Interpretable Predictive Model and Multi-Factor Coupling Mechanism of Convective Heat Transfer on Heated Cylinders in Polar Marine Environments

Abstract

In response to the problems of high energy consumption and difficulty in precise regulation of electric tracing anti-icing systems for polar marine engineering equipment in low-temperature, strong-wind, and high-humidity environments, this paper conducts experimental measurement and predictive modeling research on the convective heat transfer characteristics of electric heat-traced circular cylinders in cross-flow. First, a controllable environmental experimental system was set up to obtain 144 sets of steady-state convective heat transfer data under different combinations of wind speed, temperature, humidity, and heat flux density. Based on this, a Nusselt number (Nu) prediction model using a fully connected Deep Neural Network (DNN) was constructed, and its performance was evaluated through five-fold cross-validation. The results show that the DNN model can effectively capture nonlinear mapping relationships among multiple factors, and its prediction accuracy ($R^2$ = 0.9828) is superior to that of traditional machine learning models. Furthermore, the Shapley Additive Explanations (SHAP) method was introduced to analyze the multi-factor coupling mechanisms, quantify the contribution of each input variable to the Nu prediction, and provide a data-driven reference for the optimization of engineering parameters under extreme polar conditions.

1. Introduction

The safe operation of polar marine equipment is frequently threatened by atmospheric icing under extreme conditions of sub-zero temperatures, strong winds, and high humidity. Fully exposed circular components, such as handrails and piping systems, are particularly vulnerable to severe ice accretion. To mitigate this hazard, Electric Heat Tracing (EHT) is widely employed as an active anti-icing strategy [1]. The thermal design and dynamic control of EHT systems rely heavily on accurately estimating the convective heat loss from the cylinder surface to the environment, which is conventionally characterized by the dimensionless Nusselt number (Nu) [2]. Given the extreme polar conditions, precise quantification of this heat transfer process is essential to balance anti-icing effectiveness with energy consumption [3].

In polar environments, the heat transfer process is exceptionally complex due to the simultaneous fluctuations in wind speed, ambient temperature, and relative humidity. Furthermore, the constant heat flux boundary condition imposed by EHT, coupled with temperature-dependent fluid properties, leads to strong nonlinear interactions. Inaccurate Nu prediction can result in either excessive energy consumption or insufficient de-icing performance, both of which are critical for marine safety in harsh environments.

Convective heat transfer from a circular cylinder in cross-flow has been extensively studied as a fundamental problem in fluid mechanics. As comprehensively reviewed by Sparrow et al. [4], numerous classic correlations have been established to formulate the relationship between Nu, Reynolds number (Re), and Prandtl number (Pr), serving as textbook standards for engineering design. However, the applicability of these classic models to polar EHT scenarios is severely limited by extreme environmental factors that induce significant thermophysical variations and complex moisture effects. As highlighted by Zhao et al. [5], predictions using traditional correlations deviate substantially from actual values when severe temperature differences cause fluid property variations. Furthermore, in high-humidity polar environments, the coupling of velocity and moisture alters convective characteristics. As demonstrated by Chen et al. [6], the presence of phase change or moisture interaction exhibits a highly nonlinear dependence on flow regimes, a complexity entirely absent in standard dry-air correlations.

Secondly, the nonlinear coupling of dynamic boundary conditions and specific geometries is rarely considered. Existing studies often adopt a decoupled approach [7]. Fernández-Seara et al. [8] highlighted the necessity of evaluating heat transfer under real operating heating conditions, as transient or varying heating power fundamentally alters local convection compared to idealized steady states. Similarly, while Zhang et al. [9] and Rafi et al. [10] demonstrated that geometric variations (e.g., wall curvature) profoundly dictate heat transfer capacity, these studies were mostly conducted under uncoupled conditions. In actual EHT applications, the constant heat flux boundary interacts synergistically with specific geometries and extreme environmental cooling, creating a coupled thermal response that cannot be captured by superimposing isolated factors.

To address these multi-parameter nonlinearities, a diverging paradigm has recently emerged in the research field. On one hand, traditionalists argue for modifying semi-empirical correlations to maintain physical interpretability [11]. On the other hand, many researchers contend that traditional regression methods suffer from severely limited extrapolation capabilities when applied to unseen, complex multi-factor conditions. Consequently, data-driven approaches like Deep Neural Networks (DNNs) are increasingly viewed as indispensable for capturing underlying physics where theoretical models struggle [12,13]. Nonetheless, machine learning models often function as ‘black boxes,’ limiting their engineering utility. To enhance transparency, Shapley Additive Explanations (SHAP) has been integrated into various thermal studies. For instance, Zhong et al. [14] used SHAP to interpret flow-induced heat transfer on marine cylinders, while Li et al. [15] quantified thermal responses in aerospace cooling channels. For polar environments, Liu et al. [16] recently utilized SHAP to analyze surface thermal loss based on experimentally validated numerical simulations. However, these models primarily rely on simulated data and often overlook the coupled effects of high humidity and constant-power heat tracing (EHT). Consequently, there is still a significant need for interpretable predictive models derived directly from comprehensive physical experimental data to capture these complex multi-factor interactions in polar environments.

