Thermal management is becoming a bottleneck in several industries such as high-power density electronics [1], electric vehicles [2] , data centers [3], etc. Traditional single-phase methods are insufficient to meet the high-efficiency cooling demands of high-energy-density devices. Due to their high latent heat, two-phase methods have emerged as an alternative solution for advanced cooling systems and are in some cases implemented (e.g. immersion cooling of data centers [4]). As a representative method of two-phase 1 Corresponding author. E-mail: hanhu@uark.edu (H. Hu), heat dissipation, pool boiling has shown to be highly effective, as it operates with a high heat flux while maintaining a relatively low superheat in the nucleate boiling regime. However, there are several drawbacks when considering the feasibility of utilizing boiling for cooling. Boiling is a complex phenomenon involving intricate interactions among heat transfer, fluid dynamics, and phase change. Variations in surface conditions, fluid properties, and heat flux can lead to unpredictable safety issues, such as localized dry-out or boiling hysteresis. Additionally, there are instabilities associated with boiling that can have detrimental impacts on cooling.
One important instability is the critical heat flux (CHF) which marks the transition from the nucleate regime. At this point, a vapor layer begins forming over the heating surface, acting as an insulator.
Consequently, the temperature of the cooled equipment can rise hundreds of degrees within a short period causing overheating or burnout [5,6]. To avoid the triggering of CHF, current pool boiling applications usually operate under a high safety factor, which means the benefit of nucleate boiling is not fully realized.
Extensive research has been conducted to improve pool boiling heat transfer efficiency and reduce the risks of CHF, including surface modification [7], additives to coolants [8], enhancing fluid dynamics [9], and developing thermal management systems [10]. Among them, thermal management systems stand out for its real-time adjustments and versatile adaptability to diverse scenarios, which can prevent CHF while ensuring the cooling process operates within its maximum efficiency range. These features make thermal management particularly advantageous in handling transient conditions and heat load fluctuations. To realize the full potential of thermal management systems, accurate monitoring of the boiling process is essential, as it underpins the system's ability to respond effectively to dynamic boiling states.
Traditional Nomenclature Abbreviations l Number of neurons AE Acoustic Emission m Amount of hits in an experiment CNN Convolutional Neural Network n Number of thermocouples CHF Critical Heat Flux N Sequence length FFT Fast Fourier Transform p Number of specified units GPR Gaussian Process Regression q Heat flux HDT Hit Definition Time σ Activation function LSTM Long Short-Term Memory Ti Temperature MLP Multilayer Perceptron 𝑡 𝑖 ′ Time relative to thermocouple MSE Mean Squared Error 𝑡 𝑖,1 Time of Hi start RFR Random Forest Regression 𝑡 𝑖,𝑤𝑙 Time at end of Hi Symbols Wl Length of AE waveform A Total amount of sequences zi Thermocouple position C LSTM cell state LHP Length of hydrophone spectrum Fl Length of AE waveform spectrum R² Coefficient of performance ft LSTM forget gate α Stride h LSTM cell output W Weight matrix Hi AE Hit recorded waveform b Bias vector 𝐻 𝑡 ′ Frequency intensity of Hi LMAE Loss function it LSTM input gate n Number of thermocouples k Thermal conductivity N Sequence length boiling monitoring methods (e.g. thermocouples, thermistors) are intrusive which can lead to interference with boiling dynamics and lead to replacement difficulties. Nonintrusive methods (e.g. optical and acoustic) have been explored for analyzing and monitoring boiling.
Boiling image data has been shown to encode heat transfer information. McHale and Garimella [11] found heat flux correlations from physical descriptors (e.g. bubble departure diameter, void faction)
extracted from optical images. The introduction of machine learning and computer vision techniques have aided the speed and complexity of analysis. It has led to the utilization of optical data for various boiling monitoring tasks such as CHF detection. Hobold and Silva [12] reported that dimensionality reduction of bubble images using principal component analysis (PCA) allows the extracted principal components to be effectively utilized by support vector machine (SVM) and multilayer perceptron (MLP) classifiers for distinguishing between natural convection, nucleate boiling, and film boiling states. Hobold and Silva [13],
then, proposed using Bayesian statistics to further improve the accuracy of film boiling detection based on convolutional neural networks (CNNs) and boiling images. To improve model generalizability, Al-Hindawi et al. [14] developed a GAN-based domain adaption framework for improving regime classification accuracy on cross-domain image datasets.
