Determining the Factors Affecting the Boiling Heat Transfer Coefficient of Sintered Coated Porous Surfaces

The boiling heat transfer performance of porous surfaces greatly depends on the morphological parameters, liquid thermophysical properties, and pool boiling conditions. Hence, to develop a predictive model valid for diverse working fluids, it is necessary to incorporate the effects of the most influential parameters into the architecture of the model. In this regard, two Bayesian optimization algorithms including Gaussian process regression (GPR) and gradient boosting regression trees (GBRT) are used for tuning the hyper-parameters (number of input and dense nodes, number of dense layers, activation function, batch size, Adam decay, and learning rate) of the deep neural network. The optimized model is then employed to perform sensitivity analysis for finding the most influential parameters in the boiling heat transfer assessment of sintered coated porous surfaces on copper substrate subjected to a variety of highand low-wetting working fluids, including water, dielectric fluids, and refrigerants, under saturated pool boiling conditions and different surface inclination angles of the heater surface. The model with all the surface morphological features, liquid thermophysical properties, and pool boiling testing parameters demonstrates the highest correlation coefficient, R2 = 0.985, for HTC prediction. The superheated wall is noted to have the maximum effect on the predictive accuracy of the boiling heat transfer coefficient. For example, if the wall superheat is dropped from the modeling parameters, the lowest prediction of R2 (0.893) is achieved. The surface morphological features show relatively less influence compared to the liquid thermophysical properties. The proposed methodology is effective in determining the highly influencing surface and liquid parameters for the boiling heat transfer assessment of porous surfaces.


Introduction
Due to rapid advancements in the machining industry, microelectronics devices have gained popularity. These devices produce large amounts of heat. For the system safety and health of these devices, the removal of high heat fluxes in minimal space thickness of 428 µm. Within their tested range, both HTC and CHF increased as the particle size or coating thickness increased. Another research group [13] prepared honeycomb porous structures with and without radial gradient by electrodeposition and sintering in a reducing environment. They tested the samples in the pool boiling of DI water and found that the samples with radial gradient, and smaller pores at the center than near the edges, showed better heat transfer than the sample with uniform structure. They attributed this enhancement to the higher K/Reff, and as per their investigation, a radial gradient from edges to center assisted in quicker rewetting. Furthermore, through visualization experimentation, they also found that the radially distributed porous sample generated larger bubbles at a faster rate, which helped in removing more heat than in other samples. Recently, Pastsuzko et al. [14] investigated the boiling heat transfer of water, FC-72 and Novec-649 on microporous sintered surfaces with and without mesh. They postulated that mesh coverings on the sintered micro-finned surfaces are only effective at low heat flux for liquids with low surface tension, because at the high heat flux (50 kW/m 2 for FC-72 and 100 kW/m 2 for ethanol), maximum performance was recorded in the sample with no mesh or covering.

Aim and Motivation of the Study
As reported by researchers, porous surfaces' performance greatly depends on the morphological parameters, and these parameters are affected by changes in the working fluid. So, to develop a predictive model valid for diverse working fluids, it is necessary to incorporate the effect of these parameters and the thermophysical properties of the working fluid. The predictability of the empirical correlations is usually hampered by these parameters, because these correlations were developed on a limited database. To cover a wide range of data, an artificial intelligence (AI) model, based on advance algorithms and new libraries, can be developed with high accuracy [1,15]. Keeping this in view, the objective of this study is to develop Bayesian optimized deep neural network models to perform a sensitivity analysis for finding the most influential parameters in the boiling heat transfer of sintered coated porous surfaces fabricated on copper substrate, subjected to a variety of high-and low-wetting working fluids, including water, dielectric fluids, and refrigerants, under saturated pool boiling conditions and different surface inclination angles of the heater surface. The detailed impacts of combined and individual surfaces, liquids, and boiling condition parameters have been assessed for a range of tested data. In line with this, the most impactful parameters are highlighted by representing their predictions. The proposed method can help to assess the strong morphological parameters of sintered coated porous surfaces subjected to a range of working fluids and pool boiling conditions. The proposed methodology is effective in determining the highly influential surface and liquid parameters for boiling heat transfer assessment.

