Introduction

In current trends, renewable energy sources gain more research focus because they are alternative sources for fossil fuels. Solar energy is one of the renewable energy sources, and it has been used to fulfil global energy demands as a future option [1]. Furthermore, the International Energy Agency (IEA) reported that concentrated solar energy needs would hit around one thousand GW by the end of 2050. Solar collectors are used to converting solar energy into electricity [2]. Several numerical and experimental studies have been conducted on solar collectors’ performance and efficiency improvement [3]. Dileep et al. [4] numerically analysed the solar flow system interconnected with the Rankine cycle, a parabolic trough collector, and the medium-temperature textile industry's co-generator. Solar flow systems have been identified that there is a possibility of improvement in thermal and electrical performance.

Lamnatou et al. [5] discussed the drying phenomenon of agricultural products through the evacuated tube solar collector. Hot air from the evacuated air collectors was utilized for drying apples, apricots, and carrots. The air collectors have been identified for vast dry volumes of agricultural products and are recommended for large-scale industry. Liu et al. [6] measured the glass solar air collector’s thermal efficiency using the energy analytical approach. The outcome shows that the current hierarchical model agrees with the first-order model. The new dynamic test model exhibited better performance in diverse climates and is recommended for extreme weather conditions. Kumar et al. [7] experimentally analyzed the performance of the evacuated solar tube collector using air as a working fluid to enhance the heat transfer rate. The evacuated solar tube consists of fifteen glass tubes, including two tubes of concentrate made of solid borosilicate glass. They carried out four separate experimental analyses on the evacuated solar tube with and without a reflector. Their experimental outcomes show that 58 and 50% maximum efficiencies were attainable at a flow rate of 13.28 kg h−1 with and without reflector, respectively. Their results indicate that the use of reflectors in the evacuated solar tube significantly improves the overall efficiency. Zhen et al. [8] verified the performance of the solar tubular air collector, operated in high and moderate temperature air with a simplified compound parabolic focus. A water-based CuO nanofluid-specific thermosiphon has been used as a working fluid. The experimental findings show that even during the winter season, the air outlet temperature could exceed 170 °C. Kim et al. [9] utilized the nanofluid multiwall (MWCNT) as a working fluid to improve the solar U-tube collector’s thermal efficiency. They observed that the annual emissions of CO2 and SO2 decreased by 1600 and 5.3 kg, respectively.

Mahian et al. [10] investigated the use of four different nanofluids (Al2O3/water, TiO2/water, SiO2/water, and Cu/water) in a mini-channel-based solar collector. They found that Cu/water nanofluid exhibits the optimum temperature at the exit and the lowest entropy at the source with an increase in thermal efficiency. Verma et al. [11] inspected the effect of thermal performance of a flat-plate solar collector, employing nanofluids of Al2O3, CuO, SiO2, TiO2, and graphene, with MWCNT. They reported that thermal efficiency was enhanced by 23.5% with MWCNT nanofluid as the working medium. Kaya et al. [12] discussed the semicircular cross-sectioned microchannel filled with TiO2/water nanofluid; the researchers evaluated the entropy production, heat transmission, and friction factor for forced convection flow in the channel. The nanofluid volume concentrations ranged from 1 to 4 percent. They observed an improvement in heat transfer efficiency with TiO2 nanofluid. Ekiciler et al. [13] investigated the three-dimensional flow and heat transfer properties of hybrid nanofluids in a compound parabolic collector by using C++code to investigate non-uniform distribution heat flux, temperature distribution, and thermal efficiency from the outer surface of the receiver. They observed an increase in heat flux of about 31% and a rise in fluid temperature with the use of hybrid nanofluid systems and nanoparticles of different volume concentrations. Thansekhar et al. [14] studied the effect of nanofluids on the improvement of heat transfer rate in a microchannel heat sink. They identified that a higher volume concentration of Al2O3 nanoparticles in the nanofluid exhibits an enhanced heat transfer rate compared to SiO2 nanoparticles.

Furthermore, involving the nanoparticles for efficient convective heat transfer, many researchers employed a constant thermal flux system with TiO2 blended into the working fluid to improve thermal efficiency. The literature on the use of nanoparticles to improve thermal efficiency is summarized in Table 1.

Table 1 Literature on the use of nanoparticles
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From the extensive literature review on the relevance to the present works, it was observed that many authors had adopted various methods to improve the performance of solar water heaters by using the following techniques such as varying nanoparticles size and varying nanoparticle concentration, adoption of multiwall carbon nanotube, evacuated solar tube collector using various design of solar collectors. It was clear that none of the researchers had used dimple texture tubes with TiO2 nanoparticles for performance improvement analysis. As well as any, researchers have not documented computational analysis for dimpled tube texture using CFD.

