Enhanced energy transport in high-mass flow solar parabolic trough collectors using Fe₂O₃-laden nanofluids

Abstract

This study investigates an improved energy transport fluid for high-mass flow solar parabolic trough collectors. Fe₂O₃ nanoparticle-laden water-based nanofluids with various concentrations were studied, covering Reynolds numbers from 3 × 10⁴ to 2.6 × 10⁵. Heat transfer coefficient, Nusselt number, and friction factor were analyzed and compared to the base fluid. A performance evaluation criterion is employed to assess the benefits. Notably, a significant enhancement in energy transport and performance evaluation was observed. Specific heat of nanofluids decreased due to nanoparticles, while thermal conductivity notably increased. Viscosity slightly increased. Doping with Fe₂O₃ nanoparticles led to decreased specific heat, increased thermal conductivity and viscosity. At flow rate of 1.5 kg s^-1, nanofluid exhibited 49.21% higher surface heat transfer coefficient for 0.5% nanoparticle concentration that goes up to 57.97% 1.0%. The average Nusselt number and the performance evaluation criterion also increase with nanoparticle concentration in nanofluid and Reynolds number. Correlations for heat transfer coefficient, friction factor, and Nusselt number were proposed as a function of Reynolds number and nanoparticle concentration.

Introduction

Due to the mitigation of conventional energy resources, the burgeon in pollution, and hence the boom in the carbon footprint, researchers have shifted their interest toward alternative energy resources. Mitigation of fossil fuel reserves, air pollution, increasing electrical costs, and global warming can be controlled by reducing carbon footprints. Solar energy plays a pivotal role in the domain of renewable energy sources due to its ample availability.

A solar collector is an arrangement that converts solar radiation into heat, which is further transferred and utilized. Flat plate collectors (FPCs), compound parabolic collectors (CPCs), parabolic trough collectors (PTCs), Fresnel lens concentrating collectors, etc., are most commonly used types of solar collectors [1]. The thermal efficiency of these collectors is inherited by their concentration ratio (C_R), which is the ratio of the aperture area (A_p) to the surface area of the absorber tube (A_r) [2]. FPC is used domestically because of its low-temperature output range (120–140 °C). For a high-temperature output (350–400 °C), the parabolic trough collector (PTC) is the most suitable contender among available solar collectors in which the thermal losses are relatively lower than the other collectors [3]. Solar parabolic trough collector (SPTC) is one of the most promising types of collectors. Air, water, oil, organic solvents, etc., are typical working fluids incorporated with these solar collectors [4]. The thermal performance behavior of PTC was first investigated by Arinze et al. [5] using water as a working fluid. A dynamic computer program was developed, and the thermal performance of water storage systems under varying charging and discharging circumstances was determined. A detailed one-dimensional heat transfer analysis on the SPTC was presented by Padilla et al. [6] by dividing the receiver tube into several segments. Collector efficiency and heat losses were computed under different operating conditions. A similar study for long and short receiver tubes was carried out by Forristall [7]. As solar collectors experience thermal losses, a method to reduce the thermal losses was reported by Ratzel et al. [8]. A two-dimensional unified numerical model for the coupled heat transfer process in the SPTC, including all modes of heat transfer, is nicely covered in the literature [9]. The various performance parameters and effects of the temperature field, flow field, local Nusselt number (Nu_x), and local temperature gradient were examined. An extensive review of PTC's modeling, simulation, and thermal performance investigation was nicely covered in the literature presented by Yilmaz et al. [10]. A detailed discussion about the single- and two-phase analysis based on steady and transient states was presented. Further, novel PTC designs and active and passive efficiency intensification techniques were deliberated collectively.

These collectors have relatively low outlet temperatures and poor efficiency that can be addressed by improving the thermal properties of the heat transport medium. Nanofluids (NFs) are assumed to have superior thermal and optical properties compared to base fluid and can be a suitable candidate for collectors to enhance their thermal performance [11]. Choi and Eastman [12] were the first to define a nanofluid (NF). Any base fluid, either organic or inorganic, with solid nanoparticles (NPs) of size 1–100 nm suspended in it is called nanofluids [13]. It has superior thermal properties, and hence heat transfer rate increases anomalously.

Regarding this, the first theoretical model was given by Maxwell [14], in which he added some micron-sized solid particles to the base fluid to increase the thermal conductivity (k_f). The heat transfer capability of NF depends upon various thermal properties, namely thermal conductivity (k_f), density (ρ), specific heat (C_p), viscosity (μ), and the volume concentration of nanoparticles (ϕ). The volume concentration of nanoparticles is a quantitative measure that represents the proportion of the volume occupied by suspended solid nanoparticles to the overall volume of the nanoparticle–fluid mixture. Mathematically, it is expressed as the ratio of the volume of nanoparticles to the total volume of the NF mixture [15]. In order to increase the heat transfer capabilities of the base fluid, it can be replaced with NF [16]. Mixing the base fluid with NPs to enhance the thermal and optical performance of the solar collectors incorporated has been widely reported in the open literature [17]. Some of the widely used NPs by researchers are Cu, CuO, Al, Al₂O₃, TiO₂, SiO₂, and Fe, with Al₂O₃ being the most reported one [18]. A wide range of literature has covered the effect of NFs on the various kinds of solar collectors tabulated in Table 1.

