In this study, the pressure compensation mechanism of a reducer bellows is analyzed. This device is typically used to reduce the size of undersea instruments and improve related pressure resistance and sealing capabilities. Here, its axial stiffness is studied through a multi-fold approach based on theory, simulations and experiments. The results indicate that the mechanical strength of the reducer bellows, together with the oil volume and temperature are the main factors influencing its performances. In particular, the wall thickness, wave number, middle distance, and wave height are the most influential parameters. For a certain type of reducer bellows, the compensation capacity attains a maximum when the wave number ratio is between 6:6 and 8:4, the wall thickness is 0.3 mm, and the wave height is between 4–5 mm and 5–6 mm. Moreover, the maximum allowable ambient pressure of the optimized reducer bellows can reach 62.6 MPa without failure, and the maximum working water depth is 6284 m.

As human move towards the deep-sea field, mounting number of equipment and instruments have been applied. As an example, the diving depth of China Jiaolong reached 7020 m, and the pressure of ambient is greater than 70 MPa. This poses a grand challenge regarding undersea instruments. In order to reduce the size of submarine instruments, improve the performance of pressure and sealing, it is necessary to design a compensation structure of pressure.

In recent years, there have been many structures to compensate for pressure, such as the piston, bladder,diaphragm and bellows type [

Sun et al. [

Currently, the research mainly focuses on the non-reducing bellows, and the research on the reducer bellows is less. This paper will study the compensation mechanism of pressure fixed at both ends for the reducer bellows through theoretical research; verify the reliability of simulation model for the reducer bellows through theoretical and experimental methods. For a certain undersea testing instrument, the reducer bellows is studied and analyzed based on the principle of liquid compressibility, and the pressure changes are discussed to determine its ability of compensation.

The reducer bellows is composed of two U-shaped bellows with different diameters, and the two ends are respectively fixed on the device to realize the compensation of pressure. One surface of the reducer bellows is in contact with insulating oil, and the other surface is in contact with seawater. The inside of the reducer bellows contains a test instrument and a insulating oil (

where _{out} represents external pressure, _{b} represents the force generated by deformation of the reducer bellows, and _{oil} represents internal insulating oil pressure.

When the reducer bellows is lowered with the instrument in the sea, _{out} increases gradually. At the same time, internal pressure of the reducer bellows will be lower than external pressure. In order to balance internal and external pressure and achieve the purpose of pressure compensation, the reducer bellows will move down from location of the middle joint when pressure increases, that, the large-diameter bellow is compressed, and the small-diameter bellow is stretched. Since both ends are fixed, the total length in axial direction is unchanged. Nevertheless, the length of each bellow with either small-diameter and large-diameter varies differently (_{b} of the reducer bellows becomes smaller and reduction amount is _{b}, insulating oil volume _{oil} decreases, and _{oil} increases. In a word, when external pressure of the reducer bellows changes, the displacement of the reducer bellows with different diameters will change, resulting the volume of internal insulating oil changes, which in turn changes internal pressure, realizes balance of internal and external pressure, and achieves compensation of pressure. Among them, the change of pressure _{oil} of insulating oil is passive:

(1) When _{b} becomes smaller, _{oil} will become smaller and _{oil} will become bigger.

(2) When _{b} becomes larger, _{oil} will become bigger and _{oil} will become smaller.

For _{b}, when external pressure changes, reducer bellows will deform, then _{b} will also change, and the change of volume _{b} determines its performance of compensation. When pressure of the reducer bellows changes, its movement direction is primarily moving in the axial direction. The axial displacement determines the size of _{b}, and the axial stiffness generally expresses the axial displacement of the reducer bellows. The smaller the axial stiffness is, the easier it is for the reducer bellows to move and deform in the axial direction, and the larger the change in axial displacement, the larger _{b}, and the better performance of compensation. At the same time, the mechanical strength of the reducer bellows will also effect its axial change of displacement, which will affect performance of compensation. Under the same external pressure, the smaller the equivalent stress of the reducer bellows, the greater the mechanical-strength, the larger the allowable change of axial displacement, the larger _{b}, and the better ability of pressure compensation. Therefore, the reducer bellows performance of pressure compensation was studied in following two aspects: axial stiffness and mechanical strength.

For insulating oil inside the reducer bellows, its volume change is a passive change, but the pressure and other forces generated by volume change will act on the reducer bellows in turn. Its volume change is affected by external pressure and temperature, and formulas are shown as follows [

where Δ_{oil} represents the variable of internal insulating oil pressure, _{oil} represents the compression coefficient, Δ_{p} represents volume change of insulating oil concerning pressure, _{y} represents the expansion coefficient of insulating oil, _{q} represents air expansion coefficient, _{t} is the volume change of insulating oil with the change of temperature, Δ_{t} represents the volume change of internal insulating oil with temperature changes.

