Optimization of a solar collector with evacuated tubes using the simulated annealing and computational fluid dynamics

Energy Conversion and Management 166 (2018) 343–355

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Optimization of a solar collector with evacuated tubes using the simulated annealing and computational fluid dynamics

T

⁎

J. Arturo Alfaro-Ayalaa, , Oscar A. López-Núñeza, F.I. Gómez-Castroa, J.J. Ramírez-Minguelaa, A.R. Uribe-Ramíreza, J.M. Belman-Floresb, Sergio Cano-Andradeb a

Department of Chemical Engineering, University of Guanajuato, DCNE, Col. Noria Alta s/n, C.P. 36050 Guanajuato, Gto., Mexico Department of Mechanical Engineering, University of Guanajuato, DICIS, Carretera Salamanca-Valle de Santiago km. 3.5+1.8, Comunidad de Palo Blanco, C.P. 36885 Salamanca, Gto., Mexico b

A R T I C LE I N FO

A B S T R A C T

Keywords: Design of experiments Simulated annealing Computational fluid dynamics Optimal solar collector Thermal performance Thermal efficiency

In this work, the optimization of a low temperature, water-in-glass, evacuated tubes solar collector is presented. The process of optimization combined the simulated annealing method and a computational fluid dynamics model. The numerical study was carried out in three dimensions, steady-state and laminar regime. A design of experiments study via computational fluid dynamics was carried out with two levels and five parameters, 25, the parameters with significance in the performance of the collector were found from a commercial collector. This collector was used as base case in the process of optimization. In the optimization process, the absorber area was analyzed under three different cases because of the combination of geometrical parameters: length, diameter and number of tubes. Thus, 259 different collector geometries were constructed and modeled. Results from the design of experiments showed that the significant parameters on the thermal performance of the solar collector are: the diameter of the tubes, the absorber area, and the mass flow rate. Results of the optimization process showed that the minimum absorber area for an optimal geometry is 2.49 m2, which is 19.4% lower than the commercial geometry considering the same outlet temperature. The diameter of the tubes increased around 30%, the length of the tubes decreased 40%, the cost of the optimal geometry and the number of evacuated tubes decreased 38.9% and the thermal efficiency increased 26.3%, compared to the commercial geometry. The results of this work can be helpful in further specific applications where the maximum performance and the minimum costs are important, such as: the design of low temperature, water-in-glass, evacuated tubes solar collector networks for heating water in swimming pools, buildings, hospitals and industries.

1. Introduction Solar energy is the most appropriate source that could satisfy the growing demand for energy worldwide [1]. It has an important effect in the reduction of the greenhouse gas emissions [2] and can be converted directly into thermal energy by using solar collectors. These equipments collect energy from the solar radiation by absorbing and transferring it to a working fluid. This is the case of the water-in-glass evacuated tubes solar collectors (ETSC), which remain stationary regardless of the sun’s position and are suitable for low temperature applications (lower than 100 °C). The water-in-glass tubes are made of borosilicate glass, each one consisting of two concentrical tubes: the inner tube is coated with a selective coating, while the outer tube is transparent. Between the inner and the outer tubes, there is an annular space where the air is evacuated at a vacuum pressure. This vacuum leads to an important improvement of the thermal performance of these solar collectors due to the ⁎

Corresponding author. E-mail address: [email protected] (J.A. Alfaro-Ayala).

https://doi.org/10.1016/j.enconman.2018.04.039 Received 3 October 2017; Received in revised form 3 April 2018; Accepted 10 April 2018 0196-8904/ © 2018 Elsevier Ltd. All rights reserved.

reduction of heat losses. Low temperature water-in-glass ETSCs are generally used in serial and/or parallel networks to satisfy a duty of warm water in pools, buildings or industrial processes [3]. An adequate selection of the equipment through the optimized ETSCs would decrease the costs of the networks and the space occupied by them. The thermal performance and efficiency of the water-in-glass ETSC by means of experimental and theoretical methods have been reported in the literature. Experimental works of ETSCs were focused to obtain a better thermal performance: Recalde et al. [4] analyzed the behavior of the natural circulation of the water inside of a glass evacuated tubes solar collector under the action of the solar radiation in the Equator Andean high lands, the data collected during the experiments showed that the highest value of the water temperature is obtained with small inclinations, it occured due to the increase incidence area of solar radiation and captured energy. Tang and Yang [5] carried out an experimental work considering different tilt angles to find the thermal

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qR radiative flux, W m−2 unit vector into a given direction, dimensionless ŝ si ̂ direction vector, dimensionless T temperature, °C Tamb ambient temperature, °C inlet temperature, °C Tin CFD outlet temperature, °C ToutCFD experimental outlet temperature, °C Toutexp v velocity vector, m/s Greek letters η wave number, m−1 ηther thermal efficiency, dimensionless κη, κ absorption coefficient, m−1 μ dynamic viscosity, Pa s ρ density, kg m−3 ρg ground reflectivity, dimensionless σsη scattering coefficient, m−1 σ Stefan–Boltzmann constant, W m−2 K−4 Φη scattering phase function, dimensionless incident solid angle, sr Ωi

