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J. Ocean Eng. Technol. > Volume 38(6); 2024 > Article
Jung, Lee, and Shin: A Model Experimental Study on Water flows out from the Ballast Tank for Emergency Rising of Submarines

Abstract

Accurate assessment models are crucial for ensuring the safe operation of submarines during emergency rising, highlighting the need for a precise mathematical model to optimize ballast tank operations. The emergency rising process involves the rapid release of high-pressure compressed air from the ballast tank, a complex sequence influenced by variations in depth and pressure. This study focuses on developing and validating a mathematical model to accurately simulate this operation. Building upon prior studies of the air-blowing process within the ballast tank was performed. An experimental setup was also established to measure discharge rates under varying depths and pressures, facilitating a comparative analysis of experimental and simulated data. The findings reveal that pressure loss is a critical factor influencing the discharge flow rate. Consequently, a new formula is proposed to dynamically adjust pressure loss based on changes in depth and pressure. This model significantly improves the accuracy of flow rate calculations across diverse operating conditions, surpassing earlier models that relied on fixed pressure loss coefficients. The proposed approach is expected to enhance predictions of submarine attitude changes during emergency rising maneuvers.

1. Introduction

Submarine design is widely considered a complex task that demands a comprehensive analysis of its hydrodynamic characteristics. This analysis primarily focuses on factors such as speed performance, maneuverability, and specific mission capabilities. Submarines must operate effectively both on the water’s surface and underwater, making their motion characteristics in varying fluid environments critical design considerations.
In the case of surface ships, extensive data have been gathered through years of research and development at domestic shipyards and research institutes. These data are extensively applied in both the initial and detailed design phases. However, for submarines, access to similar data is significantly restricted. Designers must often rely on a limited set of resources, including model test data, trial run data, and numerical simulation results, which poses challenges to achieving optimal designs.
The specialized maneuverability of submarines is closely tied to their operational objectives, with stable and agile underwater performance being particularly crucial. To achieve this, six-degree of-freedom (6DOF) motion must be considered, as it is a key factor in accurately predicting submarine maneuverability. Generally, methods similar to those used for surface ships have been adapted to estimate submarine maneuverability. However, the increased degrees of freedom in submarine maneuvers have significantly complicated the estimation process, exposing its limitations. Notably, the 6DOF mathematical models proposed by Gertler and Hagen (1967) and Feldman (1979) are widely recognized as seminal approaches for predicting submarine maneuvering motion. Studies utilizing these models require the acquisition of hydrodynamic coefficients, which are typically obtained through captive model tests to measure the hydrodynamic forces acting on the hull. However, these model-based predictions are primarily applicable to steady-state motion. Extending their use to abnormal or specialized maneuvers, such as emergency rising, remains a significant challenge.
The emergency rising of a submarine is a highly complex process involving the rapid discharge of water from the ballast tank using high-pressure compressed air. This operation generates intricate flow phenomena due to the interaction of water and air within the ballast tank, enabling the submarine to ascend. To accurately predict changes in submarine attitude and ensure stability during emergency rising, it is crucial to mathematically model the ballast tank’s discharge process.
Bettle et al. (2009) conducted an early study on this subject, assuming the injection of high-pressure air from a compressed air tank into the ballast tank during emergency rising. They represented this process using an exponential function model to simulate attitude changes. Building on this, Font et al. (2010) proposed a more refined mathematical model by incorporating air flow, flow rate, and pressure variations within the ballast tank. Their model divided the flow of high-pressure compressed air into supersonic and subsonic regimes, allowing the discharge flow rate to account for volume changes in the ballast tank relative to external pressure.
Subsequent work by Font et al. (2013) introduced mathematical control models for both blowing and venting processes, facilitating research on various operational methods. Dehghani et al. (2013) numerically analyzed weight changes and shifts in the center of buoyancy resulting from compressed air impacts using mathematical models for air injected into the ballast tank. Ruijie et al. (2013) experimentally examined pressure changes when air was discharged into the ballast tank under static conditions without propulsion, providing valuable data for further modeling efforts. Collectively, these studies advanced the understanding of flow phenomena in ballast tanks and their effects on submarine attitude, significantly enhancing the precision of mathematical models.
Building on these foundational studies, the present work aims to develop a more precise simulation model for reproducing the submarine emergency rising process. The discharge flow rate was calculated under varying depth and pressure conditions by expanding the mathematical model for ballast tank blowing operations proposed by Font et al. (2010). The validity of the discharge flow rate was then verified through experimental testing. A scale model of the ballast tank was constructed, and experiments were conducted in the Deep Ocean Engineering Basin (DOEB) at the Korea Research Institute of Ships and Ocean Engineering (KRISO), which features a pressure control system powered by a commercial compressor. Discharge flow rate and time were measured under various initial conditions, and the results were compared with the mathematical model, providing critical insights for further refinement. The experimental results revealed that pressure loss significantly affects the discharge flow rate. To address this, a new formula was developed to dynamically adjust pressure loss based on depth and pressure conditions. This formula is more realistic than the conventional method of using fixed pressure loss coefficients, offering improved accuracy in flow rate calculations. This study demonstrates the potential of the proposed model to predict submarine attitude changes during emergency rising with greater precision, contributing to the safe and efficient operation of submarines in the future.
The structure of this1) paper is as follows: Chapter 2 details the modeling process and methodology, while Chapter 3 provides a comparison of experimental and simulation results. Finally, Chapter 4 discusses conclusions and outlines directions for future research.

