A hyper-optimisation method based on a physics-informed machine learning and point clouds for a flat plate solar collector

Abstract

This paper presents a new way to hyper-optimise a flat plate solar collector using a combination of regenerated point clouds, constructal theory, and physics-informed machine learning (PIML). The behaviour of the flat plate solar collector is studied as solar radiation and ambient temperature change, using both precise numerical simulation and PIML. The novel hyper-optimisation method integrates these two approaches to improve the performance of the solar collector. In this method, the volume of fluid and solid structure of the flat plate solar collector (FPSC) is transformed into point clouds based on constructal theory. The point clouds are then regenerated into a continuous and uniform 3D geometry using optimised parameters. To put the modified version of the flat plate solar collector (FPSC) into practice, a computational method is used to generate a training data set for machine learning, specifically for neural networks. After thoroughly verifying the computational results, the PIM is trained using the generated training data set. This study marked the first time that a regular computational method is replaced with PIML output to reduce the computational cost of prediction. In the second layer of calculation, a deep neural network is used to make predictions based on the outputs generated by PIML. Seven independent parameters are used to predict heat transfer and efficiency over time, including time, heat flux, mass flow rate, inlet temperature, number of pairs and clusters, and volume fraction of nanofluid, while 16 hidden layers and 63 learnable neurons are engaged in this prediction layer. The geometry matrix is redefined by constructal theory principles in a series of iteration loops to generate the first flat plate solar collector based on constructal theory (CTFPSC). The results indicated that the hyper-optimisation method could reduce calculation costs by 18.31% compared with the regular computational method. In addition, the results reveal that maximum outlet temperatures are possible when N_c > 3 and N_p> 5.

Introduction

Undoubtedly, the energy crisis has profoundly impacted every aspect of human life in the last decade, whether it is the economy [1]. As a result of our energy requirements, we had to look for new energy sources and conduct an energy efficiency audit. Considering the limitations associated with the use of traditional sources of energy, we are searching for new renewable energy sources and increasing our focus on efficiency [2]. Developing green energy requires new roles from governments to reduce reliance on fossil fuels [3]. Research has been conducted for over four decades on replacing fossil fuels with renewable energy sources [4]. Recent years have also seen an increase in the popularity of solar energy equipment such as solar air heaters (SAH)[5] and flat plate solar collectors (FPSC) [6]. According to a study by Verma et al. [7], it was demonstrated how flat plate solar collectors could be effectively utilised for water heating in both the domestic and industrial sectors. The results showed that the solar collector was highly efficient, with thermal efficiency improving by 21.94% when using a high Reynolds number (Re).

This highlights the potential of flat plate solar collectors as a viable solution for water heating in various applications. According to Moravej et al. [8], rutile TiO2–water nanofluids were evaluated as the fluid for use in an asymmetric flat plate solar collector (FPSC). Based on the results, TiO2–water nanofluids led to a higher thermal efficiency than pure water. Nanofluids can have positive effects on the overall thermal efficiency of solar collectors and are a potential benefit of this study. The increasing popularity of solar equipment, specifically solar collectors, led to optimisation techniques to improve efficiency and size reduction[9]. Solar collectors have been optimised using many techniques throughout the decades, including differential evolution [10], genetic algorithm (GA) [11], and machine learning (ML) [12]. Jeyadev et al. [13] investigated the potential of implementing differential evolution to optimise an FPSC. As a result, optimal performance occurs when Re, glass covers, and illumination are at the maximum value. Due to its modelling flexibility and fast convergence, differential evolution outperforms other evolutionary algorithms by comparing the results obtained through different algorithms. Compared with other techniques, they show that the differential evolution approach can achieve a high-quality resolution in a much shorter amount of time.

In his research, Kalogirou [14] used artificial neural networks (ANN) to predict the performance of flat plate solar collectors (FPSC). The results showed that when unknown data were presented to the networks, excellent predictions were obtained, demonstrating that the proposed method is effective for predicting the performance characteristics of FPSC. This approach has several advantages over conventional testing methods, including speed, simplicity, and the ability to learn from examples. Thermodynamic optimisation methods, such as entropy generation minimisation (EGM) [15] and constructal theory (CT) [16], may also offer a new perspective on the optimisation process. By considering these parameters, engineers and researchers can gain a deeper understanding of the thermodynamic performance of a system and identify areas for improvement. Energy and exergy analysis provides a comprehensive evaluation of the energy inputs and outputs of a system, while fluid shape configuration analysis considers the physical arrangement of components, and how it affects the overall efficiency of the system. Ganjehkaviri and Jaafar [17] highlighted the potential benefits of using constructal theory in the design and analysis of FPSC. By incorporating this theory, the authors achieved higher thermal efficiency and lower total annual cost compared to conventional FPSC. Ojeda and Messina [18] developed a solar collector with dendritic pipes filled with a nanofluid of alumina nanoparticles and water to enhance thermal energy harvest. The optimal size of the network was determined for minimal thermal resistance, and increasing the nanoparticle concentration resulted in higher outlet fluid temperatures.

Hyper-optimisation methods, including machine learning, optimise a system using algorithms such as neural networks and decision trees. Machine learning-based optimisation methods can be trained on large data sets to find the best solution to a problem [19, 20]. Physics-informed machine learning (PIML) is a method that combines machine learning algorithms and principles of physics to solve problems [21]. In this approach, the machine learning models are trained on a combination of physical laws and data, with the aim of capturing the underlying physical phenomena in the data [22,23,24]. The physics-informed aspect of the method allows the models to produce physically meaningful and interpretable results [25]. It has the potential to tackle problems that traditional physics-based approaches are unable to address, especially in cases where the underlying physical laws are unknown or incomplete. By incorporating physics into the machine learning process, PIML can produce more accurate and reliable results compared to purely data-driven methods [26].

In the present study, a hyper-optimisation method based on PIML and computational method, CT and GRA, were developed to open a new window to the effect of CT on FPSC over time. We demonstrated how PIML could be used in the prediction of thermal behaviour of FPSC over time. The present study utilises a combination of PIML and a numerical method to identify the best configuration for the flow of Al₂O₃ nanofluid and tube geometry throughout the day. The study aims to investigate the impact of changes in solar radiation and ambient temperature on the performance of a collector, whilst utilising the principles of CT to guide the optimisation process. This involves the integration of PIML and numerical simulations to generate new and optimised geometries. The thermal efficiency and other relevant parameters will be evaluated based on the boundary and environmental conditions. This methodology can be applied to enhance the design of thermal systems and analyse large data sets.