Nomenclature Aa c cb cp GD Gd Gr G G gy h Iη jη k ṁ Ntub P ,p q″

absorber area, m2 specific heat of water, J kg−1 °C−1 specific heat of borosilicate glass, J kg−1 °C−1 specific heat of polyurethane, J kg−1 °C−1 direct radiation, W m−2 diffuse radiation, W m−2 reflected radiation, W m−2 total radiation, W m−2 incident radiation = direction-integrated intensity gravity in y-direction, m s−2 convective heat transfer coefficient, W m−2 °C−1 radiation intensity, W m−2 sr−1 emission coefficient, W m−3 sr−1 conductivity, W/m °C−1 mass flow rate, kg s−1 number of tubes, dimensionless pressure, Pa solar heat flux, W m−2

a mass transfer study into an electronic enclosure were obtained. However, a work that obtains the significant parameters of a low temperature, water-in-glass, evacuated tubes solar collector through using the computational fluid dynamics in the design of experiments cannot be found in the literature. Also, the literature review indicated that few works were focused on the process of optimization of solar collectors and there is not a study focused in the optimization of low temperature, water-in-glass, evacuated tubes solar collectors. Moreover, the optimization of a low temperature, water-in-glass, evacuated tubes solar collector combining different techniques such as design of experiments, simulated annealing and computational fluid dynamics is not reported. In this work, an optimization process is applied to the geometry of a low temperature, water-in-glass, evacuated tubes solar collector. The geometrical and operational parameters with significance in the thermal performance of the evacuated tubes solar collector are obtained with the help of the computational fluid dynamics in a design of experiments study. The combination of the simulated annealing method and the computational fluid dynamics modeling allow finding the optimal geometry and optimal thermal and hydraulic performance of the solar collector. Integrating these methods, it is possible to consider details in the geometrical and operational parameters, such as: diameter, length, absorber area and number of the tubes and mass flow rate respectively, which are of outmost importance in the behavior of the fluid flow and the transfer of heat. The numerical study was carried out in three dimensions, steady-state and laminar regime. The design of experiments study via computational fluid dynamics was carried out with two levels and five parameters, 25, the parameters with significance in the performance of the collector were found from a commercial collector. This collector was used as base case in the process of optimization. In the optimization process, the absorber area was analyzed under three different cases because of the combination of geometrical parameters: length, diameter and number of tubes. Thus, 259 different collector geometries were constructed and modeled. Comparison of the thermal performance between the optimal geometry and the commercial geometry was carried out for different experimental test (mass flow rate, inlet temperature, ambient temperature and solar radiation). The optimal geometry of the low temperature, evacuated tubes, solar collector achieves higher thermal performance and lower costs than the commercial geometry used as reference at any operational and weather conditions.

performance of the collector, it was concluded that a higher thermal performance is obtained with the optimal tilt angle and it dependes of the specific geographical location. On the other hand, computational fluid dynamics (CFD) works were reported in the literature. Wang et al. [6] studied a novel design of a low temperature solar collector by using CFD. The effects of parameters, such as the tilt angle of the solar collector, the mass flow rate and the air gap distance, on the thermal performance and efficiency of the collector were obtained. Yao et al. [7] made CFD numerical simulations with a twist tape inserted inside the evacuated tubes. It was concluded that the twist tape insertions allowed having a higher dissipation of mechanical energy and a temperature field more uniform to improve the thermal performance of the equipment. Essa and Mostafa [8] showed the ability of the CFD modeling by using the radiative transfer equation (RTE). The weather conditions were set up in the simulations by considering the location, the date and the corresponding hours of a day. It was concluded that the use of CFD can be considered as a powerful tool for calculating the thermal performance of the low temperature solar collector under real climate conditions. In the review of the literature, just few works were focused on obtaining an optimum solar collector. Cheng et al. [9] applied the particle swarm optimization (PSO) method to find the optimal performance of optical parameters of a high temperature parabolic solar collector. Mohammad et al. [10] presented the optimization of the thermal performance of a high temperature parabolic solar collector with a nanofluid. The process of optimization was carried out by the genetic algorithms (GA). The results showed that the increase of heat transfer is directly related to the amount of nano-particles. Facão [11] made the optimization of the geometry and fluid flow distribution to a flat plate solar collector by the use of Nelder-Mead simplex algorithm. It was concluded that the manifold of the outlet of the flat plate solar collector should have a bigger size than the manifold of the inlet to obtain a better performance. Kulkarni et al. [12] applied genetic algorithms (GA) with CFD techniques to a solar air heater. The maximum transfer of heat and the minimum pressure drop were found. The optimal configuration of the geometry and the thermal performance were obtained. On the other hand, some works applied CFD modeling in the design of experiments (DOE) to obtain the parameters that had significance in the performance of the case study. Sant'Anna et al. [13] applied CFD modeling and a DOE study with a factorial design of 23 to find that the diameter of the biomass particles had significance in the bubbling fluidization condition. Shojaee et al. [14] applied a two-level factorial design and four parameters, 24. The parameters that had significance in 344

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2. Methods

description of the RTE solution and the coupling of Eqs. (1)–(3) can be found in [16,18].

The computational fluid dynamics model (Section 2.1) was used for predicting the fluid flow and the thermal performance of the low temperature, water-in-glass, evacuated tube solar collector. Moreover, the fluid flow and the thermal performance of the solar collector could be affected for different geometrical and operational parameters; thus, in this work a design of experiments (Section 2.2) was carried out to find the significant parameters. These parameters were considered in the optimization process, using the combination of the simulated annealing and the computational fluid dynamics techniques (Section 2.3), to find the optimal geometry of the solar collector. These techniques are described below.

