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Design and simulation of air-solar-finned reheating unit: An innovative design of a parabolic trough solar collector - S. N. Nnamchi, O. A ...
Design and simulation of air-solar-finned
reheating unit: An innovative design of
a parabolic trough solar collector
S. N. Nnamchi, O. A. Nnamchi, M. O. Onuorah, K. O. Nkurunziza and S. A. Ismael

Cogent Engineering (2020), 7: 1793453

                                                                                 Page 1 of 30
Design and simulation of air-solar-finned reheating unit: An innovative design of a parabolic trough solar collector - S. N. Nnamchi, O. A ...
Nnamchi et al., Cogent Engineering (2020), 7: 1793453
https://doi.org/10.1080/23311916.2020.1793453

                                         MECHANICAL ENGINEERING | RESEARCH ARTICLE
                                         Design and simulation of air-solar-finned
                                         reheating unit: An innovative design of
                                         a parabolic trough solar collector
Received: 21 May 2020                    S. N. Nnamchi1, O. A. Nnamchi2, M. O. Onuorah3, K. O. Nkurunziza2 and S. A. Ismael2
Accepted: 02 July 2020
                                         Abstract: Design and simulation of air-solar-finned reheating unit, an innovative
Corresponding author: S. N. Nnamchi,
Department of Mechanical                 design of a parabolic trough solar collector (PTSC) has been demonstrated in this
Engineering, Kampala International       work. Fundamentally, the design equations were formulated on the optical and
University, Kampala P.O.B 20000,
Uganda                                   thermal principles. The fundamental optical equations were transformed and
E-mail: stephen.nnamchi@kiu.ac.ug
                                         equated with the original optical equations to realize the optical design functions.
Reviewing editor:                        The design variables appear in the design function as the unknowns. The design
Zafar Said, Sustainable and
Renewable Energy Engineering,            functions were differentiated with respect to the design variables to form design
University of Sharjah, Sharjah United
Arab Emirates                            simulatory matrices. Prior to the simulation, the design functions were made to
                                         approach zero by the introduction of convergent factors which guarantee the
Additional information is available at
the end of the article                   convergence of the simulatory matrices whose final output defines the design
                                         variables. The design was algorithmized with a flowchart to justify the design
                                         procedures. A slight obtuse-angled rim design was adopted in the design of the
                                         reheating unit (RU) which yielded optimum; rim angle of 94°, collector, optical and

                                         ABOUT THE AUTHORS                                           PUBLIC INTEREST STATEMENT
                                         NNAMCHI, S. N. Nnamchi is a Senior Lecturer in              The design of parabolic trough collector (PTSC)
                                         the Department of Mechanical Engineering (ME)               falls into three facets; the acute-angled rim
                                         at Kampala International University (KIU),                  design, which the focal distance is above the
                                         Uganda. Has made prolific research contribution             aperture axis and equally greater than the
                                         in thermofluids, renewable and non-renewable                trough’s height, the right-angled rim design,
                                         energy systems; design, modelling and simula­               which the focal distance lies on the aperture axis
                                         tion.                                                       and equal to trough’s height, and the obtuse-
                                            NNAMCHI, O. A. Nnamchi is a postgraduate                 angled rim design, which the focal distance is
                                         student of Bio-processing and Food Engineering              below the aperture axis and less than the
                                         in the Department of Agricultural Engineering               trough’s height. The first two facets of the
                                         and Bio resources, Michael Okpara University,               designs are prone to misalignment problems and
                                         Nigeria. Her fast rising profile in Bioprocessing,          colossal thermal losses. However, the third
                                         Food and Chemical Engineering is valuable to this           design facet is not vulnerable to the aforemen­
   S. N. Nnamchi                         project.                                                    tioned problems but cannot raise the tempera­
                                            ONUORAH, M. O. Onuorah is an Associate                   ture of heat transfer fluid as the first two design
                                         Professor of Applied Mathematics with the                   facets. Strategically, the present design adopts
                                         Physical Sciences Department, KIU with ample                slight obtuse-angled rim design (SOARD) and
                                         publications in Biological, Ecological and                  finning the reheating unit (innovative PTSC); to
                                         Dynamical systems modelling.                                enhance heat transfer without impairing the
                                            NKURUNZIZA, K. O.Nkurunziza is                           efficiencies. Based on the enormous advantages
                                         a postgraduate student of ME at KIU. He’s                   associated with SOARD, it is recommended for
                                         developing a solar reheating unit with                      industrial application.
                                         a distinction in air-solar-finned absorber design.
                                            ISMAEL, S. A. Ismael is a dynamic postgraduate
                                         student of ME at KIU. He’s developing a solar
                                         preheating unit with excellence in air-solar-
                                         finned absorber design.

                                                                © 2020 The Author(s). This open access article is distributed under a Creative Commons
                                                                Attribution (CC-BY) 4.0 license.

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Design and simulation of air-solar-finned reheating unit: An innovative design of a parabolic trough solar collector - S. N. Nnamchi, O. A ...
Nnamchi et al., Cogent Engineering (2020), 7: 1793453
https://doi.org/10.1080/23311916.2020.1793453

                                       thermal efficiencies of 0.44, 0.72 and 0.31, respectively, and an optimum exit fluid
                                       temperature of 110o C sequel to the simulation of the design equations. Besides, the
                                       apparent tradeoffs among the design variables were useful in making design deci­
                                       sions. Considering the pitfalls of the traditional acute-angled rim design (AARD), the
                                       present work is advocating for the adoption of slight obtuse-angled rim design
                                       (SOARD) technique which will shield the PTSCs from the misalignment issues and
                                       equally minimize the thermal losses prone to the acute-angled rim design techni­
                                       que. Also, premium on material selection is recommended for the effective opera­
                                       tion of RU.

                                       Subjects: Mechanical Engineering; Heat Transfer; Fluid Mechanics; Power & Energy; Clean
                                       Tech; Design

                                       Keywords: design; design equations; simulation; air-solar-reheating unit; PTSC and design
                                       facets

                                       1. Introduction
                                       The planet Earth is cosmically supplied with enough electromagnetic radiation or wave energy to
                                       support the terrestrial life. However, the technological quests and advancement are demanding
                                       more energy than supported by the nature. Thus, mankind is seriously searching for the different
                                       ways of concentrating the electromagnetic radiation on the planet Earth to provide the technolo­
                                       gical and domestic energy demands (Abadal et al., 2014; Gwania et al., 2015). One of the ways of
                                       exploiting more energy from the sun is through the helio-thermal process via the application of
                                       solar concentrators; the parabolic, compound and dish troughs solar collectors.

                                         Therefore, having the in-depth knowledge of the physics, thermodynamics and heat transfer
                                       principles of the smart technologies and efficient processes of harnessing solar energy and
                                       subduing all odds associated with the technology is vital for exploiting more energy from the
                                       sun (Siqueira et al., 2014; Upadhyay et al., 2019). This has been the concern of recent researches in
                                       the renewable energy field. In alliance, the present work is aimed at designing a reheating unit (a
                                       parabolic trough solar collector, PTSC), which serves the thermodynamic purpose of raising the
                                       temperature of heat transfer fluid (HTF; air at 100°C), which is appropriate for drying highly
                                       moisturized agricultural products (Macedo-Valencia, 2014). Conventionally, concentrating power
                                       farm with HTF of high thermodynamic storage capacity (water) can raise steam of high tempera­
                                       ture for electricity generation in order to satisfy the industrial and domestic power requirements
                                       without posing any significant threats to the environment (Kumar et al., 2013; Tijani & Bin Roslan,
                                       2014). According to Abdelhady et al. (2017), there are three types of solar concentrators employed
                                       in hi-tech exploitation of electromagnetic radiation; the parabolic troughs, power towers and
                                       parabolic dishes. Effectively, the parabolic trough is widely used in exploiting the electromagnetic
                                       radiation because of the commensurate efficiency of the trough with a high HTF temperature
                                       (Abbood & Mohammed, 2019; Ghodbane & Boumeddane, 2018; Izweik et al., 2016; Tijani & Bin
                                       Roslan, 2014).

