Solar energy is the cleanest, renewable and most abundant source of energy available on earth. The amount of electricity that can be produced by solar energy is 4 × 1015 megawatt which is 200 times more then the global consumption of the electricity . The main use of solar energy is to heat and cool buildings, heat water and to generate electricity [2,3,4,5]. There are two types of solar energy collection systems, the photovoltaic systems and the solar thermal collectors. The solar thermal collectors consists of three main parts, the solar energy collection system, the heat storage medium and the heat circulation system. Most popular types of solar thermal collectors are parabolic dish collectors, parabolic trough and power tower systems . The recent research in the field of solar energy has been focused to increase the efficiency of solar thermal collector systems. The efficiency of any solar thermal system depend on two key parameters, the thermophysical properties of the operating fluid and the geometry/length of the system in which fluid is flowing. The properties of the operating fluids include viscosity, density, thermal conductivity and specific heat at high temperature as well as the velocity of the flow [6, 7].
The use of nanofluids instead of ordinary fluids is the key area of research to improve the performance of solar thermal collectors. However it is important to select the type of the nanoparticles, nanoparticles volumetric concentration in the base fluid and the nanofluid thermophysical properties. Chaji et al.  experimented on flat thermal solar collectors using TiO2-water nanofluid with the aim to study the collectors efficiency corresponding to nanoparticles concentration and the flow rate. They found that by adding the nanoparticles to water, the collector efficiency increases between 2.6% and 7% relative to the base fluid. Ghasemi and Ahangar  numerically investigated the thermal field and thermal efficiency of parabolic trough collectors with Cu-water nanofluid and conclude that the solar collectors with nanofluid is more efficient when compared with conventional collectors. The inclusion of copper nanoparticles considerably increase the heat gain capacity of solar collector. Sharma and Kundan  in their experimental setup for parabolic solar collector compare the efficiency of ordinary fluid with aluminium-water nanofluid with copper oxide nanofluid. It is concluded that the aluminium water nanofluid improve the efficiency of solar collector between 1 − 2.55% whereas the use copper oxide nanofluid the efficiency is improved by 0.95 − 3.05%. Bellos et al.  presented that the efficiency of parabolic trough collectors with sine geometry improved by 4.25% if nanofluids are used as operating fluids instead of thermal oil or pressurized water. Recently [12, 13] independently used carbon nanotube nanofluids as working fluids to examine the efficiency of U-tube thermal solar collectors. The use of carbon nanotube nanofluids not only improve the efficiency of solar collectors they also reduce the CO2 emissions. Kim et al.  also compared the efficiency of carbon nanotube nanofluids with Al2O3, CuO, SiO2 and TiO2 nanofluids. Their results indicated that the greatest efficiency was obtained at 62.8% when when carbon nanotube nanofluids are used. Finally the review article of Muhammad et. al  covered almost all the literature of past ten years on use of nanofluids and enhancement in thermal efficiency of solar collectors.
In all aforementioned studies authors considered Newtonian fluid model for convective transport of nanofluids. However, in real situation nanofluids do not have the characteristics of Newtonian fluids hence it is more reasonable to consider them as non-Newtonian fluids. Ellahi et al.  used OHAM and investigated the exact solution of mixed convection Power-law nanofluid of the copper nanoparticles and using the Brinkman nanofluid model. They found that the velocity profile of shear thinning fluids falls when nanoparticle volume fraction is increases. Ellahi et al.  considered the Brinkman nanofluid model to investigate the impact of HFE − 7100 fluid over a wedge the influence of porous medium, entropy generation and three different type of shape nanoparticles such as needle-shaped, disk-shaped, sphere-shaped are taken into consideration. They concluded that needle-shaped nanoparticles results is the greatest temperature in the boundary layer while the lowest temperature are observed in the case of sphere-shaped nanoparticles. It is also obtained when one chose disk-shaped particles and HFE − 7500 fluid shows the greater heat transfer ability as compared with rest of the nanoparticles and the highest entropy is found by the needle-shaped nanoparticles as compared to other nanoparticle shapes. Sheikholeslami et al.  employed Koo-Kleinstreuer-Li correlation for MHD nanofluid flowing over two vertical permeable sheets for Al2O3-water under the influence of free convection is investigated. They used the Runge-Kutta method for the numerical solutions and discussed the effects of various physical parameters on nanofluids. The results indicated that the enhancement in heat transfer is an increasing function of Hartman number. Esfahani et al.  investigated an entropy generation for the Copper-water nanofluid flow through a wavy channel over heat exchanger plat, The results indicated that the enhancement in viscous entropy generation with greater Reynolds number becomes more prominent as non-dimension amplitude rises. In addition to the above, the non-Newtonian nanofluid models are well discussed in [19,20,21].
