1 Introduction

The reduction of fossil fuels consumption and global climate change is critical a worldwide transition to an economy based on renewable energy. Global policies must be driven and included to develop this kind of technology [1]. Usage of renewable energy sources has growth in the las years, specifically the energy consumption on European Union (EU) where it showed an increase of almost a 10%, between 2004 to 2018. It is expected to increase by an additional 20% in 2020 and a 32% by 2030 [2]. To achieve this goal, several reach works were conducted [3,4,5]. Energy management has as a key role in energy consumption, turning it into a highly studied subject [6].

Within renewable energies, solar heating trough collectors systems is one of the most convenient energy to transfer the fossil fuel energy to a clean energy production to be used in numerous applications as building heating [7], agriculture [8], dehydration [9], photocatalysis [10], energy production [11], water heating [12], among many others. Solar energy system is highly studied to improve the thermal efficiency by mechanical designs, nanotechnologies and materials properties. In the first case, the location, environment conditions and heat exchanger design are involved to guaranty a proper operation in all solar collector, so many architecture can be studied to find the best arrangement on a solar field.

Some nature structures are currently used to increase the strength and energy absorption capacity on mechanical systems [13,14,15]. The honeycomb structure is highly studied due to their simplicity geometry and the possibility of being adapted in engineering applications [16, 17] such as aerospace, automotive [18], transportation, [19] biomedicine [20] and biomechanics [21]. A honeycomb conjecture was proposed by Pappus of Alexandria and proved by Thomas Hales who described the structure mathematically as ‘‘any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal grid’’, [22].

Due to the mechanical properties was studied and analyzed experimentally under certain conditions impact resistance [23], absorption [24, 25] and compression [26, 27]. In all studies was proved that the honeycomb structure has a great energy absorbing capacity and load bearing, so the regular hexagonal cell has been researched in a microstructure and macrostructure scale and the relation between both, [28]. In order to enhancement the function and the mechanical properties of a honeycomb some approaches have been discussed about embedded honeycomb structures [29, 30], tandem honeycombs [31] and honeycomb‐filled structures [32, 33].

In [34] was designed a 3D-printed auxetic honeycomb structure and their mechanical properties were analyzed by theoretical calculations, computer simulations, and experimental evaluation of Poisson's ratio, in this paper is presented an optimal cell determined for 3D printed samples. Results show that the structure based on auxetic hexagonal cell can withstand almost 1.75 times more very low cycle fatigue cycles than the similar non-auxetic structure, and the structure is a promising technology for applications in sports and medicine.

Similarly [35] exposes the mechanical enhancement of graphene resin using a constructing honeycomb conductive ring. The principles of Lorentz-like and Drude-type negative permittivity resulted in an increase of negative permittivity since filling multi-walled carbon nanotubes (MWCNTs) with a mass ratio of 2:1, lead to the honeycomb in case to propose a mechanism for metamaterials with low dielectric loss and high mechanical property.

On the other hand, absorption capacity was researched in [36] the paper presents an experimental study of the relationship between initial single-pass efficiency and structural parameters of honeycomb adsorption filters. The study established a void ratio ranged from 0.293 to 0.919, a face air velocity between 0.1–1.8 m/s, air temperature of 25 °C, and a relative humidity of 40%. Results show that the method relates and acceptable precision when the void ratio is used instead of the hydraulic diameter of the granular adsorbent in the empirical equation to calculate the initial single-pass efficiency.

Furthermore, the adsorption efficiency of a thin honeycomb adsorption filter with efficiency of less than approximately 30% should be the same along the flow channel, regardless of the effects of changes in the concentration. The analytical optimization of polymeric honeycomb structures made from thermoplastic polyurethane to achieve density-graded structures with combined desired mechanical properties was described in [37].

A novel tube‐reinforced absorbent honeycomb sandwich structure was studied in [38]. Excellent properties as mechanical, absorption and electromagnetic guarantee that the novel honeycomb sandwich structure proposed in this work is much more competitive by electromagnetic wave stealth application and load‐carry structures. The study determined that the energy absorption was superior than an empty absorbent and CFRP pipe by 30.34%. Also, peak stress, energy absorption and elastic modulus was increased by 621.69%, 327.86% and 55.74%, respectively.