                                          A prudent survey of literature has shown that the design of the parabolic trough concentrator
                                       could be influenced by the size of the rim angle, which categorizes the trough designs; the acute-
                                       angled rim design (AARD), the right-angled rim design (RARD) and obtuse-angled rim design
                                       (OARD) of a parabolic trough solar collector (PTSC). Systematically, the acute-angled rim design
                                       is characterized with a flattened trough (the focal distance is greater than the height of the trough)
                                       with an elevated focal distance above the aperture axis of the PTSC. Equitably, the right-angled rim
                                       design has the focal distance equal to the height of the PTSC with the focal point coinciding with
                                       the aperture axis of the PTSC. Practically, the obtuse-angled rim design is characterized by

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Design and simulation of air-solar-finned reheating unit: An innovative design of a parabolic trough solar collector - S. N. Nnamchi, O. A ...
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https://doi.org/10.1080/23311916.2020.1793453

                                       a depressed focal distance below the aperture axis of the PTSC with the height of the trough
                                       greater than the focal distance. Aphoristically, the depth of the depression for the same aperture
                                       size is governed by the height of the trough; the more the height of the trough, the more the focal
                                       point is depressed towards the apex of the trough and vice versa. Notably, the following workers
                                       (Abdelhady et al., 2017; Gaitan, 2012; Sup et al., 2015) have adopted the acute-angled rim design
                                       approach in the design and performance analysis of the parabolic trough, this facet of design and
                                       performance analysis is susceptible to significant convectional heat loss due to the free flow of
                                       wind around the elevated absorber and if the absorber tube is not properly enveloped with a glass
                                       tube, the efficiency of the collector is surely retarded. In retrospect, they recorded the following
                                       collector design efficiencies of 0.65 and 0.70, respectively. Classically, other researchers (Ghodbane
                                       & Boumeddane, 2018; Kumar et al., 2013; Mohamed, 2013; Montesa et al., 2013) applied the right-
                                       angled rim design technique, which incurs less convectional thermal loss since the trough partially
                                       screens the absorber from the cooling effect of the winds. Consequently, they declared several
                                       collector design efficiencies of 0.37, 0.60, 0.61, and 0.71, respectively. Irrespective of the difference
                                       in the design techniques, the design efficiency published by Abdelhady et al. (2017) is in con­
                                       cordance with those of Montesa et al. (2013), and Ghodbane and Boumeddane (2018). Similarly,
                                       the collector design efficiencies by Gaitan (2012) and Kumar et al. (2013) equally concurred with
                                       each other despite the difference in the design methodologies. However, Mohamed (2013) collec­
                                       tor efficiency disagrees with the results achieved by both techniques; acute-angled rim design and
                                       right-angled rim design methods. Probably, the difference in the design yardsticks could be
                                       attributed to the difference in the capacity of their troughs and prevailing environmental condi­
                                       tions. However, Mohamed (2013) is outstanding in the sense that moderate collector efficiency
                                       (0.37), which indicates that the thermal efficiency is significant or higher exit fluid temperature
                                       could be attained compared to design efficiency (>0.60), which optical efficiency dominates. Based
                                       on the literature survey a good design of PTSC should preserve both optical and thermal efficien­
                                       cies, which is feasible by careful design of the focal distance and the height of the trough. Notably,
                                       the three facets of the design of the reheating unit (PTC) as reviewed in the literature were carried
                                       out without a simulator, which does not encourage a tradeoff among the design variables and
                                       could limit the performance of the reheating unit (PTC). However, the present work is pivoted on
                                       a simulatory design technique to ensure proper tradeoff among the design variables, which
                                       engenders optimum optical and thermodynamic performance of the reheating unit (PTC).

                                          Characteristically, Macedo-Valencia et al. (2014) pivoted their design on the obtuse-angled rim
                                       criterion thereon the height of the trough is greater than the focal distance and the absorber is
                                       totally screened by the trough against the wind flow on evacuating the trough with a glass cover,
                                       the risk of thermal loss becomes negligible. However, the more concentration ratio is gained with
                                       the extreme obtuse-angled rim compared to the right-angled rim and acute-angled rim design
                                       techniques. Pertinently, Macedo-Valencia et al. (2014) recorded collector design efficiencies ran­
                                       ging from 0.3649 to 0.5057, which are in alignment with that of Mohamed (2013). Generally, the
                                       differences in the collector design efficiencies reviewed could be strongly attributed to the differ­
                                       ences in the design considerations and locations. The present work is fascinated by the exclusive
                                       advantages of obtuse-angled rim design to adopt the slight obtuse-angled rim design (SOARD)
                                       technique in the design and simulation of the parabolic trough solar collector (PTSC). Moreover, the
                                       aim of the present design is to diversify the application of PTSC to hi-tech drying technology, by
                                       substituting the heat transfer fluid (HTF; water) in the conventional PTSC designs with air (Huanga
                                       et al., 2016). Superficially, this idea may appear to be impracticable using air as the HTF, which has
                                       a lower thermal storage capacity, but can be directly used in drying operation. However, the
                                       application of air as the HTF with a low thermal storage capacity will be ameliorated by loading
                                       the absorber with a lot of fins, which obviously reduce the cross-sectional area available to the HTF
                                       and enhance the heat transfer phenomenon between the finned absorber and HTF. Thus, the
                                       reduction in mass flowrate of HTF engenders a rise in the exit fluid temperature of the HTF (air),
                                       which could be employed in the direct drying operation. Therefore, the present design considers;
                                       the application of finned absorber in lieu of unfined absorber, the use of air as the HTF against
                                       water and the adoption of slight obtuse-angled rim design (SOARD) technique in the bid to raise

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Design and simulation of air-solar-finned reheating unit: An innovative design of a parabolic trough solar collector - S. N. Nnamchi, O. A ...
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                                       the temperature of HTF (>100°C) and to achieve substantial collector and thermal efficiencies
                                       (>0.30) through the formulations and simulations of the design equations.

                                         Furthermore, emphasis is laid on the selection of premium materials for the optimum perfor­
                                       mance of the finned absorber and in the selection of trough material with a high reflectivity to
                                       enhance the illumination and concentration of solar irradiance on the finned absorber (Ricardo,
                                       2011). Essentially, the present work is algorithmized both in design and simulation processes,
                                       which distinguish it from other designs in the literature.

                                         Subsequently, other sections of this paper will include; materials and method articulated in
                                       a flow chart, further characterized with the formulation of design equations and their simulations,
                                       presentation of results and their discussion, and lastly, conclusions and recommendations.

                                       2. Materials and method
                                       The design of the reheating unit is adapted to the following methodologies:

                                         The design functions or equations were formulated on the fundamental (optical or thermal)
                                       principles. The design variables were identified and incorporated into the design functions or
                                       equations and parameters as the unknown (symbolic) variables.

                                         The simulatory matrices were made of the coefficient (n × n) matrix and column (n × 1) matrix
                                       whose elements were pivoted on the partial derivatives of the design functions or equations with
                                       respect to the design variables (the unknowns) and the design functions, respectively.

                                          Then, the initialization of values of the design variables and the provision of other essential input
                                       data was insightfully done to prepare the simulation process. Prior to the simulation, a check on
                                       the convergence of the design functions has to be carried out; to ascertain whether they are
                                       approaching zero or not (which is an inevitable design condition). Once there is a tendency of
                                       convergence, the simulation is then executed. Otherwise, the odd or nonconvergent terms of the
                                       design functions are identified and multiplied with the convergent factors such that the design
                                       functions have the propensity to approach zero or stand a chance of being converged.
                                       Consequently, the simulation is characterized with a quick convergence as the convergence
                                       criterion is readily attained. The optimal design variables are commensurate with the final output
                                       values that are capable of making the design functions to naturally approach zero.

                                          Thus, the design approach is strongly based on the multiple input and multiple output (MIMO)
                                       approach, which is appropriate for system design (Stoecker, 1989) rather than on single input and
                                       single output (SISO) technique that may not guarantee significant harmony among the design
                                       variables for the optimum performance of the designed system. The entire process or procedure is
                                       carefully algorithmized in Figure 1.

                                       2.1. Formulation of optical design equations
                                       Technically, the design equations are to be governed by optical and thermal behaviours of the
                                       parabolic trough solar collector (PTSC). The optical characteristics of the PTSC are expected to
                                       influence the thermal characteristics of the PTSC. Thus, the overall or collector efficiency of the
                                       system (PTSC) is defined as an attenuated difference in the efficiencies; optical and thermal. The
                                       attenuation factor is equivalent to the heat removal factor. Algorithmically, the optical design is
                                       illustrated in Figure 1, the optical design flowchart describing the design sequence.

                                         The focal point (fpt (m)), the aperture or the width of the parabolic trough solar collector (wpt (m))
                                       and the height (hpt (m)) between the apex and the rim of the parabolic trough collector is
                                       expressed in (Abdelhady et al., 2017; Borah et al., 2013; Ghodbane & Boumeddane, 2018;
                                       Pavlović et al., 2014) as

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Figure 1. Optical design
flowchart.

                                                 w2pt
                                        fpt ¼           9 fpt
Design and simulation of air-solar-finned reheating unit: An innovative design of a parabolic trough solar collector - S. N. Nnamchi, O. A ...
Nnamchi et al., Cogent Engineering (2020), 7: 1793453
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                                       Transforming with Equation (1) gives

                                            �                                       �2 �0:5
                                       Rpt ¼ 4fpt hpt þ hpt                   fpt

                                                �                �2                         �2 �0:5      �                          �2 �0:5
                                       gop2 ¼       0:5wpt            þ hpt           fpt                    4fpt hpt þ hpt   fpt                9 gop2 ! 0;                  (2)

                                       where gop2 is the second optical design function.

