Abstract
In this paper, we compared copper-engine oil Casson–Maxwell, Casson–Jeffrey, and Casson–Oldroyd-B binary nanofluids in a parabolic trough solar collector. Using appropriate similarity variables, the partial differential equations governing nanofluid flow were converted into ordinary differential equations. The resulting nonlinear systems were solved using the shooting method. The numerical results were presented in graphical and tabular forms. We investigated the effects of different parameters controlling the flow on the velocity, temperature, entropy generation, skin friction, and local Nusselt number of the nanofluids. Overall, the Casson–Maxwell and Casson–Jeffrey nanofluid models had better efficiency than the Casson–Oldroyd-B nanofluid model.
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Introduction
In recent years, renewable energy, in general, and solar energy, in particular, have gained considerable attention owing to their environmental friendliness and effectiveness in generating electrical and thermal energy. Solar energy is usually harnessed using photovoltaic technology or solar collectors (SCs). A SC is preferable owing to its cost efficiency and energy collection capabilities [1]. The suspension of nanoparticles in the base fluid is a major advancement in developing SCs with enhanced efficiency. Choi [2] was the first to observe improved heat transfer using nanoparticles. Over the years, studies investigating nanofluids in parabolic trough solar collectors (PTSCs) have been conducted [3,4,5,6]. Shahzad et al. [7] studied the thermal characterization of a solar-powered ship using Oldroyd hybrid nanofluids in a PTSC. Jamshed et al. [8] performed a thermal examination of renewable solar energy in a PTSC using a Maxwell nanofluid. Jamshed et al. [9] conducted a computational study on implementing renewable solar energy in the presence of a Maxwell nanofluid in a PTSC. Nabwey et al. [10] investigated the effect of resistive and radiative heats on enhanced heat transfer of a PTSC using Maxwell–Oldroyd-B nanofluids.
Many researchers have studied the fractional calculus methods, the effects of convective heat transfer and thermal radiation processes on nanofluid efficiency, and the applications of nanofluids in the presence of oxytactic microorganisms in different engineering problems [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28].
Currently, binary nanofluids are being studied for various applications as they have better energy transfer than mono nanofluids [29,30,31,32]. Yu [33] employed a decoupled wavelet approach for multiple physical flow fields of binary nanofluids in double-diffusive convection. Lee et al. [34] conducted a theoretical study on the performance comparison of various SCs using binary nanofluids. Reddy et al. [35] studied the effects of viscous dissipation and thermal radiation on electrically conducting Casson–Carreau nanofluid flow using the Cattaneo–Christov heat flux model. Oyelakin et al. [36] investigated the effect of double-diffusion convection on three-dimensional magnetohydrodynamic (MHD) stagnation point flow of a tangent hyperbolic Casson nanofluid. Yousef et al. [37] explored the chemical reaction impact on MHD dissipative Casson–Williamson nanofluid flow over a slippery stretching sheet through a porous medium.
Jamshed and Nisar [38] compared copper/alumina-engine oil Williamson nanofluids in a PTSC. They concluded that copper was considerably better than alumina in terms of heat transfer. Other studies [39,40,41] have also suggested the use of copper rather than other materials owing to its high efficiency to achieve enhanced thermal properties. Notably, the shape of nanoparticles can affect nanofluid heat transfer ability [42, 43].
Based on the literature review, we believe that a direct comparison of binary nanofluids in PTSCs is lacking, and we hope to contribute toward mitigating this lack of knowledge. Notably, the Jeffrey and Casson–Jeffrey nanofluid models have not been investigated in PTSC settings. The other novel aspects of the current study are as follows:
-
The comparison between the Casson–Maxwell, Casson–Jeffrey, and Casson–Oldroyd-B binary nanofluids in PTSCs.
-
The consideration of thermophoretic diffusivity.
-
The consideration of a constant inclined magnetic field.
-
The consideration of different nanoparticle shapes.
The practical significance of this study lies in the use of the considered models in PTSCs [10, 41, 44], solar water pumps [45], and solar-powered ships [7, 8].
Formulation of the problem
Consider the velocity and temperature of a nonuniform stretching insulated sheet as follows [46]:
where \(a\), \(a^{*}\), \(T_{{\text{W}}}\), and \(T_{\infty }\) denote the primary stretching rate, thermal discrepancy rate, heat at the wall, and heat at the free stream, respectively. Figure 1 presents a schematic of a PTSC.
Stress tensor of binary fluids
Herein, we studied three binary liquids: Casson–Maxwell, Casson–Jeffrey, and Casson–Oldroyd-B.
The rheological equation of state of an incompressible and isotropic flow of a Casson fluid is as follows [47]:
where \(\tau_{{{\text{ij}}}}\) denotes the stress tensor, \(e_{{{\text{ij}}}}\) denotes the \(\left( {i,j} \right)\)th component of the deformation rate, \(\pi = e_{{{\text{ij}}}} e_{{{\text{ij}}}}\), i.e., the product of the component of deformation with itself, \(\pi_{{\text{c}}}\) denotes a critical value of this product based on the non-Newtonian model, \(\mu_{{\text{B}}}\) denotes the plastic dynamic viscosity of the Casson fluid, and \(p_{{\text{y}}}\) denotes the yield stress of the fluid.
The Maxwell model is a simple linear model with one elastic parameter, which is \(\lambda_{1}\), the fluid relaxation time. This model combines the concepts of fluid viscosity and solid elasticity to arrive at the following relation:
The Jeffrey model extends the Maxwell model by adding a time derivative of the strain rate to yield the following equation:
where \(\lambda_{3}\) denotes the retardation time, which is a measure of the time required by the material to respond to deformation.
