1 Introduction

Temperature control of electronic equipments, batteries and photovoltaic (PV) modules is important for safety operation and efficiency of the systems [1,2,3]. In PV module, solar radiation that cannot be converted into electricity results in temperature rise of the cell. In PV modules, the efficiency is inversely proportional to the cell temperature. Therefore, by using a suitable cooling system, it is possible to reduce the temperature and efficient operation of the PVs is performed [4,5,6]. Thermal management by using cooling systems is required for effective operation of those, and recently many advancements have been achieved by using different cooling technologies. In PV application, utilization of fins, phase change materials (PCMs), channel cooling and jet impingement cooling are some of the available passive/active methods [7,8,9]. Bayrak et al. [10] performed an experimental work on the effectiveness of using different cooling techniques for PVs such as PCM, aluminum fins and thermoelectric (TE). Under the same ambient conditions, PV with fins generated the highest power production. Maleki et al. [11] performed a comprehensive review for the application of different active and passive techniques for PV cooling. They noted that efficiency increase of PVs was dependent upon the amount of solar irradiation, operating conditions and employed cooling technique. They also concluded that using thermal management methods in hot climates was favorable as the operating temperatures of PVs became high. In another review work, Pathak et al. [12] considered different PV cooling techniques including heat pipe, TE, PCM, immersion cooling, airflow cooling, micro-channel cooling and jet impingement. They also presented a systematic view for the challenges and benefits of artificial intelligence in PV/T systems.

Nanofluid (NF) technology has been used in many energy applications including electronic cooling, energy storage, solar power, refrigeration, battery thermal management and many more [13,14,15,16,17,18,19]. The advancements in production of novel NFs and simulation tools for analyzing the NF behavior make this technology promising in different areas. In solar power, NF technology has been implemented in various forms. Hamzat et al. [20] conducted a detailed review for applications of NF in solar energy harvesting devices. They considered various devices such as solar dishes, evacuated tubes, solar cookers and PV/T systems. Their performances when operated with NFs were compared with the system working with conventional fluids. They noted that when operated with NFs, highest performance was achieved. For solar-powered desalination systems, when NFs were used, low energy input was required with more freshwater. [21] reviewed the performance of solar collectors by using hybrid NFs. They considered both concentrating and non-concentrating collectors. When operated with hybrid NFs, profound improvements in thermal and optical characteristics were noted, while the cost and stability were found as the main challenges. Ahmadinejad and Moosavi [22] performed a numerical work on the effects of using a novel baffled-NF-based PV/T module on the energy and exergy performance improvements. CuO/water NF and CNT/water NF were considered as the cooling medium. It was observed that baffled system with CNT-water NF has the highest power generation among different cases, while 14\(\%\) thermal power increment was achieved when comparisons were made with the non-baffled PV/T system using only water. Mohammadpour et al. [23] performed a numerical work for thermal management of PV by using impinging jet with NF of water having SiC NPs. They considered different operating and geometric parameters such as inlet temperature of NF, mass flow rate and jet-surface distance. They considered the NP loading of 4\(\%\). It was observed that inlet temperature of NF significantly reduced the PV temperature. Different aspects of NF such as non-Newtonian behavior, NP shape and size on the cooling performance have been considered in various studies. Shape factor of NPs in convective HT has been contributed to the thermal performance significantly as it has been shown in several studies.

TE energy conversion can be considered in energy generation and cooling applications [24,25,26]. They offer many advantageous such as being compact, without any moving parts, high reliability, low noise characteristics. In solar applications such as in PVs, TEGs can be installed. The integration of TEG with PV provides the temperature variation of TEG operation, while efficiency loss may be prevented. In channel flow applications, TEGs have been considered. TEG material is one of the important factors in the overall effectiveness, while other operating/geometric parameters such as flow rate, location of TEG, installation of other active/passive methods have also been shown to affect the interface temperature variation, which will change the generated power characteristics [27,28,29,30]. NFs have been shown to contribute positively to the TEG operation. In the review work of Abdelkareem et al. [31], utilization of NFs in TEG-related systems has been discussed especially for waste heat recovery. They noted the improved performance features of TEGs with NF, while more research was necessary for operational and material part. The NF shape, type and loading amount in the base BF have been considered as the most important influencing factors when NFs are used with TEG-installed systems. In channel flow applications, installing TEG with NFs significantly improved the TEG performance at the highest NP loading. In PV applications, utilization of TEG on the thermal management and performance improvements has been considered in many studies. In the work by Akbar et al. [32], convection of air and different fluids including NF with Ag, and SiO\(_2\) on the performance features of PVT/TE system were explored. When compared to convective flow of air, they showed the performance improvement of PVT/TE system by about 40.54–50.53\(\%\) when using NF. Soltani et al. [33] showed the performance increments of PV/TEG system considering different NFs, while they obtained the best results when using SiO\(_2\) NF. Lekbir et al. [34] proposed a novel PV/T-TEG hybrid system that uses NF-based cooling. The electrical energy of the proposed model was found higher as compared to other hybrid system models. Khalili et al. [35] considered the utilization of hybrid NF (water+Fe\(_3\)O\(_4\)+MWCNT) in a TEG combined PVT module. They also considered different cross section of the duct where NF is used. They obtained improvement of electrical efficiency by about 16.2\(\%\) when comparisons were made with the un-cooled system.