                                        The internal surface area of the parabolic trough collector As,pt (m2) is given in Equation (3) by
                                       Macedo-Valencia et al. (2014) as

                                                       0                                          !0:5            0                        !0:5 11
                                                                          �           �2                                        �     �
                                                        wpt                   4hpt                                 4hpt          4hpt 2
                                       Apt ¼ spt lpt ¼ @                                    þ1           þ 2fpt ln@     þ               þ1      AA lpt
                                                         2                    wpt                                   wpt           wpt

                                       The present work presents a transform of Macedo-Valencia et al. (2014) as

                                                       0                                          !0:5               0                            !0:5 11
                                                                          �          �2                                       �            �2
                                                        wpt                   wpt                                   wpt             wpt
                                       Apt ¼ spt lpt ¼ @                                    þ1           þ 2fpt ln@     þ                       þ1     AA lpt                 (3)
                                                         2                    4fpt                                 4fpt             4fpt

                                       The rim angle of the parabolic trough collector, ψr (degrees) is defined in Equation (4) according to
                                       (Abdelhady et al., 2017; Alfelleg, 2014; Mohamed, 2013; Macedo-Valencia et al., 2014) in Equation
                                       (4) as

                                                         �        �
                                                     1    wpt
                                       ψ r ¼ 2tan
                                                          4fpt

                                       but the present work redefines the rim angle as

                                                         �     �
                                                     1    4hpt
                                       ψ r ¼ 2tan
                                                           wpt

                                       Considering equality in both definitions of rim angle, results in a third independent optical design
                                       equation, gop3

                                                             �        �                         �     �
                                                         1    w                             1    4hpt
                                       gop3 ¼ 2tan                            2tan                      ¼0                                                                    (4)
                                                             4fpt                                 wpt

                                       where              Rpt                 (m)                 is           the        parabolic                radius       of   curvature.

                                         The geometrical concentration ratio (CR) is expressed in Equation (5) (Ghodbane & Boumeddane,
                                       2018; Kuo et al., 2014; Lovegrove & Pye, 2012; Macedo-Valencia et al., 2014)

                                              wpt
                                       CR ¼
                                              dabo

                                       according to the present work, CR is defined as

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                                              "      � �2                     �2       �         ��2 #0:5
                                                  wpt 2 þ hpt           fpt    1 þ dabo wpt dabo
                                       CR ¼                2                  �2     �        ��2                                               (5)
                                                  ðdabo =2Þ þ hpt          fpt dabo wpt dabo

                                       where dab (m) is the outer diameter of the absorber.

                                         The optical efficiency, ηopt (-) is given in Equation (6) (Alfelleg, 2014; Pierucci et al., 2014;
                                       Vasquez, 2011)

                                       ηopt ¼ ρpt τg αab γ κðθi Þχend                                                                           (6)

                                       where ρpt (-) is the reflectance of the polished surface of the parabolic trough, τg (-) is the
                                       transmissivity of the glass cover, αab (-) is the absorptivity of the absorber pipe, γ (-) is the
                                       alignment or intercept factor of the absorber pipe (γ ≤ 1), κ (θi) is the incidence angle modifier and
                                       χend (-) is the end loss defined by Alfelleg (2014), Kuo et al. (2014) in Equation (7) as

                                                      fpt
                                       χend ¼ 1           tan θi                                                                                (7)
                                                      lpt

                                       and according to Alfelleg (2014) the incidence angle modifier, κ(θi) in Equation (8) is correlated as

                                       κðθi Þ ¼ 1:2257       0:0072θi þ 0:00003θ2i                                                              (8)

                                       According to Dudley et al. (1994) κ(θi) in Equation (9) is given as

                                                                       θi                         θ2i
                                       κðθi Þ ¼ 1:0 þ 0:000884                      0:00005369                                                  (9)
                                                                     cos θi                      cos θi

                                       Goswami and Kreith (2008) proposed the incidence angle modifier in Equation (10) as

                                       κðθi Þ ¼ 1:0     0:00022307θi               0:0001172θ2i þ 0:00000318596θ3i     0:00000001θ4i          (10)

                                       Similarly, Kalogirou (2004) defined κ(θi) in Equation (11) as follows:

                                                                          θi                       θ2i
                                       κðθi Þ ¼ 1:0 þ 0:0003178                      0:00003985                                               (11)
                                                                        cos θi                    cos θi

                                       In the same vein, Montes et al. (2009) presented κ(θi) in Equation (12) as

                                                                                                     θi             θ2
                                       κðθi Þ ¼ 1:0     0:000525097                0:00002859621          þ 0:00001 i                         (12)
                                                                                                   cos θi          cos θi

                                       The concentrated solar power in Equation (13) is given as

                                       Qsol;ab;con ¼ ηopt Aab G                                                                               (13)

                                       whereas the directly absorbed or non-concentrated solar power in Equation (14) is defined as

                                       Qsol;ab;ncon ¼ τg αab Aab G                                                                            (14)

                                       The subscript g represents the glass cover.

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                                          The total solar power absorbed, Qsol,ab (W) is expressed as

                                                                             �                       �
                                       Qsol;ab ¼ Qsol;ab;con þ Qsol;ab;ncon ¼ ηopt Aabo þ τg αab Aabo G                                                        (15)

                                       The subscripts con, ncon and g designate the concentrated, non-concentrated heat on the absor­
                                       ber and the glass, respectively.

                                       2.1.1. The Simulation of the optical design equations
                                       The optical functions; gop1, gop2, and gop3 (in Equations (1), (2) and (4), respectively, are differen­
                                       tiated with respect to fpt, hpt and wpt, respectively, leading to the optical simulatory matrices in
                                       Equation (16):

                                       2 @gop1    @gop1    @gop1   3
                                                                    2      3 2        3
                                          @fpt
                                       6 @gop2
                                                  @hpt      @w
                                                                   7 Δfpt        gop1
                                       6          @gop2    @gop2   74 Δhpt 5 ¼ 4 gop2 5
                                       4 @fpt     @hpt      @w     5                                                                                           (16)
                                         @gop3    @gop3    @gop3      Δwpt       gop3
                                          @fpt    @hpt      @w

                                       The detailed partial derivative of Equation (16) is presented in the supplementary file. The future
                                       values of the optical design variables and present values are defined in Equation (17) as follows:

                                        fpt;iþ1 ¼ fpt;i þ Δfpt ; hpt;iþ1 ¼ hpt;i þ Δhpt ;
                                                                                                                                                               (17)
                                        wpt;iþ1 ¼ wpt;i þ Δwpt ; i ¼ 0; 1; 2

                                       The final optical design variables are established the moment the set convergence criterion

                                          (ζop ¼ 10 3 ) in Equation (18) is satisfied

                                        fpt;iþ1   fpt;i � ζop ; hpt;iþ1         hpt;i � ζop ;
                                                                                                                                                               (18)
                                        wpt;iþ1     wpt;i � ζop ; i ¼ 0; 1; 2

                                       2.2. The formulation of the thermal design equations
                                       The thermal analysis of the PTSC is illustrated in Figure 2 with the symmetric thermal gradients
                                       and fluxes (conduction, convection and radiation) from the absorber through ambient to the sky.
                                       Basically, the conduction flux is defined by Fourier’s first law of conduction, the convection flux is
                                       governed by Newton and Fourier’s law, whereas the radiation flux is based on Stefan radiation
                                       laws (Cheng & Fujii, 1998).

                                         The thermal balance on the glass cover (g) is based on the steady state assumption, which gives
                                       the first independent thermal design equation in Equation (19) as

                                        gth1 ¼ Qsol;ab;g þ Qcv;abo        gi   þ Qr;abo    gi       Qcv;go a Qr;go sky Qr;gi ti       9      gth1 ! 0;
                                                                                                        �                          �                      �
                                        ) gth1 ¼ αg Ag G þ hcv;abo             gi Aabo   Tabo        Tgi þ hr;abo gi Aabo Tabo Tgi    hcv;go a Ago Tgo Ta
                                                                                  �                              �
                                                  hr;go   sky Ago   Tgo    Tsky          hr;gi   ti Agi Tgi  Tti                    9 gth1      !0
                                                                                                                                                               (19)

                                       Similarly, the thermal balance on the absorber (ab) gives the second independent thermal design
                                       equation in Equation (20)

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Figure 2. Temperature gradi­
ents and thermal fluxes across
the parabolic trough solar col­
lector for the thermal analysis.