2. Numerical Simulation for Blowing Operation of Ballast Tank in Submarine

2.1 Mathematical Model

Mathematical modeling for the blowing operation of the ballast tank in submarines was summarized based on the theory presented by Font et al. (2010, 2013). It consists of a ballast tank, a high-pressure bottle, and a discharge operation valve. Table 1 shows the variables and symbols used in Fig. 1.
The mathematical model for the ballast tank blowing operation encompasses three key conditions: air flow from the high-pressure bottle, the flow rate of water discharged from the ballast tank, and the pressure changes within the ballast tank. These conditions are detailed as follows:
  • (1) Air flow from the high-pressure bottle flows into the ballast tank through a valve acting as a nozzle. In this scenario, pressure loss and heat transfer within the pipe connecting the high-pressure bottle to the ballast tank are neglected. Research focuses on the one-dimensional steady flow of ideal compressible gas under these conditions, with references available in Crowe et al. (2004) and other traditional fluid dynamics resources. At the start of the blowing operation, the significant pressure difference between the high-pressure bottle and the ballast tank generates supersonic flow. Over time, as the pressure difference diminishes, the flow transitions from supersonic to subsonic. This transition occurs when the pressure in the high-pressure bottle, PF, and the pressure inside the ballast tank, PB, meet Eq. (1). Here, Y represents the isentropic constant, and rr is the adiabatic index, which is approximately 1.4 for air. This model forms the foundation for analyzing the dynamics of the ballast tank blowing operation, providing insights into the flow behavior and pressure interactions.