Problem statement

According to Fig. 1a, the thermal behaviour of CTFPSC was determined using a standard FPSC based on [27]. The FPSC is composed of a glass cover, an absorber plate, and a piping system (as shown in Fig. 1a). The system continuously pumps nanofluid (AL₂O₃, φ = 1) into the FPSC while it is exposed to uniform solar irradiation during the day (from 8:00 am to 3:00 pm). The temperature distribution within the FPSC is influenced by various factors, including the properties of the materials used, such as thermal conductivity, the velocity of the nanofluid, and the boundary conditions. The design of the optimal geometry for the FPSC is crucial because it can impact the thermal behaviour and pressure drop of the nanofluid flow. The goal is to achieve maximum heat transfer while minimising pressure drop. Figure 1a depicts two concepts in the CTFPSC, pairs and clusters. Pairs are the first branches of the main tube and are characterised by a constant mass flow rate due to the constant diameter of the tube. To improve thermal performance and reduce the pressure drop, the pairs are transformed into clusters, as a result of the L-shaped configuration of the fluid and shape. The study considered different combinations of the number of pairs (1 < Np < 10) and the number of clusters (10 < Nc < 30) and divided the ranges into two groups. The results of the numerical simulation were analysed for the range (1 < Np < 10 and 1 < Nc < 10), while the machine learning layer was used to predict the outcome based on the results obtained from PIML. A wide range of combinations of pairs and clusters, volume fraction, and mass flow rate were studied, as shown in Table 1. The prediction of the behaviour of the parameters for solar irradiation and ambient temperature is a function of time during the prediction process. The CT can be used to define a new geometry for the FPSC by considering the minimisation of pressure drop and the maximum interaction between fluid and solid, as guided by the GRA method [28]. The application of CT principles to the geometry of FPSC can lead to the development of a new optimised geometry[16]. To reach the optimum geometry, these steps are required to transfer FPSC to CTFPSC.

• 1. Define the fluid and solid matrix based on the generated grid for the fluid and tube zones.

• 2. Define limitation functions such as CT and physical conditions.

• 3. Using iteration methods and a convergence loop based on a deformable of boundaries (limitation),

• 4. Convert the optimal results of the iteration loop to a point cloud (point cloud)

• 5. Transform a cloud point into a geometry matrix (fluid and solid zones).

• 6. Generate an optimised geometry.

Fig. 1

a Comparison between FPSC and CTFPST, b CTFPSC types and assumptions, and c Detail of geometry regeneration algorithm

Table 1 Ranges of optimisation and data mining

Additional details for the point cloud method are available in [29]. A point cloud is a collection of data points arranged spatially. The points could represent a three-dimensional shape or object. Each point position is associated with a distinct set of cartesian coordinates (X, Y, and Z)[30]. The STL file contains geometry information in the form of x, y, and z coordinates. Mathematical operations will be applied on the point cloud for both the fluid and solid zones. Figure 1b shows three combinations of Nc and NP used in this case study. During the calculation, various assumptions were made in order to compare different types of CTFPSCs, such as constant absorber surface areas in all types of CTFPSCs.

Figure 1c presents a data flow diagram for this study. Modified geometries can be generated from physical geometry (3D data set of FPSC tube) by GRA via conversion into a matrix of dimensions. This case study assumed that FPSC geometry was defined by five matrices (volume, external surface, internal surface, thickness matrix, and fluid matrix). The matrices can define all geometrical characteristics of FPSC. By considering the physical limitations and CT principles, a series of iteration loops convert the FPSC matrices into the modified matrices in GRA [31]. In this study, the initial data for the verification and training of PIML were provided by a commercial computational method (Ansys Fluent). It's worth mentioning that computational methods are used in conjunction with PSC. After thorough verification, the results from the commercial method were utilised by the GRA method to regenerate the initial geometry, taking into account the CT algorithm. This generated the CTFPSC, and the computational method then used this new geometry to provide comparison and training data for PIML. During the PIML stage, the goal is to comprehend the relationship between outputs and inputs. To enhance the FPSC convergence slope, additional information, such as boundary conditions, will be added to the calculation process due to the specifications and characteristics of PIML.

Considering the limitations and CT principles, a modified matrix can be converted into a solid body with high accuracy. These methods reduce the computation cost for generating solid domains by a substantial amount. The GRA can also be optimised to maximise heat transfer and minimise pressure drop (maximum pressure drop). For each configuration, GRA transfers data to PIML via the tensor-based data transport protocol (TBDTP) to calculate heat transfer and pressure drop. Big data analysis is time-consuming due to data transfer delays[32].

This case study uses a tensor-based data transfer method inside the TBDTP [33] due to the large size of the generated data for each parameter based on time and location. Using this method, GRA and PIML work in real time. The number of regeneration loops is determined by Nc, Np, and additional limitations. Fluid flow external diameter, tube internal diameter, and tube thickness are the convergence indices for all types of Np and Nc configurations (residual < $1\times {10}^{-3}$). This case study examined five distinct types of limitations.

• Constructal theory

• Physical constraints such as fluid and solid zone overlap

• Continues geometry

• Interface between fluid and solid zone

• Constant external dimension of CTFPSC

These constraints reform the point cloud, resulting in the generation of new geometry. Appendix A1 provides more details regarding this method. The initial dimension of FPSC is based on [27]. For specific configurations (Np = 6 and Nc = 12), the dimension of CTFPSC is presented in Table 2 as an example. As a result of the GRA, FPSC's dimensions and geometry will be changed to CTFPSC, but the external geometry will remain the same. Nanofluid flows inside a tube in an unsteady state. There were minimal changes in nanofluid properties during this period. Additionally, the thermal conductivity and viscosity of nanofluids are determined by temperature. Simulating nanoparticles (Al₂O₃ $,\varphi =1$) using a multiphase approach is used in this case study.