The Oldroyd-B model is the nonlinear equivalent of the linear Jeffrey model, because it considers the frame invariance in the nonlinear regime. It can be constructed by replacing the partial time derivatives in the differential form of the Jeffrey model with the upper convective time derivatives to achieve the following form:
where \(\tilde{\tau }\) and \(\widetilde{{\dot{\gamma }}}\) denote the upper convected derivatives of the stress and strain tensors, respectively, and are expressed as follows:
For more details about the different aspects of various non-Newtonian fluids we refer the reader to the comprehensive work conducted by Sochi [48].
Governing equations
The principles of conservation of mass, momentum, and energy are applied to two-dimensional, incompressible, laminar, steady flow of binary nanofluids. We also consider the porous medium along with the thermophoresis effect and a constant inclined magnetic field as follows [44, 49]:
The suitable boundary constraints are as follows:
where \(u\) and \(v\) denote the velocity components in the x and y directions, respectively; \(\mu_{{{\text{nf}}}}\), \(\rho_{{{\text{nf}}}}\), \(\sigma_{{{\text{nf}}}}\), \(k_{{{\text{nf}}}}\), and \(C_{{{\text{p}}_{{{\text{nf}}}} }}\) denote the nanofluid viscosity, density, electrical conductivity, thermal conductivity, and specific heat, respectively; \(\lambda_{2}\) denotes the ratio of the relaxation to retardation times; \(\beta\) denotes the Casson parameter; \(k_{{\text{p}}}\) denotes the porosity; \(B_{0}\) denotes the strength of the magnetic field perpendicular to \(\overline{OH}\), and \(\Gamma\) denotes the angle of inclination of the magnetic field; \(T\) denotes the nanofluid temperature; \(Q_{0}\) denotes the heat source; \(q_{{\text{r}}}\) denotes the radiative heat flux; \(\chi\) denotes the ratio of operative heat capability; \(D_{{\text{T}}}\) denotes the thermophoresis diffusion coefficient; \(N_{{\text{w}}}\) denotes the slip length; \(V_{{\text{w}}}\) denotes the surface permeability; \(\kappa_{0}\) denotes the surface thermal conductance; and \(h_{{\text{f}}}\) denotes the heat transfer coefficient. Figure 2 presents the boundary layer flow in a PTSC.
The following conditions describe the different case studies:
-
(1)
\(\lambda_{1} \ne 0\), \(\lambda_{2} = 0\), \(\lambda_{3} = 0\) represents the Casson–Maxwell model.
-
(2)
\(\lambda_{1} = 0\), \(\lambda_{2} \ne 0\), \(\lambda_{3} \ne 0\) represents the Casson–Jeffrey model.
-
(3)
\(\lambda_{1} \ne 0\), \(\lambda_{2} = 0\), \(\lambda_{3} \ne 0\) represents the Casson–Oldroyd-B model.
The radiative heat flux under the Rosseland approximation [21, 50] takes the form \(q_{r} = - \frac{{16\sigma^{*} T_{\infty }^{3} }}{{3k^{*} }}\frac{\partial T}{{\partial y}}\) in which \({\upsigma }^{*}\) is the Stefan–Boltzmann constant and \(k^{*}\) is the mean absorption coefficient.
Table 1 presents the thermophysical aspects of the nanofluids [7, 51], Table 2 specifies the nanoparticle shape factor (m) values for various particle shapes [52, 53], and Table 3 lists the substance properties of engine oil and copper [52].
Solution of the problem
To transform the partial differential equations to ordinary differential equations (ODEs), we introduce the following similarity variables [44]:
This transforms Eqs. (9–10) along with boundary conditions (11–12) to the following forms:
where \(\beta_{1} = \lambda_{1} a\) and \(\beta_{3} = \lambda_{3} a\) denote Deborah numbers, \(K = \frac{{\mu_{{\text{f}}} }}{{a{ }\rho_{{\text{f}}} { }k_{{\text{p}}} }}\) denotes the porosity parameter, \(M = \frac{{\sigma_{{\text{f}}} { }B_{0}^{2} { }}}{{a{ }\rho_{{\text{f}}} }}\) denotes the magnetic number, \(\alpha = \frac{{k_{{\text{f}}} }}{{\left( {\rho C_{{\text{p}}} } \right)_{{\text{f}}} }}\) denotes the thermal diffusion rate, \(\nu = \frac{{\mu_{{\text{f}}} }}{{\rho_{{\text{f}}} }}\) denotes the kinematic viscosity, \({\text{Pr}} = \frac{\nu }{\alpha }\) denotes the Prandtl number, \(N_{{\text{r}}} = \frac{{16\sigma^{*} T_{\infty }^{3} }}{{3k^{*} \nu \left( {\rho C_{{\text{p}}} } \right)_{{\text{f}}} }}\) denotes the thermal radiation parameter, \(Q = \frac{{Q_{0} }}{{a \left( {\rho C_{{\text{p}}} } \right)_{{\text{f}}} }}\) denotes the heat source parameter, \({\text{Ec}} = \frac{{U_{{\text{W}}}^{2} }}{{C_{{{\text{p}}_{{\text{f}}} }} \left( {T_{{\text{W}}} - T_{\infty } } \right)}}\) denotes the Eckert number, \(N_{{\text{t}}} = \frac{{\chi D_{{\text{T}}} \left( {T_{{\text{W}}} - T_{\infty } } \right)}}{{\nu T_{\infty } }}\) denotes the thermophoretic parameter, \(\Lambda = N_{{\text{w}}} \sqrt {\frac{a}{\nu }}\) denotes the velocity slip parameter, \(S = - V_{{\text{w}}} \sqrt {\frac{1}{\nu a}}\) denotes the mass transfer parameter, and \({\text{Bi}}_{{\text{T}}} = \frac{{h_{{\text{f}}} }}{{\kappa_{0} }}\sqrt {\frac{\nu }{a}}\) denotes the thermal Biot number.