In the present investigation, the authors have presented a numerical and experimental analysis of a parabolic plate solar water heater using a tube in tube type heat exchanger, with a dimple inner tube having a P/D ratio of 3. Also, the authors have used di-ionized water and nanoparticles of TiO2 of size 10–15 nm at volume concentrations of 0.1–0.5% in steps of 1 to prepare the nanofluids. During the investigation, the mass flow rate of the water in the tube varied in the range of 0.5–3.0 kg min−1 in steps of 0.5 kg min−1 to analyse the absolute energy of the parabolic collector, heat loss from the dimpled tube, parabolic collector efficiency, gradient temperature of the dimpled tube, friction factor for the working fluid, Reynolds number, Nusselt number, convective heat transfer coefficient, pressure drop and collector performance index of the water heater. Also, the velocity, temperature, and pressure contour of the PTSC were analyzed by using CFD and presented.

Materials and method

In this research work, a PTSC solar water heater was fabricated to carry out the experiments. Dimple texturing was carried out on the tube of the solar water heater to create turbulence behaviour during its flow through the tube. The solar water heater tube had a length of 1200 mm and a diameter of 50 mm. There were five dimple texture tubes parallelly connected. One of the common ends was connected to the water reservoir of 1500 L capacity, and another was connected to the water out wall. The detailed specifications of the solar water heater are given in Table 2. The layout of the experimental setup is shown in Fig. 1. The photograph of the experimental setup is shown in Fig. 2. Nanoparticles of TiO2 and DI–H2O were used to prepare nanofluids of different concentrations. The nanoparticle concentration varied in the range of 0.1–0.5% in steps of 0.5%. Dimples of diameter 5 mm were fabricated on the tube that can improve turbulence behaviour and increase the heat transfer rate [3, 21]. The solar radiation intensity was measured with the help of a sun meter during the experiment, and it was found to be 847 W m−2. The turbulent fluid flow behaviour was identified and analyzed in the proposed model.

Table 2 Detail specifications of solar water heater [3]
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Fig. 1
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Layout of the experimental setup

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Fig. 2
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Photograph of the experimental setup

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During experiments, the solar water heater was placed at an inclination angle of 30° and 60° to get the highest solar radiation intensity. For the measurement of temperature at different locations of the experimental setup, K-type thermocouples were installed with the accuracy of ± 0.5 °C. This K-type thermocouple has a minimum and maximum operating temperature of − 15 to 250 °C, respectively. The flow rate of the water during the experiment was measured with the help of a flowmeter. The positive displacement pump was used to circulate the water through the PTSC. The flow of the working fluid was controlled with the help of valves connected at various locations on the experimental setup. In the present investigation, the flow rate varied from 0.5 to 3.0 kg min−1 in the steps of 0.5 kg min−1.

Nano fluid preparation

In the present investigation, nanofluid was prepared from TiO2 nanopowder and DI water. The nanopowder was 99.7% purity, 13 nm average size, pH of 7, and density of 4260 kg m−3. The detailed specifications of the nanoparticles are given in Table 3. The volume concentration of nanofluid varied in the range of 0.1–0.5% volume percent in steps of 0.1%. An ultra-sonic dispersion apparatus (model: UP400s) was used to prepare the nanofluid. During the nanofluid preparation, the neutral pH value of 7 was maintained, which directly influenced the durability of the setups. The pH value of 7 indicates minimal contamination of the solar water heater for long use. The photograph of the ultrasonic dispersion apparatus is shown in Fig. 3. The process layout for the preparation of nanofluids is shown in Fig. 4. The thermophysical properties of the prepared nanofluids are given in Table 4.