Examination of solar collectors in terms of their exergy and its efficiency and entropy generation has been carried out in [22,23,24]. The effects of heat transfer irreversibilities on the account of exergy obtained from the solar collectors systems was investigated by Bejan . A comparison between the flat plate and evacuated tube collectors as a function of exergy was carried out by Suzuki. Farzad et al. scrutinized both exergy analysis and energy of a flat plate solar collector. Luminosu and Fara  suggested exergy analysis of a flat plate solar collector based on the assumption that fluid inlet temperature is equal to ambient temperature. The analysis optimum values of mass flow rate, absorber plate area, and maximum exergy efficiency of a flat plate collector was carried out by Farahat et al.. Nasrin et al. considered radiative heat flux effects on the direct absorption of the solar collector and studied the heat transfer and collector efficiency. The use of graphene based nanofluid in the flat plate solar collector exergy efficiency analysis was carried out by Said et al.. Further detail regarding the entropy generation analysis in solar collectors can be found in [32,33,34]. In all aforementioned studies authors considered Newtonian fluid model for convective transport of nanofluids. However, in real situation nanofluids do not have the characteristics of Newtonian fluids hence it is more reasonable to consider them as non-Newtonian fluids. The flow and heat transfer analysis of non-Newtonian models of nanofluids in solar collectors will provide researchers the better understanding of thermal characteristics of nanofluids and efficiency measurement. Some authors do considered non-Newtonian models for the nanofluids under different thermophysical situations, for example, [35,36,37,38,39,40,41]. To the best of author’s knowledge no research is conducted to study the combined effect of MHD non-Newtonian Maxwell nanofluid with slip and convective boundary conditions, thermal radiation and temperature dependent thermal conductivity on velocity, temperature distribution and entropy generation of the system. The flow is induced by a non-uniform stretching of the porous sheet and the uniform magnetic field is applied in the transverse direction to the flow. The numerical computations are carried for Cu-water and TiO2-water nanofluids and results are concluded on the effect of various governing parameters on the motion, temperature and entropy generation of Maxwell nanofluid.
2 Mathematical formulation
We consider an unsteady, two-dimensional laminar flow with heat transfer of an incompressible electrically conducting non-Newtonian Maxwell nanofluid over a porous stretching sheet. The flow is generated due to the stretching of sheet with non-uniform velocity
where c is the initial stretching rate and is the effective stretching rate. The surface of the plate is insulated and partial slip and convective conditions has been invoked at the boundary. For convenience, the leading edge of the plate is assumed at x = 0 and is considered along the x-axis. A uniform magnetic field of strength is applied in the transverse direction to the flow and the induced magnetic field is considered negligible as compared to applied magnetic field. The temperature of the plate is and T∞ is the temperature outside the boundary layer. Thermal conductivity of the nanofluid is to vary as a linear function of temperature, T. This assumption is valid because thermal properties of nanofluids change significantly with rise in temperature, type of nanoparticles, pressure etc. Finally, the non-Newtonian Maxwell nanofluid is considered optically thick and radiation only travel a short distance. Therefore radiative heat transfer is taken into account and Rosseland approximation is utilized for the radiation effects. The schematic diagram of the mathematical model under consideration is presented in Figure (1).