    (1)
    (pFpB)=(ϒ+12)rr1

    When high-pressure air flows into the ballast tank, the pressure inside the high-pressure bottle decreases while its temperature increases. If heat transfer is ignored, this process can be treated as adiabatic. The instantaneous pressure and temperature are described by Eq. (2). Here, mF (t), PF (t), and TF (t) represent the mass, pressure, and temperature of the air within the high-pressure bottle, respectively. mF0 (t) and PF0 (t) are the initial mass and pressure inside the high-pressure bottle, respectively. VF is the volume of the high-pressure bottle, and Rg is the gas constant.
    (2)
    pF(t)=(mF(t)mF0)ϒpF0,TF(t)=pF(t)VFRgmF(t)
    Based on Eq. (2), when the high-pressure air from the high-pressure bottle enters the ballast tank, the mass change of the high-pressure bottle (F (t)) is divided into supersonic flow and subsonic flow as in aforementioned Eq. (1) and expressed as Eq. (3).
    (3)
    m˙F(t)=AC(ϒ(2ϒ+1)ϒ+1ϒ1pF0mF0ϒVF(mF(t))ϒ+1)12,if(pFpB)(ϒ+12)rr1m˙F(t)=A(pF0mF(t)ϒ+1mF0ϒVF2ϒϒ1)12((pB(t)pF0(mF(t)mF0)ϒ)2ϒ(pB(t)pF0(mF(t)mF0)ϒ)ϒ+1ϒ)12,if(ϒ+12)rr1(pFpB)1m˙F(t)=0,if(pFpB)1
    In Eq. (3)C = A/Ah holds. A is the area of the entire outlet and Ah is the cross-sectional area of the hole inside the outlet.
    (2) Flow rate out of the ballast tank: When the pressure inside the ballast tank exceeds the external pressure due to the air supply, water inside the tank is expelled through the flood port at the bottom. If the pressure loss at the outlet is negligible, the volumetric flow rate from the ballast tank can be determined using Bernoulli’s equation. The discharge flow rate is expressed as Eq. (4). Here, ρ is the density of water, pB (t) is the pressure inside the ballast tank, PSEA is the external pressure, vh is the flow velocity, and Cn is the flow coefficient.
    (4)
    qB=CnAhvh=CnAh2(pBPSEA)ρPSEA=Patm+ρg(z+zhxbsinθ)
    xbsinθ was included in Eq. (4) to consider the change in water height inside the ballast tank caused by the pitch of the submarine.
  • (3) Pressure change inside the ballast Tank: The air introduced at high speed is rapidly mixed with the water inside the tank, causing effective heat transfer between the water and air. Therefore, it can be assumed that the air immediately reaches the temperature in the tank, and this condition can be considered an isothermal process. Since the ideal gas law can be applied, the pressure change inside the ballast tank is expressed as Eq. (5).

    (5)
    p˙B(t)=m˙B(t)RgTBV˙B(t)pB(t)VB(t)

In Eq. (5)TB, VB, and mB are the temperature, volume, and mass of the air inside the ballast tank. B (t) is the mass flow rate over time, and B (t) is the rate of volume change of the ballast tank.

2.2 Numerical Simulation

Using the theoretical formulas described earlier, a numerical simulation model of the ballast tank’s blowing operation was implemented in Matlab, a commercial software program. The model calculates the mass flow rate (F) from the high-pressure bottle by distinguishing between supersonic and subsonic air flows. Based on these calculations, the volume changes of the high-pressure bottle and ballast tank were modeled over time. Additionally, the time required for the water in the ballast tank to be fully discharged and replaced with air was determined, accounting for the pressure changes in both the high-pressure bottle and the ballast tank.
The numerical simulation was conducted under initial pressure conditions of 3, 4, and 5 bar for the high-pressure bottle at depths of 1, 5, and 10 meters. The results for each condition were summarized, with specific analytical parameters detailed in Table 2. Fig. 2(a) illustrates the outcomes for a subset of these conditions.
Fig. 2(a) presents the simulation results for the case where the initial pressure of the high-pressure bottle is PF0 = 3 bar at a depth (z) of 1 m. The upper section of the figure displays, from left to right, the mass flow rate (F (t)), the mass of air in the high-pressure bottle (mF (t)), and the mass of air in the ballast tank (mB (t)) from the left side, respectively. The lower section illustrates, from left to right, the pressure inside the high-pressure bottle (PF (t)), the pressure inside the ballast tank (PB (t)), and the ratio of the air volume in the ballast tank (Vair (t)) to the total tank volume (VTotal). From the bottom-left graph in Fig. 2(a), it is evident that the pressure inside the high-pressure bottle (PF (t)) decreases from the initial set pressure over time, influenced by the mass flow rate (F (t)). Concurrently, as depicted in the upper-left graph, the mass of air in the high-pressure bottle (mF (t)) gradually decreases due to the mass flow rate, while the mass of air in the ballast tank (B (t)) tends to increase, as can be seen from the top right of Fig. 2(a). The pressure inside the ballast tank (PB (t)) increases as high-pressure air flows into the tank, as shown in the graph at the bottom center. This pressure gradually decreases due to water being discharged and eventually approaches the external pressure. The Vair /VTotal graph at the bottom right illustrates the process of the ballast tank filling with air as air is injected. The point in time when the water is completely discharged corresponds to when the air volume inside the ballast tank equals the total tank volume (Vair /VTotal = 1) .
Fig. 2(b) presents numerical simulation results showing the time required to completely discharge water from the ballast tank under varying conditions: depths of 1, 5, and 10 m and initial pressures of 3, 4, and 5 bar. While the discharge time shows a relatively consistent trend across pressures at depths of 1 and 5 m, it is significantly longer at a depth of 10 m when the initial pressure is 3 bar compared to other conditions. These results highlight the need to carefully evaluate discharge times based on depth conditions.