Table 3 Specification of the nanoparticle
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Fig. 3
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Photograph of the ultrasonic dispersion apparatus

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Fig. 4
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Nanofluid preparation process layout

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Table 4 Thermo-physical properties of nanofluids
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Governing equation for heat transfer

The heat transfer coefficient is a quantitative characteristic of convective heat transfer between a fluid medium and the surface (wall) streamed over by the fluid. The log means temperature difference (LMTD) was used to determine the temperature driving force for heat transfer in the flow systems, most notably in heat exchangers. The LMTD is a logarithmic mean average of the temperature difference between the plain and dimple tubes. Based on the literature [4, 22], the problem statement can be expressed as the following governing equations [23]:

$$\frac{{{\text{d}}u}}{{{\text{d}}x}} + \frac{{{\text{d}}v}}{{{\text{d}}y}} + \frac{{{\text{d}}w}}{{{\text{d}}z}} = 0$$
(1)

As explored in Eq. 1, \(u , v, w -\) are velocity vectors, and x, y, and z represent the dimensions. The momentum equations for heat transfer are given as follows:

$$\rho_{{{\text{TiO}}_{{2}} }} \left( {\frac{\partial u}{{\partial t}} + u\frac{\partial u}{{\partial x}} + v\frac{\partial u}{{\partial y}} + w\frac{\partial w}{{\partial z}}} \right) = - \frac{\partial p}{{\partial x}} + \mu \left( {\frac{{\partial^{2} u}}{{\partial x^{2} }} + \frac{{\partial^{2} u}}{{\partial y^{2} }} + \frac{{\partial^{2} u}}{{\partial z^{2} }}} \right) - \rho_{{{\text{TiO}}_{{2}} }} \propto_{{{\text{TiO}}_{{2}} }} g_{{\text{x}}} \left( {T - T_{{{\text{ref}}}} } \right)$$
(2)
$$\rho_{{{\text{TiO}}_{{2}} }} \left( {\frac{\partial v}{{\partial t}} + u\frac{\partial v}{{\partial x}} + v\frac{\partial v}{{\partial y}} + w\frac{\partial v}{{\partial z}}} \right) = - \frac{\partial p}{{\partial x}} + \mu \left( {\frac{{\partial^{2} v}}{{\partial x^{2} }} + \frac{{\partial^{2} v}}{{\partial y^{2} }} + \frac{{\partial^{2} v}}{{\partial z^{2} }}} \right) - \rho_{{{\text{TiO}}_{{2}} }} \propto_{{{\text{TiO}}_{{2}} }} g_{{\text{x}}} \left( {T - T_{{{\text{ref}}}} } \right)$$
(3)
$$\rho_{{{\text{TiO}}_{{2}} }} \left( {\frac{\partial w}{{\partial t}} + u\frac{\partial w}{{\partial x}} + v\frac{\partial w}{{\partial y}} + w\frac{\partial w}{{\partial z}}} \right) = - \frac{\partial p}{{\partial x}} + \mu \left( {\frac{{\partial^{2} w}}{{\partial x^{2} }} + \frac{{\partial^{2} w}}{{\partial y^{2} }} + \frac{{\partial^{2} w}}{{\partial z^{2} }}} \right) - \rho_{{{\text{TiO}}_{{2}} }} \propto_{{{\text{TiO}}_{{2}} }} g_{x} \left( {T - T_{{{\text{ref}}}} } \right)$$
(4)

In Eqs. 2, 3, and 4, \(u, v, w\) is a velocity vector, and \(\rho_{{{\text{TiO}}_{{2}} }}\) is the density of titanium dioxide (TiO2) in kg m−3. \(T\) represents the initial temperature of nanofluid in °C. \(T_{{{\text{ref}}}}\) represents overall temperature in °C. \(\mu\) denotes the momentum parameter. \(g_{{\text{x}}} ,g_{{\text{y}}} ,g_{{\text{z}}}\) represents asymmetrical gravity component.

The energy equation is given as follows [22, 24]:

$$\rho_{{{\text{TiO}}_{{2}} }} \left( {\frac{\partial T}{{\partial t}} + u\frac{\partial T}{{\partial x}} + v\frac{\partial T}{{\partial y}} + w\frac{\partial T}{{\partial z}}} \right) = \frac{{k_{{{\text{eff}}}} }}{{(C_{{\text{P}}} .\rho )_{{{\text{TiO}}_{{2}} }} }}\left( {\frac{{\partial^{2} T}}{{\partial x^{2} }} + \frac{{\partial^{2} T}}{{\partial y^{2} }} + \frac{{\partial^{2} T}}{{\partial z^{2} }}} \right)$$
(5)

