# Insights into the Estimation of the Enhanced Thermal Conductivity of Phase Change Material-Containing Oxide Nanoparticles using Gaussian Process Regression Method

Authors: Tzu-Chia Chen; Hasan Sh. Majdi; Aras Masood Ismael; Jamshid Pouresmi; Danial Ahangari; Saja Mohammed Noori
Journal: International Journal of Chemical Engineering
 (2022)

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## Abstract

Thermal conductivity (TC) of a phase change material (PCM) may be enhanced by distributing nanostructured materials (NSMs) termed nano-PCM. It is critical to accurately estimate the TC of nano-PCM to assess heat transfer during phase transition processes, namely, solidification and melting. Here, we propose Gaussian process regression (GPR) strategies involving four various kernel functions (KFs) (including exponential (E), squared exponential (SE), rational quadratic (RQ), and matern (M)) to predict TC of n-octadecane as a PCM. The accessible computational techniques indicate the accuracy of our proposed GPR model compared to the previously proposed methods. In this research, the foremost forecasting strategy has been considered as a GPR method. This model consists of the matern KF whoseR2 values of training and testing phases are 1 and 1, respectively. In the following, a sensitivity analysis (SA) is used to explore the effectiveness of variables in terms of outputs and shows that the temperature (T) of nanofluid (NF) is the most efficient input parameter. The work describes the physical properties of NFs and the parameters that should be determined to optimize their efficiency.


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## Body

## 1. Introduction

PCMs are extensively utilized in thermal storage devices to store heat [1]. They may be categorized chemically as organic PCMs (fatty acids and paraffin), inorganic PCMs (salt hydrates and metallic), or eutectics PCMs [2]. Though PCMs exhibit higher latent fusion heat at relatively constant T, they have a relatively poor TC [3]. This has a remarkable and considerable effect on the pace of PC [4, 5]. To improve the thermal storage of PCMs, their thermal conductance must be increased. As a result, many improvement techniques have been suggested, including incorporating highly thermally conductive materials, including metallic and graphene porous foams [6–9] into the PCM, microencapsulation of the PCM [10, 11], high-TC fins [12], and integrated heat pipes PCM [13].High-TC NSMs have been developed and marketed during the last several decades due to nanotechnology advancements. These nanostructures are made of a variety of materials, including metallic, carbon-based, and nonmetallic compounds [14]. NFs are a new category of heat transfer fluids made via incorporating nanoscale components into a base fluid [15]. Among the alluring properties of NFs is the increased TC as opposed to the base fluid. The concept of enhancing the NFs’ TC may be used to introduce PCMs with scattered NSMs that had increased TC compared to the basic PCMs [16]. In contrast to stationary enhancer structures, nanomaterials can retain the PCM’s fluidity in the liquid state and prevent contact conductance issues; furthermore, nano-PCMs would be recyclable [16]. Numerous researchers have performed significant studies on the increased TC of PCMs incorporating various nanomaterials, including oxide nanoparticles (NPs) [17–19], carbon nanomaterials [20–22], and metallic NPs [23, 24].Along with standard measures of the thermophysical characteristics of nano-PCMs, several academics have lately proposed various experimental studies to estimate these parameters [20, 25, 26]. Algorithms for machine learning (ML), such as artificial neural networks (ANNs), are founded on human neurons. These techniques have been widely employed in the last few years to forecast the thermophysical characteristics of new materials, particularly NFs [27]. Toghraie et al. [28] investigated the dynamic viscosity (μ) of an EG/AG NF. Ahmadi et al. [29] used the ANN intelligent technique to determine the dynamic μ of a SiO2/EG-H2O NF. Hemmat Esfe et al. [30], using an ANN, predicted the μ of a MWCNTs-ZnO/5W50 nanolubricant under various shear stresses, Ts, and volume concentrations. Chen et al. [31] used an ANN to determine the μ of MWCNTs-TiO2/SAE50 hybrid NFs at various shear rates, volume fractions (VFs), and Ts. Hemmati-Sarapardeh et al. [32] have published a paper focused on intelligent techniques for the prediction of the NFs’ TC. Rostamian et al. [33] used experimental data, ANNs, and nonlinear regression method to determine the TC of Cu-SWCNTs/EG hybrid NFs at various VFs and Ts. Shahsavar et al. [34] used oleic acid as a surfactant to investigate an alumina-liquid paraffin NF’s μ and TC. Safaei et al. [35] used an ANN to predict the increased TC of ZnO/TiO2 in ethylene glycol as a hybrid NF. The authors trained the ANNs using experimental data. Using a database containing 715 experimental data, Adun and his colleagues [36] estimated the TC of hybrid NFs. They accomplished this by developing support vector regression and multiple linear regression models. Hemmat Esfe et al. [37] utilized ANNs and response surface methodology for predicting the TC of TiO2 in H2O NFs. Peng et al. [38] used the ANN technique and a suggested correlation to model the TC of an alumina/Cu-ethylene glycol NF. Li et al. [39] presented novel correlations and an optimum ANN for measuring μ and TC of alumina-ethylene glycol NFs based on several sets of experiments. VFs and Ts of NPs are input to a feedforward ANN. Çolak [40] studied the TC of ZrO2-H2O NFs at a range of concentrations and Ts. Barewar et al. [41] examined the thermophysical characteristics of Ag/ZnO-EG NFs with VFs ranging from 0.05 to 0.2 vol. % to ZnO-EG NFs. Ghazvini et al. [42] presented a two-layered ANN for determining the TC ratio of a CuFe2O4-H2O NF based on experimentally determined thermal conductivities. He and his colleagues [43] used ANNs and correlation methods to approximate the TC of a ZnO/Ag-H2O hybrid NF. The authors discovered that the most precise model using an ANN model combined with the Levenberg–Marquardt method. Pare and Ghosh [44] studied the TC of Al2O3, CuO, and ZrO2 in H2O NFs. The TC of NFs was determined at weight concentrations ranging from 0.02 to 2% and Ts ranging from 20 to 90°C. Rostami et al. [45] used curve fitting and ANNs to estimate the TC ratio of graphene oxide and copper oxide antifreeze NF.The purpose of this research is to expand our understanding in a variety of ways. To begin, while the TC of NFs has been modeled using ANN, to the authors’ information, GPR methods for projecting the TC of nano-PCMs had not been investigated. Secondly, since experimental TC measurements of different solid-liquid nano-PCMs are costly and time-consuming, GPR may be employed to correctly estimate the TC of nano-PCMs. To train these models, 122 experimental data from previous research are examined, with Ts ranging from 5°C to 60°C and mass fractions of NPs ranging from 0.5 to 12 wt. % [14, 25, 26, 46]. We used three quarters of this data in the model making stage and the rest in the model testing stage. Using this data, models with four various function kernels are developed and then using different statistical parameters, the accuracy of these models is examined and the best model to predict the target parameter is identified.


