Study on the Optimization of Heat Transfer Coefficient of a Rare Earth Wrought Magnesium Alloy in Residual Stress Analysis

Abstract

To investigate the heat transfer coefficient (HTC) of a newly developed rare-earth wrought magnesium alloy under different cooling rates, the experiment of solution treatment followed by water quenching or air cooling process was carried out for calculation by lumped capacitance method (LCM) and optimized by inverse heat transfer method (IHTM), and cooling temperature curves were simulated afterward. In water quenching, the larger the temperature difference between the sample and water, the larger the maximum HTC, and the earlier it reached the maximum value, and in air cooling the HTC became larger with the airflow speeds increased. In LCM, the peak values of the HTC were 2840 W/(m²·°C) in water quenching and 54 W/(m²·°C) in air cooling. The corresponding HTC was 2388 W/(m²·°C) in IHTM. The maximum absolute average relative error (AARE) of temperature simulation in water quenching decreased from 8.46% in LCM to 2.45% in IHTM. The residual stress(RS) of a large conical component was simulated using both non-optimized and optimized HTC, the RS in the IHTM was ~30 MPa smaller than that in the ILCM, because the corresponding HTC was smaller, and the comparison of the simulation results with the measurements revealed that the RS using HTC in the IHTM is more accurate.

1. Introduction

Magnesium alloys have wide application prospects because of their low density and high specific strength [1,2]. Solution treatment followed by water quenching or air cooling to obtain the supersaturated solid solution state [3], with subsequent artificial aging [4,5,6] is the conventional heat treatment for magnesium alloys. In the process, the temperature changes itself and residual stress (RS) induced by the temperature gradient in the cooling process will affect the properties of the material, for instance, the RS could reduce dimensional stability [7], mechanical properties [8], corrosion resistance [9], and fatigue properties [10].

In the heat treatment above, the heat transfer coefficient (HTC) is a key parameter to characterize the heat exchange ability between metal alloys and cooling media which is relevant to the thermal conductivity, the specific heat capacity of the sample, the flow velocity of the media, etc. To avoid wasting time and cost in material research and development, finite element method (FEM) is often employed to reveal the evolution of temperature and RS field. Wang et al. [11] employed the HTC of both water quenching and air cooling to simulate the RS of Mg-5Zn-3.5Sn-1Mn-0.5Ca-0.5Cu alloy, the surface RS of the latter decreased by 43%, and the simulation results, whose difference was due to the variation of the HTC, are in agreement with the experimental ones, in this regard, HTC, which is an important boundary condition in the FEM, serves as a crucial factor to ensure the accuracy of the simulation results. On the other hand, the HTC is closely related to the cooling rate, thus the RS during quenching could be reduced by controlling the HTC [12,13], which provides a theoretical basis for the formulation of heat treatment. The heat transfer model is usually used to simulate the temperature value in heat treatment, Jia et al. [14] and Le et al. [15] used the heat transfer model to simulate the air-cooling transport process of the AZ31B plate whose average HTC was 20 W/(m²·°C). Furthermore, the coupled temperature–displacement model is widely applied in RS analysis [16,17], the intense heat exchange brings uneven deformation inside and outside the material which induces RS.

There are several ways to calculate the HTC, Zhang et al. [18] measured the specific heat capacity of the battery under both natural and forced convection conditions by using the HTC in the lumped capacitance model (LCM), which is based on the conservation of heat, and considers that temperature difference inside a thermal system is negligible. Yuan [19] supposed the temperature gradient cannot be ignored during quenching heat treatment, the temperature field in the sample was created by quadratic fitting according to the one-dimensional thermal conductivity characteristics of single-sided quenching, and the HTC of magnesium alloy was obtained, which is called the improved lumped capacitance method (ILCM). The inverse heat transfer method (IHTM) [14] first determines some initial HTC values, and then iteratively calculates the temperature compared with the actual value, and limits the error to get the simulated HTC. There are also the least-square regularization method [20], the neural network method [21], etc. to get HTC, whereas, LCM and IHTM have been commonly applied in recent years.