Likewise, energy absorption on structures was analyzed under nature geometric configurations, in this case the bionic honeycomb tubular nested structure (BHTNS) was inspired by the micro-architecture of bamboo vascular bundles, results showed that the specific energy absorption (SEA) of BHTNS was as high as 29.3 J/g, a theoretical model was also developed to predict the mean crush force Pm, which was in good agreement with the numerical simulation [39].

Furthermore, solar energy applications also involved the honeycomb structure in certain technologies, first in [40] is presented the numerical and experimental studied of a solar collector designed based on a honeycomb structure to conform a blackbody heat absorbing core. The design was tested under a solar radiation of 200 W/m2 to 600 W/m2 with PV coverage ratio of 15%, 30%, 45%, 60%, 75% and 90%. In the result was evident that the honeycomb solar air collector integrated with well-designed PV configuration would be able to achieve suitable thermal efficiency, so the maximum instantaneous efficiency reached 64% when PV coverage ratio was 45%.

Similarly [41], a solar application as phase change material (PCM) was proposed a novel solar air collector combined the PCM-Rubitherm RT54HC with aluminum honeycomb. The effect of using honeycomb core on the collector thermal performance against heat storage with PCM under natural convection conditions were compared. The test showed a production of useful energy in honeycomb collector lasted 469 min more than the PCM collector. Also, the thermal efficiencies under daylight obtained were 10.1% on honeycomb collector and 10.9% on PCM collector. As conclusion researcher mentioned that the use of honeycomb core as heat conductivity enhancer material is very functional, during the discharge period.

The aim of this paper is present the results of the thermal analysis of a novel solar collector designed based on the Honey-Comb conjecture studied under different connections, with the purpose of enhancing the heat transfer on the solar collector. Several structures were proposed considering a serial and parallel connection, each one was designed and simulated in SolidWorks® software Flow Simulation tool, the study consider different boundary conditions as mass flow and Solar radiation on collectors surface.

The paper has been organized as follows: the first section includes introduction and nomenclature section. Second, the methods section describes the model and thermal description as well as boundary conditions. The third section includes results where validation process during CFD simulation is presented. Finally, the paper closes with conclusions and acknowledgment sections.

2 Methods

Parameter as useful heat [\(\dot{Q}\)], collector area [\({A}_{c}\)] and total incident radiation on collector [\(I\)] are related to calculate the thermal efficiency the mathematical ratio is presented in Eq. 1.

$${\eta }_{t}=\frac{\dot{Q}}{I\times {A}_{c}}$$
(1)

Due to the collector geometry design the collector is calculated as Eq. 2, where [\(P\)] is total perimeter, [L] hexagon side length and [\(h\)] the hexagonal height, the last one can be determined by Eq. 3

$${A}_{c}=\frac{P\times h}{2}$$
(2)
$$h=\sqrt{{L}^{2}-{\left(\frac{L}{2}\right)}^{2}}$$
(3)

Useful heat involve parameters as flow mass [\(\dot{m}\)] in the working fluid, specific heat [\({C}_{p}\)] and delta temperature [\(\Delta T\)] achieved at outlet collector, Eq. 4.

$$\dot{Q}=\dot{m}{C}_{p}\Delta T$$
(4)

In Eq. 5 is presented a different mathematical expression to calculate the useful heat in this case, available energy and loss energy are involve, where [\({F}_{R}\)] is the removal heat factor and [\(Ul\)] the general loss in the collector.

$$\dot{Q}={A}_{c}{F}_{R}\left[I-Ul\Delta T\right]$$
(5)

The same parameters as flow mass, specific heat, collector area, loss coefficient and efficiency factor [\(F{\prime}\)] are related in removal heat factor calculation as Eq. 6.

$${F}_{R}=\frac{\dot{m}{C}_{p}}{{A}_{c}Ul}\left[1-{e}^{\left(-\frac{{A}_{c}Ul F^{\prime}}{\dot{m}{C}_{p}}\right)}\right]$$
(6)

Furthermore, the loss coefficient can be determined based on convection and radiation on receiver surface, the mathematical expression is shown in Eq. 7. Where [\({h}_{c,c-a}\)], is the coefficient of convection between the cover and the environment; [\({h}_{r,c-a}\)] is the radiation coefficient between the cover and the environment; and [\({h}_{r,r-c}\)] is the radiation coefficient between the receiver and the cover.

$$UL={\left[\frac{{A}_{r}}{\left({h}_{c,c-a}+{h}_{r,c-a}\right)\times \left({A}_{r}\right)}+\frac{1}{{h}_{r,r-c}}\right]}^{-1}$$
(7)

