The three physics involved in the problem will be solved in the following way: the thermal and microstructure evolution will be solved with a two-way coupling. This is necessary because thermal evolution governs the metallurgical phases while, at the same time, the thermal properties of the piece (thermal conductivity and specific heat) are strongly dependent on the metallurgical phases. Moreover, important latent heat is generated during the evolution of the metallurgical phases. Once the thermal and microstructure evolutions are solved, the mechanical problem will be solved with a one-way coupling. The International System of Units is used to express all constants and variables involved in the following equations.
3.1. Thermal Model
The PDE to be solved to obtain the piece temperature field
in the domain
corresponds to the unsteady heat diffusion Equation (1), a heat transfer boundary condition at the fluid-solid interface
defined in (2) is used, and the appropriate temperature initial condition is defined by (3):
where
,
, and
are the material density, thermal conductivity, and specific heat, and
is the local outgoing normal unit vector at the piece surface
.
stands for the piece’s surface temperature. For the steel material, density, in
, is made dependent on temperature as follows:
Thermal properties
and
are defined as functions of the proportion of the phase
, where
can be austenite
(hot phase) or ferrite
, pearlite
, bainite
, and martensite
(cold phases).
and
are extracted from [
24]. In Equation (1), the term
represents the heat source associated with the heat released by the metallurgical transformations during the cooling and depends on the transformation rates of each solid phase and its enthalpy of solid phase change
([
24]):
The thermal boundary condition (2) involves the determination of the heat flux , or alternatively, the global heat transfer coefficient defined as , which accounts for all types of heat transfer between the surrounding fluid and the piece. The heat transfer coefficient in Equation (2) represents only the convective contribution.
After the heating in the furnace, two different stages will be considered in this model:
A first one, where the pieces are transported from the outlet of the furnace to the quenching bath, being in contact with the surrounding air and the transportation tray.
A second one, with the pieces submerged in the fluid.
In turn, this second stage can be divided, depending on the quenching fluid properties and the bath agitation, into the well-known three consecutive regimes [
6]: film boiling, nucleate boiling, and single-phase regime. If film boiling appears, a vapor film surrounds the wall, insulating the piece. Thus, heat fluxes are moderate. When a specific temperature is reached, named the Leidenfrost point (LDF), the vapor destabilizes, and a new regime with the generation of vapor bubbles over the wall appears. As the cooling continues, a fully developed boiling appears, with the heat transfer reaching a maximum value,
, usually named critical heat flux (CHF). Henceforth, the heat flux decreases as fewer and fewer bubbles are formed at the piece’s surface (what is called partial boiling). Finally, when the production of bubbles stops, only single-phase heat transfer for a liquid takes place.
In this work, we propose a functional surface heat flux that depends, among others, on the characteristic dimensions of the piece:
Piece length , diameter and thickness , and on the following quenching parameters:
Bulk quenching liquid temperature
Velocity of the quenching liquid upstream of the piece
Thermophysical properties of the quenching fluid: liquid and vapor densities, viscosities, conductivities, and specific heats , , , , , , , , saturation temperature , vapor surface tension , and latent heat of vaporization .
Each color represents one type of mechanism or heat flux at a specific regime. LDF, CHF, FDB (end of fully developed boiling regime), and ONB (onset of nucleate boiling) points are also highlighted.
Figure 3 shows the surface heat flux dependency on the fluid velocity
for a quenching bath temperature
35
. The Leidenfrost point LDF moves to higher temperatures when increasing the agitation velocity, which is consistent with the fact that the bigger the velocity, the sooner the destabilization of the vapor film occurs. In addition, for more intense agitations,
is higher; that is, it enhances the nucleate boiling regime and shifts it to smaller temperatures. The following
Section 3.1.1,
Section 3.1.2,
Section 3.1.3,
Section 3.1.4,
Section 3.1.5 and
Section 3.1.6 describe in detail the functional
definition.
3.1.1. Heat Flux in Surrounding Air
During this stage, two different heat transfer phenomena occur. Firstly, the free convection to surrounding air (at a reference temperature
298
) is modeled by a heat transfer coefficient
(where the thermal conductivity of the air is denoted as
and
refers to the characteristic piece length), which uses a free convection correlation for vertical cylinders extracted from [
25]:
and
respectively, represent the Rayleigh and Prandtl dimensionless numbers of air,
is a curvature correction factor; and
is the spindle diameter (using an averaged value).
The radiative heat flux,
, will be modeled by a common Stefan-Boltzmann expression, where the steel emissivity
will follow a nonlinear dependency on the piece surface temperature
, as defined in [
26]. This regime lasts until the piece is completely submerged in the fluid tank.