In addition to these cases where boiling images are used for classification/detection, they are also commonly employed to map heat flux. For instance, Hobold and Silva [15] used pool boiling images and machine learning models (e.g. MLP and CNN) to develop a real-time heat flux prediction module. Suh et al. [16] used CNN to extract abstract features from images, combined them with physical features of bubbles (e.g. bubble size, count) extracted using Mask R-CNN, to predict heat flux. They demonstrated that the inclusion of physical features led to model improvement. Dunlap et al. [17] developed several models for monitoring heat flux using image sequences and demonstrated that the PCA-CNN model achieves the best performance. Despite the widespread use of boiling images in experimental studies, their practical application faces several limitations. First, acquiring optical images imposes strict requirements on the experimental environment, such as lighting conditions, viewing angles, and background interference.
Second, under high heat flux or intense boiling conditions, multiple bubbles may overlap or occlude each other, making feature extraction challenging and reducing detection accuracy. Additionally, the processes of bubble formation, growth, and collapse during boiling are very rapid, making it difficult to capture the full dynamics using static images or low-frame-rate videos. High-quality high-speed cameras are required to overcome this limitation; however, these devices are expensive and prone to damage in harsh environments, such as those with high temperatures and humidity. Therefore, the challenges faced by optical imaging in boiling monitoring highlight acoustic monitoring as a superior alternative.
Acoustic monitoring has more recently emerged as an approach for boiling understanding. In addition to offering a new perspective on boiling, it is compact, non-intrusive, lightweight, and cost-effective, making it a more practical option for potential industrial applications. Hydrophones, acoustic emission (AE) sensors, and microphones [18][19][20] all belong to this category. Hydrophones have been used for uncovering boiling acoustic characteristics [21]. Tang et al. [22] used Fourier transform and wavelet transform to analyze boiling acoustic frequency and boiling modes. They found key takeaways such as the boiling sound intensity in microbubble emission boiling was significantly higher than other modes. Sinha et al. [23] used hydrophone and optical data to study boiling under different subcooling conditions, identifying distinct peak frequencies for the nucleate boiling regime and a unique explosive boiling mode. Hydrophone data has also been used for CHF detection. Sinha et al. [24] identified a peak frequency shift at the boiling crisis and then later developed a CNN [25] using hydrophone spectrograms to classify boiling regimes, enabling an advanced prediction system to power down and prevent CHF. Ueki and Ara [26] developed a classification model based on hydrophone data to distinguish boiling state transitions. Their proof of concept shows the possibility of monitoring systems via acoustic signals. Dunlap et al. [27] demonstrated the feasibility of using hydrophone data for acoustic heat flux monitoring but was limited by a small dataset.
Ono et al. [28] used hydrophone frequency and cepstrum data with classification and regression models to predict microbubble emission boiling heat flux, finding cepstrum data to be more accurate and noise resistant.
While hydrophones are non-intrusive to the heater, they still need to be submerged in the liquid pool, which presents practical limitations. In contrast, AE sensors offer non-intrusive sensing by being mounted externally to the setup, recording acoustics from solid materials. However, AE sensors are high frequency sensors and due to the high sampling rates not all the data is saved. Instead, shorter waveforms or features are extracted from detected hits are stored. This data type will be further described in the next section.
Several fields have adopted transient sampling methods for fault detection and system monitoring. Signals from AE sensors are used for rail system fault detection [29], crack growth prediction [30], pipe leak detection [31] and predicting the useful life of materials [32]. They have also been used in boiling research to analyze the acoustic characteristics of boiling [33,34]. Lim and Bang [34] provided an in-depth study on boiling AE acoustics, where they determined the mechanism for signal generation in boiling and found that the AE hit count correlated to heat flux. Baek et al. [35] compared boiling at 1 bar and 130 bar via AE signals and found differences in frequency ranges, AE energy, etc. Alhashan et al. [36] demonstrated the use of AE sensors for condition monitoring of the boiling bubbles. They found that specific AE parameters (e.g. amplitude, rise time, etc.) encode information about the occurrence and propagation of the bubbles.
Other fields have adopted AE sensing for fault detection and system monitoring. Signals from AE sensors are used for detecting pipe leakage [37], gearbox fault detection [38,39], or predicting the useful life of materials [32]. However, to the best of our knowledge, AE has not yet been applied for heat flux prediction.