Table 1 Thermal performance improvement of various solar collectors using NFs

Computational analysis of NF-augmented thermal performance investigation on PTCs was reported in the literature. It was found that NFs are suitable candidates for the efficiency enhancement of PTC. In this regard, Sokhansefat et al. [45] did a numerical study on a three-dimensional SPTC for fully developed turbulent flow using Al₂O₃/synthetic oil NF. It was concluded that the convective heat transfer coefficient (h) is directly proportional to the nanoparticle concentration. Kaloudis et al. [46] numerically investigated the thermal performance of a SPTC using Al₂O₃-water NF. A two-phase model for a circular tube under constant wall temperature conditions was taken. Up to 10% improvement in collector efficiency for Al₂O₃ NPs of 4% by mass was reported. In another work, Xiong et al. [47] used Cu/water to access the heat transfer and flow characteristics of PTC. The study revealed that using shear-thinning NF instead of shear-thickening one is more beneficial and helps enhance the system's thermal performance. Nandan et al. [48] modeled the PTC by considering the constant heat-flux condition for the outer surface in both circumferential and longitudinal directions using copper oxide/therminol NF. Nazir et al. [49] used water–boehmite alumina (ALOOH) NF in a semicircular PTC. The highest energy efficacy of the system was reported at ϕ = 4%. Hussein et al. [50] numerically investigated the thermo-hydraulic performance of the PTC. Three different kinds of hybrid NFs (Ag-MWCNT, Ag-SWCNT, and Ag-MgO) were applied with a mixing ratio of 50:50 in Syltherm oil 800 base fluid. An increment of 11.5% in thermal efficiency was reported. Karimi et al. [51] used the population balance method to study heat transfer using various nanoparticles. Alnaqi et al. [52] inserted the two twisted tapes in the PTC using MgO-MWCNT/thermal oil hybrid NF to improve the thermal–hydraulic performance. Performance evaluation criterion (PEC) for the thermal system was calculated, and optimum system performance was reported at ϕ = 2%. Ekiciler et al. [53] utilized a combination of nanoparticles to investigate the thermal performance of PTC. Among all other NFs, Ag–MgO/Syltherm 800 at a volume fraction of 4% showed maximum improvement in the thermal efficacy of the system.

The literature review clarifies that significant research was carried out on the experimental and numerical studies of SPTC. However, studies on NF-augmented trough collectors are scarce. Moreover, in most studies, constant heat flux on the periphery of the receiver tube was considered to model SPTC. In actuality, the bottom half of the periphery of the receiver tube deals with the beam radiation, whereas the upper half of the periphery deals with global radiation. The range of Re reported in the literature mainly pertains to the laminar region. Literature pertaining to the high turbulent regime Reynolds number (Re) > 3 × 10⁴ is scarce to the best of the author's knowledge. Also, the literature review provides convincing proof that Fe₂O₃ nanoparticles have been underutilized in solar applications despite their superior thermal properties compared to several other metallic oxide nanoparticles. In support of this standpoint, the experimental assessment of the cooling capacity of NFs incorporating TiO₂ and Fe₂O₃ was conducted by Babar et al. [54] to cool electronic components. The comparative analysis of the results revealed that NFs formulated with Fe₂O₃ had enhanced performance compared to NFs formulated with TiO₂. Also, from an economic standpoint, Fe₂O₃ NPs are considered one of the most cost-effective options among other types of nanoparticles. This is evident from the price comparison of different nanoparticle types per 100 g, as illustrated in Fig. 1. Therefore, based on the aforementioned research gaps, an attempt has been made to cover all these areas. The key objectives are listed below.

• CFD-based numerical investigation of Fe₂O₃-H₂O NF-augmented solar parabolic trough collector for fully developed turbulent flow.

• Despite the fact that ferric oxide (Fe₂O₃) has the potential to facilitate efficient heat transport, its volume concentration (ϕ) is chosen between 0.5 and 1.0%. The rationale for this phenomenon is attributed to the limited duration of suspension stability exhibited by Fe₂O₃ [55].

• To analyze the flow behavior characteristics of heat transfer fluid (HTF) under a high flow regime. For this, the range of Re is taken as 3 × 10⁴ to 2.6 × 10⁵. The key reason behind choosing this range is the enhanced heat transfer rate in the high turbulent flow regime. As a conclusive remark, the observed disparity in thermal boundary layer growth at various axial locations along the receiver tube between turbulent and laminar flow is around tenfold [56].

Fig. 1

Price of different nanoparticles ($/100 g) [57]

The first section of this paper discusses the background and the working principle of PTC. The following section mentions problem-statement and basic governing equations related to solar collectors. It is followed by the boundary conditions applied on SPTC. Also, the grid sensitivity test and validation of the numerical model are presented. After that, the numerical findings pertaining to flow parameters are presented in the results and discussion part. In addition, the contours of velocity, temperature, and turbulent kinetic energy are also mentioned. For working engineers, easy-to-use simplified correlations have been presented.