(

∇ ·qR = κ 4σT 4−

∇ ·(ρ v) = 0

∇ ·(ρ vv) = −∇p + μ∇2 (v) + ρgy

ρ = 1001−0.0834·T −0.0035·T 2

c = 4215−2.3787·T +

̂ η (si,̂ s )̂ dΩi ∫4π Iη (si)Φ

+2×

10−6·T 4

(8) (9)

The boundary conditions of the solar collector are: inlet temperature 20 °C, inlet mass flow rate 0.02475 kg/s and 0.07425 kg/s (Table 1 shows the search space of the manipulated parameters for the geometry of the ETSC for the DOE study) and atmospheric pressure at the outlet. The convective heat transfer due to the air of the environment (surroundings of the solar collector) was modeled by a convective boundary condition [15] on all the walls of the solar collector (tubes and manifold). The ambient temperature, Tamb = 25 °C, and the convective heat transfer coefficient, h = 5 W/m2 K , allowed taking into account the appropriate thermal resistance by convection. The conduction heat transfer through the insulation material and through the glass tubes thickness were modeled by the boundary condition shell conduction approach [15]. The thicknesses, t , for the walls of the glass tubes and the polyurethane insulation were taken as 0.0018 m and 0.050 m respectively, thus the appropriate thermal resistances by conduction across the walls were taken into account. A general flow chart of the solution of the CFD model is shown in Fig. 1. The solar load model was fed by parameters such as: the geographical location, the date and the hour of the day. The solar load model allowed computing the direct, diffuse and reflected radiation (GD,Gd,Gr ) and the solar heat flux (q″) on the ETSC by solving the RTE equation to obtain the source term of the energy equation by means of solar ray tracing method (Table 2). The fluid properties, operational conditions and boundary conditions are essential information required for solving Eqs. (1)–(9). The outlet temperature, Tout CFD, the pressure drop, ΔP, and the temperature and velocity distributions inside the low temperature, water-in-glass, ETSC were obtained from this model (thermal and hydraulic performance). A design of experiments (DOE) was carried out before starting the process of optimization to find the parameters of the geometry with

(1)

On the left hand, the term represents energy transfer due to convection. On the right hand, the first term represents energy transfer due to conduction and the second term is a source term, it represents energy transfer due to radiation by the radiative transfer equation (RTE) [16,18]. The radiative transfer equation in a participating medium consists of three fundamental phenomena; the first two are the emission and absorption of electromagnetic energy by matter, the third one includes the effects of scattering, which encompasses diffraction, interference, reflection and transmission [9]. The RTE governs the behavior of radiative heat transfer in the presence of an absorbing, emitting and scattering medium, it describes the radiative intensity field within the enclosure as a function of location, direction and spectral variable. The RTE equation is represented as following:

σsη

(7)

0.0528·T 2−0.0005·T 3

μ = 0.0017−5 × 10−5·T + 9 × 107·T 2−8 × 10−9·T 3 + 3 × 10−11·T 4

The heating of water inside the solar collector due to the solar radiation can be represented by the energy equation in the steady-state form:

4π

(6)

k = 0.5634 + 0.002·T −8 × 10−6·T 2

sorber area of the tubes.

= jη −κ η Iη−σsη Iη +

(5)

The fluid properties of the water, such as density (ρ) , thermal conductivity (k ) , specific heat (c ) and viscosity (μ) [19], are set as temperature polynomials, Eqs. (6)–(9), to account the buoyancy effects. The validity range of the polynomials is from 5 °C to 95 °C (278–368 K).

• Steady state flow. • Laminar flow regime. • Solar radiation absorption, reflection and transmission on the ab-

∂s

(4)

and momentum equation,

The computational fluid dynamics (CFD) simulations of the low temperature water-in-glass ETSC are carried out using a commercial CFD software, ANSYS-FLUENT®, which uses the finite volume method. The thermal and hydraulic analysis of the ETSC requires the mathematical representation of the physical domain. The governing equations applied to the computational model include the conservation equations of mass, momentum, energy and radiation [15–18]. The CFD model considers the following assumptions:

∂Iη

(3)

The flow of the fluid is solved through the continuity equation

2.1. Computational fluid dynamics model

v·∇ (ρcT ) = ∇·(k∇T)−∇ ·qR

∫4π IdΩ) = κ (4σT 4−G)

(2)

On the left hand, the term represents the change in the radiation intensity at the direction s ̂ , (∂Iη/ ∂s ) . On the right hand, the first term represents the contributions from emission ( jη ), the second term represents the contributions from absorption (κ ηI η), the last two terms represent the contributions from scattering, scattering away (out-scattering, σsηI η) from the direction s ̂ and scattering into (in-scattering, σsη ̂ η (si,̂ s )̂ dΩi from the same direction s .̂ Solving the RTE ∫ I (s )Φ 4π 4π η i equation by integrating over the direction and spectral variable (wavenumber), the radiation intensity is obtained. Once the radiation intensity is known, the source term of the energy equation is obtained under the following assumptions: (i) the scattering is not considered due to scattering only redirects the stream of photons; it does not affect the energy content of any given unit volume and also scattering is assumed isotropic (relatively few investigations have dealt with the case of anisotropic scattering) and (ii) a gray medium (κ η = κ) . An extended

Table 1 Search space of the manipulated parameters of the geometry of the ETSC.

345

Name

Symbol

Search space

Length of tube Diameter of tube Absorber area

Lturb ϕtub Aa

1–3 m 0.0127–0.0762 m

Number of tubes

Ntub

2.5–4 m2 f (A,Ltub,ϕtub)

Length of the manifold Volume ratio Diameter of the manifold Mass flow rate Distance between tubes

Lm Vr φm ṁ Stub

f (Ntub,Stub) 0.25,0.5771,0.75 f (Vr ,Ltub) 0.02475–0.07425 kg/s f (Sper ,ϕtub)

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Fig. 1. Flow chart of the solution of the CFD model.

Diameter of the manifold,

Table 2 Constant parameters of the geometry of the ETSC. Name

Symbol

Value

Inlet temperature Total solar radiation

Tin G

20 °C

Direct radiation

GD

707.51 W/m2

Diffuse radiation

Gd

Diameter of the water entrance at the manifold Distance between external diameters of tubes

φinm Sper

91.16 W/m2 0.01905 m 0.0254 m

ϕm =

798.67 W/m2

4(Vm) πLm

(11)

Volume of water in the manifold,

Vm = Vtub ∗ (Vr )

(12)

Volume of water inside the tubes,

significance in the thermal and hydraulic performance of the low temperature, water-in-glass, ETSC.