The primary challenges preventing AE data from being used for heat flux prediction lie in its low signalto-noise ratio, the intrinsic discontinuity of AE hits, and limited dataset availability. To address these issues, we propose Hit2Flux, a novel approach that departs from traditional signal-to-heat flux point prediction methods. Leveraging a sequence-to-sequence architecture, our method can effectively capture the temporal dependencies between AE waveforms and heat flux variations while addressing the discontinuity of AE hits. This innovative solution enables accurate and robust heat flux monitoring under complex boiling conditions. Compared to conventional methods with thermistors, optical images, or hydrophones, our approach offers non-intrusive monitoring with superior application potential, making it a promising advancement for industrial and experimental boiling heat flux prediction. Additionally, since transient sampling from AE sensors is not unique to boiling applications, the Hit2Flux framework could easily be applied to other important areas of research and development.
The remainder of this paper is organized as follows: Section 2 provides a detailed description of the experimental setup, signal acquisition, data preprocessing, and the Hit2Flux model. Section 3 presents the testing results along with a corresponding discussion. The conclusions are summarized in Section 4.
This section provides a detailed description of the experimental setup and the data collection. The data preprocessing and comprehensive structure of the machine learning model are also presented.
The experimental setup shown in Figure 1 threshold are saved to suppress white noise. For these experiments, a threshold of 45 dB was applied. An AE hit starts when the signal passes the threshold and lasts until the signal does not cross the threshold for a specified amount of time (i.e., hit definition time, HDT). From each AE hit, the amplitude is defined as the highest absolute value signal reading during a hit. The rise time is defined as the amount of time from the first threshold passing to the amplitude. The count is defined as the number of times the signal passes the threshold during a hit. The duration is defined as the amount of time from the first threshold passing to the last in a hit [40]. For each hit, a waveform of length 7410 samples was saved from the AEWin software.
Note, based on the characteristics of AE acquisition, a real hit may exceed the length of the saved waveform.
The heat flux data are obtained by Fourier's law. Specifically, taking the direction vertically upward from the lowest heater thermocouple as the positive direction, the positions of the four thermocouples are
𝑧 1 , 𝑧 2 , 𝑧 3 , 𝑧 4 ], and their temperature readings [𝑇 1 , 𝑇 2 , 𝑇 3 , 𝑇 4 ]. The heat flux 𝑞 of the copper foam surface is extrapolated by:
Where 𝑘 is the thermal conductivity, in this experiment 𝑘 = 392𝑊/(𝑚 • 𝐾).
An overview of the data preparation is shown in Where each 𝐻 𝑖 = [𝑠(𝑡 𝑖,1 ), 𝑠(𝑡 𝑖,2 ), … , 𝑠(𝑡 𝑖,𝑊𝑙 )] is the recorded waveform that starts at time t=ti,1 and Wl is the recorded waveform length previously defined as 7410. The temporal spacing between each hit is not constant. From our past work [27], it was found that converting acoustic signal from the time domain to the frequency domain using fast Fourier transform (FFT) significantly improved the model performance.
Therefore, in this work the FFT is used to convert each AE waveform (hit) into a vector of frequency intensities. To prepare the data, first each hit waveform is converted to the frequency domain using length of the frequency vectors is 𝐿 𝐻𝑃 = 37. Then, a rolling sampling method was used to generate hydrophone clip and heat flux sequences of length N. The hydrophone frequency vectors are used to train and test the same network used for AE. Additionally, the hydrophone data was also prepared based on our best performance model from our past work [27]. The comparison results and discussion will be provided in the following section.
The developed Hit2Flux model is a sequence-to-sequence long short-term memory (LSTM) network for heat flux prediction, incorporating dense and dropout layers alongside the LSTM layers, as detailed in Table 1. The model explanation is provided as follows.
1) Dense layer. The Dense layer, also known as a fully connected layer, is a fundamental building block of neural networks. In this layer, each neuron is fully connected to all neurons in the previous layer or directly to the input data if it serves as the network's first layer. Each layer's operation consists of a linear transformation followed by a nonlinear activation function. Mathematically, consider an input consisting of N frequency vectors, each denoted as 𝐻′ 𝑖 with a length of 𝐹𝑙 (as previously defined). These vectors form an input matrix 𝑯 ′ of shape (𝑁, 𝐹𝑙). When this input passes through a dense layer with l neurons, the output 𝑯 ′′ is computed as 𝑯 ′′ = 𝜎(𝑯 ′ 𝑊 + 𝑏). Here, W is the weight matrix with shape (𝐹𝑙, 𝑙), b is the bias vector (broadcasted across rows) with shape (𝑙, ), σ is the activation function, and 𝑯 ′′ is the output matrix with shape (𝑁, 𝑙). ReLU is a common activation function defined as 𝜎(𝑥) = 𝑚𝑎𝑥(0, 𝑥) and is used in several of the dense layers. These dense layers perform a similar operation for inputs from any preceding layer.
