||3-D Temperature Distribution and Longitudinal Residual Warpage Analysis of Steel Strip in Continuous Annealing Line
||Department of Mechanical Engineering
continuous annealing line (CAL)
finite element method (FEM)
energy balance method (EBM)
virtual layer method (VLM)
連續退火線（Continuous Annealing Line，簡稱CAL）製造的電磁鋼片於沖製E型和I型鋼片時會有明顯的翹曲現象。此幾何缺陷主要歸咎於鋼帶通過產線內爐輥時產生寬度與厚度方向不均勻塑性變形及溫度分佈所導致的殘留應力有關。且越寬的鋼帶翹曲越嚴重並可能降低產品的品質。本論文研究探討各種不同的輸入參數對鋼帶翹曲的效應。採用三種理論技術，包括有限元素法（Finite Element Method，簡稱FEM）、能量平衡法（Energy Balance Method，簡稱EBM）以及虛擬層法（Virtual Layers Method，簡稱VLM）分別計算鋼帶的應力、溫度以及塑性應變分佈。鋼帶的縱向殘留翹曲量可由此算出。我們發現，通過冷卻區爐輥時的鋼帶橫向溫度分佈與降伏強度對鋼帶的翹曲量最敏感，可能可以藉由適當的冷卻方式將翹曲量控制在可接受的範圍。
Electrical steel (ES) is mainly used for motors and transformers whose iron loss has a great effect on the efficiency of electrical products and may be related to the residual warpage of the strip, generally defined as a deviation from flatness on unloading. According to statistics from the International Energy Agency (IEA), global electricity production was about 19000 TWh, 46% of which is consumed by motors, leading to about 6,040 Megatonnes (Mt) of CO2 emissions. Therefore, improving the efficiency of motors is the most effective way to save energy in industry.
The electrical steel produced by continuous annealing line (CAL) exhibit a significant phenomenon of warpage during punching into the E- and I-type sheets. This geometric defect is mainly attributed to the residual stress induced by the nonuniform temperature and nonuniform plastic deformation along both the width and the thickness of strip when it passes through the rolls in the line. It becomes more serious for the wider strip and may degrade the quality of the products. In the thesis, the effects of various input parameters on the warpage of strip were investigated and discussed. Three theoretical techniques, including finite element method (FEM), energy balance method (EBM), and virtual layers method (VLM), were adopted to evaluate the distributions of stress, temperature, and plastic strain of the strip, respectively. The longitudinal residual warpage of strip can then be calculated accordingly. It was found that the warpage of strip is sensitive to the transverse temperature distributions and the yielding strength of strip as passing through the rolls in CS, which is possibly to be controlled within an accepted range by applying a suitable cooling scheme in this section.
摘 要 I
誌 謝 IV
LIST OF TABLES VIII
LIST OF FIGURES IX
Chapter 1 INTRODUCTION 1
1.1 Motivation 2
1.2 Literature Review 4
1.3 Objectives 6
1.4 Dissertation Organization 7
Chapter 2 CONTINUOUS ANNEALING LINE 9
2.1 Annealing treatment 10
2.2 Shapes of roll in CAL 13
2.3 Phase transformation 14
2.4 Electrical steel 18
2.5 Measurement of warpage 19
Chapter 3 MATHEMATICAL MODELS 21
3.1 Finite element method 21
3.1.1 Mechanical model of strip 22
3.1.2 Estimation of the emissivity 28
3.1.3 Thermal model of roll 30
3.1.4 Thermal model of strip 33
3.1.5 FEM Computational procedure 35
3.2 Energy balance method 36
3.2.1 Energy balances model 41
3.2.2 EBM Computational procedures 47
3.3 Virtual layers method 49
3.3.1 Virtual layers model 52
3.3.2 Residual stress of elements in straight status on unloading 56
3.3.3 Calculation of strip residual warpage 59
3.3.4 VLM Computational procedures 60
3.4 Computational procedures 61
Chapter 4 NUMERICAL RESULTS AND DISCUSSION 63
4.1 Mechanical model of strip 63
4.1.1 Contact pressure and thermal contact resistance 63
4.1.2 Tangential stress distribution of strip 65
4.2 Temperature of roll surface 66
4.3 Equivalent heat convective coefficient 67
4.4 Thermal model of strip 68
4.4.1 History of strip temperature 68
4.4.2 Effect of phase transformation 70
4.4.3 Distribution of transverse temperature 71
4.5 VLM 73
4.5.1 Reliability test 74
4.5.2 Convergence test 74
4.5.3 Final residual warpage 76
4.5.4 History of warpage 76
4.5.5 Final accumulated plastic strain 78
4.5.6 Final residual stress 79
4.5.7 History of accumulated plastic strain 79
4.5.8 Effect of strip temperature at the outlet of CAL on warpage 80
4.5.9 Effect of cooling condition in CS on warpage 82
4.5.10 Effect of strip tension on warpage 82
4.5.11 Effect of crown and cooling effect in CS on final warpage 83
4.5.12 Effect of Young’s modulus and yielding strength of strip material on warpage 85
Chapter 5 CONCLUSIONS AND FUTURE STUDIES 87
5.1 Conclusions 87
5.2 Future studies 89
A. Simplified finite element models  97
B. Influence of centrifugal force 100
C. Distribution of thermal contact resistances 101
D. Simplification of roll surface temperature  103
D.1 2D model 105
D.2 Comparison with 3-D model 106
E. Emissivities 110
F. Enclosure of sections 111
G. Rolls specification 114
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