## 4 All-Metal Screw Rod Motor Modelling
### Effect of a High-Temperature Environment on All-Metal Screw Motor Clearance  
All-metal screw motors used in ultra-deep and extra-deep wells are subject to expansion and deformation of metallic materials due to the complex deep-ground environment and the high-temperature extremes they are subjected to, which can significantly affect the clearance fit of the stator and rotor. Therefore, the study of the clearance variation in all-metal screw motors in deep-ground high-temperature environments is of great significance for designing motor clearance values to cope with drilling environments at different depths. In this study, the initial temperature of the stator was assumed to be the ground ambient temperature of 20 \({}^{\circ}\)C. When the temperature rose uniformly to a certain value, a certain deformation and corresponding thermal stress were generated inside the stator. In particular, the following assumptions were made in the finite element analysis of the deformation of the metal motor stator as well as the rotor under the action of the temperature field: (1) The ambient temperature did not affect the material parameters and thermodynamic parameters of the metal. (2) There was no temperature gradient in the motor stator in the axial direction. (3) The temperature increased by 3 \({}^{\circ}\)C for every 100 m increase in depth. As above, well depths of 9000-13,000 m were simulated, and thus, the corresponding ambient temperatures were 290 \({}^{\circ}\)C, 320 \({}^{\circ}\)C, 350 \({}^{\circ}\)C, 380 \({}^{\circ}\)C and 410 \({}^{\circ}\)C.  
Material parameters from Section 2 and the standard were combined, that is, density \(\rho\) = 7.19, modulus of elasticity according to test results taken at room temperature was 225,765.3 MPa, with Poisson's ratio of 0.295, thermal conductivity of 0.937, specific heat capacity of 0.461 J/(g.\({}^{\circ}\)C), and volume thermal expansion coefficient of \(6.2\times 10^{-6}\). To investigate the effect of temperature on the clearance of all-metal screws, thermal loads were applied separately to stator and rotor models fitted with different original clearance values without considering the internal pressure. According to the 410 \({}^{\circ}\)C thermal load applied to seven different sizes of stator and one size of rotor, corresponding stator and rotor deformation and stress clouds were obtained, as shown in Figure 13.  
Figure 12: Variation in the displacement of the metal stator for one week with different internal pressures. Figure 13: Thermal expansion displacement nephograms of the stator and rotor. The deformation of the metal stator and rotor of the screw motor under the action of an ambient temperature of 410 \({}^{\circ}\)C and the change in the clearance are shown in Table 3.  
From Figures 13 and 14, it can be seen that (1) the deformation of the all-metal screw motor under the action of ambient temperature was somewhat larger compared to the deformation under the action of internal pressure, and the all-metal screw motor expanded outward on both the inner and outer surfaces of its metal stator and rotor with a displacement of close to 0.2 mm under the heat load of 410 \({}^{\circ}\)C. (2) For the metal stator-rotor clearance model of 0.02 mm, the fit clearance reached \(-\)0.01512 mm, and the relative change rate of the clearance reached 175.58%, the stator-rotor overfilling state due to expansion. (3) The rate of change in the fit clearance decreased with increasing initial clearance of the stator-rotor, and when the clearance increased to 0.1 mm, the relative rate of change rate in the clearance was 34.607%, and when the initial clearance was 0.2 mm, the relative rate of change was only 16.9%, i.e., basically no change. (4) As the ambient temperature field continued to increase, the clearance value after expansion continued to increase, but the relative rate of change rate in clearance continued to decrease, and after the initial clearance value of 0.2 mm, the rate of change in clearance started leveling off.