Thermal Performance Estimation of Nanofluid-Filled Finned Absorber Tube Using Deep Convolutional Neural Network

Abstract

Numerical simulations are usually used to analyze and optimize the performance of the nanofluid-filled absorber tube with fins. However, solving partial differential equations (PDEs) repeatedly requires considerable computational cost. This study develops two deep neural network-based reduced-order models to accurately and rapidly predict the temperature field and heat flux of nanofluid-filled absorber tubes with rectangular fins, respectively. Both network models contain a convolutional path, receiving and extracting cross-sectional geometry information of the absorber tube presented by signed distance function (SDF); then, the following deconvolutional blocks or fully connected layers decode the temperature field or heat flux out from the highly encoded feature map. According to the results, the average accuracy of the temperature field prediction is higher than 99.9% and the computational speed is four orders faster than numerical simulation. For heat flux estimation, the R² of 81 samples reaches 0.9995 and the average accuracy is higher than 99.7%. The same as the field prediction, the heat flux prediction also takes much less computational time than numerical simulation, with 0.004 s versus 393 s. In addition, the changeable learning rate strategy is applied, and the influence of learning rate and dataset size on the evolution of accuracy are investigated. According to our literature review, this is the first study to estimate the temperature field and heat flux of the outlet cross section in 3D nanofluid-filled fined absorber tubes using a deep convolutional neural network. The results of the current work verify both the high accuracy and efficiency of the proposed network model, which shows its huge potential for the fin-shape design and optimization of nanofluid-filled absorber tubes.

1. Introduction

Solar energy, one of the clean energies with the largest source on the planet’s surface, has attracted increasing attention due to the enormous challenges of climate change and resource depletion. Concentrated solar power (CSP), one of the well-established solar energy technologies [1,2,3], has added almost 0.6 TWh of CSP generation from 2019 to 2020. However, based on the report of IEA, to achieve net zero power CSP generation of 204 TWh in 2030, the additions are much lower than required [4]. Therefore, huge efforts are urgently demanded to increase the efficiency of the CSP thermal system. Parabolic trough collectors (PTC), as one of the commercial CSP technologies [5,6,7,8], have become a research hotspot. The PTC commonly consists of mirrors with a curved shape and an absorber tube. The former serves to concentrate the solar energy on the latter, then the circulation inside the tube is heated up, and the generated steam turns a turbine for electricity. The absorber tube is usually made of steel, which is a good conductor of heat. To reduce its thermal loss, a concentric glass envelope is employed. Furthermore, the space between the glass envelope and the steel tube can be under a vacuum for better isolation.

Therefore, to enhance the heat transfer performance of the absorber tube, there are usually two ways: one consideration is to improve the conductivity of the working fluid, for which nanofluids are commonly used; the other concentrates on the shape modification of the tube to increase the turbulence and consequently to enhance the heat transfer, such as using inserts and fins. A huge volume of papers has been published to describe the PTC performance enhancement by using nanofluids [9,10,11,12,13,14]; the total number is 34,230 from 2010 to 2021 [9]. According to the review [12], the most significant nanofluids are 8% Al₂O₃/Syltherm 800, which can achieve an increment of 76% in heat transfer coefficient [15]. Furthermore, studies are not limited to mono nanofluids; hybrid nanofluids have also attracted scientists’ attention due to the superior thermal characteristics and stability [16,17,18,19], which are even better than mono nanofluids.

At the same time, a considerable amount of literature has been published to investigate the influence of various fins on the performance of PTC absorber tubes as well. Bellos et al. [20,21] reported two investigations of PTC absorber tubes, both with eight internal rectangular longitudinal fins around the tube. Twelve cases with different fin lengths and thicknesses were tested and compared with the smooth case. Based on various criteria, the optimal case is not the same. Unlike Bellos et al., Adel et al. [22] proposed the longitudinal fins in the lower half of the receiver tube, which receives most of the reflection of the sun. Twelve of the same cases with Bellos et al. were investigated, and the results showed that the fin with the length of 15 mm and the thickness of 6 mm is optimal. Another part of the literature proposed different shapes for fins, such as a triangle fin [23,24], a trapezoidal fin [24], pin fin arrays [25], and a fin coupled with insert [26]. These studies focused more on comparing the efficiency of new PTC tubes to the reference smooth tube and did not carry out the analysis on fin-shape parameters.

