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Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq

Nanoparticles effects on MHD ﬂuid ﬂow over a stretching sheet with solar radiation: A numerical study Seiyed E. Ghasemi a, M. Hatami b,c, D. Jing b,⁎, D.D. Ganji d a

Young Researchers and Elite Club, Sari Branch, Islamic Azad University, Sari, Iran International Research Center for Renewable Energy, State Key Laboratory of Multiphase Flow in Power Engineering, Xi'an Jiaotong University, Xi'an 710049, China Department of Mechanical Engineering, Esfarayen University of Technology, Esfarayen, North Khorasan, Iran d Department of Mechanical Engineering, Babol University of Technology, Babol, Iran b c

a r t i c l e

i n f o

Article history: Received 8 December 2015 Received in revised form 17 February 2016 Accepted 24 March 2016 Available online 26 April 2016 Keywords: Numerical study Solar radiation Magneto-hydrodynamic Nanoﬂuid Stretching sheet

a b s t r a c t In this research, a numerical study on the solar radiation effect on the magneto-hydrodynamic (MHD) nanoﬂuid ﬂow over a stretching plate is performed. A different application of Rosseland approximation for thermal radiation is introduced in this study and Keller-Box numerical method is applied to solve the problem. An excellent agreement was found between our calculated results and those obtained by Mushtaq et al using Runge-Kutta method. Also, the inﬂuences of different parameters such as magnetic ﬁeld, local radiation, Brownian motion and thermophoresis parameters as well as Eckert, Lewis and Biot numbers are explained through graphs for velocity, temperature and nanoparticles concentration. As a main outcome, our results show that, regardless of the selected value of radiation parameter Rd, the temperature increases as the Brownian motion and thermophoretic effects are simultaneously increased. Furthermore, the nanoparticles volume fraction is found to decrease upon increasing the Brownian motion parameter. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Nowadays, increasing attention has been paid to renewable energy resources which are expected to replace fossils fuels and reduce the emission of hazardous greenhouse gases (GHG). One of the clean renewable energy is solar energy which can be used for heating and cooling the building, hot water in domestic and industrial applications, heating the swimming pools, dry production and electricity generation, etc. Solar collectors may work with air, water or other ﬂuids such as nanoﬂuids or combination of these working ﬂuids. Solar heaters are the solar thermal technology with which the energy from the sun, namely, solar insolation, is captured by an absorbing medium and used to heat air or water. The performances of solar collectors are mainly inﬂuenced by the meteorological parameters (direct and diffuse radiation, ambient temperature and wind speed), design parameters (type of collector, collector materials) and ﬂow parameters (ﬂuid ﬂow rate, mode of ﬂow). One of the principal requirements of these designs is a large contact area between the absorbing surface and air [1]. Solar air heating as a renewable energy heating technology is used to heat or condition air for buildings or processes of heat applications. It is typically the most cost-effective out of all the solar technologies, especially in commercial and industrial applications, and it addresses the ⁎ Corresponding author. E-mail addresses: [email protected] (S.E. Ghasemi), [email protected] (M. Hatami), [email protected] (D. Jing).

http://dx.doi.org/10.1016/j.molliq.2016.03.065 0167-7322/© 2016 Elsevier B.V. All rights reserved.

