Thixoforming : Semi-solid Metal Processing

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material properties have to be constant and not, as in this problem, temperature dependent. In the following, the approach for the feed-forward control, which uses the inverted numerical model, is described. The temperature in the middle of the billet is assumed to be the output of the system and can be predefined. This temperature can be used to calculate the temperatures next to the middle and so on until the last scale at the boundary of the billet. Due to the skin effect, we can assume that the power is only induced in the outer scale. If we discretize Equation 10.3 via the radius and convert the equation to the temperature in the i þ 1 scale, we obtain the equation with Wi þ 1 ¼ c i ¼ r i cpi li ðDr2riÞ ðDrþ2riÞ Wi 1 þ 4ri Dr þ 2ri Wi þ 2c 2 iriDr qWi ðDrþ2riÞ qt ð10:10Þ To calculate the temperature W2 next to the middle of the billet, we have to consider the first boundary condition. With Equation 10.10, this yields the following equation: W2 ¼ W1 þ c 1 Dr2 2 qW1 qt ð10:11Þ The temperatures in the third to the nth scale can be calculated with Equation 10.10. At the outer scale n we have to consider the second boundary condition and the induced power, which yields the following equation for the induced power: with PðtÞ ¼ABillet 8 < : RB þ 1 Dr Dr 2 2RB qloss ¼ aðWnWeÞþes W 4 n W 4 e qWn ln rcpn þ 2 qt Dr2 Wn Wn 1 9 = ð Þ þ qloss; ð10:12Þ If we consider the equation above, we can conclude that we need the temperature in the outer scale and the first time derivative to calculate the induced power. If we go backwards to the temperature in the middle of the billet, we can conclude that the desired temperature trajectory has to be n times differentiable. The manipulated variable of the induction furnace is the electric power of the converter. Therefore, we have to integrate Equation 10.2 from 0 to RB. This yields the equation PðtÞ ¼ ffiffiffi p N 2p 2 Sp kdL 2 Sp ^I 2 Sp ðtÞRB ber ffiffi ffiffi p 2RB d ber0 p 2RB d ber2 pffiffi 2RB d 10.3 Heating and Forming Operationsj379 bei þ bei2 ffiffi ffiffi p 2RB d bei 0 p 2RB d pffiffi 2RB d ð10:13Þ
380j 10 Thixoforging and Rheoforging of Steel and Aluminium Alloys where ber() and bei() are Kelvin functions. To ensure that the needed power is induced into the billet, the coil current has to be controlled. We can convert Equation 10.13 to achieve the coil current peak value: ^I ðtÞ ¼ Sp LSp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kd ber PðtÞ pffiffiffi NSp 2pRB 2 ffiffi pffiffi 2RB þ bei2 d pffiffi 2RB ber ber0 pffiffi 2RB þ bei bei 0 v u t pffiffi 2RB d p 2RB d pffiffi 2RB d d d ð10:14Þ The converter is controlled with a simple proportional–integral–derivative (PID) controller to ensure that the actual current match the desired coil current trajectory and thus the needed power is induced into the billet. Control with a Pyrometer In the following, we will discuss another approach to heating steel into the semi-solid state. For a more detailed description of the control scheme, see [39]. Because direct measurement of the temperature via thermocouples is not feasible in a production environment, a radiation pyrometer was used as a contactless measurement device. The accuracy of the pyrometer depends heavily on the exact knowledge of the radiation coefficient, which can vary from billet to billet due to different surface properties and which is subject to change during the heating process. These uncertainties prohibit the implementation of a closed-loop control scheme since the exact temperature cannot be measured with the required accuracy. In order to be independent of the measurement errors, the proposed control scheme relies only on the slope of the temperature. By detecting the distinct change of slope that occurs when the solidus temperature is crossed, the beginning of the melting process can be determined. The energy fed into the billet from this point onward determines the resulting liquid fraction. By feeding the same amount of energy to each billet, it is guaranteed that the billets reach the desired liquid fraction despite the uncertain absolute value of the temperature and small variations of the alloy composition. During the heating of X210CrW12, there are two distinct changes in the slope of the heating curve, one at about 800 C where the billet loses its magnetic properties and therefore the efficiency falls considerably. The second appears around 1230 C and this is the beginning of melting of the alloy. By detecting the slope change and controlling the energy fed to the billet after this point, the liquid fraction can be controlled. After the required melting energy has been induced into the billet, the billet temperature needs to be homogenized since the core temperature lags behind the surface temperature. This is achieved by reducing the converter power to a level, which ensures that the surface temperature remains nearly constant and only the radiation losses are compensated. The proposed heating control scheme consists of four steps [39]: 1. heating of the billet with constant converter power until the crossing of the solidus temperature; 2. automatic detection of the entry into the melting region by evaluation of the temperature slope;
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Thixoforming Semi-solid Metal Proce
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Further Reading Herlach, D. M. (ed.
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The Editors Prof. Dr. G. Hirt Insti
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VI Contents 1.3.5.2 Other Aluminium
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VIII Contents 4.6.1 Thixocasting 13
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X Contents 7.4.2 Isothermal Non-ste
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XII Contents 9.4.1.3 Simulation Res
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Preface Semi-solid forming of metal
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XVIII List of Contributors Andreas
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XX List of Contributors Alexander S
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2j 1 Semi-solid Forming of Aluminiu
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10j 1 Semi-solid Forming of Alumini
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Table 1.1 Overview of semi-solid pr
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Part One Material Fundamentals of M
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32j 2 Metallurgical Aspects of SSM
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44j 3 Material Aspects of Steel Thi
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Temperature (ºC) 1550 1500 1450 14
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100j 3 Material Aspects of Steel Th
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106j 4 Design of Al and Al-Li Alloy
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5 Thermochemical Simulation of Phas
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here it can be assumed that the dat
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Figure 5.2 Calculated temperature v
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Figure 5.4 Composition profiles of
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It is clear that C has by far the s
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Figure 5.7 The density of X210CrW12
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5.3 Calculations for the Bearing St
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Figure 5.11 Composition profiles of
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Figure 5.14 The density of 100Cr6.
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area which is available for heat tr
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Part Two Modelling the Flow Behavio
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170j 6 Modelling the Flow Behaviour
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7 A Physical and Micromechanical Mo
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3. Macroscopic averages or homogeni
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displacement boundary conditions or
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and liquid are both assumed to be i
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are the strain rate concentration t
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Semi-solid viscosity (Pa s) into cl
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Load (kN) strain to rupture, leadin
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Load (kN) 7 6 5 4 3 2 1 0 0 7.5 Con
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adius R radius of the spherical inc
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Part Three Tool Technologies for Fo
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242j 8 Tool Technologies for Formin
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9 Rheocasting of Aluminium Alloys a
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Figure 9.2 Processes for the semi-s
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Two Italian companies gained the fi
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9.2 SSM Casting Processesj317 For t
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Figure 9.7 Oil pump cover for Honda
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Figure 9.9 Middle temperature cours
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Figure 9.12 Components of cartridge
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Figure 9.15 Process adapted multipa
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step-shot experiments. It is also s
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430j 11 Thixoextrusion Figure 11.17
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432j 11 Thixoextrusion shell format
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436j 11 Thixoextrusion Table 11.6 P
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440j 11 Thixoextrusion solidificati
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442j 11 Thixoextrusion GPa pressure
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444j Index arbitrary volume element
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446j Index equilibrium tests 141 eq
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448j Index - importance 273 ion flu
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450j Index oxygen-sensitive element
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452j Index - thixoforming 241 semi-
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454j Index - high pressure 131 - sc
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