On the other hand, the factor efficiency defined in Eq. 8 cosist on the relation between loss coefficient and tranfer heat coefficient [\({h}_{fi}\)] and other parameter as [\({{D}_{i}, D}_{o}\)], receiver internal and external diameter resectively, pipe thermal conductivity [\(k\)].

$$F{\prime}=\frac{1/UL}{\frac{1}{UL}+\frac{{D}_{o}}{{h}_{fi}{D}_{i}}+\frac{{D}_{o}}{2k}\times \mathit{ln}\frac{{D}_{o}}{{D}_{i}}}$$
(8)

2.1 Heat exchanger

The solar collector proposed in this paper consist primarily on a heat exchanger array conformed by 11 copper pipes, which is coated by a lower and upper cover, the last consist on a glass cover coupled with Fresnel lens, the geometry of the collector is based on the bees conjecture construction where hexagonal shape are assembled to formed a honey-comb, so each collector represent one space in the honey-comb construction. The principal dimensions of the single collector can be consulted in Table 1.

Table 1 Array Dimensions
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The 3D model design of the solar collector is observed in Fig. 1.

Fig. 1
figure 1

Single collector

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To study the influence of several configurations to connect the solar collector, different architectures were established, one collector, arrays with two or three collectors in series and parallel were analyzed under the conditions presented in Table 2.

Table 2 Coupled Connections
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Figure 2 shows the collectors connection in series Fig. 2A, collectors connection in a parallel configuration Fig. 2B and finally a combined connection where collectors are connected in series and parallel conformed a simple honey-comb conjecture in Fig. 2C.

Fig. 2
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Collectors Connection

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2.2 Working fluid

Physical characteristics of working fluid (Syltherm 800) used in the proposed collector in this paper, are presented in Table 3, where parameters as density, specific heat, thermal conductivity, viscosity were stablished in the analysis. The parameters correlate at fluid properties at 300 K, which correspond to inlet temperature in all studies.

Table 3 Syltherm 800 Physical characteristics
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2.3 Boundary conditions

The boundary conditions presented in this paper study the thermal performance of each proposed collector with different configurations in the connection, under several parameters as inlet temperature, flow mass and incident radiation on collector’s surfaces. The working fluid inlet temperature was defined in 300 K, the flows mass conducted inside pipes were set as (0.051 kg/s, 0.10 kg/s and 0.17 kg/s), finally incident radiation was set as (800 \(W/{m}^{2}\), 950 \(W/{m}^{2}\) and 1050 \(W/{m}^{2}\)). Environment and other conditions involved in the study are registered in Table 4.

Table 4 Simulation Environment conditions
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SolidWorks® software with the module Flow Simulation was used to carry out all simulations presented in Results sections. The Computational Fluid Dynamics (CFD) analysis is based on Navier–Stokes Equations, which consist on a mathematical formulation of mass, momentum and energy conservation laws for fluid flows, as presented Eqs. 9–11.

$$\frac{\partial \rho }{\partial t}+\frac{\partial }{\partial {x}_{i}}\left(\rho {u}_{i}\right)=0$$
(9)
$$\begin{aligned}\frac{\partial \rho {u}_{i}}{\partial t}+\frac{\partial }{\partial {x}_{j}}\left(\rho {u}_{i}{u}_{j}\right)&+\frac{\partial p}{\partial {x}_{i}}=\frac{\partial }{\partial {x}_{j}}\left({\tau }_{ij}+{{\tau }^{R}}_{ij}\right)+{S}_{i}\\&i=\mathrm{1,2},3\end{aligned}$$
(10)
$$\begin{aligned}\frac{\partial \rho H}{\partial t}+\frac{\partial \rho {u}_{i}H}{\partial {x}_{i}}=&\;\frac{\partial }{\partial {x}_{i}}\left({u}_{j}\left({\tau }_{ij}+{{\tau }^{R}}_{ij}\right)+{q}_{i}\right)\\&+\frac{\partial p}{\partial t}-{{\tau }^{R}}_{ij}\frac{\partial {u}_{i}}{\partial {x}_{j}}+\rho \varepsilon +{S}_{i}{u}_{i}\\&+{Q}_{H}, H=h+\frac{{u}^{2}}{2}\end{aligned}$$
(11)

The fluid velocity is represented by [\(u\)], fluid density by [\(\rho\)], the mass-distributed external force per unit mass is represented by [\({S}_{i}, h\)] is the thermal enthalpy, heat source or sink per unit volume by [\({Q}_{H},\)], the viscous shear stress tensor is [\({\tau }_{ij}]\) and [\({q}_{i}]\) is the diffusive heat flux. Results section down below presents the analysis on outlet temperature in working fluid, solid temperature, useful heat rate and the thermal distribution inside collectors already described.

