Holistic consideration of grain growth behavior of tempering steel 34CrNiMo6 during heating processes

Journal of Materials Processing Technology 229 (2016) 61–71

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Holistic consideration of grain growth behavior of tempering steel 34CrNiMo6 during heating processes Sebastian Herbst a,∗ , Hans-Bernward Besserer a , Olexandr Grydin b , Andrzej Milenin c , Hans Jürgen Maier a,d , Florian Nürnberger a,d a

Institut für Werkstoffkunde (Materials Science), Leibniz Universität Hannover, An der Universität 2, 30823, Garbsen, Germany Lehrstuhl für Werkstoffkunde (Materials Science), Universität Paderborn, Pohlweg 47-49, 33098 Paderborn, Germany c University of Science and Technology Kraków, Mickiewicza 30, 30-059 Kraków, Poland d Zentrum für Festkörperchemie und Neue Materialien, Leibniz Universität Hannover, Callinstr. 9, 30167 Hannover, Germany b

a r t i c l e

i n f o

Article history: Received 28 November 2014 Received in revised form 14 August 2015 Accepted 6 September 2015 Available online 9 September 2015 Keywords: Heating process Tempering steel 34CrNiMo6 Grain growth Model parameterization Numerical simulation Heat transfer coefficient

a b s t r a c t An easily applicable method to determine simplified heat transfer coefficients (HTCs), based on numerical modelling and heating experiments by means of response surface optimization (RSO) is introduced. The HTCs determined account for convective and radiative heat transfer. The approach leads to a phenomenological model that neglects phase transformation during heating. Consequently, the HTCs incorporate effects caused by the transformation enthalpy. For the steel 34CrNiMo6 an austenite grain growth model is parameterized and enhanced with a temperature-dependent activation energy to fit experimentally determined grain growth data. The grain growth model and the HTCs determined by RSO are combined into a numerical grain growth simulation for heating processes of large forging parts. Temperature trends of simulated heating processes using HTCs determined by RSO showed good agreement with experimental data and transferability of the HTCs to larger and more complex parts. The parameterized enhanced grain growth model was found to more accurately represent the measured data in the temperature range around 1100 ◦ C than models from literature. Comparison of measured and calculated grain size evolution for temperatures in the range between 950 ◦ C and 1100 ◦ C revealed a very good agreement considering the uncertainties in grain size measurements and also a huge improvement compared to a conservative model. The grain growth simulation of the heating process of a large semi-finished crankshaft showed a significant difference in austenite grain size after the heating and before the forging process between core and near surface areas of above 100%. Consequently, either the optimization of the heating process or the consideration of the inhomogeneous grain size distribution for such parts are relevant for the subsequent thermomechanical treatment, as austenite grain size has an impact on both flow behavior and recrystallization kinetics. © 2015 Elsevier B.V. All rights reserved.

1. Introduction The heating of large steel workpieces (e.g., forged crankshafts of marine engines) induces substantial temperature gradients due to fast rising temperatures in the heated surface whereas heating of the core is delayed by the finite heat conduction. With progressing dwell time in the furnace, heat conduction from the surface leads to the heating of the core regions while the surfaces temperature is still rising due to convection and thermal radiation. Thus, significant temperature gradients can prevail in radial direction

∗ Corresponding author. Fax: +49 511 7625245. E-mail address: [email protected] (S. Herbst). http://dx.doi.org/10.1016/j.jmatprotec.2015.09.015 0924-0136/© 2015 Elsevier B.V. All rights reserved.

during the continuing heating process, depending on the relative magnitude of convection/heat radiation and heat conduction. Temperature gradients and local differences in timetemperature path at different (radial) positions in the workpiece may not only lead to thermal strains but also austenite grain size gradients as austenite grain growth is a highly temperaturedependent process. Since austenite grain size before a hot-working process has a noticeable effect on both flow curve and dynamic recrystallization properties, its knowledge is required for a computational based microstructure design using models that consider hot-working and tempering processes. For example, the strain p at peak stress of a flow curve depends on initial grain size (Sellars, 1990) and p itself is a key factor for the description of the dynamically recrystallized volume fraction (Karhausen and Kopp, 1992).

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Table 1 Chemical composition of 34CrNiMo6 used in the present study determined by means of glow discharge optical emission spectroscopy (wt.%). C

Si

Mn

Cr

0.32

0.25

0.68

1.54

Ni 1.32

Mo

Fe

0.19

Bal.

Fig. 1. Microstructure of the 34CrNiMo6 steel (as delivered).

In addition, the knowledge of the grain size in spatial and chronological evolution allows the determination of optimal heating routes with minimum durations to generate homogeneous grain size distribution while avoiding unnecessary grain growth and energy input. In order to predict austenite grain size development, a holistic approach including determination of temperature-dependent heat transfer coefficients for the heating process, selection and parameterization of the austenite grain growth model is presented. The low alloyed heat treatable steel 34CrNiMo6 (1.6582) selected for the present study is a material typically used for the manufacturing of e.g., big crankshafts. The chemical composition of the material used is listed in Table 1 and the microstructure of the initial (as-delivered) state of the material is depicted in Fig. 1.