Most of the above studies investigated the influence of the nanofluids and fin-shape parameters on the thermal performance of absorber tubes. Therefore, many numerical simulations have been performed, which have required huge computational resources. Furthermore, this problem occurs in other parametrical studies as well [27,28]. Fortunately, these burden calculations could be avoided by profiting from the machine learning or deep learning methods, which show great potential in building a reduced-order model (ROM) for highly nonlinear problems.

Machine learning has been employed to predict the properties of mono or hybrid nanofluids [29,30,31,32,33] as well as the performance of thermal systems with nanofluids, including the Nusselt number, pressure drop, heat transfer coefficient of heat exchangers [34,35], and the outlet temperature, collector efficiency of solar energy systems [36,37]. It is worth noting that all these studies of thermal system prediction achieve high accuracy with R² > 0.99. Moreover, compared to conventional methods [38,39,40], estimation time is largely saved. Recently, several authors have investigated deep convolutional neural networks (CNNs) to predict the physical field of heat transfer systems due to their great capacity for accurate and quick image processing. Peng et al. [41,42] successively developed two deep CNN-based models to predict heat conduction and then steady-state heat convection around objects. They directly used the signed distance function image to present problem geometry, and set it as the input of the neural network model. The results show that the computational speed is one and two orders faster, respectively, than the numerical simulations, and the accuracies of the temperature field prediction are both higher than 98.75%. Furthermore, the same research group also applied deep CNN to predict the velocity field [43,44,45] around airfoils or different objects. When it comes to nanofluid field reconstruction using deep learning, not many articles are available. Liu et al. proposed a GAN network [46] and then a CNN network [47] to predict the global flow field in a two-dimensional nanofluid-filled microchannel; the results also showed the accuracy and efficiency of the proposed model.

In this study, we apply deep CNN to build two ROMs to predict the nanofluid heat transfer in a PTC absorber tube with rectangle fins of various heights, widths, and relative positions. The first ROM consists of the convolutional and deconvolutional layers, and the second ROM consists of the convolutional and fully connected layers. We set the SDF-presented cross-sectional geometry of the absorber tube directly into the convolutional neural network, and then the temperature field and outlet heat flux are predicted. The CFD method is used to generate the training, validation and testing dataset, and is validated with experimental data in literature. According to our literature review, this is the first study to estimate the temperature field and heat flux of the outlet cross section in 3D nanofluid-filled finned absorber tubes using a deep convolutional neural network. The rest of the paper presents the detailed method, including the dataset preparation, the network architecture, the operations in neural networks, and the training algorithm. Then, we show the performance of the developed ROMs through the temperature field and outlet heat flux prediction of the absorber tube with random fin heights, widths, and relative positions.

2. Methods

The detailed method of building the two deep CNN-based ROMs is presented in this section, including the preparation of the dataset, the CNNs architecture, and the model training. The overall workflow of this study is described in Figure 1. The two gray boxes present the generated data: one is for the geometry information of the problem presented by signed distance function (SDF); the other is for the numerical simulation results of OpenFOAM. We divide them into two parts: one part consists of 90% of the data (black dashed-line box) and is set to train the model; the rest (green dashed line box) is used to validate the model. The proposed framework in the paper uses the convolutional-deconvolutional and convolutional fully connected layers to construct the end-to-end mappings from the input geometry to the temperature field and outlet heat flux, respectively.

2.1. Preparation of Dataset

2.1.1. Physical Model and Dataset Design

The cross section of the studied absorber tube is shown in Figure 2a. The inner radius r_i is 33 mm and the external radius r_o is 35 mm; the reference module of the parabolic trough collector is LS-2 [48]. Five longitudinal internal rectangular fins in the lower part of the tube are designed. The absorbent tube with length L = 1 m is considered.

In order to enable the ROMs to learn the heat transfer mechanism with adaptive fin height h, width w, and layout, 2000 data points are generated for each absorber tube. Among them, 1800 data points are for the training and 200 data points are for the error analysis. The height and width of all rectangular fins are randomly between 1 mm and 20 mm; the angle between any two adjacent fins α is same, the value is randomly between 0 and $\mathsf{\pi }/3$, and the position of the third fin is unchanged at the center of the lower part of the tube. Furthermore, there are two geometrical constraints: 1—the angle α is considered superior to θ (h,w) to avoid the overlap of the fins (Figure 2a); 2—the fin height h is considered to be greater than h_min (w) (Figure 2b).