largest usage of building energy in heating climates, such as space heating and industrial process heating [2]. The solar air heating system often consists of two parts: a solar collector and air distribution system. Solar-energy air-heating collectors can be classiﬁed generally into two types ,that is, bare-plate and covered-plate solar-energy collectors. Bare-plate collectors are the simplest forms of solar collectors which consist simply of an air duct, absorber plate and an insulated surface [3]. Covered-plate collectors were used for minimizing the upward heat losses which prevents convective heat losses from the absorbing plate, reduces long-wave radiative heat losses and protects the absorber plate against cooling by irregular rainfall [4]. This kind of collectors can operate at higher efﬁciencies than bare-plate collectors at moderate temperatures. However, due to the use of cover materials the cost of construction for covered-plate collectors is increased [5]. A novel solar air collector of pin-ﬁn integrated absorber was designed to increase the thermal efﬁciency by Peng et al. [6]. Their design has many advantages such as achieving high thermal efﬁciencies, large ﬂexibility, long durability and acceptable costs. A review of the various designs and the performance evaluation technique of ﬂat-plate solar-energy airheating collectors for low temperature solar-energy crop drying applications and the appropriateness of each design and the component materials selection guidelines are presented in [4]. Leon and Kumar [7] developed a mathematical model for predicting the thermal performance of unglazed transpired collectors over a range of design and operating conditions suitable for drying of food products. Other mathematical models in air-heater solar collectors are presented by Gorji et al.

S.E. Ghasemi et al. / Journal of Molecular Liquids 219 (2016) 890–896

[8] and Ghasemi et al. [9] which considered temperature-dependency for air properties and used analytical methods to obtain the collector efﬁciency and studied the dimension effect on the collector thermal efﬁciency. Most types of solar collectors use water or water based nanoﬂuids as their working ﬂuids. A review of researches on nanoﬂuid working ﬂuid in solar collectors is presented by Javadi et al. [10]. Youseﬁ et al. [11] used Al2O3-water nanoﬂuid, as working ﬂuid, to improve the efﬁciency of a ﬂat-plate solar collector. Their experimental results showed that using the nanoﬂuids as working ﬂuid increased the efﬁciency of the solar collector by about 28.3%. Faizal et al. [12], by using numerical methods and data from literatures, estimated the potential to design a smaller solar collector, using various nanoﬂuids, to produce the same desired output temperature as the larger one. Liu et al. [13] compared the results of water and CuO-water nanoﬂuid in thermal performance of a tubular solar collector. Furthermore, some studies used other methodologies to increase heat transfer in collectors, for instance, using ﬁns and porous structures were presented by Fudholi et al. [14] and Alkam and Al-Nimr [15], respectively. Taherian et al. [16] simulated dynamics of thermosyphon solar water heater collector considering the weather conditions of a city in north of Iran experimentally. Recently, Mushtaq et al. [17] investigated the effect of radiation of solar energy on stagnation point ﬂow of nanoﬂuid using Runge-Kutta numerical method and they found that the excessive movement of nanoparticles in the base ﬂuids results in the deeper absorption of solar radiations in the liquids. Other applications of nanoﬂuids have been investigated by authors [18–25]. In this study, authors intend to obtain the solution of governing equation of solar radiation effect on the nanoﬂuid ﬂow treatment over a ﬂat plate by the Keller-Box numerical method.

2. Problem description As shown in Fig. 1, we consider a two-dimensional incompressible boundary layer ﬂow of nanoﬂuid over a convectively heated stretching sheet located at y = 0. The sheet is stretched through two equal and opposite forces along x-axis by keeping the origin ﬁxed with the velocity Uw = ax. Let U∞ (x) = bx be the ﬂuid velocity outside the boundary layer. A uniform magnetic ﬁeld of strength H0 is applied perpendicular to the direction to the ﬂow. The induced magnetic ﬁeld is neglected upon the assumption of small magnetic Reynolds number. Further, it is also assumed that the external electrical ﬁeld is zero and the electric ﬁeld due to the polarization of charges is negligible. Heat transfer analysis is carried out in the presence of thermal radiation, Joule heating and viscous dissipation effects. The combined effects of Brownian motion and thermophoresis due to the presence of nanoparticles are considered. Tf denotes the convective surface temperature while T∞ is the

891

ambient ﬂuid temperature. The steady boundary layer equations governing the two-dimensional incompressible stagnation-point ﬂow of nano-ﬂuid can be written as [17] ∂u ∂v þ ¼0 ∂x ∂y