To bridge these gaps, this study presents a systematic investigation of the convective heat transfer around heated circular cylinders under simulated polar conditions. A comprehensive dataset was generated using a controlled low-temperature experimental setup, incorporating variables such as wind speed, ambient temperature, relative humidity, heating power, and geometric curvature ratio. Based on this baseline dataset, a DNN model is developed to accurately predict Nu, and the SHAP method is integrated to quantitatively reveal the relative importance and synergistic coupling mechanisms among the inputs.

2. Materials and Methods

2.1. Heat Transfer Principles

Constant-power silicone–rubber electric heating tapes were used as the internal heat source. The heating tape consisted of an alloy resistance heating wire and a silicone–rubber high-temperature insulating sheath. To facilitate the analysis of surface heat transfer, the heating power was converted into an equivalent surface heat flux density acting on the external surface of the circular tube. The convective heat transfer rate was determined from the heat balance:

$$Q_h=Q_c-Q_r$$

(1)

where $Q_h$ is the convective heat transfer rate (W), $Q_c$ is the heating power supplied by the EHT tape (W), and $Q_r$ is the radiative heat transfer rate (W).

The equivalent surface heat flux density was calculated by dividing the convective heat transfer rate by the external heat transfer area of the heated pipe section:

$$q={\displaystyle \frac{Q_h}{A}}={\displaystyle \frac{Q_c-Q_r}{\pi DL}}$$

(2)

where $A$ is the external heat transfer area of the heated circular tube section, $D$ is the outer diameter of the circular tube, $L$ is the effective heated length, and $q$ is the heat flux density (W/m²). Since the radiative heat loss was relatively small in this study, $Q_r$ was neglected, and the imposed heat flux density was obtained as:

$$q\approx {\displaystyle \frac{Q_c}{\pi DL}}$$

(3)

The convective heat transfer coefficient can be calculated using Newton’s cooling formula, which is expressed as:

$$h={\displaystyle \frac{q}{\left(T_w-T_a\right)}}$$

(4)

where $h$ is the convective heat transfer coefficient, (W/(m²·K)); $T_w$ and $T_a$ are the wall temperature and free-stream air temperature (K).

To compare heat transfer efficiency across different operating conditions, h is nondimensionalized to obtain the Nu:

$$Nu={\displaystyle \frac{hD}{k\left(T_f\right)}}$$

(5)

where $k\left(T_f\right)$ is the thermal conductivity of the fluid ($\mathrm{W}/(\mathrm{m}\cdot \mathrm{K})$), evaluated at the film temperature $T_f$, i.e., $T_f=(T_a+T_w)/2$.

2.2. Experimental System

The experiment was conducted on a controlled environmental test system set up in a low-temperature environment laboratory, as shown in Figure 1. The detailed specifications and validation of this experimental system have been described in [17]. The system is mainly composed of an air conditioning unit, a straight-through air duct and test section, a humidification and control module, and a data acquisition system. Variable-frequency fans provide the incoming flow, which passes through rectification and contraction sections before entering the test section, ensuring a uniform and stable velocity profile. Ambient temperature is controlled by a high-power refrigeration unit, covering a range from approximately −40 °C to 0 °C. By coupling an adjustable humidifier with the supply air duct, the relative humidity can be regulated between 10% and 95% to simulate the low-temperature and high-humidity conditions characteristic of polar atmospheres.

Standard test pieces, representing real handrail components from the upper structure of polar vessels, were installed in the test section. Two geometric configurations were evaluated: a horizontal handrail arrangement (Figure 2a) and a curved handrail arrangement (Figure 2b). To isolate the influence of geometry on heat transfer, the wind-induced projected areas of both components were kept identical for a controlled comparison. Constant-power heating tapes were placed inside the tubes to impose a uniform heat flux boundary condition. To minimize parasitic heat loss, the heating tapes were wrapped with expanded nitrile rubber insulation, and both ends of the circular pipe were heavily insulated, ensuring that the generated heat dissipated almost entirely through the exposed outer surface to the ambient air.

Parameter measurements were conducted using a distributed sensing scheme. Five K-type thermocouples were evenly attached along the circumference of the circular tube to monitor the wall temperature distribution [18]. The thermocouple signals were collected by an 18-channel paperless temperature recorder (Sinomeasure, Hangzhou, China), as shown in Figure 3a. The recorder has a measurement range of −50 to 200 °C and an accuracy of ±0.2%. The temperature data were recorded at a frequency of 1 Hz. The free-stream temperature, relative humidity, and wind speed were measured independently using a temperature probe, a relative humidity monitor, and an NK1000 anemometer (Kestrel, Nielsen-Kellerman, Boothwyn, PA, USA), respectively. The relative humidity monitor, shown in Figure 3b, has a relative humidity measurement range of 5–100% RH and an accuracy of ±3% RH. The NK1000 anemometer, shown in Figure 3c, has a wind-speed measurement range of 0.4–40 m/s and an accuracy of ±3%.