Solar parabolic trough collector (SPTC)

A parabolic trough collector is shown in Fig. 2, consisting of a straight receiver tube and a parabolic reflecting surface usually made of a skinny silver layer. The straight receiver tube is placed along the focal axis of the parabolic reflecting surface. Due to manufacturing constraints, the receiver tube length is usually kept below 5 m. When the reflected solar radiation concentrates onto the tube, the circulating fluid gets heated by absorbing the solar energy. Based on the application, the straight receiver comes with an evacuated or non-evacuated tube. For high-temperature applications (above 300℃), evacuated tube is suggested because it has a high vacuum between the glass cover and pipe, which helps to reduce the thermal losses. For low-temperature applications, a non-evacuated tube is used where the heat losses are not a big concern.

Fig. 2

Solar parabolic trough collector

Problem statement

The schematic diagram of the receiver tube of the PTC is illustrated in Fig. 3. The tube length is 2000 mm, and the inner and outer diameters of the tube are 60 mm and 64 mm. The diameter and range of ϕ of ferric oxide (Fe₂O₃) NPs are 40 nm and 0.5% to 1%, respectively. The fragmented heat-flux condition is applied by dividing the receiver tube into two parts: the upper and bottom half periphery. Accordingly, solar heat flux is applied on the surface of the absorber tube, and the fluid flow characteristics are analyzed for PTC incorporated with Fe₂O₃/water NF at six different volume concentrations (ϕ) ranging between 0.5 and 1.0%.

Fig. 3

Schematic diagram of the solar receiver tube

Governing equations

The governing equations for the case of the steady-state fully developed turbulent flow for an incompressible fluid in cylindrical coordinates can be summarized as follows [58]:

Continuity equation:

$$\frac{1}{r}\frac{\partial }{\partial \theta }\left(\rho u\right)+\frac{1}{r}\frac{\partial }{\partial r}\left(\rho rv\right)+\frac{\partial }{\partial Z}\left(\rho w\right)=0$$

(1)

or

$$\nabla \cdot \left(\rho V\right)=0$$

(2)

where V = (V_r, V_θ, V_z) is defined as the velocity field for different spatial coordinates.

Momentum equations:

$$\nabla \cdot \left(\rho V{V}_{\rm i}\right)=\frac{-\partial p}{\partial {x}_{\rm i}}+\nabla \cdot \left(\mu \Delta {V}_{\rm i}\right)+{S}_{\rm i}$$

(3)

where ${X}_{\rm i}=R, \theta , Z$ for different cylindrical coordinates. The viscous term associated with the above equation for different types of spatial coordinates can be given below;

For radial direction $(i=1)$;

$${S}_{1}=\rho {V}_{\theta }\frac{{V}_{\theta }}{r}-\mu \left[\frac{{V}_{\rm R}}{r}+\left(\frac{2}{{r}^{2}}\right)\frac{\partial {V}_{\theta }}{\partial \theta }\right]$$

(4)

For tangential direction $\left(i=2\right)$;

$${S}_{2}=\mu \left[\left(\frac{2}{{r}^{2}}\right)\frac{\partial {V}_{\rm R}}{\partial \theta }-\frac{{V}_{\theta }}{{r}^{2}}\right]-\rho {V}_{\rm R}\frac{{V}_{\theta }}{r}$$

(5)

For axial direction $\left(i=3\right)$;

$${S}_{3}=0$$

(6)

Energy equation:

$$\nabla \cdot \left(\rho V{C}_{\rm p}T\right)=\nabla \cdot \left({k}_{\rm f}\nabla T\right)$$

(7)

where $\rho , \mu , {C}_{\rm p}, {\mathrm{and}} {k}_{\rm f}$ are the fluid density, dynamic viscosity, specific heat, and thermal conductivity of the circulating fluid, respectively.

$k-\varepsilon RNG$ turbulence model is employed to capture the flow behavior. The governing equations for the turbulence model are given in Eqs. (8) and (9), respectively [59].

$$\nabla \cdot \left(\rho Vk\right)=\nabla \cdot \left[\left(\mu +\frac{{\mu }_{\rm t}}{{\sigma }_{\rm k}}\right)\nabla k\right]+{G}_{\rm k}-\rho \varepsilon$$

(8)

$$\nabla \cdot\left(\rho V\varepsilon \right)=\nabla \cdot\left[\left(\mu +\frac{{\mu }_{\rm t}}{{\sigma }_{\varepsilon }}\right)\nabla \varepsilon \right]+{C}_{1\varepsilon }\left(\frac{\varepsilon }{k}\right){G}_{\rm k}-{C}_{2\varepsilon }^{*}\times \rho \left(\frac{{\varepsilon }^{2}}{k}\right)$$

(9)

where $\varepsilon {\mathrm{and}} k$ is the dissipation rate and turbulent kinetic energy, respectively. Turbulent kinetic energy generation caused by mean velocity gradients is represented by ${G}_{\rm k}$. ${\sigma }_{\varepsilon} {\mathrm{and}} {\sigma }_{\rm k}$ are the Prandtl number for turbulent flow for $\varepsilon {\mathrm{and}} k,$, respectively. Constants are shown by ${C}_{1\varepsilon }$ and ${C}_{2\varepsilon }$ and eddy or turbulent viscosity is written as ${\mu }_{\rm t}$. The empirical constants associated with the $k-\varepsilon RNG$ turbulent model are listed below [60];

$${\mu }_{\rm t}=\rho {C}_{\mu }\left({k}^{2}/\varepsilon \right),\quad {C}_{\mu }=0.0845,\quad {C}_{1\varepsilon }=1.42$$