The governing equations under boundary layer approximation for the flow of Maxwell nanofluid along with heat transfer are obtained as (see for example, Mukhopadhyay )
The boundary conditions for the present model are
Here, u and v are velocities in x and y directions respectively, t is the time, μnf is the dynamic viscosity of the nanofluid, ρnf is the nanofluid density, σnf is the electrical conductivity of nanofluid. λ = λ0(1 − αt) is the thermal relaxation time of the period, λ0 is a constant. qr is the radiative heat flux, (Cp)nf and are the specific heat capacity and the thermal conductivity of nanofluid respectively. Vw represents the porosity of the stretching surface, W1 = W0 is the velocity slip factor with W0 is an initial slip parameter.
Here, ϕ and knf are nanoparticle volume concentration coefficient and thermal conductivity respectively. μf, ρf and (Cp)f, σf and κf are dynamic viscosity, density, specific heat capacity, electrical conductivity and thermal conductivity of the base fluid. ρs, (Cp)s, σs and κs are the density, specific heat capacity, electrical conductivity and thermal conductivity of the nanoparticles. Using Rosseland approximation (Brewster ), we may write
where, σ* is the Stefan Boltzmann constant and k* is the mean absorption coefficient. It is assumed that the difference in temperature within the flow is such that T4 may be expanded as a Taylor series about free stream temperature T∞ and considering only the linear terms, we get
3 Similarity solution of the problem
Here ψ(x, y) is the stream function and the similarity variables are defined as
and boundary conditions
In above equations, prime denotes the differentiation with respect to is the unsteadiness parameter, β = cλ0 is the Maxwell parameter, is the magnetic parameter,
is the porous medium parameter, is the Prandtl number, is the thermal diffusivity parameter,
is the thermal radiation parameter, is the mass transfer parameter,
is the velocity slip parameter and is the sheet convection parameter or so-called Biot number.
The nonlinear system of ordinary differential equations (3.3)-(3.4), arising from mathematical modeling of physical system of nanofluid flow in solar collector are difficult to solve analytically. Therefore, Keller box method  scheme is employed to find the approximate solution of system. This method is unconditionally stable with a second order convergence.
4 Numerical solution of the problem
The flow chart of the Keller box method  is as follows
With the newly introduced variables, the boundary condition becomes:
Then the derivatives are approximated by the central differences and averages centered at the midpoints of the mesh, defined by,
In the above discussion, for the (i + 1)-th iterate, we write
subject to the boundary conditions
Where A is J × J block tridiagonal matrix with each block size of 5 × 5. Whereas δ and b are column matrices with J rows. We, then apply the LU factorization to find the solution of δ.
The physical quantities of interest governed the flow are the Skin frication (Cf) and the local Nusselt number (Nux), and are defined as: (See for example Abel et al.)
where τw and qw are the heat flux which is given by
Applying the non-dimensional transformations (3.2), one obtains
where Cf and Nur are the reduced Skin frication and Nusselt number respectively. is the local Reynolds number based on the stretching velocity Uw.
To check the validity of our numerical scheme we compare our results to those already available in the literature [49,50,51,52] as the especial case for our study. The test case is the natural convection boundary layer flow of fluid over a flat plate with Newtonian slip. Results have been obtained for β = 0, A = 0, M = 0, K = 0 ϕ = 0, Λ = 0, ϵ = 0, S = 0, Nr = 0 and Bi → ∞. The comparison is shown in Table (1) and are found to be in an excellent agreement. Thus, we are very much confident that the present results are accurate.
5 Entropy generation analysis
Wastage of useful energy ia a part of concern for scientists and engineers. It becomes important to analyze the entropy generation in the system that involves to the irreversibility of useful energy, magnetohydrodynamics is one of the non-ideal effects which is responsible for increasing the entropy of the system. In our case, the actual entropy generation in the nanofluids is given by (See for example Das et al.)