Physiological concepts of interest
Considerable attention is directed toward the skin friction constant \(C_{{\text{f}}}\), the local Nusselt number \(Nu_{{\text{x}}}\), and the entropy generation equation [45, 46, 52, 54]:
where
Using the dimensionless parameters defined in Eq. (13), we get the following:
where \({\text{Re}}_{{\text{x}}} = \frac{{x U_{{\text{W}}} }}{\nu }\) denotes the local Reynolds number depending on the stretching velocity, \(N_{{\text{G}}} = \frac{{E_{{\text{G}}} T_{\infty }^{2} a^{2} }}{{k_{{\text{f}}} \left( {T_{{\text{W}}} - T_{\infty } } \right)^{2} }}\) denotes the entropy generation dimensionless factor, \({\text{Re}} = \frac{{U_{{\text{W}}} a^{2} }}{\nu x}\) denotes the Reynolds number, \(\Omega = \frac{{\left( {T_{{\text{W}}} - T_{\infty } } \right)}}{{T_{\infty } }}\) denotes difference in temperature parameter, and \(B_{{\text{K}}} = \frac{{\mu_{{\text{f}}} U_{{\text{W}}}^{2} }}{{k_{{\text{f}}} \left( {T_{{\text{W}}} - T_{\infty } } \right)}}\) denotes the Brinkman number.
Numerical solution procedure
Shooting method
Obtaining the exact analytical solutions from highly nonlinear problems is considerably difficult. Thus, the shooting method is employed via bvp4c in MATLAB, which is a finite difference code that implements the three-stage Lobatto–Illa formula, which is fourth-order accurate uniformly in the interval of integration. The mesh selection is based on the residual of the continuous solution. More details on the theoretical and practical backgrounds of bvp4c can be found in [55].
Before using the shooting method, higher-order ODEs must be converted into a set of first-order ODEs. Let
Validation of the numerical code
To authenticate the current study, the current findings were tested against previous studies [8, 44, 49, 56,57,58,59]; the comparisons are listed in Tables 4 and 5. Overall, the agreement with previous studies provides confidence to proceed with the current study.
Results and discussion
The effects of the parameters controlling the flow on Casson–Maxwell, Casson–Jeffrey, and Casson–Oldroyd-B copper-engine oil binary nanofluids in a PTSC are presented in Figs. 3–27 and Tables 6, 7. The objectives of this study are to compare the three nanofluids and to determine the parameters that improve the efficiency of the device. The default parameter values are taken as follows: \(\beta = 1; \beta_{1} = \beta_{3} = \lambda_{2} = 0.5;K = 0.6;M = 0.2; \Gamma = \frac{\pi }{2}; S = 0.2,\Lambda = 0.3494; \varphi = 0.2;{\text{Pr}} = 7; N_{{\text{r}}} = {\text{Ec}} = 0.2; Q = 0.1; N_{{\text{t}}} = 0.15; {\text{Bi}}_{{\text{T}}} = 0.1;{\text{Re}} = B_{{\text{K}}} = 5; \;{\text{and}}\;{ }\Omega = 0.5; \;{m = 3}\)
Effect of Casson parameter \(\beta\)
When the Casson parameter \(\beta\) is increased, the resistance to the flow also increases; consequently, a reduction in velocity is presented in Fig. 3. The Casson–Maxwell nanofluid exhibited the greatest resistance among the three binary nanofluids. The decrease in velocity extends the time allowed for the fluid to gain more heat, resulting in improved heat transferability. As presented in Fig. 4, the Casson–Maxwell nanofluid is the most enhanced in terms of heat transfer capabilities of the three binary nanofluids. Figure 5 shows that the entropy generation decreased with the increasing \(\beta\) as the increase in \(\beta\) reduces the rheological features, resulting in a faster shearing along the surface [60, 61]. At the wall, the Casson–Maxwell nanofluid had the highest \(N_{{\text{G}}}\), followed by the Casson–Jeffrey nanofluid, and then the Casson–Oldroyd–B nanofluid, whereas far from the wall, the Casson–Oldroyd–B nanofluid had the highest \(N_{{\text{G}}}\), followed by the Casson–Jeffrey nanofluid, and then the Casson–Maxwell nanofluid.
Effect of the porosity parameter \(K\)
Large values of the porosity parameter contribute against the fluidity, resulting in a reduction in the velocity profile and an increase in both heating and irreversible energy (Figs. 6–8). Physically, the increase in the porosity parameter decreases nanoparticle collision owing to an increase in the size of the flow pore, inhibiting heat generation. The viscous force controls the buoyancy force, thereby decreasing the flow magnitude. The entropy increased owing to existence of considerable temperature gradient at the surface [8, 38]. The most affected of the three nanofluids was the Casson–Jeffrey nanofluid, with the highest heat transfer and an entropy level close to that of the Casson–Maxwell nanofluid. The Casson–Oldroyd-B nanofluid exhibited the lowest heat transfer and highest entropy. The effect of \(K\) on the velocity, temperature, and entropy profiles is the same as that reported by previous studies [8, 38, 45].
Effect of the velocity slip parameter \(\Lambda\)
The friction force between the flow and the surface increases with increasing \(\Lambda\), leading to an increase in viscosity and hence a decrease in velocity (Fig. 9). The decrease in velocity is accompanied by an increase in temperature owing to less heat being transported from one point to another (Fig. 10). Figure 11 clearly shows that the entropy decreases with increasing \(\Lambda\). Generally, no-slip boundary conditions produce large velocity and temperature gradients, causing high entropy values. However, slip boundary conditions as in this study cause reduction in entropy close to the stretching wall [7, 8]. Although the Casson–Jeffrey nanofluid exhibited a substantial heat profile enhancement, the Casson–Oldroyd-B nanofluid continued to exhibit low heat transfer and a high irreversible energy drain. The effect of \(\Lambda\) on the velocity, temperature, and entropy profiles is the same as that reported by previous studies [7, 8, 38, 45, 52].