During the experiment, the ambient temperature, relative humidity, and wind speed were first set and allowed to stabilize. Once environmental fluctuations were within target ranges, the EHT system was activated. Thermal equilibrium was considered achieved when the fluctuation of each wall temperature channel remained within 0.5 °C cover a 10 min period. Following this, data were recorded. By systematically varying the controlling variables, experimental data covering the typical polar operating envelope were obtained for both handrail geometries. Each condition was repeated three times to minimize random errors, and the average was used as the representative data. Thus, the 144 samples in the final dataset correspond to 144 distinct averaged operating conditions, with no repeated measurements included.

A comprehensive uncertainty analysis was performed based on the law of error propagation to evaluate the reliability of the calculated Nu. The relative uncertainties of the independent measurable variables are listed in

Table 1. Uncertainty Analysis of Measured Physical Quantities.

The Physical Quantity Being Measured	The Symbol of the Physical Quantity	Relative Uncertainty	Sources of Uncertainty
Heating power (W)	$Q_c$	2%	Power fluctuations
Outer diameter of the round pipe (m)	D	0.5%	Pipe diameter tolerance
Wall temperature (K)	$T_w$	0.6%	K-type thermocouple accuracy
Fluid temperature (K)	$T_a$	0.6%	Environmental inhomogeneity

.

Based on the above measurements, the heat flux density at the surface of the circular pipe, the temperature difference, and ultimately the Nu can be calculated [19]. These quantities are parameters calculated based on the measurements, and the uncertainty is derived from the propagation of the measurement error through a functional relationship.

$${\displaystyle \frac{u\left(q\right)}{q}}=2.1\%, {\displaystyle \frac{u(T_w-T_a)}{T_w-T_a}}=0.85\%$$

(6)

By substituting Equation (4) into Equation (5), the Nusselt number can be expressed as a function of the heat flux density, tube diameter, and temperature difference:

$$Nu={\displaystyle \frac{qD}{k(T_w-T_a)}}$$

(7)

According to the uncertainty propagation method, the combined standard uncertainty of $Nu$ can be written as:

$$u(Nu)=\sqrt{{\left({\displaystyle \frac{\partial Nu}{\partial q}}u(q)\right)}^2+{\left({\displaystyle \frac{\partial Nu}{\partial D}}u(D)\right)}^2+{\left({\displaystyle \frac{\partial Nu}{\partial \Delta T}}u(\Delta T)\right)}^2}$$

(8)

The temperature difference is defined as $\Delta T=T_w-T_a$.

The corresponding partial derivatives are:

$${\displaystyle \frac{\partial Nu}{\partial q}}={\displaystyle \frac{D}{k\Delta T}}={\displaystyle \frac{Nu}{q}},{\displaystyle \frac{\partial Nu}{\partial D}}={\displaystyle \frac{q}{k\Delta T}}={\displaystyle \frac{Nu}{D}},{\displaystyle \frac{\partial Nu}{\partial \Delta T}}=-{\displaystyle \frac{qD}{k\Delta T^2}}=-{\displaystyle \frac{Nu}{\Delta T}}$$

(9)

Therefore, the relative uncertainty of Nu is obtained as:

$${\displaystyle \frac{u\left(Nu\right)}{Nu}}=\sqrt{{\left({\displaystyle \frac{u\left(q\right)}{q}}\right)}^2{+\left({\displaystyle \frac{u\left(D\right)}{D}}\right)}^2{+\left({\displaystyle \frac{u(T_w-T_a)}{T_w-T_a}}\right)}^2}$$

(10)

Substituting the results into the equation gives ${\displaystyle \frac{u\left(Nu\right)}{Nu}}=2.3\%$.

The results show that within the working conditions of this study, the relative synthetic uncertainty of Nu is approximately 2.3%, which is consistent with the typical range of similar convective heat transfer experiments, indicating that the test system and measurement method can meet the requirements for data accuracy in convective heat transfer studies in polar environments and can serve as a reliable basis for machine learning model training and interpretability analysis.

2.3. Dataset and Feature Selection

Despite abundant research on convective heat transfer under conventional conditions, experimental datasets reflecting polar conditions—especially those accounting for the coupled effects of low temperature, high humidity, strong wind, and component geometry—are practically non-existent in the public domain. Consequently, the dataset used in this study comprises 144 sets of valid experimental data exclusively generated from the aforementioned experiments. The experimental parameter ranges are listed in

Table 2. Experimental parameter ranges covered by the dataset.