Methodology
Training and testing data are divided 80-20% as is commonly practiced in the literature [35,36]. Firstly, two models were developed by considering the surface parameters, liquid thermophysical features, and surface inclination angle. In one of the models, other than the aforementioned parameters, heat flux was taken as the added input parameter, while in the second model, wall superheat was taken as the added input parameter. It was noticed that the wall superheat model showed much better prediction ability compared to the heat flux model. Based on these findings, we decided to consider the wall superheat model for the further assessment of boiling heat transfer coefficient.

Bayesian Optimization (BO)
Bayesian optimization is used to select the hyper-parameters for evaluation in the true objective function by building a probability model of the objective function. This employs the Bayes theorem for determining the maximum and minimum of an objective function. There are different techniques available based on the BO. Here, the two most common BO methods, GPR and GBRT, are considered for hyper-parameter optimization.

Gaussian Process Regression (GPR)
The GPR model is an ML framework employed for classification and regression problems. This is a nonparametric Bayesian method for regression. GPR can work well for small databanks. The GPR model predicts by including prior knowledge, and provides predictions uncertainty measures.

Gradient Boosting Regression Trees (GBRT)
Decision trees are machine learning algorithms that are most famous for their feature selection. These can be used for classification and regression problems. Decision trees for regression problems are known as regression trees. The model starts learning with an increase in the iterations. The training process stops when the hyper-parameters are triggered. GBRTs are iterative algorithms in which each tree takes account of the error in the previous one, and their final outcome is the mean of all trees' predictions.
These algorithms require comparatively less effort in preprocessing data, and can process incomplete data. However, they are expensive in terms of the computational training time.
In order to achieve the optimized deep learning model, the hyper-parameters were tuned by using the Bayesian optimization method. Here, two different Bayesian optimization methods, namely, Gaussian process regression (GPR) and gradient boosting regression trees (GBRT), were used for hyper-parameter tuning. For the models' performance evalua- tion, different error metrics, such as the correlation coefficient (R 2 ), mean absolute error (MAE), and absolute average relative deviation (AARD), were used. Figure 1a,b represents the procedure and flow charts for fine-tuning the hyperparameters by using two different Bayesian optimization methods, such as GPR and GBRT.
training time.
In order to achieve the optimized deep learning model, the hyper-parameters were tuned by using the Bayesian optimization method. Here, two different Bayesian optimization methods, namely, Gaussian process regression (GPR) and gradient boosting regression trees (GBRT), were used for hyper-parameter tuning. For the models' performance evaluation, different error metrics, such as the correlation coefficient (R 2 ), mean absolute error (MAE), and absolute average relative deviation (AARD), were used. Figure 1a,b represents the procedure and flow charts for fine-tuning the hyper-parameters by using two different Bayesian optimization methods, such as GPR and GBRT.
Once the hyper-parameters are optimized, then a deep neural network was developed based on the acquired information of the optimized hyper-parameters.

Hyper-Parameters
These are the parameters whose values are used in controlling the learning process. In order to achieve an optimal model with a short computational time, the hy- Once the hyper-parameters are optimized, then a deep neural network was developed based on the acquired information of the optimized hyper-parameters.