Table 2 Comparing the dimensions of FPSCs and CTFPSCs

Governing equation

This case study is developed based on transient, multiphase flow along the day. In addition, we considered the (liquid–particle) model for a nanofluid. The continuity equation for a multiphase is defined as follows in this case study [34]:

$$\frac{\partial \left({\alpha }^{{\text{m}}^{*}}{\rho }^{{\text{m}}^{*}}\right)}{\partial t}+\frac{\partial \left({\alpha }^{{\text{m}}^{*}}{\rho }^{{\text{m}}^{*}}{u}^{{\text{m}}^{*}}\right)}{\partial x}+\frac{\partial \left({\alpha }^{{\text{m}}^{*}}{\rho }^{{\text{m}}^{*}}{v}^{{\text{m}}^{*}}\right)}{\partial y}+\frac{\partial \left({\alpha }^{{\text{m}}^{*}}{\rho }^{{\text{m}}^{*}}{w}^{{\text{m}}^{*}}\right)}{\partial z}={S}_{{\text{m}}^{{\text{m}}^{*}}}^{\mathrm{int}}$$

(1)

where ${\rho }^{{m}^{*}}$ is density of mth phase and t is time. For multiphase flow, the momentum equations in each direction are defined as follows:

$$\frac{\partial \left({\alpha }^{{\text{m}}^{*}}{\rho }^{{\text{m}}^{*}}{u}^{{\text{m}}^{*}}\right)}{\partial t}+\frac{\partial \left({\alpha }^{{\text{m}}^{*}}{\rho }^{{\text{m}}^{*}}{u}^{{\text{m}}^{*}}{u}^{{\text{m}}^{*}}\right)}{\partial x}+\frac{\partial \left({\alpha }^{{\text{m}}^{*}}{\rho }^{{\text{m}}^{*}}{v}^{{\text{m}}^{*}}{u}^{{\text{m}}^{*}}\right)}{\partial y}+\frac{\partial \left({\alpha }^{{\text{m}}^{*}}{\rho }^{{\text{m}}^{*}}{w}^{{\text{m}}^{*}}{u}^{{\text{m}}^{*}}\right)}{\partial z}=\frac{\partial }{\partial x}\left[{\alpha }^{{\text{m}}^{*}}\left({\mu }^{{\text{m}}^{*}}+{\mu }_{\text{T}}^{{\text{m}}^{*}}\right)\frac{\partial {u}^{{\text{m}}^{*}}}{\partial x}\right]+\frac{\partial }{\partial y}\left[{\alpha }^{{\text{m}}^{*}}\left({\mu }^{{\text{m}}^{*}}+{\mu }_{\rm T}^{{\text{m}}^{*}}\right)\frac{\partial {u}^{{\text{m}}^{*}}}{\partial y}\right]+\frac{\partial }{\partial z}\left[{\alpha }^{{\text{m}}^{*}}\left({\mu }^{{\text{m}}^{*}}+{\mu }_{\text{T}}^{{\rm m}^{*}}\right)\frac{\partial {u}^{{\text{m}}^{*}}}{\partial z}\right]+{S}_{{\text{u}}^{{\text{m}}^{*}}}^{{\text{m}}^{*}}$$

(2)

$$\frac{\partial \left({\alpha }^{{\text{m}}^{*}}{\rho }^{{\text{m}}^{*}}{v}^{{\text{m}}^{*}}\right)}{\partial t}+\frac{\partial \left({\alpha }^{{\text{m}}^{*}}{\rho }^{{\text{m}}^{*}}{u}^{{\text{m}}^{*}}{v}^{{\text{m}}^{*}}\right)}{\partial x}+\frac{\partial \left({\alpha }^{{\text{m}}^{*}}{\rho }^{{\text{m}}^{*}}{v}^{{\text{m}}{*}}{v}^{{\text{m}}^{*}}\right)}{\partial y}+\frac{\partial \left({\alpha }^{{\text{m}}^{*}}{\rho }^{{\text{m}}^{*}}{w}^{{\text{m}}^{*}}{v}^{{\text{m}}^{*}}\right)}{\partial z}=\frac{\partial }{\partial x}\left[{\alpha }^{{\text{m}}^{*}}\left({\mu }^{{\text{m}}^{*}}+{\mu }_{\text{T}}^{{\text{m}}^{*}}\right)\frac{\partial {v}^{{\text{m}}^{*}}}{\partial x}\right]+\frac{\partial }{\partial y}\left[{\alpha }^{{\text{m}}^{*}}\left({\mu }^{{\text{m}}^{*}}+{\mu }_{\text{T}}^{{\text{m}}^{*}}\right)\frac{\partial {v}^{{\text{m}}^{*}}}{\partial y}\right]+\frac{\partial }{\partial z}\left[{\alpha }^{{\text{m}}^{*}}\left({\mu }^{{\text{m}}^{*}}+{\mu }_{\text{T}}^{{m}^{*}}\right)\frac{\partial {v}^{{\text{m}}^{*}}}{\partial z}\right]+{S}_{{\text{v}}^{{\text{m}}^{*}}}^{{\text{m}}^{*}}$$

(3)

$$\frac{\partial \left({\alpha }^{{\text{m}}^{*}}{\rho }^{{\text{m}}^{*}}{w}^{{\text{m}}^{*}}\right)}{\partial t}+\frac{\partial \left({\alpha }^{{\text{m}}^{*}}{\rho }^{{\text{m}}^{*}}{u}^{{\text{m}}^{*}}{w}^{{\text{m}}^{*}}\right)}{\partial x}+\frac{\partial \left({\alpha }^{{\text{m}}^{*}}{\rho }^{{\text{m}}^{*}}{v}^{{\text{m}}^{*}}{w}^{{\text{m}}^{*}}\right)}{\partial y}+\frac{\partial \left({\alpha }^{{\text{m}}^{*}}{\rho }^{{\text{m}}^{*}}{w}^{{\text{m}}^{*}}{w}^{{\text{m}}^{*}}\right)}{\partial z}=\frac{\partial }{\partial x}\left[{\alpha }^{{\text{m}}^{*}}\left({\mu }^{{\text{m}}^{*}}+{\mu }_{{\text{T}}}^{{\text{m}}^{*}}\right)\frac{\partial {w}^{{\text{m}}^{*}}}{\partial x}\right]+\frac{\partial }{\partial y}\left[{\alpha }^{{\text{m}}^{*}}\left({\mu }^{{\text{m}}^{*}}+{\mu }_{\text{T}}^{{\text{m}}^{*}}\right)\frac{\partial {w}^{{\text{m}}^{*}}}{\partial y}\right]+\frac{\partial }{\partial z}\left[{\alpha }^{{\text{m}}^{*}}\left({\mu }^{{\text{m}}^{*}}+{\mu }_{\text{T}}^{{\text{m}}^{*}}\right)\frac{\partial {w}^{{\text{m}}^{*}}}{\partial z}\right]+{S}_{{\text{w}}^{{\text{m}}^{*}}}^{{\text{m}}^{*}}$$