3. Model Test of Blowing Operation

3.1 Information on the Target Submarine and Ballast System

The model test was conducted to validate the numerical simulation by employing the MARIN BB2, an open hull form model, as the reference ship. The specifications of the MARIN BB2 are detailed in Table 3. To ensure the feasibility of the model test within the experimental setup, the scale ratio was determined to be 1/20, allowing the simulation to correspond to a maximum depth of 200 meters on a full scale.
The ballast tank model was designed in a simplified form while maintaining fidelity to the MARIN BB2 real ship geometry, as shown in Fig. 3(a). The design also referenced the research findings of Lee et al. (2023), illustrated in Fig. 3(b). The resulting model, depicted in Fig. 3(c), was fabricated using steel plates to endure the high pressures encountered during testing. The front portion of the model included a glass window to facilitate the observation of internal flow dynamics. Additionally, a nozzle was incorporated at the top of the model for air supply, while the bottom featured a flood port geometry to enable water discharge.

3.2 Configuration of Model Test

To implement the blowing operation model system of the ballast tank, the configuration of the ballast system depicted in Fig. 3(b) was utilized as a reference. A commercial compressor system, comprising a high-pressure compressor and a storage tank, was selected for this purpose. This system compresses atmospheric air and stores it in the storage tank. The initial pressure PF0 was regulated using a pressure regulator to ensure compatibility with the initial pressure conditions specified in the numerical simulations.
To monitor and measure the supply flow rate of the compressed air, an air flowmeter was installed downstream of the pressure regulator. The components were interconnected using urethane hoses, and valves were integrated along the connection paths to control and adjust the direction of air flow as needed. Using this setup, the blowing and discharge operations of the ballast tank were successfully implemented, as illustrated in Fig. 4. The specifications and detailed information of the devices utilized in the system, including the compressor, pressure regulator, flowmeter, and connecting hoses, are summarized in Table 4.

3.3 Pre-Test of Blowing Operation

A pre-test was carried out using a simple preparation tank to assess the operational performance of the ballast tank blowing operation model system described in Section 3.2 and to evaluate the model test procedure and associated test errors. The preparation tank measured 6 m in length, 2 m in width, and 1.5 m in depth, with glass windows forming its outer walls, allowing for the direct observation of internal behaviors, as shown in Fig. 5. Fig. 6 illustrates the configuration of the ballast tank blowing operation model system as operated within the preparation tank. The setup included a line length of approximately 20 m extending from the air supply valve to the ballast tank model, ensuring that the system could replicate operational conditions at depths of 10 m or greater during the experiment.
The test in the simple preparation tank involved placing the ballast tank model inside the preparation tank as depicted in Fig. 6, with adjustments to the blowing operation system managed from the ground. Before initiating the blowing operation, air was compressed and stored at 7 bar in the storage tank using the commercial compressor. This compressed air was then supplied to the ballast tank at a pressure set by the regulator, which was activated by opening the air supply valve. To gather test data, images of the blowing operation were captured through the glass windows of the preparation tank. Additionally, an underwater camera was installed at the front of the ballast tank to record images in preparation for experiments under deeper water conditions. These captured images (Fig. 7) were analyzed to determine key operational times: the air supply time to the ballast tank based on the line length during the pre-test phase (T1), the air entry time into the ballast tank (T2), and the time when the ballast tank was fully filled with air due to the complete discharge of water.
The T1 time was measured by analyzing the interval from the opening of the air supply valve to the T2 time point in Fig. 7. The air flow rate supplied through the valve was also monitored using a sensor. Furthermore, the pressure changes within the high-pressure bottle were recorded to evaluate the proper execution of the air blowing operation model test.
The video recorded, as shown in Fig. 7, was slowly played back at a 0.2× speed for analysis. These low-speed images revealed the time when the air was supplied to the nozzle of the ballast tank (T2) and the spread of the air inside the ballast tank with complete discharge of water (T3). Table 5 shows the pre-test results obtained through the analyzed images and measurement sensors.
A detailed analysis of the pre-test results in Table 5 revealed that the air supply time to the ballast tank (T1) varied depending on the pressure conditions. However, video analysis confirmed that there were no issues in measuring the flow-out time. Additionally, the Vair (t)/VTotal value, representing the duration when the ballast tank is filled with air, was calculated based on the analyzed T3T2 (s) value. The supply flow rate, as measured by the air flowmeter, increased slightly with pressure but remained stable within a similar range.
Under conditions where the initial pressure of the regulator was set to 7 bar, the high-pressure bottle pressure dropped to 4.8 bar by the end of the test. This suggests that the air supply could be insufficient for future operations under deep-water conditions. Consequently, the initial pressure for the test was adjusted to a range of 3 to 5 bar. Errors in various measurements, including time discrepancies from video analysis, sensor precision, and pressure gauge accuracy, were accounted for and analyzed, as detailed in the note accompanying Table 5.