The computational fluid dynamics (CFD) simulation of 3D fully coupled system of PV module with TEG and wavy channel cooling system requires large computational time. When considering a parametric study of the 3D coupled system such as varying wavy cooling channel parameters, methods are needed to reduce the computational cost. In this work, a hybrid method that uses artificial neural networks (ANNs) and CFD together has been utilized for performance predictions of PV module. In energy systems, methods have been offered to reduce the computational cost coupled thermofluid systems [36,37,38,39]. The outputs that will be coupled to the PV module from the parametric CFD part of the cooling system are predicted by using ANN-based method. It is then used as boundary condition for the PV module, while 3D computational study of the PV is conducted. Applications of ANN method in performance predictions of PV systems have been considered in many studies [40,41,42,43]. We propose a hybrid method where 3D computational part of PV module is conducted. This enables us to consider different operating and boundary conditions for the PV module as only cooling part output to the PV system is estimated by using the ANN-based method. In Fig. 1, a schematic view of computational procedure is shown.

In order to enhance the performance of the PV module, a unique cooling channel design that combines hybrid NF and TEG is proposed. Three-dimensional coupled numerical simulations are conducted for coupled cooling channel and PV module system. The channel is designed in a sinusoidal wave pattern, and the alumina-water NF, which is used as the cooling medium, contains NPs with varying shapes. To the best of the authors’ knowledge, this work is the first to take into account the use of various shaped nanoparticles and channels with sinusoidal wave forms in a PV/TEG coupled module. In addition to that, a hybrid computational method for PV+TEG+cooling channel is proposed for fast and accurate predictions of PV module performance, which enables to consider complicated cooling methods and different operating conditions of the PV module. The need for energy-efficient products has made research and efficient computational methods for innovative and alternative cooling systems applicable to PV and PV/T systems a hot topic these days. The results will be helpful for the development and optimization of systems that can be used in PVs and for improving their performance.

Fig. 1
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Efficient computational model procedure with ANN-assisted CFD

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2 Mathematical Model of Coupled PV, TEG and Cooling Channel System

2.1 Model Description and Governing Equations

In this study, cooling features of using a corrugated channel with NF of different-shaped NPs are explored when used with PV panel coupled with a TEG module. A schematic view of the coupled PV+TEG with cooling system is presented in Fig. 2. The TEG module with legs and 2D view of the coupled system with some of the geometric parameters are shown in Fig. 3a, b. The PV panel has five components, which are silicon cell (polycrystalline) and two layers of EVA, one of which is glass and the other of which is TPT. For thermal-to-electric conversion, a TEG module is mounted on the rear side of the TPT layer. This is performed as the heat from the PV panel induces a temperature difference across the TEG module. Thermophysical features of TEG components are presented in Table 1. The properties of the semiconductor material forming the TEG legs depend on temperature. The material properties of the ceramic layers and conductors are at constant material properties, and they are not dependent upon the temperature. In the numerical analysis, the material properties of the legs are taken from the code depending on the temperature. These are Seebeck coefficient, electrical and thermal conductivity values. A wavy channel below the TEG module is constructed with LC and Hc as the length and height of the cooling channel. Both flat and wavy channel configurations are considered. The wavy channel has a sinusoidal form of Amp sin (2\(\pi \) Nf s) where Amp and Nf are the amplitude and frequency of the corrugation, which are varied during the study. Aluminum oxide is utilized for the ceramic plates, while copper is considered for the electrodes. For appropriate cooling of the system, an absorbent layer is mounted between the TEG and cooling channel. In TEG module, 196 legs (equal in size) are used. The dimensions of PV/TEG system’s components are given in Table 2.

Fig. 2
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3D schematic sketch of coupled PV-TEG modules with sinusoidal wavy cooling channels

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Fig. 3
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Schematic of TEG module with legs (a) and 2D view of the system with some of the geometric parameters (b)

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Table 1 Material thermophysical properties of TEG module
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Table 2 Dimension of components of PV-TEG coupled system
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As the cooling medium, water+alumina of different-shaped NPs (spherical, cylindrical and brick) are used up to solid volume fraction (SVF) of 0.03. NP sizes of different shapes are presented in Fig. 4. NP and base fluid (BF) properties are given in Table 3.