ð1Þ

2

u

∂u ∂u du∞ ∂ u σ e H 20 ðu−u∞ Þ þ vf 2 − þv ¼ u∞ ρf ∂x ∂x ∂y ∂y

ð2Þ

where x and y are the coordinates along and normal to the sheet, respectively, vf is the kinematic viscosity, σeis the electrical conductivity of ﬂuid, H0 is uniform magnetic ﬁeld along y-axis, u and v are the velocity components along the x and y directions respectively. The boundary conditions for the considered problem are u ¼ U W ðxÞ ¼ ax ; v ¼ 0 at y ¼ 0 u→u∞ ðxÞ ¼ bx as y→∞:

ð3Þ

Using the dimensionless variables [17] η¼

sﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ a 0 y ; u ¼ axf ðηÞ ; v ¼ − av f f ðηÞ: vf

ð4Þ

Eq. (1) is identically satisﬁed and Eqs. (2) and (3) take the forms ‴

″

f þ f f −f

02

0 þ λ2 þ M λ−f ¼ 0

0

0

f ð0Þ ¼ 0 ; f ð0Þ ¼ 1 ; f ðþ∞Þ→λ where M ¼

σH 20 ρ f a is

ð5Þ

ð6Þ

the magnetic parameter and λ ¼ ba is ratio of the rates

of free stream velocity to the velocity of the stretching sheet. For M = 0 and l = 1, Eq. (5) can be reduced to the classical problem ﬁrst formulated by Hiemenz. Furthermore when l = 0 the exact solution of Eq. (5) is pﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ given by f ¼ ð1−e− 1þMη Þ= 1 þ M .

u

2 2 ∂T ∂T ∂ T v f ∂u 1 ∂qr σ e H 20 − ðu∞ −uÞ2 þ þv ¼α 2þ ðρCÞ f ∂y C f ∂y ðρCÞ f ∂x ∂y ∂y " 2 # ∂T ∂C DT ∂T þ τDB þ ∂y ∂y T ∞ ∂y

2

u

ð7Þ

2

∂C ∂C ∂ C DT ∂ T þv ¼ DB 2 þ T ∞ ∂y2 ∂x ∂y ∂y

ð8Þ

where T is the temperature, C is the nanoparticles concentration, a is the thermal diffusivity, Cf is the speciﬁc heat of the ﬂuid, DB is the Brownian motion coefﬁcient, DT is the thermophoretic diffusion coefﬁcient, τ ¼ ðρCÞp ðρCÞ f

is the ratio of the effective heat capacity of the nanoparticle material

to the heat capacity of the ﬂuid and qr is the radiative heat ﬂux. Using the Rosseland approximation for thermal radiation and applying to optically thick media, the radiative heat ﬂux is given by (Raptis [26], Brewster [27] and Sparrow and Cess [28])

Fig. 1. Physical model and coordinate system.

qr ¼ −

4σ ∂T 4 16σ 3 ∂T ¼− T 3k ∂y 3k ∂y

ð9Þ

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S.E. Ghasemi et al. / Journal of Molecular Liquids 219 (2016) 890–896

Fig. 2. Comparison between Runge-Kutta (Mushtaq et al. [17]) and Keller-Box method (present study) for a) nanoparticles concentration and b) temperature proﬁle.

where σ∗ and k∗ are the Stefan–Boltzman constant and the mean absorption coefﬁcient, respectively. Eq. (7) can now be expressed as u

"

αþ

16σ T 3 3ðρCÞ f k

∂T ¼ h T−T f ; C ¼ C W ∂y T→T ∞ ; C→C ∞ as y→∞:

ð14Þ

with the boundary conditions ð10Þ

Different from the previous studies on thermal radiation, the nonlinear Rosseland approximation for radiative heat ﬂux has been considered. The relevant boundary conditions for convective heat transfer can be written as, −k

Nt ″ θ ¼0 Nb

!