Variable	Symbol	Range	Unit
Free-stream air temperature	$T_a$	−40 to −5	°C
Wind speed	$U$	2 to 9.5	m/s
Relative humidity	$\Phi$	10 to 95	%
Heat flux density	$q$	758, 1517, 2807	W/m2
Curvature ratio	$\delta$	0, 0.336	/

.

The objective of feature selection is to extract key factors that govern the convective heat transfer process while balancing physical significance with engineering measurability. Therefore, the input features are categorized into two groups. The first group includes external environmental factors determining the flow field and thermal driving force: inflow wind speed ($U$), ambient temperature ($T_a$), and relative humidity ($\Phi$). The second group encompasses component-specific factors defining the thermal boundary and local flow structures: surface heat flux density ($q$) and the geometric curvature ratio ($\delta$, defined as the ratio of the pipe’s outer diameter to its radius of curvature).

The target output is the Nu. Consequently, the machine learning task is formulated as a regression prediction problem: $Nu=f(U,T_a,\Phi ,q,\delta )$. Since $U$, $T_a$, $\Phi$, and $q$ can be monitored and adjusted in real time during vessel operation, and $\delta$ is known from the design phase, a predictive model based on these features possesses excellent practical applicability for EHT control.

2.4. Machine Learning Algorithms

To systematically assess the capability of modeling the nonlinear multi-factor relationships, models of varying complexity were trained and compared. These include Linear Regression (LR), Decision Tree (DT), Random Forest (RF) [12], and a fully connected Deep Neural Network (DNN).

To ensure a fair comparison, all models underwent a consistent data preprocessing pipeline. After data cleaning, input features were standardized to accommodate scale-sensitive models. Model performance was evaluated using five-fold cross-validation, utilizing the coefficient of determination ($R^2$), Mean Absolute Error (MAE), and Root Mean Square Error (RMSE) as quantitative metrics. In this process, the 144 samples were randomly divided into five non-overlapping subsets; each fold iteratively served as the validation set while the remaining four were used for training.

Hyperparameters for the DT and RF models were optimized using Bayesian optimization to prevent manual bias and control model complexity. The DNN architecture comprised an input layer, four hidden layers (utilizing the Rectified Linear Unit (ReLU) activation function with a progressively decreasing number of neurons: [256, 128, 64, 32], and an output layer. Batch normalization and dropout regularization (with a rate of 0.1) were implemented to mitigate overfitting. The Adam optimizer was employed with an initial learning rate of 0.001, which was dynamically reduced when the validation loss plateaued. This rigorous framework ensures that the optimized model provides a highly accurate and stable foundation for the subsequent SHAP analysis.

It should be noted that the present DNN model was trained and validated within the experimental parameter ranges listed in Table 2; predictions for conditions substantially beyond these ranges may require additional experimental data for model recalibration.

All algorithms were implemented in Python 3.10.19 using PyTorch 2.5.1 for the DNN model, scikit-learn 1.7.2 for the traditional machine learning models (LR, DT, RF), and SHAP 0.49.1 for interpretability analysis. Data processing and visualization were performed using NumPy 2.0.1, Pandas 2.3.3, Matplotlib 3.10.8, and tqdm 4.67.1.

2.5. SHAP Analysis Methods

Limited interpretability remains a key obstacle to the widespread adoption of machine learning algorithms in practical engineering applications. This study uses the SHAP method to decompose the model output. Based on the concept of Shapley value in cooperative game theory, SHAP assigns a numerical value to each input variable corresponding to its contribution, providing an additive interpretation of the model’s predictions. SHAP, which provides information on global feature importance, local feature contribution, and interaction between features, is one of the mainstream methods for interpreting machine learning models at present [20].

The mathematical expression for calculating the Shapley value of a specific feature in a given model is represented as

$${\varnothing }_i=\sum _{S\subseteq F\backslash \left\{i\right\}}{\displaystyle \frac{\left|S\right|!\left(\left|F\right|-\left|S\right|-1\right)!}{\left|F\right|!}}\left[f\left(S\bigcup \left\{i\right\}\right)-f\left(S\right)\right]$$

(11)

where $F$ denotes the set of all features; $F\setminus \left\{i\right\}$ is the set obtained by excluding feature $i$ from $F$; $S$ is a subset of $F\setminus \left\{i\right\}$; $\mid F\mid$ and $\mid S\mid$ denote the cardinalities of $F$ and $S$, respectively, and $\mid F\mid !$ and $\mid S\mid !$ are their factorials; $f\left(S\right)$ is the predicted value generated by the model using feature subset $S$; and $f(S\cup \{i\left\}\right)$ is the predicted value generated by the model using subset $S$ augmented with feature $i$.