3.1.2. Film Boiling
As in this stage the piece is at a very high temperature, radiative phenomena are still relevant, so that, a heat transfer coefficient
, which includes this effect, is proposed ([
11,
27]):
where
is the saturation temperature (373
for pure water) and
, set as a constant value, is the bulk temperature of the quenching bath, measured far enough from the solid surface. The reference temperature will be
. Heat transfer coefficients
and
stand for radiation and convection mechanisms, respectively. The first one, following [
11], is expressed as
where
y are the fluid and the piece emissivities, and
is the Stefan-Boltzmann constant. The convection coefficient is taken from [
27]:
The vapor properties are denoted, using the subscript
, as
,
and
(viscosity, conductivity, and density, respectively). The characteristic length of the film effects is defined as
, being
the surface tension between the liquid and the vapor,
is the gravity acceleration, and
is the liquid density. A modified latent heat, used to model the energy invested in heating the liquid from
until evaporation temperature and then vaporizing it, is defined and named as
.
and are the fluid specific heat of the vapor and liquid phases, respectively, and is the latent heat of evaporation.
In Equations (5) and (6), vapor properties are taken at and liquid properties are taken at .
3.1.3. Transition Boiling
The vapor blanket starts to destabilize at the Leidenfrost temperature
, calculated using the expression proposed by [
28]:
In solving the film boiling Equation (4) at a wall temperature equal to that Leidenfrost temperature (
), the heat flux
is obtained. Previous laboratory quenching experiments [
26] have been used to adjust the coefficient defined as
in Equation (7). The characteristic liquid flux velocity of the quenching bath (increased by the immersion velocity in the first seconds) is referenced as
, while
represents the buoyancy-induced velocity associated with the presence of vapor bubbles in the liquid:
Again, the liquid density
in Equation (8) is evaluated at saturation temperature
.The heat flux dependency on the surface piece temperature
in this region has been assumed to be linear:
The maximum value of the heat flux and its corresponding temperature are detailed in the following subsection.
3.1.4. Fully Developed Boiling
This stage starts once the maximum value of the heat flux, named Critical Heat Flux and referenced as
, is reached [
29],
Then, the heat flux is reduced as the surface piece temperature decreases [
30]:
being
a single-phase heat transfer coefficient, defined in Equation (11) by the Dittus-Boelter correlation [
31]:
where the bulk temperature
is used to evaluate the dimensionless numbers
and
(the Reynolds and Prandtl numbers of the liquid). The Reynolds number is based on the previously defined fluid velocity
and the characteristic dimension perpendicular to the fluid flow,
. As the pieces remain in vertical position (as can be seen in
Figure 1), the averaged piece thickness
was taken as the characteristic length in Equation (11).
is calculated by solving the temperature value in Equation (10) for the heat flux obtained in Equation (9):
.
3.1.5. Partial Boiling
In this regime, as the vapor bubble nucleation phenomenon decreases in intensity, the contribution of the nucleate boiling becomes of the same order as the monophasic forced convection heat flux. As a result, a smooth transition between these two regimes previously characterized by (10) and (11) is sought. As in [
32],
points, named fully developed boiling (FDB) and onset of nucleate boiling (ONB), delimit this transition stage: The estimation of the first point follows
where the heat flux
is the intersection of the Dittus–Boelter correlation (11) with the fully developed boiling Equation (10), and is solved iteratively in Equation (12):
can be obtained using Equation (10) and substituting the previously calculated value
. The end of the partial boiling regime is set by the temperature
, which is obtained as follows ([
32]):
where liquid properties are evaluated at
. The heat flux value at the ONB point (
) is calculated using the Dittus–Boelter single-phase correlation of Equation (11) for
.
Finally, partial boiling heat flux is modeled according to [
32] using the following equation.
where
is obtained for each point following
, starting with
at
and ending with
at
.
3.1.6. Single-Phase Heat Flux
The last regime, where the vapor bubbles have completely disappeared, is characterized by forced convection between the piece and the monophasic fluid. The single-phase correlation (11) is used (
) for wall temperatures below
. However, for temperatures smaller than
, Dittus–Boelter’s (Equation (11)) significantly over-predicts the heat flux values. To adjust a new convective heat transfer equation sensitive to fluid velocity and capable of retaining the variation of the liquid viscosity with temperature, CFD techniques and a least-squares adjustment have been used [
24], obtaining the following expression:
3.2. Validation of the Thermal Model
The surface heat flux function
described in
Section 3.1 was solved using the software Matlab and then imported as an external tabular wall temperature-dependent function in the software COMSOL-Multiphysics (version V 3.5a), where the complete thermal model (1)–(3) was implemented (see description in
Section 3.5), solved, and validated by comparison with experiments on a standard lab probe, as the one shown in
Figure 4a.