# 2 v f ∂u ∂T þ C f ∂y ∂y " 2 # 2 σ e H0 ∂T ∂C DT ∂T þ ðu∞ −uÞ2 þ τDB þ : ðρCÞ f ∂y ∂y T ∞ ∂y

∂T ∂T ∂ þv ¼ ∂x ∂y ∂y

ϕ″ þ Le f ϕ0 þ

at y ¼ 0

ð11Þ

We now deﬁne the non-dimensional temperature θ(η)=T−T∞/Tf − T∞ with T = T∞(1+ (θw − 1)θ) and θw = Tf/T∞, the ﬁrst term on the right ∂ ∂T hand side of Eq. (10) can be written as α ∂y ½∂y ð1 þ Rd ð1 þ ðθw −1Þθ3 Þ

θ0 ð0Þ ¼ −γ½1−θð0Þ ; ϕð0Þ ¼ 1; θðþ∞Þ→0; ϕðþ∞Þ→0

ð15Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where γ ¼ h=k v f =α is the Biot number, N b = τD B (C w − C ∞ )/v f is the Brownian motion parameter, N t = τD T (T w − T ∞ )/T ∞ v f is the thermophoresis parameter and EC ¼ U W 2 =C p ðT w −T ∞ Þis the local Eckert number. Here, x-coordinate could not be eliminated from the energy equation. Thus the availability of local similarity solutions has been attempted. The surface heat and mass ﬂuxes can be deﬁned by the following equations sﬃﬃﬃﬃﬃ i ah ∂T qw ¼ −k 1 þ Rd θw 3 θ0 ð0Þ ; þ ðqr Þw ¼ −k T f −T ∞ vf ∂y y¼0 sﬃﬃﬃﬃﬃ ∂C a 0 ϕ ð0Þ: ¼ −DB ðC W −C ∞ Þ jw ¼ −DB vf ∂y y¼0 ð16Þ

where Rd ¼ 16σ T ∞ 3 =3kk denotes the radiation parameter, and Rd = 0 when there is no thermal radiation effect. The last expression can be further reduced to i0 α T f −T ∞ h ð1 þ Rd 1 þ ðθw −1Þθ3 θ0 Pr

ð12Þ

where Pr=vf/αis the Prandtl number. Eqs. (8) and (10) take the following forms i0 1 h 2 ″2 ð1 þ Rd 1 þ ðθw −1Þθ3 θ0 þ f θ0 þ N b θ0 ϕ0 þ Nt θ0 þ Ec f Pr 0 2 þMEc λ−f ¼0

ð13Þ

Considering local Nusselt number Nux = xqw/k(Tf − T∞)and local Sherwood number Sh=xjw/DB(CW − C∞) , one obtains h i Nux Sh pﬃﬃﬃﬃﬃﬃﬃﬃ ¼ − 1 þ Rd θw 3 θ0 ð0Þ ¼ Nur ; pﬃﬃﬃﬃﬃﬃﬃﬃ ¼ ‐ϕ0 ð0Þ ¼ Shr: Rex Rex

ð17Þ

where Rex = UW(x)/v is the local Reynolds number. 3. Mathematical modeling Mathematical modeling is a vantage point to reach a solution in an engineering problem, so the accurate modeling of engineering

Table 1 Compared results forθ(η) with those of Mushtaq et al. [17] when Pr = 7, M = 0.5, λ = 0.5, Ec = 0.2, Nt = 0.1, Le = 1, γ = 0.5, θW = 1.5 and Rd = 0. η

Present work (Nb = 0.3)

Mushtaq et al. (Nb = 0.3)

Present work (Nb = 0.5)

Mushtaq et al. (Nb = 0.5)

Present work (Nb = 0.7)

Mushtaq et al. (Nb = 0.7)