3.4 Model Test in DOEB (KRISO)

The model test was conducted at LOEB (KRISO). Fig. 8(a) presents an overview of the LOEB facility along with an introduction to its internal features. To simulate various depth conditions for the model test, three specific depths were selected, utilizing the maximum depth capability of LOEB, as illustrated in Fig. 8(b). In the model test procedure, an air discharge system was installed on the ground, and the ballast tank height was adjusted using a crane to achieve the designated depth conditions (A, B, and C), as shown in Fig. 8(c). This setup was based on the method validated during the pre-test. Once the depth conditions were established, air was supplied to the ballast tank by setting the initial discharge pressure of the air discharge system. The flow-out time of air from the ballast tank under high-pressure discharge was recorded on video. The recorded footage was analyzed using the same methodology as in the pre-test. Table 6 and Fig. 9 present the test conditions and results.
As shown in Fig. 9, the flow out time according to the depth tended to be clearly distinguished. The general phenomenon that the flow out time decreases alongside the increase in initial discharge pressure was confirmed. In addition, as the depth increased, the difference in flow out time by pressure difference increased.

3.5 Analysis of Numerical Simulation and Model Test

Fig. 10 compares the numerical simulation results described in Chapter 2 with the model test results. At a depth of 1 m, the flow-out time in the numerical simulation was approximately 3 s longer than in the model test. Conversely, at a depth of 5 m, the flow-out time in the numerical simulation was about 2.5 s shorter than in the model test. At a depth of 10 m, significant discrepancies were observed in the flow-out times at 4 and 5 bar, except for the 3-bar condition.
The numerical simulation results indicate that the flow-out time was very short and showed minimal dependence on the submerged depth under the 10 m condition. To further investigate this behavior, Eq. (6), which describes the discharge flow rate through the flood port, as presented in Mathematics ArXiv (Font et al., 2013) was examined.
(6)
pB(t)+ρghwc(t)Δploss=pSEA(t)+12ρvh2(t)Δploss=ζh12ρvh2qB(t)=ChAhvh=CnAh2(pB(t)+ρghwc(t)PSEA(t))ρ(1+ζh)
Eq. (6) calculates the flow rate by accounting for the pressure loss generated when water is discharged from the ballast tank. The equation incorporates factors such as the pressure difference between the ballast tank and the external environment, the density of water, and the pressure loss coefficient (ζh). The pressure loss is represented as a square root function of the flow rate. In previous studies, a certain value (ζh) was assumed for the pressure loss coefficient (∆Ploss). However, in the model test results, the discharge flow rate varied depending on the depth. This observation led to the conclusion that pressure loss increases proportionally with depth.
Accordingly, the pressure loss model used in Eq. (6) was modified in Eq. (7) so that pressure loss could vary depending on depth and pressure changes.
(7)
Δploss=(z+ζh)12ρvh2qB(t)=CnAh2(pB(t)+ρghwc(t)PSEA(t))ρ(z+ζh)
In Eq. (7), the pressure loss coefficient was set to (z + ζh ) to include the ζh value that varies depending on the depth z instead of fixed ζh . This makes it possible to reflect the pressure loss according to depth and pressure conditions more accurately, thereby increasing the accuracy of flow rate calculation.
Eq. (7) was reflected to the numerical simulation model. Fig. 11 compares the results of numerical simulation with those of the model test.
As shown in Fig. 11, the difference in flow-out time depending on depth is evident, unlike the previous numerical simulation results. The flow-out time was generally longer than the model test results at a depth of 1 m, but was nearly identical at a depth of 5 m. At a depth of 10 m, the difference in flow-out time became more pronounced as pressure increased, a trend consistent with the model test results. Based on these findings, the conclusion that pressure loss increases proportionally with depth was deemed valid, prompting a reexamination of the appropriateness of the pressure loss coefficient.