According to the international standard ISO 9950 and the American standards ASTM D 6200-01 and ASTM D 6482-99, cylindrical test probes (12.5
in diameter and 60
in length) have been used. A data acquisition unit connected to a
type thermocouple, located at the center of the probe, has been used to save temperature data every 0.01
. Since it is intended to validate the thermal model, the alloy Inconel 600, which does not undergo any metallurgical changes when cooled, is used to fabricate the test probes. The quenching container, of section 125
× 60
and height 205
, was provided by Swerea/IVF (
https://www.ri.se/en/what-we-do/services/ivf-smartquench-for-control-of-cooling-curve-measurement (accessed on 25 May 2023)) and filled with 1.2 L of water. The container has an agitation device whose velocity can be set from 0 to a maximum of 1.2 m/s. The probe was heated in an electric furnace until a uniform temperature
was reached. To assure homogenization, the piece remained inside the furnace at the objective temperature for at least 5 min. Once heated, the piece was submerged in the experimental container in less than 1.5
. Five different experimental tests were carried out with three agitation velocities,
0.34,
0.5, and
0.75 m/s, and different bulk fluid temperatures,
,
and
, as indicated in
Table 2.
For these test conditions, the thermal model (1)–(3) with
was solved in a 2D-axisymmetric domain (of dimensions equal to the radius and length of the cylinder probe) using the unstructured mesh of
Figure 4b formed by 2636 triangles. The initial uniform piece temperature and fluid bulk temperature were selected to be equal to those of the experiment. In these cases, heat transfer during the transportation of the fluid was neglected. A sensitivity analysis for the simulations with the standard probe was carried out but omitted for the sake of brevity. The computational time for each case (processed in serial) took 20
on a workstation with 128
of RAM equipped with Intel-Xeon processors. As an example,
Figure 4b shows the snapshot of the probe temperature field at t = 5 s for Test #5.
Table 2 shows the relative deviations between the numerical results and the experimental measurements for the five tests: averaged and maximum relative deviations in temperature, relative deviations on the predicted maximum cooling rate, and on their predicted temperature
. Averaged deviations between the numerical prediction and the measured temperatures remain lower than 7.5% for all the tests. An example of the comparison between the numerical and experimental thermal evolutions is shown in
Figure 4b for Test
. Solid lines correspond to the cooling curve at the center of the probe,
vs. time
, while dashed lines represent the cooling rate at the center of the probe vs. its temperature
. Black lines show the experimental measurements, while the numerical predictions are plotted in blue.
To illustrate the accuracy of the thermal model with the experimental results when compared with other simplified thermal models based on correlations extensively used in the microstructure prediction in quenching processes, problems (1)–(3) have been numerically integrated assuming the correlation approach of Smoljan [
14], which approximates the heat flux at the piece surface (Equation (2)) by a triangular function. The green lines in
Figure 4b show these numerical results for Test
, where poor agreement with the experimental measurements is observed. This behavior is consistently obtained for all tests. Therefore, despite the simplicity of the thermal model proposed in
Section 3.1, its numerical results provide a huge improvement in accuracy when compared to the common approaches used to predict the thermal history and, therefore, the final microstructure of the piece after treatment.
3.3. Metallurgical Model
During the cooling process, the evolution of the different micro-structural phases of the steel should be determined and coupled to the thermal problem described in
Section 3.1. The transformations dominated by carbon diffusion processes (that potentially lead to ferrite
, pearlite
and bainite
phases from the austenite
phase) have been modeled by an Avrami-type equation as described in [
24].
where the proportion of each microconstituent
is named as
,
is the maximum proportion of microconstituent
at a given temperature, and
is the proportion of austenite at the beginning of the transformation. In addition, the model uses two material parameters, which are extracted from the TTT diagram:
and
. As mentioned, the composition of the industrial steel is quite similar to the 25 M 6 Steel, so its TTT diagram, extracted from [
33], was used. Parameter
stands for the fraction between the piece’s austenitic grain size (taken in this study as 6.5
) and the reference austenitic grain size used in the TTT diagram. The discretization of the cooling in small intervals of constant temperatures has been done by applying the additive rule of Scheil [
24], used for continuous cooling. The following expression shows the criterion that determines the beginning of transformation:
where
is the incubation time of the microconstituent
for each temperature, given by the TTT diagram.
