. [1] showed that the

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PTR accounts for about 28.3% share of the system’s total cost. Therefore, the development of PTR is of great

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importance to the parabolic trough power plant. Numerous studies have been dedicated to optimize/improve the

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PTR performances from various aspects either by developing new structural materials, coatings and HTFs or by

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optimizing the optical and thermal designs.

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The PTR always encounters severe operating condition. Generally, the PTR has to work at elevated temperatures

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sometimes higher than 400 °C for long duration. Meanwhile, the heat load imposed on absorber tube usually shows

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high non-uniformity, which leads to large temperature gradient in tube structure resulting in large thermal stress.

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These factors may have adverse effects on the adhesion of deposed coating and on the service life of PTR. From

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the thermodynamic point of view, it is a practicable strategy to improve PTR thermal and mechanical performances

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by applying certain heat transfer enhancement techniques (HTET) to the absorber tube. By using the HTET, heat

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transfer between absorber tube and HTF can be enhanced, which is expect to reduce the tube structure temperature

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and thus decreasing the heat loss and the thermal stress.

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A recently emergent interest on this topic is to probe the practicability of using nanofluid to enhance the thermal

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performance of PTR. Sokhansefata et al. [2], Kasaeian et al. [3], Zadeh et al. [4] and Mwesigye et al. [5, 6] adopted

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various nanofluids to enhance the heat transfer inside the absorber tube taking advantages of the relatively higher

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thermal conductivity of nanofluids [2-6]. For instance, Mwesigye et al. [5] reported a 76% enhancement in heat

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transfer coefficient by using oil–Al2O3 nanofluid with 8% nanoparticle concentration. However, it was also found

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that the flow resistance enlargement due to the nanoparticle addition was remarkable particularly at relatively high

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flow rate. When the Reynolds number exceeded a certain threshold value, nanofluids even showed inferior overall

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performance compared with pure base fluid. In addition, nanofluid technologies are still in developing stage and

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not ready for widespread application in engineering fields [7]. In contrast, conventionally mature tube HTETs such

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as adding internal fins or inserts, or reconstructing tube configurations seem to be much more applicable to the PTR.

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If the HTETs can be properly deployed to absorber tube, the performance of PTR is expected to be greatly improved.

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Based on the literature survey, the state-of-art studies [8-24] on this topic are chronologically summarized in Table

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1. Most of these works emerged in recent years. It can be seen that many researchers preferred to apply the porous

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mediums or tape inserts to the PTR absorber tube to enhance the heat transfer because these methods are in passive

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form and easy to be implemented.

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For PTR, the solar irradiation is usually concentrated on optically focused positions. The corresponding surface

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zones are normally heated to a very high temperature, and thus resulting in large heat loss and thermal

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expansion/stress in these positions. Therefore, ideally, the absorber tube of PTR requires a customized HTET, which

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should better aim to those heat concentrated zones, while the rest zones of the tube experienced with lower heat

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density actually does not need the aid of HTET due to the fact that HTET inevitably causes the increase of pressure

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loss. So technically, it is expected that the goal of reducing surface temperature and temperature gradient on

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absorber tube can be achieved at the expense of a minimum flow resistance penalty if the HTET can be deployed

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on the tube rationally. For instance, Cheng et al. [12] and Wang et al. [15] placed their heat transfer enhancement

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elements on the intensively heated side of the tube. Both of them obtained advanced PTR performances.

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Nevertheless, other pipe HTETs like concentric twisted-tape inserts or nanofluids generally produce a full tube heat

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enhancement effect, which may add redundant flow resistance due to the unnecessary heat transfer enhancement

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for low heat-density regions of the tube.

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In recent, we proposed a novel wavy-tape insert configuration for pipe heat transfer augmentation [25]. It has

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been found that wavy-tape inserts can produce highly localized heat transfer enhancement effect on tube surface,

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which is expected to perfectly fit to the PTR’s absorber tube to cope with the highly concentrated heat load

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contrapuntally. Therefore, in this study, wavy-tape insert is introduced to the absorber tube for the purpose of

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improving the PTR performance effectively. (Computational Fluid Dynamics) CFD simulations for a full-sized

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PTR are carried out to probe the effects of wavy-tape insert on PTR’s thermal and hydraulic performances. In

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addition, static mechanical analysis is also performed to evaluate the PTR’s mechanical performance. Finally, the

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effects wavy-tape insert on PTR’s overall performances are assessed from different points of view including the

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heat loss, the thermal stress, the entropy generation and the gain in HTF specific enthalpy.

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2. Model descriptions

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The basic PTR frame in this paper refers to an existent PTR prototype in Spain reported by Wirz et al.[26], which

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mainly consists of an absorber tube and a glass envelop. Particularly, in this paper, a wavy-tape is placed at the

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center of the absorber tube as half shown in Fig. 1. The tape waviness is in sinusoidal shape, which can be expressed

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as 𝑊𝑡 = A𝑡 ∙ sin(2π/𝑃𝑡 ∙ 𝑧). Dimensions of PTR and wavy-tape are summarized in Table 2. The tape is assumed to

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be rigid without deformation or vibration during operation. The thickness of wavy-tape is neglected for the

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convenience of grid production. Thermal conductivities of LS2 absorber tube and the glasses are prescribed as 25

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W/(m K) and 1.04 W/(m K), respectively. The emissivity of glass envelope is constant. In order to reduce the heat

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loss, it is presupposed that the selective coating is adhered to the absorber tube and its emissivity is evaluated by 𝜀𝑠𝑐

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= 0.109 ‒ 2.98 × 10 ‒ 4𝑇 + 6.4 × 10 ‒ 7𝑇2 [27]. PTR is assumed to operate in an invariant surrounding condition

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with constant wind velocity and ambient temperature. The HTF we use is Syltherm-800 and its temperature-

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dependent properties are conveyed in Table 3 [28]. As for the solar irradiation condition, we adopt one of the

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calibrated solar irradiation profiles from the work of Mwesigye et al. [29], which was calculated by using Monte

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Carlo ray tracing method. The referred solar irradiation profile was obtained with the rim angle equals to 120° and

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the direct normal irradiance equals to 1000 W/m2, which is fitted into Fourier formulas as shown in Table 4. It

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serves as the solar irradiation thermal boundary condition for the absorber tube in present study.

