Analysis of the Drive Torque for the Spiral Pipe Section of an Earth Pressure Balanced Shield Tunnel Screw Conveyor

The driving torque of the shield machine is intricately linked to the soil silo muck load and the mechanical properties of both the muck and the spiral structure. Considering the shield screw conveyor’s mechanical structure and operational conditions, based on the momentum moment balance conditions, the separate calculation method was used to derive the spiral driving torque in the helical casing, the friction torque in the bearing and the friction torque of the sealing ring through mechanical analysis, and finally a torque calculation expression for the spiral pipe section was established. The research results provide technical guidance for the design and manufacture of shield screw conveyors and safe tunneling during construction.


Introduction
Shield screw conveyors have been widely used in the test of soil conditioning scheme.In a study conducted by Talebi et al. (2015) [1], the muck was treated as a Bingham fluid.The researchers employed computational fluid dynamics to model the muck flow within a screw conveyor and subsequently determined the rheological parameters of the muck through inverse calculation.In order to solve problems such as the rush of pipeline layout and space congestion in the shield machine.Huang (2016) [2] optimized the relevant parameters of the screw machine, and a screw conveyor in the form of a spherical hinge was developed without affecting the requirements and efficiency of the working conditions.Hou (2017) [3] introduced two installation schemes for the screw shaft sliding bearing of the shield machine screw conveyor, and gave an engineering case.Li (2018) [4] introduced and analyzed the remanufacturing process of Herrenknecht 6280 series EPB shield screw conveyor in more detail, and formed a set of EPB shield screw conveyor remanufacturing technology.Hu (2019) [5] analyzed the working process of EPB shield excavation and soil discharge system, and gave the design basis and parameter calculation equation of the muck discharge system.Considering the impracticability of the screw conveyor system during the operation of the three shield machines, Zhang (2020) [6] discussed the specific technical transformation ideas of the spiral conveying system of the shield machine around the three levels: the improvement of the rear gate structure, the screw reconstruction technology and contraction functions.Y. Yu and P. C. Arnold (1997) [7] conducted theoretical modeling and tests of driving torque requirement for a single screw feeder.Theoretical analysis and experimental research were carried out to improve the conveying efficiency and operation stability of a screw conveyor by Wulantuya et al (2020) [8].Merritt and Mair (2008) [9] analyzed the movement of the conditioned soil in the spiral groove, based on the equilibrium conditions of the muck section during steady operation, and proposed a method for calculating the shear driving torque of the muck in the spiral tube.The dynamic characteristics of the screw feeder during operation and the conveying mechanism of bulk materials were studied by W. Kenzou et al (1990) [10].A. J. Carleton et al (1969) [11] experimentally studied the influence of spiral geometry, screw shaft speed, screw groove filling rate and the properties of the conveyed material on the performance of the screw conveyor.J. Leitner and F. Kessier (1999) [12] analyzed the driving torque requirements of vertical screw conveyors in the event of blockage between the spiral blades and the screws.A. S. Merritt and R. J. Mair (2006) [13] carried out the test work using the 1:10 scale shield screw conveyor, and concluded that the constant shear stress between the screw groove and the modified muck and the linear distribution of pressure along the spiral pipe direction were obtained when the screw conveyor was running in a steady state.Tan et al (2022) [14] compared the design standards of the USA, Germany and China on screw conveyors from both theoretical calculation and quantitative analysis, it was pointed out that screw conveyor designers still relied heavily on their respective national tests and standards related to empirical data, which led to different design solutions for the same design task.
By analyzing the mechanical structure of the screw conveyor, we delve into the components that contribute to the driving torque.Through mechanical analysis, we derive the calculation equation for each component torque.Subsequently, we establish a theoretical calculation model for the driving torque of the screw conveyor during the regular tunneling operation of the shield machine.

Problem Statement
The shield spiral structure parameters are illustrated in figure 1, with e representing the thickness of the spiral blade, s denoting the pitch, 2r indicating the diameter of the spiral shaft, and 2R referring to the diameter of the spiral blade.The spiral angle is defined as the angle between the tangent line of any point on the spiral blade and its horizontal projection.Equation (1) defines the spiral angle at the outer edge of the spiral blade, while equation ( 2) defines the spiral angle at the spiral axis.