The martensite proportion
is modeled according to the Koïstinen–Marburger time-independent algebraic Equation (13) (see [
24]).
where the temperature for the beginning of martensitic transformation (
) is extracted from the CCT (Continuous Cooling Transformation) diagram of the 25 M 6 Steel.
,
y
are the ferrite, pearlite, and bainite proportions, and
is a material parameter.
3.4. Mechanical Model
The mechanical model provides the residual stresses and final deformations of the piece induced by the heat treatment. The quasi-static and small deformations mathematical model used relates the stress tensor
and the strain tensor
, the latter being:
where
,
,
, and
are the elastic, plastic, thermal, and transformation-induced plasticity (TRIP) contributions. The elasticity problem was characterized by an isotropic Young’s modulus
and Poisson coefficient
that depend on the temperature of the material [
20]. The Von Mises criteria have been used to define the fluence function
that determines the plasticity region (
):
where
represents the temperature,
the microstructure proportion (in vectorial form),
the Von Mises stress,
and
the yield stress, and the hardening law for the multiphasic material (both dependent on the temperature and the microstructure). Thus, the strain plastic tensor will be defined as
where
stands for the plastic multiplier. Thermal strains, which include deformations associated with the volumetric change generated by the metallurgical transformations, are described by the following Equation ([
34]):
where
is the thermal dilatation coefficient for the microconstituents ferrite, pearlite, bainite, and martensite, while
stands for austenite.
is a reference temperature for the cold phases (the reference state), where the term
depicts the difference in compactness between the crystallographic structures at the reference temperature.
Finally, the TRIP deformation is modeled with Equation (14) as in [
35], which involves the stress state and the evolution rate of the transformations:
being
the deviatoric stress tensor,
the austenite proportion,
the positive value of the velocity rate of the cold microconstituents,
characteristic constants associated to each phase
, and
representing the derivatives of a normalized function
that fulfills
and
. These last two terms are characteristic of each material [
35].
3.5. Numerical Implementation
The complete thermal-metallurgical-mechanical model described in
Section 3 has been numerically solved for a 3D computational domain of a complete spindle. As each piece is placed on a supporting device, the corresponding contact surface with the tray is differentiated in the thermal–metallurgical model of the spindle, assuming a constant heat transfer coefficient of ([
36]). In addition, at the contact boundary between the supporting device and the spindle, the displacements of the piece were blocked. A uniform temperature of
, which corresponds to the homogenizing furnace temperature, was considered the initial condition. Spindle geometry and mesh, formed by 391,530 hexahedrons, can be seen in
Figure 5b, while part of the supporting tray and the vertical symmetry plane, which divides the piece and the supporting device, are depicted in
Figure 5a.
The metallurgical equations, described in
Section 3.3, have been programmed in the software Matlab (version V 11)following the resolution algorithm shown in the block diagrams of
Figure 6 and
Figure 7. The complete thermal–metallurgical model described in
Section 3.1 and
Section 3.3 has been programmed using finite element methods (FEM) in Matlab, using its connection with Comsol Multiphysics v3.5a (
https://www.comsol.com/, accessed on 25 May 2023) (check [
37] for more information on Comsol-Matlab LiveLink). First-order elements
were used, linear systems were solved by the generalized minimal residual (GMRES) method, and a backward differentiation formula (BDF) scheme (of order 5) was used for time integration. Once the piece temperature and microstructure evolutions have been determined, these evolutions are exported and introduced as inputs in the mechanical model (defined in
Section 3.4). Free software Code-Aster v11.6, developed by Electricité de France (Electricité de France.
http://www.code-aster.org (accessed on 25 May 2023)) has been used to integrate and solve this mechanical problem.
A sensitivity analysis of the results was carried out for three different meshes and three step sizes for the thermal–metallurgical model.
Table 3 shows the relative errors of the maximum cooling velocity
(evaluated at point 0
S, indicated in
Figure 8) and of the maximum final bainite content in the piece
(evaluated at point 0 of
Figure 8). The final selected mesh was Mesh
of
Table 3 with the corresponding step size of
0.025
, while for the mechanical problem, the use of a step size of 1
was sufficient.
A workstation with of RAM and two Intel-Xeon processors ( nodes and ) was used. The computational times were 50 for the thermal–metallurgical model (run in serial) and 10 for the mechanical model. The difference is justified because the thermal–metallurgical model needs a finer temporal discretization to retain the high temperature gradients that appear in the first seconds of the quenching.