3 Results

Results section is divided by collector’s connection, first single connection analysis is presented, then series and parallel connection thermal results and finally results section close with the combined connection study.

3.1 Single collector-A0

A0 Collector outlet temperature, delta temperature and useful heat are registered in Table 5. As is evident the maximum temperature achieved in working fluid was 307 K at the highest solar radiation of 1050 \(W/{m}^{2}\) and the lowest mass flow of 0.051 kg/s, the total temperature increase was near to 7 K. Furthermore, the minor temperature at outlet collector was 301.82 K means a delta temperature of 1.82 K the temperature was obtained at 800 \(W/{m}^{2}\) and 0.17 kg/s as solar radiation and mass flow respectively.

Table 5 A0 Results
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Solid temperature distribution in collector A0 at 800 \(W/{m}^{2}\) is shown in Fig. 3A, the maximum temperature in this case was near to 327.6 K the highest temperature is concentrated in less than 50% on the collector surface. A better distribution was achieved at 950 \(W/{m}^{2}\) in Fig. 3B where the maximum temperature was near to 334 K, at the highest Solar radiation heat results show a solid temperature of 336 K is evident a thermal distribution superior that 80% on collector surface, Fig. 3C. The total increase in solid temperature for each radiation was 37.6 K, 44 K and 46 K respectively.

Fig. 3
figure 3

Solid Temperature in A0 Collector

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Useful heat results in collector A0 at is presented graphically in Fig. 4, as expected the maximum useful heats were obtained at the maximum solar radiation in the study, in all cases the useful heat is increased regard to mass flow, which means a greater flow and radiation involve a higher useful heat in the capture system. A total of 787W was achieved in collector A0 at 800 \(W/{m}^{2}\) and 0.051 kg/s, almost a 40% was increased in the useful heat obtained at 0.10 kg/s and a 49% was the increase between the lowest and the highest mass flow. At 950 \(W/{m}^{2}\) the useful heat at 0.051 kg/s was 1190W the result at 0.10 kg/s was 19.34% superior and at 0.17 kg/s was 27% higher. Likewise, results show a useful heat of 1377W at 1050 \(W/{m}^{2}\) and 0.051 kg/s an increase of 9% was obtained at 0.10 kg/s and a total of 39% at 0.17 kg/s the highest in this connection. As is evident the increase percentage decrease at the solar radiation increase.

Fig. 4
figure 4

Useful Heat in A0 Collector

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Internal fluid (Syltherm 800) conduction inside A0 collector cooper pipes can be observed in Fig. 5.

Fig. 5
figure 5

Fluid Temperature in A0 Collector. Collectors in Series-AS1

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Temperatures higher than 315 K were obtained in the internal environment around pipes which relates the air temperature inside the collector. Specifically, Syltherm outlet temperature obtained at 800 \(W/{m}^{2}\) and 0.051 kg/s was 304 K in Fig. 5A, at the same solar radiation and 0.10 kg/s the outlet temperature in Fig. 5B was round to 303 K, Syltherm temperature was 301.82 K with a mass flow of 0.17 kg/s in Fig. 5C.

On the other hand, at 950 \(W/{m}^{2}\) results shows an outlet temperature of 306.15 K with a mass flow of 0.051 kg/s in Fig. 5D, at 0.10 kg/s the resultant temperature was 303.67 K Fig. 5E and 302.34 K at 0.17 kg/s, Fig. 5F. Thermal distribution on A0 collector with a total incident radiation of 1050 \(W/{m}^{2}\) is presented in Fig. 5G–I, where the maximum temperature in fluid was 307.12 K, 304 K and 303 K at 0.051 kg/s, 0.10 kg/s and 0.17 kg/s, respectively.

Similarly as collector A0, the system AS1 was simulated and analyzed, Table 6 register the thermal results of this collector series connection. In this study the lowest outlet temperature of 304.26 K was achieved with 800 \(W/{m}^{2}\) and 0.17 kg/s, the temperature increase was near to 3 K. On the other hand, the maximum temperature at working fluid was 318.12 K means a delta temperature of 18.12 K this result was achieved at 1050 \(W/{m}^{2}\) and 0.051 kg/s. The temperature increase regard to a single connection maximum result was a delta of 11 K.