𝜑 = 𝑎𝑟𝑐𝑡𝑎𝑛 𝑠 2⋅𝜋⋅𝑅
(1) When the earth pressure balance shield tunnel boring machine operates under steady conditions, it conveys conditioned, uniform-flow plastic soil in a full pipe, maintaining a constant muck volume conveying rate for the screw conveyor.Figure 2 illustrates the parameters defining the object of investigation, where each parameter holds the following significance: dy represents the axial micro increment, the top length of the muck section along the spiral groove direction is denoted as d(ft)l, the average length as d(fm)l, and the bottom length as d(fb)l; the top width of the micro-element body in the groove is given by (ct)w, the average width as (cm)w, the bottom width as (cb)w, and the groove depth as d.According to the definition of the muck section parameters, the sine of the helix angle and the average sine of the helix can be described by equations ( 3) and ( 4), respectively.
Equations ( 5) and ( 6) describe the relationship between the top length d(ft)l, the bottom length d(fb)l, the average length d(fm)l and the spiral structure parameters, respectively.
By considering the interplay between the top width (ct)w, the average width (cm)w, and the bottom width (cb)w with the pitch, the thickness of the spiral blade, and the pitch angle, we can derive equations (7) to (9).
Through the decomposition of the shield screw conveyor assembly structure and the mechanical analysis during soil discharge, the driving torque is composed of the frictional shear force torque in the spiral pipe T1, the bearing friction torque T2 and the seal ring friction torque T3.

The Frictional Shear Torque in the Spiral Tube
By analyzing the forces of the muck section in the shield spiral pipe, as shown in figure 3. It can be seen that the rotation torque in the direction of the vertical spiral axis is balanced by the frictional shear torque in the spiral tube.According to the moment of momentum balance condition, equation (10) can be obtained.
) ) (10) In the given context, τc represents the shear stress between the spiral shell and the muck section, τs denotes the shear stress between the screw shaft and the muck section, (τf)t and (τf)b indicate the shear stresses between the soil plug and the upper and lower spiral blades in contact.Qn represents the combined force exerted by the upper and lower spiral blades on the plug section.φm represents the average blade helix angle, φs represents the shaft helix angle, θ represents the soil flow conveying angle, and R represents the radius of the spiral blade.Additionally, dP represents the pressure difference in the spiral groove direction of the muck section, φ represents the angle of inclination during installation of the screw conveyor, ρ represents the average density of conditioned soil, and g represents the constant acceleration of gravity.
The overall torque exerted in the rotational direction is equivalent to the torque needed by the screw conveyor to counteract the frictional shear stress generated by the spiral shell.As a result, we can derive equation (11).
where x (0≤x≤L) is the length of the spiral shaft in the spiral tube.
After integrating the equation ( 12), the equation ( 13) can be obtained.

Bearing Friction Torque
The muck outlet end of the shield screw conveyor is fixed with a sliding bearing to fix the screw shaft, and the end extended into the working chamber is installed in a suspended state.The actual situation is shown in figure 4. On the one hand, the sliding bearing supports the screw shaft and blades to fix and locate the output end of the muck.On the other hand, it transmits the driving torque required for dumping to ensure the normal implementation of the pressure maintaining and pressure regulating function.The calculation model of bearing friction torque is shown in equation ( 14).
where F is the bearing load (determined by the spiral shaft and spiral blades in the spiral tube and the muck inside the tube), rb is the radial roller contact radius, μb is the rolling friction coefficient, and φ is the angle of inclination when the shield screw conveyor is installed.

Friction Torque of Seal Ring
To prevent the leakage of muck and conditioner from the end bearing of the screw conveyor and to minimize friction and wear on components, it is typical to install sealing rings.The calculation for the friction torque of the sealing ring is detailed in equation ( 15).
In this context, the variable rs represents the radius of the sealing ring, ws corresponds to the width of the sealing ring, Ps denotes the sealing pressure applied to the sealing ring, ns indicates number of sealing rings utilized, and μs represents the friction coefficient between the sealing ring and the shaft.

Driving Torque of Shield Screw Conveyor for the Spiral Pipe Section
Considering the above torque components comprehensively, the theoretical calculation model is obtained, as shown in equation ( 16).

Conclusion
(1) Based on the mechanical structure and operating conditions of the shield screw conveyor, a separate calculation method was employed to derive the driving torque of the spiral in the helical casing, the bearing friction torque, and the sealing ring friction torque by analyzing the momentum balance conditions.Through mechanical analysis, an expression for calculating the torque of the spiral pipe section was established.
(2) The torque calculation model for for the spiral pipe section shows that the torque is related to the geometry of the screw converyor (such as helix angle, inclination angle for screw installation, helix radius, etc.) and screw motion conditions (such as rotaing speed, muck transfering rate, etc.).

Figure 3 .
Figure 3. Force analysis of muck section.