1.1. Determination of heat transfer coefficients For the prediction of time and temperature depending austenite grain size evolution during heating, knowledge of the timetemperature evolution in the whole part and consequently the heat transfer over its surface is essential. In a batch furnace the heat transfer into the cold workpiece occurs by free convection and thermal radiation. Both can be calculated by basic heat transfer relations, but the exact heat transfer characteristics need to be determined experimentally. In order to characterize the heat transfer problem during heating in a batch furnace, the heat transfer by radiation and convection is calculated using temperaturedependent heat transfer coefficients h (HTCs) . The overall heat flow q into the workpiece through its surface A is defined as q = h × A × (T − T∞ ) with T being the workpiece surface temperature and T∞ the furnace temperature. A simplified approach for the description of transient thermal processes is the “lumped-heat-capacity” method (LHCM). Assuming uniform temperature of a body, i.e., infinite heat conduction, the change in the inner energy of a body may occur due to heat transfer over its surface (Holman, 2002): h × A (T − T∞ ) + c × m

 dT  dt

=0

(1)

with the body mass m and the specific heat capacity c. Generally, this model is able to reproduce the real conditions within 5% error when: h × V × k−1 × A−1 ≤ 0.1

(2)

with the body volume V and the thermal conductivity k (Holman, 2002). Typically, this is valid for small bodies with low heat transfer coefficients and high thermal conductivities. If the progression of the bodies temperature T over the time t is known, one can easily determine h(t) and consequently h(T) from Eq. (1). This method has been successfully used by e.g., Bach et al. (2006) to determine temperature dependent HTCs for water-air spray cooling. Hamed (2000) used the LHCM in water spray cooling and points out that the HTCs determined were about 20% lower than those calculated by a series solution of the transient heat conduction problem. However, the criterion stated in Eq. (2) was clearly not met for the steel cylinders (diameters of 25.4 and 38.1 mm, 304.8 mm long) employed in that study. In that case, the transient heat conduction problem for the specific geometry needs to be solved, leading to a series solution of the differential equation that describes the problem (Holman, 2002). With this approach, however, the HTC cannot be determined directly. Instead, an iterative approach of assuming a HTC, determining the surface temperature by solving the series solution, calculating the difference to measured data and minimizing that difference is used. Previously, this iterative approach was employed for determination of HTCs in spray cooling by Krause (2008), and it was pointed out that the algorithm used is only applicable if no local maximum of HTCs is present over the range investigated. Sugianto et al. (2009) used a quenching experiment of a stainless steel cylinder and a corresponding FEM-based model of the experiment to determine the HTCs during quenching. Thermocouple measurements were done at 6 measuring points in the cylinder during the quenching. The boundary conditions of the FEM model (namely the HTCs) were then adapted iteratively on the basis of the LHCM until the model reproduced the cooling and cooling rate curves. By dividing the surface of the cylinder in the model into 5 sub-surfaces it was possible to determine the HTCs for these sub-surfaces. The use of those HTCs in a simulation of the quenching experiment led to a good agreement between simulation and experiment in the 6 measuring points for the cooling and cooling rate curves. Another group of methods to determine HTCs from temperature measurements are inverse methods. A heat transfer problem is called inverse if its purpose is not to specify the cause-effect relationship but to determine causal characteristics from the temperature field (Alifanov, 1994). By employing a mathematical model of the heat transfer process—which is most often based on equations with partial derivatives—different causal characteristics may be determined. For the case of HTCs—or more general: heat flux—determination, a boundary inverse heat transfer problem (IHTP) has to be solved. A detailed introduction in the methods for solving IHTPs can be found in literature (e.g., Alifanov, 1994). Some of the first methods to calculate the heat flux through the surface using the temperature history of internal measurements are attributed to J.V. Beck (Beck, 1968). Lately, the boundary inverse problems for determining HTCs that can be solved have become more complex. As a recent example, temperature dependent HTCs and their spatial distribution over a plate cooled by a water spray were determined by means of the inverse method assisted by finite element models (Malinowski et al., 2014). 1.2. Austenite grain growth Grain growth is activated by temperature and causes a reduction of the grain boundary area, and thus, the corresponding energy

S. Herbst et al. / Journal of Materials Processing Technology 229 (2016) 61–71

resulting in a degradation of mechanical properties (toughness as well as strength). A model which describes the evolution of the actual austenite grain size dt after a time t was first proposed by Burke and Turnbull (1952): n

dt = K × t + d0

n

(3)

where K is a constant. According to Hougardy and Sachova (1986), K can be described by an Arrhenius equation, leading to: dt = C × exp(−Qapp × R−1 × T −1 ) × t + d0 n

n

(4)

where d0 is the initial grain size, Qapp is the activation energy for grain growth, R the gas constant and C is a constant. The activation energy for grain growth Qapp is generally described as the activation energy for self-diffusion of iron in austenite that is assumed to be 270 kJ/mol (Hornbogen et al., 2012). The grain growth exponent n is 2 based on a model of atom transport across the grain boundary induced by a pressure due to surface curvature (Burke and Turnbull, 1952). Experiments typically yield n values in the range from 2 to 4, depending on metal and rate controlling mechanism (e.g., second phases or impurities inhibiting boundary migration) (Atkinson, 1988). For pure iron (total amount of impurities < 30 ppm) Hu (1974) observed a temperature dependence of the grain growth exponent, varying between 2 and 4. As recent theoretical analyses only confirmed the parabolic relationship established by Burke and Turnbull (1952), it appears that experimental data leading to higher n are always related to material impurities (Mittemeijer, 2011). Nonetheless, since a steel grade with about 4.3 wt.% of alloying elements was investigated in the present study, models with n > 2 and n(T) are included in the analysis. Krawczyk and Adrian (2010) measured the austenite grain growth of the 34CrNiMo6 steel for temperatures between 840 ◦ C and 1100 ◦ C and dwell times between 20 min and 24 h. They identified a zone of abnormal austenite grain growth for temperatures below 950 ◦ C and dwell times above 2 h. However, they did not use the measured data to establish or parameterize a grain growth model. The results of their measurements will be compared to the results presented later on.

2. Methods 2.1. Measurement of austenite grain growth For parameterizing the austenite grain growth models the grain sizes of small cuboidal specimens (20 × 10 × 10 mm) were determined for various temperatures and dwell times based on metallographic examinations. For the experimental recording of the grain growth process the temperature was set to 950 ◦ C, 1100 ◦ C and 1250 ◦ C with dwell times of 15, 30, 60, 180 and 360 min. Additionally, for the temperatures 1150 ◦ C and 1200 ◦ C in each case two more specimens were analyzed (15 min and 30 min). After quenching, the specimens were annealed for 90 min at 400 ◦ C. The specimens were then ground, polished and subsequently etched multiple times with a saturated water solution of picric acid with addition of a wetting agent at 50 ◦ C in order to reveal the former austenite grain boundaries. The grain size was measured at three points of each specimen by using an intercept procedure according to EN ISO 643:2003. To calculate the initial grain size D0 , the grain sizes measured for 950 ◦ C after 15 min and 30 min were used. The gradient in grain size between these measuring points was determined and a linear extrapolation to t = 0 min performed. The thereby calculated grain size was set as the initial grain size.