3. Results and Discussion

3.1. Experimental Heat-Transfer Characteristics

The experimental dataset covered a Reynolds number range from $6.76\times {10}^3$ to $3.87\times {10}^4$, and the corresponding Nusselt number ranged from approximately 42.7 to 399.0. This indicates that the experiments included a wide range of forced-convection conditions around the heated circular tube. The Reynolds number increased mainly with the free-stream velocity, and the Nusselt number showed a clear increasing trend as the airflow became stronger. For example, at $T_a=263.15$ K and $\Phi =0.95$, the Nusselt number increased from 54.66 to 159.40 when the air velocity increased from 2 m/s to 8 m/s. At $T_a=233.15$ K and $\Phi =0.95$, the Nusselt number increased from 64.42 to 209.35 over the same velocity range. These results demonstrate that air velocity is the dominant factor affecting convective heat transfer from the tube surface.

The ambient temperature and relative humidity also influenced the heat-transfer characteristics. At a fixed air velocity and relative humidity, lower ambient temperature generally resulted in a higher Reynolds number and a larger Nusselt number due to the variation in air thermophysical properties. For instance, at $U=8$ m/s and $\Phi =0.95$, the Nusselt number increased from 159.40 at $T_a=263.15$ K to 209.35 at $T_a=233.15$ K. In addition, increasing relative humidity generally enhanced the measured Nusselt number. At $U=5$ m/s and $T_a=263.15$ K, the Nusselt number increased from 81.89 to 109.09 as $\Phi$ increased from 0.10 to 0.95. These experimental trends indicate that wind speed, ambient temperature, and relative humidity should all be considered when evaluating the convective heat-transfer behavior of EHT-heated tubes.

The influence of the surface or installation condition represented by $\delta$ was also observed. Under comparable operating conditions, the cases with $\delta =0.336$ generally produced higher Nusselt numbers than those with $\delta =0$, suggesting that the corresponding surface or installation configuration may alter the local thermal boundary layer. These experimental observations provide the physical basis for the subsequent empirical correlation and data-driven prediction of the Nusselt number.

3.2. Predictive Model Performance

Based on valid experimental data, this paper compares the predictive performance of LR, DT, RF and DNN models.

Table 3. Predictive performance comparison of different machine learning models.

Model	R2	MAE	RMSE
LR	0.892	26.22	31.79
DT	0.973	10.93	15.85
RF	0.977	9.51	14.86
DNN	0.983	7.96	11.58

presents the results of each model under a uniform evaluation metric.

It can be seen from the results that the R² and error metrics of the LR model are significantly worse than those of the nonlinear model, with R² at 0.892 and MAE and RMSE at 26.22 and 31.79, respectively. This indicates that the convective heat transfer process of tubular components in polar environments is difficult to describe precisely through simple linear relationships, and there is a significant nonlinear coupling effect among wind speed, heat flux density, geometric curvature ratio and environmental parameters, which requires more complex nonlinear models for characterization.

After the introduction of the decision tree model with nonlinear partitioning ability, the prediction performance was significantly improved, R² increased to 0.973, and the error index decreased significantly, indicating that by segmenting the feature space, DT has been able to capture the main nonlinear laws in the convective heat transfer process. However, a single decision tree is vulnerable to local noise in the training data, and its generalization performance still has some uncertainty.

The random forest based on the ensemble idea smoothed the prediction results by averaging multiple decision trees, with R² increasing to 0.977 and MAE and RMSE decreasing to 9.51 and 14.86, respectively, indicating enhanced stability of the model across the entire operating range. The RF model demonstrated that overfitting could be effectively suppressed and the robustness of multi-factor coupled heat transfer prediction could be improved by introducing feature and sample randomness on a medium-sized dataset.

The DNN model demonstrated the best overall performance, with a coefficient of determination R² of 0.983 and MAE and RMSE reduced to 7.96 and 11.58, respectively. The RMSE index of the DNN model was reduced by approximately 22% compared to the ensemble tree model. This result confirms that deep networks, through multi-layer nonlinear mapping, can more precisely fit the variation of Nu with parameters such as wind speed and heat flux density. DNN is sensitive to training strategies and regularization Settings in small sample conditions [21]. Therefore, this paper controls the training process by means of BN (Batch Normalization), Dropout, and adaptive attenuation of the learning rate, and uses cross-validation to evaluate performance stability.

To further demonstrate the stability of the model and the characteristics of the training process, a box plot of the five-fold cross-validation results is presented in Figure 4.

The median within the box represents the typical predictive performance of the DNN model, while the box height and whisker length reflect the range of variation in performance metrics across different cross-validation folds. The red diamond markers indicate the mean value of the respective metric across the five folds. The box plot indicates that the distribution of all performance metrics (R², MAE, RMSE) for the DNN model is relatively concentrated across both the training and validation sets, suggesting that the model’s predictions are insensitive to random partitioning of the dataset.