2.1.2. Governing Equations and Numerical Simulation

We suppose an incompressible, viscous, and single-phase nanofluids. The continuity equation and the momentum equation are given as follows:

$$\mathrm{div} \mathit{u}=0$$

(1)

$$\frac{\partial \mathit{u}}{\partial t}+\left(\mathrm{grad}\mathit{u}\right)\mathit{u}=-\frac{1}{\rho }\mathrm{grad}P+\frac{\mu }{\rho }\mathrm{div}\left(\mathit{D}\right)+\mathit{b}$$

(2)

where $\mathit{u}$ is the velocity tensor of nanofluid, $t$ is the time, $P$ is the fluid pressure, $\rho$ is the nanofluid density, and $\mu$ is the nanofluid viscosity. $\mathit{b}$ is the body force per unit volume, which is supposed to be zero in this study. $\mathit{D}$ is the symmetric part of the velocity gradient:

$$\mathit{D}=\frac{1}{2}\left[\mathrm{grad} \mathit{u}+{\left(\mathrm{grad} \mathit{u}\right)}^T\right]$$

(3)

The energy transport equation for incompressible flow is defined as follows:

$$\frac{\partial T}{\partial t}+\left(\mathrm{grad}T\right).\mathit{u}=\frac{\lambda }{\rho c}\mathit{D}:\left(\mathrm{grad}\mathit{u}\right)+\frac{S}{\rho c}$$

(4)

The parameter $S$ represents the volumetric heat source. $c$, $\lambda$ represent the specific heat capacity and thermal conductivity of nanofluid, respectively, and both are supposed to be constant at the reference temperature.

We use 4% CuO nanofluids in this study; the values of the material parameters are listed

Table 1. Physical properties of 4% CuO nanofluid.

ρ (kg/m3)	μ (Pa·s)	λ (W/(m·K))	c (J/(kg·K))
1212.8	8.635 × 10−5	0.724	3419.6

, according to [49].

The boundary and initial conditions are listed as below:

The uniform fixed velocity values are applied at the inner boundary. The zero pressure is applied at the outlet boundary. Nonslip conditions on the walls are considered.

A temperature of 300 °C is initially applied to the fluid computational domain, 320 °C is initially applied to the solid domain, and the temperature of the inlet cross section is fixed at 300 °C. The thermal boundary condition on the external wall of the absorbent tube is nonuniform heat flux [22].

Gmsh is used to generate the 3D absorber tube mesh of our problem, as presented in Figure 3. We first mesh the 2D cross section of the structure with unstructured triangular elements, and then convert it into 3D. The element density of near-wall regions in fluid domain is increased where there are high-velocity gradients.

The mesh independence tests are shown in

Table 2. Mesh independence tests.

Mesh	Average Temperature of Outlet Cross Section	Average Pressure of Inlet Cross Section	Computational Time
290,000	324.59 K	100,005.35 Pa	480 s
540,000	324.77 K	100,005.39 Pa	16,252 s
780,000	324.80 K	100,005.39 Pa	29,434 s

, which are carried out for average temperature of the outlet cross section and average pressure of the inlet cross section of three sizes of grids. For all the cases, the temperature and pressure deviations are no more than 0.1%. Mesh 1 meets the precision requirements and takes the least time, so it is adopted to handle the simulation calculations.

The kOmegaSST model [50] in open-source software OpenFOAM is used to solve the turbulence flow problem.

Furthermore, we validate our CFD method with the experiment [51] and numerical simulation [48] in literature, as shown in

Table 3. A comparison between the experimental data [51], numerical simulation data [48] in literature, and the CFD calculation in our study.

	Dudley et al. (1994) [51]	Abed et al. (2001) [48]		OpenFOAM
Flow Rate (L/min)	Average Outlet Temperature (K)	Average Outlet Temperature (K)	Deviation (%)	Average Outlet Temperature (K)	Deviation (%)
39.8	120.8	127.859	−5.844	126.198	4.469
48.4	166.2	168.337	−1.286	168.075	−1.128
51.1	314.2	313.809	0.124	313.48	−0.229

.

2.1.3. Input Presentation

Each dataset contains the geometry information and the simulation results of the outlet cross sections. The former works as the input information, set to be a 240 × 240-pixel image; the latter guides the training direction.