Vtub =

2 πϕtub

4

Ltub Ntub

(13)

Number of tubes, 2.2. Design of experiments

Ntub =

The design of experiments refers to the process of planning, designing and analyzing the experiment so that valid and objective conclusions can be drawn effectively and efficiently. The design of experiments allows studying the effect of the parameters (geometrical and operational parameters of the ETSC) on a response of the process (the outlet temperature and thermal efficiency of the ETSC) so the parameters with significance are obtained and therefore the studies can be delimited. Two types of parameters are defined to be analyzed in the DOE study. The first type of parameters are called manipulated parameters of the geometry: the length of tube, Ltub , the diameter of tube, ϕtub , the absorber area, Aa , the number of tubes, Ntub , the length of the manifold, Lm , the volume ratio, Vr , the diameter of the manifold, ϕm , the mass flow rate, ṁ , and the distance between tubes, Stub . The search space of these nine manipulated parameters are shown in Table 1. The search space of the manipulated parameters of the geometry (Table 1) was set according to the minimum and maximum values reported in the literature [7]. The length of the manifold, Lm , the diameter of the manifold, ϕm , the number of tubes, Ntub and the distance between tubes, Stub , are functions of other parameters, shown in Eqs. (10)–(15): Length of the manifold,

1 Lm = 2 ⎡25.4 + ⎛ ϕtub⎞ ⎤ + [(25.4 + ϕtub)(Ntub−1)] 2 ⎝ ⎠⎦ ⎣

ATotal 2Aa = Atub π (ϕtub)(Ltub)

(14)

Distance between tubes,

Stub = Sper + ϕtub

(15)

The second type of parameters are called constant parameters of the geometry of ETSC, these included geometrical and operational parameters. The constant parameters are: the inlet temperature, Tin, the total solar radiation, G, the direct radiation, GD, the diffuse radiation, Gd, the diameter of the water inlet at the manifold, ϕinm, and the distance between external diameters of tubes, Sper, see Table 2. The DOE study considered the weather conditions of January 15th, 2017, at 10:00 h in the town of Guanajuato, Guanajuato, Mexico [20], latitude of 21.02° and longitude of −101.25°. Some manipulated and constant parameters used in the DOE study are shown in Fig. 2. A factorial design is carried out with two levels 2k and five kparameters. The five parameters are the length of the tubes, Ltub , from 1 m to 3 m, the diameter of the tube, ϕtub , from 0.0127 m to 0.0762 m (0.5–3 in), the absorber area, Aa , from 2 m2 to 4 m2, the volume ratio, Vr , 0.25–0.75 and the mass flow rate, ṁ , from 0.02475 kg/s to 0.07425 kg/s (1.5–4.5 L/min). The use of this factorial design resulted in 32 different geometries for the ETSCs. These geometries were modeled with the computational fluid dynamics.

(10) 346

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2.3. Process of optimization applying the simulated annealing method and computational fluid dynamics The simulated annealing is a method of optimization based in the analogy of heating and slow cooling a system to change its properties. The heating process takes the system to an elevated level of energy, under a “hot bath” that is called annealed. Then, the slow cooling process minimizes the energy of the system at this condition, allowing the change of the microstructure and then the properties of the system. A condition is that the cooling process should be slow to obtain an appropriate solution in the SA method. If the cooling process is performed quickly, this procedure would probably lead to a local optimum. On the other hand, a very slow cooling would lead us to having high computational requirements [21]. A detailed description of the SA method can be found in [22]. The simulated annealing method requires an initial geometry and its solution (base case). A commercial geometry of a low temperature water-in-glass ETSC, consisting of 36 evacuated tubes made of borosilicate glass connected directly to a manifold, was considered. The inner and outer nominal diameters of the glass tubes are 0.047 m and 0.058 m, respectively. The concentric tubes have a thickness of around 0.0018 m. The length of the external tubes is 1.5 m. The manifold length is 2.5 m and its internal diameter is 0.130 m. The commercial geometry is tilting 30° with respect to the horizontal. The material of the evacuated tubes and the insulation inside the manifold are borosilicate glass and polyurethane, respectively. The properties of the borosilicate glass and the polyurethane material are: density ρb = 2230 kg/m3 ρp = 30 kg/m3 , and thermal conductivity kb = 1.2 W/m °C kp = 0.06 W/m °C and and specific heat cb = 800 J/kg °C and cp = 1000 J/kg °C, respectively. The optical properties of absorbance and transmittance of the borosilicate glass are 0.92 and 0.95, respectively. The objective function, Z, was stablished as the maximization of outlet temperature of the low temperature ETSC in the process of optimization, Eq. (16), and the constrains were defined with a search space for the diameter of the tubes, the absorber area and the mass flow rate as following:

gy Fig. 2. Parameters of the geometry of the ETSC.

Objective Function Max Z = Tout Constrains 0.0127 ⩽ ϕtub ⩽ 0.0762 (m) 2.5 ⩽ Aa ⩽ 4 (m2) 0.02475 ⩽ ṁ ⩽ 0.07425 (kg/s)

(16)

A flow chart of the process of optimization, including the SA method and the CFD model, is shown in Fig. 3. The steps for the process of optimization are: 1. Obtain an initial solution of the CFD model, it is the starting point for the process of optimization. The initial geometry S is the commercial geometry of the low temperature water-in-glass ETSC described in the above sections, and Z is the outlet temperature obtained by the CFD model. It is important to recall that the manipulated variables are the diameter of the tubes, ϕtub , the mass flow rate, ṁ , and the absorber area, Aa, previously obtained in the DOE study. 2. Stablish the initial simulated annealing temperature, T = Ti , this temperature has no meaning in the thermal performance of the low temperature water-in-glass ETSC and it only has meaning in the SA method. The simulated annealing temperature was considered with a value of 110, tested with 10 initial movements and achieving the acceptance criterion of 80%, according the SA theory [22]. 3. Perturb the geometry, S′, see Section 2.4, then, obtain the outlet temperature, Z′, in the CFD model. Then comparing Δ = Z′−Z , the new value of the objective function Z′ with the value on the previous solution, Z. 4. Select the new best point, if the solution is improved, Δ > 0 . If the

Fig. 3. Flow chart for the process of optimization integrating SA-CFD methods. 347

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Fig. 4. Mesh analysis.

