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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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
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. Page 2 of 30
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 Page 3 of 30
Nnamchi et al., Cogent Engineering (2020), 7: 1793453 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 Page 4 of 30
Nnamchi et al., Cogent Engineering (2020), 7: 1793453 https://doi.org/10.1080/23311916.2020.1793453 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 Page 5 of 30
Nnamchi et al., Cogent Engineering (2020), 7: 1793453 https://doi.org/10.1080/23311916.2020.1793453 Figure 1. Optical design flowchart. w2pt fpt ¼ 9 fpt
Nnamchi et al., Cogent Engineering (2020), 7: 1793453 https://doi.org/10.1080/23311916.2020.1793453 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 Page 7 of 30
Nnamchi et al., Cogent Engineering (2020), 7: 1793453 https://doi.org/10.1080/23311916.2020.1793453 " � �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. Page 8 of 30
Nnamchi et al., Cogent Engineering (2020), 7: 1793453 https://doi.org/10.1080/23311916.2020.1793453 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) Page 9 of 30
Nnamchi et al., Cogent Engineering (2020), 7: 1793453 https://doi.org/10.1080/23311916.2020.1793453 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 Page 10 of 30
Nnamchi et al., Cogent Engineering (2020), 7: 1793453 https://doi.org/10.1080/23311916.2020.1793453 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 Page 11 of 30
Nnamchi et al., Cogent Engineering (2020), 7: 1793453 https://doi.org/10.1080/23311916.2020.1793453 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 Page 12 of 30
Nnamchi et al., Cogent Engineering (2020), 7: 1793453 https://doi.org/10.1080/23311916.2020.1793453 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): Page 13 of 30
Nnamchi et al., Cogent Engineering (2020), 7: 1793453 https://doi.org/10.1080/23311916.2020.1793453 � � 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) Page 14 of 30
Nnamchi et al., Cogent Engineering (2020), 7: 1793453 https://doi.org/10.1080/23311916.2020.1793453 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) Page 15 of 30
Nnamchi et al., Cogent Engineering (2020), 7: 1793453 https://doi.org/10.1080/23311916.2020.1793453 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 Page 16 of 30
Nnamchi et al., Cogent Engineering (2020), 7: 1793453 https://doi.org/10.1080/23311916.2020.1793453 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 Page 17 of 30
Nnamchi et al., Cogent Engineering (2020), 7: 1793453 https://doi.org/10.1080/23311916.2020.1793453 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. Page 18 of 30
Nnamchi et al., Cogent Engineering (2020), 7: 1793453 https://doi.org/10.1080/23311916.2020.1793453 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. Page 19 of 30
Nnamchi et al., Cogent Engineering (2020), 7: 1793453 https://doi.org/10.1080/23311916.2020.1793453 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 Page 20 of 30
Nnamchi et al., Cogent Engineering (2020), 7: 1793453 https://doi.org/10.1080/23311916.2020.1793453 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. Page 21 of 30
Nnamchi et al., Cogent Engineering (2020), 7: 1793453 https://doi.org/10.1080/23311916.2020.1793453 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. Page 22 of 30
Nnamchi et al., Cogent Engineering (2020), 7: 1793453 https://doi.org/10.1080/23311916.2020.1793453 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. Page 23 of 30
Nnamchi et al., Cogent Engineering (2020), 7: 1793453 https://doi.org/10.1080/23311916.2020.1793453 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 Page 24 of 30
Nnamchi et al., Cogent Engineering (2020), 7: 1793453 https://doi.org/10.1080/23311916.2020.1793453 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. Page 25 of 30
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