Hyper-Parameters
These are the parameters whose values are used in controlling the learning process. In order to achieve an optimal model with a short computational time, the hyper-parameters' tuning should be optimized, which is a major goal of the BO. In the developed model, the hyper-parameters are activation function, learning rate, Adam decay, number of nodes, no. of hidden layers, no. of neurons in each hidden layer, and batch size, as shown in Figure 1. A pair plot showing the fine-tuned hyper-parameter can be seen in Figure 2. The red circles represent the optimal value for each hyper-parameter. In Figure 2, the yellow region shows a strong relationship (those values can be selected for the hyper-parameter) while the green region represents a weak relationship (those values are not recommended for selection of that hyper-parameter). The optimal hyper-parameters in terms of the partial dependence are highlighted in Figure 2. It can be seen that the no. of dense layers should be 6, the activation function should be ReLU, and so on. The range of hyper-parameters considered in this study can be witnessed in Table 2.
per-parameters' tuning should be optimized, which is a major goal of the BO. In the developed model, the hyper-parameters are activation function, learning rate, Adam decay, number of nodes, no. of hidden layers, no. of neurons in each hidden layer, and batch size, as shown in Figure 1. A pair plot showing the fine-tuned hyper-parameter can be seen in Figure 2. The red circles represent the optimal value for each hyper-parameter. In Figure 2, the yellow region shows a strong relationship (those values can be selected for the hyper-parameter) while the green region represents a weak relationship (those values are not recommended for selection of that hyper-parameter). The optimal hyper-parameters in terms of the partial dependence are highlighted in Figure 2. It can be seen that the no. of dense layers should be 6, the activation function should be ReLU, and so on. The range of hyper-parameters considered in this study can be witnessed in Table  2.

Hyper-Parameter
Range Learning rate 0.0001 to 0.1 Adam decay 0.000001 to 0.01  The method for achieving the best solution and minimizing the error with respect to the number of calls can be seen in Figure 3. It is obvious that the error reaches a minimum level after 40 calls, although the full convergence occurs at the 18th iteration. This represents the potential of the Bayesian optimization technique when considering a large number of input variables for a wide range of data. For instance, timely convergence was achieved for an optimal solution. The architecture of the optimized models is provided in Table 3, which clearly shows that both of the Bayesian optimization techniques, GPR and GBRT, yield the same values of correlation coefficient (R2). Other details regarding learning rate, no. of input and dense nodes, activation function, batch size, Adam decay, and no. of dense layers for both of the algorithms, can be found in Table 3.

Factors Affecting the Boiling Heat Transfer Coefficient
Pool boiling heat transfer mainly relies on the surface morphological features, liquid thermophysical properties, and pool boiling testing conditions (see Figure 4).