(4)

The multiphase model's (mixture's) energy equations are written as follows [35]:

$$\frac{{\partial \left( {\rho ^{\text{m}} H^{\text{m}} } \right)}}{{\partial t}} + \nabla\cdot \left( {\rho ^{\text{m}} U^{\text{m}} w^{\text{m}} } \right) = \nabla\cdot \left( {\lambda ^{\text{m}} \Delta T^{\text{m}} } \right) - \nabla\cdot q_{\text{H}}^{{\mathop {\mathop m\limits^{{\prime }} }\limits^{{\prime }} }} + \zeta - \nabla\cdot \sum\limits_{{{\text{m}}^{*} = 1}}^{2} {\left( {\alpha ^{{{\text{m}}^{*} }} \rho ^{{{\text{m}}^{*} }} U^{{{\text{dr}},{\text{m}}^{*} }} H^{{{\text{m}}^{*} }} } \right)}$$

(5)

$${\rho }^{\text{m}}=\sum_{{m}^{*}=1}^{2}\left({\alpha }^{{\text{m}}^{*}}{\rho }^{{\text{m}}^{*}}\right)$$

(6)

$${\lambda }^{\text{m}}=\sum_{{m}^{*}=1}^{2}\left({\alpha }^{{\text{m}}^{*}}{\lambda }^{{\text{m}}^{*}}\right)$$

(7)

The $k-\varepsilon$ turbulence model is used to solve the heat transfer and fluid mechanics problems in this study. $k-\varepsilon$ equations are considered as follows [36]:

$$\frac{\partial }{\partial {x}_{\text{i}}}\left(\rho\cdot k.{u}_{\text{i}}\right)+\frac{\partial }{\partial t}\left(\rho\cdot k\right)=\frac{\partial }{\partial {x}_{\text{j}}}\left(\left(\mu +\frac{{\mu }_{\text{t}}}{{Q}_{\text{k}}}\right)\frac{\partial k}{\partial {x}_{\text{j}}}\right)+{G}_{\text{k}}+{G}_{\text{b}}-\rho\cdot \varepsilon -{Y}_{\text{M}}+{S}_{\text{K}}$$

(8)

$$\frac{\partial }{\partial {x}_{\text{i}}}\left(\rho\cdot k.{u}_{\text{i}}\right)+\frac{\partial }{\partial t}\left(\rho\cdot \varepsilon \right)=\frac{\partial }{\partial {x}_{\text{j}}}\left(\left(\mu +\frac{{\mu }_{\text{t}}}{{Q}_{\varepsilon }}\right)\frac{\partial \varepsilon }{\partial {x}_{\text{j}}}\right)+{C}_{1\varepsilon }\frac{\varepsilon }{k}\left({G}_{\text{K}}+{C}_{3\varepsilon }{G}_{\text{b}}\right)+{C}_{2\varepsilon }\rho \frac{{\varepsilon }^{2}}{k}+{S}_{\varepsilon }$$

(9)

In this case study, different types of non-dimensional numbers, such as the Nusselt number (Nu) and thermal efficiency ($\eta$), are defined as follows [37]:

$$Nu=\frac{{h}_{\mathrm{fluid}-\mathrm{tube}}{L}_{\mathrm{total}}}{{\lambda }_{\mathrm{nano\,\,fluid\,\, at }{\text{T}}={\text{T}}_{\text{i}}}}$$

(10)

$$\eta =\frac{{Q}_{\mathrm{U}}}{{A}_{\mathrm{C}}{G}_{\mathrm{t}}}$$

$${Q}_{\mathrm{U}}={F}_{\mathrm{R}}{A}_{\mathrm{c}}\left({G}_{\mathrm{t}}\left(\tau \alpha \right)-{U}_{\mathrm{l}}\left({T}_{\text{i}}-{T}_{\mathrm{a}}\right)\right)$$

(11)

Physics-informed machine learning

Dynamic systems can be used to represent fluid behaviour in patterns and predictions. The Navier–Stokes equations are the outcome of empirical and semi-empirical laws of force boundary interactions. Various numerical methods have been used to solve Navier–Stokes equations over the years [38]. In recent years, machine learning has provided a new perspective for understanding parameter behaviour and finding the hidden relationship between inputs and outputs [39]. PIML was developed based on differentiable physics (PD) [40]. Therefore, domain knowledge is converted into model equations, and then, discretised versions of these models are integrated into the training process. Figure 2a describes the procedure of PIML. To support neural network training, it is essential to have differentiable formulations, as implied by the name. It is necessary to find a function (f*) which generates solutions from a space (Y) taking inputs from a space (X), i.e. f*: X → Y. The differential equation (*: Y → Z) encodes a property of the solution, such as some physical property [41].

Fig. 2

Unfolded form of customised physics-informed machine learning

All approaches are either forward target simulation [42] (predicting state or temporal evolution) or inverse problems [24]. DL techniques that use domain knowledge, usually in the form of partial differential equations (PDEs), are the key to separating the following topics, whether the problem is forward or backward. Based on input parameters such as temperature and heat flux, PIML attempts to present a pattern for (f*) in the space of the FPSC and CTFPSC. Using a modified version of the Navier–Stokes equation in this section also reduces the calculation cost and presents the pattern between input and output depending on real physics. Automatic differentiation plays a crucial role in the core of PIML for solving PDEs [43].

Automatic differentiation is a method in computational mathematics that allows for the efficient computation of the derivative of a given function. It involves the use of algorithms that automatically generate the derivative of a function without the need for manual differentiation by a user. This technique is useful in various applications such as optimisation, machine learning, and numerical simulations, where derivatives are required for gradient-based methods. Automatic differentiation differs from symbolic and numerical differentiation, where the derivatives are calculated symbolically or numerically, respectively. In automatic differentiation, the derivative information is obtained by tracing the operations performed during the function evaluation, making it more efficient and accurate compared to other methods.

