1 Introduction

The Economic Simplified Boiling Water Reactor (ESBWR) is a modern boiling water reactor with a 1,535 MWe capacity, classified as generation III+ [1]. It incorporates advanced passive safety features to handle emergencies, specifically addressing station blackouts (SBOs), and loss of coolant accidents (LOCA) as per U.S. guidelines. During an SBO or LOCA, the ESBWR can manage for over 72 h without AC power or operator intervention, utilizing passive condensers to remove decay heat [1]. In prolonged SBOs or LOCA, the passive condenser system remains the primary mechanism, capable of safely cooling the reactor for over 7 days [1].

The innovative cooling system of ESBWR utilizes gravity to condense water vapor, effectively dissipating heat from the containment vessel in the event of an accident and transferring the heat into the secondary cooling system. The passive condenser system is pivotal in managing decay heat and preventing the containment from exceeding pressure limits in emergency scenarios. Comprising six distinct low-pressure loops, each featuring a steam condenser immersed in a water pool with atmospheric venting. Each condenser receives steam from the drywell region around the reactor pressure vessel, which then flows downward to its vertically submerged upper part in a connected water pool located externally and above the containment structure. Positioned in sub-compartments of the water pool, all condensers interconnect at lower elevations, facilitating the use of the collective water inventory regardless of the operational status of a specific condenser loop. The effective operation of the passive condenser is essential, extracting ample energy from the reactor containment to prevent it from surpassing its designed pressure and temperature during emergencies [2].

Steam condensation is vital for ensuring the efficient functioning of the nuclear power plant’s (NPP) cooling system under both routine operational conditions and unforeseen emergencies. Generally, the utilization of steam condensation is observed across multiple components of NPPs, including condensers and steam generators for cooling purposes during normal operations as well as heat exchangers and emergency cooling systems during transient conditions. Furthermore, the presence of non-condensable gases (NCGs) like air and nitrogen, along with lighter gases such as hydrogen or helium, is a factor to consider. When steam interacts with these gases, condensation ensues, resulting in the expansion of the steam/NCG layer and the formation of a diffusion layer near the condensed liquid layer. This diffusion layer acts as an obstacle, impeding the efficiency of the heat transfer process during condensation [3]. Studies indicate that the steam condensation heat transfer coefficient (HTC) can decrease by up to 50% when the air concentration is at 0.5% [4]. Research conducted by Ma et al. [5] revealed a significant decrease in condensation HTC from 1.393 to 0.165 kW/(m2 K) at a pressure of 0.517 MPa with varying nitrogen mass fractions from 5 to 85%. Furthermore, lighter gases, such as hydrogen and helium, ascend due to their lower densities compared to steam, possibly causing stratification. This stratification has the potential to decrease the HTC by approximately 50% during condensation in the upper part of the enclosure [6].

Assessing the HTC in condensation processes involves employing diverse methods such as empirical, semi-empirical, and computational fluid dynamics (CFD) models, each with its own unique limitations. Empirical models depend on specific experimental conditions, which have their own problems, like making it hard to control all variables, working on a small scale, thinking about safety, staying within budget and time limits, and not having easy access to operational data [7]. Semi-empirical models, on the other hand, incorporate some physical and chemical mechanisms into the models. They are more versatile than empirical models but are still limited in their ability to predict behavior accurately in the presence of NCG. The accuracy of these models depends on the accuracy of the assumptions made in their development, and they may not be effective for systems with varying conditions [7]. Conversely, the CFD approach requires substantial resources and rigorous verification [8]. These obstacles underscore the importance of continuous research to advance our understanding of condensation in NPPs, especially in the presence of NCGs.