0 0.5 1 1.5 2 2.5 3

0.433131 0.192426 0.031337 0.002061 0.000090 0.000005 0.000000

0.433378 0.192910 0.031472 0.002183 0.000096 0.000005 0.000000

0.559314 0.302830 0.061149 0.004317 0.000144 0.000006 0.000000

0.559820 0.302897 0.061267 0.004429 0.000148 0.000006 0.000000

0.713093 0.459378 0.116772 0.009590 0.000284 0.000007 0.000000

0.713220 0.459814 0.116978 0.009621 0.000290 0.000007 0.000000

S.E. Ghasemi et al. / Journal of Molecular Liquids 219 (2016) 890–896

893

Table 2 Compared results for ϕ(η) with those of Mushtaq et al. [17] when Pr = 7, M = 0.5, λ = 0.5, Ec = 0.2, Nb = 0.1, Le = 1, γ = 0.5, θW = 1.5 and Rd = 0. η

Present work (Nt = 0.3)

Mushtaq et al. (Nt = 0.3)

Present work (Nt = 0.5)

Mushtaq et al. (Nt = 0.5)

Present work (Nt = 0.7)

Mushtaq et al. (Nt = 0.7)

0 1 2 3 4 5

1.000000 0.887265 0.247857 0.040873 0.004035 0.000000

1.000000 0.887329 0.247973 0.040981 0.004057 0.000000

1.000000 1.269700 0.367775 0.060668 0.005989 0.000000

1.000000 1.270348 0.367910 0.607173 0.006101 0.000000

1.000000 1.715403 0.517634 0.085411 0.008432 0.000000

1.000000 1.715880 0.517911 0.085503 0.008601 0.000000

problems is an important step to obtain accuratre solutions [29–33]. Most differential equations of engineering problems do not have exact analytic solutions so approximation and numerical methods must be used. Recently some different methods have been introduced to solving these equations, such as the Variational Iteration Method (VIM) [34], Homotopy Perturbation Method (HPM) [35], Parameterized Perturbation Method (PPM) [36], Differential Transformation Method (DTM) [37,38], Modiﬁed Homotopy Perturbation Method (MHPM) [39], Least Square Method (LSM) [40,41], Collocation Method (CM) [42], Galerkin Method (GM) [43], Optimal Homotopy Asymptotic Method (OHAM) [44] and Differential Quadrature Method (DQM) [45]. In this study, Keller Box Method (KBM) has been employed as an efﬁcient numerical method for solving the above problem using Maple 15.0 software. The Keller Box scheme is a face-based method for solving partial differential equations that has numerous attractive mathematical and physical properties. The scheme discretizes partial derivatives exactly and only makes approximations in the algebraic constitutive relations appearing in the Partial Differential Equation (PDE). The exact Discrete Calculus associated with the Keller-Box scheme has been found to be fundamentally different from all other mimetic (physics capturing) numerical methods. Actually, Keller Box is a variation of the ﬁnite volume approach in which unknowns are stored at control volume faces rather than at the more traditional cell centers. Its name alludes to the fact that in space-time, the unknowns sit at the corners of the space-time control volume which is a box in one space dimension on a stationary mesh. The original development of the method [46] dealt with parabolic initial value problems such as the unsteady heat equation. The method was made better by Cebeci and Bradshaw [47] as a method for the solution of the boundary layer equations. 4. Results and discussions As described above, solar radiation effect on a nanoﬂuid boundary layer over a stretching plate in presence of magnetic ﬁeld has been