## 2. Methodology (GPR Model)

We follow a nonparametric approach, to model various inconsistent complex systems [47, 48]. In fact, we want to use one of the benefits of this method which is the flexibility of its algorithm to describe the uncertainty sources making it most attractive to researchers following prediction issues through that [49]. It is noted that the uncertainty sources are detected by GPR models [50]. As an instance, these models distribute the values predicted rather than only a predicted one. In addition, this model can have the capability to add characteristics and instruction on forms of models by employing different KFs manually. So, a covariance function (CovF), kx.x′, and mean function, mx, are used to model time series as follows [51–53]:(1)y=fx∼Nmx,kx,x′,where output and input are depicted by y and x for training, respectively. Also, fx describes the hidden variable of this algorithm [54]. It is popular to select the mean function as zero. CovF function is used to show the similarity between input variables. Because similar inputs may make similar outputs that are not useful for our database. The KFs of this study are described as follows [55, 56]:Exponential(2)kx,x′=θ12exp−rθ2,(3)r=x,x′Tx−x′.Squared exponential(4)kx,x′=θ12expd22θ22,Where,θ1 and θ2 are the hyperparameters and must be optimized. Also, d is the Euclidean distance of x and x′.Matern(5)kx,x′=1Γv2v−12vrlvkv2vrl.Where the modified Bessel function is shown byKv, and a positive parameter by ν [57].Rational quadratic(6)kx,x′=θ121+r22∝θ22,where a covariance positive parameter is shown by α.Before training the algorithm, the negative log marginalized likelihood (NLML) must be minimized and this is done by hyperparameters of KFs as follows [58]:(7)NLML=−logpy|x,θ=−12logk+σn2I−12yTk+σn2I−1y−n2log2π.This is an optimization process that must reach the minimum ofθ. This process is explained as follows:(8)θ^=argmin−logpy|x,θ.During this process, NLML is minimized by optimization methods known as off-the-shelf methods. These methods utilize a convex function. Then, the testing phase is done to predict distribution as follows [59]:(9)f∗|x,y,x∗∼Nf∗¯,covf∗,f¯∗=mx,x∗+Kx,x∗Kx,x+σn2I−1y−mx,covf¯∗=Kx∗,x∗−Kx∗,xKx,x+σn2I−1Kx,x∗,where f∗¯ and covf∗¯ are the prediction results and prediction uncertainty, respectively. The average GPR distribution is linearly when mx=0. In this case, this distribution is specified as a linear function of y used for training (see equation (10)) and defined as follows:(10)f¯∗=Kx,x∗Kx,x+σn2I−1y=WGPRY,where WGPR is the weighting matrix.