During the investigation of flow condensation heat transfer to ensure the safety of NPPs, researchers have gathered an extensive dataset encompassing diverse fluids, geometric configurations, and flow conditions. Leveraging advanced machine learning (ML) models enables the identification of relationships between input parameters and their significance in determining output parameters [9]. For example, Zhou et al. [10] employed four ML models, specifically Artificial Neural Networks (ANN), Random Forest, Adaptive Boost, and Extreme Gradient Boosting, to analyze a dataset comprising 4,882 data points related to heat transfer during condensation in mini/micro-channels. Their findings indicated that the ANN model exhibited the highest accuracy. Similarly, Zhu et al. [11] employed an ANN model to predict condensation HTCs under various conditions, achieving relatively low mean absolute relative deviations compared to experimental data.

The outcomes indicated that the ML models exhibited more effective predictive capabilities in comparison to currently available correlations. However, using ML models to predict condensation heat transfer encounters difficulties in visually representing and articulating a clear mathematical relationship that directly connects input and outputs. To overcome these challenges, the integration of Multi-Gene Genetic Programming (MGGP) can offer valuable assistance. MGGP represents a progression in Genetic Programming (GP) by employing a linear combination of low-depth GP blocks to enhance solution accuracy [12]. The model's complexity, as developed by MGGP, is notably more straightforward compared to monolithic GP, thanks to the utilization of smaller trees [13]. It offers a straightforward and clear mathematical formulation that illustrates the connection between input features and output. Furthermore, training MGGP does not require an extensive database, and the training process generates concise, clear models that can be directly applied [14, 15].

Accurate prediction methods are essential for understanding the significance of co-current steam condensation in ESBWR as a passive safety feature. This necessity arises due to the presence of NCGs, requiring precise prediction techniques to determine condensation HTC. Because of this, the main goal of this research is to find correlations that can accurately predict in-tube co-current steam condensation when NCGs are present. Notably, extensive literature does not exist which utilizes ML techniques for the specific realm of in-tube co-current flow condensation HTC, particularly within the intricate milieu of NCG-infused environments, mirroring the requisites of passive condenser in the context of the nuclear reactors or more specifically Economic Simplified Boiling Water Reactor (ESBWR) apart from the recent study of Tang et al. [16]. In this work the authors utilized MGGP to develop comprehensive correlations for predicting the HTC during steam condensation around a vertical tube in the presence of NCG. The study by Tang et al. substantiates the efficacy of the MGGP framework in innovatively formulating correlations, elucidating intricate physical nuances inherent in data pertaining to the amalgamation of heat transfer complexities and multiphase flow phenomena. This application, delineated within the purview of steam-NCG condensation, signifies the pioneering initiative employing MGGP for HTC prediction under such conditions. It is imperative to underscore the singular nature of this scholarly contribution, as Tang et al.'s work remains unparalleled in its utilization of MGGP for prognosticating condensation HTC.

In the present study, we harnessed the potential of the MGGP methodology to formulate empirical correlations delineating HTC in the specific domain of in-tube co-current flow condensation. This was executed within the distinctive contextual parameters of NCG influence, catering specifically to the exigencies of ESBWR application scenarios. The MGGP models were rigorously trained utilizing a robust database, comprising a comprehensive assembly of 2913 data points meticulously curated from diverse sources, bolstering the reliability and comprehensiveness of our predictive models.

2 Methods

2.1 Empirical Correlations for In-Tube Co-current Flow Condensation

Typically, assessments of the HTC in the co-current flow condensation process within a vertical tube are approached through two distinct types of correlations. In the first category, the local HTC is represented as a degradation factor (f), reflecting the ratio of the experimental HTC (in the presence of NCG) to the HTC with pure steam. These correlations generally depend on the local NCG mass fraction and either the mixture Reynolds number or condensate Reynolds number. The second category of correlation articulates the local HTC using dimensionless numbers. In these correlations, the local Nusselt number is formulated as a function that encompasses the mixture Reynolds number, Jacob number, NCG mass fraction, condensate Reynolds number, and other pertinent parameters. Table 1 offers a summary of these correlations, particularly in situations where forced convection dominates in the secondary loop.