(10)

$${C}_{2\varepsilon }^{*}={C}_{2\varepsilon }+\frac{{C}_{\mu }{\eta }^{3}\left(1-\eta /{\eta }_{0}\right)}{1+\beta {\eta }^{3}},\quad { C}_{2\varepsilon }=1.68,\quad {\eta }_{0}=4.38$$

(11)

$$\eta ={\frac{k}{\varepsilon }\left(2{E}_{\rm ij}{E}_{\rm ij}\right)}^{1/2},\quad {\sigma }_{\rm k}=1,\quad {\sigma }_{\varepsilon }=1.3$$

(12)

$${G}_{\rm k}=2{\mu }_{\rm t}{E}_{\rm ij}{E}_{\rm ij},\quad {E}_{\rm ij}=\frac{1}{2}\left(\frac{\partial {u}_{\rm i}}{\partial {x}_{\rm j}}+\frac{\partial {u}_{\rm j}}{\partial {x}_{\rm i}}\right),\quad \beta =0.012$$

(13)

Thermophysical properties of nanofluid

For the current numerical study, co-relations used for the calculation of density and specific heat of NF are given below [61]:

$${\rho }_{\rm nf}=\left(1-\phi \right){\rho }_{\rm bf}+(\phi {\rho }_{\rm nf})$$

(14)

$${\left(\rho {C}_{\rm p}\right)}_{\rm nf}=\left(1-\phi \right){\left(\rho {C}_{\rm p}\right)}_{\rm bf}+\phi {\left(\rho {C}_{\rm p}\right)}_{\rm nf}$$

(15)

The density (${\rho }_{{\rm nf}}$) and specific heat (${{C}_{{\rm p}}}_{{\rm nf}}$) of NF are assumed to be a linear function of the volume concentration of NPs. The other two essential thermal properties of NF, i.e., thermal conductivity (${k}_{{\rm eff}}$) and dynamic viscosity (${\mu }_{{\rm eff}}$), depend not only upon volume concentration (ϕ) but also strongly dependent on mixture combination, size of the nanoparticle, surfactant mixed, slip mechanism, etc. For the current study, ${\mathrm{Fe}}_{2}{\mathrm{O}}_{3}$ (40 nm) NP is chosen. The material properties of the H₂O and Fe₂O₃ NPs are mentioned in Table 2.

Table 2 Thermophysical properties of the base fluid and nanoparticle [62]

The expression for the thermal conductivity employed by Xuan et al. [63] is given as:

$$\frac{{k}_{\rm eff}}{{k}_{\rm bf}}=\left[\frac{\left({k}_{\rm np}+2\times {k}_{\rm bf}\right)-2\times \phi \left({k}_{\rm bf}-{k}_{\rm np}\right)}{\left({k}_{\rm np}+2\times {k}_{\rm bf}\right)+\phi \left({k}_{\rm bf}-{k}_{\rm np}\right)}\right]+\frac{{\rho }_{\rm nf}\times \phi \times {{C}_{\rm p}}_{\rm bf}}{2\times {k}_{\rm bf}}\times {\left(\frac{2\times {K}_{\rm b}\times {T}_{\rm avg}}{3\times \pi \times {d}_{\rm p}\times {\mu }_{\rm f}}\right)}^{0.5}$$

(16)

where d_p is the nanoparticle diameter, and K_b is the Boltzmann constant. The effect of Brownian motion and the aggregation of the NPs is considered. Further, it is considered that the thermal conductivity of NF is a function of the size of the NPs, the volume fraction of NPs, the thermal conductivity of NPs (${k}_{{\rm np}}$), the average temperature of the mixture, and various other thermal properties of base fluid, including viscosity, specific heat capacity, and thermal conductivity.

Based on the literature, it is found that in most of the numerical studies, the effect of viscosity is shown through either the Einstein model [64] or the Brinkman model [65]. All these models have considered only the effect of the addition of volume concentration of NPs. In the present study, the Corcione model [66] for viscosity is chosen in which it is shown that effective NF viscosity depends upon the molecular diameter of base fluid and nanoparticles, i.e., (d_{base fluid}) and (d_np), respectively, and is given as:

$$\frac{{\mu }_{{\rm eff}}}{{\mu }_{{\rm bf}}}=\frac{1}{1-34.87\times {\left(\frac{{d}_{{\rm np}}}{{d}_{{\rm bf}}}\right)}^{-0.3}\times {\phi }^{1.03}}$$

(17)

where d_bf can be calculated by knowing the molecular mass of the base fluid (M) and is given as:

$${d}_{{\rm bf}}={0.1\left[\frac{6\times M}{N\times \pi \times {\rho }_{{\rm bf},0}}\right]}^{1/3}$$

(18)

where N is the Avogadro number (6.022 × 10²³), and the density of the fluid is calculated at 293K.