investigated numerically using Keller-Box numerical method. The boundary layer equations are non-dimensionalized through appropriate similarity transformations and the resulting differential system has been solved for the numerical solutions. Also, heat transfer analysis was performed in the presence of thermal radiation, Joule heating and viscous dissipation by using the Rosseland approximation for thermal radiation. The thus obtained results were then compared with that obtained over the fourth-ﬁfth order Runge-Kutta method as reported by Mushtaq et al. [17]. Fig. 2 shows this comparison for both temperature and nanoparticle concentration proﬁles with different radiation, Brownian motion and thermophoresis parameters. As seen, the results obtained with Keller-Box method shows excellent agreement with those with RungeKutta in wide range of constant parameters. Tables 1 and 2 conﬁrm this agreement in a special case for temperature and nanoparticle concentration values, respectively. The effect of radiation parameter (Rd) and Brownian motion parameter (Nb) on proﬁles are presented in Fig. 3. It can be concluded that in the absence of radiation effects,Rd = 0, the increasing values of Nb accompany with the ﬂatter proﬁles near the stretching wall. According to Brownian motion deﬁnition, by increasing the Nb, the intensity of this chaotic motion increases the kinetic energy of the nanoparticles and as a consequence the nanoﬂuid's temperature rises. On the other hand, from the deﬁnition of thermophoretic parameter, Nt, it is obvious that larger values of Nt correspond to the larger temperature difference and shear gradient. Thus increase in Nt leads to the larger temperature inside the boundary layer ,as shown in Fig. 4. Fig. 5-a and -b demonstrate the effect of Prandtl (Pr) and Eckert (Ec) numbers,respectively, on nanoﬂuid thermal boundary layer. An increase in Pr corresponds to a decrease in thermal diffusivity which thereby increases variations in the thermal characteristics and decreases the ﬂuid's temperature and thermal boundary layer thickness while Eckert number has an inverse effect on the proﬁles. Effect of Biot number (γ) was investigated through Fig. 6. As can be seen in this ﬁgure, the increase in γ results in an increase in both temperature and particles concentrations. Furthermore, γ = 0 corresponds to the

Fig. 3. Effect of radiation parameter (Rd) and Brownian motion parameter (Nb) on a) temperature proﬁle and b) nanoparticles concentration proﬁle.

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Fig. 4. Effect of radiation parameter (Rd) and thermophoresis parameter (Nb) on a) temperature proﬁle and b) nanoparticles concentration proﬁle.

Fig. 5. Effect of radiation parameter (Rd) and a) Prandtl and b) Eckert numbers on temperature proﬁle.

Fig. 6. Effect of radiation parameter (Rd) and Biot number (γ) on a) temperature proﬁle and b) nanoparticles concentration.

S.E. Ghasemi et al. / Journal of Molecular Liquids 219 (2016) 890–896

Fig. 7. Effect of radiation parameter (Rd) and Lewis number (Le) on a) temperature proﬁle and b) nanoparticles concentration.

Fig. 8. Effect of radiation parameter (Rd) and magnetic parameter (M) on a) temperature proﬁle and b) nanoparticles concentration.

Fig. 9. Effect of radiation parameter and a) thermophoresis and b) Brownian motion parameter on θ′(η).

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S.E. Ghasemi et al. / Journal of Molecular Liquids 219 (2016) 890–896