Table 1 Empirical correlations for HTC during co-current flow condensation inside a tube in the presence of NCG (forced convection in the secondary loop)
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2.2 Machine Learning Approach

Evolutionary algorithms (EAs) showcase proficiency in mimicking complex models by employing stochastic search methods. GP models, a subgroup of EAs, delve into the realm of nodes—constituents derived from either a function (tree) or a terminal (leaf) set. The diversity within function sets is notable, including a spectrum of elements like arithmetic operators, mathematical functions, Boolean operators, logical expressions, or any user-defined functions. Simultaneously, terminal sets embrace variables, constants, or a blend of both. This underscores the versatility of these algorithms in replicating intricate structures through a combination of dynamic functions and diverse components [25]. The GP tree is formed by randomly selecting functions and terminals. Essentially, the root node, which is functional, initiates the growth of one or more branches that extend and terminate in terminal nodes, as illustrated in Fig. 1. Initially, an array of GP trees is constructed in a random manner, incorporating functions and terminals specified by the user [25]. The number of GP trees generated initially determines the initial population size, known as the initial GP gene. This size is based on the user-defined maximum allowable number of genes. Solutions in MGGP models are provided by combining individual genes or sub-genes linearly. A sub-gene represents a mathematical solution that connects some or all of the input and output parameters, and each sub-gene contributes to the overall solution for its respective generation. In Fig. 1, a multigene model is depicted, predicting an output variable (\({\text{y}}\)) by considering input variables \({m}_{1}, {m}_{2},\mathrm{ and }{m}_{3}\). This model configuration incorporates non-linear elements, such as cosine and square root functions, while maintaining linearity concerning the coefficients \({a}_{0}, {a}_{1},\mathrm{ and }{a}_{2}\). The amalgamation of these genes is denoted as a multigene, and the weights of these genes, along with the model bias, are derived from the training data through ordinary least squares regression. Equation (1) is the representation of the general mathematical expression of MGGP model [25].

$$Y={a}_{0}+\sum_{i=1}^{m}{a}_{i}{G}_{i},$$
(1)

where \({G }_{i}\) represents the expression of a gene tree, \(m\) is the total number of genes, \({a}_{0}\) is the bias coefficient, and \({a}_{i}\) is the weight coefficient used to adjust the scale of the gene tree.

Fig. 1
figure 1

Illustration depicting a symbolic model with multiple genes

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The MGGP model initiates its population with individuals possessing randomly evolved genes. Subsequently, the MGGP population undergoes evolutionary processes that involve reproduction, crossover, mutation, and operations that alter the architecture. These processes aim to produce an improved generation, and this enhanced population takes the place of the existing one. These evolutionary mechanisms mimic biological evolution, particularly adhering to the concept of "survival of the fittest". Throughout the genetic programming process, candidate solutions, often represented as mathematical expressions or trees, undergo generation and evaluation based on their ability to yield precise predictions. Due to the initially low accuracy of the model, the algorithm refines solutions iteratively through selection, crossover, and mutation cycles, progressively reducing RMSE [26]. The process of crossover involves exchanging nodes and branches between two parents to create new offspring. During reproduction, individuals with superior performance are directly copied from the initial population to the offspring population. In contrast, mutation introduces changes in the genetic composition of individuals to generate new offspring. This iterative procedure continues until specific termination conditions are met, as defined by either the user-specified fitness function or a predetermined maximum number of generations [26]. Users define the probabilities of these evolutionary events in the form of categories, ensuring that the sum of reproduction, crossover, and mutation probabilities equals 1. In the context of the current work, Fig. 2 illustrates the MGGP flowchart for the in-tube condensation of steam/ NCG.