                                        gth2 ¼ Qsol;ab;ab Qcv;abo gi Qr;abo                     gi      Qcv;abo    ti    Qr;abo    ti        Qgain           9 gth2 ! 0;
                                                   �                     �                                                               �                                         �
                                         ) gth2 ¼ ηopt Aabo þ τg αab Aabo G                          hcv;abo   gi Aabo   Tabo      Tgi         hr;abo   gi Aabo    Tabo      Tgi
                                                                                                                                                                                             ��
                                               hcv;abo ti Aabo ðTabo            Tti Þ      hr;abo      ti Aabo ðTabo       Tti Þ   hcv;abi      hair Aabi   Tabi     0:5 Thair;o þ Thair;i
                                              9 gth2 ! 0
                                                                                                                                                                                               (20)

                                       Also, the thermal balance on the parabolic trough collector (t) is represented in Equation (21)
                                       provides the third independent thermal design equation in Equation (21)

                                        gth3 ¼ Qcv;abo   ti   þ Qr;abo    ti   þ Qr;gi     ti        Qcv;to    a   Qr;to   sky                   9 gth3 ! 0;
                                                                                                                                                                         �
                                        ) gth3 ¼ hcv;abo       ti Aabo ðTabo            Tti Þ þ hr;abo     ti Aabo ðTabo  Tti Þ þ hr;gi ti Agi Tgi                 Tti                         (21)
                                                                                                                     �
                                               hcv;to a Ato ðTto         Ta Þ      hr;to    sky Ato      Tto Tsky        9 gth3 ! 0

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                                         Adding Equations (19) and (21) or summing the thermal power functions; gth1 and gth3 gives the
                                       fourth independent thermal design equation in Equation (22)

                                                                                                    �                                              �                                            �
                                        gth4 ¼ αg Ag G þ hcv;abo        gi Aabo    Tabo          Tgi þ hr;abo        gi Aabo       Tabo      Tgi         hcv;go a Ago Tgo                  Ta
                                                                              �
                                               hr;go    sky Ago   Tgo     Tsky þ hcv;abo            ti Aabo ðTabo          Tti Þ þ hr;abo      ti Aabo ðTabo             Tti Þ
                                                                                                                      �
                                               hcv;to a Ato ðTto        Ta Þ      hr;to    sky Ato    Tto      Tsky                                            9 gth4 ! 0
                                                                                                                                                                                                           (22)

                                       Hypothetically considering the equality of thermal conduction and convection on the absorber and
                                       heat transfer fluid, respectively, gives the fifth independent thermal design equation in Equation (23)

                                                          �          �
                                                           Tabo Tabi                                                                                    ��
                                       gth5 ¼ kab Ac;ab                                hcv;abi   air;i Aabi   Tabi        0:5 Thair;o þ Thair;i                                  9 gth5
                                                               δab
                                            !0                                                                                                                                                             (23)

                                       Pertinently, subtracting Equation (23) from Equation (20) or thermal power function gth5 from gth2
                                       yields    the    sixth    independent     thermal    design     equation   in    Equation     (24)

                                              �                       �                                                             �                                                  �
                                        gth6 ¼ ηopt Aabo þ τg αab Aabo G                   hcv;abo     gi Aabo   Tabo         Tgi          hr;abo      gi Aabo     Tabo          Tgi
                                                                                                                                                    �          �
                                                                                                                                                     Tabo Tabi
                                               hcv;abo    ti Aabo ðTabo        Tti Þ      hr;abo    ti Aabo ðTabo          Tti Þ        kab Ac;ab                                              9 gth6 ! 0
                                                                                                                                                         δab
                                                                                                                                                                                                           (24)

                                       Also, considering the effectiveness of the absorber, the seventh independent thermal design
                                       equation is obtained in Equation (25)

                                       gth7 ¼ Qsol;ab;ab ΦQu      9 gth7 ! 0;
                                              �                      �     �                �                                                             ��                             ��
                                       gth7 ¼ ηopt Aabo þ τg αab Aabo G Φ ρair cpair ublower 0:25πd2abi                            n δfin 2lfin þ bfin             Thair;o       Thair;i            9 gth7 ! 0
                                                                           0                                                                                                           !1
                                                                                                             � 1010:13412 0:03977 Thair;o
                                             �                       �              B 1:6843 0:0015 Thair;o                      2                                                         C
                                                                                    B                             þ0:000105 Thair;o                                                        C
                                       gth7 ¼ ηopt Aabo þ τg αab Aabo G            ΦB                                                                                                      C        9 gth7 ! 0
                                                                                    @         �                               ��                 �                                         A
                                                                                      �ublower 0:25πd2abi n δfin 2lfin þ bfin    Thair;o Thair;i
                                                                                                                                                                                                           (25)

                                       where Φð Þ � 1:0 is the effectiveness of the absorber, ρair (kg/m3) is the density of air, cpair (kJ/kg) is
                                       the specific heat capacity of air, ui (m/s) is internal air velocity, n (-) is the number of rectangular
                                       fins, δfin (m) is the thickness of the fins, lfin (m) is the length of the fins and wfin (m) is the width of
                                       the fins. The variables in Equations (19) to (25) are defined as follows:

                                                                                                                                                                             �
                                         The outer surface area of the air-solar-finned absorber pipe, Aabo m2                                                                    in Equation (26) is
                                       computed as

                                       Aabo ¼ 2πr abo lpt ¼ πdabo lpt                                                                                                                                      (26)
                                                                                                                                                            �
                                       The inner surface area of the air-solar-finned absorber pipe, Aabi                                                 m2 in Equation (27) is deter­
                                       mined as

                                       Aabi ¼ 2πrabi lpt ¼ πdabi lpt                                                                                                                                       (27)

                                                                                                                                                             2
                                                                                                                                                               �
                                       The cross-sectional area of the air-solar-finned absorber, Ac;ab m                                                           available to conduction is
                                       designed in Equation (28) as

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                                                     π� 2               � π
                                       Ac;ab ¼         d           d2abi ¼ ðdabo þ dabi Þðdabo          dabi Þ                                                (28)
                                                     4 abo                4
                                                                                                          �         �
                                       The outer and inner surface areas of the glass cover (glaze) Ago m2 andAgi m2 , respectively, are
                                       geometrically defined in Equation (29) as

                                       Ago ¼ Agi ¼ wpt � lpt                                                                                                  (29)
                                                                                                       �
                                       The inner surface area of the parabolic trough collector, Ati m2 is given in Equation (30) as

                                                             0                   !0:5          0                                        !0:5 11
                                                                       �     �                                      �           �2
                                                          wpt           4hpt 2                  4hpt                     4hpt
                                        Ati ¼ spt lpt ¼ @                      þ1     þ 2fpt ln@     þ                                þ1     AA lpt
                                                           2             wpt                     wpt                     wpt

                                        or                                                                                                                    (30)
                                                             0                           !0:5           0                              !0:5 11
                                                                       �        �2                                  �          �2
                                                                 wpt     wpt                               wpt          wpt
                                        Ati ¼ spt lpt ¼ @                            þ1         þ 2fpt ln@     þ                    þ1      AA lpt
                                                                  2      4fpt                             4fpt          4fpt

                                                                                                       �
                                       The outer surface area of the parabolic trough collector, Ato m2 is given in Equation (31) as

                                                                        ��                                   �
                                                                                      �2       �2 �0:5
                                                               �                0:5wpt þ f hpt         þ δpt
                                                     Rpt þ δpt
                                       Ato ¼                   Ati ¼            �        �2       �2 �0:5      spt lpt                                        (31)
                                                        Rpt
                                                                                   0:5wpt þ f hpt

                                       The convective heat transfer coefficient, hcv;go a between the outer glass (go) and ambient (a) is
                                       expressed in Equation (32) as follows (Hammami et al., 2017; Nnamchi et al., 2020; Oko, 2011):

                                                                                 �0:5             �             �
                                                       4:392773 uw;o lpt                                                                 �
                                       hcv;go    a   ¼                                  ¼ 4:392773 u0:5
                                                                                                    w;o lpt
                                                                                                            0:5
                                                                                                                               W=m2 K                         (32)
                                                               lpt

                                       The radiative heat transfer coefficient, hr;go sk between the outer glass (go) and sky (sk) is given in
                                       Equation (33) as (Kreith et al., 2000; Nnamchi et al., 2020; Oko, 2011)

                                                                    �     �                 �4 �
                                                       5:103 � 10 8 Tgo4
                                                                            0:0552 � 2981:5
                                                                                                                               �
                                       hr;go    sk   ¼        �     �                 ��                                 W=m2 K                               (33)
                                                                Tgo   0:0552 � 2981:5

                                       The convective heat transfer coefficient, hcv;abo gi between the periphery of the absorber (abo) and
                                       inner glass (gi) is specified in Equation (34) as (Ali & Sadek, 2018; Hammami et al., 2017; Nnamchi
                                       et al., 2020; Rincón-Casado et al., 2017)