The graph in Figure 5 shows the DNN training history curve. Judging from the curve shape, the training error and validation error decreased rapidly with iteration in the early stage, indicating that the model was able to learn the main mapping rules from the data; In the middle and later stages, the two curves gradually flatten and remain relatively stable, indicating that the optimization process has entered the convergence stage. If the training error continues to decline while the validation error begins to rise at a certain stage, it usually corresponds to the risk of overfitting, that is, the model fits the training samples better, but the generalization ability for the unseen samples decreases. The curves in this paper show that the training and validation errors change in the same direction as a whole and tend to stabilize, indicating that strategies such as batch normalization, Dropout, and adaptive decay of the learning rate have suppressed overfitting to some extent, making the network training process smoother.

Figure 6 compares the experimentally measured Nusselt numbers with the out-of-fold predictions from the five-fold cross-validation. Each predicted value comes from a DNN model that was trained without seeing that specific sample. The overall coefficient of determination computed from all 144 out-of-fold predictions is $R^2=0.983$, with a MAPE of 4.3%. Most points lie close to the diagonal line and within the ±4.3% error band, indicating good generalization within the experimental parameter range. The blue dots represent individual out-of-fold predictions. A small scatter appears at higher Nu values but without systematic bias, confirming stable predictive performance.

To validate the rationality of the hyperparameter selection for the DNN model and evaluate its training robustness, a single-factor sensitivity analysis was conducted. Specifically, while holding other hyperparameters at their optimal baseline values, each individual parameter was adjusted to observe the variation patterns of the model’s predictive performance (measured by $R^2$ and RMSE). Figure 7 provides a visual representation of the model sensitivity, where Figure 7a–d illustrate the trends of $R^2$ and RMSE as functions of learning rate, dropout rate, batch size, and weight decay.

The learning rate plays a decisive role in the model’s convergence behavior and predictive accuracy. When the learning rate is too low (${10}^{-4}$), the model suffers from severe under-convergence, resulting in a negative $R^2$ and an extremely high RMSE. When increased to ${10}^{-3}$, the model reaches its optimal convergence state, characterized by a sharp peak in $R^2$ and a significant drop in RMSE. Further increasing the learning rate causes a slight degradation in performance, indicating that an excessively large learning rate makes the optimizer oscillate around the global optimum. Therefore, ${10}^{-3}$ was selected as the baseline to balance convergence speed and precision.

The dropout rate primarily determines the model’s ability to resist overfitting. At a dropout rate of 0.1, $R^2$ reaches its maximum while RMSE drops to its minimum. This setting effectively suppresses overfitting while preserving the model’s capacity to learn complex nonlinear heat transfer laws. As the dropout rate exceeds 0.1, $R^2$ gradually decreases and RMSE rises, implying that excessive dropout impairs the model’s feature extraction capabilities. Notably, within the tested range, $R^2$ consistently remains above 0.93, demonstrating excellent robustness against overfitting.

The batch size directly affects the stability of gradient estimation and the model’s generalization capability. As the batch size increases, $R^2$ exhibits a steady upward trend while RMSE continuously decreases. A larger batch size effectively stabilizes gradient calculations and enhances the model’s generalization to the experimental data. Consequently, a batch size of 64 was chosen as the baseline to achieve an optimal balance between training efficiency and predictive accuracy.

The weight decay shows a relatively minor overall impact on model performance within the tested range, with $R^2$ remaining stable and RMSE showing no massive fluctuations. This proves that the model’s parameter matrices possess strong inherent generalization capabilities. Although a weight decay of ${10}^{-5}$ yields marginally better metrics on the current dataset, such a weak regularization effect may risk reducing generalization. Considering the engineering need for hyperparameter robustness, a weight decay of $5\times {10}^{-4}$ was selected. This value maintains stable predictive performance across a broader range, achieving a better trade-off between fitting accuracy and generalization for predicting unknown conditions in extreme polar environments.

In summary, the hyperparameter sensitivity analysis in Figure 7 confirms the rationality of the baseline settings for the DNN model. It also demonstrates that the optimized model is robust to minor parameter perturbations, establishing a reliable model foundation for the subsequent SHAP interpretability analysis.

3.3. SHAP-Based Interpretability Analysis

This section systematically presents the internal data-driven mechanisms and feature dependencies of the optimized machine learning model from multiple perspectives, utilizing global SHAP values, SHAP interaction values, and stratified analysis.

3.3.1. Global Analysis

According to the global SHAP summary dot plot (Figure 8), the input variables exhibit distinct distribution widths and color gradients, visually reflecting their impact directions and magnitudes on the Nu. To quantitatively unravel the internal structure of these impacts, the total SHAP values were decomposed into independent main effects and interaction effects. Furthermore, the relative importance of each feature was calculated by normalizing its total SHAP value against the sum of all features, as detailed in

Table 4. Quantitative breakdown of mean absolute SHAP values and relative importance.