The mathematical background and details of PIML are presented in [22].

Optimisation loop

A significant advantage of the PIML approach to the FPSC is investigating various initial and boundary conditions to determine the optimal condition. By using many results, we can determine the optimal boundary conditions for each boundary, such as the optimal Nc and Np at each mass flow rate and volume fraction. Furthermore, GRA allows us to account for overdesign as well. Using data mining methods [44], we determine for the first time the minimum external geometry of CTFPSC when the targets (thermal efficiency and outlet temperature) are equal to the original FPSC under different boundary conditions (mass flow rate and nanoparticle volume fraction). Figure 3 demonstrates how four distinct layers of calculation were used to determine the variation in the input parameters affecting thermal efficiency and outlet pressure. In the model, each layer is created based on a single input, while the boundary conditions are constant. In order to minimise the effect of input variation on targets (outlet temperature and thermal efficiency), different combinations of constant parameters were considered for each input. According to the CTFPSC's manufacturing capabilities, the range of inputs and outputs was defined. As increasing Nc and Np decrease CTFPSC's practicality, these values were assumed for pattern recognition. Increasing Nc and Np increase CT's contact surface between the surface absorber and the tube and fluid network. As a result, the capability of thermal efficiency and outlet temperature while the external dimensions are smaller than the original FPSC will be studied.

Fig. 3

Optimisation algorithm based on data mining and PIML method

Machine learning and deep learning often use gradient descent as an optimisation method [45]. An algorithm that minimises a cost function by updating model parameters in accordance with the negative gradient of the cost function. A negative gradient indicates the direction of the steepest decline in the cost function or the direction of the greatest decrease in the cost function at the current point. Gradient Descent is a simple and effective optimisation algorithm that has several key features:

• 1. Iterative Optimisation: Gradient Descent is an iterative optimisation algorithm that updates the model parameters in small steps until the cost function reaches a minimum.

• 2. The direction of Descent: Gradient Descent updates the parameters in the direction of the negative gradient of the cost function, which points towards the steepest descent.

• 3. Local Minima: Gradient Descent is prone to getting stuck in local minima, which can result in suboptimal solutions. To mitigate this issue, techniques such as momentum or Nesterov Accelerated Gradient can be used.

Gradient Descent is also used in PIML to optimise the parameters of a neural network that has been trained to solve a physics-based problem. In PIML, the neural network is trained to approximate the solution of a partial differential equation (PDE) that describes a physical phenomenon. The training process involves minimising the difference between the neural network predictions and the known physics-based constraints [46].

Boundary conditions and limitations of geometry:

Figure 4 illustrates how the radiation heat flux and ambient temperature change with time [27]. In the code, the updated values of solar radiation and ambient temperature are automatically imported at each time step. This feature significantly enhances the accuracy and precision of the results. By updating the values of these parameters at each time step, the code can more closely approximate the real-world conditions and provide a more accurate simulation. The solar radiation input influences the calculation of the absorber plate's temperature, which is performed using the Fourier equation. Alternatively, the ambient temperature influences the convective heat transfer in the FPSC or CTFPSC. Table 3 displays the initial and boundary conditions for the FPSC. In this study, all cases were assumed to have a no-slip boundary condition, which means that the fluid velocity at the boundaries is zero. In addition, solar radiation was modelled as a flow of heat energy, as it was treated as a heat flux. This information is used in the calculation of the temperature of the absorber plate and the convective heat transfer in the FPSC or CTFPSC.

Fig. 4

Solar radiation and ambient temperature variations throughout the day [27]

Table 3 Boundary and initial condition

The application of CT to cylindrical tubes necessitates multiple correction loops due to the correct fit between the outer surface of the fluid and the inner surface of the tube. These correction loops are required to account for CT's limitations in geometry matrices. These restrictions include pairs and clusters that generate new wall boundary conditions for continuous walls. These loops attempt to define new boundaries within the regeneration domain, and the convergence condition is to maintain constant thickness and diameter across all calculation domains. Appendix A2 contains deviation from the original geometry.

Fluid properties

Al₂O₃/water nanofluid was considered in the analysis. Its thermophysical properties are shown in Table 4. This table shows the thermal conductivity and viscosity ratios for the range of volume fractions considered [47].

Table 4 Correlation of thermal conductivity and viscosity

Grid study and validation

Grid study and validation are important steps in the numerical simulation of fluid flows to ensure that the simulation results are accurate and reliable. The grids were analysed and modified based on Nc and Np to ensure an accurate representation of the physical system in the simulation, accounting for its unique characteristics and producing precise results.

Independency analysis of grids and modification loops

Grid quality significantly affects simulation accuracy, but increasing grids can increase costs and introduce errors. Figure 5a and b shows that the average outlet temperature of water is unaffected by varying Nc/Np. The results indicate that the average outlet water temperature is unaffected by varying Nc and Np values. In addition, the results of pressure gradient calculations based on clustering loops were analysed. Figure 5c illustrates the residual standard deviation (RSD) for two distinct zones (tube volume and internal surface) as a function of the number of optimisation loops. To attain a uniform and continuous new geometry, over 3000 optimisation loops were required (CTFPSC).

Fig. 5

Grid study for a a variation of outlet temperature based on different Np and Nc at 11:00 am, b pressure gradient at 11:00 am, c residual standard deviation of tube geometry and internal surface during optimisation loops, and d diameters ratio of the fluid zone and tube

The internal and external diameters of the tube and fluid zone were kept constant in this case study to compare FPSC and CTFPSC. As a result of the inherent nature of this method (iteration of optimisation on a series of matrices), the outer diameter of the fluid zone and the internal diameter of the tube must be the same. Figure 5d shows the variation of this ratio (D _{fluid zone, CTFPSC}/ D _{fluid zone, FPSC}) during optimisation.

Numerical simulation validation

For numerical simulation validation, the outlet temperature and pressure drop results were compared with Hawwash et al. [27]. The results indicated that the average error for numerical simulation is about 6.78% (Fig. 6).

