To calculate the flow resistance of a main cable dehumidification system, this study considers the air flow in the main cable as the flow in a porous medium, and adopts the Hagen–Poiseuille equation by using average hydraulic radius and capillary bundle models. A mathematical derivation is combined with an experimental study to obtain a semi-empirical flow resistance formula. Additionally, Fluent software is used to simulate the flow resistance across the main cable relative to the experimental values. Based on the actual measured results for a Yangtze River bridge, this study verifies the semi-empirical formula, and indicates that it can be applied in actual engineering.

In a suspension bridge, the main cable dehumidification technology plays an important role in preventing the main cable from erosion [

The main cable of a suspension bridge is composed of tens of thousands of specially-treated high-strength steel wires. These steel wires are wound with a wire wrapped with a protective layer. The structure is shown in

Many scholars have studied and optimized a main cable dehumidification system. Jensen et al. [

However, there has been no research on the flow resistance of dry air in the main cable. A main cable is made of thousands of steel wires. Air flows through the gaps of the wires, leading to an irregular cross-section. The bending wire also leads to gap bending. Both the irregular cross-section and bending wires make the calculation unique and difficult. In this study, based on a porous medium model and capillary bundle model, the flow resistance in the main cable is analyzed in depth, and a formula is obtained for the bending coefficient. Through experimental research to determine the bending coefficient, a semi-an empirical formula is obtained for the flow resistance. The accuracy of the semi-empirical formula proposed in this study is verified by simulation modeling and field measurements of a main cable.

The main cable of a suspension bridge is an accumulated porous medium, and the air flow in the main cable can thus be considered as the air flow in a porous medium, as found in nature. Models for studying the flow resistance in a porous medium include the average hydraulic radius model, capillary model, gap model, flow resistance model, and statistical model [

Air flows into the main cable from the intake hood and flows out through the exhaust hood, as shown in

The study of the air flow process when adopting the average hydraulic radius model considers the circulation channel in the main cable as a circular tube of equivalent diameter. The pressure drop of the fluid flow in the main cable passage can be computed using the Hagen–Poiseuille formula [

Here, ^{2});

Here, ^{3}), ^{3}), and ^{2}).

Each steel wire is considered to be composed of many smaller cylindrical steel wires, with a height equal to its diameter. The contact surface area of the air flowing in the main cable is the lateral area of the cylinder steel wire, and the specific surface of a cylindrical steel wire with a height equal to the diameter is the ratio between the lateral area and volume of the steel wire (the lateral area is two-thirds of the surface area of the cylindrical steel wire). If the ratio between the surface area and volume of the cylinder with the height equal to its diameter is

Thus, the relationship between

In this relationship, ^{2}/m^{3}); ^{2}/m^{3}); and ^{2}/m^{3}).

Two parameters are required to determine the character of a non-spherical particle, namely, the degree of sphericity and the equivalent diameter. The sphericity degree is also called the form factor. It expresses the difference between the particle form and a sphere, and is defined as the surface area of a sphere with the same volume as the particle divided by the surface area of the particle, as follows:

Here, ^{2}); and ^{2}).

In engineering, a non-spherical particle is commonly replaced with a “equivalent” spherical particle to make some character of the non-spherical particle equivalent to that of the spherical particle. The diameter of such a spherical particle is called the equivalent diameter, and represents the size of the non-spherical particle. According to the equivalence theory, the equivalent diameter of a flow pass can be determined by adopting an isopycnic or equal specific surface area method.

The equivalent diameter of a non-spherical particle can be expressed by adopting the diameter of a spherical particle with the same volume as that of the particle, as follows:

Here, ^{3}).

The specific surface area equivalent diameter of the non-spherical particle

Using

The geometric proportion surface area equivalent diameter of a non-spherical particle must be smaller than the isopycnic equivalent diameter. The character of a non-spherical particle can be expressed by the above form factor and equivalent diameter, as follows:

Substituting

The relationship between the apparent velocity of the main cable

According to

The flow of air in the main cable void is a curve; therefore, we should consider the impact of the bendability on the flow resistance. Thus, the pressure drop when the fluid passes through a unit length of the main cable should be determined as follows:

In the formula,

According to

Here,

According to

Hence,

Substituting

Here, ^{2}·s); ^{3});

The diameter of the main cable steel wire was 5.2 mm. The porosity of the cable was 0.2. The main cable was 6 m long, and the diameter of the main cable was 432.8 mm. We can solve for the applicable flow velocity range in

Using

According to

To obtain the flow resistance in the main cable, a laboratory table was set up, as shown in

The main purpose of the experiment was to obtain flow resistance data that can be regressed to determine the bending coefficient

Steel diameter | Wire number of single strands | Strand number | Wire number | Area | Porosity | Diameter | Length | |
---|---|---|---|---|---|---|---|---|

Model | 5.2 mm | 91 | 61 | 5551 | 1178 cm^{2} |
19.8% | 432 cm | 6 m |

The porosity of the main cable must meet the requirements of actual engineering, i.e., no greater than 20%, and the laboratory table should have good sealing. Moreover, after erecting the laboratory table, the main cable model should be inclined toward different angles, to meet the requirements for simulating different working conditions.