63

2.2. Parameterization of the grain growth model For fitting the model to the measured grain size development at different temperatures, the constant C and the grain growth exponent n were used. By setting the activation energy Qapp to 270 kJ/mol in each case, the fitting was performed by adjusting the parameters of four different grain growth models by means of multiple linear regression. The model was defined (M0 to M3, see Table 3), the measured data provided (combinations of time, temperature and grain size) and as a result the combination of parameters (e.g., C1 and n1 in model M1, see Table 3) was calculated that described the measured data with the defined model best. The simplest model proposed by Burke and Turnbull (1952) is a parabolic approach with n0 = 2. The other two models employed are using a variable or a function for the grain growth exponent n which is therefore part of the parameterization. As a second variant a constant that needs to be determined is used for n1 . In the third variant the grain growth exponent is set as a function of the temperature T with the two constant parameters m2 and n2 as follows: n (T ) = m2 × n2 T

(5)

2.3. Determination of heat transfer coefficients In order to predict austenite grain size in the heated workpiece depending on ambient temperature and heating time, knowledge of the heat transfer at the workpiece surface is indispensable. Therefore, temperature-dependent interfacial heat transfer coefficients were determined that describe the superimposed effects of radiation and surface-convection. In the following the use of HTCs is referring to these simplified heat transfer coefficients. One has to take into account that the convective and radiative heat transfer depend on the geometry and surface condition of the heated component and the furnace and their arrangement. Hence, the transferability of HTCs determined at one set-up to a different one is generally limited. However, the following method is applicable for any convective or radiative heating operation. As a first approximation, the LHCM was used to determine the HTCs from the heating curve of a thin disc (diameter ddisc = 100 mm, thickness tdisc = 2 mm) of the investigated material, following the method used by Bach et al. (2006) to determine HTCs during waterair spray cooling. The criterion stated in (2) is matched for HTCs below hmax = 2700 W m−2 K−1 (assuming a minimal thermal conductivity of k = 26 W m−2 K−1 , see Appendix 1). As higher HTCs were not expected, the LHCM was applicable. The discs were equipped with a 0.5 mm diameter thermocouple (temperature was recorded with f = 10 Hz) in its axial center and put in an electrical batch furnace at Tfurnace = 1200 ◦ C. Measurements were stopped when the disc temperature was equal to the ambient temperature. The timetemperature data recorded were smoothed and the HTC for each time-step was calculated using Eq. (1). For this purpose, T was set to be the measured disc temperature Ti of the actual time-step ti , dT to be the change in disc temperature relating to the last time step (dT = Ti − Ti-1 ) and dt to be the time-step size (dt = ti − ti-1 = 1/f). The specific heat capacity c was determined for every calculation step by linear interpolation of temperature dependent material data from literature (see Appendix 1). After every calculation step the calculated HTC hi was stored in a data table together with its associated Temperature Tm = (Ti + Ti-1 )/2. As a second method for the determination of the HTCs, response surface optimization (RSO) in combination with numerical simulation was employed. The method is depicted schematically in Fig. 2. In order to use the RSO, the temperature trend during the heating process of a massive specimen is required. A cylinder (diameter dcyl. = 29 mm, length Lcyl. = 40 mm) of the alloy used was equipped with two thermocouples (one in the center and one near-surface, cf.

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Table 2 Heating experiments for verification and validation of HTCs. ID

Specimen

HTCs determined by

Purpose

H0 H1 H2 H3 H4

Disc Cylinder (Ø29) Cylinder (Ø29) Cylinder (Ø98) “Dumbbell”

LHCM RSO LHCM LHCM, RSO RSO

Rule out methodic errors (same experiment as for HTC-determination) Rule out methodic errors (same experiment as for HTC-determination) Transferability to larger specimen Transferability to larger specimen Transferability to more complex specimen

Table 3 Results of the parameterization for the different variants of grain growth model. ID

Model

Formula

M0

Parabolic approach

= C0 × e t + D02

M1

Constant grain growth exponent

Dtn1 = C1 × e(−Qapp /RT ) × t + D0n1

M2

Temp.-dependent grain growth exponent

Dt

M3

Temp.-dependent grain growth exponent and activation energy Qapp

Dt2

m2 ×nT

2

×

= C2 × m2 ×nT

e(−Qapp /RT ) × t + D0 m3 ×nT

2

3 Dt = C3 × e(−Qapp (T )/RT ) × t +

m3 ×nT

D0

Ra 2

Parameters (−Qapp /RT )

3

T u−T (forT = 950...1250◦ C) Qapp (T ) = r − s

8

C0 = 7.19 × 10

0.93

C1 = 6.34 × 108 n1 = 2.429

0.95

C2 = 5.39 × 108 m2 = 0.01813 n2 = 1.00422

0.98

C3 = 3.34 × 106 m3 = 11.5054 n3 = 0.99904 r = 314,480 s = 13,307.2 u = 1422.28

0.98

[Qapp ] = J mol−1 ; R = 8.31446 J mol−1 K−1 ; [T] = ◦ C; [t] = min.

Fig. 2. Schematic representation of the RSO-method.

S. Herbst et al. / Journal of Materials Processing Technology 229 (2016) 61–71

65

Fig. 3. Program flowcharts of the main macro, the HTC-assignment sub-macro and the grain growth sub-macro.