Fig. 2
figure 2

Visualizing in-tube steam-NCG condensation and outlining the MGGP model's flowchart for predicting in-tube condensation HTC

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The investigation in the current work employs the MATLAB-based GPTIPS 2 program [25] to construct the MGGP model for predicting the HTC in the presence of NCGs during the condensation process. A dataset containing 2913 data points gathered from five specific research sources within defined ranges is utilized for the primary parameters that are both highly controlled and influential [7]. To ensure the reliability of the MGGP model, the dataset was partitioned into training (70%), validation (15%), and testing (15%) datasets to mitigate the risk of overfitting. It is crucial that the validation and testing sets are sufficiently large to yield accurate outcomes. The evaluation of each model's fitness relied on minimizing RMSE between predicted and actual values. The model was deemed optimal if there was a consistent absence of change in the RMSE over a consecutive series of generations or if the maximum number of generations (900) was reached. Table 2 provides a summary of the parameter configurations for the MGGP model.

Table 2 A summary of the model parameters
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2.3 Experimental Database

A comprehensive dataset was assembled, consisting of a total of 2913 experimental data points. These data were gathered from five different research sources, with a specific focus on in-tube co-current flow condensation, particularly in scenarios where forced convection prevails in the secondary loop, as outlined in Table 3.

Table 3 Identification of sources for the condensation database
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In this compiled dataset, essential geometric and operational factors were recognized and meticulously recorded. The parameters extracted from the datasets are outlined as follows:

  • Types of NCGs: air, helium and air and helium mixture.

  • Mass fraction of NCGs, \({w}_{NCG}\) = 0–0.976.

  • Local location: z = 0–2.54 m.

  • Temperature of steam/NCG mixture = 10.1–156.6 °C

  • Temperature of condenser wall, \({T}_{wall}\) = 9.8–134.1 °C

  • Coolant flow rate, \({\dot{m}}_{c}\)=400–15,00 kg/h.

  • Inner diameter of condenser, \({{\text{D}}}_{1}\) = 0.022–0.0478 m

  • Outer diameter of the condenser tube, \({{\text{D}}}_{2}\) = 0.0253–0.0508 m

  • Reynolds number of steam/NCG mixture, \({Re}_{mix}=183-\mathrm{43,600}\)

  • Total pressure, \({P}_{tot}\) = 0.0303–0.6026 MPa.

  • Condensation HTC, h = 10–33,212 W/m2 K.

Before proceeding with the analysis, an examination of the input data was conducted using the Pearson correlation coefficient, denoted as “\({R}_{AB}\)” [28]. This statistical method is commonly employed to evaluate the underlying relationships between the output and different input parameters. The Pearson correlation coefficient is a widely accepted technique for measuring the strength and direction of the linear relationship between two variables. It functions as a valuable metric for assessing the extent of association between the variables under scrutiny.

The calculation of the Pearson correlation coefficient is established through the following formula:

$${R}_{AB}=\frac{\sum_{i}^{n}({A}_{i}-\overline{\overline{A}})({B}_{i}-\overline{B })}{\sqrt{\sum_{i}^{n}{({A}_{i}-\overline{A })}^{2}} \sqrt{\sum_{i}^{n}{({B}_{i}-\overline{B })}^{2}}},$$
(2)

where \(n\) is the number of data points, \({A}_{i} and {B}_{i}\) are the individuals data points, \(\overline{A } and \overline{B }\) are the sample means of \(A\) and \(B\), respectively.

Covering the entire range from + 1 to − 1, the Pearson correlation coefficient provides a comprehensive overview of both positive and negative linear relationships. The magnitude of the coefficient (\({R}_{AB}\)) signifies the strength of the association between the two variables, where a larger absolute value indicates a more robust connection. Figure 3 depicts the associations between condensation HTC and several input parameters. Notably, the coolant mass flow rate, denoted as, (\({\dot{m}}_{c})\) demonstrates a positive correlation with a coefficient of 0.82. The Reynolds number of the mixture (\({Re}_{mix}\)) is positively correlated with a coefficient of 0.64, and the temperature of the steam/NCG mixture (\({T}_{mix}\)) exhibits a correlation coefficient of 0.60. In contrast, the condensation HTC shows a weaker correlation with wall temperature (\({T}_{wall}\)), and total pressure (\({P}_{tot}\)), with respective values of 0.05 and 0.02. Furthermore, the NCGs mass fraction, (\({w}_{NCG}\)) significant negative correlation with a coefficient of − 0.79, followed by the local location (z) with a coefficient of − 0.66, molecular weight (\(MM\)) with − 0.53, and finally inner diameter (\({D}_{1}\)) and outer diameter (\({D}_{2}\)) with − 0.09 and − 0.03.