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3. Computational fluid dynamics simulation

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The conjugate heat transfer in PTR is a multi-step processes [16], which generally can be divided into five steps:

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(1) heat convection between HTF and absorber tube; (2) heat conduction in the absorber structure; (3) heat radiation

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between absorber tube outer-face and glass envelop inner-face; (4) heat conduction in the glass envelop; (5) heat

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dissipation from the envelop outer-face to ambient and sky. It is of great importance to accurately model the heat

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current in each step in order to accurately predict the thermal performance of PTR. The numerical models and their

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solution methods for our simulation are presented in following sections.

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3.1 Modeling convective heat transfer inside absorber tube

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The convective heat transfer inside the absorber tube is calculated by solving the Reynolds-averaged Navier–

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Stokes governing equations. The solver we use is ANSYS FLUENT (15.0 Version), a finite-volume-based

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commercial CFD software. The k-ε realizable model is chosen to cope with the turbulent flow inside the tube. In

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order to well treat the near-wall flow, the enhanced wall function is adopted in present study, which is a two-layer

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model that can resolve the viscous sublayer and thus help improve the accuracy of the model. 6

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The fluid inlet velocity is assumed to be uniform and normal to the inlet boundary. The volumetric flow rate

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ranges from 240 L/min to 720 L/min, corresponding to Reynolds number ranges from 7.20104 to 2.16105.

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Turbulent intensity at the inlet is calculated by using the general equation of 𝐼 = 0.16𝑅𝑒 ‒ 1/8. We assume the

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simulated PTR module in present study is one of the cells of an entire solar concentrator loop. Fluid temperature is

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fixed at 500K for the entrance of this cell. Pressure-outlet boundary condition is assigned to the export of the tube.

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The outer-face of the absorber tube is subjected to the heat flux profile described in Table 4. No-slip condition is

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applied to all the fluid-solid interfaces. The thermal radiation between fluid and tube is neglected. The edges of

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absorber tube are defined as adiabatic walls.

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The discretization of governing equations is carried out with second order upwind scheme. Gradients are

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reconstructed with the least squares cell based approach. The SIMPLEC algorithm is used for the pressure–velocity

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coupling. The solution is considered to have converged when the residuals for all flow variables are less than 10-5.

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3.2 Modeling radiative heat transfer between absorber tube and glass envelop

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It is assumed that the gap between absorber tube and glass envelop is absolutely vacuum. Therefore, no thermal

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convection takes place in the gap. Only the radiation in the gap is considered in this study. Since the incident solar

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energy has already been prescribed previously, we directly impose it upon the absorber tube outer-face by using

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user-defined-function. In this way, the glass envelop can be treated as a gray body with opaque surfaces. The

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emissivity of the coating adhered on the absorber tube outer surface is temperature-dependent while the emissivity

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of glass envelop inner surface is set to 1. DO (Discrete Ordinates) radiation model [30] is adopted to compute the

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radiation between the two surfaces. The convergence of radiation is achieved when the residual of DO-intensity

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term is less than 10-5.

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3.3 Modeling heat transfer between glass envelop and ambient environment.

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A mixture thermal boundary condition is assigned to the external surface of glass envelop in order to calculate

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the thermal radiation and convection between the glass envelop and ambient environment. The net heat flux due to

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radiation is calculated by using Stefan–Boltzmann law. The glass external emissivity is set to 0.89. The ambient

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temperature is 296.15 K in this study while the effective sky temperature is assumed to be 8 K lower than the

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ambient temperature [31]. The convective heat transfer coefficient is determined by using the correlation ℎ𝑔 = 4

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‒ 0.42 𝑉0.58 𝑔 𝐷 𝑔 [32]. Wind velocity is prescribed to be 3.6 m/s.

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After the models and boundary conditions stated above are properly settled, the conjugate heat transfer problem

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for PTR under steady operation condition can be resolved. For each case, it takes about 240 minutes to reach the

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convergence on a 22-intel-core computer with a RAM memory of 24 GB.

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3.4 Mesh and model checking

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Due to the symmetry of the model, half of the geometric model is considered as the computational domain. The

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mesh tailored for the computational domain is shown in Fig. 2. Nodes are deployed to fulfill the requirement that

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the location of the first layer nodes satisfies the criterion of y+<1 for all of the considered Reynolds numbers and all

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fluid-solid interfaces. The mesh independence tests are carried out for the tape-inserted case to guarantee the relative

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errors of the calculated Nusselt number and the pressure drop are less than 2% by refining the mesh size

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successively. It is found that the proper mesh size contains 5695634 hexahedral cells in total.