Fig. 2a) and heated in an electrical batch furnace to Tfurnace = 1230 ◦ C. The recorded temperature data were then used as the target of the RSO and define the simulation steps by dividing the heating process in 30 K segments (referring to the near surface temperature) as shown in the table in Fig. 2a. The experimental set-up was recreated by thermal transient numerical simulation using the commercial software ANSYS® WorkbenchTM . For the first simulation step i = 1 (i.e., from a near surface temperature of RT to RT + 30 K) the HTC (hi-1 , cf. Fig. 2b “set-up”) applied as convective boundary condition was guessed by using the values taken from the predetermined LHCM results. The numerical simulation was then carried out until the time ti+1 (cf. table in Fig. 2a) was reached. In the next step, the software’s “Goal Driven Optimization”-tool was used to carry out the RSO. For this purpose, hi-1 was used as the input parameter and the simulated temperatures at the center Tsim. (r = 0 mm) and near the surface Tsim. (r = 13 mm) of the specimen as the output parameters (cf. table in Fig. 2b, “solution”). For five different HTCs in the range of 75% to 125% of the initial HTC hi-1 the simulation was repeated and by using the resulting temperatures Tsim. (r = 0 mm) and Tsim. (r = 13 mm) the response functions were calculated (cf. Fig. 2b, “response”). In the following, the measured temperatures Tmeas. (r = 0 mm) and Tmeas. (r = 13 mm) were set as target and the optimization was started. Within this optimization, the response functions are used to identify appropriate candidates for hi that lead to a good reproduction of the measured temperatures. This operation is exemplarily shown for the near surface temperature in Fig. 2c on the left. From the resulting candidates for hi , the one leading to the smallest deviation of the near-surface temperature Tns = |Tmeas. (r = 13 mm) − Tsim. (r = 13 mm)| was selected. With the selected candidate hi , the thermal simulation was repeated and the combination of hi and the resulting surface temperature is noted (cf. Fig. 2c). Based on the resulting calculated temperature distribution using this HTC hi , the procedure was repeated until the defined maximum temperature in the specimen’s center was achieved. The HTCs determined by this method were assigned to the mean surface temperature of the specimen during the corresponding time interval and are depicted in Fig. 6. The parameters used for the RSO with ANSYS® WorkbenchTM are listed in Appendix 2. 2.4. Verification and validation of heat transfer coefficients For verification, the two different temperature-dependent HTCs were applied to simulate the heating of the specimens (disc and

cylinder) that were used during their determination to rule out methodic errors (Table 2, H0 and H1). Afterwards, the HTCs determined by LHCM were also used to simulate the heating of the small cylindrical specimen from the RSO-method in order to validate the transferability from plain to cylindrical geometries (H2). Both temperature-dependent HTCs were finally used to simulate the heating processes of a larger cylindrical specimen (diameter dcyl. = 98 mm, length lcyl. = 120 mm) to validate transferability to larger geometries (H3). Finally RSO-HTCs were used to simulate the heating of a dumbbell-like specimen to test transferability to more complex geometries by comparison to measured data (H4). 2.5. Simulation of austenite grain growth In order to compute the austenite grain growth behavior during the heating process, an APDL-macro (Ansys Parametric Design Language) was developed. Fig. 3 depicts a flow chart of the grain growth macro (right) along with the HTC-assignment macro and the main macro that have been developed to simulate the heating process within a thermal transient analysis. After the preprocessing and definition of the simulation time (main macro), for every time step i the HTC-assignment sub-macro is executed. In that the sub-macro, for every surface node j of the model-mesh a certain HTC depending on the nodes temperature Tj is assigned as a surface load together with the ambient temperature (i.e., furnace temperature). After all surface loads are set, the current time step is solved and the resulting node temperatures are used for the actual grain growth macro. As temperatures are solely provided at nodes and grain development is a three-dimensional problem, the temperatures of the nodes adjacent to a volume element are used to calculate the volume element temperature Tk , i for every volume element k at the current time step i. The Tk , i are saved in a data field and a mean volume element temperature Tk ,m is determined by forming the arithmetic mean of the volume element temperature of the current and the previous time-step. Therefore, grain growth is treated as an isothermal process during the short time difference between two time steps. The grain size d versus time t development for the temperature Tk ,m (see exemplarily in Fig. 3, right, highlighted in grey) is used to determine the grain growth during the time difference t = ti − ti-1 as follows. The mean grain size of the volume element of the last time step dk , i-1 is imported from the data field and the correlating time ti-1 is determined using the isothermal grain growth model at Tk ,m . The time step size t is added to ti-1 and the resulting grain size dk , i can be calculated from the isother-

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S. Herbst et al. / Journal of Materials Processing Technology 229 (2016) 61–71

Fig. 4. Results of the investigated grain growth evolution (top left) and examples of the metallographic analysis.

1.2

1.2 1

grain size d, mm

grain size d, mm

1 0.8 0.6 0.4 0.2

0.8 0.6 0.4 0.2

0

0 1200

300 1100

temperature T, °C

1200

300 1100

200 100

1000 0

200 100

1000

time t, min temperature T, °C

0

time t, min

Fig. 5. Graphic representation of the parabolic approach (M0, left) and the approach with a temperature-dependent grain growth exponent (M2, right). Measured data points are depicted by triangles pointing to the modelled surfaces.

3. Results 3.1. Measurement of austenite grain growth The results of the grain size measurements are depicted in Fig. 4. The initial grain size d0 was determined to be 28.3 ␮m. While at a temperature of 950 ◦ C even after a holding time of 300 min almost no grain growth is measurable, the grains grow significantly at 1250 ◦ C within the first 15 min.

500 LHCM (Disk) RSO (Cylinder)

400

HTC h , Wm -2K-1

mal grain growth model and stored in the data field for further use in the following time step. The grain size for every volume element in every time step stored in the data base can finally be used during postprocessing for either analyzing grain size distribution for a certain time step or grain size evolution over time for certain volume elements.

α˧γ

300 200 100 0 0

300

600

900

1200

Temperature T , °C 3.2. Parameterized grain growth models The results of the parameterization of the three different grain growth models are summarized in Table 3. For the second approach

Fig. 6. Temperature-dependent HTC determined by Lumped Heat Capacitance Method (LHCM) and Response Surface Optimization (RSO).

