CLADDING TECHNOLOGY SHANXI CO., LTDMathematical Model Establishment for Automatic Overlay Welding of Spherical Heads
Literature Overview
This 2003 paper by Yu Zhonghai from the School of Mechanical Engineering, Yanshan University, published in Transactions of the China Welding Institution, addresses the challenge of developing a mathematical model for automatic overlay welding of spherical heads. Spherical heads are used in pressure vessels, spherical storage tanks, and reactor vessels where uniform stress distribution is required. Overlay welding of spherical heads presents unique challenges compared to flat plates due to the complex curvature, the need for multi-pass coverage of a doubly curved surface, and the difficulty of maintaining consistent weld bead geometry and deposition rate across the entire surface.
Core Technical Content and Methodology
The authors developed a mathematical model that relates the weld bead geometry, deposition rate, and coverage pattern to the welding process parameters and the geometric configuration of the spherical head. The model framework includes:
- Geometric modeling: The spherical head surface is discretized into a series of parallel or spiral paths, with the weld gun trajectory defined by spherical coordinate parameters (radius R, polar angle θ, azimuthal angle φ).
- Weld bead geometry model: The bead width W and height H are expressed as functions of welding current I, voltage U, travel speed v, and wire feed speed. For submerged arc welding (SAW), the typical relationships are: W = k1·(I/v)^0.5·U^0.3, H = k2·(I/v)^0.7·U^0.2, where k1 and k2 are empirical constants.
- Coverage model: The overlap between adjacent passes is calculated based on the bead width and the path spacing. The model ensures complete coverage of the spherical surface without gaps or excessive overlap.
- Deposition rate model: The volumetric deposition rate Q is given by Q = η·I·U·ρ/(ρ·Tm), where η is the arc efficiency, ρ is the density of the filler metal, and Tm is the melting temperature. For SAW, η is typically 0.8–0.9.
- Heat input and cooling rate model: The linear heat input q = η·I·U/v determines the cooling rate, which affects the microstructure and residual stress in the overlay layer.
The model was validated experimentally on 304 stainless steel overlay on carbon steel spherical heads, with a typical overlay thickness of 6–10 mm deposited in 3–5 passes.
Process Parameter Optimization
The mathematical model enables systematic optimization of welding parameters for spherical head overlay welding. The following table summarizes the typical parameter ranges and their effects:
| Parameter | Typical Value | Effect on Bead Geometry | Effect on Coverage |
|---|---|---|---|
| Welding current I | 400–600 A | Increases bead width and height | Reduces number of passes |
| Travel speed v | 150–300 mm/min | Decreases bead width | Increases path spacing |
| Wire feed speed | 6–12 m/min | Increases deposition rate | Affects bead height |
| Arc voltage U | 28–36 V | Increases bead width | Affects bead profile |
| Path spacing | 0.6–0.8·W | — | Ensures overlap |
| Number of passes | 3–5 | — | Determines total thickness |
The critical challenge in spherical head overlay welding is maintaining a consistent travel speed despite the varying surface geometry. As the weld gun traverses the spherical surface, the effective travel speed changes due to the curvature, which must be compensated by adjusting the wire feed speed and travel speed in real time. The mathematical model provides the basis for developing a control algorithm that maintains constant heat input and consistent bead geometry across the entire surface.
Engineering Practice Integration
In practice, automatic overlay welding of spherical heads requires a multi-axis robotic welding system or a specialized welding head that can follow the spherical contour. The following practical considerations are important:
- Weld gun trajectory planning: The trajectory should be planned to minimize the number of direction changes and to maintain a consistent torch angle relative to the surface normal. A spiral path is often preferred over parallel passes because it provides more uniform coverage and reduces the number of terminations.
- Interpass temperature control: The interpass temperature should be maintained between 150–250 °C for stainless steel overlay. On a spherical head, the heat accumulation effect is more pronounced due to the reduced surface area, and the interpass temperature may rise faster than on flat plates.
- Distortion control: The spherical geometry provides inherent constraint that reduces distortion compared to flat plate overlay. However, the non-uniform heating during overlay welding can cause local deformation that affects the dimensional accuracy of the vessel.
- Post-weld treatment: The overlay layer on spherical heads is typically ground to the required thickness and surface finish. For critical applications, the entire vessel may require PWHT after overlay welding to reduce residual stresses.
Key Questions and Reflections
The mathematical model developed in this study provides a theoretical foundation for the rational design of overlay welding processes on curved surfaces. However, several limitations remain. The model assumes idealized welding conditions and does not fully account for the dynamic interaction between the arc and the curved surface, which can cause arc drift and bead shape variation. Additionally, the model does not consider the effect of residual stress on subsequent passes, which can alter the effective heat input and bead geometry.
A significant practical challenge is the automation of the welding process on spherical heads. The weld gun must maintain a precise stand-off distance and torch angle relative to the curved surface, which requires sophisticated position sensing and control algorithms. The mathematical model serves as the basis for developing such control systems, but the actual implementation requires careful calibration and validation for each specific vessel geometry.
Study Insights and Implications
This research demonstrates the value of mathematical modeling in the rational design of overlay welding processes for complex geometries. The approach can be extended to other curved surfaces, including cylindrical shells, torispherical heads, and dished heads, by adapting the geometric model and trajectory planning algorithms. For engineering practice, the key insight is that the overlay welding process on curved surfaces requires a systematic approach that integrates geometric modeling, process parameter optimization, and trajectory planning to ensure consistent weld quality and complete surface coverage. The mathematical model provides a quantitative basis for process development and qualification, reducing the reliance on trial-and-error methods and improving the reproducibility of the overlay welding process.