Boundary conditions

Fragmented heat flux is applied over the periphery of the receiver tube by applying beam radiation at the bottom half of the tube, whereas the top half periphery receives the global radiation. The following mentioned boundary conditions are applied to the receiver tube of the parabolic trough collector:

• Inlet boundary condition: At the inlet of the receiver tube, a uniform velocity corresponding to a particular Reynolds number is taken. The fluid temperature at the inlet is taken as 300 K.

  $$u = U_{\rm inlet}, T_{\rm fluid} = T_{\rm inlet} = 300\,{\text{K}}, {{\rm at}} \,L = 0$$

• Wall boundary condition: No-slip conditions are applied for the wall-fluid interface.

  $$u = 0, {\text{at}} r = d/2$$

• The outer periphery of the receiver tube is split into two parts.

  (a) ${q}_{{\rm top }}^{{\prime\prime} }={{I}}_{{\rm g}}$

  (b) ${q}_{{\rm down}}^{{\prime\prime} }={{I}}_{{\rm b}}\times {C}_{{\rm R}}$

  Here, C_R = concentration ratio $={A}_{{\rm p}}/{A}_{{\rm r}}$, ${I}_{{\rm g}}$ = global radiation (800 W m^-2), ${I}_{{\rm b}}$ = beam radiation (600 W m^-2), A_p = aperture area of trough collector and A_r = absorber tube area.

• Outlet boundary condition: a zero-pressure gradient is employed across the outer boundary at the receiver tube outlet.

Dimensionless parameters

Nusselt number (Nu) is assessed to study the thermal performance of the PTC. The Nu is used to contrast the convective and conductive heat transfer mechanisms and can be defined as:

$${\text{Nu}}=\frac{h\times {L}_{{\rm c}}}{{k}_{{\rm f}}}$$

(19)

where L_c stands for the characteristic length of the absorber tube of PTC, k_f is the thermal conductivity of the HTF, and h is the convection heat transfer coefficient. It is defined as below:

$$h=\frac{q}{\left({T}_{{\rm wall}}-{T}_{{\rm f}}\right)}$$

(20)

where T_wall is the average surface temperature of the receiver tube of PTC.

Reynolds number (Re) is given as:

$${\text{Re}}= \frac{U\times {L}_{{\rm c}}}{{\vartheta }_{{\rm f}}}$$

(21)

The dimensionless friction factor is a crucial metric to analyze the pressure drop via the receiver tube. It illustrates how operational factors affect pumping efficiency and is given as:

$$f= \frac{2\times \Delta P\times {L}_{\rm c}}{L\times \rho \times {U}_{\rm in}^{2}}$$

(22)

where ∆P, L, and U_in stand for the pressure drop associated with the receiver tube, the length of the receiver tube, and the inlet velocity of the HTF successively.

The balance between thermal improvement and the requisite source of pumping power must be taken into account when assessing the efficacy of the nanofluid-augmented PTCs. For this, a non-dimensional number is suggested called the Performance evaluation criterion (PEC) [67]. The need for PEC for any thermal system in which nanofluid is used is vital to assess the overall efficacy of the system. In this regard, PEC is evaluated according to Eq. (23), which tells whether the thermal system employed with nanofluid is feasible. Generally, greater than unity is taken as a benchmark for this purpose.

$${\mathrm{PEC}}= \frac{\frac{{\text{Nu}}}{{{\text{Nu}}}_{0}}}{{\left[\frac{f}{{f}_{0}}\right]}^{1/3}}$$

(23)

where Nu₀ and f₀ stand for the Nusselt number and friction factor of the base fluid, respectively.

If the PEC > 1: this means that heat transfer capabilities are more than the increased pressure drop, and hence the system is feasible.

If the PEC ≤ 1: this implies that heat transfer enhancement is lower than or equal to the pumping power, which negates the need to use the NF in place of any traditional base fluid.

Numerical methodology

A finite volume-based algorithm is employed in the current study to model nanofluid flows in the system using a single-phase model. A semi-implicit approach for pressure-linked equations consistent (SIMPLEC) algorithm is used to solve the pressure–velocity coupled equations. Moreover, the second-order upwind technique is used to solve the diffusion and convection terms included in the governing equations. The discretized form of the energy equation is then solved after the continuity and momentum equations are solved. The algebraic equations are then solved using the Jacobi conjugate gradient technique. While examining the convergence characteristics of continuity and momentum equations, the residual values are assumed to be 10^–4. However, the maximum residual for the energy equation is set at 10^–6.

Grid generation and mesh sensitivity test

A three-dimensional structured grid is employed to solve the present numerical model. The simulated geometry of the receiver tube of PTC is made in the design modular of ANSYS R19.2 commercial solver. At the inlet and outlet of the receiver tube, a combination of tetrahedral and hexahedral meshing is considered, whereas hexahedral meshing with eight nodes is considered for the rest of the section. Uniform-sized cells are considered along the length of the tube due to constant velocity and temperature gradients in the z-direction. Figure 4 shows the grid generation for the computational domain. Near the pipe wall, layers have been inflated. To keep the computation time and cost in control, finer grids only near the boundary walls have been considered. It also helps to capture the gradients given at the wall surfaces efficiently and provides accurate results.

Fig. 4

Grid generation for the computational domain: a outer periphery of receiver tube, b inlet and outlet section of the receiver tube

A mesh sensitivity test is carried out to ensure the grid independency. As shown in Fig. 5. Five cases with total cell counts of 513,882, 756,154, 1,432,722, 1,804,442, and 2,198,430 are taken into consideration. The surface heat transfer coefficient (h_sur) is determined in each of the five cases, and the results are compared to the values obtained in case 5 with 2,198,430 cells. For the last two cells, values obtained for h_sur are nearly constant, with a difference of 0.185%. Hence, a grid size of 1,804,442 cells is considered.