S. Herbst et al. / Journal of Materials Processing Technology 229 (2016) 61–71

1200

r = 0mm 1000

Temperature T , °C

(M1) a constant grain growth exponent n1 = 2.65 was calculated, which is in the range of to the growth exponent values for steel between 2 and 4 described by Atkinson (1988). The temperaturedependent function (4) of the grain growth exponent of model M2 increases with the computed parameters in the temperature-range of 950 ◦ C to 1250 ◦ C from 1.16 to 3.68. Thus, the empirically determined temperature-dependent grain growth exponent has a much greater influence on the course of the model over temperature as the approaches with constant grain growth exponents. In comparison to the parabolic approach (M0) the adaptation of the grain growth exponent to the parameterization also shows a significant improvement of the model-predicted values of grain size compared to the experimental measured values (Fig. 5). Furthermore the temperature dependent grain growth exponent achieved a significant reduction of the percentage mean deviation of the measured values as compared to the parabolic approach (M0: 17.8%; M2: 11.6%). Thus, for the description of the austenite grain growth the temperature-dependent grain growth exponent (M2) was used in the following. The reason for the development and use of Model M3 will be described later on.

67

800

Measurement T (r =0mm) Measurement T (r =13mm) Simulation T (r =0mm); LHCM Simulation T (r =13mm); LHCM Simulation T (r =0mm); RSO Simulation T (r =13mm); RSO

r = 13mm

600

400

200

0 0

2

4 Time t, min

6

8

Fig. 7. Measured and computed time-temperature-diagrams of the heating process for the small cylindrical specimen (d = 29 mm, l = 40 mm).

1200

r = 0mm

Temperature T , °C

Measurement T (r =0mm) Measurement T (r =47mm) Simulation T (r =0mm); LHCM Simulation T (r =47mm); LHCM Simulation T (r =0mm); RSO Simulation T (r =47mm); RSO

r = 47mm

800

600

400

200

0 0

10

20 Time t , min

30

Fig. 8. Measured and computed time-temperature-diagrams of the heating process for the big cylindrical specimen (d = 98 mm, l = 120 mm).

1200 45 A

1000 20

B

800

D

C 30

25

Fig. 6 depicts the temperature-dependent HTCs determined by LHCM and RSO, respectively. Below surface temperatures of 700 ◦ C the HTCs determined by RSO are up to 35% lower than those determined by LHCM. Between 700 ◦ C and 800 ◦ C the HTCs for both methods show similar qualitative characteristics. The descent at about 800 ◦ C derives from the phase transformation of ferrite to austenite and is present for both methods. However, the characteristic descent appears at slightly lower temperatures for the RSO-method. Between 800 ◦ C and 900 ◦ C, the RSO-HTCs are quite similar to those determined by LHCM. For temperatures above 900 ◦ C, the HTCs determined by RSO increase up to 1100 ◦ C followed by a descent while those determined by LHCM show a descent between 850 ◦ C and 1050 ◦ C followed by an ascent. The difference in evolution in this temperature range leads to up to 40% smaller HTCs for the LHCM-Method. The direct comparison of the measured time-temperaturediagrams for the disc and the small cylinder with their respective simulation (LHCM-HTCs were used for the simulation of the disc, RSO-HTCs for the simulation of the small cylinder, cf. Table 2, H0 and H1) showed both excellent agreements. While the results for the disc are not depicted, the results for the small cylindrical specimen can be found in Fig. 7 (dashed line). The Figure also shows the results of a simulation of the heating process for the small cylinder when LHCM-HTCs were used (cf. Table 2, H2). Deviations of up to 150 K can be found. To analyze the scalability of the HTCs determined using the thin disc and comparatively small cylinder, a heating process of a bigger cylinder (d = 98 mm, l = 120 mm) was carried out, measured and computed using both HTC-variants (cf. Table 2, H3). The results are depicted in Fig. 8, showing fairly good agreement for the simulation with RSO-HTCs while the simulation with LHCM-HTCs revealed deviations from the measured trend as high as for the smaller cylinder (Fig. 7). As the LHCM-HTCs were not able to reproduce time-temperature-diagrams of the cylindrical parts, only the RSOHTCs were used for the investigations of the dumbbell specimen (cf. Table 2, H4). The results are depicted in Fig. 9. The simulation covers the process well below 600 ◦ C and above 850 ◦ C. The highest deviations of below 100 K can be found in the temperature range of the ferritic-austenitic transformation as seen in the other time-temperature-diagrams.

1000

Temperature T , °C

3.3. Heat transfer coefficients

Simulation A (RSO) Simulation B (RSO)

600

Simulation C (RSO) Simulation D (RSO)

400

Measurement A Measurement B

200

Measurement C Measurement D 0 0

2

4

6 Time t, min

8

10

12

Fig. 9. Measured and computed time-temperature-diagrams of the heating process for the dumbbell specimen (see upper left inlay for geometry).

S. Herbst et al. / Journal of Materials Processing Technology 229 (2016) 61–71 1096

1200

274

Temperature (measured)

400

Temperature (simulated)

0.04

200

0.02

0 0

60

120

180 Time t, min.

240

300

360

El. 5189 El. 1092

1050

0.6

800

0.4 )

600

Grain size (simulated, M3)

ce

Grain size (simulated, M2)

0.06

rna

Grain size (measured)

0.8 T d (M2) d (M3)

550

0.2

T ( Fu

0.08

1300

Temperature T , °C

800

El. 1092

Ø446

1000

0.10

Temperature T , °C

Austenite grain size d , mm

El. 5189

Fig. 10. Measured and computed temperature and austenite grain size trends for small cubic specimens (10 × 10 × 10 mm).