To prevent cracking, magnesium alloys are usually air-cooled [22] or quenched in hot water [23], the magnesium alloy used in this study is a newly developed high-strength heat resistant rare-earth wrought magnesium alloy Mg-Gd-Y-Zr-Ag-Er, which already made it possible to quench in room temperature water. However, the HTC in quenching or air cooling remains unknown, and that is vital both in the heat transfer model and the coupled temperature–displacement model to predict temperature and RS. A solution treatment followed by either water quenching or air cooling for a sample was conducted. The HTC between the alloy and water or air was investigated, and the water quenching HTC was calculated by the lumped capacitance method (LCM) and then optimized by the inverse heat transfer method (IHTM). Finally, the RS of a large conical component was simulated using both non-optimized and optimized HTC, and the comparison of the simulation results with the measurements revealed that the RS using HTC in the IHTM is more accurate.

2. Theory and Methods

2.1. Conventional Lumped Capacitance Method

The essence of the LCM is to use the conservation law of energy to derive the temperature field of the unsteady heat conduction process of the body, that is, the change of thermal energy of the body per unit of time is equal to the heat exchange between the surface of the body and the convection of fluid, this equation can be written as

$$\rho ·c·V\frac{\partial T}{\partial t}=\frac{\partial }{\partial x}(\lambda \frac{\partial t}{\partial x})$$

(1)

where the left side of the equation is the time-dependent unsteady state term and the right side is the diffusion term, ρ, c, V, T, t, λ, and x are density, specific heat capacity, volume, temperature, time, thermal conductivity coefficient, and distance, respectively.

Biot coefficient B_i characterizes the relative magnitude between the thermal resistance of the heat conduction and the surface heat transfer [14]. The application condition of the conventional lumped capacitance method (CLCM) is B_i ≦ 0.1, and its characteristic is that there is no temperature difference inside the object. This method is suitable for air cooling. Bi [18] is computed in Equation (2), Equation (1) can be derived as Equation (3).

$$B_i=hL/\lambda ,$$

(2)

$$\rho ·c·V\frac{\partial T}{\partial t}=-hA\left(To-T_{\infty }\right)$$

(3)

where h, L, A, T_o, and T_∞ are the HTC, characteristic length, heat transfer area, initial temperature, and ambient temperature, respectively.

2.2. Improved Lumped Capacitance Method

Different from the CLCM, the improved lumped capacitance method (ILCM) considers that the internal temperature of the instantaneous object is not uniformly distributed. The ILCM is applicable to the case of single-side quenching, so the quenching process can be simplified as a one-dimensional unsteady heat conduction problem. Position and time variation is introduced based on steady heat conduction, so the numerical solution needs to be discretized in space and time, as illustrated in Equation (4).

$$\rho ·c·V\frac{\partial T(x,t)}{\partial t}=-hA\left(T(\delta ,t)-T_{\infty }\right)$$

(4)

where T(x,t) and T(δ,t) mean that temperature is a function of position and time in ILCM, which is suitable for water quenching with a large temperature gradient.

As shown in Figure 1, for the air cooling condition, lifted the sample so that all the six surfaces could transfer heat with air; for the water quenching condition, the sample was adjusted to an appropriate height to make sure that the bottom surface had just been immersed in water, given that the ratio of the side length of the quenching surface to thickness is equal to 3, and other surfaces are air cooling which should be much less intense than water quenching; therefore, the sample can be simplified as a one-dimensional unsteady heat conduction model in which only the bottom surface exchanges heat with water, that is, the temperature at the same depth is uniformly distributed. In this case, the heat transfer area was the quenching surface of the sample.

The temperature field inside the sample was fitted using the temperatures in the R, P, and Q planes, corresponding to distances r, p, and q from the bottom. T_n1, T_n2, and T_n3 were the temperatures at time n in the R, P, and Q planes. Through quadratic fitting and coefficient matrix inversion in Equation (5), the quadratic term coefficient R_n, the first term coefficient P_n, and the constant term coefficient Q_n are computed.