We propose the signed distance function (SDF) as the boundary and geometrical representation of the problem for the input of both the field and scalar prediction ROMs. The geometry is embedded in a matrix, then we create the zero-level set to represent the location of the boundary, whose property is ‘wall’, which means the absorbent tube and fins in our study. After that, each special point of the fluid domain is presented by the distance to the nearest wall. Moreover, we mask the space out of the tube. The SDF-presented geometry of the absorber tube with rectangular fins is shown in Figure 4.

2.2. Architecture Design of ROM Networks for Field and Scalar Prediction

The network of the field prediction ROM is presented Figure 5. Inspired by Unet [52], the proposed network consists of two paths: the downsampling path and upsampling path, which form a U shape. Additionally, the residual blocks are inventively employed to replace all skip connections to enhance the performance of Unet.

The downsampling path compresses the image, and at the same time extracts the different features of the image, also called a contracting path. On the contrary, the upsampling path expands the image to the original size, referred to expanding path. In both paths, there are four repeated blocks, each consisting of one dark-gray box and two light-gray boxes. The light-gray box represents the normal 3 × 3 convolution operations followed with an activation function. In the downsampling path, the dark-gray box represents the output of the max pooling operation and in the upsampling path, the dark-gray box represents the output of a deconvolutional operation. The residual blocks take the place of skip connections, which allows part of the feature map to be convolved three more times before passing from the contracting path to the expending path. The advantages are first to provide more types of feature maps with different degrees of compression to the downsampling path and simultaneously to avoid the gradient vanishing and exploding problems in the deep neural network. The residual block first appeared in the ResNet, and there are two typical types of residual block [53]: basic block and bottleneck block. The latter is used in Unet, which consists of two 3 × 3 normal convolutional layers and one 1 × 1 convolutional layer in the block, which allows us to transform the input into the desired dimension. Furthermore, each convolutional layer is followed with a ReLU activation function, as presented in Figure 6.

The network of the scalar prediction ROM is presented in Figure 7. The first half is the same as the ROM network of field prediction: four repeated blocks with one 2 × 2 max pooling operation and two 3 × 3 convolution operations. Meanwhile, for the remaining part, two fully connected layers replace the upsampling path, and finally, one scalar is output.

2.3. Operations in Convolutional and Fully Connected Layers

The light-gray box represents the normal 3 × 3 convolution operations followed with a RELU6 activation, which can be expressed as the equation below:

$$a^l=\sigma \left(w^l\ast a^{l-1}+b^l\right)$$

(5)

where $a^l$ is the output of the l-th convolutional layer, $a^{l-1}$ is defined as the output of the (l − 1)-th layer and also is the input of the l-th convolutional layer, $w^l$ are the weight, and $b^l$ is the bias. $\sigma (·)$ is a nonlinear activation function. The Relu6, a modification of Relu (rectified linear Unit) activation function is used in the convolutional layers.

A simple example of 2D convolution operation with a 3 × 3 convolutional filter weight 1 × 1 stride and 1 layer zero-padding is shown in Figure 8a. We fill the 0 around the 3 × 3 input matrix, making it become a 5 × 5 matrix. Then, we move the filter over the top left part of the input matrix with the same size of 3 × 3 and perform an element-wise product. The accumulation of the nine products is the first element of the output matrix. Then, we move the filter all around the input matrix to calculate the other values of the output matrix. The size of the feature matrix generated after convolution can be expressed as:

$$N_l=\frac{\left(N_{l-1}+2p-k\right)}{s}+1$$

(6)

$N_{l-1}$ represents the feature size of (l − 1) th layer and also the input feature size of l th layer; $k$ presents the size of convolution kernel. $s$ is the stride size, and $p$ is the number of zero-padded layers.

The transpose convolution layers can be viewed as the inverse process of the convolution layers [54]. Figure 8b presents a simple example of 2D deconvolution operation with a 2 × 2 filter weight, 1 × 1 stride, and 1 layer zero-padding. The size of the feature matrix increases after the operation and can be calculated as:

$$N_l=\left(N_{l-1}-1\right)\times s+k$$

(7)

Max pooling operation is often used to reduce the size of the representation, and a 2D example of 2 × 2 max pooling operation with stride 2 is presented in Figure 8c. We first take the top left 2 × 2 block of input matrix and store the maximum value in the output matrix. Then, we move the block by 2 and fill the other elements in output matrix.