Effect of the nanoparticle concentration parameter \(\varphi\)
The fluid viscosity increases with increasing \(\varphi\) owing to an increase in nanoparticle concentration and friction escalations, resulting in a decrease in velocity, as shown in Fig. 12. Because fluid velocity is critical for heat transfer, the reduction in velocity causes heat to accumulate, and the temperature profile increases as observed in Fig. 13. It is expected that the entropy increases owing to heat accumulation [7, 8], which is confirmed in Fig. 14. Of the three nanofluids, Casson–Maxwell nanofluid benefited the most, as it underwent the highest improvement in terms of heat transfer. The effect of \(\varphi\) on the velocity, temperature, and entropy profiles is the same as that reported by previous studies [7, 8, 38, 45, 52].
Effect of the nanoparticle shape factor \(m\)
Increasing shape factor values \(m\) by changing the nanoparticle shapes improves heat transferability. Figures 15 and 16 demonstrate that a lamina shape is the most effective in terms of heat transfer while causing a negligible increase in irreversible energy. Among the three nanofluids, the Casson–Maxwell nanofluid exhibited the highest heat transfer after increasing the shape factor. The effect of \(m\) on the temperature and entropy profiles is the same as that reported by a previous study [52].
Effect of the magnetic number \(M\)
Figure 17 shows the effect of \(M\) on the velocity profile. As the strength of the magnetic field increases, the drag force increases, which reduces the velocity of the nanofluid flow. Figure 18 confirms the fact that temperature distribution increases with increasing magnetic field because less heat can be transferred from one point to another with decreasing fluid movement. The lack of heat transfer in the system causes a natural increase in entropy [7], which is seen in Fig. 19. The Casson–Oldroyd-B nanofluid exhibited the lowest heat transfer. The effect of M on the velocity, temperature, and entropy profiles is the same as that reported by a previous study [7].
Effect of the magnetic angle of inclination \(\Gamma\)
As \(\Gamma\) increases from \(\frac{\pi }{6}\) to \(\frac{\pi }{2}\), the normal magnetic forces increase, resulting in a decrease in velocity. The reduced velocity will cause the temperature distribution to increase owing to low heat relocation from the surface toward the flow. The heat accumulation will intensify entropy generation in the system. Heat transferability and entropy are therefore increased, with the Casson–Maxwell nanofluid exhibiting the highest heat transferability as seen in Figs. 20–22. The effect of \(\Gamma\) on the velocity, temperature, and entropy profiles is the same as that reported by a previous study [7].
Effect of the thermal Biot number \({\text{Bi}}_{{\text{T}}}\)
The thermal Biot number describes the ratio between thermal resistances inside the fluid and at the surface. Hence, large values of \({\text{Bi}}_{{\text{T}}}\) amplify the thermal resistance inside the flow, which increases the fluid temperature. The increase in \({\text{Bi}}_{{\text{T}}}\) values leads to an increase in the heat transfer rate and the entropy generated by the system [7], which is illustrated in Figs. 23 and 24. Of the three nanofluids, the Casson–Oldroyd-B nanofluid exhibits the lowest heat transfer. The effect of \({\text{Bi}}_{{\text{T}}}\) on the temperature and entropy profiles is the same as that reported by previous studies [7, 45].
Effect of thermophoretic parameter \(N_{{\text{t}}}\)
Higher values of the thermophoretic parameter indicate larger thermophoretic force, which appears in suspended mixtures of particles and fluids such as nanofluids. This phenomenon is caused by the temperature gradient which leads to a movement of fluid particles in the hot zone toward the cold region. Hence, heat transfer capabilities increase as seen in Fig. 25. The highest heat transfer was observed in the Casson–Maxwell and Casson–Jeffrey nanofluids.
Effects of the Reynolds number Re and the Brinkman number \(B_{{\text{K}}}\)
At high values of Re, the resistive forces replace the viscous forces. Further, the Brinkman number measures the dominant effects responsible for variation in irreversible energy drain. It determines whether heat is transferred or dissipated through conduction at the surface. The increase in \(B_{{\text{K}}}\) values as in this case, causes an increase in entropy generation which indicates the dominant effect of dissipation [7]. Notably, increasing Re or \(B_{{\text{K}}}\) increases the irreversible energy drain, as seen in Figs. 26 and 27. The difference among the three nanofluids was found to be marginal. The effects of Re and \(B_{{\text{K}}}\) on the velocity, temperature, and entropy profiles are the same as that reported by previous studies [7, 8, 38, 45].
Conclusions
The copper-engine oil Casson–Maxwell, Casson–Jeffrey, and Casson–Oldroyd-B binary nanofluids were compared in PTSC settings. Porosity, constant inclined magnetic field, heat source, thermal radiation, viscosity dissipation, and thermophoresis were considered. Analysis was conducted by formulating the continuity, momentum, and energy equations. Subsequently, similarity variables were used to transform the partial differential equations into ODEs, which were then solved using the shooting method. There are various practical applications for the presented models, including solar water pumps, solar-powered ships, solar street lights, solar energy plates, and photovoltaic cells. Overall, an increase in PTSC efficiency can be achieved by decreasing the nanofluid velocity and boosting the nanofluid temperature while maintaining reasonable entropy generation. The key conclusions of this numerical study are as follows:
-
The parameters that reduced the velocity profile are \(\beta\), \(K\), \(\Lambda\), \(\varphi\), \(M\), and \(\Gamma\).
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The parameters that enhanced heat transfer are \(\beta\), \(K\), \(\Lambda\), \(\varphi\), \(m\), \(M\), \(\Gamma\), \({\text{Bi}}_{{\text{T}}}\), and \(N_{{\text{t}}}\).
-
Entropy generation increased with \(K\), \(\varphi\), \(m\), \(M\), \(\Gamma\), \({\text{Bi}}_{{\text{T}}}\), Re, and \(B_{{\text{K}}}\) but decreased with \(\beta\) and \(\Lambda\).
-
The skin friction values increased with \(\varphi\), \(K\), \(M\), \(\Gamma\), and \(S\) but decreased with \(\beta\) and \(\Lambda\).