Notably, the observation that the numerical simulation results at a depth of 1 m were generally longer than the model test results supports the idea that the pressure loss coefficient should be treated as a variable dependent on depth, rather than as the fixed pressure loss coefficient ζh =2.5 proposed by Font. Additionally, it was found that the ζh value also varies with pressure, as evidenced by the results at a depth of 10 m. Consequently, the ζh values were adjusted to align the numerical simulation results with the model test flow-out times. The adjusted values are presented in Table 7. The newly obtained ζh was expressed as ζh*.
When the pressure loss coefficient is changed from ζh to ζh* by reflecting the results in Table 7, Eq. (7) can be expressed as Eq. (8).
(8)
Δploss=(z+ζh*)12ρvh2qB(t)=CnAh2(pB(t)+ρghwc(t)PSEA(t))ρ(z+ζh*),(ζh*z,pF)
To utilize the ζh* data obtained above in numerical simulation, an estimation formula dependent on the depth and pressure is necessary. This formula was derived using a nonlinear multiple regression model for ζh*, a two-variable function, based on the data of Table 7 as shown in Eq. (9). The newly calculated ζh* based on Eq. (9) was expressed as ζhLSM*. Fig. 12 shows the ζhLSM* data along with the ζh* data.
(9)
ζhLSM*(z,p)=5.01+1.04z0.22z2+0.49zp0.09zp2+0.05z2p+2.74p0.33p2
As shown in Fig. 12, the estimated z+ζhLSM* is almost identical to z+ζh*, which was considered to match the model test results. The influence of z+ζhLSM* on the numerical simulation results was examined, and the results are summarized in Table 8. The conditions of 8 and 12 m were included for future analysis of whether Eq. (9) can be used under various depth conditions.
As shown in Table 8, the difference between z+ζh* and z+ζhLSM* (A − B) is ±0.33 or less. The flow out time difference of numerical simulation that reflected this (C − D) is ±1.1 seconds or less. It was expected that this estimation error can be used for a comparative analysis with the model test.

3.6 Validation of the Proposed Mathematical Model

To verify whether z+ζhLSM* presented in Table 8 can be properly used in numerical simulation on the air-blowing operation of the ballast tank, an additional model test was conducted as shown in Table 9. To evaluate whether z+ζhLSM* was properly estimated, the conditions of the additional model test were set similarly to the range of the previous model test, and depths of 8, 10, and 12 m were selected. Fig. 13 shows the results of the additional model test for the flow out time.
As shown in Fig. 13, the data at a depth of 10 m show the results of both the previous (10 m old) and additional model tests. The results of the additional model test are almost identical to those of the previous model test. In addition, the flow out time according to the depth was clearly distinguished as in the results of the previous model test, and the flow out time tended to decrease as the pressure increased at the same depth.
Fig. 14 provides the results of all model tests and the results of performing numerical simulation using z+ζhLSM* of Table 8 in one graph.
Fig. 14 compares the flow-out time tendencies across various depth and pressure conditions by combining all model test data and numerical simulation results. The model test results and numerical simulation outcomes show good agreement under most depth and pressure conditions, with the flow-out time generally decreasing as the depth increases. This consistency demonstrates that the proposed numerical model effectively predicts flow-out time based on depth and pressure. However, some discrepancies are observed between the numerical simulation and model test results, particularly at a depth of 1 meter under a pressure of 5 bar and at a depth of 12 meters under a pressure of 3 bar. These differences appear to arise from errors in the estimated z+ζhLSM* values used in the numerical simulations, as described in Section 3.3. This suggests that further optimization is necessary to refine the pressure loss coefficient for conditions outside the experimental scope of this study.