In fully connected layers, the neurons in current layer are connected to all the neurons of previous and next layers. Each neuron can be expressed as:

$$a_j^l=\sigma \left({\theta }_{jk}^l\ast a_k^{l-1}+b_j^l\right)$$

(8)

where $a_j^l$ is the output of the jth neuron in lth layer; $a_k^{l-1}$ is the output of the kth neuron in (l − 1)th layer and also one of the inputs of jth neuron in lth layer; ${\theta }_{jk}^l$ is one element of the learnable weight in lth layer; $b_j^l$ denotes the bias; $\sigma (·)$ represents nonlinear activation function; ReLU is used in the fully connected layers. Defined as the input of the l-th layer, $w^l$ are the weights, and $b^l$ is the bias. $\sigma (·)$ is a nonlinear activation function.

2.4. Model Training and Evaluation Methods

Loss function is calculated to present the difference between the numerical simulation (considered as reference values) and the prediction of the proposed ROMs. We applied the mean absolute error (MAE) loss, defined as:

$$loss\_T=\frac{1}{n_{val}}{\displaystyle \sum }_{n=1}^{n_{val}}{\displaystyle \sum }_{i=1}^H{\displaystyle \sum }_{j=1}^W\frac{\left|\widehat{y}-y\right|}{H\times W}$$

(9)

$$loss\_q=\frac{1}{n_{val}}{\displaystyle \sum }_{n=1}^{n_{val}}\left|\widehat{y}-y\right|$$

(10)

where $loss\_T$ is the loss function for temperature field prediction and $loss\_q$ is for heat flux prediction. $n_{val}$ is the amount of data in validation set. H and W are the height and width of the reconstructed field, respectively. $\widehat{y}$ is the predicted result of the ROM network, and $y$ is the corresponding simulated result. In this study, model training is implemented in PyTorch, a popular deep learning framework using Python environment. The loss value is optimized with Adam optimizer, a gradient descent optimization algorithm. When the loss is closer to 0, it indicates that the prediction effect is better, and the ROM network is more robust.

The field prediction accuracy of the proposed model is defined as:

$$\mathrm{Accuracy}=1-{\displaystyle \sum }_{n=1}^{n_{val}}{\displaystyle \sum }_{i=1}^H{\displaystyle \sum }_{j=1}^W\frac{\left|\widehat{y}-y\right|}{n_{val}\times H\times W}$$

(11)

For scalar prediction of the trained ROM, the relative error is defined as:

$$RE=\frac{\widehat{y}-y}{y}\times 100\%$$

(12)

The determination coefficient ${\mathrm{R}}^2$ is used for the error measurement of the predicted scalar:

$${\mathrm{R}}^2=1-\frac{{\sum }_{n=1}^N{\left(y^n-{\widehat{y}}^n\right)}^2}{{\sum }_{n=1}^N{\left(y^n-\overline{y}\right)}^2}$$

(13)

where $\overline{y}$ represents the average of the simulated result, N is the amount of test data. The value of ${\mathrm{R}}^2$ is closer to 1, which indicates that the prediction is more accurate.

3. Results

The results of the temperature field and outlet heat flux predictions of the nanofluid-filled absorber tube are successively presented in this section.

3.1. The Performance of ROM for Nanofluid Temperature Field Prediction

After training with 1800 data points, the temperature field predictions of the outlet cross section with random fin size and position in the validation dataset are executed. The prediction performance of the proposed model is evaluated by the MAE loss, computed according to Equation (5). Three representative results are chosen with the rectangular height, width, and adjacent fin angle [h, w, α] = [12 mm, 5 mm, 45°], [12 mm, 6 mm, 18°], [10 mm, 21 mm, 52°], as presented, respectively, in Figure 9a–c. The ROM prediction, numerical simulation, and MAE loss are presented respectively in the first, second, and third columns. As can be seen, the CNN-based ROM learns the mechanism of heat transfer from heated absorber tube to the working nanofluid. The temperature is highest in the lower part of the absorber wall, which is reasonable with our heat flux setting. From the lower part to the center and top part of the absorber tube, the temperature decreases. Furthermore, a great temperature gradient occurs on the contact surface of the solid tube and the working fluid. From the results in the third column, the largest relative error is about 1.9%, which only appears on the middle fin’s contact surface in the third case. Most areas in the cross section show an error value less than 0.2%, leading to the high average accuracy value of 99.935%. All the results above demonstrate that the proposed CNN-based ROM has excellent performance for field prediction of nanofluid-filled absorber tubes with various rectangular fin sizes and positions.