300

Grain size d , mm

0.12

Ø527

68

0 0

60

120 180 Time t , min

240

3.4. Computed austenite grain growth The collective data (grain growth model with temperature dependent grain growth exponent and simplified HTCs determined by the RSO-method) were finally applied with the simulation routines depicted in Fig. 3 for three different simulations. Firstly, small specimens consisting of just one volume-element were set to a constant temperature (950 ◦ C, 1100 ◦ C and 1250 ◦ C) and a transient simulation was carried out for 300 min. The time-austenite grain size diagrams were then compared to the corresponding isothermal trends from the grain growth model used (Fig. 5, right). The calculated values matched the model exactly. Secondly, the above mentioned one-volume-element specimen was hold at a certain time-temperature trend. This was to verify the capability of the routine used (utilization of small isothermal time steps etc.) to handle non-isothermal temperature-trends and predict the resulting grain sizes correctly. The results are depicted in Fig. 10. In addition to the simulation four small (10 × 10 × 10 mm) specimens of 34CrNiMo6 were placed sequentially in two different furnaces to generate the same temperature trend as simulated (see Fig. 10, solid line). Each was quenched at a different time and the grain size was measured as described above. The drop offs in the measured temperature trend are due to the opening of the furnaces door to unload a specimen (smaller drops) or due to the transfer from one furnace to the next (drop at min 282). The trend of the simulated austenite grain size (using model M2, chain line in Fig. 10) qualitatively represents the measured grain size development. A huge growth of the grain size after a few minutes at 1100 ◦ C can be found for the computations as well as for the experimental data. The grain size measured at the end of the temperature rise deviates by nearly 67% from the calculations (using model M2, chain line), at the end of the 1100 ◦ C plateau by about 12%. The calculated grain size trend depicted by the dotted line will be described later on. In order to show grain size deviations in a huge forging part, the heating of a big semi-finished crankshaft according to Walczyk et al. (2011) was modelled and simulated. The modeled part was a cutout of a semi-finished crankshaft (shaft-diameter: 527 mm, overall length: 4822 mm, see sketch in Fig. 11). The temperature trend of the furnace is depicted in Fig. 11 as well. It is heated from 300 ◦ C to 1250 ◦ C in the first 120 min, followed by isothermal holding for an additional 150 minutes. The temperature and austenite grain size trends for two representative volume-elements from the surface (black lines, Fig. 11) and the core (grey lines, Fig. 11) are also depicted. Obviously, the large local temperature gradients (>300 K) result in austenite grain sizes ranging from 0.278 mm to 0.612 mm at the end of the heating process when using model M2 (chain lines, Fig. 11). The calculated trends using model M3 will be discussed later on.

Fig. 11. Temperature and grain-size evolution for two elements during the computed heating process of a semi-finished crankshaft.

4. Discussion The overall precision of the austenite grain-growth calculation may be estimated by the measured and computed temperature and austenite grain size trends for the small cubic specimens (Fig. 10). The maximum deviations of the simulated grain size from the measured did exceed 66% at the end of the temperature raise from 900 ◦ C to 1100 ◦ C. All inaccuracies from the grain size measurement, simplifications and regression analysis are summed up in this calculation. In the following, the results and inaccuracies of each single step will be described and discussed. 4.1. Measurement of austenite grain growth Austenite grain sizes in a quenched microstructure cannot be measured directly and needs to be visualized by an adapted etching. From the optical micrographs in Fig. 4 it is obvious that a clear determination of the former grain boundaries was not always possible. Moreover, during measurement by the intercept method the counted grain boundaries per millimeter varied between 35 and 3. The precision in this grain size range is affected either by too few intercepted boundaries (statistic uncertainty) and/or issues related to etching. For every measuring point shown in Fig. 4, the grain size was determined three times along the corresponding specimen. The maximum and minimum deviations in these three measurements from its mean value are depicted by the error bars. On average, a deviation of ±6.7% was determined. A direct comparison to the grain sizes measured by Krawczyk and Adrian (2010) can only be made for t = 120 min and T = 950 ◦ C and T = 1100 ◦ C, respectively. For both cases, the grain sizes described by Krawczyk and Adrian (2010) were more than two times larger. The grain growth observed in their work was significantly higher, especially for the lower temperatures (T < 1100 ◦ C) and short dwell times (t < 120 min) in which only very small grain growth could either be measured or predicted by the grain growth models in the present study. The differences observed are attributed to two factors. On the one hand, the chemical compositions of the materials tested do not match exactly, as they are differently positioned in the tolerance band for the 34CrNiMo6 grade. On the other hand, the initial state of the material in the study of Krawczyk and Adrian (2010) gives the impression of a much finer microstructure that may be the result of a thermomechanical treatment. The finer initial microstructure as well as a possible recrystallization due to residual stresses may have affected