$$\left[\begin{array}{c}{T}_{n1}\\ T_{n2}\\ T_{n3}\end{array}\right]=\left[\begin{array}{ccc}{r}^2& r& 1\\ p^2& p& 1\\ q^2& q& 1\end{array}\right]\left[\begin{array}{c}\begin{array}{c}{R}_n\\ P_n\end{array}\\ Q_n\end{array}\right]\to \left[\begin{array}{c}\begin{array}{c}{R}_n\\ P_n\end{array}\\ Q_n\end{array}\right]={\left[\begin{array}{ccc}{r}^2& r& 1\\ p^2& p& 1\\ q^2& q& 1\end{array}\right]}^{-1}\left[\begin{array}{c}{T}_{n1}\\ T_{n2}\\ T_{n3}\end{array}\right]$$

(5)

The temperature at a certain time and depth z can be expressed by Equation (6), when z equals the sample thickness d, the temperature of the quenching surface T_nd is calculated.

$$T_{nz}=R_n{z}^2+P_{n}z+Q_n=\left[\begin{array}{ccc}{z}^2& z& 1\end{array}\right]\left[\begin{array}{c}\begin{array}{c}{R}_n\\ P_n\end{array}\\ Q_n\end{array}\right]z\in (0,d)$$

(6)

Through a series of thermodynamic operations in Equation (7), the HTC h can be acquired.

$$\left\{\begin{array}{c}d{Q}_{nz}=C_{nz}\cdot T_{nz}\cdot dm=C_{nz}\cdot T_{nz}\cdot \rho \cdot Adz\\ Q_n={\int }_0^{d}d{Q}_{nz}=\rho \cdot A\cdot {\int }_0^d{C}_{nz}\cdot T_{nz}dz\\ {\Phi }_n=\frac{-\Delta Q_n}{\Delta t}=\frac{Q_n-Q_{n+1}}{t_{n+1}-t_n}\\ q_n=\frac{{\Phi }_n}{A}=h\left(T_{nd}-T_{\infty }\right)\\ h=\frac{q_n}{T_{nd}-T_{\infty }}\end{array}\right.$$

(7)

where dQ_nz, C_nz, and d_m denote the heat, heat capacity, and mass in d_z at time n, respectively. Q_n, Φ_n, and q_n are heat, heat flux, and density of heat flux, respectively.

As for the air cooling conditions in the CLCM, there is no temperature gradient in the sample at the same time, which means that T_n1, T_n2, T_n3, and T_nd are equivalent, making the solution rather simple.

2.3. Inverse Heat Transfer Method

The measured temperature curves were utilized, together with the governing equations and boundary conditions of the transient temperature field and the sensitive coefficient field, for the IHTM to calculate the surface temperature field and the surface heat flux.

The norm of the difference between the calculated temperature and the measured temperature at each iteration was used as the convergence criterion [24,25].

$$F\left(h\right)=\sum _{i=1}^{N_t}\sum _{j=1}^{N_m}\frac{1}{{\delta }_T^2}{[T_{ij}^m-T_{ij}^c(h\left)\right]}^2+\sum _{k=1}^{N_h}\frac{1}{{\delta }_k^2}{[h_k-h_k^0]}^2$$

(8)

To calculate the HTC, F(h) should be minimized in the iteration process.

$$\frac{\partial F\left(h\right)}{\partial h_l}=\sum _{i=1}^{N_t}\sum _{j=1}^{N_m}\frac{(-2)}{{\delta }_T^2}[T_{ij}^m-T_{ij}^c(h)]X_{ijl}+\frac{2}{{\delta }_l^2}\left[h_l-h_l^0\right]=0$$

(9)

where X_ijl is the sensitivity coefficient, it is expanded by the Taylor formula at h_l.

$$X_{ijl}=\frac{\partial T_{ij}^c\left(h\right)}{\partial h_l}\cong \frac{T_{ij}^t\left(h_1,\dots ,h_l+{\delta h}_l,\dots ,{h_N}_h\right)-T_{ij}^c\left(h_1,\dots ,h_l,\dots ,{h_N}_h\right)}{{\delta h}_l}$$

(10)

where δh_l is the change of h_l in the previous iteration, which is used to solve the sensitivity coefficient, it is also necessary to linearize the temperature $T_{ij}^c\left(h^{v+1}\right)$ of the next iteration when calculating the temperature field.

$$T_{ij}^c\left(h^{v+1}\right)\cong T_{ij}^c\left(h^v\right)+\sum _{k=1}^{N_k}{X}_{ijk}^v·∆h_k$$