Fig. 5

Mesh sensitivity test for surface heat transfer coefficient

Validation

Validation of the present numerical model is illustrated in Figs. 6 and 7. Numerical values of Nusselt number, Nu and friction factor, f are calculated and compared with the available results given by Ghasemi et al. [68] and from the Petukhov correlations for fully developed turbulent flow [69] as given in Eqs. (24) and (25). Results show excellent agreement with earlier results having a variation of not more than 6.0%.

Fig. 6

Validation of surface heat transfer coefficient with Ghasemi et al. [68] and Petukhov correlation

Fig. 7

Validation of friction factor with Ghasemi et al. [68] and Petukhov correlation

Petukhov correlation [69] for Nu and f are mentioned below:

$${\text{Nu}}=\frac{\left(\frac{f}{8}\right){\rm Re}\cdot{\rm Pr}}{1.07+12.7{\left(\frac{f}{8}\right)}^{0.5}\left({\rm Pr}^{2/3}-1\right)} \quad{\mathrm{for}}\quad {10}^{4}\le {\rm Re}\le 5\times {10}^{6}, 0.5\le {\rm Pr}\le 2000$$

(24)

$$f={\left(0.79\ln\left({\rm Re}\right)-1\right)}^{-2}$$

(25)

Results and discussion

Variation of thermal properties of nanofluids

Based on the correlations presented from Eqs. (15) to (19), the thermal properties variation of NFs with the volume concentration (ϕ) of NPs ranging between 0.5 to 1.5% is depicted in Fig. 8. In corresponds to Fig. 8a, the density of NF increased linearly with ϕ. At ϕ = 0.5%, the density of the mixture is increased by 1.94% compared to the base fluid. For a change in ϕ from 0.5 to 1.5%, the density of the NF mixture increased by 6.38%. In Fig. 8b, the specific heat of the NF mixture decreases with the concentration of nanoparticles. It is found that the C_p of the NF is decreased by 2.547% compared to base fluid with the suspension of a tiny amount of NPs at ϕ = 0.5%. The C_p of the NF is found to be 4003.825 at ϕ = 1%. Further, it decreased by 4.18% when the volume concentration of the NPs varied from ϕ = 0.5% to 1.5%. A crucial factor in improving heat transport is thermal conductivity (k_f), and its variation with ϕ is depicted in Fig. 8c. At ϕ = 0.5%, the improvement in k_f was 3.57%, whereas, for ϕ = 1.0%, the percentage enhancement in the k_f was found to be 5.567%. The maximum enhancement in k_f is recorded as 6.423% at ϕ = 1.5%, which proves that the NFs have remarkably high k_f and can be used as a HTF in various applications. Viscosity is crucial because it resists fluid mobility. The base fluid becomes more viscous when solid particles are introduced, because of which pressure drop increases. For a rise in the concentration of NPs from 0.5 to 1.5%, an increase in the viscosity of the test fluid by 2.56% is obtained, as shown in Fig. 8d.

Fig. 8

Variation of thermal properties for Fe₂O₃/water NF with volume concentration (ϕ); (a) density, (b) specific heat, (c) thermal conductivity, (d) viscosity

Contours of velocity, temperature, and thermal kinetic energy

The contours of streamwise velocity distribution at the receiver tube outlet at ϕ = 1% of Fe₂O₃ NPs at five different Re values are depicted in Fig. 9. As shown in Fig. 9 (a–e) while moving from the center of the receiver tube to the pipe wall; the velocity decreased with the maximum at the center of the tube and zero at the pipe wall. In addition, velocity increases with the increase in the Re value, which is evident as Re is directly proportional to velocity.

Fig. 9

Velocity contours of receiver tube at the outlet for different Re and at ϕ = 1%; (a) Re ≈ 3 × 10⁴, (b) Re ≈ 6 × 10⁴, (c) Re ≈ 1.2 × 10⁵, (d) Re ≈ 2 × 10⁵, (e) Re ≈ 2.6 × 10⁵

The temperature distribution for the receiver tube at ϕ = 1% and different Re are shown in Fig. 10a–e. Due to the fragmented heat-flux heat condition at the periphery of the receiver tube, non-uniformity in the temperature distribution can be seen. Since the solar insolation, after reflecting from the polished parabolic reflector, concentrates on the bottom periphery of the receiver, the temperature around the bottom half periphery of the receiver tube is more than the upper half periphery. In addition, the temperature of the working fluid is reduced with the increase in the Re. This is due to the increase in the mass flow rate, because of which the time to exchange heat between the fluid and the surface of the receiver tube is decreased. Furthermore, uniformity in the contours can be seen with the increase in Re, which means the temperature distribution of the absorber tube is becoming uniform and extending through wider regions.