Fig. 3
figure 3

Assessment of Pearson correlation coefficients pertaining to the input parameters of the MGGP model

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A potential explanation is that an increased coolant mass flow rate (\({\dot{m}}_{c})\) is associated with a diminished thickness of the boundary layer in the secondary system directly interfacing with the condenser. This suggests that the heightened flow rate effectively reduces the insulating effect of the boundary layer, thereby promoting enhanced heat transfer efficiency and resulting in a higher condensation HTC. Moreover, higher coolant flow rates play a role in enhancing heat transfer by sustaining higher temperature differences between the steam inside the tube and the coolant in the annulus. On the other hand, as the Reynolds number (\({Re}_{mix}\)) rises, the tube flow may transition from laminar to turbulent. Turbulent flow generally exhibits higher HTC due to improved fluid mixing, disrupting the boundary layer and enhancing thermal contact between condensing vapor and the tube surface. This disruption leads to increased heat transfer rates in turbulent regimes. Since the condensation is the transformation of vapor into liquid, influenced by steam/NCGs mixture temperature (\({T}_{mix}\)). Higher mixture temperatures result in the liberation of additional latent heat, thereby causing an increase in both energy release and heat transfer rates, consequently leading to elevated HTC. Finally. While higher wall temperature (\({T}_{wall}\)) and the total pressure (\({P}_{tot}\)) usually boost heat transfer, their impact may lessen in certain conditions. When the steam/NCG mixture temperature is high enough, the dominant influence shifts to latent heat transfer, making the condensation HTC less responsive to changes in wall temperature and total pressure.

As anticipated, the mass fraction (\({w}_{NCG}\)) and molecular weight (\(MM\)) exert a detrimental influence on the condensation HTC. These findings can be clarified as follows: When investigating the presence of light gases like helium along with other NCG such as air, only two factors related to condensation rely on the gas composition, namely the coefficient \([\propto {D}^{2/3}]\) and the driving force of total densities \({[\alpha ({\rho }_{b}-{\rho }_{i})}^{1/3}]\) [7]. Air has a molecular weight of 28.0134 g/mol, surpassing the molecular weight of steam (18.015 g/mol). Conversely, helium has a molecular weight of 4.0026 g/mol and is lighter than steam. The mechanism of steam diffusion through gases is contingent upon their respective molecular masses and viscosities. In comparison to helium, air, predominantly composed of nitrogen and oxygen, possesses a higher molecular mass and viscosity. As a result, the diffusion of steam through air experiences a relative deceleration. On the contrary, helium, characterized by a lower molecular weight and reduced viscosity, facilitates faster steam diffusion, leading to an enhanced condensation rate.

The existing literature indicates that gas species with lower mass diffusivity, such as air, exhibit a more pronounced concentration gradient than those with higher mass diffusivity. Consequently, this results in a thicker gas boundary layer at the interface, forming a thermal resistance layer that hinders the condensation process by impeding steam diffusion and condensation on the system's surface. Conversely, gases with higher mass diffusivity have a less steep concentration gradient [29]. Therefore, an increase in the mass fraction exacerbates this effect on condensation.

The local position (z) also exhibits a detrimental effect, as indicated in existing literature [30, 31]. Reports suggest the presence of a brief passive condensation zone near the entrance, followed by an active condensation zone after a specific distance from the entrance. Subsequently, another passive zone emerges, signifying that beyond a certain distance from the entrance, the local position significantly hampers the condensation process. Ultimately, the majority of the literature indicates that smaller condenser diameters, within the range considered in this study, exhibit a less noticeable effect on condensation HTC [32, 33].