Fig. 6

Comparison of results of the numerical study and Hawwash et al. [27] for a outlet temperature and b pressure drop during the day

Neuron sensitivity analysis for PIML

Neuron sensitivity analysis refers to the process of analysing how changes in the input of a single neuron in a neural network affect the output of the network. This analysis is used to identify which neurons have the most significant impact on the output of the network, which can be used to better understand the network's behaviour and improve its performance. Neuron sensitivity analysis can also be used to identify neurons that are redundant or unnecessary, which can help to simplify the network and reduce its computational complexity. The quality of results in PIML is highly dependent on input, hidden layers, and training data quality. Table 5 shows how hidden layer variation affects results based on MAE and R². To achieve an MAE of 0.0017 and an R² of 0.9993 for 11 input parameters, 210 layers of calculation are required for measuring CTFPSC thermal efficiency at 12:00 am. While the number of hidden layers can increase calculation costs, finding an accurate model between layers could reduce maximum error. Additional details about the optimum geometry compared to the original FPSC are presented in Appendix A2.

Table 5 Comparison of the effect of the number of hidden layers on the results (thermal efficiency)

Appendix A3 contains additional information about the verification process.

Results and discussion

The contact surface area between the heated tube and fluid is a key factor that determines how effectively the heat is transferred from the solar collector to the fluid. Increasing the contact surface area by using a tube with a larger diameter or by incorporating fins or other features that increase the surface area can lead to improved efficiency. The flow network of the collector is important, as the combination of pairs and clusters can affect heat transfer and pressure drop. The collector's thermal behaviour will also vary throughout the day due to changes in solar radiation and ambient temperature.

Thermal behaviour of FPSC and CTFPSC

The temperature at the outlet of a solar collector is influenced by the surface area between the heated tube and fluid flow, properties of the nanofluid, and turbulence intensity. Np and Nc have a direct relationship in generating temperature patterns and vorticities inside the tube. Figure 7 shows different CTFPSC configurations based on outlet temperature with nanoparticles. Results indicate that the maximum outlet temperature is achieved when Np > 4 and Nc > 8, resulting in a 12.86% average outlet temperature increase compared to FPSC. Moreover, the pattern of outlet temperature did not differ significantly between FPSC and CTFPSC. As a result, when Np = 6 and Np = 12, the average Nu will increase by 13.46%. This means that for values of Np and Nc below 3 and 6, respectively, there is a significant reduction in the average outlet temperature in the CTFPSC compared to the FPSC. Specifically, when Np is equal to 3, 2, and 1, the contact surface between the nanofluid and tube is reduced by 8.65%, 12.87%, and 23.14%, respectively, which leads to a reduction in the outlet temperature. Based on the results, it appears that the outlet temperature pattern will remain constant when the value of Np is greater than 6. This suggests a threshold value of Np beyond which the outlet temperature will not vary significantly.

Fig. 7

Variation of outlet temperature over time for a AL₂O₃, $\boldsymbol{\varphi }$=1%, and b pure water

Furthermore, it appears that the behaviour of heat transfer is strongly influenced by velocity. This means that changes in velocity can result in variations in heat transfer. Additionally, changes in Np and Nc can affect pressure drop and velocity, as indicated by the Bernoulli equation. For instance, a comparison between Np = 10 and FPSC shows that the average velocity can drop by -7.11%. The outcomes provided suggest that the relationship between pressure, velocity, and geometry in the system is affected by turbulence intensity. Changes in the system's geometry can alter the fluid flow and increase the turbulence intensity. The number of 90-degree bends in the fluid flow can impact the turbulence intensity, which can further affect the system's performance. Increasing the values of Np and Nc can cause an escalation in the average turbulence intensity, which can affect the system's efficiency. Moreover, increasing Np and Nc can also result in an increase in the pumping power required to maintain the fluid flow and cause a rise in the system's pressure drop. Therefore, while making changes to the system's geometry or values of Np and Nc, it is important to consider the potential impact on turbulence intensity, pumping power, and pressure drop.

Figure 8b compares different configurations of Np and Nc when pure water is used as the working fluid. According to Fig. 8a, the behaviour of the outlet temperature remains constant after Np reaches a value of 6. This indicates that increasing Np beyond this value may not lead to further improvements in outlet temperature value. Figure 8b, on the other hand, shows the variation in thermal efficiency with respect to different values of Np. The results suggest that the use of a CTFPSC can improve thermal efficiency by up to 16.51% when Np is greater than 6. This suggests that the use of nanofluids can lead to significant improvements in thermal efficiency under certain conditions. However, the study also found that when Np is less than 6, the thermal efficiency of the system decreases dramatically. This is due to the reduced contact surface area between the tube and fluid, which can limit the heat transfer between the fluid and the tube. Therefore, it is important to carefully consider the choice of Np when designing a system that uses nanofluids to ensure optimal thermal performance.

Fig. 8

a Variation of pressure drop and outlet temperature at different Np and thermal efficiency variation

Nanofluids can improve heat transfer and thermal efficiency. Table 6 provides additional information about the effects of nanoparticles. Table 6 shows that nanoparticles influence pressure drop, heat transfer, average velocity, and turbulence intensity. Based on the results, the average variation of pressure drop ratio and outlet temperature with respect to different ranges of nanoparticle volume fraction over a day. The study found that increasing the value of Np can increase the outlet temperature ratio by up to 10% when compared to a FPSC. However, the results demonstrated that increasing Np could result in a significant increase in pressure drop. When Np reaches 10, the pressure drop was observed to increase by 30%.

Table 6 Study of the effect of adding nanoparticles on different parameters while Nc = 2 $\times$ Np

Predicting CTFPSC heat transfer and fluid behaviour using the DNN layer

The optimal condition is to achieve maximum thermal efficiency while minimising pressure drop. This is because a high thermal efficiency means that the device is able to convert a greater proportion of the input energy into the desired output, whether that is heat transfer or some other form of energy. Minimising pressure drop, on the other hand, is important because it reduces the energy required to pump fluid through the device, which can be a significant source of inefficiency. Figure 9 shows a comparison of different parameters as Np varies, while Nc is kept constant at 2. The parameters being compared could include thermal efficiency, average velocity, flow rate, temperature distribution, and other relevant variables related to thermal behaviour and flow characteristics. It is important to note that when Np is increased, the average velocity of the particles in the system is also increased, which can have a significant impact on the system's overall behaviour.