The shape impacts of NPs on the viscosity and thermal conductivity are stated as in the following [44]:

$$\begin{aligned} \frac{k_\textrm{nf}}{\mu _f}= & {} 1+C_k \phi , \end{aligned}$$
(1)
$$\begin{aligned} \frac{\mu _\textrm{nf}}{\mu _f}= & {} 1+A_{1} \phi +A_{2} \phi ^2, \end{aligned}$$
(2)

where Table 4 presents the \(C_{k}\), \(A_{1}\) and \(A_{2}\) constants considering different NP shapes.

Fig. 4
figure 4

Different-shaped NPs used in the base fluid and their sizes

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When dealing with the spherical NPs, impacts of Brownian motion are taken into account. Thermophysical relations for NF viscosity and thermal conductivity are stated as in the following: [45]:

$$\begin{aligned}{} & {} \begin{aligned}&\mu _\textrm{nf}=\mu _\textrm{static}+\mu _\textrm{Brownian}=\\&=\frac{\mu _f}{(1-\phi )^{2.5}}+5 \times 10^{4} \beta \phi _f (C_p)_f \frac{\mu _f}{k_f \text {Pr}}\sqrt{\frac{k_b T}{\rho _p d_p}}\\&\quad f (T, \phi ). \end{aligned} \end{aligned}$$
(3)
$$\begin{aligned}{} & {} \begin{aligned}&k_\textrm{nf}=k_\textrm{static}+k_\textrm{Brownian}=\frac{k_{p}+2k_f-2(k_f-k_p)}{k_p+2k_f+(k_f-k_p)\phi }k_f \\&+5 \times 10^{4} \beta \phi _f (C_p)_f \sqrt{\frac{k_b T}{\rho _p d_p}}f (T, \phi ) \end{aligned} \end{aligned}$$
(4)

Spherical-shaped NP with 47 nm diameter is used, while the function f and \(\beta \) are stated as in the following [45]:

$$\begin{aligned} \begin{aligned} f (T, \phi )&=\left( 2.8217 \times 10^{-2} \phi + 3.917 \times 10^{-3} \right) \left( \frac{T}{T_0} \right) \\&\quad +\left( -3.0669 \times 10^{-2} \phi -3.91123 \times 10^{-3} \right) , \\ {}&\quad \beta =8.4407 (100 \phi )^{-1.07304} \end{aligned} \end{aligned}$$
(5)

Density and specific heat of NF are described as in the following:

$$\begin{aligned} \rho _\textrm{nf}{} & {} =(1-\phi ) \rho _f +\phi \rho _p, \ \ \ (\rho C_p)_\textrm{nf}=(1-\phi )(\rho C_p)_f\nonumber \\{} & {} \quad +\phi (\rho C_p)_p. \end{aligned}$$
(6)

PV module layer equations such as EVA, TPT, and silicon cell are stated as [46] in the following:

$$\begin{aligned} \rho _l C_{p,l} \frac{\partial T}{\partial t} -\nabla . \left( k_l \nabla T \right) =\dot{q}_{s,l}-\dot{P}_{e,l}, \end{aligned}$$
(7)

where \(\rho _l, C_{p,l}\) and \(k_l\) are the density, thermal conductivity and heat capacity of the PV layer, while \(\dot{P}_{e,l}\) denotes the energy absorption of the layers. Here, electrical power of PV (per unit volume) is represented by the \(\dot{P}_{e,l}\). By utilizing the intensity of solar radiation (\(G_0\)), absorbed energy amount by each layer can be obtained. The following equations are valid [47, 48]:

$$\begin{aligned} \dot{q}_{s,l}= & {} \frac{G_{r,l} \times \alpha _l \times A_l}{V_l}, \end{aligned}$$
(8)
$$\begin{aligned} G_{r,l}= & {} G_{r, l-1} \left[ \left( 1- \alpha _{l-1} \right) - \gamma _{l-1} \right] , \end{aligned}$$
(9)

where \(A_l\) and \(V_l\) denote the layer-l area and volume. The terms \(\alpha _l\), \(\gamma _l\) and \(G_{r,l}\) are the absorptivity, reflectivity and radiation intensity of layer l. Table 4 gives the optical characteristics of PV components. PV system efficiency is described as in the following:

$$\begin{aligned} \eta _{p}=\eta _\mathrm{{ref}, \textrm{si}} \left[ 1- \beta \left( T_{c, \textrm{si}}- T_\mathrm{{ref}, \textrm{si}} \right) \right] . \end{aligned}$$
(10)

In the above equation, average silicon cell and temperature coefficient are given by the terms \(T_{c,si}\) and \(\beta \).