## 3. Evaluating the Accuracy of the Gathered Databank

Here, outlier data are considered as data points. They have various behaviors compared to other data. These outlier data return data to the faults that occurred in the experimental method. These data result in false estimations of our proposed models. So, to improve the efficiency and integrity of our models, we must identify these kinds of data points. We can use the leverage method to enhance the databank quality. This method identifies the outlier data points by a Hat matrix defined as the following [60, 61]:(11)H=UUTU−1UT,where U refers to a i∗j dimensional matrix. The i and j show the number of model parameters and training points, respectively. To test this process, a critical leverage limit is considered as a parameter to identify the outlier data from others. This limit is defined as the following [62–64]:(12)H∗=3j+1i.Afterward, William’s plot is used to evaluate the authenticity of the TC databank. As you see, the standard residuals are shown against the Hat values in Figure1. In Figure 1, to define a reliable zone to use the dataset, a bounded area is considered between the critical limitations of leverage and standard residuals of −3 to 3. Then, William’s plot is used to show the reliability of resulted TC data points. The value of outlier data for M, SE, E, and RQ models, was obtained 2, 3, 1, and 0, respectively. So, they are appropriate to test and train models.Figure 1
Detection of suspected data points using Hat analysis for models: (a) M, (b) E, (c) SE, and (d) RQ.
(a)(b)(c)(d)


## 4. Results and Discussion

### 4.1. SA

It is noted that researchers and engineers always track the way to identify the impact of input numbers on the TC to suggest an accurate model. So, a SA is the best way to achieve this goal. In this regard, they follow the relevancy factor,r, for every input variable as follows [65–67]:(13)r=∑i=1nXk,i−X¯kYi−Y¯∑i=1nXk,i−X¯k2∑i=1nYi−Y¯2,where Xk.i and Yi are the input and output, respectively. Xk¯ and Y¯ are means of inputs and outputs, respectively.The impact of every parameter on the TC is shown in Figure2. Here, when the absolute value, r, of an input number increases, it influences on the TC more and conversely. Additionally, here the positive value depicts that every input variable has a direct relationship with TC. Moreover, the results show that Ts with positive r values, e.g., 0.98, are the most efficient variables to detect the TC.Figure 2
SA on the input parameters using relevancy factor for models: (a) M, (b) E, (c) SE, and (d) RQ.


### 4.2. Modeling Results

In this part, to explore the fulfillment of suggested models, many more attempts have been performed to predict the TC. Here, the assessment of the performance of our models is conducted in two main ways employing graphical comparisons and matching parameters. To find out a match between the actual and predicted databank, the matching parameters have been utilized [68].Table1 reports the statistical parameters calculated for testing, training, and the whole dataset. You can see in this table that GPR models with RQ, E, M, and SE kernel functions have R2 values of 0.999, 1, 1, and 0.999, respectively. In addition, lower values of other parameters including STD, MSE, MRE, and RMSE during the training phase show that they have had acceptable precision. In particular, this acceptable value is very important regardless of the model performance for estimating unseen TC points.Table 1
Determining statistical parameters in different phases for models (a) M, (b) E, (c) SE, and (d) RQ.
ModelSetR2MRE (%)MSERMSESTDMTrain1.0000.6102.77298E − 060.00170.0011Test1.0000.4651.0165E − 060.00100.0006Total1.0000.5742.34106E − 060.00100.0010ETrain0.9990.7326.37106E − 060.00250.0018Test1.0000.6394.38691E − 060.00210.0015Total1.0000.7095.88315E − 060.00210.0017SETrain0.9990.8476.65673E − 060.00260.0018Test0.9990.9987.82742E − 060.00280.0019Total0.9990.8846.94461E − 060.00280.0018RQTrain0.9991.0781.11893E − 050.00330.0022Test0.9990.8987.14543E − 060.00270.0018Total0.9991.0341.01949E − 050.00270.0021Figure3 compares the experimental and predicted TC values of these models simultaneously. In the GPR models, there is a superior agreement between various GPR models and real TC.Figure 3
Visual and simultaneous comparison of actual and modeled output data using various KFs: (a) M, (b) E, (c) SE, and (d) RQ.
(a)(b)(c)(d)Here, the experimental TC of all proposed models is exactly covered by the estimated TC. So, these models have the reliability to perform and predict TC. In the following, Figure4 depicts the cross plots of 4 GPR proposed models. They illustrate that the whole estimated TC is placed in its real values. We see their fitting lines and the bisector line of the first quarter have near similarity. Generally, the bisector line is used to measure the precision of the proposed models. Precision is higher when the data are very close to the 45°line.Figure 4
Cross plot analysis to determine the accuracy of different models in predicting target values using various KFs: (a) M, (b) E, (c) SE, and (d) RQ.
(a)(b)(c)(d)Next, Figure5 shows the relative deviations (RDs) between predicted TC and real values for all suggested models. In this study, M and KF have absolute deviation points less than 3%, whereas various KFs of RQ, SE, and E have less than 4%.Figure 5
RD analysis to determine the accuracy of different models using various KFs: (a) M, (b) E, (c) SE, and (d) RQ.
(a)(b)(c)(d)


