# Application of Digital Twin in the Industry of Axial Hollow-Wall Pipes
## 3 Establish a Digital Twin of the Pipe End-Forming Process
### Pre-Production Test  
PVC-U pipes are commonly connected using various methods, with the socket-and-spigot connection being widely preferred due to its convenience and ease of construction. Among the stages of the end-forming process for PVC-U pipes, thermoforming plays a critical role. This process involves deforming one end of the pipe to facilitate piping connection. Numerous researchers have conducted investigations on different techniques for the end-forming process of pipes, as referenced in studies [10; 11; 12]. These techniques can be categorized, based on the employed method, such as incremental forming [13] and punch press forming [14; 15]. They can also be distinguished by the conditions applied, such as cold flaring [16] and hot flaring [17; 18]. Furthermore, the number of forming cycles can differentiate between single-cycle flaring and multi-cycle flaring. Considering the material properties of PVC, the socket formation for PVC-U axial hollow-wall pipes typically involves single-cycle punch press forming under heated conditions. The end-forming equipment used in the experimental setup is illustrated in Figure 3, and the step of the end-forming process is depicted in Figure 4.  
The quality of the pipe's end-forming process is influenced by several factors, as illustrated in Figure 4. These factors include the forward rate of the equipment's punching head, the design of the punching equipment, the friction coefficient between the pipe and the equipment, the end-forming conditions (presence of sealing ring), the heating temperature of the pipe, the heating time, the preheating temperature of the punching head, the ambient temperature, and the cooling temperature of the pipe after end-forming. Among these factors, the change in the pipe's cross-sectional structure has minimal impact on the first four factors. To minimize the design and construction costs of the axial hollow pipe production line, it is advisable to keep these four factors unchanged, compared to the existing design of the company's solid-wall pipe end-forming production line. Specifically, the equipment forward rate is set at 100 mm/s, the punching head has a rounded corner design of 3\({}^{\circ}\) to reduce stress concentration, lubricating oil is applied to the outer surface of the equipment and rubber ring to reduce the friction coefficient, and a sealing ring is used during the end-forming process to simplify subsequent construction steps and to facilitate the insertion of a steel ring inside the sealing ring to improve its stiffness. However, the change in the pipe's cross-sectional structure has a significant impact on the last five factors, which require further research to determine the optimal design. The experimental site for this research is located in Shandong Province, China. The experiments were conducted in November, when the ambient temperature in the factory was approximately 15 \({}^{\circ}\)C. Under the conditions of an ambient temperature of 15 \({}^{\circ}\)C, with heating temperatures of 225 \({}^{\circ}\)C on both the inner and outer sides, the heating time required for a solid-wall PVC-U pipe with an outer diameter of 200 mm and a wall thickness of 8 mm was determined to be 180 s. Considering that the axial hollow-wall pipe has a lower cross-sectional rigidity, compared to the solid-wall pipe, it requires a lower end-forming process temperature. Furthermore, considering that the axial hollow pipe has the same material composition as the solid-wall pipe, but with a thinner wall thickness (the hollow pipe has a wall thickness of 7.5 mm, with 1.5 mm for the outer wall and 2 mm for the inner wall, for an outer diameter of 200 mm, as illustrated in Figure 5), it can be inferred from Equation (1) that the optimal heating time for the hollow pipe, under the same heating temperature conditions, should fall within the range of 50 to 70 s. Therefore, in the pre  
Figure 4: Steps of the end-forming process. Figure 3: Production line of the end-forming process.