Fig. 9

a Variation in Nusselt number ratio with Np and internal surface area ratio, and b pressure drop and turbulence ratio with Np

This is because higher velocities can result in greater heat transfer rates, as well as increased fluid flow and turbulence, which can affect factors such as pressure drop, mixing, and heat distribution. The exact predicted thermal behaviour and flow characteristics for different combinations of Np and Nc will depend on the specific system and its operating conditions. However, by studying the variations in these parameters, it may be possible to optimise the system's performance and identify the most efficient and effective operating conditions. Based on the results, increasing Np can increase the internal surface for CTFPSC up to six times compared to FPSC. Figure 9a shows that increasing the contact surface between fluid and tube while reducing the average velocity by increasing Np could increase the Nu ratio. The pairs determine the pattern of escalating the intensity of turbulence and pressure drop in the first branch.

Figure 9a illustrates that increasing Np could affect flow patterns, especially near the outlet. Since the absorber plate is connected to the tube, the temperature distribution is non-uniform, which ensures a temperature gradient for heat transfer. In Fig. 9b, increasing Np could increase turbulence intensity by 6.5 times for CTFPSC compared to FPSC. Figure 9a and b indicates that the optimal value for Np is 6, as exceeding this value results in a significant increase in pressure drop despite increased heat transfer. One must understand the target behaviour patterns at various boundary conditions to find the optimal combination of Nc and Np.

The effect of variation of Nc and Np on flow pattern and heat transfer (DNN results)

Big data analysis in neural networks involves training neural networks on large data sets to learn patterns and make accurate predictions. With a large amount of data, neural networks can detect complex relationships between input and output variables, which may not be apparent with smaller data sets. Big data analysis can involve a variety of techniques, including deep learning. These techniques can handle a large number of input features and can be used for a wide range of applications, such as image classification, natural language processing, and anomaly detection. To train neural networks on big data, specialised hardware such as graphics processing units (GPUs) and tensor processing units (TPUs) may be used to accelerate computations. In addition, data preprocessing and feature engineering are critical steps in big data analysis, as they can affect the quality and accuracy of the trained model. Overall, big data analysis in neural networks can provide powerful tools for solving complex problems and making accurate predictions. If we deal with a large amount of data and the variation in Np and Nc is significant, then using traditional methods to analyse the data can be time-consuming and computationally intensive. However, there are several strategies and techniques that can be used to make the analysis more efficient and manageable such as parallel computing, use data compression techniques, and deep neural methods.

PIML analysed over 186 million data points in 63 series to determine the optimum range for Np and Nc. Figure 10 presents the results of PIML in terms of variation of both Np and Nc. Figure 10a shows the outlet temperature of CTFPSC in different combinations of Nc and Np, compared to FPSC, when solar radiation and ambient temperature are functions of time. In the results, a maximum outlet temperature is achieved when Np exceeds 5 and Nc > 3. In cases where there is a wide range of Nc (3 < Nc < 30), Np > 5 provides the maximum outlet temperature. The results reveals that the compound CTFPSC is capable of providing outlet temperatures that are 1.025–1.455 times higher than those achieved by the FPSC in the range of 3 < Nc < 30 and Np > 6.

Fig. 10

The effect of variation of Nc and Np on a outlet temperature and b thermal efficiency

This implies that the CTFPSC is generally more effective at converting solar energy into thermal energy than the FPSC, particularly when configured with a larger number of collector units in series and parallel. It's worth noting that the specific performance of a solar thermal system will depend on a wide range of factors, including the type of solar collector, the materials used in the system, the efficiency of the heat transfer fluid, and the design of the system as a whole. However, the results of this study suggest that in certain situations, a CTFPSC may be a more effective option for generating high outlet temperatures than a FPSC. A thermal efficiency analysis at Nc > 9 reveals that increasing Nc above this range does not affect thermal efficiency. The mathematical approach of increasing Nc can improve heat transfer by increasing the contact surface area between fluid and tube and absorber plate, but increasing Nc > 12 generates merged geometry between clusters due to physical limitations. Using the PIML results, it can be seen that CTFPSC reduces thermal efficiency and outlet temperature during the day for the low ranges of Np and Nc. When Np = 1 and Nc = 1, FPSC can increase outlet temperature by more than 58% compared with CTFPSC during the day. Figure 11a and b illustrates the optimal combination of Np and Nc based on average Nu and pressure drop. According to Fig. 11a, different combinations of Np and Nc can cause a different Nu ratio variation pattern. The maximum Nu ratio will be provided when Nc > 6 and Np > 5. Increasing Np plays a larger role in increasing Nu ratio than increasing Nc. In the high range of Nc and Np, the behaviour of Nu ratio is similar to that of thermal efficiency and outlet temperature.

Fig. 11

Effect of variation of Nc and Np on a Nu and b pressure drop (Pa)

In general, pressure drop in a fluid flow system is influenced by a number of factors, including the geometry of the system and the turbulence intensity of the fluid flow. The pressure drop can be thought of as the difference in pressure between two points in the flow system, and it is typically caused by the resistance of the fluid to flow through the system. Figure 11b shows the variation of pressure drop ratios as a function of Np and Nc. Increasing Np and Nc will change the pressure drop as a mathematical approach. Furthermore, nanofluid's pressure drop is a nonlinear temperature function due to the relation between temperature and viscosity. The pattern of the pressure gradient is affected by geometry and temperature-dependent viscosity.

Conclusions

The proposed approach in this study is aimed at developing a new generation of flat plate solar collectors (CTFPSC) based on constructal theory by employing physics-informed machine learning (PIML) and numerical simulation. The approach involves using a computational method to generate a training data set to determine the thermal behaviour of both the FPSC and the CTFPSC. After careful verification of the data set, a physics-based neural network is used to generate the thermal model of CTFPSC based on a constant number of pairs (Np) and clusters (Nc). Using a deep neural network layer, this model can predict the thermal behaviour and variation in pressure drop for specific ranges of Np and Nc. One novel aspect of this approach is the use of the point cloud method to regenerate new geometries for both the solid and fluid zones of the collector. The study investigates combinations of clusters and pairs while considering the variable solar radiation heat flux and ambient temperature throughout the day. The results indicate that:

• 1. The combination of numerical simulation and physics-based machine learning could affect calculation costs, the quality of results, and a number of predictions. In comparison with pure numerical simulations, PIML reduced calculation time by 18.31%.