Table 3 Thermophysical properties of water and alumina at 25 \(^\text {o}\)C [62]
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Table 4 Constant values for different shapes of Al\(_2\)O\(_3\) NPs [44]
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Electric and thermal field are coupled in the TEG module. The related coupled equations are stated as in the following: [49]:

$$\begin{aligned} \begin{aligned}&\nabla \cdot J=0, \ E=\rho J+ \alpha \nabla T, \ q=\Pi J-k \nabla T, \\&\Pi =\alpha T, \ J=\sigma \left( E- \alpha \nabla T\right) , \ E=-\nabla V. \end{aligned} \end{aligned}$$
(11)

where current density and Peltier coefficient are given by the terms J and \(\Pi \), while q represents the heat flux term. Energy equation is stated as in the following [49]:

$$\begin{aligned} \nabla \left( k \nabla T \right) +\frac{J^2}{\sigma }-T J. \nabla \alpha =0 \end{aligned}$$
(12)
Table 5 PV component optical features [63]
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Seebeck coefficient (\(\alpha \)) is given as:

$$\begin{aligned} \alpha =-\frac{\Delta V}{\Delta T}. \end{aligned}$$
(13)

Figure of merit (ZT) is stated as:

$$\begin{aligned} \text {ZT}=\frac{\alpha ^2 \sigma }{k} T, \end{aligned}$$
(14)

where electrical and thermal conductivity is given by \(\sigma \), and k.

In the cooling channel with flat and wavy walls, the flow is 3D and laminar. Governing equations are stated as in the following:

$$\begin{aligned}{} & {} \nabla . \left( \rho {\varvec{u}} \right) =0 \end{aligned}$$
(15)
$$\begin{aligned}{} & {} {\nabla . \left( \rho {\varvec{u}} {\varvec{u}} \right) =-\nabla p+\nabla . \tau } \end{aligned}$$
(16)
$$\begin{aligned}{} & {} {\nabla .\left( {\varvec{u} T} \right) =\alpha \nabla ^2T} \end{aligned}$$
(17)

The shear stress \(\tau \) is defined as:

$$\begin{aligned} \tau =\mu \left( \nabla {\varvec{u}}+ \left( \nabla {\varvec{u}}\right) ^T \right) , \end{aligned}$$
(18)

where dynamic viscosity is given by the term \(\mu \).

2.2 Boundary Conditions

Radiation and convective HT take place for the bottom and top surfaces of PV panel, which are stated as [50, 51]:

$$\begin{aligned}{} & {} {Q_\textrm{rad}=\varepsilon \sigma \left( T^2-T_\textrm{amb}^2 \right) \left( T^2+T_\textrm{amb}^2 \right) A}, \ \ \nonumber \\{} & {} Q_\textrm{conv}=h \left( T-T_\textrm{amb} \right) A \end{aligned}$$
(19)

where Stefan–Boltzmann constant, emissivity, and PV panel temperature are given by the terms \(\sigma , \varepsilon \) and T. The definitions of the sky temperature and HT coefficient are given as [52]:

$$\begin{aligned} T_\textrm{sky}=T_\textrm{amb}-6 (\text {K}), \ \ \ h=5.82+4.07v, \end{aligned}$$
(20)

where wind velocity and HT coefficient are given by the terms v and h. Heat flux continuity is assumed for the interfaces of the PV module layers, which can be stated as:

$$\begin{aligned} \dot{q}_{l}=\dot{q}_{l+1}. \end{aligned}$$
(21)

The value of conversion efficiency for polycrystalline silicon cell is 16.8\(\%\) at reference temperature of 298.15 K. The value of concentration ratio is taken as C = 2. In the TEG module, \(V=0\) at the ground while at the terminal current is equal to zero. Surfaces of TEG (except for ceramics) are electrically insulated (\(n \cdot J=0\)). PV component optical features are stated in Table 5.