Ignoring the effect of temperature, _{out} increases, and _{b} decreases. When _{oil} increases, the change of pressure is Δ_{oil}, and the insulating oil change of volume is Δ_{p}. When _{b} and _{oil} decrease, the purpose of bellows pressure compensation is achieved. Considering the effect of temperature, provided that the external temperature is raised without the change of _{out}, _{t} increases, _{oil} increases, and _{b} increase, which decreases the deformation volume of the reducer bellows and increases the range of variation in volume. The ability of pressure compensation will increase due to the increase in temperature.

Through research on the compensation of pressure excitation of the reducer bellows, it is determined that the axial stiffness has a great influence on it, so it is necessary to discuss and analyze the axial stiffness of the reducer bellows. The empirical formula for calculating the axial elastic stiffness _{a} of non-reducing bellows is:

where _{a} represents the axial stiffness of the bellows, N/mm; _{m} represents average diameter of the bellows, _{m} = (_{b}+^{t}_{b}_{p} represents nominal thickness of a layer of material of the forming bellows, mm; _{f} represents the correction factor for calculation of U-shaped bellows;

For the designed reducer bellows, the empirical formula of bellows cannot be directly used for calculation. It is necessary to calculate the axial stiffness of the large-diameter bellow and the small-diameter bellow separately, and then use the equivalent stiffness principle to calculate the axial stiffness of the reducer bellows.

When the reducer bellows is fixed on one end face, a load is applied to the other end face; at this time both the large-diameter bellow and the small-diameter bellow are stretched, and the two bellows are both subjected to the same force. It can be determined that the combination of the reducer bellows is in series, and its equivalent stiffness is:

When the reducer bellows is in a fixed state at both ends, a pressure load is applied to its surface, the large-diameter bellow is compressed, the small-diameter bellow is stretched, and the displacement generated by the two bellows alone is the same. The combination mode of the reducer bellows is parallel, and its equivalent stiffness is:

For the reducer bellows, due to its innovative structure, it does not directly apply previous theories. It is necessary to use the combination of finite element simulation method, theoretical and experimental method to verify the simulation model correctness. On this basis, the sensitivity analysis of parameters affecting the pressure compensation is carried out, and the optimal parameter combination is determined. Firstly, perform a pressure analysis on the reducer bellows to determine the maximum pressure it can withstand. Furthermore, the bidirectional Fluid-Solid Interaction simulation is performed on the reducer bellows and the insulating oil inside. Then the deformation and maximum depth of the ocean determined under the change of external pressure and temperature. Now the specific structure of the reducer bellows can be obtained is simulated and analyzed.

The material of bellows is 304 stainless steel, its 06Cr19Ni10, the elastic modulus is 1.94 × 10^{11} Pa, the Poisson’s ratio is 0.3, the density is 7930 kg/m^{3} . The bellows for experiments are mature products, which standard is GB/T 16749-2018 and the specifications are shown in

Bellows | _{b}/mm |
^{t}_{b} |
_{f} |
||||
---|---|---|---|---|---|---|---|

Small-diameter bellow | 16 | 12 | 0.2 | 1.8 | 1.94 × 10^{11} |
2.517 | 12 |

Large-diameter bellow | 20 | 15 | 0.25 | 2.25 | 1.94 × 10^{11} |
2.186 | 12 |

Finite element analysis of the axial stiffness of bellows is extensive. The research results show that it is feasible to use finite element method to study the axial stiffness of bellows. In this paper, finite element analysis software ANSYS Workbench is used to analyze the axial stiffness of bellows. Solidworks is used for 3D modeling of bellows, and then the model is imported to ANSYS Workbench for analysis. The simulation model uses a C3D8R mesh, which supports nonlinear parameters such as large deformation, stress strengthening and plasticity. The simulation model adopts the C3D8R grid, and

The axial stiffness of the large-diameter bellow and the small-diameter bellow can be calculated by

Large-diameter bellow: _{r1} = 368.69 N/mm;

Small-diameter bellow: _{r2} = 256.16 N/mm.

In the experiment, the fixed method of the reducer bellows is to fix one end face and apply a load to the other end face. It can be determined that the combination method is a series connection, and its equivalent axial stiffness is calculated by the

_{r} = 151.15 N/mm.

Before the experiment, the large-diameter bellow and the small-diameter bellow are welded together. In order to reduce errors caused by welding, a set of comparative experiments are now performed. There are three samples in the experiment, that is to say number1, number2 and number3 (

The axial stiffness of the reducer bellows can be obtained through linear regression equation:

Number 1: _{p1} = 147.05 N/mm;

Number 2: _{p2} = 146.34 N/mm;

Number 3: _{p3} = 136.13 N/mm.