Fig. 5. Parameters with significance in the thermal performance of the ETSC.

Fig. 6. Solution profile of the process of optimization.

reaches the freezing temperature, T ⩽ Tfreezing .

solution is worst, ΔΔ < 0 , it is selected if it accomplishes the Δ Metropolis criterion, P ⩾ Pacceptance , where Pacceptance = e− T . Then the new simulated annealing temperature is reduced using T = αTold, where α is set as 0.9. 5. Finish the process of optimization if the objective function reaches a steady condition, Δ = 0 , or the simulated annealing temperature

2.4. Minimum absorber area The minimum absorber area of the ETSC with the maximum outlet temperature leads to obtaining the optimal solution through the 348

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Fig. 7. Field of solutions of case a and case c with lower absorber area than the commercial geometry. (a) Outlet temperature, (b) Thermal efficiency.

Table 3 Optimal geometries with standard dimensions and absorber areas lower than the commercial geometry. Optimal geometries

Aa , m2

ϕtub, m

Lm , m

Ntub

ToutCFD, °C

ηther

Costs of tubes (US $)

% cost decrease

commercial geometry 1s 5s 54s

3.09 2.48 2.71 2.94

0.0370 0.0480 0.0480 0.0480

1.5 1.5 1.5 1.5

36 22 24 26

29.78 30.00 30.35 31.52

0.411 0.518 0.520 0.505

$642.58 $392.68 $428.38 $464.09

– 38.9 33.3 27.8

Case a: The geometries were considered with a constant number of tubes, Ntub = 22. This constant number of tubes in the process of optimization was taken into account since it is the lowest number of tubes found in a collector for industrial applications. The diameter of the tubes, ϕtub , and the length of the tubes, Ltub , were varied with an interval of 0.0254 m and 0.05 m respectively. Case b: The geometries were stablished with a diameter of the tubes, ϕtub = 0.0127 m. This constant diameter for the tubes in the process of optimization was defined since it is the lowest value of standard dimensions. The number of tubes, Ntub , and the length of the tubes,

application of the SA method and the CFD modeling. The absorber area is obtained as:

Aa =

(Ntub) π (ϕtub)(Ltub) 2

(17)

Geometrical parameters such as the length of the tube, Ltub , the diameter of the tube, ϕtub , and the number of tubes, Ntub , are needed to compute the absorber area; thus, three cases called case a, case b and case c are defined in this work: 349

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Fig. 8. Field of solutions with higher absorber areas than the commercial geometry. (a) Outlet temperature, (b) Thermal efficiency. Table 4 Optimal geometries with standard dimensions for absorber areas higher than the commercial geometry. Optimal Geometries

Aa , m2

ϕtub, m

Lm , m

Ntub

ToutCFD, °C

ηther

Costs of tubes (US $)

% cost decrease

Commercial geometry 25s 27s 39s 86s 87s 109s

3.09 3.16 3.25 3.83 3.63 3.86 3.71

0.0370 0.0480 0.0480 0.0480 0.0480 0.0480 0.0370

1.5 1.5 1.8 1.8 1.5 1.5 2.0

36 28 24 28 32 34 32

29.78 31.54 31.65 32.75 33.65 34.08 31.95

0.411 0.472 0.464 0.43 0.486 0.472 0.417

642.58 499.78 428.38 499.78 571.18 606.88 571.18

– 22.2 33.3 22.2 11.1 5.6 11.1

and hydraulic performance of optimal geometries and the commercial geometry for different experimental test are described.

Ltub , were varied with an interval of 1 tube and 0.05 m, respectively. Also, in this case b the maximum number of tubes were constrained to Ntub = 62. Case c: The geometries were assigned with a length of the tubes, Ltub = 1 m. This constant length for the tube in the process of optimization was specified since it is the lowest value of standard dimensions. The number of the tubes, Ntub , and the diameter of the tubes, ϕtub , were varied with an interval of 1 tube and 0.0127 m, respectively.

3.1. Mesh analysis The mesh analysis considered a full study of the grid size, the number of unstructured cells (tetrahedrons and hexahedrons) was taken between 0.25 million and 1.75 million (Fig. 4), avoiding any unnecessary computational effort required for the calculations with a large number of cells. The highest temperature found between the coarse and the fine meshes was around 7.14%. The time required to reach a CFD solution of an individual with 1.75 million cells was around 5 h in a computer with a quad-core CPU (4th generation i7 processor) and 8 GB in RAM. It was observed that for a number higher than 0.88 million cells, the outlet temperature is independent of the mesh.

3. Results and discussion In the next sections, the results of the mesh analysis, the design of experiments, the optimization using the simulated annealing method and computational fluid dynamics and the comparison of the thermal 350

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The results obtained with the CFD model were validated with experimental data using the same low temperature water-in-glass evacuated tube solar collector. These data were reported in a previous published work [23]. The 259 geometries constructed and modeled in the optimization process had a size around 1.15 million of cells.