3 Results and Discussion

The current MGGP model relies on a set of input parameters, including \(z, {T}_{mix},{T}_{wall},{\dot{m}}_{c},{P}_{tot},{D}_{1},{D}_{2},MM,{w}_{NCG},{Re}_{mix}\) while the output, \(h\), is used for the model's development. Following an iterative process involving the generation of an initial random population and evaluating its fitness using the RMSE, the RMSE reaches a point where it remains constant across generations. This consistency fulfils the termination criteria, leading to the generation of the mathematical model. The graphical representation in Fig. 4 depicts the fluctuation in RSME concerning different generations. The average error decreases quickly across all trained correlations. The selected successful correlations achieve the lowest error in each new generation as the generation number increases. Beyond approximately 100 generations, the influence of additional generations on correlation performance diminishes, suggesting that the MGGP model has reached a point of sufficient generational refinement.

Fig. 4
figure 4

The change in RSME across generations

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The following outlines four specific correlations that have undergone training:

$$h={a}_{0}+{a}_{1}{MM}^{0.05}+{a}_{2}{w}^{2.1}$$
(3)
$$h={b}_{0}+{b}_{1}{z}^{0.1}+{b}_{2}{{T}_{mix}}^{3.85}+{b}_{3}{{\dot{m}}_{c}}^{2.25}+{b}_{4}{MM}^{0.05}+{b}_{5}{w}^{0.52}+{b}_{6}{{Re}_{mix}}^{1.2}$$
(4)
$${\text{ln}}\left(h\right)={c}_{0}+{c}_{1}{z}^{(1/8)}+{c}_{2}{{T}_{mix}}^{(9/2)}+{c}_{3}{MM}^{(1/9)}+{c}_{4}{w}^{(5/8)}+{c}_{5}{{Re}_{mix}}^{(3/4)}$$
(5)
$${\text{ln}}\left(h\right)={d}_{0}+{d}_{1}MM+{d}_{2}w$$
(6)

The coefficients of the new correlations are presented in Table 4.

Table 4 The parameters of the new correlations
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Figure 5 illustrates a comparison of the performance of MGGP-based correlations, utilizing the current consolidated database, in relation to both the performance index and the coefficient of determination. The data presented in Fig. 5 shows that the third correlation displays the highest values for both the performance index and the coefficient of determination. This suggests that this particular correlation captures the most significant variability in the dependent variable and exhibits the strongest linear relationship between the input features and the output compared to the correlations being evaluated. Consequently, based on these metrics, the correlation model associated with the third correlation is deemed to offer the most optimal fit to the data.

Fig. 5
figure 5

Evaluation of MGGP correlation performance based on their performance index and determination coefficient

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In Fig. 6, an assessment of the performance of MGGP correlations is showcased, employing the existing consolidated database. The value \({\eta }_{30\%}\) represents the percentage of data points predicted within a deviation range of \(\pm 30\%.\)

Fig. 6
figure 6

Evaluation of the performance of MGGP correlations

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The third correlation achieved the highest predicted values within a deviation range of ± 30, registering at 90.9%. Subsequently, the second correlation followed with 64.48%, trailed by the first correlation at 58.69%, and lastly, the fourth one at 47.15%.

The subsequent stage involves evaluating the third correlation of MGGP, utilizing insights derived from previous experimental investigations presented in Table 5. Notably, Kuhn's [17] work demonstrates the highest predicted value, reaching 91.8% within a deviation range of ± 30 and 94.3% within a deviation range of ± 50. In contrast, the lowest prediction is attributed to Siddique [21] recording percentages of 41.0% and 64.8% within deviation ranges of ± 30 and ± 50, respectively. It is worth mentioning that the variation in these percentages across diverse experimental configurations underscores the inherent uncertainty associated with each experimental condition. This significant uncertainty extends to both the experimental setup, the correlation itself, and its applicability to the experimental parameter ranges.