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The flow and heat transfer performances of the hollow absorber tube subject to the heating from considered solar

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heat flux are computed at first in order to validate numerical model and computational procedures. The computed

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Nusselt number is compared with the Dittus-Boelter correlation, while the calculated friction factor is compared

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with Filonenko correlation. Both two empirical correlations can be acquired from the textbook [33]. The comparison

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with respect to the Reynolds number range of interest are presented in Fig. 3. As shown, good agreements are

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achieved both for Nusselt number and friction factor, the deviation of Nusselt number is within 10% while the

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maximum deviation of friction factor is only about 2%, which demonstrate the high fidelity of the numerical model

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and methods used in this paper.

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3.5 Parameter definitions

160 161 162 163

In order to present the simulation results in a more effective way. The data is reduced and the parameters of interest are defined as follows. The circumferentially-average temperature 𝑇𝑐𝑖𝑟 and the circumferentially-average Nusselt number 𝑁𝑢𝑐𝑖𝑟 of the absorber tube’s external surface are defined as Eq. (1) and Eq. (2), respectively.

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𝑇𝑐𝑖𝑟 =

1 𝐿∆𝜃

𝑁𝑢𝑐𝑖𝑟 =

𝐿

𝜃𝑖 + ∆𝜃

0

𝜃𝑖

∫∫

1 𝐿∆𝜃

𝑇𝑡𝑢𝑏𝑒,𝑖𝑛𝑑𝜃𝑑𝑧

𝐿

𝜃𝑖 + ∆𝜃

0

𝜃𝑖

∫∫

(1)

𝑁𝑢𝑑𝜃𝑑𝑧

(2)

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The volume-average entropy generation rates due to irreversible viscous flow and heat transfer irreversiblility

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are evaluated by using Eq. (3) and Eq. (4) [34], respectively. The total entropy generation rate 𝑆ge𝑛 inside abosrber

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𝐹 𝑇 + 𝑆𝑔𝑒𝑛 tube during PTR operation is therefore given as a sum of two terms, 𝑆ge𝑛 = 𝑆𝑔𝑒𝑛 . 𝐹 = 𝑆gen

1 𝑉

∭ [𝑇[2{(∂𝑥) 𝜇

∂𝑢

2

+

𝛺

𝑇 𝑆gen =

167 168

4.

( ) ( )} ( ∂𝑣 ∂𝑦

1 𝑉

2

+

∂𝑤 ∂𝑧

∭( 𝛺

2

1+

+

) (

∂𝑢 ∂𝑣 + ∂𝑦 ∂𝑥

𝛼𝑇 𝑘 ∂𝑇 𝛼 𝑇2 ∂𝑥

2

+

) (

∂𝑢 ∂𝑤 + ∂𝑧 ∂𝑥

2

+

) [( ) ( ) ( ) ] 2

+

∂𝑇 ∂𝑦

2

+

∂𝑇 ∂𝑧

2

𝑑𝑉

)]

∂𝑣 ∂𝑤 + ∂𝑧 ∂𝑦

2

+

]

𝜌𝜀 𝑑𝑉 𝑇

(3)

(4)

Results and discussion 4.1 Thermal-hydraulic performance

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The contribution of wavy-tape insert to the PTR can be directly observed from Fig. 4, wherein the tube temperature

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profiles for the two cases, one without insert and another with wavy-tape insert, are compared under a same

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operating condition. Here we tag the two cases that with wavy-tape insert and without insert as CASE-W and CASE-

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H (Hollow absorber tube), respectively. It can be seen that under a same HTF flow rate, the tube temperature of

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CASE-W is apparently lower than that of CASE-H. When the wavy-tape insert is fitted into the tube, the peak

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temperature on the tube (𝑇𝑚𝑎𝑥) decreases from 559.6 K to 518.3K, and the average temperature of the tube outer

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surface (𝑇𝑎𝑣𝑔) decreases from 530.5 K to 508.9K. For a prescribed heat flux and coolant inlet condition, the decrease

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of tube temperature can only attribute to the enhancement of heat transfer between the HTF and the tube wall. As

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certified in Fig. 5, the Nusselt number for CASE-W is apparently higher than that of CASE-H. Notably the Nusselt

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numbers on the two sides of CASE-W’s tube, closing to the two flanks of the wavy-tape, are significantly improved.

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This is due to the production of strong secondary vortexes near the flanks of the wavy-tape, who break the thermal

180

boundary layer in those regions. The relevant mechanism has been detailed in our prior work [25]. The peak of

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circumferentially-average Nusselt number for CASE-W ups to 4899.2 while it is only 872.1 for CASE-H. Overall,

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in terms of the global-average Nusselt number, CASE-W is about 4.10 times higher than CASE-H. Fig. 6

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quantitatively plots the circumferentially-average temperatures 𝑇𝑐𝑖𝑟 and the circumferentially-average Nusselt

184

numbers 𝑁𝑢𝑐𝑖𝑟 for the two cases mentioned above, wherein 𝑇𝑐𝑖𝑟 and 𝑁𝑢𝑐𝑖𝑟 are calculated with the ∆θ = 5°.

185

Meanwhile, the circumferential heat flux distribution 𝑞𝑐𝑖𝑟 is also conveyed in Fig. 6a. Obviously, the wavy-tape

186

causes a substantial enhancement in Nusselt number in the θ range from 80° to 100°, which exactly covers the area

187

where the peak circumferential heat fluxes situate (at θ=94°). As a result, the high-density heat flux on those areas

188

is efficiently carried away by the vortex provoked by the wavy-tape. Therefore, the peak tube temperature (see Fig.