Outlet

Inlet

3.2. Design of experiments results

(a) The significant parameters in the thermal performance of the low temperature ETSC are shown in Fig. 5. The length of the tube, Ltub , and the volume ratio, Vr , were not significant in the thermal performance within the range of the search space stablished in this work. The diameter of the tubes, ϕtub , the mass flow rate, ṁ , and the absorber area, Aa, resulted with significance in the thermal performance. The DOE study indicated that the geometry with the highest diameter of the tubes ϕtub = 0.0762 m, the lowest flowrate ṁ = 0.0248 kg/ s (1.5 L/min) and the highest absorber area Aa = 4 m2 could have a better thermal performance. A geometry using these values was made and modeled, including the following additional parameters: length of tubes, Ltub = 1.5 m, number of tubes, Ntub = 23, length of the manifold, Lm = 2.362 m, diameter of the manifold, ϕm = 0.222 m and volume ratio, Vr = 0.577. An outlet temperature, ToutCFD = 33.95 °C, was obtained and a higher thermal performance than with the previous 32 geometries of the DOE study was achieved. However, the process of optimization applying simulated annealing method and computational fluid dynamics modeling was required to ensure obtaining the optimal ETSC geometry.

(b)

(c)

Fig. 9. Temperature inside the manifold of the ETSC (a) commercial geometry, (b) no. 1s, (c) no. 27s. (°C).

3.3. Results of the optimization process using the simulated annealing method and computational fluid dynamics The geometries obtained for the case a, case b and case c in the optimization process were 318, 70 and 320 respectively. However, making and simulating all these different geometries would consume a lot of human and computational resources, thus a representative sample was chosen, the criterions were: an interval and a level of confidence of 8 and 95%, respectively, resulting in a total number of 259 different geometries: case a with 92, case b with 42 and case c with 125. These geometries were made and modeled via CFD. In case a, the average temperature at the outlet was 31.1 °C and the average pressure drop was 9.21 Pa. Analyzing the results of the 92 geometries, it can be observed that the length of the tubes has no significance (as it was concluded in the DOE study, Fig. 5). In case b, a high number of tubes was required to reach the absorber area because of the small diameter. The total number of geometries for case b is lower than for case a. The average temperature obtained was 20.87 °C in case b, it indicates that many of these geometries are not capable to increase one Celsius degree, the behavior of the small diameter of tubes had unfavorable significance in the thermal performance as it can be observed in the DOE study, see Fig. 5. The outlet temperature is lower when the diameter of the tube is small. These cases were rejected due to its low capacity to increase the outlet temperature. The pressure average of the drop was 15.92 Pa. Pressure drop for case b is higher than for case a, because the fluid flows through a smaller diameter increasing the velocity and then the pressure drop. However, the allowed pressure drop should not be higher than the value allowed: ΔP ≤ 1 MPa [24]. In case c, the number of geometries is higher than for cases a and b due to the wide search space and the low interval of variation of the parameters made in the process of optimization. The average temperature was 31.86 °C for case c and this value is higher than the one for case a and case b, thus a better thermal performance was found for the geometries of case c. The average pressure drop was 10.52 Pa; this value is higher than for case a and lower than for case b, being in the range of allowance, ΔP ⩽ 1 MPa [24]. The outlet temperatures for the three cases, case a, case b and case c

Fig. 10. Temperature inside the tubes. (a) Commercial geometry, (b) no. 1s, (c) no. 27s. (°C).

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Fig. 11. Velocity and temperature profiles. (a) Commercial geometry, (b) no. 1s and (c) no. 27s.

obtained due to the wide search space of the absorber area considered in this work. The geometries of the water-in-glass ETSC for case b presented the lowest outlet temperatures, around than 21 °C (with an inlet temperature of 20 °C), therefore no relevant information can be obtained. Case a and case c have the most relevant information in this study. A total of 63 geometries for case a and case c with lower absorber areas than the commercial geometry are shown in Fig. 7. This plot includes the absorber area, the outlet temperature, and the thermal efficiency. The outlet temperature is shown in Fig. 7a and the thermal efficiency is shown in Fig. 7b. In this field of solutions, the geometries no. 27 and no. 28 have similar absorber areas to the commercial geometry, and these geometries achieved higher outlet temperature, an increment of ΔT27 = 1.81 °C and ΔT28 = 2.34 °C respectively. Geometries no. 2, no. 38 and no. 53 have lower absorber areas than the commercial

are shown in Fig. 6, including the case of reference called commercial geometry (red dot with an absorber area of 3.09 m2 and a temperature of 29.78 °C)1. This was the initial solution of the process of optimization. It is observed that case a and case c have higher outlet temperatures than commercial geometry. Optimal geometries are those that achieved both criterions, (i) higher outlet temperature than commercial geometry and (ii) lower absorber area than commercial geometry or equal absorber area to commercial geometry. On the other hand, some of the geometries of case a and case c with higher absorber area than commercial geometry resulted in a higher temperature than the commercial geometry. These geometries were

1 For interpretation of color in Fig. 6, the reader is referred to the web version of this article.

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efficiency. The outlet temperature is shown in Fig. 8a and the thermal efficiency is shown in Fig. 8b. In general, this field of solutions resulted in higher temperatures and higher thermal efficiencies than the commercial geometry. Geometry no. 75 with slightly higher absorber area than the commercial geometry resulted in a higher outlet temperature: an increment of ΔT75 = 2.77 °C . The average thermal efficiency for all these geometries is 47.9% and the average outlet temperature is 33.04 °C, these values are higher than the values of the commercial geometry of 41.1% and 29.78 °C respectively. This means that the field of solutions of case a and case c with higher absorber areas than the commercial geometry results in higher thermal performance. The geometries no. 25, no. 27, no. 39, no. 86, no. 87 and no. 109 (green squares) had dimensions of ϕtub and Ltub close to the standard values of commercial geometries. To obtain the standard dimensions of the tubes, the diameter was adapted to standard values from 0.04826 m to 0.048 m for all these six geometries, also, for the length, it was kept constant with a value of 1.5 m. These new geometries were named no. 25s, no. 27s, no. 39s, no. 86s, no. 87s and no. 109s, see Table 4. These geometries were made and simulated to obtain the outlet temperature and the thermal efficiency. An average cost per tube of around 9.3 USD, estimated in July 2017, was considered. The percentage of cost decreases due to the tubes, the decrements for the optimal geometries no. 25s, no. 27s, no. 39s, no. 86s, no. 87s and no. 109s with standard dimensions are 22.2%, 33.3%, 22.2%, 11.1%, 5.6%, 11.1%, respectively (Table 4). Geometry no. 27s presents the highest percentage of cost decrease of 33.3% with an outlet temperature of 31.65 °C (1.87 °C higher than the commercial geometry). Geometry no. 87s resulted in the highest outlet temperature of 34.08 °C (an increment of 4.3 °C related to the commercial geometry) and a cost decrease of 5.6%.