(11)

where Δh_k is the change of the interfacial HTC, it is calculated in Equations (12)~(14).

$$\sum _{k=1}^{N_k}{A}_{lk}·∆h_k=f_l$$

(12)

$${\mathrm{A}}_{lk}=\sum _{i=1}^{N_t}\sum _{j=1}^{N_m}\frac{X_{ijk}·X_{ijl}}{{\sigma }_T^2}+\frac{{\delta }_{lk}}{{\sigma }_l^2}$$

(13)

$$f_l=\sum _{i=1}^{N_t}\sum _{j=1}^{N_m}\frac{1}{{\delta }_T^2}\left[T_{ij}^m-T_{ij}^c\left(h^v\right)\right]·X_{ijl}-\frac{1}{{\delta }_l^2}(h_l^v-h_l^0)$$

(14)

3. Experiments and Simulation

3.1. Material

The magnesium alloy studied in this work is high strength heat resistant rare-earth wrought magnesium alloy Mg-Gd-Y-Zr-Ag-Er. Its chemical composition (wt.%) is listed in

Table 1. Chemical composition (wt.%).

Gd	Y	Zr	Ag	Er	Mg
8.0~9.6	1.8~3.2	0.3~0.7	0.02~0.50	0.02~0.30	Bal

. The alloy was melted and cast above 700 °C and was die forging at ~460 °C, then immersed into 20 °C water within ~60 s to prevent grain growth. The mechanical properties can be referred to the previous research [17].

3.2. Sample for the HTC Experiment and Simulation

As illustrated in Figure 1, the dimension of the sample was 120 mm × 120 mm × 40 mm, four vertical holes were evenly distributed around a circle with a diameter of 28 mm, which was 4 mm in diameter and a depth of 39 mm, 27 mm, 14 mm, and 25 mm from the top surface, respectively, four k-type thermocouples labeled 1#, 2#, 3#, and 4# (spare) were sealed inside the hole with high-temperature resistant glue, as a result, the distance between the quenching surface and R, P, and Q planes were 26 mm, 13 mm, and 1 mm, respectively. The 4# (spare) thermocouple was prepared in case any thermocouple may get damaged in the experiment process. B_i in these experimental conditions has been estimated to be less than 0.1 in air cooling and greater than 0.1 in water quenching, thus the CLCM and the ILCM are suitable for air cooling and water quenching, respectively. The surface of the sample was milled to make it smooth, thereby preventing the influence of the surface state on the HTC, such as the possibility of forming an oxide layer and the roughness was different.

Solution treatment at 500 °C was conducted for 8 h in a resistance furnace with a fan and then quenched in 20 °C, 40 °C, 60 °C, and 80 °C water only by the bottom surface, with a transfer time of fewer than 20 s. While cooled in air, speeds of 0 m/s, 3 m/s, and 5 m/s were employed with a large adjustable fan and an anemometer. The temperature data of the thermocouples were saved by a temperature paperless recorder with a frequency of 1 Hz.

A 3D full-size model using the finite element method (FEM) was established, and density, thermal conductivity, and expansion coefficient under different temperatures were tested as boundary conditions, together with the HTC calculated under different conditions above. The heat transfer model was used in the FEM software ABAQUS 2017 to verify the accuracy of the HTC.

Repeated iterations and optimizations through Equations (8)~(14) made the simulation rather slow, considering the symmetric structure of the sample, a 2D axisymmetric model using the IHTM was established in the FEM software DEFORM in case that a full-size model would cost too much time and computational resources. The measured temperature curves were fitted by b-spline interpolation, 10 control points at an incremental temperature of 50 °C were set at the range of 50–500 °C.

3.3. Residual Stress Measurement and Simulation of a Large Magnesium Alloy Component

The dimension of the as-received large magnesium alloy component is shown in Figure 2. The hole drilling method and X-ray diffraction (XRD) method were employed to measure the residual stress along two generatrixes which were 180° apart on the outer surface of the cone, as shown in Figure 3. Residual stresses were analyzed in ABAQUS using a one-fourth size geometry and a coupled temperature–displacement analysis model, 31,482 finite elements C3D8T were established, and the elasto-plastic model and the Mises yield criterion were applied. A hot compression test with a reduction of 20% was carried out at a temperature of 50~500 °C at an interval of 50 °C and room temperature, and a strain rate of 0.01 s⁻¹ to mainly obtain the yield strength at different temperatures [26]. The air-cooling HTC was used during transfer and the 20 °C water quenching HTC was used after immersion in water. Material parameters of density, thermal conductivity, expansion coefficient, Yong’s modulus, Poisson’s ratio, etc. at different temperatures were tested, respectively, the above measurement and simulation methods are more detailed in the previous research [17].