-
The local Nusselt number values increased with \(\varphi\), \(\Lambda\), Pr, \(N_{{\text{r}}}\), \(Bi_{{\text{T}}}\), and \(m\) but decreased with \(\beta\), \(K\), \(M\), \(Q\), \(N_{{\text{t}}}\), and \(E_{{\text{c}}}\).
-
The Casson–Oldroyd-B nanofluid consistently exhibited the lowest heat transfer and highest generated entropy.
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The Casson–Jeffrey and Casson–Maxwell nanofluids responded differently to the change in parameters. Casson–Maxwell had better heat transfer with \(\beta\), \(\varphi\), \(m\), \(M\), \(\Gamma\), \(N_{{\text{t}}}\), and \({\text{Bi}}_{{\text{T}}}\), whereas Casson–Jeffrey had better heat transfer with \(K\) and \(\Lambda\).
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From the aforementioned discussion, we conclude that the Casson–Maxwell and Casson–Jeffrey nanofluids exhibit similar performance in a PTSC setting, with a slight preference to use the Casson–Maxwell nanofluid.
Limitations and further research
This study is limited to the three binary nanofluids and PTSC settings mentioned herein. As any different nanofluid will have different properties and can lead to different results. Further, different settings will require different assumptions and boundary conditions, which will also lead to different results.
Further research can be directed toward theoretical investigation of different nanofluid models (e.g., Carreau, Walter-B, etc.) in PTSC settings and their comparison. Additionally, the presented model can be generalized by considering the impact of time-dependent flow and temperature-dependent viscosity, conductivity, and porosity on the performance of PTSCs. It is also important to explore the effect of hybrid nanoparticles on the PTSC performance using different binary nanofluid models.
Data availability
Correspondence and requests for materials should be addressed to F.N.I.
Abbreviations
- \(a\) :
-
Primary stretching rate
- \(a^{*}\) :
-
Thermal discrepancy rate
- \(B_{0}\) :
-
Strength of the constant magnetic field
- \(B_{{\text{K}}}\) :
-
Brinkman number
- \({\text{Bi}}_{{\text{T}}}\) :
-
Thermal Biot number
- \(C_{{\text{f}}}\) :
-
Skin friction constant
- \(C_{{\text{p}}}\) :
-
Specific heat
- \(D_{{\text{T}}}\) :
-
Thermophoretic diffusion coefficient
- \({\text{Ec}}\) :
-
Eckert number
- \(e_{{{\text{ij}}}}\) :
-
\(\left( {i,j} \right)\)Th component of the deformation rate
- \(f^{\prime}\) :
-
Dimensionless velocity
- \(h_{{\text{f}}}\) :
-
Heat transfer coefficient
- \(k\) :
-
Thermal conductivity
- \(k_{{\text{p}}}\) :
-
Porosity
- \(k^{*}\) :
-
Mean absorption coefficient
- \(K\) :
-
Porosity parameter
- m :
-
Nanoparticle shape factor
- \(M\) :
-
Magnetic number
- \(N_{{\text{G}}}\) :
-
Entropy generation dimensionless factor
- \(N_{{\text{r}}}\) :
-
Thermal radiation parameter
- \(N_{{\text{t}}}\) :
-
Thermophoretic parameter
- \({\text{Nu}}_{{\text{x}}}\) :
-
Local Nusselt number
- \(N_{{\text{w}}}\) :
-
Slip length
- \({\text{Pr}}\) :
-
Prandtl number
- \(p_{{\text{y}}}\) :
-
Yield stress
- \(Q\) :
-
Heat source parameter
- \(Q_{0}\) :
-
Heat source
- \(q_{{\text{r}}}\) :
-
Radiative heat flux
- \({\text{Re}}\) :
-
Reynolds number
- \({\text{Re}}_{{\text{x}}}\) :
-
Local Reynolds number
- \(S\) :
-
Mass transfer parameter
- \(t\) :
-
Time
- \(T\) :
-
Temperature
- \(u\) :
-
\(v\), Velocity components
- \(V_{{\text{w}}}\) :
-
Surface permeability
- x, y :
-
Dimensional space coordinates
- \(\alpha\) :
-
Thermal diffusion rate
- \(\beta\) :
-
Casson parameter
- \(\beta_{1}\) :
-
\(\beta_{3}\), Deborah numbers
- \(\dot{\gamma }\) :
-
Rate of strain tensor
- \(\widetilde{{\dot{\gamma }}}\) :
-
Upper convected derivative of the strain tensor
- \(\Gamma\) :
-
The magnetic field’s angle of inclination
- \(\eta\) :
-
Dimensionless space variable
- \(\theta\) :
-
Dimensionless temperature
- \(\kappa_{0}\) :
-
Surface thermal conductance
- \(\lambda_{1}\) :
-
Fluid relaxation time
- \(\lambda_{2}\) :
-
Ratio of the relaxation to retardation times
- \(\lambda_{3}\) :
-
Fluid retardation time
- \(\Lambda\) :
-
Velocity parameter
- \(\mu\) :
-
Viscosity
- \(\mu_{0}\) :
-
Low-shear viscosity
- \(\mu_{{\text{B}}}\) :
-
Plastic dynamic viscosity of the Casson fluid
- \(\nu\) :
-
Kinematic viscosity
- \(\pi_{{\text{c}}}\) :
-
Critical value of the deformation rate
- \(\rho\) :
-
Density
- \(\sigma\) :
-
Electrical conductivity
- \({\upsigma }^{*}\) :
-
Stefan–Boltzmann constant
- \(\tau\) :
-
Extra stress tensor
- \(\tau_{ij}\) :
-
Stress tensor
- \(\tilde{\tau }\) :
-
Upper convected derivative of the stress tensor
- \(\varphi\) :
-
Nanoparticle concentration
- \(\chi\) :
-
Ratio of the operative heat capability
- \(\Omega\) :
-
Difference in temperature parameter
- \({\text{f}}\) :
-
Base fluid
- \({\text{nf}}\) :
-
Nanofluid
- \({\text{s}}\) :
-
Solid particles
- \({\text{W}}\) :
-
Wall
- \(\infty\) :
-
Free stream
References
Kalogirou SA. Solar thermal collectors and applications. Prog Energy Combust Sci. 2004;30:231–95. https://doi.org/10.1016/j.pecs.2004.02.001.