189

6c) existed in CASE-H (at θ≈96°) is eliminated when the wavy-tape insert shows up (see CASE-W in Fig. 6c). In

190

addition to that, the circumferential temperature gradient for CASE-W is also decreased accordingly (see Fig. 6c),

191

which certainty benefits to the decrease of stress and deformation in tube structure [35].

192

The global-average Nusselt numbers for CASE-H and CASE-W for different HTF flow rate are presented in Fig.

193

7a. As can be seen, the global-average Nusselt number increases with HTF flow rate. The global-average Nusselt

194

number of CASE-W is about 3.61 to 4.10 times higher than that of CASE-H in considered Reynolds number range.

195

When concerning the flow resistance, as presented in Fig. 7b, the wavy-tape insert brings significant flow resistance

196

to the system. Within the considered Reynolds number ranges, the friction factor of CASE-W is about 4.82-5.05

197

times higher than that of CASE-H. For the sake of evaluating the overall effect of wavy-tape insert on the tube

198

thermal-hydraulic performance, the performance evaluation criteria (PEC) factor [36], defined as PEC =

199

(𝑁𝑢𝑊/𝑁𝑢𝐻) (𝑓 /𝑓 )1 3, is calculated and presented in Fig 7c. PEC gives a roughly assessment of the performance 𝑊 𝐻

200

of CASE-W in comparison to the base case (CASE-H), which are shown to be greater than 2.11 in considered flow

201

rate ranges, indicating about 2.11-2.43 times higher thermal performances of CASE-W than that of CASE-H under

202

same pumping power consumption.

203

Four representative temperatures including 𝑇𝑡𝑢𝑏𝑒,𝑚𝑎𝑥, 𝑇𝑡𝑢𝑏𝑒,𝑎𝑣𝑔, 𝑇𝑔𝑙𝑎𝑠𝑠,𝑎𝑣𝑔 and 𝑇𝑔𝑙𝑎𝑠𝑠,𝑎𝑣𝑔 are in association with

204

the heat loss and thermal stress on the tube and the glass envelop. Therefore, Fig. 8 gives an inclusively comparison

205

of these representative temperatures between CASE-H and CASE-W. It can be recognized from the figure that

206

wavy-tape enables a remarkable decrease in the maximum and mean temperatures of both absorber tube and glass

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207

envelop. As a result, the heat loss (𝑄𝑙𝑜𝑠𝑠) due to the heat dissipation from glass envelop to ambient environment can

208

be reduced. As shown in Fig. 9, the 𝑄𝑙𝑜𝑠𝑠 significantly decreases in the presence of wavy-tape. The reduce-rate of

209

heat loss 𝜉(𝑄𝑙𝑜𝑠𝑠) due to the wavy-tape insert is in the range of 0.175 to 0.331, which decreases with the increase

210

of HTF flow rate.

211

4.2 Analysis on thermal stress and structure deformation

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212

A static structural analysis for the absorber tube is carried out followed the CFD simulation in order to analyze

213

the stress that the absorber tube experiences when suffering from the non-uniform heat load. The finite element

214

method (FEM) is employed to analyze the stress distribution in absorber tube structure. The temperature and

215

pressure histories computed by the CFD simulation are used as the input load for mechanical analysis. The study of

216

Wu et al. [37] indicated that the deflection and stress of the absorber tube depends not only on the thermal load, but

217

also mainly on its installation restrictions. Therefore, the boundary constraints for the structural analysis should be

218

carefully defined according to the practical situation. In present study, the installation mode of the absorber tube is

219

shown in Fig. 10, which refers to the previous studies [37-38]. One end of the tube is defined as zero-displacement

220

in axial direction while another end is under free-supporting condition that allows the tube expands along axial

221

direction for the purpose of releasing the structural stress. The tube is free to rotate axially. Acceleration of gravity

222

is taken into account, which is -9.8 m/s2 along the y-direction. The symmetry panel is defined as frictionless

223

supporting condition. The temperature histories computed by the CFD are interpolated to the finite element model

224

serving as the input thermal load. The absolute static pressure histories on fluid-solid interface obtained from the

225

CFD simulation are interpolated to the tube internal surface while the pressure on tube outer surface is assumed to

226

be 0 Pa. For each case, the ambient temperature of 296.15K is used as the base temperature to evaluate the thermal

227

expansion and stress. The thermal expansion coefficient and Young’s Modulus of the tube structure are 1.210-5

228

K-1 and 21011 Pa, respectively. The grid for structural analysis is partially presented in Fig. 11. The finite element

229

analysis is performed on ANSYS workbench. A grid test was also conducted for structural analysis model. It is

230

found that the grid with 208624 elements is fine enough to guarantee the independence.

231

The mechanical analysis results including the Von-Mises stress and total structure deformation for CASE-H and

232

CASE-W are presented in Fig. 12. The tube is both bended and stretched as it experiences non-uniform heat load

233

and gravity force. The deformation may have adverse effect on the PTR’s optical performance and service life. Both

234

the von-Mises equivalent stress and the total deformation of CASE-H are predictably higher than those of CASE-

235

W due to the much larger temperature gradient suffered by the former (see Fig. 4). Specifically, the discrepancy of

236

maximum stress is tremendous. The comparison of the maximum stress between two cases over the entire flow rate

237

ranges are plotted in Fig. 13. As can be seen, with the aid of wavy-tape insert, the maximum stress of the tube can 12

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238

be reduced by 61.2-69.9%, depending on the HTF flow rate. The decrease of stress is found to be more remarkable

239

at relatively lower HTF flow rates.