                                                              kair       1
                                        hcv;abo      gi   � 0:54   Ralpt 4 ; 104 � Ralpt � 107 ;
                                                              lpt
                                                 �                                           ��                                  �
                                                   3 2
                                        Ralpt   ¼ lpt ρair cpair ðg cos ϕÞβabo gi Tabo Tgi =ðμair kair Þ; βabo gi ¼ 2= Tabo þ Tgi ;
                                                 2                                                                       � 3
                                                   l3pt ð2:1313 0:003Tair Þ2 1031:31 0:2047 Tair þ 0:00042 Tair       2
                                                                                                                            �                                 (34)
                                                 4               �         �            �                                      5
                                                   ðg cos ϕÞ Tabo2þTgi Tabo Tgi
                                        Ralab   ¼ �                                                  �
                                                        1:03 � 10 6 þ7 � 10 8 Tair 4 � 10 11 Tair  2     0:0121eð0:0025 Tair Þ

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                                       The radiative coefficient, hr;abo gi between the periphery of the absorber (abo) and inner glass (gi)
                                       is defined in Equation (35) as (Kreith et al., 2000; Nnamchi et al., 2020; Oko, 2011)
                                                                                                   �            �
                                                                                                       4     4
                                                                                                 σ Tabo     Tgi
                                        hr;abo       gi    ¼�                                            �      �
                                                                                                                  A
                                                                                                                         �                            �
                                                                      1 εabo                1
                                                                       εabo     þ Fabo   gi þFabo air
                                                                                                      þ ε1gi 1 As;abo
                                                                                                                    s;gi
                                                                                                                           Tabo                 Tgi

                                                                                 �            �
                                                                                   4       4
                                                                               σ Tabo     Tgi
                                                     ¼�                             �         �           �                                                            (35)
                                                       1 εabo           1              1        πdabo lpt                                              �
                                                        εabo  þ                   þ   εgi  1     wpt lpt    Tabo                                 Tgi
                                                                ðð12þ1πÞþð12 1πÞÞ

                                                                                   �            �
                                                                                      4       4
                                                                                  σ Tabo    Tgi                                                            �
                                        ) hr;abo                gi   ¼�       �       �       �                             �                   W=m2 K
                                                                         1      1
                                                                        εabo þ εgi   1 πdwabo
                                                                                          pt
                                                                                                Tabo                  Tgi

                                       The radiative coefficient, hr;gi ti between the inner; glass (gi) and parabolic trough collector (ti) and
                                       is well defined in Equation (36) as (Kreith et al., 2000; Nnamchi et al., 2020; Oko, 2011).
                                       Introduction to heat transfer: an algorithmic approach.

                                                                                                �        �
                                                                                             σ Tgi4 Tti4
                                        hr;gi   ti   ¼ �1              εgi
                                                                                                  �      � �
                                                                                                           A                          �
                                                                                        1
                                                                     εgi     þ Fgi   ti þFti ti
                                                                                                þ ε1ti 1 As;gi
                                                                                                             s;ti
                                                                                                                  Tgi           Tti

                                                                        �         �
                                                                      σ Tgi4 Tti4
                                                 ¼�                        �       �         �                                                                         (36)
                                                   1 εgi         1           1       wpt lpt                                                �
                                                    εgi  þ               þ   εti 1   spt lpt   Tgi                                    Tti
                                                           ðð2πÞþð1 2πÞÞ

                                                                             �         �
                                                                           σ Tgi4 Tti4                                                  �
                                        ) hr;gi            ti   ¼�      �        � �                              �               W=m2 K
                                                                    1     1       w
                                                                   εgi þ εti   1 sptpt Tgi                  Tti

                                       The convective heat transfer coefficient, hcv;abo ti between the periphery of the absorber (abo) and
                                       inner trough (ti) is empirically correlated in Equation (37) as (Ali & Sadek, 2018; Nnamchi et al.,
                                       2020; Rincón-Casado et al., 2017)

                                                             kair 1
                                        hcv;abo       ti    � 0:54 Ra4l ; 104 � Ralpt � 107 ;
                                                              ld       pt
                                                 �                                           �
                                                   3 2
                                        Ralpt   ¼ lpt ρair cpair ðg cos ϕÞβabo ti ðTabo Tti Þ =ðμair kair Þ;
                                        Tair ¼ 0:5ðTabo þ Tti Þ; βabo ti ¼ 2=ðTabo þ Tti Þ;
                                                2                                                                  � 3                                                 (37)
                                                  l3pt ð2:1313 0:003Tair Þ2 1031:31 0:2047 Tair þ 0:00042 Tair  2
                                                                                                                      �
                                                4            �       �                                                   5
                                                                 2
                                                  ðg cos ϕÞ Tabo þTti ðTabo Tti Þ
                                        Ralpt ¼ �                                                �
                                                       1:03 � 10 6 þ7 � 10 8 Tair 4 � 10 11 Tair
                                                                                             2     0:0121eð0:0025 Tair Þ

                                       The radiative heat transfer coefficient, hr;abo ti between the periphery of the air-solar-finned
                                       absorber (abo) and the inner trough (ti) is given in Equation (38) as follows (Kreith et al., 2000;
                                       Nnamchi et al., 2020; Oko, 2011):

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                                                                                                 �             �
                                                                                                     4
                                                                                               σ Tabo     Tti4
                                        hr;abo        ti   ¼�                                          �        �
                                                                                                                  A
                                                                                                                         �
                                                                    1 εabo                1
                                                                     εabo    þ Fabo    ti þFabo air
                                                                                                    þ ε1ti 1 As;abo s;ti
                                                                                                                           ðTabo   Tti Þ

                                                                                         �          �
                                                                                             4
                                                                                        σ Tabo Tti4
                                                  ¼�                                     �     �           �                                                                     (38)
                                                                1 εabo         1                 πd l
                                                                 εabo    þ             þ ε1gi 1 sptabolptpt ðTabo          Tti Þ
                                                                             ð12þ12Þ

                                                                                  �             �
                                                                                     4        4
                                                                                σ Tabo      Tgi                                              �
                                        ) hr;abo               gi   ¼�       �        �       �                                    W=m2 K
                                                                        1      1
                                                                       εabo þ εgi   1 πdsptabo ðTabo               Tti Þ

                                       The radiative coefficient, hr;abi hair between the inner absorber (abi) and hot air stream (hair) is
                                       precisely defined in Equation (39) as (Ali & Sadek, 2018; Hammami et al., 2017; Nnamchi et al.,
                                       2020)

                                                                                                   0                                                                             114
                                                                                                  ðcos ϕÞ�
                                                                                                  0                                                         2                  1
                                                                                                B     426976:7297    655:4854153T    hair;o þ 2:195492077T                       C
                                                                                                B                                                           hair;o               C
                                                                                                B B 0:001293428T3                             7 4
                                                                                                                                                       1:18125 � 10 10 Thair;o
                                                                                                                                                                        5      CC
                                                                                                BB                 hair;o þ 2:748173 � 10 Thair;o                              CC
                                                                                                BB                                                                  2          C C
                                                                                                B @ þ2865:615635Tabi 5:216928875Thair;o Tabi þ 0:002771627Thair;o      Tabi A C
                                                                                                B                                                                                C
                                                                                                B                  7 3                            10 4
                                                                                                      6:20037 � 10 Thair;o Tabi þ 2:3625 � 10 Thair;o Tabi                       C
                                                                                                B                                                                                C
                                       hcv;abi   hair;o        � 0:01996 labi0:25 e0:0009375Tfo B                                                                     !          C
                                                                                                B                             5
                                                                                                           5:53265377 � 10 þ 3:37634802 � 10 Thair;o   7                         C
                                                                                                B                                                                                C
                                                                                                B                               10 2
                                                                                                           þ4:57651074 � 10 Thair;o 1:75614380 � 10 Thair;o    13 3              C
                                                                                                B                                                                                C
                                                                                                B                                                                                C
                                                                                                B                                                                                C
                                                                                                B                                                                                C
                                                                                                @                                                                                A

                                                                                                                                                                                  (39)

                                       The convective heat transfer coefficient, hcv;to a between the periphery of the parabolic trough
                                       collector (to) and the ambient (a) is obtained by considering the wind velocity to be one third
                                       of its velocity in the windward direction according to (Nnamchi et al., 2020) Equation (40)

                                                                                                   �0:5             �            �
                                                               4:392773 0:33uw;o ld                                                                       �
                                       hcv;to     a        ¼                                              ¼ 2:523456 u0:5
                                                                                                                      w;o ld
                                                                                                                             0:5
                                                                                                                                                 W=m2 K                          (40)
                                                                        ld

                                       The radiative heat transfer coefficient, hr;to sk between the periphery of the parabolic trough
                                       collector (to) and the sky (sk) is expressed in Equation (41) as (Kreith et al., 2000; Nnamchi et al.,
                                       2020; Oko, 2011)