In the above energy Eq. 5, \(k_{{{\text{eff}}}}\) represents effective thermal conductivity, \((C_{{\text{P}}} .\rho )_{{{\text{TiO}}_{{2}} }}\) represents a collector heat gain, \(C_{{\text{P}}}\) represents specific heat capacity of nanofluid in (kJ kg−1 K−1). The volume fraction and densities of the base fluid and the TiO2 nanoparticles may be used to create the mass of nanofluids. The density of nanofluids can be calculated from the following equation, which represents the density of DI water and the density of nanopowder. The correlation for the nanofluid density is given as follows:

$$\rho_{{{\text{TiO}}_{{2}} }} = \varphi \left( {C_{{\text{P}}} .\rho } \right) + \left( {1 - \varphi } \right)(\beta \rho )_{{{\text{TiO}}_{{2}} }}$$
(6)

The nanofluid thermal conductivity can be calculated using the Maxwell model as follows.

$$\frac{{k_{{{\text{TiO}}_{{2}} }} }}{{k_{{\text{f}}} }} = \frac{{2k_{{\text{f}}} + k_{{\text{s}}} + 2\varphi k_{{\text{f}}} - 2\varphi k_{{\text{s}}} }}{{2k_{{\text{f}}} + k_{{\text{s}}} - \varphi k_{{\text{f}}} - \varphi k_{{\text{s}}} }}$$
(7)

It has been decided to use Brinkman’s very effective dynamic viscosity relationship in this investigation [23, 25].

$$\mu_{{{\text{TiO}}_{{2}} }} = \frac{{\mu_{{\text{f}}} }}{{(1 - \varphi )^{2.5} }}$$
(8)

The solar radiation is only imposed on the upper part of the dimple tube, which is expected to receive 50–57% of solar radiation. Hence, the bottom part of the dimple tube indirectly experiences solar radiation from the reflector panel placed below the tube. In this investigation, this solar radiation intensity was recorded from 8 am to 5 pm using a sun meter. A maximum solar radiation intensity of 952 W m−2 was observed during the afternoon at about 12.30 pm (Indian Standard Time). After 12.30 pm, the intensity decreases gradually to 847 W m−2 at about 5 pm. Hence, in the present investigation, the average solar radiation intensity of 847 W m−2 is considered for the analysis.

Simulation

Grid independent test

Fine mesh size and shape are the critical parameters to ensure the accuracy and quick computational time of the CFD numerical calculation. In this analysis, 3D meshes are utilized in the ANSYS platform to analyse the influence of grid numbers on the maximum temperature of the tube. The convergence was observed at a maximum tube temperature of 308 K. Figure 5 portrays the CFD simulation for plain, dimple, and dimple tubes with TiO2. During the formation of the computational domain, the element size was maintained at 586,275 million elements.

Fig. 5
figure 5

CFD simulations, a Plain tubes, b Dimple tubes, and c Dimple tubes with TiO2

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Performance analysis

The performance parameters for the PTSC were carried out in experimental and numerical analysis using CFD. During the experiment, the concentration of nanofluid and the flow rate of water were varied to find the effect of dimple texture design and nanoparticles on the performance parameters of the PTSC. The experiments are carried out in six stages to achieve consistency in results, and each stage consists of experiments for about 30 min. The data collection time was constant at 64.25% of sky cleanness as per ASHRAE requirements. After collecting data, linear regression analysis was used to find the mean value of the temperature and velocity by considering 20 data points to make a steady-state model.

Analysis of collector efficiency

The collector efficiency reveals the entire radiation of the incident from the opening area, and the collector produces the available heat gain as given as follows [3]

$$Q_{{\text{g}}} = C_{{{\text{P}}_{{{\text{TiO2}}}} }} \left( {T_{{\text{O}}} - T_{{\text{I}}} } \right)$$
(9)

where \(Q_{{\text{g}}}\) represents collector efficiency, \(T_{{\text{O}}}\) represents the total amount of radiation opening region, \(T_{{\text{I}}}\) represents the total amount of radiation incidents,\(C_{{{\text{P}}_{{{\text{TiO}}_{{2}} }} }}\) represents the collector heat gain. The following methods were used to calculate the efficiency of a PTSC solar water heater [26,27,28].

$$\eta_{{\text{c}}} = \frac{{Q_{{\text{g}}} }}{{A_{{\text{a}}} }} = \frac{{C_{{{\text{P}}_{{{\text{TiO2}}}} }} \left( {T_{{\text{O}}} - T_{{\text{I}}} } \right)}}{{A_{{\text{a}}} }}$$
(10)

where \(\eta_{{\text{c}}}\) represents the thermal efficiency.\(C_{{\text{p}}}\) represents the specific heat capacity of working. The following equation denotes the collector’s efficiency.