Choi S. Enhancing thermal conductivity of fluids with nanoparticles. IMECE. 1995;66:99–105.
Ekiciler R, Arslan K, Turgut O, Kursun B. Effect of hybrid nanofluid on heat transfer performance of parabolic trough solar collector receiver. J Therm Anal Calorim. 2021;143:1637–54. https://doi.org/10.1007/s10973-020-09717-5.
Bellos E, Tzivanidis C, Said Z. A systematic parametric thermal analysis of nanofluid-based parabolic trough solar collectors. Sustain Energy Technol Assess. 2020. https://doi.org/10.1016/j.seta.2020.100714.
Razmmand F, Mehdipour R, Mousavi SM. A numerical investigation of nanofluids on heat transfer of the solar parabolic trough collectors. Appl Therm Eng. 2019;152:624–33. https://doi.org/10.1016/j.applthermaleng.2019.02.118.
Ajbar W, Parrales A, Huicochea A, Hernandez JA. Different ways to improve parabolic tough solar collectors’ performance over the last four decades and their applications: a comprehensive review. Renew Sustain Energy Rep. 2022. https://doi.org/10.1016/j.rser.2021.111947.
Shahzad F, Jamshed W, Safdar R, Hussain SM, Nasir NAAM, Dhange M, Nisar KS, Eid MR, Sohail M, Alsehli M, Elfasakhany A. Thermal analysis characterization of solar-powered ship using Oldroyd hybrid nanofluids in parabolic trough solar collector: an optimal thermal application. Nanotechnol Rev. 2022;11:2015–37. https://doi.org/10.1515/ntrev-2022-0108.
Jamshed W, Sirin C, Selimefendigil F, Shamshuddin MD, Altowairqi Y, Eid MR. Thermal characterization of coolant Maxwell type nanofluid flowing in parabolic trough solar collector (PTSC) used inside solar powered ship application. Coatings. 2021;12:1552. https://doi.org/10.3390/coatings11121552.
Jamshed W, Shahzad F, Safdar R, Sajid T, Eid MR, Nisar KS. Implementing renewable solar energy in presence of Maxwell nanofluid in parabolic trough solar collector: a computational study. Waves Random Complex Media. 2021. https://doi.org/10.1080/17455030.2021.1989518.
Nabwey HA, Reddy MG, Kumar BKN, Sandeep N. Effect of resistive and radiative heats on enhanced heat transfer of parabolic trough solar collector. J Process Mech Eng. 2022. https://doi.org/10.1177/09544089221117317.
Kumar S, Ghosh S, Samet B, Goufo EFD. An analysis for heat equations arises in diffusion process using new Yang–Abdel–Aty–Cattani fractional operator. Math Methods Appl Sci. 2020;43:6062–80. https://doi.org/10.1002/mma.6347.
Kumar S, Kumar A, Baleanu D. Two analytical methods for time-fractional nonlinear coupled Boussinesq–Burger’s equations arise in propagation of shallow water waves. Nonlinear Dyn. 2016;85:699–715. https://doi.org/10.1007/s11071-016-2716-2.
Kumar S, Nisar KS, Kumar R, Cattani C, Samet B. A new Rabotnov fractional-exponential function based fractional derivative for diffusion equation under external force. Math Methods Appl Sci. 2020;43:4460–71. https://doi.org/10.1002/mma.6208.
Veeresha P, Prakasha DG, Kumar S. A fractional model for propagation of classical optical solitons by using nonsingular derivative. Math Methods Appl Sci. 2020. https://doi.org/10.1002/mma.6335.
Alshabanat A, Jleli M, Kumar S, Samet B. Generalization of Caputo-Fabrizio fractional derivative and applications to electrical circuits. Front Phys. 2020;8:1–10. https://doi.org/10.3389/fphy.2020.00064.
Shafqat H, Zehba R, Abdelraheem MA. Thermal radiation impact on bioconvection flow of nano-enhanced phase change materials and oxytactic microorganisms inside a vertical wavy porous cavity. Int Commun Heat Mass Transf. 2022. https://doi.org/10.1016/j.icheatmasstransfer.2022.106454.
Shafqat H, Abdelraheem MA, Noura A. Bioconvection of oxytactic microorganisms with nano-encapsulated phase change materials in an omega-shaped porous enclosure. J Energy Storage. 2022. https://doi.org/10.1016/j.est.2022.105872.
Shafqat H, Noura A, Abdelraheem MA. Natural convection of a water-based suspension containing nano-encapsulated phase change material in a porous grooved cavity. J Energy Storage. 2022. https://doi.org/10.1016/j.est.2022.104589.
Shafqat H, Abdelraheem MA, Hakan FÖ. Magneto-bioconvection flow of hybrid nanofluid in the presence of oxytactic bacteria in a lid-driven cavity with a streamlined obstacle. Int Commun Heat Mass Transf. 2022. https://doi.org/10.1016/j.icheatmasstransfer.2022.106029.
Shafqat H, Muhammad J, Zoubida H, Arıcı M. Numerical modeling of magnetohydrodynamic thermosolutal free convection of power law fluids in a staggered porous enclosure. Sustain Energy Technol Assess. 2022. https://doi.org/10.1016/j.seta.2022.102395.
Kumar KG, Ramesh GK, Gireesha BJ, Gorla RSR. Characteristics of Joule heating and viscous dissipation on three-dimensional flow of Oldroyd B nanofluid with thermal radiation. Alex Eng J. 2018;57:2139–49. https://doi.org/10.1016/j.aej.2017.06.006.