The average axial stiffness of the reducer bellows is:

_{p} = (_{p1} + _{p2} + _{p3})/3 = 143.17 N/mm.

The reducer bellows is analyzed statically and large deformation is opened. The 8-node hexahedral element is used in the reducer bellows, and its mesh size independence is verified. The linear force is applied to the reducer bellows in the elastic deformation range and the displacement of the reducer bellows in different mesh sizes is observed. Now set the order of the reducer bellows mesh size is 0.3, 0.35 and 0.4 mm, 0.45 and 0.5 mm, the grid number is 168590, 102568, 76041, 66633, 52699.

Force/N | Mesh size/mm | ||||
---|---|---|---|---|---|

0.3 | 0.35 | 0.4 | 0.45 | 0.5 | |

9 | 0.0607 | 0.0553 | 0.0573 | 0.0542 | 0.0561 |

18 | 0.1214 | 0.1107 | 0.1147 | 0.1084 | 0.1121 |

27 | 0.1823 | 0.1663 | 0.17249 | 0.1627 | 0.1681 |

36 | 0.2431 | 0.2217 | 0.2301 | 0.217 | 0.2241 |

45 | 0.3037 | 0.2774 | 0.2893 | 0.2716 | 0.2802 |

54 | 0.3647 | 0.332 | 0.3462 | 0.3257 | 0.3362 |

The model with mesh size of 0.5 mm is selected for the simulation calculation, and the axial stiffness of the reducer bellows only needs to be calculated in linear range, which can be obtained by Hooke’s law:

where _{a} represents the axial force, _{a} represents the axial stiffness in the linear phase and

_{a} = 160.61 N/mm.

Force/N | 9 | 18 | 27 | 36 | 45 | 54 |
---|---|---|---|---|---|---|

Displacement (mm) | 0.0561 | 0.1121 | 0.1681 | 0.2241 | 0.2802 | 0.3362 |

Axial stiffness (N·mm^{−1}) |
160.43 | 160.57 | 160.62 | 160.64 | 160.59 | 160.62 |

Through theoretical calculation, experiment and simulation, the axial stiffness of the reducer bellows obtained is: 151.15, 143.17, and 160.61 N/mm, respectively. The error between simulation and theoretical calculation is 6.3%, and the error between simulation and experiment is 10.9%. The error between theoretical calculation and simulation is small, which shows that the calculation method of equivalent stiffness can be used to calculate the axial stiffness of the reducer bellows. The error between experiment and simulation is large, which may be due to the following reasons: the wall thickness of the selected bellows is relatively thin, the shell is perforated during welding, which cannot achieve the ideal welding quality, and is different from the simulation model; The axial stiffness of different corrugated pipes cannot be guaranteed, resulting in a discrepancy with the real situation when calculating the displacement at the end. Through experiments and theoretical calculations, the final comparison shows that the error obtained by the simulation method is within the acceptable range, which verifies the reliability and authenticity of the simulation analysis.

In order to obtain the reducer bellows performance of compensation, it is necessary to carry out a sensitivity analysis on the parameters affecting the performance of compensation, such as wall thickness, wave number, and wave height. In this experiment, an undersea test instrument was selected as the research object, and the structural size of the reducer bellows was initially determined (_{b} is 0–0.2 MPa.

Bellows | _{b}/mm |
|||||
---|---|---|---|---|---|---|

Small-diameter bellow | 111 | 104 | 0.2 | 1 | 3.5 | 6 |

Large-diameter bellow | 117 | 110 | 0.2 | 1 | 4.5 | 6 |

The wall thickness of the reducer bellows has a significant impact on its axial stiffness and mechanical strength, and it has an enormous impact on the ability to resist and compensate for pressure in the deep ocean. Now analyze the variation law of the displacement and equivalent stress of the reducer bellows with the external pressure _{b} when the wall thickness is 0.2, 0.25, 0.3, 0.35 and 0.4 mm.

It can be seen from

The wavenumber combination of the reducer bellows has an influence on its axial stiffness and mechanical strength, and the influence on the capacity of the compensation cannot be ignored, so it is necessary to select a right combination. Now change the wavenumber combination mode of the small diameter-bellow and the large-diameter bellow, and analyze the variation law of the displacement and the equivalent stress of the reducer bellows with the external pressure. When the total wavenumber is 12, the displacement and equivalent stress of the reducer bellows with a wavenumber ratio of 4:8, 5:7, 6:6, 7:5 and 8:4 are analyzed.