Table 5 Assessing the MGGP's second correlation by drawing on findings from prior experimental research
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To validate the established third correlation based on MGGP, a parametric analysis was conducted. The outcomes of the analysis for NCG mass fraction are illustrated in Fig. 7. It is observed that the condensation HTC diminishes with the rise in NCG fraction and escalates with an increase in total pressure, consistent with findings reported in earlier studies [34,35,36]. On the other hand, increasing the value of \({T}_{mix}\) leads to the release of extra latent heat. Consequently, this enhances the condensation HTC, as illustrated in Fig. 8.

Fig. 7
figure 7

Exploration of NCG mass fraction and total pressure through parametric analysis under conditions of Tmix = 120 °C, Remix = 32,800, and z = 0.17 m

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Fig. 8
figure 8

Exploration of total pressure and mixture temperature through parametric analysis under conditions of \({w}_{NCG}\)=0.4, Remix = 15,000, and z = 0.996 m

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Ultimately, the third correlation's ability to capture the influence of NCG was tested by applying it to the experimental data conducted by Lee and Kim [20] specifically involving Nitrogen as a NCG. It is noteworthy that these data were not incorporated in the correlation's development, and the outcomes indicate that 92.94% of the predicted data align within a deviation range of ± 30% as shown in Fig. 9. In summary, the highly accurate correlation's predictive performance is crucial for exploring worst-case scenarios during emergencies such as SBO or LOCA, especially when NCGs are present. In these situations, the ESBWR is required to operate for 72 h without AC power or operator intervention, relying on passive condensers to manage decay heat removal. It is important to note that the presence of NCGs significantly impacts condensation performance, leading to the expansion of the steam/NCG layer and the formation of a diffusion layer near the condensed liquid layer. This diffusion layer acts as an impediment, hindering the efficiency of the condensation heat transfer process. The high-performance correlation can be employed in system codes such as TRACE or RELAP5 to predict the system's performance. Emphasizing the scarcity of studies on co-current condensation heat transfer for the ESBWR application, both experimentally and computationally, underscores the necessity to validate such a model for more reliable and robust predictions.

Fig. 9
figure 9

Prediction results for the Lee and Kim’s experimental data

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4 Conclusions

This research focuses on heat transfer during condensation in in-tube co-current flow, considering the presence of NCGs in the context of the passive condenser application for the ESBWR. To explore the condensation process, 2,913 data points were gathered from five different literature sources. These data points specifically pertain to NCGs such as air, helium, and a mixture of air and helium, in addition to data for pure steam. The collected data covers various ranges of geometric and operational parameters, including \(0\le {w}_{nc}\le 0.976\), \(0\le z\le 2.54\,\mathrm{m}\), \(0.022\le {D}_{1}\le 0.0478\,\mathrm{mm}\), \(0.0253\le {D}_{2}\le 0.0508\mathrm{ mm}, 0.0303\le {P}_{tot}\le 0.6026\mathrm{ MPa}\), \(183\le {R}_{mix}\le {43{,}600},\) \(10.1\le {T}_{mix}\le 156.6\,{^\circ{\text{C}} }\), and \(10.1\le h\le \mathrm{33,212 } {\text{W}}/{{\text{m}}}^{2} {\text{K}}\). The research introduces four specific correlations derived from the model, with the third correlation standing out by yielding the highest expected values within a deviation range of ± 30%, achieving an accuracy of 90.9%. This correlation demonstrates strong performance across both the performance index and coefficient of determination. Another validation of the third correlation aimed to assess its accuracy in predicting the physical variations in the inverse relationship between mass fraction and condensation HTC, as well as the proportional relationship between the temperature of the mixture and condensation HTC. The validation confirmed its ability to accurately predict these relationships. Finally, the data from Lee and Kim [20], featuring nitrogen as a NCG, was employed as input for the third correlation. Notably, these data were not utilized in the development of the correlation. The results reveal that 92.94% of the predicted data fall within a deviation of ± 30%. Conclusively, this study contributes to the broader understanding of passive safety features in ESBWR by employing ML techniques to accurately predict co-current steam condensation in the presence of NCGs, addressing a notable gap in existing literature.