240

4.3 Overall performance assessment

241

It is quite interesting to assess the quality of the thermal hydraulic system from the entropy generation point of

242

view. Fig. 14 shows the comparison of the thermal and flow entropy generation rates between CASE-H and CASE-

243

𝑇 W. As can be seen, for CASE-H, the 𝑆𝑔𝑒𝑛 always accounts for an overwhelmingly large share of the total S𝑔𝑒𝑛

244

𝑇 because the heat transfer irreversibility is dominant in considered Reynold number range. In contrast, the 𝑆𝑔𝑒𝑛 of

245

CASE-W is greatly reduced compared with CASE-H because of the substantial heat transfer enhancement lead by

246

𝐹 the wavy-tape insert that decreases the thermal irreversibility. However, the 𝑆gen in the absorber tube of CASE-W

247

increases rapidly with the increase of flow rate. Notably that the minimum total S𝑔𝑒𝑛 of CASE-W is obtained when

248

𝐹 flow rate is around 360 L/min. As flow rate exceeds this value, the increase rate of 𝑆gen due to the block of wavy-

249

𝑇 tape insert becomes greater than the reduce rate of 𝑆gen , which results in the augmentation of total S𝑔𝑒𝑛 with

250

successive increase of flow rate. It also can be found from Fig. 14 that the wavy-tape insert reduces the total S𝑔𝑒𝑛

251

by about 30.2-81.8%, depending on the flow rate. The reduction of S𝑔𝑒𝑛 is more remarkable when the flow rate is

252

relatively smaller.

253

Another assessment is performed based on the criteria of consuming same amount of pumping power. In Fig. 15,

254

the heat loss per unit length 𝑄𝑙𝑜𝑠𝑠 𝐿, the maximum stress 𝜎𝑚𝑎𝑥, the total volumetric entropy generation rate S𝑔𝑒𝑛

255

and the gain of specific enthalpy per unit length ∆𝐻𝑓 𝐿 are compared between the two cases under the situation of

256

identical pumping power consumption, wherein the pumping power consumption 𝑃𝑤 is calculated in a general form

257

by multiplying the volumetric flow rate 𝑉𝑟 and total pressure drop ∆𝑃. When comparing the performance variables

258

of interest, including 𝑄𝑙𝑜𝑠𝑠 𝐿, 𝜎𝑚𝑎𝑥 and S𝑔𝑒𝑛 that represent the operation penalties of PTR from different aspects, it

259

can be found that all of them for CASE-W are evidently smaller than those for CASE-H in pumping power

260

overlapped region (3.3W < Pw < 24.5W), which altogether demonstrate the tremendous advantages that wavy-tape

261

insert brings to the PTR. As pumping power goes up successively, the contributions of wavy-tape insert to the

262

reduction of 𝑄𝑙𝑜𝑠𝑠 𝐿, 𝜎𝑚𝑎𝑥 and S𝑔𝑒𝑛 declines gradually. So it indicates that the wavy-tape would provide more

263

efficient improvement to PTR’s performance at a relatively lower HTF flow rate (or lower pumping power). On the

264

other hand, the gain of HTF specific enthalpy ∆𝐻𝑓, which denotes the promotion of HTF energy grade, is expected

13

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265

to be as larger as possible. As can be seen in Fig. 15d, the ∆𝐻𝑓 𝐿 of CASE-W is significantly larger than that of

266

CASE-H because the HTF flow rate of the former is much smaller than the latter for an identical pumping power

267

consumption, which further discloses the merit of wavy-tape to PTR from another aspect.

268

Conclusions

269

In this paper, a wavy-tape insert is applied to the solar absorber tube to ameliorate the PTR thermal and

270

mechanical performances. The conjugate heat transfer in the PTR equipped with wavy-tape insert is modeled by

271

using CFD method. The simulation results are compared with the results of the regular case without insert in order

272

to demonstrate the contribution of wavy-tape insert to the PTR. The main findings are summarized as follows.

273

The wavy-tape insert can improve the heat transfer performance inside the tube by generating swirl flow. In our

274

considered condition, the global-average Nusselt number is enhanced by 261-310%, which benefits to the decreases

275

of both PTR structure temperature and total heat loss. The heat loss is found to be reduced by 17.5-33.1%, depending

276

on the HTF flow rate. In Particular, the wavy-tape insert differs from other heat transfer enhancement techniques

277

in a way that it can lead to highly-localized heat transfer enhancement effect in the vicinity of its two flanks. This

278

is very applicable to the PTR system to cope with the highly-concentrated solar load contrapuntally. When the tape

279

flanks aim at the solar energy focused area, the high-density heat load can be carried away efficiently by the

280

intensive vortexes deployed by the wavy-tape. Consequently, the structure temperature of both absorber tube and

281

glass envelop can be greatly decreased. Mechanical analysis also reveals that the wavy-tape insert makes a

282

remarkable contribution to relieving the thermal stress in tube structure.

283

However, in the presence of wavy-tape insert, the friction factor is increased by 382-405%. This penalty has been

284

taken into consideration when we assess the overall thermal-hydraulic performance. The wavy-tape leads to a PEC

285

factor greater than 2.11. With regard to the entropy generation, the wavy-tape insert contributes to about 30.2-81.8%

286

decrease of total entropy generation rate inside the absorber tube within considered flow rate range. When

287

comparing the performance between the case with insert and the one without insert under the situation of identical

288

pumping power consumption, it is found that the heat loss, the maximum stress and the total entropy generation

289

rate of the former are far smaller than those of the latter. Besides, the working fluid gains much more specific

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290

enthalpy with the aid of wavy-tape insert. However, these merits brought by wavy-tape insert gradually decline as

291

the HTF flow rate increases due to the rapid increase of flow resistance, which indicates that the wavy-tape insert

292

is more effective and applicable to be used in PTR when the system is operated at a relatively lower HTF flow rate.