4. Conclusion

This study conducted a detailed analysis of the discharge flow rate of the ballast tank during submarine emergency rising. Specifically, the influence of high-pressure compressed air inflow on the hydrodynamic conditions in the ballast tank was mathematically modeled, and the theoretical numerical simulation results were validated through comparisons with model test results. The main findings are as follows:
First, the discharge flow rate from the ballast tank was identified as a critical variable significantly influenced by pressure loss. To address this, a new formula was proposed to dynamically adjust pressure loss based on depth and pressure. This formula enhanced the accuracy of flow rate calculations by incorporating a variable pressure loss coefficient instead of relying on fixed coefficients.
Second, the validity of the numerical simulation model was confirmed through comparisons with model test data. The flow-out time under various depth and initial pressure conditions demonstrated practical accuracy compared to existing models. Furthermore, the model’s ability to predict attitude changes during submarine emergency rising was validated, as it showed strong agreement with experimental results.
Third, a system capable of predicting discharge rates under diverse depth and pressure conditions was developed based on the proposed model and formula. This system is expected to provide valuable technical insights for the design and operation of submarines and similar marine structures.
The findings of this study present a method for more accurately predicting hydrodynamic changes during submarine emergency rising and are expected to serve as essential foundational data for future research in related fields. For further studies, it is necessary to expand the application range of the proposed model by conducting tests under more varied depth and pressure conditions and by incorporating real sea conditions to enhance the model’s applicability.

Conflict of Interest

No potential conflict of interest relevant to this article was reported.

Funding

This study was conducted by the “Deep Ocean Engineering Basin-based standard ocean structure performance evaluation technology development (3/5)”, one of the major projects of KRISO (PES5140) and this research was supported by “Regional innovation Strategy (RIS)” through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2023RIS-007).

Notes

1) DOEB (KRISO) is an experimental facility designed to simulate marine and underwater environments. It is located in Saenggok-dong, Gangseo-gu, Busan, Korea, and used to test the performance of various marine structures or ship models in a deep-sea environment. The facility has a length of 100 m, a width of 50 m, and a depth of 15 to 50 m. It is equipped with environmental reproduction equipment (e.g., waves, algae, wind, and tow) to reproduce various marine conditions.

Fig. 1.
Schematic of blowing process in submarine
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Fig. 2.
Results of numerical simulation
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Fig. 3.
Information of BB2 submarine
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Fig. 4.
Configuration of model test
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Fig. 5.
Simple preparation tank in DOEB and model of ballast tank
ksoe-2024-084f5.jpg
Fig. 6.
Configuration of pre-test in DOEB
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Fig. 7.
Photo of pre-test
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Fig. 8.
Model test configuration of DOEB
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Fig. 9.
Results of model test
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Fig. 10.
Comparison of numerical analysis and model test results
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Fig. 11.
Comparison of numerical analysis and model test results (reflecting Eq. (7))
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Fig. 12.
z+ζh* & z+ζhLSM* Data of depth and bottle pressure in simulation
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Fig. 13.
Add-model test result for z+ζhLSM* data validation
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Fig. 14.
Final result comparison graph of all model tests and simulations ( z+ζhLSM*)
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Table 1.
Variables and symbols of ballast tank
Symbols Description
A Area in exit area (m2)
Ah Outlet hole area (m2)
Cn Outlet hole coefficient
g Gravitational acceleration (m/s2)
Htk Ballast tank height (m)
mB (t) Mass of air in ballast tank (kg)
mF (t) Mass of air in pressure bottle (kg)
F (t) Mass flow rate from pressure bottle (kg/s)
B (t) Mass flow rate from ballast tank (kg/s)
PB (t) Pressure in ballast tank (Pa)
PF (t) Pressure in pressure bottle (Pa)
PSEA (t) Pressure outside the outlet hole (Pa)
qB (t) Water flow through outlet hole (m2/s)
TB Water temperature in ballast tank (K)
TF(t) Temperature in pressure bottle (K)
VB0 Initial air volume in ballast tank (m3)
VB(t) Volume of air in ballast tank (m3)
VBB Ballast tank volume (m3)
VF Pressure bottle volume (m3)
vh Velocity of water flow in outlet hole (m/s)
x, y, z Location of geometrical center of ballast tank (m)
Zh Outlet hole distance from origin (m)
ρ Density of water (kg/m3)
ϒ Isentropic constant
Table 2.
Simulation conditions of blowing operation
Z (depth) (m) PF0 (bar) Value
1 3 F(t), mF(t), mB(t), PF(t), PB(t), Vair/VTotal
1 4
1 5
5 3
5 4
5 5
10 3
10 4
10 5
Table 3.
Principal dimensions of MARIN BB2
Dimensions Real ship
Length between perpendiculars, Lpp (m) 70.2
Design beam, B (m) 9.6
Depth, D (m) 10.6
Displacement (m3) 4365
Wetted surface area (m2) 2095.6
Propeller diameter (m) 5
Table 4.
Information of device in model test
Item quantity Specification
Compressed air tank 1 F.A.D 360 l/min, Tank volume 150 l, Maximum pressure 8 bar
Pressure regulator 1 Pressure setting range 1–10 bar
Air flowmeter 1 Flow rate measurement range 10–210 l/min
Air/vent supply valve 2 straight ball valve
Air supply line 1 urethane hose (outlet size 10 mm)
Table 5.
Result of pre-test
Item Regulator initial pressure (mF0) Note