                                                                      �     �                 �4 �
                                                         4:536 � 10 8 Tto4    0:0552 � 2981:5
                                                                                                                                                    �
                                       hr;to     sk    ¼        �     �                 ��                                                 W=m2 K                                (41)
                                                                  Tto   0:0552 � 2981:5
                                       2.2.1. Simulation of thermal design equations
                                       Similarly, the thermal design functions; gth1(W), gth2(W), gth3(W), gth4(W), gth5(W), gth6(W) and
                                       gth7 are differentiated with respect to temperatures; Tgo(K), Tgi(K), Tabo(K), Tabi(K), Tti(K) and
                                       Tto(K) and Thair (K), respectively, resulting in the simulatory matrices in Equation (42)

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                                                                                                                                2          3
                                                                                                   2                     3          gth1
                                                                                                          ΔTgo
                                                                                                                           6             7
                                       2 @gth1   @gth1   @gth1    @gth1   @gth1                    36                    7 6             7
                                                                                   0        0        6    ΔTgi           7 6        gth2 7
                                          @T     @Tgi    @Tabo    @Tabi    @Tti                      6                   7 6             7
                                       6 go      @gth2   @gth2    @gth2   @gth2            @gth2 7   6                   7 6             7
                                       6 0                                         0               7 6                   7 6             7
                                       6         @Tgi    @Tabo    @Tabi    @Tti           @Tair;o 76
                                                                                                         ΔTabo           7 6        gth3 7
                                       6         @gth3   @gth3            @gth3   @gth3            76                    7 6             7
                                       6 0       @Tgi    @Tabo     0       @Tti   @Tto      0 76                         7 6             7
                                       6                                                           76                    7 6             7
                                       6 @gth4                                                                                      gth4 7
                                       6 @Tgo
                                                 @gth4
                                                 @Tgi
                                                         @gth4
                                                         @Tabo     0      @gth4
                                                                           @Tti
                                                                                  @gth4
                                                                                  @Tto      0 7    76
                                                                                                     6    ΔTabi          7 6
                                                                                                                         7¼6             7                               (42)
                                       6                                                           76                    7 6             7
                                       6 0        0      @gth5    @gth5
                                                                           0       0       @gth5 76                      7 6        gth5 7
                                       6                 @Tabo    @Tabi                   @Thair;o 76     ΔTti           7 6             7
                                       6         @gth6   @gth6    @gth6   @gth6
                                                                                                   76                    7 6             7
                                       6 0                                         0        0 7      6                   7 6             7
                                       4         @Tgi    @Tabo    @Tabi    @Tti                    56                    7 6        gth6 7
                                                                                           @gth7     6    ΔTto           7 6             7
                                           0      0       0        0       0       0      @Thair;o 4
                                                                                                     6                   7 6             7
                                                                                                                         5 6             7
                                                                                                                           4        gth7 5
                                                                                                         ΔThair;o

                                       Essentially, the thermal design procedure is akin to that of the optical design flowchart in
                                       Figure 1 except that the thermal design variables are seven against the three optical design
                                       variables.

                                         Also, the detailed partial derivative of Equation (42) is given in the supplementary file. The
                                       future values of the thermal design variables and the present values are defined in Equation
                                       (43) as

                                        Tk;iþ1 ¼ Tk;i þ ΔTk ; i ¼ 0; 1; 2; � � � ; N        1;
                                                                                                                                                                         (43)
                                        k ¼ f1; 2; 3; 4; 5; 6; 7g ¼ fgo; gi; abo; abi; ti; to; hairog

                                                                                                                                                                         3
                                       The final thermal design variables are established once the set convergent criterion (ζth ¼ 10                                        ) in
                                       Equation (44) is attained

                                        Tk;iþ1   Tk;i ¼ ζth ; i ¼ 0; 1; 2; � � � ; N       1;
                                                                                                                                                                         (44)
                                        k ¼ f1; 2; 3; 4; 5; 6; 7g ¼ fgo; gi; abo; abi; ti; to; hairog

                                       In accordance with Abbood and Mohammed (2019), Macedo-Valencia (2014) and Alfelleg (2014)
                                       the thermal efficiency of the air-solar-finned PTC, ηth (-) is stated in Equation (45) as

                                                                                                                 �
                                                    HTF output power m  _ air cp air Tf ;o               Tf ;i
                                       ηthermal ¼                     ¼                                              9       Tf ;o ¼ Thair;o ; Tf ;i ¼ Thair;i           (45)
                                                    Input Solar power             Ag G

                                       The overall collector efficiency in Equation (46) is approximated as the difference between the
                                       attenuated optical and thermal efficiencies, which is in concordance with Mohamed (2013) find­
                                       ings (0.3649 ≤ ηcollector ≤ 0.5057)

                                                      �                        �
                                       ηcollector � FR ηoptical        ηthermal ¼ 0:90ð0:7193309                     0:310727Þ � 0:44                                    (46)

                                       2.3. Design of the air-solar-finned absorber
                                       According to Nnamchi et al. (2020); the fin length, lf ðmÞ is expressed as a function of the outer
                                       diameter of the air-solar-finned absorber in Equation (47)

                                       lf ¼ 0:13588π dabo                                                                                                                (47)

                                       Also, Nnamchi et al. (2020) the fin width, wf ðmÞ designed as a function of the outer diameter of the
                                       air-solar-finned absorber in Equation (48)

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                                       wf ¼ 0:0027 þ 0:0620836πdabo              0:0557077π2 d2abo                                        (48)

                                       In the same vein, the number of fins, nf ð Þ is related to the outer diameter of the air-solar-finned
                                       absorber in Equation (49) according to Nnamchi et al. (2020) as

                                                                                                            !
                                                                                 πdabo
                                        nf ¼ 0:7117 þ 0:39525
                                                               0:0027 þ 0:0620836πdabo þ 0:0557077π2 d2abo
                                                                                                   !2                                     (49)
                                                                          πdabo
                                              þ 0:0035                                                ; Integer ðnf Þ
                                                       0:0027 þ 0:0620836πdabo þ 0:0557077π2 d2abo

                                       The design, formulation of the three facets of the designs (optical, thermal and fin) is covered in
                                       Equations (1) to (49); the optimum design variables of the trio-designs on the specification of the
                                       design input data are given in Equations (18), (44) and (46–49), respectively.

                                       3. Results and discussion
                                       The results comprise pertinent tables and informative graphs which are germane to discussion.

                                       3.1. Results presentation
                                       Apparently, some of input data were originated from the existing design data (Mohamed, 2013);
                                       thus, they were not arbitrarily set. Moreover, the tradeoffs among the design variables guided in
                                       the selection of the final design input data, which were subjected to an overall tradeoff in the
                                       simulatory matrices (Equations (16 and 42)) leading to the final design variables.

                                         Tables 1–5 inclusively contain the design input data and the output results; Precisely, Table 1
                                       holds the input data for optical design equations (Equations (1)–(15)), Table 2 contains the input
                                       data for thermal design equations (Equations (16)–(46)), Table 3 encompasses the physical char­
                                       acteristics of the PTSC, Tables 4 and 5 cover the simulated optical and thermal design variable
                                       results. The output results were engaged in Equation (46) to determine the collector efficiency of
                                       0.44 based on the slight obtuse-angled rim design.

                                         Deeply, Figures 1–15 give insight into the design results by revealing the influence of design
                                       variables on the key design parameters (optical efficiency, thermal efficiency, rim angle, and
                                       concentration ratio). Sequentially, Figure 3 shows the dependency of optical efficiency on the
                                       incidence angle and the design was pivoted on the minimum incidence angle. Figure 4 portrays
                                       the reliance of optical efficiency on the rim angle and aperture. Figure 5 indicates the reliability of
                                       optical efficiency on the aperture and the height of the trough. Figure 6 presents the dependence
                                       of optical efficiency on the concentration ratio and absorber outer diameter. Figure 7 depicts the
                                       sensitiveness of concentration ratio of the aperture and absorber sizes. Figure 8 describes the

                                        Table 1. Input data to the optical design equations
                                       S#                                    Description                          Symbol   Unit      Value
                                       1.           The initial absorber tube outer diameter                       dabo    (m)       0.0700
                                       2.           The initial focal distance                                     fpt0    (m)       0.2502
                                       3.           The initial height of the trough                               hpt0    (m)       0.2897
                                       4.           The initial aperture of the trough                             wpt0    (m)       1.1997
                                       5.           The absorber length                                             lab    (m)       1.9800
                                       6.           Convergent factor of the optical design Equation (1) (gop1)     Γ1      (-)      0.8269
                                       7.           Convergent factor of the optical design Equation (2) (gop2)     Γ2      (-)      0.9530
                                       8.           Convergent factor of the optical design Equation (3) (gop3)     Γ3      (-)      0.8271