293

Reference

294

[1]

295 296

analysis of parabolic trough collector in Dhahran, Saudi Arabia. Energy Conv Manag 2014;86:622-33. [2]

297 298

[3]

[4]

Zadeh PM, Sokhansefat T, Kasaeian A, Kowsary F, Akbarzadeh A. Hybrid optimization algorithm for thermal analysis in a solar parabolic trough collector based on nanofluid. Energy 2015;82:857-64.

[5]

303 304

Kasaeian A, Daviran S, Azarian RD, Rashidi A. Performance evaluation and nanofluid using capability study of a solar parabolic trough collector. Energy Conv Manag 2015;89:368-75.

301 302

Sokhansefat T, Kasaeian A, Kowsary F. Heat transfer enhancement in parabolic trough collector tube using Al2O3/synthetic oil nanofluid. Renew Sust Energ Rev 2014;33:636-44.

299 300

Mokheimer EM, Dabwan YN, Habib MA, Said SA, Al-Sulaiman FA. Techno-economic performance

Mwesigye A, Huan Z, Meyer JP. Thermodynamic optimisation of the performance of a parabolic trough receiver using synthetic oil–Al 2 O 3 nanofluid. Appl Energy 2015;156:398-412.

[6]

Mwesigye A, Huan Z, Meyer JP. Thermal performance and entropy generation analysis of a high

305

concentration ratio parabolic trough solar collector with Cu-Therminol® VP-1 nanofluid. Energy Conv

306

Manag 2016;120:449-65.

307

[7]

308 309

Renew Sust Energ Rev 2010;14(2):629-41. [8]

310 311

314

Reddy K, Satyanarayana G. Numerical study of porous finned receiver for solar parabolic trough concentrator. Eng Appl Comp Fluid Mech 2008;2(2):172-84.

[9]

312 313

Godson L, Raja B, Lal DM, Wongwises S. Enhancement of heat transfer using nanofluids—an overview.

Kumar KR, Reddy K. Thermal analysis of solar parabolic trough with porous disc receiver. Appl Energy 2009;86(9):1804-12.

[10]

Aldali Y, Muneer T, Henderson D. Solar absorber tube analysis: thermal simulation using CFD. International Journal of Low-Carbon Technologies 2013;8(1):14-9.

15

ACCEPTED MANUSCRIPT

315

[11]

316 317

Appl Energy 2011;88(11):4139-49. [12]

318 319

Cheng Z, He Y, Cui F. Numerical study of heat transfer enhancement by unilateral longitudinal vortex generators inside parabolic trough solar receivers. Int J Heat Mass Transf 2012;55(21):5631-41.

[13]

320 321

Muñoz J, Abánades A. Analysis of internal helically finned tubes for parabolic trough design by CFD tools.

Delussu G. A qualitative thermo-fluid-dynamic analysis of a CO 2 solar pipe receiver. Sol Energy 2012;86(3):926-34.

[14]

Mwesigye A, Bello-Ochende T, Meyer JP. Heat Transfer Enhancement in a Parabolic Trough Receiver Using

322

Wall Detached Twisted Tape Inserts. ASME 2013 International Mechanical Engineering Congress and

323

Exposition: American Society of Mechanical Engineers; 2013. p. V06BT7A031-V06BT07A.

324

[15]

325 326

generation with parabolic trough by inserting metal foams. Appl Energy 2013;102:449-60. [16]

327 328

Song X, Dong G, Gao F, Diao X, Zheng L, Zhou F. A numerical study of parabolic trough receiver with nonuniform heat flux and helical screw-tape inserts. Energy 2014;77:771-82.

[17]

329 330

Wang P, Liu D, Xu C. Numerical study of heat transfer enhancement in the receiver tube of direct steam

Mwesigye A, Bello-Ochende T, Meyer JP. Heat transfer and thermodynamic performance of a parabolic trough receiver with centrally placed perforated plate inserts. Appl Energy 2014;136:989-1003.

[18]

Waghole DR, Warkhedkar R, Shrivastva R. Experimental investigations on heat transfer and friction factor

331

of silver nanofliud in absorber/receiver of parabolic trough collector with twisted tape inserts. Energy

332

Procedia 2014;45:558-67.

333

[19]

334 335

absorber with a three-dimensional heat flux density distribution. Sol Energy 2015;122:873-84. [20]

336 337

340

Reddy K, Kumar KR, Ajay C. Experimental investigation of porous disc enhanced receiver for solar parabolic trough collector. Renew Energy 2015;77:308-19.

[21]

338 339

Demagh Y, Bordja I, Kabar Y, Benmoussa H. A design method of an S-curved parabolic trough collector

Fuqiang W, Qingzhi L, Huaizhi H, Jianyu T. Parabolic trough receiver with corrugated tube for improving heat transfer and thermal deformation characteristics. Appl Energy 2016;164:411-24.

[22]

Mwesigye A, Bello-Ochende T, Meyer JP. Heat transfer and entropy generation in a parabolic trough receiver with wall-detached twisted tape inserts. Int J Therm Sci 2016;99:238-57. 16

ACCEPTED MANUSCRIPT

341

[23]

342 343

Bellos E, Tzivanidis C, Antonopoulos K, Gkinis G. Thermal enhancement of solar parabolic trough collectors by using nanofluids and converging-diverging absorber tube. Renew Energy 2016;94:213-22.