Table 6 AS1 Results
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In Fig. 6, solid temperature in connection AS1 was analyzed at each incident radiation. Specifically at 800 \(W/{m}^{2}\), Fig. 6. A show a maximum solid temperature of 331 K on collector surface, however the most part of the total surface remain at lower temperatures. A better distribution on surface temperature were obtained at 950 \(W/{m}^{2}\) and 1050 \(W/{m}^{2}\) on Fig. 6B and C where the maximum temperature obtained in the analysis at the lowest mass flow was 338 K and 343 K, respectively, which means a delta temperature of 48 K and 53 K. The last represent an increase of 7 K regard to A0 collector at the same boundary conditions.

Fig. 6
figure 6

Solid Temperature in AS1 Collector

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Useful heat results in collectors AS1 is illustrated graphically in Fig. 7, likewise results in A0 connection the maximum useful heats were obtained at the maximum solar radiation.

Fig. 7
figure 7

Useful Heat in AS1 Collector

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At 800 \(W/{m}^{2}\) and 0.051 kg/s the useful heat obtained was 2704W, at 0.10 kg/s the useful heat was increased a 0.14%, instead at 0.17 kg/s an increase of 1.51% was evident. The useful heat with 950 \(W/{m}^{2}\) at the lowest mass flow was 3160W the result at 0.10 kg/s was 0.61% higher and at 0.17 kg/s was 1.01% superior. Similarly, at 1050 \(W/{m}^{2}\) and 0.051 kg/s the study show a useful heat of 3505W, the increase regard to 0.10 kg/s was near to 2.42% and a total increase of 2.64% was obtained at 0.17 kg/s. Comparing A0 connection the useful heat in AS1 is higher on a 88.51%, so coupled two collectors in a series connection increase the useful heat by approximately 89%.

The thermal profile inside the collectors in AS1 connection under several boundary conditions can be observed in Fig. 8. The outlet temperature obtained in Syltherm at 800 \(W/{m}^{2}\) and 0.051 kg/s was 314 K Fig. 8A, at the same solar radiation and 0.10 kg/s the temperature achieved was 307 K in Fig. 8B and 304 K was the resultant temperature at 0.17 kg/s in Fig. 8C. At 950 \(W/{m}^{2}\) the outlet temperature achieved at the lowest flow mass was 316.34 K Fig. 8D, at 0.10 kg/s the temperature obtained was 308.22 K Fig. 8E and 304.97 K at0.17 kg/s in Fig. 8F. Thermal distribution on AS1 collectors with a total incident radiation of 1050 \(W/{m}^{2}\) and a flow mass of 0.051 kg/s is presented in Fig. 8G, where the maximum temperature in fluid was 318.12 K, Fig. 8H illustrated results temperature at 0.10 kg/s where the maximum temperature in Syltherm was 309.28 K and a total of 305.58 K was the outlet temperature at 0.17 kg/s in Fig. 8I. As is evident in the figures the thermal distribution is more uniform when solar radiation is increased.

Fig. 8
figure 8

Fluid Temperature in AS1 Collector

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3.2 Collector in parallel-AP1

AP1 Collector outlet temperature, delta temperature and useful heat are described in Table 7. In this connection the maximum temperature achieved in Syltherm 800 was 340 K at the highest solar radiation of 800 \(W/{m}^{2}\) with a mass flow of 0.0086 kg/s, the delta temperature regard to inlet temperature was near to 40 K. The lowest outlet temperature was 304.32 K which means an increase in fluid temperature of 4.32 K this result was obtained at 800 \(W/{m}^{2}\) and 0.17 kg/s.

Table 7 AP1 Results
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Furthermore, solid temperature results in connection AP1 at 800 \(W/{m}^{2}\) is shown in Fig. 9A, the maximum temperature obtained was 338.67 K, a total of 349.20 K was obtained at 950 \(W/{m}^{2}\) in Fig. 9B. In Fig. 9C is observed the surface temperature distribution at 1050 \(W/{m}^{2}\), where the maximum temperature in solid was 353 K. The total increase in solid temperature for each radiation was 49 K, 50 K and 63 K respectively. Comparing the best result in solid temperature of AP1 analysis and connection A0 results the difference between both is near to 17 K, regard to AS1 connection the difference is near to 9.7 K.