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the grain growth kinetics. However, the differences observed also show that even if nominally the same steel grades are used, grain growth may vary substantially and thus chemical composition and thermo-mechanical treatment have to be taken into account. 4.2. Parameterization of the grain growth model The grain growth model according to Atkinson (1988) with a temperature dependent grain growth exponent was identified to reproduce the measurements best (Table 3, Model M2). An adjusted coefficient of determination of 0.98 and an average relative amount of deviation of the calculated grain sizes from the measured of 11.6% characterize the model. However, the deviation has a non-uniform distribution along the models surface (see Fig. 5). For T = 950 ◦ C, the average deviation is 9.2%, for T = 1250 ◦ C it is 8.7% and along T = 1100 ◦ C the highest average deviation of 16.9% can be determined. This indicates a problem that was found during the model selection, adjustment and parameterization: the models proposed in literature cannot produce a change in the slope of the models surface along the time-axis (e.g., for ascending temperatures) above 1100 ◦ C that is as strong as required by the measured data—even when a temperature-dependent grain growth exponent is applied. Therefore, the calculated grain sizes for T = 1100 ◦ C are much larger than the measured ones. Otherwise the calculated grain sizes for all other temperatures would be too low by trend, resulting in higher overall average deviations and a lower coefficient of determination. This fact may be explained with the necessary activation energy for grain growth which seems to change above 1100 ◦ C (considering Fig. 4). Therefore, grain growth strongly increases above that temperature. The initially implemented activation energy of 270 kJ/mol was based on the activation energy for iron self-diffusion in austenite. As the steel investigated contains 4.3 wt.% of alloying elements and the solution of different carbides at elevated temperatures may have an impact on the activation energy, one could assume that an implementation of a temperature-dependent activation energy would improve the model. Hence, a model with a temperaturedependent activation energy was developed and parameterized (M3, Table 3). The overall relative amount of deviation from the measured data was reduced to 5.9%. Most important, it was substantially reduced for the former critical high temperature region, leading to an average deviation of only 6.0% for T = 1100 ◦ C (3.0% for T = 950 ◦ C; 8.7% for T = 1200 ◦ C). The improvements achieved by using the M3-model are clearly seen in Fig. 10 (dotted line). A much higher representation of the measured grain sizes for the variable temperature trend could be achieved; the largest deviation was below 9.0%. The simulation of the grain size development in the semifinished crankshaft using the enhanced M3-model revealed grain sizes between 0.2 and 0.52 mm. While the absolute values are slightly lower than for the calculations based on model M2, the deviations between the largest and smallest grain sizes are of the same magnitude. 4.3. Heat Transfer Coefficients The transferability of the simplified HTCs determined by the LHCM at thin discs for the heating simulation of massive cylindrical geometries was found to be insufficient. Especially in the temperature range between 400 ◦ C and 900 ◦ C it was not possible to reproduce the temperature trend in both cylindrical specimens (see Figs. 5 and 6) as the calculated LHCM-HTCs were too high in that range (Fig. 6). The maximum HTC calculated by LHCM was 350 W m−2 K−1 at 1150 ◦ C, leading to h × V × k−1 × A−1 = 0.011 (with k (T = 1150 ◦ C) = 29.48 W m−1 K−1 , see Appendix 1). Hence, the condition (2) is clearly met, leading to the conclusion that the temperature throughout the disc may be assumed to be uniform

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and that the error made by application of the LHCM is under 5% (Holman, 2002). A comparison of the LHCM-HTCs to the RSO-HTCs shows up to 41% higher and down to 68% lower values for the LHCM-HTCs (Fig. 6). As the simulations with the HTCs determined by RSO lead to very small deviations from the temperature trends (for the small and bigger cylinder, Figs. 7 and 8, respectively), one can assume these to represent the situation in the specimens very well. The highly insufficient transferability of HTCs determined by LHCM to cylindrical specimens seem to arise only from the geometric differences as the external factors (material, furnace) and systematic errors due to simplification by the LHCM are not able to explain the extent of deviation. Obviously, the high dependence of the heat radiation on the geometric conditions make the use of a specimen necessary that is geometrically similar to the workpiece to be simulated. Additionally, the emissivity that is mainly influencing the radiative heat transfer, depends on the surface condition. The LHCM specimen (disc) and the cylinders (including the RSO-specimen) indeed both had blank surfaces but were machined differently: by grinding and fine turning, respectively. This may also be a reason for the lower precision when the LHCM-HTCs were used for computation of the cylindrical specimens. Another reason may be found in the heating process influencing the surfaces itself. During the heating in the furnace under atmosphere, the surfaces start to oxidize. Since this is a time and temperature dependent process, the longer heating time and consequently longer time at elevated temperatures of the cylindrical specimens have probably led to a denser oxidation (scaling) of their surfaces. This scale increases the emissivity of the surfaces (compared to the blank ones) and may therefore explain the higher HTCs calculated for the RSO specimens, especially for the higher temperatures (cf. Fig. 6). Additional work has to be done to clarify this issue. This would require a separate analysis of the convective and radiative heat transfer by carrying out heating experiments in vacuum for both scaled and blank specimens. The use of the above presented methodology using a RSO to determine simplified HTCs from the measured data of a small cylindrical specimen by stepwise numerical simulation turned out to be much more effective. Both the scalability and the increase of complexity of the heated specimen could be reproduced properly within simulations using the RSO-HTCs (see Figs. 6 and 7). A minor exception is the temperature range around 800 ◦ C during which ferritic-austenitic transformation occurs. This is expected as the transformation was neglected in the numerical simulation. The endothermic transformation process starts to have a substantial impact for increasing specimen dimensions. Further improvements would necessitate a microstructural simulation that covers austenite evolution combined with HTCs determined on austenitic steel specimens. In addition, reproducibility tests will be part of further investigations to confirm the differences between the HTCs measured by LHCM and those measured by RSO to back up the aforementioned theories concerning the influences on the HTCs. In summary the holistic consideration of austenite grain growth during heating processes of a heat treating steel grade described in this paper covered:

(a) the determination of simplified heat transfer coefficients by means of a simplistic, simulation-supported methodology (“RSO”) using a small cylindrical specimen; (b) the evaluation of these HTCs for use with heating simulations of larger and more complex geometries; (c) the selection and parameterization of a suitable austenite grain growth model that represents the measured grain size data and its enhancement by introduction of a temperature-dependent activation energy;

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(d) the implementation of the above stated simplified HTCs and a grain size evolution routine based on the parameterized grain growth model into a thermal transient numerical simulation and the evaluation of the overall model; (e) the simulation of the heating process and grain size development of a Ø > 0.5 m crankshaft cutout. 4. Conclusions The results of the present study can be summarized as follows: a) The heat transfer coefficients (HTCs) determined by response surface optimization (RSO) on a small cylinder were significantly different to the ones determined by an approach based on the lumped heat capacitances method (LHCM) using a thin disc. The fact that neither the condition for applicability of the LHCM was violated nor the external factors were different from those of the RSO-experiment leads to the conclusion that the geometric differences between the two specimens induced the large deviations in HTCs. b) With the RSO-HTCs it was possible to predict the temperature trends in similar but both larger and more complex workpieces by means of thermal transient numerical simulations. This has not been the case by using the LHCM-HTCs. It can be concluded that a geometric similarity is a prerequisite for a precise HTCdetermination in heating processes due to a strong dependence of radiation on geometric conditions. c) The grain growth model that initially fit best for the reproduction of the measured grain sizes is based on the suggestion of implementing a temperature-dependent grain growth exponent. However, this model was insufficient in the medium temperature range and it was not able to reproduce grain growth well for a variable temperature trend. Thus, it was enhanced by the introduction of a temperature-dependent activation energy. The average relative amount of deviation of the calculated from the measured grain size of 5.9% indicates a very good reproduction of the measured data. d) A verification of the developed grain growth subroutine by simulating isothermal experiments showed an exact reproduction of the underlying models. In a validation step the grain size trend over a non-isothermal process was simulated for the enhanced model M3 stated in (c) and compared to the measured grain size at four significant points. The overall trend and the maximum relative deviation of the measured from the simulated grain size of below 9% indicate a very good reproduction and a huge improvement compared to the model without temperature-dependent activation energy (maximum relative deviation > 66%). e) The simulation of the heating process of a huge crankshaftcutout (process and workpiece based on real conditions) showed a wide difference in the grain sizes of two representative volume elements for both the literature-based grain growth model M2 and the enhanced model M3.