2) LSTM layer. Unlike dense layers, which transmit information in only one direction, LSTM layer incorporates gating mechanisms equipped with recurrent connections between its output and input, forming a feedback loop. These gated recurrent connections enable LSTM layer to feed their outputs back into their next input, allowing the network to remember and utilize information over time. As shown in the LSTM
Where the [~] operator is concatenating, 𝑊 𝑓 is the weight matrix with shape (𝑝, 𝑝 + 𝑘) when the shape of ℎ 𝑡-1 is 𝑝 (p is the defined number of units in the LSTM) and shape of 𝑥 𝑡 is 𝑘, 𝑏 𝑓 is the bias vector with shape (p,), 𝜎 is the sigmoid activation function, and 𝑓 𝑡 is the output of forget gate also has shape (p,). The
Where the 𝑊 * is the weight matrix, 𝑏 * is the bias vector, tanh is the hyperbolic tangent function. After the operation of forget gate and input gate, the current cell state can update as
The output gate controls what information the cell output based on the current input 𝑥 𝑡 , last output ℎ 𝑡-1 , and the current cell state 𝐶 𝑡 . The computing equation of current output ℎ 𝑡 are provide below:
For the next feature vector 𝑥 𝑡+1 , the LSTM layer repeat the above-mentioned procedure to extract the deep feature with temporal dependence. In this work, model was trained with an Adam optimizer and mean absolute error loss function 𝐿 𝑀𝐴𝐸 .
Where A is the total number of sequences, N is the length of each sequence, 𝑞 𝑖,𝑗 is the true heat flux value, and 𝑞 𝑖,𝑗 ̂ is the predicted heat flux value. The model was set to run for 1000 epochs. Early stopping was used to stop the training process after 100 epochs of the validation loss not decreasing. The model weights at the epoch with the lowest validation loss were stored and used for testing. The model predicts a heat flux sequence of length N. To make comparisons, the last heat flux in the sequence was taken as the models output for testing. This operation ensures each model has the same number outputs regardless of the input sequence length. This model was built and trained in python using the TensorFlow library.
In this section, the acoustic signal collected by the AE sensor during the testing transient pool boiling experiment is used to test the effectiveness of the proposed Hit2Flux model. Moreover, to illustrate the superiority of Hit2Flux in heat flux prediction, the prediction results are also compared with different machine learning models, signal input types, and signal preprocessing methods.
Several models using the Hit2Flux architecture as previously described were trained and tested using AE sequences of varying lengths to evaluate whether samples spanning longer durations yield higher prediction accuracy. The sequence lengths N considered were 3, 5, 10, 15, 20, and 25 AE hits. shows some sequence examples with 𝑁 = 1, 3,20 where (a)~(c) are the raw AE waveforms, (d)~(f) are their frequency spectrum. Figure 3(a) highlights the high-frequency and short-duration characteristics of an AE waveform, while (b) and (c) demonstrate the discontinuity of the waveforms. These characteristics make direct heat flux prediction using AE waveform more challenging compared to the continuous natured hydrophone acoustics. Consequently, most studies on boiling AE have focused on exploring AE characteristics during the boiling process or analyzing changes in AE features at key moments, rather than heat flux monitoring. In contrast, this work introduces the "sequences of sequences" approach, enabling the effective utilization of AE waveforms for heat flux prediction. Based on the raw AE sequences, the FFT is applied to transfer them to frequency domain as shown in (d) to (f). These frequency vectors are feed into the proposed Hit2Flux model. 𝑅 2 = 1 -∑ (𝑞 𝑖 -𝑞 𝑖 ̂)2 𝐴 𝑖=1 ∑ (𝑞 𝑖 -𝑞 ̅) 2 𝐴 𝑖=1 (8) 𝑀𝑆𝐸 = 1 𝐴 ∑ (𝑞 𝑖 -𝑞 𝑖 ̂)2 𝐴 𝑖=1 (9) The R 2 score is a value in the range from 0 to 1 which demonstrates how well a model fits the data. A higher R 2 value indicates a better fit. The MSE is a measure of the spread of the data points around the mean. The smaller the MSE the more densely clustered around the mean. Based on these metrics the best model will be the one with the highest R 2 and lowest MSE. Similar to the visual comparison conclusion from Figure 4, the model trained with 25 AE hits sequence length performs the best; it achieved an R 2 value of 0.97 and a MSE of 89. It is also seen that in general as the sequence length increases these metrics both improve. These results also demonstrate that the Hit2Flux model can effectively monitor boiling heat flux using data from AE sensors.