The cooling mini-channels are wavy, which have a sinusoidal form. Cold BF/NF enters the mini-channels with velocity of uc and temperature of Tc. Pressure outlet is used at the exit of the channels. The relevant physical parameters of interest are the Reynolds number and Prandtl number, which are given as:

$$\begin{aligned} \text {Re}_\text {c}=\frac{\rho u_c D_h}{\mu }, \ \ \text {Pr}=\frac{\nu }{\alpha }. \end{aligned}$$
(22)

2.3 Solution Method and Code Validation

The solution method employs Galerkin-weighted residual FEM approach where several sources can be mentioned for modeling procedures and basic principles in flow and HT problems [53, 54]. In the method, field variables of interest such as velocity components, pressure, temperature and electric potential are approximated by using various ordered Lagrange FEs. When they are used in the related governing equations, residual (R) is formed. Weighted average of the residual is set to be zero in average as in the following by using a weight function (\(W_f\)):

$$\begin{aligned} \int W_f R \textrm{d}V=0. \end{aligned}$$
(23)

To deal with the numerical instability problem, streamline upwind Petrov–Galerkin (SUPG) method is utilized. Biconjugate gradient stabilized (BICGStab) solver is considered in the code. The criteria for the convergence are defined as:

$$\begin{aligned} {\left| \frac{\Gamma ^{i+1}-\Gamma ^i}{\Gamma ^{i+1}} \right| \le \varepsilon }, \end{aligned}$$
(24)

where i and \(\Gamma \) represent the iteration step and field variable, while \(\varepsilon \) is set to \(10^{-7}\). A commercial CFD code Comsol [55] is used for the numerical simulation.

To reduce the computational time of high-fidelity 3D coupled system, mesh independence studies are carried out. An optimal mesh is required to reduce computational cost and to obtain accurate results. Figure 5a shows the average PV-cell temperature variation for different grid sizes when using BF and NF having cylinder-shaped NPs and SVF = 0.03. Case 5 with 18590514 number of elements is selected and considered in the subsequent computations. An unstructured grid system with tetrahedral elements is used. Boundary layer mesh is also considered with 3 number of layers while stretching factor and thickness adjustment factor as set to 1.2 and 5, respectively. Grid distribution is shown in Fig. 5b.

Fig. 5
figure 5

Average PV-cell temperature for various grid sizes considering BF and NF with cylindrical-shaped NPs (a, Amp = 0.1, Nf = 8) and its distribution (b)

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Fig. 6
figure 6

Average Nu comparisons with respect to changes in Re by using experimental results from a converging–diverging channel in Ref. [56]

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Fig. 7
figure 7

Variation of local Nu for various locations of a micro-scale BFS using SiO\(_2\) NF in the base fluid (loading of 0.5\(\%\)). Comparisons are made by using the experimental work of [57]

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Fig. 8
figure 8

Solar irradiation variation for different time instances (a) and average PV-cell temperature variations by considering the experimental results in Ref. [59]

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Fig. 9
figure 9

Effects of wavy channel amplitude on the variation of temperature of whole system (a–d) and electric potential of TEG (e, f) (Nf = 8, Cylindrical-shaped NPs, SVF = 0.03)

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Different validations studies are conducted to show the reliability of the code for solving TEG/PV coupled systems having NF-enhanced wavy channel cooling. In the first work, experimental results from Ref. [56] were used where convective HT from converging–diverging channel was explored. In the reference work, a correlation for the Nu was provided. Figure 6 presents the average Nu comparisons for several Re in between 2000 and 6000. The maximum deviation between the results is found below 5.5\(\%\).

In another validation work, impacts of NF on convective HT are micro-channel with area expansion considered, and experimental results in Ref. [57] were used. The micro- BFS geometry has step size of 0.6 mm, while sizes of upstream and downstream parts are 0.1 m and 0.15 m. SiO2–water NF is used with loading of 0.5\(\%\). Local variation of Nu is presented in Fig. 7 for different locations. In the numerical study, thermophysical features are adopted from Ref. [58]. The highest deviation between the results is obtained as 10. 64\(\%\).

Fig. 10
figure 10

Average cell temperature (a), temperature uniformity (b) and TEG power generation (c) with respect to changes in the wavy form amplitude (Nf = 8, cylindrical-shaped NPs, SVF = 0.03)

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Fig. 11
figure 11

Impacts of frequency of wavy form of cooling channel on the average cell temperature (a) and temperature uniformity (b) variation (Amp = 0.08, cylindrical-shaped NPs, SVF = 0.03)

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In the last validation work, experimental work in Ref. [59] is considered where PV module with novel passive cooing was analyzed. Variation of solar irradiation for different time instances available in the study is given in Fig. 8a. Comparisons of the average cell temperature variation for the time instances (Elazig /Turkey—time between 9:00 and 15:00) are shown in Fig. 8b. The value of the highest temperature is obtained as 50 \(^\text {o}\)C in references study, while it is 48 \(^\text {o}\)C in the current work. It is seen that temperature difference is less than 1 \(^\text {o}\)C for all points except for 13:00. The overall agreement between the results is satisfactory.