Fig. 10

Temperature contours of receiver tube at the outlet for different Re and at ϕ = 1%; a Re ≈ 3 × 10⁴, b Re ≈ 6 × 10⁴, c Re ≈ 1.2 × 10⁵, d Re ≈ 2 × 10⁵, e Re ≈ 2.6 × 10⁵

The contours of turbulent kinetic energy (TKE) at the outlet of the receiver tube with increasing Re are presented in Fig. 11a–e. It represents the mean kinetic energy per unit mass associated with the eddies current formed in a turbulent flow. Physically, it can be expressed as the computed root-mean-square (RMS) velocity irregularities. For a constant volume concentration of NPs at ϕ = 1.0%, the TKE is found to be increasing in nature with the increase in mass flow rate. As the mass flow rate increases, the outer boundary regions of the contours get thicker, which shows that the intensity of the turbulence is increasing.

Fig. 11

Turbulent kinetic energy contours of receiver tube at the outlet for different Re and at ϕ = 1%; a Re ≈ 3 × 10⁴, b Re ≈ 6 × 10⁴, c Re ≈ 1.2 × 10⁵, d Re ≈ 2 × 10⁵, e Re ≈ 2.6 × 10⁵

Flow parameters variation with the Re and volume concentration of NPs (ϕ)

In Fig. 12, the variation of surface heat transfer coefficient (h_sur) with the Re for both base fluid and NF at six different volume concentrations of NPs ranging between 0.5 to 1% is depicted. The presented graph shows that the h_sur increases with the increase in the Re. With the increase in Re, the formation of eddies and vortices enables thermal energy transfer between the fluid with higher temperatures and the surface with lower temperatures. The continuous mixing process promotes the displacement of stagnant fluid at the surface by introducing fresh fluid, improving the heat transfer rate. Moreover, with the increased fluid velocity, the boundary layer experiences a reduction in its thickness due to the turbulent and disorderly motion. This reduction in thickness leads to a decrease in heat transfer resistance, facilitating an augmented heat transfer across the boundary layer. Besides, with the increase in the volume concentration of NPs (ϕ), the h_sur is increasing. Critical examination of the graph signifies that the relative increment in the h_sur, at the mass flow rate of 1.5 kg s^-1, is 49.21% compared to base fluid at ϕ = 0.5%, whereas at ϕ = 1%, the relative increment in h_sur is nearly 57.97%. This enhancement in the h_sur is due to the suspended NPs, which possess very high thermal conductivity compared to the water. In addition, in conventional base fluids like water, the existence of a boundary layer may have an undesirable effect on heat transfer efficiency. However, the presence of nanoparticles can potentially impede the establishment of this layer due to their capacity to agitate and homogenize the fluid. Consequently, a reduction in resistance to heat transmission occurs, resulting in an improvement in the heat transfer rate.

Fig. 12

Variation of surface heat transfer coefficient with Re and addition of volume concentration of NPs

The variation of the Nusselt number (Nu) with the increasing Re for both base fluid and NF at six different volume concentrations of NPs ranging from 0.5 to 1.0% is depicted in Fig. 13. It is found that for Re≈ 6 × 10⁴ and at ϕ = 0.5%, the relative increment in the Nu is 50.13%, whereas at ϕ = 1.0%, it went up to 59.09%. For Re≈ 2 × 10⁵ and ϕ = 0.5%, the relative increment in the Nu is 42.87%. With the increase in ϕ, the Nu is increasing, and a maximum relative increment of 51.11% is found at the same Re and ϕ = 1.0%.

Fig. 13

Variation of surface Nusselt number with Re and addition of volume concentration of NPs

Figure 14 shows the variation of relative Nusselt number with Re at six different volume concentrations of NPs. The graph depicts the relative enhancement in Nu by replacing the base fluid with NF. The relative Nu increases with the addition of NPs. The nature of the graph rises until Re≈ 6 × 10⁴; after that, it decreases. Furthermore, the percentage increment in the relative Nu is 4.64% for a mass flow rate of 1.5 kg s^-1 and ϕ = 0.5–0.6%. The percentage increment is reduced to 2.67% at the same mass flow rate and ϕ = 0.9–1.0%. A similar trend is found at Re≈ 2.6 × 10⁵, where the percentage increment in the Relative Nu is found to be a maximum of 3.47% at ϕ = 0.5–0.6%, and then it kept on decreasing to a value of 2.04% at ϕ = 0.9–1.0%.

Fig. 14

Variation of relative Nusselt number with Re and addition of volume concentration of NPs

In Fig. 15, the variation of friction factor (f) with the increasing Re for base fluid and NF at six different volume concentrations of NPs ranging from 0.5 to 1.0% is depicted. An inverse relationship between Re and friction factor led to a decreased friction factor with increased fluid inlet velocity. Furthermore, a significant rise in the friction factor can be observed with the rise in nanoparticle concentration. Rise in the density of the whole mixture and increased viscosity is the cause behind this phenomenon. The relative increment in the friction factor compared to the base fluid is maximum when the mass flow rate is minimum. At this point, the increment is about 30.30% for ϕ = 0.5%, whereas for ϕ = 1.0%, the friction factor increases to 30.67%. The percentage increment in the friction factor with adding NPs is also examined. It is found that the percentage increment is minimal. For ϕ = 0.9 and 1.0%, a rise in the friction factor of 0.12% is obtained.

Fig. 15

Variation of friction factor with Re and addition of volume concentration of NPs

Figure 16 depicts the variation of the relative friction factor with Re at six different volume concentrations of NPs. The suspension of NPs results in increased viscosity, which further assists friction losses to occur predominantly. Investigation reveals that the relative friction factor decreases continuously until Re≈ 2 × 10⁵; thereafter, it increases further. The increment in the f is 20.83% at ϕ = 0.5%. It continued increasing with ϕ and reached a maximum of 21.51% compared to the base fluid for ϕ = 1.0%.