Fig. 9
figure 9

Solid Temperature in AS1 Collector

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In Fig. 10 shows the useful heat results in collector AP1, similar to previous results the maximum useful heat was obtained at the maximum solar radiation. At 800 \(W/{m}^{2}\) and 0.051 kg/s the total useful heat obtained was 2708W, near to 1.43% was the increase in the useful heat at 0.10 kg/s and a near to 2.86% was the increase between the lowest and the highest mass flow. At 950 \(W/{m}^{2}\) the useful heat at 0.051 kg/s was 3288.506W the heat at 0.10 kg/s was 0.12% superior and at 0.17 kg/s was 0.6% higher. Furthermore, a useful heat of 3657W was evident at 1050 \(W/{m}^{2}\) and 0.051 kg/s an increase of 0.16% was obtained at 0.10 kg/s and a 0.65% at 0.17 kg/s. In contrast to A0 connection the useful heat in AP1 is higher on a 92.91%, and regard to AS1 connection the maximum useful heat is superior by 2.33%, which means that two collectors coupled in a parallel connection increase the useful heat by approximately 93% regard to one single collector, and 2% regard to two collectors in series.

Fig. 10
figure 10

Useful Heat in AP1 Collector

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Syltherm flow conduction inside AP1 connection in the collectors is presented in Fig. 11. Working fluid outlet temperature obtained at 800 \(W/{m}^{2}\) and 0.051 kg/s was 314 K in Fig. 11A, at the same solar radiation and 0.10 kg/s the outlet temperature in Fig. 11B was near to 307.1 K, Syltherm temperature was 304.32 K with a mass flow of 0.17 kg/s in Fig. 11C.

Fig. 11
figure 11

Fluid Temperature in AP1 Collector

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Likewise, at 950 \(W/{m}^{2}\) results shows an outlet temperature of 317 K with a mass flow of 0.051 kg/s in Fig. 11D, at 0.10 kg/s the resultant temperature was 308.51 K Fig. 11E and 305.13 K at 0.17 kg/s, Fig. 11F. The thermal profiles in fluid of AP1 connection with a total incident radiation of 1050 \(W/{m}^{2}\) can be observed in Fig. 11G–I, where the maximum temperature in fluid was 318.91 K, 309.47 K and 305.71 K at each flow mass proposed.

3.3 Honey-comb connection-AC1

The last configuration of the honey-comb solar collectors was AC1 connection. The study results related to the outlet temperature, delta temperature and useful heat are presented in Table 8. The maximum temperature obtained in Syltherm was 344.45 K at the highest solar radiation of 1050 \(W/{m}^{2}\) and the lowest mass flow 0.0086 kg/s, the total temperature increase was around to 44.45 K. The minor temperature at outlet collector was 310.45 K with a delta temperature of 10.45 K this temperature was obtained at 800 \(W/{m}^{2}\) and 0.17 kg/s as mass flow.

Table 8 AC1 Results
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Solid temperature distribution in collector AC1 at 800 \(W/{m}^{2}\) is presented in Fig. 12A, the maximum temperature was near to 330 K as is evident the highest temperature is concentrated in more than 35% on the collector surface. A better distribution was achieved at 950 \(W/{m}^{2}\) in Fig. 12B where the maximum temperature was near to 360 K. Furthermore, at the highest Solar radiation was possible to obtain a solid temperature of 365 K also is evident a thermal distribution superior that 90% on collector surface, Fig. 12C. The total increase in solid temperature for each radiation was 35 K, 61 K and 67 K respectively.

Fig. 12
figure 12

Solid Temperature in AC1 Collector

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Useful heat results in collector AC1 at is showed graphically in Fig. 13, as expected the maximum useful heats were obtained at the maximum solar radiation. A total of 4062.27W was achieved in collector AC1 at 800 \(W/{m}^{2}\) and 0.051 kg/s, almost a 0.86% was increased in the useful heat obtained at 0.10 kg/s and a 1.11% was the increase in the useful heat obtained at 0.17 kg/s. At 950 \(W/{m}^{2}\) the useful heat at 0.051 kg/s was 4836W the result at 0.10 kg/s was 1.84% superior and at 0.17 kg/s was 2.13% higher. Similarly, results show a useful heat of 5416W at 1050 \(W/{m}^{2}\) and 0.051 kg/s an increase of 0.36% was obtained at 0.10 kg/s and a total of 1.19% at 0.17 kg/s the highest in this connection.