Fig. 12
figure 12

Effects of NP shape on the distribution of temperature of whole system (a–d) and TEG electric potential (e, f) (Amp = 0.08, Nf = 8, SVF = 0.03)

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Fig. 13
figure 13

Effects of NP loading amount in the BF on the average cell temperature variation for flat (Amp = 0, a) and corrugated (Amp = 0.08, b) channel considering various NP shapes (Nf = 8)

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Fig. 14
figure 14

Temperature uniformity variation with respect to changes in the NP loading amount in the BF for flat (Amp = 0, a) and corrugated (Amp = 0.08, b) channel considering various NP shapes (Nf = 8)

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Fig. 15
figure 15

Impacts of SVF on the average cell temperature (a) and temperature uniformity (b) variation for flat (Amp = 0) and corrugated (Amp = 0.08) channel (cylindrical-shaped NPs (Nf = 8)

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3 Results and Discussion

A novel wavy cooling channel is designed, and its performance is tested for a PV module coupled with TEG. In the cooling channel, NF of alumina–water having different-shaped NPs is utilized. Numerical simulations are conducted for various values of corrugation amplitude (\(0 \le \text {Amp} \le 0.1\)), frequency of waveform (\(2 \le \text {Nf} \le 16\)), NP loading amount (\(0 \le \text {SVF} \le 0.03\)) and NP shape (spherical, brick and cylindrical). Average temperature and temperature uniformity of the PV module are examined for varying parameters of interest. The Reynolds number is taken as 25. An efficient computational method for estimation of PV-cell temperature is proposed by using ANN, which significantly reduces the computational cost of fully coupled CFD simulations.

3.1 Computational Study of Fully Coupled PV+TEG System with Nano-Enhanced Cooling Channel

Figure 9a–d presents the variation of temperature for the whole system considering different amplitudes of corrugation of the cooling channel. Temperature gradients across the TEG module and PV module are affected by the variation of Amp. Maximum temperature will drop, and at the highest Amp value, the amount of drop is 1.8 \(^\text {o}\)C as compared to a flat cooling channel. The electric potential of the TEG module increases as well as shown in Fig. 9e, f. When corrugation amplitude is increased, local velocity in the cooling channel will rise and thermal transport in the vicinity of those regions will increase, which leads to higher cooling effects and TEG power generation.

Figure 10 shows impacts of wavy form amplitude on the variation of average surface temperature of the PV panel, temperature uniformity and generated power of TEG. Both BF and NF are used as the cooling medium. As the NF, alumina with cylindrical-shaped particles at SVF of 0.03 is used. Higher amplitudes of corrugation result in average temperature drop for both BF and NF due to the favorable effects on enhanced thermal transport with higher Amp. When cases of flat cooling channel and corrugated channel with Amp = 0.1H are compared, average temperature drops of 1.73 \(^\text {o}\)C and 1.88 \(^\text {o}\)C are achieved. When NF is used instead of BF, average temperature is 0.70 \(^\text {o}\)C lower for flat and 0.84 \(^\text {o}\)C for corrugated channel. Utilizing both NF and higher corrugation amplitudes of the channel leads to more reduction of the average panel surface temperature. In terms of temperature uniformity over the panel surface (difference between the maximum and minimum of the PV surface temperature), there is very slight impact of corrugation amplitude. The increments in the TEG power with higher corrugation amplitude become 9.5\(\%\) for both BF and NF when used as cooling medium as compared to a flat channel case. Corrugation frequency is also another influential geometric parameter of the cooling channel. A sinusoidal wave form is used for the channel. Impacts on the frequency on the PV surface temperature, temperature uniformity and TEG power output are given in Fig. 11. For both BF and NF cases, there is very slight contribution of the frequency of the corrugation on the average temperature and temperature uniformity while the best values are seen at Nf = 4. When corrugation frequency is increased, near the TEG interface mini-curved cavities are formed, and these zones are prone to vortex formation. Therefore, thermal performance is deteriorated for those regions. On the other hand, some parts at the interface will experience higher velocities due to the area reduction. These two effects are responsible for cooling performance enhancement within the channel while best case is found at Nf = 4.

Fig. 16
figure 16

Comparison of various cases of PV cooling methods in terms of average cell temperature drop (TD)