Fig. 16

Variation of relative friction factor with Re and addition of volume concentration of NPs

The variation of performance evaluation criterion (PEC) with Re for concentration (ϕ) ranging between 0.5 and 1.0% is depicted in Fig. 17. At any particular mass flow rate, the PEC is found to be more than unity. In addition, it kept on increasing with the increasing ϕ. Initially, at a low mass flow rate, the overall enhancement in PEC is 18.84% when the ϕ changes from 0.5% to 1.0%. At Re ≈ 6 × 10⁴, a maximum PEC of 2.206 is obtained for ϕ = 1.0%; after that, it decreased successively with the increase in mass flow rate of HTF. At Re ≈ 2.6 × 10⁵, the PEC is assessed, and an increment of 15.42% is obtained with an increase in ϕ from 0.5 to 1.0%.

Fig. 17

Variation of PEC with Re and addition of volume concentration of NPs

Correlation for ${h}_{{\rm sur}}, {\mathrm{Nu}}_{{\rm sur}}$, f and correlation matrix

A set of correlations are developed for the surface heat transfer coefficient (${h}_{{\rm sur}}$), surface Nusselt number (${Nu}_{{\rm sur}}$), and friction factor (f), respectively, to maximize the utilization of the obtained data. Such simplified easy-to-use correlations come handy to working engineers. It also helps to quantify the dependency among the variables. Their dependency on Re and volume concentration of NPs (ϕ) are created using regression analysis. Various indices and constants are computed using user-defined code based on partial least square regression methodology. A correlation matrix is also presented in Table 3, clarifying the variables' dependency. Except for the friction factor (f), all other properties have a positive attitude, whereas f has a negative tendency toward the variables.

Table 3 Correlation matrix

Correlation for surface heat transfer coefficient $({h}_{{\rm sur}})$

The correlation for the surface heat transfer coefficient is given in terms of Re and ϕ in Eq. (26). It is valid for 3 × 10⁴ ≤ Re ≤ 2.6 × 10⁵ and 0.5 ≤ ϕ ≤ 1.0. The developed correlation has a value of 0.9998 and 0.9995 for R and R², respectively.

$$h=2.6698{\times \mathrm{Re}}^{0.7389}\times {\phi }^{0.2485}$$

(26)

Correlation for surface Nusselt number $({\rm Nu}_{{\rm sur}})$

The correlation for surface Nusselt number is given in terms of Re and ϕ in Eq. (27), which is valid for 3 × 10⁴ ≤ Re ≤ 2.6 × 10⁵ and 0.5 ≤ ϕ ≤ 1.0. The developed correlation has a value of 0.9998 and 0.9995 for R and R², respectively.

$${\mathrm{Nu}}=0.2541\times {\mathrm{Re}}^{0.7389}\times {\phi }^{0.2287}$$

(27)

Correlation for friction factor $(f)$

The correlation friction factor is given in terms of Re and ϕ in Eq. (28). This correlation is valid for 3 × 10⁴ ≤ Re ≤ 2.6 × 10⁵ and 0.5 ≤ ϕ ≤ 1.0. The developed correlation has a value of 0.9974 and 0.9948 for R and R², respectively.

$$f=0.7109\times {\mathrm{Re}}^{-0.2979}\times {\phi }^{0.0092}$$

(28)

Conclusions

A computation investigation has been carried out to propose an improved working fluid for energy transport in solar parabolic trough collectors, especially for applications requiring a high-mass flow rate. The thermofluid behavior of Fe₂O₃ nanoparticles-laden water-based nanofluid at six different volume concentrations (ϕ) ranging from 0.5 to 1.0% has been assessed. The flow Reynolds number (Re) varies between 3 × 10⁴ to 2.6 × 10⁵. The heat transfer coefficient, Nusselt number, and friction factor are evaluated and compared to the base fluid. A significant enhancement in energy transport and performance evaluation criterion has been observed.

The specific heat of nanofluids decreases with the inclusion of nanoparticles, whereas the thermal conductivity increases significantly. A small but significant rise in viscosity has been observed. By doping the base fluid with 1.50% of Fe₂O₃ nanoparticles, the specific heat decreases by 4.18%, thermal conductivity increases by 6.42%, and viscosity increases by 2.56%.

At a mass flow rate of 1.5 kg s^-1 and ϕ = 0.5%, a 49.21% increment in surface heat transfer coefficient was observed for nanofluid. This enhancement further rises to 57.97% for nanofluid with ϕ = 1.0%, and the corresponding average Nusselt number rises by 59.09%. The friction factor increases with the rise in flow Reynolds number as well as nanoparticle concentration.

A performance evaluation criterion (PEC) has been employed to assess the overall efficacy of the system due to the inclusion of nanoparticles in the base fluid. A peak PEC of 2.206 was obtained at ϕ = 1.0% at Re ≈ 6 × 10⁴. In the end, correlations for surface heat transfer coefficient, friction factor, and surface Nusselt number have been proposed as a function of Reynolds number and nanoparticle volume concentration in the nanofluid.