case of insulated sheet. The concentration ﬁeld is driven by the temperature gradient and since temperature is an increasing function of γ, thus one would expect an increase in ϕ with an increase in γ. According to Fig. 7, as Lewis number Le gradually increases, a larger temperature difference and a thinner concentration boundary layer can be found, assuming to be due to the weaker molecular diffusivity. Effects of magnetic parameter (M) on described proﬁles are shown in Fig. 8. The increase in temperature and nanoparticles concentration due to magnetic and radiation parameters can be easily noticed. Finally, Fig. 9 are presented to show the effect of temperature derivative proﬁles in different Brownian motion and thermophoresis parameters. The results are useful for the calculation of the reduced Nusselt number using Eq. (17) which has a direct relation to its value when η = 0. 5. Conclusion In this paper, the effect of solar radiation on the laminar twodimensional nanoﬂuid ﬂow over a stretching plate in the presence of magnetic ﬁeld is investigated numerically using Rosseland approximation for thermal radiation and Kelller-Box method for solving the governing equations. An excellent agreement was found between the obtained results from present study using Kelller-Box method and previous study using Runge-Kutta method. As an important outcome, regardless of the selected value of radiation parameter Rd, the temperature increases as the Brownian motion and thermophoretic effects are simultaneously increased. Furthermore, the nanoparticles volume fraction is found to decrease upon increasing the Brownian motion parameter. Acknowledgments D. Jing gratefully acknowledges the ﬁnancial support from the National Natural Science Foundation of China (no. 51422604, 21276206) and China Fundamental Research Funds for the Central Universities. References [1] S. Kalogirou, Solar Energy Engineering-Processes and Systems, Elsevier, USA, 2009. [2] H.Y. Andoh, P. Gbaha, B.K. Koua, P.M.E. Kofﬁ, S. Toure, Thermal performance study of a solar collector using a natural vegetable ﬁber, coconut coir, as heat insulation, Energy Sustain. Dev. 14 (2010) 297–301. [3] A.A. Hegazy, Thermohydraulic performance of air heating solar collectors with variable width, ﬂat absorber plates, Energy Convers. Manag. 41 (13) (2000) 1361–1378. [4] O.V. Ekechukwu, B. Norton, Review of solar-energy drying systems III: low temperature air-heating solar collectors for crop drying applications, Energy Convers. Manag. 40 (1999) 657–667. [5] M.S. Sodha, N.K. Bansal, K. Kumar, P.K. Bansal, M.A.S. Malik, Solar Crop Drying, vol. 1C. R. C. Press, Palm Beach, Florida, USA, 1987. [6] D. Peng, X. Zhang, H. Dong, K. Lv, Performance study of a novel solar air collector, Appl. Therm. Eng. 30 (2010) 2594–2601. [7] M.A. Leon, S. Kumar, Mathematical modeling and thermal performance analysis of unglazed transpired solar collectors, Sol. Energy 81 (2007) 62–75. [8] M. Gorji, M. Hatami, A. Hasanpour, D.D. Ganji, Nonlinear thermal analysis of solar air heater for the purpose of energy saving, Iran. J. Energ. Environ. 3 (4) (2012) 361–369. [9] S.E. Ghasemi, M. Hatami, D.D. Ganji, Analytical thermal analysis of air-heating solar collectors, J. Mech. Sci. Technol. 27 (11) (2013) 3525–3530. [10] F.S. Javadi, R. Saidur, M. Kamalisarvestani, Investigating performance improvement of solar collectors by using nanoﬂuids, Renew. Sust. Energ. Rev. 28 (2013) 232–245. [11] Tooraj Youseﬁ, Farzad Veysi, Ehsan Shojaeizadeh, Sirus Zinadini, An experimental investigation on the effect of Al2O3-H2O nanoﬂuid on the efﬁciency of ﬂat-plate solar collectors, Renew. Energy 39 (2012) 293–298. [12] M. Faizal, R. Saidur, S. Mekhilef, M.A. Alim, Energy, economic and environmental analysis of metal oxides nanoﬂuid for ﬂat-plate solar collector, Energy Convers. Manag. 76 (2013) 162–168. [13] Zhen-Hua Liu, Ren-Lin Hu, Lin Lu, Feng Zhao, Hong-shen Xiao, Thermal performance of an open thermosyphon using nanoﬂuid for evacuated tubular high temperature air solar collector, Energy Convers. Manag. 73 (2013) 135–143. [14] Ahmad Fudholi, Kamaruzzaman Sopian, Mohd Yusof Othman, Mohd Haﬁdz Ruslan, B. Bakhtyar, Energy analysis and improvement potential of ﬁnned double-pass solar collector, Energy Convers. Manag. 75 (2013) 234–240. [15] M.K. Alkam, M.A. Al-Nimr, Solar collectors with tubes partially ﬁlled with porous substrate, Sol. Energy Eng. 121 (1999) 20–24. [16] H. Taherian, A. Rezania, S. Sadeghi, D.D. Ganji, Experimental validation of dynamic simulation of the ﬂat plate collector in a closed thermosyphon solar water heater, Energy Convers. Manag. 52 (1) (2011) 301–307.

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