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Using NF instead of water and varying shapes of the NPs will change the favorable transport features of cooling medium, which will influence the temperature distribution of the PV and TEG modules. Figure 12 shows the temperature variation of the whole coupled system considering BF and NF having different-shaped NPs. Maximum temperature has its highest and lowest value when using BF and NF with cylindrical-shaped NPs. Favorable contribution of using NF with cylindrical shape is seen to the electrical potential distribution of TEG. Variation of average panel surface temperature considering different-shaped NP loading is shown in Fig. 13 for flat and wavy channel configurations. Using NF and increasing the loading amount will increase the viscosity, and at the same Re, fluid velocity will be higher. In addition to that, thermal conductivity rise will also contribute to the enhanced cooling performance, which leads to lowering the surface temperature with higher NP loading (SVF). But with higher loading, the impact becomes significant using cylindrical-shaped NPs followed by brick- and spherical-shaped NPs. The BF case is represented with SVF value of 0. When flat cooling channel is used, including spherical-, brick- and cylindrical-shaped particles in BF results in temperature drop of 0.1 \(^\text {o}\)C, 0.4 \(^\text {o}\)C and 0.7 \(^\text {o}\)C, while these values become 0.14 \(^\text {o}\)C, 0.5 \(^\text {o}\)C and 0.9 \(^\text {o}\)C for corrugated channel case. In terms of reducing PV surface temperature, cylindrical-shaped NPs work best in combination with corrugated channels. It is followed by NPs with spherical and brick shapes, while system performance improves with increasing NP loading. The superior performance of this NP in convective HT has been shown in several studies as compared to other particle shapes such as spherical and cubic. In terms of temperature uniformity over PV panel, NP loading and shape effects are very slight as shown in Fig. 14. Comparison of flat and wavy channel cooling performances with varying NP loading of cylindrical shape in the BF is presented in Fig. 15. Average PV temperature varies almost linearly with loading amount of NP (SVF). Average temperature for corrugated cooling channel is 1.75 \(^\text {o}\)C, 1.8 \(^\text {o}\)C and 1.9 \(^\text {o}\)C lower at SVF of 0, 0.01 and 0.03 as compared to flat cooling channel.

In Fig. 16, comparison of different cases in terms of temperature drop (TD) is shown where the reference case of PV without TEG and cooling system is considered. When only TEG is used, TD value of 43.2 \(^\text {o}\)C is obtained. Including flat cooling channel with water as the cooling medium improves the performance and TD value becomes 57 \(^\text {o}\)C. When NF is added having cylinder-shaped NPs with SVF = 0.03, an additional 0.7\(^\text {o}\)C increment of TD is seen. Corrugation of the cooling channel even operated with BF results in TD increment of 1.7 \(^\text {o}\)C as compared to Case 2 (flat cooling channel with BF). The best performance is achieved by using corrugation of the channel and inclusion of NF as the cooling medium, which results in TD value of 59.8 \(^\text {o}\)C. It is seen that using cylindrical-shaped NP in the corrugated channel provides more benefit in terms of cooling performance as compared to flat cooling channel. Combined utilization of TEG with cooling channel provides significant reduction of the PV module surface temperature as compared to un-cooled case.

Fig. 17
figure 17

Network architecture and block diagram with different layers and inputs/outputs

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Fig. 18
figure 18

ANN performance during iteration process (a) and comparisons of ANN and CFD outputs for interface temperature estimation between TEG and PV (b)

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3.2 Efficient Modeling of Coupled PV-TEG Cooling Channel System Assisted with ANN

The computational cost of fully coupled 3D PV-TEG and wavy cooling channel system is expensive. Parametric CFD study computations takes larger computations times. Therefore, in order to address this issue a new hybrid method is proposed by using coupled CFD and ANN method. The cooling system output, which is the base temperature of the PV panel, is estimated by using ANN. The estimated temperature is the used as boundary condition for the 3D computational model of the PV module.

The following relation describes the neuron model equation in the ANN model [60]:

$$\begin{aligned} s_n=\sum _{j=1}^m w_{jn} x_{j}, \ \ y_n=f(s_n+b_n) \end{aligned}$$
(25)

where \(x_j\) and \(w_{jn}\) are the input and nth neuron weights. The terms \(b_n\), f(.) and \(y_n\) denote the bias, activation function and neuron output. ANN is a feed-forward network with different layers such as input, output and hidden layers [60, 61]. The learning algorithm is Levenberg–Marquardt with backpropagation, while hyperbolic tangent sigmoid is chosen as the activation function.

The CFD study of the coupled TEG and wavy cooling channel system is conducted for various values of sinusoidal channel parameters such as varying amplitude of the wavy form and NP solid volume fraction. In total, 500 different cases from high-fidelity CFD are used, while the range of amplitude is between 0 and 0.01 and SVF range is taken between 0 and 0.03. These are used as the input to the network and top side of TEG, which is the interface between the TEG and PV, is considered as the output. Data are separated into training, validation, and testing samples of 70\(\%\), 15\(\%\) and 15\(\%\), respectively. Schematic view of the network architecture with different layers and input/output data sets is presented in Fig. 17.