Feature	Mean Absolute Main Effect	Mean Absolute Interaction Effect	Total Mean Absolute SHAP Value	Proportion of Main Effect (%)	Relative Importance (%)
$U$	60.9	9.8	70.7	86.1	51.1
$q$	26.5	7.6	34.1	77.7	24.6
$T_a$	10.1	3.7	13.8	73.2	10.0
$\delta$	10.8	2.0	12.8	84.4	9.2
$\Phi$	3.7	3.4	7.1	52.1	5.1

.

Based on the quantitative results in Table 4, wind speed ($U$) is unequivocally the paramount factor governing convective heat transfer, accounting for 51.1% of the overall predictive importance. Its independent main effect constitutes 86.1% of its total influence, indicating that its contribution primarily manifests as a direct action. This aligns perfectly with the classical boundary layer theory, wherein forced convection dictates the baseline heat transfer intensity. As clearly shown in Figure 8, the high-value points for wind speed (red dots) are distributed far to the right on the positive SHAP axis, indicating a strong positive correlation with heat transfer enhancement.

Heat flux density ($q$) emerges as the second most important feature, explaining 24.6% of the model’s global variance. Interestingly, while its main effect accounts for 77.7%, the overall trend observed in Figure 8 indicates an inverse effect: the red dots (high heat flux) mostly correspond to negative SHAP values, while the blue dots (low heat flux) result in positive SHAP values. Physically, this phenomenon reflects that a rapid rise in wall temperature—induced by high heat flux—causes an increase in the kinematic viscosity and a decrease in the density of the near-wall air film. This elevates the local thermal resistance and slightly degrades the dimensionless heat transfer capacity ($Nu$) when the bulk convective flow strength is constrained.

Ambient temperature ($T_a$) and geometric curvature ratio ($\delta$) serve as critical secondary regulatory factors, contributing 10.0% and 9.2% to the global importance, respectively. The curvature ratio exhibits a highly independent positive contribution (red dots on the right in Figure 8), which is fundamentally associated with the perturbation of the main flow field and the modulation of the boundary layer structure induced by geometric bending. Conversely, the negative correlation trend of ambient temperature ($T_a$) is consistent with the thermodynamic principle that a lower ambient temperature yields a higher thermal driving force ($\Delta T$) under constant heat flux conditions.

Relative humidity ($\Phi$) demonstrates the weakest overall impact (5.1%). Notably, its interaction effect constitutes nearly half of its contribution (47.9%), indicating that its absolute main effect is insufficient to independently alter the trend of $Nu$ within the current spectrum of operating conditions. The minor influence of humidity is largely coupled with other thermal parameters in the absence of significant phase changes.

3.3.2. Interaction Effects

By quantifying the pairwise coupling between features based on the SHAP interaction heatmap (Figure 9), the interaction between wind speed and heat flux density ($U-q$) emerges as the most significant term. Its mean absolute interaction value reaches 5.19, which is substantially higher than any other feature pair. This robust interaction reflects a profound synergistic enhancement between flow kinematics and thermal loading. Specifically, wind speed enhances the convective capacity by thinning the thermal boundary layer, while heat flux determines the thermal potential of the system. When both parameters increase simultaneously, their combined positive effect on $Nu$ is amplified profoundly.

In addition to this dominant pair, several secondary interactions with distinctly smaller but still non-negligible magnitudes are observed, including $U-T_a$ (1.87), $U-\Phi$ (1.60), $U-\delta$ (1.21), and $\Phi -q$ (1.16). These interactions are of substantially lower magnitude than $U-q$ and function as auxiliary coupling pathways that modify how $Nu$ responds to the dominant variables under specific operating conditions. The $U-T_a$ interaction reflects the fact that the cooling effect of airflow depends partly on the ambient thermal background, with lower temperatures amplifying the effective thermal gradient. The $U-\Phi$ indicates that humidity modifies the convective heat transfer response to wind speed indirectly, through its influence on the thermal conductivity, density, and heat capacity of the air. The $U-\delta$ interaction indicates that geometric curvature alters the local flow organization and therefore changes how wind speed translates into convective enhancement. Meanwhile, the $\Phi -q$ interaction suggests that humidity exerts a more noticeable influence under elevated heat flux conditions, likely because the associated rise in wall temperature amplifies the contrast in near-wall thermophysical properties between moist and dry air.

To complement the quantitative interaction matrix in Figure 9, Figure 10 presents a SHAP interaction network that visualizes the relative strength of couplings among variables in a more intuitive format. In this network, each node corresponds to an input feature, with larger nodes indicating stronger main effects. Edges between nodes represent pairwise SHAP interactions, with thicker and more opaque edges denoting larger interaction magnitudes. The color of each edge indicates the interaction direction: red for positive synergistic coupling and blue for negative antagonistic coupling. In this representation, the $U-q$ edge stands out as both the thickest and most vividly colored connection, visually confirming its role as the primary interaction. The noticeably thinner and paler edges linking $T_a$, $\Phi$, and $\delta$ to the dominant features make the distinction between primary and secondary couplings immediately apparent.