Results and Discussions Factors Affecting the Boiling Heat Transfer Coefficient
Pool boiling heat transfer mainly relies on the surface morphological features, liquid thermophysical properties, and pool boiling testing conditions (see Figure 4).
Generally, the porosity of a material is defined by the ratio of void volume to total volume, and various hypotheses about the influence of porosity on the boiling surface have been reported, for instance it can influence the heat transfer through the thickness of the porous coating, the diameter of the coating particles, the number of pores, and pore connectivity. By increasing the porosity, the wetted area is increased, and more nucleation sites become active as it becomes easier to supply the liquid to the nucleation sites, and this increases the number of bubbles and the bubble generation frequency, which result in heat transfer enhancement. At low heat fluxes, boiling performance is improved with the addition of the porous coated layer because of the formation of a large number of bubbles. However, at high heat flux, increases in coating thickness may result in the formation of a vapor blanket and heat transfer deterioration, and this is why the ratio of the coating thickness to particle diameter is calculated and an optimal value is reported [37]. Furthermore, the heat transfer coefficient can also be augmented by increasing the roughness of the boiling surface. This effect was reported for the first time by Jackob in 1931 [38]. A surface is roughened to create bubbles' seeding cavities in order to produce a large number of bubbles, and due to this, the heat transfer coefficient is improved [39,40].  Generally, the porosity of a material is defined by the ratio of void volume to total volume, and various hypotheses about the influence of porosity on the boiling surface have been reported, for instance it can influence the heat transfer through the thickness of the porous coating, the diameter of the coating particles, the number of pores, and pore connectivity. By increasing the porosity, the wetted area is increased, and more nucleation sites become active as it becomes easier to supply the liquid to the nucleation sites, and this increases the number of bubbles and the bubble generation frequency, which result in heat transfer enhancement. At low heat fluxes, boiling performance is improved with the addition of the porous coated layer because of the formation of a large number of bubbles. However, at high heat flux, increases in coating thickness may result in the formation of a vapor blanket and heat transfer deterioration, and this is why the ratio of the coating thickness to particle diameter is calculated and an optimal value is reported [37]. Furthermore, the heat transfer coefficient can also be augmented by increasing the roughness of the boiling surface. This effect was reported for the first time by Jackob in 1931 [38]. A surface is roughened to create bubbles' seeding cavities in order to produce a large number of bubbles, and due to this, the heat transfer coefficient is improved [39,40].
Some of the most influential liquid thermophysical properties in boiling heat transfer are liquid density, specific heat, vapor density, the latent heat of vaporization, surface tension, boiling point, and liquid thermal conductivity. Similarly, for microporous coated surfaces, the morphological parameters of porosity, coating thickness, particle diameter, and surface roughness have a strong influence on the boiling heat transfer coefficient.
The surface morphological parameters affecting the boiling heat transfer coefficient or porous surfaces are porosity, coating thickness, particle diameter, and surface roughness. The impact of individual morphological parameter on the boiling heat transfer assessment of various working fluids is provided in Figure 5, followed by the error density analysis in Figure 6. The impact of liquid thermophysical properties on the BHTC prediction along with error density analysis is given in Figures 7 and 8, while Figure 9 shows the sensitivity analysis of surface inclination and wall superheat. Some of the most influential liquid thermophysical properties in boiling heat transfer are liquid density, specific heat, vapor density, the latent heat of vaporization, surface tension, boiling point, and liquid thermal conductivity. Similarly, for microporous coated surfaces, the morphological parameters of porosity, coating thickness, particle diameter, and surface roughness have a strong influence on the boiling heat transfer coefficient.
The surface morphological parameters affecting the boiling heat transfer coefficient or porous surfaces are porosity, coating thickness, particle diameter, and surface roughness. The impact of individual morphological parameter on the boiling heat transfer assessment of various working fluids is provided in Figure 5, followed by the error density analysis in Figure 6. The impact of liquid thermophysical properties on the BHTC prediction along with error density analysis is given in Figures 7 and 8, while Figure 9 shows the sensitivity analysis of surface inclination and wall superheat. Table 4 presents the individual and combined impacts of surface morphological features, liquid thermophysical properties, and pool boiling conditions on the assessment of heat transfer coefficient. Apparently, the model with all the surface morphological features, liquid thermophysical properties, and pool boiling testing parameters demonstrated the highest accuracy for HTC prediction. The wall superheat is noted to have the maximum impact on the predictive accuracy of the boiling heat transfer coefficient. More specifically, if the wall superheat is dropped from the modeling parameters, the lowest predictive R 2 (0.893%) is achieved. Among the surface morphological parameters, the particle diameter exhibits the strongest influence on the heat transfer coefficient. On the contrary, the surface roughness and coating thickness do not seem to have a strong impact on the pool boiling data of sintered coated porous surfaces for water, refrigerants, and dielectric liquids. In general, the surface features show relatively less influence compared to the liquid thermophysical properties. This is because liquids with totally different thermophysical properties result in very different boiling phenomena. However, liquid thermal conductivity and specific heat cause a noticeable impact on the boiling heat transfer coefficient of porous surfaces. From the above results, it can be stated that liquid thermophysical properties have much more of an effect on the pool boiling phenomenon of sintered coated porous surfaces compared to the morphology of the heater surfaces. In addition, the BHTC is strongly influenced by the surface inclination angle of the heater surface. (c) (d) Figure 9. Impact of (a) surface inclination, (b) wall superheat and error density analysis of (c) surface inclination and (d) wall superheat in the BHTC prediction. Table 4 presents the individual and combined impacts of surface morphological features, liquid thermophysical properties, and pool boiling conditions on the assessment of heat transfer coefficient. Apparently, the model with all the surface morphological features, liquid thermophysical properties, and pool boiling testing parameters demonstrated the highest accuracy for HTC prediction. The wall superheat is noted to have the maximum impact on the predictive accuracy of the boiling heat transfer coefficient. More specifically, if the wall superheat is dropped from the modeling parameters, the lowest predictive R 2 (0.893%) is achieved. Among the surface morphological parameters, the particle diameter exhibits the strongest influence on the heat transfer coefficient. On the contrary, the surface roughness and coating thickness do not seem to have a strong impact on the pool boiling data of sintered coated porous surfaces for water, refrigerants, and dielectric liquids. In general, the surface features show relatively less influence compared to the liquid thermophysical properties. This is because liquids with totally different thermophysical properties result in very different boiling phenomena. However, liquid thermal conductivity and specific heat cause a noticeable impact on the boiling heat transfer coefficient of porous surfaces. From the above results, it can be stated that liquid thermophysical properties have much more of an effect on the pool boiling phenomenon of sintered coated porous surfaces compared to the morphology of the heater surfaces. In addition, the BHTC is strongly influenced by the surface inclination angle of the heater surface.   The choice of the pool boiling parameters is a very important aspect of this study. For instance, to predict the boiling heat transfer coefficient of engineered surfaces subjected to different working fluids of completely different thermophysical properties for a wide range of pool boiling conditions, it is extremely important to include the most influential parameters in the architecture of the model. This is the major limitation of the existing correlations in the literature-that these correlations cannot account for all of the important surface, liquid, and testing parameters, resulting in a very poor predictability for different surface liquid combinations and testing ranges. Other than considering the important surface and liquid features, the proposed model accounts for important pool boiling testing parameters. For example, in the proposed model, wall superheat and surface inclination angle were considered as the inputs. However, the predictions can be done by replacing the wall superheat with the applied heat flux. However, for the investigated data, the predictability of the model with the heat flux was relatively lower than the model with the wall superheat. Hence, wall superheat was chosen as the input parameter.
With the help of the proposed methodology, a highly accurate BHTC can be estimated by incorporating the strongly influencing liquid, surface, and pool boiling testing parameters into the Bayesian optimization-based neural network model. Owing to improved heat transfer performance, microporous surfaces are the prime candidate to remove the high heat fluxes generated through various applications. For instance, 3M copper powder is widely used in industry to form a boiling coated surface. These boiling surfaces can be used to cool microprocessors, LEDs, power electronics, MOSFETs and IGBTs. Other than that, sintered coated surfaces can be used to manufacture high-flux surfaces, employed to reduce plant size by decreasing the number of reboilers required in petrochemical plants.