In the entropy equation the first term represents the heat transfer irreversibility and the second term is due to the fluid friction and the third term is because of magnetohydrodynamic effect and porous medium. The dimensionless entropy generation is represented by NG and is defined as
From Equation (3.4), we can achieve the entropy equation in the dimensionless form as
here Re represents the Reynolds number, Br represents the Brinkmann number and the dimensional less temperature gradient that can represented by Ω, which is defined by
6 Numerical results and discussion
Numerical results in the form of graphs and tables for chosen values of some physical parameters are presented in this section. The results are produce for the Cu-water and TiO2-water non-Newtonian Maxwell nanofluids. The discussion is focused on the thermal enhancement in nanofluids and comparison can be drawn on the behavior of two different type of nanofluids. The numerical results are presented in Figs. (2)-(22) and in Table (3). The material properties of Cu-water and TiO2-water nanofluids are tabulated in Table 2 (Sharma et al. ).
The effects of Maxwell parameter β on velocity, temperature and entropy generation profiles of Cu-water and TiO2-water non-Newtonian Maxwell nanofluids are presented in Figures (3)-(5). Computations are performed for β = 1.0, 5.0, 10.0 at uniform nanoparticle concentration of 0.2. The velocity profiles in Figure (3) decreases with an increasing values of β and hence decreases the thickness of momentum boundary layer. Moreover, for the fixed value of β = 0.3 the boundary layer thickness of TiO2-water nanofluid is relatively more than the Cu-water nanofluid. The decreasing trend in velocity profiles is due to increase of resistance in fluid and also corresponds to increase in skin friction coefficient (velocity gradient) at the boundary. It can be seen from Figure (4) that the temperature of nanofluids rises with the increasing values of parameter β. This increasing trend indicate the enhancement in the thickness of thermal boundary layer and reduction in the rate of heat transfer. The reason behind this behaviour of temperature profiles is the increase in the elasticity stress parameter. Figure (5) shows the impact of Maxwell parameter β on the entropy of the system. It is noticed that increasing Maxwell parameter increases the entropy of the system. Finally, it is observed from Table (3), the rate of heat transfer at the boundary (Nusselt number) decreases for both Cu and TiO2 water based nanofluids.
Figures (6)-(8) depicted the influence of unsteady parameter A on velocity, temperature and entropy generation profiles of Maxwell nanofluid. It is found that the fluid flow slowly (Figure (6)) and its temperature decrease within boundary layer with ascending values of parameter A (Figure (7)). The effect of increasing values of parameter A is to decrease the thickness of both momentum and thermal boundary layer thickness. Figure (8) displays the influence of variation of unsteadiness parameter A on the entropy generation. It is observed (at about η = 0.3) the entropy profiles show the cross-over point. Before this point the entropy is increasing and after this it starts decreasing. From Table (3), the increasing trends are observed for the velocity and temperature gradients at the boundary. The boundary layer energy is absorbed due to unsteadiness resulting the increase in the rate of heat transfer at the boundary.
Figures (9)-(14) exhibited the behaviours of nanofluids motion, temperature distribution and entropy generation with increasing strength of applied transverse magnetic field and the porosity of the medium respectively. Similar behaviours are observed in profiles of velocity, temperature and entropy with increasing values of parameter M and K. The magnetic field applied normal to the flow direction, produces a resistive force known as Lorentz force which reduces the fluid motion within the boundary layer. The Lorentz force impact in the form of decreasing trend in velocity profiles are clearly visible in Figure (9). Whereas the increase in permeability is to decrease the magnitude of the resistive Darcian body force, therefore a continuous less drag is experienced by the fluid and flow reduces thereby declines the velocity within boundary layer (12). The parameters M and K are inversely proportional to the density of nanofluid hence the increase in the strength of applied magnetic field or the permeability of the medium reduces the density and as a result the temperature of the fluid rises within boundary layer (Figs. (10), (13)). This will increase the thickness of thermal boundary layer and reduces the rate of heat transfer. The influence of Lorentz or the Darcian body force at the boundary is presented in Table (3). The Skin friction coefficient increases but the Nusselt number decreases with increasing strength of parameters M and K. Figure (11), (14) demonstrated that the entropy of the system increases with increase in the strength of applied transverse magnetic field and the permeability of the medium.