4. Results and Discussions

4.1. Temperature Curves

The measured (1#, 2#, and 3#) and calculated (quenching surface) temperature curves at 20 °C, 40 °C, 60 °C, and 80 °C have been illustrated in Figure 4, the general trend of the curves was that the temperature drops quickly at first and then slowly, and almost flat within 200 s, which was consistent with the decrease in temperature difference between the sample and water. In addition, the temperatures at the same time basically obeyed the following orders: 3# > 2# > 1# > surface, this was because the quenching surface of the sample first exchanged heat with water in single-side quenching, and then the internal heat was transferred to the surface and exchanged heat with water; therefore, the farther the thermocouple was from the quenching surface, the slower the cooling rate was.

However, by comparing the four conditions, the cooling rates are different. Taking the time required for cooling down to 150 °C on the quenching surface as an example, 45 s, 52 s, 63 s, and 65 s were needed, respectively. What’s more, the curves at 20 °C and 40 °C were concave, while the curves at 60 °C and 80 °C were convex at the beginning and then became concave, this implied that the higher the water temperature, the slower the cooling rate.

The temperature differences in air cooling were supposed to be small, therefore, only the measured temperature curves of 1# and 3# thermocouples at speeds of 0 m/s, 3 m/s, and 5 m/s have been illustrated in Figure 5. The temperature differences between 1# and 3# were almost zero, which indicated that there was no temperature gradient inside the sample in air cooling conditions and that CLCM was applicable in these cases.

The general trend of the temperature curves was that the cooling rates were fast at first and then slowed down, and the curves became flat at around 1500 s at 3 m/s and 5 m/s, while at about 6000 s at 0 m/s.

4.2. Calculation of the HTC in ILCM

The HTC in water quenching with various times and temperatures is shown in Figure 6 through the calculation in the ILCM. The overall trend of HTC first increased then decreased, and finally became oscillatory. This process could be divided into three stages, namely, film boiling, nucleate boiling, and single-phase cooling [27]. A large amount of steam was initially formed on the quenching surface, and isolated heat transfer between the surface and water, which made the HTC quite small at first. A large number of bubbles were then generated and broken as the sample temperature was lowered, at which point the heat flow reached its maximum, So near the peak is the boundary between film boiling and nucleate boiling. After that, the surface temperature cooled down and entered the single-phase cooling regime. Thus the HTC in 20 °C water reached its peak value of 2840 W/(m²·°C) at 21 s at 254 °C, and the corresponding values in 40 °C, 60 °C and 80 °C were 2605 W/(m²·°C) at 27 s at 230 °C, 2540 W/(m²·°C) at 47 s at 200 °C, and 2252 W/(m²·°C) at 79 s at 164 °C, respectively, which meant the larger the temperature difference between the sample and water, the larger the maximum HTC, and the earlier it reached the maximum value. From Figure 6a of the HTC curves against time, the greater the temperature difference, the shorter the duration of the film boiling stage which isolated heat transfer between sample and water, led to more heat exchange per unit time and a larger cooling rate. That’s the physical mechanism of the curves at 20 °C and 40 °C were concave, while the curves at 60 °C and 80 °C were convex.

The HTC in air cooling with various times and temperatures is shown in Figure 7 through calculation in CLCM. Similar to water quenching, the HTC in air cooling initially increased and then decreased, while at 3 m/s and 5 m/s, it rose again and oscillated vigorously at the end. The HTC at 5 m/s reached its peak value of 54 W/(m²·°C) at 44 s at 461 °C, and the corresponding values in 3 m/s and 0 m/s were 45 W/(m²·°C) at 57 s at 454 °C and 21 W/(m²·°C) at 70 s at 485 °C, respectively, the HTC became larger with the speeds increased. So the faster the airflow speed of the workpiece surface, the better the heat dissipation, for the sample was exposed to more air and exchanged more heat per unit time.