Gireesha BJ, Kumar KG, Ramesh GK, Prasannakumara BC. Nonlinear convective heat and mass transfer of Oldroyd-B nanofluid over a stretching sheet in the presence of uniform heat source/sink. Results Phys. 2018;9:1555–63. https://doi.org/10.1016/j.rinp.2018.04.006.
Kumar KG, Gireesha BJ, Krishanamurthy MR, Rudraswamy NG. An unsteady squeezed flow of a tangent hyperbolic fluid over a sensor surface in the presence of variable thermal conductivity. Results Phys. 2017;7:3031–6. https://doi.org/10.1016/j.rinp.2017.08.021.
Reddy MG, Rani MVVLS, Kumar KG, Prasannakumara BC. Cattaneo-Christov heat flux and non-uniform heat-source/sink impacts on radiative Oldroyd-B two-phase flow across a cone/wedge. J Braz Soc Mech Sci Eng. 2018. https://doi.org/10.1007/s40430-018-1033-8.
Sureshkumar RS, Kumar KG, Rahimi-Gorji M, Khan I. Darcy-Forchheimer flow and heat transfer augmentation of a viscoelastic fluid over an incessant moving needle in the presence of viscous dissipation. Microsyst Technol. 2019;25:3399–405. https://doi.org/10.1007/s00542-019-04340-3.
Ramesh GK, Kumar KG, Shehzad SA, Gireesha BJ. Enhancement of radiation on hydromagnetic Casson fluid flow towards a stretched cylinder with suspension of liquid-particles. Can J Phys. 2017;96:18–24. https://doi.org/10.1139/cjp-2017-0307.
Kumar KG, Rudraswamy NG, Gireesha BJ, Krishnamurthy MR. Influence of nonlinear thermal radiation and viscous dissipation on three-dimensional flow of Jeffrey nano fluid over a stretching sheet in the presence of Joule heating. Nonlinear Eng. 2017;6:207–19. https://doi.org/10.1515/nleng-2017-0014.
Rudraswamy NG, Kumar KG, Gireesha BJ, Gorla RSR. Combined effect of joule heating and viscous dissipation on MHD three dimensional flow of a Jeffrey nanofluid. J Nanofluids. 2017;6:300–10. https://doi.org/10.1166/jon.2017.1329.
Nourafkan E, Asachi M, Jin H, Wen D, Ahmed W. Stability and photo-thermal conversion performance of binary nanofluids for solar absorption refrigeration systems. Renew Energy. 2019;140:264–73. https://doi.org/10.1016/j.renene.2019.01.081.
Zeng J, Xuan Y. Enhanced solar thermal conversion and thermal conduction of MWCNT-SiO2/Ag binary nanofluids. Appl Energy. 2018;212:809–19. https://doi.org/10.1016/j.apenergy.2017.12.083.
Ma M, Xie M, Ai Q. Study on photothermal properties of Zn-ZnO/paraffin binary nanofluids as a filler for double glazing unit. Int J Heat Mass Transf. 2022. https://doi.org/10.1016/j.ijheatmasstransfer.2021.122173.
Nejad AS, Barzoki MF, Rahmani M, Kasaeian A, Hajinezhad A. Simulation of the heat transfer performance of Al2O3-Cu/water binary nanofluid in a homogenous copper metal foam. J Therm Anal Calorim. 2022;147:12495–512. https://doi.org/10.1007/s10973-022-11487-1.
Yu Q. A decoupled wavelet approach for multiple physical flow fields of binary nanofluids in double-diffusive convection. Appl Math Comput. 2021. https://doi.org/10.1016/j.amc.2021.126232.
Lee M, Shin Y, Cho H. Theoretical study on performance comparison of various solar collectors using binary nanofluids. J Mech Sci Technol. 2021;35:1267–78. https://doi.org/10.1007/s12206-021-0238-4.
Reddy KV, Reddy GVR, Chamka AJ. Effects of viscous dissipation and thermal radiation on an electrically conducting Casson-Carreau nanofluids flow with Cattaneo-Christov heat flux model. J Nanofluids. 2022;11:214–26. https://doi.org/10.1166/jon.2022.1836.
Oyelakin IS, Lalramneihmawaii PC, Mondal S, Sibanda P. Analysis of double-diffusion convection on three dimensional MHD stagnation point flow of a tangent hyperbolic Casson nanofluid. Int J Ambient Energy. 2022;43:1854–65. https://doi.org/10.1080/01430750.2020.1722964.
Yousef NS, Megahed AM, Ghoneim NI, Elsafi M, Fares E. Chemical reaction impact on MHD dissipative Casson-Williamson nanofluid flow over a slippery stretching sheet through porous medium. Alex Eng J. 2022;61:10161–70. https://doi.org/10.1016/j.aej.2022.03.032.
Jamshed W, Nisar KS. Computational single-phase comparative study on a Williamson nanofluid in a parabolic trough solar collector via the Keller box method. Int J Energy Rep. 2021;45:10696–718. https://doi.org/10.1002/er.6554.
Sharafeldin MA, Grof G, Abu-Nada E, Mahian O. Evacuated tube solar collector performance using copper nanofluid: energy and environmental analysis. Appl Therm Eng. 2019. https://doi.org/10.1016/j.applthermaleng.2019.114205.
Dogonchi AS, Sheremet MA, Ganji DD, Pop I. Free convection of copper-water nanofluid in a porous gap between hot rectangular cylinder and cold circular cylinder under the effect of inclined magnetic field. J Therm Anal Calorim. 2019;135:1171–84. https://doi.org/10.1007/s10973-018-7396-3.
Olia H, Torabi M, Bahiraei M, Ahmadi MH, Goodarzi M, Safaei MR. Application of nanofluids in thermal performance enhancement of parabolic trough solar collector: state of the art. Appl Sci. 2019. https://doi.org/10.3390/app9030463.
