1. Reference problem#
1.1. Geometry#
Cylindrical bar:
Overall height: \(\mathrm{2H}=150\mathrm{mm}\)
Diameter: \(\mathrm{2L}=\mathrm{100mm}\)
1.2. Material properties#
The bar is made of 16 MND5 steel. The thermal properties (conductivity and enthalpy of the material) vary with temperature and are given in [bib1].
Coefficient for metallurgy:
TRC « standard »
\(\mathrm{AR3}=830°C\), \(\mathrm{alpha}=-0.0249\)
\(\mathrm{MS0}=415°C\), \(\mathrm{AC1}=724°C\), \(\mathrm{AC3}=846°C\)
\({\tau }_{1}=0.034\), \({\tau }_{3}=0.034\)
Microhardness of the various metallurgical phases:
for ferrite \(d=184.\mathrm{HV}\)
For perlite \(d=184.\mathrm{HV}\)
For the bainite \(d=309.\mathrm{HV}\)
for martensite \(d=450.\mathrm{HV}\)
for austenite \(d=100.\mathrm{HV}\)
1.3. Boundary conditions and loads#
Coefficient of heat exchange with the outside for 14 sec. (before tempering) based on temperature measurements by thermocouples:
Time \((s)\) |
|
13 |
|
\({T}_{\infty }=25°C\)
Surface heat flow density as a function of temperature \(T\) on the borders (during tempering) based on temperature measurements:
\(T(°C)\) |
250 |
||||||||
Flow \((\mathrm{W.m}-2)\) |
|
-0.03 E+06 |
-0.08 E+06 |
-0.08 E+06 |
-0.71 E+06 |
-1.51 E+06 |
-1.51 E+06 |
-1.99 E+06 |
-1.99 E+06 |
\(T(°C)\) |
||||||
Stream \((\mathrm{W.m}-2)\) |
-2.53 E+06 |
-3.34 E+06 |
-3.78 E+06 |
-6. E+06 |
-7. E+06 |
-7.9 E+06 |
1.4. Initial conditions#
\(T(r,z)=906°\mathrm{C.}\)
Bar entirely austenitized in the initial state i.e. \(\mathrm{Zf}(r,z)=0.\), \(\mathrm{Zp}(r,z)=0.\), \(\mathrm{Zb}(r,z)=0.\) and \(\mathrm{Zm}(r,z)=0.\)