4.3. Verification of the Heat Transfer Coefficient

To quantify the accuracy, relative errors η, the absolute average relative error (AARE), and the correlation coefficient (R) can be calculated as follows [28]:

$$\mathsf{\eta }=\frac{P_i-E_i}{E_i}\times 100\mathrm{\%}$$

(15)

$$AARE=\frac{1}{N}\sum _{i=1}^N\left|\frac{E_i-P_i}{E_i}\right|\times 100\mathrm{\%}$$

(16)

$$R=\frac{\sum _{i=1}^N(E_i-\overline{E})(P_i-\overline{P})}{\sqrt{\sum _{i=1}^N{(E_i-\overline{E})}^2\sum _{i=1}^N{(P_i-\overline{P})}^2}}$$

(17)

where E_i and P_i stand for the experimental (EXP) and simulated (FEM) values, respectively; $\overline{E}$ and $\overline{P}$ are the mean values of the experimental and simulated (FEM) values, respectively; N is the number of data, and the maximum relative errors denote η_m.

The HTCs calculated in Section 4.2 were used in the FEM to verify the accuracy, based on Figure 4 and Figure 5, the temperatures of the corresponding simulated results were added in the form of dashed curves, as illustrated in Figure 8. The accuracy of air cooling, with AARE of 4.47% and η_m of 8.36%, was satisfactory because the tiny HTC led the cooling process slow without fluctuation, thus further analysis was unnecessary. In the water quenching, however, the data were slightly larger, especially at the beginning of the quenching. Nevertheless, the AAREs were still within 10%, which would be acceptable in engineering applications.

The reason why quenching simulation values differ larger can be explained as follows: during the quenching experiment, a small part of the side walls was exposed to water to ensure that the bottom surface of the sample was submerged, and the boiling bubbles would inevitably splash onto the side walls as well, the higher the water temperature was, the longer the nucleate boiling stage maintained and the more heat exchange between side walls and water, these two aspects made the actual heat transfer area more than the calculated value, led to a larger HTC by Equation (7). These could also explain why AARE of 60 °C and 80 °C (~8%) water quenching were larger than that of 20 °C and 40 °C (~5%).

Cooling rates are important factors in solution and quenching heat treatments. Keeping it from getting too low in a certain quenching sensitivity temperature range is the key to maintaining the mechanical properties of aluminum alloys [12], while no similar phenomenon is reported in magnesium alloys as the diffusion rate of elements is considerably slower. Thus taking the relative linear temperature range of 450–150 °C into consideration to calculate cooling rates, the experimental data, the simulated data, and AARE are shown in

Table 2. Experimental, simulated cooling rates, and AARE by ILCM (unit: °C/s).

	1# (EXP)	1# (FEM)	2# (EXP)	2# (FEM)	3# (EXP)	3# (FEM)	S(EXP)	S(FEM)
20 °C	6.81	6.38	5.58	5.14	5.28	4.64	6.97	8.04
AARE	6.31%		7.89%		12.12%		15.35%
40 °C	6.17	5.94	5.29	4.39	4.54	4.04	6.42	6.27
AARE	3.73%		17.01%		11.01%		2.34%
60 °C	5.92	5.51	4.96	4.96	4.26	3.91	6.06	5.74
AARE	6.93%		0.00%		8.22%		5.28%
80 °C	5.41	4.55	4.82	3.68	3.88	3.47	5.46	4.69
AARE	15.90%		23.65%		10.57%		14.10%

.

The cooling rates decreased with increasing water temperature and with increasing distance from the bottom surface both in experiments and simulations. An analysis of this phenomenon could refer to Section 4.2. However, the AARE of 80 °C was significantly larger than other conditions, which was consistent with the analysis above.

4.4. Calculation of the HTC in IHTM

Given that the initial input parameters have a large influence on the calculation results and the errors calculated by ILCM were not that ideal, the HTC calculated in Section 4.2 was probably a better choice as an initial value for the IHTM optimization.