Lin Y, Li BT, Zheng L, Chen G. Particle shape and radiation effects on Marangoni boundary layer flow and heat transfer of copper-water nanofluid driven by an exponential temperature. Powder Technol. 2016;301:379–86. https://doi.org/10.1016/j.powtec.2016.06.029.
Ellahi R, Zeeshan A, Hassan M. Particle shape effects on marangoni convection boundary layer flow of a nanofluid. Int J Numer Methods Heat Fluid Flow. 2016;26:2160–74. https://doi.org/10.1108/HFF-11-2014-0348.
Alkathiri AA, Jamshed W, Devi SSU, Eid MR, Bouazizi ML. Galerkin finite element inspection of thermal distribution of renewable solar energy in presence of binary nanofluid in parabolic trough solar collector. Alex Eng J. 2022;61:11063–76. https://doi.org/10.1016/j.aej.2022.04.036.
Ouni M, Ladhar L, Omri M, Jamshed W, Eid MR. Solar water-pump thermal analysis utilizing copper-gold/engine oil hybrid nanofluid flowing in parabolic trough solar collector: thermal case study. Case Stud Therm Eng. 2022;30:101756. https://doi.org/10.1016/j.csite.2022.101756.
Hussain SM, Jamshed W. A comparative entropy based analysis of tangent hyperbolic hybrid nanofluid flow: implementing finite difference method. Int Commun Heat Mass Transf. 2021;129:105671. https://doi.org/10.1016/j.icheatmasstransfer.2021.105671.
Pramanik S. Casson fluid flow past an exponentially porous stretching surface in presence of thermal radiation. Ain Shams Eng J. 2014;5:205–12. https://doi.org/10.1016/j.asej.2013.05.003.
Sochi T. Flow of non-Newtonian fluids in porous media. J Polym Sci B Polym Phys. 2010;48:2437–767. https://doi.org/10.1002/polb.22144.
Sandeep N, Sulochana C. Momentum and heat transfer behaviour of Jeffrey, Maxwell and Oldroyd-B nanofluids past a stretching surface with non-uniform heat source/sink. Ain Shams Eng J. 2018;9:517–24. https://doi.org/10.1016/j.asej.2016.02.008.
Lim YJ, Shafie S, Isa SM, Rawi NA, Mohamad AQ. Impact of chemical reaction, thermal radiation and porosity on free convection Carreau fluid flow towards a stretching cylinder. Alex Eng J. 2022;61:4701–17. https://doi.org/10.1016/j.aej.2021.10.023.
Reddy NB, Poornima T, Sreenivasulu P. Influence of variable thermal conductivity on MHD boundary layer slip flow of ethylene-glycol based Cu nanofluids over a stretching sheet with convective boundary convection. Int J Eng Math. 2014. https://doi.org/10.1155/2014/905158.
Jamshed W, Devi SSU, Safdar R, Redouane F, Nisar KS, Eid MR. Comprehensive analysis on copper-ion (II, III)/oxide-engine oil Casson nanofluid flowing and thermal features in parabolic trough solar collector. J Taibah Univ Sci. 2021;15:619–36. https://doi.org/10.1080/16583655.2021.1996114.
Xu X, Chen S. Cattaneo-Christov heat flux model for heat transfer of Marangoni boundary layer flow in a copper-water nanofluid. Heat Trans Asian Res. 2017;46:1281–93. https://doi.org/10.1002/htj.21273.
Reddy M, Makinde OD. Magnetohydrodynamic peristaltic transport of Jeffrey nanofluid in an asymmetric channel. J Mol Liq. 2016;223:1242–8. https://doi.org/10.1016/j.molliq.2016.09.080.
Keskin AÜ. Solution of BVPs Using bvp4c and bvp5c of MATLAB. In: Boundary value problems for engineers. Springer, Cham; 2019. p. 417–505. https://doi.org/10.1007/978-3-030-21080-9_10.
Wang CY. Free convection on a vertical stretching surface. J Appl Math Mech. 1989;69:418–20. https://doi.org/10.1002/zamm.19890691115.
Ferdows M, Uddin MJ, Afify AA. Scaling group transformation for MHD boundary layer free convective heat and mass transfer flow past a convectively heated nonlinear radiating stretching sheet. Int J Heat Mass Transf. 2013;56:181–7. https://doi.org/10.1016/j.ijheatmasstransfer.2012.09.020.
Rashidi MM, Rostami B, Freidoonimehr N, Abbasbandy S. Free convective heat and mass transfer for MHD fluid flow over a permeable vertical stretching sheet in the presence of the radiation and buoyancy effects. Ain Shams Eng J. 2014;4:901–12. https://doi.org/10.1016/j.asej.2014.02.007.
Gorla RSR, Sidawi I. Free convection on a vertical stretching surface with suction and blowing. Appl Sci Res. 1994;52:247–57.
Sahoo A, Nandkeolyar R. Entropy generation and dissipative heat transfer analysis of mixed convective hydromagnetic flow of a Casson nanofluid with thermal radiation and Hall effect. Sci Rep. 2021;11:3926. https://doi.org/10.1038/s41598-021-83124-0.
Kumar A, Tripathi R, Singh R, Sheremet MA. Entropy generation on double diffusive MHD Casson nanofluid flow with convective heat transfer and activation energy. Indian J Phys. 2021;91:1423–36. https://doi.org/10.1007/s12648-020-01800-9.
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Raafat, P.B., Ibrahim, F.N. Entropy and heat transfer investigation of Casson–Maxwell, Casson–Jeffrey, and Casson–Oldroyd-B binary nanofluids in a parabolic trough solar collector: a comparative study. J Therm Anal Calorim 148, 4477–4493 (2023). https://doi.org/10.1007/s10973-023-12003-9
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DOI: https://doi.org/10.1007/s10973-023-12003-9