Model type | Condition 1 | Condition 2 | Condition 3 | Condition 4 | Condition 5 | Condition 6 | Condition 7 | Condition 8 | |
---|---|---|---|---|---|---|---|---|---|

Model | Velocity m/s | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | 0.09 |

To determine the air parameters in the main cable, pressure sensors were embedded at 30 testing points in the main cable model. One testing point was set in the intake hood on the left side, and the other was positioned in the plenum chamber on the right side. Seven testing points were set in each of the Sections 1-1, 2-2, 3-3, and 4-4 in the model. The positions of the sensors in each section are shown in

A McMillan Model 50 flowmeter was used to obtain the flow measurements, and the pressure measurements were made using a DST6800-DR micro-differential pressure gauge. The characteristics of the testing devices are listed in

Parameter | Equipment | Range | Accuracy | Adaptability |
---|---|---|---|---|

Q | McMillan Model 50 flowmeter | 0–100 L/minute | ±1% | High repeatability, with almost no drift. Fast response. Linear signal output. |

P | DST6800—DR | 0–1000 Pa | ±0.075%FS | Air or similar non-gaseous conductor |

As the gas flowed from the right side to the left side, the flow resistance loss was invariable. Under the same flow velocity, the unit length loss of the flow resistance was the same, and the pressure drop curve should be a straight line. For the main cable model, during the airflow process, the unit length flow resistance was determined as follows:

Here,

After fitting the experimental data using the least squares method, a computational formula for the unit length flow resistance pressure drop of the air flow in the main cable was obtained, as follows:

Here,

When the air flow velocity in the main cable is equal, the computational results of

Fluent software was used for a simulation of air flow resistance. The simulation model was the same as the experimental model. The simulation adopted a porous medium model in Fluent to simulate the air flow in the main cable; the porosity was 19.8%, and the air flow velocity ranged from 0.02 m/s to 0.09 m/s (see

The simulation model was divided into six steps, as described below:

1. The length of the main cable model was set as 6 m, the diameter as 432.8 mm, and the porosity as 19.8%. In the Fluent software, the coordinates were X_{min} = 100 mm, X_{max} = 6100 mm, Y_{min} = −216.4 mm, Y_{max} = 216.4 mm, Z_{min} = −216.4 mm, and Z_{max} = 216.4 mm. The resistance numerical simulation of the main cable model was conducted with air velocities of 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9 m/s. The grid spacing was 0.04 m, the main cable model node number was 33968, and the grid number was 158837.

2. The non-coupled implicit solver was selected for the steady-state conditions to solve for the flow field, and the physical velocity in the porous formulation area of the solver panel was selected for calculation. The laminar calculation model was also selected.

3. The desired fluid name was selected in the Material Name. The editing function was used (if necessary) to change the fluid parameter settings.

4. The inlet, outlet, and solid boundary conditions were set, the velocity inlet boundary condition and pressure outlet boundary condition were selected, and the fluid area was set as a porous area.

5. The monitoring volume and number of iterations were set, the calculation model was initialized, and the file was saved.

6. Iterative calculations were conducted, and Tecplot software was used to post-process the calculation results.

The simulation results are shown in

As shown in

The simplified model for the porous media does not need to draw all the main cable wires in the gambit, and the parameter settings for the simplified model of porous media can be determined according to the research situation. Some qualitative parameters can be calculated and determined based on experimental data, such as the viscosity resistance coefficient. Using the simplified model in the Fluent software to calculate the air flow resistance in the main cable can lead to an improved prediction of the change law of the air flow pressure drop in a main cable with the air velocity at low flow rates.

The simplified model can only describe the change in the air flow velocity in the main cable approximately, and cannot reflect the actual change. In the simulation process, the simulated air flow is not identical to the actual flow process. Thus, the simulation results need to be compared with the test results, and a small gap should be allowed.