$$\eta_{{\text{c}}} = \alpha \tau \left( {F_{{\text{R}}} } \right) - \frac{{U_{{\text{L}}} F_{{\text{R}}} }}{C}\left( {\frac{{T_{{\text{O}}} - T_{{\text{I}}} }}{{T_{{{\text{ref}}}} }}} \right)$$
(11)

In Eq. 11, \(F_{{\text{R}}} \left( {\eta_{{\text{c}}} } \right) = \alpha \tau \left( {F_{{\text{R}}} } \right)\) represents the energy parameter,\(\frac{{{\varvec{U}}_{{\text{L}}} {\varvec{F}}_{{\text{R}}} }}{{\varvec{C}}}\user2{ }\) represents the parameter of removal energy, \(\frac{{T_{{\text{O}}} - T_{{\text{I}}} }}{{T_{{{\text{ref}}}} }}\) represents the parameter of heat loss. The relationship between the heat loss parameter and collector efficiency is represented in Eq. 11. The \(F_{{\text{R}}} \left( {\eta_{{\text{c}}} } \right)\) and \(\alpha \tau \left( {F_{{\text{R}}} } \right)\) intercept component with a linear regression approach.

Data reduction

The factors such as friction factor, uncertainty analysis, Reynolds number, collector efficiency, Nusselt number, and convective heat transfer coefficient were determined in the solar PTSC system. Nusselt number represents heat transfer by pure convection. The Nusselt number is the ratio of convective to conductive heat transfer across a boundary. A value between 1 and 10 is characteristic of slug flow or laminar flow. Further, a larger Nusselt number corresponds to more active convection, with turbulent flow typically in the 100–1000 range [29]. The convection and conduction heat flows are parallel to each other and the surface normal of the boundary surface and are all perpendicular to the mean fluid flow in the simple case. Solar PTSC performance depends explicitly on normal direct irradiance, fluid inlet temperature, mass flow rate, and outlet temperature (\(\eta_{{\text{c}}}\), Re, Nu f,) (T, \(Q_{{\text{g}}}\)). \({ \ni }_{{\text{V}}}\) represents flow rate of velocity (m s−1), \({ \ni }_{{\text{V}}}\) represents pressure drop, the following equation gives clarity regarding the performance of the solar collector:

$$\eta_{{\text{c}}} = f\left( {T_{{\text{O}}} ,T_{{\text{I}}} ,{\text{T}},{ }Q_{{\text{g}}} } \right)$$
(12)
$${ \ni }_{{\eta _{{\text{c}}} }} = \sqrt {\left[ {\left( {\frac{{\partial \eta _{{\text{c}}} }}{{\partial Q_{{\text{g}}} }}{ \ni }_{{T_{O} }} } \right) + \left( {\frac{{\partial \eta _{{\text{c}}} }}{{\partial T_{{\text{I}}} }}{ \ni }_{{T_{I} }} } \right) + \left( {\frac{{\partial \eta _{{\text{c}}} }}{{\partial T_{{\text{O}}} }}{ \ni }_{{Q_{g} }} } \right) + .........\left( {\frac{{\partial \eta _{{\text{c}}} }}{{\partial T}}{ \ni }_{{\text{T}}} } \right)} \right]}$$
(13)

The change in velocity with flow rate can be expressed as

$${ \ni }_{{\text{V}}} = \sqrt {\frac{{\partial V}}{{\partial m}}{ \ni }_{{\text{m}}} }$$
(14)

Dittus–Boelter equation shows the variation in the Nusselt number and is stated below which indicates the variance of the Nusselt number [30, 31].

$${\text{Nn}} = 0.023\,\text{Pr} ^{{0.3}} \text{Re} ^{{0.8}}$$
(15)

where \(\Pr\) represents the Prandtl number, \({\text{Re}}\) represents the Reynolds number. The expressions for \(\Pr {\text{and}} {\text{Re}}\) are detailed in Eqs. (16) and (17),

$$\Pr = \frac{{C_{{\text{p}}} \mu }}{{k_{{\text{t}}} }}$$
(16)
$${\text{Re}} = \frac{{\rho D_{{\text{h}}} V}}{\mu }$$
(17)

where \(C_{{\text{p}}}\) represents the specific heat capacity of nanofluid, \(k_{{\text{t}}}\) represents the thermal conductivity of the nanofluid, \(\mu\) represents the dynamic viscosity of the nanofluid. The field under curves was eventually used to compare cases for the collector’s overall efficiency index.