[24]

Jaramillo O, Borunda M, Velazquez-Lucho K, Robles M. Parabolic trough solar collector for low enthalpy

344

processes: An analysis of the efficiency enhancement by using twisted tape inserts. Renew Energy

345

2016;93:125-41.

346

[25]

347 348

Energy Conv Manag 2016;127:140-8. [26]

349 350

[27]

Wu Z, Li S, Yuan G, Lei D, Wang Z. Three-dimensional numerical study of heat transfer characteristics of parabolic trough receiver. Appl Energy 2014;113:902-11.

[28]

353 354

Wirz M, Petit J, Haselbacher A, Steinfeld A. Potential improvements in the optical and thermal efficiencies of parabolic trough concentrators. Sol Energy 2014;107:398-414.

351 352

Zhu XW, Fu YH, Zhao JQ. A novel wavy-tape insert configuration for pipe heat transfer augmentation.

Delgado-Torres AM, García-Rodríguez L. Comparison of solar technologies for driving a desalination system by means of an organic Rankine cycle. Desalination 2007;216(1):276-91.

[29]

Mwesigye A, Bello-Ochende T, Meyer JP. Minimum entropy generation due to heat transfer and fluid friction

355

in a parabolic trough receiver with non-uniform heat flux at different rim angles and concentration ratios.

356

Energy 2014;73:606-17.

357

[30]

358 359

method. Numer Heat Tranf B-Fundam 1993;23(3):269-88. [31]

360 361

[32]

366

Mullick S, Nanda S. An improved technique for computing the heat loss factor of a tubular absorber. Sol Energy 1989;42(1):1-7.

[33]

364 365

Forristall R. Heat transfer analysis and modeling of a parabolic trough solar receiver implemented in engineering equation solver: National Renewable Energy Laboratory; 2003.

362 363

Chui E, Raithby G. Computation of radiant heat transfer on a nonorthogonal mesh using the finite-volume

Bergman TL, Incropera FP, DeWitt DP, Lavine AS. Fundamentals of heat and mass transfer: John Wiley & Sons; 2011.

[34]

Herwig H, Kock F. Direct and indirect methods of calculating entropy generation rates in turbulent convective heat transfer problems. Heat Mass Transf 2007;43(3):207-15. 17

ACCEPTED MANUSCRIPT

367

[35]

368 369

solar parabolic trough concentrator and comparison with analytical results. Sol Energy 2016;125:1-11. [36]

370 371

374

R.L. Webb, Performance evaluation criteria for use of enhanced heat transfer surfaces in heat exchanger design, Int. J. Heat Mass Transf. 1981; 24: 715–726.

[37]

372 373

Khanna S, Sharma V, Kedare SB, Singh S. Experimental investigation of the bending of absorber tube of

Wu Z, Lei D, Yuan G, Shao J, Zhang Y, Wang Z. Structural reliability analysis of parabolic trough receivers. Appl Energy 2014;123:232-41.

[38]

Akbarimoosavi S, Yaghoubi M. 3D thermal-structural analysis of an absorber tube of a parabolic trough collector and the effect of tube deflection on optical efficiency. Energy Procedia 2014;49:2433-43.

375

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List of Figure captions Fig. 1.

Schematic of the PTR equipped with wavy-tape insert.

Fig. 2.

The mesh tailored for the computational domain of the PTR with wavy-tape insert.

Fig. 3.

Validation of numerical model and computational procedures.

Fig. 4.

Comparison of tube temperature contours between CASE-H and CASE-W under a same flow rate of 360 L/min.

Fig. 5.

Comparison of Nusselt number contours between CASE-H and CASE-W under a same flow rate of 360 L/min.

Fig. 6.

Quantitative comparisons of circumferential variables between CASE-H and CASE-W under a same flow rate of 360 L/min. (a) circumferential heat flux profile; (b) circumferentially-average Nusselt number profiles; (c) circumferentially-average temperature profiles.

Fig. 7.

Comparison of (a) global-average Nusselt number and (b) global-average friction factor between two cases at different flow rate; (c) performance evaluation criteria factor.

Fig. 8.

The decrease of representative temperatures due to the effect of wavy-tape insert. 𝑇𝑡𝑢𝑏𝑒,𝑚𝑎𝑥 and 𝑇𝑡𝑢𝑏𝑒,𝑎𝑣𝑔 represent the maximum and the mean temperature on absorber tube structure, respectively; 𝑇𝑔𝑙𝑎𝑠𝑠,𝑎𝑣𝑔 and 𝑇𝑔𝑙𝑎𝑠𝑠,𝑎𝑣𝑔 represent the maximum and the mean temperature on glass envelop structure, respectively.

Fig. 9.

Comparison of heat losses between two cases at different volume flow rate.

Fig. 10.

The supporting arrangement for the absorber tube.

Fig. 11.

Grids on the absorber tube designed for structural analysis.

Fig. 12.

Mechanical analysis results for CASE-H and CASE-W under a same flow rate of 360 L/min. (a) VonMises stress distribution; (b) Total displacement.

Fig. 13.

Comparisons of maximum stress of the tube structure between CASE-H and CASE-W.

Fig. 14.

Comparison of the entropy generation rate between CASE-H and CASE-W.

Fig. 15.

Comparisons of (a) heat loss, (b) maximum stress, (c) entropy generation rate and (d) gain of HTF specific enthalpy between CASE-H and CASE-W under the situation of identical pumping power consumption.