It can be seen from

In the reducer bellows structure, the intermediate distance between the small diameter bellow and the large diameter bellow significantly impacts its axial stiffness and mechanical strength. Therefore, it is necessary to determine a suitable distance of intermediate to improve the reducer bellows ability of compensation. Now change the middle distance of the reducer bellows, and analyze the changing law of the displacement and equivalent stress of the reducer bellows with the external pressure when the middle distance of the reducer bellows is 0, 1, 2, 3, 4 mm.

It can be observed from

The change of the wave height and the combination of the wave height of the reducer bellows will affect its axial stiffness and mechanical strength, and affect its ability of pressure compensation, so it needs to be discussed and analyzed. Two groups of analyses will be performed: one group changes the wave height of the small-diameter bellow and the large-diameter bellow at the same time, and the other group changes the wave height of the large-diameter bellow alone, and analyzes the variation law of the reducer bellows performance of pressure compensation under different wave heights.

Change the wave height of the small diameter bellow and the large diameter bellow at the same time, analyze the variation law of displacement and equivalent stress of the reducer bellows with external pressure when the wave height is 3.5–4.5 mm, 4–5 mm, 4.5–5.5 mm, 5–6 mm and 5.5–6.5 mm.

It can be seen from

Now we keep the wave height of the small-diameter bellow at 3.5 mm, analyze the variation law of the displacement and equivalent stress of the reducer bellows with external pressure when the wave height of the large-diameter bellow at 4.5, 5, 5.5, 6 and 6.5 mm.

It can be seen from

It can be seen from

Through the sensitivity analysis of the parameters of the reducer bellows, the preferred parameters of the reducer bellows is: the wall thickness is 0.3 mm, the wavenumber ratio is 7:5, the intermediate distance is 2 mm, the inner diameter of the small bellow is 104 mm, the wave height is 5 mm; the inner diameter of the large bellow is 111 mm, the wave height is 5 mm. The inside of the reducer bellows is insulating liquid; the density of the insulating liquid is 920 kg/m^{3}, the bulk modulus is 6 × 10^{8} Pa, and the thermal expansion coefficient is 0.0008. The density of seawater is 1020 kg/m^{3}. The reducer bellows initial volume of internal is _{b} = 671180 mm^{3}, the volume of insulating oil is _{oil} = 442986 mm^{3}, the volume of solid is _{s} = 228194 mm^{3}, and the initial temperature is set to 25°C. The boundary conditions are: fix both ends of the bellows and apply pressure of 0.89 MPa to the outer surface.

When the external pressure _{b} is 0.89 MPa, the axial motion position of the large-diameter bellow reaches the limit, and the reducer bellows capacity of compensation reaches the maximum at this time. The maximum displacement is the middle position of the deformed bellows, and the maximum value is 1.1504 mm (^{3}, and the volume reduction is 45611 mm^{3}. From the compressibility _{b} and _{oil} into _{out} = 62.6 MPa and the maximum allowable ocean depth is 6284 m, which means that the maximum allowable ambient pressure without failure of the reducer bellows under this environmental condition.

The model is simulated and verified by two-way Fluid-solid Interaction (^{3} (^{3}. According to

Considering the principle of compressibility of liquids, the following measures can be taken to improve the reducer bellows performance of pressure compensation: reduce the initial volume _{oil} of the internal insulating oil; choose insulating oil with a larger bulk modulus

The following conclusions can be drawn from the study:

(1) It is determined that the axial stiffness and mechanical strength are the main factors affecting the reducer bellows compensation of pressure.

(2) Sensitivity analysis of the reducer bellows parameters to clarify the influence of wall thickness, wave number ratio, intermediate distance and wave height on performance of pressure compensation.

(3) A certain type of bellows used in testing instruments of the deep ocean is analyzed, and the following conclusions are drawn: the capacity of compensation is best when the wavenumber ratio is between 6:6 and 8:4, the wall thickness is 0.3 mm, and the wave height is between 4–5 mm and 5–6 mm. The reducer bellows can be used in the deep sea with a pressure of 62.6 MPa and a working water depth up to 6284 m after optimization in the example analysis. Moreover, it will improve the resistance of bellows to external pressure when increasing the axial stiffness _{c}, the volume of inner _{b} and the bulk elastic modulus of the internal insulating oil _{oi}_{l}, or lowing the insulating oil volume _{oil}.

(4) The research results not only have pertinence and significance for the design and optimization components of pressure balance used in deep-sea instruments, but also provide reference value for other structures such as a large high-pressure deformation and fluid thermal deformation of volume.

Key Laboratory of Petroleum and Natural Gas Equipment of Ministry of Education.

The authors declare that they have no conflicts of interest to report regarding the present study.