## 2 Analysis of the Reverse Torque and Vibration in the Mud Pump
### Working Principle and Structure of the Mud Pump Module
#### 2.2.1 Theoretical Calculations of the Reverse Torque for a Single-Stage Mud Pump  
According to the principle of energy conservation, the torque work conducted by the single-stage mud pump on the mud can be divided into two parts: one portion is converted into an increase in the energy of the mud and another portion is lost due to friction in the pipeline. The portion that contributes to the increase in mud energy is considered useful, while the portion lost due to friction in the pipeline is considered useless. The energy of the mud includes pressure energy, potential energy, and kinetic energy. Since the single-stage mud pump selected has the same inlet and outlet pipe diameters, the kinetic energy of the mud at the inlet and outlet remains constant. The calculation formula for the increase in the pressure energy of the mud within a time unit is shown as follows [38], where \(\Delta E_{p}\) is the pressure energy from the increase in mud within a time unit, \(\Delta P\) is the pressure difference between the inlet and outlet, and \(Q\) is the mud volume flow. \[\Delta E_{p}=\Delta PQ \tag{1}\]  
The formula for calculating the increase in potential energy of mud per unit time is shown as follows [38], where \(\Delta E_{h}\) is the potential energy from the increase in mud within a time unit, \(\Delta h\) is the height difference between the inlet and outlet, \(\rho\) is the density of the mud, and \(g\) is the gravitational acceleration. \[\Delta E_{h}=Q\rho g\Delta h \tag{2}\]  
Based on Equations (1) and (2), the formula can be derived for calculating the useful work of the single-stage mud pump per unit time, as shown in the following equation, where \(W_{e}\) is the useful work of the single-stage mud pump within a time unit. \[W_{e}=\Delta E_{p}+\Delta E_{h} \tag{3}\]  
In the three formulas above, the mud flow rate \(Q\) is 2000 L/min, the mud density \(\rho\) is 1200 kg/m\({}^{3}\), the acceleration of gravity \(g\) is 9.81 m/s\({}^{2}\), the height difference \(\Delta h\) between the inlet and outlet is 0.32 m, and the speed is 1500 rpm. In addition, in order to find the useful  
Figure 3: Exterior diagram of the mud pump module. work value, it is necessary to know the pressure difference between the inlet and outlet of the single-stage pump. For this reason, a three-dimensional model of the single-stage pump flow channel was established by using the three-dimensional modeling software Solidworks (Version 2018), and its cutaway view is shown in Figure 4a. The model was meshed by using the Mesh tool in ANSYS Workbench (version 2020 R1), and automatic meshing was used for the division. A total of 94,412 grids were divided by setting the mesh size to 0.005 m, as shown in Figure 4b. It is worth mentioning that when it is more finely meshed, the analysis results are consistent with those obtained using the current mesh; hence, the mesh independence is verified. Finally, the fluid model was analyzed by using the Fluent tool in ANSYS Workbench (Version 2020 R1). The pressure inlet-mass flow outlet boundary was used for this analysis. Based on the mud parameters shown in Table 1 and Equation (4), the outlet mass flow \(Q_{m}\) can be obtained, which is equal to 40 kg/s. In order to calculate the inlet and outlet pressure difference, the inlet pressure was 0 MPa, and the pressure difference between the inlet and outlet was the outlet pressure. Since the impeller was set as the rotating domain and the guide shell was set as the stationary domain, the interface between the rotating domain and stationary domains needed to establish the interaction surface so as to achieve the exchange of data between the rotating and stationary domains. Domain surfaces are all wall surfaces. \[Q_{m}=Q\,\rho \tag{4}\]  
Based on the fluid analysis, the pressure difference \(\Delta p\) between the inlet and outlet of the single-stage mud pump at 1500 rpm is 0.756 MPa, as shown in Figure 4c, which represents the pressure distribution along the flow path of the pump.