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                                         Table 2. Input data to the thermal design equations
                                        S#                                         Description                        Symbol      Unit        Value
                                        1.              Intercept factor                                                 γ         (-)         1.00
                                        2.              Incidence angle                                                 θi         (-)         0.00
                                        3.              The absorptivity of glass                                       αg         (-)       0.0023
                                        4.              The emissivity of the inner trough                              εti        (-)         0.25
                                        5.              The transmittance of the inner glass                            τg         (-)         0.90
                                        6.              The absorptivity of the absorber                                εgi        (-)         0.90
                                        7.              The emissivity of the outer absorber                           εabo        (-)         0.23
                                        8.              The absorptance of the absorber tube                           αab         (-)         0.90
                                        9.              The reflectance of absorber tube                                ρab        (-)         0.38
                                        10.             The aperture of the trough                                     wpt        (m)          1.20
                                        11.             The height of the trough                                        hpt       (m)         0.279
                                        12.             The focal distance of the trough                                fpt       (m)         0.254
                                        13.             The curve length of the trough                                  spt       (m)         1.667
                                        14.             The length of the trough                                        lab       (m)          1.20
                                        15.             The ratio of top to base velocity                              λb-t        (-)         0.50
                                        16.             The initial outer temperature of the glass                     Tgo0       (K)        303.15
                                        17.             The initial inner temperature of the glass                     Tgi0       (K)        308.15
                                        18.             The initial outer temperature of the absorber tube             Tabo0      (K)        408.15
                                        19.             The initial inner temperature of the absorber tube             Tabi0      (K)        398.15
                                        20.             The initial inner temperature of the trough                     Tti0      (K)        303.15
                                        21.             The initial outer temperature of the trough                    Tto0       (K)        298.65
                                        22.             The initial outer temperature of the heat transfer fluid      Thair,o0    (K)        363.15
                                        23.             The initial exit fluid temperature                             Tfo0       (K)        339.15
                                        24.             The ambient temperature                                         Ta        (K)        298.15
                                        25.             The sky temperature                                             Tsk       (K)        284.18
                                        26.             The wind speed, outside the PTSC                               uwo       (m/s)        1.200
                                        27.             The wind speed inside the PTSC                                  uwi       (m)         0.400
                                        28.             Number of fins                                                  nf         (-)        8.000
                                        29.             Width of the fins                                               ωf        (m)        0.0199
                                        30.             Length of the fins                                               lf       (m)         0.030
                                        31.             The thickness of the absorber                                   δab       (m)        0.0010
                                        32.             The thickness of the fin                                        δf        (m)        0.0005
                                        33.             Thermal conductivity of the absorber                            kab      (W/mK)       0.163
                                                        Solar irradiance                                                 G       (W/m2)      740.15
                                        34.             Thermal conductivity of the HTF                                kair      (W/mK)       0.012
                                        35.             Convergent factor in the thermal design Equation (1) (gth1)     Ψ1         (-)       0.0383
                                        35.             Convergent factor in the thermal design Equation (2) (gth2)     Ψ2         (-)       0.0357
                                        37.             Convergent factor in the thermal design Equation (3) (gth3)     Ψ3         (-)       0.0215
                                        38.             Convergent factor in the thermal design Equation (4) (gth4)     Ψ4         (-)       0.0853
                                        39.             Convergent factor in the thermal design Equation (5) (gth5)     Ψ5         (-)       0.9075
                                        40.             Convergent factor in the thermal design Equation (6) (gth6)     Ψ6         (-)       0.0922
                                        41.             Convergent factor in the thermal design Equation (7) (gth7)     Ψ7         (-)       0.0853

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                                         Table 3. Physical characteristics of the parabolic trough solar collector (PTSC)
                                        S#                                         Description                            Symbol            Unit        Value
                                        1.              The radius of curvature,                                           Rpt(m)           (m)        0.667
                                        2.              The internal surface area of the parabolic trough collector         As,pt           (m2)       2.456
                                        3.              The rim angle of the PTSC                                            ψr         (degree)      94
                                        4.              The effective geometrical concentration ratio,                       CR              (-)      17
                                        5.              The optical efficiency,                                             ηopt             (-)       0.719
                                        6.              The end loss                                                        χend             (-)       1.000
                                        7.              The incidence angle modifier,                                       κ(θi)            (-)       1.000
                                                                                                                                               2
                                        8.              The outer surface area of the absorber pipe,                        Aabo            (m )       0.2640
                                        9.              The inner surface area of the absorber pipe, Aabi ðm2 Þ             Aabi            (m2)       0.257
                                        10.             The cross sectional area of the absorber,                           Ac;ab           (m2)       0.00022
                                                                                                                                               2
                                        11.             The outer surface areas of the glass cover (glaze                   Ago             (m )       1.206
                                        12.             The inner surface areas of the glass cover (glaze) and               Agi            (m2)       0.256
                                        13.             The inner surface area of the parabolic trough collector,            Ati            (m2)       2.000
                                                                                                                                               2
                                        14.             The outer surface area of the parabolic trough collector,            Ato            (m )       2.134

                                         Table 4. The simulated results of the optical design variables
                                                                         Optical geometry                                 Net power function
                                        Iteration, i              fpt              hpt            wpt             gopt1             gopt2             gopt3
                                                                 (m)               (m)            (m)             (W)               (W)               (W)
                                        0                      0.2502             0.2897         1.1997       1.420E-08      1.6587E-08            −4.7459E-06
                                        1                      0.2505             0.2891         1.1992       1.420E-08      3.7311E-08            −9.4940E-06
                                        2                      0.2511             0.2878         1.1980       1.420E-08      9.1185E-08            −1.8997E-05
                                        3                      0.2524             0.2851         1.1955       1.420E-08      2.4869E-07            −3.8030E-05
                                        4                      0.2552             0.2791         1.1898       1.420E-08      7.6326E-07            −7.6203E-05
                                        5                      0.2552             0.2791         1.1898       1.108E-09      1.4902E-04            −4.6102E-05

                                       dependency of rim angle on the focal distance and height of the trough. Figure 9 exhibits the
                                       responsiveness of rim angle on the aperture and focal distance. Figure 10 shows the sensitivity of
                                       thermal efficiency on the solar irradiance and aperture area.

                                          Figure 11 displays the susceptibility of thermal efficiency on the exit fluid temperature and inlet
                                       fluid temperature; Figure 12 explains the susceptibleness of thermal efficiency on the absorber
                                       inner diameter and HTF velocity; Figure 13 expounds the response of exit fluid temperature on the
                                       solar irradiance and aperture area; Figure 14 explains the reaction of exit fluid temperature on the
                                       HTF velocity and absorber diameter and lastly, Figure 15 shows the dependency of thermal
                                       efficiency on the overall heat transfer coefficient and the concentration ratio.

                                         The main objective of every design is to increase the efficiency or performance of the systems.
                                       Individually, Figures 3–15 give a clearer picture on how to achieve the premium values of the four
                                       design parameters (optical efficiency, thermal efficiency, rim angle, and concentration ratio).

                                         Clearly, Figure 3 shows that the maximum optical efficiency (0.719355) could be attained when
                                       the incidence angle, which is the angle between the sun ray and normal from the trough is zero;
                                       thus, the slight obtuse angle rim design of the present work was carried out at zero incidence
                                       angle which coincides with 12:00noon.

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                                         Table 5. The simulated results of the thermal design variables
                                                                                       Thermal design variables
                                        i                  Tgo         Tgi            Tabo          Tabi       Thair,o       Tti         Tto
                                                           (K)         (K)            (K)           (K)         (K)          (K)         (K)
                                        0.              305.15000   308.15000     418.15000      413.15000   408.15000   303.15000   299.65000
                                        1.              305.15004   308.15000     418.14990      413.16438   408.15000   303.15021   299.65016
                                        2.              305.14611   308.15000     418.14822      413.17717   408.15000   303.15329   299.64928
                                        3.              305.12935   308.15000     418.14232      413.18574   408.15001   303.16402   299.65036
                                        4.              305.12114   308.15237     418.13892      413.40484   408.16205   303.17051   299.64852
                                        5.              305.12114   308.15237     418.13892      413.40484   408.16205   303.17051   299.64852
                                                                                The corresponding thermal design functions
                                        i                  gth1        gth2           gth3          gth4        gth5         gth6        gth7
                                                           (W)         (W)            (W)           (W)         (W)          (W)         (W)
                                        0.                0.0003192 −0.0052588       0.0000928     0.0003107 −0.0004957 −0.0001592      0.0001910
                                        1.                0.0003192 −0.0052588       0.0000928     0.0003107 −0.0004957 −0.0001592      0.0001910
                                        2.               −0.0001019 −0.0195252 −0.0125094 −0.0021633 −0.0004970 −0.0007241              0.0007472
                                        3.                0.0587205 −0.0258267 −0.0260340          0.0497415 −0.0005020 −0.0012826      0.0019453
                                        4.                0.0546539 −0.0621507 −0.0525982          0.0457129 −0.0068348 −0.0027437      0.0032894
                                        5.                0.0546539 −0.0621507 −0.0525982          0.0457129 −0.0068348 −0.0027437      0.0032894

Figure 3. Dependency of optical
efficiency on the incidence
angle.