377

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List of Table captions Table 1

Review of recent studies on the application of heat transfer enhancement techniques in PTR.

Table 2

Design parameters of the PTR and the wavy-tape insert.

Table 3

Physical properties of Syltherm-800 [28].

Table 4

Mathematical expression of heat flux profile [29].

379 380

20

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381

Fig. 1

382 383

21

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384

Fig. 2

385 386

22

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387

Fig. 3

388 389

23

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390

Fig. 4

391 392

24

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393

Fig. 5

394 395

25

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396

Fig. 6

397 398

26

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399

Fig. 7

400 401

27

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402

Fig. 8

403 404

28

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405

Fig. 9

406 407

29

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408

Fig. 10

409 410

30

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411

Fig. 11

412 413

31

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414

Fig. 12

415 416

32

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417

Fig. 13

418 419

33

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420

Fig. 14

421 422

34

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Fig. 15

424 425

35

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426

Table 1 Reference

Heat transfer

Benefits to PTR

enhancement techniques Reddy and

Internal porous fins

Satyanarayana [8]

The decrease of heat loss ups to 31.8% compared with tubular receiver.

Kumar and Reddy

Wall-attached porous

[9]

discs inserts

Aldali et al. [10]

Internal helical fins

Significantly improving the thermal efficiency of PTR. Making the temperature distribution on tube surface more uniform.

Muñoz and

Internal helical fins

Abánades [11]

Temperature difference on the tube was reduced by 1641.6%; Thermal loss was reduced by 7-18.6%. Overall, 2% improvement of the plant efficiency was achieved, which ideally prevailed the 0.5% increase of total costs.

Cheng et al. [12] Delussu [13] Mwesigye et al.

Unilateral longitudinal

Wall temperature was reduced. Thermal loss was reduced by

vortex generators

2.23–13.62%.

Inlet blades with internal

Maximum temperature on tube surface can be reduced by 8%

fins

and temperature gradient can be reduced by 35%.

Twisted tape inserts

The reduction of tube circumferential temperature difference

[14] Wang et al. [15]

up to 76%. Metal foam inserts

Heat transfer enhancement ratio up to 1.4-3.2 (optimal); The maximum circumferential temperature difference on the outer surface of receiver tube was decreased by about 45% (optimal).

Song et al. [16]

Helical screw-tape inserts

Heat transfer performance was enhanced. Peak temperature, temperature difference and heat loss were significantly decreased.

Mwesigye et al.

Perforated semicircle

Thermal efficiency increased in the range of 3-8%;

[17]

plate inserts

Temperature gradient was reduced by 67%; Total entropy generation was decreased.

Waghole et al.

Twisted tape inserts

Thermal-hydraulic efficiency was enhanced by 135-205%.

[18]

36

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Demagh et al.

S-curved absorber tube

[19]

Heat flux distribution on tube surface became more uniform. Temperature gradient in tube structure was reduced accordingly.

Reddy et al. [20]

Porous disc inserts

Thermal efficiency was significantly enhanced. Heat loss was reduced. Angular thermal gradient in the receiver was relieved.

Wang et al. [21]

Corrugated tube

Effective heat transfer coefficient can be increased by up to 8.4%; Thermal strain can be reduced by 13.1% (max.); Tube wall temperature was decreased.

Mwesigye et al

Twisted tape inserts

[22]

The reduction of circumferential temperature difference was about 4-68%. thermal efficiency increased by 5-10%. Total entropy generation can be reduced.

Bellos et al. [23] Jaramillo et al. [24]

Converging-diverging

4.55% mean efficiency improvement of heat absorption was

absorber tube

found. Heat loss was decreased.

Twisted tape inserts

Both thermal efficiency and exergy efficiency were improved.

427 428

37

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Table 2 Dimension

𝐿𝑃𝑇𝑅

Value, mm 4096

𝐷𝑡𝑢𝑏𝑒 𝛿𝑡𝑢𝑏𝑒 𝐷𝑔𝑙𝑎𝑠𝑠 t𝑔𝑙𝑎𝑠𝑠 70

3

125

430 431

38

3

𝑊𝑡

𝐴𝑡

𝑃𝑡

28.55 3.2 128

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432

Table 3 properties

𝑎 × 𝑇4 + 𝑏 × 𝑇3 + 𝑐 × 𝑇2 + 𝑑 × 𝑇 + 𝑒 a

b

c

d

e

𝜇, Pa.s

6.6723310-10 -1.5660010-6 1.3882910-3

𝑘, W m-1K-1

0

0

-5.7535010-10 -1.8752710-4 1.9002110-1

𝑐𝑝, J kg-1K-1 0

0

0

1.70800

𝜌, kg m-3

0

-6.0616610-4

-4.1535010-1 1.10570103

0

433

39

-5.5412810-1 8.48661101 1.10780103

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434

Table 4 𝜃 range

𝑞 = 𝑎0 + 𝑎1cos (𝜔𝜃) + 𝑏1sin (𝜔𝜃) + 𝑎2cos (2𝜔𝜃) + 𝑏2sin (2𝜔𝜃) 𝜔

𝑎0

𝑎1

𝑏1

𝑎2

𝑏2

0° ≤ 𝜃 < 18°

0

300

0

0

0

0

18° ≤ 𝜃 < 94°

4.41210-2 3.146104 -8.090103 -3.123104 -7.178103 -3.119103

94° ≤ 𝜃 ≤ 180°

4.23710-2 3.363104 -1.162104 -1.899104 3.648103

435

40

7.571103