## 4.1. SA

It is noted that researchers and engineers always track the way to identify the impact of input numbers on the TC to suggest an accurate model. So, a SA is the best way to achieve this goal. In this regard, they follow the relevancy factor,r, for every input variable as follows [65–67]:(13)r=∑i=1nXk,i−X¯kYi−Y¯∑i=1nXk,i−X¯k2∑i=1nYi−Y¯2,where Xk.i and Yi are the input and output, respectively. Xk¯ and Y¯ are means of inputs and outputs, respectively.The impact of every parameter on the TC is shown in Figure2. Here, when the absolute value, r, of an input number increases, it influences on the TC more and conversely. Additionally, here the positive value depicts that every input variable has a direct relationship with TC. Moreover, the results show that Ts with positive r values, e.g., 0.98, are the most efficient variables to detect the TC.Figure 2
SA on the input parameters using relevancy factor for models: (a) M, (b) E, (c) SE, and (d) RQ.


## 4.2. Modeling Results

In this part, to explore the fulfillment of suggested models, many more attempts have been performed to predict the TC. Here, the assessment of the performance of our models is conducted in two main ways employing graphical comparisons and matching parameters. To find out a match between the actual and predicted databank, the matching parameters have been utilized [68].Table1 reports the statistical parameters calculated for testing, training, and the whole dataset. You can see in this table that GPR models with RQ, E, M, and SE kernel functions have R2 values of 0.999, 1, 1, and 0.999, respectively. In addition, lower values of other parameters including STD, MSE, MRE, and RMSE during the training phase show that they have had acceptable precision. In particular, this acceptable value is very important regardless of the model performance for estimating unseen TC points.Table 1
Determining statistical parameters in different phases for models (a) M, (b) E, (c) SE, and (d) RQ.
ModelSetR2MRE (%)MSERMSESTDMTrain1.0000.6102.77298E − 060.00170.0011Test1.0000.4651.0165E − 060.00100.0006Total1.0000.5742.34106E − 060.00100.0010ETrain0.9990.7326.37106E − 060.00250.0018Test1.0000.6394.38691E − 060.00210.0015Total1.0000.7095.88315E − 060.00210.0017SETrain0.9990.8476.65673E − 060.00260.0018Test0.9990.9987.82742E − 060.00280.0019Total0.9990.8846.94461E − 060.00280.0018RQTrain0.9991.0781.11893E − 050.00330.0022Test0.9990.8987.14543E − 060.00270.0018Total0.9991.0341.01949E − 050.00270.0021Figure3 compares the experimental and predicted TC values of these models simultaneously. In the GPR models, there is a superior agreement between various GPR models and real TC.Figure 3
Visual and simultaneous comparison of actual and modeled output data using various KFs: (a) M, (b) E, (c) SE, and (d) RQ.
(a)(b)(c)(d)Here, the experimental TC of all proposed models is exactly covered by the estimated TC. So, these models have the reliability to perform and predict TC. In the following, Figure4 depicts the cross plots of 4 GPR proposed models. They illustrate that the whole estimated TC is placed in its real values. We see their fitting lines and the bisector line of the first quarter have near similarity. Generally, the bisector line is used to measure the precision of the proposed models. Precision is higher when the data are very close to the 45°line.Figure 4
Cross plot analysis to determine the accuracy of different models in predicting target values using various KFs: (a) M, (b) E, (c) SE, and (d) RQ.
(a)(b)(c)(d)Next, Figure5 shows the relative deviations (RDs) between predicted TC and real values for all suggested models. In this study, M and KF have absolute deviation points less than 3%, whereas various KFs of RQ, SE, and E have less than 4%.Figure 5
RD analysis to determine the accuracy of different models using various KFs: (a) M, (b) E, (c) SE, and (d) RQ.
(a)(b)(c)(d)


## 5. Conclusions

In this research, we reach a high-degree match between real and predicted TC numbers with the help of evaluation of our suggested models and the collected databank. Our visual and mathematical comparisons show that our GPR models have an excellent capability to determine the enhanced TC of PCM-containing oxide NPs. Moreover, the SA illustrates that there is a direct relationship between all input parameters and TC. This study is a road map for the engineering communities to forecast the behavior of the heat exchanger and that of refrigeration systems even though having little knowledge about artificial intelligence techniques and nanoscience.



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*Source: 7119336-2022-06-09.xml*