Additionally, we carry out the model analysis on learning rate, a hyperparameter controlling how quickly the model is updated to the solution. A relatively large learning rate ensures the training within a reasonable time, while a relatively small learning rate helps the convergence of the solution. Therefore, a larger learning rate is usually applied in the early stage of training to accelerate the computation. On the contrary, a relatively small learning rate is employed at the end of training to make the training converge in the desired point. In this study, we set the learning rate multiplied by 0.1 per 10 epochs, to achieve a satisfactory learning rate at both the beginning and end of training. Figure 10 shows the good performance of the strategy of changeable learning rate (abbreviated as “lr” in the figure) in the field prediction of the nanofluid-filled absorber tube. All MSE loss curves with various initial learning rates sharply decrease before 10 epochs; the greater learning rate value shows higher convergence speed. Then after 10 epochs, we multiply the learning rate by 0.1 for every 10 epochs; after that, the curves start to oscillate and converge to the similar level. For the learning rate values of 5 × 10⁻⁷, 2 × 10⁻⁷, and 5 × 10⁻⁸, the average accuracies in the validation set are 0.9967, 0.9966, and 0.9951, respectively. Therefore, the 5 × 10⁻⁷ initial learning rate gives the highest accuracy and the fastest calculation speed, and the strategy of 90% reduction in learning rate per 10 epochs is applied in this study.

3.2. The Performance of ROM for Heat Flux Prediction

The results of Section 3.1 show that the proposed ROM enables the reconstruction of the temperature field of the whole cross section of the nanofluid-filled absorber tube, which allows us to clearly see the temperature distribution in the nanofluid area and tube area, in order to alarm us if there is a locality with high temperature. According to our literature review, no researchers evaluate heat transfer performance directly from the physical field. Therefore, the second ROM is proposed to predict the scalar performance parameter of the nanofluid-filled absorber tube.

Useful heat production, as one of the most important physical quantities for PTC absorber tube, measures the heat captured by the working fluid passing through the absorber tube. It can be calculated using the energy balance in the flow volume, equal to the heat variation of the inlet and outlet cross section:

$$Q_u=\left(q_{out}-q_{in}\right)\times t$$

(14)

where t presents the time, and the expression of the heat flux in a cross section is defined as:

$$q={\iint }_A^{}cTu·dA$$

(15)

where A represents the cross-sectional area, c represents the specific heat capacity of the working fluid, $u$ is the flow velocity on the cross section, and T is the temperature. The temperature, velocity of the inlet section, and the fluid cross-sectional area for each type of finned absorber tube are fixed. Therefore, to obtain $Q_u$, $q_{out}$ is the only scalar that needs to be predicted. Thus, the second ROM is designed to predict the outlet heat flux $q_{out}$.

In the following three figures, from Figure 11, Figure 12 and Figure 13, we present the comparison of the predicted outlet heat flux $q_{out}$ and the true value. Furthermore, the size of the datasets has been investigated.

It is well known that the size of datasets plays an important role in training: a small training dataset size generally leads to a poor approximation, while a big dataset size requires more training time, and sometimes it is expensive to generate raw data or the data are just limited. In our study, numerical simulation in OpenFOAM is applied to generate training data. Thereby, repeatedly solving large numbers of partial differential equations consumes huge computational resources. Therefore, we attempt to find the most appropriate dataset scale to avoid the waste of the computational resources.

Figure 11a–d display the comparison of outlet heat flux $q_{out}$ between ROM prediction and numerical simulation with four sizes of dataset: 500, 1000, 1500, and 2000 data points, respectively. The 81 red points present sampled results; the blue line illustrates the regression line. All four regression lines almost coincide with the line y = x, which means that the predicted results with all four sizes of datasets agree well with the CFD results. Furthermore, we evaluate our ROM with the coefficient of determination R², which measures how well the proposed model predicts the outlet heat flux. All four R² values are higher than 0.999, and the differences are very small; the peak value occurs with the dataset of 1000 and 1500.