3.3.3. Stratified Analysis

Because heat flux density and geometric curvature may alter the interaction hierarchy under different operating regimes, a stratified SHAP analysis was further conducted to examine how the primary and secondary interactions are locally reshaped. The purpose is not to overturn the global ranking, but to reveal how the interaction patterns shift under specific thermal and geometric conditions.

Under moderate heat flux and straight geometry ($\delta =0$), wind speed and heat flux density remain the leading influencing factors, and the primary interaction is still $U-q$, followed by $U-T_a$. This indicates that under relatively regular operating conditions, the convective heat transfer process is primarily controlled by the balance between external flow intensity and thermal driving, with ambient temperature playing a secondary regulatory role. In the curved handrail configuration ($\delta =0.336$), the importance of geometric curvature rises markedly, and the $U-\delta$ interaction becomes the strongest coupled pair within this regime, surpassing even $U-q$. This local shift indicates that, under curved geometry, the interaction between wind speed and curvature outweighs that between wind speed and heat flux, likely because modifications of the flow field by the curved surface alter the way convective transport responds to the external flow.

Under high heat-flux conditions, the $U-q$ interaction is further intensified, confirming that strong thermal loading amplifies the coupling between aerodynamic cooling and wall heating. At the same time, humidity-related interactions such as $U-\Phi$ and $\Phi -q$ become more noticeable, implying that moisture effects are more likely to emerge when the thermal state is sufficiently elevated.

Overall, the stratified results confirm that the global SHAP hierarchy remains broadly valid, while the relative prominence of secondary interactions can shift substantially under specific geometric or thermal regimes. This demonstrates that the heat-transfer response is governed not only by a stable dominant mechanism, but also by context-dependent local interaction reorganization.

3.4. Practical Implications for EHT Control

To further connect the heat-transfer results with practical EHT operation, the convective heat loss from the pipe surface was estimated based on the measured heat-transfer characteristics. Compared with the Nusselt number alone, the estimated heat loss provides a more direct engineering reference for determining the required heating tape output. For a typical straight handrail segment ($D=0.042$ m, $L=0.25$ m), the required power to compensate for convective heat loss increased from approximately 9 W to 23 W as the wind speed increased from 2 to 8 m/s, representing an increase of about 150%. This result confirms that stronger airflow substantially elevates the heating demand of EHT systems.

From a control perspective, the heating tape should therefore be adjusted according to the local convective cooling condition rather than operated at a fixed output. Under high wind-speed conditions, the heating tape power or duty cycle should be increased to compensate for the enhanced heat loss. Under low wind speed or mild ambient conditions, the heating output can be reduced to avoid overheating and unnecessary energy consumption. For practical pipeline systems, ambient temperature, wind speed, and relative humidity can be used as feed-forward inputs, while the measured pipe surface temperature can be used as feedback to correct the heating tape output. For long pipelines exposed to nonuniform outdoor environments, zoned control is recommended to improve temperature uniformity and energy efficiency.

4. Conclusions

This study investigated the convective heat transfer characteristics of electric heat-traced circular cylinders under simulated polar marine conditions. Based on the steady-state experimental data, a DNN model was developed to predict Nu, and SHAP analysis was used to interpret the learned nonlinear relationships. The main novelty of this work is the establishment of an experimentally based DNN framework that is interpretable by SHAP for both predicting and explaining convective heat transfer of EHT circular cylinders under coupled polar environmental conditions. The principal conclusions are drawn as follows:

(1)

The proposed DNN model achieved high prediction accuracy for Nu, with an R² of 0.983, an MAE of 7.96, and an RMSE of 11.58 on the testing set. Compared with conventional machine learning methods, the DNN model better captured the nonlinear relationship between environmental/operating parameters and convective heat transfer.

(2)

SHAP analysis identified wind speed as the dominant factor affecting Nu, with its independent main effect accounting for 86.1% of its total contribution. Heat flux density and geometric curvature ratio were secondary but important factors, whereas ambient temperature and relative humidity showed relatively weaker global effects.

(3)

SHAP interaction analysis further revealed that the strongest coupling effect occurred between wind speed and heat flux density, with a peak average absolute interaction value of 5.19. This indicates that the combined increase in flow velocity and thermal load significantly enhances convective heat transfer.

(4)

The present work provides a transferable data-driven framework for heat loss estimation and thermal management of EHT systems on polar marine equipment, but its validity is limited to the experimental parameter ranges investigated. Extrapolation to substantially different conditions requires further validation.

Future work will extend the model to broader conditions, incorporate unsteady thermal processes including dynamic icing and melting, and include phase-change effects, including condensation and frosting, to improve completeness and extrapolation capability.