Uncertainty analysis

The uncertainties of the parameters from the derived variables (\(\eta_{{\text{c}}}\), Re, Nu, and f) were discovered in the experimental calculation of independent variables (T,\(Q_{{\text{g}}}\)). The standard deviation has been derived and given below Eq. 18 since each variable has been measured over a minimum of three intervals [30].

$${ \ni }_{{\text{R}}} = \sqrt {\left[ {\left( {\frac{{{ \ni }_{1} \partial S}}{{\partial X_{1} }}} \right) + \left( {\frac{{{ \ni }_{2} \partial S}}{{\partial X_{2} }}} \right) + \left( {\frac{{{ \ni }_{3} \partial S}}{{\partial X_{3} }}} \right) + .........\left( {\frac{{{ \ni }_{{\text{N}}} \partial S}}{{\partial X_{{\text{N}}} }}} \right)} \right]}$$
(18)

Based on the above equation, the proportional error is calculated as

$$E_{{\text{R}}} = \frac{{{ \ni }_{{\text{R}}} }}{S}\%$$
(19)

Results and discussion

Analysis flow velocity

The dimple tube’s performance was analysed by comparing the expected water velocity magnitude and the performed test results. Furthermore, there is a positive correlation between the experimental value and CFD analysis. The velocity magnitude was determined in steady-state conditions and utilized to determine the numerical model’s prediction efficiency. The velocity magnitude displays the velocity contour’s effects in the dimples tube. The flow behaviour was almost identical in both experimental and CFD analysis, as illustrated in Fig. 6. This flow behaviour demonstrates a fair and accurate estimation process and numerical procedure. In the experimental analysis, nanoparticles of TiO2 containing a volume of concentration 0.1–0.5 percent were used to absorb the heat energy from the solar panel and the dimple tube. The base fluid, nanoparticles, and nanofluid thermophysical properties are input parameters for the analysis.

Fig. 6
figure 6

Fluid flow behaviour, a Flow of water in dimple tube, b Low velocity with low flow rate c High velocity with a high flow rate

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The dimple tube temperature contour at different inclination angles with different nanofluid levels for PTSC is shown in Fig. 7. The dimple tube flow is turbulent, where TiO2 nanofluid is filled outer core of the dimple tube. The cold water from the tank is passed through the inner core of the dimple tube. The inclination angle of the PTSC rises from 30° to 60°. The magnitude plot of 0.1–0.5% nanofluid and three distinct inclination angles at the tube inlet are analyzed. Furthermore, the greatest velocity and temperature have been achieved at a volume concentration of 0.3% at inclination angles of 45°. Results showed that the dimple tube’s flow velocity at the tilt angle of 45o is recorded to be high. TiO2 fluid has been assumed to significantly influence thermosiphon phenomena at the analysis phase, contributing to the above results.

Fig. 7
figure 7

Temperature contour, a Analysis of temperature with low flow rate b Analysis of temperature with high flow rate

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The disparity between the pressure contour and dimple texture design concerning change in flow velocity and volume concentration is shown in Fig. 8. The pressure drop increases for all inclination angle values, with the increment in the nanoparticle’s volume fraction and the dimple texture design. However, the rising angle of inclination reduces the mean amount of pressure drop for the entire volume fraction concentration. This is induced by developing and reducing the mean pressure drop of the weak secular recirculation at the higher inclination angles. The adjustment in the pressure drop in the heated plain tube and dimple tube for 0.1–0.5% volume concentration at different angles is observed. The experimental outcomes confirmed that a higher pressure drop was attained at a small inclination angle of the PTSC collector.

Fig. 8
figure 8

Pressure contour, a Low-pressure contour for high inclination angle b High-pressure contour low inclination angle

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Effect of volume concentration

Figures 9 shows the variation in absolute energy and rise in temperature with respect to different concentrations of nanoparticles and different flow rates.

Fig. 9
figure 9

Variation in absolute energy and rise in temperature, a Absolute energy-volume concentration, b Rise in temperature-volume concentration