The Yangtze River Bridge comprises three towers and two spans, and its span is divided into sections of 390 m + 1080 m + 1080 m + 390 m. There are two main cables in the bridge, each with an average length of 3109 m. The cables are made of hot-dip galvanized high-strength steel wire with a diameter of 5.0–5.5 mm, constructed using air spinning or prefabricated parallel wire strands. Each main cable contains 184 parallel steel wire cables. Each wire cable has an average length of 2580.8 m and contains 127 hot-dip galvanized high-strength steel wires with diameters of 5.3 mm; each main cable contains 23268 wires in total. Each main cable weighs 1,0583.6 tons, and the internal void ratio of the main cable is approximately 20%. These steel wires are wound with wrapping and an outer sheath. To verify the accuracy of the semi-empirical formula and simulation, a field test was performed on the Yangtze River Bridge. The results were compared to flow resistance test data calculated using the semi-empirical formula.

To ensure the accuracy of the experiment, two sets of experiments were conducted, and the measurement regions were the same length (AB) and (CD). AC is the air inlet, and BD is the exhaust; AB and CD were both 18 m long. Pressure sensors were set at Points A, B, C, and D, and the pressure values were recorded. Field photographs are shown in

The length of the main cable in the experiment was 18 m, and the diameter was 906 mm. The main cable included 184 strands of parallel steel wires, each containing 127 parallel steel wires (each with a diameter of 5.3 mm). The total volume of the air supply was Q, the sectional area of the main cable was S = 6447 cm^{2}, and the porosity was 0.2. Based on the testing time, the average velocity of the air flow in the main cable was calculated.

Experiment 1: The experimental results obtained when the Roots blower frequency converter was adjusted to 54.6 Hz are shown in

Experimental working conditions | Time | ||||
---|---|---|---|---|---|

1^{st} testing |
5’52” | 270 | 130 | 130 | 272 |

2^{nd} testing |
5’40” | 268 | 129 | 130 | 271 |

3^{rd} testing |
5’58” | 273 | 130 | 130 | 270 |

Average value | 5’50” | 270.3 | 129.7 | 130 | 271 |

Experiment 2: The experimental results obtained when the Roots blower frequency converter was adjusted to 38.2 Hz are shown in

Experimental working conditions | Time | ||||
---|---|---|---|---|---|

1^{st} testing |
3’56” | 420 | 206 | 208 | 426 |

2^{nd} testing |
3’40” | 422 | 208 | 210 | 430 |

3^{rd} testing |
3’30” | 430 | 211 | 210 | 419 |

Average value | 3’42” | 424 | 208.3 | 209.3 | 425 |

The average airflow velocity

Using formula

The average air flow velocity

Using

The numerical simulation results for the flow resistance in the main cable of the bridge are shown in

The calculated values (according to the semi-empirical formula) and numerical simulation results for the main cable flow resistance in Experiments 1 and 2 are shown in

Practical value (Pa) | Calculated value (Pa) | Simulated result (Pa) | Error | |
---|---|---|---|---|

Experiment 1 | 140.8 | 130.6 | 131.7 | <8% |

Experiment 2 | 215.7 | 206.7 | 208.4 | <5% |

The results in

Based on a porous medium model and capillary bundle model, the Hagen-Poiseuille formula is adopted. An average hydraulic radius resistance model is used to perform a mathematical derivation of the air flow resistance in the main cable, to thereby obtain an identical formula. This indicates that the equivalence of the air flow in the main cable to that in a porous medium is correct. The air flow in the void of the main cable follows a curve; therefore, we should consider the impact of the bendability on the flow resistance. The bending coefficient

The study determines the air flow resistance value in the main cable using an empirical model. It solves for the bending coefficient in the air flow resistance computational formula of the main cable to obtain a semi-empirical formula for the flow resistance in the main cable.

Using Fluent, the air flow resistance in the main cable is computed, along with the change law of the air flow pressure drop with changes in the air flow velocity in the main cable.

To verify the correctness of the semi-empirical formula, a field test was conducted on the air flow resistance in the Yangtze River bridge. The measured resistance value was basically identical to that calculated through the semi-empirical formula, providing validation for the formula.

The air flow resistance in the main cable has a very important influence on the drying of the main cable. Based on the semi-empirical formula determined in this study, a designer can choose an appropriate air supply fan pressure and air volume under different main cable parameters. Furthermore, it is possible to determine the annual operating energy consumption of the main cable dehumidification system, which has significant practical significance in energy saving control.

We gratefully acknowledge the kind assistance of Dr. Luyan Sui and Dr. Guanzhong Peng in the preparation of this article.