• 2. According to the results, as Np is reduced from 3 to 2 to 1, the average contact surface area between the nanofluid and tube is reduced by -8.65%, 12.87%, and 23.14%, respectively. This suggests that the geometry of the collector system, as defined by the values of Nc and Np, has a significant impact on the contact surface area between the nanofluid and the tube, which, in turn, can affect the heat transfer performance of the system.

• 3. Based on the results, CTFPSC and Np > 6 and Nc > 10 increased thermal efficiency by up to 16.51%.

• 4. As a result, maximum outlet temperature is available if Np > 5 and Nc > 3. When Np > 5, the maximum outlet temperature will be offered for a wide range of Nc (3 < Nc < 30).

• 5. When comparing the results of PIML, it is clear that CTFPSC reduces thermal efficiency and outlet temperature during the day for low-range Np and Nc. The results indicate that for Np = 1 and Nc = 1, FPSC can increase outlet temperature by 58% during the day compared to CTFPSC

• 6. Based on the results of data mining, it's clear that for the maximum flow rate ($\dot{{\varvec{m}}}=$ 1 l/min) and $\boldsymbol{\varphi }$ =1, the absorber plate could be reduced by 21% without losing thermal efficiency.

Appendices

Appendix A1: geometry regeneration algorithm (point cloud method)

The point cloud method is a technique used in computer graphics, computer vision, and 3D modelling to represent three-dimensional objects as a collection of points in space. Point cloud data are typically generated by scanning an object using a 3D scanner or by extracting 3D information from multiple 2D images using photogrammetry. In the point cloud method, each point in the point cloud represents a single location in 3D space and may contain additional information such as colour, intensity, or surface normal. These points can be used to reconstruct the original 3D object and can also be used for various other tasks such as object recognition, scene analysis, and motion tracking [48, 49].

One advantage of the point cloud method is that it can represent objects of arbitrary shape and complexity without the need for a predefined mesh or surface representation. This makes it useful in applications such as archaeology, architecture, and geology, where the objects being studied may have irregular or complex shapes. One challenge with point clouds is that they can be very large and dense, requiring significant computational resources to process and analyse. As a result, various techniques have been developed to efficiently store and process points cloud data, such as octrees, kd-trees, and voxel grids. In this paper, we used pyntcloud (https://github.com/daavoo/pyntcloud) to convert surfaces to the point. One sample of codes is presented as follows:

import pyntcloud from pyntcloud import PyntCloud # Load the surface point cloud from a PLY file cloud = PyntCloud.from_file("surface.ply") # Create a mesh using Poisson surface reconstruction mesh = cloud.get_mesh("Poisson") # Save the mesh to a file in STL format mesh.to_file("mesh.stl")

In this code, PyntCloud.from_file is used to load the surface point cloud from a PLY file. To generate a volume point cloud for fluid in Python, you can use the pyvista library (https://docs.pyvista.org/index.html). Here's an example code:

import pyvista as pv # Create a surface mesh (you can load a surface from a file if needed) surface = pv.Sphere(radius = 1) # Generate a tetrahedral volume mesh from the surface mesh tet_mesh = surface.compute_implicit_distance().threshold(0.0).delaunay_3d() # Save the mesh to a file (you can use a different format if needed) tet_mesh.save('volume.vtk')

This code first creates a surface mesh (in this case, a sphere) using the ‘pyvista.Sphere’ function. Then, it generates a tetrahedral volume mesh from the surface mesh using the ‘compute_implicit_distance’ =  ‘threshold’ and ‘delaunay_3d’ methods. Finally, it saves the mesh to a file (in VTK format in this example) using the ‘save’ method. Note that the compute_implicit_distance method is used to ensure that the resulting mesh is watertight. This method computes the signed distance from every point in the mesh to the surface and sets the distance to zero at the surface. The ‘threshold’ method then removes all cells whose vertices have the same sign of the distance, leaving only the cells that intersect the surface. The ‘delaunay_3d’ method[50] then generates a tetrahedral mesh from the remaining cells.

Appendix A2: geometrical analysis of CTFPSC

Deviation from the original geometry of a solar collector system can have a significant impact on its performance. Changes to the geometry of the system, such as altering the dimensions of the collector tubes or the spacing between them, can affect the flow behaviour of the fluid and the heat transfer performance of the system. Table

Table A1 Deviation from the original geometry

A1 presents details of new geometry in comparison with original FPSC model.

Appendix A3: additional validation details

The effect of the number of hidden layers on the results of predictions in a neural network model depends on the complexity of the problem being solved and the size and quality of the available training data. In general, increasing the number of hidden layers in a neural network can increase the model's capacity to learn complex patterns in the data and improve its accuracy in making predictions. However, adding too many hidden layers can lead to overfitting [51], where the model becomes too specialised to the training data and does not generalise well to new data. Therefore, it is important to strike a balance between model complexity and generalisation when selecting the number of hidden layers in a neural network. This can be achieved by using techniques such as cross-validation and regularisation to evaluate the performance of the model on a validation set and prevent overfitting. Additionally, other factors such as the choice of activation functions, learning rate, and regularisation strength can also affect the performance of a neural network model, and these should be carefully tuned and optimised in conjunction with the number of hidden layers to achieve the best results. Figure 

Fig. A1

Comparison of the impact of a number of hidden layers on dimensionless temperature for a numbers of hidden layers (Nh) = 210 and b numbers of hidden layers (Nh) = 150

A1 presents a comparison of the impact of the number of hidden layers on results.

In machine learning, standard deviation is used as a measure of how much the data points in a particular feature vary from the mean of that feature. It is often used to measure the spread or dispersion of data in a distribution and can help identify outliers and anomalies. Table

Table A2 Standard deviation for Hawwash et al. [27] and numerical and PIML results in different numbers of hidden layers (Nh) based on pressure drop ($\boldsymbol{\Delta }$ P_t=i/$\boldsymbol{\Delta }$ P_max)

A2 presents details of standard deviation in this study.