Conclusions
The objective of this study is to develop Bayesian optimized deep neural network models to perform a sensitivity analysis for finding the most influential parameters in the boiling heat transfer assessment of sintered coated porous surfaces subjected to a variety of high-and low-wetting working fluids, including water, dielectric fluids, and refrigerants, under saturated pool boiling conditions and different surface inclination angles of the heater surface. Some of very specific conclusions are provided as follows.

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The model with all the surface morphological features, liquid thermophysical properties, and pool boiling testing parameters demonstrates the highest R 2 = 0.985 for HTC prediction.

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The wall superheat is noted to have the maximum impact on the predictive accuracy of the boiling heat transfer coefficient. For example, if the wall superheat is dropped from the modeling parameters, the lowest prediction of R 2 (0.893) is achieved.

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The surface morphological features show relatively less influence compared to the liquid thermophysical properties, e.g., liquid thermophysical properties are much more sensitive to the pool boiling phenomenon of sintered coated porous surfaces compared to the morphology of the heater surfaces.

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Particle diameter showed the strongest influence on the heat transfer coefficient compared to the rest of the morphological parameters.

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The BHTC is strongly influenced by the surface inclination angle of the heater surface. • By dropping the surface inclination angle from the modeled parameters, R 2 is reduced to 0.967.

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The proposed methodology can be applied to a wider range of data in order to determine the highly influential surface and liquid parameters for boiling heat transfer assessment.