The error distribution density maps of four size datasets with 81 samples are presented in Figure 12, indicating that the ROM trained with 1000 data points shows the lowest number of outliers. The relative errors of 1000 data points have the narrowest range in outlet heat flux prediction of the nanofluid-filled absorber tube: the prediction errors of 95% samples are lower than 0.99% and greater than −0.84% (presented with short vertical lines); 50% of the samples fall within the range between −0.1% and 0.39% (presented with blue box). At the same time, the smallest dataset with 500 data points has the largest relative error range, between −4.7% and +5.4%.

The average accuracies of all the validation data with the four sizes of dataset are shown in Figure 13. Although based on the results of the 81 samples, the 1000- and 1500- data-point datasets show higher coefficients of determination and a narrower range of relative errors, the differences are very small. In addition, the ROM trained by 2000 data points shows the highest average accuracy on the validation dataset, which means it performs better in heat flux prediction compared to the ROMs trained by 500, 1000 or 1500 data points. All the results in the section show a huge potential of the proposed ROM to replace the CFD solver for outlet heat flux prediction of nanofluid-filled absorber tubes.

Furthermore, we compare the heat flux of the absorber tube for four fin geometries, as listed in

Table 4. Predictions results of absorber tube for four fin geometries with fin adjacent angle of 45°.

(Width, Height) (mm)	qout (CFD) (J/s)	qout (ROM) (J/s)	Relative Error
5, 5	351,411.6	350,942.1	0.13%
5, 10	338,082.8	337,573.8	0.15%
5, 15	324,877.0	324,254.9	0.19%
5, 20	311,527.1	311,089.1	0.14%

. The fin adjacent angle is fixed at 45°, the fin width is set at 5 mm, and the fin height is from 5 mm to 20 mm. Firstly, the four cases’ relative errors are very small, less than 0.2%, which confirms the accuracy of the proposed model. Then, the q_out values vary with the change in fin height. In detail, the q_out decreases when the height of the fin augments. This is reasonable, as we fixed the inlet velocity. Thus, the flow rate is proportional to the cross-sectional area of the fluid domain. When the height of the fin increases, so does the surface of the fin. Consequently, the cross section of the fluid domain decreases, then the rate of flow decreases. Therefore, there is not enough working fluid to carry the heat out, and the q_out reduces.

3.3. Time Consumption Comparison with CFD Simulators

In Section 3.1 and Section 3.2, we evaluated the performance of the proposed two ROMs on temperature and outlet heat flux prediction and we obtained very satisfied results. In this subsection, we focus on analyzing the computational resources required by the ROM and numerical simulation for field and scalar prediction. All the calculations of CFD have been run with a CPU AMD Ryzen 7 5700X 8-core Processor, and a GPU NVDIA GeForce PTX 2060 has been used to accelerate the model training and predictions.

The related data of time consumption are presented in

Table 5. Comparison of the time cost for temperature field and heat flux predictions by the proposed ROMs and the CFD simulation in OpenFOAM.

		CFD	ROM
Temperature field prediction	Model training	-	1.14 h
	Input preparation	1.68 s (mesh)	0.062 s (SDF)
	Prediction	451s	0.011 s
Outlet heat flux prediction	Model training	-	0.8 h
	Input preparation	1.68 s (mesh)	0.062 s (SDF)
	Prediction	393 s	0.004 s

. As can be seen, the great advantage of the CFD solver is that it does not need any training time, while about one hour of training is indispensable for each ROM. For prediction, the ROM requires converting the cross-sectional geometry of the absorber tube into SDF presentation. Similarly, the CFD model also needs some time to mesh the geometry. Furthermore, it frequently takes a long time to find a good mesh strategy. Although the proposed ROMs need a great amount of time for training, the prediction time is much less than the calculation in the CFD solver, even can be ignored. Each prediction of the ROM for the temperature field costs just 0.011 s, and the value is less—only 0.004 s—for heat prediction. Meanwhile, for the CFD solver, due to repeatedly solving a large number of partial differential equations, each temperature field calculation takes more than 400 s, which is four orders of magnitude slower than the ROM prediction.