Fig. 13
figure 13

Useful Heat in AC1 Collector

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Working fluid (Syltherm 800) inside of AC1 collector can be observed in Fig. 14. Specifically, Syltherm outlet temperature obtained at 800 \(W/{m}^{2}\) and 0.051 kg/s was 321 K in Fig. 14A, at the same solar radiation and 0.10 kg/s the outlet temperature in Fig. 14B was round to 310.6 K, Syltherm temperature was 306 K with a mass flow of 0.17 kg/s in Fig. 14C. On the other hand, at 950 \(W/{m}^{2}\) results shows an outlet temperature of 325 K with a mass flow of 0.051 kg/s in Fig. 14D, at 0.10 kg/s the resultant temperature was 313 K Fig. 14E and 308 K at 0.17 kg/s, Fig. 14F. Thermal distribution on AC1 collector with a total incident radiation of 1050 \(W/{m}^{2}\) is presented in Fig. 14G–I, where the maximum temperature in fluid was 328 K, 314 K and 308.5 K at 0.051 kg/s, 0.10 kg/s and 0.17 kg/s, respectively.

Fig. 14
figure 14

Fluid Temperature in AC1 Collector

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In contrast to A0 connection the useful heat in AC1 is higher on a 187%, regard to AS1 and AP1 connections the maximum useful heats is superior by 52.33% and 48.86% respectively, which means that a honeycomb connection conformed by 3 collectors increase the useful heat by approximately 187% regard to one single collector, 52.33% regard to two collectors in series and 48.86% regard to two collectors in parallel connection.

4 Summary results

Outlet temperature in the working fluid (Syltherm 800) and useful heat obtained in each collector’s connection was conformed on a summary results table presented in Table 9. The best result as expected was achieved with honey-comb connection which involves three collectors, the outlet temperature is superior by 3% and 2.85% regard to the maximum temperature with AS1 and AP1 connection, respectively. Regard to A0 one single collector the difference on the maximum fluid temperature is almost 6.8%. The maximum temperature in the analysis was obtained at the highest solar radiation of 1050 \(W/{m}^{2}\) and the lowest flow mass of 0.052 kg/s. On the other hand, the best result in useful heat thermal parameter in all the study was achieved at the highest solar radiation of 1050 \(W/{m}^{2}\) and the highest flow mass of 0.17 kg/s.

Table 9 Summary Results
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5 Conclusions

In the analysis was possible to find that the best result was obtained with honey-comb connection AC1, the outlet temperature was higher by 3% and 2.8% regard to the maximum temperature with AS1 and AP1 connection, respectively. Comparing one single collector outlet temperature results the delta on the maximum fluid temperature was near to 7%. The maximum temperature was achieved at the highest solar radiation of 1050 \(W/{m}^{2}\) and the lowest flow mass of 0.0086 kg/s. On the other hand, the best result in useful heat thermal parameter in all the study was achieved at the highest solar radiation of 1050 \(W/{m}^{2}\) and the highest flow mass of 0.127 kg/s.

Thermal distribution on collector’s surface shows different scales of temperature along pipes, in the first case A0 structure the highest temperature was concentrated in less than 50% on the collector surface when solar radiation was lower, when the radiation was increased was evident a thermal distribution superior that 80% on collector surface. In AS1 and AP1 analysis a temperature distribution at the highest value on surface was less that 25% when the collectors were under a solar radiation of 800 \(W/{m}^{2}\), at 950 \(W/{m}^{2}\) and 1050 \(W/{m}^{2}\), the highest temperature was distributed on a 90% of the total surface. Similarly, AC1.

Shows a highest temperature concentrated in more than 35% on the collector surface at 800 \(W/{m}^{2}\) and thermal distribution superior that 90% on collector surface was obtained at higher solar radiations.

Comparing useful heat results in each connection was possible to determinate that AS1 honey-comb structure which coupled two collectors in a series connection increase the useful heat by approximately 89% regard to one single collector as A0 structure. Furthermore, two collectors coupled in a parallel connection as AP1 increase the useful heat by 93% regard to A0 connection, and 2% regard AS1 connection. Finally, a honey-comb structure conformed by three collectors (AC1) shows an increase near to 187%, regard to one single collector, comparing the other structures AS1 and AP1 connections the total increase in the useful heat obtained with AC1 was 52% and 49% respectively.