Figures (15)-(17) displayed respectively the nature of fluid motion, temperature distribution and entropy generation within boundary layer for Maxwell nanofluids due to variation in nanoparticle volume concentration parameter ϕ. It is found that the nanofluid velocity decreases and the temperature increases with increasing values of parameter ϕ. These figures are in agreement with the physical behavior that the denser nanoparticle volume fraction causes thinning of momentum boundary layer and the rate of heat transfer also reduces within boundary layer. The increase in volume of nanoparticles increases the overall thermal conductivity of nanofluids because the solid particles have higher thermal conductivity as compared to base fluid. The increase in thermal conductivity is responsible for decrease in thickness of momentum boundary layer. Whereas the increase in overall thermal conductivity of nanofluids raises the temperature and boundary layer thickness. The increasing and decreasing trend in velocity and temperature gradient respectively at the boundary are observed with increase in parameter ϕ Table (3). In Figure (17) depicts increasing nanoparticle volume fraction Parameter ϕ the entropy of the system is also increases.
Figures (18)-(20) showed the positive values of slip parameter Λ, slows down the fluid motion and entropy generation but raise the temperature of Maxwell nanofluids. The decrease in velocity is obvious because the increase in lubrication and slipperiness at the surface retards the flow and the stretching pull can be only partly transmitted to the fluid. As a result the thickness of momentum boundary layer will decreases with increase in parameter Λ. On the other hand, slipperiness affects the temperature of the fluid inversely; that is, the temperature of the nanofluid enhances within the boundary layer. Table (3) shows the increase in Λ, leads to decrease in skin friction coefficient for both Cu and TiO2 water based nanofluids. This was expected to happen due to the fact that slip effects reduces the friction at the solid-fluid interface and thus reduce skin friction coefficient. The reduction in Nusselt number or the heat transfer rate at the boundary is observed from Table (3) with increase in slip. Increasing velocity slip parameter Λ is analyzed in Figure (20). It is noteworthy that the entropy decreases by increasing the values of Λ.
Figure (21) shows the impact of Reynolds number on the entropy of the system. It is noticed that increasing Reynolds number Re increase the entropy of the system. The physical reason for this observation is that for higher Reynolds number the inertial forces dominates the viscous forces thus the entropy of the system rises. Figure (22) discusses the influence of Brinkman number on the entropy of the system. It is observed that increasing the values of Br increases the temperature and hence the entropy of the system.
The present research analyzed the heat transfer capabilities and the entropy generation of non-Newtonian Maxwell nanofluid in the presence of slip and convective boundary conditions. Thermal radiation and the temperature dependent thermal conductivity are also considered in the present model and the uniform magnetic field is applied in the transverse direction to the flow. The mathematical formulation is carried out through a boundary layer approach and the numerical computations are carried out for Cu-water and TiO2-water nanofluids. The results are summarized on the basis of variation in nanofluid’s motion, temperature distribution and entropy generation within boundary layer.
The key parameters such as, non-Newtonian Maxwell fluid parameter, strength of the applied magnetic field, permeability, nanoparticle volumetric concentration, variable thermal conductivity, velocity slip and thermal radiation increases the temperature distribution within the boundary layer. This will increase the thickness of thermal boundary layer and reduce the rate of heat transfer at the surface. This will increase the overall entropy of the system and decreases the fluid motion within boundary layer. The unsteadiness parameter and the suction parameter at the boundary reduces the thickness of the thermal boundary layer and increases the rate of heat transfer at the surface.
The highlight of the study, entropy of the system is observed to enhance with the increase in the values of Reynolds number Re, Brinkmann number Br, unsteadiness parameter A, magnetic parameter M, permeability parameter K, nanoparticle volume fraction Parameter ϕ and suction parameter S > 0 but reduce with increase in the values of injection parameter S < 0 and velocity slip Parameter Λ. Finally, the Cu-water based nanofluid is observed as better thermal conductor than TiO2-water based nanofluid.
In future the present qualitative analysis can be quantified to calculate the thermal efficiency of solar collectors and can be generalized to include the effects of variable viscosity, variable porosity, multidimensional MHD slip flow and heat transfer of non-Newtonian and regular nanofluids.
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Published Online: 2018-04-18