Figure 1.
Solution domain for the steady-state indirect extrusion problem. Only a small segment of the billet near the die is considered, where the effects of the process are significant; far from the die the billet is relatively unaffected by the process. The domain consists of three subdomains, Ω1, Ω2 and Ω3 (billet, container and die respectively), with their corresponding boundaries denoted as ∂iΩ1, ∂iΩ2 and ∂iΩ3.
Figure 2.
Computational mesh consisting of ~100,000 elements. The mesh is refined near the container and die walls, with 20 layers that span ~1.5 mm; the wall adjacent layer has a thickness of 10 μm.
Figure 3.
(a) Optical micrograph of the microstructure of a typical dual-phase brass on transverse section. (b) Detail of (a). Bright areas represent α-phase and dark areas represent β-phase. In (a) there is an appreciable amount of Pb particles, which appeared as black dots. Immersion etching in FeCl3-based solution.
Figure 4.
Contours of the equivalent strain rate (values larger than 15 s−1 are clipped to the red color contour). (a) Solution with no friction f = 0 Ns/m3, S = 1.5 cm/s, fβ = 0.5, Tin = 715 °C, h = 1.7 W/(m2 K). (b) Solution with friction f = 5 × 108 Ns/m3, S = 1.5 cm/s, fβ = 0.5, Tin = 715 °C, h = 1.7 W/(m2 K).
Figure 5.
Contours of temperature (values smaller than 700 °C are clipped to the blue color contour). (a) Solution with no friction f = 0 Ns/m3, S = 1.5 cm/s, fβ = 0.5, Tin = 715 °C, h = 1.7 W/(m2 K). (b) Solution with friction f = 5 × 108 Ns/m3, S = 1.5 cm/s, fβ = 0.5, Tin = 715 °C, h = 1.7 W/(m2 K). Notice the formation of a small hot spot—due to high friction—near the outlet and on the outer layer of the extruded product.
Figure 6.
(a) Streamlines. (b) Pressure contours. Parameter values: f = 5 × 108 Ns/m3, S = 1.5 cm/s, fβ = 0.5, Tin = 715 °C, h = 1.7 W/(m2 K).
Figure 7.
Sensitivity chart of the output parameters (designated by column—horizontal axis) with respect to the input parameters (designated by color bars). The analysis is performed in the neighborhood of a reference point with input parameter values: f = 5 × 108 Ns/m3; S = 1.5 cm/s, fβ = 0.5, Tin = 715 °C, h = 1.7 W/(m2 K).
Figure 8.
Process charts showing the dependence of the process output parameters on the input parameters. (a) Extrusion pressure (P1) vs. extrusion speed (S). (b) Extrusion pressure (P1) vs. β-phase fraction (fβ). (c) Max. extrusion temperature (P2) vs. extrusion speed (S). (d) Max. extrusion temperature (P2) vs. β-phase fraction (fβ). (e) Average extrusion temperature (P9) vs. extrusion speed (S). (f) Average extrusion temperature (P9) vs. β-phase fraction (fβ). (g) Container temperature (P10) vs. heat transfer coefficient (h). (h) Container temperature (P10) vs. inlet temperature (Tin). The input parameters have the default values: f = 5 × 108 Ns/m3, S = 1.5 cm/s, fβ = 0.5, Tin = 715 °C, h = 1.7 W/(m2 K), unless otherwise stated by each chart.
Figure 9.
(
a) Optical micrograph (as-polished/bright field illumination) showing extensive surface cracking occurred in extruded brass CW614N-CuZn39Pb3 (hexagonal rod of 6 mm across flats), (
b) and (
c) scanning electron microscope (SEM) fractographs (secondary electron imaging) showing a typical intergranular fracture mode and secondary, transverse cracking due to overheating. The white arrow shows the extrusion direction. This failure case history was presented in detail in Ref. [
32].
Table 1.
Parameters of Equation (20) for the single-phase brasses, CuZn36 (α-brass) and CuZn44 (β-brass).
Brass | Q (J/mol) | A (s−1) | n (Dimensionless) | a (MPa−1) |
---|
CuZn36 | 157,000 | 6.60 × 109 | 4.5 | 6.58 × 10−3 |
CuZn44 | 92,000 | 1.44 × 107 | 3.0 | 9.21 × 10−3 |
Table 2.
Input parameters.
Input Parameter | Notation | Alternate Notation (Reduced Order Model) | Units |
---|
Extrusion speed | S | P3 | m/s |
Friction factor | f | P4 | Ns/m3 |
Heat transfer coefficient on the liner outer walls | h | P7 | W/(m2 K) |
β-phase volume fraction | | P13 | dimensionless |
Inlet (initial) temperature of the billet | Tin | P14 | K |
Table 3.
Output parameters.
Output Parameter | Notation | Alternate Notation (Reduced Order Model) | Units |
---|
Extrusion pressure | extrpress | P1 | Pa |
Maximum temperature (hot spots—see Section 3.2) | maxTemp | P2 | °C |
Average temperature of the extruded product | aveTemp | P9 | °C |
Average temperature of the container liner | contemp | P10 | °C |
Average temperature of the die | dietemp | P11 | °C |
Average temperature of the brass shell | sleevetemp | P12 | °C |
Table 4.
Thermo-physical properties. The specific heat and thermal conductivity of CW626N is approximated with that of CuZn30. Typical thermo-physical properties of tool steels were used.
Property | Brass—CW626N | Steel (Die & Container) |
---|
Density (kg/m3) | 8500 | 8030 * |
Specific heat (J/(kg·K)) | 355 + 0.136T [27] | 502.48 * |
Thermal conductivity (W/(m·K)) | 140.62 + 0.011214T [27] | 25 ** |
Viscosity (Pa·s) | Equation (3) | N/A |
Table 5.
Validation experiments in the production process. The measurements of the hydraulic pressure are juxtaposed with the model results. The measured hydraulic pressure is obtained via a pressure gage with an estimated precision of 2.5 bar. The inlet temperature is the average of three measurements at the front, middle and back of the billet, using an optical pyrometer with estimated precision of 0.5 °C.
Billet Number | Inlet Temperature (°C) | Extrusion Speed (cm/s) | Hydraulic Pressure (bar) | Calculated Hydraulic Pressure (bar) |
---|
1 | 713.5 | 1.45 | 155 | 154.2 |
2 | 716.0 | 1.45 | 150 | 152.9 |
3 | 717.0 | 1.50 | 155 | 157.6 |
4 | 721.5 | 1.45 | 150 | 150.1 |
Table 6.
Input parameter intervals.
Input parameter | Interval | Units |
---|
S | [0.01, 0.02] | m/s |
f | [0, 109] | Ns/m3 |
h | [0, 3] | W/(m2 K) |
fβ | [0, 1] | dimensionless |
Tin | [900, 1050] | K |