Fig. 12. Thermal efficiency curves of the optimal geometry no. 1s and the commercial geometry.

geometry and obtained around the same outlet temperature. The field of solutions (Fig. 7) had a reduction of around 9.1% of the average absorber area and an increment around 27.2% of the outlet temperature of the ETSC. In addition, the average thermal efficiency of all the geometries was 52.1%, this value is higher than the value of the commercial geometry of 41.1%. The geometries no. 1, no. 5 and no. 54 (green squares) have dimensions of ϕtub and Ltub close to the standard values of the commercial geometries of ETCs found in the market. To obtain the standard dimensions of tubes, the diameter was adapted to the standard values from 0.04826 m to 0.048 m and the length was kept with a constant value of 1.5 m. These new geometries were called no. 1s, No. 5s and no. 54s, the letter “s” stands for “standard dimensions”. These geometries were made and modeled to obtain the outlet temperature and the thermal efficiency, see Table 3. An average cost of around USD 9.3 per tube, estimated in July 2017, was considered. The percentage of cost decreases due to the tubes, the decrements for the optimal cases no. 1s, no. 5s and no. 54s with standard dimensions are 38.9%, 33.3% and 27.8%, respectively. Geometry no. 1s resulted in the highest percentage of costs decrease of 38.9%, with the same outlet temperature as the commercial geometry. Geometry no. 54s resulted with the highest outlet temperature of 31.52 °C (an increment of 1.74 °C related to the commercial geometry) and a cost decrease of 27.8%. A total of 113 geometries of case a and case c with higher absorber areas than the commercial geometry are shown in Fig. 8. This plot includes the absorber area, the outlet temperature and the thermal

3.4. Comparison of the thermal and hydraulic performance of optimal geometries A comparison of the temperature distributions for the optimal geometries with standard dimensions found in the market, is shown in Fig. 9. The temperature distributions are depicted in a vertical plane located in the middle of the manifold. It is shown the commercial geometry, the geometry no. 1s with a lower absorber area than the commercial geometry and the geometry no. 27s with a higher absorber area than the commercial geometry. It can be observed that the optimal geometry no. 1s and no. 27s achieve higher temperatures inside the manifold than the commercial geometry. The temperature distributions inside the tubes for the commercial geometry, the geometry no. 1s and the geometry no. 27s are shown in

Fig. 13. Comparison of the thermal efficiency between optimal geometry no. 1s and commercial geometry for different operational and weather conditions. 353

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rate as an operational parameter.

Fig. 10. It is observed that higher temperatures are found in the bottom of the tubes for the commercial geometry due to the stagnation of water, increasing the heat losses to the ambient. The optimal geometries no. 1s and no. 27s resulted in better temperature distributions inside the tubes. These geometries also presented higher temperatures at the top of each tube, just coming back from the manifold. The optimal geometries showed a temperature increase in the direction of the fluid flow from the first tube at the inlet to the last tube at the outlet of the ETSC and the stagnation was not obtained. The velocity and temperature profiles are shown in Fig. 11. The profiles were obtained in a vertical line representing the normalized diameter, D∗, ranging from 0 to 1, located at the top of the tube where it is connected with the manifold. These profiles are shown for three tubes, the first tube is located at the inlet of the collector, the second tube is located at the middle and the third tube is located at the outlet of the collector. In general, low velocities that enter and leave the tubes were found, ranging from 0.003 m/s (positive values mean that the inlet of the fluid flow to the tube, is coming from the manifold), to −4 mm/s (negative values mean that the outlet of the fluid flow in the tube, is backing to the manifold). In the case of the commercial geometry, it generally presents higher velocity values at the inlet and the outlet of the tubes than those for the optimal geometries. Different velocities are found inside each of the tubes. The outlet temperature of the first tube is 26.7 °C and the last tube is 34 °C, closer outlet temperature was observed between the middle tube and the last tube, both of them around 34 °C. The optimal geometries show better thermal and hydraulic performance. A more uniform velocity in each tube is obtained (avoiding the stagnation at the bottom of the tubes) and a gradual increment of temperature from the first tube to the last tube of the collector. For example, the optimal geometry no. 1s achieved a temperature of 32.5 °C for the first tube, 33.8 °C for the middle tube and 35 °C for the last tube. The thermal efficiency curves for the optimal geometry no. 1s and the commercial geometry are illustrated in Fig. 12. It can be observed that the optimal geometry no. 1s has the highest thermal efficiency at any mass flow rate with any solar radiation - e.g., for the case with solar radiation of 798 W m−2, for a mass flow rate of 0.015 kg/s an increase of 23% is obtained, and for a mass flow rate of 0.04 kg/s an increase of 27.5% is obtained. The thermal efficiency of the low temperature water-in-glass evacuated tube solar collector is compared with experimental data as shown in Fig. 13. It is observed that the thermal efficiency of the solar collector decreases as the difference of temperature (Tin-Tamb) increases. It can be observed that the optimal geometry no. 1s has the highest thermal efficiency at any operational and weather condition (inlet temperature, ambient temperature and solar radiation) - e.g., the increase of the thermal efficiency for the Test 1 (Tin = 15 °C, Tamb = 15 °C, G = 798 W m−2, and ṁ = 0.02475 kg/s), Test 2 (Tin = 20 °C, Tamb = 15 °C, G = 798 W m−2, and ṁ = 0.02475 kg/s) and Test 3 (Tin = 25 °C, Tamb = 15 °C, G = 798 W m−2, and ṁ = 0.02475 kg/s) are 23.7%, 26.5% and 29.8%, respectively.