Different number of neurons are tested, while 10 neurons are used in the hidden layer. The weights are updated during training, and network performance is shown in Fig. 18a. The MSE (mean square error) is reduced with higher epochs, while at epoch 68 best results are obtained. Table 6 shows the MSE and R\(^2\) (coefficient of determination) for different data sets. These are their definitions:

$$\begin{aligned} \text {MSE}=\frac{1}{M} \sum _{i=1}^{M} \left( y^{\text {CFD}}_{i}-y^{\text {NNET}}_{i} \right) ^2, \end{aligned}$$
(26)
$$\begin{aligned} \text {R}^{2}=1-\frac{\sum _{i=1}^{M} \left( y^{\text {CFD}}_{i}- y^{\text {NNET}} \right) ^{2} }{\sum _{i=1}^{M} \left( y^{\text {CFD}}_{i}-{\bar{y}}^{\text {NNET}} \right) ^2} \end{aligned}$$
(27)

where total number of simulation data is denoted M and average value is represented by \({\bar{y}}\). The TEG+cooling channel system output from CFD and ANN model output comparisons are shown in Fig.  18b. The overall agreement and model accuracy are satisfactory.

Table 6 MSE and R\(^2\) for different data sets
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Table 7 Comparison of average PV-cell temperature variation with different SVF and Amp obtained by using fully coupled CFD model and proposed ANN-assisted model
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In the second stage, the temperature values from the ANN model are used as boundary conditions for the PV module where CFD computations are conducted. Computational time for the fully coupled TEG+wavy channel+PV is time-consuming. However, by using the proposed model which is assisted by the ANN, the computational time is reduced by a factor of 75. Table 7 shows the comparison of PV-cell temperature, which are calculated by using the fully coupled CFD and obtained by sing ANN-assisted 3D-PV model considering different amplitude of the wavy channel and NP solid volume fraction. The PV-cell temperature from fully coupled CFD is well captured by using the ANN-assisted PV system model. The method does not only provide the accurate PV-cell temperature, but it is also very computationally efficient which makes it very promising for different complicated cooling channel configurations. The TEG module operating and geometric parameters such as leg dimensions, shape of the legs, material of the TEG, arrangement of the legs can also be incorporated in the ANN model. The proposed model can be used considering different operating conditions such as effects of solar irradiation, wind velocity, and ambient temperatures that can easily be taken into account.

4 Conclusions

A novel cooling channel design is proposed for thermal management of coupled PV-TEG system. Different-shaped NPs are used in the water–alumina NF in the cooling channel, while a sinusoidal wavy form from the channel is considered. Some of the important outcomes can be listed as in the following:

  • The average panel surface temperature is reduced more when NF and larger corrugation amplitudes of the channel are used. As compared to flat channel, cell temperature drop of 1.88 \(^\text {o}\)C can be obtained by using wavy form at the highest corrugation amplitude. The TEG power increment amount with higher corrugation amplitude is achieved as 9.5\(\%\).

  • The average temperature and temperature uniformity are only slightly influenced by the frequency of the corrugation for both BF and NF situations, with Nf = 4 showing the best results.

  • When corrugated cooling channels are utilized, adding spherical-, brick-shaped, and cylindrical particles to BF causes temperature drops of 0.14 \(^\text {o}\)C, 0.5 \(^\text {o}\)C and 0.9 \(^\text {o}\)C. When utilized with corrugated channel, cylinder NPs exhibit the best decrease in PV surface temperature performance.

  • The variation in average PV temperature with NP loading (SVF) is roughly linear. When compared to flat cooling channels with BF, the average temperature for corrugated cooling channels 1.75 \(^\text {o}\)C, 1.8 \(^\text {o}\)C and 1.9 \(^\text {o}\)C is lower at SVF of 0, 0.01 and 0.03, respectively.

  • A temperature drop (TD) value of 43.2 \(^\text {o}\)C is achieved when just TEG is employed. Adding a flat cooling channel and using water as the cooling medium increases performance and raises the TD value to 57 \(^\text {o}\)C. Corrugation of the channel and the use of NF as a cooling medium produce the highest performance, resulting in a TD value of 59.8 \(^\text {o}\)C.

  • An efficient computational strategy is proposed to predict the PV-cell temperature by using feedforward ANN-based model for the cooling part of the coupled system. Computational time is reduced by a factor of 75 by using the proposed method as compared to fully coupled PV-TEG-cooling channel system.

The current study can be extended to include exergy analysis, different NF types, and different shapes of wavy form for cooling channel. Other aspects of NFs such as dispersal of NPs in base fluid, non-Newtonian fluid behavior and multi-phase modeling of NF can also be considered as some of the possible extensions of the present work. Experimental study is needed for better analyzing the shape effects of NPs in the base fluid, cost analysis and agglomeration kinetics of NPs on the long-term performance of the cooling system.