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Figures 10 shows variation in mass flow rate for different modes of operation and variation in collector efficiency for experiment and CFD with different volume concentrations. The collector efficiency improved by 11%, and that convective heat transfer increased by 34.25% relative to the base fluid. The collector efficiency and heat transfer have been improved due to an increased TiO2 nanoparticle concentration due to higher absorbance and absorption coefficient. Consequently, a pressure drop in the gradient temperature increased the convective heat transfer coefficient for nanofluids as expressed as \(\Delta T=({T}_{\mathrm{O}}-{T}_{\mathrm{I}})\). For higher volume concentrations (0.1–0.5%), there is an 11% improvement in solar PTSC quality compared with base fluid. It can happen observed from Fig. 10b that the collector efficiency has been improved by 0.4%, while the volume concentration was rising by 0.5% at the various flow rates. Since solar PTSC performance increased by an average of 0.5%, the volume’s optimum concentration decreased to 0.3%. The nanoparticle volume concentration of 0.3% gives optimum collector efficiency for PTSC compared to other concentrations. This may be due to the effective balance between de-ionized water and nanoparticles, which provides superior flow behaviour by considering balanced density and higher heat transfer at maximum turbulence induced by the dimple texture design of the tube. The flow rate is often shown to be inversely proportional to the temperature gradient. As a result, the flow rate increased, and the heat transfer coefficient improved in PTSC with the dimple tube, resulting in a significant improvement in the collector’s performance.

Fig. 10
figure 10

Variation in mass flow rate and collector efficiency, a Mass flow rate for different modes, b Collector efficiency for experiment and CFD-volume concentrations

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Figures 11 shows the temperature gradient maps at varying flow rates and concentrations of TiO2 nanoparticles and convective heat transfer coefficients at different flow rates.

Fig. 11
figure 11

Variation in convective heat transfer coefficient and gradient temperature, a Convective heat transfer coefficient-volume concentration b Gradient temperature-mass flow rate

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From Fig. 11a, it is observed that with variation in mass flow rate and variation in volume concentration of TiO2, the convective heat transfer coefficient gradually increases. This increase in convective heat transfer coefficient with an increase in mass flow rate is due to added advantages of an increase in Reynolds number and Nusselt number influenced by the quantum of flow and the dimple texture design to the inner tube of PTSC. Apart from this, the heat transfer coefficient at a flow rate of 2.5 kg min−1 and a flow rate of 3 kg min−1 does not give any remarkable change with variation in the concentration of TiO2. However, TiO2 concentration of 0.3% indicates a higher heat transfer coefficient compared to other concentrations considered in this analysis. Mass flow rate of 2.5 kg min−1 at volume concentration of 0.3% gives higher convective heat transfer coefficient of 60, 45, 34, and 18% in comparison with 0.5, 1.0,1.5, and 2.0 kg min−1, respectively. From Fig. 11b, it is observed that gradient temperature increases with the use of dimple texture and dimple texture with TiO2 in comparison with the plain tube. This may be due to the effect of high turbulence and the effect of TiO2 superior thermal conductivity. Apart from this, the temperature gradient in CFD analysis and experimental analysis are in similar trend, however, in CFD analysis, mass flow rate of 3.0 kg min−1 gives 42, 36, 22, 20, 16, and 10% of increase in temperature gradient in comparison with mass flow rate of 0.5, 1.0, 1.5,2.0,2.5, and 3.0 kg min−1 respectively.

The variation in mass flow rate and variation in volume concentration of nanoparticles concerning variation in Reynolds number (for experimental and simulation) and Nusselt number (for experimental and simulation) are illustrated in Fig. 12. An error value of ± 3.12% has been observed between the CFD analysis and experimental values. Empirical correlation calculations are performed based on varying volume concentrations and mass flow rates as per Eq. 15 and Eq. 17 for different Reynolds numbers and Nusselt numbers. The new model significantly improved the collector performance by 62.32% for the Reynolds number range of (3256 < Re < 9685) and Prandtl (6.324–9.254). The findings were compared with CFD and experimental analyses and identified that the variation in Nusselt number of 11% for the Reynolds number range (4568 > Re > 2.36 × 105) and concentration of particle is \(0<\varphi <11\mathrm{\%}\). The experimental and numerical values for the Nusselt number and Reynolds number are given in Table 5.

Fig. 12
figure 12

Variation in Nusselt number and Reynolds number, a Nusselt number (Experimental)-Nusselt number (CFD), b Reynolds number -Nusselt number

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Table 5 Experimental and numerical values for Nusselt number and Reynolds number
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The experimental and CFD analysis plots for friction factors with various concentrations and flow rates are illustrated in Fig. 13.

Fig. 13
figure 13

Variation in friction factor and Reynolds, a Friction factor (Experimental)-friction factor (CFD), b Reynolds number- friction factor

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The friction factor is measured as a product of pressure drop and the surface roughness of the dimple tube. Furthermore, the average pressure drop is recorded as 2.36 kPa for the solar PTSC system. In the current model, the deviation from the estimated friction factor is roughly ± 4.62%. The expected friction factor for a higher Reynolds number can deviate in a range of ± 11%. Table 6 gives the experimental and CFD values for friction factor and Reynolds number.