• The process of optimization was carefully carried out to obtain a • • •

Finally, the results can be helpful in the design of networks of low temperature evacuated tubes solar collectors due to the number of collectors that could be reduced, so the cost could be reduced according to a specific application. Acknowledgements This work was supported by “Programa para el Desarrollo Profesional Docente (PRODEP)”. Folio: UGT-PTC-379 and UGTO-PTC-477. DSA/ 103.5/14/10539 and DSA/103.5/15/7007, Secretaría de Educación Pública (SEP, Secretary of Public Education), Mexico. The authors gratefully acknowledge to the National Council of Science and Technology (CONACyT), Mexico, under the SNI program for financial support. References [1] Mazarron FR, Porras-Prieto CJ, Garcia JL, Benavente RM. Feasibility of active solar water heating systems with evacuated tube collector at different operational water temperatures. Energy Convers Manag 2016;113:16–26. http://dx.doi.org/10.1016/ j.enconman.2016.01.046. [2] Wiser R, Millstein D, Mai T, Macknick J, Carpenter A, Cohen S, et al. The environmental and public health benefits of achieving high penetrations of solar energy in the United States. Energy 2016;113:472–86. http://dx.doi.org/10.1016/j. energy.2016.07.068. [3] Picón-Núñez M, Martínez-Rodríguez G, Fuentes-Silva AL. Solar thermal networks operating with evacuated-tube collectors. Energy 2017:1–8. http://dx.doi.org/10. 1016/j.energy.2017.04.165. [4] Recalde C, Cisneros C, Avila C, Logroño W, Recalde M. Single phase natural circulation flow through solar evacuated tubes collectors on the equatorial zone. Energy Proc 2015;75:467–72. http://dx.doi.org/10.1016/j.egypro.2015.07.424. [5] Tang R, Yang Y. Nocturnal reverse flow in water-in-glass evacuated tube solar water heaters. Energy Convers Manag 2014;80:173–7. http://dx.doi.org/10.1016/j. enconman.2014.01.025. [6] Wang N, Zeng S, Zhou M, Wang S. Numerical study of flat plate solar collector with novel heat collecting components. Int Commun Heat Mass Transf 2015;69:18–22. http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.10.012. [7] Yao K, Li T, Tao H, Wei J, Feng K. Performance evaluation of all-glass evacuated tube solar water heater with twist tape inserts using CFD. Energy Proc 2015;70:332–9. http://dx.doi.org/10.1016/j.egypro.2015.02.131. [8] Essa MA, Mostafa NH. Theoretical and experimental study for temperature distribution and flow profile in all water evacuated tube solar collector considering solar radiation boundary condition. Sol Energy 2017;42:267–77. http://dx.doi.org/ 10.1016/j.solener.2016.12.035. [9] Cheng Z-D, He Y-L, Du B-C, Wang K, Liang Q. Geometric optimization on optical performance of parabolic trough solar collector systems using particle swarm optimization algorithm. Appl Energy 2015;148:282–93. http://dx.doi.org/10.1016/j. apenergy.2015.03.079. [10] Mohammad ZP, Sokhansefat T, Kasaeian AB, Kowsary F, Akbarzadeh A. Hybrid optimization algorithm for thermal analysis in a solar parabolic trough collector based on nanofluid. Energy 2015;82:857–64. [11] Facão J. Optimization of flow distribution in flat plate solar thermal collectors with riser and header arrangements. Sol Energy 2015;120:104–12. [12] Kulkarni Kishor, Afzal Arshad, Kim Kwang-Yong. Multi-objective optimization of solar air heater with obstacles on absorber plate. Sol Energy 2015;114:364–77. [13] Sant'Anna SMC, dos Santos Cruz WR, Da Silva GF, Medronho EdeA, Lucen S. Analyzing the fluidization of a gas-sand-biomass mixture using. Powder Technol 2017;316:367–72. [14] Shojaee NP, Jabbari M, Hattel JH. CFD simulation and statistical analysis of moisture transfer into an electronic enclosure. Appl Math Model 2017;44:246–60. [15] ANSYS Inc. Fluent, ANSYS-FLUENT Theory and User’s Guides. Release 15.0. Canonsburg, PA: ANSYS Inc; 2015. [16] Modest FM. Radiative heat transfer. third ed. Elsevier Inc.; 2013. ISBN: 978-0-12386944-9. [17] Hachicha AA, Rodríguez I, Capdevila R, Oliva A. Heat transfer analysis and numerical simulation of a parabolic trough solar collector. Appl Energy

4. Conclusions In this work, an optimization process was applied to the geometry of a low temperature, water-in-glass, ETSC. This optimization was carried out through the combination of a simulated annealing method and a CFD model. The use of these methodologies allowed predicting the optimal performance of water-in-glass evacuated tube solar collectors.

• The use of the design of experiments and the CFD model allowed •

field of solution of 259 geometries, these were made and modeled by CFD. Higher thermal efficiency and thermal performance were found for some of the geometries of the case a and the case c with absorber areas lower or higher than the commercial geometry. The optimal geometry showed a cost decrease of 38.9% because it has 14 tubes less than the commercial geometry. The optimal geometry had a higher thermal efficiency at any operational and weather condition than the commercial geometry.

finding the geometrical and operational parameters that had significance in the thermal performance of the ETSC. The parameters that had significance were: the diameter of the tubes and the absorber area as geometrical parameters, and the mass flow 354

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