3 (bar) 5 (bar) 7 (bar)
T1 (s) 0.60 0.48 0.42 There is a measurement error of ±0.1 s The video was played back at 0.2× speed and analyzed
T2 (s) 0.60 0.48 0.42
T3 (s) 9.81 6.66 6.19
T3T2 (s) 9.21 6.18 5.71 Calculated of water-flow-out time from BWT
Air flowmeter (l/m) 35 36 37 There is a measurement error of ±1 l/m
Regulator pressure (bar) 3.0 4.9 7.0 There is a measurement error of ±0.1 bar
Air tank pressure (bar) 5.0 4.8 4.8
Table 6.
Matrix of model test
Z (Depth) (m) PF (bar) Time for water outflow (s) Air Flowmeter (l/m)
1 3 9.6 32
1 4 7.6 33
1 5 6.4 33
5 3 18.4 40
5 4 15.2 43
5 5 12.0 43
10 3 28.8 50
10 4 21.6 50
10 5 19.0 53
Table 7.
Modified z+ζh* data matrix according to experimental conditions
Depth (m) PF (bar) ζh ζh* z+ζh*
1 3 2.5 1.0 2.0
1 4 2.5 1.0 2.0
1 5 2.5 0.8 1.8
5 3 2.5 2.0 7.0
5 4 2.5 3.0 8.0
5 5 2.5 2.7 6.7
10 3 2.5 0.0 10.0
10 4 2.5 3.5 13.5
10 5 2.5 5.0 15.0
Table 8.
z+ζh* & z+ζhLSM* data
Z (Depth) (m) PF (bar) z+ζh*(A) z+ζhLSM*(B) A − B Simulation Result Time for water outflow (s)

z+ζh*(C) z+ζhLSM*(D) C−D
1 3 2.00 1.86 0.14 9.5 9.3 0.2
1 4 2.00 2.20 −0.2 7.8 8.1 −0.3
1 5 1.80 1.71 0.09 6.6 7.7 −1.1
5 3 7.00 6.78 0.22 18.7 18.4 0.3
5 4 8.00 7.67 0.33 15.1 14.9 0.2
5 5 6.70 6.97 −0.27 11.9 12.1 −0.2
8 3 - 8.98 - - 24.2 -
8 4 - 11.36 - - 19.1 -
8 5 - 11.59 - - 15.9 -
10 3 10.00 9.74 0.26 28.8 28.4 0.4
10 4 13.50 13.63 −0.13 22.0 22.1 −0.1
10 5 15.00 14.99 0.01 18.6 18.6 0
12 3 - 9.93 - - 34.0 -
12 4 - 15.74 - - 25.2 -
12 5 - 18.64 - - 21.4 -
Table 9.
Results for add model test for z+ζhLSM* data validation
Depth (m) PF (bar) Time for water outflow (s) Air Flowmeter (l/m)
8 3 23.7 32
8 4 18.9 33
8 5 16.1 33
10 3 28.3 40
10 4 22.2 43
10 5 18.1 43
12 3 32.0 50
12 4 24.8 50
12 5 21.5 53

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