Table 6 Experimental1and CFD values for friction factor and Reynolds number
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Figures 14 displays the performance index of experimental and CFD analysis with various flow rates and concentrations of nanoparticles, and the deviations can be recorded as approximately ± 7.42% percent. It shows efficiency index combinations for various Reynolds numbers and concentrations of nanoparticles. At 0.1–0.5% of the volume concentrations and a volume flow rate of 2.5 kg min−1, the maximum output index is 2.42. The TiO2 nanofluid output index is larger than 1; in the PTSC application, the heat transfer increased in this research. The efficiency variations are based on pH changes and thermo-physical nanofluid properties of TiO2. Particle size affects the solar performance of PTSCs. Nanoparticles of greater sizes tend to spread and absorb radiation. The measurement of the TiO2 nanoparticles should be 50 nm to provide efficient heat transfer rates. In this analysis, the average size of the TiO2 nanoparticles is 50 nm. The heat transfer has been improved in this research, and TiO2 nanoparticles were efficient for PTSC solar applications. Table 7 gives the experimental and CFD values for the performance index and Reynolds number.

Fig. 14
figure 14

Variation in performance index, a Performance index (Experimental)-Performance index (CFD), b Reynolds number- performance index experimental-performance index CFD

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Table 7 Experimental and CFD values for performance index and Reynolds number
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Figure 15 portrays the variation in pressure drop with the variation in nanoparticle volume concentration and mass flow rate. From the figure, it is observed that with an increase in the concentration of nanoparticles, the pressure drop increases irrespective of the mass flow rate of the water. This is due to an increase in volume density of the nanoparticles in the nanofluid, which resulted in a more viscous flow of the nanofluid due to an increase in density. The figures also observed that dimple tubes with TiO2 nanoparticles expedited a higher drop in pressure compared to the plain tube with TiO2. This is due to increased obstacles to fluid flow due to dimple texturing on the tube surface. Apart from this, the dimpled texture offers more drag force to be experienced by the molecules adjacent to the inner tube layer during its dynamic condition. The pressure drop of the solar water heater has increased gradually with the increase in the mass flow rate of water. At a maximum mass flow rate of 3.0 kg min−1 and 0.5% volume concentration, the pressure drop has been increased by around 5.68% compared to the mass flow rate of 2.5 kg min−1. The error observed in the experiment and simulation are 6.82 and 2.67%, respectively, indicating the linear relationship between the experiment and simulation outcomes. Furthermore, the pressure drop results for the experimental condition are significantly higher than the simulation results.

Fig. 15
figure 15

Variation in pressure drop with variation in nanoparticle volume concentration and mass flow rate

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Conclusions

The research analysed TiO2 efficiency in heat transfer for solar PTSC applications. Tests were performed using various concentrations of nanoparticles and mass flow speeds. The numerical prediction was made on the TiO2/water nanofluid collector, evacuated by thermosiphon, using commercial ANSYS tools. For various conduit angles, the effect of nanoparticle volume fraction on the collector’s thermal efficiency was studied. The most significant conclusions achieved in this study are as follows:

  1. 1.

    The nanoparticle volume fraction increases the heat conductivity and Nusselt Number of solar evacuation pipes.

  2. 2.

    The inclination of 45° plays a significant role in enhancing the thermal efficiency and increases the evacuated tunnel’s thermosiphon effect.

  3. 3.

    At the nanoparticles volume fraction and angle of inclination of 0.5 percent and 45° degrees, respectively, the optimum speed and heat transfer change was observed.

  4. 4.

    The nanofluid’s output index is 2.42 with a 0.3% mass flow rate and concentration of 2.5 kg min−1. The PTSC with TiO2 nanofluid has a maximum overall efficiency of 34.25%, whereas 11% higher than the overall efficiency of the base fluid.

  5. 5.

    At a mass flow rate of 3.0 kg min−1 and 0.5% volume concentration, the pressure drop has been increased by around 5.68% compared to the mass flow rate of 2.5 kg min−1.

  6. 6.

    The temperature gradient in CFD analysis and experimental analysis are in similar trend, however, in CFD analysis, mass flow rate of 3.0 kg min−1 gives 42, 36, 22, 20, 16, and 10% of increase in temperature gradient in comparison with mass flow rate of 0.5, 1.0, 1.5, 2.0, 2.5 and 3.0 kg min−1, respectively.