Acknowledgement The authors thank the German Research Foundation (DFG) for financial support within the project NU297/2-1. Appendix 1. Temperature-dependent material data used in the simulations (Spittel and Spittel, 2009).

Temperature (T, ◦ C)

Specific heat capacity (c, J kg−1 K−1 )

Thermal conductivity (kW m−1 K−1 )

20 100 200 300 400 500 600 700 800 900 1000 1100 1200

476.91 490.43 504.26 524.54 549.15 576.4 605.73 637.03 699.26 696.75 694.25 691.75 689.27

34.1 34.94 36.34 36.4 35.8 34.8 33.6 32.28 26.05 27.25 28.09 28.95 30.02

Appendix 2. Settings used for “Goal Driven Optimization” in ANSYS® WorkbenchTM (15.0). Property

Value

Design of experiments type:

Central composite design (type: Auto-defined) Heat transfer coefficient (design variable) 5 T0 = T (r = 0 mm); T13 = T (r = 13 mm) Standard response surface (full 2nd-order polyn.) Manual (not used) Screening (shifted-hammersley sampling) 1000 (default) 3

Input parameter(s): Number of design points: Output parameter(s): Meta-model type: Meta-model refinement: Goal driven optim. method: Number of samples: Maximum number of candidates:

References Alifanov, O.M., 1994. Inverse Heat Transfer Problems. Springer-Verlag, Berlin, New York, pp. 1–32. Atkinson, H.V., 1988. Theories of normal grain growth in pure single phase systems. Acta Metall. 36, 469–491, http://dx.doi.org/10.1016/0001-6160(88)90079-x. Bach, F.-W., Schaper, M., Nürnberger, F., Krause, C., Broer, C., 2006. Simulation des Abschreckhärtens mittels Sprühkühlung—Wärmeübergang, Gefüge und Härte. HTM Härtereitechnische Mitteilungen 61, 142–147, http://dx.doi.org/10.3139/ 105.100376. Beck, J.V., 1968. Surface heat flux determination using an integral method. Nucl. Eng. Des. 7, 170–178. Burke, J.E., Turnbull, D., 1952. Recrystallization and grain growth. Prog. Metal Phys. 3, 220–292, http://dx.doi.org/10.1016/0502-8205(52)90009-9. Hamed, M.S., 2000. Evaluation of heat transfer coefficients in water spray quenching systems. In: Bell, T., Cohen, J.B., Funatani, K., Totten, G.E. (Eds.), Heat Treating: Proceedings of the 20th Conference, 9–12 October 2000. St. Louis, Missouri, pp. 785–790. Holman, 2002. Heat Transfer, ninth Ed. McGraw-Hill, New York, pp. 131–135. Hornbogen, E., Eggeler, G., Werner, E., 2012. Werkstoffe, 10 Ed. Springer Verlag, Berlin, Heidelberg, pp. 125. Hougardy, H.P., Sachova, E., 1986. Ways to influence the size of austenite grains. Steel Res. 57, 188–198. Hu, H., 1974. Grain growth in zone-refined iron. Can. Metall. Q. 13, 275–286. Karhausen, K., Kopp, R., 1992. Model for integrated process and microstructure simulation in hot forming. Steel Res. 63, 247–256. Krause, C., 2008. Wärmeübergangs- und Tropfencharakteristik für eine Spraykühlung im Temperaturbereich von 900–100 ◦ C. Forschung im Ingenieurwesen 72 (3), 163–173, http://dx.doi.org/10.1007/s10010-008-00797. Krawczyk, J., Adrian, H., 2010. The kinetics of austenite grain growth in steel for wind power plant shafts. Arch. Metall. Mater. 55, 91–99. Malinowski, Z., Cebo-Rudnicka, A., Telejko, T., Hadała, B., Szajding, A., 2014. Inverse method implementation to heat transfer coefficient determination over the plate cooled by water spray. Inverse Prob. Sci. Eng. 23, 518–556. Mittemeijer, E.J., 2011. Fundamentals of Materials Science. Springer, Berlin, Heidelberg, pp. 488, http://dx.doi.org/10.1007/978-3-642-10500-5.

S. Herbst et al. / Journal of Materials Processing Technology 229 (2016) 61–71 Sellars, C.M., 1990. Modelling microstructural development during hot rolling. Mater. Sci. Technol. 6, 1072–1081. Spittel, M., Spittel, T., 2009. Steel symbol/number: GS-34CrNiMo6/1.6582. In: Warlimont, H. (Ed.), Springer Materials—The Landolt-Börnstein Database. Springer, Berlin, Heidelberg, http://dx.doi.org/10.1007/978-3-540-44760-3 154. Sugianto, A., Narazaki, M., Kogawara, M., Shirayori, A., 2009. A comparative study on determination method of heat transfer coefficient using inverse heat

71

transfer and iterative modification. J. Mater. Process. Technol. 209, 4627–4632. Walczyk, W., Milenin, A., Pietrzyk, M., 2011. Computer aided design of new forging technology for crank shafts. Steel Res. Int. 82, 187–194, http://dx.doi.org/10. 1002/srin.201000121.