                                         Vividly, Figure 4 indicates the ephemeral tradeoff between the aperture and rim angle in the
                                       bid to maximize the optical efficiency. The intercept of the two curves may not be the true
                                       balance until overall superposition is carried out. Similarly, Figure 5 presents a tentative tradeoff
                                       between the aperture and the height of the trough in an attempt to maximize the optical
                                       efficiency. Figure 6 presents a zero tradeoff between the absorber outer diameter and concen­
                                       tration ratio (the relative area of the aperture to the surface area of the absorber) in an
                                       endeavour to optimize the optical efficiency. The step size change in Figure 6 indicates the
                                       point at which further increment in both variables amounts to drastic drop in the optical
                                       efficiency. Moreover, the transition points coincide with the design output variables. Alike,
                                       Figure 7 depicts a momentary tradeoff between the aperture and the height of the trough (the
                                       distance between the rim and the apex of the trough) in striving to enhance the concentration
                                       ratio.

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Figure 4. Reliance of optical
efficiency on the rim angle and
aperture.

Figure 5. Reliability of optical
efficiency on the aperture and
height of the trough.

Figure 6. Dependence of optical
efficiency on the concentration
ratio and absorber outer
diameter.

                                         In the same vein, Figure 8 represents a brief tradeoff between the height of the trough and focal
                                       distance (a point of convergence of infinite sun rays) in a stride to boost the rim angle. Likewise,
                                       Figure 9 epitomizes a transitory tradeoff between the focal distance and aperture (the distance

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Figure 7. Sensitiveness of con­
centration ratio of the aperture
and absorber sizes.

                                       between the rims) in an advance to increase the rim angle (the angle between the focal axis
                                       and rim).

                                          Contrarily, the design variables in Figures 10–14 absolutely lack tradeoff with respect to thermal
                                       efficiency. Systematically, Figure 10 depicts the noncompeting and the inverse behaviour of
                                       aperture area and solar irradiance with respect to thermal efficiency. The absence of equilibrium
                                       in Figure 10 is because both variables form the denominator of the thermal efficiency and cannot
                                       compete against each other.

                                          Precisely, Figure 11 describes none competing and the apparent convergence behaviour of exit
                                       fluid temperature and inlet fluid temperature with respect to thermal efficiency. The absence of
                                       equilibrium in Figure 11is because both variables appear in the numerator of the thermal efficiency
                                       and the inlet fluid temperature can never equalize the exit fluid temperature; otherwise, the
                                       thermal efficiency becomes zero.

                                         Specifically, Figure 12 designates none challenging and progressive behaviour of internal air
                                       velocity and an absorber cross-section with respect to thermal efficiency. The absence of equili­
                                       brium in Figure 12 is because both variables form the numerator of the thermal efficiency and
                                       cannot compete against each other.

Figure 8. Dependency of rim
angle to the focal distance and
height of the trough.

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Figure 9. Responsiveness of rim
angle of the aperture and focal
distance.

Figure 10. Sensitivity of ther­
mal efficiency on the solar
irradiance and aperture area.

Figure 11. Susceptibility of
thermal efficiency on the exit
fluid temperature and inlet
fluid temperature.

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Figure 12. Susceptibleness of
thermal efficiency on the
absorber inner diameter and
HTF velocity.

                                          Explicitly, Figure 13 defines none opposing and progressive behaviour of aperture area and solar
                                       irradiance with respect to exit fluid temperature. The nonexistence of equilibrium in Figure 12 is
                                       because both variables form the denominator of the thermal efficiency. Thus, they cannot com­
                                       pete against each other.

                                          Similarly, Figure 14 expresses none contending and inverse behaviour of internal fluid velocity and
                                       absorber cross-section with respect to exit fluid temperature. The absence of equilibrium in Figure 14
                                       is because both variables form the denominator of the thermal efficiency. Thus, would not compete
                                       against each other.

                                         Contrarily, Figure 15 delineates a tradeoff between the overall heat transfer coefficient and
                                       concentration ratio with respect to thermal efficiency. Hence, the emerging of equilibrium in
                                       Figure 15 signifies that tradeoff sets in if the design variables are separated in the denominator
                                       and the numerator of the objective function. Notably, the optical design and performance of
                                       PTSC influence the thermal performance through the concentration ratio. Remarkably, high
                                       concentration ratio diminishes the thermal efficiency and vice versa.

                                         Collectively, the optimum values corresponding to the fleeting balance in Figures 3–5, 7–9 may
                                       not hold as a result of internal conflicts or competition among the design variables leading to

Figure 13. Response of exit
fluid temperature on the solar
irradiance and aperture area.

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Figure 14. Reaction of exit fluid
temperature on the HTF velo­
city and absorber diameter.

Figure 15. The Effect of con­
centration ratio and overall
heat transfer coefficient on the
thermal efficiency.

                                       a final tradeoff among the design variables. Practically, the ultimate tradeoff among the design
                                       variables is manoeuvred by the optical and thermal simulatory matrices at the point of conver­
                                       gence of each simulation. Pertinently, the design was made feasible by the introduction of optical
                                       (Γs) and thermal Convergent factors (Ψs) in Tables 1 and 2, respectively.

                                       3.1.1. Design input data
                                       Tables 1–3 furnish the absolute input data required for both optical and thermal simulations in
                                       Equations (16) and (42), respectively:

                                       3.2. Discussion
                                       Generally, the collector efficiency is dependent on the heat removal factor, the optical and thermal
                                       efficiencies, it is worthy of noting that as the collector efficiency approaches the optical efficiency;
                                       then, the thermal efficiency becomes insignificant and this phenomenon occurs. When the con­
                                       centration ratio is very high and the overall heat transfer coefficient (thermal conductance)
                                       balances heat transport resistance. Thus, the obtuse-angled rim design ought to be developed
                                       with a small rim angle such that the thermal component of the collector efficiency is preserved
                                       and the exit fluid temperature equally raised according to Mohamed (2013). Hence, caution must

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                                       be traded not to design at an extremely obtuse-angled rim. However, the obtuse-angled rim
                                       design has the enablement to evacuate the absorber, by screens the absorber from the cooling
                                       effect of the ambient air. Consequently, thermal loss is drastically minimized by this method of
                                       design compared to the acute-angled and right-angled rim designs, which are exposed to the
                                       ambient cooling resulting in immense thermal losses. However, with the enveloping of the absor­
                                       ber tube in the acute-angled rim design, the performance of the PTSC would be boosted. Although,
                                       the high temperature achieved poses a great threat to the operation of the system like misalign­
                                       ment and the associated problems, which distort the optical performance and at large reduce the
                                       collector performance.

                                         Peculiarly, the right-angled rim design has its merits and demerits; the thermal efficiency may be
                                       higher relative to obtuse angle rim angle design because the concentration ratio is smaller but the
                                       thermal loss is more in the right-angled rim design since the absorber is partially screened from
                                       the cooling effect of the surrounding air.

                                         Thus, gaining high thermal efficiency automatically risks the optical and collector efficiencies of
                                       a PTSC, which is the probable design outcome of the acute-angled rim design (AARD). Also, gaining
                                       high collector efficiency is at the detriment of diminishing the thermal efficiency, this is a likely
                                       design outcome of extreme obtuse-angled rim design (EORAD), which is characterized with high
                                       concentration ratio. Notably, the right-angled rim design (RARD) seems to be suited midway AARD
                                       and OARD, since it does not encourage the risk of neither optical nor thermal efficiencies. RARD
                                       appears to be the most attractive in the design and development of future PTSC but with the pitfall
                                       of appreciable thermal loss is inevitable.

                                         Notwithstanding, the current design for slight OARD (SOARD) serves as an eye-opener to the
                                       enterprising designers of PTSCs to know that they have two primary design variables in striking
                                       balance between the optical and thermal efficiencies; these are the focal distance and the height
                                       of the trough. Equal height of the trough and focal distance support RARD whereas having the
                                       focal distance higher than the height of the trough supports AARD. Lastly, having the focal
                                       distance less than the height of the trough encourages OARD.

                                         Being aware of these intricate outcomes; the future designs should be rested on making, the
                                       collector efficiency to be possibly half of the optical efficiency by selecting an appropriate focal distance
                                       and the height of the trough. This work candidly recommends that slight obtuse-angled rim design
                                       (SOARD) should not be in the excess of 5° ahead of RARD for the efficient performance of the PTSCs.

                                         Furthermore, the present design gave a collector efficiency of 0.44 based on the slight obtuse-
                                       angled rim design (94°), this collector efficiency compared well with those of Macedo-Valencia
                                       et al. (2014) and Mohamed (2013) who designed at obtuse-angled rim of 96° and recorded

Figure 16. The isometric draw­
ings of the reheating unit.

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