Ten control points at an incremental temperature of 50 °C which based on the ILCM results were set at the range of 50–500 °C, the experimental and simulated temperature curves in water quenching were illustrated in Figure 9. Surface temperatures were not included in the original data, so only temperatures of 1#, 2#, and 3# were shown here. Through iteration and optimization in the IHTM, the temperature accuracy between experiments and simulations improved prominently to the extent of below 2.5% of AARE and 8% of η_m.

As shown in Figure 10, a line at an angle of 45 degrees from the horizontal direction was drawn and all the data should be on the line for a perfect fitting [29]. The R² between the experimental and simulated temperature of 1# in CLCM is 0.9549, while that in ITHM grows to 0.9962; the corresponding values of 2# are 0.9994 and 0.9847, respectively; the corresponding values of 3# are 0.9941 and 0.9980, respectively. The R² at high temperatures in CLCM is not that satisfactory. The improvement effect of R² gradually weakens as the distance between the thermocouple and the quenching surface increases, because the intense heat transfer and bubble boiling of the quenched surface have limited effects on 2# and 3# thermocouples. However, from the distribution of points, it is necessary to optimize the HTC, especially for 1# and 2# thermocouples.

The experimental, simulated cooling rates and AARE were listed in

Table 3. Experimental, simulated cooling rates and AARE by IHTM (unit: °C/s).

	1# (EXP)	1# (FEM)	2# (EXP)	2# (FEM)	3# (EXP)	3# (FEM)
20 °C	6.81	6.55	5.58	5.60	5.28	5.36
AARE	3.82%		0.36%		1.52%
40 °C	6.17	5.78	5.29	5.06	4.54	4.85
AARE	6.32%		4.35%		6.83%
60 °C	5.92	5.48	4.96	4.83	4.26	4.68
AARE	7.43%		2.62%		9.86%
80 °C	5.41	5.07	4.82	4.54	3.88	4.38
AARE	6.28%		5.81%		12.89%

, Except for a few conditions where the error increased slightly, most of the AARE declined, especially in the case of 80 °C. This optimization was of great significance for the control of cooling rates in quenching sensitivity temperature range and thus may help to more precisely improve the mechanical properties of the alloy.

The simulated HTCs were plotted in Figure 11, the data declined compared with the initial ones, the HTC in 20 °C water reached its peak value of 2388 W/(m²·°C) at 300 °C, and the corresponding values in 40 °C, 60 °C and 80 °C were 2244 W/(m²·°C) at 250 °C, 1951 W/(m²·°C) at 200 °C, and 1565 W/(m²·°C) at 150 °C, respectively.

The temperature corresponding to the HTC peak decreased with increasing water temperature, which is consistent with the previous calculations by the ILCM, with the drawback that the HTC data is discrete, and the calculation results would be affected by the number of initial control points.

4.5. Residual Stress

HTCs at 20 °C calculated by IHTM in Figure 11 were input into the FEM as the interaction conditions, and the simulated distribution of RS in the radial, circumferential, and axial directions is shown in Figure 12, with the radial RS being relatively small except for the outer and inner tops, where the outer top has a compressed RS of −39.46 MPa and the inner top has a tensile RS of 89.44 MPa. Circumferential and axial RS have similar distributional regulation, that is, the RS on the outer surface is compressed and the inner surface is tensile, due to the cooling sequence from the outer to the inner surface, when the outer surface had cooled and the inner surface was still shrinking. The difference is that the axial RS approaches zero near the top and bottom, while the circumferential RS reaches its maximum compressive RS at the bottom. The largest circumferential RS is the compressive RS of −117.60 MPa and the tensile RS of 100.20 MPa both near the bottom, and the largest axial RS is the compressive RS of −89.50 MPa near the upper surface and the tensile RS of 141.10 MPa near the inner tip. In this case, the magnitude of RS in a certain direction is positively correlated with the dimension of the direction, more specifically, radial RS is much less than that of circumferential and axial as radial size is ~200 mm or below while circumferential and axial size are both more than 1000 mm. The uneven deformation caused by the huge temperature difference between inside and outside at the beginning of quenching (similar to Figure 4) accumulated more as dimensions became larger, thus the RS maintained at room temperature was also larger. The part with a large RS needs to be noted in the subsequent treatment.