The core characteristic of the proposed method lay on the data organization, signal processing, and the developed model. It can be summarized as AE+FFT+sequences-of-sequences-input+sequence-tosequence-LSTM.
To further validate the superiority of the proposed method, serval different methods with different signal processing and state-of-arts model combination are used here for comparison. The raw signal, spectrum, cepstrum, and AE features (i.e., hit, amplitude, count, etc.) are paired with Multilayer perceptron (MLP), Random Forest Regression (RFR), Gaussian Process Regression (GPR), Convolutional neural network (CNN), and LSTM. Only combinations that obtained R 2 values above 0.1 are reported.
Table 2 summarizes the performance evaluation results on the AE signals. MLP has the best result with R2 of 0.94 when FFT is applied and sequence length is 25. Among all the feature types, AE features only works well when and RFR model is used. GPR is functional when the input consists of a single hit and the spectrum or cepstrum is used as the input representation. However, its best R2 and MSE are only 0.82 and 518 respectively. CNN works for the raw signal, spectrum, or cepstrum inputs. The CNN+Spectrum+25sequence achieves the best performance with 0.95 of R2 and 142 of MSE. This result indicates the effectiveness of FFT and sequence input. However, when compare with the LSTM model, other methods are far inferior. The LSTM+Spectrum+25-sequence achieves the highest R2 at 0.97 and lowest MSE at 89.
At the same hit sequence length, LSTM performs better when using the spectrum obtained via FFT as input compared to using the raw signal or cepstrum as input. In summary, the highest R2 score and the lowest MSE values proves the superiority of the proposed Hit2Flux model. It also highlights the importance of FFT, the sequences-of-sequences-input, and the temporal information extraction for heat flux prediction. Our past work [27] found the FFT-GPR model to be superior for hydrophone heat flux prediction.
Therefore, GPR models were developed using the raw AE and hydrophone data to compare their performance in heat flux prediction. The comparison results, as shown in Figure 7 (a) and (b), indicate that hydrophone data outperform AE data as inputs. These findings also align with the observations in Figure 6 where the PC-1 of AE signals are unstable compared to hydrophone signals, making heat flux prediction more challenging when using AE signal as input for classic regression model. However, when using the proposed method, the situation is different. As shown in Figure 7 (c) and (d), the AE signal input for the proposed model have more concentrated distribution compare to hydrophone especially as the heat flux increases. All the qualitative results are provided in Figure 8, which shows that the Hit2Flux model has a R 2 score of 0.97 and a MSE at 89 where the hydrophone has lower R 2 score at 0.88 and higher MSE at 519. These results indicate that the proposed model exhibits superior noise suppression and feature extraction capabilities. It can also be inferred that AE signals carry more information due to their significantly higher sampling rate. This is particularly evident in the high heat flux regime, where predictions based on AE signals exhibit a more consistent distribution, reflecting the characteristics of AE data collection. As boiling becomes more intense, AE signals are more likely to exceed the threshold and be recorded. Consequently, within the same time frame, AE signals capture more critical information compared to hydrophone signals.
In this paper, we propose a novel method for predicting transient pool boiling heat flux with a nonintrusive sensor. The proposed method uses acoustics acquired by AE sensor as the driven data, i.e., AE hits (waveforms). Firstly, the AE hits are organized in sequence to form the sequences of sequences pattern.
Then, the fast Fourier transform is employed to transfer the time domain data to frequency domain. are shown to improve the performance on test data. And iii) the best AE signal model outperformed the hydrophone-based predictions for the test data. Which demonstrates the improved generalizability of the Hit2Flux model compared to past work. Future work will focus on extending the boiling experiments to include variations in heater surfaces, operating pressures, and working fluids to further enhance the generalizability of the proposed method.