4. Discussion

Modifying the internal surface of the PTC receiver tube with fins and using nanofluids as the working fluid are two popular techniques to enhance the performance of the PTC systems. The turbulence of the working fluid is increased by fins, and the conductivity is improved using nanofluids. Thereby, the heat transfer of the receiver tubes is enhanced. However, using the two techniques makes the receiver systems more complex; consequently, much more cost is required for the analysis and optimization investigations using experimental or CFD methods. Therefore, to address this problem, two ROMs based on CNNs were developed in this study.

According to the results of Section 3.1, the first ROM could accurately and quickly predict the temperature field of the outlet cross section, including the fluid and solid domain, which could provide some interesting local temperature information for the designer. As commonly known, local high temperature could cause local thermal stress and deformation, further leading to the PTC structure failure. In our problem, the sunlight concentrates on the lowest part of the tube wall, then logically, as shown in Figure 9, the lower part of the tube shows the highest temperature. Furthermore, the larger fins heat more nanofluid, and its own temperature also rises. This may be due to the fact that in our study, the stream velocity is fixed, so a larger cross-sectional area of fins results in a smaller fluid cross-sectional area, and thus a smaller fluid flow. Therefore, there is insufficient working fluid to carry the heat away from the absorption tube. Hence, if we use the proposed temperature field predictor in the fin-shape optimization process, the local high temperature could be known and avoided.

The second predictor is developed to estimate the heat flux of the outlet cross section, then calculate the useful heat production of the PTC receiver, which is one of the most important physical quantities for PTC absorber tubes, measuring the heat captured by the working fluid passing through the absorber tube. According to the comparison in Table 4, the size of the fin has a significant effect on the outlet heat flux, and an increased height of the fins leads to a decrease q_out in the outlet cross section. This is probably because the increase in the height of the fin reduces the nanofluid flow cross-sectional area. And in the case of a fixed inflow velocity, the flow rate of the absorber tube is reduced, and there is not enough working fluid to transfer the heat from the tube. Furthermore, we have investigated the influence of dataset size on the performance of the proposed q_out predictor. The predictions with all tested data sizes agree well with the reference, and the biggest dataset of 2000 data points leads to the highest accuracy.

Benefitting from the proposed high-performance ROMs in this study, the future investigation could be the fin-shape optimization of the PTC receiver using the performance enhancement criteria and an appropriate optimization [55,56]. Conventional optimization framework usually uses a CFD solver to estimate the observation points. Meanwhile, for a complex, highly nonlinear heat transfer system, the optimization process requires a large number of observation points, and each estimation needs to solve lots of PDEs. Therefore, the required computational time grows exponentially. Fortunately, the proposed ROMs enable fast and accurate performance prediction of the nanofluid-filled receiver tube with fins. Therefore, replacing the CFD solver with the ROMs will significantly improve the optimization efficiency.

5. Conclusions

Two deep neural network-based ROMs are proposed to predict the heat transfer of nanofluid in absorber tubes with various fin heights, widths, and positions. Both network models contain several repeated convolutional blocks to extract the features of the SDF presented problem geometry. Then, for field prediction, the deconvolutional blocks are connected to expand the highly encoded features to the original size, while for heat flux prediction, the deconvolutional blocks are replaced by two fully connected layers. The temperature field and the heat flux of the outlet cross section are estimated using the two neural network-based ROMs, respectively, to test their performance. The results can be drawn as follows:

• The temperature field of the outlet cross section of the absorber tube is reconstructed with high accuracy, more than 99.9%. Moreover, the computational speed is four orders faster than using the numerical simulation.
• For the estimation of heat flux at the outlet, various sizes of datasets are examined. The 2000-data-point dataset achieves the highest accuracy, more than 99.7%, and the determination coefficient R² of 81 samples is higher than 0.9995. Furthermore, each prediction of outlet heat flux takes very little time, only about 0.004 s, which is five orders faster than the CFD solver in OpenFOAM, which is about 400 s for each simulation.
• The strategy of learning rate decreasing is applied to provide a relatively high computational speed at the beginning of the training to reduce the time cost, and at the same time, a relatively small learning rate value at the end of training facilitates the convergence of the solution. Moreover, through the hyperparameter analysis, the initial learning rate of 5 × 10⁻⁷ is confirmed as the optimum choice.

In conclusion, the proposed network ROMs in the study enable the accurate and fast prediction of the temperature field and heat flux of nanofluid-filled absorber tube with various fin heights, widths, and positions, which show their huge potential for the fin-shape design and optimization of nanofluid-filled absorber tube.