RS of measurement (dots) and simulation results (lines) along the generatrixe were depicted in Figure 13, among them the result with the error bars was detected by X-ray while others were measured through hole drilling, black lines, and dots represented circumferential RS and axial ones were in red, solid and dashed lines were RS simulated using HTC with IHTM and ILCM, respectively. The overall trend was similar between the two methods, the distribution of RS was like an “M” in the circumferential direction and a “W” in the axial direction, and most of the middle part was relatively flat, it was only because the radial dimension first decreased and then increased (Figure 2) that cause the RS decreased first and then increased. In the circumferential direction, RS exhibits a phenomenon of first decreasing and then increasing from the middle part to the top and bottom. In the axial direction, the RS is almost zero at the top and bottom, and then increases to a maximum at the middle “platform”. The main difference was that the RS in the IHTM was ~30 MPa smaller than that in the ILCM, because the corresponding HTC was smaller. The measurements revealed that RS using the HTC in IHTM was more accurate except for the point next to the top and a few other points. Errors between the measurement and simulation results mainly lay in the aspects that the anisotropy caused by the crystal characteristics of the hexagonal system of magnesium alloy, the non-uniformity of the material inside and outside of the large component, and the existence of lubricating oil on the surface of the component after molding which caused the heat transfer coefficient to deviate from the experimental value during water quenching.

To further explain the accuracy of the optimized HTC for RS simulation, AARE was applied to quantify the deviation between the measurement and the simulation value of the two models above, as shown in

Table 4. AARE of RS under two HTC models.

Model	Circumferential	Axial
IHTM	22.13%	17.38%
ILCM	55.33%	24.71%

. The values in the circumferential and axial direction of the IHTM are 22.13% and 17.38%, respectively, which could be acceptable in engineering applications, while the corresponding values are 55.33% and 24.71% in the ILCM.

On a smaller scale, the second type of inter-grain RS model is affected by the precipitated phase, and the third type of intra-grain RS model is affected by alloy atoms; however, on a larger macroscopic scale, these effects on RS get balanced and canceled, which are hard to measure directly. The macroscopically measurable RS in engineering, often referred to as the first type of RS model, prefers to be a physically global model. This is because the difference between the quenched and aged states mainly lies in the precipitation strengthening, but the precipitated phase in this alloy is relatively small in quantity compared with the magnesium base. Although the elastic modulus is very much related to the RS [30], previous studies [17] have shown that the elastic modulus of the quenched and aged states of this magnesium alloy is basically the same, and the grain size has little correlation with the cooling rates. It could be inferred from the above that the microstructure has limited influence on the first type RS of magnesium alloy in this research. Therefore, in this case, the HTC is considered to be one of the most important factors affecting the RS in heat treatment.

5. Conclusions

• The peak values of HTC by LCM in air cooling of 5 m/s, 3 m/s, and 0 m/s were 54 W/(m²·°C), 45 W/(m²·°C), and 21 W/(m²·°C), respectively, the faster the airflow speed of the workpiece surface, the better the heat dissipation, for the sample was exposed to more air and exchanged more heat per unit time.
• The peak values of HTC by LCM in water quenching of 20 °C, 40 °C, 60 °C, and 80 °C were 2840 W/(m²·°C), 2605 W/(m²·°C), 2540 W/(m²·°C), and 2252 W/(m²·°C), respectively, and that the peak values of HTC by IHTM in water quenching of 20 °C, 40 °C, 60 °C, and 80 °C were 2388 W/(m²·°C), 2244 W/(m²·°C), 1951 W/(m²·°C), and 1565 W/(m²·°C), respectively. Compared with the ILCM, the HTC decreased and the accuracy between experiments and simulations of temperature curves and cooling rates increased after optimizing by the IHTM because the boiling bubbles would inevitably splash onto the side walls, the higher the water temperature was, the longer the nucleate boiling stage maintained and the more heat exchange between side walls and water, made the actual heat transfer area more than the calculated value by LCM.
• The RS in the IHTM was ~30 MPa smaller than that in the ILCM, the AARE of RS decreased from 55.33% to 22.13% in the circumferential direction, and decreased from 24.71% to 17.38% in the axial direction in IHTM compared with in ILCM, because the HTC in the IHTM was smaller and more accurate.